A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr\"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.
A lossless beam splitter (a dielectric interface, a passive interferometer, or a linear coupler) changes the quantum state of two incident modes by an SU(2) transformation. Apart from phase shifting, the argument of the quadrature wave function of the system undergoes a rotation. Quasiprobabilities are changed by the inverse mode transformation. The use of balanced beam splitting allows the simultaneous measurement of conjugate quadrature components via homodyning the emerging beams with two strong coherent reference fields that differ in their phases by \ensuremath{\pi}/2. The measured probability distribution is given by a generalized Q function. It depends on the state of the field entering the second beam-splitter port. For a vacuum, the Q function will be obtained. The use of unbalanced beam splitting allows the measurement of a squeezed Q function without using squeezed states. Dissipation in Gaussian reservoirs corresponds exactly to a heuristic beam-splitter model. As a mathematical tool, the Fokker-Planck equation of damping in phase-sensitive reservoirs and the corresponding quantum master equation were solved. The dissipative decay of a Schr"odinger-cat state was studied as an example. The sensitivity of quantum coherence with respect to damping can be interpreted geometrically.
A comprehensive formalism for a two-mode SU(1,1) interferometer (a nondegenerate parametric amplifier, a degenerate four-wave mixer, or a phase-conjugating mirror) is presented. The device changes the quantum state of the two incident modes by an SU(1,1) transformation. Apart from phase shiftings, the quadrature wave functions are Lorentz transformed. A transformation formula in the Fock space of the two modes is derived. Hybrids of s and -s quasiprobability distributions are changed by the inverse-mode transformation. It is related to quantum noise in phase conjugation. Linear amplification caused by Gaussian reservoirs corresponds exactly to a heuristic nondegenerate-parametric-amplifier model. It allows a simple geometrical explanation for the sensitivity of squeezing or quantum superposition with respect to phase-sensitive amplification.
For optical homodyning, the matrix representation of a lossless beam splitter belongs to the SU(2) group of unimodular second-order unitary matrices. The connection between this group and the rotation group in three dimensions permits the field density operators at the input and output ports of the beam splitter to be related by means of well-known angular-momentum transformations. This, in turn, provides the joint output photon-number distribution, which may be written as a Fourier series in the relative phase shift imparted by the beam splitter, for a general joint state at its inputs. The series collapses to a single term if one of the input fields is diagonal in the number-state representation. If the inputs to both ports are further restricted to be pure number states, the joint, as well as the marginal photon-number distributions, turn out to be directly proportional to the square of Jacobi polynomials in the beam-splitter transmittance. These photon-number probabilities are invariant to a set of physical and time-reversal symmetries. When one of the input photon-number states is the vacuum, the beam splitter simply deletes photons from the other port in Bernoulli fashion, as if they were classical particles. The output photon number is then described by the binomial distribution. If the inputs at the two ports are different number states, neither of which is the vacuum, the photon-number distribution is expressible in terms of summed and weighted products of the results for photomixing with the vacuum. If the inputs at the two ports are identical number states, and a beam splitter of transmittance \ensuremath{\tau}=(1/2 is used, the photon-number distribution assumes a simple but interesting form. It vanishes for odd photon numbers, indicating that the photons assemble in pairs at each output port. Finally, it is shown that homodyning quantum fluctuations can be reduced by using a balanced photomixer for arbitrary input states.
The superposition of input fields in a lossless beam splitter is studied in the Schr\"odinger picture by using the convolution of the positive P representations, and the convolution law for these representations is extended to other quasiprobability functions such as the Wigner and Q functions. We show that the reservoir can be modeled by an infinite array of beam splitters, and we use the convolution law and this model to derive the Fokker-Planck equation for a system coupled with a phase-sensitive reservoir. Solving this equation shows that a phase-sensitive attenuation and amplification can be described by the superposition of two independent quantum fields, one of which is the initial signal field and the other the squeezed thermal noise field representing the reservoir.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring this output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal relative nonclassicality, which can be achieved by comparing two entanglement measures for (i) arbitrary two-qubit states and (ii) those which can be generated from a photon-number qubit via a balanced lossless beam splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal relative nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal relative nonclassicality, measured by the NP for a given value of the REEP, can be increased by phase damping (dephasing). Note that the entanglement itself is not increased by these losses (since they act locally), but the possible ratios of different measures are affected. Moreover, we show that partially dephased states can be more nonclassical than both pure states and completely dephased states, by comparing the NP for a given value of the REEP. Thus, one can conclude that not all standard entanglement measures can be used as entanglement potentials. Alternatively, one can infer that a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. The application of a lossy beam splitter can solve this problem, at least for the cases analyzed in this paper.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring the output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal nonclassicality, which can be achieved by comparing two entanglement measures for arbitrary two-qubit states and those which can be generated from a photon-number qubit via a balanced lossless beam-splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal nonclassicality, measured by the NP for a given value of the REEP, can be increased by dephasing. The entanglement itself is not increased by these local losses, but the possible ratios of different measures are affected. Moreover, we show that partially-mixed states can be more nonclassical than both pure states and completely-mixed states, by comparing the NP for a given value of the REEP. This implies that not all entanglement measures can be used as entanglement potentials. Alternatively, a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. Applying a lossy beam splitter can solve this problem at least for the cases analyzed in this paper.
Using the quantum theory of a lossless beam splitter, we determine the joint and marginal photon-number distributions at its output ports, for two arbitrary (not necessarily independent) input fields. The joint probability at the output of the beam splitter turns out to be a Fourier series in the relative phase between the output beams. Photomixing with the number states exhibits no phase sensitivity. An elementary example of phase sensitive dependence in the photon-number domain is obtained with the Bernoulli states, which are a subclass of the binomial states.1 For a joint input Bernoulli state, the probability of detecting one photon (or zero photons) at either output port varies sinusoidally with the relative phase. For independent Bernoulli states incident on the beam splitter, the results differ and depend on whether the input is in a pure or mixed state. Photomixing of other nonclassical states exhibit higher orders of phase-induced modulation of the output photon-number probabilities.