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International Conference on Quantum Information

Effect of Dispersion on Fidelity of a Quantum Interferometer

Thomas B. Bahder, Alexander V. Sergienko, David S. Simon

Published Jan 1, 2008• 0 citations
Publisher

Abstract

We analyze the effect of frequency dispersion on the Shannon mutual information of a Mach-Zehnder interferometer by comparing input states consisting of monochromatic photons and photons having a bandwidth that are produced by parametric downconversion.
DOI
10.1364/icqi.2008.jmb57

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We consider Mach-Zehnder and Hong-Ou-Mandel interferometers with nonclassical states of light as input, and study the effect that dispersion inside the interferometer has on the sensitivity of phase measurements. We study in detail a number of different one- and two-photon input states, including Fock, dual Fock, maximally path-entangled $\ensuremath{\mid}N,0⟩+⟨{0}_{,}N\ensuremath{\mid}$ (``N00N'') states, and photon pairs from parametric down-conversion. Assuming there is a phase shift ${\ensuremath{\phi}}_{0}$ in one arm of the interferometer, we compute the probabilities of measurement outcomes as a function of ${\ensuremath{\phi}}_{0}$, and then compute the Shannon mutual information between ${\ensuremath{\phi}}_{0}$ and the measurements. This provides a means of quantitatively comparing the utility of various input states for determining the phase in the presence of dispersion. In addition, we consider a simplified model of parametric down-conversion for which probabilities can be explicitly computed analytically, and which serves as a limiting case of the more realistic down-conversion model.
For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome mmm will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes mmm. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.
A pair of light quanta with different colors (155.9-nm difference in center wavelength) generated from parametric down-conversion was injected collinearly into one input port of a Mach-Zehnder interferometer. Coincidence interference behavior was studied over a wide range of optical path differences of the interferometer. A measurement of 75% interference visibility with oscillation of the pump frequency for a large optical path difference of the interferometer (43 cm) is the signature of a quantum two-photon entangled state, which reflects both particle and wave nature of the light quanta in one experiment.
In a Franson interferometer two-photon interference fringes occur despite the photons never encountering each other [1]. This phenomenon is a fundamental demonstration of the nonlocal nature of quantum mechanics. It serves also as a direct proof of frequency-time-entanglement. Furthermore, it can benefit not only from local, but also nonlocal dispersion cancellation techniques [2], which is a very useful property when working with spectrally broad photon pairs. More specifically, one has to distinguish between dispersion occuring before the individual Mach-Zehnder interferometers (MZIs), which constitute the Franson interferometer, and within them. Analysis of influences up to second order has shown that no loss of contrast can be expected from dispersion before the Franson interferometer. Differential second-order dispersion inside the interferometer, however, can cause a reduction of contrast in the interferogram [3].