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Progress of Theoretical Physics Supplement

Dual Spaces Structure of Quantum Kinetic Equations

Buang Ann Tay, Tomio Yamakoshi Petrosky

Published Jan 1, 2010• 1 citation
Publisher

Abstract

The relation between the Hilbert space structure and the generalized spaces structure represented by dual states for dissipative kinetic equation is discussed for quantum systems. As working examples, we consider the systems of a harmonic oscillator or a particle interacting with a thermal reservoir and construct analytic solutions to the eigenvalue problem of the quantum collision operators of these systems. The generalized spaces structure of the eigenfunctions indicates that dissipation destroys the Hilbert space structure of the undamped system. In the Wigner representation where the quantum collision operators closely resemble the classical kinetic operators in phase space, the Hilbert space structure can be restored to certain extent by introducing a weighted norm or a similarity transformation on the operators. However, in the position space where the collision operators have no classical counterpart, generalized spaces description cannot be avoided.

DOI
10.1143/ptps.184.533

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The relation between the Hilbert space structure and the generalized spaces structure represented by dual states for dissipative kinetic equation is discussed for quantum systems. As working examples, we consider the systems of a harmonic oscillator or a particle interacting with a thermal reservoir and construct analytic solutions to the eigenvalue problem of the quantum collision operators of these systems. The generalized spaces structure of the eigenfunctions indicates that dissipation destroys the Hilbert space structure of the undamped system. In the Wigner representation where the quantum collision operators closely resemble the classical kinetic operators in phase space, the Hilbert space structure can be restored to certain extent by introducing a weighted norm or a similarity transformation on the operators. However, in the position space where the collision operators have no classical counterpart, generalized spaces description cannot be avoided.
Generalised Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on the phase-space are studied. Using such transformations, quantum linear evolution equations are given a phase-space representation. In particular this is done for the general kinetic equation of the Lindblad type. The resulting expressions are better suited for the passage to the classical limit and for a general comparison of classical and quantum systems. In this context a preliminary discussion of a number of problems of kinetic theory of open systems is given, whereas explicit applications are made in the next paper of the series.
The formalism of generalized Wigner transformations developped in a previous paper, is applied to kinetic equations of the Lindblad type for quantum harmonic oscillator models. It is first applied to an oscillator coupled to an equilibrium chain of other oscillators having nearest-neighbour interactions. The kinetic equation is derived without using the so called rotating-wave approximation. Then it is shown that the classical limit of the corresponding phase-space equation is independent of the ordering of operators corresponding to the inverse of the generalized Wigner transformation, provided the latter is involutive. Moreover, this limit equation, which conserves the probabilistic nature of the distribution function and obeys an H-theorem, coincides with the kinetic equation for the corresponding classical system, which is derived independently and is distinct from that usually obtained in the litterature and not sharing the above properties. Finally the same formalism is applied to more general model equations used in quantum optics and it is shown that the above results remain unaltered.
Six major theories of quantum dissipative dynamics are compared: Redfield theory, the Gaussian phase space ansatz of Yan and Mukamel, the master equations of Agarwal, Caldeira-Leggett/Oppenheim-Romero-Rochin, and Louisell/Lax, and the semigroup theory of Lindblad. The time evolving density operator from each theory is transformed into a Wigner phase space distribution, and classical-quantum correspondence is investigated via comparison with the phase space distribution of the classical Fokker-Planck (FP) equation. Although the comparison is for the specific case of Markovian dynamics of the damped harmonic oscillator with no pure dephasing, certain inferences can be drawn about general systems. The following are our major conclusions: (1) The harmonic oscillator master equation derived from Redfield theory, in the limit of a classical bath, is identical to the Agarwal master equation. (2) Following Agarwal, the Agarwal master equation can be transformed to phase space, and differs from the classical FP equation only by a zero point energy in the diffusion coefficient. This analytic solution supports Gaussian solutions with the following properties: the differential equations for the first moments in p and q and all but one of the second moments (q2 and pq but not p2) are identical to the classical equations. Moreover, the distribution evolves to the thermal state of the bare quantum system at long times. (3) The Gaussian phase space ansatz of Yan and Mukamel (YM), applied to single surface oscillator dynamics, reduces to the analytical Gaussian solutions of the Agarwal phase space master equation. It follows that the YM ansatz is also a solution to the Redfield master equation. (4) The Agarwal/Redfield master equation has a structure identical to that of the master equation of Caldeira-Leggett/Oppenheim-Romero-Rochin, but the two are equivalent only in the high temperature limit. (5) The Louisell/Lax HO master equation differs from the Agarwal/Redfield form by making a rotating wave approximation (RWA), i.e., keeping terms of the form ââ†,â†â and neglecting terms of the form â†â†,ââ. When transformed into phase space, the neglect of these terms eliminates the modulation in time of the energy dissipation, modulation which is present in the classical solution. This neglect leads to a position-dependent frictional force which violates the principle of translational invariance. (6) The Agarwal/Redfield (AR) equations of motion are shown to violate the semigroup form of Lindblad required for complete positivity. Considering the triad of properties: complete positivity, translational invariance and asymptotic approach to thermal equilibrium, AR sacrifices the first while Lindblad’s form must sacrifice either the second or the third. This implies that for certain initial states Redfield theory can violate simple positivity; however, for a wide range of initial Gaussians, the solution of the AR equations does maintain simple positivity, and thus for these states appears to be distinctly more physical than the solution of the semigroup equations.
In the paper we discuss possible approaches to the problem of the rigorous derivation of quantum kinetic equations from underlying many-particle dynamics. For the description of a many-particle evolution we construct solutions of the Cauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitable Banach spaces. In the framework of the conventional approach to the description of kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchy solution is constructed. We develop also alternative approaches. One method is based on the construction of the solution asymptotics of the initial-value problem of the quantum dual BBGKY hierarchy. One more approach is based on the generalized quantum kinetic equation that is a consequence of the equivalence of the Cauchy problems of such evolution equation and the BBGKY hierarchy with initial data determined by the one-particle density operator.
In this second in a series of four articles we create a mathematical formalism sufficient to represent nontrivial hamiltonian quantum dynamics, including resonances. Some parts of this construction are also mathematically necessary. The specific construction is the transforming of a pair of quantized free oscillators into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, thereby creating dynamical representations of numerous semi-groups. The quantum free oscillators are the quantum analogue of classical action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances represented by Gamow vectors are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodated in a well defined formalism which represents dynamics, and these may be met by using the commutative real algebra $\C (1,i)$ as the ring of scalars in place of the field of complex numbers. These mathematical constraints compel use of fundamental (spinor) representations rather than UIR's. By including distributional solutions to the Schr{ö}dinger equation, placing us in a rigged Hilbert space, and by using the Hamiltonian as a generator of canonical transformation of the space of states, the Schr{ö}dinger equation is the equation for parallel transport of generalized energy eigenvectors, explicitly establishing the Hamiltonian as the generator of dynamical time translations in this formalism.
The eigenvalue problem of Kossakowski-Linblad’s kinetic equation associated with the reduced density matrix of a harmonic oscillator interacting with a thermal bath in equilibrium is solved. The solution gives rise to a complete orthogonal eigenbasis endowed with Hilbert space structure that has a weighted norm. We find that the eigenfunctions at finite temperature can be obtained from the eigenfunction at zero temperature through a hyperbolic rotation on the position variables. This transformation enables the extension of the simple harmonic oscillator density matrix to that of a finite temperature. We further investigate the decay of these extended states under our dissipative kinetic equation. Furthermore, the Hilbert space structure enables the proof of a H-theorem in this system. We apply the eigenbasis expansion of an initial state to analyze decohorence as well as coherence processes. We find that coherence process occurs at a longer time scale compared to decoherence process. The time scales of both processes are estimated with the eigenbasis expansion. In the same way we analyze the evolution of the coherent state. We show that in addition to the ordinary decay time, we found another time scale which is defined by the time when the motion of the peak of the coherent state become comparative to the width of the coherent state. In contrast to the ordinary decay time this new relaxation time depends on the initial value of the momentum of the oscillator. We also find that our eigenbasis is applicable to a class of non-linear interactions, with a slight extension of the form of transport coefficients due to the non-linear interactions.