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Physica Scripta

A Theory of the Relative Phase and Number Difference of Two Quantized Harmonic Oscillators

G. Björk, Jonas S derholm, Alexei S. Trifonov, Tedros Tsegaye

Published Jan 1, 2002• 4 citations
Publisher

Abstract

We present a comprehensive and self-consistent theory of relative-phase measurements and the associated Hermitian relative-phase operator of two harmonic oscillators. We find that since Nature does not favor any particular initial condition of the two oscillators, the relative-phase operator is not unique. We show that the relative-phase eigenstates are maximally entangled. Therefore, most relative-phase operators lack a classical correspondence, even in the high-excitation limit. Furthermore, we find that the relative phase and the excitation number difference are noncommuting, noncanonical observables and we derive a commutation relation.

DOI
10.1238/physica.topical.102a00133

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We present a comprehensive and self-consistent theory of relative-phase measurements and the associated Hermitian relative-phase operator of two harmonic oscillators. We find that since Nature does not favor any particular initial condition of the two oscillators, the relative-phase operator is not unique. We show that the relative-phase eigenstates are maximally entangled. Therefore, most relative-phase operators lack a classical correspondence, even in the high-excitation limit. Furthermore, we find that the relative phase and the excitation number difference are noncommuting, noncanonical observables and we derive a commutation relation.
We present a comprehensive and self-consistent theory of the relative phase between fundamental and harmonic modes in the nonlinear process of second-harmonic generation. We introduce an adequate polar decomposition of the field amplitudes that leads to a truly Hermitian relative-phase operator, whose eigenstates correctly describe the phase properties of the fields. We find the probability distribution for the relative phase and, by using a numerical diagonalization of the interaction Hamiltonian, we study its time evolution. This evolution shows quite different asymptotic behaviours for long times depending on the total number of quanta.
From the experimentalist’ spoint of view, it is natural to think of aphase measurement as a measurement that involves a reference time. As a part of the system, the reference time generating “clock” must be taken into account in the quantum treatment of the measuring process. This could be argued to be the reason why there does not exist any observable corresponding to the absolute phase in the unrestricted Hilbert space.Luis and Sánchez-Soto [Phys. Rev. A 48, 4702 (1993)] have investigated the case where the “clock” is another harmonic oscillator and defined a Hermitian phase-difference operator, which is valid in the infinite Hilbert space.In the present work we propose and experimentally demonstrate a setup for direct measurement of the phase-difference. We derive the phase-difference distributions for some two-mode states and compare them with the experimental results.
An apparent paradox is resolved that concerns the existence of time operators which have been derived for the quantum harmonic oscillator. There is an apparent paradox because, although a time operator is canonically conjugate to the Hamiltonian, it has been asserted that no operator exists that is canonically conjugate to the Hamiltonian. In order to resolve the apparent paradox, we work in a representation where the phase operator is diagonal. The boundary condition on wave functions is such that they be periodic in the phase variable, which is related to the (continuous) eigenvalue of the time operator. Matrix elements of the commutator of the time operator with the Hamiltonian involve the phase variable itself in addition to periodic functions of the phase variable. The Hamiltonian is not hermitian when operating in space that includes the phase variable itself. The apparent paradox is resolved when this non-hermeticity is taken into account correctly in the evaluation of matrix elements of the commutation relation.
The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.