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arXiv:1905.05088 · Physica Scripta

Schmidt-mode analysis of quadrature entanglement in superpositions of two-mode multiphoton states

Mikhail Vladimirovich Fedorov

Published Feb 27, 2020• 3 citations
arXivHTMLPDF

Abstract

Abstract Schmidt-decomposition formalism is proposed to be used for evaluation of the degree of quadrature entanglement in two-mode multiphoton states. A series of examples are considered, including analysis of the quadrature entanglement of the two-mode squeezed vacuum. Single-mode and two-mode coherent states with stochastic phases are analyzed by the same method though in this case features of quadrature states are interpreted in terms of decoherence rather than entanglement.

DOI
10.1088/1402-4896/ab7aa7

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Abstract Schmidt-decomposition formalism is proposed to be used for evaluation of the degree of quadrature entanglement in two-mode multiphoton states. A series of examples are considered, including analysis of the quadrature entanglement of the two-mode squeezed vacuum. Single-mode and two-mode coherent states with stochastic phases are analyzed by the same method though in this case features of quadrature states are interpreted in terms of decoherence rather than entanglement.
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 examined a two-mode coherent and subharmonic light beam within a cavity that is coupled to a two-mode vacuum reservoir through a single port mirror. Utilizing the solutions of the quantum Langevin equations, we developed Q-functions for the two-mode coherent and subharmonic cavity light beams. Subsequently, we calculated the Q-function for the superposed two-mode light beams in question. By leveraging the relevant Q-functions for the superposed cavity light beams, we investigated the quadrature statistics and entanglement. To conduct our analysis, we focused on a quantum system characterized by Gaussian operators with a zero mean. We have identified that the superposed two-mode coherent and subharmonic cavity light beams are in a squeezed state, with the squeezing occurring in the plus quadrature. Moreover, our analysis revealed that at steady state and at threshold, the superposed two-mode coherent and subharmonic light beams exhibit a maximum squeezing of 25% below the two-mode vacuum-state level. We have also clearly illustrated, using continuous variable entanglement criteria, that the superposed two-mode cavity light beams are entangled at both steady-state and threshold, with the entanglement being manifested in the highly correlated squeezed photons, which show a 75% degree of entanglement. Our analysis, applying Cauchy-Schewarz, Hillary-Zubiary, and Logarithmic Negativity entanglement criteria, further confirmed that the superposed cavity light beams in question are entangled. The Logarithmic Negativity criteria validated the entanglement of the superposed light, indicating a 50% degree of entanglement.
With the aid of the solutions of the Quantum Langevin equations, we have calculated the Q-functions for the two-mode coherent and subharmonic cavity light beams. We have then determined the Q-function for the superposed two-mode coherent and subharmonic cavity light beams. With the help of the pertinent Q-functions for cavity light beams, we determined the photon statistics, quadrature statistics, and entanglement. we have obtained that the superposed two-mode coherent and subharmonic cavity light beams are in a squeezed state and the squeezing occurs in the plus quadrature. Besides, our analysis shows that at steady state and at threshold, the superposed two-mode coherent and subharmonic light beams have a maximum squeezing of 25%25\%25% below the two-mode vacuum-state level. We have also clearly shown that the superposed two-mode light beam is entangled at steady-state and threshold and the entanglement turned out to be observed in the highly correlated squeezed photons with 75%75\%75% degree of entanglement.
Motivated by their frequent use in quantum mechanical studies of entanglement, we give a brief overview of Schmidt modes and Schmidt decompositions of two-particle wave functions. We discuss methods of their derivation and include a little-known approach used in the original work by E. Schmidt [Math. Annalen, 63 (1906), 433]. This employs the bipartite wave function itself rather than the more complicated two-party reduced density matrix. As an illustration, Schmidt modes for two-photon polarisation qutrits are derived in a general form. The derivation is accompanied by a series of simple examples with special choices of parameters. Relationships between Schmidt modes, polarisation Stokes vectors and entanglement are also discussed.