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arXiv:0810.4501 · Physical Review A

Dispersion and fidelity in quantum interferometry

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

Published Nov 26, 2008• 7 citations
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Abstract

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.

DOI
10.1103/physreva.78.053829

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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∣N,0⟩+⟨0,N\ensuremath∣\ensuremath{\mid}N,0⟩+⟨{0}_{,}N\ensuremath{\mid}\ensuremath∣N,0⟩+⟨0,​N\ensuremath∣ (``N00N'') states, and photon pairs from parametric down-conversion. Assuming there is a phase shift \ensuremathϕ0{\ensuremath{\phi}}_{0}\ensuremathϕ0​ in one arm of the interferometer, we compute the probabilities of measurement outcomes as a function of \ensuremathϕ0{\ensuremath{\phi}}_{0}\ensuremathϕ0​, and then compute the Shannon mutual information between \ensuremathϕ0{\ensuremath{\phi}}_{0}\ensuremathϕ0​ 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.
When a photon interferes with itself while traversing a Mach-Zehnder interferometer, the output port where it emerges is influenced by the phase difference between the interferometer arms. This allows for highly precise estimation of the path length difference (delay) but is extremely sensitive to phase noise. By contrast, a delay between the arms of the two-photon Hong-Ou-Mandel interferometer directly affects the relative indistinguishability of the photon pair, affecting the rate of recorded coincidences. This likewise allows for delay estimation, notably less precise but with the advantage of being less sensitive to perturbations of the photons' phase. Focusing on two-photon input states, we here investigate to what degree of noise Mach-Zehnder interferometry retains its edge over Hong-Ou-Mandel interferometry. We also explore the competing benefits of different two-photon inputs for a Mach-Zehnder interferometer and under what parameter regimes each input performs best.
We have studied the sensitivity of four different phase shift measurement schemes with a Mach-Zehnder interferometer. The input light is considered to be in a coherent state and the detectors are assumed to be ideal with the quantum efficiency of unity. It is shown by direct calculation of the operators corresponding to the measurement schemes that the uncertainty of the phase-shift measurement is limited to the classical one 1m\frac{1}{\sqrt{m}}m​1​ (m is the average number of the photons in the input state) regardless of the phase-shift measurement schemes.
Determining the phase in one arm of a quantum interferometer is discussed taking into account the three nonideal aspects in real experiments: nondeterministic state preparation, nonunitary state evolution due to losses during state propagation, and imperfect state detection. A general expression is written for the probability of a measurement outcome taking into account these three nonideal aspects. As an example of applying the formalism, the classical Fisher information and fidelity (Shannon mutual information between phase and measurements) are computed for few-photon Fock and N00N states input into a lossy Mach-Zehnder interferometer. These three nonideal aspects lead to qualitative differences in phase estimation, such as a decrease in fidelity and Fisher information that depends on the true value of the phase.