Statistical mechanics of Floquet systems with regular and chaotic states

Statistical mechanics of Floquet systems with regular and chaotic states

Abstract

We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori and the parameters temperature and driving frequency. For a constant winding number within a regular island it simplifies to Boltzmann-like weights , similar to time-independent systems. For this we introduce the regular energies of the quantizing tori and an effective winding-number-dependent temperature , different from the actual bath temperature. Furthermore, the occupations of other typical Floquet states in a mixed phase space are studied, i.e. regular states on nonlinear resonances, beach states, and hierarchical states, giving rise to distinct features in the occupation distribution. Avoided crossings involving a regular state lead to drastic consequences for the entire set of occupations. We introduce a simplified rate model whose analytical solutions describe the occupations quite accurately.

pacs:
05.45.Mt, 05.30.-d, 05.70.Ln

I Introduction

The response of a dynamical system to a time-periodic driving force is ubiquitous in both classical and quantum mechanics and plays a fundamental role in many physical and technical applications. It opened the field for the coherent control of atoms and molecules RicZha2000 (), the optimal control of chemical reactions BruSha2003 (), or the manipulation of semiconductor-nanodevices and heterostructures in solids Phi1994 (). Under realistic, nonidealized conditions real physical systems interact with their environment. If the environment contains a vast number of degrees of freedom, the full dynamics of the composite system is not traceable. The system is then interpreted as an open subsystem in mutual contact with a heat bath and the dynamics of the subsystem is characterized by its reduced density operator . To evaluate the evolution of for an open quantum system in a time-varying, strong external field is a nontrivial task, as it is permanently driven out of equilibrium. For only very few systems exact analytical solutions of the damped dynamics are feasible, in particular a driven two-level system GriSasStoWei1993_GriSasHaeWei1995 () and a driven harmonic oscillator GraHue1994 (); ZerHae1995 (). In general systems it is studied numerically, e.g. with focus on tunneling, see Ref. GriHae1998 () and references therein. Especially in the regime of weak interaction with the environment, standard methods, originally established for time-independent quantum systems, have been adapted to the demands of time-periodic systems BluETAL1991 (); KohDitHae1997 (); BreHubPet2000 (); Koh2001 (); HonKetKoh2009 ().

The final state of the relaxation process has so far not received as much attention as transient phenomena, although this can be ranked as even more fundamental and is in fact a core question of statistical mechanics. The usual thermodynamic concepts for the equilibrium state of time-independent systems are not applicable, such as the canonical distribution of Boltzmann weights, reached in the stationary limit of a time-independent system that is weakly coupled to a heat bath. The Boltzmann weights of the eigenstates are unique functions of the eigenenergy with the temperature of the heat bath as the only relevant parameter, whereas microscopic details of the weak coupling play no role. Such a stationary limit, in the sense of convergence to time-independent values for all dynamical variables, is not encountered in a periodically driven system, where energy is permanently exchanged between the driven system and the environment. Instead, the relaxation process finally leads to an asymptotic state that adopts the periodicity of the driving, and that in general depends on the microscopic details of the coupling. The density operator of the time-periodic subsystem is best represented in the Floquet state basis. The Floquet states are quasi-periodic solutions of the Schrödinger equation for the time-periodic Hamiltonian without the coupling to the environment. In the Floquet basis the evolution equation for the density matrix can be approximated within the Floquet-Markov approach BluETAL1991 (); KohDitHae1997 (); BreHubPet2000 (); Koh2001 (); HonKetKoh2009 () by a Markovian quantum master equation. In the long-time limit of the evolution the Floquet states are populated with asymptotic occupation probabilities that can be determined from a system of rate equations. Beyond the numerical evaluation of such a master equation, an intuitive understanding of these Floquet occupations is still lacking.

A related problem occurs at avoided crossings, which are ubiquitous in the quasienergy spectra of generic Floquet systems, since the quasienergies are bounded within a finite interval, , where is the driving frequency and is the driving period. As a consequence, the number of avoided crossings grows without limit for increasing Hilbert-space dimension, leading to a breakdown of the adiabatic theorem HonKetKoh1997 (). In the presence of a heat bath this problem is approached in Ref. HonKetKoh2009 () where it is shown that the reduced density operator is not affected by a small avoided crossing, provided that it is smaller than a specific effective coupling parameter and so is not ‘resolved’ by the heat bath. These findings justify the unevitable truncation of the, in general, infinite Hilbert space dimension in numerical implementations.

One way to tackle the general challenge of finding the Floquet occupations beyond their numerical evaluation is to study the semiclassical regime of one-dimensional driven systems. In their classical limit regular and chaotic motions generically coexist. This is most clearly reflected in phase space, where regular trajectories evolve on invariant tori and chaotic trajectories fill the remaining phase-space regions. According to the semiclassical eigenfunction hypothesis Per1973_Ber1977_Vor1979 () almost all Floquet states can be classified as either regular or chaotic, provided that the phase-space regions are larger than the Planck constant. The regular states localize on the regular tori and the chaotic states typically spread out over the chaotic region. For a driven particle in a box coupled to a heat bath the Floquet occupations of regular and chaotic states were found to follow different statistical distributions BreHubPet2000 (). The regular states, which in this example differ only slightly from the eigenstates of the undriven system, carry almost Boltzmann weights, whereas all chaotic states have nearly the same occupation probability.

In this paper we concentrate on situations characteristic for strong driving, where both phase space and Floquet states are strongly perturbed compared to the originally time-independent system. We demonstrate that the Floquet occupations of the states in a regular island under these conditions deviate considerably from the Boltzmann result. For kicked systems, making use of some reasonable assumptions, we derive an analytical expression for the regular occupations. In many cases it can be well approximated by weights of the Boltzmann type . This requires the introduction of the regular energies , which are semiclassical invariants of the quantizing tori of the regular island, and the parameter , which is an effective temperature depending on the winding number of the regular island. Furthermore, we give an overview and interpretation for the occupations of other typical Floquet states in a mixed phase space, such as states on a resonance island chain, beach states, and hierarchical states.

Avoided crossings in the Floquet spectrum can lead to severe changes in the occupations if they are larger than an effective coupling parameter. The effect can be intuitively explained by a set of effective rate equations with an additional rate between the states of the avoided crossing HonKetKoh2009 (). It can be exploited for a switching mechanism in driven quantum systems, e.g. a weakly driven bistable system KetWus2009 (). For the above occupation distributions for regular and chaotic states we demonstrate drastic consequences if a regular state has an avoided crossing with either a regular or a chaotic state. We introduce a simplified rate model whose analytical solution describes the Floquet occupations accurately.

The paper is organized as follows: in Sec. II the microscopic model of driven dissipative systems and the Floquet-Markov description of its asymptotic state are sketched and the relevant coupling operators are introduced. Section III presents general occupation characteristics for the example of a driven quartic oscillator (Sec. III.1) and a kicked rotor (Sec. III.2). They are related to the corresponding rate matrices (Sec. III.3). In Sec. IV we derive an analytical expression for the regular occupations of kicked systems, depending on the winding numbers of the regular tori. For a constant winding number a simplification to the Boltzmann-like weights is shown (Sec. IV.3). An example where this is not possible is also discussed (Sec. IV.4). Section V gives an overview of the occupation characteristics of other types of Floquet states. In Sec. VI the influence of avoided crossings on the Floquet occupations is demonstrated, which are compared to the analytical solutions of a simplified rate model (Sec. VI.1). Section VII summarizes the paper.

Ii Master equation in time-periodic systems

The coupling of a quantum system with the Hamiltonian to a heat bath is modelled in a standard way by the composite Hamiltonian Wei1999 ()

(1)

Herein, the bath Hamiltonian describes an ensemble of noninteracting harmonic oscillators coupled via the interaction Hamiltonian to the system. In spatially extended systems this interaction is commonly assumed to be bilinear,

(2)

with some coupling operator of the system. The properties of the system-bath coupling are specified by the spectral density of the bath . In the continuum limit the spectral density is assumed to be a smooth function which is linear for an Ohmic bath. An exponential cutoff beyond the spectral mode leads to , where is proportional to the classical damping coefficient.

In the absence of the heat bath the solutions of the time-dependent Schrödinger equation for the isolated system with the -periodic Hamiltonian

(3)

are the Floquet states . These can be factorized into a product

(4)

of a phase factor with the quasienergy and a periodic state vector ,

(5)

with the period of the Hamiltonian.

In the presence of the heat bath the state of the system is described by the reduced density operator . Its equation of motion for time-periodic quantum systems has been derived within the Floquet-Markov approach BluETAL1991 (); KohDitHae1997 (); BreHubPet2000 (); Koh2001 (); HonKetKoh2009 (): herein the Floquet formalism ensures a non-perturbative treatment of the coherent dynamics of the driven system. The density operator is represented in the set of the time-periodic state vectors , which form a complete orthonormal basis at all times . The coupling to the heat bath is treated perturbatively in second order of , which is valid in the limit of weak coupling between the driven system and the bath. This approximation requires a rapid decay of bath correlations compared to the typical relaxation time of the system. In this paper we use a cutoff frequency , which is large compared to the frequency of the driving. In the following we restrict the discussion to the limit of large times, much larger than the relaxation time. In this limit the density-matrix elements are approximated as time-independent KohDitHae1997 (); HonKetKoh2009 (). Note that the corresponding density operator, , is still time-periodic because of the inherent time-dependence of the . In this paper we restrict to the weak-coupling regime, where the system-bath coupling is small compared to all quasienergy spacings of a truncated Hilbert space (see discussion below). The Floquet occupations then obey the set of rate equations

(6)

which are independent of the damping coefficient . Note that the rate equations beyond this weak-coupling regime would also contain the non-diagonal elements (). The rates

(7)

that describe bath-induced transitions between the Floquet states, use the Fourier coefficients

(8)

of the time-periodic matrix elements

(9)

The correlation function of the bath coupling operator contains the spectral density and the thermal occupation number of the boson bath with temperature .

The reduction to the set of rate equations (6) seems possible only for systems with a finite dimension of the Hilbert space, since otherwise the quasienergies densely fill the interval . However, as demonstrated in Ref. HonKetKoh2009 (), near degeneracies much smaller than the coupling strength are not resolved by the interaction to the heat bath and do not influence the asymptotic density operator. The Hilbert dimension can therefore be truncated, keeping only those Floquet states of non-negligible occupation.

Equation (6) is formally identical to the familiar system of rate equations describing the equilibrium state in time-independent systems. In contrast, however, specifics of the time-periodic system are present in the rates, Eq. (7), whose structure does in general not allow a detailed balance relation Koh2001 ().

ii.1 Coupling operator for extended and for cyclic systems

For extended systems we assume as usual Wei1999 () the linear coupling operator

(10)

in Eq. (2) for the interaction Hamiltonian with the heat bath.

For cyclic systems defined on the unit interval with periodic boundary conditions in the coupling operator would be discontinuous at the borders of the interval and the coupling to the heat bath would therefore not be homogeneous. An adapted coupling scheme with the interaction Hamiltonian

(11)

has been proposed in Ref. Coh1994 () for such situations. The angles characterize the individual bath oscillators and are equidistributed in the interval . The new spectral density

(12)

hence factorizes into independent factors, , with the spectral density as defined above and the homogeneous angular density . The spatially periodic interaction Hamiltonian (11) is continuous and, by virtue of the equidistributed angles , models the interaction with a homogeneous environment, where no position is singled out. Making use of the trigonometric addition theorem , the interaction Hamiltonian (11) leads to the same system of rate equations, Eq. (6). But now the rates

(13)

are composed of two independent contributions from the simpler coupling operators

(14)
(15)

respectively, to be used in the interaction Hamiltonian in Eq. (2). For this result we use that the mixed second-order term vanishes, while the other two terms give rise to the operators in Eqs. (14) and (15).

Iii Occupations of regular and chaotic Floquet states

In this section we study the Floquet occupations for two classes of periodically driven systems, the additively driven quartic oscillator as a representative of a continuously driven system and the quantum kicked rotor from the class of kicked systems. In both cases we demonstrate that the Floquet occupations of regular and chaotic Floquet states follow tremendously different distributions. Similar observations have been made for a driven quantum particle in a box BreHubPet2000 (). While there it was found that regular states carry Boltzmann weights, we find typically significant deviations. Numerical results will be presented in this section and the quantitative analysis of the occupations of the regular states will be deferred to Sec. IV.

iii.1 Driven Quartic Oscillator

We consider as an example of a continuously driven system the additively driven quartic oscillator with the Hamiltonian

(16)

We introduce the dimensionless quantities , , with , , , and also , the ratio of the Planck constant to a typical phase-space area. In the following we omit the overtilde and then the dimensionless Hamiltonian reads

(17)

At and the stroboscopic Poincaré section of the phase space at integer multiples of the driving period, , features a chaotic domain, see Fig. 1(a). Furthermore, there are two distinct regular regions: first the highly excited tori, which are only slightly influenced by the driving, and second a regular island embedded in the chaotic sea.

The Floquet states are determined using the -technique PfeLev1983_PesMoi1993 (). Their energy expectation value, , oscillates with the period of the driving. It is convenient to introduce the cycle-averaged energy

(18)

An energy shift is determined by the classical periodic orbit at the center of the regular island, such that there the cycle-averaged energy is zero.

Figure 1: (Color online) (a) Stroboscopic Poincaré section of the classical phase space of the driven osillator, Eq. (17). The size of the chosen dimensionless Planck constant is indicated in the lower right corner. (b) Floquet occupations vs. cycle-averaged energies compared to the Boltzmann-like prediction (dashed line). The insets show Husimi representations of a regular Floquet state localized in the central island, a chaotic Floquet state, and a regular state on a surrounding torus. The parameters are , , , and .

For a sufficiently small value of almost all Floquet states can be classified as either regular or chaotic according to the semiclassical eigenfunction hypothesis Per1973_Ber1977_Vor1979 (). The regular states are localized on the regular island and can be ordered by a quantum number , whereas the chaotic states typically spread over the entire chaotic phase-space area and fluctuate irregularly. The different types of states are visualized in phase space by means of their Husimi representation , i.e. the projection onto the coherent states centered at the phase space points , see insets in Fig. 1(b). In the example of Fig. 1 the small central island is about times larger than the dimensionless Planck constant , indicated in the lower right corner of Fig. 1(a). Thus the central island supports regular states. Besides, there are chaotic states, spreading over the chaotic region. It is surrounded by regular tori, on which a further group of infinitely many regular states are localized. The cycle-averaged energies of these three different types of Floquet states form distinct intervals with only small overlaps, as indicated by the arrows in Fig. 1(b). The regular states of the central island are lowest in cycle-averaged energy, followed by the chaotic states and finally the high excited regular states of the surrounding tori.

The oscillator coupled to a heat bath is treated as explained in Sec. II with the linear coupling operator for this spatially extended system. We restrict the consideration to the weak-coupling regime, where the occupations are well described by the rate equation (6). In Fig. 1(b) the resulting Floquet occupations are shown as functions of the cycle-averaged energies . The monotonously falling occupations at low values of belong to the central regular island, with the state in the center of the island having the highest occupation. At intermediate values of one finds the occupations of the chaotic states, that fluctuate around a mean value , with a very small variance compared to the range of occupations of the regular states. This is similar to the observation for chaotic states in Ref. BreHubPet2000 () for a driven particle in a box. At high values of , there are again monotonously falling occupations belonging to the regular states of the surrounding tori.

The observed characteristics of the occupations are clearly in contrast to the naive expectation motivated by the Boltzmann weights of equilibrium thermodynamics. Note that even the occupations of the (low-energy) regular states notably differ from the Boltzmann result, indicated by the dashed line in Fig. 1(b). A quantitative analysis of these observations will be presented in Sec. IV for the numerically more convenient kicked rotor.

iii.2 Kicked rotor

Figure 2: (Color online) (a) Stroboscopic Poincaré section of the classical phase space of the kicked rotor, Eq. (20), for . The size of the chosen dimensionless Planck constant is indicated in the lower right corner. (b) Floquet occupations vs. cycle-averaged energies compared to the Boltzmann-like prediction (dashed line). The insets show Husimi representations of two regular and a chaotic Floquet state. (c) Floquet occupations of regular states vs. regular energies , defined in Eq. (21), compared to the Boltzmann-like prediction Eq. (40) with the inverse effective temperature of Eq. (39) (red solid line), compared to the inverse bath temperature (dashed line). The parameters are and .

Kicked quantum systems feature all essential phase-space characteristics of periodically driven systems. They allow for a simplified numerical and conceptual treatment. As a paradigmatic model for a driven system with a mixed phase space we consider here the quantum kicked rotor. Its dynamics is generated by the Hamiltonian

(19)

with the kinetic energy and the potential acting at the kicks. We study it on a two-torus with dimensions and . We make the Hamiltonian dimensionless by a similar transformation as in Sec. III.1, where now and the dimensionless kick period is . With the rescaled kick strength the Hamiltonian reads

(20)

In an intermediate regime of the kick strength the Poincaré section of the phase space features a regular island embedded in the chaotic sea, see Fig. 2(a) for .

The Floquet states are evaluated as eigenstates of the time evolution operator over one period , which factorizes into a potential and a kinetic part . The quantization on the two-torus relates the effective Planck constant to the dimension of the Hilbert space by the condition . For the area of the regular island supports regular states.

The asymptotic state of the kicked rotor weakly coupled to a heat bath is again determined from Eq. (6), with the composite rates from Eq. (13) appropriate for a cyclic system. The resulting Floquet occupations are shown in Fig. 2(b) as functions of the cycle-averaged energies , with . The regular and chaotic states again are ordered with respect to this quantity, as indicated by the arrows in Fig. 2(b). The regular states have small values of , since both kinetic and potential energies are minimal in the center of the regular island, whereas the chaotic states have a stronger overlap with regions of phase space with higher energies. Similarly as for the driven oscillator, the regular occupations depend monotonously on , while the occupations of the chaotic states seem uncorrelated with the cycle-averaged energy and form a plateau with only weak fluctuations around a mean value .

iii.3 Rate matrix

The Floquet rate matrix determines the Floquet occupations via Eq. (6). In Fig. 3 we show for both the driven oscillator in Fig. 1 (Fig. 3(a)) and the kicked rotor in Fig. 2 (Fig. 3(b)). Employing as the ordering parameter for the entries , the regular and chaotic parts are well-separated, revealing a distinct block structure of the matrix BreHubPet2000 (). There are only few rates between the regular and the chaotic subspaces. Similarly, the rates between the two different regular subspaces in the case of the driven oscillator in Fig. 3(a) are practically zero. Also by virtue of the chosen ordering, the regular domains feature a band structure with particular dominance of the first off diagonals (nearest-neighbor rates). The rates in the subspace of the chaotic states on the contrary fluctuate strongly.

Figure 3: Density representation of the rate matrix , with entries sorted by increasing for (a) the driven oscillator, Eq. (17), and (b) the kicked rotor, Eq. (20), with enlarged domain of regular island states, revealing strong nearest-neighbor rates. For parameters see Figs. 1 and 2, respectively.

One can thus observe a close relation between the structure of the rate matrix and the resulting set of occupations. First, the almost independent behaviour of the occupations of regular states and chaotic states owes to the relatively weak rates connecting the corresponding subspaces. Furthermore, the random character of the chaotic rate submatrix gives rise to the equally random character of the set of chaotic Floquet occupations. Note that for the kicked rotor we find in the semiclassical limit , that the mean value decreases, since more and more regular states emerge. Also the relative variance of the chaotic occupations decreases in this limit and we observe a universal scaling in the semiclassical limit which can be analyzed with the help of a random-rate model Wus2010 (). These aspects of the chaotic occupations are not explored in this paper, instead the focus is set on the regular occupations.

Iv Regular States

The observations in Figs. 1 and 2 indicate that the asymptotic state of a time-periodic system in weak interaction with a heat bath carries signatures of the classical phase-space structure. The Floquet occupations of the regular and the chaotic states behave very differently, e.g. as functions of the cycle-averaged energy . In this section we focus on the asymptotic occupations of the regular states, which we label by their quantum number starting with for the state in the center of the island. Figures 1(b) and 2(b) suggest a roughly exponential dependence for the regular occupations as functions of the cycle-averaged energies . However, the regular occupations are different from the Boltzmann weights with the true inverse bath temperature . In fact, there is no physical reason for a coincidence with the Boltzmann distribution when expressed in terms of the, qualitatively suitable but arbitrary, energy measure  Koh2001 (). In the following sections we therefore make use of an alternative energy measure for the regular states, the regular energy (Sec. IV.1), allowing us to consistently parametrize the regular occupations as functions of (Sec. IV.2). Often, this functional dependence is approximately exponential (Sec. IV.3). Examples are presented in Sec. IV.4.

iv.1 Regular energy

A time-periodic system is equivalent to an autonomous system with the time as an additional coordinate, leading to the Hamiltonian in the extended phase space, which has periodic boundary conditions in . This allows the application of Einstein-Brillouin-Keller(EBK)-quantization rules for the regular tori and the determination of semiclassical Floquet states on the quantizing tori and their associated semiclassical quasienergies BreHol1991 (); BenKorMirBen1992 ().

We introduce the regular energies

(21)

Herein, is the winding number, i.e. the ratio of the winding frequency of a trajectory on the th torus around the central orbit to the driving frequency . Furthermore, is the long-time average of the Lagrangian for an arbitrary trajectory on the th torus. For convenience we add the time-averaged Lagrange function of the central orbit of the island. The regular energies of Eq. (21) are related to the semiclassical quasienergies BreHol1991 (); BenKorMirBen1992 () by

(22)

whereas there is no relation to the cycle-averaged energies . The time-averaged Lagrangian varies only slowly inside the island. The winding number likewise varies slowly across the island. To determine we have applied the frequency map analysis LasFroeCel1992 () being based on a Fourier decomposition of the quasiperiodic orbits within a stable regular island. Note that due to nonlinear resonances and small chaotic layers within a regular island the semiclassical quantization might require interpolations of the quantities and or the introduction of a fictitious integrable system BenKorMirBen1992 (); BohTomUll1993_BaeKetLoeSch2008 ().

Figure 4: (Color online) Ratio between the occupations and an exponential fit for (a) the kicked rotor and (b) the continuously driven oscillator using the regular energies (red circles) and the cycle-averaged energies (black diamonds). For parameters see Figs. 2 and 1, respectively.

Figure 2(c) shows the occupations of the regular states of the kicked rotor as functions of regular energies . The functional dependence of the occupations is close to exponential, but also different from the Boltzmann weights with the true bath temperature . However, the assumption of an exponential dependence of vs.  is fulfilled far better than vs. . This is demonstrated in Fig. 4, where the ratio between the occupations and the respective exponential fit is shown for being the regular energy (red circles) and the cycle-averaged energy (black diamonds). The fit involves the parameter . For the kicked rotor, Fig. 4(a), the considered ratio for is close to for the majority of regular states, whereas the ratio for systematically deviates from already for smaller values of . This indicates that the exponential scaling is far better fulfilled by using the regular energies .

Likewise, Fig. 4(b) shows the same ratio for the regular states of the central island in the driven oscillator, Eq. (17). The regular energies are again determined according to the above semiclassical quantization, where the frequency map analysis is applied to the solutions of the classical equations of motion, evaluated in the Poincaré section. Here, the quality of the fit with respect to is only marginally better as the the fit with respect to , see Fig. 4(b). We have evidence, that the existence of next-nearest-neighbor rates is responsible for this. Note that for other examples of continuously driven systems we typically find a better quality of the fit with respect to , similar to the situation in Fig. 4(a).

iv.2 Restriction to nearest-neighbor rates

In this section, the ratio of the rates and between two neighboring regular states and is analyzed for kicked systems. With the help of a detailed balance condition the occupations can be related to the winding numbers of the regular tori.

In the lower part of Fig. 3(b) the rate matrix , Eq. (13), for the regular subspace of the kicked rotor is shown. We remind the reader that the indices are ordered by increasing , coinciding with the natural order of growing quantum number . Figure 3(b) illustrates that the nearest-neighbor rates are dominant among the regular states. These nearest-neighbor rates are mainly contributed by the rates originating from the coupling operator , whereas the rates between next-nearest neighboring states are mainly due to the contribution of the coupling operator . In the following analytical considerations we will neglect the contribution of and in addition approximate the coupling operator inside the regular island at by the linear coupling operator . Using the resulting rate matrix in Eq. (6), we observe almost the same regular occupations as in Fig. 2(b), which in first approximation differ from the latter only by a tiny -independent factor. Only the occupations of the chaotic states are strongly affected by the different coupling scheme, as for them the discontinuity of the coupling operator at the border of the unit cell is not negligible.

In the rate matrix due to the linear coupling operator, , the nearest-neighbors dominate strongly, and the next-nearest-neighbor rates of regular states and are zero for symmetry reasons. We will neglect higher-order rates in the following analysis. Thus, the total rate balance among the regular states can be reduced to the detailed balance condition

(23)

between two neighboring regular states and .

Assuming Eq. (23) and using the definition of the rates in Eq. (7) the occupation ratio between neighboring regular states becomes

(24)
(25)

where the properties and have been used, and where we introduced the short-hand notations

(26)

for the (regular) nearest-neighbor matrix elements of the operator and

(27)

in the arguments of the correlation function .

If there were just a single Fourier component , which is approximately the case for a weakly driven system, then the occupation ratio would simplify to , resulting in Boltzmann-like occupations. In general, however, several components have to be considered.

Multiplying numerator and denominator of the fraction in Eq. (25) with ,

(28)

we introduce ratios of the matrix elements and of the correlation functions. The ratio of the correlation functions reads, using their definition in Sec. II,

The last factor in Eq. (IV.2) is close to and will be omitted in the following, such that the -dependence is neglected. This is possible since we are interested here in the case and we use the fact that only small integers contribute significantly to the sums in Eq. (28).

For the required ratio of the matrix elements we approximate the evolution of the coupling matrix elements for by

(30)

using the factorization of the time evolution operator for kicked systems and the approximate commutation relations with the operator on the two-torus, and . These commutation relations, which are exact in the infinite Hilbert space, apply here in very good approximation to the regular states, as these are almost independent of the periodic boundary conditions on the two-torus. The coupling matrix elements are time-periodic and have the Fourier components

(31)

whose ratio simplifies to

(32)

Finally, inserting Eqs. (IV.2) and (32) into the occupation ratio of Eq. (28) yields

(33)

with the function

(34)

It is invariant under an integer shift of the first argument, , with comment1 (). We choose the shift , such that

(35)

is fulfilled, which is possible according to Eq. (22). This allows us to replace in Eq. (33) with the regular energy spacing , leading to

(36)

Based on Eq. (21) we approximate this energy difference by the winding number

(37)

which is exact for a harmonic oscillator-like island with -independent winding number and and is a reasonable approximation even for more generic islands. The occupation ratio then becomes a function of the winding number,

(38)

With that an analytical prediction for the occupation of the regular states of a kicked system is found, valid under the assumption of dominant nearest-neighbor rates . It is a function of the winding number of the th quantizing torus and of the parameters temperature and driving frequency.

iv.3 Assumption of constant winding number

Figure 5: Inverse effective temperature according to Eq. (39) vs. winding number for and three different temperatures.

The function in Eq. (38) becomes independent of if the winding number is constant throughout the regular island. It is then appropriate to introduce an effective temperature by

(39)

With this new parameter the occupation ratios are expressed in a form analogous to the Boltzmann weights,

(40)

The ratio is shown in Fig. 5. Its value is smaller than and it is symmetric in . For , where the kicked system approaches its static limit, the true bath temperature is retained. A substantial deviation from the true bath temperature, , takes place around .

For generic islands with non-constant winding number, where the regular energy spacings are -dependent, the exponential scaling, Eq. (40), is still approximately valid if varies only moderately. Figure 5 indicates that this is fulfilled especially good for values . But even beyond this interval we observe a good agreement of Eq. (40) with the regular occupations. This breaks down for islands with winding numbers varying close to , where is particularly sensitive to variations of . As develops a pronounced -dependence there, the approximation (40) is no longer adequate and the general equation (38) has to be used instead, as demonstrated in Fig. 6 below.

iv.4 Examples

Figure 2(c) shows the regular Floquet occupations of the kicked rotor, Eq. (20), vs. the regular energies and demonstrates excellent agreement with the above predicted exponential weights (red solid line) for almost all regular states. The regular energies are here slightly -dependent, as the winding number decreases in the island monotonously from for the first () regular state to () for the outermost regular state. Deviations from the exponential distribution occur for the outermost regular states, , only. These have a stronger weight outside the regular island and are thus coupled stronger to the chaotic states. The non-negligible rates between these regular states and the chaotic states, see Fig. 3, enforce a gradual adaptation between the outermost regular occupations and the occupation level of the chaotic states. Besides, Fig. 2(c) shows the large discrepancy of the Boltzmann weights with the true bath temperature .

Figure 6: (Color online) (a)–(c) Analogous to Fig. 2 for the kicked rotor with . The insets in (b) show Husimi representations of two regular states. In (c) the occupations of the innermost regular states, which are not yet affected by phase-space structures at the border of the island, are well described by the analytical prediction (red solid line) of Eq. (38). The parameters are and .

The exponential distribution, Eq. (40), requires a constant or moderately varying winding number inside the island. If however varies close to , where is particularly sensitive to variations of , see Fig. 5, the exponential distribution is no longer adequate. For the kicked rotor this is the case for , where the central periodic orbit bifurcates and the regular island hence splits into two islands. As an example, Fig. 6 shows the occupations for the kicked rotor for , with and . The regular occupations in Fig. 6(c) are well described by the general prediction, Eq. (38) (red line), but clearly deviate from the exponential approximation, Eq. (40).

The derivation of the analytical occupation ratios (38) and (40) is based on assumptions which are justified for kicked systems only. For continuously driven systems an analogous prediction remains a future challenge. Figure 1(b) demonstrates that a significant and systematic deviation of the occupations from the Boltzmann result is observed also for continuously driven systems.

For another continuously driven system, the driven particle in a box, the regular occupations are almost identical to the Boltzmann weights with the true temperature  BreHubPet2000 (). However, in this particular example the regular region is almost identical to the undriven system, leading to Boltzmann weights for the regular states by the following reasoning: the regular states of the driven box potential, which emerge from the highly excited eigenstates of the undriven box, still strongly resemble the latter and change only slightly during the driving period . By that, one dominant Fourier contribution in the coupling matrix elements is singled out, for . In this situation the occupation ratio (25) simplifies to and the detailed balance among the regular states, Eq. (23), is fulfilled accurately. Since at the same time their cycle-averaged energies are close to the eigenenergies in the undriven potential, the occupations are close to the Boltzmann weights. Tiny deviations for the lowest regular states close to the chaotic region are visible in Fig. 1 of Ref. BreHubPet2000 (), which we attribute to rates occurring between the regular and the chaotic Floquet states.

As substantiated in this section, systematic and much stronger deviations from the Boltzmann behavior can occur in generic situations, especially in situations characteristic for strong driving, where the phase-space structure and the Floquet states are strongly perturbed compared to the original time-independent system. The strong driving allows us to study the Floquet occupations far from the thermodynamic equilibrium situation, encountered in the time-independent system, and at the same time to have dominant regular structures present in the classical phase space. These host, a sufficiently semiclassical regime presumed, a series of regular states, which under the condition of pronounced nearest-neighbor rates and only small rates to the subspace of the chaotic states are occupied with weights given by Eq. (38). For constant or slowly varying winding number the occupations even simplify to the exponential weights, Eq. (40), with the effective temperature of Eq. (39).

A generic modification of the analytical predictions of this section, takes place as a consequence of avoided crossings. We will go back to this point in Sec. VI.

V Implications of additional classical phase-space structures

The set of Floquet states in the examples of the last section are dominated by regular states in large regular islands and chaotic states. Apart from these, other types of Floquet states can exist, depending on the structures in the classical phase space and the size of the effective Planck constant . The following section gives an overview of the fingerprints of such additional types of Floquet states on the distribution of the Floquet occupations .

v.1 Nonlinear resonance chains

Apart from the islands centered at stable elliptic fixed points of period 1, there are nonlinear :-resonances consisting of regular islands around stable periodic orbits of period , see Fig. 7(a). A trajectory on such a resonance chain passes from an island to the th next island and returns after periods to the island, where it initially started. Considering the –fold iterated map instead of the map itself, the trajectory always remains on one and the same island.

The semiclassical quantization is done with respect to this –fold map of period  MirKor1994 (). To each principal quantum number there exist regular Floquet states of different quantum numbers with equidistant quasienergy spacing . We refer to these states as regular resonance states. Each of them has equal weights in each of the dynamically connected resonance islands, but with different phases.

Similarly as in Eq. (21) we derive from the semiclassical quasienergies the corresponding regular energies

(41)

which are independent of the quantum number . The winding number refers to the –fold iterated map.

Figure 7(b) shows the Floquet occupations vs. the cycle-averaged energy for the kicked rotor with , where the phase space features in addition to the main regular island a :-resonance around the periodic orbit of period , see Fig. 7(a). The entire resonance chain hosts regular resonance states for . The Floquet occupations of both the regular states of the central island and the chaotic states resemble those of Fig. 2(b). In addition, one finds a branch belonging to the regular resonance states. Interestingly, it has a positive slope stemming from the fact that the cycle-averaged energies of the regular resonance states decrease with increasing quantum number , in contrast to the regular states of the central island. This is due to the asymmetry of the resonance tori around their respective island center in phase space. This is another clear evidence, that the cycle-averaged energy does not serve as a suitable measure to quantify the regular occupations by exponential weights in analogy to the Boltzmann distribution.

Figure 7: (Color online) (a) and (b) Analogous to Fig. 2 for the kicked rotor with in presence of a :-resonance chain. The insets in (b) show Husimi representations of a regular state from the main island, a regular resonance state, and a chaotic state. (c) Floquet occupations of the regular resonance states vs. regular energies , the Boltzmann-like prediction Eq. (40) with (red solid line) compared to the inverse bath temperature (dashed line). The parameters are and .

The regular resonance states of fixed quantum number have almost the same cycle-averaged energy . As long as the coupling to the heat bath does not disturb the equivalence of the resonance islands, the occupations of the regular resonance states of fixed principal quantum number are independent of the quantum number . In Fig. 7(b) the corresponding four branches of the occupations therefore lie almost on top of each other and cannot be distinguished on the scale of the figure. Small deviations from this degeneracy exist only for the outermost regular resonance states. These can be attributed to the occurrence of avoided crossings, which break the degeneracy of for and fixed , as well as the degeneracy of and .

Now we want to explain that the occupations of the regular resonance states are likewise distributed as , according to the exponential weights (40) with the effective temperature of Eq. (39). We note that Eq. (IV.2) for the ratio of the correlation functions and Eq. (32) for the ratio of the coupling matrix elements apply without restriction also to the regular resonance states. The assumed detailed balance relation, Eq. (23), however, is no longer adapted to the structure of the rate matrix since here, in addition to the nearest-neighbor rates , also ‘internal’ rates exist, i.e. rates in the subspace of the equivalent regular resonance states with fixed quantum number . Nonetheless, the total rate balance approximately decouples for each principal quantum number into the balance relations for

(42)

They have the same structure as Eq. (23) and turn out to be approximately -independent, leading to approximately -independent occupations . In Eq. (42) the tiny rates with and are neglected and one can show that the contribution vanishes as a consequence of the equidistant quasienergy spacing for the regular resonance states of the same  Wus2010 ().

The decoupling into the equivalent balance relations (42) finally allows us to approximate the occupations by the exponential weights of Eq. (40) with the effective temperature of Eq. (39). Figure 7(c) shows the occupations of the regular resonance states vs. the regular energies . Even on the magnified scale of this subfigure, compared to Fig. 7(b), the tiny differences of the occupations with different quantum numbers are not visible. The effective temperature is nearly indistinguishable from the actual temperature , because the winding number of the resonance islands is small and yields a value of very close to , compare Fig. 5. Note that it differs, although weakly, from of the main island. The parameter is the same for each of the four independent occupation branches .

The phase space of a generic time-periodic system contains a hierarchy of nonlinear resonance chains and islands of all scales. If is sufficiently small, one has Floquet states on these islands and we expect that the entire set of Floquet occupations becomes increasingly structured by the branches originating from each nonlinear resonance.

v.2 Beach states

Figure 8: (Color online) (a) and (b) Analogous to Fig. 2 for the kicked rotor with in the presence of a :-resonance surrounded by a series of strong partial barriers. The insets in (b) show Husimi representations of a regular resonance state () and a beach state. (c) Floquet occupations of the regular resonance states vs. regular energies , the Boltzmann-like prediction Eq. (40) with (red solid line) compared to the inverse bath temperature (dashed line). The parameters are and .

The transition between regular phase-space regions and the chaotic sea is usually not sharp, but shaped by a multitude of small island chains and cantori, the fractal remains of broken Kolmogorov-Arnol’d-Moser tori. These additional phase-space structures can strongly inhibit the classical flux of trajectories toward and away from the regular island and, depending on the size of , can give rise to the formation of quantum beach states, a term introduced in Ref. FriDor1998 (). These reside on the transition layer around the regular islands and have little overlap with the remaining chaotic sea. Typically, beach states have very similar appearance and properties like the regular states of the adjacent island. Due to the proximity they partly even allow a quantization similar to the EBK-quantization rules FriDor1998 (); BohTomUll1990 ().

At the central island of the kicked rotor bifurcates into a resonance around a stable periodic orbit of period . It is accompanied by a series of partial barriers with a reduced classical flux toward and away from the islands. This is indicated for in the stroboscopic Poincaré section of Fig. 8(a) by the relatively high density of the chaotic orbit in the vicinity of the island. Figure 8(b) shows the Floquet occupations vs. the cycle-averaged energy . The highest occupations belong to the regular states of the resonance. The occupations of the beach states form a separate, nearly monotonous set in the transition region between the occupations of the regular resonance states and the chaotic states. This is a consequence of the structure in the coupling matrix , where typically the nearest-neighbor rates dominate, similarly as for the regular states.

Furthermore, the regular occupations of the regular resonance states are shown vs.  in Fig. 8(c). In this example the winding number in the resonance islands yields a stronger deviation between and , with , than in the example presented in Fig. 7.

v.3 Hierarchical states

As mentioned above, in the vicinity of regular islands typically many partial barriers with a limited classical flux toward and away from the island can be found, e.g. in the form of cantori or based on stable and unstable manifolds MaKMeiPer1992_Mei1992 (). Depending on the ratio of to the classical flux, partial barriers can prevent Floquet states from spreading over the entire chaotic domain, apart from tunneling tails. If the phase-space area enclosed by the island and the partial barrier exceeds , these states locally resemble chaotic states. For decreasing values of they resolve and occupy the hierarchy of the classical phase space better and better and are therefore called hierarchical states KetHufSteWei2000 (). The existence of these states does not contradict the semiclassical eigenfunction hypothesis Per1973_Ber1977_Vor1979 (), as their fraction vanishes with in the semiclassical limit. We apply an overlap criterion to determine, whether a Floquet state is hierarchical or not: it is identified as a hierarchical state, if it is not a regular state but comparably strongly localized, such that its Husimi weight within a large chaotic phase-space area away from the regular island falls below , compared to a state that is uniformly spread over the entire phase space.

Figure 9: (Color online) (a) and (b) Analogous to Fig. 2 for the kicked rotor with in the presence of a :-resonance surrounded by a partial barrier. The inset in (b) is the Husimi representation of a hierarchical state. The branch with positive slope belongs to the regular resonance states like in Fig. 7(b). (c) Magnification of (b) with emphasized data points of the hierarchical states (large green crosses), which are determined by the overlap criterion from the shaded phase-space area in (a). The parameters are and .

Figure 9(a) shows the Poincaré section and Fig. 9(b) shows the Floquet occupations for the kicked rotor with , where the fraction of hierarchical states is comparatively high KetHufSteWei2000 (). In Fig. 9(c), the occupations of the hierarchical states are emphasized. The figure indicates that their occupations are distributed analogously to the chaotic states which explore the entire chaotic phase-space region. Again, the fluctuation pattern of the occupations has its origin in the randomly fluctuating rates in the subspace of the hierarchical states, as is the case for the chaotic states.

To conclude this section, the occupation characteristics of the beach states and the hierarchical states again confirm the influence of the classical phase-space structure not only on the spectrum and on the Floquet states, but eventually also on the Floquet occupations and hence on the asymptotic state of the system.

Note that in the above examples, Figs. 8 and 9, either of the two types is predominant, but still representatives of the other are present. In general, hierarchical and beach states coexist. For example, a few of the states of intermediate cycle-averaged energy that are indicated in Fig. 9 as hierarchical by the above overlap criterion had rather to be classified as beach states or as states with scarring behavior, i.e. localized on hyperbolic fixed points or on a family of parabolic fixed points.

Vi Avoided crossings

Since the spectrum of Floquet systems is restricted to a finite interval , a multitude of avoided level crossings typically emerges under the variation of a parameter and gives rise to the hybridization of the involved Floquet states. In the case of an infinite dimensional Hilbert space the quasienergy spectrum is dense and there is no longer an adiabatic limit, i.e. any tiny parameter variation will hybridize infinitely many Floquet states in a complex way HonKetKoh1997 (). However, as shown in Ref. HonKetKoh2009 (), the asymptotic density operator is not affected by a small avoided crossing, provided that it is smaller than a specific effective coupling strength to the heat bath. Thus, the interaction with the heat bath resolves the difficulties of the dense quasienergy spectrum. In this section we focus on the opposite limit, where a single isolated avoided crossing strongly influences the entire set of Floquet occupations.

Figure 10: (Color online) Influence of an avoided crossing between the regular states and on Floquet occupations. (a) Floquet occupations vs. cycle-averaged energies for (black circles) and (red dots) close to the center of the avoided crossing. The insets show the Husimi representations of states , at and of a corresponding hybridized state at (‘ac’). (b) Occupations of regular states vs. regular energies at (red dots) and comparison to the analytical solution (47) of the rate model (solid gray line) and the model with from HonKetKoh2009 () (dashed gray line). The arrow indicates the effective rate between states and according to Eq. (44). Note that and are measured in the diabatic basis, in contrast to in (a). The parameters are and .

Figure 10 presents a typical example of the kicked rotor. In Fig. 10(a) the Floquet occupations are shown vs.  for two values of the kick strength near , very close to the parameter realization in Fig. 2. The difference of these -values is sufficiently small, such that the classical phase space and almost all regular states vary only marginally. For two of the regular Floquet states, which we will denote as states and , and which are initially identical to the semiclassical modes with the quantum numbers and , this is however not the case. Under the variation of they undergo an avoided crossing at , where they hybridize.

The tiny -variation strongly affects the Floquet occupations, and most prominently all the regular occupations. Away from the avoided crossing (), the regular occupations monotonously decrease with , similar as in Fig. 2(b). When approaching the center of the avoided crossing () the monotonous behaviour is locally disturbed: the states and , as a consequence of their hybridization, have shifted mean energies and as well as modified occupations . In Fig. 10(a) the data points of the states and at (marked as ‘ac’) are therefore found indistinguishable on top of each other. Beyond that, also the occupations of all regular states with quantum numbers from the interval change severely. They are close to the value of occupation of the hybridized states. In contrast, their mean energies do not change notably under the tiny -variation, like those of the semiclassical modes and . The relative occupations among the regular states with quantum numbers outside the range are also not affected. Only the absolute values of their occupations are shifted due to the normalization . The latter is also the origin of a shift of the chaotic occupation plateau .

This example demonstrates that the presence of avoided crossings can change the entire character of the occupation distribution. To explain this impact the authors of Ref. HonKetKoh2009 () introduced an effective rate equation,

(43)

which refers to a representation in the local diabatic basis of the avoided crossing, denoted by an overbar. In the diabatic basis the states and are replaced with states that remain invariant at the avoided crossing, i.e. the semiclassical modes and in the case of an avoided crossing of two regular states. In Eq. (43) the typically negligible rates , are replaced with a new effective rate HonKetKoh2009 ()

(44)

which acts between the states and . The gap size , i.e. the minimal quasienergy splitting of states and and the dimensionless distance from its center are characteristic properties of the avoided crossing. Unlike the rates , which are nearly constant in the vicinity of the avoided crossing, the additional rate changes dramatically. The composite rate