Memory-Critical Dynamical Buildup of Phonon-Dressed Majorana Fermions
aa r X i v : . [ qu a n t - ph ] J u l Memory-Critical Dynamical Buildup of Phonon-Dressed Majorana Fermions
Oliver Kaestle, ∗ Ying Hu,
2, 3
Alexander Carmele Technische Universit¨at Berlin, Institut f¨ur Theoretische Physik,Nichtlineare Optik und Quantenelektronik, Hardenbergstrae 36, 10623 Berlin, Germany State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan, Shanxi 030006, China (Dated: July 2, 2020)We investigate the dynamical interplay between topological state of matter and a non-Markoviandissipation, which gives rise to a new and crucial time scale into the system dynamics due to its quan-tum memory. We specifically study a one-dimensional polaronic topological superconductor withphonon-dressed p -wave pairing, when a fast temperature increase in surrounding phonons induces anopen-system dynamics. We show that when the memory depth increases, the Majorana edge dynam-ics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildupbehavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above acritical memory depth, the system can approach a new dressed state of topological superconductorin dynamical equilibrium with phonons, with nearly full buildup of Majorana correlation. Exploring topological properties out of equilibrium iscentral in the effort to realize, probe and exploit topo-logical states of matter in the lab [1–11]. A paradig-matic scenario is where the topological system is coupledto a Markovian bath, inducing open-dissipative dynam-ics that is described by a Lindblad-form master equa-tion for the time-evolved reduced system density oper-ator [12–22]. Yet, solid-state realizations of topologicalmatter often rely on nanotechnological design strategiesthat restructure the environment by fine-tuning the cor-responding frequency-dependent density of states. In thiscase, the Markovian approximation and thus the Lind-blad formalism usually fails, e.g. in condensed mat-ter with nanostructured acoustic environments. Suchnon-Markovian situations can also occur when ultracoldatoms are immersed into Bose gases [23–25]. Comparedto Markovian scenarios, key differences arise from thepresence of quantum memory effects in non-Markovianprocesses: The information is lost from the system tothe environment but flows back at a later time [26, 27].This generates a new and critical time scale into the sys-tem dynamics that is strictly absent in a Markovian con-text, and raises the challenge as to what are the uniquedynamical consequences of the interplay between topo-logical state of matter and non-Markovian dissipation.Here, we demonstrate that the quantum memory froma non-Markovian parity-preserving interaction of a topo-logical p -wave supercondutor with surrounding phononscan give rise to intriguing edge mode relaxation dy-namics with no Markovian counterpart. Our study isbased on the polaron master equation, describing open-dissipative dynamics of a polaronic topological supercon-ductor with phonon-renormalized Hamiltonian parame-ters [see Fig. 1]. In contrast to Markovian decoherencethat typically destroys topological features for long times,we show that a finite quantum memory allows for sub- stantial preservation of topological properties far fromequilibrium, even when the polaron Hamiltonian is closeto the topological phase boundary [see Fig. 2 (b)]. De-pending on the memory depth (i.e., the characteristictime scale of the quantum memory), the Majorana edgedynamics can monotonically relax to a plateau, or it un-dergoes a collapse-and-buildup relaxation [see Fig. 2 (c)].Remarkably, when the memory depth increases above acritical value, the Majorana correlation can nearly fullybuild up, corresponding to a new polaronic state of topo-logical superconductor in dynamical equilibrium withphonons. This topological polaronic steady-state goesbeyond typical frameworks where phonons act as a per-turbation in the static and weak coupling limit.Concretely, we consider the paradigmatic Kitaev p -wave superconductor [28], with a superohmic couplingto a 3D structured phonon reservoir. The total Hamil-tonian is denoted by H = H k + H b ( ~ ≡ H k = P N − l =1 [( − Jc † l c l +1 + ∆ c l c l +1 ) +H . c . ] − µ P Nl =1 c † l c l describes spinless fermions c l , c † l ona chain of N sites l , with a nearest-neighbor tunneling Figure 1. A polaronic Kitaev chain, with phonon-dressedspinless fermions, exhibits a renormalized p -wave pairing h B i ∆ at temperature T [see Eq. (1)]. In the topologicalground state, two unpaired Majorana edge modes γ L , γ R emerge. The coupling g k to the structured phonon bath fea-tures mode-dependence with a spectral width σ . The rightpanel illustrates a Gaussian profile of g k in momentum space(see text), respectively, for σ = 0 . σ = 0 . amplitude J ∈ R , pairing amplitude ∆ ∈ R , and chem-ical potential µ . When | µ | < J and ∆ = 0, the su-perconductor is in the topological regime featuring un-paired Majorana edge modes γ L = γ † L and γ R = γ † R at two ends [29], which exhibit a nonlocal correlation θ = − i h γ L γ R i = ± H b = R d k (cid:2) ω k r † k r k + P Nl =1 g k c † l c l ( r † k + r k ) (cid:3) [30, 31].Here, operator r ( † ) k annihilates (creates) phonons withmomentum k and frequencies ω k = c s k , where c s is thesound velocity of the environment. We choose a genericsuperohmic fermion-phonon coupling g k which featuresa frequency dependence modeled as a Gaussian function, g k = f ph p k/ ( V σ ) exp (cid:0) − k /σ (cid:1) with a width σ and adimensionless amplitude f ph . Such fermion-phonon in-teractions can represent the coupling of electrons withacoustic phonons in relevant solid-state experiments andprotocols [32–34], and can arise in ultracold atoms fromcoupling a fermionic lattice to a Bose-Einstein conden-sate inducing p -wave superconductivity [35–37]. We con-sider the topological superconductor and bath are ini-tially in equilibrium at low temperatures, before a fastincrease in the bath temperature induces an open-systemdynamics. Polaron master equation.
The interactions with thestructured reservoir result in non-Markovian dynamics,whose description - particularly on long time scales -is a challenge due to unfavorable scaling of the Hilbertspace dimension. At the heart of our following solu-tion lies the polaron representation of the coupled sys-tem [see Fig. 1]: We derive a master equation in second-order perturbation theory of the dressed-state system-reservoir Hamiltonian tracing out phonons, whilst retain-ing the coherent process (i.e., higher order contributions)from the fermion-phonon interaction through phonon-renormalized
Hamiltonian parameters [38–42], thus effi-ciently accounting for the non-Markovian character of thedynamics in the long-time limit not captured in typicalsecond-order Born approximation of the bare interactionHamiltonian H b [30].Defining collective bosonic operators R † = R d k ( g k /ω k ) r † k , we apply a polaron transforma-tion U p = exp[ P Nl =1 c † l c l ( R † − R )], which resultsin c † l → e − ( R − R † ) c † l that describes phonon dress-ing of fermions. The transformed total Hamiltonian H p ≡ U p H U − p is derived as H p = N − X l =1 h − Jc † l c l +1 + ∆ e − R − R † ) c † l +1 c † l + H . c . i − µ N X l =1 c † l c l + Z d k ω k r † k r k . (1)Thus the considered fermion-phonon interaction results in a polaronic Kitaev chain featuring phonon-dressed p -wave pairing, with phonon-induced quantum fluctua-tions. Note that the polaron transformation also rendersan energy renormalization which has been canceled by acounter term in Eq. (1) [38, 43]. This is justified given thecoupled system is initially in equilibrium in the presentstudy. Before tracing out the phononic degrees of free-dom, we rewrite Eq. (1) as H p = H p,s + H p,I + H p,b with H p,b = R d k ω k r † k r k for the reservoir. To recover thebare Kitaev Hamiltonian dynamics for the limiting case g k →
0, we introduce a Franck-Condon renormalizationof H p,I satisfying Tr B { [ H p,I , ρ ( t )] } = 0, with ρ ( t ) denot-ing the total density operator. The renormalized systemHamiltonian H p,s is given by H p,s = N − X l =1 h − Jc † l c l +1 +∆ h B i c † l +1 c † l +H . c . i − µ N X l =1 c † l c l , (2)where the pairing renormalization factor h B i is givenexplicitly below. The system-reservoir interaction inthe polaron picture reads H p,I = ∆ P N − l =1 [( e − R − R † ) −h B i ) c † l +1 c † l + H.c.]. Crucially, in the following we treat H p,I perturbatively in second-order Born theory, as weare interested in phonon equilibration time scales muchfaster than the system dynamics, so that dynamical de-coupling effects cannot occur [38, 40]. We then derivethe polaron master equation for the reduced system den-sity matrix ρ S ( t ) of the polaron chain, obtaining (Suppl.Mat.) ∂ t ρ S ( t ) = − i [ H p,s , ρ S ( t )] − h B i Z t dτ n (cosh [ φ ( τ )] −
1) [ X a , X a ( − τ ) ρ S ( t )] − sinh [ φ ( τ )] [ X b , X b ( − τ ) ρ S ( t )] + H . c . o . (3)Here X a = − J P N − l =1 ( c † l c † l +1 + c l +1 c l ) and X b = J P N − l =1 ( c † l c † l +1 − c l +1 c l ) denote collectivesystem operators, whose dynamics obeys a time-reversed unitary evolution governed by the renormalized Hamil-tonian H p,s , X a,b ( − τ ) ≡ e − iH p,s τ X a,b e iH p,s τ includingthe full density matrix (Suppl. Mat.). The φ ( τ )represents the phonon correlation function, φ ( τ ) = R d k | g k ( σ ) /ω k | { coth [ ~ ω k / (2 k B T )] cos ( ω k τ ) − i sin( ω k τ ) } , with k B the Boltzmann constant. Therenormalization factor h B i in Eq. (2) is determined bythe initial phonon correlation, h B i = exp[ − φ (0) / phonon-renormalized Hamiltonian H p,s de-scribing polarons which exhibit temperature-dependentpairing, and a non-Markovian dissipative term in theform of a memory kernel which involves both reservoirand system correlators φ ( τ ) and X a,b ( − τ ), respectively,with a finite memory depth determined by the lifetime -5 0 5 100 25 50J τ (a) Abs[ Φ ( τ )]Im[ Φ ( τ )]Re[ Φ ( τ )] -0.500.510 0.5 1 θ ( t ) θ ( t ) σ = 0.4 σ = 0.2 θ ( t ∞ ) σ Figure 2. Non-Markovian dynamics of the polaronic topological superconductor. (a) Phonon correlation function φ ( τ ) forbandwidth σ = 0 . σ = 0 . g k . (b) Comparisons of Majorana correlation θ ( t ) calculated using, respectively, time-independent Lindblad-type master equation for dephasing (solid black), Eq. (3) inthe Markovian limit (blue), Eq. (3) with full account of memory (orange). The dashed black line shows asymptotic θ ( t ) in acoherent quench scenario ∆ → ∆ h B i . (c) Non-Markovian dynamics of θ ( t ) for various bandwidths σ of phonon coupling. Thecorresponding steady-state value θ ( t ∞ ) is shown as a function of σ in the inset. In all plots, the amplitude of g k is taken as f ph = 0 .
1, and phonon modes within k ∈ [0 . , .
0] nm − are considered. Due to the numerically very expensive size of thedensity matrix and its memory kernel, computations are performed for N = 4 sites. of φ ( τ ). The memory depth critically depends on thebandwidth σ of fermion-phonon coupling g k [see Fig. 2(a)]: A larger σ results in a faster decay of φ ( t ) andhence a smaller memory size, but it also leads to a smaller φ (0) and thus a larger h B i in both H p,s and the prefac-tor of the memory kernel. In the limit when the systemevolves much slower than the phonons, one can approxi-mate X a,b ( − τ ) ≈ X a,b , and Eq. (3) transits into a Marko-vian type of master equation with a time-dependent de-phasing rate γ ( t ).Below we investigate the Majorana edge correlation θ ( t ) = − i Tr [ ρ s ( t ) γ L γ R ] based on Eq. (3), starting froma factorizing system-bath state. For concreteness, we as-sume the system is initially at zero temperature in theground state of a dressed Kitaev Hamiltonian with ∆ = J and µ = 0, exhibiting even parity θ (0) = 1. Then, a fastincrease of temperature to T = 4 K results in ∆ → ∆ h B i of the dressed Hamiltonian, thus inducing the dynamicsof the polaron chain for times t > Markovian limit . In the Markovian limit X a,b ( − τ ) ≈ X a,b (0) of Eq. (3), the edge correlation θ ( t ) rapidly de-cays to a very small but finite value [blue line in Fig. 2(b)]. This is expected because dressing fermions with fastphonons induces dephasing, as has been familiar fromMarkovian decoherence based on a time-independentLindblad operator [see e.g., Ref. [44] and black line inFig. 2 (b)]. Still, the non-Markovian character of thestructured bath leads to a small residue correlation. Memory: loss vs. rephasing of topological properties .However, the picture drastically changes when takinginto account the full memory including the system’s past X a,b ( − τ ), as a highly non-Markovian interplay between system and reservoir unfolds. The orange line in Fig. 2(b) shows the non-Markovian dynamics for σ = 0 . h B i = 0 .
07: The Majorana edge correlationstill decays in an oscillatory manner [orange line], but itrelaxes to a substantial value as opposed to the Marko-vian limit [blue line]. Considering the small system sizein our computation, we have verified that such asymp-totic non-local correlation is genuinely of topological ori-gin, rather than phonon-mediated long range correlations(Suppl. Mat.).The long lived and substantial Majorana correlation inFig. 2 (b) is quite remarkable, given that H p,s is near thetopological phase boundary due to a significantly sup-pressed renormalized pairing ∆ h B i ≪ ∆. Indeed, with-out the dissipation in Eq. (3), the dynamics formally re-duces to that of a coherent quench in the pairing from∆ to ∆ h B i . There, the Majorana correlation would ap-proach an asymptotic value determined by the overlapof the edge mode wave functions for the pre- and post-quench topological Hamiltonians [18], which is small ifthe post-quench Hamiltonian is close to the phase bound-ary [see dashed black line in Fig. 2 (b)]. This differs sig-nificantly from the non-Markovian behavior in Fig. 2 (b)and underscores the essential role of the memory effect.A unique feature of the memory effect is that it,because of the dependence on both φ ( τ ) and thereversed dynamics of system correlations X a,b ( − τ ), simultaneously introduces decoherence and backflowof coherence. The dynamical consequence of thesetwo competing processes can be intuitively under-stood as follows: The dressed Kitaev wire initially inits ground state is perturbed by a temperature in-crease to T , resulting in a renormalization of the po-laron chain towards the phase boundary via φ (0) = R d k | g k ( σ ) /ω k | coth ( ~ ω k / (2 k B T )), which generatessignificant bulk excitations and populates the Majoranaedge mode, changing the parity of Majorana states.Combined with phonon-assisted dephasing, this leads tostrong decoherence in the polaronic Kitaev wire. On theother hand, the reversed dynamics of X a,b ( − τ ) acts toreinstate coherence of the p -wave pairing that is the keyingredient for a topological wire. Such a rephasing ef-fect is marginal at times smaller than the characteristictime of the memory, so an irreversible loss of parity in-formation dominates the short-time dynamics. Yet φ ( τ )proceeds to decay over time, as its sine and cosine func-tions tend to cancel each other. Once φ ( τ ) approacheszero at large times, the memory reaches its full depthand the rephasing of topological properties grows dueto X a,b ( − τ ), giving considerable Majorana correlation inFig. 2 (b). Critical memory depth.
We find the edge dynamics canexhibit distinct relaxation behavior depending cruciallyon the memory depth, tunable through the bandwidth σ of fermion-phonon coupling. Fig. 2 (c) presents the non-Markovian dynamics of θ ( t ) for various σ . Compared to σ = 0 . σ results in a steeper monotonic decay of θ ( t ) and asmaller asymptotic value [blue line in Fig. 2 (c)]. How-ever, when σ decreases further, the monotonic relaxationtransits into a non-monotonic one: While the short-timedecoherence is accelerated, a buildup of edge correlationnonetheless occurs at some large times [purple line]. Suchbuildup becomes stronger with decreased σ , approach-ing an asymptotic value larger than the σ = 0 . σ surpassesa critical value, the asymptotic Majorana correlation ap-proaches θ ( t ∞ ) → θ ( t ∞ ):When σ decreases from a large value, θ ( t ∞ ) first de-creases to a minimum and then increases toward unity.Insights into above intriguing phenomena can be ob-tained from the fact that reducing σ leads to an increasedmemory size at the cost of a smaller h B i = exp[ − φ (0) / X a,b ( − τ ) and hence therephasing of pairing, whereas the latter further sup-presses the superconducting gap, rendering H p,s closerto the phase boundary as well as weakening the mem-ory strength. When σ is initially decreased from 0 . σ , however, the former rephasingeffect grows, allowing phonons and fermions to synchro-nize and hence inducing backflow of parity information.Consequently, a new dressed state manages to emerge,with buildup of Majorana correlation starting to domi-nate over decoherence. Importantly, the existence of acritical σ indicates a critical memory depth, above which -0.500.510 0.5 1 θ ( t ) U = 0.002U = 0.000 U = -0.002U = -0.004
Figure 3. Non-Markovian Majorana dynamics for renormal-ized Hamiltonian H p,s in the presence of weak p -wave interac-tion, H int = U P N − l =1 ( c † l c l − / c † l +1 c l +1 − / the system asymptotically approaches a new polaronicsteady-state, in dynamical equilibrium with phonons at T = 4 K, which can remarkably exhibit θ ≈
1. The crit-ical value of σ in our case is between 0 .
21 and 0 .
20 cor-responding to h B i = 0 .
01, but it is model specific. Wenote that the fermion-phonon coupling bandwidth canbe controlled, such as in solid state setups by nanotech-nological design, e.g. alloys, impurities and confinementpotentials [45].
Concluding discussions.
The central results of ourwork shown in Fig. 2 are found to be robust for initiallynon-ideal Kitaev chains and other forms of superohmiccoupling (Suppl. Mat.). Moreover, we find that the pres-ence of a weak attractive p -wave interaction can stronglysuppress the phonon-induced short-time parity loss, thusaugmenting the buildup of Majorana correlation for sub-critical memory depth. In Fig. 3, we calculate the non-Markovian evolution of θ ( t ) by including a weak p -waveinteraction, H int = U P N − l =1 ( c † l c l − / c † l +1 c l +1 − / | U | ≪
1, in H p,s of Eq. (3) for σ = 0 .
6. Compared to the U = 0 case [blue line], adding aweak attractive interaction U < | U | , the asymptotic Majorana correlationcan be significantly increased [orange and red lines]. Anintuitive understanding can be obtained by noting thatthe attractive interaction is energetically favorable for co-herent formation of superconductive pairing, which pro-vides a mechanism to counteract aforementioned phonon-induced dephasing. This is consistent with the observa-tion that for U > θ ( t ) significantly declines from the U = 0 case at long times [purple line], as repulsive inter-actions energetically suppress pairing.Summarizing, we have demonstrated memory-criticaledge dynamics in a topological superconductor with non-Markovian interaction with phonons. We show this in-triguing phenomenon uniquely arises from the interplaybetween the phonon-renormalized topological Hamilto-nian and the quantum memory effect that simultaneouslyinduces dephasing and information backflow. Our analy-sis is based on the Kitaev chain, but we expect the essen-tial physics to occur for a wide class of topological materi-als coupled to a superohmic reservoir. These discussionsare different from recent work [46–49], e.g., where an im-purity or a qubit acts as a non-Markovian quantum probeof a topological reservoir. Our result is relevant to ongo-ing efforts aimed at realizing topological computations inexperimentally realistic conditions where non-Markovianeffects are inevitable. It further opens an appealing newprospect as to the explorations and control of memory-dependent topological phenomena.O.K. and A.C. gratefully acknowledge support fromthe Deutsche Forschungsgemeinschaft (DFG) throughSFB 910 project B1 (project number 163436311). Y.H.acknowledges National Natural Science Foundation ofChina (Grant No. 11874038). ∗ [email protected][1] N. Lindner, G. Refael, and V. Galitski,Nature Physics , 490 (2011).[2] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner,E. Berg, I. 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