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CRC Press eBooks

- Coherent and Squeezed States

S. Rajasekar, R. Velusamy

Published Dec 10, 2014• 0 citations
Publisher

Abstract

Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.

DOI
10.1201/b17827-8

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Coherent and Squeezed States

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Cambridge University Press eBooks•Sep 29, 1995•6 citations
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.
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.
Using only elementary quantum-mechanical concepts, the statistical properties of various harmonic oscillator states that are linear superpositions of its energy eigenfunctions are described. These superpositions include coherent states and squeezed (or two-photon coherent) states. The resulting Gaussian, minimum-uncertainty wave packets are shown to oscillate back and forth for both coherent and squeezed states, but with an oscillating ‘‘width’’ for the squeezed states. Also examined are the principles underlying the production of squeezed electromagnetic waves via parametric amplification or four-wave mixing, their measurement by homodyne detection, and the connection between squeezing and non-Poissonian counting statistics.
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.
A single closed-form expression is obtained for the probability amplitude ψ(x,t) for a harmonic oscillator whose initial state ψ(x,0) has a minimum value for the uncertainty product of position x and momentum p, Δx Δp=ℏ/2. This form describes both coherent and squeezed states. Oscillations in time for uncertainties in position and momentum, characteristic of squeezed states, are shown to follow directly from the derived form of ψ(x,t) and from Heisenberg equations of motion for operators x̂ and p̂. Expressions for average photon number and its uncertainty are obtained from calculations of average oscillator energy and fluctuations about this energy.
Introduction In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field. We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states. Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.