Physica E 9 (2001) 498–501www.elsevier.nl/locate/physe

Chaos in dissipative quantum maps:a Wigner function description(

Daniel Braun ∗

FB7, Universit�at-GHS Essen, 45 117 Essen, Germany

Abstract

We present a semiclassical analysis of dissipative quantum maps. We show that in the presence of a small amount ofdissipation the propagator of su�ciently smooth Wigner functions reduces to the classical Frobenius–Perron propagator ofthe phase space density for the corresponding dissipative classical map. As a consequence, the invariant Wigner functionis a smeared out classical attractor. Time-dependent expectation values and correlation functions of observables are givenby quantum–classical hybrid formulae, where the quantum mechanical character enters only through the initial Wignerfunction. If this function is classical or if the map is iterated many times, classical periodic orbit theories can be applied(L. Graham, T. T�el, Z. Phys. B 60 (1985) 127). ? 2001 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Sq; 05.45.Mt

Keywords: Dissipation; Quantum chaos; Semiclassical methods

1. Introduction

A dissipative quantum map P is a map of a reduceddensity matrix � from a discrete time t to a time t + 1,

�(t + 1) = P�(t) : (1)

The reduced density matrix describes a system in con-tact with an environment; the environmental degreesof freedom have been traced out of a larger densitymatrix for both system and environment. Dissipative

( This work was supported by the Sonderforschungsbereich 237,“Unordnung und gro�e Fluktuationen”.

∗ Fax: +49-201-183-2831.E-mail address: daniel@indy1.theo-phys.uni-essen.de

(D. Braun).

quantum maps have been studied for many years inorder to understand the interplay between chaos, dis-sipation, and decoherence in quantum mechanics [1–5]. Whereas it is clear that the ordinary de�nition ofchaos in classical mechanics via exponentially spread-ing trajectories does not apply in quantum mechanics[6], decoherence tends to make systems more clas-sical and one expects classical features of chaos tobecome important again. It has been shown in manyexamples that the Wigner function (or other suitablec-number representations of the density matrix in clas-sical phase space like the Q- or P-functions known inquantum optics) are propagated as if they were classi-cal phase space densities up to the so-called Ehrenfesttime [2,7–9]. This is a time of order �−1 ln ˜ (where �

1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(00)00252 -6

D. Braun / Physica E 9 (2001) 498–501 499

is the Lyapunov exponent) at which the exponentiallystretching and folding phase space structures reach ascale comparable to a Planck cell. Further reductionof the scales in the Wigner function is prohibited byHeisenberg’s uncertainty principle. The classical evo-lution continues to produce ever �ner structures andleads ultimately to a strange attractor as invariant state.Quantum mechanically the �ne structure is washedout, an e�ect that was often described by di�usiveterms in a Fokker–Planck equation for the time evolu-tion of the Wigner function, related both to quantumand thermal noise.

In the present work, we present a semiclassicalpropagator of the density matrix for certain dissipa-tive quantum maps that contains both the e�ect ofclassical chaos as well as quantum noise [10,11]. Wederive from it the propagator of the Wigner functionand show that the dissipative classical propagator gov-erns the time evolution of the Wigner function up tothe Ehrenfest time. Similar results have been obtainedwith very di�erent methods before for the quantizedHenon map [1] and in the theory of superradiance[12]. Whereas our earlier papers [10,11] focussed onspectral properties, we show in the present work thattime-dependent expectation values of observables andtheir correlation functions can be obtained from hy-brid quantum–classical formulae. The quantum me-chanical character enters only via the initially preparedWigner function, whereas its propagation is classical.If the initial Wigner function is a classical phase spacedistribution or if the map is iterated many times, theformulae become entirely classical, and classical pe-riodic orbit theories can be applied. The present workonly summarizes the main results; readers interestedin the details of the derivations are invited to consultRefs. [10,11,13,14].

2. The Wigner function and its propagator

We assume for our maps that the dissipation (de-scribed by a propagator D) be well separated froma remaining purely unitary evolution (described by aunitary Floquet matrix F) [9,10]

�(t + 1) = D(F�(t)F†) ≡ P�(t) : (2)

To have a speci�c model at hand, we consider a dis-sipative kicked top, which is an angular momentum J

with components Jx, Jy, and Jz. The unitary evolutionis generated by

F = e−i(k=2J )J 2z e−i�Jy ; (3)

and the dissipative one by D = exp(��), where � is agenerator de�ned in continuous time � by the Marko-vian master equation

dd�

�(�) =1

2J([J−; �(�)J+] + [J−�(�); J+])

≡ ��(�) ; (4)

J± = Jx ± iJy are the usual raising and lowering op-erators, and J = j + 1

2 was introduced to simplifysemiclassical formulae. The integer j de�nes thelength of the angular momentum, J 2 = j(j + 1). Thelength is conserved by both unitary evolution anddissipation. The map has a classical limit formallyattained for j → ∞. One can show that the Blochsphere limj→∞J 2=J 2 = 1 is the classical phase space[6]; � = Jz=J plays the role of classical momentumand �, the azimuthal angle of J reckoned against theJx axis, the role of the conjugate coordinate. The map�rst rotates J by an angle � about the y-axis and thensubjects it to a torsion, whose strength is controlledby the parameter k. Subsequently, the dissipation letsthe angular momentum sink down towards the southpole � = −1 of the Bloch sphere. The larger �, thefurther it can sink down, and consequently � mea-sures the dissipation strength. In the chaotic regimecharacterized by su�ciently large values of k and �(k ∼ 10, � ∼ 1) and not too strong damping (�. 1)a strange attractor arises in phase space. The dampingmechanism (4) is known to describe certain superra-diance experiments. The angular momentum operatorJ is then the Bloch vector of the collective excitationof a large number of two-level atoms that radiate col-lectively in a cavity of bad quality [15,16]. Explicitforms of D can be found in Refs. [12,15,17].

In the following we denote matrix elements of �(t)in the usual (J 2; Jz) representation (Jz|m〉 = m|m〉) by〈m1|�|m2〉 = �m(k; t) with m = (m1 + m2)=2 and k =(m1 − m2)=2, and go over to variables � = m=J and� = k=J that become continous in the limit of j → ∞,�(�; �; t) = �m(k; t). The Wigner function is de�nedas usual as a Fourier transform with respect to theskewness,

�W (�; �; t) =J 2

2�

∫d� eiJ���

(�;

�2; t)

: (5)

500 D. Braun / Physica E 9 (2001) 498–501

It is normalized as∫d� d��W (�; �; t) = J

∫d��(�; 0; t)

'∑m

�mm(t) = 1 : (6)

The inverse transformation reads

�(�; �; t) =1J

∫d�e−i2J���W (�; �; t) : (7)

Denoting phase space points by y = (�; �) and x =(�′; �′), the propagator of the Wigner function is de-�ned by

�W (y; t + 1) =∫

dxPW (y; x)�W (x; t) (8)

as an integral over the total phase space. A semiclas-sical approximation of PW can be derived by substi-tuting the de�nition (5) for �W (y; t + 1), expressing�(�; (�=2); t + 1) as a propagated initial density ma-trix,

�(�; �; t + 1) = 2J 2∫

d�′ d�′∞∑

s1 ; s2=−∞�(�′; �′; t)

×ei2�J(s1(�′+�′)+s2(�′−�′))P(�; �; �′; �′) ;

and then transforming �(�′; �′; t) back to the Wignerfunction �W (�; �; t). A Poisson summation over in-tegers s1 and s2 was used in order to transform thesummations over discrete indices m and k to integralsthat were subsequently parametrized by � and �. Adecisive ingredient is the propagator P(�; �; �′; �′) ofthe density matrix. A semiclassical approximation ofit was derived in great detail in Ref. [11] in the formof a van Vleck propagator. It contains a double sumover paths � of the classical dissipative map y = f(x).Along each path an action is accumulated which isa function of the initial and �nal points of the pathsde�ned by the arguments of P,

P(�; �; �′; �′) =∞∑

l=−∞

∑�1 ;�2

∑��

eJ (�;�;�′ ; �′)

×B(�; ��; �)C�1 ( �� + �; �′ + �′)

×C∗�2

( ��− �; �′ − �′) :

The real part of the action arises from D, the imaginarypart from F and F†, both of which have a van Vleckapproximation themselves,

(�; �; �′; �′) = R(�; ��; �) + i(2�l ��

+S�1 ( �� + �; �′ + �′) − S�2 ( ��− �; �′ − �′)):

The sum over �� in the epxression for P(�; �; �′; �′) isover all solutions of @ �� = 0. The prefactors B andC� are given by partial derivatives of the correspond-ing actions R and S�. They have important generatingproperties that can be found in Ref. [11]. It shouldbe noted that the semiclassical approximation of P isonly valid for �& 1=J . On the other hand, P containsthe deterministic chaotic classical motion as well asthe quantum noise that leads to a washing out of theever �ner classical phase space structures.

Inserting all ingredients in the expression for�W (�; �; t + 1) and performing the integrals bysaddle–point approximation (SPA), one is lead to thecentral result

�W (y; t + 1) =∫

dx�(y − f(x))�W (x; t);

which identi�es the propagator of the Wigner functionas classical propagator of the phase space density,

PW (y; x) = Pcl(y; x) = �(y − f(x)) : (9)

An important condition for the validity of this formulais that �W (�; �; t) does not contain any structure on ascale 1=J , since it enters as a pre–exponential factor inthe SPA. Since such structure develops even out of aninitially smooth Wigner function after the Ehrenfesttime, iterations of Eq. (9), Pt

w = Ptcl, are restricted to

t.�−1 ln J . Note that this restriction comes from theadditional SPA in the derivation of PW . The semiclas-sical propagator P(�; �; �′; �′) itself is valid up to theHeisenberg time, t . J .

3. Consequences

Eq. (9) allows in a straightforward way to calcu-late time-dependent expectation values and correlationfunctions in semiclassical approximation. The expec-tation value of any observable A at time t is de�nedas

〈A(t)〉 ≡ tr(A�(t))

=∫

dx AW (x)�W (x; t) ; (10)

where AW (x) is the Weyl symbol associated with theoperator A. The de�nition of AW is analogous to thede�nition of �W [9]. To lowest order in 1=J AW (x)

D. Braun / Physica E 9 (2001) 498–501 501

equals the classical observable A(x) that correspondsto A. Using Eq. (9) we obtain

〈A(t)〉 =∫

dx A(x)Ptcl�W (x; 0) ; (11)

up to corrections of order 1=J . Thus, quantum me-chanical expectation values can be obtained from theknowledge of the classical propagator and the clas-sical observable for any initial Wigner function thatcontains no structure on the scale 1=J . Eq. (11) is ahybrid classical–quantum formula, since the initialWigner function can be very non–classical, e.g. cancontain regions where �W (x; 0)¡ 0.

A similiar expression can be derived for a correla-tion function of two observables A and B,

K(t2; t1) = 〈B(t2)A(t1)〉0

= tr(BPt2APt1�(0)

): (12)

The result is again a hybrid formula

K(t2; t1) =∫

dx B(ft2 (x))A(x)Pt1cl�w(x; 0) ; (13)

valid up to O(1/J ) corrections. This formula is mani-festly real; the imaginary part of K(t2; t1) can indeedbe seen to be at least one order in 1=J smaller than thereal part.

If �W (x; 0) is chosen as a classical phase space den-sity, both Eqs. (11) and (13) become entirely clas-sical formulae. The same is true if in Eq. (11) t →∞ or in Eq. (13) t1 → ∞ with t2 − t1 �xed. For inthis case, �W (x; t) (respectively �W (x; t1)) approachesthe invariant state �W (x;∞) which is a smeared outclassical strange attractor [4,5]. To see this note that�W (x;∞) is an eigenstate of PW to the eigenvalueunity. But in view of Eq. (9), eigenstates of PW andof Pcl agree up to corrections of order 1=J whichhave to be understood as a smearing out on a scaleof that order. Consequently, powerful classical traceformulae can be used to calculate expectation val-ues and correlation functions in the invariant state[14,18,19].

4. Conclusion

We have shown that for certain dissipative quan-tum maps the propagator of Wigner functions that are

smooth on a scale 1=J can be approximated up tothe Ehrenfest time by the classical propagator of thephase space density for the corresponding classicaldissipative map. This leads to quantum–classical hy-brid formulae for expectation values and correlationfunctions of observables in which the quantum me-chanical character only enters via the initial state.The invariant state has a Wigner function that is asmeared out classical strange attractor. Expectationvalues or correlation functions with respect to thisinvariant state can be calculated with classical traceformulae.

Acknowledgements

I would like to thank Petr A. Braun and F. Haakefor stimulating discussions. This work was supportedby the Sonderforschungsbereich 237, “Unordnung undgro�e Fluktuationen”.

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