In Chapter 4 we defined different representations of the elec- tromagnetic (EM) field: the Fock, thermal and coherent state representations, and discussed in details properties of the EM field in these representations. We have seen that in the coherent state the fluctuations in the two quadratures of the EM field amplitudes are equal and minimize the uncertainty product given by the Heisenberg’s uncertainty relation. In other words, the quantum fluctuations of the field in a coherent state are equal to the zero- point fluctuations and are randomly distributed in phase. These zero-point fluctuations represent the standard quantum limit to the reduction of noise in a signal. Even an ideal laser operating in a pure coherent state would still possess quantum noise due to zero-point fluctuations.
In Chapter 4 we defined different representations of the elec- tromagnetic (EM) field: the Fock, thermal and coherent state representations, and discussed in details properties of the EM field in these representations. We have seen that in the coherent state the fluctuations in the two quadratures of the EM field amplitudes are equal and minimize the uncertainty product given by the Heisenberg’s uncertainty relation. In other words, the quantum fluctuations of the field in a coherent state are equal to the zero- point fluctuations and are randomly distributed in phase. These zero-point fluctuations represent the standard quantum limit to the reduction of noise in a signal. Even an ideal laser operating in a pure coherent state would still possess quantum noise due to zero-point fluctuations.
We showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise . Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states , provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.
In this chapter, we will introduce different representations of the electromagnetic (EM) field. One may ask, why do we need different representations for the EMfield? The answer is that usually we do not know the state of the EM field which in quantum optics is represented by the annihilation and creation operators. However, results of measurements of the field are given in terms of real variables, for example, intensity. The representations allow us to recognize the state of the field from values of the measured quantities such as average amplitude, intensity, and correlation functions. The nature of the state is present in terms of the interpretation of the apparently classical (measured) variables. We will discuss two basic types of representations often used in quantum optics, Fock state (photon number) representation and coherent states representation, the later one introduced by Glauber.a We also discuss properties of fields with thermal and Poisson distribution of photons. The photon number states are very often used as a basis for quantum optics problems, and despite of many difficulties have recently been generated experimentally [11-13].
Experimental chapter devoted to quantum observables endowed with continuously varying quantum fluctuations, such as position and momentum, quadrature operators, or phase and amplitude of electromagnetic fields . It shows that one can manipulate this quantum noise by generating squeezed states of light, always within the limits imposed by the Heisenberg inequality, and create strong correlations between these observables to conditionally generate quantum states having intensity quantum fluctuations below the "shot noise" limit imposed by the existence of vacuum fluctuations. Describes an experiment dealing with macroscopic mechanical oscillators displaying motional squeezing below the zero point fluctuations, and another one dealing with macroscopic superconducting exhibiting a whole spectrum of strongly nonclassical states, generated by using the strong anharmonicity of the Josephson potential.
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.
Having developed a formalism for the quantum theory of radiation, we will find it both interesting and profitable to determine what wave function most nearly describes the classical electromagnetic field given by Eo cos (vt •f $) and H0 sin (vt 4-j)- This field is supposedly known with absolute cer tainty, whereas the quantum operators for the electric and magnetic fields are associated with nonzero uncertainties obeying the relation (symbols defined in Sec. 14-1), just as the position q and momentum p obey the relation Aq Ap ^ We propose that the wave function which corre sponds most closely to the classical field must have minimum uncertainty [equality in (1)] for all time when subject to the appropriate simple harmonic potential. We will find in Appendix H that minimum uncertainty merely at one time is not sufficient, for if the width of the wave packet is initially chosen too small, it becomes too large at a later time, as illustrated in Fig. 15-16. Furthermore, it fails to satisfy the conditions for minimum uncertainty except at isolated points in time. If the right width is chosen for the potential, how ever, the packet bounces back and forth sinusoidally without changing width, as shown in Fig. 15-26. This latter packet “coheres,” always has minimum uncertainty, and resembles (Fig. 15-2a) the classical field as closely as quan tum mechanics permits.
Abstract In the preceding two chapters we have explored the consequences of quantizing the energy of a light beam in terms of the number of photons. This enabled us to classify light according to either the photon statistics or the second-order correlation function. In this chapter we shall consider the effects of quantizing the electric and magnetic fields that comprise the light. We shall make use of our knowledge of the quantum harmonic oscillator, and adopt a predominantly intuitive approach, leaving the rigorous mathematics to the next chapter.