Symmetry entanglement in polarization biphoton optics

We study kinematic and dynamic ways of forming entangled states of quantum light fields due to their local and global polarization SU(2) symmetries. The kinematic entanglement is shown to be associated with particular polarization bases in the spaces of quantum states of multi-mode radiation, which are generated by the global SU(2) — symmetry. Dynamic entanglement is due to SU(2) symmetries of the Hamiltonians of the matter-radiation interaction. We also define some entanglement measures, which are related to characteristics of light depolarization. Applications of results obtained in biphoton optics are briefly discussed.     

Operationalistic orthogonality condition for single-mode biphotons (qutrits)

An arbitrary polarization state of a single-mode biphoton is considered. The operationalistic criterion is formulated for the orthogonality of these states. It can be used to separate a biphoton with an arbitrary degree of polarization from a set of biphotons orthogonal to it. This is necessary for the implementation of quantum cryptography protocol based on the three-level systems. The experimental test of this criterion amounts to the observation of the anticorrelation effect for a biphoton with an arbitrary polarization state.     

Polarization States of Light and Their Quantum Tomography

The notion and main features of polarization states of light are discussed within the framework of classical and quantum optics. This notion is shown to be correctly defined for arbitrary light beams only within quantum optics by using the P-quasispin formalism developed earlier. Polarization states of quantum light are shown to be fully described by a polarization density operator (PDO) obtained via reducing the total field density operator. Theoretical foundations are given for quantum tomography of polarization states of light fields considered as a way of measuring PDO. Herewith, the main attention is paid to a method where proper polarization tomographic observables (PDO “measurers”) are used. The method is shown to be adequately formulated by means of quasi-spectral tomographic expansions of PDO in special operator bases (given by finite sums of partially orthogonal projectors), which determine probability distributions of tomographic observables as expansion coefficients. Matrix versions of such “tomographic” PDO representations are obtained. In particular, projections of these expansions on quasiclassical operator bases, determining polarization quasiprobability functions, are given. An example of experimental implementation of polarization tomography of unpolarized light (biphoton radiation with hidden polarization) is analyzed.     

Biphoton light with hidden polarization and its polarization tomography

Polarization properties of biphoton light, emitted by an optical parametric oscillator, are studied both theoretically and experimentally. The theoretical treatment is performed using the Stokes operator formalism; polarization density matrix of the light fields; and the quasi-probability function, which represents an analog of the field Wigner function. The hidden polarization of biphoton light is demonstrated experimentally with the help of elements of polarization tomography.     

Qutrits and ququarts in spontaneous parametric down-conversion, correlations and entanglement

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.     

Temporally Correlated Narrowband Biphoton Interference Based on Mach–Zehnder Interferometer

Since early 1990s, Mach–Zehnder interferometer has been used to investigate the interference of biphoton wave packets. Due to subpicosecond time coherence of biphoton generated by spontaneous parametric downconversion process, some physical processes are ignored in the interferometer, most likely the biphoton time‐domain interference. Here, the two‐photon interference phenomenon based on the Mach–Zehnder interferometer is theoretically studied, where the correlated photon pairs are produced by the four‐wave mixing in atomic system. In particular, the quantum interference effect to effectively control the coherent time of two‐photon by adjusting the input delay is used. In the damped Rabi oscillation regime, two‐photon bunching and antibunching effects are observed. In addition, in the group‐delay regime, the interference between biphoton precursor, slow‐light wave packets and also in between the precursor and the slow‐light wave packets is observed, which had never been reported before. These results may have potential applications in the fields of biphoton shaping and quantum information processing.     

Reconstruction of the polarization states of a biphoton field

A method for reconstructing an arbitrary quantum state of an optical system in Hilbert space with dimension d = 4 is discussed. Such states can be realized using a collinear frequency-nondegenerate regime of generating spontaneous parametric down-conversion. The method has been tested for a number of polarization states of a biphoton field. The high accuracy of the reconstruction of the states (above 99%) indicates that the procedures proposed for reconstructing the quantum state of the system are adequate.     

Invariant-algebraic approach to problems of quantum optics

A general approach is formulated for analyzing algebraic models of quantum composite systems with an internal symmetry described by group G. The case G = SU(2) is examined in detail as applied to the analysis of polarization invariance in quantum optics. A new class of fully depolarizable quantum states of light (scalar biphotons) is defined and investigated. Certain interactions of scalar biphoton light with material media are considered in the context of Dicke and Jaynes-Cummings models.Landsberg Optics Division, Lebedev Physics Institute. Translated from Preprint No. 137 of the Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow, 1990.     

Observation of two-photon coherence in plasmon-assisted transmission

We experimentally study two-photon coherence in plasmon-assisted transmission with a two-photon Mach–Zehnder (MZ) interferometer. Two collinear photons of identical or orthogonal polarization are simultaneously incident on one optically thick metal film, perforated with a periodic array of subwavelength holes. Interference fringes with two-photon de Broglie wavelength are observed, which indicates that the quantum coherence of biphoton is preserved in surface plasmon assisted transmission.     

Realization of four-level qudits using biphotons

The novel experimental realization of four-level optical quantum systems (ququarts) is presented. We exploit the polarization properties of the frequency nondegenerate biphoton field to obtain such systems. A simple method that does not rely on am interferometer is used to generate and measure the sequence of states that can be used in quantum key distribution protocol.     

Projection of two biphoton qutrits onto a maximally entangled state

Bell state measurements, in which two quantum bits are projected onto a maximally entangled state, are an essential component of quantum information science. We propose and experimentally demonstrate the projection of two quantum systems with three states (qutrits) onto a generalized maximally entangled state. Each qutrit is represented by the polarization of a pair of indistinguishable photons-a biphoton. The projection is a joint measurement on both biphotons using standard linear optics elements. This demonstration enables the realization of quantum information protocols with qutrits, such as teleportation and entanglement swapping.     

Polarization transformations of multimode light fields

The concept of P-quasispin is used as a framework for developing a formalism for polarization transformation by means of lossless polarization-changing optical elements for both classical and quantum multimode light fields. As an example, a classification of transformations of this kind is presented for two-mode biphoton fields that implement the information-theoretic concepts of qutrit and ququart (quantum systems in three-and four-dimensional Hilbert states, respectively). Original Russian Text ? V.P. Karassiov, S.P. Kulik, 2007, published in Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2007, Vol. 131, No. 1, pp. 37–53.     

The influence of the icosahedral percolation transition in supercooled liquid iron on the diffusion mobility of atoms

An experiment on preparation of entangled photon pairs (biphotons) in an arbitrary polarization state is described. The biphotons are qutrits (three-state quantum systems). They can be used in ternary quantum cryptography protocols. A theoretically derived orthogonality criterion for the prepared biphotons is validated experimentally. The criterion can be used to identify orthogonal biphoton states.     

A Four-Dimensional Measurement Scheme for Single Photon Time-Polarization

We propose an experimental scheme to realize the four-dimensional projective measurements tor a single photon. The photon polarization and time-energy provide the four-dimensional Hilbert space. Based on this scheme, we suggest an experiment to test the violation of Bell inequalities of four-dimensional systems. In addition, by virtue of a maximally entangled biphoton state, we also show that it is possible to construct a quantum key distribution channel that can provide two-bit key with one pair of entangled photons.     

Polarization tomography of a narrow-band biphoton field

Narrow-band orthogonally polarized collinear frequency-degenerate biphotons are generated in the process of the spontaneous parametric down-conversion of light in a nonlinear BBO crystal placed in an optical resonator. Quantum polarization tomography and polarization transformation of the generated biphoton states are performed. The results from our experiment agree well with theoretical calculations.     

Measurement of qutrits

A method of optical realization of a protocol of restoration of the density matrix of an unknown state of a three-level system that represents an arbitrary polarization state of a single-mode biphoton field is proposed. The method is applied to a set of pure states of qutrits, with this set being determined by the properties of SU(2) transformations performed by polarization transformers (retardation plates).     

Occurrence conditions for two-dimensional Borromean systems

Features of biphoton polarization-frequency ququarts are considered. Their wave functions are defined as functions of both polarization and frequency variables of photons with the symmetry obligatory for two-boson states taken into account. In experiments, biphoton ququarts can display different features in dependence on whether experiments involve purely polarization or (alternatively) polarization-frequency measurements. If in experiments one uses only polarization measurements, the originally pure states of ququarts can be seen as mixed biphoton polarization states. Features of such states are described and discussed in details. Schemes of coincidence measurements for reconstruction of the ququart’s parameters are suggested and described.     

On the distinguishability of downconverted modes with orbital angular momentum

Assuming two quantum states of spontaneous parametric downconversion carrying orbital angular momentum, one may ask the question what is the minimum probability of error in identifying between two of these biphoton states by an arbitrary physical measurement over the biphoton state generated. While correctly chosen geometries may lead to perfect distinguishability of modes, it is worth noticing that experimental subtleties may lead to a poor mode distinguishability. We discuss the case where a restricted range instead of the needed range of wave vectors is collected by the experimental setup. These considerations may be useful for some applications, e.g., cryptography.     

Scalable scheme for entangling multiple ququarts using linear optical elements

We report a scalable linear optical scheme for generating entangled states of multiple ququarts in which the individual single-ququart state is prepared with the biphoton polarization state of frequency-nondegenerate spontaneous parametric down-conversion. The output state is calculated with the full consideration of the higher order effect (double-pair events) of spontaneous parametric down-conversion. Scalability to multiple-ququart entanglement is demonstrated with examples: linear optical entanglement of three and four individual biphoton ququarts.