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Group theoretical approach to the analysis of interferometers

Bernard Yurke, S. L. McCall, John R. Klauder

Published Jun 9, 1986• 0 citations
OpenAlex

Abstract

A geometric approach to the analysis of interferometers is presented in which beam splitters and phase shifters are described as performing rotations in an abstract three space on the state vector of the light. This formulation allows one to take full advantage of representation theory for the group SU(2) in analyzing interferometer sensitivity. With these techniques it is easy to show that in conventional interferometry, where light enters only one input port, the phase sensitivity scales as the inverse square root of the number of photons that pass through the interferometer during measurement time.

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A geometric approach to the analysis of interferometers is presented in which beam splitters and phase shifters are described as performing rotations in an abstract three space on the state vector of the light. This formulation allows one to take full advantage of representation theory for the group SU(2) in analyzing interferometer sensitivity. With these techniques it is easy to show that in conventional interferometry, where light enters only one input port, the phase sensitivity scales as the inverse square root of the number of photons that pass through the interferometer during measurement time.
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.
Interferences in multi-path systems for single and multiple particles are theoretically analyzed. A holistic method is presented, which allows to construct the unitary transition matrix describing interferometers for any port number d and particle number n. This work has two centerpieces. First, a solution algorithm for realizing any desired unitary transformation in an interferometer for the single particle case is demonstrated. Secondly, a simple scheme for calculating the rotation matrices of beam splitters and phase shifters for multiple particles is shown. Such linear optical devices are especially important for quantum circuits and quantum information processing.
In a conventional interferometer one shines light only into one interferometer input port while the second input port is unexcited. This allows vacuum states to enter the unexcited port, leading to imperfect correlation between both the amplitude and the phase fluctuations of the light in the upper and lower interferometer arms. This, in turn, limits the interferometer sensitivity regardless of its operating panel (its arm-length difference, which we in the following will translate into a phase shift θ). The sensitivity of an ideal conventional interferometer is 1 / N. That is, a phase shift of 1 / N radians can be detected with a signal-to-noise ratio of unity by using N photons. Since N typically is a large number, it is tempting to try to find some scheme approaching the fundamental (Heisenberg) limit of phase-shift detection, 1/N. One way of doing that is to inject squeezed vacuum into the unexcited input port of the interferometer.1
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 SU(2)\text{SU}(2)SU(2) and SU(1,1)\text{SU}(1,1)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.
As was originally pointed out by Caves, squeezed light can be used to increase the accuracy of interferometric phase measurements. This analysis was carried further by Yurke, McCall, and Klauder, who used group theory to examine the ultimate sensitivity of conventional interferometers and also those constructed from four-wave mixers. Here we consider interferometers composed of mixtures of beam splitters, four-wave mixers, and degenerate parametric amplifiers. We show that each interferometer can be characterized by an uncertainty relation whose minimum uncertainty states can be used to explore its ultimate accuracy. The variables in the uncertainty relation are quadratic in the mode operators, which allows us to connect the study of improved interferometric measurements to recent work on higher-order squeezing.
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.