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CRC Press eBooks

COHERENT STATES

Murray, Iii Sargent

Published Mar 5, 2018• 0 citations

Abstract

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.

DOI: 10.1201/9780429493515-22

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Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.

Quantum electrodynamics is a very cumbersome theory, and one tries to avoid it in applied problems. Experience shows that the semiclassical theory of radiation, which operates with a nonquantized elec� tromagnetic field, provides, as a rule, required accu� racy of calculations. The authority of the semiclassical theory of radiation has grown greatly, because it describes adequately the operation of quantum devices such as lasers. In this work we present an example that demon� strates the inadequacy of the semiclassical theory of radiation. It is shown that, upon resonance scattering of an electromagnetic field by excited systems, quanti� zation of the electromagnetic field is necessary and the consequences of this operation manifest themselves at the macroscopic level. Suppose that resonance radiation is scattered by a certain system, the initial state of which is described in the interaction representation by the wave function . The total wave function of the electromagnetic field and scattering system after the scattering is denoted by . Expansion of the function in terms of the basis of the wave functions of the scatterer has the form

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