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Journal of Modern Optics

Phase Fluctuations and Squeezing

John A. Vaccaro, Stephen Mark Barnett, David Thomas Pegg

Published Mar 1, 1992• 9 citations
Publisher

Abstract

We investigate the relationship between squeezing and reduced phase fluctuations for various states of the single-mode electromagnetic field, including the strongly-squeezed vacuum and phase states. We find that, although squeezing the fluctuations of the electric field that arise from the vacuum guarantees a more well-defined phase, reducing phase fluctuations does not guarantee a squeezed electric field. We also investigate the evolution of the electric field and its fluctuations for a phase state. Our results show that even though the electric field fluctuations never vanish for a phase state, the times when the electric field changes sign are precisely defined. We also discuss why it is not always possible to attribute physical properties to certain states, such as simple superpositions of phase states.

DOI
10.1080/09500349214550571

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Physical Review A•Sep 22, 2009•4 citations
We investigate the relationship between squeezing and reduced phase fluctuations for various states of the single-mode electromagnetic field, including the strongly-squeezed vacuum and phase states. We find that, although squeezing the fluctuations of the electric field that arise from the vacuum guarantees a more well-defined phase, reducing phase fluctuations does not guarantee a squeezed electric field. We also investigate the evolution of the electric field and its fluctuations for a phase state. Our results show that even though the electric field fluctuations never vanish for a phase state, the times when the electric field changes sign are precisely defined. We also discuss why it is not always possible to attribute physical properties to certain states, such as simple superpositions of phase states.
In Section 11.5 we showed that, although the complex amplitude of the electromagnetic field has a well-defined value in any coherent state, yet the real and imaginary (Hermitian and anti-Hermitian) parts of the field fluctuate with equal dispersions. The phenomenon of vacuum fluctuations is a manifestation of this effect, because the vacuum state is an example of a particular coherent state. This behavior is quite different from that of an ordinary, classical field. In a squeezed state, which is even more non-classical, as we shall see, one part of the field fluctuates less and another part fluctuates more than in the vacuum state. In general, a squeezed state is one in which the distribution of canonical variables over the phase space has been distorted or ‘squeezed’ in such a way that the dispersion of one variable is reduced at the cost of an increase in the dispersion of the other variable. In the following we shall examine the properties of squeezed states when the two canonical variables are two quadratures of the electromagnetic field. Although the squeezing terminology is sometimes applied to variables other than the two field quadratures, it is less meaningful in those cases. A number of review articles on squeezing have been published and can be consulted for more details (Walls, 1983; Schumaker, 1986; Loudon and Knight, 1987; Teich and Saleh, 1989, 1990; Kimble, 1992).
The nature of phase noise in quantum optics is analyzed. In an experiment involving the measurement of the electromagnetic field the two quantities of interest are the energy and phase of the field. However, measurements of the quantities produce quantum fluctuations. The quantum fluctuations are regarded as noise in the treatment presented here. The quantum system is represented by a probability distribution, the Wigner function, and the quantum fluctuations are treated as stochastic noise associated with the quantity being measured. The difficulties of associating a quantum operator with the phase of the system are reviewed and the related energy-phase uncertainty relation is discussed. The alternate interpretation of the phase noise of a quantum system as being the classical phase noise of the Wigner function is presented. In particular the energy and phase noise of the vacuum state, the coherent state, the squeezed state and the squeezed vacuum are discussed in this way. The squeezed states of light are minimum uncertainty states with respect to the quadrature operators and exhibit noise of one quadrature below the noise level associated with the vacuum. The reduced noise level in one quadrature of the field underlies the importance of squeezed states in many practical applications where there is a need to reduce the quantum noise of one quadrature of coherent light. The periodic phase operator eliminates the difficulties associated with the multivalued nature of phase. The analysis of the vacuum and intense coherent state of Carruthers and Nieto by employing periodic phase operators is reviewed, particularly with respect to the energy-phase uncertainty relations and we generalize the approach to develop a phase operator analysis of the squeezed state in the intense field and vacuum limits. We demonstrate here for the first time that the phase operator is simply related to the phase of the squeezed state in the intense field limit and that the squeezed state is approximately an energy-phase minimum uncertainty state in the low-squeezing limit. Also we enlarge on previous work to demonstrate that the phase operator corresponds simply and unambiguously to the phase of the squeeze parameter for the strongly squeezed vacuum and the intensely squeezed vacuum is an energy-phase minimum uncertainty state for some values of phase. The occurrence of squeezing for the case of two coupled quantum oscillators is presented. The system consisting of one mode of the electromagnetic field coupled to a spinless nonrelativistic electron subjected to an harmonic potential is represented by two coupled harmonic oscillators. The dynamics are compared for the case that the rotating wave approximation is employed and for the case that the counter-rotating terms are included. These calculations have not been performed before. The parametric amplifier Hamiltonian with a nonresonant coupling is also studied in order to provide insight into the effects of the counter-rotating terms. Squeezing of the field produced by the electron is a consequence of the inclusion of the counter-rotating terms. The case of a spinless nonrelativistic electron subject to an harmonic potential and coupled to a continuum of electromagnetic field modes is also considered. The case of two coupled oscillators discussed above is generalized by replacing the oscillator which represents the single-mode field by a bath of oscillators. The effects of including counter-rotating terms and of ignoring the counter - rotating terms in the Hamiltonian are compared. The interaction is assumed to produce a frequency shift and an exponential damping term for the oscillating electron. The frequency shift is assumed to be small in either case and so the Wigner-Weisskopff approximation is employed to solve the equations of motion. We demonstrate the new results that dissipation-induced phase-dependent noise is a consequence of including the counter-rotating terms and that the noise is phase-independent for the case that the counterrotating terms are excluded. The relation between these results and recent work on quantum tunnelling in superconducting quantum interference devices is discussed. We conclude by suggesting further research related to the work in this thesis.
The amplitude of the electric field of a mode of the electromagnetic field is not a fixed quantity: there are always quantum mechanical fluctuations. The amplitude, having both a magnitude and a phase, is a complex number and is described by the mode annihilation operator a. It is also possible to characterize the amplitude by its real and imaginary parts which correspond to the Hermitian and anti-Hermitian parts of a, X sub 1 = 1/2(a(sup +) + a) and X sub 2 = i/2(a(sup +) - a), respectively. These operators do not commute and, as a result, obey the uncertainty relation (h = 1) delta X sub 1(delta X sub 2) greater than or = 1/4. From this relation we see that the amplitude fluctuates within an 'error box' in the complex plane whose area is at least 1/4. Coherent states, among them the vacuum state, are minimum uncertainty states with delta X sub 1 = delta X sub 2 = 1/2. A squeezed state, squeezed in the X sub 1 direction, has the property that delta X sub 1 is less than 1/2. A squeezed state need not be a minimum uncertainty state, but those that are can be obtained by applying the squeeze operator.
We study the eigenstates of the square of the phase operator for a single-mode field, and show that they are given as superpositions of macroscopically distinguishable phase states, which we call Schrödinger phase cats, the phase states being eigenstates of the phase operator. Those solutions that are also eigenstates of the parity operator with even parity are shown to be very similar to the squeezed vacuum states over a range of relevant parameters. Thus, it is possible to consider some squeezed vacuum states as approximate Schrödinger phase-cat states. We discuss the connections to the phase states of the SU (1, 1) phase operators for various realizations and representations.
We show that phase-coherent states of a single-mode quantized electromagnetic field are well approximated by certain superpositions of the squeezed vacuum and one-photon states. We further show that the squeezed vacuum and one-photon states can be understood, approximately, as superpositions of phase-coherent states. Our approximate phase-coherent states may be generated by a degenerate parametric down-converter acting on a prepared input state of the form $|0⟩+\ensuremath{\alpha}|1⟩$. The input superposition state itself can be prepared from a coherent state $|\ensuremath{\alpha}⟩$ by a quantum scissors device.