In this chapter we will consider some simple systems arising out of the Hamiltonians in the previous chapters. Here, we will look at these systems without considering damping or reservoirs. These systems involve only a small number of field modes – typically one, two or three. We will revisit them in later chapters to consider what happens when the fields are confined to a cavity with damping, including possible input and output coupling. Before discussing the models, however, we need to describe a number of features of fields consisting of a small number of modes. We will begin with a discussion of the quantum theory of photo-detection and optical coherence, due to Mandel and Glauber. This makes use of the microscopic model for atom–field interactions in the previous chapter, to allow us to introduce a theory of coherence and photon counting. We will also treat the representation for a single-mode field known as a P -representation. This will be the first quasi-distribution function we will encounter. It is a c-number representation of the quantum state of a field mode. As we shall see in subsequent chapters, quasi-distribution functions are very useful in describing the dynamics of interacting quantum systems coupled to reservoirs. We will then go on to use the P -representation to define the notion of a nonclassical state of the field; such states are signified by the fact that the P -representation is no longer a well-behaved, positive distribution. Nonclassical states are natural results of nonlinear optical interactions, and we will show how that comes about for some of our models. It is also possible to define P -representations for multi-mode fields. This leads to a discussion of nonclassical correlations between modes.
In this chapter we will consider some simple systems arising out of the Hamiltonians in the previous chapters. Here, we will look at these systems without considering damping or reservoirs. These systems involve only a small number of field modes – typically one, two or three. We will revisit them in later chapters to consider what happens when the fields are confined to a cavity with damping, including possible input and output coupling. Before discussing the models, however, we need to describe a number of features of fields consisting of a small number of modes. We will begin with a discussion of the quantum theory of photo-detection and optical coherence, due to Mandel and Glauber. This makes use of the microscopic model for atom–field interactions in the previous chapter, to allow us to introduce a theory of coherence and photon counting. We will also treat the representation for a single-mode field known as a P -representation. This will be the first quasi-distribution function we will encounter. It is a c-number representation of the quantum state of a field mode. As we shall see in subsequent chapters, quasi-distribution functions are very useful in describing the dynamics of interacting quantum systems coupled to reservoirs. We will then go on to use the P -representation to define the notion of a nonclassical state of the field; such states are signified by the fact that the P -representation is no longer a well-behaved, positive distribution. Nonclassical states are natural results of nonlinear optical interactions, and we will show how that comes about for some of our models. It is also possible to define P -representations for multi-mode fields. This leads to a discussion of nonclassical correlations between modes.
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.
As we have seen in the previous chapters, there are quantum fluctuations associated with the states corresponding to classically welldefined electromagnetic fields. The general description of fluctuation phenomena requires the density operator. However, it is possible to give an alternative but equivalent description in terms of distribution functions. In the present chapter, we extend our treatment of quantum statistical phenomena by developing the theory of quasi-classical distributions. This is of interest for several reasons. First of all, the extension of the quantum theory of radiation to involve nonquantum stochastic effects such as thermal fluctuations is needed. This is an important ingredient in the theory of partial coherence. Furthermore, the interface between classical and quantum physics is elucidated by the use of such distributions. The arch type example being the Wigner distribution. In this chapter, we introduce various distribution functions. These include the coherent state representation or the Glauber–Sudarshan Prepresentation. The P -representation is used to evaluate the normally ordered correlation functions of the field operators. As we shall see in the next chapter, the P -representation forms a correspondence between the quantum and the classical coherence theory. This distribution function does not have all of the properties of the classical distribution functions for certain states of the field, e.g., it can be negative. We also discuss the so-called Q -representation associated with the antinormally ordered correlation functions. Other distribution functions and their properties are also presented.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem. In this chapter, we present a theory of damping based on the density operator in which the reservoir variables are eliminated by using the reduced density operator for the system in the Schrödinger (or interaction) picture. We also present a ‘quantum jump’ approach to damping. In the next chapter, the damping of the system will be considered using the noise operator method in the Heisenberg picture. An insight into the damping mechanism is obtained by considering the decay of an atom in an excited state inside a cavity. The atom may be considered as a single system coupled to the radiation field inside the cavity. Even in the absence of photons in the cavity, there are quantum fluctuations associated with the vacuum state. As discussed in Chapter 1, the field may be visualized as a large number of harmonic oscillators, one for each mode of the cavity.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
“The word ‘classical’ means only one thing in science: it's wrong!” We have previously emphasized the fact that all states of light are quantum mechanical and are thus nonclassical, deriving some quantum features from the discreteness of the photons. Of course, in practice, the nonclassical features of light are difficult to observe. (We shall use “quantum mechanical” and “nonclassical” more or less interchangeably here.) Already we have discussed what must certainly be the most nonclassical of all nonclassical states of light – the single-photon state. Yet, as we shall see, it is possible to have nonclassical states involving a very large number of photons. But we need a criterion for nonclassicality. Recall that in Chapter 5 we discussed such a criterion in terms of the quasi-probability distribution known as the P function, P (α). States for which P (α) is positive everywhere or no more singular than a delta function, are classical whereas those for which P (α) is negative or more singular than a delta function are nonclassical. We have shown, in fact, that P (α) for a coherent state is a delta function, and Hillery has shown that all other pure states of the field will have functions P (α) that are negative in some regions of phase space and are more singular than a delta. It is evident that the variety of possible nonclassical states of the field is quite large.