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Journal of Modern Optics

Revival Hamiltonians, Phase Operators and Non-Gaussian Squeezed States

C V Sukumar

Published Dec 1, 1989• 11 citations
Publisher

Abstract

The phase operators P ± of the electromagnetic field, whose classical analogues are exp (∓ iθ), θ being the classical phase, and the number operator N associated with a harmonic oscillator mode are used to construct a class of operators T m = N(P −)m and T + m = (P +)m N, m = 1, 2,…. It is shown that T m, T + m and N satisfy a simple algebra. T m and T + m are used to construct model Hamiltonians H m describing the coupling of a two-level atom to a field that leads to decay and exact revivals in the atomic dynamics. In the second part of the paper, a family of unitary operators U m is constructed in terms of T m and T + m. For m = 1 and m = 2 the states generated by U m from the vacuum are non-Gaussian squeezed states. The variance properties of the squeezed states so constructed are analysed and compared with those of the Gaussian squeezed states. It is shown that the non-Gaussian squeezed states constructed in this paper have small values for the variance product that rise only logarithmically with for large values of , the mean number of quanta.

DOI
10.1080/09500348914551711

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American Journal of Physics•Aug 1, 1989•69 citations
The phase operators P ± of the electromagnetic field, whose classical analogues are exp (∓ iθ), θ being the classical phase, and the number operator N associated with a harmonic oscillator mode are used to construct a class of operators T m = N(P −)m and T + m = (P +)m N, m = 1, 2,…. It is shown that T m, T + m and N satisfy a simple algebra. T m and T + m are used to construct model Hamiltonians H m describing the coupling of a two-level atom to a field that leads to decay and exact revivals in the atomic dynamics. In the second part of the paper, a family of unitary operators U m is constructed in terms of T m and T + m. For m = 1 and m = 2 the states generated by U m from the vacuum are non-Gaussian squeezed states. The variance properties of the squeezed states so constructed are analysed and compared with those of the Gaussian squeezed states. It is shown that the non-Gaussian squeezed states constructed in this paper have small values for the variance product that rise only logarithmically with for large values of , the mean number of quanta.
The phase properties of the quantized electromagnetic field have been studied using operators ${P}^{\ifmmode\pm\else\textpm\fi{}}$ of the harmonic oscillator whose classical analogs are exp(\ensuremath{\mp}i\ensuremath{\theta}), \ensuremath{\theta} being the classical phase. P+{P}^{+}P+ and ${P}^{\mathrm{\ensuremath{-}}}$ are used to construct a unitary operator Up{U}_{p}Up​(\ensuremath{\alpha})=exp(\ensuremath{\alpha}P+{P}^{+}P+-${\ensuremath{\alpha}}^{\mathrm{*}}$${P}^{\mathrm{\ensuremath{-}}}$) and the properties of the state generated by Up{U}_{p}Up​ from a number state are studied. It is shown that by tuning \ensuremath{\alpha} it is possible to produce a pure state ${\ensuremath{\psi}}_{0}$=Up{U}_{p}Up​(\ensuremath{\alpha})\ensuremath{\Vert}0〉 in which a specific number-state component is suppressed.
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 noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.