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Ergodicity and Mixing in Quantum Dynamics Dongliang Zhang(张东良),1 H. T. Quan (全海涛),1,2 and Biao Wu(吴飙)3,2,4,∗ 1School of Physics, Peking University, Beijing 100871, China 2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 3International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 4Wilczek Quantum Center, College of Science, Zhejiang University of Technology, Hangzhou 310014, China After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system’s energy levels and spacings. Our definitions are consistent with usual understanding of 6 ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings 1 are introduced. They are computed for right triangular billiards and the results indicate a very 0 close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue 2 that, besides ergodicity and mixing, there may exist a third class of quantum dynamics which is characterized by a maximized entropy. l u J 7 I. INTRODUCTION authorsinthesecondpaper[9]. Inourpaperwedoboth: 2 we first introduce our definitions for quantum ergodicity and mixing and then illustrate them with examples. Ergodicity and mixing are of fundamental importance ] h in statistical mechanics. Ergodicity justifies the use of c microcanonical ensemble and mixing ensures that a sys- e tem approach equilibrium dynamically [1]. However, it II. QUANTUM ERGODICITY AND MIXING m is difficult to prove with mathematical rigor that a clas- - t sical dynamical system is ergodic or mixing. As a result, A. History a the microcanonical ensemble in textbooks is still estab- t s lishedwithpostulates[2]. Moreimportantly,theconcept . Ergodicity was introduced by Boltzmann in 1871 as t of ergodicity and mixing is now obsolete in the follow- a a hypothesis to understand thermodynamics microscop- ing sense: they are defined for classical dynamics while m ically [1]. Mixing was first discussed by Gibbs [10] and the dynamics of microscopic particles are fundamentally its mathematical definition was introduced by von Neu- - quantum. To establish statistical mechanics with quan- d mann in 1932 [11]. Both concepts concern the long time n tum dynamics, we need to define ergodicity and mixing behavior of dynamical systems and are of fundamental o in quantum dynamics. importance to statistical mechanics. They are now the c In this work we define quantum ergodicity and mix- focus of a fully-developed branch of mathematics called [ ing using the structure of the system’s energy levels and ergodic theory[12, 13]. However, ergodicity and mixing 3 spacingswithoutanyassumption. Withtheearlyresults are becoming less interesting to physicists for two rea- v byvonNeumann[3,4]andReimann[5],itcanbeshown sons: (1) After decades of research with many meaning- 2 thatourdefinitions,whichappearverymathematical,do ful results[12, 13], it is still not rigorously proved that 1 5 leadtotheusualunderstandingofergodicityandmixing. many of the physical systems existing in nature are ei- 0 We introduce two parameters to characterize the degen- ther ergodic or mixing. (2) Both ergodicity and mixing 0 eracy in energy levels and spacings, respectively. They are only defined for classical systems while the micro- . are computed numerically for right triangular billiards, scopic particles are fundamentally quantum. Therefore, 1 0 whose classical dynamical properties have been studied it is imperative to define both ergodicity and mixing to 6 in great detail [6, 7]. The numerical results indicate that quantum systems. 1 thereisaverycloserelationbetweenquantumchaosand The first physicist who discussed quantum ergodicity v: quantumergodicormixing: mostofnon-integrablefinite was von Neumann. In 1929, von Neumann proved two i quantumsystemsarebothergodicandmixing. Itisclear inequalities, which he named quantum ergodic theorem X fromthisexamplethatasystemwhosequantumdynam- and quantum H-theorem [4], respectively. His ergodic r ics is ergodic may not be ergodic in the corresponding theoremensuresthatnotonlythelongtimeaverageofa a classical dynamics. macroscopic observable equals to its microcanonical en- We draw a parallel between our paper and Peres’s two semble average but also has small fluctuations. In other papers [8, 9]. Peres introduced his definitions for quan- words, the observable deviates considerably from its av- tum ergodicity and mixing in the first paper [8] then il- eragedvalueonlyrarely. So,byergodicityvonNeumann lustrated with examples these two concepts with his co- meantactuallybothergodicityandmixing. Interestingly, mixing as defined for classical dynamics was only introduced three years later in 1932 by von Neumann [11]. In addition, according to von Neumann’s H-theorem, once ∗Electronicaddress: [email protected] the quantum system dynamically relaxes to its equilib- 2 rium state, where macroscopic observables have small and expressed in the language of pure quantum mechan- fluctuations, this state also has a maximized entropy. icswithoutreferringtoclassicalmechanics. Neartheend VonNeumann’sresultshadbeencriticizedbymany[8, of this paper, based on a recent work [15], we argue that 14]. We share the view by Goldstein et al. [14] that we may be able to expand our definitions to include a the criticism was mostly misguided; von Neumann had third class of quantum dynamics, which is charactering captured the essence of quantum ergodicity and mixing by a maximized entropy. and his results are inspirational. Nevertheless, there do exit some issues with von Neumann’s results. Most of the variables involved in the two theorems are not com- B. Definitions putable in principle [15]. The reason is as follows. To prove his theorems, von Neumann introduced a coarse- Consider a quantum system with discrete eigen- graining,whichgroupsthePlanckcellsinquantumphase energies {E } and corresponding energy eigenstates n space into some big cells. All the microscopic states in {|φ (cid:105)}. n one group of Planck cells correspond to a single macro- scopicstate. Thiskindofcoarse-grainingiscertainlyrea- Ergodicity. A quantum system is ergodic if its eigen- sonable. However, no one knows how to technically es- energies satisfy tablish such many-to-one mapping between macroscopic state and microscopic state. This makes many of von δEm,En =δm,n. (I) Neumann’s variables in Ref. [4] uncomputable. Though there may be some revisions to the theorem [16], this Mixing. A quantum system is mixing if its eigen- difficulty is not overcome. energies satisfy both the above condition (I) and the following condition More recent definitions of ergodicity and mixing in quantum mechanics were given by Peres [8]. Peres re- δ =δ δ , for k (cid:54)=l,m(cid:54)=n. (II) calledthebehaviorofdynamicalvariablesinclassicaler- Ek−El,Em−En k,m l,n godicandmixingsystemsandexpectedthatthereshould Condition (I) indicates that there is no degenerate be analogous behavior in quantum ergodic and mixing eigenstate. Condition(II)impliesthatthereisnodegen- systems. In accordance with von Neumann’s results, eracy in energy gaps between any pair of eigen-energies. Peresdefinedergodicityasthetimeaverageofanyquan- Itisclearthataquantumsystemthatismixingmustbe tum operator equal to its microcanonical ensemble aver- ergodic, similar to classical dynamics. As we shall show age and mixing as any quantum operator having small in the following, condition (I) can lead to the usual intu- fluctuations. However, Peres’s definitions were based on itive understanding of ergodicity: the long time average his own definition of quantum chaos [8], which is a sub- equalstotheensembleaverage. Withbothconditions(I) jectofdebateitself. Todefinequantumchaos,Peresused and(II),onecanshowthattheso-definedmixingindeed an ambiguous concept of pseudorandom matrix. These means a small time fluctuation for an observable. two steps that are not mathematically very rigorous , Supposethatthequantumsystemisinaninitialstate alongwithotherreasonablebutambiguousassumptions, (cid:80) |ψ(0)(cid:105) = c |φ (cid:105). After evolving for a period of time render Peres’s definitions not satisfactory. n n n t,thequantumsystemisinastatedescribedby|ψ(t)(cid:105)= a dIniffleirteenrattpureersqpueacntitvuem, werhgeoreditchiteychoanscbepeetn“sqtuuadniteudmfroemr- (cid:80)ncne−iEnt/(cid:126)|φn(cid:105). ForanobservableAˆ, itsexpectation value at time t is given by godicity” is regarded as a branch of quantum chaos [17– 26]. Thisgroupofresearchersmainlyfocusedonhowthe (cid:104)Aˆ(t)(cid:105)≡(cid:104)ψ(t)|Aˆ|ψ(t)(cid:105)=trAˆρˆ(t), (1) eigenfunctions of a Hamiltonian converges to equidistri- bution in classical phase space in the semi-classical limit where ρˆ(t)≡|ψ(t)(cid:105)(cid:104)ψ(t)| is the density matrix at time t. or high energy limit with little discussion on dynami- Its long time average is cal behavior of a quantum system. Their definitions of quantum ergodicity and mixing rely on the correspond- 1 (cid:90) T imnegchclaansisciacal.l cSarseedsniacnkdi’sthwuosrkaroenneoitgegnefnuunicnteiolynqtuhaernmtuaml- (cid:104)Aˆ(cid:105)T ≡Tl→im∞T 0 (cid:104)Aˆ(τ)(cid:105)dτ. (2) izationfollowsalongthislineandhaslittlediscussionon We now introduce a density matrix quantum dynamics [27]. Inthefollowingsubsectionwedefinequantumergodic- ρˆ ≡(cid:88)|c |2|φ (cid:105)(cid:104)φ |. (3) mc n n n ity and mixing using the energy structure of the system, n that is, eigen-energies and their spacings. Our definitions are mathematically precise and have no assump- Thisdensitymatrixdoesnotchangewithtimeanditcan tions. Furthermore, by following von Neumann [3] and be regarded as describing a micro-canonical ensemble [3, Reimann [5], we can show that our definitions lead to 5, 15]. This allows us to define the ensemble average as the usual physical understanding of ergodicity and mixing. Our definitions are based on quantum dynamics (cid:104)Aˆ(cid:105) ≡trAˆρˆ . (4) E mc 3 For a quantum system satisfying condition (I), it is easy an initial state such that |c |2 = 1/N for E ∈ [E,E + n n to check that [3, 8] (see also Appendix B) δE] and |c | = 0 otherwise. In this way, we recover the n textbook micro-canonical ensemble ρˆ . That is, ρˆ is (cid:104)Aˆ(cid:105)E =(cid:104)Aˆ(cid:105)T , (5) just a special case of ρˆmc. tb tb that is, the long time average of Aˆ equals to its micro- Remark 5. Our definitions of ergodicity and mixing for canonical ensemble average. quantum systems are independent of initial conditions. For a quantum system satisfying both conditions (I) Nevertheless,tothoroughlyunderstandthem,wedoneed and(II),onecanprove[5,28]thatthelong-timeaveraged to consider initial conditions as the density matrix ρˆmc fluctuation F2 satisfies (see also Appendix B) depends on initial conditions. While Eq.(5) and Eq.(6) A hold for an arbitrary initial condition, not all of the ini- (cid:28)(cid:12)(cid:12)(cid:104)Aˆ(t)(cid:105)−(cid:104)Aˆ(cid:105) (cid:12)(cid:12)2(cid:29) tial conditions are of physical interest. If we choose an (cid:12) E(cid:12) initial state where only a few eigenstates are occupied, F2 ≡ T ≤trρˆ2 , (6) A (cid:107)Aˆ(cid:107)2 mc not only ρˆmc is no longer sensible to be regarded as a micro-canonical ensemble but also the fluctuation F2 in A where (cid:107)Aˆ(cid:107)2 = sup{(cid:104)ψ|Aˆ†Aˆ|ψ(cid:105) : |ψ(cid:105) ∈ H} is the upper Eq.(6) is not small. However, this kind of initial condi- limit of the expectation value of Aˆ2 in the Hilbert space. tions are hard to realize in experiment or to be found in nature for a many-body quantum system. Physically, Thisdemonstratesthatamixingquantumsystemindeed whenamany-bodyquantumsystemisexcitedbyaprac- has small time fluctuations. ticalmeans,itusuallyentersintoaquantumstatewhere A few remarks are warranted here to put our defini- a large number of eigenstates are occupied. This is also tions in perspective. the reason that the standard micro-canonical ensemble Remark 1. Our definitions of ergodicity and mixing for ρˆ ,whichlooksquiteartificial,workswellaslongasthe quantumsystemsaremathematicallyveryprecise. They tb system is large. do not involve any concepts and assumptions, which are This aspect is quite similar to classical systems. In mathematically ambiguous. Peres made many assump- an ergodic or mixing classical system there always exist tions in his definitions [8], which are reasonable but am- solutions which are not ergodic or mixing, for example, biguous mathematically. In particular, we do not need the periodic orbits. However, these non-ergodic or non- to define quantum chaos first as Peres did [8]. mixingsolutionsarerareorhavemeasurezeroinrigorous Remark 2. Although our definitions appear very mathe- mathematical language so that the overall properties of matical, as we have shown, they are consistent with the the system are not affected. familiar physical pictures that we have had with ergod- Remark 6. Manyquantumsystemshavecertainsymme- icity and mixing in classical dynamics: ergodicity means tries and correspondingly some good quantum numbers. that long time average equals to ensemble average; mix- Energy degeneracy can easily occur between the eigen- ing implies small time fluctuations. Moreover, similar to states of different symmetric sectors. As a result, these the classical case, a quantum mixing system is ergodic quantum systems are in general not ergodic and mixing. but not vice versa. However, if one focuses only one symmetric sector, the Remark 3. Our definitions have their roots in the 1929 quantum system can be ergodic or mixing. In this case, paper, where von Neumann proved a quantum ergodic wemaysaythatthequantumsystemisergodicormixing theorem [3]. However, von Neumann in 1929 did not in a sub-Hilbert space. distinguish between ergodicity and mixing. His view of Remark 7. Although classical ergodicity and mixing are ergodicity at that time is closer to the current view of offundamentalimportanceinstatisticalmechanics,their mixing. In other words, his quantum ergodic theorem definitionscanbeappliedtosingle-particlesystems. Sim- may be better called quantum mixing theorem. ilarly, our definitions can be applied to single-particle It is worthwhile to note two interesting points: (i) von quantum systems. Neumann used both condition (I) and (II) to prove his quantumergodictheorem;(ii)mixinginclassicaldynam- To conclude our definitions we offer two simple and ics was introduced three years later in 1932 by von Neu- illustrative examples. The first is one dimensional har- mann himself [11]. monic oscillator. There is no energy degeneracy so it is Remark 4. The density matrix ρˆ is used as the micro- ergodic. There is a great deal of degeneracy in energy mc canonical ensemble in the above discussion. It is not spacings so that it is not mixing. Interestingly, for the the standard micro-canonical ensemble found in text- classical dynamics, the one dimensional harmonic oscil- books [2], lator is similar: it is ergodic but not mixing. The second example is a particle in a one dimensional 1 (cid:88) box system, where there is neither degeneracy in energy ρˆ = |φ (cid:105)(cid:104)φ |, (7) tb N n n levels nor in energy spacings. According to our defini- En∈[E,E+δE] tions, it is both quantum ergodic and mixing. It is not where N is the number of energy-eigenstates in energy difficult to find that its classical counterpart is indeed interval [E,E +δE]. However, we can certainly choose both ergodic and mixing [29]. 4 C. Degeneracy parameters withm(cid:54)=n,E −E andE −E giverisetotwoenergy m n n m level spacings rather than one. We also emphasize that There has been a tremendous amount work on quan- our definitions of ζ and ξ are for a given set of energy tum chaos or quantum non-integrability [30]. It is inter- levels not for all the energy levels in the system. The esting to see how our quantum ergodicity or mixing is reasonisthatonlyafinitesetofenergylevelsareinvolved related to quantum chaos. In other words, are the two in any meaningful physical process. conditions(I)and(II)easytosatisfyinquantumchaotic According to these definitions, the two conditions (I) systems? and (II) are equivalent to ζ ≡ 0 and ξ ≡ 0, respectively. On the other hand, it is also interesting to know how The larger ζ(or ξ) is the stronger the non-degenerate- infrequentexceptionstothesetwoconditionsaffectquan- energy condition (I) (or the non-degenerate-gap condi- tumdynamics. Whenaquantumsystemhasarelatively tion (II)) is violated. We anticipate that for small ζ and small number of degeneracies, then almost all its states ξ quantum systems can still be regarded as ergodic or contain either no degenerate eigenstates or only a few. mixing. For systems where ζ and ξ are strictly equal to For the former, Eq.(5) still holds; for the latter, the zero, we say that they are ideal ergodic systems or ideal left hand side in Eq.(6) differs the right hand side only mixing systems. slightly. So, in a practical sense, this quantum system is ergodic. The situation is similar for mixing: infrequent III. MODEL degeneracy in energy gap is not important. Von Neu- mann had a similar point of view [3]. Short and Farrelly showedquantitativelyhowinfrequentdegeneracyarenot Inthissectionweuseanexampletoillustrateourcon- important [31]. cepts of quantum ergodicity and mixing. We consider To address the above two issues, we introduce two pa- the motion of one particle with mass m in a right trian- rametersζ andξ, whichdescribetheaveragedegeneracy gular billiard, as is shown in Fig.1. Mathematically, this in energy levels and average degeneracy in energy level billiard is described by the following potential spacings, respectively, foragivenfinitesetofenergylev- (cid:26)0 0<x<l,0<y <αx els. The parameter ζ is defined as V(x,y)= . (13) ∞ otherwise 1 (cid:88) ζ = N (δEm,En −δm,n), (8) Without loss of generality, we restrict ourselves to α≥1 m,n or 0 < θ ≤ π/4 (α = cotθ). It is interesting to note that this billiard system is equivalent to the system of where N is the number of energy levels in the set. The twohard-coreparticlesmovinginone-dimensionalsquare other parameter ξ is defined as potential with infinite walls [7]. 1 (cid:88) We choose this simple model for two reasons. (1) We ξ = (δ −δ δ ). (9) N(N −1) Ek−El,Em−En k,m l,n can study both the integrable cases and chaotic cases by k(cid:54)=l,m(cid:54)=n adjusting α. (2) Many meaningful results on classical Furthermore, it is useful to define two distribution ergodicity and mixing in this model have been obtained functions, f((cid:15)) and g(∆). f((cid:15)) is the probability of the previously [6, 7], and we can compare them to our quan- eigen-energies having value (cid:15); g(∆) is the probability of tum results. Other models such as the Bose-Hubbard the energy level spacings at ∆. With the aid of these model [32] do not have the above advantages. distributionfunctions,wecanreformulatethedefinitions The classical integrability of this model is well known. of ζ and ξ, respectively, as The system is integrable only √when θ = π/4 or θ = π/6 (equivalently, α = 1 or α = 3). When θ = πM/N, (cid:88) ζ =N f2((cid:15))−1, (10) where M and N are two coprime integers and (M,N)(cid:54)= (cid:15) (1,4),(1,6), the system has two independent invariants. However, it is regarded as pseudointegrable[33–35] be- (cid:88) cause the invariant surface of classical motion in phase ξ =N(N −1) g2(∆)−1. (11) space has a genus 2≤g <∞ (it is integrable only when ∆ the genus g = 1). For all other values of θ, the triangle These two functions are clearly related; their explicit re- system has only one invariant and is generally regarded lation is as chaotic. Tostudythequantumdynamicsofthismodel,weneed g(∆)= (cid:88)f((cid:15))Nf((cid:15)+∆)−δ∆,0 to calculate the eigenenergies and eigenstates. This can N −1 (cid:15) be done only numerically for an arbitrary value of pa- = N (cid:88)f((cid:15))f((cid:15)+∆)− δ∆,0 . (12) rameterα. Weusetheexactdiagonalizationmethod(see N −1 N −1 Appendix A for details). In our calculation, we choose (cid:15) h = m = l = 1. In addition, to avoid confusion, we We clarify that in our definition ∆ can be negative. In usethesingleparameterα(insteadofθ)torepresentthe other words, for two arbitrary energy levels E and E shape of the triangle billiard in following discussion. m n 5 α ζ ξ 1∗ 0.588 300.89 1.007846 0.032 14.39 1.015675 0 8.86 1.077744 0 8.65 1.154062 0 9.25 1.376382 0 11.04 1.461725 0 11.72 1.662013 0.002 13.33 1.700000 0.004 13.89 1.718079 0.010 18.84 1.725067 0.084 40.85 1.732051∗ 0.414 167.03 TABLEI:Degeneracyparametersζ andξ fordifferentvalues of α. The first 1000 energy levels are used in the calculation. Integrable cases are marked by superscript ∗. Clearly, integrable cases have much larger ζ and ξ. FIG. 1: A right triangle billiard. Without loss of generality, we take α≥1 or, equivalently, 0<θ≤π/4. pseudointegrable and chaotic cases have few. IV. DISTRIBUTIONS OF ENERGY LEVELS With the obtained eigen-energies we can compute ζ andξ,thetwoparametersthatweintroducedtodescribe quantitatively how well the two conditions (I) and (II) In Section II we have defined quantum ergodicity and for ergodicity and mixing are satisfied in a given system. mixing with two conditions (I) and (II) that regard the We first construct the two distribution functions f((cid:15)) distributionofthesystem’senergylevels. Inthissection, and g(∆) and then compute ζ and ξ using Eq.(10) and weshallexaminetowhatextentthesetwoconditionsare Eq.(11). The distribution function f((cid:15)) together with satisfiedbythetrianglebilliardandhowtheyarerelated g(∆) is constructed by binning the energy levels with a to the integrability of this model via parameters ζ and widthδ(cid:15)=(cid:126)/T,whereT isthetotaltimeofadynamical ξ. In the next section, we shall show that the quantum evolution. For a dynamical evolution of time T, energy dynamics of the triangle billiard are dictated by these levelsorspacingsseparatedbyδ(cid:15)=(cid:126)/T canberegarded two conditions. as the same. In our calculation, we use T =40 in accor- The study of quantum chaos has revealed that the dance with our numerical study of quantum dynamics in structure of quantum energy levels of a system is closely the next section. relatedtothe classicalintegrability ofthesystem[8, 36]. One often uses the nearest spacing distribution(NSD) The results are shown in Table.I, where we see clearly p(s) of a system to describe its structure of quantum the values of ζ and ξ are strongly correlated to the clas- energy levels, where s > 0 is the spacing between two sical integrability of the system. For integrable systems, nearest energy levels. The following feature of NSD is both ζ and ξ are large. As α changes and the system well known. For a system whose classical dynamics is becomes more chaotic, ζ decreases almost to zero while integrable, its NSD is Poisson-like with a peak distribu- ξ is reduced by about two orders of magnitude. These tion at s = 0. For a system whose classical dynamics is numericalresultsstronglysuggestthatconditions(I)and chaotic,theNSDofitsquantumenergylevelsisWigner- (II) are largely satisfied by chaotic systems. like: an almost zero probability density at s = 0 and a peak density at s = sm (cid:54)= 0. This feature indicates that The pseudointegrable systems are subtle. As one condition (I) is always satisfied by a quantum chaotic may have already noticed in Fig.2(c) and Table I, the system. Thereisnoclearconclusionforcondition(II)as pseudointegrable case α = cotπ/5 behaves very much the peak at nonzero s in the Wigner distribution seems like a chaotic system. However, not all pseudointe- to suggest that condition (II) is not satisfied by a quan- grable systems has a chaotic NSD. Some pseudointe- tum chaotic system. However, our following numerical grable triangle billiards, such as the triangle with angles results show that condition (II) is also largely satisfied (π/5,2π/5,2π/5), have Possion-like NSDs [37]. Because by a quantum chaotic system. the peaks of their NSDs p(s) are at s = 0 which in- The Hamiltonian matrix of the triangle billiard is di- dicates large degeneracy, these triangle billiards should agonalized numerically for a set of α. Its NSDs for the have large ζ and ξ, and they are not ergodic or mixing. first1000energylevelsareshowninFig.2forfourtypical Thisdifferenceshowsthattherelationbetweenquantum valuesofα. Asexpected, twointegrablecasesα=1and ergodicity and mixing and classical integrability is very √ α = 3 have lots of degenerate energy levels while the subtle in the case of pseudointegrable systems. 6 √ FIG. 2: Nearest spacing distribution of eigenenergies. (a)α = 1 and (b)α = 3 are two integrable cases. (c) α = cot(π/5) is pseudointegrable. (d) is chaotic. Calculations are done in first 1000 energy levels. V. QUANTUM DYNAMICAL BEHAVIOR once we have obtained the expansion coefficients c , we k cangeneratethewavefunctionatanytime. Inourstudy, In this section we shall study the quantum dynamics we choose a Gaussian wave packet as an initial state of the triangle billiard for a set of typical values of α 1 to see whether it exhibits ergodic or mixing behavior as ψ(x,y)= √ e−4σ12[(x−x0)2+(y−y0)2]e−i2π(pxx+pyy), 4πσ2 describedbyEq.(5)orEq.(6),respectively,andhowthese (17) dynamical behaviors are dictated by conditions (I) and where x = 0.5,y = 0.3,p = 5cos(eπ),p = 5sin(eπ), (II) via parameters ζ and ξ. 0 0 x y and σ =0.02. This initial state mainly occupies the first To study the quantum dynamical behavior, we need 1000 energy eigenstates. to calculate the time evolution of a wave function. We It is sufficient to focus on the momentum of the sys- usethemethodofeigenstateexpansion. Foranarbitrary tem. For the initial condition in Eq.(17) we have ex- initial wave function ψ(x,y,0), we expand it in terms of actly (cid:104)p(cid:126)(cid:105) =0. The quantum dynamical evolutions of p(cid:126) the energy eigenstates of the Hamiltonian φ (x,y) E k are shown in Fig.3 for four typical values of α: α = 1, √ √ (cid:88) α = 3, α = cotπ/5, and α = tan 5−1π. It is clear ψ(x,y,0)= c φ (x,y). (14) 4 k k that the evolution of p(cid:126) varies greatly with different α. k Beforewediscussitindetail, letusfirstrecalltheclassi- According to the Schördinger equation, the time evolu- cal dynam√ics for these four cases. The cases with α = 1 tion of this initial wave function is given by and α = 3 are integrable; α = cotπ/5 is pseudointegrable and has only finite directions of p(cid:126) in classical ψ(x,y,t)=(cid:88)cke−iE(cid:126)ktφk(x,y). (15) dynamics√which means nonergodicity. The case with k α = tan 5−1π is nonintegrable but classically noner- 4 godic [7]. As the expansion coefficients can be calculated easily as Let us come back to the quantum dynamics in Fig.3. (cid:90)(cid:90) Forthetwointegrablecases,thelong-timeaverageisap- ck = ψ(x,y,0)φ∗k(x,y)dxdy, (16) parently not equal to its microcanonical ensemble average, and the fluctuation is large as well. For other cases, Ω1 7 FIG. 4: Averaged relative fluctuations of the momentum via α. The vertical axis F2 indicates the averaged relative fluctuation as defined in Eq.(6) and Eq.(19). The solid points arethenumericalresults;thesolidlineisjustaguidance;the dashed line is the upper bound trρ2 in Eq.(6). mc In order to check quantitatively whether a quantum systemismixing,weneedtocalculatethetime-averaged relative fluctuation. The averaged deviation of the momentum operator p(cid:126) in a given evolution time T is (cid:12) (cid:12)2 1 (cid:90) T (cid:12) (cid:88) (cid:12) (cid:104)σ2(cid:105) = (cid:12)(cid:104)ψ(t)|p(cid:126)|ψ(t)(cid:105)− |c2|(cid:104)φ |p(cid:126)|φ (cid:105)(cid:12) dt. p(cid:126) T T (cid:12) k k k (cid:12) 0 (cid:12) k (cid:12) (18) Considering E = p(cid:126)2, the relative fluctuation of p(cid:126) is 2 (cid:104)σ2(cid:105) (cid:104)σ2(cid:105) F2 = p(cid:126) T = p(cid:126) T. (19) |p(cid:126)|2 2E The relaxation time scale is ∼ 10−1 and the oscillating period for integrable systems is ∼101. Considering that longer evolution time may lead to large numerical error andunreliableresult,wechooseT =40. Thistimelength is much larger than relaxation time, and it is also long enough for us to see if there are frequent recurrences. FIG. 3: Evolution of the momentum for different α. α = 1 The results of relative fluctuation for different α are √ √ and α = 3 are two integrable cases. α = tan 5−1π is shown in Fig.4. The dashed line is trρ2 , the upper 4 mc chaoticbutnonergodicinclassicalmechanics[7]. α=cotπ/5 bound in Eq.(6). It can be clearly seen that for α away √ is pseudointegrable. from 1 or 3, the averaged fluctuation is small, and the inequality Eq.(6) is satisfied. This confirms the intuitive picture in Fig. 3 that the quantum dynamics is mixing. the momentum quickly relaxes to its microcanonical en- When α approaches the two integrable cases, α=1 and √ semble average and has only small fluctuations. The re- α = 3, the averaged fluctuation becomes much larger, laxationtimeisveryshortandisabout∼10−1 forthese andtheinequalityEq.(6)isviolated. Infact,atthesetwo cases. Up to t = 40, we do not observe a revival or integrablecases, theaveragedfluctuationreachestwolo- largedeviationfromtheequilibriumvalue. Theseresults calmaxima. Thereisaratherrapidtransitionfrommix- demonstratethatthequantumdynamicsforα=cotπ/5 ing to nonmixing while the system is tuned from nonin- √ (the pseudointegrable regime) and α = tan 5−1π are tegrable to integrable. 4 not only ergodic but also mixing. This is in stark con- The quantum dynamic behavior shown in Fig. 3 are trast with their classical dynamics which are not even dictated by conditions (I) and (II). This can be seen ergodic. This suggests that it is easier to have quantum clearly in Fig.5, where the square of time averaged mo- ergodicity and mixing than their classical counterparts. mentum (cid:104)p(cid:126)(cid:105)2 and its relative fluctuations are plotted T 8 It is possible to expand this quantum ergodic hierarchy to three. We define a quantum system is equilibrable if the system satisfies E +E −E −E =E +E −E −E and m n k l m(cid:48) n(cid:48) k(cid:48) l(cid:48) {m,n}∩{k,l}=∅⇒{m,n}={m(cid:48),n(cid:48)}{k,l}={k(cid:48),l(cid:48)}. (III) This condition impliesthat there is no degeneracy inthe gaps of energy gaps. One can find the full implication of this condition in Ref.[15]. Here we briefly summarize. The entropy for a quantum pure state ρˆ≡ |ψ(cid:105)(cid:104)ψ| is defined as [15] (cid:88) S ≡− (cid:104)ψ|W |ψ(cid:105)ln(cid:104)ψ|W |ψ(cid:105) w qi,pj qi,pj qi,pj (cid:88) ≡− tr(ρˆW )lntr(ρˆW ), (21) qi,pj qi,pj FIG. 5: (a)The square of time-averaged p(cid:126) at T = 40 with qi,pj differentζ. Theverticalaxisisinlogscale. Notethat(cid:104)p(cid:126)(cid:105) = E where W ≡ |w (cid:105)(cid:104)w | is the projection onto 0. (b)Averaged relative fluctuation vs. ξ. No mixing when qi,pj qi,pj qi,pj Planck cells in quantum phase space at position q and F2/trρ2 >1. i mc momentump and{|w (cid:105)}isacompletesetofWannier j qi,pj functions. ThisentropyS willchangewithtime. Anin- w equalityregardingtherelativefluctuationofentropyS , against the two degeneracy parameters ζ and ξ, respec- w similar to Eq.(6), was proved in Ref.[15] with condition tively. It is clear from the figure that for systems with (III).Thisinequalitymeansthataquantumsystemwith significantly non-zero ζ, the time average of the momen- smallentropyS willrelaxdynamicallytoastatewhose tum significantly deviates from the microcanonical en- w entropy S is maximized and stay at this maximized semble value. We can see a strong positive correlation w valuewithsmallfluctuations. Thisisillustratedwiththe between the relative fluctuation F2/trρ2 and ξ as well. mc trianglebilliardinFig.6. Inthisfigureweseethattheen- These results illustrate that systems with small ζ and tropyS oftheintegrablecases,forwhichcondition(III) ξ have ergodic and mixing quantum dynamics, respec- w is not satisfied, fluctuates periodically with large ampli- tively. Ourdefinitionsofquantumergodicityandmixing tudeanddoesnotstayatthemaximumvalue. Forcases with conditions (I) and (II) are legitimate. √ with α = tan 5−1π and α = cotπ/5, where condition 4 (III) is largely satisfied, the entropy quickly relaxes to the maximum value and stays there with small fluctua- VI. DISCUSSION AND CONCLUSION tions. Theseresultsdemonstratethataquantumsystem thatsatisfiescondition(III)iscapableofequilibratingto Let us summarize what we have done. We have given a state where not only its observables fluctuate around our own definitions of ergodicity and mixing for quan- its equilibrium value with small amplitude but also its tum systems with conditions (I) and (II). It can be rig- entropy is maximized. This is the reason that we call orously proved that these two conditions lead to quan- such a quantum system equilibrable. In this way we have tum dynamical behaviors which are described by Eq.(5) a quantum ergodic hierarchy and Eq.(6) and are reminiscent of classical ergodic and mixingdynamics,respectively. Throughanexample,the Equilibrable ⊂Mixing⊂Ergodic. (22) triangle billiard, we have further shown that although both conditions (I) and (II), which are characterized by We do not call it quantum Kolmogorov as we do not see ζ and ξ, are related to classical integrability, there are anapparentconnectiontotheclassicalKolmogorovmix- differences. The most important is that a system whose ing system at this moment. It would be very interesting classical dynamics is neither ergodic nor mixing can be to find a quantum system which is equilibrable but not both ergodic and mixing in its quantum dynamics. mixing. Classicaldynamicshasanergodichierarchy[1,12,13], Finally we conclude with expectations of more future which is worktofollow. Wehavegivenprecisedefinitionsofquan- tum ergodicity and mixing which are in accordance with Bernoulli⊂Kolmogorov⊂Mixing⊂Ergodic. (20) our usual understanding of ergodicity and mixing. We have illustrated with single-particle billiard systems. It Now mixing and ergodicity have their quantum counter- wouldbeveryinterestingtofurtherexaminethemintrue parts. In particular, we have similar relation: quantum many-bodyquantumsystems[32,38],wherethethermo- mixingsystemsareasubsetofquantumergodicsystems. dynamics limit can be considered. 9 2012CB921300) and the National Natural Science Foun- dation of China (Grants No. 11274024, No. 11334001, and No. 11429402).. Appendix A: Calculation of eigen-energies and eigenstates in the triangle billiards We use the exact diagonalization method to calculate the eigen-energies and eigenstates: (i) choose an appro- priate set of basis; (ii) calculate the Hamiltonian matrix elementsinthebasis;(iii)numericaldiagonalizationthat results the eigenenergies and eigenstates. Inordertoreducethenumericalerrortoanacceptable range, we choose the basis as follows FIG. 6: Evolution of the entropy S for different α. 2 mπx nπy nπx mπy w |m,n(cid:105)= √ (sin sin −sin sin ), l α l αl l αl (A1) VII. ACKNOWLEDGMENTS This choice is similar to that in Ref.[37]. This basis is complete and orthogonal. It is easy to check that all We thank Zongping Gong for helpful discussion. This these base functions |m,n(cid:105) are zero on the boundaries work is supported by the National Basic Research Pro- of the triangle. The elements of the Hamiltonian matrix gram of China (Grants No. 2013CB921903 and No. can be computed analytically (cid:104)m ,n |Hˆ|m ,n (cid:105) = h2 (cid:110)(cid:18)m2+ n22(cid:19)(cid:2)I(m ,n ,m ,n )−I(n ,m ,m ,n )(cid:3) 1 1 2 2 2ml2 2 α2 1 1 2 2 1 1 2 2 −(cid:18)n2+ m22(cid:19)(cid:2)I(m ,n ,n ,m )−I(n ,m ,n ,m )(cid:3)(cid:111), (A2) 2 α2 1 1 2 2 1 1 2 2 where (cid:90) 1 (cid:90) x I(m,n,p,q)= dx dysin(mπx)sin(nπy)sin(pπx)sin(qπy) 0 0 1 88π1,2(12−(−1)m+n+2p+q)(cid:20)(cid:21)−{p−m+n+q}−1+{q+n−p+m}−q1+−n{p+m+n+q}−1−{ifnm+=q−p p&−nm=}q−1 = + − , if m(cid:54)=p & n=q (m+p)2 (m−p)2 8+π1{2p(1−−m(−+1q)m−+nn+}−p+1q+)(cid:20){−q−{pn−−mp++nm+}−qq1}−−−1n{+m{+q+p+n−q−p+n}m−1}−−q1{+−qn−{pm+−mn+−np+}−q1}(cid:21)−1, −{ifnm+(cid:54)=q−p &p−nm(cid:54)=}q−1 (A3) with the curly braces {·}−1 representing This notation is used only in Eq.(A3) to simplify the expression.  0, if z =0, After the above derivation, we take a cutoff in  {z}−1 = 1 (A4) n1,n2,m1,m2 andchooseh=m=l=1tocalculatethe  , if z (cid:54)=0. elements of Hamiltonian matrix. The eigenenergies and z 10 eigenstates can be obtained after diagonalization of the This is the proof of Eq.(5). Hamiltonianmatrix. AstheelementsofHamiltonianma- We now compute the standard deviation of Aˆ. trixareexplicit, theerroroftheeigenenergiesandeigen- states mainly arises from the cutoff of n ,n ,m ,m . 1 2 1 2 In our calculation, the number of basis is set as 8500. (cid:28)(cid:12) (cid:12)2(cid:29) (cid:104)σ2(cid:105) = (cid:12)(cid:104)Aˆ(t)(cid:105)−(cid:104)Aˆ(cid:105) (cid:12) Changing this number from 6000 to 10000 only cause a A T (cid:12) E(cid:12) T ∼0.01%relativevariationofeigenenergies(intheunitof (cid:28)(cid:12) (cid:12)2(cid:29) (cid:12) (cid:12)2 h2/ml2). This indicates that the error in the numerical = (cid:12)(cid:104)Aˆ(t)(cid:105)(cid:12) −(cid:12)(cid:104)Aˆ(cid:105) (cid:12) (cid:12) (cid:12) (cid:12) E(cid:12) results of eigenenergies is around 0.01%, which is accu- T rate enough for our analysis. = (cid:88) ρ∗lkρnm(cid:68)e−i[(En−Em)−(El−Ek)]/(cid:126)(cid:69) AmnA∗kl T k(cid:54)=l,m(cid:54)=n (B6) Appendix B: Proofs of Long-time Quantum Ergodic and Mixing Behaviors, Eq.(5) and Eq.(6) When the quantum system is mixing, that is, both con- In this appendix we provide the proofs of Eq.(5) and ditions (I) and (II) are satisfied, we have Eq.(6), which concern the long-time ergodic and mixing behavior in quantum systems, respectively. The original (cid:88) versions of the proofs can be found in Ref.[3, 5, 8, 28]. (cid:104)σ2(cid:105) = ρ ρ δ A A A T kl nm En−Em,El−Ek mn lk Consider a quantum system that starts with the fol- k(cid:54)=l,m(cid:54)=n lowing initial condition, (cid:88) = ρ ρ δ δ A A kl nm mk nl mn lk (cid:88) k(cid:54)=l,m(cid:54)=n |ψ(0)(cid:105)= c |φ (cid:105) , (B1) n n (cid:88) n = ρmnρnmAmnAnm m(cid:54)=n where |φ (cid:105)’s are the energy eigenstates. At time t, the n (cid:88) (cid:88) wave function becomes = |cm|2|cn|2AmnAnm− |cn|4|Ann|2 m,n n |ψ(t)(cid:105)=(cid:88)c e−iEnt/(cid:126)|φ (cid:105) . (B2) (cid:88) n n ≤ ρ ρ A A mm nn mn nm n m,n The corresponding density matrix is then =trAˆρˆ Aˆ†ρˆ =tr(cid:104)(ρˆ Aˆ†)†(Aˆ†ρˆ )(cid:105) , mc mc mc mc ρˆ(t)=|ψ(t)(cid:105)(cid:104)ψ(t)| (B7) = (cid:88)ρ e−i(En−Em)t/(cid:126)|φ (cid:105)(cid:104)φ | . (B3) nm n m m,n where we have used A = (cid:104)φ |Aˆ|φ (cid:105). We define a mn m n whereρ =c∗ c . Inanergodicsystemwherecondition scalar product for two operators Pˆ,Qˆ as tr(Pˆ†Qˆ). Using nm m n (I) is satisfied, we have the Cauchy-Schwartz inequality for operators with such scalar product, we have [28] (cid:42) (cid:43) (cid:104)ρˆ(t)(cid:105) = (cid:88)ρ e−i(En−Em)t/(cid:126)|φ (cid:105)(cid:104)φ | T nm n m (cid:114) (cid:104) (cid:105) (cid:104) (cid:105) m,n T (cid:104)σ2(cid:105) ≤ tr (ρˆ Aˆ†)†(ρˆ Aˆ†) tr (Aˆ†ρˆ )†(Aˆ†ρˆ ) = (cid:88)ρ (cid:104)e−i(En−Em)t/(cid:126)(cid:105) |φ (cid:105)(cid:104)φ | A T mc mc mc mc nm T n m (cid:113) m,n = tr(Aˆ†Aˆρˆ2 )tr(AÂˆ†ρˆ2 ) mc mc (cid:88) = ρnmδEn,Em|φn(cid:105)(cid:104)φm| ≤(cid:107)Aˆ(cid:107)2trρˆ2mc, (B8) m,n (cid:88) = ρ δ |φ (cid:105)(cid:104)φ | nm n,m n m where (cid:107)Aˆ(cid:107)2 = sup{(cid:104)ψ|Aˆ†Aˆ|ψ(cid:105) : |ψ(cid:105) ∈ H} is the upper m,n limit of the expectation value of Aˆ2 in the Hilbert space. (cid:88) = ρmm|φm(cid:105)(cid:104)φm| Finally, we have for the fluctuation m = ρˆ , (B4) mc (cid:28)(cid:12) (cid:12)2(cid:29) (cid:12)(cid:104)Aˆ(t)(cid:105)−(cid:104)Aˆ(cid:105) (cid:12) which is exactly the micro-canonical ensemble that we (cid:104)σ2(cid:105) (cid:12) E(cid:12) introduced in Eq.(3). Therefore, for an observable Aˆ, FA2 ≡ (cid:107)AAˆ(cid:107)2T = (cid:107)Aˆ(cid:107)2 T ≤trρˆ2mc. (B9) (cid:104)Aˆ(cid:105) =(cid:104)trAˆρˆ(t)(cid:105) =tr[Aˆ(cid:104)ρˆ(t)(cid:105) ]=trAˆρˆ =(cid:104)Aˆ(cid:105) . T T T mc E (B5) This is the proof of Eq.(6).