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arXiv:2011.08099 · Optics Express

Broadband complex two-mode quadratures for quantum optics

Leon Bello, Yoad Michael, M. Rosenbluh, Eliahu Cohen, Avi Pe’er

Published Nov 17, 2021• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1364/oe.432054

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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.
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.
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type of quantum squeezing, namely planar quantum squeezing. This has the property that it squeezes two orthogonal spin directions simultaneously, which is possible owing to the fact that the SU(2) group that describes spin symmetry has a three-dimensional parameter space, of higher dimension than the group for photonic quadratures. The practical advantage of planar quantum squeezing is that, unlike conventional spin-squeezing, it allows noise reduction over all phase-angles simultaneously. The application of this type of squeezing is to quantum measure- ment of an unknown phase. We show that a completely unknown phase requires two orthogonal measurements, and that with planar quantum squeezing it is possible to reduce the measurement uncertainty independently of the unknown phase value. This is a different type of squeezing to the usual spin-squeezing interferometric criterion, which is only applicable when the measured phase is already known to a good approximation, or can be measured iteratively. As an example, we calcu- late the phase-entanglement of the ground state of a two-well, coupled Bose-Einstein condensate, similar to recent experiments. This system demonstrates planar squeezing in both the attractive and repulsive interaction regimes.
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.
We derive a phase-entanglement criterion for two bosonic modes which is immune to number fluc- tuations, using the generalized Moore-Penrose inverse to normalize the phase-quadrature operator. We also obtain a phase-squeezing criterion that is immune to number fluctuations using similar techniques. These are utilized to obtain an operational definition of relative phase-measurement sensitivity, via analysis of phase measurement in interferometry. We show that these measures are proportional to enhanced phase-measurement sensitivity. The phase-entanglement criterion is a hallmark for a new type of quantum squeezing, namely planar quantum squeezing. This has the property that it squeezes two orthogonal spin directions simultaneously, which is possible owing to the fact that the SU(2) group that describes spin symmetry has a three-dimensional parameter space, of higher dimension than the group for photonic quadratures. The practical advantage of planar quantum squeezing is that, unlike conventional spin-squeezing, it allows noise reduction over all phase-angles simultaneously. The application of this type of squeezing is to quantum measure- ment of an unknown phase. We show that a completely unknown phase requires two orthogonal measurements, and that with planar quantum squeezing it is possible to reduce the measurement uncertainty independently of the unknown phase value. This is a different type of squeezing to the usual spin-squeezing interferometric criterion, which is only applicable when the measured phase is already known to a good approximation, or can be measured iteratively. As an example, we calcu- late the phase-entanglement of the ground state of a two-well, coupled Bose-Einstein condensate, similar to recent experiments. This system demonstrates planar squeezing in both the attractive and repulsive interaction regimes.
In recent decades there has been a resurge of interest in the foundations of quantum theory, partly motivated by new experimental techniques, partly by the emerging field of quantum information science. Old questions, asked since the seminal article by Einstein, Podolsky and Rosen (EPR), are being revisited. The work of John Bell has changed the direction of investigation by recognising that those fundamental philosophical questions can have, after all, input from experiment. Abner Shimony has aptly termed this new field of enquiry experimental metaphysics. The objective of this Thesis is to contribute to that body of research, by formalising old concepts, proposing new ones, and finding new results in well-studied areas. Without losing from sight that the appeal of experimental metaphysics comes from the adjective, every major result is followed by clear experimental proposals with detailed analysis of feasibility for quantum-atom optical setups. After setting the appropriate terminology and the basic concepts, we will start by analysing the original argument of Einstein, Podolsky and Rosen. We propose a general mathematical form for the assumptions behind the EPR argument, namely those of local causality and completeness. That formalisation entails what was termed a Local Hidden State model by Wiseman et al., which was proposed as a formalisation of the concept of steering first introduced by Schrodinger in a reply to the EPR paper. Violation of any consequences that can be derived from the assumption of that model therefore implies a demonstration of the EPR paradox. We will show how one can then re-derive the well-known EPR-Reid criterion for continuous-variables correlations, and derive new ones applicable to the spin setting considered by Bohm. The spin set-up of the EPR-Bohm was used by Bell to derive his now famous theorem demonstrating the incompatibility of the assumption of local causality and the predictions of quantum mechanics. The inequalities which bear his name can be derived for any number of discrete outcomes, but so far there has been no derivation which can be directly applied to the continuous-variables case of the original EPR paradox. We close the circle by deriving a class of inequalities which make no explicit mention about the number of outcomes of the experiments involved, and can therefore be used in continuous-variables measurements with no need for binning the continuous results into discrete ones. Apart from that intrinsic interest, these inequalities could prove important as a means to perform an unambiguous test of Bell inequalities since optical homodyne detection can be performed with high detection efficiency. The technique, which is based on a simple variance inequality, can alsobe used tore-derive alarge class of well-known Bell-type inequalities and at the same time find their quantum bound, making explicit from a formal point of view that the non-commutativity of the local operators is at the heart of the quantum violations. Finally, we address the issue of macroscopic superpositions originally sparked by the infamous cat paradox of Schrodinger. We consider macroscopic, mesoscopic and ‘S -scopic’ quantum superpositions of eigenstates of an observable, and develop some signatures for their existence. We define the extent, or size S of a (pure-state) su-perposition, with respect to an observable X, as being the maximum difference in the outcomes of X predicted by that superposition. Such superpositions are referred to as generalised S -scopic superpositions to distinguish them from the extreme superpo-sitions that superpose only the two states that have a difference S in their prediction for the observable. We also consider generalised S -scopic superpositions of coherent states. We explore the constraints that are placed on the statistics if we suppose a system to be described by mixtures of superpositions that are restricted in size. In this way we arrive at experimental criteria that are sufficient to deduce the existence of a generalised S -scopic superposition. The signatures developed are useful where one is able to demonstrate a degree of squeezing.