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Physical Review A

Quantum-optical master equations: An interaction picture

Berthold‐Georg Englert, M. Naraschewski, Axel Schenzle

Published Sep 1, 1994• 43 citations
Publisher

Abstract

Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.

DOI
10.1103/physreva.50.2667

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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.

Cambridge University Press eBooks•Apr 16, 2009•0 citations
Quantum-optical master equations---exemplified by the Jaynes-Cummings model with damping---are turned into numerical partial differential equations of first order for phase-space functions, which are generalizations of the Wigner function and its relatives. The time dependence of these phase-space functions originates solely in the atom-photon interaction; all other time dependences, in particular the dissipative contribution of the photon damping, are accounted for by the time-dependent operator bases to which the phase-space functions refer. The judicious choice of operator basis also effects the absence of second-order derivatives in the partial differential equation. Our first-order equations are hyperbolic and can be integrated conveniently along their characteristics. As an illustrative application we study how the Jaynes-Cummings revivals are affected by photon damping. We show how to handle squeezed reservoirs and how to apply the method to laser cooling.
We have in this thesis looked at two very different problems related to quantum optics. In Part I we have studied the Jaynes-Cummings model with cavity damping and a thermal background. To give an introduction to the field we started Chapter 2 by repeating how one in general derives the master equations for a system coupled to a reservoir. We then repeat the Jaynes-Cummings model and show extensions of the Jaynes-Cummings model where cavity-damping and a thermal background are taken into account. We proceed, in Chapter 3, by finding an analytical solution to the Jaynes-Cummings model with cavity damping in a thermal background when there is no detuning in the system, i.e. we have resonance between the energy of the photons in the cavity and the energy difference in the excited and ground state of the atom. We derive analytical expressions for P+(t), the probability of finding the atom in an excited state after some transition time t given that the atom was prepared in an excited state, and P++(t), the probability of finding the atom in an excited state after some transition time t when a previous atom was measured to be in the excited state. Both atoms where initially prepared in the excited state. We also, in Section 3.7 make use of the Poisson summation formula to rewrite the analytical solution P+(t) in terms of a Fourier series, from which we can predict the location of the revivals in the system. In Section 3.8 we derive stationary results, i.e. when the transition time t→∞, for the probabilities P+(t) and P++(t) and the mean number of photons ‹ n(t) › in the cavity. From our analytical results one is, for instance, led to believe that the probability P+(t) approaches 0 as the transitiontime increases, when it in reality approaches the value nb/(1 + 2nb). The results of our study of the extended Jaynes-Cummings model is presented in the article “Macroscopic Interference Effects in Resonant Cavities” and is published in Physica Scripta [5]. The article is reprinted as Article 1 at the end of this thesis. In Chapter 4 we present the result of a numerical solution of the Jaynes-Cummings model with cavity damping in a thermal background. Here we consider both the resonant case and also the case where detuning is present in the system. When we compare our numerical simulation with our analytical solutions we find that the correlation between the two are close to 1, at least when experimental values of Walther et al. [22] and Brune et al. [15, 16] are used. We also present some figures that show how the revival times and probabilities previously discussed is affected by detuning. In Section 4.2 we find an analytical formula for the revival time where detuning is embedded. In Chapter 4 we also, shortly, discuss how the detuning changes the effective number of photons in the cavity and what criteria should be present in order to separate the revival times.In Part II we study some aspects of quantum mechanical phase. We start, in Section 5.1, by repeating the definition and some results of the Susskind-Glogower phase operator. We then turn our attention to the Pegg-Barnett phase operator [26, 27, 28] in Section 5.4. We describe how to evaluate expectation-values and variances of the Pegg-Barnett phase. In Section 5.5 we establish an upper and lower bound on the phase based on the Pegg-Barnett phase description.In Section 5.9 we repeat some details of a new formalism of the phase which we have conveniently called the NFM-formalism after its inventors Noh, Foug`eres and Mandel [29, 30, 31]. We then describe how to find expectation-values using the Pegg-Barnett phase operator definition applied to the NFM-experiment described in Section 5.9. Our work on the quantum mechanical phase operator is presented in the paper “On the Quantum Phase Operator for Coherent States” and is published in Physica Scripta [6]. This paper is reprinted as Article II at the end of this thesis.
In this paper, we study a Jaynes-Cummings model (JCM) with phase damping, by introducing a master equation describing phase damping under the Markovian approximation. With the help of the superoperator technique, we find an analytic solution of the master equation for a nonresonant multiphoton JCM with phase damping. We investigate the influence of phase damping on nonclassical effects in the JCM, such as revivals of the atomic inversion, oscillations of the photon-number distribution, the quadrature squeezing of the cavity field, and sub-Poissonian photon statistics. It is shown that phase damping suppresses the revivals of the atomic inversion and nonclassical effects of the cavity field in the JCM.
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.