Topics in modeling, analysis and simulation of near-term quantum physical systems with continuous monitoring

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Content TOPICS IN MODELING, ANALYSIS AND SIMULATION OF NEAR-TERM QUANTUM PHYSICAL SYSTEMS WITH CONTINUOUS MONITORING by Shesha Shayee K. Raghunathan A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2010 Copyright 2010 Shesha Shayee K. Raghunathan Dedication To my parents and to Deepa. ii Acknowledgments I am greatly indebted to my advisor, Todd Brun, for his kindness and patience. He has been a great source of help and inspiration over the years that I have known him. His intuitive and `hand-wavy' explanations of important concepts were quite useful to me, a person with non-physics background; it helped me see physics in an intuitive way, and in its full color, depth and in its various dimensions, rather than merely as a set of mathematical equations. I have also had many opportunities to discuss wide-ranging non-academic topics with him|politics, art, movies etc|which has always been fun. It also made my academic experience that much more fullling. Knowing him has been my privilege, and working with him a great learning experience. I would like to thank Prof. Gandhi Puvvada for having me as his Teaching Assistant over the years. Prof. Puvvada's knowledge of computer hardware and design is immense and deep, and I have learnt a lot working with him. His dedication to work is truly inspiring. I thank Prof. Hsi-Sheng Goan for collaborating with me in one of my projects. His inputs were very important to the project's success. I thank Profs. Daniel Lidar, Stephan Haas, Tony Levi and Paolo Zanardi for kindly agreeing to be part of my qualifying exam. Special thanks to Daniel and Stephan for agreeing to be on my defense committee. I thank all for their valuable feedback. I thank all my colleagues in my group|Hari Krovi, Bilal Shaw, Min-hsu Hsieh, Martin Varbonov, Ognyan Oreshkov, Mark Wilde, Jos e Alonso, Ching-Yi Lai, Yicong Zheng|for contributing in my development as a researcher. I also thank all the sta in iii the EE department for their help and support, particularly Milly Montenegro, Mayumi Trasher, Anita Fung, Gerrielyn Ramos and Diane Demetras; they have all been very kind to me, and forthcoming with their help whenever I sought it. My research was supported in part by NSF Grant No. ECS-0507270 and NSF CAREER Grant No. CCF-0448658. I thank NSF for the support. Life as a doctoral student is challenging. Having wonderful people as room-mates makes it a touch easier. I thank Ganesha Bhaskara and Jay Jayanna for being such fantastic room-mates; I will cherish the debates/discussions that I have had, and also many lighter moments that I have shared with them. Finally, I thank my parents for their un-conditional love and support; needless to say, I owe it all to them. And, to my wife Deepa, what can I say... thank you! I am lucky indeed! iv Table of Contents Dedication ii Acknowledgments iii List of Figures vii Abstract xi Chapter 1: Introduction 1 1.1 Single-photon source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Single-photon probability . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 MRFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2: Single-photon source: Indistinguishability 1 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 AMMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Electronic monitoring in Quantum Dots . . . . . . . . . . . . . . . . . . . 14 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Chapter 3: Single-photon source: Probability 2 18 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Single-photon light emitting diode . . . . . . . . . . . . . . . . . . . . . . 21 3.3 System modeling and parameter regime . . . . . . . . . . . . . . . . . . . 22 3.3.1 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 Parameter regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 This chapter is adapted from our work in [36]. 2 This chapter is adapted from our work in [37]. v 3.4 Change detection algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.1 Sequential CUSUM procedure . . . . . . . . . . . . . . . . . . . . . 29 3.4.2 Bayesian solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.5.1 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.2 Deterministic solution . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5.3 CUSUM performance . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5.4 Measurement Eciency . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 4: Single-spin measurement in OSCAR MRFM with spin noise 3 44 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.1 Chapter overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 OSCAR MRFM - Model and Operation . . . . . . . . . . . . . . . . . . . 47 4.3 Eective Hamiltonian using the adiabatic approximation . . . . . . . . . . 48 4.4 Modeling decoherence in OSCAR MRFM system . . . . . . . . . . . . . . 52 4.4.1 Decoherence in the cantilever . . . . . . . . . . . . . . . . . . . . . 52 4.4.2 Modeling spin noise . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.5 Continuous measurement and feedback in OSCAR protocol . . . . . . . . 55 4.6 Moment equations for OSCAR MRFM with Gaussian approximation . . . 56 4.6.1 Unitary evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.6.2 Open system evolution with continuous monitoring . . . . . . . . . 58 4.6.2.1 First-order moment equations . . . . . . . . . . . . . . . 60 4.6.2.2 Second-order moment equations and Gaussian approxi- mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7.1 Accuracy of Gaussian approximation . . . . . . . . . . . . . . . . . 66 4.7.2 OSCAR MRFM and single-spin measurement . . . . . . . . . . . . 67 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Chapter 5: Conclusions 75 5.1 Single photon source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.2 Single-photon probability . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Single-spin measurement using MRFM . . . . . . . . . . . . . . . . . . . . 76 Bibliography 78 3 This chapter is adapted from our work in [38]. vi List of Figures 2.1 Continuous monitoring with feed forward involves 3 components: (i) emit- ter + cavity, (ii) AMMSE-based transition time estimation, and, (iii) variable delay element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Energy level diagram of the emitter + cavity model, including the signi- cant dynamical processes with their rates. Solid arrows indicate incoherent processes, and broken arrows indicate coherent evolution. . . . . . . . . . 7 2.3 Canonical experiment to calculate indistinguishability. SPS 1 andSPS 2 are identical and independent sources. A (50 : 50) beam-splitter takes the photons from SPS 1 and SPS 2 as input, with output modes 3 and 4; a coincidence counter records the output of the detectors in modes 3 and 4. If the photons in modes 1 and 2 are identical, they will always either both go into mode 3 or both into mode 4, implyingp c = 0 and = 1. 11 2.4 Numerical Results. We plot indistinguishability () as a function of 2 with system parameters: g = 0:1; = 1; = 0:1; 1 = 0:001; the X-axis is plotted in logscale. The 5 cases are: (i) = 0:0, (ii) = 0:1; = 0:0, (iii) = 0:1; = 1:0, (iv) = 0:1; = 0:1, and, (v) = 0:2; = 0:5. Cases (iii), (iv) and (v) involve the continuous monitoring feed forward technique, and those curves have small deviations due to statistical uncertainties in the Monte Carlo calculation (5000 trajectories). 12 2.5 Quantumdot p-ndiodeor p-i-nheterojunction. A quantum dot p- n diode consists of an insulator sandwiched between p- and n-type silicon. A quantum dot is fabricated inside the insulator, and this is contained within an optical microcavity. The diode is biased in the forward direction, such that a single electron (e ) tunnels through from the n-side to the dot. The electron remains in the dot until a hole (e + ) tunnels through to the dot from the p-side. The electron-hole pair in the dot recombines spontaneously to emit a photon into the cavity, which subsequently leaks out to an external mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 vii 3.1 Quantumdot p-ndiodeor p-i-nheterojunction. A quantum dot p- n diode comprises an insulator sandwiched between p- and n-type silicon. A quantum dot is fabricated inside the insulator, and this is contained within an optical microcavity. The diode is biased in the forward direction, such that a single electron (e ) tunnels through from the n-side to the dot. The electron remains in the dot until a hole (e + ) tunnels through to the dot from the p-side. The electron-hole pair in the dot recombines to emit a photon into the cavity, which subsequently leaks out to an external mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Single-photonLEDoperation: Schematicmodel. A single-photon LED consists of three quantum objects: an emitter (QD), a microcavity and an external mode. Electrical pumping involves tunneling of an elec- tron (e ) from the n-side to the dot; an e e + recombination leads to an emission of a photon into the cavity, which then leaks out to the exter- nal mode. The e tunneling occurs when the bias voltage in the diode is favorable to the event. We control the bias voltage externally; we keep the bias `on' to allowe tunneling, and turn it `o' to stop tunneling. We monitor the state of the dot continuously, and use a sequential algorithm called CUSUM and its associated decision circuit to decide whether to keep the bias `on' or to turn it `o.' . . . . . . . . . . . . . . . . . . . . . 23 3.3 Energy level diagram. A state of the system is described by three quantum numbers: the dot, the cavity and the external mode. In this diagram, we show those states with at most 2 quanta of energy and their signicant dynamical processes: broken arrows indicate incoherent pro- cesses, and solid arrows indicate coherent evolution. If the pumping dura- tion is nite,jG; 0; 0i,jG; 0; 1i andjG; 0; 2i are the possible nal states of this system, and their corresponding probabilities represent the zero photon, 1 photon and 2 photon probabilities, respectively. . . . . . . . . . 25 3.4 Markov chain approximation. We approximate the energy level dia- gram as a Markov chain. The individual states represent the state of the dot along with some number of photons in the external mode. The state of the dot is represented by x, and n is the number of photons in the external mode;x is 0 if there is noe in the dot and 1 if ane is present; n is a positive integer. The parameters r p and r e are the pumping and emission rates, respectively. We combine x and n into a single variable k = 2n +x; note that oddk states contain ane while evenk states have no e in the dot. The system starts initially in the state k = 0 with no e in the dot and no photon in the external mode. . . . . . . . . . . . . 31 viii 3.5 Deterministic evolution. We plot photon probabilities|p(0), p(1) and p(2+)|as a function of the pumping duration; the parameter values are given in Eq. (3.34) and = 0:1. The system is initially assumed to be in the statejG; 0; 0i. Initiallyp(0) = 1 and decreases as pumping duration is increased;p(1), on the other hand, starts at 0, increases initially, reaches a maximum and then starts to decrease; p(2+) increases with pumping duration (more gradually thanp(1)) and starts to dominate at long times. The maximum ofp(1) is 0:73 and it does so att 19:5; however,p(2+) is about 12%, which is quite high. Imposing the constraint in Eq. (3.33), the best time to stop pumping at these parameter values is t 8, where p(1) 53% and p(0) 46%. . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.6 CUSUMperformance. We plot the bestp(1) given constraint Eq. (3.33) for dierent pumping rates 2 [0:01; 0:1]; we nd a close-to-optimal value for the threshold h by numerical exploration. There are 4 cases corre- sponding to dierent measurement qualities = ( ) 1=2 (here = 1): (i) deterministic (no measurement), (ii) low-noise ( = 10), (iii) interme- diate noise ( = 1:0), and (iv) high-noise ( = 0:1); we plot the deter- ministic case for comparison. Low-noise (ii) naturally leads to the best performance while high-noise (iv) performs the worst; in fact, (iv) per- forms worse than the deterministic case (i); in the intermediate regime (iii), CUSUM does better than cases (i) and (iv). In cases (ii) and (iii), the performance improvement is higher for lower and the improvement reduces as is increased. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.7 Monitoring efficiency and performance. In this plot, we explore the eect of monitoring eciency on performance. The 4 cases considered are: (i) ( = 1:0; = 0), (ii) ( = 1:0; = 1:0), (iii) ( = 1:0; = 0:1), and (iv) ( = 2:0; = 0:5). Case (i) performs consistently better than (iii) while (ii) dominates both (i) and (iii). However, case (iv) performs even modestly better than (ii). Hence, a small increase in monitoring strength can compensate for lower eciency. . . . . . . . . . . . . . . . . 42 4.1 MRFM system: model and operation. A cantilever with a ferromagnetic tip oscillates in close proximity to a substrate with a free spin. A uniform magnetic eldB 0 is applied in thez direction and a microwave (RF) eld is applied in the x-y plane. The position of the cantilever is monitored continuously by shining a laser on the cantilever tip and using optical interference to make measurements. A feedback mechanism (not shown in the gure) maintains the amplitude of the cantilever at a pre-determined xed value, making the cantilever a frequency-determining element. The direction of the frequency shift reveals the orientation of the spin. . . . . 47 ix 4.2 Comparing full quantum SME and Gaussian approximation (1). We plot h ^ Zi as a function of time for both the quantum SME as well and the Gaussian approximation at initial times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Comparing full quantum SME and Gaussian approximation (1). We plot h ^ Zi as a function of time for both the quantum SME as well and the Gaussian approximation at intermediate times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. . . . . . . . . . . . . . . . . . . . . . . . . 68 4.4 Comparing full quantum SME and Gaussian approximation (2). We plot h ^ Zi as a function of time for both the quantum SME as well and the Gaussian approximation at long times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.5 Comparing full quantum SME and Gaussian approximation (4). We plot the evolution of spin-up probability r u =h ^ P " i.The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. We assume that the cantilever-spin system is initially decoupled, with the cantilever at the lowest point in its oscillation, while the spin is in an equal superposition of up and down states. . . . . 70 4.6 Spin probability for dierent s . We plot the time evolution of spin-up probability, r u =h ^ P " i, for 3 dierent values of spin noise rate s : (i) 10 3 , (ii) 10 4 , and (iii) 10 5 . The number of spin ips decreases as s is reduced. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Spin-up probability evolution of two trajectories. We show the time evo- lution of the spin-up probability, r u , for the two trajectories; in the rst, the spin relaxes to its up state, while in the second, it relaxes to its down state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.8 Frequency shift in OSCAR MRFM. We show the frequency shift in the OSCAR MRFM protocol, for two dierent trajectories (Fig. 4.7). We plot the Fourier amplitude (in arbitrary units) as a function of cantilever frequency, !, corresponding to the two trajectories in Fig. 4.7. The spin noise rate s used in this plot is 10 5 . The Fourier amplitude is calculated using a standard FFT algorithm; the number of samples is N = 2 19 , and the sample spacing is t = 0:02. . . . . . . . . . . . . . . . . . . . . . . . 73 x Abstract In this work, we study two important problems in the quantum world: single-photon source and single-spin measurement. Single-photonsource. We consider a quantum dot that is inside a cavity operating in a weak coupling regime. We model the system in the cavity QED setting and use Jayes- Cummings Hamiltonian to describe the dot-cavity interaction. The decoherence eects that we consider include spontaneous emission, cavity leakage and dephasing, along with continuous monitoring of the dot; we describe the system using a stochastic master equa- tion. There are 2 measure of `goodness' for a single-photon state|indistinguishability and single-photon probability. 1. Indistinguishability: We propose an engineering technique using continuous quan- tum measurement in conjunction with feed forward to improve indistinguishability of a single-photon source. The technique involves continuous monitoring of the state of the emitter, processing the noisy output signal with a simple linear esti- mation algorithm, and feed forward to control a variable delay at the output. In the weak coupling regime, the information gained by monitoring the state of the emitter is used to reduce the time uncertainty inherent in photon emission from the source, which improves the indistinguishability of the emitted photons. 2. Single-photon probability: An engineering technique using continuous quantum measurement in conjunction with a change detection algorithm is proposed to xi improve probability of single photon emission for a quantum-dot based single- photon source. The technique involves continuous monitoring of the emitter, inte- grating the measured signal, and a simple change detection circuit to decide when to stop pumping. The idea is to pump just long enough such that the emitter + cavity system is in a state that can emit at most 1 photon with high probability. Continuous monitoring provides partial information on the state of the emitter. This technique is useful when the system is operating in the weak coupling regime, and the rate of pumping is smaller than, or comparable to, the emitter-cavity coupling strength, as can be the case for electrical pumping. Single-spin measurement. A promising technique for measuring single electron spins is magnetic resonance force microscopy (MRFM), in which a microcantilever with a permanent magnetic tip is resonantly driven by a single oscillating spin. The most eective experimental technique is the OScillating Cantilever-driven Adiabatic Reversals (OSCAR) protocol, in which the signal takes the form of a frequency shift. If the quality factor of the cantilever is high enough, this signal will be amplied over time to the point that it can be detected by optical or other techniques. An important requirement, however, is that this measurement process occur on a time scale short compared to any noise which disturbs the orientation of the measured spin. We describe a model of spin noise for the MRFM system, and show how this noise is transformed to become time-dependent in going to the usual rotating frame. We simplify the description of the cantilever-spin system by approximating the cantilever wavefunction as a Gaussian wavepacket, and show that the resulting approximation closely matches the full quantum behavior. We then examine the problem of detecting the signal for a cantilever with thermal noise and spin with spin noise, deriving a condition for this to be a useful measurement. xii Chapter 1 Introduction Modeling of physical systems is an important task of abstraction where only necessary details are retained and unnecessary ones ignored. Abstraction gives a deeper under- standing of a system. Once a system is captured in a model, the next step is analysis. Here, we consider the model along with its parameters and try to simplify the model in certain limits of parameter space. Approximations, transformations etc. are carried out, and a more simplied, yet still robust model is obtained. Though modeling and analysis generally yield a model that is much simpler than the actual physical system it represents, the dynamics of these systems are still so compli- cated that, in many cases, the solution cannot be obtained analytically; one has to make do with numerical simulations. Modeling, analysis and simulation of near-term systems is essential if quantum com- puting has to become a reality. Quantum systems in the real-world have noise in them and modeling decoherence eects is vital. As with most engineering solution, monitoring the system can help improve the performance, as access to information about the sys- tem state is fundamental to any feedback mechanism. Hence incorporating continuous monitoring in the modeling of the system is essential. In this thesis, I describe the modeling, analysis and numerical simulation of two important physical systems that I have studied: A. Single photon source; B. Magnetic resonance force microscopy (MRFM). These two physical systems, and my contributions, are explained in greater detail in the forthcoming chapters. Here, however, I will highlight some of my contributions. 1 1.1 Single-photon source Generation of a single photon at a given time with high probability is the goal of a single-photon source. Using continuous monitoring of the emitter to improve temporal indistinguishability of the photon [36], and to enhance the probability of single-photon emission, are the two fundamental contributions of my work. 1.1.1 Indistinguishability In [36], we combine continuous quantum measurements with feed forward to reduce the uncertainty in the time at which a photon leaks out of a source. The idea is to continuously monitor the state of the emitter, and to use this information to make a time correction at the end, so that the time uncertainty of the single-photon source is reduced. We process the signal with the simple technique of Ane Minimum Mean Squared Error estimation (AMMSE) to estimate the time at which the emitter returned to its ground state. This time estimate is fed forward to control a variable delay at the output of the single-photon source. We show that this approach works well in the weak coupling regime, where the emission time of the photon time into the cavity closely controls the emission time of the photon from the cavity. 1.1.2 Single-photon probability In [37], we use continuous quantum measurements to improve single-photon probability. The idea is simple: to monitor the state of the emitter continuously, and determine when to stop pumping energy into the system based on the information obtained. Unfortu- nately, the output from monitoring such a microscopic system is intrinsically noisy, and little time is available to process the received signal. This complicates the procedure. We utilize a sequential statistical technique called CUmulative SUM (CUSUM) as the deci- sion making process. We show numerically that this mechanism substantially improves 2 single-photon probability in the weak coupling regime, particularly when the pumping rate is comparable to the QD-Cavity coupling strength. 1.2 MRFM In [38], we analyze the MRFM system in generality within a Markovian framework. We describe the system using a density matrix, and evolve it using a stochastic master equation [7, 23], including feedback of the continuous measurement output. We consider decoherence eects acting on both the cantilever and the spin. The stochastic master equation, though it gives a quantum-mechanical description of the system, is numerically expensive. To ameliorate this, we show that the state of the cantilever can be approx- imated as a Gaussian wave packet, leading to a description of the MRFM system by a closed set of 11 coupled stochastic dierential equations. We show that the Gaussian approximation is valid in the parameter regime of interest, by numerically comparing the evolution of the Gaussian equations with the fully quantum stochastic master equation. Further, we use the Gaussian equations to analyze the OSCAR protocol as a tool for single-spin measurement. We consider the constraints set by the spin noise on a single-spin measurement. For the parameter values chosen, we calculate a bound on the spin noise time scale. We choose a rate for spin noise that satises this time scale, and numerically show that the OSCAR protocol can indeed be used to do single-spin measurement. The underlying theme of my work is the use of quantum trajectories in the numerical studies of the physical systems mentioned earlier. Quantum trajectories is an impor- tant technique that has been developed over the last two decades, and has found wide applications in simulating quantum systems that are measured continuously. The use of stochastic master equations in modeling physical systems is another underlying theme of my work. 3 Chapter 2 Single-photon source: Indistinguishability 1 2.1 Introduction Developing a scalable model of computation is paramount for practical quantum com- putation, and has generated consuming interest over the last decade or more. In 2001, Knill et al. [26] proposed linear optical quantum computation (LOQC), based on linear optics and single photon measurements, and showed that it was scalable, albeit at the cost of very high overheads and unrealistic component reliabilities. This scheme has been further developed in ways that have reduced both the overhead and the reliability requirements drastically [27]. LOQC, however, assumes the availability of good quality single photon states on demand, which is technologically challenging. Such single-photon sources would also have many other applications, such as quantum imaging, metrology, communication and cryptography [32]. Semiconductor quantum dot-based implementations of a single-photon source are of particular interest, as they scale well upon integration, and are amenable to commercial fabrication techniques [32, 24]. Typically, implementation of these devices involves a quantum dot (QD or dot) inside a microcavity, with the dot acting as the photon source: the presence of the cavity increases the collection eciency due to the spatial connement of the photons [32, 24, 30]. A wide variety of cavities with dierent sizes, shapes, and quality factors have been designed and fabricated [30]. 1 This chapter is adapted from our work in [36]. 4 A quantum dot in a microcavity can be pumped either optically or electrically. Opti- cal pumping is more straightforward experimentally, but electrical pumping may be better suited to large scale integration. Also, as it does not directly insert photons into the cavity, it opens up the possibility of pumping directly into an energy level resonant with the cavity mode. It also allows a channel for continuous measurement of the dot. In electrical pumping, a bias voltage is applied across a quantum dot pn diode (Fig. 2.5) that enables an electron to tunnel through from n-type onto the dot (present at pn junction). Once an electron tunnels through to the dot, no new electron can tunnel through due to the Coulomb blockade eect [22]. A further increase in the bias voltage then enables a hole from thep-type to tunnel to the dot. Recombination of electron and hole in the dot follows, leading to a photon emission into the cavity. Recombination also leads to a drop in the potential across the quantum dot pn diode. By observing this change in the potential across the quantum dot pn diode, we can gain information about the state of the dot. This observation can be seen as a weak quantum measurement that provides partial information about the system [28]. Variable Delay SPS AMMSE Emitter + Cavity Figure 2.1: Continuous monitoring with feed forward involves 3 components: (i) emitter + cavity, (ii) AMMSE-based transition time estimation, and, (iii) variable delay element. 5 In this work, we combine continuous quantum measurements with feed forward to reduce the uncertainty in the time at which a photon leaks out of a source (Fig. 2.1). The idea is to continuously monitor the state of the emitter, and to use this information to make a time correction at the end, so that the time uncertainty of the single-photon source is reduced. We process the signal with the simple technique of Ane Minimum Mean Squared Error estimation (AMMSE) to estimate the time at which the emitter returned to its ground state. This time estimate is fed forward to control a variable delay at the output of the single-photon source. As we will see, this approach works well in the weak coupling regime, where the emission time of the photon time into the cavity closely controls the emission time of the photon from the cavity. 2.2 The system Figure 2.2 shows the energy level diagram of the model under consideration [25]. The emitter + cavity system consists of an emitter (dot) with 3 energy levels: jGi,jX 1 i andjX 2 i, where G is the ground state, and, X 1 and X 2 represent the rst and second excited state, respectively; we assume that X 1 is resonantly coupled to the cavity mode. The interaction between the emitter and the cavity is given by the Jaynes-Cummings Hamiltonian: ^ H I = i~g ^ a y ^ 1 ^ a^ + 1 , where g is the interaction strength, and ^ 1 = jGihX 1 j; we operate in the interaction picture, and henceforth set the total Hamiltonian ^ H to ^ H I . It is also assumed that the cavity can contain at most one photon. Further, in this work, we do not model the pumping process explicitly, and we assume that the system is initially injX 2 ; 0i. This assumption is reasonable if the pumping is strong. Incoherent processes considered in this model include spontaneous decay, cavity leak- age and dephasing. 1 and 2 are spontaneous emission rates for the X 1 ! G and X 2 !X 1 transitions, respectively; is the rate of photon leakage from the cavity, and 6 ! " # 1 !G, 0" !X 1 , 0" !G, 1" !X 2 , 0" # 2 g Figure 2.2: Energy level diagram of the emitter + cavity model, including the signicant dynamical processes with their rates. Solid arrows indicate incoherent processes, and broken arrows indicate coherent evolution. is the dephasing rate betweenjX 1 ; 0i andjG; 1i. The evolution of the system is described by a stochastic master equation (SME) [8]: d =L = i ~ [ ^ H;]dt + 1 H[^ 1 ] + 2 H[^ 2 ] dt + H[^ a] + H[ ^ O] dt + p D[ ^ O]dW t : (2.1) H andD are superoperators: H[ ^ A] = ^ A ^ A y ( ^ A y ^ A + ^ A y ^ A)=2; D[ ^ O] = ( ^ Oh ^ Oi) +( ^ O y h ^ O y i); whereh ^ Oi =Trf ^ Og. Also, ^ 2 =jX 1 ihX 2 j, ^ a is the annihilation operator acting on the cavity mode, and ^ O = ^ IjGihGj is the dephasing or \observer" operator acting on the quantum dot. Dephasing has two potential sources: the interaction of the emitter with other degrees of freedom, for instance phonon modes in the dot [30], and the back action 7 of measuring devices coupled to the dot. A measuring device will allow us to recover some information about the system of interest, but it is unlikely that we can tap into internal modes of the dot; we use 0 1 to denote the measurement eciency with which we (the \observer") recover information lost in dephasing. We operate in the bad cavity limit, so the parameters satisfy the following conditions: 1 <<g; <: (2.2) 2.3 AMMSE The output signal obtained from our continuous measurement is given (in rescaled units) by J(t) =h ^ Oi(t) + (t) (2.3) where = ( ) 1=2 and (t) = dW t =dt is Gaussian white noise with zero mean, i:e: E[dW t ] = 0 and dW 2 t = dt, whereE is the expectation of a random variable [8]. The transition time of the emitter into its ground state is = Z T 0 h ^ Oi(t)dt: (2.4) where T is a suciently long period of time. However, we have access to J(t) and not h ^ Oi(t). Integrating J(t) over time gives = Z T 0 J(t)dt = + Z T 0 dW t : (2.5) To allow for estimation error, we pass the integrated signal through an AMMSE lter [33], whose output ^ =K +m; (2.6) 8 is the optimal linear estimate of the transition time, where K and m are chosen to minimize the mean squared errorE[j ^ j 2 ]. Making the approximation that the signal and noise () are not correlated, we get K andm in terms of the total integration time T and the mean m and variance 2 of : K = 2 2 + 2 T ; m =m 2 T 2 + 2 T ; (2.7) In the bad cavity limit, the the emitter displays a quasi-exponential behaviour [18]. Since the variance of an exponential distribution is the square of the mean, we have 2 m 2 . The mean, m , is calculated by numerically integrating the deterministic (Lindblad) master equation, Eq. (2.1) with = 0:0. (In practice, of course, this would be measured experimentally.) 2.4 Indistinguishability Indistinguishability is a measure of how temporally identical a photon that leaks out is to other photons produced by the same source. It is most strongly aected by the time uncertainty of when a photon leaks out of the source. This uncertainty is unavoidable, as the process of generating a photon involves incoherent processes; for a photon to leak out of the cavity, the system represented by Fig. 2.2 has to undergo the transitions jX 2 ; 0i 2 !jX 1 ; 0i g $jG; 1i !jG; 0i, where 2 is an incoherent process. In the weak coupling regime, dominates g, and hence the reabsorption of the energy in the cavity by the emitter is unlikely. TheX 1 !G transition of the emitter thus largely determines when a photon leaks out of the cavity, so that knowing the quantum dot emission time should reduce the overall time uncertainty. To calculate the indistinguishability, consider a Hong-Ou-Mandel-like [20] experi- mental setup as shown in Fig. 2.3. The setup has two independent, but identical, single-photon sources, SPS 1 and SPS 2 ; the sources have the same parameter values (g;; ; 2 ; 1 ), and the noise acting on them is independent. The sources SPS 1 and 9 SPS 2 each have an emitter + cavity system and a variable delay. The delay is deter- mined by the information gained from continuous monitoring, and is T ^ where ^ is given by Eq. (2.6) for each source and T is a large pre-determined constant that is the same for both sources. The sources emit photons into modes 1 and 2, respectively, which then pass through a (50 : 50) beam-splitter with output modes 3 and 4, ending at two perfect photodetectors. The indistinguishability is the probability that only one detector clicks: = 1p c , where p c is the coincidence probability. The coincidence probability is the normalized second-order correlation function of the output of the beam-splitter [25]: p c = R T 0 dt R Tt 0 d D ^ a y 3 (t) ^ a y 4 (t +) ^ a 4 (t +) ^ a 3 (t) E R T 0 dt R Tt 0 d D ^ a y 3 (t) ^ a 3 (t) ED ^ a y 4 (t +) ^ a 4 (t +) E: (2.8) The output operators ^ a 3 (t) and ^ a 4 (t) can be expressed in terms of the input operators ^ a 1 and ^ a 2 by ^ a 3 (t) = (^ a 1 (t) ^ a 2 (t))= p 2 and ^ a 4 (t) = (^ a 1 (t) + ^ a 2 (t))= p 2. Assuming that the photons from sources SPS 1 and SPS 2 do not scatter into modes other than 1 and 2, ^ a 1 (t) and ^ a 2 (t) are in turn proportional to the annihilation operator for the cavity mode in SPS 1 and SPS 2 , respectively. 2.5 Results To numerically calculate indistinguishability, we calculate the coincidence probability p c , which in turn requires us to calculate the two-time correlation functions (Eq. (2.8)). The coincidence probability in Eq. (2.8) is expressed in the Heisenberg picture with operators|^ a 3 and ^ a 4 |evolving in time. One can represent the same in the Schr odinger picture with the state(t) evolving, and operators being applied on the state at dierent instants in time; for example,h^ a y (t +) ^ a(t)i can be expressed in the Schr odinger pic- ture as Trf^ a y e L [^ ae Lt [(0)]]g, whereL is the Liouvillian superoperator that represents time evolution due to the stochastic master equation (2.1). Since the stochastic master 10 (50:50) Beam Splitter Variable Delay Variable Delay Mode 2 Mode 3 Mode 4 Co-incidence Counter SPS_2 SPS_1 DETECTOR 1 DETECTOR 2 Mode 1 AMMSE AMMSE Figure 2.3: Canonical experiment to calculate indistinguishability. SPS 1 and SPS 2 are identical and independent sources. A (50 : 50) beam-splitter takes the photons from SPS 1 and SPS 2 as input, with output modes 3 and 4; a coincidence counter records the output of the detectors in modes 3 and 4. If the photons in modes 1 and 2 are identical, they will always either both go into mode 3 or both into mode 4, implying p c = 0 and = 1. equation is nonlinear (due toh ^ Oi in the stochastic partD[ ^ O] in Eq. (2.1)), we cannot utilize the quantum regression theorem to simplify our calculations. Calculating explicit trajectories is necessary, and the nal coincidence probability is obtained by averaging over the coincidence probabilities of all individual trajectories. To analyze the performance of the continuous monitoring feed forward technique, we consider ve cases: (i) no dephasing ( = 0:0), (ii) dephasing with no feed forward ( = 0:1; = 0:0), (iii) feed forward with high detection eciency ( = 0:1; = 1:0), (iv) feed forward with low detection eciency ( = 0:1; = 0:1), and, (v) feed forward with higher dephasing and modest detection eciency ( = 0:2; = 0:5). The evolution of the system in cases (i) and (ii) is given by a deterministic (Lindblad) master equation, 11 Eq. (2.1) with = 0:0. Cases (iii), (iv) and (v) involve continuous monitoring and feed forward, and their evolution is given by the stochastic master equation (2.1) with nonzero and . In these cases we averaged 5000 trajectories of a Monte Carlo calculation. Figure 2.4 plots indistinguishability () as a function of 2 for all ve cases. We see that case (ii) performs the worst, while case (i) is best for higher values of 2 ; for smaller values of 2 , however, case (iii) performs the best. We also note that case (iv) performs consistently better than case (ii), though its advantage diminishes as 2 increases. 0.5 0.6 0.7 0.8 0.9 1 0.2 0.16 0.12 0.1 0.08 0.06 0.04 0.02 ! " 2 (i) (ii) (iii) (iv) (v) Figure 2.4: Numerical Results. We plot indistinguishability () as a function of 2 with system parameters: g = 0:1; = 1; = 0:1; 1 = 0:001; the X-axis is plotted in logscale. The 5 cases are: (i) = 0:0, (ii) = 0:1; = 0:0, (iii) = 0:1; = 1:0, (iv) = 0:1; = 0:1, and, (v) = 0:2; = 0:5. Cases (iii), (iv) and (v) involve the continuous monitoring feed forward technique, and those curves have small deviations due to statistical uncertainties in the Monte Carlo calculation (5000 trajectories). 12 2.5.1 Discussion We can understand the potential usefulness of monitoring by comparing these cases. Case (i) illustrates the loss of indistinguishability from timing uncertainty due to the inco- herent X 2 ! X 1 decay. As the decay rate 2 becomes large, this uncertainty becomes unimportant, but for low 2 it is critical. This loss of indistinguishability can be substan- tially oset by continuous monitoring, as is shown in cases (iii) and (iv). In both cases, the presence of the measuring device produces dephasing between states X 1 and G in the quantum dot. Nevertheless, for the high-eciency ( = 1) measurement in case (iii), the timing uncertainty is substantially reduced by the measurement and feed forward technique. Even the low eciency ( = 0:1) measurement shows some improvement for very small 2 ( 0:02). However, cases (i), (iii), and (iv) make the unrealistic assumption that there is no intrinsic decoherence causing dephasing in the quantum dot. It is not easy to completely eliminate dephasing in most physical systems, and this kind of intrinsic dephasing would lead to additional timing uncertainty. With additional timing uncertainty due to dephas- ing and no monitoring|case (ii)|the indistinguishability remains low even for large 2 . In principle, the information lost in intrinsic dephasing could be recovered by mea- suring the internal degrees of freedom of the quantum dot, but for most physical systems this is very hard to do. The dephasing process is likely to have coupled into many degrees of freedom, and as a result the that can realistically be achieved this way is likely to be small. We can think of (iv) as a case where the quantum dot has intrinsic dephasing, and a small amount of this information can be recovered. We observe that even with less ecient detection there is a lot to be gained with our technique, as it consistently outperforms case (ii). However, if a system with very small inherent dephasing could be engineered, then our technique is not likely to be very useful. With small and , in Eq. (2.3) is large, making our observation noisy. A more realistic picture includes both intrinsic dephasing in the quantum dot and additional dephasing due to the presence of monitoring. This is case (v): the total 13 dephasing is higher than in case (ii), and is assumed to come from both intrinsic deco- herence and continuous monitoring. The detector eciency is moderate, re ecting the fact that only the information from the monitoring is available, and not the information lost to the quantum dot's internal degrees of freedom. The numerical results indicate that such an approach performs quite well, and is comparable to that of case (iii)|the trade-o of increasing the dephasing by adding a monitoring device is more than recom- pensed by the increased information about timing uncertainty. For small 2 , the feed forward technique outperforms a system with no inherent dephasing|case (i)|by a rea- sonable margin. The improvement in indistinguishability is even greater if we compare case (v) with case (ii), the system with inherent dephasing and no monitoring. We make a brief remark about the type of monitoring assumed in our model. We have assumed that the observer can determine whether or not the emitter is in its ground state (G), but not which excited state it is in. We made this assumption with the idea that in electically pumped devices the same physical circuitry used in pumping can provide this information. Though this model is restrictive, in that it cannot distinguish between X 1 and X 2 , we have seen that it still gives sucient information (at least for some parameters) for us to know when a photon was emitted with reasonable accuracy. If the monitoring gives more detailed information about the quantum dot state this technique should perform even better. 2.6 Electronic monitoring in Quantum Dots The type of single-photon source that we are considering would be implemented as a solid-state semiconductor device, comprising a quantum dot p-n diode, wherein a thin layer of insulator (containing the dot) is sandwiched between a p-type and a n-type silicon (Fig. 2.5). The dot in the insulator is embedded inside an optical microcavity. The presence of the cavity in close proximity to the dot increases the collection eciency of the single-photon source: it alters the spontaneous emission rate of the dot into the 14 cavity mode with respect to other modes [43]. This is due to the Purcell eect [35], a process fundamental to cavity QED experiments. Figure 2.5 shows a schematic of a quantum dot p-n diode, sometimes referred to as a p-i-n heterojunction [21]. An electron from the n-type silicon tunnels through to the dot when there is a bias voltage across the diode. When the bias voltage is increased further, it then becomes energetically favorable for a hole to tunnel through to the dot from the p-type silicon. The electron-hole pair spontaneously recombine, emitting a photon into the cavity. This photon then leaks out of the cavity to an external mode (for example, a waveguide). In the weak coupling regime, each recombination event leads to a single photon emission from the quantum dot p-n diode. p n i Quantum dot embedded in a cavity e- e+ Photon Figure 2.5: Quantum dot p-n diode or p-i-n heterojunction. A quantum dot p-n diode consists of an insulator sandwiched between p- and n-type silicon. A quantum dot is fabricated inside the insulator, and this is contained within an optical microcavity. The diode is biased in the forward direction, such that a single electron (e ) tunnels through from the n-side to the dot. The electron remains in the dot until a hole (e + ) tunnels through to the dot from the p-side. The electron-hole pair in the dot recombines spontaneously to emit a photon into the cavity, which subsequently leaks out to an external mode. The quantum dot p-n diode in Fig. 2.5 behaves as a single-photon source if the tunneling event is controlled such that only one tunneling event (electron and hole) occurs per pumping cycle. This is achieved by applying a dc bias voltage and utilizing 15 the Coulomb blockade eect. The Coulomb blockade regime is when the energy due to a single electron charge is greater than the thermal energy of the charge carriers in the bath, i.e. E = e 2 =2C > k B T , where e is the electron charge, C is the junction capacitance, k B is the Boltzmann's constant and T is the temperature. A dc bias voltageV (t) =V bias +v(t) is applied across the diode. When v(t) = 0, the bias voltageV bias favors a single electron to tunnel through to the dot; this voltage is not sucient, however, for a hole to tunnel through. The pulse being high,i:e:v(t) = E=e, however energetically favors a hole to tunnel through to the dot. The electron and hole recombine, emitting a photon into the cavity; this recombination leads to a drop in the potential (of E=e) across the junction. This voltage drop across the p-n junction signals photon emission. One can detect this change in potential by capacitively coupling the dot (in the p-n junction) to a charge meter, such as single electron transistor [45]. However, the measured signal will have (thermal) noise in it. To measure the signal of interest reliably, it is imperative that the coupling capacitance|C cpl |be small, and that it satisfy the condition e 2 =2C cpl >k B T . Current implementations of electronic measurement in quantum dot sources do not yet have the time resolution required for the AMMSE-feed forward technique developed in this work. Measurable time scales have been reported of approximately 8s [15, 45]. This is several orders of magnitude away from what is required for our technique to be useful. A major impediment in improving the time-resolution is the signal-to-noise ratio, with instrumentation noise being the signicant contributor [45]. Better instrumentation engineering should lead to a better time-resolution; also, some of the components operate at room temperature while the quantum dot operates in the Coulomb blockade regime. Moving more components to low temperature may lead to better time resolution. 16 2.7 Summary The continuous monitoring feed forward technique is a simple yet powerful method that can provide great improvements in indistinguishability of a single-photon source. We have numerically shown that, for small transition rates 2 , the eect on indistinguisha- bility of timing uncertainty in the quantum dot emission can be signicantly reduced. This improvement in indistinguishability holds true even when there is intrinsic dephas- ing in the quantum dot. The feed forward technique requires continuous monitoring, a simple linear estimation algorithm, and the ability to introduce variable delays. Though the algorithm is straightforward, the time resolution of continuous monitoring accessible with current technology is an issue. Better instrumentation, and moving more compo- nents to lower temperature, are possible ways forward to improve the time resolution. We believe that techniques similar to those described in this paper will lead to great improvements in single-photon sources in the near future. 17 Chapter 3 Single-photon source: Probability 1 3.1 Introduction Generation of single-photon states has wide-ranging applications, spanning quantum computing, quantum imaging, metrology, communication, and cryptography, amongst others [32]. These applications are important to scientic and technological progress in many important areas. For the current work, we are especially interested in Linear Optical Quantum Computation (LOQC) [26, 27]. A key requirement of LOQC is the availability of high quality single-photon states on demand. Semiconductor quantum dot-based implementations of single-photon sources are of particular interest, as they scale well upon integration and are amenable to commercial fabrication techniques [32, 24]. Typically, implementation of these devices involves a quantum dot (QD or dot) inside a microcavity, with the dot acting as the photon source: the presence of the cavity increases the collection eciency due to spatial connement of the photons [32, 24, 30]. A wide variety of cavities with dierent sizes, shapes, and quality factors have been designed and fabricated [30]. A quantum dot in a microcavity can be pumped either optically or electrically. Opti- cal pumping is more straightforward experimentally, but electrical pumping may be better suited to large scale integration. Also, as it does not directly insert photons into the cavity, it opens up the possibility of pumping directly into an energy level resonant 1 This chapter is adapted from our work in [37]. 18 with the cavity mode, which may reduce timing uncertainties in photon emission [36]. It also allows a channel for continuous measurement of the dot. In electrical pumping, a bias voltage is applied across a quantum dot pn diode that enables an electron to tunnel through from n-type onto the dot (present at pn junction). Once an electron tunnels through to the dot, no new electron can tunnel through due to the Coulomb blockade eect [22]. A further increase in the bias voltage then enables a hole from the p-type to tunnel to the dot. Recombination of electron and hole in the dot follows, leading to a photon emission into the cavity. Recombination also leads to a drop in the potential across the quantum dotpn diode. By observing this change in the potential across the quantum dotpn diode, we can gain information about the state of the dot. This observation can be seen as a weak quantum measurement that provides partial information about the system [28]. A good single-photon source should be able to produce exactly one photon at the required time, in a specied state. Indistinguishability is a measure that captures the specicity of the photon state (due, e.g., to time-uncertainty of photon emission), while single-photon probability determines the likelihood that a single photon is indeed emit- ted. We have considered continuous monitoring as a tool to improve indistinguishability elsewhere [36]. We focus on single-photon probability here. There are various processes that aect single-photon probability: collection eciency (emitting into the cavity mode), pumping, cavity leakage to non-waveguide modes, and photon loss, to name a few. Many of these processes depend on the fabrication techniques and the materials that go with it, and hence can only be improved by building better sources. Some, however, may be improved by better control. The most obvious process to treat from this point of view is pumping: how long should we pump at a given pumping strength to maximize the single-photon probability, given that other parameters are xed? In the case of strong pumping|as optical pumping often is|the answer is straight forward: pump for a duration short compared to the emission rate of the dot. However, 19 weakly pumped systems (like electrical pumping) present a more complicated scenario. Since the pumping strength is weak, we might have to pump for times comparable to, or even longer than, the emission rate of the dot. We thus may increase the likelihood of multi-photon emission. Since electrical pumping with semi-conductor QD based imple- mentations is of active experimental interest [22], understanding the behavior of weakly pumped systems is critical. In this work, we use continuous quantum measurements to improve single-photon probability. The idea is simple: to monitor the state of the emitter continuously, and determine when to stop pumping energy into the system based on the informa- tion obtained. Unfortunately, the output from monitoring such a microscopic system is intrinsically noisy, and little time is available to process the received signal. This complicates the procedure. We utilize a sequential statistical technique called CUmu- lative SUM (CUSUM) as the decision making process. We show numerically that this mechanism substantially improves single-photon probability in the weak coupling regime, particularly when the pumping rate is comparable to the QD-Cavity coupling strength. 3.1.1 Chapter overview The emitter we consider is a p-n diode operated as a single-photon LED, though the techniques we discuss are probably applicable to other systems. We discuss the emitter in Sec. 3.2. We then capture the essential features of the LED's operation in a schematic model in Sec. 3.3; we present details regarding the system in a cavity-QED setting, and give its energy-level diagram; we present a stochastic master equation including the relevant physical processes, and discuss the parameter regime of operation. Section 3.4 presents the change detection algorithm and the decision circuit used to stop pumping; we consider sequential CUSUM technique in Sec. 3.4.1 and present Bayesian approach in Sec. 3.4.2. We give results in Sec. 3.5: we discuss the deterministic case rst in Sec. 3.5.2, before proceeding to analyze the numerical performance of the CUSUM-based technique 20 in Sec. 3.5.3. Finally, we scrutinize the eect of monitoring eciency on performance in Sec. 3.5.4. 3.2 Single-photon light emitting diode p n i Quantum dot embedded in a cavity e- e+ Photon Figure 3.1: Quantum dot p-n diode or p-i-n heterojunction. A quantum dot p-n diode comprises an insulator sandwiched between p- and n-type silicon. A quantum dot is fabricated inside the insulator, and this is contained within an optical microcavity. The diode is biased in the forward direction, such that a single electron (e ) tunnels through from the n-side to the dot. The electron remains in the dot until a hole (e + ) tunnels through to the dot from the p-side. The electron-hole pair in the dot recombines to emit a photon into the cavity, which subsequently leaks out to an external mode. Figure 3.1 shows a quantum dot p-n diode that acts as a single-photon source. A dot is present at the p-n junction and is assumed to be inside an optical microcavity in the weak coupling regime. We assume that the diode is forward biased and is in the Coulomb blockade regime. When biased at an appropriate level, an electron (e ) tunnels through to the dot from the n-side; this electron remains in the dot until a hole (e + ) tunnels through from the p-side. This leads to recombination of e and e + , and a photon is emitted into the cavity. The photon subsequently leaks out to an external mode, such as an outgoing waveguide. 21 While photon generation as described above is intuitively straightforward, we need to control the pumping so that the system generates at most one photon with high probability. Since the dot is inside a microcavity, there is a non-zero probability of multi- photon emission; this happens when there is more than one recombination event in a pumping cycle. Determining the length of a pumping cycle should ideally be determined by the knowledge of when the rst e tunneling event occurs. This is dicult to know, as the tunneling process is stochastic. In the absence of specic information about tunneling times, the best that can be done is to time the pumping a priori, either to maximize the probability of a single photon, or keep the multi-photon probability below a given threshold. (These are not necessarily the same thing.) In principle, though, we can do better. We continuously monitor the state of the dot by measuring the voltage across the heterojunction; the output record gives information about whether an e has tunneled onto the dot. We use this information to stop the pumping cycle when the dot is in its excited state (equivalently, after an e tunneling event). If done correctly, the dot is excited only once, and the diode emits at most one photon. However, the measurement signal is noisy, and the decision of whether the dot is in its excited state or not is not necessarily easy. We use a well known sequential statistical decision technique known as CUmulative SUM (CUSUM), a decision circuit with a simple implementation that accommodates noisy observations (Sec. 3.4). 3.3 System modeling and parameter regime Figure 3.2 schematically models the single-photon LED operation described in Sec. 3.2. The system comprises three quantum objects: a quantum dot, an optical microcavity, and an external mode (e.g., a waveguide). We assume that the dot is electrically pumped: this allows an electron to tunnel through from the n-side to the dot, and recombine with a hole from the p-side, when the diode bias voltage is favorable. The bias voltage is 22 tunneling Par,ally reflec,ng mirror Photon emission Electron (n-‐side) Con,nuous monitoring Decision circuit CUSUM Algorithm Con,nue pumping Figure 3.2: Single-photon LED operation: Schematic model. A single-photon LED consists of three quantum objects: an emitter (QD), a microcavity and an external mode. Electrical pumping involves tunneling of an electron (e ) from the n-side to the dot; an e e + recombination leads to an emission of a photon into the cavity, which then leaks out to the external mode. The e tunneling occurs when the bias voltage in the diode is favorable to the event. We control the bias voltage externally; we keep the bias `on' to allowe tunneling, and turn it `o' to stop tunneling. We monitor the state of the dot continuously, and use a sequential algorithm called CUSUM and its associated decision circuit to decide whether to keep the bias `on' or to turn it `o.' controlled externally, giving us some control over the e tunneling event. To exert this control eectively, we continuously monitor the state of the dot and use the output to decide when to stop pumping. We include the external mode in our description in order to calculate the various photon emission probabilities|p(0), p(1), etc. These probabilities are the quantitative measure of \goodness" of the system as a single-photon source. Obviously, we want the single-photon emission probability, p(1), as close to unity as possible; unfortunately, in 23 many parameter regimesp(1) is not very close to 1, especially if it is electrically pumped. In this case we must explore a more complex trade-o landscape, as it is no longer sucient to just maximizep(1); we must also consider how high a zero-photon probability p(0) and multi-photon probability p(2+) we can tolerate. For certain applications it is tolerable for the single-photon source sometimes to emit no photon, but multi-photon emission must be strongly suppressed. Thus, in our trade-o analysis we impose an additional constraint on p(2+), and try to maximize p(1) subject to this constraint (see Sec. 3.5). 3.3.1 The system The dot in our model has 2 energy levels: the ground statejGi, and the excited statejXi. The cavity and external mode contain some number of photons, and are represented by the usual photon number state notation|j0i,j1i,j2i etc. Thus, any state in this system has the formjG=X; 0=1=2:::; 0=1=2:::i where the order corresponds to the dot, the cavity and the external mode, respectively. We assume that the dot's excited statejXi is resonantly coupled to the cavity mode. The interaction between the dot and the cavity is given by the Jaynes-Cummings Hamil- tonian: ^ H I = i~g ^ a y ^ ^ a^ + , where g is the interaction strength, ^ a y (^ a) is the cre- ation (annihilation) operator acting on the cavity mode, and ^ =jGihXj; we operate in the interaction picture, and henceforth set the total Hamiltonian ^ H to ^ H I . The system is initially decoupled, in the statejG; 0; 0i. Incoherent processes included in this model are pumping, spontaneous decay of the quantum dot, cavity leakage and dephasing. is the rate of pumping; is the spon- taneous emission rate for the X! G transition; is the photon leakage rate from the cavity to the external mode, and is the dephasing rate between emitter and the cavity mode. We treat electrical pumping as a incoherent process [17]. The process of pumping involves an electron tunneling through to the dot when the bias voltage at the diode 24 junction is favorable. Since electron hopping (n-side! dot) happens at random times, we model pumping as an incoherent G!X transition with rate . G,0,0 X,0,0 G,1,0 G,0,1 X,1,0 G,2,0 X,0,1 G,1,1 G,0,2 Figure 3.3: Energy level diagram. A state of the system is described by three quantum numbers: the dot, the cavity and the external mode. In this diagram, we show those states with at most 2 quanta of energy and their signicant dynamical processes: broken arrows indicate incoherent processes, and solid arrows indicate coherent evolution. If the pumping duration is nite,jG; 0; 0i,jG; 0; 1i andjG; 0; 2i are the possible nal states of this system, and their corresponding probabilities represent the zero photon, 1 photon and 2 photon probabilities, respectively. Figure 3.3 shows the energy level diagram of the single-photon LED system. The system moves up the energy ladder as we pump longer. For nite pumping time, the system eventually evolves to its possible nal statesjG; 0; 0=1=2:::i, with correspond- ing probabilities p(0=1=2::: ). Both the cavity mode and the external photon mode in principle have innitely-many energy levels; however, since our goal is to generate a single-photon statejG; 0; 1i with high probability (p(1)! 1), and we are assuming that 25 the coupling between the cavity and external mode is stronger than the coupling between the dot and the cavity, we truncate the higher energy states (> 2) as shown in Fig. 3.3. By truncating the state space, we make the statejG; 0; 2i now representjG; 0; 2+i|that is, a state with 2 or more photons in it|and its corresponding probability becomes the multi-photon probability p(2+). Note that this approximation neglects some eects that could in principle contribute to the single-photon probability, such as a series of reabsorptions and spontaneous emissions. Since the probability of the system being in the higher energy states is small in the rst place, and the spontaneous emission rate is assumed to be quite low, the neglected eects should have little impact on the estimates of single-photon probability. The evolution of the system is described by a stochastic master equation (SME) [8]: d = i ~ [ ^ H;]dt + H[^ ] +H[^ a ^ b y ] + H[ ^ P X ] + H[^ + ] dt + p D[ ^ P X ]dW t : (3.1) H andD are superoperators: H[ ^ A] = ^ A ^ A y ( ^ A y ^ A + ^ A y ^ A)=2; D[ ^ P X ] = ^ P X + ^ P X 2h ^ P X i; whereh ^ P X i =Trf ^ P X g is the quantum expectation; dW t is a Brownian motion with E[dW t ] = 0 and E[dW t dW s ] = (ts)dsdt; which characterizes a Wiener process; here E is the expectation of a random variable. Also, ^ + =jXihGj, ^ b y is the creation operator acting on the external mode, and ^ P X = jXihXj is the dephasing or \observer" operator acting on the quantum dot. Dephasing has two potential sources: the interaction of the emitter with other degrees of freedom, 26 for instance phonon modes in the dot [30], and the back action of a measuring device coupled to the dot. A measuring device will allow us to recover some information about the system of interest, but it is unlikely that we can tap into internal modes of the dot; we use 0 1 to denote the measurement eciency with which we (the \observer") recover information lost in dephasing. 3.3.2 Parameter regime We operate in the weak coupling regime, and assume that is the dominant system parameter. We require that spontaneous emission be small, for higher implies lower single-photon probability. The technique developed in this work is based on continuous monitoring of the dot, and will be useful only when the pumping rate is comparable to g. This is because it takes time to gather information using continuous monitoring, and with higher the chances are that the decision to turn o the pumping will be too late, increasing the multi-photon probability above tolerable levels. In fact, the scenario where g << has a much simpler solution: turn on pumping for time much less than the emission time scale ( (g 2 =) 1 ); this will work quite well because we move up the energy ladder (Fig. 3.3) only when the dot makes multiple X! G transitions; since is very small, the transition time scale is thus set by g. We therefore are interested in the case where the parameters satisfy the following conditions: << ;g <; (3.2) for our technique to be useful. 27 3.4 Change detection algorithm The output signal obtained from our continuous measurement is given (in rescaled units) by J(t) =h ^ P X i(t) + (t) (3.3) where = ( ) 1=2 and (t) = dW t =dt is Gaussian white noise with zero mean, i:e: E[dW t ] = 0 and dW 2 t = dt, where E is the expectation of a random variable [8]; ^ P X =jXihXj is the dot's excited state projection operator, whileh ^ P X i(t) is its quantum expectation Trf ^ P X (t)g. The excited state of the dot isjXi, and the unexcited state isjGi. The measurement output conveys the information about this state: E[J(t)] =h ^ P X i(t) = 8 < : 0 if the dot is injGi; 1 if the dot is injXi; (3.4) asE[(t)] = 0. If the output contained no noise (! 0) then the state of the dot would tend to be localized at eitherjXi orjGi, and transitions between would be readily detectable. The presence of noise complicates matters, since it can mask the state of the dot|indeed, if the noise is high, it is easy to miss the transitions. Note that if we average the output signal over a short interval t, its variance is a constant: 2 = 2 =t. When an electron tunnels through, there is a change in the mean of the output signal. By detecting this change, we could detect when a tunneling event takes place, and turn o the pumping. However, we cannot access the meanE[J(t)] directly, but have to infer it from J(t) [Eq. (3.3)]. We obtain the output signalJ(t) continuously in time, and the decision to turn o the pumping must be made in real time. This imposes practical restrictions on the kind of algorithms that are feasible; for instance, Bayesian machine learning-type algorithms [2] may be too slow to be useful in practice. Sequential algorithms [2] are procedures that use only the output signals gathered to the present time, and not a priori information, 28 to infer the probability density function (pdf). As such, they are generally suboptimal, and will succeed only in certain parameter regimes, but they have the virtue of being easier to implement in practice. One such algorithm is Cumulative Sum (CUSUM). 3.4.1 Sequential CUSUM procedure We present a discrete version of CUSUM here for simplicity, which can straightforwardly be extended to the continuous case. Let y 1 = J(t 1 );y 2 = J(t 2 );::: represent the time series of the output signal, averaged over a succession of intervals of size t. We know that Y k is a 2 parameter Gaussian random variable with a variable mean () and a constant variance ( 2 = 2 =t). We know from Eq. (3.4) that if the dot is localized onto a single energy state, the mean has two possible values: 0 = 0 or 1 = 1. Detecting a change in the mean is therefore equivalent to a simple hypotheses testing problem: H 0 : = 0 H 1 : = 1 9 = ; : (3.5) CUSUM is based on a sequential probability ratio test (SPRT). SPRT, in turn, is based on the concept of log-likelihood ratios. These are dened by S n =S(y n 1 ) = ln p 1 (y n 1 ) p 0 (y n 1 ) ; (3.6) wherep 0 ; 1 (y) is the probability density function (pdf) given means 0 and 1 , respec- tively;S is called the sucient statistic in the parlance of statistics [2]; note y n 1 includes all output signals (y 1 :::y n ). Further, assuming that the random variablesY 1 ;:::;Y n are independent and identically distributed (iid), S n = n X k=1 ln p 1 (y k ) p 0 (y k ) = n X k=1 s k ; (3.7) 29 s k is the sucient statistic for random variableY k . Utilizing the fact thaty k is Gaussian with pdf p 0=1 (y k ) = 1 p 2 2 exp (y k 0=1 ) 2 2 2 ! ; we have s k = 1 0 2 y k 0 + 1 2 = 1 0 2 y k 0 + 1 2 t: (3.8) Since the bias voltage is on at the start of a pumping cycle, and the algorithm has to decide when to turn it o, CUSUM has to detect the change 0 ! 1 . Ignoring the constant pre-factor in Eq. (3.8), it is easy to see that S n [Eq. (3.7)] has a negative drift if the dot is in its ground state (y k 0 = 0), and a positive one when in excited state (y k 1 = 1); a tunneling therefore should cause a V-shaped prole inS n . As the signal is noisy, we use an appropriate threshold to mitigate false positives. To this end, we dene m k = min 1jk S j : (3.9) We calculate m k using a simple procedure: at each step, set m k = S k if S k < m k1 , otherwise it retains its previous value. In hardware, this can be done with a single register and a comparator. The decision rule, at each time step, is d k = 8 < : H 0 if (S k m k )h; H 1 otherwise; (3.10) whereh is the threshold, chosen based on the parameters of the system to mitigate false positives. We stop pumping if d k =H 1 . 30 3.4.2 Bayesian solution The primary dierence between a Bayesian and sequential solution is the use of prior probabilities in the former case. We approximate the system described in Fig. 3.3 as a Markov chain, shown in Fig. 3.4. This approximation is good because the system is assumed to operate in the weak coupling regime [Eq. (3.2)], where the cavity decay rate dominates the emittercavity coupling strength g. k = 0 k = 1 k = 2 r p r e k = 3 r p k = 4 k = 5 r p r e Reset Figure 3.4: Markovchainapproximation. We approximate the energy level diagram as a Markov chain. The individual states represent the state of the dot along with some number of photons in the external mode. The state of the dot is represented by x, and n is the number of photons in the external mode; x is 0 if there is no e in the dot and 1 if an e is present; n is a positive integer. The parameters r p and r e are the pumping and emission rates, respectively. We combine x andn into a single variablek = 2n + x; note that odd k states contain an e while even k states have no e in the dot. The system starts initially in the state k = 0 with no e in the dot and no photon in the external mode. 31 A state in the Markov chain represents the state of the dot and the external mode. The state of the dot is represented by x, and takes values 0 or 1 depending on whether an electron has tunneled through to the dot or not, with 1 indicating the presence of an electron. The external mode contains n photons, where n is a non-negative integer. We combine the state of the dot (x) and cavity (n) into a single variable using, k = 2n + x: (3.11) Evenk indicates the absence of an electron, and oddk the presence of an electron in the dot. The parametersr p andr e are the pumping and eective emission rates, respectively; here, r p = and r e = 4g 2 =. (We ignore spontaneous emission in this approximation as its contribution is small, see Eq. (3.2).) Let p k (t) be the probability of state k at time t. The system starts in state k = 0, sop 0 (0) = 1 and p k (0) = 0 for all k> 0. The Markov chain in Fig. 3.4 can be described by a set of coupled dierential equations: dp 0 dt = r p p 0 dp 1 dt = r p p 0 r e p 1 (3.12) dp 2 dt = r e p 1 r p p 2 . . . Using these initial conditions, we can solve the system of equations analytically; we rst solve for p 0 , then use this solution to solve for p 1 , and so on. Equation (3.13) describes the Markov chain in Fig. 3.4 when no measurement output is available. To incorporate the information from the continuous measurements, we rst consider a xed time step t, and then go to the continuum limit. During each t the system evolves according to Eq. (3.13); then at the end of the interval t we update the 32 probabilities by conditioning on the measured output. We now derive update formulas (conditional probabilities) for p k 's using the Bayes rule. Let y(t + t) =h ^ P X i(t + t) t + W t+t ; (3.13) where W t+t is a Wiener process withE[W t+t ] = 0 andE[W 2 t+t ] = t;h ^ P X i(t + t) is the quantum expectation of the state of the dot. We dene Q 0 p(x = 0) = p 0 + p 2 +:::; (3.14) Q 1 p(x = 1) = p 1 + p 3 +:::; (3.15) where Q 1 and Q 0 represent the probabilities of an electron to be present in the dot or not (corresponding to x being 1 or 0). They obey the simple equations dQ 0 = (r p Q 0 + r e Q 1 )dt; (3.16) dQ 1 = (+r p Q 0 r e Q 1 )dt; (3.17) with initial conditions Q 0 (0) = 1 and Q 1 (0) = 0. We have used Eq. (3.13) in the above derivation. Since y is Gaussian random variable with means 0=t, we have p(yjx = 0) = 1 p 2 2 expfy 2 =2 2 g; (3.18) p(yjx = 1) = 1 p 2 2 expf(y t) 2 =2 2 g; (3.19) where, 2 =E[y 2 ] 2 t is its variance. Also, p(y) = p(yjx = 0)p(x = 0) + p(yjx = 1)p(x = 1) = 1 p 2 2 Q 0 expfy 2 =2 2 g +Q 1 expf(y t) 2 =2 2 g ; (3.20) 33 so that p(yjx = 0) p(y) = Q 0 +Q 1 exp y 2 (y t) 2 2 2 1 ; which using the relation 2 = 2 t becomes = Q 0 +Q 1 exp y 2 exp t 2 1 : Assuming that t 2 , we get Q 0 +Q 1 1 + y 2 + t 2 2 1 t 2 2 +O(t 3=2 ) 1 1 + Q 1 y 2 +O(t 3=2 ) 1 1 Q 1 (y Q 1 t)= 2 : (3.21) In the above derivation, we used the fact that the total probability is conserved, and hence Q 0 +Q 1 = 1 at all times. Similarly, p(yjx = 1) p(y) 1 + Q 0 (y (1Q 0 )t)= 2 : (3.22) Therefore the update formula for Q 0 is Q 0 ! p(yjx = 0) p(y) Q 0 = Q 0 Q 0 Q 1 (y Q 1 t)= 2 (3.23) and that of Q 1 is Q 1 ! p(yjx = 1) p(y) Q 1 = Q 1 + Q 0 Q 1 (y (1Q 0 )t)= 2 : (3.24) 34 We now combine the system dynamics given by Eq. (3.17) with the update formulae due to observation from Eqs. (3.23) and (3.24) to get Q 1 (t + t) = Q 1 (t) (r p +r e )Q 1 (t) t + r p t +Q 1 (t)(1 Q 1 (t)) y(t) Q 1 (t)t 2 : (3.25) We can infer Q 0 straightforwardly using Q 0 +Q 1 = 1. Since x only takes values 0 and 1, its expected value x is x =E[x] = 1p(x = 1) =Q 1 : (3.26) We can now make precise the question of when to stop pumping. We need to calcu- late the individual probabilities p k including the prior probabilities [Eq. (3.13)] and the update formula due to continuous monitoring of the dot. Here, we use (n;x) notation instead ofk to represent a state in the Markov chain (Fig. 3.4), for clarity. Applying the Bayes rule we get p(n;xjy) = p(yjn;x) p(y) p(n;x) = p(yjx) p(y) p(n;x): (3.27) We have used the fact that the observation y is independent of number of photons n in the external mode. Switching back to the variable k we get p(kjy) = p(yjx) p(y) p(k): (3.28) 35 Using the above relation with update formulas in Eqs. (3.21) and (3.22), along with prior probabilities given in Eq. (3.13), we get p 0 (t + t) = p 0 (t) r p p 0 (t) t p 0 (t) x(t) (y x t) 2 ; (3.29) p 1 (t + t) = p 1 (t) (r e p 1 (t) r p p 0 (t)) t +p 1 (t) (1 x(t)) (y(t) x(t) t) 2 ; (3.30) p 2 (t + t) = p 2 (t) (r p p 2 (t) r e p 1 (t)) t p 2 (t) x(t) (y(t) x(t) t) 2 ; (3.31) . . . Here, represents the quality of measurement, eectively the inverse of the signal-to- noise ratio (SNR): the higher the value of, the lower the signal quality. Note that as is reduced (better signal quality), the terms from the (Bayesian) update formula dominates, indicate that our estimate of the probabilities p k derives mostly from our observation; while as!1 the probabilityp k converges to its a priori solution Eq. (3.13). Therefore, the Bayesian technique should perform no worse than the a priori solution, at least as long as the Markov chain approximation remains good. Observe that states k 2 (0; 1; 2) can emit at most 1 photon, while states k > 2 emit 2 or more photons. If we bound the tolerable multi-photon probability by , then our decision circuit becomes straight-forward: we continue to pump until p(k > 2) = p 3 + p 4 + ::: < , and stop pumping as soon as the inequality is violated. The above condition can be re-written in more convenient form: p(k 2) = p 0 + p 1 + p 2 (1 ); (3.32) and as before, we stop pumping when the inequality is violated. In this form it suces to keep track of just 3 probabilities|p 0 ;p 1 ;p 2 |together with the expectation value x. 36 3.5 Results In Sec. 3.4, we presented two dierent approaches|sequential and Bayesian|to improve single-photon probability in the presence of continuous monitoring. The Bayesian solu- tion (Sec. 3.4.2) is computationally expensive, but provides a smooth transition from low-noise limit to the high-noise one; since in the high-noise limit the Bayesian updates converges to an a priori solution, we will do no worse than the deterministic solution (where no measurement is done). Sequential CUSUM (Sec. 3.4.1) on the other hand, is simple and requires less computational resources (an integrator, a comparator and 2 registers). This is important because the decision to turn o the pumping has to be taken in real time. Though the Bayesian solution is very useful for the insight it provides, particularly in high-noise settings, in this section we explore sequential CUSUM, as it is far easier to implement in real time. At high SNR it should approach the performance of the Bayesian solution. We rst discuss the numerical values of the system parameters. Then we consider the benchmark against which we compare the CUSUM technique: the a priori evolution of Eq. (3.13) without continuous monitoring (Fig. 3.5). We compare the results of CUSUM to this deterministic solution, and plot the single-photon probability as a function of pumping rate (Fig. 3.6); we consider 3 cases in CUSUM corresponding to dierent measurement quality () regimes: (i) low-noise, (ii) intermediate-noise, and (iii) high- noise. The monitoring eciency has a strong eect on on the performance of the CUSUM algorithm, and we explore this dependence. For low (= 0:1), CUSUM is not useful, and in fact is detrimental. In this regime, the measurement output is so noisy that CUSUM cannot recognize when the dot becomes excited. In such a case, increasing the monitoring strength (and hence the dephasing rate) nominally (and thereby improving ) can lead to regimes where the technique performs better (Fig. 3.7). Note that in all our simulations, we use the constraint p(2+) 1%; (3.33) 37 That is, we require that all solutions satisfy the condition that the multi-photon proba- bility can at most be 1%. We use the fourth-order Runge-Kutta integrator rk4 [34] to numerically integrate the stochastic master equation (3.1). 3.5.1 Parameter values In our model, ve parameters characterize the system: g, , , , and . We rescale all the parameters with respect to the cavity decay rate, which establishes the dimension- less (frequency) units for the simulation. In these dimensionless units, the parameter values are: g = 0:1; = 0:001 and = 1:0: (3.34) The value of in physical (frequency) units is 95 KHz (in [19], f = 220 MHz and Q = 2300; in frequency units, =f=Q) . The physical units of other parameters can be obtained straightforwardly by rescaling with respect to . Pumping rate and measurement strength are interesting from our standpoint, in that the eectiveness of CUSUM as compared to an a priori strategy is strongly aected by them; determines the time-window for the decision circuit while measurement quality ( = ( ) 1=2 ) in uences the ability to make the right decision, that is, to turn o the pumping at the right time. The a priori solution also improves with higher , further eroding the benets of the CUSUM protocol. 3.5.2 Deterministic solution To evolve the system deterministically, we integrateE[d] whered is dened in Eq. (3.1); due to the expectationE[:], the stochastic contributions vanish and the equation reduces to the usual deterministic Lindblad master equation [8]. Figure 3.5 shows the photon number probabilities|p(0),p(1) andp(2+)|as a func- tion of pumping duration for parameters values dened in Eq. (3.34) and = 0:1. The probabilities are obtained by integrating the Lindblad master equation with pumping 38 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 Probability Pumping duration 0 photon 1 photon 2+ photons 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 Probability Pumping duration Prob = 1% Figure 3.5: Deterministic evolution. We plot photon probabilities|p(0), p(1) and p(2+)|as a function of the pumping duration; the parameter values are given in Eq. (3.34) and = 0:1. The system is initially assumed to be in the statejG; 0; 0i. Ini- tiallyp(0) = 1 and decreases as pumping duration is increased; p(1), on the other hand, starts at 0, increases initially, reaches a maximum and then starts to decrease; p(2+) increases with pumping duration (more gradually than p(1)) and starts to dominate at long times. The maximum of p(1) is 0:73 and it does so at t 19:5; however, p(2+) is about 12%, which is quite high. Imposing the constraint in Eq. (3.33), the best time to stop pumping at these parameter values is t 8, where p(1) 53% and p(0) 46%. turned on up to time t, and continuing the simulation for a suciently long time after the pumping is turned o. The initial state isjG; 0; 0i. We see that p(0) falls exponen- tially as the pumping time is increased, whilep(1) increases initially, reaches a maximum, and then falls o;p(2+) increases slowly but grows to 1 at long times. The maximum of p(1) is about 73% at t 20; however, p(2+) is about 12%, which is unacceptably high. Imposing the constraint Eq. (3.33), we nd that the best solution is p(1) 53%, and is achieved for a pumping time t 8. 39 3.5.3 CUSUM performance We analyze the performance of CUSUM by integrating the stochastic master equa- tion (3.1) for the parameter values in Eq. (3.34), with 2 [0:01; 0:1]; we set = 1, and impose Eq. (3.33). We assume that the system is initially decoupled and starts in jG; 0; 0i. The decision rule given in Eq. (3.10) is used to stop pumping. This rule requires us to specify the threshold value h, to avoid false positives; we nd a close-to-optimal value for h by numerical exploration. For each , we continue to integrate each trajec- tory for suciently long time after the decision circuit stops pumping to calculate the photon probabilities. We repeated this procedure for 1000 trajectories and averaged to obtain p 1 ( ). We plot this in Figure 3.6. In Fig. 3.6, we consider 4 cases corresponding to dierent measurement quality = ( ) 1=2 : (i) deterministic (no measurement), (ii) low-noise = 10, (iii) intermediate noise = 1, and (iv) high-noise = 0:1. The better the quality of measurement (lower ), the more accurate is our knowledge of the state of the dot; naturally, we expect CUSUM to perform well for lower and to fare badly when is high. This is borne out in Fig. 3.6 where we observe case (ii) performing the best and (iv) the worst. In fact, (iv) does worse than the deterministic case; this means that we are better-o not using CUSUM in the high-noise limit. Observe that cases (ii) and (iii) provides greater improvement for lower , and the improvement reduces as is increased. Physically, a smaller pumping rate means that the successive e tunneling events are spread out in time, giving sucient time for the decision circuit to make the right decision; higher means less time between successive e tunneling events, and consequently a tighter time-window for the decision circuit. Also, as we have discussed above, for high the optimal strategy is strong pumping for a short time, and we do not expect monitoring and feedback to yield much improvement. 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 p(1) (single-photon probability) 1 (pumping rate) (i) (ii) (iii) (iv) Figure 3.6: CUSUM performance. We plot the best p(1) given constraint Eq. (3.33) for dierent pumping rates 2 [0:01; 0:1]; we nd a close-to-optimal value for the thresh- oldh by numerical exploration. There are 4 cases corresponding to dierent measurement qualities = ( ) 1=2 (here = 1): (i) deterministic (no measurement), (ii) low-noise ( = 10), (iii) intermediate noise ( = 1:0), and (iv) high-noise ( = 0:1); we plot the deterministic case for comparison. Low-noise (ii) naturally leads to the best performance while high-noise (iv) performs the worst; in fact, (iv) performs worse than the determin- istic case (i); in the intermediate regime (iii), CUSUM does better than cases (i) and (iv). In cases (ii) and (iii), the performance improvement is higher for lower and the improvement reduces as is increased. 3.5.4 Measurement Eciency The results in the previous section assumed that the monitoring eciency was = 1. This is unrealistic even in principle, because any realistic quantum dot will have multiple sources of dephasing, and we cannot have access to information that is lost to the dot's internal degrees of freedom. On top of this diculty in principle, ecient monitoring is hard to do in practice. An obvious question is: how does lower aect performance? To understand the eect of we simulated 4 cases: (i) deterministic ( = 1:0; = 0), (ii) low monitoring strength but high eciency ( = 1:0; = 1:0), (iii) low monitoring 41 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 p(1) (single-photon probability) 1 (pumping rate) (i) (ii) (iii) (iv) Figure 3.7: Monitoring efficiency and performance. In this plot, we explore the eect of monitoring eciency on performance. The 4 cases considered are: (i) ( = 1:0; = 0), (ii) ( = 1:0; = 1:0), (iii) ( = 1:0; = 0:1), and (iv) ( = 2:0; = 0:5). Case (i) performs consistently better than (iii) while (ii) dominates both (i) and (iii). However, case (iv) performs even modestly better than (ii). Hence, a small increase in monitoring strength can compensate for lower eciency. strength and low eciency ( = 1:0; = 0:1), and (iv) higher monitoring strength and moderate eciency ( = 2:0; = 0:5). Case (i) represents the deterministic evolution, where we do not monitor the dot, though some natural dephasing occurs (Sec. 3.5.2). Cases (ii), (iii) and (iv) involve continuous monitoring and implement the CUSUM-based technique (Sec. 3.5.3). Case (ii) is the scenario of high monitoring eciency (which, as we argue above, is likely unrealistic in practice), while in case (iii) the eciency is low. Finally, in case (iv) the monitoring eciency is moderate, but we have boosted the measurement strength to compensate. Physically, dephasing is intrinsic in any realistic system, as information about the state of the dot is lost to various internal modes, such as phonon modes. In addition, 42 there will be additional dephasing due to the presence of monitoring. In these terms, case (i) has intrinsic dephasing but no monitoring; case (ii) has ecient monitoring and no intrinsic dephasing; case (iii) has monitoring with low eciency, but no additional intrinsic dephasing. In case (iv) we assume that there is both monitoring and intrinsic dephasing, so the total dephasing rate is higher than in case (ii), and the detector e- ciency = 0:5 is moderate, re ecting the fact that only information from the monitoring is available, and not the information lost to the internal degrees of freedom. Figure 3.7 displays the single-photon probability as a function of for these four cases. The parameter values are dened in Eq. (3.34), and we have set g = 0:1. This plot explores just a representative slice of the (g, ) space that we considered earlier. We see that case (i) performs consistently better than (iii), while (ii) unsurprisingly outperforms both (i) and (iii). However, case (iv) performs even better, and consistently dominates (ii). 3.6 Summary CUSUM is a simple yet powerful method that can signicantly improve single-photon probability using continuous monitoring. The protocol itself requires only simple com- ponents like an integrator and subtractor, along with 2 registers and a comparator. This technique is useful in the weak coupling regime when the pumping rate is comparable to coupling strength. In regions of strong coupling, or strong pumping, it is ineective and can be worse than no monitoring at all. We included various decoherence processes in our simulations, including spontaneous emission, dephasing and cavity leakage. We modeled the electrical pumping and continuous monitoring as a stochastic master equa- tion. Numerical simulations showed that CUSUM performs quite well in regions of low , and signicant improvements in single-photon probability were observed. We also studied the eect of imperfect monitoring eciency on performance. 43 Chapter 4 Single-spin measurement in OSCAR MRFM with spin noise 1 4.1 Introduction Sidles, in 1991, rst proposed the use of magnetic resonance and mechanical oscillators to sense a weak force [42]. Ever since this proposal, there has been considerable progress in the experimental implementation of such a technique [39, 44, 40], culminating in single spin detection in 2004 [40]. That the force detected is of the order of attonewtons, high- lights the usefulness of magnetic resonance force microscopy (MRFM) based techniques. MRFM based techniques have a wide range of applications. One such application is imaging; in fact, imaging was the original motivation for proposing MRFM; imaging at the nanoscale like that of biological molecules such as proteins, viruses etc., is of immense value to the society at large [12]. Another application, and the one that is of interest to us, is the use of MRFM for single spin measurements. Numerous models of quantum computing involving spins have been proposed [29], and these models require the ability to do single spin measurements, in addition to purely scientic value of such a measurement capability. In 2001, Stipe et al. [44] rst implemented the OScillating Cantilever-driven Adiabatic Reversals (OSCAR) protocol. The frequency shift that takes place in this protocol, can be measured experimentally with high precision. The idea of this spin manipulation protocol is to transform the cantilever-spin interaction force into a shift in the resonant 1 This chapter is adapted from our work in [38]. 44 frequency of the oscillating cantilever, by using a gain-controlled feedback mechanism; the interaction force between the cantilever and the spin, which is either attractive or repulsive depending on the orientation of the spin, gets transformed to a positive or a negative shift in frequency; by measuring this shift one can determine the orientation of the spin. Theoretical analysis quickly followed with a series of work by Berman et al. [6, 3, 4, 5] and Brun and Goan [9, 10]. Berman et al. [6, 3, 4, 5] use a quantum analysis, wherein they describe the cantilever as a harmonic oscillator in the coherent state basis (since its behavior remains close to classical), coupled to a spin-1=2. Brun and Goan [9, 10] provide a fully quantum description of the system, and also consider decoherence eects acting on the cantilever in their analysis, including continuous measurement of the cantilever due to optical interferometric techniques. Though this modeling and analysis is an important theoretical development, Brun and Goan [9, 10] describe the system as a pure quantum state, instead of a density matrix, and use quantum (pure) state diusion techniques to describe their evolution, instead of a stochastic master equation evolution [7, 23]. While in principle this is a perfectly valid approach|after all, density matrices can be decomposed into mixtures of pure states|because of the nonlinearity of the trajectory equations, a partial coarse-graining (as done in that work) can disagree with the results from the appropriate stochastic master equation [16]. Also, that work does not include decoherence acting on the spin. In this work, we analyze the MRFM system in generality within a Markovian frame- work. We describe the system using a density matrix, and evolve it using a stochastic master equation [7, 23], including feedback of the continuous measurement output. We consider decoherence eects acting on both the cantilever and the spin. The stochas- tic master equation, though it gives a quantum-mechanical description of the system, is numerically expensive. To ameliorate this, we show that the state of the cantilever can be approximated as a Gaussian wave packet, leading to a description of the MRFM system by a closed set of 11 coupled stochastic dierential equations. We show that 45 the Gaussian approximation is valid in the parameter regime of interest, by numerically comparing the evolution of the Gaussian equations with the fully quantum stochastic master equation. Further, we use the Gaussian equations to analyze the OSCAR protocol as a tool for single-spin measurement. We consider the constraints set by the spin noise on a single-spin measurement. For the parameter values chosen, we calculate a bound on the spin noise time scale. We choose a rate for spin noise that satises this time scale, and numerically show that the OSCAR protocol can indeed be used to do single-spin measurement. 4.1.1 Chapter overview Sec. 4.2 gives an overview of the MRFM system and the OSCAR protocol. In Sec. 4.3, we use the adiabatic approximation and derive an eective Hamiltonian for the OSCAR MRFM system. We consider the decoherence eects acting on both the cantilever and the spin, in Sec. 4.4; the sources of decoherence include the thermal bath, continuous measurement of the cantilever using optical interferometry, and spin noise due to mag- netic sources. We then discuss the feedback mechanism used to implement the OSCAR protocol (Sec. 4.5). We derive equations for the moments in Sec. 4.6; we consider the unitary case rst (Sec. 4.6.1), before proceeding to the more general open system evolu- tion (Sec. 4.6.2). Finally, we present the numerical results in Sec. 4.7. We rst discuss the parameter values that are used in our simulations; we consider the eectiveness of the Gaussian approximation by comparing it to the fully quantum stochastic master equation evolution (Sec. 4.7.1); and lastly, we ask what it takes for an OSCAR MRFM system to be an useful single-spin measurement device. 46 4.2 OSCAR MRFM - Model and Operation The basic model of MRFM involves a cantilever oscillating in close proximity to a spin that is contained in a substrate (Fig. 4.1). A ferromagnet that is attached at the tip of the cantilever interacts with the spin. This interaction changes the amplitude and frequency of the oscillator, which is measured to determine the orientation of the spin. ! # Figure 4.1: MRFM system: model and operation. A cantilever with a ferromagnetic tip oscillates in close proximity to a substrate with a free spin. A uniform magnetic eld B 0 is applied in thez direction and a microwave (RF) eld is applied in the x-y plane. The position of the cantilever is monitored continuously by shining a laser on the cantilever tip and using optical interference to make measurements. A feedback mechanism (not shown in the gure) maintains the amplitude of the cantilever at a pre-determined xed value, making the cantilever a frequency-determining element. The direction of the frequency shift reveals the orientation of the spin. 47 There are three dierent sources of magnetic eld present in the system: (i) a constant eld, B 0 , in the z direction, (ii) an oscillating ac microwave (RF) eld, B ac , that is applied in the xy plane, and, (iii) a dipole eld due to the ferromagnetic tip, B d . The constant eld B 0 polarizes the spin in the z direction. The rotating microwave (RF) eld, together with the dipole eld from the ferromagnet, causes the spin precession axis to periodically reverse with the same frequency as the cantilever. When in resonance, the spin oscillates with the same frequency as that of the cantilever. Under the resonant condition, the force due to interaction of the ferromagnet at the tip of the cantilever and that with the magnetic moment of the spin gets amplied: this amplication leads to a change in the resonant amplitude and frequency of the cantilever. This change can be observed by monitoring the cantilever, usually through optical interference. In the OSCAR protocol, the oscillation of the cantilever (ferromagnetic tip) causes adiabatic reversals of the spin, which in turn interacts with the cantilever to change the resonant amplitude and frequency of the cantilever [44]. In this protocol, feedback is used to shift the resonant frequency of the cantilever; the change in resonant frequency| positive or negative|of the cantilever is used to determine the orientation of the spin. The feedback mechanism used is positive gain controlled, and is used to maintain the cantilever amplitude at a pre-determined xed value. The cantilever thus behaves as a frequency determining element [1]. 4.3 Eective Hamiltonian using the adiabatic approxima- tion The cantilever and spin form the two parts of the system, and are described by their respective Hamiltonians: ^ H C (t) = ^ p 2 =2m + m! 2 ^ Z 2 =2 f(t) ^ Z ^ H S (t) = ^ S x ^ Z ^ S z (4.1) 48 where ^ H C (t) and ^ H S (t) are the Hamiltonians of the cantilever and spin, respectively [4]; f(t) is the positive gain controlled feedback mechanism that implements the OSCAR protocol (refer Sec. 4.5); and, and represent the strength of the microwave (RF) eld and spin cantilever interaction, respectively. Note that we have included the interaction term ^ Z ^ S z as part of the spin Hamiltonian ^ H S (t). In the parameter regime of the OSCAR protocol, the time scales of the spin and the cantilever are well separated. The frequency of spin precession is much greater than the oscillation frequency of the cantilever. On the time scale of the spin, the cantilever part in the interaction term in the Hamiltonian, ^ H I = ^ Z ^ S z can be treated as a constant; thus, the quantum cantilever position term, ^ Z, on the time scale of the spin can instantaneously, in the adiabatic limit, be treated as a classical function Z. The time dependent spin Hamiltonian in the adiabatic limit is given by ^ H S (t) = ^ S x Z ^ S z ; (4.2) whose instantaneous eigenstates are jv (t)i = p 2 + ( +Z) 2 j"i + +Z p 2 + ( +Z) 2 j#i; (4.3) with eigenvalues (t) = p 2 + 2 Z 2 : (4.4) If the initial state of the spin is j S (0)i = (0)jv + (0)i + (0)jv (0)i; (4.5) 49 then the state of the spin at later times can be written as j S (t)i = (t)jv + (t)i + (t)jv (t)i; (4.6) wherejv (t)i is the instantaneous eigenstate of the time-dependent spin Hamiltonian at time t. The time evolution of the state is given by the Schr odinger equation d dt j S (t)i = i ~ ^ H S (t)j S (t)i: (4.7) Using Eq. (4.6) in Eq. (4.7), and solving for (t) and (t) using the adiabatic approxi- mation, we get (t) = (0) exp i ~ Z t 0 (s)ds ; (4.8) (t) = (0) exp i ~ Z t 0 (s)ds : (4.9) The spin experiences an eective magnetic eld comprising a constant uniform eld, an applied microwave (RF) eld, and the eld of the permanent magnet on the cantilever tip. In the rotating frame, these produce the two terms of the spin Hamiltonian Eq. (4.2). The probabilities of the spin being up or down in the reference frame of the eective magnetic eld are constants of motion, and are given byj(0)j 2 andj(0)j 2 , respectively. In the solution above, we have used the fact that the time scale of cantilever motion is much slower than the time scale of spin precession, i.e., the rate of change of the cantileverjdZ=dtj ( 2 =), and consequently, we have ignored terms proportional to dZ=dt in deriving Eqs. (4.8) and (4.9). Assuming the cantilever and spin are initially decoupled, the initial state of the cantilever-spin system is j (0)i = j C (0)i j S (0)i; (4.10) 50 wherej C (0)i is the initial state of the cantilever, andj S (0)i is the initial state of the spin as dened in Eq. (4.5). Further, we assume that the cantilever is initially in a localized wavepacket, and therefore that the subsequent motion of the cantilever can be described by two wave packets, one corresponding to each spin orientation. This state description is very reasonable, as we will see later in the paper: the cantilever is maintained in a highly localized state by the continuous position measurements. As we know,jv (t)i are the eigenstates of the spin Hamiltonian, and satisfy ^ H S (t)jv (t)i = (t)jv (t)i; where (t) is its eigenvalue [Eq. (4.4)]. We assume that the microwave (RF) eld, , is much stronger than the interaction strength between the cantilever and the spin Z. Expanding (t) as a binomial series, we get (t) = 1 + (Z=) 2 =2 +O((Z=) 3 ) : (4.11) As the microwave (RF) eld dominates the interaction strength, we can ignore third| and higher order|contributions in this expansion. Since the rst term in the expansion is a constant, it amounts to a phase factor, and can be ignored as well. Let us now consider the action of the total Hamiltonian on the joint cantilever and spin states: ( ^ H C + ^ H S (t)) (jzi jv (t)i) = ( ^ H C (t)) (jzi jv (t)i); (4.12) 51 wherejzi is the position of the cantilever at time t. We now quantize the position function to obtain the eective Hamiltonian acting on the cantilever: ^ H 0 C = 8 < : ^ H C + ^ Z 2 ; if spin-up, i.e.jv + (t)i ^ H C ^ Z 2 ; if spin-down, i.e.jv (t)i = ^ H C + 2 ^ Z 2 ^ 0 Z (4.13) where we have set = 2 =2; (4.14) and ^ 0 Z is the rotating operator that represents the spin with respect to a reference frame that moves with the eective magnetic eld. The total eective Hamiltonian acting on the cantilever + spin system is, thus, ^ H 0 = ^ H 0 C : (4.15) 4.4 Modeling decoherence in OSCAR MRFM system In reality, quantum systems are never completely isolated. The interaction of the system with the rest of the universe is seen as decoherence on the system of interest. There are various decoherence processes that act on both the cantilever and the spin part of the system. 4.4.1 Decoherence in the cantilever Two primary sources of decoherence act on the cantilever: (a) the thermal environment, and, (b) the continuous measurement of the external read out. Brun and Goan [9] use the Caldeira-Leggett master equation for a harmonic oscillator coupled to a high- temperature bath [11], as modied by Di osi to form an explicit Lindblad equation [13, 14]. 52 They showed that these noise processes can be modeled by Lindblad operators of the form ^ L k = A k ^ Z + iB k ^ p; (4.16) where ^ L 1 captures the eect of thermal environment and ^ L 2 is due to continuous moni- toring of the cantilever position. The coecients are A 1 = p 4 m mk B T=~ 2 ; B 1 = p m =4mk B T A 2 = p 8 2 E 2 = 3 c ; B 2 = 0: (4.17) Modeling the thermal environment as a Lindblad operator ^ L 1 requires an additional damping term to be added to the Hamiltonian (refer [9] Sec. IV), and the new eective Hamiltonian of the cantilever becomes ^ H eff = ^ H 0 + m ^ R; (4.18) where ^ H 0 is the eective Hamiltonian from Eq. (4.15), m is the damping rate from the thermal environment, and ^ R 1 2 ( ^ Z ^ p + ^ p ^ Z): (4.19) 4.4.2 Modeling spin noise The source of noise acting on the spin could either be dipole-dipole interactions of the spin with other spins in the lattice, or noise in the magnetic eld. The dipole-dipole interaction acts in the laboratory frame while the noise due to magnetic sources can be treated in the reference frame of the eective magnetic eld. Thermal noise is one of the dominant decoherence process in the MRFM system. This noise aects the cantilever motion, which in turn aects the interaction between the cantilever and the spin. Thus, noise in the cantilever motion means that there is 53 noise in the dipole magnetic eld due to cantilever-spin interaction. Also, the external microwave (RF) eld is not free of noise. Stipe et al. [44] have shown that the spin-lattice relaxation time is quite long, on the order of seconds. This means that spin decoherence is dominated by magnetic noise. As a result, we consider spin relaxation due to magnetic noise and ignore spin-lattice relaxation. In the reference frame of the eective magnetic eld, spin relaxation due to magnetic noise corresponds to simple spin- ip noise. The Lindblad operator corresponding to spin noise can thus be modeled as, ^ L 3 = p s ^ 0 X : (4.20) s is the rate of spin noise due to magnetic sources, and ^ 0 X = jv + (t)ihv (t)j +jv (t)ihv + (t)j; (4.21) where jv (t)i are the instantaneous eigenstates of the spin Hamiltonian at time t [Eq. (4.3)]; note that spin-up or spin-down here corresponds to spin in the direction of, or opposite to, the eective magnetic eld. For MRFM to be useful as a single spin measurement, it is necessary that the spin decoherence rate s be small compared to other parameters in the cantilever-spin system. As we measure the cantilever, we learn something about the state of the spin, and eventually, the spin relaxes to one of its eigenstates, either in the direction of the eective eld or in the opposite direction. There are two dierent time scales at work here: (i) the spin localization time, and, (ii) our observation time; the spin localization time scale is the time it takes, on average, for the spin to collapse to one of its eigenstates due to continuous monitoring of the cantilever; on the other hand, the observation time scale is the time it takes for us to `know' the state of the spin. Typically, the observation time scale is greater than the spin localization time scale as it would take a little longer after the spin is completely relaxed to `know' the state of the spin; however, one can also conclude the state of the spin based on the trend (depending on the signal-to-noise 54 ratio) in the frequency shift. In the latter case, we do not wait for the spin to localize completely to guess the state of the spin. Irrespective of the method used, the time scale of spin noise ( 1 s ) has to be longer than both spin localization and observation time scales for MRFM to be useful in single-spin measurement. 4.5 Continuous measurement and feedback in OSCAR protocol In the OSCAR protocol, the spin orientation is measured by measuring the frequency shift of the cantilever. A positive gain-controlled feedback mechanism maintains the amplitude of the cantilever at a pre-determined constant, leading to a change in cantilever frequency. The continuous monitoring of the cantilever motion is done by optical interferometry. As shown in Fig. 4.1, a laser is placed close to the tip of the cantilever; an optical micro- cavity is formed with the cantilever on one side and the cleaved end of the ber (laser) on the other. Since the motion of the cantiever is slow compared to the optical frequency, this system can be analyzed in the adiabatic limit. A homodyne measurement is carried out on the light that escapes this cavity. The output of the homodyne measurement corresponds to the position of the cantileverh ^ Zi, and is given by [9] I c (t) = 8e d E c h ^ Zi + p c e d dW t dt : (4.22) Here, I c (t) is the output photocurrent of the homodyne measurement, is the coupling between the cantilever and the cavity, c is the cavity loss rate and e d is the detector eciency. Brun and Goan [9] (Sec. V) have analyzed this system in detail and we refer to that paper for more information. 55 The feedback mechanism in the OSCAR protocol is positive gain-controlled, and its objective is to maintain the cantilever amplitude at a pre-determined xed value. The feedback has the form f(t) = g (AMP Amp(t)) I c (t ); (4.23) where g is the feedback gain, AMP is the pre-determined set point amplitude, Amp(t) is the cantilever amplitude at time t, I c (t ) is the delayed output photocurrent, and corresponds to a delay of =2 radians, or equivalently, a fourth of the cantilver oscillation time period. The current amplitude of the cantilever,Amp(t), is derived from the measured output photocurrent I c (t), using simple signal processing techniques. 4.6 Moment equations for OSCAR MRFM with Gaussian approximation In this section, we derive moment equations for the cantilever and spin. We consider the unitary case rst and then move to open system evolution. We use the Schr odinger equation to describe the unitary evolution, and a full quantum stochastic master equation for the open system evolution, with continuous monitoring of the cantilever motion. Finally, we make Gaussian approximation for the cantilever degree of freedom and show that the moments then obey a closed set of coupled equations. 4.6.1 Unitary evolution The evolution of the cantilever-spin system is given by a Schr odinger (or a von Neumann) equation: d dt = i ~ [ ^ H 0 ;]: (4.24) ^ H 0 is the eective Hamiltonian of the cantilever-spin system [Eq. (4.15)]. 56 To derive the moment equation for an arbitrary operator ^ O, we apply the Schr odinger equation to obtain equation forh ^ Oi: dh ^ Oi dt = Tr ^ O d dt = i ~ h[ ^ H 0 ; ^ O]i: (4.25) We dene (instantaneous) projectors on to the spin-up and spin-down states as: ^ P " = ^ I jv + (t)ihv + (t)j ^ P # = ^ I jv (t)ihv (t)j ; (4.26) where ^ I acts on the cantilever. The probability of spin-up is dened as r u = h ^ P " i; (4.27) and, the probability of spin-down is r d = h ^ P # i = (1r u ): (4.28) The weight of the spin-up and spin-down wave packets is given byr u andr d , respectively. Though we can derive the value of r d from r u straightforwardly using equation (4.28), we use r d to represent spin-down probability for notational convenience. To keep track of the cantilever degree of freedom, we dene variables that capture the mean position and momenta of the two wave packets as: Z u := h ^ Z ^ P " i=r u ; p u := h^ p ^ P " i=r u Z d := h ^ Z ^ P # i=r d ; p d := h^ p ^ P # i=r d 9 = ; : (4.29) 57 We can now apply Eq. (4.25) to derive rst-order moment equations: dZ u =dt = p u =m; dp u =dt = (m! 2 + 2 )Z u + f(t); dZ d =dt = p d =m; dp d =dt = (m! 2 2 )Z d + f(t); dr u =dt = 0: (4.30) The equations derived above close, which means that higher-order moment terms can be ignored. The expected position of the cantilever is obtained by the identity h ^ Zi = r u Z u + r d Z d : (4.31) We observe that the moment equations actually are identical to the equations of motion of a driven (classical) Harmonic oscillator; the resonant frequency of the two wave packets is shifted by an amount ! =m!, from the natural resonant frequency ! of the cantilever. We also note that spin-up probability, r u , is a constant of motion. This is consistent with our derivation section 4.3. Spin probabilities, however, are not a constant of motion for the non-unitary evolution considered in the next section. 4.6.2 Open system evolution with continuous monitoring As discussed in Sec. 4.4, the cantilever-spin system is not isolated, and there are many sources of decoherence that aect both the cantilever and the spin. The sources include thermal noise, noise due to the process of continuous monitoring of the cantilever and noise in the magnetic eld. We model these decoherence eects in terms of Lindblad operators. There are three Lindblad operators that model the important sources of deocherence, two of which| ^ L 1 and ^ L 2 |capture the decoherence acting on the cantilever [Eq. (4.16)], 58 while ^ L 3 models the noise on the spin [Eq. (4.20)]. ^ L 1 is the noise due to the thermal environment and ^ L 2 is the noise due to continuous monitoring of the cantilever position. The evolution of the cantilever-spin system is given by a quantum stochastic master equation [7, 23]: d = i ~ [ ^ H eff ;]dt + 3 X k=1 ^ L k ^ L y k 1 2 f ^ L y k ^ L k ;g dt + p e d ( ^ L 2 h ^ L 2 i) ( ^ L y 2 h ^ L y 2 i) dW t ; (4.32) where e d is the detector eciency; dW t is a stochastic (white) noise process with the property: M[dW t ] = 0 and dW 2 t =dt; and ^ H eff is the eective Hamiltonian acting on the system [Eq. (4.18)]. As in the unitary case, we consider an arbitrary operator ^ O acting on the system (with evolution described in Eq. (4.32)), and derive its moment equations. Using Eq. (4.32), the rst-order moment equation forh ^ Oi = Trf ^ Og is dh ^ Oi = i ~ h[ ^ H eff ; ^ O]idt + 3 X k=1 h ^ L y k ^ O ^ L k i 1 2 hf ^ L y k ^ L k ; ^ Ogi dt + p e d h ^ O ^ L 2 ih ^ Oih ^ L 2 i +h ^ L y 2 ^ Oih ^ L y 2 ih ^ Oi dW t : (4.33) We describe the (mean) position and momentum of the cantilever in terms of nor- malized wave packets with spin up or down, respectively [Eq. (4.29)]. They are of the form: h ^ Oi u =h ^ O ^ P " i=r u andh ^ Oi d =h ^ O ^ P # i=r d ; (4.34) where r u and r d are the spin probabilities as dened in equations (4.27) and (4.28). As in the unitary case, the meanh ^ Oi is dened as h ^ Oi = r u h ^ Oi u + r d h ^ Oi d : 59 The system evolution described in Eq. (4.32) is stochastic. This means that the moments likeh ^ Oi u andh ^ Oi d evolve stochastically as well; the moment equation (4.33) is a stochas- tic dierential equation, unlike its unitary counterpart Eq. (4.25). These equations follow stochastic calculus described by It^ o rules [31]. To give the evolution ofh ^ Oi u andh ^ Oi d , we apply It^ o rules and get, dh ^ Oi u = dh ^ Oi=r u h ^ Oidr=r 2 u dh ^ Oidr=r 2 u +h ^ Oi(dr) 2 =r 3 u ; (4.35) and similarly forh ^ Oi d . 4.6.2.1 First-order moment equations We now get to the task of deriving moment equations for the cantilever and the spin. As in the unitary case, we rst derive the equation for spin-up probability r u =h ^ P " i, before we derive equations for the wave packets that describe the cantilever motion. Using Eq. (4.33), we derive the equation for spin-up probability dr u = s (1 2r u )dt + 2 p e d A 2 r u r d (Z u Z d )dW t : (4.36) Here, Z u and Z d are mean positions of the cantilever wave packets for spin up or down, respectively; s is the rate of spin noise [Eq. (4.20)]; and A 2 is part of the Lindblad operator for cantilever decoherence due to continuous measurement [Eqs. (4.16) and (4.17)]. We used the identity in Eq. (4.31) in our derivation. Note that unlike the unitary case, Eq. (4.30), the spin-up probability r u is not a constant of motion. However, since r d = (1r u ), the single variabler u is enough to capture the evolution of spin probabilities. The motion of the cantilever is described in terms of two wave packets corresponding to the spin being in the up or down state, respectively. Each wave packet is described 60 by its position and momentum. To derive the moment equation for these variables, we apply Eqs. (4.35) and (4.33), to obtain dZ u = (p u =m) 4e d A 2 2 h ^ Z 2 i u r d (Z u Z d ) s (r d =r u ) (Z u Z d ) dt + 2 p e d A 2 h ^ Z 2 i u dW t ; (4.37) and dp u = (m! 2 + 2 )Z u 2 p u + f(t) 4e d A 2 2 h ^ Ri u r d (Z u Z d ) dt s (r d =r u ) (p u p d )dt + 2 p e d A 2 h ^ Ri u dW t : (4.38) Similarly, dZ d = (p d =m) + 4e d A 2 2 h ^ Z 2 i d r u (Z u Z d ) dt + s (r u =r d ) (Z u Z d )dt + 2 p e d A 2 h ^ Z 2 i d dW t ; (4.39) and dp d = (m! 2 2 )Z d 2 p d + f(t) + 4e d A 2 2 h ^ Ri d r u (Z u Z d ) dt + s (r u =r d ) (p u p d )dt + 2 p e d A 2 h ^ Ri d dW t : (4.40) In the unitary case, the rst-order moment equations (4.30) formed a closed set. In the open system evolution however, the rst order moment equations (4.374.40) do not form a closed set, and depend on second-order terms likeh ^ Z 2 i u andh ^ Ri u : h ^ Z 2 i u := h ^ Z 2 i u Z 2 u ; (4.41) h ^ Ri u := h ^ Ri u Z u p u ; (4.42) h ^ Z 2 i d := h ^ Z 2 i d Z 2 d ; (4.43) h ^ Ri d := h ^ Ri d Z d p d ; (4.44) 61 where ^ R was dened in (4.19) above. Since the set of rst-order equations do not close, we need to include second-order terms, and derive equations for them as well. 4.6.2.2 Second-order moment equations and Gaussian approximation We will need equations for six second-order equations:h ^ Z 2 i u ,h^ p 2 i u ,h ^ Ri u ,h ^ Z 2 i d , h^ p 2 i d andh ^ Ri d . We begin with the derivation ofh ^ Z 2 i u . To derive the moment equation, we rst expand h ^ Z 2 i u in terms of its deni- tion (4.41), then apply It^ o rules where necessary as shown in Eq. (4.35) for an arbitrary operator. Finally, we expand using the moment equation (4.33) to get: dh ^ Z 2 i u = 2h ^ Ri u =m + B 2 1 ~ 2 dt 4e d A 2 2 h ^ Z 2 i u + r d (Z u Z d )h ^ Z 3 i u dt s (r d =r u ) h ^ Z 2 i u h ^ Z 2 i d (Z u Z d ) 2 dt + 2 p e d A 2 h ^ Z 3 i u dW t : (4.45) We have used some of the rst order equations in the above derivation. We derive the equation forh ^ Ri u , dened in Eq. (4.42), by the same procedure used to derive Eq. (4.45), and obtain: dh ^ Ri u = (m! 2 + 2 )h ^ Z 2 i u +h^ p 2 i u =m 2 h ^ Ri u dt 4e d A 2 2 h ^ Ri u h ^ Z 2 i u + r d (Z u Z d )hf ^ Z 2 ; ^ pgi u dt s (r d =r u ) h ^ Ri u h ^ Ri d (Z u Z d )(p u p d ) dt + p e d A 2 hf ^ Z 2 ; ^ pgi u dW t : (4.46) Note that a new second order termh^ p 2 i u appears in the above equation, and is dened as h^ p 2 i u := h^ p 2 i u p 2 u : (4.47) 62 Following the same procedure, we construct the moment equation forh^ p 2 i u : dh^ p 2 i u = 2 k=1 A 2 k ~ 2 4( (m! 2 =2 + )h ^ Ri u + h^ p 2 i u ) dt 2e d A 2 2 2h ^ Ri 2 u + r d (Z u Z d )hf ^ Z; ^ p 2 gi u dt s (r d =r u ) h^ p 2 i u h^ p 2 i d (p u p d ) 2 dt + p e d A 2 hf ^ Z; ^ p 2 gi u dW t ; (4.48) Similarly, we derive equations for the second-order moment terms of the spin-down wave packet: dh ^ Z 2 i d = 2h ^ Ri d =m + B 2 1 ~ 2 dt 4e d A 2 2 h ^ Z 2 i d r u (Z u Z d )h ^ Z 3 i d dt + s (r u =r d ) h ^ Z 2 i u h ^ Z 2 i d + (Z u Z d ) 2 dt + 2 p e d A 2 h ^ Z 3 i d dW t ; (4.49) dh ^ Ri d = (m! 2 2 )h ^ Z 2 i d + h^ p 2 i d =m 2 h ^ Ri d dt 4e d A 2 2 h ^ Ri d h ^ Z 2 i d r u (Z u Z d )hf ^ Z 2 ; ^ pgi d dt + s (r u =r d ) h ^ Ri u h ^ Ri d + (Z u Z d ) (p u p d ) dt + p e d A 2 hf ^ Z 2 ; ^ pgi d dW t ; (4.50) dh^ p 2 i d = 2 k=1 A 2 k ~ 2 4 ( (m! 2 =2 )h ^ Ri d + h^ p 2 i d ) dt 2e d A 2 2 2h ^ Ri 2 d r u (Z u Z d )hf ^ Z; ^ p 2 gi d dt + s (r u =r d ) h^ p 2 i u h^ p 2 i d + (p u p d ) 2 dt + p e d A 2 hf ^ Z; ^ p 2 gi d dW t : (4.51) 63 We have now derived equations for all the second order terms that appear in the equations. However, there are new terms, of third order, that appear in the second order equations (4.454.51): h ^ Z 3 i u;d ;hf ^ Z; ^ p 2 gi u;d andhf ^ Z 2 ; ^ pgi u;d : The second order equations (4.454.51) do not close as they depend on terms of third order. We now make the Gaussian approximation for the cantilever. Assuming that the wave packets were initially Gaussian, and that they remain approximately Gaussian at later times, then the third order terms vanish. The system of rst-order equations along with the second order equations now close, giving us a set of 11 coupled equations. We will now verify the validity of this approximation by numerically comparing the approximate solution to that of the full quantum stochastic master equation (4.32). 4.7 Numerical results We simulated the full quantum stochastic master equation using aC++ quantum master equation library developed by Brun and Shaw [46]. We use a fth order Runge-Kutta integrator [34] to simulate the set of coupled equations (4.364.40) and (4.454.51) for the Gaussian approximation. We choose parameters based on those used by Berman et al. [4] and Brun and Goan [9, 10]. The values of parameters, in dimensionless units, are: ~ =! =m = 1 = 0:6; = 100; = !=Q = 10 5 ; e d = 0:85 A 1 = 0:2; B 1 = 5 10 5 ; A 2 = 0:07; B 2 = 0: (4.52) 64 We assume the physical units in which! = 10 5 s 1 ,m = 10 12 kg, = 310 7 T/m, and = 300T, consistent with current experiments. The values of A 1 and B 1 are dierent from the ones used in [9]. We assume that, in dimensionless units, k B T = 10 3 ; in our present calculations. This value of k B T corresponds to a temperature of about 10 mK; though this is lower than the temperature accessible currently (which is around 300 mK [12]), given the considerable progress made on the experimental front in the last few years, the temperature we have used should be accessible in the near future. The output of the homodyne measurement that measures the cantilever position is given by Eq. (4.22), and can be expressed (in rescaled units) as I c (t)dt = h ^ Zidt dW t ; (4.53) where = 50 8EQ s 3 c e d : (4.54) Here, Q is the quality factor of the cantilever, and the other parameters are due to the cantilever-light interaction, which is part of the optical interferometry process (see Sec. 4.5). Since the driving force applied to the cantilever is part of a feedback process, the driving frequency is resonant with the cantilever frequency. We know that, in the resonant case, the steady state amplitude of a Harmonic oscillator is proportional to its quality factor [1]. In our simulation, we assume that the cantilever is in its steady state before it interacts with the spin, and that the steady state amplitude is, in dimensionless units, 50, which corresponds to 32 nm in physical units (in [40], the peak amplitude is 16 nm and the quality factor is 50; 000); due to this rescaling of amplitude, the can- tilever quality factor Q appears in the above equation (4.54), and we set =50=Q in Eq. (4.22). 65 We can nd the value of the coecient straightforwardly, by noting that (in dimen- sionless units) 8E c = 1:9 10 3 ; and c = 1:4 10 8 : See [9] (Sec. VII) for more information. The positive gain controlled feedback function f(t) is dened in Eq. (4.23), and depends on the feedback gain g and the pre-determined set point amplitude AMP . In our simulation, we set these parameters to g = 0:0001 and AMP = 50: (4.55) We chose a small value for g to avoid numerical instability when simulating the fully quantum stochastic master equation. However, the accuracy of the approximation is unaected by larger values of g. 4.7.1 Accuracy of Gaussian approximation We now numerically demonstrate the accuracy of the Gaussian approximation by com- paring it to a fully quantum stochastic master equation (4.32). The parameters used to generate these plots are given in equations (4.52) and (4.55); the spin noise rate, s , is set to 0:001. We assume that the cantilever and spin are initially decoupled; we also assume that the spin is in an equal superposition of up and down state, while the cantilever is at its lowest position (AMP ). Figs. 4.2, 4.3 and 4.4, shows the time evolution of the expected position of the cantileverh ^ Zi, as dened in Eq. (4.31), at dierent times, while Fig. 4.5 plots the time evolution of the spin-up probability r u =h ^ P " i. Figures 4.2, 4.3, 4.4 and 4.5 show that the Gaussian approximation tracks the quantum stochastic master equation very closely. 66 The spin probability does not relax completely (to either 0 or 1); after initial uctua- tions, the spin probability gets closer to 1 (spin-up), the spin ` ips' and the probability is then closer to zero (spin-down), and nally, another ` ip' takes it back closer to 1 (spin- up). The spin exhibits this behavior because the value of s is relatively high (= 0:001), leading to multiple ` ips' in spin probability. -60 -40 -20 0 20 40 60 0 1 2 3 4 5 6 < Z > time (ms) SME Gaussian Figure 4.2: Comparing full quantum SME and Gaussian approximation (1). We plot h ^ Zi as a function of time for both the quantum SME as well and the Gaussian approx- imation at initial times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. 4.7.2 OSCAR MRFM and single-spin measurement We now address the question of how high s can be, for OSCAR MRFM to be a useful single-spin measurement device. Henceforth we use the numerically ecient Gaussian approximation equations to describe the OSCAR MRFM system. 67 -60 -40 -20 0 20 40 60 19 20 21 22 23 24 25 < Z > time (ms) SME Gaussian Figure 4.3: Comparing full quantum SME and Gaussian approximation (1). We ploth ^ Zi as a function of time for both the quantum SME as well and the Gaussian approximation at intermediate times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. In Figure 4.6, we plot the time evolution of spin-up probability for dierent values of s : (i) 10 3 , (ii) 10 4 and 10 5 in dimensionless units (16 Hz, 1:6 Hz and 160 mHz, respectively, in physical units); the parameters and the initial conditions are the same as in Figs. 4.2, 4.3, 4.4 and 4.5. We observe that the number of spin ips decreases with the value of s . Intuitively, one can see from Fig. 4.6 that smaller values of s are preferable, as we must track the cantilever position for a sucient length of time to determine the orientation of the spin with required accuracy. If s is higher, the likelihood of a spin ip increases, corrupting the samples used to determine the spin orientation. In the OSCAR protocol, we measure the spin by measuring the resonant frequency of the cantilever; more precisely, the shift in the cantilever frequency. To determine the 68 -60 -40 -20 0 20 40 60 38 39 40 41 42 43 < Z > time (ms) SME Gaussian Figure 4.4: Comparing full quantum SME and Gaussian approximation (2). We plot h ^ Zi as a function of time for both the quantum SME as well and the Gaussian approx- imation at long times. The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. cantilever frequency numerically, we use the Fast Fourier Transform (FFT) algorithm dfour1 from [34]. The (numerical) frequency resolution of this FFT is f 1 N t 1 T Sampling ; (4.56) where N is the number of samples, t is the sample spacing, and f is the frequency resolution. T Sampling is the total sampling time. 69 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 40 r u time (ms) SME Gaussian Figure 4.5: Comparing full quantum SME and Gaussian approximation (4). We plot the evolution of spin-up probability r u =h ^ P " i.The parameter values used in this simulation are dened in equations (4.52) and (4.55); the rate of spin noise s is set to 0:001. We assume that the cantilever-spin system is initially decoupled, with the cantilever at the lowest point in its oscillation, while the spin is in an equal superposition of up and down states. We assume that the sample spacing t is xed. The shift in the resonant frequency is proportional to [Eq. (4.14)]. For the parameters chosen in our simulation [Eq. (4.52)], the value of is 1:8 10 3 . Thus, the frequency resolution to be achieved should satisfy f =2 2:8 10 4 : (4.57) The sample spacing used in our simulation is t = 0:02; setting the frequency resolution f = 10 4 , and using Eq. (4.56), one can easily derive N = 5 10 5 . It is numerically convenient to the FFT algorithm if N is a power of 2; we thus choose N = 2 19 (the 70 0.25 0.5 0.75 1 0 200 400 600 800 1000 r u g s = 10 -3 0 0.25 0.5 0.75 1 0 200 400 600 800 1000 r u g s = 10 -4 0.5 0.75 1 0 200 400 600 800 1000 r u time (ms) g s = 10 -5 Figure 4.6: Spin probability for dierent s . We plot the time evolution of spin-up probability, r u =h ^ P " i, for 3 dierent values of spin noise rate s : (i) 10 3 , (ii) 10 4 , and (iii) 10 5 . The number of spin ips decreases as s is reduced. power of 2 closest to 5 10 5 ) for our simulation, and the frequency resolution becomes f = 9:5 10 5 . Given the sample spacing, t, and the number of samples, N, the total sampling period in dimensionless (time) units is T Sampling =Nt 10; 500: (4.58) The time scale of spin noise, 1 s , must be longer than the sampling periodT Sampling , for OSCAR MRFM system to be useful in single-spin measurement. Thus, we set s = 10 5 in our simulation. In Figure 4.8 we show the frequency shift achieved in the OSCAR protocol. We plot two dierent trajectories: one in which the spin relaxes to the spin-up state, and another 71 in which it relaxes to the spin-down state. Figure 4.7 plots the time evolution of the spin-up probabilityr u for the two trajectories, and Fig. 4.8 displays their corresponding Fourier amplitudes. It is clear from Fig. 4.8 that there is a marked shift in resonant frequency: to the right in the case of spin-up, and to the left in the case of spin-down; we also observe that this shift is proportional to (= 1:8 10 3 ). (Note that the natural resonant frequency of the cantilever, in dimensionless units, is ! = 1.) A shift to the right in the resonant frequency of the cantilever implies that the spin is in the spin-up state, while a shift to the left indicates that the spin is in the spin-down state. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 100 200 300 400 500 600 700 800 r u time (ms) Spin-up Spin-down Figure 4.7: Spin-up probability evolution of two trajectories. We show the time evolution of the spin-up probability, r u , for the two trajectories; in the rst, the spin relaxes to its up state, while in the second, it relaxes to its down state. The value of sampling period, T Sampling , in Eq. (4.58) is in dimensionless units. In our calculation of parameter values for our simulations, we assumed that the cantilever frequency, in physical units, is f phy c =! phy c =2 16 kHz: 72 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.994 0.996 0.998 1 1.002 1.004 1.006 Fourier Amplitude t (units of frequency) Spin-up Spin-down Figure 4.8: Frequency shift in OSCAR MRFM. We show the frequency shift in the OSCAR MRFM protocol, for two dierent trajectories (Fig. 4.7). We plot the Fourier amplitude (in arbitrary units) as a function of cantilever frequency, !, corresponding to the two trajectories in Fig. 4.7. The spin noise rate s used in this plot is 10 5 . The Fourier amplitude is calculated using a standard FFT algorithm; the number of samples is N = 2 19 , and the sample spacing is t = 0:02. Using this frequency, the duration of sampling, in physical units, is T phy Sampling = T Sampling =f phy c 656 ms; and the shift in the cantilever frequency is phy = f phy c 29 Hz: Thus, it takes about 650ms for the OSCAR protocol to determine a shift of around 30Hz in cantilever frequency, and consequently, to ascertain the orientation of the spin. The 73 time scale of the spin noise must be longer than the sampling duration to use OSCAR MRFM as a single-spin measurement device. 4.8 Summary In this work, we modeled the OSCAR MRFM operation including decoherence eects on the cantilever (both thermal and monitoring), as well as spin noise due to magnetic sources. We describe the evolution of the system using a quantum stochastic master equation, the most general description in a Markovian framework. We then simplied this description using a series of approximations, and arrived at a semi-classical description of the system based on a Gaussian approximation of the cantilever state. We numerically compared the Gaussian approximation to the fully quantum stochas- tic master equation, and found that the Gaussian approximation tracked the fully quan- tum stochastic master equation very closely. Thus, we conclude that the Gaussian approximation of the cantilever is useful for our range of parameter values, and that the MRFM system implementing the OSCAR protocol can be described by a closed set of 11 coupled stochastic dierential equations. Finally, we used the Gaussian approximation equations to numerically verify OSCAR as a useful single-spin measurement protocol. The critical element in the OSCAR proto- col is the time (sampling) it takes to determine the shift in resonant cantilever frequency. For the parameter values chosen in our simulations, we found that the sampling duration is around 650 ms, and the frequency shift is about 30 Hz. The sampling duration thus places a bound on the time scale of spin noise that the OSCAR protocol, implementing a single-spin measurement, can tolerate. 74 Chapter 5 Conclusions In this work, I have studied two important physical systems: (a) single photon source, and, (b) single spin measurement using MRFM. The single photon source is critical to the implementation of quantum computing based on linear optics, while single spin measurement is essential for spin based systems. These systems are modeled in its generality within a Markovian framework, that included modeling decoherence eects and continuous quantum measurements, amongst others. The parameters considered are either already achieved by current experiments (MRFM), or should be accessible in the near future. 5.1 Single photon source 5.1.1 Indistinguishability The continuous monitoring feed forward technique is a simple yet powerful method that can provide great improvements in indistinguishability of a single-photon source [36]. We have numerically shown that, for small transition rates 2 , the eect on indistinguisha- bility of timing uncertainty in the quantum dot emission can be signicantly reduced. This improvement in indistinguishability holds true even when there is intrinsic dephas- ing in the quantum dot. The feed forward technique requires continuous monitoring, a simple linear estimation algorithm, and the ability to introduce variable delays. Though the algorithm is straightforward, the time resolution of continuous monitoring accessible with current technology is an issue. Better instrumentation, and moving more compo- nents to lower temperature, are possible ways forward to improve the time resolution. 75 We believe that techniques similar to those described in this paper will lead to great improvements in single-photon sources in the near future. 5.1.2 Single-photon probability CUSUM is a simple yet powerful method that can signicantly improve single-photon probability using continuous monitoring [37]. The protocol itself requires only simple components like an integrator and subtractor, along with 2 registers and a compara- tor. This technique is useful in the weak coupling regime and when the pumping rate is comparable to coupling strength. In regions of strong coupling, or strong pumping, it is ineective and can be worse than no monitoring at all. We modeled various deco- herence processes including spontaneous emission, dephasing and cavity leakage. We modeled the electrical pumping and continuous monitoring as a stochastic master equa- tion. Numerical simulations showed that CUSUM performs quite well in regions of low , and signicant improvements single-photon probability were observed. We also studied the eect of imperfect monitoring eciency on performance. 5.2 Single-spin measurement using MRFM In [38], we modeled the OSCAR MRFM operation including decoherence eects on the cantilever (both thermal and monitoring), as well as spin noise due to magnetic sources. We describe the evolution of the system using a quantum stochastic master equation, the most general description in a Markovian framework. We then simplied this description using a series of approximations, and arrived at a semi-classical description of the system based on a Gaussian approximation of the cantilever state. We numerically compared the Gaussian approximation to the fully quantum stochas- tic master equation, and found that the Gaussian approximation tracked the fully quan- tum stochastic master equation very closely. Thus, we conclude that the Gaussian approximation of the cantilever is useful for our range of parameter values, and that 76 the MRFM system implementing the OSCAR protocol can be described by a closed set of 11 coupled stochastic dierential equations. Finally, we used the Gaussian approximation equations to numerically verify OSCAR as a useful single-spin measurement protocol. The critical element in the OSCAR proto- col is the time (sampling) it takes to determine the shift in resonant cantilever frequency. For the parameter values chosen in our simulations, we found that the sampling duration is around 650 ms, and the frequency shift is about 30 Hz. The sampling duration thus places a bound on the time scale of spin noise that the OSCAR protocol, implementing a single-spin measurement, can tolerate. Many engineering solutions in the classical realm use continuous monitoring and control as a fundamental component: from something as simple as controlling an elevator or a thermostat, to very complex operations such as building a micro-processor or the navigation system of an airplane. For quantum systems to be built in the real world, and for quantum computing to be a reality, the study of physical systems is but a rst step. To better build and control quantum systems, a deeper understanding of continuous monitoring in physical quantum systems is essential|be it modeling, analysis or numerical simulations. I hope this work is a step in that direction. 77 Bibliography [1] T.R. Albrecht, P. 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Asset Metadata

Creator Raghunathan, Shesha Shayee K. (author)

Core Title Topics in modeling, analysis and simulation of near-term quantum physical systems with continuous monitoring

Contributor Electronically uploaded by the author (provenance)

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/16/2010

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

Tag continuous monitoring,Gaussian approximation,indistinguishability,magnetic resonance force microscopy,OAI-PMH Harvest,quantum continuous measurements,quantum trajectories,single-photon probability,single-photon source,single-spin measurement,spin-relaxation,stochastic master equation

Language English

Advisor Brun, Todd A. (committee chair), Haas, Stephan (committee member), Lidar, Daniel (committee member)

Creator Email shesha@gmail.com,sraghuna@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3529

Unique identifier UC1186673

Identifier etd-Raghunathan-4212 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-397852 (legacy record id),usctheses-m3529 (legacy record id)

Legacy Identifier etd-Raghunathan-4212.pdf

Dmrecord 397852

Document Type Dissertation

Rights Raghunathan, Shesha Shayee K.

Type texts

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

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email uscdl@usc.edu

Abstract (if available)

Abstract In this work, we study two important problems in the quantum world: single-photon source and single-spin measurement.