We solve the master equation for the coordinate-coordinate damped harmonic oscillator for initial superpositions of coherent states. In the zero-temperature case the solution remains a simple superposition of coherent states. While the underdamped oscillator evolves all initial superpositions into mixtures of coherent states the overdamped oscillator does so selectively. For finite temperatures coherent states are no longer preserved, and we find a decrease in the variance of the off-diagonal coordinate-basis density-matrix elements below the coherent-state value. This variance decreases with increasing bath temperature. In the overdamped case there is negligible associated spreading of the diagonal coordinate-basis density-matrix elements. Thus the coordinate basis is an example of Zurek's pointer basis and the coordinate damped oscillator models the coordinate-basis density-matrix diagonalization which occurs in a coordinate measurement.
We solve the master equation for the coordinate-coordinate damped harmonic oscillator for initial superpositions of coherent states. In the zero-temperature case the solution remains a simple superposition of coherent states. While the underdamped oscillator evolves all initial superpositions into mixtures of coherent states the overdamped oscillator does so selectively. For finite temperatures coherent states are no longer preserved, and we find a decrease in the variance of the off-diagonal coordinate-basis density-matrix elements below the coherent-state value. This variance decreases with increasing bath temperature. In the overdamped case there is negligible associated spreading of the diagonal coordinate-basis density-matrix elements. Thus the coordinate basis is an example of Zurek's pointer basis and the coordinate damped oscillator models the coordinate-basis density-matrix diagonalization which occurs in a coordinate measurement.
We use a Markovian master equation to solve for the dynamics of a free particle damped by the coupling of its coordinate to a bath of harmonic oscillators. In the coordinate basis the reduced density matrix for the free particle tends to be diagonalized. Considering the evolution of an initial superposition of two plane waves, we find that the interference-pattern fringe contrast is decreased by the damping. However, for fixed times and sufficiently large damping, the fringe contrast increases with increasing damping. Considering an initial coordinate-space Gaussian wave function, we find that for sufficiently large damping the coupling to the environment models a quantum measurement of the particle's coordinate.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes-grand canonical ensemble, number states, and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and momentum show a Brownian-like behavior. We derive an exact master equation for the characteristic function of the oscillator valid for arbitrary factorized initial conditions. In the case of the equilibrium bath this equation reduces to an equation of the Hu-Paz-Zhang type, while for the coherent states bath it leads to an exact stochastic master equation with a multiplicative noise.
Some aspects of quantum damped harmonic oscillator (DHO) obeying a Markovian master equation are considered in the absence of thermal noise. The continuity equation is derived and Bohmian trajectories are constructed. As a solution of the master equation, we take a single coherent state and compute analytically the relative entropy of coherence, [Formula: see text], in the energy, position and momentum bases. Although [Formula: see text] is constant in both the position and the momentum bases, it is a decreasing function of time in the energy basis becoming zero at long times, revealing its role as the preferred basis. Then, quantum coherence is computed for a superposition of two coherent states, a cat state, and also a superposition of two cat states in the energy basis as a function of separation, in the complex plane, between the two superposed states. It is seen that the quantum coherence increases with this separation. Furthermore, quantum coherence of superposition is compared to that of decomposed states in the superposition. Finally, by considering a system of two noninteracting DHOs, the effect of quantum statistics is studied on the coherence of reduced single-particle states, the joint detection probability and the mean square separation of particles. Our computations show that the single-particle coherence for antisymmetric states is always less than that of symmetric ones. Furthermore, boson anti-bunching and fermion bunching is seen in this open system. This behavior of bosons is the matter-wave analogue of photon anti-bunching seen in a modified Hanbury Brown–Twiss (HBT) interferometer.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2(t)〉, 〈p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.
We investigate the exact dynamics of the damped quantum harmonic oscillator under the (un)correlated initial conditions. The master equation is generalized to the cases of the arbitrary factorized state and/or Gaussian state. We show that the variances of the factorized Gaussian state do not sensitively depend on the initial oscillator-bath correlation, which however can remarkably affect the mean values even at high temperature. We also illustrate that the correlations among the factorized states still give rise to the initial dips during the purity evolutions, which can be smoothed out by increasing the amount of correlation to some extent. We finally study the effects of repeated measurements on the time evolution of the damped oscillator analytically, which are compared with the weak coupling results to indicate that they give rather different transient behaviors even for an intermediate coupling.
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 P representation, the antinormal ordering Q representation and the Wigner W 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.