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Laser Physics Letters

Entanglement of multiphoton two-mode polarization Fock states and of their superpositions

Stephen Vintskevich, D. A. Grigoriev, Nikolai Miklin, Mikhail Vladimirovich Fedorov

Published Feb 24, 2020• 4 citations
Publisher

Abstract

Abstract Density matrices of pure multiphoton Fock polarization states and their resultant reduced density matrices of mixed states are expressed in similar ways in terms of matrices of correlators, defined as averaged products of equal numbers of creation and annihilation operators. The degree of entanglement of considered states is evaluated for various combinations of parameters of states and characters of their reduction.

DOI
10.1088/1612-202x/ab72a1

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Abstract Density matrices of pure multiphoton Fock polarization states and their resultant reduced density matrices of mixed states are expressed in similar ways in terms of matrices of correlators, defined as averaged products of equal numbers of creation and annihilation operators. The degree of entanglement of considered states is evaluated for various combinations of parameters of states and characters of their reduction.
We investigate features of biphoton qutrits and ququarts with the symmetry of biphoton wavefunctions fully taken into account. Entanglement quantifiers of such states are found as functions of qutrit and ququart parameters. For biphoton qutrits, a general relation between degrees of entanglement and polarization is established. In the case of biphoton ququarts, photon frequencies are considered as variables independent of polarization variables, with both polarization and frequency degrees of freedom equally contributing to the total ququart entanglement. Averaging over one of these two degrees of freedom is shown to reduce pure ququart states to mixed two-qubit states. The degree of polarization and degrees of correlations in mixed polarization states are investigated.
In a general form, three-photon polarization states are superpositions of contributions from four three-photon polarization modes with four arbitrary complex coefficients. Entanglement of such states is defined as entanglement between their one-photon and two-photon parts. The degree of entanglement is found explicitly in terms of such entanglement quantifiers as the generalized concurrence, entropy of reduced states and the Schmidt parameter. Polarization Stokes vectors, as well as the Schmidt decompositions of three-photon states are found and discussed. Similarity and differences with respect to biphoton polarization states are discussed too.
A two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modes
We analyze features of mixed biphoton polarization states, which arise from pure states of polarization-frequency biphoton ququarts after averaging over frequencies of photons. For mixed states, we find their concurrence CCC, Schmidt parameter KKK, degree of polarization PPP, as well as the von Neumann mutual information III. In some simple cases, we also find the relative entropy Srel{S}_{\mathrm{rel}}Srel​ and the degree of classical correlations Ccl{C}_{\mathrm{cl}}Ccl​. In mixed states, the Schmidt parameter does not characterize the degree of entanglement anymore, as it does in pure states. Nevertheless, the Schmidt parameter remains useful even in the case of mixed states because it remains directly related to the degree of polarization. We compare results occurring in the cases of full pure polarization-frequency states of ququarts and mixed states (averaged over frequencies). Differences between these results can be seen in experiments with and without frequency filters in front of a detector.