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Physical Review A

Entangled coherent states

Barry C. Sanders

Published May 1, 1992• 485 citations
Publisher

Abstract

The nonlinear Mach-Zehnder interferometer is presented as a device whereby a pair of coherent states can be transformed into an entangled superposition of coherent states for which the notion of entanglement is generalized to include nonorthogonal, but distinct, component states. Each mode is directed to a homodyne detector. We show that there exist nonclassical intensity correlations at the output ports of the homodyne detectors which facilitate a test of local realism. In contradistinction to previous optical schemes which test local realism, the initial state used here possesses a positive Glauber-Sudarshan representation and is therefore a semiclassical state. The nonlinearity itself is responsible for generating the nonclassical state.

DOI
10.1103/physreva.45.6811

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The nonlinear Mach-Zehnder interferometer is presented as a device whereby a pair of coherent states can be transformed into an entangled superposition of coherent states for which the notion of entanglement is generalized to include nonorthogonal, but distinct, component states. Each mode is directed to a homodyne detector. We show that there exist nonclassical intensity correlations at the output ports of the homodyne detectors which facilitate a test of local realism. In contradistinction to previous optical schemes which test local realism, the initial state used here possesses a positive Glauber-Sudarshan representation and is therefore a semiclassical state. The nonlinearity itself is responsible for generating the nonclassical state.
We suggest an all-optical scheme to generate entangled superposition of a single photon with macroscopic entangled states for testing macroscopic realism. The scheme consists of source of single photons, a Mach-Zehnder interferometer in routes of which a system of coupled-down converters with type-I phase matching is inserted, and a beam splitter for the other auxiliary modes of the scheme. We use quantization of the pumping modes, depletion of the coherent states passing through the system, and interference effect in the pumping modes in the process of erasing which-path information of the single-photon on exit from the Mach-Zehnder interferometer. We show the macroscopic fields of the output superposition are distinguishable states. This scheme generates macroscopic entangled state that violates Bell's inequality. Moreover, the detailed analysis concerning change of amplitudes of entangled superposition by means of repeating this process many times is accomplished. We show our scheme works without photon number resolving detection and it is robust to detector inefficiency.
Nonlinear interferometers are Mach-Zehnder interferometers with Kerr media in either one or both arms. We refer to these devices, respectively, as the asymmetric and symmetric nonlinear interferometers. In the asymmetric case, with one input mode in the vacuum, it is possible to generate maximally entangled photonic states or superpositions of such states. We consider the device as a resource of entangled states for applications to Heisenberg-limited interferometry. Interferometry with the maximally entangled states cannot be performed by simply subtracting the output photocounts as in standard interferometry. Instead, one must perform parity measurements on only one of the output beams. We show that the symmetric nonlinear interferometer, with one input mode in the vacuum, may be used to perform such parity measurements. The same device is shown to produce, with an input coherent state and upon projective measurements, even or odd coherent states, examples of the Schr\"odinger-cat states.
The nonlinear χ (2) Mach-Zehnder interferometer is proposed as a device for conditional generation of a modified coherent nonclassical state. We show that the generated macroscopic state exhibits nonclassical effects, such as squeezing, photon antibunching, and sub-Poissonian statistics. The modified coherent state generates a macroscopic entangled state. The scheme works without the photon number resolving detection but requires high-efficiency photodetectors. We explain the mechanism of generation of the modified coherent non-classical state.
A method for producing macroscopic quantum superposition states (generally known as Schr\"odinger cat) states for optical fields is presented. The proposed method involves two modes of the field interacting dispersively in a Kerr medium where one of the modes is an arm of a Mach-Zehnder interferometer and the other mode is external to it. If the external mode initially contains a macroscopic quantum state, such as a coherent state, and the vacuum and a single photon state are the inputs to the interferometer, the external field state becomes entangled with the number states associated with the two paths of the interferometer. Selective measurement at the output ports of the interferometer project the external mode into the desired cat states. It is pointed out that the method can also be used to generate cat states out of multimode states initially containing correlations.
We demonstrate several results for the nonlinear interferometer, that emerge from a formalism which elegantly describes the output field of the nonlinear interferometer as two-mode entangled coherent states. We clarify the relationship between squeezing and entangled coherent states, since a weak nonlinear evolution produces a squeezed output, while a strong nonlinear evolution produces a two-mode, two-state entangled coherent state. In between these two extremes exist superpositions of two-mode coherent states manifesting varying degrees of entanglement for arbitrary values of the nonlinearity. The cardinality of the basis set of the entangled coherent states is finite when the ratio \ensuremath{\chi}/\ensuremath{\pi} is rational, where \ensuremath{\chi} is the nonlinear strength. We also show that entangled coherent states can be produced from product coherent states via a nonlinear medium without the need for the interferometric configuration. This provides an important experimental simplification in the process of creating entangled coherent states.
We describe a nonlocal method for generating entangled coherent states of a two-mode field wherein the field modes never meet. The proposed method is an extension of an earlier proposal [C. C. Gerry, Phys. Rev. A 59, 4095 (1999)] for the generation of superpositions of coherent states. A single photon injected into a Mach-Zehnder interferometer with cross-Kerr media in both arms coupling with two external fields in coherent states produces entangled coherent states upon detection at one of the output ports. We point out that our proposal can be alternatively viewed as a ``which path'' experiment, and in the case of only one external field, we describe the implementation of a quantum eraser.