Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum superpositions

We discuss a device capable of filtering out two-mode states of light with mode populations differing by more than a certain threshold, while not revealing which mode is more populated. It would allow engineering of macroscopic quantum states of light in a way which is preserving specific superpositions. As a result, it would enhance optical phase estimation with these states as well as distinguishability of "macroscopic" qubits. We propose an optical scheme, which is a relatively simple, albeit non-ideal, operational implementation of such a filter. It uses tapping of the original polarization two-mode field, with a polarization neutral beam splitter of low reflectivity. Next, the reflected beams are suitably interfered on a polarizing beam splitter. It is oriented such that it selects unbiased polarization modes with respect to the original ones. The more an incoming two-mode Fock state is unequally populated, the more the polarizing beam splitter output modes are equally populated. This effect is especially pronounced for highly populated states. Additionally, for such states we expect strong population correlations between the original fields and the tapped one. Thus, after a photon-number measurement of the polarizing beam splitter outputs, a feed-forward loop can be used to let through a shutter the field, which was transmitted by the tapping beam splitter. This happens only if the counts at the outputs are roughly equal. In such a case, the transmitted field differs strongly in occupation number of the two modes, while information on which mode is more populated is non-existent (a necessary condition for preserving superpositions).Comment: 11 pages, 12 figure

Orthogonality of Biphoton Polarization States

Orthogonality of two-photon polarization states belonging to a single frequency and spatial mode is demonstrated experimentally, in a generalization of the well-known anti-correlation 'dip' experiment.Comment: Submitted to Phys.Rev.Let

Conditioned Unitary Transformation on biphotons

A conditioned unitary transformation ($90^o$ polarization rotation) is performed at single-photon level. The transformation is realized by rotating polarization for one of the photons of a polarization-entangled biphoton state (signal photon) by means of a Pockel's cell triggered by the detection of the other (idler) photon after polarization selection. As a result, polarization degree for the signal beam changes from zero to the value given by the idler detector quantum efficiency. This result is relevant to practical realization of various quantum information schemes and can be used for developing a new method of absolute quantum efficiency calibration

Biphoton compression in standard optical fiber: exact numerical calculation

Generation of two-photon wavepackets, produced by spontaneous parametric down conversion in crystals with linearly chirped quasi-phase matching grating, is analyzed. Although being spectrally broad, two-photon wavepackets produced this way are not Fourier transform limited. In the paper we discuss the temporal compression of the wavepackets, exploiting the insertion of a standard optical fiber in the path of one of the two photons. The effect is analyzed by means of full numerical calculation and the exact dispersion dependencies in both the crystal and the fiber are considered. The study opens the way to the practical realization of this idea.Comment: 10 pages, 16 figure

Generation of different Bell states within the SPDC phase-matching bandwidth

We study the frequency-angular lineshape for a phase-matched nonlinear process producing entangled states and show that there is a continuous variety of maximally-entangled states generated for different mismatch values within the natural bandwidth. Detailed considerations are made for two specific methods of polarization entanglement preparation, based on type-II spontaneous parametric down-conversion (SPDC) and on SPDC in two subsequent type-I crystals producing orthogonally polarized photon pairs. It turns out that different Bell states are produced at the center of the SPDC line and on its slopes, corresponding to about half-maximum intensity level. These Bell states can be filtered out by either frequency selection or angular selection, or both. Our theoretical calculations are confirmed by a series of experiments, performed for the two above-mentioned schemes of producing polarization-entangled photon pairs and with two kinds of measurements: frequency-selective and angular-selective.Comment: submitted for publicatio

Statistical Reconstruction of Qutrits

We discuss a procedure of measurement followed by the reproduction of the quantum state of a three-level optical system - a frequency- and spatially degenerate two-photon field. The method of statistical estimation of the quantum state based on solving the likelihood equation and analyzing the statistical properties of the obtained estimates is developed. Using the root approach of estimating quantum states, the initial two-photon state vector is reproduced from the measured fourth moments in the field . The developed approach applied to quantum states reconstruction is based on the amplitudes of mutually complementary processes. Classical algorithm of statistical estimation based on the Fisher information matrix is generalized to the case of quantum systems obeying Bohr's complementarity principle. It has been experimentally proved that biphoton-qutrit states can be reconstructed with the fidelity of 0.995-0.999 and higher.Comment: Submitted to Physical Review

Symbolic approach and induction in the Heisenberg group

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.Comment: 18 page

Polarization state of a biphoton: quantum ternary logic

Polarization state of biphoton light generated via collinear frequency-degenerate spontaneous parametric down-conversion is considered. A biphoton is described by a three-component polarization vector, its arbitrary transformations relating to the SU(3) group. A subset of such transformations, available with retardation plates, is realized experimentally. In particular, two independent orthogonally polarized beams of type-I biphotons are transformed into a beam of type-II biphotons. Polarized biphotons are suggested as ternary analogs of two-state quantum systems (qubits)

Measurement of qutrits

We proposed the procedure of measuring the unknown state of the three-level system - the qutrit, which was realized as the arbitrary polarization state of the single-mode biphoton field. This procedure is accomplished for the set of the pure states of qutrits; this set is defined by the properties of SU(2) transformations, that are done by the polarization transformers.Comment: 9 pages, 9 figure