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Physical Review A

New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation

Bonny L. Schumaker, Carlton M. Caves

Published May 1, 1985• 575 citations
Publisher

Abstract

This paper provides the mathematical foundation for the two-mode formalism introduced in the preceding paper. A vector notation is introduced; it allows two-mode properties to be written as compactly as the comparable properties for a single mode. The fundamental unitary operators of the formalism are described and their properties are examined; particular attention is paid to the two-mode squeeze operator. Special quantum states associated with the formalism are considered, with emphasis on the two-mode squeezed states.

DOI
10.1103/physreva.31.3093

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arXiv:quant-ph/0308081•Physical Review A•Mar 19, 2004•8 citations
This paper provides the mathematical foundation for the two-mode formalism introduced in the preceding paper. A vector notation is introduced; it allows two-mode properties to be written as compactly as the comparable properties for a single mode. The fundamental unitary operators of the formalism are described and their properties are examined; particular attention is paid to the two-mode squeeze operator. Special quantum states associated with the formalism are considered, with emphasis on the two-mode squeezed states.
A new formalism for analyzing two-photon devices, such as parametric amplifiers and phase-conjugate mirrors, is proposed in part I, focusing on the properties and the significance of the quadrature-phase amplitudes and two-mode squeezed states. Time-stationary quasi-probability noise is also detailed for the case of Gaussian noise, and uncertainty principles for the quadrature-phase amplitudes are outlined, as well as some important properties of the two-mode states. Part II establishes a mathematical foundation for the formalism, with introduction of a vector notation for compact representation of two-mode properties. Fundamental unitary operators and special quantum states are also examined with an emphasis on the two-mode squeezed states. The results are applied to a previously studied degenerate limit (epsilon = 0).
This paper introduces a new formalism for analyzing two-photon devices (e.g., parametric amplifiers and phase-conjugate mirrors), in which photons in the output modes are created or destroyed two at a time. The key property of a two-photon device is that it excites pairs of output modes independently. Thus our new formalism deals with two modes at a time; a continuum multimode description can be built by integrating over independently excited pairs of modes. For a pair of modes at frequencies \ensuremath{\Omega}\ifmmode\pm\else\textpm\fi{}\ensuremath{\epsilon}, we define (i) quadrature-phase amplitudes, which are complex-amplitude operators for modulation at frequency \ensuremath{\epsilon} of waves ``cos[\ensuremath{\Omega}(t-x/c)]'' and ``sin[\ensuremath{\Omega}(t-x/c)]'' and (ii) two-mode squeezed states, which are the output states of an ideal two-photon device. The quadrature-phase amplitudes and the two-mode squeezed states serve as the building blocks for our formalism; their properties and their physical interpretation are extensively investigated.
Quantum squeezing, a major resource for quantum information processing and quantum metrology, is best analyzed in terms of the field quadratures - the quantum optical analogues of position and momentum, which form the continuous-variable formalism of quantum light. Degenerate squeezing admits a very helpful and simple description in terms of the single-mode quadrature operators, but the non-degenerate case, i.e. when the squeezing involves pairs of modes, requires a more complicated treatment involving correlations between the quadratures of the different modes. We introduce a generalized set of complex quadrature operators that treats degenerate and non-degenerate squeezing on equal footing. We describe the mode-pairs (and photon-pairs) as a single entity, generalizing the concept of single-mode quadrature operators to two-mode fields of any bandwidth. These complex operators completely describe the SU(1,1) algebra of two-photon devices and directly relate to observable physical quantities, like power and visibility. Based on this formalism, we discuss the measurement of optically-broad squeezed signals with direct detection, and present a compact set of phase-dependent observables that completely and intuitively determine the two-mode squeezed state, and quantify the degree of inseparability and entanglement between the modes.
The phase operator for a two-mode photon system is defined in terms of the relative-number states, and its property is investigated in detail. The phase operator thus defined has a unitary exponential form. It is shown that the average value and fluctuation of the phase operator, where one mode is in some physical state, such as a coherent or a squeezed state, and the other is in a vacuum state, are equivalent to those obtained by means of the Pegg–Barnett phase operator for a single-mode photon. The average value and fluctuation of the phase operator are calculated for coherent and squeezed states of a two-mode photon system.
Abstract These lecture notes provide a comprehensive exploration of coherent and squeezed states, which are central to quantum optics and continuous-variable quantum information. We begin with the traditional formalism, where displacement and squeezing operators are defined via the creation and annihilation operators. Coherent states are presented as quantum analogues of classical fields with minimal uncertainty, while squeezed states are introduced as nonclassical resources that reduce quantum noise below the vacuum limit, enabling next-generation quantum technologies. A central feature of these notes is a parallel development from a group-theoretical standpoint. Here, these operators are defined abstractly by their algebraic properties, revealing their structure as bosonic realizations of the Weyl–Heisenberg and SU(1,1)SU(1,1)SU(1,1) S U ( 1 , 1 ) structures, respectively. This advanced perspective not only provides a more rigorous mathematical foundation but also leads to new insights, including a proof of the uniqueness of the single-mode squeezing operator and a no-go theorem for certain multi-mode squeezing configurations. Designed for both pedagogical clarity and advanced study, these notes bridge the foundational operator algebra of quantum optics with the modern group-theoretical structures essential for research in continuous-variable quantum systems.
We introduce a formalism of nonlinear canonical transformations for general systems of multiphoton quantum optics. For single-mode systems the transformations depend on a tunable free parameter, the homodyne local-oscillator angle; for nnn-mode systems they depend on nnn heterodyne mixing angles. The canonical formalism realizes nontrivial mixing of pairs of conjugate quadratures of the electromagnetic field in terms of homodyne variables for single-mode systems, and in terms of heterodyne variables for multimode systems. In the first instance the transformations yield nonquadratic model Hamiltonians of degenerate multiphoton processes and define a class of non-Gaussian, nonclassical multiphoton states that exhibit properties of coherence and squeezing. We show that such homodyne multiphoton squeezed states are generated by unitary operators with a nonlinear time evolution that realizes the homodyne mixing of a pair of conjugate quadratures. Tuning of the local-oscillator angle allows us to vary at will the statistical properties of such states. We discuss the relevance of the formalism for the study of degenerate (up-)down-conversion processes. In a companion paper [F. Dell'Anno, S. De Siena, and F. Illuminati, 69, 033813 (2004)], we provide the extension of the nonlinear canonical formalism to multimode systems, we introduce the associated heterodyne multiphoton squeezed states, and we discuss their possible experimental realization.