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Physical Review A

Quantum tunneling in dispersive optical bistability

H. Risken, Craig M. Savage, Fritz Haake, Daniel F. Walls

Published Feb 1, 1987• 58 citations
Publisher

Abstract

Quantum tunneling times for the model of Drummond and Walls describing dispersive optical bistability are investigated for small cavity damping. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator in eigenstates of this Hamilton operator the stationary solution as well as the lowest eigenvalues are obtained from a Pauli master equation for the diagonal elements of the density matrix for small cavity damping. The tunneling time follows from the lowest nonzero eigenvalue of this master equation. Expectation values as well as the Q function for the stationary case are also presented.

DOI
10.1103/physreva.35.1729

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Journal of the Optical Society of America B•Nov 1, 1993•19 citations
Quantum tunneling times for the model of Drummond and Walls describing dispersive optical bistability are investigated for small cavity damping. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator in eigenstates of this Hamilton operator the stationary solution as well as the lowest eigenvalues are obtained from a Pauli master equation for the diagonal elements of the density matrix for small cavity damping. The tunneling time follows from the lowest nonzero eigenvalue of this master equation. Expectation values as well as the Q function for the stationary case are also presented.
For the model of Drummond and Walls describing dispersive optical bistability, quantum tunneling rates are calculated in the low-cavity-damping limit. Without damping the system can be described by an appropriate Hamilton operator. By expanding the density operator into eigenstates of this Hamilton operator the equation of motion of the density operator is cast into an equation of motion for the density matrix. In this equation we take into account only diagonal elements and those nondiagonal elements for which the corresponding eigenstates of the Hamilton operator are nearly degenerate. The tunneling rates are then determined by the eigenvalues of the truncated density-matrix equation.
For the model of Drummond and Walls describing dispersive optical bistability, stationary solutions as well as quantum-tunneling rates are calculated by solving a pseudo-Fokker-Planck equation (PFPE), i.e., a Fokker-Planck-like equation with a non-positive-definite diffusion matrix, in terms of matrix continued fractions. Due to the quantum nature of the model, an interesting oscillating variation of the decay rates of the PFPE as a function of the scaling factor for the photon number is found for small cavity damping.
Introduction In this chapter we will turn to the arena of quantum optics for illustrative examples of the use of the master equation discussed in Chapters 3 and 6. In fact, quantum optics examples have already been utilized in Chapter 2 as introduction to the density matrix. The principal focus here will be quantum damping in these systems, that is, the damping effect on an atom in interaction with the electromagnetic field as a reservoir. Damping is discussed extensively in Chapter 17 in connection with decay-scattering systems. For this system general phase space distribution functions will be reexamined. To some degree, this has already been done in Chapter 2 with the introduction of the Glauber–Sudarshan P (α a * ) function. The micromaser will be discussed as a modern and interesting example of the dynamic interaction of an atom with an electromagnetic cavity not in equilibrium. For use of the student, an appendix to this chapter will briefly review the quantization of the free electromagnetic field and its atomic interaction. There is no possibility of reviewing this extensive and growing field here. Our desire in this chapter is to connect the general topics of this book to this example. The books of Louisell (1973) and Scully and Zubairy (1997) are excellent. We are also indebted to the work of Nussenzweig, Schleich, and Mandel and Wolf (Nussenzweig, 1973; Mandel and Wolf, 1995; Schleich, 2001). We also recall the fine early introduction to this topic by Agarwal (1973). Atomic damping: atomic master equation In this section we shall consider the so-called quantum optics master equation for the reduced atomic density operator, ρ A ( t ), in the Born approximation. The elements of this derivation have already been discussed in Chapter 3 with the derivation of the Pauli equation for 〈α| ρ A |α〉. Here the generalization to off-diagonal contributions only adds complication.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.
We solve analytically the master equation for a quantized cavity mode, finding the time-dependent reduced density operator. The formalism applies to both amplifiers and attenuator, permitting the results to be converted between them by introducing a temperature that can be negative. We obtain a quasi-probability distribution function similar to that obtained by Glauber [ GlauberR. J., ed., Quantum Optics ( Academic, New York, 1969)] from our reduced density matrix and equate his parameters with the coefficients appearing in the master equation. We illustrate the formalism by calculating variances, expectation values, and photon statistics for the damped simple harmonic oscillator and the linear amplifier.