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Topological protection of quantum coherence in a dissipative, disordered environment
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Topological protection of quantum coherence in a dissipative, disordered environment
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Topological Protection of Quantum Coherence in a Dissipative, Disordered Environment Zhengzhi Ma A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) DECEMBER 2019 Topological Protection of Quantum Coherence in a Dissipative, Disordered Environment Author: Zhengzhi Ma Thesis Advisor: Stephan Haas Abstract One dimensional topological insulators are characterized by edge states with exponentially small energies. According to one generalization of topo- logical phase to non-Hermitian systems, a nite system in a non-trivial topo- logical phase displays surface states with exponentially long life times. In the rst half of this thesis we explore the possibility of exploiting such non-Hermitian topological phases to enhance the quantum coherence of a ducial qubit embedded in a dissipative environment. We rst show that a network of qubits interacting with lossy cavities can be represented, in a suitable super-one-particle sector, by a non-Hermitian \Hamiltonian" of the desired form. We then study, both analytically and numerically, one- dimensional geometries with up to three sites per unit cell, and up to a topological winding number W = 2. For nite-size systems the number of edge modes is a complicated function ofW and the system sizeN. However we nd that there are preciselyW modes localized at one end of the chain. In such topological phases the quibt's coherence lifetime is exponentially large in the system size. We verify that, forW > 1, at large times, the Lindbladian evolution is approximately a non-trivial unitary. For W = 2 this results in Rabi-like oscillations of the qubit's coherence measure. The second half the this thesis, we investigate the robustness of the topo- logical protection of coherence against disorder. A projector formalism is in- troduced to study open quantum many-body systems with non-trivial topol- ogy. Within this approach, the spectral composition with complex eigenval- ues of the non-Hermitian Lindblad operator is determined, and a generalized ii open-system density of states is obtained. The local density of states is then used to examine the edge states of the system, indicating its topological state. This method is applied to construct the phase diagram of the dissipative Su-Schrieer-Heeger model in the presence of diagonal disorder. It is shown that in the topologically non-trivial sector, the characteristic edge modes survive up to a threshold level beyond which the system is Anderson local- ized, whereas in the topologically trivial sector Anderson localization sets in immediately upon introducing disorder. A nite-size scaling analysis reveals that the sharp topological phase transition driven by the dimerization in the pure case gives way to a crossover in the presence of disorder. iii Acknowledgements I would like to give special thanks to my research advisor, Professor Stephan Haas. Stephan is extremely helpful and responsive in advising my research work. He is patient with me, always keeping an open mind with my pro- posals, and commending me even on the smallest accomplishments. He does not hesitate to give me counsel, not only on my research work, but also on the challenging quest of pursuing a doctorate degree in general. Most impor- tantly to me, Stephan has the warmest personality. He has created a friendly work environment, and truly respects me and cares about me. I want to thank our collaborator, Professor Lorenzo Campos Venuti. He is meticulous with every single calculation and every sentence in our manuscript. He inspires us to ask the hard questions, to strive for the most complete result, and to forge the most convincing narrative. His boyish de- meanor also gives us a sense of lightness in the sometime dull academic life. I thank every Professor who has taught me, counseled me, and encouraged me during my years at the physics department. I also thank my classmates who have studied, lived and played with me. I am thankful that at the age when most people are faced with the cold realities of the world, I have mentors and friends who still have passion for knowledge and comradeship. iv Contents Chapter 1 Introduction 1 Chapter 2 Topological Protection of Coherence in a Dis- sipative Environment 4 2.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Mapping to a non-Hermitian tight-binding model . . . . . . . 6 2.3 Single impurity . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Topological classication of dissipative systems . . . . . . . . . 11 2.5 Non-Hermitian SSH model . . . . . . . . . . . . . . . . . . . . 13 2.5.1 N odd . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.2 N even . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Three-site unit cell . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 Eect of Noise on Coherence Decay . . . . . . . . . . . . . . . 22 2.7.1 Noise Eect on the Non-Hermitian SSH Model . . . . . 23 2.7.2 Noise Eect on the Three-Site Unit Cell System . . . . 24 Chapter 3 Projector Method: Phase Diagram of the Dis- ordered Dissipative Su-Schrieer-Heeger Model 28 3.1 Projector Formalism . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 Projector Formulation . . . . . . . . . . . . . . . . . . 30 3.1.3 Density Operator . . . . . . . . . . . . . . . . . . . . . 31 3.1.4 Bin Counting Method . . . . . . . . . . . . . . . . . . 33 3.1.5 Number of Edge Modes . . . . . . . . . . . . . . . . . 34 3.2 Disordered non-Hermitian SSH Model . . . . . . . . . . . . . . 35 3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . 35 3.2.2 Distribution of Eigenvalues . . . . . . . . . . . . . . . . 36 3.2.3 Construction of Topological Phase Diagram via Edge Mode Counting . . . . . . . . . . . . . . . . . . . . . . 41 v 3.2.4 Interpretation of the Phase Diagram . . . . . . . . . . 45 3.2.5 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . 46 Chapter 4 Conclusions 48 Bibliography 50 Appendix A \Single impurity" case 53 Appendix B A note on timescales 55 Appendix C Conditions to optimize the coherence decay 57 vi Chapter 1 Introduction Topological features are potentially useful, as they tend to be robust with respect to small perturbations and local noise sources. In particular, consider a one-dimensional Hermitian topological insulator such as the Su-Shrieer- Heeger (SSH) model, H =J 1 N X m1 (jm;Bihm;Aj + h:c:) +J 2 N1 X m=1 (jm + 1;Aihm;Bj + h:c:); (1.1) which is a dimerized tight-binding model consisting with sites A and B with alternating hopping potentials J 1 and J 2 , where J 1 denotes the hop- ping within the unit cells, and J 2 denotes the the hopping between adjacent unit cells. If J 1 < J 2 , this model permits surface states that are protected by the translational symmetry of the lattice. The surface states are not only localized at the edge of the lattice, but also have exponentially small energies, making it robust against time evolution. Imagine we assemble the SSH model with optical cavities, and couple a qubit to its edge states, then the symmetry of this lattice chain will protect the coherence of the qubit even if the cavities are dissipative. There is a growing interest in the study of non-Hermitian gen- eralizations of topological phases of matter [21, 22, 8, 6, 1, 28, 18, 20, 23, 16] which can be observed in dissipative systems. In the rst part of this thesis we explore the possibility of exploiting such, non-trivial, non-Hermitian topo- logical phases to protect the coherence of a preferential qubit in a network of dissipative cavities. Since eigenvalues of non-Hermitian matrices are complex there are at least two possible denitions of topological phases in non-Hermitian systems 1 [8, 23]. These denitions dier in how one generalizes the Hermitian notion of gap: namely one can consider either the real or the imaginary part of the eigenvalues. According to the imaginary-part classication of Ref. [23], as a consequence of a generalized bulk-edge correspondence, a non-trivial topological dissipative phase is characterized at nite size by the presence of quasi-dark states localized at the boundary of the system. By quasi-dark states we mean eigenstates of the system that have a decay time exponentially large in the system size, just like their counterparts in the Hermitian case. It is natural to expect that this feature may be useful to protect quantum coherence. Indeed, as we will show, if a ducial qubit is placed at one end of a linear system, both these features, localization and darkness, conspire to preserve its coherence in a well dened way. In recent experiments such non-Hermitian systems { in fact essentially non-Hermitian quantum walks { can be observed in classical waveguides using the analogy between Helmoltz and Schr odinger equation [28]. In Ref. [21] it was proposed that a non-Hermitian version of the Su-Schrieer-Heeger (SSH) model [25] could emerge from a single resonator described by a Jaynes- Cummings model in the semi-classical, large-photon number regime. Here we consider a network of dissipative cavity resonators interacting a- la Jaynes-Cummings. This model is known to describe the physics of many experimental quantum platforms, ranging from superconducting qubits to arrays of microcavities [12]. In such dissipative, non-Hermitian system, we need to consider the density matrix, instead of the states. The time evolution of the density matrix is given by the Lindblad master equation _ =i [H;] + N 2 1 X l=1 l a l a y l 1 2 n a y l a l ; o ; (1.2) where N is the dimension of the original Hilbert space, H is the Hamilto- nian of the system. The rst term is the usual Hermitian time evolution of the system, while the second term denotes the level transitions, therefore the non-Hermitian time evolution of the system. We show that, in an ap- propriate super-one-particle sector, the Lindbladian is precisely given by a non-Hermitian quantum walk determined by the network geometry. More- over, the coherence of a preferential qubit in the network is exactly described by the Schr 0dinger evolution with such a \non-Hermitian Hamiltonian". Having in mind the goal of prolonging the coherence, we analyze ana- lytically, and conrm numerically, the behavior of the coherence for various 2 nite size networks. The simplest of such a networks is a non-Hermitian tight-binding model with a single, both diagonal and o-diagonal, impurity. We then consider topologically non-trivial models, such as a non-Hermitian SSH model, that can have topological charge zero or one. In nite size, there are always two dark modes for N odd while there is one quasi-dark mode in the topologically non-trivial sector for N even. However there is always (irrespective of N) a dark or quasi-dark mode localized at one end of the chain. An analogous situation is found in models with three sites per unit cell, were the topological winding number W can be zero, one or two. The exact number of quasi-dark modes is not a simple function ofW alone. How- ever we nd precisely W dark or quasi-dark modes localized at one end of the chain. In the caseW = 2, the long-time dynamics of the dissipative net- work becomes unitary, spanning a two-dimensional space were the coherence shows Rabi-like oscillations. One characteristic of symmetry protected topological phases is that they are robust against small perturbations and local noise sources[10, 11, 14]. In the second half of this thesis, we carefully study the consequences of diagonal disorder on the topological edge states of the dissipative Su-Shrieer-Heeger (SSH) model. Using the number of edge states, we dene the boundary between trivial and non-trivial topological phases, which also signies the threshold of eective topological prolongation of the coherence time. We observe that up to a noise rate comparable to the tunneling strength be- tween the cavities, the edge states of the system are still protected, but the transition between the dierent topological sectors is not sharp anymore. 3 Chapter 2 Topological Protection of Coherence in a Dissipative Environment 2.1 Setting the stage Our model is a network of dissipative cavities (modes) interacting with two- level systems (qubit) in a Jaynes-Cummings fashion. To make it more gen- eral, we allow qubits to interact with more than one cavity, although this may be experimentally challenging to realize. We imagine a network of M qubits interacting with K cavity modes. Excitations can hop from mode to mode and also from qubit to mode. At this stage we don't include hopping from qubit to qubit, as this is denitely harder to realize. Our goal will be to monitor, and possibly enhance, the coherence of a ducial qubit in this network. We assume the standard rotating-wave approximation, such that the co- herent part of the evolution is given by the following Hamiltonian: H = M X i=1 ! 0 i z i + K X l;m=1 J l;m (a y l a m + h:c:) (2.1) + K X l=1 ! l a y l a l + M X i=1 K X l=1 l;i (a y l i + h:c:); (2.2) wherea y l anda l are the creation and annihilation operators for the cavity 4 mode l and i are the ladder operators for qubit i. On top of this, cavities leak photons at rate l . A Lindblad master equation for the system can be written as _ =L[] withL =K +D. The coherent term isK =i [H;] and the dissipative part reads D[] = K X l=1 l [a l a y l 1 2 fa y l a l ;g]; (2.3) i.e., we assume suciently low temperatures such that no photons are excited via interaction with the bath. This form of the dissipation is con- sistent with the cavity physics whereby essentially only the cavity modes decay whereas the two level-systems (corresponding to some hyperne level of an atom in the cavity) are extremely long-lived and decay only indirectly through interaction with the cavity. An example of such a dissipative net- work withM = 4 andK = 5 is schematically depicted in Fig. 2.1. Let i = 1 indicate the ducial qubit. In order to study the evolution of the qubit's coherence, we initialize it in a pure state 0 j"i + 0 j#i, while we require that all cavities be empty and all other qubits in thej#i state. We denote withj0i the overall vacuum (cavities with no photons and qubits in thej#i state) andjji, j = 1;:::;N M +K the state with an excitation, either bosonic or spin-like, at positionj, withj = 1 denoting the ducial qubit and j = 2; 3;:::;N the remaining cavities/qubits. With this initial condition the relevant Hilbert space isH = Spanfj0i;jji;j = 1;:::;Ng, and the dynamics are restricted to the spaceV =L(H). A density matrix inV has the form = 0;0 j0ih0j + N X j=1 0;j j0ihjj + h:c: ! (2.4) + N X i;j=1 i;j jiihjj: (2.5) After tracing out all but the ducial qubit degrees of freedom, the reduced qubit density matrix reads qubit = 0;0 + N X i=2 i;i ! j#ih#j + ( 0;1 j#ih"j + h:c:) + 1;1 j"ih"j: (2.6) 5 A coherence measure of the qubit can be dened as [2] C(t) = X i;j(i6=j) qubit i;j (t) : (2.7) Using equation (2.6) we obtainC = 2j 0;1 j. Figure 2.1: A general network of qubits interacting with lossy cavities. Wavy lines indicate coherent hopping and straight arrows incoherent decay. White dots represent (leaky) cavities while black dots are (long-lived) two-level sys- tems (qubit). 2.2 Mapping to a non-Hermitian tight-binding model If we initialize the system with at most one excitation, the Lindbladian generates states with at most one excitation and the dynamics are con- tained in the sectorV. We are then led to consider the following linear spaces V 0;0 = Span (j0ih0j), V 0;1 = Span (fj0ihjj;j = 1;:::;Ng), V 1;0 = Span (fjjih0j;j = 1;:::;Ng) andV 1;1 = Span (fjiihjj; i;j = 1;:::;Ng). The 6 Hamiltonian conserves the number of excitations so the coherent partK is block diagonal in the reduced spaceV =V 0;0 V 0;1 V 1;0 V 1;1 . Moreover D(j0ih0j) = 0 (2.8) D(j0ihjj) = j 2 j0ihjj (2.9) D(jiihjj) = i i;j j0ih0j 1 2 ( i + j )jiihjj: (2.10) Note that i = 0 fori = qubit site, as we are ignoring the spontaneous decay of the qubits (typically much smaller than cavity loss rate). This implies that onV the Lindbladian has the following block-structure (asterisks denote the only non-zero elements) inV =V 0;0 V 0;1 V 1;0 V 1;1 Lj V = 0 B B B B B B B B B B @ 0 1 C C C C C C C C C C A : (2.11) We also call ~ L = Lj V 0;1 the restriction ofL toV 0;1 and, in this basis, one hasLj V 1;0 = ~ L (overline indicates complex conjugate). Clearly the vacuum j0ih0j is a steady state (with eigenvalue zero). We use the following notation for the Hilbert-Schmidt scalar product inV: hhxjyii = Tr(x y y) and use the identicationjjii$j0ihjj for j = 1;:::;N which denes a basis ofV 0;1 . According to Eq. (2.7) we need the matrix element [(t)] 0;1 =h0j(t)j1i = hh1j(t)ii. Because of the block-structure of the Lindbladian one obtains [(t)] 0;1 =hh1je tL j(0)ii =hh1je t ~ L j~ (0)ii;where we indicated with ~ (0) the pro- jection of (0) toV 0;1 according to the above direct sum decomposition of V. Note that if the qubit is initialized in the state 0 j"i + 0 j#i, we have ~ (0) = 0 0 j0ih1j or equivalentlyj~ (0)ii = 0 0 j1ii. In the following we will always consider 0 0 = 1=2, i.e. maximal initial coherence, such that C(t) = hh1je t ~ L j1ii : (2.12) As usual we can identifyV 0;1 'C N , and the Hilbert-Schmidt scalar product carries over to the` 2 scalar product. We also use the the normkxk = p hhxjxii 7 for x2V 0;1 and the induced norm for elements of L(V 0;1 ). Since the basis jjii is orthonormal, Hilbert-Schimdt adjoint simply corresponds to transpo- sition and complex conjugation in this basis. With these identications the setting resembles that of standard one-particle quantum mechanics, with the important dierence that operators are not Hermitian. For example, for the case of a single qubit, M = 1, interacting with a single cavity and cavities connected on a linear geometry J i = J i;i+1 (see Figure 2.2 for a schematic picture), the matrix ~ L becomes ~ L =i 0 B B B B B @ ! 0 1 0 0 ! 1 i 1 2 J 1 0 0 J 1 ! 2 i 2 2 0 . . . . . . . . . . . . J K 0 0 0 J K ! K i K 2 1 C C C C C A iH; (2.13) where we also dened the matrix H which is a non-Hermitian generalization of a tight-binding chain. Remark. The ` 2 scalar product (and corresponding norm) inV 0;1 is natural in that, via Hilbert-Schmidt, allows to move from Schr 0dinger to Heisenberg representation. However in this setting, the ` 2 moduli square are not probabilities. Conservation of quantum-mechanical probabilities is enforced by the complete positivity and trace preserving property of the full map e tL for t 0. Trace conservation in turn implieshh1 IjL = 0, wherehh1 Ij corresponds to the identity operator on the Hilbert spaceV. This property, however, does not carry over to the restricted generator ~ L. What can still be said is that the eigenvalues of ~ L, since they are a subset of those ofL, fulll Re() 0. In generalC(t) will decay in time starting form its maximum value 1 at t = 0. From Eq. (2.12) we realize that our goal is to make a particular matrix element of the restricted evolution e t ~ L , have large absolute value for possibly large times. In fact, ideally we would like: i) ~ Lj1ii = 1 j1ii and ii) Re( 1 ) = 0. Both of these conditions can be trivially achieved simply setting l;1 = 0,8l. However this entirely decouples the qubit from the rest of the network which means one does not have a way to address the qubit anymore - in fact experimenters generally try to increase the qubit-mode coupling. In view of this we replace the two conditions above with the more physical requirements, i') ~ Lj1ii 1 j1ii and ii') Re( 1 ) as small as possible. 8 Condition ii') (that there exist an eigenvalue of ~ L with almost zero real part) resembles the condition for having an approximate zero mode familiar in (Hermitian) topological insulators. More generally, in a linear geometry, a way to fulll conditions i') and ii') is to nd, approximate, non-Hermitian, topological zero mode of ~ L. Non-Hermitian generalization of topological insulators have been studied to some extent (see e.g., [21, 8, 28, 17]). In particular we will be concerned with nite size systems which have not been discussed in the literature so-far. Before turning to topological models let us rst consider what seems to be the simplest geometry. 2.3 Single impurity Figure 2.2: The \single impurity model": a qubit in a cavity connected to a linear array of cavities. The simplest case is that of linear geometry with a single impurity (see Fig. (2.2)), i.e. we setJ i =J, i = and also! i =! 0 1 for alli (no detuning) in Eq. (2.13): H = 0 B B B B B @ 0 0 0 i 2 J 0 0 J i 2 0 . . . . . . . . . . . . J 0 0 0 J i 2 1 C C C C C A ; (2.14) where H has been transformed to the rotating frame of frequency ! 0 1 . This is a non-Hermitian generalization of a single impurity in a tight binding chain [7]. For N = 3 this model has been investigated in [24, 19], where it was established that adding one auxiliary cavity to a dissipative optical cavity coupled to a qubit can signicantly increase the coherence time of the qubit. An equation for the eigenvalues can be found using the techniques to diagonalize tridiagonal matrices. The eigenvalues of the matrix (2.14) ~ L 9 can be written as k =i2J cos(k) =2, where k is a (possibly complex) quasi-momentum that satises the following equation [2 cos(k) +ia] sin(kN) 2 sin(k(N 1)) = 0; (2.15) where a = =(2J), = =J. In order to look for a localized state we look for a solution of the above equation with complex k = x +iy. Essentially the localization length is given by = y 1 : More details are provided in Appendix A. Neglecting terms of order O e Njyj the eigenvalues of ~ L of such localized modes are given by = 4 2 p 16(J 2 2 ) + 2 +O e Njyj : (2.16) This formula is valid in regions where Re( )< 0. Because the wave vector k is complex, a plane wave trial solution will decay like e yn = e n= which denes the localization length. In such cases the localization length is given by = 1= ln 4J p 16(J 2 2 ) + 2 : (2.17) For= small (strong dissipative regime), using a perturbative argument (more details in Appendix C), one can show that the coherence has approx- imately the form of a single exponential decay e t= 0 , with 1 0 = 2 2 =. Using Eq. (2.16) the eigenvalue connected with 1 0 is + . By continuity, we can now we can use the expression for the localized mode outside from the strict perturbative region. In other words we have C(t)e t= (2.18) = Re h + p 16(J 2 2 ) + 2 4 2 i : (2.19) The above equations are extremely accurate in the region of small but sur- prisingly are quite accurate also for large. Increasing one starts observing non-Markovian oscillations 1 in the coherence also noted in [19] at an energy scale of the order of J 2 + 2 =16 (when the square root term in Eq. (2.19) becomes imaginary). In this regime Eq. (2.18) describes well the envelope of the coherence. See Fig. 2.3 for comparisons with numerics. 1 Obviously a Lindblad master equation that ignores non-Markovian eects between the system and the bath is perfectly able to encompass non-Markovian features between a qubit and the rest of the system. This should be no source of confusion. 10 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 Figure 2.3: (Color online) Behavior of the coherence for the single impurity model. Here and in the following we compute Eq. (2.12) by numerical di- agonalization of the corresponding reduced Lindbladian. Continuous lines are numerical simulation and dashed lines are analytical approximation of Eqns. (2.18-2.19). Dissipation is xed to = 4J. The results for N = 4 are indistinguishable from those at N = 400. 2.4 Topological classication of dissipative sys- tems We recall here for completeness the basics of the topological classication of models of Ref. [23] (see also [8]). Since eigenvalues are now complex, there are at least two ways to generalize this notion to the non-Hermitian world. Namely one may extend the role played by the Hermitian gap to either the imaginary or the real part of the eigenvalues. Two points in parameter space are dened to be in the same phase if the corresponding (non-Hermitian) Hamiltonians can be smoothly connected without closing the imaginary (resp. real) part of the eigenvalues. For the \imaginary-gap" classication of Ref. [23], according to a generalized bulk-edge correspon- dence, a non-trivial phase at nite size would have edge modes with innite or exponentially large life-time. Clearly this is the relevant classication in our context. We assume a periodic linear chain with n sites per unit cell such that, in the thermodynamic limit, the Hamiltonian is given by 11 H = I dk=(2) X ; H ; (k)jk;iihhk;j; (2.20) and we simply need to focus on the n n Bloch matrix H(k). The dissipation has the special form shown in Sec. 2.2 which consists of imaginary terms on the diagonal (of negative imaginary part). Without constraint such models are topologically trivial if the number of leaky sites per cell is greater than one [23]. We then focus on the case where there is only one leaky site per cell. As shown in [23], any such H(k) that does not admit a dark state can be written in the following way H(k) = U(k) 0 0 1 ~ h(k) ~ v k ~ v y k (k)i U(k) y 0 0 1 ; (2.21) where ~ h(k) is an (n1)(n1) diagonal matrix with real eigenvalues,U(k) is an (n 1) (n 1) unitary matrix that diagonalizes ~ h(k) and also makes the (n 1) dimensional vector ~ v k real and positive. Any U(k) satisfying the above criteria can be chosen without aecting the following result. In Ref. [23] it is further shown that the winding number of H then reduces to the winding number of U(k) which is given by W = I dk 2i @ k ln det(U(k)): (2.22) From what we have said, in a non-trivial topological phase, at nite size one expects to observe dark states localized at the edges. Such a dark (or quasi-dark) statejii fullls ~ Ljii = jii with Re()' 0. However, given the structure of the spaceV 0;1 all such states are e.g. traceless. Hence these are not strictly quantum states, they are in fact o-diagonal elements of a quantum state. In the quantum-chemistry community these are sometimes called coherences. We would like to conclude this section by reminding a general result for completely positive maps/semigroups. We assume here nite dimensionality. Let the Jordan decomposition ofL beL = P k k P k +D where D is the nilpotent part. Dene the projector onto the dark states sector as P ds = X k;Re( k )=0 P k : (2.23) 12 Decomposing the Liouville space as 1 I = P ds (1 I P ds ) one has e tL = W t R t whereW t is the part of the evolution inside the dark-state sector: W t = P ds W t =W t P ds and the remaining termR t can be made as small as one wishes in norm, by taking largert. It can be shown (see Theorem 6.16 of [27]) thatW t is a unitary evolution, more preciselyW t [ 0 ] = U t ~ 0 U y t where the state ~ 0 is partly determined by the initial state 0 . In other words, the time evolution inside the dark state sector is unitary. 2.5 Non-Hermitian SSH model To start we consider the model given by the following non-Hermitian gen- eralization of the SSH Hamiltonian (for simplicity we rename all hopping constants J i both for qubit-mode and mode-mode hopping) H = 0 B B B B B @ 0 J 1 0 0 0 J 1 i J 2 0 0 0 J 2 0 J 1 0 0 0 J 1 i . . . 0 0 0 . . . . . . 1 C C C C C A : (2.24) One may obtain an intuitive understanding of the model by considering the periodic boundary conditions version of the above. In that case it suces to consider the 2 2 Bloch Hamiltonian H(k) = 0 v k v k i ; (2.25) with v k = J 1 +J 2 e ik . Model (2.25) is, up to a constant term, pseudo-anti- Hermitian, in that ~ H(k) := H(k) +i(=2)1 I satises z h ~ H(k) i y z = ~ H(k). Moreover ~ H(k) is a linear combination of the matricesf x ; y ;i z g which span the Lie algebra ofSU(1; 1) (S(1; 1) in turn is the group of 22 complex matrices U satisfying U y z U = z and det(U) = 1). Model (2.25) is then also referred to as SU(1; 1) model [8]. The more familiar, Hermitian, SSH model being a SU(2) model. 13 The eigenvalues of Eq. (2.25) are simply k; =i 2 r jv k j 2 2 4 (2.26) =i 2 r J 2 1 +J 2 2 + 2J 1 J 2 cos(k) 2 4 ; (2.27) with momenta given byk = 4n=N (N even). For example, if 2 =4<v 2 min (J 2 1 +J 2 2 2jJ 1 J 2 j), the square root term above is real and all the modes decay at a rate =2. This model admits a topological phase characterized by a winding number according to the \imaginary gap" classication of [23]. The winding number W Eq. ((2.22)) turns out to be analogous to that of the Hermitian SSH model, and it simply counts the number of times the vector J 1 +J 2 e ik winds around the origin as k moves around the Brillouin zone [0; 2). Consequently W = 1 forjJ 2 j>jJ 1 j while W = 0 forjJ 2 j<jJ 1 j 2 . This picture gets modied for an open chain. Most importantly, as a con- sequence of the topological character of the model and the so-called bulk-edge correspondence, there will appear edge state(s) localized at the boundary of the chain. The calculations are dierent depending on whether N is even or odd. We x the geometry by xing the dissipation to act only on the even sites as in Eq. (3.29). 2.5.1 N odd Figure 2.4: Non-Hermitian SSH model Eq. (3.29) for N odd. For N odd the conguration of the bonds is given in Fig. 2.4. For N odd there is always one edge state irrespective of the values of J 1 ;J 2 . In this case the edge-mode has exactly zero eigenvalue i.e., is a dark state. The edge 2 It turns out that this model is topological also according to the (real-gap) classication proposed in [8]. More precisely the shifted matrix ~ H(k) has exactly the same (real-gap) classication [8]. 14 mode is localized at the site where the weak link is (whether it is J 1 or J 2 ). Clearly the transition is at J 1 =J 2 . If J 1 is the weak link we can write such an edge mode as j L ii =A 0 B B B B B B B B B @ e ik 0 e 3ik 0 e 5ik . . . e Nik 1 C C C C C C C C C A (2.28) where A is a normalization factor. One nds that Hj L ii = 0 provided J 1 + J 2 e 2ik = 0. Under this conditionj L ii is a dark state. From this equation we see that jhhnj L iij 2 =A 2 e n for n odd, where ln(jJ 2 =J 1 j)> 0 was assumed to be positive. Hence we call ` 1= ln(jJ 2 =J 1 j) the localization length of the edge mode. Fixing the normalization one nds A 2 = 1x 2 xx N+2 ; (2.29) with x =jJ 1 =J 2 j< 1. The casejJ 2 j<jJ 1 j can be reduced to the previous one by a left-right sym- metry transformation. Under this transformation the dark state is mapped ontoj R ii which is localized at the opposite end of the chain. Recalling the result for the periodic case one sees that, in general, the other, non-localized, modes decay on a relaxation time-scale given by relax 1 O(1). Coming to the behavior of the coherence we see that, after a time relax all but the modej L ii will have decayed. Hence the coherence, for t> relax , is approximately given by C(t) = X k e k t hh1jP k 1ii jhh1j L iihh L j1iij =jhh L j1iij 2 = 1x 2 1x N+1 : (2.30) Note that, since x< 1, this is a decreasing function of N. The largest value with N > 1, odd, is obtained for N = 3. 15 ForjJ 2 j<jJ 1 j the role ofj L ii andj R ii are reversed. Hence now the dark state is localized at the end of the chain. After a time relax the coherence drops to a valueC(t)'jhh R j1iij 2 =jhh L jNiij 2 = z N1 (1z 2 )(1z N+1 ) 1 , where z is now z = jJ 2 =J 1 j, i.e. an exponentially small value. The two asymptotic expressions are in fact the same and can be combined in a single expression valid for all J 1 ;J 2 C(t)' ( J N1 2 J 2 2 J 2 1 J N+1 2 J N+1 1 J 1 6=J 2 2 N+1 J 1 =J 2 : (2.31) To summarize, forN odd there is always an exact localized dark state for all values of parameters and consequently an innite lifetime of the coher- ence's qubit. However, in the topologically trivial phase W = 0 (jJ 1 j>jJ 2 j) the edge mode is localized at the opposite end of the chain, and the asymp- totic value of the coherence is exponentially small. The numerical simulations conrm that a non-trivial topological winding number has a strong eect on the coherence time of the qubit, as illustrated on Fig. 2.5. To connect with the previous discussion we see that, in general we satisfy the requirement ii') (there is an eigenmode with Re() = 0), but not nec- essarily i') . In other words, in generalj L iihh L j is not close toj1iihh1j. We progressively enter this regime when the localization length becomes very short (or very large). Clearly this happens whenjJ 2 jjJ 1 j. 2.5.2 N even ForN is even the conguration of the links is depicted in Fig. 2.6. When N is even,j L ii of Eq. ((2.28)) does not satisfy the last row of the eigenvalue equation but rather one has Hj L ii =J 1 e ik(N1) jNii. This is consistent with our expectation of an exponentially small eigenvalue. The exact diagonal- ization of the model can be found in [15] (see also [9, 3]). For N even edge modes appear for dJ 2 =J 1 > 1 + 2=N. This is an interesting eect as one can in principle enter the topologically non-trivial phase for xed values of the parameters by only changing N. The eigenvalues of the edge modes are given by [15] =i 2 r J 2 1 +J 2 2 + 2J 1 J 2 cosh(y) 2 4 (2.32) 16 � �� �� �� �� ��� ��� ��� ��� ��� ��� Figure 2.5: (Color online) Behavior of the coherence in the non-Hermitian SSH model with an odd number of sites. Continuous lines are results in the topological phase (W = 1) with parameters J 1 = 1; J 2 = 1:8 and = 0:5. Dashed lines are for the topologically trivial phase (W = 0 , J 1 = 1; J 2 = 0:5 = 0:5). The thin dashed lines are the asymptotic values given by Eq. (2.31). The qubit has innite lifetime for all values of parameters, but the asymptotic coherence is exponentially small in the topologically trivial region. The intrinsic coherence lifetime of the qubit (N = 2) is added for comparison. We observe that the lattice of cavities with W = 1 vastly improves the lifetime of the coherence. where y satises sinh( N 2 y) =x sinh ( N 2 + 1)y : (2.33) For N large the solution of Eq. (2.33) approaches e y = d. Up to rst order in d N one obtains that the solution of Eq. (2.33) is e y =d +d N d 1 d +O(d 2N ): (2.34) Plugging the above into Eq. (2.33) one nds + =i J 2 1 d N d 1 d 2 (2.35) =i +i J 2 1 d N d 1 d 2 : (2.36) 17 Figure 2.6: Non-Hermitian SSH model Eq. (3.29) for N even. The + eigenvalue corresponds to the mode localized at the rst site of the chain. Moreover, even if there are two localized modes, only one of them has exponentially large life-time in the system size. So for N even the the left edge mode has a coherence time of coh = J 2 1 d N (d 1 d) 2 . The eigenvalue corresponds to edge mode localized at the end of the chain, with fastest decay time. In order to compute the coherence we need the rst component of the edge modej + ii. It turns out that (see [15]) jhh1j + iij 2 = 4 sinh 2 (Ny=2) h sinh[(N+1)y] sinh(y) (N + 1) i + +i 2 + +i : (2.37) Since + is exponentially small, the last fraction is exponentially close to 1 and can be evaluated up to d N using Eq. (2.35). For the remaining terms we plug in the asymptotic value y = ln(d) and obtain jhh1j + iij 2 = 1 + J 2 1 2 x N (xx 1 ) 2 (1x 2 ) 1 x N (N + 1) +O(x 2N ) = 1x 2 +x N 1x 2 2 (N + 1) 1x 2 x 2 J 2 1 2 +O(x 2N ) (2.38) In this case the statej + ii is not an exact dark state and it will start decaying at a time around coh . As for the odd case, the other states decay after a time relax = 1 O(1). Hence, whenever there is a separation of time- scales coh > relax , one will observe a coherence ofC(t)jhh1j + iij 2 for times roughly in the window t2 [ relax ; coh ]. Numerical experiments for the even case are shown in Fig. 2.7. In table 2.1 we show comparisons of the numerics with the analytic expressions. For comparison, theW = 0 case is also shown in Fig. 2.7, where the coherence is from bulk modes only, and the decay is given by the bulk relaxation time 1 . 18 � �� �� �� �� ��� ��� ��� ��� ��� ��� Figure 2.7: (Color online) Behavior of the coherence in the non-Hermitian SSH model with an even number of sites. Continuous lines are results in the topological phase (W = 1) with parameters J 1 = 1; J 2 = 1:8 and = 0:5. IncreasingN has the eect of exponentially increasing the (coherence) time- scale coh at which the approximate dark state starts decaying. Dashed lines are for the topologically trivial phase (W = 0 , J 1 = 1; J 2 = 0:5 = 0:5). For N = 10; 20 the plot is indistinguishable from that of N = 8. The thin dashed lines is the asymptotic value given by Eq. (2.38). The intrinsic coherence lifetime of the qubit (N = 2) is added for comparison. We observe that the lattice of cavities with W = 1 vastly improves the lifetime of the coherence. 2.6 Three-site unit cell We now turn to a case where the unit cell consists of three sites. According to the prescription of Ref. [23] for the existence of a topological phase we consider only one leaking site per cell. We allow for nearest neighbor hopping and also between the rst and third site in the cell (see Fig. (2.8)). As we will see, this geometry will allow us to have topological number of 0; 1 and 19 N coh jhh + j1iij 2 Exact Theory Exact Theory 6 6.9367 10.9813 0.5355 0.6638 8 31.8117 35.5794 0.6715 0.6915 10 111.1859 115.2774 0.6888 0.6941 20 4:1153 10 4 4:1159 10 4 0.6914 0.6914 Table 2.1: Comparison of exact numerics with the approximate theoretical formulae. Parameters are J 1 = 1; J 2 = 1:8 and = 0:5. Figure 2.8: Model (2.39) with a three-site unit cell. 2. The Hamiltonian is H = X x J 1 jx; 1iihhx; 2j +J 2 jx; 2iihhx; 3j +J 3 jx; 3iihhx + 1; 1j +Jjx; 1iihhx; 3j + h:c: + X x 1 jx; 1iihhx; 1j + 2 jx; 2iihhx; 2jijx; 3iihhx; 3j : (2.39) For periodic boundary conditions the corresponding Bloch Hamiltonian reads H(k) = 0 @ 1 J 1 J 3 e ik +J J 1 2 J 2 J 3 e ik +J J 2 i 1 A : Using Eq. (2.22) it can be shown that the winding number is given by W = (jJ 3 j>jJ +J 2 tan(#=2)j) + (jJ 3 j>jJJ 2 cot(#=2)j); (2.40) where (true) = 1, (false) = 0 and # = arccos[( 1 2 )= p 4J 2 1 + ( 1 2 )]. The above quantity can assume the valuesW = 0; 1; 2. The valueW = 2 can 20 be obtained, for example, by taking J 3 suciently large. When W = 2 the open, nite size chain has two edge modes per end. This gives the possibility to encode a qubit in the dark state manifold of the model. In the following we restrict to the case 2 = 1 = for which W = (jJ 3 j>jJ +J 2 j) + (jJ 3 j>jJJ 2 j): (2.41) As we can see from the above the presence of the two-sites hopping J is not necessary for having W = 2 but it allows to have W = 1. As we have seen in section 2.5, at nite size the exact number of edge modes can be a complicated function of N and the other parameters of the models. For the model of Eq. (2.39), we have veried numerically that for N mod 3 = 2 there are always (irrespective of W ) two edge modes with imaginary part of the eigenvalues exactly equal to zero. In other words there are always two exact dark states. However, we have also checked that essentially only W of them are localized on the qubit site. For N mod 36= 2 our simulations suggest that there are W edge modes with life-time exponentially large in the system size (see Fig. 3.8). Moreover, precisely W of them are localized at the qubit site. This picture is consistent with what we have found analytically in sec. 2.5. In other words, there are always (for all N) W dark or quasi-dark modes localized at the qubit site. Since, as we have seen, the behavior of the coherence is not only dictated by the number of localized modes, but rather by the modes localized at the qubit, the value of W has a strong impact on the coherence. From what we have said so far, the behavior of the coherence of the rst qubit is now clear. For W = 0 the coherence decays to zero after a time relax = 1 O(1) or it saturates to an exponentially small value in N if N = 3p + 2. For times relax .t. coh , forW = 1 it saturates to an amount given byC(t)'jhh1j 1 iij 2 wherej 1 ii is the dark state localized at the left of the chain. For W = 2 the coherence will oscillate between two values in a similar way as in Rabi oscillations,C(t)' e i! 1 t jhh1j 1 iij 2 +e i! 2 t jhh1j 2 iij 2 wherej 1;2 ii are the two dark states localized at the left with (real) eigenvalues ! 1;2 . The time-scale coh is innite forN = 3p + 2 and exponentially large in N otherwise. A plot of the behavior of the coherence in dierent topological sectors is shown in Fig. 2.10. Finally, let us comment on the long-time behavior of the full Lindbladian evolution. For N = 3p + 2 there is an exact, non-trivial dark space and so, for what we have said at the end of chapter 2.4, the evolution inside this 21 10 20 30 40 50 -12 -10 -8 -6 -4 -2 Figure 2.9: (Color online) Scaling of the imaginary part of the eigenval- ues of the edge modes, for dierent topological sectors and dierent values of N mod 3. For N = 3p + 2 we have observed always two exact dark states(Im( k ) = 0) for all parameters values. This simulations suggest that, forN mod 36= 2 there areW edge modes with exponentially large life-time. Parameters are 1 = 2 = 0, J 1 = 1:4; J 2 = 0:3; J = 0:7, = 1:5 and J 3 = 1 for W = 1 while J 3 = 3 for W = 2. dark space is unitary. When N mod 36= 2 andW = 2 there are two modes with life-time coh exponentially large inN. In this case an exact dark space sector cannot be dened, however we have veried that the dynamics are approximately unitary for times t in the window relax . t . coh . In this sense the term Rabi oscillations is accurate. 2.7 Eect of Noise on Coherence Decay In this section we explore the eect of disorder on the coherence time of our topologically protected systems. Specically we consider random (real) detuning of the qubits with respect to the cavity modes. This amounts to add a diagonal term to our one-particle eective \Hamiltonians" with on- site \chemical potentials" i where i are i.i.d. random variables with zero mean and uniform, distribution in [;] . We compute the corresponding coherence averaging over many (1000 in numerical simulations) realization. 22 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 Figure 2.10: (Color online) Behavior of the coherence in the linear chain with three sites per cell Eq. (2.39). We chose N = 8, but the numerical results are not sensitive to N mod 3. The winding number can assume values W = 0; 1; 2. W also counts the number of edge modes localized near the rst qubit. ForW = 2 the dark state manifold is a physical qubit and one sees Rabi oscillations in the coherence. Parameters are J 1 = 1; = 0; = 0:5; J 2 = 0:3; J = 0:7 and J 3 xes the value of W : J 3 = 0:2 (W = 0), J 3 = 0:7, (W = 1), and J 3 = 2, (W = 2). 2.7.1 Noise Eect on the Non-Hermitian SSH Model We rst consider the topological model of Eq. (3.29). For odd system size N, the topologically protected W = 1 phase has a true dark state, and the coherence of the qubit saturates to a nite value (Fig. 2.5). We observe that the system is quite robust against disorder (see Fig. 2.11). Even for a noise strength comparable to the tunneling strength J 1 , the qubit's coherence remains signicant over a long period of time. In addition, a shorter chain of cavities better protects the system against noise. For even system sizeN, the imaginary part of the dark state in theW = 1 phase is not exactly 0, but for a large enough N, the qubit's coherence still saturates to a nite value for times in a an exponentially large (inN) window. (Fig. 2.7) We observe that for N = 8 (Fig. 2.12 (a)), where the coherence of the clean system itself decays to 0, noise does not change the time evolution much. On the other hand, for N = 20 (Fig. 2.12 (b)), where the coherence 23 saturates, the eect of noise is similar to odd N, and again, when the chain of cavities gets longer, the disruptive eect of noise gets more pronounced. 2.7.2 Noise Eect on the Three-Site Unit Cell System Here we consider the topological model of Eq. (2.39) where we set 1 = 2 = 0. For this model (Fig. 2.8) there are three distinct topological phases, W = 0, W = 1, and W = 2, and the latter two protect the qubit's coherence from decaying. For the W = 1 phase (Fig. 2.13 (b)), the eect of noise is similar to that of the W = 1 phase of the non-Hermitian SSH model. The coherence of the qubit no longer saturates to a nite value, but decays to 0. We again note that the topologically protected system is quite robust against the introduction of noise. A noise strength comparable to the rst tunneling rate ( =J 1 ) does not decrease the coherence much even over a long time. For the W = 2 phase (Fig. 2.13 (a)), adding detuning noise vastly alters the oscillatory behavior of the clean system. In this case the time evolution of the coherence under noise resembles that of the W = 1 case, in other words the coherence saturates to a nite value. Yet larger noise strength seems to be able to eventually drive the system to a W = 0-like phase where the coherence decays to zero (see Fig. 2.13 (a), = 5J 1 ). Further investigations are needed to clarify the nature of this noise-induced, dissipative, topological phase transition [5]. 24 � �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (a) � �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (b) Figure 2.11: (Color online) The eect of noise on the coherence in the non- Hermitian SSH model with (a) N = 3 and (b) N = 7 respectively. We take the W = 1 phase, where J 1 = 1, J 2 = 1:8, = 0:5, and the noise rate is taken between 0 and 0:8J 1 . The nal result is averaged over 1000 runs of randomly generated systems with the respective noise rates. 25 � �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (a) � �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (b) Figure 2.12: (Color online) The eect of noise on the coherence in the non- Hermitian SSH model with (a)N = 8 and (b)N = 20 respectively. We take the W = 1 phase, where J 1 = 1, J 2 = 1:8, = 0:5, and the noise rate is taken between 0 and 0:8J 1 . The nal result is averaged over 1000 runs of randomly generated systems with the respective noise rates. 26 � �� �� �� �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (a) � �� �� �� �� ��� ��� ��� ��� ��� ��� ��� ��� ��� (b) Figure 2.13: (Color online) The eect of noise on the coherence in the three- site unit cell model with N = 8. (a) Phase W = 2 with J 1 = 1, J 2 = 0:3, J 3 = 2, J = 0:7, and = 0:5. (b) Phase W = 1 with J 1 = 1, J 2 = 0:3, J 3 = 0:7, J = 0:7, and = 0:5. The noise rate is taken to be 0, 0:5J 1 and J 1 . The nal result is averaged over 1000 runs of randomly generated systems with the respective noise rates. 27 Chapter 3 Projector Method: Phase Diagram of the Disordered Dissipative Su-Schrieer-Heeger Model 3.1 Projector Formalism 3.1.1 Background To study the non-Hermitian Su-Schrieer-Heeger model with diagonal noise, we consider an array of coupled qubits and cavities, as shown in Fig. 3.1. Within the super-one-particle sector with at most one excitation in the entire system, the corresponding Hilbert spaceH is spanned by the basis states:j0i andjji (for j = 1; 2; ;N). Here, thej0i state is special, as it represents the vacuum with no excitation on any qubit or cavity. The remaining states jji correspond to an excitation on the jth site. Depending on whether the jth site represents a qubit or an optical cavity,jji can thus be interpreted either as an excited qubit or as an excited cavity hosting a photon. This system can be described by a density matrix of the form = 0;0 j0ih0j + N X j=1 ( 0;j j0ihjj + h.c.) + N X i;j=1 i;j jiihjj; (3.1) which is a superposition of matrix basis statesjiihjj (i;j = 0; 1; 2; ;N). 28 µ1 µ2 µ3 µ4 µ5 µN Figure 3.1: Non-Hermitian Su-Schrieer-Heeger (SSH) model with open boundary conditions. The alternating qubit-cavity couplings J 1 , and J 2 are used to tune between the topologically trivial (J 2 < J 1 ) and non-trivial (J 2 >J 1 ) sectors. Dissipation is controlled by the parameter , and diagonal disorder is controlled via the random on-site chemical potential i . The matrix basis spans the superoperator Hilbert spaceH 2 . must satisfy the normalization condition Tr = 1, which can be written ashh1jii = 1, wherej1ii$ 1 =j0ih0j + P N j=1 jjihjj: We are interested in the coherence of the qubit located at one of the edges of the system, e.g. at the rst site, which is given by C =j 0;1 j =jh0jj1ij =j Trj1ih0jj =jhh1jiij; (3.2) wherej1ii$j0ih1j. This coherence measure can be derived from the reduced density matrix by tracing out all other sites Tr rest = 0;0 + N X i=2 i;i j0ih0j + 1;1 j1ih1j + 0;1 j0ih1j + h.c.: (3.3) In the previous chapter we have shown that topological band structure of the qubit-cavity system helps to protect quantum coherence of the ducial qubit.[4] In this chapter, we will focus on topological protection against di- agonal disorder and derive a phase diagram in the parameter space spanned by dimerization J 2 =J 1 and disorder i 2 [;]. The decay in time of the coherence is governed by the non-unitary time evolution of the density matrix of the system. This dynamics is described by the quantum master equation@ t =L[], whereL[] =K[] +D[] is the Lindblad operator.K[] =i[H;] describes the Hamiltonian dynamics and D[] describes the dissipation. Since we are primarily interested in the time dependence of the matrix element C(t) =j 0;1 (t)j =jhh1j(t)iij =jhh1je tL j(0)iij; (3.4) 29 and the Lindblad operatorL only mixesj1ii$j0ih1j withjjii$j0ihjj in the superoperator Hilbert spaceH 2 , we only need to consider the representation ofL within the subspaceV 0;1 spanned by the basisjjii (forj = 1; 2; ;N). Therefore, by restrictingL to ~ L =Lj V 0;1 and projecting the initial density matrix to ~ (0) =(0)j V 0;1 , the ducial qubit coherence can be simply evalu- ated from C(t) =hh1je t ~ L j~ (0)ii: (3.5) The problem therefore boils down to studying the spectrum of the restricted Lindblad operator ~ L. In general, ~ L takes the form of ~ L = X i;j ~ L ij jiiihhjj; (3.6) where the o-diagonal elements ~ L ij 2 R (i6= j) are real, but the diagonal elements ~ L ii 2C can be complex. This makes ~ L a non-Hermitian operator, such that the familiar formulation for the spectral decomposition of closed- system Hermitian Hamiltonians must be generalized to non-Hermitian cases in order to understand the Lindbladian dynamics. 3.1.2 Projector Formulation A Hermitian operatorH always admits a spectral decomposition of the form H = X a ju a i a hu a j; (3.7) where a 2R are real eigenvalues, andju a i are the eigenvectors. For reasons that will become apparent in Eq. (3.29), it is neater to factor out ai from ~ L, and work in terms of H, dened by ~ L =iH: (3.8) For non-Hermitian operators H, the spectral decomposition must be gen- eralized, H = X a jv a i a hu a j; (3.9) where, compared to the Hermitian case, there are two main dierences. First, the eigenvalues a 2 C can be complex. Second, the left-eigenvectorhu a j 30 and the right-eigenvectorjv a i are no longer related via complex conjuga- tion. But they still share the same eigenvalue, such thathu a jH =hu a j a and Hjv a i = a jv a i. Furthermore, the left- and right-eigenvectors satisfy the orthonormality conditionhu a jv b i = ab . This motivates us to introduce the projection operator P a =jv a ihu a j; (3.10) such that the spectral decomposition of the Lindblad operator H can be written as H = X a a P a : (3.11) In the Hermitian limit, P a simply reduces to P a =ju a ihu a j. A discussion of several salient properties ofP a are in order. First, as a projection operatorP a satises P 2 a =P a , as can be seen from P 2 a =jv a ihu a jv a ihu a j =jv a ihu a j =P a . Second, TrP a = 1 because TrP a = Trjv a ihu a j =hu a jv a i = 1. Third, P a satises the matrix equation HP a =P a H = a P a , which follows readily from the equations for the left- and right-eigenvectors. Finally, we can arrange the column vectorsjv a i and the row vectorsju a i into a matrix V and its inverse, such thatjv a i = Vjai andhu a j =hajV 1 , hence P a = VjaihajV 1 . Therefore, the matrix H can be decomposed as H = X a a P a =V X a jai a haj V 1 =VDV 1 ; (3.12) where D = diag( 1 ; ; N ) is a diagonal matrix. In the Hermitian case, V becomes a unitary matrix, i.e. V 1 = V y , such the above equation reduces to H =VDV y . 3.1.3 Density Operator The introduction of the projection operator P a allows us to generalize the notion of density of states and of density operator from the Hermitian case to the open-quantum-system case. Let us rst review the closed quantum formalism, where the Hermitian operator H = P a ju a i a hu a j = P a a P a admits a spectral decomposition with real eigenvalues a . The density of states (DOS) is dened as (!) = X a (! a ): (3.13) 31 To further gain spatially resolved information, one can dene the local density of states (LDOS) as (!;i) = X a (! a )p a (i); (3.14) where p a (i) is the probability density on site i when the system is in the eigenstateju a i, p a (i) =jhiju a ij 2 =hiju a ihu a jii =hijP a jii: (3.15) At this point, if we introduce the density operator (!H) = X a (! a )P a ; (3.16) both the DOS and LODS can be represented in a unied way by (!) = Tr(!H); (!;i) =hij(!H)jii: (3.17) The DOS is the trace of the density operator and the LDOS is the diagonal element of the density operator corresponding to site i. Generalizing to the non-Hermitian case, when H = P a a P a no longer admits a real spectrum, the DOS can still be dened as (!) = X a (! a ): (3.18) The only dierence compared to Eq. (3.13) is that the eigenvalues a 2C are generally complex now, so the energy !2 C is also complex and the delta function(! a ) is dened in the complex plane, such that R C (!)d! = 1. To dene the LDOS, we also need to generalize the denition of the probabil- ity densityp a (i). We may still follow the same denition as Eq. (3.15), using the projection operator P a , i.e. p a (i) =hijP a jii =hijv a ihu a jii. But since the left and right eigenvectors are no longer conjugate to each other,hijv a ihu a jii is complex in general, which therefore cannot be interpreted as a probability density anymore. In order to solve this problem, we take the absolute value and dene the probability density on site i for the eigenmode a to be p a (i) =jhijP a jiij =jhijv a ihu a jiij: (3.19) 32 Now, following an analogous denition of the LDOS to Eq. (3.14), we can generalize it to the non-Hermitian case as (!;i) = X a (! a )jhijP a jiij: (3.20) With these generalizations, we can introduce the density operator for the non-Hermitian Lindblad operator H as (!H) = X a (! a )P a ; (3.21) such that the DOS and LDOS are given by (!) = Tr(!H); (!;i) =jhij(!H)jiij; (3.22) in close analogy to the Hermitian case, Eq. (3.17). At the rst glance, Eq. (3.20) and Eq. (3.22) seems incompatible, as the sum of the absolute values is not the absolute value of the sum in general. However, here we have encountered a special case where the sum of the absolute values happens to coincide with the absolute value of the sum. To demonstrate this, we consider two scenarios: (i) when ! = b coincide with some eigenvalue b labeled by b, then for any functionf a ofa we have P a (! a )f a = P a ( b a )f a = P a (0) ab f a =(0)f b , hence X a (! a )jhijP a jiij =(0)jhijP b jiij = X a (! a )hijP a jii ; (3.23) (ii) when ! does not coincide with any eignvalue, X a (! a )jhijP a jiij = 0 = X a (! a )hijP a jii : (3.24) Thus in any case, we can pull the absolute value out of the summation, so Eq. (3.20) can be equivalently written as Eq. (3.22). 3.1.4 Bin Counting Method Performing a numerical analysis of the DOS and the LDOS for a specic system, we can divide the energy domain into small bins. Each bin ! = 33 (!!;! +!] is a small box region in the complex plane, centered around a complex energy !. The complex bin size ! is chosen to be small enough to guarantee that at most only one mode falls into each bin. With this set up, we can approximate the delta function (! a ) by (! a )' 1 (!) 2 ( a 2 ! ); (3.25) where the function is dened as (true) = 1 and (false) = 0. The measure (!) 2 of ! is divided to ensure the normalization R C d!(! a ) = 1. In this way, P a ( a 2 ! ) counts the number of states in the ! bin. We gradually decrease! until the condition P a ( a 2 ! ) 1 is met. In this manner we determine the optimal bin size. Given a non-Hermitian Hamiltonian H, we can construct the density op- erator by binning the projection operators according to its eigenvalues (!H)' 1 (!) 2 X a ( a 2 ! )P a : (3.26) In this construction, we rst nd the spectral decomposition ofH = P a a P a , then distribute the projection operator P a to the bin ! that a falls into. After collecting all eigenmodes, the density operator is constructed. Once we have obtained the density operator, the DOS and LDOS follow directly from Eq. (3.22). 3.1.5 Number of Edge Modes The density operator is a useful tool to identify the number of localized edge modes, which serves as an indicator of the topological sector. Given the density operator (! H), the numerical procedure we follow is outlined here. First, we t the LDOS prole(!;i) =jhij(!H)jiij with a spatially decaying exponential function for every given !, (!;i) =A ! e i=! ; (3.27) whereA ! and ! are t parameters. Ther 2 statistics of the tting is recorded as r 2 ! . We then count the number of localized edge modes via N [(!H)] = X ! (r 2 ! >r 2 min )( ! < max )((!; 1)> 0 ); (3.28) 34 which is a functional of the density operator(!H). It is worth mentioning that, although notational-wiseN [(!H)] contains a symbol!, it does not imply thatN [(!H)] is !-dependent. By saying thatN is a functional, it maps the density operator (!H) to an integer, such thatN [(!H)] does not have explicit dependence on either ! or i. This approach is based on the following considerations. The localized edge modes must satisfy both the localization condition and the edge mode condition. The localization condition requires the density prole of the mode to follow an exponential decay with a localization length, ! . In order to check if the LDOS (!;i) is describing a localized mode, we can t it with an exponentially decay function. If the t is good, as characterized by a large (close to 1) r 2 statistics r 2 ! > r 2 min , and if the localization length is bounded by ! < max , then the mode of energy ! can be considered a localized mode. To make sure the mode is localized at the edge, we further require that the LDOS (!; 1) at the rst site to be suciently large, i.e. (!; 1) > 0 , such that it dominates the density distribution. Putting all these conditions together, we arrive at Eq. (3.28), which precisely counts the number of modes that simultaneously satisfy these three conditions. This way, we dene the edge mode counting functionalN [(!H)], which takes the density operator as input and returns the number of edge modes. Note, however, that this method does not indicate whether the such identied edge modes are of topological origin. 3.2 Disordered non-Hermitian SSH Model 3.2.1 Model Description We now apply the above approach to study topological edge states in the disordered non-Hermitian SSH model. This model can describe an open quantum system of coupled qubits and optical cavities arranged in an alter- nating manner, as shown in Fig. 3.1. In the super-one-particle sector, the corresponding restricted Lindblad operator ~ L (Eq. (3.6)) is represented by 35 the following matrix ~ L =i 0 B B B B B @ 1 J 1 0 0 0 J 1 2 i J 2 0 0 0 J 2 3 J 1 0 0 0 J 1 4 i . . . 0 0 0 . . . . . . 1 C C C C C A iH: (3.29) We have dened the non-Hermitian Hamiltonian H by factoring out ai from ~ L to make it easier for later calculations. In this non-Hermitian Hamil- toninan, J 1 and J 2 describe the alternating couplings between neighboring qubits and optical cavities, is the dissipation rate of the cavity, and the i along the diagonal describe the characteristic on-site frequencies of the qubits and cavities. Here, odd indices i label the qubits, and even indices label the cavities. Considering that each qubit and cavity experiences uctuations in frequency, this leads to a diagonally disordered model. In particular, in our study i are a set of independent random variables drawn from a uniform distribution within the range [; +], such that parametrizes the strength of the randomness. The clean limit ( = 0 limit) of this model has been analyzed in previous work, where it was found that the model exhibits two topological phases: a trivial phase (J 2 <J 1 ) and a non-trivial phase (J 2 >J 1 ). In the topologically non-trivial phase, the open-ended system supports robust edge modes whose decoherence is close to zero, i.e. quasi-dark states, as the edge modes are approximately decoupled from the bulk. It has been argued that this eect can be used to protect the quantum coherence of the qubit on that boundary. In this work, we investigate the eects of quenched diagonal disorder on these topological edge modes, and discuss the stability of the topological phase transition in the presence of disorder. 3.2.2 Distribution of Eigenvalues We start by inspecting the distribution of eigenvalues of the non-Hermitian Hamiltonian H in the disordered non-Hermitian SSH model. In general, for a complex valued matrix such as the representation of H in Eq. (3.29), its eigenvalues are complex as well. Here we would like to study how these eigenvalues are distributed in the complex plane. We therefore apply the bin counting method detailed in subsection 3.1.4 to obtain the eigenvalue 36 statistics of this model. The density of states, which is the distribution of the eigenvalues on the two-dimensional complex energy mesh, can be used to identify topological features that connect this non-Hermitian model to known Hermitian ones. Fig. 3.2 shows distributions of complex eigenvalues collected from 100 random realizations of the disordered dissipative SSH chain with a moderate level of diagonal disorder in both the (a) trivial and (b) non-trivial phase. In Fig. 3.2 (a) we observe that in the non-Hermitian SSH Hamiltonian, the two energy bands of the trivial (J 1 >J 2 ) Hermitian SSH model become two dis- tinct islands within the complex energy plane. The band broadening along the Re! direction is due to the band dispersion, and the band broadening along the Im! direction is caused by the diagonal disorder. Specically, one can see that all states have imaginary energy components Im!'=2 distributed around the average dissipation rate, meaning that in the topo- logically trivial phase they dissipate with roughly the same rate. In addition, in the case of the topologically non-trivial (J 1 <J 2 ) phase, shown in Fig. 3.2 (b), the topological zero-energy edge modes of the Hermitian SSH model form a \ridge" that lies between these two islands. These in-gap states originate from a pair of edge modes due to the non-trivial band topology, localized at the left and right system boundaries. As seen in Fig. 3.1, the left edge mode is localized on the qubit at site 1, which has almost no dissipation and a long coherence time. In contrast, the right edge mode is localized on a cavity which has a large dissipation rate. The eigenvalues corresponding to these topological edge modes are sensitive to the on-site disorder because they are spatially localized such that the disorder potential cannot average out. This explains why the topological edge modes have a rather broad span in com- plex energy space along the Im! direction, forming an elongated ridge inside the band gap. In particular, the left edge states in the topologically non- trivial sector of the SSH chain can be considered to be quasi-dark because of their near-vanishing dissipation rate. They are therefore located close to the center of the in-gap ridge in complex energy space in Fig. 3.2 (b), i.e. these states have eigenvalues that have an almost vanishing imaginary part. In this sense these quasi-dark states possess ideal characteristics for topological qubit protection. Fig. 3.3 shows the blurring of the distribution of eigenvalues with increas- ing on-site disorder. In the clean limit (Fig. 3.3(a)), we observe two distinct in-gap states, the top one being localized at the leftmost ducial qubit, which have eigenvalues with an almost zero imaginary part. As the strength of the 37 -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 20 40 (a) -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 20 40 (b) Figure 3.2: Density of states of the non-Hermitian SSH chain with a moder- ate level of on-site (diagonal) disorder, averaged over 100 realizations, shown as intensity plots and as smoothed histograms of bin counts: (a) the topo- logically trivial sector, with N = 8, J 1 = 1, J 2 = 0:6, = 0:2, and = 0:5 and (b) the topologically non-trivial sector with N = 8, J 1 = 1, J 2 = 1:4, = 0:2, and = 0:5 38 -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 50 100 (a) -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 50 100 (b) -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 50 100 (c) -3 -2 -1 0 1 2 3 -0.20 -0.15 -0.10 -0.05 0.00 Re ω Im ω 0 50 100 (d) Figure 3.3: Blurring of the density of states with increasing on-site (diagonal) disorder in the topologically non-trivial sector of the non-Hermitian SSH chain (N = 20, J 1 = 1, J 2 = 1:4, = 0:2): (a) = 0, (b) = 0:5, (c) = 1 and (d) = 1:5. Results are averaged over 100 random realizations for a given noise level. 39 quenched on-site disorder is increased Fig. 3.3(b,c), both, the two edge states and the bands (\islands"), smear out into blurry blobs, and the ideal edge state with a small dissipative part becomes harder and harder to come by. When the noise level reaches 1.5 times the characteristic coupling strength (J 1 ) Fig. 3.3(d), the distribution of eigenvalues is all over the place, and the \ridge" that signies the in-gap states becomes undetectable. This illus- trates how the topological nature of the non-trivial phase is destroyed by strong disorder, i.e. when the disorder strength is larger than the band gap. In this strong disorder limit, there is essentially no dierence between the topological and trivial phases. However, if the disorder strength is weaker than the band gap, as in Fig. 3.3(b), the topological edge states are still well protected. This illustrates the extend of robustness of the quasi-dark states. 0.0 0.5 1.0 1.5 2.0 -0.035 -0.030 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 disorder strength μ Im λ 0 Figure 3.4: Dependence of the imaginary part of the eigenvalue of a quasi- dark topological edge state on the strength of the on-site disorder, , for a dissipative SSH chain with 60 sites, and J 1 = 1, J 2 = 1:4, and = 0:2. In the pure limit, = 0, it disappears, and it saturates to a nite value in the opposite limit of Anderson localization. An important feature of the topological edge modes in the clean limit is that their eigenvalues approach zero in the thermodynamic limit, such that they are dark states that do not dissipate. We can use this property to detect the topological edge mode in the clean limit. However, this property no longer holds in the disordered case. Fig. 3.4 shows that with increasing diagonal noise, the quasi-dark states no longer have an imaginary eigenvalue that is strictly zero. Here, Im 0 is dened to be the most negative imaginary 40 part of the eigenvalues of states localized at the left boundary. We observe that Im 0 decreases with increasing , leading to a decrease of coherence of the leftmost qubit. Therefore in the presence of disorder, the topological edge mode is no longer dark, and hence we cannot use the Im 0 condition to diagnose the topological edge state. 3.2.3 Construction of Topological Phase Diagram via Edge Mode Counting Having examined the spectral properties of the topological edge modes, we can now use them to identify the topological phases by counting the topo- logical edge modes. This will allow us to map out the phase diagram of the disordered non-Hermitian SSH chain. To this end, we will adopt the projector formalism detailed in Sec. 3.1 to determine the number of edge modes. 0.0 0.5 1.0 1.5 2.0 0 50 100 150 200 μ M Figure 3.5: Dependence of the optimal bin numberM (along each side of the window in complex energy space) on the strength of the on-site disorder . The data shown here was collected for a 40-site dissipative SSH chain, with J 1 = 1, J 2 = 0:6, and = 0:2, averaged over 100 random realizations. The numerical calculation starts with an appropriate choice of the bin size. We x the bin size! = (! R + i! I )=M by an integer numberM, where we choose ! R = 6 and ! I = 0:2 to be the range for the complex energy region of interest. ThereforeM indicates the number of bins along each side of this region. Fig. 3.5 shows the dependence of the optimal choice of the 41 bin number M with respect to the disorder strength . Generally, as the disorder increases, the energy levels are more spread out along the imaginary axis, and thus a smaller number of bins is required to satisfy the condition P a ( a 2 ! ) 1. From Fig. 3.5, one can see that about 200 bins along each side are sucient for the study of the disordered non-Hermitian SSH chain. Given a non-Hermitian Hamiltonina H, we rst nd its spectral decom- position H = P a a P a . With the mesh set up in the complex energy plane, we then collect the projection operators falling into each bin and use them to construct the density operator (!H) following Eq. (3.26), from which we identify and count the edge modes using Eq. (3.28). Instead of considering a single realization of the non-Hermitian Hamil- tonina H, we study the topological properties of an ensemble of Lindblad operators. Therefore, we need to specify how to take the ensemble average. Let H (s) be the sth random realization of the Lindblad operator. There are two natural ways to take the ensemble average: N I =N h P s (!H (s) ) P s Tr(!H (s) ) i ; N II = 1 N s X s N [(!H (s) )]: (3.30) The two approaches dier on the order of taking random ensemble average and counting the localized edge mode number.N I takes the ensemble average of the density operator before counting the edge mode. N II rst counts the edge mode, then take the ensemble average of the edge mode number. In the rst approach,N I is guaranteed to be an integer, but its value scales with the number of realizations, as observed in Fig. 3.6. We rst collect the projectors obtained from all random realizations in each bin. After a sucient number of realizations has been sampled, we go to each bin and average the projectors within the bin. If the bin is empty, we assign it to a zero matrix. We analyze the diagonal elements of each averaged projector to identify the edge modes. Finally, the total number of edge mode is added up over the complex energy mesh grid. Because the mode counting functionalN always returns an integer,N I is ensured to be integer. It collects all the edge modes that have appeared in all of the random realizations and merges the edge modes that fall in the same bin. In the clean limit,N I will match the topological index and switch from 0 to 1 as we tune from the topologically 42 50 100 100 100 150 150 200 200 250 300 350 400 450 500 550 600 650 700 700 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 μ/J 1 J 2 /J 1 Figure 3.6: Phase diagram of the disordered dissipative SSH Model with 40 sites, J 1 = 1, = 0:2, while varying J 2 and , usingN I . This simulation is run on 1000 realizations. 43 0.1 0.3 0.5 0.5 0.7 0.7 0.9 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.5 2.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 μ/J 1 J 2 /J 1 Figure 3.7: Phase diagram of the disordered, dissipative SSH model with 40 sites, J 1 = 1, = 0:2, while varying J 2 and , constructed by usingN II . This simulation is run on 1000 realizations. 44 trivial phase to the topologically non-trivial phase. Because in the clean limit all the edge modes fall precisely into the same bin centered around ! = 0, only the average projector in this particular bin will be non-zero. As we introduce the disorder, the edge modes no longer have denite complex energy. Instead, they spread out from the ! = 0 bin into the in-gap ridge. Therefore many other bins will start to collect edge modes, and the mode count grows with disorder strength. Because we do not divide the counting by the number of samples,N I scales linearly with the sample numberN s and hence does not have a nite limit in the thermodynamic limit (N s !1). In the second approach,N II is simply counting the number of edge modes for each random realization, and then averages over all samples. N II is not guaranteed to be an integer, because we divide by the sample number N s when taking the ensemble average. The resulting phase diagram is shown in Fig. 3.7. In the clean limit,N II also matches the topological index, which jumps from 0 to 1, as J 2 =J 1 increases across the transition point. N II = 0 indicates the trivial phase without topological edge modes, andN II = 1 indicates the topologically non-trivial phase where there is one edge mode localized on the left chain edge. As we increase the disorder strength,N II = 1 remains well quantized deep into the topological phase for moderate noise level. The integer counting eventually collapses in the strong disorder limit. The jump ofN II is smeared out with increasing disorder strength, indicating that the topological transition in the clean limit is no longer sharply dened in the presence of disorder, i.e. the phase transition becomes a crossover. 3.2.4 Interpretation of the Phase Diagram Based on the above analysis, we can identify the trivial and non-trivial topo- logical phases by the behavior ofN I andN II . The phase boundary is de- termined by the region whereN drops suddenly. More quantitatively, we can use the criterionN II = 0:5 to dene the phase boundary, becauseN II has a well-dened limit as N s !1. We have marked out this boundary in Fig. 3.7(a). Because our criterion for the edge mode is an exponentially localized mode on the left edge, this cannot distinguish between a topological edge mode and an Anderson localized mode which accidentally appears near the left edge. For 1D systems with unbrokenZ 2 symmetry, disorder is a relevant perturbation, such that all modes will be Anderson localized in the presence of disorder.[26] If they appear near the left edge, they are also counted by the 45 projection method. So a non-zero countingN does not necessarily indicate a topological phase. In particular, for small J 2 =J 1 with strong disorder, we can see a peak in bothN I andN II inside the trivial phase, which needs to be ascribed to Anderson localization. Nevertheless, if we focus along the topological transition line, the mode counting method is still a good indicator of topological edge modes, and can thus be used to map out the topological phase diagram. 3.2.5 Finite-Size Scaling To further scrutinize the nature of the transition around J 2 =J 1 ' 1 at nite, but moderate, disorder strength , we x the disorder strength = 0:4 and scanN II along J 2 =J 1 for dierent system sizes N = 20; 40; 60; 80; 100. The result is shown in Fig. 3.8. One observes that the number of detected edge modes increases from 0 to 1 as J 2 becomes larger than J 1 , signifying a transition from the topologically trivial phase to the non-trivial phase. As the system size N is increased, the curve saturates to its thermodynamic limit, indicating that the transition is not sharply dened in the presence of disorder. This is expected as the topological phases of the SSH model in the clean limit was originally protected by the sublattice symmetry. The disorder breaks the sublattice symmetry and hence the sharp topological phase transition of the clean system is smoothed out, and becomes a crossover transition instead. 46 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.0 0.2 0.4 0.6 0.8 1.0 J 2 /J 1 II N 20 40 60 80 100 Figure 3.8: Finite size scaling of theN II for a non-Hermitian SSH model with J 1 = 1, = 0:2, and = 0:4, varying system size and J 2 . This simulation is obtained by averaging 2000 random realizations. 47 Chapter 4 Conclusions Non-Hermitian topological phases in a nite system permit the construction of states whose decay time is either innite or exponentially large in the system size. This feature is extremely appealing from the point of view of creating long-lived quantum bits. In this work we have shown that networks of qubits interacting with lossy cavities may be congured to possess non- trivial topological structure. For networks with a simple linear geometry, we have found that localization and long-livedness of the topological edge modes both concur to increase dramatically the coherence of a qubit sitting at the end of the chain. Specically, a non-zero topological winding number W results in an exponentially long lived qubit. Although at nite size the exact number of edge modes is a complicated function ofW andN, there are always W edge modes localized at one end of the chain. For W = 2 we nd that the long-time dissipative, Lindbladian evolution becomes approximately unitary, and the coherence of the qubit displays long-lived Rabi-oscillations. In general, such long-lived, topological edge modes, are not legitimate quan- tum states, but rather they are o-diagonal elements of quantum a states or, coherences. The possibility of using such long-lived coherences for quantum computation is an interesting and challenging task for future studies. To study the robustness of the topological protection of quantum coher- ence against random noise, we have formulated a projector method that can be used to identify localized modes in open quantum systems. It is based on an analysis of the distribution of eigenvalues in complex energy space. Here the presence of quasi-dark (near zero-energy) edge states signies topologi- cally non-trivial phases. A counting method is devised that captures topo- logical phase transitions. This approach was then applied to investigate the 48 dissipative SSH chain with on-site (diagonal) disorder, and a phase diagram was mapped out. The projector method correctly identies theJ 2 =J 1 -driven topological phase transition in this system and also correctly captures the crossover nature transition of this transiton at nite disorder. However, it also picks up spurious (non-topological) localized states at the system edges that are caused by Anderson localization. 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Rudner, Mordechai Segev, and Alexander Sza- meit. Observation of a Topological Transition in the Bulk of a Non- Hermitian System. Phys. Rev. Lett., 115(4):040402, July 2015. 52 Appendix A \Single impurity" case Through shift and rescaling ~ L = iJH 0 (=2)1 I we are led to consider the following matrix H 0 = 0 B B B B B @ ia 0 0 0 1 0 0 1 0 0 . . . . . . . . . . . . 1 0 0 0 1 0 1 C C C C C A : (A.1) We write the eigenvalues as 2 cos(k). It can be shown that k satises the following equation (both for N even and odd) [2 cos(k) +ia] sin(kN) 2 sin(k(N 1)) = 0; (A.2) and one can restrict oneself to 0< Re(k)<. In order to look for localized states we look for a complex root of Eq. ((A.2)). Hence we set k = x +iy. Plugging this in the above and forgetting terms e Njyj we obtain (2 cos(k) +ia) = 2 sin(k(N 2)) sin(k(N 1)) ' 2 ( e ix e y y> 0 e ix e y y< 0 (A.3) We also set ik =q. Case y> 0. The equation is 2 cosh(q) +ia = 2 e q setting z =e q one nds q = ln (i) a p a 2 + 4(1 2 ) 2(1 2 ) ! 53 and the corresponding eigenvalues =i 2 2 (1 2 ) a p a 2 + 4(1 2 ) ia: (A.4) We need to make sure thaty =Re(q)> 0. From this we obtain Re(ln(z)) = lnjzj< 0 orjzj< 1. Case y< 0. Now the equation is 2 cosh(q) +ia = 2 e q setting z = e q one nds the same equation as for y > 0. This means that the eigenvalues have the same from (A.4), but now y< 0 impliesjzj> 1. Going back to the eigenvalues of ~ L = iJH 0 (=2)1 I, remembering a = =(2J) and ==J we get nally = 4 2 p 16(J 2 2 ) + 2 +O e Njyj ; (A.5) as shown in the main text. 54 Appendix B A note on timescales Here we would like to dene a time-scale associated to the coherence decay. This time scale should measure the time after which the coherence has de- graded to an unacceptable value. Several denition of such (de-)coherence time are possible. For example one may take the smallest such that C() =C(0). According to Eq. (2.12)C(t) has the formC(t) = P j c j e j t , where j are (a subset of) Lindbladian eigenvalues satisfying Re( j ) 0. Let us say that one is interested in very small . In this limit the coherence time becomes proportional to . A meaningful denition then would be lin = =Re P j c j j (the name stemming from the fact thatC(t) is ap- proximately linear for t. lin ). A more conservative denition is given by the shortest time-scale associated with the setfRe( j )g i.e. the timescale dened as 1 min = max j Re( j ). If min is large one is guaranteed that the coherence will be close to maximal for all 0 t. min for any initial state. This is a very pleasant feature which makes min quite attractive. Let us also dene the the slowest decay time ofC(t) by 1 max = min j Re( j ). Clearly max can be much larger than any meaningful denition of coherence time 1 . Obviously all these time-scales agree if the coherence decays as a sin- gle exponential. Quite surprisingly in all the situation we considered in the text, we veried that indeedC(t) can be well approximated with a single exponential over a wide range of J= ( dissipation scale and J coherent energy scale, see main text). Some arguments why this is so will be given in the next section. In all the cases considered the coherence timescale coh 1 The meaning of max is that for times t & max the coherence is guaranteed to be essentially zero for any initial state. But we may have lost interest inC long before. 55 dened in the main text coincides with what is commonly called Purcell rate in the cavity QED community. Through topological protection we are able to exponentially increase the Purcell rate. 56 Appendix C Conditions to optimize the coherence decay In the following we will identify sucient conditions for the requirements i') and ii'). For simplicity we assume that ~ L can be diagonalized 1 with spectral resolution ~ L = P j j P j . We start analyzing the following consequence of i'): Fact 1. Assume i'), i.e. ~ Lj1ii = 1 j1ii+jeii withkek =O(1). Then, up to an error, the evolution of the coherence is governed by a single exponential, in particular C(t) = e t 1 +O(): (C.1) Proof. We start with the identity e t ~ L j1ii =e t 1 j1ii + e t ~ L e t 1 ~ L 1 jeii: We then obtainhh1je t ~ L 1ii = e t 1 +, and taking the modulus hh1je t ~ L 1ii = e t 1 + 0 +O( 2 ) withj 0 j hh1j e t ~ L e t 1 ~ L 1 eii = O(1): Moreover, assuming that ~ L can be diagonalized,j 0 j does not blows up with t, rather j 0 jkek ( ~ L 1 ) 1 X j e t j kP j k + e t 1 ! = (c + 1)kek ( ~ L 1 ) 1 ; (C.2) 1 This is not a serious limitation as the set of non-diagonalizable ~ L has measure zero in the space of parameters, J i ; i ;:::. 57 having set c = P j kP j k, since Re( j ) 0 and t 0. The same conclusion holds, not surprisingly, using a slightly relaxed as- sumption P 1 = j1iihh1j + X. Using the normalization of the projectors P i P j = i;j P j one obtainshh1jP j j1ii = Tr(P j j1iihh1j) = Tr[P j (P 1 X)]. The latter expression equals 1 for j = 1 and O() otherwise. Hence hh1je t ~ L 1ii = X j e t j Tr(P j j1iihh1j) (C.3) =e t 1 h Tr(P 1 X) + X j6=1 e t j Tr(P j X) i ; (C.4) and the result holds withj 0 j P j jTr(P j X)j. As can be seen from the absence of the resolvent term, in this case the error can be made signicantly smaller. Note that if Re() = 0 the leading term of the coherence does not decay. This fact will be important when discussing dark or quasi-dark states in topological models. To gain further insight we analyze the weak and strong dissipative limits. Fact 2 (strong dissipative limit). Assume a linear geometry and a hopping to dissipation ratiojJ 1 = 2 j = suciently small. Then conditions i') and ii') are satised and in particularC(t) = e t= coh +O () with 1 coh = 2J 2 1 = 2 + J 1 O( 2 ). Proof. We consider the o-diagonal terms of Eq. (2.13) as a perturbation. The spectrum of the unperturbed system isf0; i =2;i = 2;:::;Ng, and the zero eigenvalue has eigenprojectorj1iihh1j. Using (non-Hermitian) perturba- tion theory, the rst correction to the zero eigenvalue occurs at second order and is given by (2) 1 =2J 2 1 = 2 . The corresponding eigenprojector is given byP 1 =j1iihh1j+O () so that we are in the condition for fact 1 and the result follows. Note that, since eigenvalues are continuous in their parameters, as long as there is no level crossing, 1 is the eigenvalue with real part closest to zero. In other words the coherence time is given by the slowest time-scale of ~ L. We dene H = H 0 + D where H 0 (D) is the Hermitian (anti-Hermitian) part of H. Note that the matrix D is diagonal in the \position" basisjjii. Since H 0 is Hermitian it can be written as H 0 = P k (0) k jkiihhkj, wherejkii are 58 the unperturbed eigenvectors. Up to rst order, the eigenvalues of ~ L become k =i (0) k ihhkjDjkii (C.5) =i (0) k 1 2 N X j=2 j jhhkjjiij 2 : (C.6) In the isotropic case where all the cavities are equal j = and the above becomes k =i (0) k 2 1jhhkj1iij 2 : Moreover, assume now that the Hamiltonian H 0 has a state localized at the rst site: 9k 0 : jk 0 iij1ii. This means thatj1iihh1j is an approximate eigenprojector of H 0 : P k 0 jk 0 iihhk 0 j =j1iihh1j +Y with a small . Sur- prisinglyj1iihh1j is also an eigenprojector of H up to the same order. In fact the rst correction to the eigenprojectors of H is P (1) =P k 0 DSSDP k 0 where S is the reduce resolvent [13]. Plugging in D =i(=2) (1 Ij1iihh1j) we obtain P (1) = i(=2) [Y ((1 Ij1iihh1j)S +S (1 Ij1iihh1j)Y ]. Thenj1iihh1j is a projector of ~ L up to an errorO(). In other words we have the following Fact 3 (weak dissipative limit). Assume that the Hamiltonian H 0 has a state localized at the rst site: P k 0 =j1iihh1j + 0 Y with a small 0 . For small both hypothesis i') and ii') hold. Moreover Fact 1 holds with = 0 : C(t) = e t= coh +O( 0 ): The coherence time is given in this case by 1 coh = (=2)(1jhhk 0 j1iij 2 ) +O( 2 ). 59
Abstract (if available)
Abstract
One dimensional topological insulators are characterized by edge states with exponentially small energies. According to one generalization of topological phase to non-Hermitian systems, a finite system in a non-trivial topological phase displays surface states with exponentially long life times. ❧ In the first half of this thesis we explore the possibility of exploiting such non-Hermitian topological phases to enhance the quantum coherence of a fiducial qubit embedded in a dissipative environment. We first show that a network of qubits interacting with lossy cavities can be represented, in a suitable super-one-particle sector, by a non-Hermitian “Hamiltonian” of the desired form. We then study, both analytically and numerically, one-dimensional geometries with up to three sites per unit cell, and up to a topological winding number W = 2. For finite-size systems the number of edge modes is a complicated function of W and the system size N. However we find that there are precisely W modes localized at one end of the chain. In such topological phases the quibt’s coherence lifetime is exponentially large in the system size. We verify that, for W > 1, at large times, the Lindbladian evolution is approximately a non-trivial unitary. For W = 2 this results in Rabi-like oscillations of the qubit’s coherence measure. ❧ The second half the this thesis, we investigate the robustness of the topological protection of coherence against disorder. A projector formalism is introduced to study open quantum many-body systems with non-trivial topology. Within this approach, the spectral composition with complex eigenvalues of the non-Hermitian Lindblad operator is determined, and a generalized open system density of states is obtained. The local density of states is then used to examine the edge states of the system, indicating its topological state. ❧ This method is applied to construct the phase diagram of the dissipative Su-Schrieffer-Heeger model in the presence of diagonal disorder. It is shown that in the topologically non-trivial sector, the characteristic edge modes survive up to a threshold level beyond which the system is Anderson localized, whereas in the topologically trivial sector Anderson localization sets in immediately upon introducing disorder. A finite-size scaling analysis reveals that the sharp topological phase transition driven by the dimerization in the pure case gives way to a crossover in the presence of disorder.
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Ma, Zhengzhi
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Topological protection of quantum coherence in a dissipative, disordered environment
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Doctor of Philosophy
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11/14/2019
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