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Annual Meeting Optical Society of America

What is a broadband squeezed state?

Bonny L. Schumaker

Published Jan 1, 1985• 5 citations
Publisher

Abstract

A broadband or two-mode (nondegenerate) squeezed state is the natural two-mode analog of a single-mode (degenerate) squeezed state. The squeezing is a result of correlations between photons in the two modes. Two-mode squeezed states are produced by the same kind of physical process and the same kinds of physical device that produce single-mode squeezed states (e.g., a parametric amplifier), simply by moving away from degeneracy. They are to be contrasted with another kind of two-mode state, one produced by separately squeezing two single modes. The latter are produced by a different kind of physical process and different physical devices from those that produce two-mode squeezed states. States that are products of two single-mode squeezed states do not exhibit squeezing, for the photons in the two modes are not correlated. There is a formal sense, however, in which these different kinds of two-mode states are (unitarily) equivalent. This formal equivalence tells one that, to achieve the desired squeezing by separately squeezing two modes, one would have to use a frequency-converting device before and after squeezing the two modes separately. This process produces the required correlations, hence the squeezing, but it Is clearly not the natural way to obtain a broadband squeezed state.

DOI
10.1364/oam.1985.fm2

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A broadband or two-mode (nondegenerate) squeezed state is the natural two-mode analog of a single-mode (degenerate) squeezed state. The squeezing is a result of correlations between photons in the two modes. Two-mode squeezed states are produced by the same kind of physical process and the same kinds of physical device that produce single-mode squeezed states (e.g., a parametric amplifier), simply by moving away from degeneracy. They are to be contrasted with another kind of two-mode state, one produced by separately squeezing two single modes. The latter are produced by a different kind of physical process and different physical devices from those that produce two-mode squeezed states. States that are products of two single-mode squeezed states do not exhibit squeezing, for the photons in the two modes are not correlated. There is a formal sense, however, in which these different kinds of two-mode states are (unitarily) equivalent. This formal equivalence tells one that, to achieve the desired squeezing by separately squeezing two modes, one would have to use a frequency-converting device before and after squeezing the two modes separately. This process produces the required correlations, hence the squeezing, but it Is clearly not the natural way to obtain a broadband squeezed state.
In this paper we discuss a new, two-mode formalism1 for analyzing two-photon optical devices (e.g., parametric amplifiers and four-wave mixers)–devices whose output light is generated by the simultaneous emission of two photons into two of the output modes. Such devices can produce in principle squeezed-state light, in which the two modes of each pair of output modes have correlated noise. Two-photon devices are to be contrasted with one-photon devices, such as the laser, whose output light is generated by the emission of independent single photons.
Continuous variable entanglement between two modes of a radiation field is usually studied at optical frequencies. Here we demonstrate experiments that show the entanglement between microwave photons of different energy in a broadband squeezed beam. We use a Josephson parametric amplifier to generate the two-mode correlated state and detect all four quadrature components simultaneously in a two-channel heterodyne setup using amplitude detectors. Analyzing two-dimensional phase space histograms for all possible pairs of quadratures allows us to determine the full covariance matrix, which is in good agreement with the one expected for a two-mode squeezed state.
Squeezed vacuum with large photon numbers is an example of a macroscopic system with nonclassical properties. These properties are manifested in pairwise intensity correlations, also called twin-beam squeezing. We produce squeezed vacuum in a traveling-wave pulse-pumped frequency non-degenerate optical parametric amplifier (NOPA). Twin-beam squeezing is measured via direct detection and measurement of the difference-signal variance for two balanced detectors. It is shown that the study of the intensity correlations using this method requires a collection of a large number of modes, both longitudinal and transverse.
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.
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.
In their seminal paper, Caves and Schumaker presented a new formalism for quantum optics, intended to serve as a building block for describing two-photon processes, in terms of new, generalized qudratures. The important, revolutionary concept in their formalism was that it was fundamentally two-mode, i.e. the related observables could not be attributed to any single one of the comprising modes, but rather to a generalized complex quadrature that could only be attributed to both of them. Here, we propose a subtle, but fundamentally meaningful modification to their important work. Unlike the above proposal, we deliberately choose a frequency-agnostic definition of the two-mode quadrature, that we motivate on physical grounds. This simple modification has far-reaching implications to the formalism – the real and imaginary parts of the quadratures now coincide with the famous EPR variables, and our two-mode operators transform trivially under two-mode and single-mode squeezing operations. Their quadratic forms, which we call the “quadrature powers” are shown to succinctly generate the SU(1, 1) algebra of squeezing Hamiltonians, and correspond directly to important, broadband physical observables, that have been directly measured in experiment and are explicitly related to properties like squeezing and entanglement. This new point of view gives a fresh perspective on two-mode processes that is completely agnostic to the bandwidth, and reveals intriguing new ways for understanding and measuring broadband two-mode squeezing.