This chapter is devoted to coherent states. Generally speaking, a wave function of a system of identical particles does not have to have a certain number of particles. It can be a superposition of states with different numbers of particles. We have already encountered this in Chapters 5 and 6 when we studied superconductivity and superfluidity. Another important example of states with no well-defined number of particles is given by coherent states . In a way, these states are those which most resemble classical ones: the uncertainty in conjugated variables (such as position and momentum) is minimal for both variables, and their time-evolution is as close to classical trajectories as one can get. Coherent states of radiation are relatively easy to achieve. They arise if we excite the electromagnetic field with classical currents. We look in detail at the coherent state of a single-mode oscillator and all its properties. The coherent state turns out to be an eigenfunction of the annihilation operator, the distribution of photon numbers is Poissonian, and it provides an optimal wave function describing a classical electromagnetic wave. We come back to our simple model of the laser and critically revise it with our knowledge of coherent states, and estimate the time at which the laser retains its optical coherence. We then derive Maxwell–Bloch equations that combine the master equation approach to the lasing (as outlined in Chapter 9) with the concept of coherence.
This chapter is devoted to coherent states. Generally speaking, a wave function of a system of identical particles does not have to have a certain number of particles. It can be a superposition of states with different numbers of particles. We have already encountered this in Chapters 5 and 6 when we studied superconductivity and superfluidity. Another important example of states with no well-defined number of particles is given by coherent states . In a way, these states are those which most resemble classical ones: the uncertainty in conjugated variables (such as position and momentum) is minimal for both variables, and their time-evolution is as close to classical trajectories as one can get. Coherent states of radiation are relatively easy to achieve. They arise if we excite the electromagnetic field with classical currents. We look in detail at the coherent state of a single-mode oscillator and all its properties. The coherent state turns out to be an eigenfunction of the annihilation operator, the distribution of photon numbers is Poissonian, and it provides an optimal wave function describing a classical electromagnetic wave. We come back to our simple model of the laser and critically revise it with our knowledge of coherent states, and estimate the time at which the laser retains its optical coherence. We then derive Maxwell–Bloch equations that combine the master equation approach to the lasing (as outlined in Chapter 9) with the concept of coherence.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
In this chapter we shall be using the formalism of Chapter 4 to describe some properties of the quantized free electromagnetic field, emphasizing the analogies and differences with classical electromagnetism. The analogies explain why classical optics was so successful, to the extent that, even though light was not quantized, it was able to describe almost all known optical phenomena up until the 1970s. The differences highlight typical quantum behaviour, with no classical equivalent, which lies at the heart of modern quantum optics. The chapter begins by presenting formal ways of describing the main components of quantum optics experiments, namely, photodetectors and semi-reflecting mirrors. One can then understand homodyne detection, used to measure field quadrature components, even at optical frequencies for which no detector is fast enough to follow the oscillations of the field itself. Section 5.2 deals with the quantum vacuum where, in contrast to the classical vacuum, radiation has properties, in particular, fluctuations, with which one can associate physical effects. We then discuss in Section 5.3 some quantum radiation states in the simple case where a single mode l of the field is excited (non-vacuum). We introduce the idea of a quasiclassical state, or coherent state, used to make the link with classical optics. We shall also study other states, those with a definite number of photons, or squeezed states, which exhibit typically quantum properties, quite inconceivable in classical electromagnetism, and we shall show how these properties can be understood by reference to the Heisenberg relations, with an extremely useful graphical representation.
In Chapter 4, we introduced the concept of the photon number representation of the electromagnetic (EM) field that is based on the number states, the eigenstates of the Hamiltonian of the EM field. However, this is not the only possible representation, in fact not always the most convenient. In this chapter, we introduce the concept of phase space representations of a quantum system which is in a state described by the density operator ρˆ. The density operator of a given system encodes classical as well as non-classical (quantum) properties of the system. How to distinguish these two kinds of properties is of basic importance in quantum optics. Therefore, we shall address this issue by using representations of the density operator that are based on the parametric space of complex eigenvalues of the annihilation operator aˆ in a coherent state |α〉. We first determine what we mean by the density operator of a quantum system and discuss its basic properties. Next, we introduce different representations of the density operator in terms of coherent state projectors |α〉〈α|. The representations show that a state of the EM field or of an arbitrary quantum system may be regarded as a mixture of coherent states. Along the way, we will learn how to find the P , Q and Wigner representations of the density operator, what are their properties, how to interpret the properties and how to calculate relations between the different representations. Most importantly, certain field states exhibit non- classical features and these non-classical features can be manifested in the phase space representations. Accordingly, we will learn how the representations are very convenient tools to describe quantum states of simple systems.
This chapter presents some basic aspects of coherent and squeezed states. Certain superpositions of energy Eigen functions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. The chapter obtains the Heisenberg uncertainty relation for the harmonic oscillator states and shows that it is a minimum for ground state. It defines coherent states and shows that they are the minimum uncertainty product states. Coherent states were constructed for many interesting potentials including infinite well and Poschl-Teller potentials and Morse potential. A particular quantum state of a spin system which most closely resembles a classical spin is termed as a spin coherent state. Use of squeezed states in quantum cryptography, quantum distribution of keys, extended coherence time, quantum engineering, teleportation, optics, detection of gravitational waves, quantum metrology and laser interferometers have been analyzed.
Photons and electrons One of the great technical achievements of the second half of the 20th century was the invention of the laser. With its help extremely intense light could be produced. Furthermore, it even became posssible to excite a single resonator mode so that both the frequency and the propagation direction of the emitted radiation field are extraordinarily sharp. The mere existence of such high-intensity single-mode fields – the real problem that was ingeniously solved was to generate them – is no matter of surprise. We know from classical electromagnetic theory that the amplitude of a monochromatic electromagnetic field can take on arbitrarily large values. This is a direct consequence of the linearity of Maxwell's equations. What does this mean for quantized radiation fields? Certainly, the mentioned property of classical fields must also be retained in the quantum mechanical description. Since the electromagnetic energy is quantized in ‘packets’ of magnitude h ν (photons), the number of photons in a given mode must be unlimited, in principle. All this is very well known. However, it becomes noteworthy when we compare the behaviour of photons with that of electrons. Amazingly, it turns out that an electronic quantum state can be occupied by one electron, at maximum. (Strictly speaking, in the description of the electronic state the spin variable must be included.) This drastic difference in the collective behaviour of photons and electrons could, in fact, not be expected from what we know from single photons and electrons. Both of them have particle-like as well as wave-like properties. However, when many of them are present, they exhibit quite different a demeanour.