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Feedback control of optical cavity systems

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Feedback control of optical cavity systems

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Content FEEDBACK CONTROL OF OPTICAL CA VITY SYSTEMS by Walter Unglaub A Thesis Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (PHYSICS) December 2015 TableofContents Personal acknowledgements: page 3 Abstract: page 4 Introduction: page 5 Chapter 1: Delayed Feedback in Semiconductor Lasers: page 7 References: page 30 Chapter 2: Control of Particle Statistics in Mesoscale Lasers: page 31 References: page 65 Chapter 3: Single-Photon Dynamics and Control in Cavity QED: page 66 References: page 141 Summary: page 142 Bibliography: page 143 2 PersonalAcknowledgements I would like to thank the various people who have supported me academically during these last few years, as well as provided useful insights and discussions surrounding the variety of topics presented in this work. In particular, I would like to thank my advisor, Anthony Levi, for his guidance and support, as well as Stephan Haas. I would also like to thank Kaushik Roy-Choudhury for baseline code used in solving the master equations describing the solitary mesolaser and open cavity QED models. Finally, I would like to thank Tameem Albash and Lorenzo Campos Venuti for their contributed insights and discussions. 3 Abstract Non-Markovian processes are of particular interest when studying the dynamical evolution and control of quantum systems. While non-Markovianity is an inherent property of time-evolving unitary systems, the transient evolution of open systems can also exhibit memory resulting from delayed feedback. By understanding the various system timescales and subsystem correlations, transient control may be exerted by exploiting time delays and other mechanisms external to the system of interest. In this thesis, the dynamics of various open and closed optical cavity sys- tems are examined, including semiconductor laser diodes, microcavity diodes, and coupled-cavity systems, with a focus on delayed feedback control. 4 Introduction This thesis concerns itself with three distinctly different approaches to dealing with feedback con- trol in optical systems. Emphasis is placed on delayed feedback schemes to control laser dynam- ics and particle statistics, as well as matter-light interactions in cavity quantum electrodynamics (cQED). The first chapter of this thesis concerns itself with several rate equation approaches to semicon- ductor laser diodes. This chapter is included because it introduces the concept of non-Markovian dynamics through delayed feedback in a classical setting, as well as giving insights into the rich topic of chaotic systems and nonlinear dynamics in driven systems. This chapter provides a classi- cal framework for understanding cavity formation, as well as coherent feedback control in external cavity semiconductor lasers. The second chapter introduces a mesoscopic model of a Fabry-Perot semiconductor microlaser, in which energy is quantized through quantization of particle number. A master equation is used to describe the evolution of probability corresponding to all possible microstates in the corresponding Fock state basis describing electronic excitation (electron) and photon particle numbers. Unlike the electric field described by the coupled rate equation approach in the preceding chapter, the amplitude and phase corresponding to each particle are ignored due to computational constraints. Thus, without a complex Hilbert space to describe the amplitude and phase corresponding to each particle, any and all delayed photon feedback incorporated into the model will generally be incoherent. Regardless, however, such feedback can significantly impact particle statistics in the diode, including the mutual information between the two particle subsystems, which may be interpreted as a measure for how strongly coupled the subsystem distributions are. The third chapter is divided into three sections, and describes the nonrelativistic dynamics of light-matter interaction in one-dimensional cavities. The first section derives and reproduces the Jaynes-Cummings model, which describes the interaction dynamics between an initially-excited two-level system and a photon field inside a one-dimensional cavity, subject to closed boundary conditions. The model is extended by including multiple cavity modes that can interact with a single atom placed in the cavity, as well as spatially-resolving the cavity in order to study the spatial distribution of the single photon energy. Using the broadband approximation, the feed- back dynamics corresponding to spontaneous decay and re-excitation of the atom are examined in parameter space for different cavity sizes, revealing a transition from strongly-coupled behavior 5 in small cavities (where the cavity length is comparable to the photon wavelength), to weakly- coupled dynamics in the large cavity limit in which the single photon energy emitted from the atom is localized in the form of two pulses traveling away from the atom, reflecting from the cavity walls, and partially re-exciting the atom within a cavity round-trip time. The second section describes an open cavity that is coupled to an external, passive cavity, which may be used to store the emitted photon. This section describes a method for controlling the amplitude and phase of the emitted pulses to coherently control single photon dynamics in a spatio-temporal manner. Insertion of a beam splitter in the primary cavity allows the atom to couple to two sets of modes, and the atom is diabatically-detuned after spontaneously emitting the photon in order to remove pulse reshaping resulting from feedback interactions. A delayed, control pulse from one of the cavities can then be used to quench the photon field excitations in the external cavity. Finally, the third section explores a cavity involving one or more atoms coupled to a single mode. The cavity system is coupled to the external environment by means of an external pump and dissipation to an external reservoir comprised of a bath of harmonic oscillators at zero temperature. The the statistics of the emitted photons are calculated in terms of the second-order correlation function, and transitions between thermal and non-classical light properties are explored in the parameter space of the model. A brief summary of the various optical systems and control schemes presented in this thesis is given at the end. 6 Chapter1: DelayedFeedbackin SemiconductorLasers I. CH.1INTRODUCTION The dynamics of a semiconductor laser diode have been studied for decades and can be de- scribed by coupled delayed differential equations for carrier density and electric field in the active region, which is typically considered as a gain medium inside a resonant cavity. Coupling an external cavity to the diode introduces an extra degree of freedom via the feedback of light, and allows for the exploration of many nonlinear phenomena, including chaos and cavity formation. The behaviour of a solitary laser diode can be described by coupled rate equations [1] that describe the behaviour of the laser diode, dn dt = I eV n n GS (1) ds dt = (G)s +r spon ; (2) wheren ands are respectively the carrier and photon densities, and where G = cG slope n (nn 0 ) (1" bulk s) (3) is the gain of the medium, with being the optical confinement factor,G slope the optical gain- slope coefficient, n 0 the transparency carrier density, " bulk the gain saturation coefficient, andV the diode volume. is the loss rate, and is calculated as the sum of the total mirror loss and an internal loss. Finally, we can expand the sink term in (1) describing the loss of carrier density over the carrier lifetime n , in terms of three coefficients: n n =A nr n +Bn 2 +Cn 3 ; (4) 7 withA nr the nonradiative recombination coefficient,B the radiative recombination coefficient, and C the nonlinear recombination coefficient. r spon in equation (2) describes the contribution to the photon number due to spontaneous emission, and is formulated in terms of B, such that r spon = Bn 2 . Finally, the factor is the spontaneous emission coefficient, and is equal to the fraction of the spontaneous emission into the lasing mode. Using a 4 th -order Runge-Kutta integrator, one can solve these coupled differential equations, and study the evolution of photon density, output light intensity, and carrier density in time, as well as the relation between output power and carrier density. The results are shown in figures (1) - (4), using typical Fabry-Perot semiconductor parameters (given in the Appendix). 0 1 2 3 4 5 0 5 10 15 20 25 30 Time, t (ns) Current, I (mA) Figure 1: Injected step current vs. time, with a time delay of 2 ns (r 1 =r 2 = 0:32). Furthermore, one can find the threshold current for any given set of system parameters by loop- ing the DC input, and plotting the steady-state value of the output light as a function of injection current. Using the same values from earlier (in which the mirror reflectivities are both set to 0.32), we find the threshold current to be approximately 5.75 mA, as evidenced by figure (5). For current values greater than this threshold, we get lasing characteristics. 8 0 1 2 3 4 5 0 2 4 6 8 10 12 14 16 18 20 Time, t (ns) Power, L (mW/facet) Figure 2: Light intensity from facet 1 vs. time, with a time delay of 2 ns (r 1 =r 2 = 0:32). 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 Time, t (ns) Carrier density, n (10 18 cm −3 ) Figure 3: Carrier density vs. time, with a time delay of 2 ns (r 1 =r 2 = 0:32). II. BASICMODEL We can extend this model to describe an external cavity semiconductor laser (ECSL) by cou- pling passive, external cavity coupled to the active region, as shown in figure (6). 9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 0 2 4 6 8 10 12 14 16 18 20 Carrier density, n (10 18 cm −3 ) Power, L (mW/facet) Figure 4: Light intensity from facet 1 vs. carrier density (r 1 =r 2 = 0:32). 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 8 Current, I (mA) Light, L (mW) Figure 5: Light intensity vs. current (r 1 =r 2 = 0:32). We can describe this system by incorporating feedback terms into the rate equations. The equation for photon density is modified with terms that are time-delayed by the round-trip time, , between the coupled diode facet and the external mirror. The rate equation describing the time evolution of the photon density now includes two terms that characterize a time-delay by half a round-trip and a time-delay of exactly one full round-trip. The respective terms are and such 10 Figure 6: External cavity schematic that = 2, so thatt = is the time it takes for a photon to leave and re-enter the diode cavity after being reflected off the external cavity mirror. Fort, equations (1) and (2) are used, since photons that have left the active region don’t re-enter untilt. After a timet , the photons that were transmitted through mirror 2 will either be reflected by mirror 3 or leave the external cavity; the fraction depends on the value of r 3 , which we set initially set to unity so that all light continually being lost from facet 2 returns to the same facet at a time later, and a fraction (1r 2 ) re-enter the active region. Furthermore, the photons that stay within the fiber for multiple reflections must be accounted for, as they could re-enter the diode after several round-trips. A. Stepresponse To numerically implement this transient formulation of delayed photons, we can utilize a sum that accounts for photons delayed by integer multiples of the round-trip time. Fort, we must then modify (2) to incorporate these effects, thus yielding the full “delayed" rate equation (5); we can just as well formulate a transient rate equation describing the time evolution of the fiber photon number (6) and the light intensity emanating from facet 3 (the other end of the fiber), delayed by half a round-trip time (7): 11 dS dt = (G)S +r spon V + 2 " n X j=1 (1r 2 )r (j1) 2 r j 3 S (tj) # (t); (5) dS c dt = 2 S 2 " m X j=1 (1r 3 )r (j1) 2 r (j1) 3 S (t (2j 1)) # (t) 2 " n X j=1 (1r 2 )r (j1) 2 r j 3 S (tj) # (t); (6) and the light intensity from facet 3 is given by L 3 = hc 2 " m X j=1 (1r 3 )r (j1) 2 r (j1) 3 S (t (2j 1)) # (t) (7) wherem is the number of trips completed to the end of the fiber,n is the number of round-trips completed back to the diode, and 2 is the rate at which the photons leave the diode from facet 2 (a fraction of the inverse cavity photon lifetime); it is given by 2 = (1r 2 ) (2r 1 r 2 ) c 2L d n r ln 1 r 1 r 2 (8) One way of implementing this time delay is to calculate an array of photon density values, using the original rate equations, for each step in time up until time. Then, the array values are introduced into the delayed rate equation, for t 2, while the new values of the photon density replace the old values. Due to the sum in the delay term for the photon number rate equation, a new array is generated each integer multiple of in order to keep track of the photons that remain in the external cavity. The process is then repeated for the desired length of time to solve the rate equations. The same holds for keeping track of the light exiting the system from facet 3. It is not difficult to see that computing time will rise exponentially as one runs the code for many round-trip times, as each pass generates an additional array that the computer needs to keep track of. While there are several methods one could employ to speed up computation, such as pre- allocation of arrays, if one assumes the limit of no or negligible reflectivity at facet 2 (r 2 1), 12 then we can approximate the actual code by ignoring all but the first term in the sum. Physically, this corresponds to effectivelyall the photons reflected fromr 3 returning back to the diode. Hence in this limit, our delayed rate equation for photon number becomes dS dt = (G)S +r spon +r 3 (1r 2 ) 2 S (t); (9) Figure (8) shows the results for the light intensity from the facet 1 plotted as a function of time, for injection current values of 5, 10, 20, and 30 mA, wherer 2 = 1 10 5 andr 3 = 1 Likewise, figure (9) displays the evolution of the carrier density as a function of time, forI = 30 mA. Figure (10) displays the light intensity as a function of the injection current, for multiple round trips. Figure 7: Injected current vs. time. Before going further with the limited case, it is worthwhile to check the results of each code and see whether they agree in the limit of small r 2 . For a given set of typical InGaAsP diode parameters and a step current value of 30 mA, figure (11) displays the light output from facet 1 for several round-trips, and compares the results calculated the different approaches. In the case for which r 2 = 1 10 5 , we readily see that they agree rather well - effectively all photons are re-introduced. By setting r 2 = 0:5, however, the results begin to deviate after two round-trips 13 Figure 8: Light intensity from facet 1 vs. time (r 1 =r 2 = 0:32;r 3 = 1). Figure 9: Carrier density vs. time (r 1 =r 2 = 0:32;r 3 = 1). (20 ns). The results from the full code demonstrate an enhancement over the approximated code in the light output due to the surviving photons in the fiber re-entering the active region multiple round-trips later. Thus, in the limitr 2 ! 0, the codes yield the same results. From figures (8) and (9), we can see that steady-state conditions begin to be reached after about 10 round-trips, and that by increasing the current, the steady-state value of light intensity from facet 1 is reached faster. From figure (7), we see that many round-trips are required to go from LED characteristics to lasing. Steady-state conditions are achieved for ni60, in which the 14 Figure 10: Optical power vs. injected current (r 1 = 0:32;r 2 = 2 10 6 ;r 3 = 1; = 0:001). lasing cavity has formed due to the high reflectivity of mirror 3. Finally, if we turn our attention to the relation between light power and carrier density, we see similar behaviour in comparison to the case when the reflectivity of facet 2 is significantly higher (r 2 = 0:32); however, just as there is a step response in figures (8) and (9), we observe jumps in the intensity-carrier density curve associated with each round-trip. This is shown in figure (12), forn = 5. In order to keep track of the external cavity photons, we must now write down a third differ- ential equation which describes the time evolution of photon density in the passive fiber region. Obviously, the photons transmitted through the diode facet on the right act as a source, and even though the external mirror is 100% reflecting, the diode facet now has a finite reflectivity so that there is some loss after a round-trip and some retention of external cavity photons. We can go further and assume the general case in which r 3 < 1:0, such that another sink term in the fiber photon density rate equation is introduced, describing the fractional loss of light after half a fiber round-trip time. Equation (9) is modified such that only a fraction of the photons originally lost 15 0 20 40 60 80 100 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Time, t (ns) r2 = 0.00001 r2 = 0.25 0 20 40 60 80 100 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Time, t (ns) Power, log(L) (mW/facet) r2 = 0.00001 r2 = 0.25 Figure 11: Intensity vs. time (code comparison). Figure 12: Light intensity from facet 1 vs. carrier density. from the active region are re-introduced; the fraction is given by (1r 4 ), wherer 4 represents the outer facet reflectivity on the external cavity side of the diode cavity. For simplicity, we simply set r 4 = r 2 . The third differential equation can be used to examine the time evolution of the photon density in the external cavity, given by 16 dS c dt = 2 S (1r 3 ) 2 S(t=2)r 3 (1r 4 ) 2 S(t); (10) and dS dt = (G)S +r spon +r 3 (1r 4 ) 2 S (t); (11) whereS (t=2) andS (t) represent terms that are respectively delayed by=2 and. If we plot the time evolution of the photon number inside the diode, and the photon number inside the fiber, we can see that steady-state conditions are reached after many round-trip times. Since the optical fiber acts as a passive region (in this case, with absolutely zero internal loss and little to no transmittance through mirror 3), the main loss occurs from the photons that return to the diode after a time. The results are shown below in figures (13) and (14) forr 3 = 1 andI = 30 mA. Figure 13: Diode photon density vs. time. 17 Figure 14: Fiber photon density vs. time. B. FiniteCurrentPulse Now we look at the case where a current pulse is introduced into the system, so that there is an immediate step in current to a given DC value (e.g., 30 mA), and at some specified time later, is turned off. Figures (15) - (19) display the system response to a pulse of finite length, in the case where r 1 = 0:32;r 2 = 0:20;r 3 = 0:80, so that the mirror 3 reflectivity is less than unity and light may be observed from the other end of the fiber. Since the mirror reflectivity is now not negligible, we must use the full code, which takes longer to run, since new arrays are created during each round-trip to keep track of the photons in the fiber; this is crucial in this case, since an appreciable fraction of fiber photons reflect from facet 2 each round-trip. A step current is introduced, the diode photon density and carrier densities reach their steady-state values after a finite turn-on delay, and as soon as the current is switched off (here, att = 2), the system begins turning off in response. Here we observe periodic behaviour as the system shuts off. This is due to the fact that as photons are lost from facet 2 (hence decreasing the diode photon density), delayed photons return to the diode after each round-trip and provide a small boost to the density. Without a driving 18 Figure 15: Step current vs. time. Figure 16: Light intensity from facet 1 vs. time (r 1 = 0:32;r 2 = 0:20;r 3 = 0:80). current, however, the number of diode and fiber photons immediately begin to drop again. This process can repeat for several external cavity round-trip times; however, the photon number in the diode drops to zero much faster than in the fiber. C. Modulatedcurrent Now, we introduce modulation in the injection current, where in addition to the frequency at which the driving current oscillates, we can also vary the amplitude. Of particular interest is 19 Figure 17: Light intensity from facet 3 vs. time (r 1 = 0:32;r 2 = 0:20;r 3 = 0:80). 0 10 20 30 40 50 60 70 0 0.5 1 1.5 2 2.5 3 Time, t (ns) Carrier density, n (10 18 cm −3 ) Figure 18: Carrier density vs. time (r 1 = 0:32;r 2 = 0:20;r 3 = 0:80). seeing how the system responds to a sinusoidally-oscillating current, driven at a frequency near the inverse of the external cavity round-trip time. Hence, the current is now given by I(t) =A sin(!t) +I 0 (12) If we choose to set ! = ! t 2= and evolve the system for several round-trip times, we 20 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 35 Carrier density, n (10 18 cm −3 ) Power, L (mW/facet) Figure 19: Light intensity from facet 1 vs. carrier density (r 1 = 0:32;r 2 = 0:20;r 3 = 0:80). observe oscillating behaviour in which the system itself oscillates, both in the case of the carrier density, and correspondingly in the light output from facet 1 as well. This is done for three sets of parameters; for the first set we have: r 1 = 0:32;r 2 = 0:32;r 3 = 2 10 6 ;A = 2;I 0 = 5:75, so effectively no external cavity. For the second set of parameters we now apply the AR coating facet 2 and set-up an external cavity: r 1 = 0:32;r 2 = 2 10 6 ;r 3 = 1;A = 1:5;I 0 = 4. The third set of parameters are the same as the second, but nowI 0 = 28 mA andA = 5 mA. These results for the first set of parameters are shown in figures (20) - (23). By sweeping the various control parameters available in the model, we can explore the available phase space of the system. III. EXTENDEDMODEL A much more common method to model the non-linear dynamics of external cavity lasers involves coupling a third differential equation to those describing the photon and carrier densities; namely, the phase associated with the average strength of the electric field in the active region. We begin with the equation of Lang and Kobayashi [2] for the complex electric field, given by 21 Figure 20: Injected current vs. time (r 1 = 0:32;r 2 = 0:32;r 3 = 2 10 6 ). Figure 21: Light intensity from facet 1 vs. time (r 1 = 0:32;r 2 = 0:32;r 3 = 2 10 6 ). dE dt = i!n(t) + 1 2 (G n (nn 0 )) E + 1 X k=1 ( p r 2 r 3 ) k1 E(tk); (13) in which multiple reflections of the electric field in the external cavity are accounted for in the feedback term, and where 22 0 10 20 30 40 50 60 70 0 0.5 1 1.5 2 2.5 Time, t (ns) Carrier density, n (10 18 cm −3 ) Figure 22: Carrier density vs. time (r 1 = 0:32;r 2 = 0:32;r 3 = 2 10 6 ). Figure 23: Light intensity from facet 1 vs. carrier density (r 1 = 0:32;r 2 = 0:32;r 3 = 2 10 6 ). = c(1r 2 ) 2L d n r r r 3 r 2 ; (14) withc the speed of light,n r the average refractive index in the active region, andL d the length of the semiconductor diode. The gainG n in equation (13) is given by 23 G = G slope c n (1" bulk S) (15) with being the optical confinement factor,G slope the optical gain-slope coefficient,n 0 the trans- parency carrier density, and" bulk the gain saturation coefficient. is the loss rate, and is calculated as the sum of the total mirror loss and an internal loss. We need to separate equation (13) into two coupled equations to study the time dependence of the electric field amplitude and its associated phase. The higher-order terms in equation (13) can be neglected in the limit of either weak feedback from the external cavity mirror or where the facet between the active and passive regions is AR-coated, respectively corresponding to either most of the light leaving the external cavity system or completely returning to the active region. Under such conditions, the feedback term in the electric field rate equation simply reduces toE(t), where is the feedback coefficient. The time-dependence of the electric field amplitude and phase can be separated usingE(t) =E 0 e i and the approximation !n(t)! 0 + 2 G n (nn th ); (16) with! 0 as the solitary laser oscillation mode frequency, as the linewidth enhancement factor, and n th as the carrier density at threshold. The carrier density generally induces change in the refrac- tive index of semiconductor lasers [3–5], and this is taken into consideration in Lang-Kobayashi models of external cavity lasers. Taking the time derivative of the electric field in terms of its amplitude and phase yields _ E = d dt E 0 e i = _ E 0 e i +iE 0 e i _ (17) and making the proper substitutions on the right-hand-side yields _ E = i ! 0 + 2 (nn th )G n E 0 e i + 1 2 (G n (nn 0 ))E 0 e i +E 0 (t)e i(t) (18) 24 multiplying through bye i and setting! 0 to zero, we have _ E 0 +iE 0 _ =i 2 (nn th )G n E 0 + 1 2 (G n (nn th ))E 0 +E 0 (t) (cos ((t))i sin ((t))) (19) Equating the real and imaginary components, we are left with two coupled differential equations for the electric field amplitude and its phase in the active region, _ E 0 = 1 2 (G n (nn 0 ))E 0 +E 0 cos ((t)) _ = 2 (nn th )G n E 0 (t) E 0 (t) sin ((t)) (20) Along with the carrier density equation in (1), the equations in (20) describe the basic dynamics of an external cavity semiconductor laser system. An equivalent formulation of these equations can be derived in terms of the photon density rather than the electric field, in order to allow for the continuous monitoring of optical intensity in the gain medium. Since the photon density is given asS(t) =jE(t)j 2 =jE 0 (t)e i(t) j 2 =jE 0 j 2 , we can write the real electric field amplitude asE 0 = p S and easily solve for a set of equations describing the time evolution of the photon density, as well as the phase of the field as a function of photon density, _ S = (G n (nn 0 ))S + 2 p S(t)S (t) cos ((t)) _ = 2 (nn th )G n s S(t) S(t) sin ((t)) (21) We can augment the photon density rate equation by incorporating spontaneous emission into the lasing mode with a positive rate termr spon in equation (22). Such a term is formulated in terms of the radiative recombinationB and carrier density, such thatr spon =Bn 2 and the spontaneous emission coefficient. We can incorporate multiple reflections from the external cavity by using the sum formulation of the feedback term in equation (13) to arrive at a final set of equations encompassing all the dynamics in our model, given as 25 _ S = (G n (nn 0 ))S +Bn 2 + 2 1 X k=1 (r 2 r 3 ) k1 p S(t)S (t) cos ((tk)) _ = 2 (nn th )G n 1 X k=1 (r 2 r 3 ) k1 s S(t) S(t) sin ((tk)) (22) Again, we repeat the same analysis as with the basic model, including sweeps through param- eter space to study the available phase space. A. Chaos Another common behaviour seen in driven systems is chaos, which can arise in semiconductor lasers due to their sensitivity to external feedback light. Although there are several parameters one may use to describe laser instabilities resulting in chaos, one typical parameter is the external facet reflectivity. These instabilities are typically categorized into several different regimes, depending on the light feedback fraction [6–14]. Typically, small feedback is required to produce chaos, on the order of 1%. Near this fraction of light feedback, the relaxation oscillation becomes undamped and the laser linewidth is broadened. Besides an enhancement in noise level, the laser system can display chaotic output depending on the operating conditions (such as driving frequency and modulation depth of the injection current), and evolves into unstable oscillations in a coherence collapse state. Since we have three coupled, non-linear differential equations describing the time evolution of three independent dynamical variables, it is possible to generate chaotic dynamics in the system. In particular, the various dynamics resulting from incorporating phase into the model can be attributed to different feedback regimes corresponding to various levels of re-injected light. When the amount of feedback is on the order of 0.01%, the laser linewidth can broaden or narrow, depending on the exact feedback fraction. For feedback on the order of 0.1%, the generation of external cavity modes can give rise to mode-hopping among the internal and external modes. Re-injecting slightly more light can send the laser into a narrow, third feedback regime in which the noise associated with mode-hopping is suppressed, and the laser again oscillates with a narrow linewidth. 26 For feedback on the order of 1%, the relaxation oscillations in the diode cavity can become undamped and the laser linewidth is broadened. Under such conditions, the laser can evolve into unstable oscillations and is typically referred to as being in a coherence collapsed state. Here, noise levels are significantly enhanced, and the optical intensity from the active region has a chaotic time series. When the model parameters satisfy such conditions, chaotic light can be generated, shown below in figures (24) - (25). 990 992 994 996 998 1000 0 0.5 1 1.5 2 2.5 3 3.5 Intensity vs. Time Time, t (ns) Optical power, L (mW/facet) Figure 24: Light output vs. time. This chaotic behaviour can be numerically checked using a higher order integrator. Because chaos is quite sensitive to initial conditions, any small deviations away from the attractor in phase space will result in a different history for the time series. Finally, the fifth feedback regime is asso- ciated with feedback levels on the order of 10% or greater. Here, the internal and external cavities conform like a single cavity and the laser linewidth is greatly narrowed as the laser oscillates with a single mode. 27 2.3 2.32 2.34 2.36 2.38 2.4 2.42 2.44 2.46 2.48 0 0.5 1 1.5 2 2.5 3 3.5 Light intensity vs. Carrier density Carrier density, n (10 18 cm −3 ) Power, L (mW/facet) Figure 25: Intensity vs. carrier density phase space. IV. APPENDIX Typicalsemiconductorlaserparametersused In table (I), we list all the typical parameters used in producing the figures above. Several of these parameters were varied in order to observe different transient and steady-state behaviour in the diode-fiber cavity system. 28 Table I: Fabry-Perot diode (active region) parameters Description Parameter Value Refractive index n r 4 Diode length L d (cm) 3:00 10 2 Active region thickness t a (cm) 1:40 10 5 Active region width w a (cm) 8:00 10 5 Integration time increment t inc (s) 1:00 10 12 Non-radiative recombination rate A nr (s 1 ) 2:00 10 8 Radiative recombination coefficient B (cm 3 s l ) 1:00 10 10 Non-linear recombination coefficient C (cm 6 s l ) 1:00 10 29 Transparency carrier density n 0 (cm 3 ) 1:00 10 18 Threshold carrier density n th (cm 3 ) 2:0576 10 18 Optical gain-slope coefficient G slope (cm 2 s 1 ) 2:50 10 16 Gain compression " bulk (cm 3 ) 3:00 10 18 Spontaneous emission coefficient 1:00 10 3 Optical confinement factor 0.25 Optical emission wavelength (m) 1.31 Mirror 1 reflectivity r 1 0.50 Mirror 2 reflectivity r 2 0.85 Internal optical loss i (cm 1 ) 40 Linewidth enhancement factor 1.0 External cavity round-trip time (ns) 5 29 Chapter1References [1] G. Björk and Y . Yamamoto, IEEE Journal of Quantum Electronics27, 11 (1991). [2] R. Lang and K. Kobayashi, IEEE Journal of Quantum Electronics16, 3 (1980). [3] B.R. Bennett, R.A. Soref, J.A. Del Alamo, IEEE Journal of Quantum Electronics26, 1 (1990). [4] G.R. Gray and G.P. Agrawal, IEEE Photonics Technology Letters4, 11 (1992). [5] L. Wu, Appl. Phys. Lett.77, 1777 (2000). [6] G.H.M. van Tartwijk and G.P. Agrawal,LaserInstabilities: AModernPerspective, (Elsevier Science, 1998). [7] J. Ohtsubo,SemiconductorLasers: Stability,InstabilityandChaos, (Springer, 2012), 3rd ed. [8] N. Kikuchi, Y . Liu, and J. Ohtsubo, IEEE Journal of Quantum Electronics33, 1 (1997). [9] R.J. Jones, P.S. Spencer, J. Lawrence, and D.M. Kane, IEEE Proceedings - Optoelectronics 148, 1 (2001). [10] A.P. Willis and D.M. Kane, Optics Communications107, 1-2 (1994). [11] A.P. Willis and D.M. Kane, Optics Communications111, 5-6 (1994). [12] J.S. Lawrence and D.M. Kane, Optics Communications167, 1-6 (1999). [13] J.S. Lawrence and D.M. Kane, IEEE Journal of Quantum Electronics38, 2 (2002). [14] B.C. Lam,etal., Journal of the Optical Society of America B12, 6 (1995). 30 Chapter2: ControlofParticleStatisticsin MesoscaleLasers In this chapter, we consider a single-mode semiconductor meso-scale laser subjected to optical feedback. Previously, solitary lasers in the meso-scale have been described by birth-death models in which particle number is quantized and transitions between discrete particle states are contin- uous in probability. Here, we modify a master equation describing the dynamics of mesolasers by introducing non-Markovianity via an external cavity in order to control the system statistics when fluctuations and correlations in particle number are significant. By projecting out the photon probability distribution of the gain medium and re-injecting it back into the laser system at time later, it is possible to explore how particle statistics are affected, including higher moments of the particle subsystems and the associated classical mutual information between the photons and excitations. The mutual information is interpreted as a potential for the system to explore the available phase space, which is bounded by the total energy at any given time. For large electron and photon numbers (n;s >> 1), the meso-mean field results of such an approach qualitatively agree with the thermodynamic limit, where correlations vanish. By varying the round-trip time and using a saturable absorber in the external cavity to quench the excitations in the active gain medium, various forms of control are demonstrated over the particle statistics and correlations. V. CH.2INTRODUCTION The physics that describe the operation of a conventional laser is well understood in the ther- modynamic limit, where there exists a large number of excited states in the system, the fraction of spontaneous emission feeding into the lasing optical mode is small ( 10 4 ), and there is a well-defined threshold between incoherent photon emission into non-lasing modes and co- herent lasing photon emission. Such behaviour can be approximately described using continuum mean-field rate equations. Beyond these relatively crude models, there are also quantum statistical theories of laser operation, which, when evaluated in the large particle number limit, success- fully reproduce the continuum mean-field results. Beyond modelling particle number fluctuations and statistics in the thermodynamic limit, such theories could be used to explore different forms of controlling optical noise, particle probability distributions, and transient dynamics in discrete 31 systems with continuous probability [1–4]. VI. MESOLASERMODEL In the limit of few particles, mean-field rate equations can not be utilized to effectively describe the dynamics of a semiconductor gain medium. In addition to the increased importance of fluctua- tions in photon number and electron number, correlations between photons and electrons prohibit factorization of average particle numbers. While typically mean-field rate equations can be used to study laser systems in the thermodynamic limit, master equations can be utilized to study the transient evolution of discrete quantum systems with continuous probability distributions. The master equation describing the evolution of probabilitiesP n;s associated with particle states (n;s) for a solitary single-mode semiconductor laser with photon emission energy~! is given as [5–7] dP n;s dt = I e (P n;s P n1;s ) (sP n;s (s + 1)P n;s+1 ) (sG n P n;s (s 1)G n+1 P n+1;s1 ) (sAP n;s (s + 1)AP n1;s+1 ) B n 2 P n;s (n + 1) 2 P n+1;s1 (1)B n 2 P n;s (n + 1) 2 P n+1;s (23) or in compact notation, _ P n;s = ~ CP n;s : (24) where the elements of the matrixC are comprised of the various coefficients from equation (23). The various processes described by equation (23) can easily be mapped onto a birth-death diagram (see figure (26), which schematically represents the flow of probability in and out of a particle state (n;s). In equation (23), the device is driven by currentI. The first term,sP n;s (t), represents cavity photon decay due to internal and mirror losses, in which transition occur from state (n;s) to state (n;s 1) and where is the optical loss coefficient. The termsG n P n;s (t) describes photons that undergo stimulated emission from state (n;s) to (n 1;s + 1) whereG n is the stimulated emission coefficient. The third set of terms containingAsP n;s represents stimulated absorption of photons, where transitions occur from (n;s) to (n + 1;s 1) and whereA is the stimulated 32 Figure 26: Probability flow diagram absorption coefficient. The termBn 2 P n;s (t) describes the spontaneous emission of photons into the lasing mode, in which a state (n;s) transitions to a state (n 1;s + 1), is the fraction of spontaneously-emitted photons, andB = B 0 =V is the radiative recombination constant divide by the device volume. The decay of electrons into non-lasing photons is represented by the term (1)Bn 2 P n;s (t), in which particle state (n;s) transitions to state (n 1;s). Finally, the last set of terms describes the pumping of the system via an injected currentI, such that one electron is added to the gain medium every interval of timet I =e=I withe being the electron charge. This causes transitions of probability from a state (n;s) to (n + 1;s), where the total probability of the system is conserved for all times. The particular values of each coefficient for the results obtained in this section are listed in the appendix section. VII. ENTROPYANDMUTUALINFORMATION Controlling the mesolaser requires coupling the system to the environment, which is repre- sented by the pump and mirror loss terms in the master equation. The microstate transition timescales associated with these processes not only introduces additional complexities to the in- 33 ternal dynamics, but allow for the injection and dissipation of total energy in the system. Such dynamics change the various entropies of the system, which can fundamentally limit the maxi- mum amount of information that can be extracted for optical signal processing. Due to the finite set of particle number microstates, the quantization of the total system energy is described by a Fock basis, so it is natural to utilize Shannon entropy in defining the time-dependent mutual in- formation between the two particle subsystems, M I (n;s)(t) H n (t) +H s (t)H n;s (t). From information theory, the entropy of a random variable is a measure of its uncertainty [8]; for the mesolaser model, then, the electron(photon) marginal entropy and the joint system entropy (mea- sured in units of bits) are H n(s) (t) = X i(j) P n(s) (i(j);t) log 2 P n(s) (i(j);t) H n;s (t) = X i;j P n;s (i;j;t) log 2 (P n;s (i;j;t)); (25) where the marginal probability distributions are projections over the respective particle spaces, P n(s) (t) = X s(n) P n;s (t); (26) and where the sum indices correspond to the allowed electron(photon) microstates in the system, 0 i(j) N n (N s ), where N n (N s ) are the maximum electron(photon) numbers possible due to the finite energy present in the system, which constrains probabilities for particle microstates greater than (N n ;N s ) to zero. In addition to the mutual information, a second measure can be formulated to study the cor- relations between the two subsystems. Utilizing the fact that the average particle numbers do not factorize, we define the correlation measure hnihsihnsi to quantify the degree of correlation between electrons and photons. A. Closedsystemdynamics The various directed transition processes described by the master equation imply that the dy- namics of the system distribution are not simply diffusive in nature. The fact that probability is forced to spread in various directions within the particle phase space over time, however, allows 34 one to explore the relaxation dynamics. Before one can even say something about the controlla- bility of mesolaser microstates, the effective time scales associated with such relaxation must first be examined. To understand entropy and mutual information within the context of laser systems, we first consider closed system dynamics, in which there is no pump or loss in the system. Since only the internal dynamics are being considered, the system must be initialized in a non-zero particle mi- crostate. This initial energy allows the system to explore the available phase space via a diffusion of the probability distributionP n;s . If the initial system at time t 0 is modelled as a cold cavity by setting P 0;0 (t 0 ) = 1 then all other microstate probabilitiesP n;s (t 0 ) = 0 forn +s 1. In order to model a laser system that is completely isolated from the environment, there must be no loss or gain of particles in the laser due to internal losses or coupling to the external environment. Such a setup may be achieved by neglecting the pump and loss terms, namely the injection current, electron loss, and mirror loss terms in the original master equation. Thus, the isolated system master equation is given as dP n;s dt = (sG n P n;s (s 1)G n+1 P n+1;s1 ) (sA n P n;s (s + 1)A n P n1;s+1 )B n 2 P n;s (n + 1) 2 P n+1;s1 : (27) Creating an initial condition represents a fixed point in the Fock phase space, since no energy is injected into, or lost from, the system. Thus, to study the effective transition rates from a single microstate into other allowed states, the system must be initiated with some energy. This is accomplished by initializing the probability distribution in a delta state, P n;s (t = 0) = n;n 0 s;s 0 ; (28) wheren 0 ands 0 are the initial excitation and photon numbers. Due to the presence of diagonal transition processes in (27), the evolution of the probability distribution is anisotropic in the par- ticle phase space, which requires examination of the coupled systemP n;s probability distribution, rather than the projected subsystem distributionsP n andP s . Furthermore, since the transition rates in the coefficient matrix depend on the number of particles in the system, one would expect the diffusion timescales to depend on the system size N. Figure (27) shows the initial probability 35 distribution of the laser system, in which the chosen initial microstate (n 0 ;s 0 ) = (20; 20), so that P 20;20 = 1. Over the course of several picoseconds, the diagonal transition processes evolve the probability until steady-state is asymptotically reached, as shown in figure (28). Such probability relaxation occurs relatively quickly in the laser, since stimulated processes dominate the internal dynamics of the isolated system above lasing threshold conditions. 0 10 20 30 0 10 20 30 Electron number, n Photon number, s 10 log 10 P n, s −50 −35 −20 Figure 27: Initial distribution configuration for isolated laser system. Initially, there are 20 electrons and 20 photons in the system, for a total system size ofN 0 = 40. Using the freedom to choose any initial microstate, we study the system’s ability to explore the surrounding phase space by fitting the exponential decay of the probability peak in time. Figure (29) shows the decay of the coupled system probability peak for different system sizes, for typ- ical Fabry-Perot laser parameters listed in Table (1). As the system size is increased, the decay timescale shortens and steady-state conditions are reached faster. Likewise, the increase in system entropy occurs over shorter timescales for larger system sizes. Figure (30) shows the increase in entropy, starting at the initial time where the entropy is exactly zero since the entire distribution is localized to a single microstate, (n 0 ;s 0 ) = (20; 20). 36 Figure 28: Steady-state probability distribution in the particle phase space for the isolated laser system occurs aftert = 10 ps for the initial distribution shown in figure(27). 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Time, t (ps) max(P n,s ) N 0 = 10 N 0 = 20 N 0 = 30 N 0 = 60 N 0 = 80 N 0 = 100 N 0 = 10 N 0 = 100 Figure 29: Decay in the probability peak for the isolated laser system, due to diffusion via diagonal tran- sitions processes from internal laser dynamics. Various different initial system sizes yield different decay timescales, so a single diffusion constant cannot be used to describe the system dynamics. 37 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 Time, t (ps) Joint entropy, H n,s (bits) N 0 = 10 N 0 = 20 N 0 = 40 N 0 = 60 N 0 = 80 N 0 = 100 N 0 = 10 N 0 = 100 Figure 30: Transient increase in joint system entropy H n;s for various initial system sizes, due to the exploration of phase space over time. Steady-state conditions are asymptotically reached faster for larger system sizes. 38 B. Opensystemdynamics Coupling to the environment via a pump I and mirror loss introduces more complexities in the dynamics of the mutual information due to the additional timescales. Incorporating the injection current and mirror loss transition terms in the master equation solution couples the laser dynamics to the external environment, which introduces additional timescales in the gain medium and complexity in the dynamics. For a given initial system size (initial energy), the pumpI and the mirror loss respectively dictate the rates at which new electrons are added to the system and photons are lost. Initializing the system at different microstates affords the laser various paths to explore the available phase space. For the same initial microstate configuration shown in figure (27), figure (31) shows the probability distribution of the open laser system near steady-state conditions. Figure 31: Steady-state probability distribution in the particle phase space for the open laser system occurs aftert = 10 ps for the initial sizeN 0 = 40. Pixilation in distribution is due to discrete and finiten ands values for a mesolaser. Coupling to the external environment allows non-diagonal transitions in the probability and provides the laser extra degrees of freedom to explore more of the particle phase space. As with the isolated system, the diffusion of the probability is studied for different system sizes in the open system, for which figure (32) shows the diffusion of probability. The dependence of the probability diffusion on such non-diagonal transition processes introduces new degrees of sensitivity with 39 respect to the environmental coupling. In general, one should expect that the extra degrees of freedom allotted by the environment shorten the probability timescales, since the probability can be distributed in a greater number of ways. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Time, t (ps) max(P n,s ) N 0 = 10 N 0 = 20 N 0 = 40 N 0 = 60 N 0 = 80 N 0 = 100 N 0 = 10 N 0 = 100 Figure 32: Decay in the probability peak for the open laser system, due to diffusion via internal laser dynamics and external couplings. Various different initial system sizes yield different diffusive timescales, which are noticeably shorter than in the closed system. Thus for example, if one wished to maintain at least 60% of a given microstate configuration for a system size of 10 particles, the level of control would require timescales on the order of 1 ps or shorter. Furthermore, the dynamics generated by coupling to the external environment can affect the increase of entropy in the system. Figure (33) demonstrates that as the system size becomes greater, the joint system Shannon entropy H n;s correspondingly increases, due to the increased exploration of phase space and hence the greater number of possible states of the laser when performing a measurement. 40 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 Time, t (ps) Joint entropy, H n,s (bits) N 0 = 10 N 0 = 20 N 0 = 40 N 0 = 60 N 0 = 80 N 0 = 100 N 0 = 10 N 0 = 100 Figure 33: Transient increase in joint system entropy H n;s for various initial system sizes, due to the exploration of phase space over time. Steady-state conditions are reached faster for larger system sizes. C. Systemsizescaling It is well known that in the thermodynamic limit, continuum mean-field models capture many essential features of laser dynamics. Furthermore, correlations between subsystems become negli- gible as the particle numbers factorize. To understand the transition between quantum and classical laser systems, it is important to study how the correlations and mutual information scale with re- spect to system size, defined as the sum of average particle number, N hni +hsi. In the open system case, pumping the laser with a step current non-linearly increases the system size below threshold. Above threshold, where the dynamics are dominated by the stimulated emission timescales, the system size increases linearly with respect to current. Figures (34) - (39) show how the correlation measure and the mutual information I(n;s) decay as the steady-state system sizeN increases, for a fixed mirror loss. System behavior towards the thermodynamic limit may be examined by fitting the decay of the correlations and mutual information. The correlation is observed to fall off exponentially with a chararcteristic system sizeN c that depends on the size of the cavity. In particular,N c is inversely proportional to the cavity lengthL d since correlations are stronger for a given system size when 41 50 100 150 200 250 8.5 9 9.5 10 System size, N ≡ 〈 n 〉 + 〈 s 〉 System correlation, Δ (V d = 5120 nm 3 ; r 1 = r 2 = 0.999) 50 100 150 200 250 15.4 15.6 15.8 16 16.2 16.4 16.6 16.8 17 17.2 System size, N ≡ 〈 n 〉 + 〈 s 〉 System correlation, Δ (V = 2 × V d ; r 1 = r 2 = 0.999) β = 1.0 β = 10 −1 β = 10 −2 β = 10 −3 β = 1.0 β = 10 −1 β = 10 −2 β = 10 −3 Figure 34: The decay of correlations with respect to steady-state system size, for two different volumes. 1 2 3 4 5 6 7 8 x 10 4 0 1 2 3 4 5 6 7 8 9 10 System size, N ≡ 〈 n 〉 + 〈 s 〉 System correlation, Δ (V d = 5120 nm 3 ; r 1 = r 2 = 0.999, β = 0.1) V = V d V = 2V d Figure 35: Extrapolation of correlation decay towards the thermodynamic limit. the cavity Q-factor is greater. This is shown in figure (34), where correlations for different laser volumes converge in the thermodynamic limit, as shown in figure (35). The mutual information follows a power law behavior with respect to system size, where the steady-state value is inversely 42 proportional to system size and is significantly more fragile than the correlations, as shown in figures (38) and (39). 40 60 80 100 120 140 0 0.05 0.1 0.15 0.2 0.25 System size, N ≡ 〈 s 〉 + 〈 n 〉 Mutual information, I(n;s) (V d = 5120 nm 3 ; r 1 = r 2 = 0.999) V = V d , β = 1.0 V = V d , β = 10 −1 V = V d , β = 10 −2 V = V d , β = 10 −3 V = 2V d , β = 1.0 V = 2V d , β = 10 −1 V = 2V d , β = 10 −2 V = 2V d , β = 10 −3 Figure 36: The decay of mutual information with respect to steady-state system size, for two different volumes. In addition to system size scaling from increasing the pump, the steady-state correlations and mutual information can be studied for a constant system size above threshold as a function of the coupling between the system and the bath, where such a coupling is represented by the loss in the system,. The decrease in information between the particle subsystems with an increase in loss physically corresponds to a stronger bath coupling since the cavity Q is lowered. 43 100 200 300 400 500 600 700 800 900 1000 0 0.05 0.1 0.15 0.2 0.25 System size, N ≡ 〈 s 〉 + 〈 n 〉 (V d = 5120 nm 3 ; r 1 = r 2 = 0.999) Mutual information, I(n;s) V = V d V = 2V d Figure 37: Extrapolation to the thermodynamic limit where the classical mutual information vanishes. 40 50 60 70 80 90 100 110 120 130 0 0.5 1 1.5 2 2.5 3 3.5 4 Cavity loss, κ (THz) Correlation, Δ N = 120 N = 140 N = 160 N = 180 Figure 38: The decay of correlations with respect to bath coupling. The characteristic loss is approximately 10 GHz. 44 40 50 60 70 80 90 100 110 120 130 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Cavity loss, κ (THz) Mutual information, I(n;s) N = 120 N = 140 N = 160 N = 180 Figure 39: The decay of mutual information with respect to bath coupling. 45 VIII. INTRODUCINGANEXTERNALCAVITY A. FeedbackControl In order to include optical feedback in a probabilistic picture, the original master equation is extended to include a feedback term describing a passive external cavity that contains memory about the photon probability distribution in the active region a time ago, where is the external cavity round-trip time. To properly incorporate delayed feedback of probabilities among all (n;s) states, the photon probability distribution P s is first projected out from the total probability distribution P n;s after each electron time-step in an additive fashion, meaning that for a given photon number s 0 , the probabilities associated with all states (s 0 ;n) are summed to give a valueP s 0, wheren ranges from 0 to the total number of electrons added to the system. This is done for 0 s 0 S, whereS is the maximum number of photons allowed in the system at that time. This probability array is then stored for every time-step, and after the external cavity round-trip time has passed, the series of photon probability distributionsP s;t are fed back into the active region and distributed among the (n;s) states with a weighting equal to the relative probability strengths of each possible electron state for each given value ofs at the present time. Thus, we define P (d) n;s (t) =P n;s (t) P n;s (t) P s P n;s (t) P s (t); (29) We can then modify equation (23) by adding a set of delayed transition rate terms which in- duce transitions from particle states (n;s 1) to (n;s), wherein the memory is contained in the probability arrayP (d) n;s ; including multiple reflections inside the external cavity, these rate terms are 2 n X j=1 [(1r 2 )r (j1) 2 r j 3 (sP n;s (tj) (s 1)P n;s1 (tj))]; (30) which simplify under the considerations of just one external cavity round-trip, weak feedback, or AR-coating the facet coupling the active and passive regions to just the first-order terms, 2 (1r 2 )r 3 (sP n;s (t) (s 1)P n;s1 (t)); (31) 46 where 2 is given in terms of the diode mirror reflectivities in (8). The delayed transition dynamics are diagrammatically represented in figure (40). Figure 40: Birth-death schematic for feedback probability transition rates. Such transitions are time-delayed in the system by the external round-trip time. In matrix representation, the master equation is thus modified to include a coefficient matrix associated with the delayed probability array: _ P n;s = ~ CP n;s + ~ C (d) P (d) n;s (t) (32) Once the delayed coefficient matrix is numerically implemented, the photon distribution is projected fromP n;s (t) after each time-step, and stored as a separate array. If we examine the time evolution of the photon and electron statistics and probability distributions in Figs. (41) - (43), we can readily see how they differ from the solitary laser case. In the case of the photon probability distribution, the average value slightly shifts to a higher value due to the flow of probability. The peak is enhanced with each round-trip, as probability is significantly drawn away from all states with zero photons. The electron probability distribution is similarly affected. The return of photons causes a suppression of probability associated with greater number of electrons, and 47 this probability flows into the main peak, causing it to raise. Immediately after the peak of the first relaxation oscillation is introduced into the device, probability is again kicked back into states with a greater number of electrons, causing a slight increase in the distribution, aroundn = 25. Hence, the rise in probability for lower number of electrons (and suppression of probability for higher number of electrons) is associated with the re-injected photons. Figure 41: (1)hni vs t (2)hsi vs t (3)Fano factor vs t (4)hsi vshni 48 Time, t (ns) Photon number, s 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 P s contour map <s> σ s σ s 2 /<s> 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 Figure 42: Time evolution of photon probability distributionP s , with feedback incorporated. It is important to note that while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 Time, t (ns) Electron number, n P n contour map <n> σ n σ n 2 /<n> 0.1 0.2 0.3 0.4 0.5 0.6 Figure 43: Time evolution of electron probability distributionP n , with feedback incorporated. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. 49 B. Transientdynamicalcontrol Beyond such expected results in the time evolution of the particle probability distributions, one can ask whether such a probabilistic model for external cavity semiconductor lasers in the meso regime can allow for the control of particle statistics. Since such a model does not include information about phase, it is not a quantum model; however, as a first cut in such a direction, it may be used to explore how the various moments, distributions, and fluctuations can be affected by varying certain control parameters. In particular, it is interesting to consider the question of how the photon and electron numbers are affected by choosing the moment at which light is fed back into the system. As a first step towards exploring the dynamics of the particle probabilities, we examine the mean field model described by equations (1) and (2). For the set of parameters given in the appendix, we begin by shortening the round-trip time of the external cavity and examine the effect on the optical power within the active region. Choosing the external cavity length to re-inject the light at different minima or maxima of the relaxation oscillations corresponds to affecting the system dynamics at different points in the dynamical phase space. As in figure (44) When = 0:15 ns, the initial burst of light hits the active region at the first minimum of the relaxation oscillations, which is when the electron density is starting to decrease. The system is periodically excited from the light feedback and the relaxation oscillations persist for several round-trips of the external cavity, causing the system to take longer to reach steady-state. At = 0:01 ns, the external cavity is short enough such that the rapid rise in photon density hits itself before it peaks, causing a higher initial relaxation oscillation peak. The external cavity is sufficiently short that the system reaches steady state relatively quickly, since the external cavity mirror acts to effectively increase the reflectivity of the mirror coupling the active and passive regions. Hence, the final steady-state optical power is higher than in the case of the solitary laser ( =1). Performing this type of external cavity control with the meso-scale model, we choose the length of the external cavity (the external cavity round-trip time) such that the first relaxation oscillation peak is re-injected at different points in the average particle phase space, as seen in figure (45). Re-injecting the photon peak at the first local minima of the photon relaxation oscillations corresponds to re-injecting light at a local maxima of the carrier oscillations; in the case of a 50 0 1 2 3 0 20 40 60 Power, L (mW/facet) 0 1 2 3 0 20 40 60 0 1 2 3 0 20 40 60 0 1 2 3 0 20 40 60 0 1 2 3 0 20 40 60 Power, L (mW/facet) 0 1 2 3 0 20 40 60 τ = ∞ τ = 0.5 ns τ = 0.4 ns τ = 0.3 ns τ = 0.2 ns τ = 0.1 ns 0 1 2 3 0 20 40 60 0 1 2 3 0 20 40 60 Power, L (mW/facet) Time, t (ns) 0 1 2 3 0 20 40 60 Time, t (ns) 0 1 2 3 0 20 40 60 τ = 0.08 ns τ = 0.06 ns τ = 0.04 ns 0 1 2 3 0 20 40 60 Time, t (ns) 0 1 2 3 0 20 40 60 Time, t (ns) τ = 0.15 ns τ = 0.02 ns τ = 0.01 ns Figure 44: Feedback control on light intensity in mean-field model. 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Mean electron number, <n> Mean photon number, <s> τ = 0.02 τ = 0.20 τ = 0.50 τ = 0.85 τ = ∞ Figure 45: Feedback introduced at different phase space locations. longer external cavity, photon re-injection at a local photon maxima corresponds to a local minima in average electron number. More importantly, we can directly compare the photon and electron distributions separately, for different external cavity round-trip times. Figures (48) - (50) display 51 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time, t (ns) mean photon number, <s> τ = 0.02 τ = 0.20 τ = 0.50 τ = 0.85 τ = ∞ Figure 46: Time evolution of average photon number for different external cavity round-trip times 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 time, t (ns) mean electron number, <n> τ = 0.02 τ = 0.20 τ = 0.50 τ = 0.85 τ = ∞ Figure 47: Time evolution of average electron number for different external cavity round-trip times 3D information about the photon distribution as it changes in time due to feedback, along withhsi, photon particle standard deviation, and photon Fano factor overlayed for comparison purposes. Figures (51) - (53) display a contour map of the electron probability distribution as a function of time, with the average valuehni, standard deviation, and electron Fano factor overlayed directly 52 upon the distribution map. Such figures represent contour maps of the type of photon and electron probability distributions shown in figures (42) and (43) for different values of. Time, t (ns) Photon number, s 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 data1 <s> σ s σ s 2 /<s> 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 Photon number, s Probability, P s t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 48: Time evolution of photon probability distribution for = 0:20 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. Time, t (ns) Photon number, s 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 P s contour map <s> σ s σ s 2 /<s> 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 Photon number, s Probability, P s t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 49: Time evolution of photon probability distribution for = 0:50 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. It’s important to note that in figures (48) - (50), most of the probability resides with zero-photon states; such data is not displayed since it can take more than ten round-trips before states with a 53 Time, t (ns) Photon number, s 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 P s contour map <s> σ s σ s 2 /<s> 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 Photon number, s Probability, P s t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 50: Time evolution of photon probability distribution for = 0:85 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. non-zero number of photons dominate the distribution; also, it is easier to study how probability flows between all other possible states in time. Upon examination, there are several features worth noting. First, there is an inherent asymmetry in the photon distribution about the average value throughout time; such an asymmetry in the distribution implies that allowed states with less photons than the average will be affected by the feedback differently than allowed states with greater photons than the average. Shortening the time delay has the effect of synchronizing the diode cavity with the external cavity such that steady-state conditions are reached faster, and the effective reflectivity of the coupling diode facet is increased. In the case of the time evolution of the electron probability distribution, the bimodal distribu- tion associated with the initial lasing transition can clearly be made out at approximatelyt = 0:20 nanoseconds. As various snapshots in time, the electron probability distribution appears on the right in figures (51) - (53); within the paradigm of round-trip time control, there is little quali- tative difference in the steady-state distribution for different values of . As the laser achieves steady-state conditions, the distribution peak associated with the higher number of electrons be- gins to shrink as probability flows towards the peak with a lower mean value of electron number. Likewise, for the photon distribution, the system starts with effectively all of the probability dom- inating the zero-photon states, and probability is drawn away from the zero-states, forming a peak at higher photon number; 54 0.5 1 1.5 2 0 2 4 6 8 10 12 Time, t (ns) Electron number, n P n contour map <n> σ n σ n 2 /<n> 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 Electron number, n Probability, P n t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 51: Time evolution of electron probability distribution for = 0:20 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6 8 10 12 Time, t (ns) Electron number, n P n contour map <n> σ n σ n 2 /<n> 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 Electron number, n Probability, P n t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 52: Time evolution of electron probability distribution for = 0:50 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. Increasing the time delay of the external cavity has similar effects on the electron distribution as it did with the photon distribution; additionally, the effect of re-injected photons causes a shift of probability toward states with a smaller number of electrons, represented by the creation of valleys in figure (43), and dips in the contour lines att = 1:25 ns andt = 2:25 ns in figure (53) toward such states. Such dips directly correspond to the shifts in photon probability towards higher photon 55 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 4 6 8 10 12 Time, t (ns) Electron number, n P n contour map <n> σ n σ n 2 /<n> 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 0 0.05 0.1 0.15 0.2 0.25 Electron number, n Probability, P n t = 0.2 ns t = 0.4 ns t = 1.0 ns t = 1.6 ns t = 2.0 ns Figure 53: Time evolution of electron probability distribution for = 0:85 ns. Note: while the probability distribution is displayed in an interpolated manner here, the distribution is actually discrete, due to the finite particle number microstates available. number states in figure (50), for example. From figures (48) - (50), we see that over time, the entire photon probability distribution tends to dominate higher photon numbers when the feedback is in resonance with the relaxation oscillations, while the electron probability distribution in this case dominates lower electron numbers. Examining figure (46), we see that in the limit of zero round-trip time, one recovers the solitary laser transient dynamics, albeit with a higher steady-state intensity due to a greater effective mir- ror reflectivity caused by the external cavity mirror. Comparing figures (48) and (50), we see that shorter round-trip time induces a greater amount of optical noise via the re-injection of light, and hence a broader photon distribution peak. Overall, observed effects were rather small for the set of modified parameters used, since relatively small volume and injected current values were used in order to achieve lasing faster and to evolve the system for enough external round-trips to observe the behaviour of the system as it approaches steady-state conditions (cavity formation). In order to exert more control over such statistics, a combination of larger computational resources as well as numerical truncation should be utilized. With larger currents and smaller values of, the relax- ation oscillation amplitudes can be further increased to explore faster changes in the distributions as well as particle number fluctuations, which are quantified by the Fano factor. Figures (54) - (55) show the particle Fano factors as a function of time for the meso device for two different external cavity round-trip times, wherein feedback significantly increases fluctuations in the system. 56 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time (ns) Photon Fano factor τ = 0.50 ns τ = 0.85 ns Figure 54: Photon Fano factor vs. time. The system is initially prepared as a cold cavity. Feedback not only affects the particle subsystem statistics, but the correlations and mutual in- formation between the two subsystems as well. Figures (56) and (57) show the effect of round-trip time control on such correlations. Matching the relaxation oscillation period maximizes oscil- lations in the correlations, and hence mutual information in the laser system, enabling the non- equilibrium system to explore more of the available phase space. 57 0 0.5 1 1.5 2 2.5 3.8 4 4.2 4.4 4.6 4.8 5 time (ns) Electron Fano factor τ = 0.50 ns τ = 0.85 ns Figure 55: Electron Fano factor vs. time Figure 56: Transient dynamics of subsystem correlations for different feedback times. 58 Figure 57: Transient dynamics of mutual information different feedback times. 59 C. Externalsaturableabsorber To exert an additional degree of control on the mutual information, the external subsystem is modified to include a saturable absorber, which serves to modify the projected photon distribution in the external cavity by reshaping it. Such a modification is modelled by P s (t)7! ~ P s (t) = 8 > < > : P s (t);ss 0 P s (t);s<s 0 ; (33) wheres 0 is the saturation photon number and 0 1 is the probability attenuation coefficient. Since absorption of light decreases with increasing light intensity (photon number), the effect of such an modification is to attenuate the probabilities for all photon microstates less than the saturation number in the external cavity. The delayed photon distribution is then renormalized so that total probability remains equal to unity, ~ P s (t)7! ~ P s (t) P s ~ P s (t) (34) The effect of such a renormalization causes the average photon number in the delayed light to increase, as the probability is increased for microstates with higher photon number. Figure (58) shows the transient effect of feedback on the particle subsystem correlations with a reshaped photon distribution. As show in figures (59) - (60), the enhanced time-delayed photon distribution dramatically quenches electron excitations while increasing average photon number; this in turn has a dramatic effect on the mutual information dynamics, suppressing the steady-state result. Shifting the photon probability towards a higher particle number has the effect of quenching the excitations in the active gain medium, which attenuates the subsystem and correspondingly drastically lowers the system correlations. Thus, such a form of feedback control can be utilized to destroy the mutual information in the laser, which severely limits the ability of the laser system to explore the available phase space by dramatically increasing the timescales associated with microstate transitions. 60 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 Time, t (ns) Average particle product <n><s> <ns> Δ Figure 58: Feedback effects on particle correlations. 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 8 9 Mean electron number, 〈 n 〉 Mean photon number, 〈 s 〉 Figure 59: Average electron number is quenched due to enhanced photon distribution. 61 0 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 Time, t (ns) Mutual information, I(n;s) τ = ∞ ns τ = 2 ns Figure 60: Due to subsystem quenching, the mutual information is significantly suppressed. 62 IX. CONCLUSIONS By solving the coupled external cavity rate equations in the continuum mean-field limit, as well as master equation for particle state probabilities, transient dynamics can be studied. By introduc- ing an external cavity and effective saturable absorbers, it is possible to control such dynamics. We have demonstrated such control in meso-scale semiconductor lasers, using the classical mean-field limit as a guide. In the limit of small number of particles, fluctuations can play a dominant role in the evolution of particle statistics and probability distributions. Such fluctuations can dramatically increase the noise in the system and by controlling various feedback mechanisms in the external cavity laser, we have shown that in principle, it is possible to control such noise. Such a model provides an ideal framework for the study of the breakdown of classical thermo- dynamics as fluctuations begin to dominate in few-particle systems. The probability distributions in the model phase space contain information about the history of the laser dynamics at any given moment in time, inherent within the distribution shape. Such information cannot be extracted in continuum mean-field models, allowing for the study of non-Markovian control of entropy in meso-scale systems. Acknowledgements All numerical computations and simulations were performed on the University of Southern California campus. This research was supported by the ARO MURI Grant W911NF-11-1-0268. 63 X. APPENDIX A. Mesolaserparameters Several of the parameters in Table (II) were varied in order to observe different transient and steady-state behaviour in the system. Table II: Meso-scale model diode parameters Description Parameter Value Refractive index n r 4 Diode length L d (cm) 3:3 10 6 Active region thickness t a (cm) 3:3 10 7 Active region width w a (cm) 3:3 10 6 Radiative recombination B (cm 3 s l ) 1 10 10 Transparency carrier density n 0 (cm 3 ) 1 10 18 Optical gain-slope coefficient G slope (cm 2 s 1 ) 7:5 10 16 Gain compression " bulk (cm 3 ) 0 Spontaneous emission 0.3 Optical confinement factor 2:5 10 3 Optical emission wavelength (m) 1.31 Internal optical loss i (cm 1 ) 1 64 Chapter2References [1] H. Haken, Handbuch der Physik, edited by L. Genzel (Springer, Berlin, 1970), V ol. XXV/2c. [2] M. O. Scully and W. E. Lamb, Phys. Rev. Lett.16, 853 (1966). [3] M. O. Scully and W. E. Lamb, Phys. Rev.159, 208 (1967). [4] L. Florescu and S. John, Phys. Rev. E69, 046603 (2004). [5] P.R. Rice and H.J. Carmichael, Phys. Rev. A50, 4318 (1994). [6] H. J. Carmichael, inLasersandQuantumOptics,inproceedingsoftheInternationalSchoolonLasers and Quantum Optics, Mar del Plata, Argentina, 22ï¿œ31 August 1988 (World Scientific, Singapore, 1990), p. 52. [7] K. Roy-Choudhury, S. Haas, and A.F.J. Levi, Phys. Rev. Lett.102, 053902 (2009). [8] F. Yu,OpticsandInformationTheory, (Wiley Interscience, 1976). 65 Chapter3: Single-PhotonDynamicsand ControlinCavityQED XI. CH.3INTRODUCTION Among the myriad of fascinating aspects of quantum mechanical systems, a particularly inter- esting property of atoms is the phenomenon of spontaneous emission in cavity quantum electrody- namics (cQED). Such a phenomenon is inherently quantum mechanical, and does not require an external field - the transition is induced by interactions with vacuum noise. Interestingly enough, this transition can depend on its local environment, and hence the decay of the atom can be con- trolled and probed by designing different types of open cavity QED environments[1]. Since the spontaneous generation of individual photons is an inherently random and quan- tum mechanical process, however, obtaining a specific photon in a predictable fashion is difficult. Despite various proofs that single photons cannot be localized in free space [2–4], discussions continue to present day regarding the quantum treatment of single photons, as well as fundamental interpretations. The dual wave-particle nature of quantum objects can still be explored within the framework of cavity QED, however, and insight into the spatial localization and phase properties of photons in a non-relativistic setting may be found in models that generalize the traditional Jaynes- Cummings Hamiltonian[5, 6]. In the following sections, we consider a multi-mode extension to the Jaynes-Cummings model[7, 8], in order to spatially resolve and control single-excitation dynamics in cavity QED. We end with a brief exploration of non-classical light in open cavity systems, which could be useful for future methods of non-Markovian control in open quantum systems. XII. EMERGENCEOFCLASSICALDYNAMICSINCQED In this section, we couple a single two-level emitter to a multimode electromagnetic field in a one-dimensional geometry to study the spatio-temporal properties of a single photon in cavity QED. Increasing the size of the environment has a dramatic effect on the space-time characteristics of atomic decay and re-absorption of a single photon unitarily evolving in a perfect cavity, as classical feedback patterns emerge in the transition from a strongly-coupled environment to a 66 weakly-coupled one. The positioning of the atom in the cavity can be used to suppress or enhance spontaneous emission and hence control Rabi oscillations. A. Physicalmodel To move beyond the single-mode Jaynes-Cummings model and explore delayed dynamics, a single bare atom with two energy levelsjgi andjei is placed within a one-dimensional cavity with perfectly-reflecting mirrors, and is coupled to a continuum of standing wave modes in one dimension, which exist in the plane wave basis with equal mode spacing k = c=(n r L), where c is the speed of light in vacuum, n r is the refractive index of the cavity medium, and L is the length of the cavity. Such a multi-mode model was initially developed by Wigner and Weisskopf to explore spontaneous emission in quantum optics [9]. The retarded dynamics of an atom interacting with a single photon come from the structure of the large cavity, which gives rise to the mode spacing and delayed feedback occurring at a time t = n r L=c later[6, 10]. As shown in Fig. 61, the single atom decays into a continuum that is shaped by the boundary conditions imposed by the perfectly-reflecting mirrors. The electromagnetic modes arising from perfectly-reflecting mirrors are given by the resonance condition for standing waves, 2L =j j , where j is the photon wavelength in thejth mode. Such a condition causes the density of states of the photon field to strongly peak at the cavity mode energies. Thus, the mirrors induce a delayed feedback effect on the atomic excitation due to the structured continuum. In deriving expressions for the observables of interest, we follow the methods employed by Ligare, et al.[6] of projecting the state vector onto the corresponding basis states of interest. This method is computationally straightforward, as it involves simply diagonalizing the total system Hamiltonian to solve for the eigenenergies and eigenvectors. Alternatively, Laplace transform methods could be used to derive time-dependent expressions for the observables of interest [11]. We begin by describing the energy of the two-level atom in terms of the gap between the valence and conduction energy levels, ~! a = ~(! e ! g ), and the spin operator in the computational basis for the two-level atom, z =jeihejjgihgj, for whichjei is an eigenbasis of the system corresponding to the excited atomic state, andjgi represents the two-level ground state, z = 0 @ 1 0 0 1 1 A : (35) 67 † = 0 = r 1 = 1 r 2 = 1 | ⟩ | ⟩ | ⟩ Figure 61: Physical system initially under consideration. A single two-level atom is placed inside a one- dimensional cavity. The atom has a gap respectively defined by the ground and first excited state energy levels, ~! g and ~! e , and interacts with a set of cavity modes with strengthG j . The atom may be placed anywhere within the cavity and interacts with the various modes during spontaneous emission. Therefore, the atom Hamiltonian is H atom = 1 2 ~! a z : (36) The photon field Hamiltonian is given in terms of a sum of the harmonic oscillator energies for every cavity mode included in the calculation, H field = J X j ~! j a y j a j + 1 2 ) J X j ~! j a y j a j ; (37) where we ignore the zero-point energy as a reference point energy. The operatorsa y j anda j re- spectively represent the creation and annihilation operators for a single photon in thejth mode, and the sum is carried over the total number of modes included in the calculation,J. The angular frequencies corresponding to the modes are easily derived from the resonance condition, ! j = ck j n r = c n r 2 j = c n r 2j 2L = jc n r L ; (38) wheren r is the refractive index of the lossless, dispersionless medium, and where the modes are equally spaced with the angular frequency separation k . While some modes will have relatively significant overlap with the two-level atom’s resonant frequency, many will not. Since the cavity modes are treated as independent quantized harmonic oscillators, and the free field eigenstates are described by the individual mode occupation numbers, 68 we can write down the basis states for the system as such. Thus, we denote the stateje; 0i as the atom being in the excited state, with no photons in the radiation field. For when the atom is in the ground state, we denote the statejg;ji to describe one photon in the radiation field, detuned from resonant mode by the separationj k . Finally, the Hamiltonian describing the interaction between the atom and the various cavity modes is H interaction = J X j ~G y a j +a y j sin [(j 0 +j)x a =L]; (39) where j 0 is the central, or resonant, mode integer. The sum over j once again represents the contribution from all modes either side of the resonant modej 0 , for whichJ represents the total number of modes considered. The atom-mode coupling rate G is given in terms of the dipole matrix elementd eg between atomic ground and excited states, the resonant mode angular frequency ! 0 , and the mode volumeV =AL in terms of mirror areaA and cavity lengthL: G =d eg p ! 0 =(2~ 0 V ): (40) The total Hamiltonian is thus built from the atom Hamiltonian, the sum of mode energies in the large cavity field, and the interaction energy: H total =H atom +H field +H interaction = 1 2 ~! a z + J X j ~! j a y j a j + J X j ~G y a j +a y j sin [(j 0 +j)x a =L]: (41) This model is a modified version of the original Hamiltonian used by Wigner and Weisskopf, which has no dependence on the atomic position and in which the interaction Hamiltonian is not mode specific [9]. The state of the system can generally be formulated as a linear combination of the excited state and a sum of states associated with the modes in which the atom is in the ground state. Thus, the general time-dependent state is given as 69 j (t)i =c e (t)je; 0i + J X j c g;j (t)jg;ji; (42) where c e (t) and c g;j are the time-dependent amplitudes,je; 0i =jei represents the atom in the excited state (zero photons in the cavity), andjg;ji = jji represents the state of the atom in its ground state and a single photon in thejth mode. Thus, one can simply populate the matrix representation for the Hamiltonian and solve for the energies and eigenvectors. In this energy eigenbasis, the time evolution of atomic excitation and photon amplitudes can be calculated di- rectly from the time-dependent state of the system. Taking the excited atom as the initial state, we have j (t)i = J+1 X q e iEqt=~ jE q ihE q je; 0i; (43) where, in terms of the time-independent coefficientsd (q) e andd (q) g;j satisfying the Schrödinger equa- tion, the Hamiltonian eigenstates are given as the linear combination jE (q) i =d (q) e je; 0i + J X j d (q) g;j jg;ji: (44) Thus, the time-dependent amplitudes for the excited state and a photon in thejth mode of the cavity are, respectively, c e (t) =he; 0j (t)i = J+1 X q e iEqt=~ jd (q) e j 2 (45) and c g;j (t) =hg;jj (t)i = J+1 X q d (q) g;j d (q) e e iEqt=~ : (46) The eigensystem is solved and the time-dependent amplitudes are calculated, in which the units of time are taken to be in terms of the cavity round-trip time, r = 2L=c. Since we have spatially resolved the quantum system, it is possible to explore the transient dynamics of the spontaneous emission event with respect to the position of the atom. In the case of an excited atom, the stored energy is released due to its interaction with the surrounding optical field. Since spontaneous emission is an inherently random process, exerting control over it is of considerable interest; for example, if the interaction could be turned off, the atom would never 70 transition from its excited state [12]. Such an effect was discovered by Purcell [13], who sought to increase the emission rate. The discovery of control over the rate of spontaneous emission has had a significant impact on reducing the threshold of lasers, as well as experimentally studied with quantum dots in microdisks, micropillars, as well as photonic crystals [14]. Placing a two- level system such as an atom or a quantum dot inside a cavity allows for the dependence of the interaction between the two-level system on position to be explored. We begin with Fermi’s golden rule, which describes the transition rate for the atom to decay to the ground state, [15] i!f = 2 ~ jV fi j 2 (E f ); (47) in which the rate i!f for the spontaneous emission of the atom is proportional to the density of cavity modes (E f ) with energy E f = ~! f at the atomic transition frequency ! f . It has been known for some time that the environment can play a special role in the phenomenon of sponta- neous emission, especially if the atom is placed within a resonant cavity [13]. Introducing bound- ary conditions changes the density of field modes, allowing for the suppression or enhancement of the transition rate. Such enhancement of photon emission within a resonant cavity is described by the Purcell factor, essentially derived as the ratio of the density of states of free space to that of the cavity, and is given by F P = 3Q 4 2 V c n r 3 ; (48) in whichQ is the quality factor for a lossy cavity, and c =n r is the photon wavelength inside the material.V is the cavity mode volume, given by [16] V = (x)jE(x)j 2 M jE(x M )j 2 ; (49) where(x)jE(x)j 2 is the position-dependent field intensity inside the cavity. In the model being studied, the atom is placed in vacuum, so the dielectric constant is set to unity everywhere. x M corresponds to the position where the field intensity is at a maximum, and M is the value of the dielectric constant at such a position. The Purcell effect, described by 48, relies on two assump- tions [17]. The first assumption is that in non-dissipative systems such as a perfect cavity, normal modes may be defined, and the density of modes described in 47 can be described locally by a dipole source, and hence the transition rate may be rewritten as [17] 71 (!;x) = d 2 ! ~ 0 X ! e T d ! E (x;!) 2 (!! 0 ); (50) wheree d is the dipole orientation. The second assumption made by the Purcell effect is that the sum given by 50 is dominated by only one term. Doing so replaces the peaks corresponding to the discrete cavity modes by a Lorentzian spectrum centered on the resonant mode of the cavity, with a width ! = !=Q [17]. Such a description is only valid for dissipative models and hence not strictly applicable to a unitary cavity system. Due to the absence of cavity loss in our model, such an approximation is invalid, and all modes must be considered in calculations. The emission dynamics depend strongly not only on the position of the atom, but also the cavity mode volume. Hence, the unitary dynamics describing the probability of emission and re- absorption of the atom may be studied as a function of time as well as atomic positioning within the cavity. A probability “landscape" can be visualized by varying the position of the cavity atom and evolving the system in time for each position, and this is explored in the following sections. B. Quantumtoclassicaltransition 1. Strongcouplingregime/single-modelimit If only a single radiation mode is considered, the fact that there is no outcoupling to the external environment results in Rabi oscillations in the system, as shown in Fig. 62. By varying the position of the atom within the cavity, an excitation landscape may be obtained. Fig. 63 shows such a single-mode landscape for the case in which the atom is coupled to the funda- mental cavity mode, which yields insight into the position-dependent dynamics of the cavity. The corresponding landscape in the frequency domain may be obtained by calculating the spectrum of the time series. Fig. 64 displays the spectrum landscape as a function of frequency and atomic position. 72 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c| 2 |c e | 2 |c g | 2 Figure 62: Time-dependence of atomic and single photon excitation probabilities for multiple round-trip timest= r . The atom is placed in the center of the cavity (x a = 0:5) and only couples to a single mode. Due to the coupling to a single radiation mode, the atom and photon probabilities oscillate with frequency 2G as energy is completely transferred between light and matter. Parameters: x a =L=2;G = 4,j 0 = 1, 1 total mode. 73 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 63: Position-time distribution of probability of atomic excitation for the smallest possible mode volume, in which the atom is strongly coupled to only a single mode, the fundamental mode of the cavity. Placing the atom by the edge of the mirror suppresses spontaneous emission due to the proximity to the edge node, while placing the atom towards the middle of the cavity produces Rabi oscillations with the highest frequency, equal to twice the coupling rate, whereas the oscillation frequency decays with respect to proximity to the mirror. The probability dynamics are symmetric aboutx a =L = 0:5 due to symmetry of the one-dimensional cavity. Parameters:G = 4,j 0 = 1, 1 total mode. 74 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 Atom position, x a /L Frequency, τ r /t |A| 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Figure 64: Frequency-position distribution for the smallest possible mode volume, in which the atom is strongly coupled to only a single mode. As expected, the Rabi frequency peaks in the center of the cav- ity, and is forced to zero at the boundaries. Such spatial “bending" of the resonant frequency is a direct consequence of perfectly-reflecting boundaries. Parameters:G = 4,j 0 = 1, 1 total mode. 75 2. Mesoregime Increasing the size of the cavity introduces additional modes, which dramatically affect the atom-cavity dynamics. Figs. 65 - 70 demonstrate the importance of location on the atom-photon interaction in the single photon limit, and how the dynamics strongly depend on mode volume. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 65: The mode volume is now increased by doubling the cavity length, such that an entire wavelength fits within the cavity. The atom is coupled to 3 modes in this case, which breaks up the monotonic depen- dence of the Rabi frequency on atomic positioning the single mode limit. We begin to observe delays in the oscillations near the nodes of the higher modes atx a = 0:5 associated with the second harmonic and x a = 0:333 associated with the third harmonic. Parameters:G = 4,j 0 = 2;J = 3. In addition to examining the number of cavity modes and boundary conditions, accessibility to isolated contours of constant excitation or de-excitation in parameter space significantly depends on the atom-field coupling strength. Figs. 71 - 76 demonstrate the dependence of the boundary- induced feedback dynamics on the atom-field interaction strength for small numbers of modes. If the atom-field coupling timescale (1=G) is large relative to the cavity round-trip time r , the probability landscape corresponding to the single-mode limit can be seen to repeat throughout the length of the cavity. Such a pattern results from the atom being on resonance with the central mode of the cavity. 76 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Atom position, x a /L Frequency, τ r /t |A| 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Figure 66: Frequency-position distribution corresponding to the 3-mode landscape. The relation between the number of cavity modes and their corresponding frequencies is made apparent by the frequency- dependence on atomic position. The number of highest-frequency peaks formed by frequency vs. posi- tion curves corresponds to the number of modes inside the cavity, of which there are 3. These frequencies peak at the positions of the anti-nodes, where the light-matter interaction is greatest. Once again, perfectly- reflecting mirrors require the frequency-position curves to terminate at the boundaries. Parameters:G = 4, j 0 = 2;J = 3. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 67: The bare atom is now resonant with the third harmonic and coupled to 5 modes. Ripples in the probability distribution become increasingly apparent, as well as chains of probability “islands" associated with delayed feedback effects. Parameters:G = 4,j 0 = 3;J = 5. 77 Figure 68: Frequency-position distribution corresponding to the 5-mode landscape. In addition to 5 fre- quency peaks, multi-mode structure is beginning to emerge from increasing the cavity length. Parameters: G = 4,j 0 = 3;J = 5. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 69: Coupling the atom to an additional set of modes in a slightly larger cavity further resolves the probability island chains. The island pattern results from feedback from the two pulses emitted in either direction; thus, the probability will peak in space-time regions in which both pulses simultaneously return to the atom. Parameters:G = 4,j 0 = 4;J = 7. 78 Figure 70: Frequency-position distribution corresponding to the 7-mode landscape. Parameters: G = 4, j 0 = 4;J = 7. 79 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 71: Atomic excitation landscape in logarithmic scale, for three distinct sets of modes. From left to right, the excitation landscapes correspond to the atom respectively coupled to J = 1; 3; and 5 total modes. The atom-field coupling is set toG = 0:5 in all three cases, corresponding to a decay rate that is slow in comparison to the feedback rate, resulting in a suppression of spontaneous emission for an entire cavity round-trip time, regardless of where the atom is placed inside the cavity. As the number of modes is increased, the single-mode behavior is effectively repeated across the length of the cavity. Such a slow decay rate allows for reliably isolated contours of constant excitation or de-excitation in parameter space. Parameters:G = 0:5,j 0 =f1; 2; 3g,J =f1; 3; 5g. 80 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 72: For the same sets of modes, the atom-field coupling is now doubled toG = 1. Increasing the decay rate has the effect of terminating contours of total excitation or de-excitation over time in parameter space, but only in the multiple-mode case (J > 1). Parameters:G = 1,j 0 =f1; 2; 3g,J =f1; 3; 5g. 81 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 73: The atom-field coupling is increased toG = 1:5. At this coupling rate, several of the constant de-excitation “trajectories" in parameter space (isolated contours of constant probability) are closed off upon the introduction of multiple modes. Isolated contours are now restricted to the center and sides of the cavity. Parameters:G = 1:5,j 0 =f1; 2; 3g,J =f1; 3; 5g. 82 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 74: At G = 2, many former trajectories of constant probability are closed off. Several stored excitation contour paths are formed into islands and most former de-excitation contour paths are now valleys of low excitation probability in the parameter space. Such contour “pinching" gives rise to vertical and horizontal chains of islands. Parameters:G = 2,j 0 =f1; 2; 3g,J =f 1; 3; 5g. 83 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 75: Doubling the interaction rate to G = 4 demonstrates a dramatic difference in the excitation landscape as the number of modes is increased. With the atom coupled to a total of 5 modes, we begin to see the semiclassical feedback structures observed in Fig. 67, as well as an increase in the regularity of oscillations throughout the entire cavity, as the former horizontal chains of probability islands coalesce into horizontal bands of excitation and de-excitation. Parameters:G = 4,j 0 =f1; 2; 3g,J =f1; 3; 5g. 84 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L Time, t/τ r 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L 0 0.5 1 0.5 1 1.5 2 2.5 3 3.5 4 Atom position, x a /L log 10 (|c e | 2 ) −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Figure 76: Excitation landscapes for few modes with G = 8. In such a fast decay regime, there exist no subluminal “trajectories" of constant probability in parameter space, as the atomic excitation dynamics are nearly completely dominated by Rabi oscillations throughout the entire cavity. Parameters: G = 8, j 0 =f1; 2; 3g,J =f1; 3; 5g. 85 3. Weakcouplingregime/classicallimit We can model a significantly larger cavity by including many more equally-spaced radiation modes and ensuring that the central, or resonant, cavity mode is significantly larger than the mode spacing (i.e., k 0 k ). By placing the atom in the center of the large cavity at an antinode of the resonant mode, coupling to modes away from resonance results in complete de-excitation of the atom, until re-absorption of the same photon occurs half a cavity round-trip time later. Fig. 77 demonstrates the transient effect of delayed feedback on the bare atom. The atomic amplitude will decay in an almost exponential fashion [6], withc e (t)/e t . The decay constant is proportional to the modulus of the atom-field coupling squared, = jGj 2 2 k : (51) 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c| 2 |c e | 2 |c g | 2 Figure 77: Time-dependence of atomic excitation probability for multiple round-trip times t= r . Due to the coupling to the various modes, the atom will decay, only to be re-excited a round-trip time later. Such feedback dynamics may be thought of as a partial Poincaré recurrence [18]. As the atom decays, the various cavity modes will become excited and a single photon will be produced. Parameters: x a = 0:5, G = 4, j 0 = 599;J = 1197. It can be shown that in the continuum limit, where the mode spacing vanishes, the excited state amplitude decays exponentially with decay constant [11]. While the decay may be ap- proximately exponential in the limit ! 0 k , however, such early-time dynamics are strictly non-unitary [11] and hence they cannot occur in a perfect cavity due to the discrete spectral 86 structure[19]. Since the energy spectrum of the electromagnetic field exhibits a lower bound in the cavity, the notion of exponential decay is simply an approximation [20], albeit a useful one. More generally, it has been shown [19] that strictly exponential decay is forbidden in any system ex- hibiting unitary evolution, as is the case here. Fig. 78 confirms this fact by showing the short-time transient behavior of the emission process, in which the decay slope approaches zero ast! 0. The constraint of unitarity requires for reciprocity to hold at early times, regardless of the validity of T invariance in the system [19]. Furthermore, whereas placing the atom in the center of the large cavity yields complete decay within a cavity round-trip, placing the atom closer to one of the mirrors prevents complete emission due to the feedback of the photon from the closest mirror, as seen in Fig. 79, where oscillation period is modified due to the atom’s proximity to the mirror. 87 1 2 3 4 5 6 7 x 10 −3 0.9 0.92 0.94 0.96 0.98 1 Time, t/τ r |c e | 2 Figure 78: Short-time transient dynamics of atomic decay, in which the decay slope approaches zero as t! 0. For sufficiently short times, reciprocity is required by unitarity, regardless of whether time invariance in the system is valid [19]. Parameters:x a = 0:5,G = 4,j 0 = 599;J = 1197. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c| 2 |c e | 2 |c e | 2 Figure 79: Decay and excitation of the atom near a mirror. Feedback-induced oscillations in the occupation probability occur due to the proximity to the mirror. Parameters: x a = 0:202L, G = 4, j 0 = 599;J = 1197. As the total number of coupled modes is increased, the probability landscape begins to acquire classical feedback structures, corresponding to localized photon pulses re-exciting the atom We can further explore the transient dynamics of the rate of decay of the atom by comparing the dynamics at the first few nodes and anti-nodes of the resonant mode away from one of the mirrors. 88 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 80: The atom is now resonant with the 11th harmonic of the meso-cavity and coupled to a total of 21 modes. Besides continuing to observe isolated islands, former island chains become probability bands that trace out delayed feedback associated with a single pulse. Intersection of the bands from either pulse induce relatively high re-excitation probabilities. Furthermore, regular valleys away from the mirror are formed due to placement of the atom at anti-nodes of the resonant mode, which enhances spontaneous emission. Parameters:G = 4,j 0 = 11;J = 21. Fig. 83 displays the position-dependent dynamics near the cavity wall, as shown in the corner of Fig. 82. Figs. 84 and 85 compare the transient behavior of the spontaneous decay process for atomic positioning that respectively suppresses and enhance emission. The time series in these figures represent slices taken out of Fig. 83. Near the perfectly-reflecting mirror, the suppression and enhancement of spontaneous emission is respectively dominated by destructive interference at nodes and constructive interference at anti- nodes of the resonant cavity mode, even though off-resonant modes still interact with the atom [18]. Beyond studying the time-dependent amplitudes and probabilities of the atom and single pho- ton, we can explore the spatial and temporal properties of the electric field operator, given by a sum over terms for each cavity mode, in which each term exhibits a sinusoidal space dependence identical to the classical mode structure[6], 89 Figure 81: Frequency-position distribution corresponding to the 21-mode landscape. Parameters: G = 4, j 0 = 11;J = 21. E(x) = J X j C j a j +a y j sin h (j 0 +j) x L i ; (52) where C j = r ~! j 0 V ; (53) withV being the effective mode volume and 0 the permittivity of free space. We assume that the atomic resonance frequency! 0 is much greater than the mode spacing, so thatC j can simply be treated as a constant for all modes. Quantum mechanically, the energy density of the photon field is given by the expectation value of the quantized electric field operator squared, E(x) 2 = J X j;j 0 C 2 sin h (j 0 +j) x L i sin h (j 0 +j 0 ) x L i (54) a j a j 0 + 2a y j a j 0 +a y j a y j 0 + j;j 0 : (55) 90 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Atom position, x a /L Time, t/τ r |c e | 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 82: Finally, the cavity mode volume is made large enough to where placing the atom in the center kills off interference effects from the cavity wall. Coupling the atom to 399 modes produces well-defined resonances and valleys in the position-time dynamics, which serve to nicely demonstrate respectively the effect of suppression and enhancement of spontaneous emission near the cavity wall. Initial decay of the atom becomes uniform everywhere in space except when placed within a few wavelengths to the cavity wall. Parameters:G = 4,j 0 = 200;J = 399. If a single-mode field-state with definite particle number is considered, the expectation value of the field,hEi, is zero. However, the expectation value of the field squared,hE 2 i, is non-zero, and in terms of the cavity mode amplitudes, may be written as h (t)jE 2 (x)j (t)i = 2C 2 J X j 0 hg;j 0 j (t)i sin h (j 0 +j) x L i 2 (56) Thus, every term in this sum is an amplitude for a photon with the detuning j k multiplied with the associated the spatial mode function. Fig. 86 displays a stroboscopic set of plots of the propagation of the emitted photon energy in space and time, described by equation (56). The approximately exponential dependence of the atomic decay on the coupling strength and mode 91 Figure 83: Excitation landscape near a cavity mirror in the limit of large numbers of modes. Near the cavity wall, excitation can be stored at nodes of the resonant mode, and de-excitation can be stored at anti-nodes of the resonant mode up until a cavity round-trip time later. As the atom is moved away from the wall towards the center of the large cavity, the atom is allowed to completely decay before becoming re-excited at a time t = x a =c later, and the dependence of the excitation dynamics on node-positioning is lost. Parameters: c = 1;G = 4,j 0 = 200;J = 399. spacing is reflected in the spatial dependence of the photon pulse intensity initially seen in Fig. 86. By comparing the transient behavior of the intensity operator for different positions of the atom, we can further explore local control over the generated photon pulse within the cavity. Figs 87 - 90 demonstrate the importance of the atom’s location within the cavity. 92 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c e | 2 Node 1 Node 3 Node 5 Node 15 Node 50 Figure 84: Transient decay of the atom near a mirror, in which the atom has been positioned at various nodes of the resonant cavity mode, away from one of the mirrors. Atomic decay will be highly suppressed at nodes, even though the atom is coupled to many other modes with equal spacing. Suppression of emission is lost as the round-trip time between the atom and the mirror is increased by positioning the atom further away from the mirror, allowing the photon to be completely emitted before being re-absorbed. Parameters: G = 4,j 0 = 200;J = 399. 93 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c e | 2 Anti−node 1 Anti−node 3 Anti−node 5 Anti−node 15 Anti−node 50 Figure 85: Transient decay of the atom near a mirror, in which the atom has been positioned at various anti-nodes of the resonant cavity mode, away from one of the mirrors. Atomic decay is now facilitated by placing the atom in regions where the resonant mode amplitudes are the largest. The anti-node dynamics asymptotically approach those of the node dynamics as the atom moved away from the mirror towards the center of the large cavity, at which point the photon is allowed to be completely emitted before re-absorption half a cavity round-trip time later. Parameters:G = 4,j 0 = 200;J = 399. 94 0 0.2 0.4 0.6 0.8 1 Position, x/L t = 0.1τ r t = 0.2τ r t = 0.3τ r t = 0.4τ r t = 0.5τ r t = 0.7τ r t = 0.8τ r t = 0.9τ r t = 1.0τ r t = 0.6τ r Figure 86: Space-time propagation of the single photon pulse energy,hE 2 i, within the cavity. As the atom decays, two pulses are produced and travel away from the source at the speed of light. After half a cavity round-trip time, the two pulses interact with the atom, causing a partial re-excitation of the atom shown in Fig. 77. The position is given in units of the cavity length L, and the propagation time t is given in units of the cavity round-trip time, r . Since the atom is placed in the center of the cavity, the emitted photon interacts with the atom twice in a single full cavity round-trip. Parameters: x a = 0:5, G = 4, j 0 = 599;J = 1197. 95 0.90 0.92 0.94 0.96 0.98 1.0 0 0.02 0.04 0.06 0.08 0.1 0.12 Space, x/L Time, t/τ r Figure 87: Initial transient dynamics of localized photon pulse intensityhE 2 i in the cavity, near the right mirror. The atom is placed at the third anti-node away from the mirror, located at -x=L = 5 0 =4, enhancing emission and inducing a rapid decay. The photon travels to either side of the atom, but due to the mirror, the right-travelling pulse is immediately absorbed by the atom and consequently reemitted. Parameters: G = 4;j 0 = 200;J = 399;x a = 0:9875L. −0.1 −0.08 −0.06 −0.04 −0.02 0 0.9 0.95 1 1.05 1.1 1.15 Space, x/L Time, t/τ r Figure 88: Feedback dynamics of localized photon pulse intensity in the cavity, near the right mirror. The two initially emitted photon pulses reflect off of the left mirror and return to the atom located at x=L = 5 0 =4, whereupon the interaction with the atom and reflectance from the right mirror decreases the time delay between the pulses. Parameters:G = 4;j 0 = 200;J = 399;x a = 0:9875L. 96 0.90 0.92 0.94 0.96 0.98 1.0 0 0.02 0.04 0.06 0.08 0.1 0.12 Space, x/L Time, t/τ r Figure 89: Initial transient dynamics of localized photon pulse intensity in the cavity, near the right mirror. The atom is placed at the third node away from the perfectly-reflecting mirror now atx=L =3 0 =2, and while the left piece of the photon travels away from the atom towards the left mirror, the right-travelling piece is trapped between the atom and the right mirror. Energy is stored between the mirror and atom, suppressing emission. Parameters:G = 4;j 0 = 200;J = 399;x a = 0:9850L. 97 0.90 0.92 0.94 0.96 0.98 1.0 0.9 0.95 1 1.05 1.1 1.15 Space, x/L Time, t/τ r Figure 90: Feedback dynamics of localized photon pulse intensity in the cavity, near the right mirror. The photon pulse returns to the atom located atx=L =3 0 =2 at timet= r = 2(Ljx a j)=(c r ) = 0:9697 later, whereupon the photon energy stored between the atom and the mirror is briefly cancelled by the incoming pulse. Due to the interaction with the atom, the photon is re-emitted and the process repeats every cavity round-trip. Parameters:G = 4;j 0 = 200;J = 399;x a = 0:9850L. 98 XIII. SPATIO-TEMPORAL CONTROL OF SINGLE PHOTONS IN COUPLED CQED SYS- TEMS In this section, we consider models that extend the work done by Weisskopf and Wigner [9] on spontaneous emission by coupling a single two-level emitter to single and multimode electromag- netic fields in a variety of one-dimensional geometries in order to spatially and temporally control a single photon in composite-cavity quantum electrodynamics. Under the sudden approximation, we detune the atom after photon emission in order to retain the emitted pulse characteristics and use an auxiliary resonator to store the photon energy. We introduce methods to control the ampli- tude and phase of single photon pulses emitted by the atom, and demonstrate coherent control of the external resonator ring-down by exploiting photon self-interference in a fully quantum model. We also exploit coherent feedback to exert control over microcavity dynamics, which can be driven from weak to strong coupling-type, or vice versa. Our approaches suggest that access to ampli- tude and phase control is sufficient for the shut-down of open quantum subsystems as a form of bang-bang control. A. Coherentresonatorcontrol The system under consideration is illustrated in Fig. 91 and consists of multiple cavities which are treated as subsystems. The primary cavity contains a two-level emitter that is initially coupled to two sets of modesfa j gf l g shaped by the end mirrors atx =L 1 andx = 0 for the former set, andx =L 2 andx = 0 for the latter. Thus, the primary resonator consists of two sub-cavities with lengths L 1 and L 2 . An external cavity is coupled to the primary subsystem by placing a perfectly-reflecting mirror atx =L 3 , and the two sets of modes interact at the center mirror via a coupling constantK. By coupling to a passive resonator with its own set of equally-spaced modes, we can construct a composite cavity-resonator system in order to exploit storage and control over single photons in space and time. 99 � = (1-r 1 ) † = r 1 † δ(t) = 0 = 3 = − 1 † = − 2 K r 1 r 2 Figure 91: Composite cavity-resonator system. A single two-level atom interacts with two sets of cavity modes with strengthsG j andM l in the primary cavity, while the primary modes couple to a second set of modes in the passive resonator via the transmission rateK. An intra-cavity beam-splitter allows the atom to couple to the sets of modesfa j g andf l g, respectively associated with the path lengthsL 1 andL 2 . The atom may be placed anywhere within the cavity and interacts with the various modes during spontaneous emission, while the composite-cavity structure exhibits delayed transient dynamics, due to the storage of the photon in the passive resonator. The time-dependent detuning of the free field modes from atomic resonance is initially zero and is switched on at a later timet 1 after the atom has emitted the photon. 1. Compositecavitymodel The ground and excited states of the atom are respectively denoted byjgi andjei, while the zero-field gap energy is~! eg , where~ is the reduced Planck constant. The electromagnetic modes in the cavity subsystems have spatial dependence E i (x)/ sin nx L i ; (57) where n is a positive integer used to denote the mode number under consideration, and where i = f1; 2; 3g for the mode setsfa j g,f l g andfb q g in the cavities with lengths L 1 , L 2 , and L 3 , respectively. The modes have angular frequencies respectively given by ! 1;n = nc=n r L 1 , ! 2;n =nc=n r L 2 , and! 3;n =nc=n r L 3 , with respective spacing 1 =c=n r L 1 , 2 =c=n r L 2 , and 3 =c=n r L 3 . The mode numbers closest to resonance with the two-level emitter are denoted asj 0 , l 0 , andq 0 , and likewise, the corresponding angular frequencies are! i;0 . In general, mode numbers will be written in terms of their separation from the central mode asn j = j 0 +j,n 0 l = l 0 +l, andn 00 q =q 0 +q wherej,l, andq are integers corresponding to the respective multiple of the 100 separation 1 , 2 , and 3 from the resonant frequency. In addition, the detuning of the resonant mode from the atomic resonance! eq is given by, such that =! 0 ! eg : (58) We treat the cavity modes as independent quantized harmonic oscillators, for which the eigenstates of the free field are labelled by the occupation numbers of the individual modes: j:::;n 2 ;n 1 ;n 0 ;n 1 ;n 2 ;:::i for the set of modesfa j g and likewise,j:::;n 0 2 ;n 0 1 ;n 0 0 ;n 0 1 ;n 0 2 ;:::i andj:::;n 00 2 ;n 00 1 ;n 00 0 ;n 00 1 ;n 00 2 ;:::i for the other mode sets. Since the total number of excitations in our model is 1, either all or all but one occupation number is 0; the former corresponding to absorp- tion by the atom, and the latter corresponding to emission into a particular mode. The free field eigenstate labels for the primary cavity are condensed to indicate either no photons in the field or a single photon in thejth mode, such thatj0; 0; 0;:::ij 1 i andj0;:::; 0;n j = 1; 0;:::; 0ijji. Likewise, for other subcavity coupled to the atom,j0; 0; 0;:::i j 2 i andj0;:::; 0;n 0 l = 1; 0;:::; 0ijli, and for the external resonator,j0; 0; 0;:::ij 3 i andj0;:::; 0;n 00 q = 1; 0;:::; 0i jqi. Thus, the coupled cavity eigenstates for the two fields arej 1 ; 2 ; 3 ij 1 i j 2 i j 3 i for no photons in any cavity,jj; 2 ; 3 ijji j 2 i j 3 i for one photon in mode j in the primary cavity,j 1 ;l; 3 ij 1 i jli j 3 i for one photon in mode l, andj 1 ; 2 ;qi j 1 i j 2 i jqi for one photon in modeq in the external resonator. Since there can only exist one photon between the cavities, the product statesjj;l;qi,jj;l; 3 i,jj; 2 ;qi, andj 1 ;l;qi are not free-field eigenstates of the coupled system. The total free-field Hamiltonian for the coupled system is given in terms of the creation and annihilation operators for every mode of each cavity, and is equivalent to sums of independent harmonic oscillators for each cavity, H field = X j ~! j a y j a j + 1 2 + X l ~! l y l l + 1 2 + X q ~! q b y q b q + 1 2 ' X j ~! j a y j a j + X l ~! l y l l + X q ~! q b y q b q ; (59) where we have ignored the zero-point (common reference) energy, and in which the photon num- ber operators are N j = a y j a j , N 0 l = y l l , and N 00 q = b y q b q ; thus, a single photon in the coupled 101 system corresponds to the constraint P j N j + P q N 0 l + P q N 00 q = 1. The transmission of the photon from thejth andlth mode in the primary resonator to theqth mode in the external resonator is made possible due to the outcoupling parameterK. The interface coupling the cavities atx = 0 is treated as a mirror with uniform reflectivity over frequency, such that the transmission rate is a constant across all modes and proportional to the transmission coeffi- cientt = p 1r 2 2 , wherer 2 is the reflectivity of the interface mirror. Therefore, the transmission rate associated with the hopping of the photon from a mode in the primary cavity to a mode in the secondary cavity is given as K =K 0 q 1r 2 2 ; (60) whereK 0 is some proportionality constant which we set equal to unity in the model, and must be real to ensure equal transmission from either side of the interface. The cavity coupling Hamiltonian can then be written as H hopping = X j;q ~K a y j b q +a j b y q + X l;q ~K y l b q + l b y q : (61) The basis states of the entire physical system are given by the direct product of the atomic states and the coupled field states, so that the basis stateje; 1 ; 2 ; 3 ijei corresponds to the atom in the fully excited state and zero photons in any of the radiation fields, the statejg;j; 2 ; 3 ijji corresponds to the atom in the ground state and the single photon in thejth mode of the primary cavity, the statejg; 1 ;l; 3 ijli corresponds to the single photon in thelth mode of the primary cavity, and the statejg; 1 ; 2 ;qijqi corresponds to the single photon in theqth mode of the external cavity. In the two sets of basis states describing the atom in the ground state, the detuning of the photon from the zero-field atomic resonance frequency is given by! 1;j ! eq =j 1 +(t), and! 2;l ! eq = l 2 +(t), and! 3;q ! eq = q 3 +(t) for cavity mode setsfa j g,f l g, and fb q g, respectively. The electric field operator for the coupled system is defined over the entire system in terms of the field operators for each mode set, E(L 1 xL 3 ) =E 1 (L 1 x 0) +E 2 (L 2 x 0) E 3 (0xL 3 ); (62) 102 where each cavity term is given as a sum over terms exhibiting sinusoidal spatial dependence for all modes in the respective mode sets: E 1 (L 1 x 0) = X j A j a j +a y j sin (j 0 +j) x L 1 ; (63) where A j = r ~! j 0 V 1 ; (64) with 0 the permittivity of free space andV 1 the effective mode volume. If the resonance frequency is significantly greater than the mode spacing thenA j will not vary much across the modes, and so can be assumed to be constant for all modes such thatA j = A = p ~! eg =( 0 V 1 ). Since the detuning between the atom and the mode nearest resonance is initially taken to be zero, we have A = p ~! 0 =( 0 V 1 ). Likewise, for the second set of modes, E 2 (L 2 x 0) = X l A 0 l l + y l sin (l 0 +l) x L 2 ; (65) where A 0 l = r ~! l 0 V 2 = r ~! 0 0 V 2 A 0 ; (66) withV 2 the effective mode volume of the primary resonator subcavity. We takeE 1 (x<L 1 ;x> 0) =E 2 (x<L 2 ;x> 0) = 0. For the field in the external resonator, E 3 (0xL 3 ) = X q B q b q +b y q sin (q 0 +q) x L 2 ; (67) where B q = r ~! q 0 V 3 = r ~! 0 0 V 3 =B; (68) withV 3 the effective mode volume of the external cavity, and outside this region,E 3 (x < 0;x > L 3 ) = 0. The quantum mechanical analogue to the energy density of the electric field is the expectation value of the operatorE 2 , thus the spatial dependence of the energy density for the single photon in the coupled cavity system is 103 E(x) 2 = X j;j 0 A 2 sin (j 0 +j) (L 1 x 0) L 1 sin (j 0 +j 0 ) (L 1 x 0) L 1 (a j a j 0 + 2a y j a j 0 +a y j a y j 0 + j;j 0) + X l;l 0 A 02 sin (l 0 +l) (L 2 x 0) L 2 sin (l 0 +l 0 ) (L 2 x 0) L 2 ( l l 0 + 2 y l l 0 + y l y l 0 + l;l 0) + X q;q 0 B 2 sin (q 0 +q) (0xL 3 ) L 3 sin (q 0 +q 0 ) (0xL 3 ) L 3 (b q b q 0 + 2b y q b q 0 +b y q b y q 0 + q;q 0) (69) We assume the dipole approximation the light-matter interaction Hamiltonian is given in terms of the electric dipole operatord, H int = [eE(x)r](xx a ) = [Ed](xx a ); (70) wherex a is the location of the interaction between the atom and the field within the cavity, and where d = ~(d eg + +d eg ); (71) withd eg the dipole matrix element between ground and excited states of the atom, and +() the corresponding raising(lowering) operators, defined as + =jeihgj (72) =jgihej (73) Thus, the interaction Hamiltonian becomes 104 H int = X j A(a j +a y j ) sin (j 0 +j) x a L 1 (d eq + +d eg ) = X l A 0 ( l + y l ) sin (l 0 +l) x a L 2 (d eq + +d eg ): (74) Under the rotating wave approximation, the energy non-conserving terms + a y j , a j , + y l , and l are dropped so that the interaction Hamiltonian simplifies to H int ' X j ~G j + a j + a y j + X l ~M l + l + y l ; (75) where G j =G j (1r 1 ); (76) M l =M l r 1 ; (77) with r 1 the reflectivity of the intra-cavity beam splitter facing the atom inside the primary res- onator, and G j =G sin (j 0 +j) x a L 1 ; (78) M l =M sin (l 0 +l) x a L 2 ; (79) where G =d eg p ! 0 =(2~ 0 V 1 ); (80) M =d eg p ! 0 =(2~ 0 V 2 ): (81) Finally, the Hamiltonian for the atom may be expressed in terms of the inversion operator, z =jeihejjgihgj, and expanded in terms of the time-dependent detuning, 105 H atom (t) = 1 2 ~! eg (t) z = 1 2 ~! 0 z + 1 2 ~(t) z : (82) The statesjei,jji,jli, andjqi form a complete set of states, and the total Hamiltonian for the multi-mode coupled system is H(t) =H atom (t) +H fields +H interactions +H hopping = 1 2 ~! 0 z + 1 2 ~(t) z + X j ~! j a y j a j + X l ~! l y l l + X q ~! q b y q b q + X j ~G j + a j + a y j + X l ~M l + l + y l + X j;q ~K a y j b q +a j b y q + X l;q ~K y l b q + l b y q (83) 2. Compositecavitydynamics a. Unitarydecay In order to demonstrate delayed feedback control over the emitted photon, we first need the ability to effectively remove the light-matter interaction within the primary cav- ity in order to avoid pulse re-shaping due to feedback with the atom. To this end, the atom-field system is detuned from resonance due to an applied external electric field significantly shifting the atom’s energy levels far from the cavity mode frequencies; thus, = (t). We wish to perform the detuning once the photon has been emitted, so that the atom will not interfere with any control schemes, so initially we have(0) = 0 and at a switching timet 1 later,(t 1 ) = 1 , where 1 is suf- ficiently strong to remove any consequential interactions between the atom and the excited modes of the primary cavity. At such a point in time, the energy eigenstates of the system instantaneously switch to a new set of eigenstates and corresponding energies, which evolve the photon dynamics without the presence of the atom. We would like to calculate the time-dependent amplitudes for the atom and both sets of cavity modes, as well as spatially resolve the energy density of the pho- ton in the coupled-cavity system, so that the system length- and time-scales may be exploited for single photon control. We begin by initializing the system with the atom in the excited state, so that at timet = 0 106 j (0)i =je; 1 ; 2 ; 3 i: (84) Since the photon does not have to be exactly resonant with the atom, the basis states are not eigenstates of the total Hamiltonian, so the initial state of the entire system will evolve into a linear combination of the basis states with time-dependent amplitudes, j (t<t 1 )i =c e (t)jei + X j c j (t)jji + X l c l (t)jli + X q c q (t)jqi: (85) We will start by finding the energy eigenstates and corresponding eigenvalues of the complete Hamiltonian, and then performing a projection of the initial state onto the eigenstates. Before the switch (t < t 1 ), the nth eigenstate of the system with energy E (1) n can be written as the linear combination jE (1) n i =d (n) e jei + X j d (n) j jji + X l d (n) l jli + X q d (n) q jqi; (86) whered (n) e ,d (n) j ,d (n) l , andd (n) q are time-independent projections of thenth energy eigenstate onto the excited state, jth-mode photon state in the primary resonator, lth-mode photon state in the primary resonator subcavity, and qth-mode photon state in the external resonator respectively, such that H(t<t 1 )jE n (t<t 1 )i =H 1 jE (1) n i =E (1) n jE (1) n i: (87) Thus, fort<t 1 , d (n) e =hejE (1) n i; (88) d (n) j =hjjE (1) n i; (89) d (n) l =hljE (1) n i; (90) d (n) q =hqjE (1) n i: (91) 107 Likewise, for timett 1 , H(t>t 1 )jE n (t>t 1 )i =H 2 jE (2) n i =E (2) n jE (2) n i; (92) with corresponding energy projection coefficients u (n) e =hejE (2) n i; (93) u (n) j =hjjE (2) n i; (94) u (n) l =hljE (2) n i; (95) u (n) q =hqjE (2) n i: (96) In terms of these detuned-energy coefficients, thenth eigenstate of the system with energyE (2) n is jE (2) n i =u (n) e jei + X j u (n) j jji + X l u (n) l jli + X q u (n) q jqi: (97) To calculate the basis state amplitudesc e (t < t 1 ), c j (t < t 1 ), c l (t < t 1 ), andc q (t < t 1 ), we project the initial system state onto the first set of energy eigenstates of the total system, j (0)i =jei = X n jE (1) n ihE (1) n j (0)i; (98) which evolves unitarily as j (0t<t 1 )i = X n e iE (1) n t=~ jE (1) n ihE (1) n jei: (99) Substitutingd (n) e and equation (86) into (99), we get: 108 j (t<t 1 )i = X n e iE (1) n t=~ jd (n) e j 2 ! jei + X n e iE (1) n t=~ d (n) e d (n) j ! jji + X n e iE (1) n t=~ d (n) e d (n) l ! jli + X n e iE (1) n t=~ d (n) e d (n) q ! jqi: (100) Thus, the time-dependent amplitudes are c e (t<t 1 ) =he; 1 ; 2 ; 3 j (t)i = X n e iE (1) n t=~ jd (n) e j 2 ; (101) c j (t<t 1 ) =hg;j; 2 ; 3 j (t)i = X n e iE (1) n t=~ d (n) e d (n) j (102) c l (t<t 1 ) =hg; 1 ;l; 3 j (t)i = X n e iE (1) n t=~ d (n) e d (n) l (103) c q (t<t 1 ) =hg; 1 ; 2 ;qj (t)i = X n e iE (1) n t=~ d (n) e d (n) q : (104) b. Diabatic detuning Not only is the light-matter interaction mode-specific in this model, but it also depends on the position of the atom within the cavity, due to the spatial structure of the cavity modes. In order to demonstrate delayed feedback control over the emitted photon, we first need the ability to effectively remove the light-matter interaction within the primary cavity in order to avoid pulse re-shaping due to feedback with the atom. To this end, the atom-field system is detuned from resonance due to an applied external electric field significantly shifting the atom’s energy levels far from the cavity mode frequencies; thus, = (t). We wish to perform 109 the detuning once the photon has been emitted, so that the atom will not interfere with any control schemes, so initially we have (0) = 0 and at a switching time t 1 later, (t 1 ) = 1 , where 1 is sufficiently strong to remove any consequential interactions between the atom and the excited modes of the primary cavity. The detuning timet 1 must be carefully chosen such that the photon has been completely emitted and not re-absorbed by the atom an atom-mirror round-trip time AM later, which is only possible in the weak coupling regime or in the large cavity limit. Thus, the criteria for atomic detuning can be quantified: 2 2 M 2 t 1 < 2jx AM jn r c ; (105) wherejx AM j =jL 2 jjx a j is the distance between the atom and the control mirror located at x = L 2 , and where the lower limit is the time-scale for the atom to decay into the control pulse modes associated with path lengthjL 2 j. Therefore at the detuning timet 1 , the energy eigenstates of the system instantaneously switch to a new set of eigenstates and corresponding energies, which evolve the photon dynamics without the presence of the atom. Such a diabatic passage is only valid under the sudden approximation [15], in which the functional form of the system remains unchanged: e =jc e (t 1 +)c e (t 1 )j! 0: (106) Thus, the approximation is valid provided the system state probability remains unchanged. We would like to calculate the time-dependent amplitudes for the atom and both sets of cavity modes, as well as spatially resolve the energy density of the photon in the coupled-cavity system, so that the system length- and time-scales may be exploited for single photon control. We begin by taking the system state at the timet 1 as the initial condition for unitary evolution once the atom has been detuned. The system will then evolve under the new HamiltonianH(t t 1 ). To calculate the new set of time-dependent amplitudes, we begin by projecting the new initial statej (t 1 )i onto the new set of energy eigenstates. j (t 1 )i = X n jE (2) n ihE (2) n j (t 1 )i; (107) which now evolves unitarily as 110 j (tt 1 )i = X n e iE (2) n (tt 1 )=~ jE (2) n ihE (2) n j (t 1 )i: (108) Substituting 97 and 85 into 108, we arrive at j (tt 1 )i = X n Z n (t 1 )e iE (2) n (tt 1 )=~ u (n) e jei + X j u (n) j jji + X l u (n) l jli + X q u (n) q jqi ! ; (109) whereZ n (t 1 ) is the amplitude transition coefficient for thenth energy eigenstate at timet 1 , Z n (t 1 ) =hE (2) n j (t 1 )i =u (n) e c e (t 1 ) + X j u (n) j c j (t 1 ) + X l u (n) l c l (t 1 ) + X q u (n) q c q (t 1 ): (110) To avoid delayed feedback interactions, we choose t 1 c=(L 2 x a ) so that c e (t 1 ) 0, provided a large enough mode volume is used in the primary sub-cavity with mode setf l g. The time-dependent amplitudes after detuning are simply projections of the system state onto the respective eigenbases: c e (t>t 1 ) =he; 1 ; 2 ; 3 j (t>t 1 )i (111) c j (t>t 1 ) =hg;j; 2 ; 3 j (t>t 1 )i (112) c l (t>t 1 ) =hg; 1 ;l; 3 j (t>t 1 )i (113) c q (t>t 1 ) =hg; 1 ; 2 ;qj (t>t 1 )i (114) Since the energy density of the single photon is proportional to the expectation value ofE(x) 2 , we have 111 h (t<t 1 )jE 2 (x)j (t<t 1 )i = 2 X n;j Ad (n) e d (n) j e iE (1) n t=~ sin (j 0 +j) (L 1 x 0) L 1 2 + X j A 2 sin 2 (j 0 +j) (L 1 x 0) L 1 = 2 X n;l A 0 d (n) e d (n) l e iE (1) n t=~ sin (j 0 +j) (L 2 x 0) L 2 2 + X l A 02 sin 2 (l 0 +l) (L 2 x 0) L 2 + 2 X n;q Bd (n) e d (n) q e iE (1) n t=~ sin (q 0 +q) (0xL 2 ) L 3 2 + X q B 2 sin 2 (q 0 +q) (0xL 3 ) L 3 ; (115) which we can simplify by removing the third, fourth, and sixth terms, since they represent the vacuum expectation value ofE 2 . Thus, in terms of (102), (103), and (104), h (t)jE 2 (x)j (t)i = 2A 2 X n c j (t) sin (j 0 +j) (L 1 x 0) L 1 2 = 2A 02 X n c l (t) sin (l 0 +l) (L 2 x 0) L 1 2 + 2B 2 X n c q (t) sin (q 0 +q) (0xL 3 ) L 3 2 : (116) 3. Transientresonatorresponse We begin by considering the transient response of the emitted photon pulse from the atom. Since the system state is initialized in the excited state of the atom,jc e (t = 0)j 2 = 1. Interaction with the cavity modesfa j g andf l g causes the atom to decay to its ground state, as shown in Fig. (92). While the photon pulse initially exhibits no wavelike behavior, self-interference becomes ev- ident within several wavelengths of the resonator mirror. Since the external cavity is an integer 112 0 0.1 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 Time, t/τ r |c e(g) | 2 = |<e(g)|ψ(t)>| 2 G = 8 G = 20 |c g | 2 |c e | 2 |c g | 2 |c e | 2 Figure 92: Decay of initially-excited atom into the ground state and total photon probability in the system, as functions of time. Due to unitary dynamics, the decay is not strictly exponential, though the decay time- scale depends on the coupling strength and is inversely proportional to the mode spacing. The atom is later detuned att= r = 1, effectively removing all future atom-field interactions. Parameters: x a =L =0:25, G =M =f8; 20g,L 1 = 1,L 2 = 1:5,r 1 = 0:57,r 2 = 0:50. number of photon wavelengths, resonance build-up and ring-down phenomena are observed as the stored energy is lost each cavity round-trip time r . Amplitude decay of the photon pulse is schematically demonstrated in Fig. (93), which demonstrates the external cavity photon ring-down in terms of an on-resonance ray trace. Fig. (94) shows the immediate transient increase in photon probability due to the transmit- ted energy, followed by characteristic ring-down with time-scale Q and stepwise behavior corre- sponding to the resonator structure. The spatio-temporal characteristics of the single-photon pulse energy incident on the resonator for two different coupling strengths are shown in Figs. (95) and (96). 113 Space Time 1 it = 0 = 3 r = r 2 -it t 2 r itr -itr t 2 r itr 2 r = 1 Figure 93: Ray trace schematic of the ring-down in photon pulse amplitude within the auxiliary cavity. The cavity boundary atx =L 3 is perfectly reflecting, so energy can only be transmitted atx = 0. 114 0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 Time, t/τ r Photon probabilit y G = 8 G = 20 P 1 + P 2 P 2 P 1 Figure 94: Photon probabilities, for an atom initially placed atx =0:25. P 1 is the probability of finding the single photon in the primary cavity, while P 2 is the probability of finding the photon in the external cavity. The total photon probability increases from 0 to 1 as the atom decays and remains at 1 due to the detuning of the atom at t = t 1 = 0:5. The probability of the photon in the external cavity increases at a time t= r = 1:25 later due to transmission of the emitted photon, and at round-trip time t= r = 2:25 later, the probability begins to decay in step-wise fashion. Increasing the atom-field coupling amplifies the exponential decay of the single photon pulse observed at every round-trip time of the external cavity. Parameters:G =M =f8; 20g,L 1 = 150 0 ,L 2 = 100 0 ,L 3 = 10 0 ,r 1 = 0:57,r 2 = 0:50. 115 −1.5 −1 −0.5 0 0.5 1 0 0.5 1 1.5 2 2.5 3 3.5 4 Position, x/L 3 Time, t/τ r Figure 95: Space-time plot of photon energy densityhE 2 i in the external resonator. The mode structure of the system becomes evident: since the single photon pulse width is several wavelengths long, self- interference occur on either side of the flat mirror. Parameters: G = M = 8,L 1 = 150 0 ,L 2 = 100 0 , L 3 = 10 0 ,r 1 = 0:57,r 2 = 0:50. −1.5 −1 −0.5 0 0.5 1 0 1 2 3 4 Position, x/L 3 Time, t/τ r Figure 96: Space-time plot of photon energy densityhE 2 i in the external resonator. Increasing the atom-field coupling 2.5x dramatically increases photon pulse localization, resulting in a removal of self interference at the boundary between the two cavities. Parameters:G =M = 20,L 1 = 150 0 ,L 2 = 100 0 ,L 3 = 10 0 , r 1 = 0:57,r 2 = 0:50. 116 4. Ring-downcontrol In the one-dimensional cavity, the photon may be described as the sum of excitations in all the cavity modes, which are non-local and structured by the end mirrors. The expectation value of the photon energy, however, is localized and propagates away from the atom at the speed of light. The two pulses emitted from the atom represent the total energy of the photon in the primary cavity, and can be used to store and control energy in the external resonator. The pulse originally heading towards the external cavity will be denoted the lead pulse, and the other pulse will be denoted the control pulse. The lead pulse will reach the resonator first, and the control pulse will be time- delayed by twice the distance between the atom and the perfectly-reflecting mirror. We wish to exact control over the shaping and timing of the lead and control components of the single photon in the coupled-cavity system, in order to allow for the exploration of novel forms of quantum control in the coupled-cavity system. We can exploit the fact that both pulses are coherent by controlling the time delay between the lead and control pulse, such that the energy in the resonator is cancelled out by the delayed control pulse. Thus, control over phase is done by placing the atom at a distanceL 3 away from the mirror, delaying the second pulse the round-trip time of the resonator. To demonstrate control over the local energy density, we perform amplitude and phase control over the control(delayed) pulse. To do so, we exploit intracavity beam splitter within the primary resonator. Controlling the coupling between the primary cavity modesa j and the control cavity modes l corresponds to control over the control pulse amplitude, in order to completely cancel out the stored energy in the passive resonator located between 0 x L 3 . This additional component introduces greater complexity and time-scales in the delayed dynamics of the entire system, and may be exploited as an additional degree of control. Ring-down elimination is dis- played schematically in Fig. (97), which describes the amplitude and phase requirements for the delay(control) pulse. Since the control pulse is initially emitted by the atom away from the lead pulse, the pulse will acquire a phase shift in amplitude upon reflection from the mirror placed at the end of the control arm at x =L 2 . Thus, the time delay for the control pulse is delay = r , where r = 2L 3 =c is the round-trip time of the external resonator, so that destructive interference between the pulses removes the energy in the resonator at the interface between the two cavities. Using similar parameters to the case of transient resonator response, ring-down of the resonator energy 117 Space Time 1 it = 0 = 3 r = r 2 -it r 2 + t 2 =1 r itr - itr = 0 -r r = 1 Figure 97: Ray trace schematic of applying amplitude and phase control to the control pulse in order to eliminate ring-down within the auxiliary cavity. The energy stored within the auxiliary cavity is completely transferred to the primary cavity. can be completely removed by exploiting amplitude and phase control. Fig. (98) shows the result of applying shut-off control for two different atom-field coupling strengths, while Figs. (99) and (100) display the corresponding space-time dynamics of the photon energy in the system. 118 4 4.5 5 5.5 6 6.5 7 0 0.2 0.4 0.6 0.8 1 Time, t/τ r Photon probability P 2 P 1 G = 8 G = 20 Figure 98: Photon probabilities, for an atom initially placed at x=L 2 =4:5. P 1 is the probability of finding the single photon in the primary cavity, while P 2 is the probability of finding the photon in the external cavity. The total photon probability increases from 0 to 1 as the atom decays and remains at 1 due to the detuning of the atom at t = t 1 = 0:5. The probability of the photon in the external cavity increases at a time t= r = 4:5 later due to transmission of the emitted photon, and at time t = 5:5, the probability is cancelled due to destructive interference of the lead pulse with the delayed pulse. Parameters: G =M =f8; 20g,L 1 = 150 0 ,L 2 = 100 0 ,L 3 = 0:5050,r 1 = 0:80,r 2 = 0:50. −0.5 0 0.5 1 4 4.5 5 5.5 6 Position, x/L 3 Time, t/τ r Figure 99: Space-time plot of photon energy densityhE 2 i being turned off in the external resonator. Param- eters:G =M = 8,L 1 = 150 0 ,L 2 = 100 0 ,L 3 = 0:1,r 1 = 0:80,r 2 = 0:50. 119 −0.5 0 0.5 1 4 4.5 5 5.5 6 Position, x/L 3 Time, t/τ r Figure 100: Space-time plot of photon energy densityhE 2 i being turned off in the external resonator. Lo- calization of the pulse removes the interference observed at the cavity interface. Parameters:G =M = 20, L 1 = 150 0 ,L 2 = 100 0 ,L 3 = 0:1,r 1 = 0:80,r 2 = 0:50. 120 B. Microcavityfeedbackcontrol 1. Microcavitymodel In addition to spatio-temporal control of a single photon inside cavities that are large relative to photon wavelengths, we would like to exert a variety of control over light-matter interaction in space and time by exploiting delayed feedback. [21] exerted a form of control in open cavity QED systems by using an external mirror and delayed feedback to induce Rabi oscillations of a single-mode field inside of a microcavity in the weak-coupling regime. In such a model, a single, initially-excited atom within a microcavity is placed inside of a much larger cavity, in which one mirror has unity reflectivity, while the other mirror may be removed either by imposing half-cavity boundary conditions or increasing the mode volume of the larger cavity. The model is described in Fig. (101). ( ) † † † † | a | Figure 101: Interaction scheme for delayed feedback control of microcavity dynamics. A single atom is placed within a microcavity and interacts with a single mode with coupling strengthM, while the micro- cavity mode couples with strengthG to a continuum of modes shaped by an external, perfectly-reflecting mirror a distancejx a j away from the atom-microcavity system. The structured continuum is modelled by placing the leaky microcavity within large cavity with mode spacing set by the cavity length. The Hamiltonian for the microcavity model is functionally similar to the coupled cavity model (83), with the key difference that the mode coupling interaction (between the external modes and the mode directly coupled to the atom) is localized within the large cavity structure. The atom Hamiltonian is once again given as H atom = ~! c E +~! v V = ~! cv 3 ; (117) whereE = A y c A c is the atomic excitation number,V = A y v A v is the ground state number, 3 = jeihej is the pseudo-spin operator for the excited atom, and! cv =! c ! v is the gap energy for the 121 two level system. Using the rotating-wave and dipole approximations, the interaction Hamiltonian is once again given as H interaction = ~M Tc y +T y c ; (118) whereM =d eg p ! cv =(2~ 0 V mc ) =d eg p ! 0 =(2~ 0 V mc ) is the coupling strength between the atom and the single optical mode on resonance in the microcavity system with mode volumeV mc . The total field Hamiltonian consists of the free field Hamiltonian for the microcavity mode, as well as the set of free field energy terms for the external modes shaped by the volume of the large external cavity: H fields = ~! 0 S +~ X j ! j a y j a j ; (119) whereS =c y c is the micro-cavity photon number anda j (a y j ) is the annihilation(creation) operator for a photon in thejth external cavity mode. Finally, the outcoupling Hamiltonian allowing for the leakage of the photon from the microcavity to the external structured continuum is H hopping = ~ X j G j c y a j +G j a y j c ; (120) whereG j is the coupling strength for a photon transition between the micro-cavity mode, given by G j =G sin(jx a =L) for thejth external cavity mode, in whichL is the length of the large cavity. Thus, the total Hamiltonian is H total = ~! cv 3 +~M Tc y +T y c +~! 0 S +~ X j G j c y a j +G j a y j c +! j a y j a j : (121) The basis states of the embedded microcavity system are described in the same shorthand no- tation as in section II; for a total of one excitation, we have the stateje; 0;i =jei representing the single atom in the fully excited state,jg; 1;i =jP j 0 i representing a photon occupying the microcavity’s single mode, andjg; 0;ji =jji representing a photon occupying thejth mode of the large cavity external containing the microcavity system. The initial state of the system is, like the other control models, equal to the atom state:j (0)i =jei, and the spatial dependence of the photon energy outside of the microcavity is the expectation value of the electric field operator squared, given in equation (56). 122 −0.9 −0.7 −0.5 −0.3 −0.1 Position, x/L t = 0.4τ r t = 1.6τ r t = 0.2τ r t = 0.6τ r t = 0.8τ r t = 1.0τ r t = 1.2τ r t = 1.4τ r t = 1.8τ r t = 2.0τ r Figure 102: Space-time propagation of the single photon pulse energy within the large cavity. As the atom decays due to spontaneous emission, a photon density initially builds up within the microcavity. A relatively large outcoupling to the cavity modes external to the microcavity system inhibits reversible exchange of energy between the single resonant mode and the atom, leading to a decay of excitation probability at the microcavity location. In one dimension, the external mode structure produces two pulses shaped by the atom-mode coupling M and the outcoupling rate G, which travel away from the source at the speed of light and are reflected by perfectly-reflecting boundaries at the ends of the large cavity. After half a cavity round-trip time, the two pulses exchange energy again with the microcavity by building up the photon density inside and re-exciting the atom. The position of the atomx a is given in units of the cavity length, and the propagation timet is given in units of the large cavity round-trip time, r . Parameters: j 0 = 500, L = 250 0 ,jx a j =L=2,M = 4,G = 1:8, r = 8. 2. Weak-to-strongcouplingcontrol In the approach taken by [21], the microcavity subsystem is operated in the weak coupling limit and delayed feedback from an external mirror is used to drive the dynamics into strong coupling- type, in which the Rabi oscillations produced from the external loop are directly connected to the interaction strength between the microcavity photon and atom. Such a model demonstrates fundamentally new delayed dynamics into the standard Jaynes-Cummings model. Fig. (103) shows the various types of dynamics exhibited by the microcavity system, and demonstrates how regular oscillations in the single-mode field can be induced with feedback. 123 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 |c| 2 0.25 0.5 0.75 1 0 0.1 0.2 |c p | 2 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 0 0.05 0.1 |c e | 2 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 Time, Mt/τ r |c| 2 |c p | 2 |c e | 2 |c p | 2 |c e | 2 (d) (a) (b) (c) Figure 103: Dynamics in the single-excitation limit. Here, the photon is confined within the primary res- onator. From top to bottom: (a) SettingG 1 = 0 represents the case with no outcoupling, so Rabi oscillations occur. Parameters: j 0 = 500,L = 250 0 ,jx a j = L=2,M = 4,G = 0; (b) SettingG 1 to a constant re- moves the structure and hence feedback effects, so the photon will leak out of the microcavity. Parameters: j 0 = 500,L = 250 0 ,jx a j =L=2,M = 4,G = 1:8; (c) Removing the microcavity, the atom will interact with various modes, and feedback from the mirrors causes a re-excitation of the atom a round-trip time later. Parameters:j 0 = 125,L = 62:5 0 ,jx a j =L=5,M = 6; (d) Placing the atom-microcavity system in front of a mirror induces Rabi oscillations, even in the weak-coupling regime. This is due to an exchange of en- ergy from the leaked photon and the delayed feedback from the mirror. Parameters:j 0 = 500,L = 250 0 , jx a j =L=8,M = 4,G = 1:8. 124 −0.5 −0.375 −0.25 −0.125 0 0.5 1 1.5 2 2.5 Space, x/L Time, Mt/τ r Figure 104: Space-time plot of the single photon energy outside of the leaky microcavity system located at jx a j in the weak-coupling regime. Rabi oscillations are induced within the resonant microcavity mode due to exchange of energy from the leaked photon and the delayed feedback from the mirror. Oscillations in the energy density become highly regular after timeMt= r = 2, which is reflected in the microcavity photon transient behavior in figure (103). The microcavity-mirror system acts like a subcavity with a very high Q-factor, sustaining oscillations with little loss over time. Parameters: j 0 = 500,L = 250 0 ,jx a j =L=8, M = 4,G = 1:8, r = 8. 125 3. Strong-to-weakcouplingcontrol Beyond using feedback to induce transitions in microcavity photon behavior from weak to strong regimes, the reverse is possible as well when exploiting amplitude and phase. Recent work by [22] demonstrates a variety of phenomena by coupling a dissipative single-mode CQED system to an external single-mode cavity, including achieving dark state resonances between the microcavity emitter and the external cavity, as well as driving a highly-dissipative microcavity subsystem into effective strong coupling. Here, we consider the scenario in which the coupling ratioG=M is not large enough to irre- versibly remove atom-photon exchange, yet non-zero, resulting in damped Rabi oscillations within the microcavity subsystem. Fig. (105) shows the dynamics associated with the various subsystems in the model described by (121), and Fig. (106) depicts the photon pulse train propagation. 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Time, Mt/τ r Excitation Probabilities |c e | 2 P j 0 Σ j P j |c g | 2 Figure 105: Without delayed feedback, oscillations occur as energy is exchanged between the atom and the microcavity photon; microcavity loss damps the oscillations.jc e j 2 is the atomic excitation probability, which is set to unity at timet = 0,P j 0 is the microcavity photon probability, j P j is the photon probability in the large, external cavity, andjc g j 2 is the overall photon probability in the total system, which is equal to 1jc e j 2 . Parameters:j 0 = 500,L = 250 0 ,jx a j =L=2,M = 4,G = 0:724, r = 8. Exploiting delayed feedback between the microcavity subsystem and one of the mirrors of the larger cavity allows for control over the strength of the oscillations. Fig. (107) depicts the subsystem probabilities for the control condition in which feedback resonantly enhances the oth- erwise damped oscillations, while Fig. (108) shows the corresponding space-time dynamics that 126 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 Space, x/L Time, Mt/τ r Figure 106: Space-time dynamics of the emitted photon energy into the external cavity in the strong- coupling regime. The microcavity subsystem is placed in the center of the external cavity. The energy exchange between the atom and the microcavity is damped by the loss rateG into the larger cavity, which results in a set of damped pulse trains propagating away from the microcavity. Parameters: j 0 = 500, L = 250 0 ,jx a j =L=2,M = 4,G = 0:724, r = 8. are qualitatively very similar to the case in which strong-coupling dynamics are induced in the weak-coupling regime with feedback, as seen in Fig. (104). Similarly, if the right phase conditions are met by setting the round-trip distance of the left- travelling photon pulse train to a half-integer multiple of the resonant photon wavelength, the microcavity atom amplitude can be re-excited close to unity. The subsystem dynamics are show in Fig. (109) and Fig. (110) shows the cancellation of the photon energy density in the external cavity, due to the transfer of nearly all the energy to the microcavity atom. Such dynamics are nearly conceptually identical to the delayed dynamics of a bare atom placed no more than a few nodes of the resonant mode away from one of the mirrors of a large cavity. Finally, if the round-trip time between the microcavity and the mirror is equal to an oscilla- tion period of the microcavity photon and the round-trip distance is set to be exactly an integer multiple of the resonant wavelength, the-phase shift induced by reflection from the mirror will cause the photon pulse train to completely shut-off the microcavity dynamics, forcing the energy to propagate away from the microcavity-mirror subsystem, as shown in Figs. (111) and (112). A comparison of the microcavity photon dynamics for the various control conditions described 127 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Time, Mt/τ r Excitation Probabilities |c e | 2 P j 0 Σ j P j |c g | 2 Figure 107: Now, the microcavity is placed closer to one of the perfectly-reflecting mirrors of the larger cavity, such that the feedback time is equal to a Rabi period and the round-trip distance is equal to a half- integer multiple of the resonant photon wavelength, resulting in an enhancement in the microcavity photon oscillation, due to reflection from the mirror at x = 0. The probability oscillations are maximized at 50%, while the remaining 50% of the energy has been carried away by the other photon pulse originally transmitted by the microcavity. Parameters: j 0 = 500,L = 250 0 ,jx a j = 24:75 0 = 0:396,G = 0:724, r = 8. appears in Fig. (113). 128 −0.2 −0.15 −0.1 −0.05 0 0.5 1 1.5 2 Space, x/L Time, Mt/τ r Figure 108: Space-time dynamics of the emitted photon energy for the optimized feedback regime, in which the energy is resonantly stored between the microcavity and one of the external cavity mirrors. Parameters: j 0 = 500,L = 250 0 ,jx a j = 24:75 0 = 0:396,M = 4,G = 0:724, r = 8. 3 3.5 4 4.5 5 5.5 0 0.2 0.4 0.6 0.8 1 Time, Mt/τ r Excitation Probabilities |c e | 2 P j 0 Σ j P j |c g | 2 Figure 109: Under oscillation-enhancement conditions between the microcavity and a nearby mirror, feed- back from the other photon pulse train originally emitted from the microcavity can nearly completely re- move the energy in the external cavity and exchange the total photon energy into the microcavity atom, an example of a Poincaré recurrence. Here, the atomic amplitude reachesc e = 0:977 just before one com- plete external cavity round-trip time, due to interaction with the other photon pulse train emitted from the microcavity. Parameters:j 0 = 500,L = 250 0 ,jx a j = 25:75 0 = 0:412,M = 4,G = 0:724, r = 8. 129 −0.2 −0.15 −0.1 −0.05 0 3 3.5 4 4.5 5 5.5 Space, x/L Time, Mt/τ r Figure 110: Space-time dynamics of external cavity photon energy, which is promptly removed upon feed- back from the other photon pulse train. The effect of feedback facilitates atomic excitation similarly to the scenario in Fig. (90), in which atomic decay is suppressed due to the storage of energy between the atom and the cavity mirror. Parameters: j 0 = 500,L = 250 0 ,jx a j = 25:75 0 = 0:412,M = 4,G = 0:724, r = 8. 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Time, Mt/τ r Excitation Probabilities |c e | 2 P j 0 Σ j P j |c g | 2 Figure 111: The microcavity is once again placed in proximity to the mirror with a cavity-mirror round-trip time approximately equal to one Rabi oscillation period, but with a round-trip distance equal to an integer multiple of the resonant wavelength. The-phase shift from the mirror causes the emitted pulse train to cancel the microcavity photon oscillations and completely remove the energy stored in the microcavity. Parameters:j 0 = 500,L = 250 0 ,jx a j = 25 0 = 0:400,M = 4,G = 0:724, r = 8. 130 −0.2 −0.15 −0.1 −0.05 0 0.5 1 1.5 Space, x/L Time, Mt/τ r Figure 112: Space-time dynamics of the external cavity photon energy, for the control case. The energy stored in the microcavity-mirror subsystem is lost to the rest of the environment due the perfectly out-of- phase feedback pulse train. Parameters: j 0 = 500,L = 250 0 ,jx a j = 25 0 = 0:400,M = 4,G = 0:724, r = 8. 131 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 Time, Mt/τ r Microcavity Photon Probability, P j 0 x a = 125λ 0 x a = 24.75λ 0 x a = 25λ 0 Figure 113: Comparison of induced Rabi oscillation conditions in the strong coupling regime with delayed feedback: (a) Damped oscillations of the microcavity photon occur with non-zero outcoupling to the exter- nal world. (b) Enhanced microcavity oscillations, due to the-phase shift induced by the reflection from the mirror atx = 0. The probability oscillations are maximized at 50%, while the remaining 50% of the energy has been carried away by the other photon pulse originally transmitted by the microcavity. (c) A round-trip distance equal to an integer multiple of the resonant wavelength, along with a-phase shift from the mirror causes the emitted pulse train to destroy microcavity oscillations. Parameters: j 0 = 500, L = 250 0 , M = 4,G = 0:724, r = 8. 132 4. Selectiveoscillationcontrol The delayed feedback control scheme may be augmented by coupling an additional external cavity with large mode volume, as described in Fig. (114). By allowing for hopping of the emitted photon from one resonator system to another atx = 0 via the flat mirror transmission rateK, an additional degrees of control can be used to selectively turn off an oscillation of the microcavity photon. ( ) † † † † | a | † | 2 | K r Figure 114: Interaction scheme for delayed feedback control of microcavity dynamics. A single atom is placed within a microcavity and interacts with a single mode with coupling strengthM, while the micro- cavity mode couples with strengthG to a continuum of modes shaped by an external, perfectly-reflecting mirror a distancejx a j away from the atom-microcavity system. The structured continuum is modelled by placing the leaky microcavity within large cavity with mode spacing set by the cavity length. An auxil- iary resonator is coupled to the primary large cavity with strength K, allowing for the storage of leaked energy from the microcavity. Delayed feedback from the external resonator allows for the disruption or enhancement of intrinsic quantum cavity electrodynamics in the microcavity. Thus, the Hamiltonian in (121) is modified by adding the field and cavity coupling Hamiltoni- ansH auxiliary andH hopping , respectively given as: H auxiliary = X q ~! q b y q b q (122) and H hopping = X j;q ~K a y j b q +a j b y q : (123) Fig. (115) shows the various subsystem dynamics, in which the auxiliary cavity is used to setup feedback conditions to completely cancel the fourth microcavity oscillation. Fig. (116) compares 133 the microcavity oscillations with and without the auxiliary cavity control. Finally, Fig. (117) depicts the dynamics of the photon energy outside the microcavity, in which feedback from the pulse train in the auxiliary cavity removes energy exchange between the microcavity subsystem and the large cavity interface atx = 0. 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 Time, Mt/τ r Excitation Probabilities |c e | 2 P j 0 Σ j P j Σ q P q |c g | 2 Figure 115: Subsystem dynamics of the damped Rabi oscillations with implementation of delayed feedback control from the auxiliary cavity to selectively turn off a microcavity oscillation. Sustained suppression of the oscillations due subsequent pulses in the train re-injecting energy into the microcavity subsystem.jc e j 2 is the atomic excitation probability, which is set to unity at time t = 0, P j 0 is the microcavity photon probability, j P j is the photon probability in the large primary cavity, while q P q is the photon probability in the large auxiliary cavity coupled to the primary cavity at the mirror, andjc g j 2 is the overall photon probability in the total system, which is equal to 1jc e j 2 . Parameters: j 0 = 500, L 1 = 250 0 , L 2 = 53:75 0 ,jx a j = 3L 1 =40 = 0:349L 2 = 18:75 0 ,M = 4,G = 0:8,K = 0:5426. 134 0.25 0.5 0.75 1 1.25 1.5 1.75 2 0 0.05 0.1 0.15 0.2 0.25 Time, Mt/τ r Microcavity Photon Probability, P j 0 r = 1.00 r = 0.84 Figure 116: Comparison of microcavity dynamics in the damped strong-coupling regime with and without feedback control from using the auxiliary cavity. In the former case, the fourth oscillation is completely killed off due to the feedback control conditions involving a pi-phase shift from the mirror reflection at x = L 2 and a round-trip time between the microcavity and the auxiliary cavity mirror at L 2 equal to three Rabi periods of the microcavity photon. Parameters: j 0 = 500, L 1 = 250 0 , L 2 = 53:75 0 , jx a j = 3L 1 =40 = 0:349L 2 = 18:75 0 ,M = 4,G = 0:8,K = 0:5426. −2 −1.5 −1 −0.5 0 0.5 1 0.5 1 1.5 2 2.5 Space, x/L 2 Time, Mt/τ r Figure 117: Space-time dynamics of selective oscillation control. The fourth left-travelling pulse in the emitted train carries all the energy stored between the microcavity and the interface between the two large cavities, prohibiting any excitations in the microcavity system. Parameters: j 0 = 500, L 1 = 250 0 , L 2 = 53:75 0 ,jx a j = 3L 1 =40 = 0:349L 2 = 18:75 0 ,M = 4,G = 0:8,K = 0:5426. 135 XIV. NON-CLASSICALLIGHTINMESOSCALELASERS Beyond the unitary models explored in the previous two sections, it is worth examining the dynamics of open cavity systems. In order to study the effect of field quantization on fluctuations and noise in mesolasers, a quantum model of an open, pumped system is required. An example of one such model is shown schematically in Fig. 118. While particle number, and hence energy, is quantized in the pumped mesolaser model introduced in Ch. 2, such a model does not account for phase and hence remains classical. Figure 118: A model of a pumped open cavity system, in which the system of interest is the atom-cavity subsystem. Here, a two-level emitter interacts with a single cavity mode with couplingg. The system is pumped at rateP , undergoes spontaneous emission at rate into non-lasing leaky modes, and couples to the external world via a total loss rate , which represents a decay rate of the laser photon field due to partially-transmitting mirrors at either end of the cavity. We would like to study non-classical light produced by such a system, and how different cou- pling regimes affect photon statistics in open cavity systems. To this end, we utilize a density matrix approach [23] describing the system shown in Fig. 118. The reduced density matrix for the subsystem of interest may be found by tracing out the environmental degrees of freedom from the total density matrix T describing the various dissipation mechanisms in the open cav- ity model. The state dynamics can then be resolved by solving the Lindblad master equation for the subsystem of interest, . Under the Born-Markov approximation, the master equation in the interaction picture for multiple atoms inside the cavity is given as 136 ^ L = d dt = i ~ [;H S ] + X k P k 2 2 y k k k y k k y k + 2 2aa y a y aa y a + X k k 2 2 k y k y k k y k k ; (124) whereH S is the Jaynes-Cummings Hamiltonian first given in Eq. (41). Solving the master equa- tion enables one to compute the dynamics corresponding to the expectation values for any opera- tors acting on the system. A particularly useful measure is the intensity-intensity correlation for light inside the microcavity, which yields information on the statistics of photon number. A variety of phenomena including anti-bunching and squeezed states can be studied with such a measure, which is also called the second-order correlation function. In terms of the cavity mode creation and annihilation operators, the unnormalized intensity-intensity correlation function is given as [8] g 2 (t;) =ha y (t)a y (t +)a(t +)a(t)i; (125) where is a time delay between photons. Since such a correlation is time-independent in steady state, it can be evaluated from the system density matrix using ha y (t)a y (t +)a(t +)a(t)i = Tr h a y ae ^ L a S a y i ; (126) which is valid in the Markovian approximation, and where S is the steady state solution for Eq. (124). In order to study the statistics of light generated from such a model, we examine the normalized second-order correlation function at zero time delay ( = 0). The normalized correlation can then be written in terms of Eq. (126), G 2 ( = 0) = ha y a y aai ha y ai 2 = P s s(s 1)( 2s;2s + 1s;1s ) [ P s s( 2s;2s + 1s;1s )] 2 ; (127) where the reduced density matrix elements kn;jm =hknjjjmi are given in the basisfj1si;j2sig, withs representing the number of cavity photons andf1; 2g representing the states of the emitter [23]. Such basis states are product states of the emitter states and cavity mode Fock states. For G 2 ( = 0)< 1, non-classical light exists in the cavity system. Figs. (119) - (121) show howG 2 (0) depends on the dissipation rates for the system excitations. A sharp transition from non-classical to classical light generally occurs whenever more than one emitter is introduced into the system. Interestingly, however, non-classical light persists in the many-body scenario provided the system 137 is in the strong coupling regime, for whichg= 1 andg= 1. This is in contrast to the single emitter case, which requires the dissipation rates to be greater than the pump rate in order to avoid photon bunching in the mesolaser cavity. 0 1.0 2.0 0 1.0 2.0 1 1.5 2 κ (meV) γ (meV) G 2 (τ = 0) (g = 2 meV; P = 1 meV) G 2 (0) = 1 n = 1 Classical Non−classical Boundary Figure 119: Numerical simulation demonstrating parameter competition for a classical-quantum boundary in the open cavity model. Here, a single atom (n = 1) is placed inside a leaky cavity, and is excited by an external pump. 138 0 1.0 2.0 0 1.0 2.0 0.6 0.8 1 1.2 1.4 1.6 1.8 κ (meV) γ (meV) G 2 (τ = 0) (g = 2 meV; P = 1 meV) G 2 (0) = 1 n = 1 n = 2 n = 3 Figure 120: Comparison of second-order correlation for different numbers of atomsn inside the cavity. As soon as more than one atom is placed inside the cavity, the non-classical behavior is strongly suppressed and we observe thermal light statistics. 0 0.25 0.5 0 0.50 1.0 0.9 1 1.1 1.2 1.3 κ (meV) γ (meV) G 2 (τ=0) (g = 2 meV; P = 1 meV) G 2 (0) = 1 n = 2 n = 3 Figure 121: Second-order correlation for 2 and 3 atoms. If the system is sufficiently in the strongly-coupled regime, non-classical light can still be achieved, even in the presence of multiple atoms. 139 XV. CONCLUSIONS In the first part of this chapter, we have studied the decay dynamics associated with a single, initially-excited atom, and how such dynamics strongly depend on the environment. We utilize the Wigner-Weisskopf approach to describe the dynamics of a single atom coupled to a multimode cavity, and demonstrate a dynamic Purcell effect by elucidating the dependence of excitation and de-excitation dynamics on atomic position within the cavity. We also explore how the characteris- tic dynamics of the system transition from strongly-coupled behavior in the limit of single-mode coupling to the weak-coupling regime in the large cavity limit. Placing the atom at the center of the large cavity induces exponential-type decay that becomes irreversible only in the continuum limit. We have also demonstrated various methods to control single photon dynamics in space and time by employing coupled-cavity systems, using fully quantized models which elucidate how the behavior of classical radiation fields is manifested in the quantum regime. We have studied the decay dynamics associated with a single, initially-excited atom, and how such dynamics strongly depend on the environment. Finally, we have demonstrated various types of control over light- matter interaction in microcavities by using delayed feedback and exploiting the amplitude and phase degrees of freedom in composite cQED systems. Our methods illustrate a variety of forms of spatio-temporal control in quantum optical systems, and can be applied towards manipulating single photons for quantum logic gate operations, as well as provide theoretical tools for explor- ing feedback control in open quantum systems by enhancing or suppressing subspace dynamics. Finally, we explored the statistics of light produced by a pumped atom-cavity system, and found regions in parameter space that define a quantum-classical boundary differentiating classical from non-classical light. While the photon statistics in the cavity behave in a thermal fashion as soon as more than one atom is introduced to the model, non-classical light can still be achieved, provided the system remains in the strongly-coupled regime. Acknowledgements All numerical computations and simulations were performed on the University of Southern California campus. This research was supported by the ARO MURI Grant W911NF-11-1-0268. 140 Chapter3References [1] P.W. Milonni and P.L. Knight, Opt. Commun.9, 2 (1973). [2] T.D. Newton and E.P. Wigner, Reviews of Modern Physics21, 400 (1949). [3] A.S. Wightman, Reviews of Modern Physics34, 845 (1962). [4] L. Mandel, Phys. Rev.144, 1071 (1966). [5] U. Dorner and P. Zoller, Phys. Rev. A66, 023816 (2002). [6] M. Ligare and R. Oliveri, Am. J. Phys.70(1) (2002) 58. [7] E.T. Jaynes and F.W. Cummings, IEEE Proceedings511 (1963)89. [8] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, Cambridge University Press, 1997). [9] V . Weisskopf and E. Wigner, Z. Physik63, 54 (1930). [10] R.J. Cook and P.W. Milonni, Phys. Rev. A35, 12 (1987). [11] G.C. Stey and R.W. Gibberd, Physica60, 1 (1972). [12] A. Kavokin, J.J. Baumberg, G. Malpuech, F.P. Laussy,Microcavities, (Oxford University Press, 2011), 1st ed. [13] E.M. Purcell, Phys. Rev.69, 681 (1946). [14] J. Gérard and B. Gayral, Journal of Lightwave Technology17, 11 (1999). [15] J.J. Sakurai, S.F. 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In chapter 1 I solved the coupled external cavity rate equations in the continuum mean-field limit to explore a variety of dynamical phenomena in solitary and external cavity semi- conductor lasers, including delayed feedback control, cavity formation, and transitions to chaos. In chapter 2 I extended the same forms of control performed on the classical, mean-field model to mesolasers described by master equations for particle state probabilities, and explored a vari- ety of transient dynamics of the particle statistics. By introducing external cavities and effective saturable absorbers, it is possible to control such dynamics by taking advantage of the system time-scales. We have demonstrated such control in meso-scale semiconductor lasers, using the classical mean-field limit as a guide. In the limit of small number of particles, fluctuations can play a dominant role in the evolution of particle statistics and probability distributions. Such fluc- tuations can dramatically increase the noise in the system and by controlling various feedback mechanisms in the external cavity laser, we have shown that in principle, it is possible to control such noise. Such a model provides an ideal framework for the study of the breakdown of clas- sical thermodynamics as fluctuations begin to dominate in few-particle systems. The probability distributions in the model phase space contain information about the history of the laser dynamics at any given moment in time, inherent within the distribution shape. Such information cannot be extracted in continuum mean-field models, allowing for the study of non-Markovian control of entropy in meso-scale systems. Finally, in chapter 3 I studied the decay dynamics associated with a single, initially-excited atom, and how such dynamics strongly depend on the environment, as well as how the characteristic dynamics transition from quantum to classical as the system size is increased. 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Abstract (if available)

Abstract Non‐Markovian processes are of particular interest when studying the dynamical evolution and control of quantum systems. While non‐Markovianity is an inherent property of time‐evolving unitary systems, the transient evolution of open systems can also exhibit memory resulting from delayed feedback. By understanding the various system timescales and subsystem correlations, transient control may be exerted by exploiting time delays and other mechanisms external to the system of interest. In this thesis, the dynamics of various open and closed optical cavity systems are examined, including semiconductor laser diodes, microcavity diodes, and coupled‐cavity systems, with a focus on delayed feedback control.

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Asset Metadata

Creator Unglaub, Walter V. (author)

Core Title Feedback control of optical cavity systems

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Physics

Publication Date 11/03/2015

Defense Date 12/10/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cavity QED,delayed feedback dynamics,mesolasers,nonclassical light,non‐Markovianity,OAI-PMH Harvest,optical cavities,photon control,photon dynamics,quantum optics,semiconductor lasers

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Levi, Anthony (committee chair), Dappen, Werner (committee member), Haas, Stephan (committee member)

Creator Email unglaub@usc.edu,wunglaub2@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-195036

Unique identifier UC11278176

Identifier etd-UnglaubWal-4009.pdf (filename),usctheses-c40-195036 (legacy record id)

Legacy Identifier etd-UnglaubWal-4009.pdf

Dmrecord 195036

Document Type Thesis

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Rights Unglaub, Walter V.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA