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Physical Review A

Phase operators and a special kind of transformation in the Hilbert space of a two-mode boson

Sixia Yu

Published Nov 1, 1997• 6 citations
Publisher

Abstract

A special kind of unitary transformation capable of changing the degrees of freedom of a quantum system, a super quantum coordinate transformation, is introduced. Two such transformations in the Hilbert space of a two-mode boson are investigated in detail. Under one transformation, this two-mode bosonic Hilbert space is transformed to a Hilbert space of a two-mode boson plus a single-mode fermion; under the other transformation, it is transformed to a Hilbert space of a single-mode boson. Their applications to phase operators and the phase difference operator are discussed.

DOI
10.1103/physreva.56.3464

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A special kind of unitary transformation capable of changing the degrees of freedom of a quantum system, a super quantum coordinate transformation, is introduced. Two such transformations in the Hilbert space of a two-mode boson are investigated in detail. Under one transformation, this two-mode bosonic Hilbert space is transformed to a Hilbert space of a two-mode boson plus a single-mode fermion; under the other transformation, it is transformed to a Hilbert space of a single-mode boson. Their applications to phase operators and the phase difference operator are discussed.
Classical linear transformations in coordinate-momentum phase space are mapped to unitary quantum-mechanical operators in Hilbert space to produce a generalization of squeeze operators. The classical to quantum transition is manifestly apparent in the derivation. The unitary operators are evaluated in the coherent-state representation using the ``integration within ordered products'' technique in the one- and two-mode cases. Application of these operators results in a new generalization of coherent states.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.
We introduce unitary quantum phase operators for material particles. We carry out a model study on quantum phases of interacting bosons in a symmetric double-well potential in terms of unitary and commonly-used non-unitary phase operators and compare the results for different number of bosons. We find that the results for unitary quantum phase operators are significantly different from those for non-unitary ones especially in the case of low number of bosons. We introduce unitary operators corresponding to the quantum phase-difference between two single-particle states of fermions. As an application of fermionic phase operators, we study a simple model of a pair of interacting two-component fermions in a symmetric double-well potential. We also investigate quantum phase and number fluctuations to ascertain number-phase uncertainty in terms of unitary phase operators.
A simple formalism is introduced for describing the quantum phase of a single-mode field (or, equivalently, a harmonic oscillator) using a particular infinite-dimensional Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This space is spanned by vectors which represent states of the single field mode containing non-negative numbers of photons. It has a symmetry in the sense that it contains vectors in the neighborhood of both the vacuum state and the infinite photon number state. A striking property of Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ is that it supports Hermitian and unitary phase operators. Indeed the Pegg-Barnett unitary phase operators are shown to converge strongly on the Hilbert space Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​ and so the correponding limit points are ordinary operators on Hsym{\mathit{H}}_{\mathit{s}\mathit{y}\mathit{m}}Hsym​. This can be compared with the situation for the infinite-dimensional Hilbert space H conventionally used in quantum optics on which the Pegg-Barnett unitary phase operators converge weakly, in general.
Any future quantum information machine will contain unitary operators and entangled particle states. The Hilbert space describing the action of the quantum information machine separates into a bosonic and a fermionic sector. Because the bosonic sector is of higher dimension, it is always possible to encode more information into a multiboson state than into a multifermion state, given the same complexity, that is unitary representation, of the quantum information machine. This is explicitly studied for the case of two particles defined in two modes. There the beam splitter is a generic representation of any U(2) matrix, and it has recently been shown that one can realize any N-dimensional unitary operator by successive application of such two-dimensional operators. The two-boson two-mode Hilbert space is of dimension three, and thus one can encode log23 = 1·57 bits of information into such an entangled state. Finally, some explicit schemes for creating and detecting the three possible, two-photon, two-mode states spanning the bosonic Bell basis are given.