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Physical Review A

Nonclassical fields in a linear directional coupler

W. K. Lai, V. Buek, P. L. Knight

Published Jun 1, 1991• 95 citations
Publisher

Abstract

In the present paper, we discuss the photon statistics of nonclassical fields in a linear directional coupler. We study the photon distributions at the output ports of the coupler for number-state inputs, as well as for arbitrary-state inputs. In particular, we show that when a number state enters one port of the coupler and the vacuum enters the other port, an SU(2) coherent state is generated at the output ports. If the same number state enters both ports of the coupler, the probability of finding an odd number of photons at either of the output ports vanishes for a particular choice of the coupler length. We show that if a number state enters port 1 and a strong coherent field enters port 2, the coupler can serve as a generator of displaced number states (in fact, in this case the coupler behaves essentially as a homodyne detector). We show that the coupler is equivalent to an imperfect photon detector with efficiency governed by the coupler parameters. We study the effect of dissipative losses within the coupler as well as the effect of scattering coupling in the coupler. The latter leads to equalization of the energy in each guide. We show that in the presence of scattering coupling, the coupler behaves neither like an imperfect photon detector nor a homodyne detector.

DOI
10.1103/physreva.43.6323

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In the present paper, we discuss the photon statistics of nonclassical fields in a linear directional coupler. We study the photon distributions at the output ports of the coupler for number-state inputs, as well as for arbitrary-state inputs. In particular, we show that when a number state enters one port of the coupler and the vacuum enters the other port, an SU(2) coherent state is generated at the output ports. If the same number state enters both ports of the coupler, the probability of finding an odd number of photons at either of the output ports vanishes for a particular choice of the coupler length. We show that if a number state enters port 1 and a strong coherent field enters port 2, the coupler can serve as a generator of displaced number states (in fact, in this case the coupler behaves essentially as a homodyne detector). We show that the coupler is equivalent to an imperfect photon detector with efficiency governed by the coupler parameters. We study the effect of dissipative losses within the coupler as well as the effect of scattering coupling in the coupler. The latter leads to equalization of the energy in each guide. We show that in the presence of scattering coupling, the coupler behaves neither like an imperfect photon detector nor a homodyne detector.
In the framework of Hamiltonian formalism nonclassical effects of an optical field propagating inside a directional coupler containing nondegenerate parametric amplification have been studied. We investigate the effect of switching between the input modes and the out going fields from the coupler. Particular attention has been paid to two mode squeezing, second-order correlation function, quasiprobability distribution functions, and photon-number distribution. Incident number and coherent states are considered. It has been shown that when one of the modes enters the coupler in the Fock state and the other modes are in vacuum states, the coupler can serve as a generator for coherent state. Furthermore, regimes for generation and transmission of squeezed and/or sub-Poissonian light are found.
In the framework of Hamiltonian formalism the quantum statistical properties of an optical field propagating inside a directional coupler operating by nondegenerate parametric amplification are studied. We investigate the effect of switching between the input modes and the outgoing fields from the coupler. Particular attention is paid to two-mode squeezing, the second-order correlation function, quasiprobability distribution functions, and photon-number distribution. Incident number and coherent states are considered. It has been shown that when one of the modes enters the coupler in the Fock state |1 and the other modes are in vacuum states, the coupler can serve as a generator of a coherent state. Furthermore, regimes for the generation and transmission of squeezed and/or sub-Poissonian light are found.
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.
In this paper we derive the quantum statistical and dynamical properties of nonlinear optical couplers composed of two nonlinear waveguides operating by second subharmonic generation, which are coupled linearly through evanescent waves and nonlinearly through non-degenerate optical parametric interaction. Main attention is paid to generation and transmission of non-classical light, based on a discussion of the squeezing phenomenon, the normalized second-order correlation function and quasiprobability distribution functions. Initially coherent, number and thermal states of optical beams are considered. In particular, results are discussed with dependence on the strength of the nonlinear coupling relatively to the linear coupling. We show that if the Fock state |1〉 enters the first waveguide and the vacuume state |0〉 enters the second waveguide, the coupler can serve as a generator of squeezed vacuum state governed by the coupler parameters. Further, if thermal fields enter initially the waveguides the coupler plays a similar role as a microwave Josephson-junction parametric amplifier to generate squeezed thermal light.
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.
We investigate the photon statistics at the outputs of a four-port beam splitter, when a strong coherent field is incident on one input port and various other fields enter at the second inner port. Some non-classical interference effects are encountered. In particular, there is a large difference between the cases when a one photon field enters at input port 2 and when a field in a coherent state with an average of one photon enters at port 2. In the former case the probability P(m) of the difference m between the detected photons at output ports 3 and 4 has a minimum at m = 0 whenever the quantum efficiency exceeds one third. As the quantum efficiency falls from one to zero, the form of the probability distribution P(m/1) changes drastically.