Exact master equation for generalized quantum Brownian motion with momentum-dependent system-environment couplings

In this paper we generalize the quantum Brownian motion (QBM) to include the momentum-dependent system-environment couplings. The resulting Hamiltonian in the particle number representation is given by H_{t}ot=ℏω_{S}a^{†}a+∑_{k}ℏω_{k}b_{k}^{†}b_{k}+∑_{k}ℏ(V_{k}a^{†}b_{k}+V_{k}^{*}b_{k}^{†}a)+∑_{k}ℏ(W_{k}a^{†}b_{k}^{†}+W_{k}^{*}b_{k}a). The conventional QBM model corresponds to the special case W_{k}=V_{k}. The generalized QBM can capture more physical applications because the single-particle transition and the two-particle pair production between the system and the environment represent two very different physical processes that usually do not have the same coupling strengths. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized system Hamiltonian after tracing out all the environmental states induced naturally a momentum-dependent potential, which shows the consistency of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and initial correlated states, the issues about the initial jolt are also reexamined. We find that the so-called “initial jolt,” which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing to do with the initial decoupled state. The exact master equation for the generalized QBM also has potential applications to photonics quantum computing.