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Acta Physica Sinica

Wigner function of N00N state and quantum interference with N00N state as input

Xue-xiang Xu, Zhang Ying-Kong, Haoliang Zhang, Chen Yuan-Yuan

Published Jan 1, 2013• 1 citation
Publisher

Abstract

Using the formula of Wigner function in coherent representation, we have obtained the analytical expression for Wigner function of N00N state. Based on phase space method, we study the quantum interference with N00N state as input. We derive the analytical expression of conditional probability related with the input parameter N and phase parameter φ and analyze it numerically. It is shown that, when φ is 0 or π, the output is just N00N state. It is also shown that, for 2002 state as input, the output must be 2002 state, which is independent of phase parameters. Moreover, as the number of input photon N increases, the phase probability distributions remain to have one, two, three and four peaks and get narrower. All these results can offer theoretical reference for experiments.

DOI
10.7498/aps.62.114204

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Using the formula of Wigner function in coherent representation, we have obtained the analytical expression for Wigner function of N00N state. Based on phase space method, we study the quantum interference with N00N state as input. We derive the analytical expression of conditional probability related with the input parameter N and phase parameter φ and analyze it numerically. It is shown that, when φ is 0 or π, the output is just N00N state. It is also shown that, for 2002 state as input, the output must be 2002 state, which is independent of phase parameters. Moreover, as the number of input photon N increases, the phase probability distributions remain to have one, two, three and four peaks and get narrower. All these results can offer theoretical reference for experiments.
Using the phase space method, we study the quantum interference through a Mach–Zehnder interferometer with an arbitrary-photon Fock state and a coherent state as two inputs. The statistical properties, in terms of the Wigner function, average photon number, photon number distribution and parity, are derived analytically for the field of the individual output port. These results indicate that the output state is actually a statistical mixture between a bunch of Fock state and coherent states. In addition, the phase sensitivity is also examined by using the detection scheme of intensity difference measurement.
A quantum-statistical description of noise in interferometers is given in terms of the Wigner distribution function. The interferometer may contain an amplifier in one of its arms. The input field can be in a variety of states such as a Fock state, a coherent state or a squeezed coherent state. The Wigner function of the output field at the detector is shown to have a general Gaussian form with non-zero complex field amplitude and with other parameters related to the characteristics of the input field, amplifier and beam splitters, etc. An explicit form of the photon-number distribution is given. Higher-order correlations of the output field can be obtained from the Gaussian property of the Wigner function.
Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states—called Wigner entropy for brevity—emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the p -norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter α ≥ 1 / 2 .
We discuss a phase-space description of the photon number distribution of non-classical states which is based on Husimi's Q (α) function and does not rely on the WKB approximation. We illustrate this approach using the examples of displaced number states and two photon coherent states and show it to provide an efficient method for computing and interpreting the photon number distribution. This result is interesting in particular for the two photon coherent states which, for high squeezing, have the probabilities of even and odd photon numbers oscillating independently.
Based on the correspondence between Collins diffraction formula (optical Fresnel transform) and the transformation matrix element of a three-parameters two-mode squeezing operator in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip both in spacial domain and frequency domain. The entangled Wigner function is introduced for recapitulating the result.