Renyi Entropy Dynamics and Lindblad Spectrum for Open Quantum System
RR´enyi Entropy Dynamics and Lindblad Spectrum for Open Quantum System
Yi-Neng Zhou, ∗ Liang Mao,
1, 2, ∗ and Hui Zhai † Institute for Advanced Study, Tsinghua University, Beijing 100084, China Department of Physics, Tsinghua University, Beijing 100084, China (Dated: January 28, 2021)In this letter we point out that the Lindblad spectrum of a quantum many-body system displaysa segment structure and exhibits two different energy scales in the strong dissipation regime. Oneenergy scale determines the separation between different segments, being proportional to the dis-sipation strength, and the other energy scale determines the broadening of each segment, beinginversely proportional to the dissipation strength. Ultilizing a relation between the dynamics of thesecond R´enyi entropy and the Lindblad spectrum, we show that these two energy scales respectivelydetermine the short- and the long-time dynamics of the second R´enyi entropy starting from a genericinitial state. This gives rise to opposite behaviors, that is, as the dissipation strength increases, theshort-time dynamics becomes faster and the long-time dynamics becomes slower. We also interpretthe quantum Zeno effect as specific initial states that only occupy the Lindblad spectrum aroundzero, for which only the broadening energy scale of the Lindblad spectrum matters and gives rise tosuppressed dynamics with stronger dissipation. We illustrate our theory with two concrete modelsthat can be experimentally verified.
For a closed quantum system, the energy spectrums ofHamiltonian fully determine the time scales of its dynam-ics. For an open quantum system, when the environmentis treated by the Markovian approximation, the couplingsbetween system and environment are controlled by a setof dissipation operators. In this case, the dynamics of thesystem is governed by the Lindblad equation which con-tains the contributions from both the Hamiltonian andthe dissipation operators [1]. Obviously, the spectrum ofthe Hamiltonian alone can no longer determine the timescales of the entire dynamics, and a natural question isthen what energy scales set the time scales of dynamicsof an open quantum system.There are various directions to approach this issue, andthe answer also relies on what type of dynamics that weare concerned with. Here let us focus on the dissipationdriven dynamics. There are still different physical intu-itions from different perspectives. One intuition is fromthe perturbation theory when the dissipation strength isweaker compared with the typical energy scales of theHamiltonian [2]. In this regime, by treating the dis-sipation perturbatively, it leads to a scenario that thedissipation dynamics becomes faster when the dissipa-tion strength is stronger. Another intuition is from thestudies of the quantum Zeno effect [3–6], which statesthat frequent measurements can slow down the dynam-ics, provided that the typical time interval between twosuccessive measurements are shorter than the intrinsictime scale of the system. Since the measurement canalso be understood in term of dissipations in the Lind-blad master equation, it provides another scenario thatthe dissipation dynamics is suppressed when the dissipa-tion becomes stronger, in the regime that the dissipationstrength is stronger compared with the typical energyscales of the Hamiltonian. It seems that these two sce-narios respectively work on different parameter regimes
FIG. 1: Schematic of the mapping between the Lindblad equa-tion (left) and the Sch¨odinger like equation in a doubled sys-tem (right). Here ˆ L ˆ ρ denotes the r.h.s. of Eq. 1. and the results are also opposite to each other. It will beinteresting to see that there actually exists a frameworkthat can unify these two scenarios.When a system is coupled to a Markovian environ-ment, the entropy of the system will increase in time.The entropy dynamics of an open quantum many-bodysystem is a subject that attracts lots of interests recently[7–12]. In this letter, we address the issue of typical timescales of the entropy increasing dynamics of a quantummany-body system coupled to a Markovian invironment,and especially, we should focus on the second R´enyi en-tropy, for the reason that will be clear below, and answerthe question whether the entropy dynamics is faster orslower when the dissipation strength increases.Our studies are based on a mapping between theLindblad master equation and a non-unitary evolutionof wave function in a doubled space, as shown in Fig.1. Let us first review this mapping [13, 14]. Con-sidering a density matrix ˆ ρ , and given a set of com-plete bases {| n (cid:105)} , ( n = 1 , . . . , D H ) of the Hilbert spacewith dimension D H (say, the eigenstates of the Hamil-tonian ˆ H with eigenenergies E n ), the density matrixˆ ρ can be expressed as ˆ ρ = (cid:80) mn ρ mn | m (cid:105)(cid:104) n | . By theoperator-to-state mapping, we can construct a wave func- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n tion Ψ ρ = (cid:80) mn ρ mn | m (cid:105) ⊗ | n (cid:105) , which contains exact thesame amount information as ˆ ρ . Here Ψ ρ is a wave func-tion on a system whose size is doubled compared to theoriginal system, and we will refer these two copies of origi-nal system as the “left” (L) and the “right” (R) systems.Under this mapping, for instance, a density matrix ofa pure state ˆ ρ = | ψ (cid:105)(cid:104) ψ | is mapped to a product stateΨ ρ = | ψ (cid:105) ⊗ | ψ (cid:105) in the double system, and a thermal den-sity matrix at temperature T as ˆ ρ = (cid:80) n e − E n / ( k b T ) | n (cid:105)(cid:104) n | is mapped to a thermofield double state at temperature T / ρ = (cid:80) e − E n / ( k b T ) | n (cid:105) ⊗ | n (cid:105) in the double system.For an open system coupled to a Markovian environ-ment, the density matrix obeys the Lindblad masterequation given by (cid:126) d ˆ ρdt = − i [ ˆ H, ˆ ρ ] + (cid:88) µ γ µ (cid:16) L µ ˆ ρ ˆ L † µ − { ˆ L † µ ˆ L µ , ˆ ρ } (cid:17) , (1)where ˆ L µ stand for a set of dissipation operators, and γ µ are their corresponding dissipation strengths. Afterthe mapping, the wave function Ψ ρ in the double systemsatisfies a Schr¨odinger-like equation i (cid:126) d Ψ ρ dt = (cid:16) ˆ H s − i ˆ H d (cid:17) Ψ ρ . (2)Here ˆ H s is the Hermitian part of the Hamiltonian deter-mined by system itself, and it is given byˆ H s = ˆ H L ⊗ ˆ I R − ˆ I L ⊗ ˆ H TR , (3)where operators with subscript “L” and “R” respectivelystand for operators acting on the left and the right sys-tems, and “T” stands for the transpose, and ˆ I representsthe identity operator. − i ˆ H d is the non-Hermitian part ofthe Hamiltonian determined by the dissipation operators,which is given byˆ H d = (cid:88) µ γ µ (cid:104) − L µ, L ⊗ ˆ L * µ, R +( ˆ L † µ ˆ L µ ) L ⊗ ˆ I R + ˆ I L ⊗ ( ˆ L † µ ˆ L µ ) *R (cid:105) , (4)where the superscript * stands for taking complex conju-gation. We can diagnolize this non-Hermitian Hamilto-nian ˆ H s − i ˆ H d , which leads to a set of eigenstates as( ˆ H s − i ˆ H d ) | Ψ lρ (cid:105) = (cid:15) l | Ψ lρ (cid:105) , (5)where (cid:15) l is in general a complex number, and we denotethem as (cid:15) l = α l − iβ l . This spectrum, originated from theLindblad equation, is referred to as the Lindblad spec-trum. The full Lindblad spectrum has been studied for anumber of models before [15–20]. Here we would like tomake several useful comments on the Lindblad spectrum.i) α l and − α l always appear in pairs in the spectrum; ii) β l is always non-negative; iii) If ˆ L µ are all hermitian,there always exists a zero-energy eigenstate with (cid:15) l = 0, FIG. 2: The dynamics of the second R´enyi entropy S (2) as afunction of tγ . γ is the dissipation strength. Different curveshave different γ in unit of J . The inset show the long-timebehavior of S (2) as functions of tJ and tJ /γ . The dashedline is a fitting of initial slop based on Eq. 13. (a) is for theBose-Hubbard model with U = J and the number of sites L = 6, and the number of bosons N = 3. (b) is for hard corebosons model with V = J , L = 8 and N = 4. The initialstate is taken as the ground state of ˆ H . and this eigenstate is labelled as l = 0 and is given by | Ψ l =0 ρ (cid:105) = √D H (cid:80) n | n (cid:105) ⊗ | n (cid:105) . R´enyi Entropy and Lindblad Spectrum.
Here we bringout a close relation between the dynamics of the secondR´enyi entropy and the Lindblad spectrum. For any den-sity matrix ˆ ρ ( t ), the second R´enyi entropy S (2) ( t ) is givenby e − S (2) = Tr(ˆ ρ ) = (cid:88) mn ρ mn ( t ) ρ nm ( t ) . (6)On the other hand, in the double system, the total am-plitude of the wave function is given by | Ψ ρ | = (cid:88) mn ρ mn ( t ) ρ ∗ mn ( t ) . (7)Since the density matrix is always Hermitian, it gives ρ nm ( t ) = ρ ∗ mn ( t ), and therefore, we have e − S (2) = | Ψ ρ | . (8)An initial state Ψ ρ (0) in the double space can be ex-panded as Ψ ρ (0) = (cid:80) l c l | Ψ lρ (cid:105) , the subsequent evolutionis given byΨ( t ) = e − i ˆ H s t − ˆ H d t | Ψ ρ (0) (cid:105) = (cid:88) l c l e − iα l t − β l t | Ψ nρ (cid:105) (9)and therefore e − S (2) = | Ψ ρ | = (cid:88) n | c l | e − β l t . (10)Since the evolution in double system is non-unitary andall β l are non-negative, the total amplitude of the wavefunction always decays in time. Hence, by this entropy-amplitude relation Eq. 8, the decaying of | Ψ ρ | gives riseto the increasing of S (2) . Note that for any initial den-sity matrix with trace unity and for hermitian ˆ L µ , c l =0 always equals 1 / √D H . This mode always does not decayin time because β l =0 = 0. If there is no other eigenmodeswith β l = 0, l = 0 mode is the only remaining mode atinfinite long time, which gives a maximum second R´enyientropy log D H . Before reaching that limit, the imagi-nary parts of the Lindblad spectrum of occupied statesdetermine the time scales of the R´enyi entropy dynamics.Our discussion below will be based on this connection. Models.
Although our discussion below is quite generalfor quantum many-body systems, we illustrate the resultswith two concrete models. The first model is the Bose-Hubbard model, which readsˆ H = − J (cid:88) (cid:104) ij (cid:105) (ˆ b † i ˆ b j + h.c.) + U (cid:88) i ˆ n i (ˆ n i − , (11)where ˆ b i is the boson annihilation operator at site- i , andˆ n i = ˆ b † i ˆ b i is the boson number operator at site- i . (cid:104) ij (cid:105) denotes nearest neighbor sites. J and U are respectivelythe hopping and the on-site interaction strengths. Forthe second model, we consider hard-core bosons, whichprevent two bosons to occupy the same site. In addi-tion, we introduce the nearest-neighbor repulsion, andthe model readsˆ H = − J (cid:88) (cid:104) ij (cid:105) (ˆ b † i ˆ b j + h.c.) + V (cid:88) (cid:104) ij (cid:105) ˆ n i ˆ n j . (12)In one-dimension, these two models are quite different,because the second model can be mapped to a spinlessfermion model with nearest neighbor repulsion, and canalso be mapped to a spin model with nearest neighborcouplings, but the first model cannot. In both cases, wetake all ˆ n i as the dissipation operators and we set thedissipation strengthes uniformly as γ . In the numericalresults shown below, we have choose J ∼ U or J ∼ V such that J sets the typical energy scale of the Hamil-tonian part, and therefore, strong and weak dissipationsrespectively mean γ/J > γ/J <
1. Below we willshow that both models exhibit similar features, whichsupports that our results are quite universal.
FIG. 3: The Lindblad spectrum for strong dissipation case(a1,a2,b1,b2) with γ = 5 J and for weak dissipation case(c1,c2) with γ = 0 . J . The red points mark the eigenstateswith significant occupation ( | c l | (cid:62) / D H ) by the initial state.For (a1) and (a2) in the first raw, the initial state is taken asΨ ρ = | ψ g (cid:105)⊗| ψ g (cid:105) , where | ψ g (cid:105) is the ground state of ˆ H . For (b1)and (b2) in the second raw, the initial states are taken as thezero-energy eigenstate of ˆ H d , that are | (cid:105) for (b1) and | (cid:105) for (b2) in Fock bases. The left column (a1,b1,c1)are for the Bose-Hubbard model with U = J and the num-ber of sites L = 6, and the number of bosons N = 3. Theright column (a2,b2,c2) are for hard core bosons model with V = J , L = 8 and N = 4. Dynamics of the R´enyi Entropy.
We first consider theshort-time behavior of the R´enyi entropy dynamics. Weapply the short-time expansion to Eq. 9 and ultilizethe relation Eq. 8, and to the leading order of entropychange, we obtainlim t → dS (2) dt = 2 (cid:104) Ψ ρ (0) | ˆ H d | Ψ ρ (0) (cid:105)(cid:104) Ψ ρ (0) | Ψ ρ (0) (cid:105) . (13)The physical meaning of the r.h.s. of Eq. 13 in originalsystem is the fluctuation of the dissipation operators. Forinstance, if the initial state is a pure state and ˆ ρ (0) = | ψ (0) (cid:105)(cid:104) ψ (0) | , then | Ψ ρ (0) (cid:105) = | ψ (0) (cid:105) ⊗ | ψ (0) (cid:105) , and Eq. 13can be rewritten aslim t → dS (2) dt =4 (cid:88) µ γ µ (cid:16) (cid:104) ψ (0) | ˆ L † µ ˆ L µ | ψ (0) (cid:105) − |(cid:104) ψ (0) | ˆ L µ | ψ (0) (cid:105)| (cid:17) . (14)Suppose all γ µ are taken as the same γ , this result showsthat the time-dependence of S (2) is governed by a dimen-sionless time γt . In other word, the larger γ is, the fasterthe R´enyi entropy dynamics increases. This γt scaling isshown in Fig. 2 for two different models, where one cansee that the short-time parts of S (2) curves with different γ collapse into a single line when plotted in term of γt .The dashed lines compare the short-time behavior withthe slope given by Eq. 13 and Eq. 14.In Fig. 2, one also finds that S (2) no longer obeysthe γt scaling when γt >
1. Moreover, in the strongdissipation regime, the insets plotted in term of tJ showan opposite trend at long-time, that is, the larger γ is,the slower the R´enyi entropy increases. In fact, the long-time behavior of S (2) exhibits a t/γ scaling. As shownin the insets of Fig. 2, when the long-time part of S (2) curves with different γ are ploted in term of tJ /γ , theyall collapse into a single curve. Lindblad Spectrum with Strong Dissipation.
This op-posite behavior between short- and long-time can be un-derstood very well in term of the Lindblad spectrum. Asone can see from Fig. 3(a,b), for strong dissipation, themain feature of the Lindblad spectrum is that it separatesinto segments along the imaginary axes of the spectrum,and the separation between segments are approximately2 γ . For each segment, the width along the imaginaryaxes is approximately given by J /γ . This feature canbe understand by perturbation treatment of ˆ H s − i ˆ H d .Since the dissipation strength is stronger than the typ-ical energy scales of the Hamiltonian, we can treat ˆ H s as a perturbation to ˆ H d . To the zeroth order of ˆ H d , thespectrum is purely imaginary and different segments areseparated by 2 γ . More importantly, it worth emphasizingthat the eigenstates of ˆ H d are usually highly degenerate,for instance, when different ˆ L µ commute with each otherand are related by a symmetry, such as ˆ L µ being ˆ n i inour examples. Usually, ˆ H s and ˆ H d do not commute witheach other, and the perturbation in ˆ H s lifts the degener-acy of the imaginary parts and gives rise to a broadeningof the order of J /γ , due to the nature of the secondorder perturbation.We call these eigenstates with imaginary energies ofthe order of a few times of γ as “high imaginary energystates”, and these eigenstates with imaginary energies ofthe order of a few times of J /γ as “low-lying imaginaryenergy states”. For a generic initial state, both two typesof eigenstates are occupied. Quite generally, the occupa-tions of the “high imaginary energy states” are signifi-cant, for instance, when the initial state is taken as theeigenstates of ˆ H s . With the relation between the R´enyientropy dynamics and the Lindblad spectrum discussedabove, it is clear that the short-time dynamics is domi-nated by these “high imaginary energy states” that givesa dynamics scaled by tγ . Nevertheless, when γt >
1, theweights on these “high imaginary energy states” mostly
FIG. 4: The dynamics of the second R´enyi entropy S (2) as afunction of tJ /γ for specific initial state. γ is the dissipationstrength. Different curves have different γ in unit of J . Theinset show the short-time behavior of S (2) as functions of tJ and tγ . (a) is for the Bose-Hubbard model with U = J and thenumber of sites L = 6, and the number of bosons N = 3. (b)is for hard core bosons model with V = J , L = 8 and N = 4.The initial states are taken as the zero-energy eigenstate ofˆ H d , that are | (cid:105) for (a) and | (cid:105) for (b) in Fockbases. decay out and the long-time dynamics is therefore dom-inated by the “low-lying imaginary energy states” thatgives a dynamics scaled by tJ /γ . Quantum Zeno Effect Revisited.
Here we consider aspecific initial state that satisfies ˆ H d | Ψ(0) (cid:105) = 0. In otherword, such initial states do not exhibit fluctuation ofdissipation operators. Thus, according to Eq. 13 andEq. 14, the initial slop of S (2) is zero. Moreover, in thestrong dissipation regime, the populations of the “highimaginary energy states” are strongly suppressed by the“gap” between different segments and their contributionbecomes negligible, and such initial states mainly popu-late the “low-lying imaginary energy states”, as we shownin Fig. 3(b). Therefore, the entire dynamics of the sec-ond R´enyi entropy is set by the energy scale J /γ andit obeys the t/γ scaling. This is shown in Fig. 4 fortwo models. To contrast such specific initial states withgeneric states discussed above, we plot in the inset of Fig.4 the short-time behavior of S (2) as a function of tγ and tJ . Unlike the results shown in Fig. 2, the short-timedynamics with tγ < tγ .For these initial states, that the dynamics is slowerwith stronger dissipation is reminiscent of the quantumZeno effect. In fact, the quantum Zeno effect can indeedbe understood in this way. Introducing {| M (cid:105)} , ( M =1 , . . . , D H ) as a set of complete and orthogonal mea-surement bases, we define the projection operators asˆ P M = | M (cid:105)(cid:104) M | , and the frequent measurement processcan also be described by the Lindblad equation Eq. 1with dissipation operator ˆ L µ given by all ˆ P M . With suchdissipation operators, the Lindblad spectrum exhibits aset of “low-lying imaginary energy states” with energyscale given by J /γ . It can be shown that, as long as theinitial state density matrix is diagonal in the measure-ment bases, the initial states satisfy ˆ H d | Ψ(0) (cid:105) = 0.
From Strong to Weak Dissipation.
Finally we showthat when γ decreases and eventually becomes weakercompared with the typical energy scales in the Hamilto-nian, the segments structure in the Lindblad spectrumdisappears, as we shown in Fig. 3(c). Thus, the entropydynamics for generic states no longer display the featureof two time scales. The quantum Zeno effect also disap-pears even for the specific initial states, and this is un-derstandable because in this regime, the typical time in-terval between two measurements is already longer thanthe intrinsic evolution time of the system. Summary.
In this work, we establish a relation be-tween the R´enyi entropy dynamics and the Lindbladspectrum in double space. At the strong dissipationregime, the Lindblad spectrum exhibits a segment struc-ture, in which we can introduce the “high imaginary en-ergy eigenstates” and the “low-lying imaginary energyeigenstates”. For a generic initial state with significantlyoccupied “high imaginary energy eigenstates”, the formerdominates the short-time dynamics and the latter dom-inates the long-time dynamics, which respectively giverise to tγ scaling and t/γ scaling. For a specific initialstate with only “low-lying imaginary energy eigenstates”significantly occupied, the dynamics is dominated by t/γ scaling, and we show the quantum Zeno effect belongsto this class. We illustrate our results with two concretemodels. The second R´enyi entropy can now been mea-sured in ultracold atomic gases in optical lattices, and infact, it has been measured in the Bose-Hubbard modelwith or without disorder [21–23]. The dissipation oper-ators and their strenghes can also now be controlled inultracold atomic gases [24], our predictions can thereforebe verified directly in the experimental setup. Acknowledgment.
We thank Lei Pan, Tian-Shu Deng,Tian-Gang Zhou and Pengfei Zhang for helpful discus-sions. This work is supported by Beijing Outstanding Young Scientist Program, NSFC Grant No. 11734010,MOST under Grant No. 2016YFA0301600.
Note Added.
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