PHYSICAL REVIEW E VOLUME 61, NUMBER 5 MAY 2 Quasistationary distributions of dissipative nonlinear quantu oscillators in strong periodic driving fields Heinz-Peter Breuer, 1 Wolfgang Huber, 2 and Francesco Petruccione 1 1 Fakultät für Physik, Albert-Ludwigs-Universität, D-7914 Freiburg i Breisgau, Gerany 2 IBM, Aladen Research Center, San Jose, California 9512 Received 14 October 1999; revised anuscript received 17 Deceber 1999 The dynaics of periodically driven quantu systes coupled to a theral environent is investigated. The interaction of the syste with the external coherent driving field is taken into account exactly by aking use of the Floquet picture. Treating the coupling to the environent within the Born-Markov approxiation one finds a Pauli-type aster equation for the diagonal eleents of the reduced density atrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, tie-periodic density atrix which describes the long-tie behavior of the syste. Taking the exaple of a periodically driven particle in a box, the stationary solution is deterined nuerically for a wide range of driving aplitudes and teperatures. It is found that the quasistationary distribution differs substantially fro a Boltzann-type distribution at the teperature of the environent. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain nuber of Floquet states. This nuber of Floquet states turns out to be nearly independent of the teperature. The plateau region is sharply separated fro an exponential tail of the stationary distribution which expresses a canonical Boltzann-type distribution over the ean energies of the Floquet states. These results are explained in ters of the structure of the atrix of transition rates for the dissipative quantu syste. Investigating the corresponding classical, nonlinear Hailtonian syste, one finds that in the seiclassical range essential features of the quasistationary distribution can be understood fro the structure of the underlying classical phase space. PACS nubers: 5.7.Ln, 3.65.Sq, 2.5.Ga I. INTRODUCTION Within the Born-Markov approxiation, autonoous open quantu systes are described by quantu dynaical seigroups with a tie-independent Lindblad generator 1. Under quite general physical conditions such systes relax in the long-tie liit to a unique stationary state, which is given by the principles of equilibriu statistical echanics 2. For exaple, requiring the condition of detailed balance for the transition rates and soe kind of ergodic property regarding the operators which describe the coupling of the syste to its environent, one finds an equilibriu stationary state which is given by the Boltzann distribution over the energy eigenvalues of the syste. For open quantu systes in tie-varying external fields, the quantu dynaics ust be described, in general, by a tie-dependent generator. In the case in which the external driving field is strong, one expects that the long-tie dynaics differs significantly fro the equilibriu stationary state. In this paper, we shall investigate the question of the existence and the basic properties of a certain quasistationary state which governs the long-tie behavior for systes in strong, tie-periodic driving fields. In our study, the interaction with the external field will be treated exactly using the Floquet representation for tieperiodic quantu systes 3,4, whereas the coupling to environent will be taken into account in the Born-Markov approxiation. It is known that for this case the diagonal eleents of the reduced density atrix in the Floquet representation obey a closed equation of otion which is forally equivalent to a Pauli-type aster equation 5 7. We shall perfor analytical and nuerical investigations of the stationary solution of the Pauli aster equation for a general class of periodically driven, nonlinear oscillators coupled to an environent at finite teperature. Our results reveal that a large class of Hailtonian systes leads to a unique, quasistationary density atrix which is diagonal in the Floquet representation. The structure of the quasistationary distribution will be discussed in detail. We shall also study the connections to the phase flow of the corresponding classical Hailtonian syste, which shows a sharp dichotoy of quasiregular and chaotic otion. II. MASTER EQUATION FOR OPEN QUANTUM SYSTEMS IN STRONG DRIVING FIELDS A. The density atrix in the Floquet representation We consider in the following a periodically driven quantu syste coupled to an environent at teperature T for a review, see Ref. 7. The coherent part of the dynaics is generated by soe Hailtonian H(t) which is periodic in tie with frequency L, that is, we have H(tT L )H(t), where T L 2/ L denotes the period. Usually, H(t) takes the for H(t)H H I (t), where H is the unperturbed syste Hailtonian and H I (t) represents a tie-periodic interaction with an external driving field. According to the Floquet theore 3,4, there exists a basis of T L -periodic wave functions u j (t)u j (tt L ), the Floquet states, such that any solution (t) of the tiedependent Schrödinger equation pertaining to the Hailtonian H(t) can be represented in the for we choose units such that 1) 163-651X/2/615/48837/$15. PRE 61 4883 2 The Aerican Physical Society
4884 BREUER, HUBER, AND PETRUCCIONE PRE 61 t a j e i j t u j t. j N e 1 1 for, N 1, for, The quantity j is the Floquet index or quasienergy corresponding to the Floquet state u j (t). The iportant point to note is that the aplitudes a j in the above representation are independent of tie. In principle, the coupling to the environent ay take any for. To be specific, we take this interaction to be of dipole for, that is, the relevant syste operator which couples to the degrees of the environent is the dipole operator D. Such a coupling occurs, for exaple, in atoic or quantu optical systes where the environent could be the quantized radiation field in theral equilibriu 1,8. Invoking the Floquet representation, one finds that the dynaics of the diagonal atrix eleents tu j ttu j t of the reduced density operator (t) of the open syste is governed by the following Pauli aster equation in the strong driving liit: 1 d dt t k w jk p k tw kj t k W jk p k t. 2 This equation ay be derived directly within the densityatrix forulation 5 or else by aking use of the stochastic wave-function ethod and by investigating the associated jup process 6. In both cases, one uses the Born-Markov approxiation for the coupling of the reduced syste to its environent, whereas the coupling to the driving field is treated exactly by invoking the Floquet representation of the tie-evolution operator. Forally, Eq. 2 represents an ordinary aster equation in the sense of classical probability theory for a stochastic jup process 9. The quantity w kj is the rate probability per unit of tie for a jup fro the Floquet state u j (t) into the Floquet state u k (t). For dipole coupling, these rates are given by the explicit expression 6 w kj w kj kj N kj D kj 2. 3 Here, the su is to be extended over all integers which label the Fourier odes D kj of the tie-periodic dipole atrix eleent u k (t)du j (t), D kj T L dt T L e i L t u k tdu j t. The kj denote the corresponding transition frequencies which are given though differences of quasienergies plus integer ultiples of L, kj k j L. Finally, ()() denotes the density of odes of the environent belonging to the frequency. For, N () is the Planck distribution for the quanta of frequency. For the sake of a copact notation we define 4 5 where 1/k B T, T is the teperature of the environent, and k B denotes the Boltzann constant. B. The quasistationary solution Any aster equation of the for 2 has at least one stationary solution * 9. This is due to the fact that the atrix W jk has always a left eigenvector (1,1,1,...) belonging to the eigenvalue zero. The corresponding right eigenvector p* then fulfills Wp*, and, when noralized, is a stationary solution of the aster equation. Once we have deterined a stationary distribution * of the Pauli-type aster equation 2, we iediately obtain a solution *(t) of the corresponding density-atrix equation which is diagonal in the Floquet representation and which is given in the Schrödinger picture by *t j u j t *u j t. 6 This equation represents a density atrix which varies periodically in tie with a period which is equal to that of the external driving field. For this reason the stationary solution * of the Pauli-type aster equation ay be called quasistationary. An iportant question is whether the diagonal part of any initial density atrix converges for large ties to the quasistationary solution. For this to be the case, the stationary solution * of the Pauli aster equation ust be unique, which eans that the Pauli aster equation ust be irreducible 9. A siilar condition is used in the study of the return to equilibriu in relaxing seigroups of autonoous open quantu systes 2. For general rates w kj the deterination of the quasistationary solution * can be a difficult task. However, for an autonoous physical syste one expects that any initial state relaxes to a state which is in theral equilibriu with the environent. Thus, without external driving, the stationary solution of Eq. 2 should represent a canonical distribution over the energy eigenvalues j of the unperturbed syste Hailtonian H. It is instructive for the considerations below to recall briefly the basic arguents which lead to this conclusion. For an autonoous syste without external driving field, the aster equation is of the sae for as Eq. 2, where, however, the transition rates are given by w kj kj N kj D kj 2. The transition frequencies are now obtained as differences of unperturbed energy eigenvalues, kj k j, and D kj k D j is the dipole atrix eleent between the corresponding eigenstates k and j of H. According to general principles of statistical echanics 9, the stationary equilibriu distribution p* j of a closed physical syste obeys the condition of detailed balance which is given by w kj p* j w jk p k *. The crucial point is that the rates w kj for the autonoous syste do not involve a su over the index
PRE 61 QUASISTATIONARY DISTRIBUTIONS OF... 4885 which labels the Fourier coponents in Eq. 3. Therefore, the density of the odes of the environent as well as the dipole atrix eleent drop out when one fors the ratio w kj N kj N kj 1 e kj, w jk where we assue without restriction that kj. As can be seen iediately, the solution * of the detailed balance condition yields, as expected, the canonical distribution * Nexp( j ), where N is a noralization factor. This is the usual arguent eployed in statistical echanics, deonstrating that the syste relaxes to a stationary equilibriu state which is given by the canonical distribution over the unperturbed energy eigenvalues at the given teperature T of the environent. It is interesting to observe that a siilar arguent applies also to another case, naely the periodically driven haronic oscillator with syste Hailtonian, Ht 1 2 p2 1 2 2 x 2 x sin L t. In this case one obtains 6 D kj, j1,k j1 j1,k j 1 2. This expression shows that for jk1 the rates w kj vanish. For jk1, however, only the ter in Eq. 3 is different fro zero. This is a very specific property which is valid only for haronic potentials. Thus, the dipole operator couples only neighboring Floquet states u j (t) and u j1 (t) and the expression for the rates w kj involves only a single Fourier coponent, as is the case for an autonoous syste. On using the sae arguents as above, we therefore get the following quasistationary distribution: *Ne j, where j is the quasienergy spectru of the haronic oscillator see, e.g., 1. Note that the quasienergies of the driven haronic oscillator differ fro the unperturbed energies just by a j-independent ter. The quasienergies of the haronic oscillator are thus equidistant for all. Equation 8 iplies that the stationary, nonequilibriu distribution of the driven oscillator is a canonical distribution over its quasienergy states, that is, the quasistationary density atrix *(t) varies periodically in tie with tie-independent occupation probabilities of the Floquet states. A ore detailed investigation of the dynaics of dissipative, periodically driven systes with quadratic potentials can be found in Refs. 11,12. We now turn to the case of a driven, nonlinear oscillator. Instead of the siple relation 7, we have, in general, the following expression for the ratio of transition rates: 7 8 w kj w jk w kj w jk w kj w jk, kj N kj D kj 2, 9 kj N kj D kj 2 where we have used the relations jk, kj,, ()(), and D jk, D kj, *, which are valid by definition. In the case of strong driving, any Fourier odes of the Floquet wave functions are excited with appreciable aplitude. This eans that D kj 2 ay be appreciably different fro zero for any s which label these odes. Equation 9 therefore shows that the ratio w kj /w jk of transition rates differs significantly fro the siple relation 7, which is valid in the zero driving liit. Therefore, the siple line of reasoning leading to a stationary state given by a canonical distribution does not apply in the present case. In the next section we shall investigate nuerically the properties of the quasistationary distribution p* j of the Pauli-type aster equation for the case of a nonlinear, strongly driven oscillator. III. NUMERICAL SIMULATIONS A. Model syste and nuerical ethods As an exaple for a strongly anharonic syste, we consider a periodically driven particle in a potential box. The tie-dependent Hailtonian is given by d 2 Ht 1 2 dx Vxx sin Lt, 2 where the potential V(x) reads for xa, Vx for xa. Scaling space, tie, and oentu coordinates as xˆ x a, tˆ L t, pˆ p a L, 1 11 the Schrödinger equation corresponding to the Hailtonian 1 can be written as i 1 d dtˆ 1 d2 2 2 dxˆ Vˆ xˆ xˆ sin tˆ, 2 where Vˆ (xˆ )V(axˆ )/a 2 L 2 is the scaled potential and we have introduced the diensionless paraeters a 2 L, a L 2. 12
4886 BREUER, HUBER, AND PETRUCCIONE PRE 61 The coherent part of the dynaics thus depends only upon the two diensionless paraeters and. The paraeter is a diensionless coupling constant which is proportional to the field aplitude. The eaning of the paraeter ay be seen by looking at the coutator of space and oentu coordinate, pˆ,xˆ i 1. Recall that we have set 1. The quantity 1 is thus a diensionless, scaled Planck constant. The liit corresponds to the classical liit, whereas sall values of iply that quantu behavior doinates in a certain region of the classical phase space 1. The dissipative part of the dynaics introduces two further paraeters, naely the density of odes and the teperature of the environent. For siplicity we chose a constant density of odes, (). deterines the relaxation tie of the process but not the stationary solution. The latter only depends on the diensionless teperature Tˆ k BT L. 13 Thus, the stationary solution * depends on three diensionless paraeters, naely,, and Tˆ. The nuerical deterination of the stationary distribution * over the Floquet states proceeds in three steps as follows. i Deterination of the quasienergies and Floquet wave functions pertaining to the Hailtonian 1. ii Calculation of the Fourier coponents D kj of the dipole operator and deterination of the atrix W of the aster equation 2. iii Deterination of the noralized right eigenvector * of W corresponding to the eigenvector zero. To perfor step i we have represented the tiedependent Schrödinger equation in a finite basis consisting of N eigenfunctions of H. The Floquet spectru is deterined by diagonalization of the onodroy operator U(T L,), that is, the tie-evolution operator U(t,t ) of the tie-dependent Schrödinger equation taken over a period of the driving field. With the help of the Floquet states, one evaluates the tie-dependent atrix eleents u j (t)du k (t) of the dipole operator. In step ii one deterines the Fourier transfor of the dipole atrix eleents to obtain the Fourier coponents D kj. We denote by ax the nuber of sapling points which are used used in the nuerical Fourier tranforation. The Fourier coponents of the dipole atrix eleents together with the quasienergies deterined in step i yield the rates w kj and the atrix W. In step iii one has to find the zero ode of the atrix W. In all cases considered the lowest eigenvalue of W turned out to be saller than the other eigenvalues by a factor of at least 1 4. This clearly excludes the possibility of a degenerate zero ode, and of a decoposable W atrix 9. In all cases we thus have a unique stationary solution *. B. Nuerical results and discussion In the following we shall represent the stationary distribution p* j as a function of the ean energies Ē j which are FIG. 1. Logarithic plot of the quasistationary distribution * for the open syste with Hailtonian 1 as a function of the ean energy Ē j for different scaled teperatures Tˆ.5,1.,1.5,...,8. sybols. The lines represent a linear fit of the nuerical data in the exponential region, which is found to be in excellent agreeent with a Boltzann-type distribution corresponding to the various teperatures. The plateau region as well as the teperature independence of the ean transition energy Ē C which separates both regions are also clearly seen. The paraeters are 2,.248, N32, and ax 248. obtained by averaging the expectation value of the tiedependent Hailtonian H(t) in the Floquet states u j (t) over a period of the external field, Ē j T L dt T L u j thtu j t. 14 We display in Fig. 1 a logarithic plot of the stationary distributions * for a fixed driving aplitude of.248 and for different scaled teperatures Tˆ.5,1.,1.5,...,8.. As can be seen fro the figure, the canonical distribution over the unperturbed energies changes significantly for strong driving fields. The ost striking feature is that the stationary distribution * exhibits two qualitatively very different and clearly separated regions. In the first region we have a nuber of states which are populated with an approxiately constant probability. This region will be called plateau region in the following. For increasing ean energy the plateau region goes over to a second region where * clearly decays exponentially with the ean energy. A siilar conclusion was found in an investigation of chaotic tunneling in a double-well potential 13. Figure 1 also deonstrates that the plateau region is separated fro the exponential region by a sharp transition at soe ean energy Ē C which is nearly independent of the teperature. The solid lines of Fig. 1 are obtained by a linear fit of the nuerical data in the exponential region. One finds that the slopes of these lines are in perfect agreeent with the chosen scaled teperatures Tˆ of the environent. In the
PRE 61 QUASISTATIONARY DISTRIBUTIONS OF... 4887 FIG. 2. Representation of the (NN)-atrix w kj fored by the various rates for the transition between the Floquet states for the Hailtonian 1. The atrix eleent w is found in the upper corner. The paraeters are 2,.418, Tˆ 4.5, N32, and ax 248. exponential region the stationary distribution therefore represents a canonical distribution over the ean energies of the Floquet states. Suarizing these results, we ay write for the stationary distribution in the plateau region and in the exponential region *const for Ē j Ē C, 15 *expē j /k B T for Ē j Ē C. 16 In order to explain this behavior of the quasistationary distribution, we plot in Fig. 2 the atrix w kj. As can be seen fro the figure, for the chosen paraeters the states j 1,...,15 are strongly ixed: Each Floquet state is coupled to all other Floquet states fro this set and the corresponding transition rates vary erratically with k and j. This explains why in the plateau region the stationary solution is nearly constant since all rows and coluns of the W atrix su up to nearly the sae value in this range. However, above a certain sharp threshold, which is given by j15 for the paraeters of the figure, the transition rates strongly decrease with increasing j and k and couple only states with jk1. This behavior is very siilar to that of the haronic oscillator and explains why for j15 the stationary solution * behaves in a way which is siilar to that of the distribution of the haronic oscillator. FIG. 3. Poincaré surface of section for the strongly driven particle in the box for a scaled driving aplitude of.66. The figure represents a faily of solutions of the classical equations of otion in the (xˆ,pˆ ) plane at ties t,t L,2T L,...,6T L, where T L denotes the period of the driving field. C. Coparison with the classical phase-space structure The appearance of two clearly separated regions in the stationary distribution p* j can also be understood fro a siple seiclassical analysis. To this end, we first investigate the classical analog of the quantu syste given by the Hailtonian 1. The classical phase flow generated by the corresponding Hailtonian function is known to be chaotic and has been extensively studied in the literature see, e.g., 14,15. For strong driving fields, that is, for driving aplitudes c which are larger than the aplitude c.625 given by the Chirikov criterion 16 for the overlap of all priary resonances, the classical phase space also consists of two clearly separated regions 15. This can be seen fro Fig. 3 which shows a Poincaré surface of section of the phase space. The first region constitutes a connected chaotic sea which eerges fro the region of the priary nonlinear resonances and which contains only sall stable, elliptic islands. The second phase-space region is densely filled with invariant tori corresponding to perpetual adiabatic invariants 15. We see fro Fig. 3 that both regions are separated by a sharp border which arks the transition fro the irregular, chaotic otion to the quasiperiodic, nearly integrable otion in the region surrounding the chaotic sea. As is shown in 1, the quantization of the invariant flux tubes in the regular region of the phase space yields an excellent seiclassical approxiation for the Floquet states and the quasienergies of the quantu syste. In view of our above results on the quasistationary probability distribution of the open, dissipative syste, it is tepting to relate the exponential region of that distribution to the nearly integrable region of the phase space. In fact, for Ē j Ē C the quasienergies rapidly approach the ean energies Ē j and one expects a Boltzann distribution over the quasienergies in this region of phase space. On the other hand, the quasienergy spectru for the states corresponding to the chaotic sea shows a coplicated avoiding crossing structure when plotted as a function of the driving aplitude. This reflects large dipole atrix eleents and a broad Fourier spectru of these eleents. Consequently, one expects that the chaotic sea corresponds to the plateau region observed in the stationary distribution *.
4888 BREUER, HUBER, AND PETRUCCIONE PRE 61 IV. SUMMARY FIG. 4. The nuber of Floquet states in the plateau region of the quasistationary distribution * as a function of the scaled driving aplitude for fixed 2. The figure shows this nuber as it is obtained fro the full quantu calculation crosses and copares it with the nuber of seiclassical Floquet states corresponding to the chaotic sea of the classical phase space triangles estiated by eans of Eq. 17. To verify this siple seiclassical picture, we study the behavior of the quasistationary distribution as a function of the scaled driving aplitude. Figure 4 shows the nuber of Floquet states in the plateau region of p* j deterined fro the full quantu calculation crosses and copares it with the nuber of seiclassical Floquet states corresponding to the chaotic sea of the classical phase space triangles. The nuber of seiclassical Floquet states corresponding to the chaotic sea is estiated as follows. First, we have deterined fro the Poincaré surface of section the area A of the chaotic sea in the scaled coordinates xˆ, pˆ. According to the quantization rules derived in 1, we then get for the nuber n of seiclassical Floquet states in that region the estiate n 2 A. 17 This relation siply expresses Weyl s rule applied to the extended phase space of the Hailtonian syste. It has already been used in Ref. 17 for an investigation of the ixed regular and chaotic dynaics of a driven rotor. Note that A and n depend on. It is this function nn() which is represented in Fig. 4 triangles. The agreeent between both quantities plotted in this figure nicely confirs our above seiclassical picture. We have studied the dynaics of open quantu systes subjected to strong, periodic driving fields. The stationary solution of the Pauli-type aster equation which governs the diagonal eleents of the reduced density atrix in the Floquet representation has been deonstrated to differ substantially fro a canonical distribution at the teperature of the environent: It exhibits a certain region in which a nuber of Floquet states is occupied with approxiately constant probability. This plateau region is clearly separated fro an exponential tail of the stationary distribution which describes a Boltzann-type distribution at the environental teperature over the ean energies of the Floquet states. The nuber of Floquet states within the plateau region is nearly independent of the teperature but strongly depends on the driving aplitude. The essential features of the stationary solution can be understood fro an investigation of the classical phase-space structure. It has been shown that the plateau region corresponds to the chaotic sea which eerges fro the region of phase space belonging to the priary nonlinear resonances. This chaotic sea is surrounded by a nearly integrable phasespace region which is densely filled with invariant flux tubes. This region which is doinated by regular otion corresponds to the exponential tail of the stationary distribution. A sharp transition border separates the chaotic sea fro the nearly integrable otion and arks the transition fro the plateau region of the stationary distribution to its exponential tail. These results have been obtained fro nuerical siulations of a siple, strongly nonlinear odel, naely fro the periodically driven particle in a potential box. It ust be ephasized, however, that the dichotoy of the classical phase space as well as the general structure of the atrix w kj describing the transition rates between Floquet states is siilar for all potentials that lead to a discrete spectru of unperturbed energy eigenvalues whose spacing increases with increasing energy. For strong driving fields our results thus describe generic features for this class of potentials. We reark finally that our forulation of the proble of periodically driven open systes also allows the deterination of the quanta radiated during the jups between Floquet states. The frequencies of these quanta are deterined by relation 5. The above properties of the quasistationary state of the open syste could thus lead to characteristic features of the radiation spectru. ACKNOWLEDGMENT Financial support by the Deutsche Forschungsgeeinschaft within the Schwerpunktprogra Zeitabhängige Phänoene und Methoden in Quantensysteen der Physik und Cheie is gratefully acknowledged. 1 C. W. Gardiner, Quantu Noise Springer, Berlin, 1991. 2 R. Alicki and K. Lendi, Quantu Dynaical Seigroups and Applications, Lecture Notes in Physics Vol. 286 Springer- Verlag, Berlin, 1987. 3 Ya. B. Zeldovich, Zh. Éksp. Teor. Fiz. 51, 1492 1966 Sov. Phys. JETP 24, 16 1967. 4 J. H. Shirley, Phys. Rev. 138B, 979 1965. 5 R. Blüel, A. Buchleitner, R. Graha, L. Sirko, U. Silan-
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