Reconstruction approach to quantum dynamics of bosonic systems

We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.

Figure 1

(a) Schematic process chart of the dynamical reconstruction method. First, the Schrödinger picture Hamiltonian is transformed into the Heisenberg picture. Subsequently, the Heisenberg equations of motion for each annihilation operator of the system are obtained and solved. Finally, the density operator of the reduced system of the discrete modes is reconstructed by inserting the solutions for the annihilation operators into the dynamical reconstruction formula, Eqs. (3) and (4). For simplicity, the process chart describes a single-mode bosonic system. (b) Schematic of a general bosonic quantum system consisting of $N$ discrete modes and $M$ continua of modes. The annihilation operators ${\widehat{a}}_j$ label the discrete modes, and the continuous-mode annihilation operators ${\widehat{B}}_j\left(\omega \right)$ label the continua of modes. Schematic illustrations of the energy-level diagrams of the discrete modes and the spectra of the continua are provided.

Figure 2

Schematic of the example system of two coupled damped quantum harmonic oscillators. The annihilation operators, the frequencies, and the decay rates of the harmonic oscillators are ${\widehat{a}}_j, {\omega }_j$, and ${\kappa }_j$, respectively, the annihilation operators of the corresponding heat baths of the harmonic oscillators are ${\widehat{B}}_j\left(\omega \right)$, and the coupling strength between the harmonic oscillators is $g$.