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arXiv:quant-ph/0407237 · Journal of Optics B Quantum and Semiclassical Optics

Quantum noise and mixedness of a pumped dissipative non-linear oscillator

Jirí Bajer, Adam Miranowicz, Mateusz Andrzejewski

Published Aug 24, 2004• 10 citations
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Abstract

Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states, evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. It is shown analytically and verified numerically that the steady state can be well approximated by a non-classical Gaussian state exhibiting quadrature squeezing and sub-Poissonian statistics for the cold thermal reservoir. A rapid increase is found in the mixedness, especially for the initial Fock states and superpositions of coherent states, during a very short time interval, and then for longer evolution times a decrease in the mixedness to the same, for all the initial states, and relatively low value of the non-classical Gaussian state.

DOI
10.1088/1464-4266/6/10/002

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Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states, evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. It is shown analytically and verified numerically that the steady state can be well approximated by a non-classical Gaussian state exhibiting quadrature squeezing and sub-Poissonian statistics for the cold thermal reservoir. A rapid increase is found in the mixedness, especially for the initial Fock states and superpositions of coherent states, during a very short time interval, and then for longer evolution times a decrease in the mixedness to the same, for all the initial states, and relatively low value of the non-classical Gaussian state.
Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. It is shown analytically and verified numerically that the steady state can be well approximated by a nonclassical Gaussian state exhibiting quadrature squeezing and sub-Poissonian statistics for the cold thermal reservoir. It is found a rapid increase in the mixedness, especially for the initial Fock states and superpositions of coherent states, during a very short time interval, and then for longer evolution times a decrease in the mixedness to the same, for all the initial states, and relatively low value of the nonclassical Gaussian state.
In this communication we investigate the quantum statistics of three harmonic oscillators mutually interacting with each other considering the modes are initially in Fock states. After solving the equations of motion, the squeezing phenomenon, sub-Poissonian statistics and quasiprobability functions are discussed. We demonstrate that the interaction is able to produce squeezing of different types. We show also that certain types of Fock states can evolve in this interaction into thermal state and squeezed thermal state governed by the interaction parameters.
The focus of the study is the free quantum evolution of Duffing oscillators which are nonlinearly coupled to bosonic environments. The work evolves from a particular system of an undamped, nonlinear graphene resonator mode to a more generalized framework of oscillator modes interacting with two different configurations of bosonic environments. In a model with no dissipation, the nonlinear Duffing oscillator allows an initial coherent state to evolve into a macroscopic superposition state, a Schrödinger Cat state. By further subjecting a Duffing mode to nonlinear damping, the parity conservation due to the two-quanta system-bath exchange, brings the system to a nonclassical steady state - an equilibrium state very much different from the ground state. The quantum features of this state are analysed for temperatures above zero and in a more realistic scenario where the interplay of linear and nonlinear decay is taken into account. The scope of the study is then extended to bipartite systems of nonlinear interacting oscillators, each nonlinearly coupled to a bosonic environment. The generation of entanglement in initially separable states and entanglement's asymptotic behaviour are investigated. One of the outcomes is the Duffing nonlinearity not affecting the entanglement asymptote. Therefore, finally a bipartite system of harmonic oscillators, with nonlinear system-bath coupling is initiated with entangled, squeezed states, and the asymptotic behaviour is evaluated in the parameter space of temperature, squeezing and dissipation rate. This is done for common and individual reservoir configurations, and either zero or non-zero inter-mode coupling.
The degenerate parametric oscillator above threshold is studied with phenomenological stochastic colored (nonwhite) pump noise for arbitrary pump to subharmonic relaxation rate. The current experimental limits of large intracavity threshold photon numbers (very small quantum noise) are considered, allowing for a semiclassical treatment of the system dynamics. A comparison between the effects of isotropic and squeezed pump noise on the internal transient and steady-state fluctuations is presented by simulation of the nonlinear semiclassical stochastic Langevin equations in the Wigner quadrature representation. It is found that the transient squeezing for the system starting in the unstable steady state (the vacuum) is not degraded by stochastic pump noise. A damped oscillatory behavior of the noise levels (periodic exchange of fluctuations between the squeezed quadrature of the signal and the pump) is observed for large damping of the signal in the turn on of pump depletion. Finally, it is shown that the limited squeezing above threshold in the steady state (50%) due to pump depletion can be enhanced if squeezed stochastic noise with sufficient significant spectral components (broadband squeezing) is fed to the pump. The above-threshold steady-state squeezing has been calculated analytically from the linearized stochastic equations and the effects of the time scale associated to the relaxation of the pump noise (the noise correlation time) compared to the dissipative time scale of the system and the pump to subharmonic loss ratio are presented.