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arXiv:1006.1645 · Physical Review A

Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state

Sergey Knysh, Vadim Smelyanskiy, Gabriel A. Durkin

Published Feb 25, 2011• 218 citations
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Abstract

Phase precision in optimal two-channel quantum interferometry is studied in the limit of large photon number $N\ensuremath{\gg}1$, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the number of photons (or losses) increase. The optimal state has a continuous form in the Fock state basis for large $N$. The loss parameter limits any precision improvement over classical light to at most a constant factor independent of $N$. We determine a crossover value of photon number ${N}_{c}$ beyond which supraclassical precision is progressively lost.

DOI
10.1103/physreva.83.021804

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Phase precision in optimal two-channel quantum interferometry is studied in the limit of large photon number N\ensuremath≫1N\ensuremath{\gg}1N\ensuremath≫1, for losses occurring in either one or both channels. For losses in one channel an optimal state undergoes an intriguing sequence of local bifurcations as the number of photons (or losses) increase. The optimal state has a continuous form in the Fock state basis for large NNN. The loss parameter limits any precision improvement over classical light to at most a constant factor independent of NNN. We determine a crossover value of photon number Nc{N}_{c}Nc​ beyond which supraclassical precision is progressively lost.
By using a systematic optimization approach, we determine quantum states of light with definite photon number leading to the best possible precision in optical two-mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit, thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
We analyse the phase space representation of the optimal measurement of a phase shift in an interferometer with equal photon loss in both its arms. In the local phase estimation scenario with a fixed number of photons, we identify features of the spin Wigner function that warrant sub-shot-noise precision, and discuss their sensitivity to losses. We derive the asymptotic form of an integral kernel describing the process of photon loss in the phase space in the limit of large photon numbers. The analytic form of this kernel allows one to assess the ultimate precision limit for a lossy interferometer. We also provide a general lower bound on the quantum Fisher information in terms of spin Wigner functions.
Interferometry with quantum light is known to provide enhanced precision for estimating a single phase. However, depending on the parameters involved, the quantum limit for the simultaneous estimation of multiple parameters may not be attainable, leading to tradeoffs in the attainable precisions. Here we study the simultaneous estimation of two parameters related to optical interferometry: phase and loss, using a fixed number of photons. We derive a tradeoff in the estimation of these two parameters which shows that, in contrast to single-parameter estimation, it is impossible to design a strategy saturating the quantum Cram\'er-Rao bound for loss and phase estimation in a single setup simultaneously. We design optimal quantum states with a fixed number of photons achieving the best possible simultaneous precisions. Our results reveal general features about concurrently estimating Hamiltonian and dissipative parameters and have implications for sophisticated sensing scenarios such as quantum imaging.
The presence of loss limits the precision of an approach to phase measurement using maximally entangled states, also referred to as NOON states. A calculation using a simple beam-splitter model of loss shows that, for all nonzero values LLL of the loss, phase measurement precision degrades with increasing number NNN of entangled photons for NNN sufficiently large. For LLL above a critical value of approximately 0.785, phase measurement precision degrades with increasing NNN for all values of NNN. For LLL near zero, phase measurement precision improves with increasing NNN down to a limiting precision of approximately 1.018Lradians1.018L\phantom{\rule{0.3em}{0ex}}\text{radians}1.018Lradians, attained at NNN approximately equal to 2.218∕L2.218∕L2.218∕L, and degrades as NNN increases beyond this value. Phase measurement precision with multiple measurements and a fixed total number of photons NT{N}_{T}NT​ is also examined. For LLL above a critical value of approximately 0.586, the ratio of phase measurement precision attainable with NOON states to that attainable by conventional methods using unentangled coherent states degrades with increasing NNN, the number of entangled photons employed in a single measurement, for all values of NNN. For LLL near zero this ratio is optimized by using approximately N=1.279∕LN=1.279∕LN=1.279∕L entangled photons in each measurement, yielding a precision of approximately 1.340L∕NTradians1.340\sqrt{L∕{N}_{T}}\phantom{\rule{0.3em}{0ex}}\text{radians}1.340L∕NT​​radians.
In quantum mechanics, the precision achieved in parameter estimation using a quantum state as a probe is determined by the measurement strategy employed. The quantum limit of precision is bounded by a value set by the state and its dynamics. Theoretical results have revealed that in interference measurements with two possible outcomes, this limit can be reached under ideal conditions of perfect visibility and zero losses. However, in practice, these conditions cannot be achieved, so precision never reaches the quantum limit. But how do experimental setups approach precision limits under realistic circumstances? In this Letter, we provide a model for precision limits in two-photon Hong-Ou-Mandel interferometry using coincidence statistics for nonperfect visibility and temporally unresolved measurements. We show that the scaling of precision with visibility depends on the effective area in time-frequency phase space occupied by the state used as a probe, and we find that an optimal scaling exists. We demonstrate our results experimentally for different states in a setup where the visibility can be controlled and reaches up to 99.5%. In the optimal scenario, a ratio of 0.97 is observed between the experimental precision and the quantum limit, establishing a new benchmark in the field.