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Annalen der Physik

A Method of Calculating the Density Matrix for a System of Interacting Bosons in Coherent‐state Basis

Tran van Trung, F.‐J. Schütte

Published Jan 1, 1977• 3 citations
Publisher

Abstract

Abstract The time‐dependent behaviour of a system of several interacting boson modes is described in coherent state basis. In the Schrödinger picture the unitary time development operator is used. The variation of the density operator for the whole system is described for arbitrary situations (even such without P‐representation). The method is applied to the parametric amplification process with strong pump field.

DOI
10.1002/andp.19774890405

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The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the $P$ representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and $n$-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative $P$ representation for the density operator for a single mode of oscillation.

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We present a numerically tractable method to solve exactly the evolution of a N boson system with binary interactions. The density operator of the system $\ensuremath{\rho}$ is obtained as the stochastic average of particular operators $|{\ensuremath{\Psi}}_{1}〉〈{\ensuremath{\Psi}}_{2}|$ of the system. The states $|{\ensuremath{\Psi}}_{1,2}〉$ are either Fock states $|N:{\ensuremath{\varphi}}_{1,2}〉$ or coherent states $|\mathrm{coh}:{\ensuremath{\varphi}}_{1,2}〉$ with each particle in the state ${\ensuremath{\varphi}}_{1,2}(x).$ We determine the conditions on the evolution of ${\ensuremath{\varphi}}_{1,2},$ which involves a stochastic element, under which we recover the exact evolution of $\ensuremath{\rho}.$ We discuss various possible implementations of these conditions. The well known positive P representation arises as a particular case of the coherent state ansatz. We treat numerically two examples: a two-mode system and a one-dimensional harmonically confined gas. These examples, together with an analytical estimate of the noise, show that the Fock state ansatz is the most promising one in terms of precision and stability of the numerical solution.

arXiv:cond-mat/0003399•Physical Review A•Jan 11, 2001•80 citations
Abstract The time‐dependent behaviour of a system of several interacting boson modes is described in coherent state basis. In the Schrödinger picture the unitary time development operator is used. The variation of the density operator for the whole system is described for arbitrary situations (even such without P‐representation). The method is applied to the parametric amplification process with strong pump field.
Abstract The time variation of the density matrix for a system of three interacting boson field modes is described in the Schrödinger picture with the help of the unitary time development operator in coherent state basis. The nonlinear c‐number equation for this operator is solved using a perturbation expansion in powers of the coupling constant. The explicit time dependence is obtained for the matrix elements of the total density matrix and of the particular density matrices of the individual modes. Photon number distributions and second order correlations are calculated for several initial situations.
The amplification of electromagnetic fields is analyzed in a quantum-mechanical context by discussing the behavior of a simple theoretical model of the parametric amplifier. The statistical properties of the amplifier fields are described by means of the time-dependent density operator for the system. In doing this, extensive use is made of the coherent states and the PPP representation of the density operator, which provide a quantum-mechanical description of the fields closely resembling their classical description. Explicit solutions are obtained for the density operator for either of the two field modes for a variety of initial states of the modes. Initial states considered include combinations of coherent states, chaotic mixtures, and nnn-quantum states. Particular attention is given the behavior of the amplifier fields in the limit of large amplification. The conditions are established under which the amplification process leads in this limit to the existence of a non-negative PPP representation for the density operator for a single mode of oscillation.
The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the PPP representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
Fermion-like coherent states are obtained as projections from the Glauber coherent states for a boson field, and the two sets of states have an interesting correspondence. It is shown that because of the correspondence the equation of motion of the density operator for a fermion interacting with a radiation field maps into a classical-like transport equation in a hypothetical boson phase space. Besides being useful in the description of systems of fermions involved in cooperative phenomena, it is shown that the use of the continuous basis offers some unique advantages even for systems which are essentially Markovian. As an illustration, the method is applied to the description of the time evolution of a two-level atom interacting with a blackbody radiation field.
We present a numerically tractable method to solve exactly the evolution of a N boson system with binary interactions. The density operator of the system $\ensuremath{\rho}$ is obtained as the stochastic average of particular operators $|{\ensuremath{\Psi}}_{1}〉〈{\ensuremath{\Psi}}_{2}|$ of the system. The states $|{\ensuremath{\Psi}}_{1,2}〉$ are either Fock states $|N:{\ensuremath{\varphi}}_{1,2}〉$ or coherent states $|\mathrm{coh}:{\ensuremath{\varphi}}_{1,2}〉$ with each particle in the state ${\ensuremath{\varphi}}_{1,2}(x).$ We determine the conditions on the evolution of ${\ensuremath{\varphi}}_{1,2},$ which involves a stochastic element, under which we recover the exact evolution of $\ensuremath{\rho}.$ We discuss various possible implementations of these conditions. The well known positive P representation arises as a particular case of the coherent state ansatz. We treat numerically two examples: a two-mode system and a one-dimensional harmonically confined gas. These examples, together with an analytical estimate of the noise, show that the Fock state ansatz is the most promising one in terms of precision and stability of the numerical solution.