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Journal of the Optical Society of America B

Photon statistics and entanglement in coherent-squeezed linear Mach-Zehnder and Michelson interferometers

Ronen Barak, Y. Ben-Aryeh

Published Feb 22, 2008• 10 citations
Publisher

Abstract

Photon statistics effects are analyzed for Mach-Zehnder and Michelson interferometers in which a strong coherent state is inserted into one input port and a squeezed vacuum is inserted in the other input port. We study the effects related to entanglement, for the general case, by using Zassenhaus's perturbation theory as a function of the squeezing parameter. For the special case in which the strong coherent state exits fully through one output port while the squeezed vacuum exits through the other “dark” output port, the Lie group disentangling method is used. For this case we show that the signal can be amplified with sub-Poissonian photon statistics.

DOI
10.1364/josab.25.000361

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Photon statistics effects are analyzed for Mach-Zehnder and Michelson interferometers in which a strong coherent state is inserted into one input port and a squeezed vacuum is inserted in the other input port. We study the effects related to entanglement, for the general case, by using Zassenhaus's perturbation theory as a function of the squeezing parameter. For the special case in which the strong coherent state exits fully through one output port while the squeezed vacuum exits through the other “dark” output port, the Lie group disentangling method is used. For this case we show that the signal can be amplified with sub-Poissonian photon statistics.
We study the entanglement between the two beams exiting a Mach-Zehnder interferometer fed by a couple of squeezed-coherent states with arbitrary squeezing parameter. The quantum correlations at the output are functions of the internal phase shift of the interferometer, with the output state ranging from a totally disentangled state to a state whose degree of entanglement is an increasing function of the input squeezing parameter. A couple of squeezed vacuums at the input lead to maximum entangled state at the output. The fringe visibilities resulting from measuring the coincidence counting rate or the squared difference photocurrent are evaluated and compared to each other. Homodynelike detection turns out to be preferable in almost all situations, with the exception of the very-low-signal regime.
Abstract We present a theoretical study of the interferences between hybrid single-photon state, coherent state, and vacuum state in the designed medium of a six-ports Mach-Zehnder Interferometer (6p-MZI). The hybrid three different quantum states were simultaneously injected into the input ports of the interferometer. The derivation result demonstrated that the output state showed the statistical blend states among a substituted single-photon state, coherent state, and vacuum state. Interestingly, our calculation results attested that the maximum probability of detected photons at the individual interferometer output ports were significantly dependents on the amplitude of the inputted coherent state.
We study a scheme for Mach–Zehnder (MZ) interferometer as a quantum linear device by injecting two-mode squeezed input states into two ports of interferometer. Two-mode squeezed states can be changed into two types of inputs for MZ interferometer: two squeezed states and Einstein–Podolsky–Rosen (EPR) entangled states. The interference patterns of the MZ interferometer vary periodically as the relative phase of the two arms of the interferometer is scanned, and are measured by the balanced homodyne detection system. Our experiments show that there are different interference patterns and periodicity of the output quantum states for two cases which depend on the relative phase of input optical fields. Since MZ interferometer can be used to realize some quantum operations, this work will have the important applications in quantum information and metrology.
Photon counting measurements are analyzed for obtaining a classical phase parameter in a linear Mach–Zehnder interferometer (MZI) by the use of phase estimation theories. The detailed analysis is made for four cases: (a) coherent states inserted into the interferometer, (b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port, (c) coherent state inserted into one input port of the interferometer and squeezed-vacuum state into the other input port, and (d) exchanging the first beam splitter of an MZI by a nonlinear system that inserts a NOON (representing a superposition of N particles in the first mode with zero particles in the second mode, and vice versa) state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.
We study the photon counting noise in optical interferometers used for gravitational wave detection. In order to reduce quantum noise a squeezed vacuum state is injected into the usually unused input port. Here, we specifically investigate the so-called “dark port case,” when the beam splitter is oriented close to 90° to the incoming laser beam, such that nearly all photons go to one output port of the interferometer, and only a small fraction of photons is seen in the other port (“dark port”). For this case it had been suggested that signal amplification is possible without concurrent noise amplification [R. Barak and Y. Ben-Aryeh, J. Opt. Soc. Am. B 25 , 361 (2008)]. We show that by injection of a squeezed vacuum state into the second input port, counting noise is reduced for large values of the squeezing factor, however the signal is not amplified. Signal strength only depends on the intensity of the laser beam.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.