Non-Markovian steady states of a driven two-level system
NNon-Markovian steady states of a driven two-level system
Andreas Ask and G¨oran Johansson
Department of Microtechnology and Nanoscience (MC2),Chalmers University of Technology, SE-41296 G¨oteborg, Sweden (Dated: February 23, 2021)We show that an open quantum system in a non-Markovian environment can reach steady statesthat it cannot reach in a Markovian environment. As these steady states are unique for the non-Markovian regime, they could offer a simple way of detecting non-Markovianity, as no informationabout the system’s transient dynamics is necessary. In particular, we study a driven two-levelsystem (TLS) in a semi-infinite waveguide. Once the waveguide has been traced out, the TLS seesan environment with a distinct memory time. The memory time enters the equations as a time delaythat can be varied to compare a Markovian to a non-Markovian environment. We find that somenon-Markovian states show exotic behaviors such as population inversion and steady-state coherencebeyond 1 / √
8, neither of which is possible for a driven TLS in the Markovian regime, where the timedelay is neglected. Additionally, we show how the coherence of quantum interference is affected bytime delays in a driven system by extracting the effective Purcell-modified decay rate of a TLS infront of a mirror.
There are no truly closed quantum systems. In one wayor another, a quantum system is always in contact witha noisy environment and will inevitably lose its quan-tum properties [1]. If the dynamics are Markovian in na-ture, the environment can be considered memoryless, andthere is no back-flow of information. Such systems aredescribed by a quantum dynamical semi-group, whosegenerator governs a master equation in Lindblad form[2, 3]. In many realistic systems, the requirements fora Markovian time evolution, such as weak interactionand short environment correlation-times, are not satis-fied, and long-time memory effects of the environmentinfluence the system dynamics. In what ways such in-teraction affects the evolution of a quantum system isnot only interesting from a fundamental perspective, butcould also prove useful to probe properties of the environ-ment [4, 5], and ultimately lead to a better understandingof the decoherence-mechanisms of quantum systems [6].Although one cannot translate the classical definitionof a Markov process directly to the quantum regime [7],several definitions and corresponding measures of non-Markovianity for open quantum systems have been intro-duced [8–13]. These measures are all constructed to de-tect deviations from Markovianity by characterizing thesystem’s transient dynamics. Irrespective of definition,it has remained an open question if non-Markovianitycan be seen in the steady state of driven systems. Thatwould not only simplify the characterization of non-Markovianity in such cases, but it is also an interestingquestion in itself.In this Letter, we show that an open quantum sys-tem coupled to a non-Markovian environment can reacha unique set of steady states that are out of reach for thesystem coupled to a Markovian environment. We callthese states “non-Markovian steady states”, as they canbe distinguished from any state in the Markovian regime.To quantify these steady states’ uniqueness, we propose a measure based on trace distance and distinguishabil-ity [14]. We note that our measure does not attempt toquantify the degree of non-Markovianity in these systemsbut rather gives a quantitative measure on how easily onecan distinguish these states from the states in the Marko-vian regime.To demonstrate when non-Markovian steady states canoccur, we study a driven two-level system (TLS) in asemi-infinite waveguide (an atom in front of a mirror)[15–23], see Fig. 1. The drive amplitude and the sys-tem’s coupling strength to the waveguide are taken asfixed parameters throughout the system evolution. Oncethe waveguide has been traced out, the distance to themirror gives the environment seen by the TLS a distinctmemory time. This memory time enters the equationsfor the system dynamics as a time delay, which can beset to zero for comparison between a Markovian and anon-Markovian environment. Thus, the physical originof any non-Markovian effects in this system has an easyinterpretation in terms of coherent quantum feedback.Additionally, an atom in front of a mirror has been real-ized with both artificial and natural atoms in a varietyof systems already [24–30], so the physics discussed inthis letter could be further investigated experimentallyimmediately. We also note that a similar system to theatom in front of a mirror (in fact, they are fully map-pable to each other in some parameter regimes) is thegiant atom [31–35], which was recently realized in boththe Markovian [36, 37] and non-Markovian regime [38].Despite being an archetypical quantum-optical model-system for decades [15], the atom in front of a mirror hasremained a hard system to simulate without resorting tosubstantial approximations. The propagation time-delaybetween the atom and the mirror prohibits a treatmentbased on Markovian master equations, and earlier workon non-Markovianity has been limited to either few exci-tations [39, 40] or short time scales [41, 42]. It was onlyrecently that Pichler et al. [21] proposed a method basedon Matrix Product States (MPS) which could allow the a r X i v : . [ qu a n t - ph ] F e b FIG. 1. (a) Schematic of an atom in front of a mirror. Photonsemitted to the left is reflected from the mirror and interactswith the atom again, giving the atom’s environment an ef-fective memory time. (b) Representation of one time step inthe evolution of the setup in (a) as a MPS. The waveguide isrepresented by time bins (gray), which moves together in aconvey belt fashion one time step, ∆ t , at a time, and interactswith the atom (turquoise) twice. system to be integrated all the way to steady state, whilestill allowing for many excitations in the feedback loopand long delay times. There, it was shown that the timedelay does in fact alter the steady state of the atom.However, that does not imply that the state is unique for the non-Markovian regime, i.e., that the same statecannot be reached using a different drive strength and ne-glecting the time delay. It could also happen that a non-Markovian environment reduces the purity of the steadystate, in which case the effect of the non-Markovian en-vironment could be captured by adding additional puredephasing to a fully Markovian treatment. In fact, wefind that the non-Markovian environment mostly pro-duces steady states one cannot distinguish from thosereachable in the Markovian environment. For some sys-tem parameters, however, we find non-Markovian steadystates that are not only unique for the non-Markovianregime but also show exotic behaviors such as popula-tion inversion in the TLS or steady-state coherence be-yond 1 / √
8, neither of which is possible in the Markovianregime.
Definition of non-Markovian steady states : In theMarkovian regime, the dynamics of the TLS is given bythe Markovian master equation in Lindblad form [2],˙ ρ M = − i [ H TLS , ρ ] + γ (cid:0) σ − ρσ + − { σ + σ − , ρ } (cid:1) + γ φ (cid:0) σ + σ − ρσ + σ − − { σ + σ − , ρ } (cid:1) , (1)where γ φ is a pure dephasing rate, γ is a renormalized de-cay rate due to the mirror γ = 2 γ (cid:48) cos( φ ), where γ (cid:48) is thebare decay rate (without the mirror), φ is a phase shift, H TLS = ∆ σ + σ − + Ω2 ( σ + + σ − ), where ∆ = ω d − ω is thedetuning between the TLS transition frequency ω and the drive frequency ω d , Ω is the amplitude of the drivingfield, and σ + ( σ − ) creates (annihilates) an excitation inthe TLS. We always consider resonant driving through-out the paper, ∆ = 0. The solutions to Eq. (1) yieldsan elliptical area in the Bloch sphere of possible steadystates, whose outer boundary is given by Ω /γ = [0 , ∞ ]and γ φ = 0, see Fig. 2. The mirror is thus irrelevant fordetermining possible steady states; it only re-scales thedecay rate. We let ρ M ( γ, γ φ ) denote the steady state so-lution to Eq. (1) for a fixed drive strength, and ρ denotethe steady-state reduced density matrix of the TLS ina non-Markovian environment. Then, the ability to dis-tinguish the non-Markovian regime from the Markovianregime can be captured by N ss = min γ,γ φ T [ ρ, ρ M ( γ, γ φ )] , (2)where T [ ρ, ρ M ] = Tr (cid:113) ( ρ − ρ M ) † ( ρ − ρ M ) is the tracedistance between ρ and ρ M . As the trace distance isclosely related to the distinguishability of quantum states[14, 43], we define a steady state as non-Markovianif N ss >
0. With this definition, a non-Markoviansteady state is a state that is unique for the system ina non-Markovian environment. For the atom in frontof a mirror, we show that most non-Markovian steadystates would correctly be classified as belonging to anon-Markovian system according to the definition of non-Markovianity in Ref. [9]. However, we note that a largedegree of non-Markovianity does not necessarily corre-spond to a large N ss . FIG. 2. Markovian versus non-Markovian regimes of steadystates for a driven TLS in front of a mirror. By neglectingthe time delay, the system evolves according to a Markovianmaster equation and can only reach states lying either on thesolid black line (for no pure dephasing) or its inside (withdephasing). If the time delay is taken into account the sys-tem can reach steady states which are, e.g., outside of theMarkovian regime (green crosses), precisely on the boundarybetween the two regimes (magenta triangles), or well insidethe Markovian regime (blue circles). In all cases the followingparameters were used: γ = γ L + γ R = 1, γ L/R = γ/
2, andΩ /γ = [0 . ,
4] (Ω /γ = [0 . , .
5] for φ = π/ Model:
To model a non-Markovian environment, weput the driven TLS in a semi-infinite waveguide, seeFig. 1(a). After the waveguide has been traced out, thedistance to the mirror gives the environment of the TLSa distinct memory time, τ . Emission towards the mir-ror enters a coherent feedback loop, in which it pick upsa propagation phase, φ , that in our calculation includesany extra phase shift imposed by the field’s boundarycondition at the mirror. In a frame rotating with thedrive frequency, ω d , the total Hamiltonian has two partsin the interaction picture H ( t ) = H TLS + H int ( t ), wherethe interaction Hamiltonian H int ( t ) = i (cid:16) √ γ L b † L ( t ) + √ γ R b † R ( t ) (cid:17) σ − + H.c. , (3)can be rewritten in terms of a single bath operator, b ( t ), since the mirror couples left and right-going modes b R ( t ) = b L ( t − τ ) e iφ , H int ( t ) = i (cid:0) √ γ L b † ( t ) + √ γ R b † ( t − τ ) e iφ (cid:1) σ − + H.c. , (4)where γ L and γ R denotes the decay rate into left (L) andright (R)-going modes in the waveguide, respectively, φ is the phase shift acquired by a photon (or phonon) trav-eling to the mirror and back. The phase shift is in factrelated to the drive-frequency, φ = ω d τ , but we keepit as an independent variable in order to study the ef-fect of the phase shift and delay time separately. Thebosonic operator, b ( t ), obeys the quantum white-noisecommutation-relation [ b ( t ) , b † ( t (cid:48) )] = δ ( t − t (cid:48) ), and is de-fined as b ( t ) = (cid:82) B dωb ( ω )exp[ − i ( ω − ω d ) t ], where b † ( ω )[ b ( ω )] creates [annihilates] a photon at frequency ω , satis-fying the commutation relation [ b ( ω ) , b † ( ω (cid:48) )] = δ ( ω − ω (cid:48) ).A full derivation of the interaction Hamiltonian in Eq. (4)can be found in Ref. [21]. The interpretation, however,is clear: the TLS interacts with a bosonic bath at time t ,after some time τ the TLS interacts with the bath again,the state of the bath at this later time has to be the stateof the bath at an earlier time t − τ , taking into accountthe traveling phase acquired during this time.The system dynamics is calculated by solving the quan-tum stochastic Shr¨odinger equation (QSSE) i ddt | Ψ( t ) (cid:105) = H ( t ) | Ψ( t ) (cid:105) , (5)using the MPS method formulated in Ref. [21]. Ma-trix product states have shown to be efficient represen-tations of 1-D many-body systems [44–50]. Solving theQSSE is turned into a many-body problem by discretiz-ing time, t n = n ∆ t , turning Eq. (5) into a dynamicalmap | Ψ( t n +1 ) (cid:105) = U n | Ψ( t n ) (cid:105) . Throughout the paper weuse a time step much smaller than all other time scalesinvolved: ∆ t (cid:28) { /γ, / Ω } . The state of the field in thewaveguide is represented by time bins, with associatedbosonic noise increments ∆ B ( t n ) = (cid:82) t n +1 t n b ( t ) dt , whichfulfills the commutation relation [∆ B ( t n ) , ∆ B † ( t n (cid:48) )] =∆ tδ n,n (cid:48) . The operator ∆ B † ( t n ) can thus be seen as acreation operator for time bin n , with a corresponding Fock state defined as: | n (cid:105) l ≡ (∆ B † l ) n √ n !∆ t n | vac (cid:105) l . The unitary, U n , is written in this time-bin formulation as U n = exp (cid:2) iH TLS ∆ t + ( √ γ L ∆ B † ( t n ) σ − + √ γ R ∆ B † ( t n − k ) e iφ σ − − H.c.) ] , (6)here t k = k ∆ t is the feedback time τ . The total quantumstate at time t n , for both the bath and the TLS, is thenwritten as | Ψ( t n ) (cid:105) = (cid:88) i T ,i ,...,i N Ψ i T ,i ,...,i N ( t n ) | i T (cid:105) ⊗ | i (cid:105) ⊗ . . . ⊗ | i N (cid:105) , (7)where t N = N ∆ t is the total integration time, i T denotesthe state of the TLS, and i j is the photon number intime-bin j . The initial state is written as an MPS ansatzΨ i S ,i ,...,i N ( t ) = M [ S ] i T M [1] i . . . M [ N ] i N , (8)where M [ j ] i j is a matrix of dimension D j × D j +1 . Themaximum matrix dimension D max in the MPS chain isreferred to as the bond dimension and sets an upper limitto the amount of entanglement in the system, which inour system depend on the length of the feedback loop[21]. We use a bond dimension of D max = 32 for themoderate time delays considered here. The state ampli-tudes are updated in each time step using standard MPStechniques [44, 51]. FIG. 3. Non-Markovianity of the steady state evaluated usingthe measure introduced in Eq. (2). A phase shift of φ = 0 wasused unless it is stated otherwise in the figure. A large N ss means that the state is easily distinguishable from any statein the Markovian regime. Non-Markovian regime:
Two parameters are impor-tant for quantifying the significance of the feedback: γτ ,where γ = γ L + γ R , and Ω τ . Only when both γτ (cid:28) τ (cid:28) γτ determines thenon-Markovian properties alone.We first study the effect of the time delay, and set φ = 0. The feedback is thus in phase with the driveand they interfere constructively. In all calculations thatfollows we use γ = 2 γ L = 2 γ R = 1. In Fig. 2 we plotthe steady-state reduced density-matrix elements of theTLS for drive-strengths in the range Ω /γ = [0 . ,
4] for γτ = 0 . γτ = 3 (magenta tri-angles). The trace distance to the closest Markoviansteady state, N ss [Eq. (2)], is plotted for a greater va-riety of time delays in Fig. 3. From these two figures wemake the following observations: (i) When the drivingis weak, the steady state cannot be distinguished from aMarkovian steady state, independent of time delay. (ii)The states start to deviate from the Markovian regimeinitially for γτ >
0, reaches a maximum deviation at γτ ≈ .
5, and then starts to approach the Markovianregime again. For sufficiently long time delays the statescannot be distinguished from a Markovian state anymore.(iii) For γτ = 0 . ρ ee > /
2, for sufficiently strong driving, and largercoherence than what is possible in the Markovian regime, | ρ eq | > / √
8. The oscillatory behavior that can be seenin both Fig. 2 and Fig. 3 is due to the additional phaseshift, Ω τ , induced by the drive, primarily noticeable forthe longer time delays.When the phase shift deviates from 0 mod 2 π , thestates either approaches or falls well inside the Marko-vian regime (blue circles in Fig. 2). Inside the Markovianregime N ss = 0 per definition.Coherent quantum interference phenomena play an im-portant role in many applications in waveguide QED. Itis well known, e.g., that the mirror doubles the TLS’ de-cay rate to the waveguide due to the Purcell effect in theMarkovian regime (for φ = 0). Increasing the distanceto the mirror reduces the coherence of the radiation thatcomes back to the TLS. The longer time it takes for thefeedback to come back, the higher chance for spontaneousemission to occur in the TLS. For long enough time delay,the TLS should behave as if it was positioned in an infi-nite waveguide instead, without the mirror present. Thisis precisely what we observe in Fig. 4. We extract the“effective” decay rate by calculating the ratio betweenthe output time-bin population and the TLS population, γ eff = (cid:104) ∆ B † out ∆ B out (cid:105) ss /ρ ee . For weak driving we observethat the decay rate is not affected by the time delay, asthe drive strength increases, however, it approaches theexpected value of γ . We note that the effective decayrate could have been extracted from the master equationin Eq. (1) if negative dephasing rates were allowed. Infact, all the non-Markovian steady states seen in Fig. 2could be described by the master equation with a nega-tive dephasing rate. Since negative dephasing rates havebeen used to describe temporary increases in quantumcoherence for non-Markovian systems [52, 53], we find itinteresting to note that such effects can persist all theway to the steady state. Non-Markovianity measure of the transient dynamics:
Finally, we make the remark that maximizing N ss isnot necessarily the same thing as maximizing the non-Markovianity of the system in a traditional sense. Forthis argument, we compare N ss to the measure proposed FIG. 4. Effective decay rate for φ = 0 as a function of drivestrength for various time delays. . . . γτ N Ω /γ = 0 . /γ = 1 . /γ = 1 . FIG. 5. Non-Markovianity according to the measure inEq. (9) as a function of time delay. Dashed lines are forthe same drive strengths as written in the figure but for φ = π/
2. By comparing with Fig. 2 we conclude that a largenon-Markovianity does not correspond to a steady state thatcan be distinguished from the Markovian regime. in Ref. [9], N = max ρ , (cid:90) σ> dtσ ( t, ρ , ) , (9)where σ ( t, ρ , ) = ddt T [ ρ ( t ) , ρ ( t )], and T [ ρ , ρ ] denotesthe trace distance. The maximum is taken over all pairsof initial states. For our system, we can safely choosethe ground and excited states as the two initial states[54]. We plot N as a function of time delay for variousdrive-strengths in Fig. 5 for both φ = 0 (solid lines) and φ = π/ N ss for γτ ≈ . −
1, which is barely non-Markovianaccording to N , whereas longer time delays and strongerdriving increases N significantly. Conclusion:
We have introduced the concept of “ non-Markovian steady states ” as a set of steady states uniquefor open quantum systems in a non-Markovian environ-ment. The system cannot reach these states if it is cou-pled to a Markovian environment and could thus offera simple way of detecting non-Markovianity as only asteady state measurement is required. Moreover, we in-troduced an appropriate measure for these states’ unique-ness based on the trace distance to the closest Marko-vian steady state. As an example, we show that non-Markovian steady states occur in a driven TLS in a semi-infinite waveguide. Among the non-Markovian steadystates, we find states with population inversion in theTLS, or steady state coherence larger than 1 / √
8, twoimpossible scenarios in the Markovian regime. Addition-ally, we showed that time delay could have a detrimentaleffect on coherent quantum interference in waveguides byextracting the effective Purcell-modified decay-rate as a function of time delay and drive strength.
Acknowledgments:
We thank Arne L. Grimsmo forvaluable discussions, and acknowledge funding from theSwedish Research Council (VR) through Grant No. 2017-04197. G.J. also acknowledges funding from the Knutand Alice Wallenberg foundation (KAW) through theWallenberg Centre for Quantum Technology (WACQT). [1] H. P. Breuer and F. Petruccione,
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