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arXiv:quant-ph/9608021 · Physical Review A

Nonclassical interferometry with intelligent light

Constantin Brif, A. Mann

Published Nov 1, 1996• 97 citations

Abstract

We study the phase sensitivity of SU(2) and SU(1,1) interferometers fed by two-mode field states which are intelligent states for Hermitian generators of the SU(2) and SU(1,1) groups, respectively. Intelligent states minimize uncertainty relations and this makes possible an essential reduction of the quantum noise in interferometers. Exact closed expressions for the minimum detectable phase shift are obtained in terms of the Jacobi polynomials. These expressions are compared with results for some conventional input states, and some known results for the squeezed input states are reviewed. It is shown that the phase sensitivity for an interferometer that employs squeezing-producing active devices (such as four-wave mixers) should be analyzed in two regimes: (i) fixed input state and variable interferometer, and (ii) fixed interferometer and variable input state. The behavior of the phase sensitivity is essentially different in these two regimes. The use of the SU(2) intelligent states allows us to achieve a phase sensitivity of order 1/N\ifmmode\bar\else\textasciimacron\fi{} (where N\ifmmode\bar\else\textasciimacron\fi{} is the total number of photons passing through the phase shifters of the interferometer) without adding four-wave mixers. This avoids the duality in the behavior of the phase sensitivity that occurs for the squeezed input. On the other hand, the SU(1,1) intelligent states have the property of achieving the phase sensitivity of order 1/N\ifmmode\bar\else\textasciimacron\fi{} in both regimes. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.4505

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Introducing squeezed states of light into interferometers can increase the phase sensitivity of the device beyond the standard quantum limit (SQL). We discuss an SU(1,1) interferometer, where nonlinear optical elements replace the beam splitters in a Mach-Zehnder interferometer. A two-mode squeezed state of light is generated inside of such an interferometer. We talk about the phase sensitivities of an SU(1,1) interferometer with different detection schemes and their improvement over the SQL. We also discuss the concept of an optimal detection scheme for phase measurement in an interferometer. We describe a modification of the SU(1,1) interferometer which reduces the experimental complexities while giving the same phase sensitivity. We call the design a truncated SU(1,1) interferometer. We show our experimental results of 4 dB improvement in phase sensitivity over the SQL using a truncated SU(1,1) interferometer. We also compare the theoretical sensitivities of vacuum-seeded configurations of a conventional and a truncated SU(1,1) interferometer, and show our experimental results for the phase sensitivity of the truncated version. We explain the dependence of phase sensitivity on the measurement of squeezing. We talk about the methods to improve the measurement of squeezing in a 4-wave mixing experiment, and our efforts in implementing them. Finally, we mention our progress in measuring a big phase shift using an adaptive algorithm in the truncated SU(1,1) interferometer, and discuss the technicalities involved in big phase measurements.

A Lie-group-theoretical approach to the analysis of interferometers is presented. Conventional interferometers such as the Mach-Zehnder and Fabry-Perot can be characterized by SU(2). We introduce a class of interferometers characterized by SU(1,1). These interferometers employ active elements such as four-wave mixers or degenerate-parametric amplifiers in their construction. Both the SU(2) and SU(1,1) interferometers can in principle achieve a phase sensitivity \ensuremath{\Delta}\ensuremath{\varphi} approaching 1/N, where N is the total number of quanta entering the interferometer, provided that the light entering the input ports is prepared in a suitable quantum state. SU(1,1) interferometers can achieve this sensitivity with fewer optical elements.

To achieve the optimal phase sensitivity in two-mode interferometers, one key problem is how to distribute photons between the two input ports, given a fixed total average photon number. In both the

\text{SU}(2)

and

\text{SU}(1,1)

interferometers with a product of coherent and Fock states as inputs, we find there exists a critical value of the total photon number, below which the optimal phase sensitivity can be achieved with the second port being the vacuum state. The critical points can be analytically fixed. We further consider a more general input state as a product of coherent and squeezed Fock states. Then we analytically identify three regions in the parameter space, similar to a phase diagram, and find that critical phenomena occur in two regions. If the total average input photon number is less than the critical value, the optimal distribution tends to minimize the average photon number in the second port. Our finding indicates that, in order to get interferometers with high sensitivity, one must carefully choose the photon number distribution according to the value of the input total average photon number. This central result will be useful in theoretical studies in the field of quantum parameter estimation.

We theoretically investigate an interferometer composed of four four-wave-mixers by Lie group method. Lie group SU(1,2) characterizes the mode transformations of this kind of interferometer. With vacuum state inputs, the phase sensitivity of SU(1,2) interferometer achieves the Heisenberg limit, and the absolute accuracy beats SU(1,1) interferometer because of higher intensity of light inside the interferometer. For different input cases, the optimal combination of output photon number for detection to obtain the best phase sensitivity is calculated. Our research on SU(1,2) interferometer sheds light on the performance of SU(1,n) interferometer in quantum metrology.

We study the phase sensitivity in the conventional SU(2) and nonconventional SU(1,1) interferometers with a coherent and squeezed vacuum input state via quantum Cramer-Rao bound. We explicitly construct the detection scheme that gives the optimal phase sensitivity. For practical purposes, we show that in the presence of photon loss, both interferometers with proper homodyne detections are nearly optimal. We also find that unlike the coherent state and the squeezed vacuum state, the effects of imperfect detector on the phase sensitivity cannot be asymptotically removed for a generic coherent-squeezed state.