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Alternative Derivation of the Hu-Paz-Zhang Master Equation for Quantum Brownian Motion

Hu, Paz and Zhang [ B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D {\bf 45} (1992) 2843] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases.


I. INTRODUCTION
Quantum Brownian motion (QBM) models provide a paradigm of open quantum systems that has been very useful in quantum measurement theory [1], quantum optics [2] and decoherence [3][4][5]. One of the advantages of the QBM models is that they are reasonably simple, yet sufficiently complex to manifest many important features of realistic physical processes.
Central to the study of QBM is the master equation for the reduced density operator of the Brownian particle, derived by tracing out the environment in the evolution equation for the combined system plus environment. A variety of such derivation have been given [6][7][8][9].
The most general is that of Hu, Paz and Zhang [10,11], who used path integral techniques and in particular, the Feynman-Vernon influence functional.
The purpose of this paper is to provide an alternative and elementary derivation of the Hu-Paz-Zhang master equation for QBM, by tracing the evolution equation for the Wigner function of the whole system.

II. MASTER EQUATION FOR QUANTUM BROWNIAN MOTION
The system we considered is a harmonic oscillator with mass M and bare frequency Ω, in interaction with a thermal bath consisting of a set of harmonic oscillators with mass m n and natural frequency ω n . The Hamiltonian of the system plus environment is given by where q, p and q n , p n are the coordinates and momenta of the Brownian particle and oscillators, respectively, and C n are coupling constants.
The state of the combined system (1) is most completely described by a density matrix ρ(q, q i ; q ′ , q ′ i , t) where q i denotes (q 1 , ...q N ), and ρ evolves according tȯ The state of the Brownian particle is described the reduced density matrix, defined by tracing over the environment, The equation of time evolution for the reduced density matrix is called the master equation.
For a general environment, Hu, Paz, and Zhang [10] derived the following master equation by using path integral techniques: The explicit form of the coefficients of the above equation will be given later on. The coefficient δΩ 2 (t) is the frequency shift term, the coefficients Γ(t) is the "quantum dissipative" term, and the coefficients Γ(t)h(t), Γ(t)f (t) are "quantum diffusion" terms. Generally, these coefficients are time dependent and of quite complicated behaviour.
We find it convenient to use the Wigner function of the reduced density matrix, Taking the Wigner transform of (4), we obtain 1 The inverse transformation of (5) is given by ρ r (q, q ′ , t) = dp e −ip(q−q ′ )/hW q + q ′ 2 , p, t .
Our strategy for deriving the master equation (4) is to derive the Fokker-Planck type equation (6) from the Wigner equation for the total system. The master equation can be obtained from the Wigner equation for the system by using the transformation (7).
We shall make the following two assumptions: (1) The system and the environment are initially uncorrelated, ie. the initial Wigner function factors where W s 0 and W b 0 are the Wigner functions of the system and the bath, respectively, at t = 0.
(2) The heat bath is initially in a thermal equilibrium state at temperature T = (k B β) −1 .
This means that the initial Wigner function of bath is of Gaussian form, where H n is the Hamiltonian of the n-th oscillator in the bath, In addition, one easily see that the initial moments of the bath are q n (0) = p n (0) = 0 , and q 2 n (0) =h 2m n ω n coth( 1 2h ω n β) , For the QBM problem described by (1) and (2), the Wigner function of the combined system plus environment satisfies By integrating over the bath variables on the both sides of the above equation , one obtains whereW (q, p) is the reduced Wigner function and it follows from Eq. (3) that This definition is equivalent to Eqs. In what follows, we shall demonstrate that the quantity appearing (differentiated with respect to p) in (17) can be expressed in terms ofW and its derivatives. To this end, we first perform Fourier transform of G(q, p) It is well known that q(t), p(t) and q n (t), p n (t) are related to the classical evolution of their initial values q(0), p(0) and q n (0), p n (0) through a canonical transformation: where Since the Hamiltonian (1) Changing the integration variables into their initial values by this canonical transformation, we obtain Here the coefficients f, g, f n , g n , α, β, a n , b n are time dependent. Their explicit values are not required.
Similarly, the Fourier transform of the reduced Wigner function is ×exp ik αq(0) + βp(0)) + n (a n q n (0) + b n p n (0)) ×exp ik ′ M αq(0) +βp(0) + n (ȧ n q n (0) +ḃ n p n (0)) Now compare G(k, k ′ ) andW (k, k ′ ). They differ by the terms linear in q(0), p(0), q n (0), p n (0) in the preexponential factor in G(k, k ′ ). Consider the factors f n q n (0) and g n p n (0) in , the terms f n q n (0)W b 0 and g n p n (0)W b 0 may be replaced by terms of the form ∂W b 0 /∂q n (0), ∂W b 0 /∂p n (0) up to time dependent factors. An integration by parts then may be performed, and these factors are then effectively replaced by multiplicative factors of Similarly, the factors f q(0), gp(0) in the prefactor in G(k, k ′ ) may be replaced by ∂/∂k, ∂/∂k ′ (plus some more factors of k and k ′ ). Hence, it is readily seen that G(k, k ′ ) is a linear combination of terms of the form k, k ′ , ∂/∂k, ∂/∂k ′ operating onW (k, k ′ ), with time dependent coefficients.
Inverting the Fourier transform, it follows that for some coefficients A(t), B(t), C(t), D(t) to be determined. This result immediately leads to the general form Wigner equation :

III. DETERMINATION OF THE COEFFICIENTS(GENERAL CASE)
Having found the functional form of the Wigner equation (26) of the Brownian particle, the next step is to determine the coefficients in the equation. Undoubtedly, there is more than one way to do this. Here we shall choose a way which is both mathematically simple and physically heuristic. Towards this direction, let us consider the time evolution of the expectation values of the system variables: q, p, q 2 , p 2 and 1 2 (pq + qp). By using Eq. (16), we have Similarly, using Eq. (26) yields Since the evolution equations of the expectation values are confined to the system variables, the above two sets of equations must be identical.
Now by comparing (28) with (33) we see that n C n q n = A q + B p .
Similarly, by comparing (31) with (36), (30) with (35), respectively, we get The coefficients A, B, C, D may now be determined from (37)-(39) by regarding the expectation values q , q n q etc. as expectation values of Heisenberg picture operators, and by solving the operator equation of motion. For simplicity, we still use ordinary notation to represent an operator without adding a hat on it.
The solution to the equation of motion may be written, q(t) = αq(0) + βp(0) + n (a n q n (0) + b n p n (0)) , for some time-dependent coefficients α n , β n , a nm , b nm , α, β, a n , b n . Note that q n (t) has been expressed in terms of the final, not initial values of q, p. By substituting Eq. (40) into Eq.
(37), keeping (11) in mind, and comparing the two sides of the resulting equation, we have Similarly, substituting (40) and (41) into (38) and (39), respectively, we get Here we have made use of p = Mq. The coefficients A, B, C, D are therefore completely determined by solving the equation of motion. We now do this explicitly.
The kernel η(s) is defined as where Here I(ω) is the spectral density of the environment: In order to get the expressions (40) where t is any given time point. In doing so, we consider the elementary functions u i (s)(i = 1, 2) introduced by Hu, Paz and Zhang [10] which satisfy the following homogeneous integrodifferential equationΣ with the boundary conditions: and u 2 (s = 0) = 0 , u 2 (s = t) = 1 . (57) The solution to equation (55) with the initial condition (53) is obtained as the linear combination of u 1 , u 2 , The solution to equation (48) with the homogeneous initial conditions can be formally written asw Where G 1 (s 1 , s 2 ) is the Green function which can be constructed in terms of u i (i = 1, 2): Note that G 1 (s 1 , s 2 ) as the function of s 1 satisfies equation (55) with Then the solution to the equation (48) with initial condition (53) reads explicitly, It can be shown that the solution to the homogeneous equation (55) with the initial conditions (54) is is the solution to the inhomogeneous equation (48) with the homogeneous initial conditions where Green function G 2 (s 1 , s 2 ) is Hence, we get the solution to Eq. (48) with the initial conditions (54) Substituting (66) into (47), one obtains By using (42) we immediately arrive at Furthermore, A, B can be written as From (62), the momentum of the Brownian particle is then Here "prime" stands for derivative with respect to the first variable of G 1 (s, τ ).
With these results, It can be easily shown that and where ν(s) is defined as It is seen that the coefficients A(t), B(t), C(t), D(t) are dependent only on the kernels η(s) and ν(s) and the initial state of the bath, not dependent on the initial state of the system.
Once the spectral density of the environment is given, the elementary functions u i (i = 1, 2) can be solved from equation (55), the Green functions G i (i = 1, 2) are then obtained. Thus the coefficients of master equation can be determined.

IV. PARTICULAR CASES
In this section, we will consider some special cases. Let us at first treat a special case in which we assume that the interaction between the system and environment is weak, so the C n are small. In this case, the coefficients are of simple forms, and the determination of these coefficients is very simple and straightforward. We shall work out these coefficients directly using the method in the last section, rather than the general formulae.
The solution to Eq. (46) may be written as q n (t) = q n (0) cos(ω n t) + p n (0) m n sin(ω n t) Using Eq. (37) and ignoring terms with higher than the second order of C n , we get We next evaluate n C n q(t)q n (t) and n C n p(t)q n (t) . After a few manipulations, we arrive at the expressions C(t) = − n C n { q(t)q n (0) cos(ω n t) + q(t)p n (0) sin(ω n t) m n ω n } , and D(t) = − n C n { p(t)q n (0) cos(ω n t) + p(t)p n (0) sin(ω n t) m n ω n } .