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The Journal of Chemical Physics

Phase space approach to theories of quantum dissipation

Daniela Kohen, C. Clay Marston, David J. Tannor

Published Oct 1, 1997• 212 citations
Publisher

Abstract

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.

DOI
10.1063/1.474887

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The Journal of Chemical Physics•Oct 1, 1997•28 citations
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.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad's theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function is also solved by using the Wang-Uhlenbeck method of transforming it into a linearized partial differential equation for the Wigner function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The Wigner functions which we obtain are two-dimensional Gaussians with different widths.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber PPP representation, the antinormal ordering QQQ representation and the Wigner WWW representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided.
We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier transform or chord function as it has been called, it becomes particularly simple to solve the dynamics of the open oscillator in the sense that the dynamics of the system are reduced to the application of an evolution matrix related to the damped motion of the oscillator.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions, is also solved by using the Wang-Uhlenbeck method. The obtained Wigner functions are two-dimensional Gaussians with different widths.
By using a rather simple algebraic approach, we revisit and further bridge between two most commonly used quantum dissipation theories, the Bloch–Redfield theory and a class of Fokker–Planck equations. The nature of the common approximation scheme involving in these two theories is analyzed in detail. While the Bloch–Redfield theory satisfies the detailed-balance relation, we also construct a class of Fokker–Planck equations that satisfy the detailed-balance relation up to the second moments in phase-space. Developed is also a generalized Fokker–Planck equation that preserves the general positivity of the reduced density operator. Both T1-relaxation and pure-T2 dephasing are considered, and their temperature dependence is shown to be very different. Provided is also an analogy between the quantum pure-T2 dephasing and the classical heat transport.
In a recent paper we showed the equivalence, under certain well-characterized assumptions, of Redfield’s equations for the density operator in the energy representation with the Gaussian phase space ansatz for the Wigner function of Yan and Mukamel. The equivalence shows that the solutions of Redfield’s equations respect a striking degree of classical-quantum correspondence. Here we use this equivalence to derive analytic expressions for the density matrix of the harmonic oscillator in the energy representation without making the almost ubiquitous secular approximation. From the elements of the density matrix in the energy representation we derive analytic expressions for Γ1n(1/T1n) and Γ2nm(1/T2nm), i.e., population and phase relaxation rates for individual matrix elements in the energy representation. Our results show that Γ1n(t)=Γ1(t) is independent of n; this is contrary to the widely held belief that Γ1n is proportional to n. We also derive the simple result that Γ2nm(t)=|n−m|Γ1(t)/2, a generalization of the two-level system result Γ2=Γ1/2. We show that Γ1(t) is the classical rate of energy relaxation, which has periodic modulations characteristic of the classical damped oscillator; averaged over a period Γ1(t) is directly proportional to the classical friction, γ. An additional element of classical-quantum correspondence concerns the time rate of change of the phase of the off diagonal elements of the density matrix, ωnm, a quantity which has received little attention previously. We find that ωnm is time-dependent, and equal to |n−m|Ω(t), where Ω(t) is the rate of change of phase space angle in the classical damped harmonic oscillator. Finally, expressions for a collective Γ1(t) and Γ2(t) are derived, and shown to satisfy the relationship Γ2=Γ1/2. This familiar result, when applied to these collective rate constants, is seen to have a simple geometrical interpretation in phase space.