SU(1,1) interferometry is a technique that uses parametric amplification for splitting and mixing of electromagnetic waves for precise estimation of phase change and achieves the Heisenberg limit of sensitivity with fewer optical elements than conventional interferometric techniques.
Introduction Interferometry is an important technique in the field of optics that have been utilised for fundamental proof of principles experiments and in the development of new technologies. This technique, primarily based on the interference of electromagnetic waves, has been widely explored in the field of quantum metrology and precision measurements for achieving sensitivity in measurements beyond what is possible with classical methods and resources. Interferometry is a desired platform for precise estimation of physical quantities because of its ability to sense small phase changes. One of the most prominent examples of the application of this property is the detection of gravitational waves (LIGO).
Conventional interferometers are based on the wave nature of the light and hence the classical interference of electromagnetic waves. Although the design and layout for these types of interferometers can vary depending upon the type of application and corresponding suitable scheme, they all can be mapped to an arrangement similar to that of a Mach-Zehnder interferometer. In this type of interferometry, the input field is split into two by a beam splitter which then propagates along different paths and acquires a relative phase difference (corresponding to a path length difference). Considering one of the beams undergoing a phase change as the probe and the other beam as the reference, the relative phase is estimated after the two beams interfere at another beam splitter. The estimation of the phase difference is done through the detection of the intensity change at the output after the interference at the second beam splitter. These standard interferometric techniques, based on beam-splitters for the splitting of the beams and linear optical transformations, can be classified as SU(2) interferometers as these interferometric techniques can be naturally characterized by SU(2) (Special Unitary(2)) group. Theoretically, the sensitivity of conventional SU(2) interferometric schemes are limited by the vacuum fluctuation noise, also called the shot-noise limit which scales as 1 / N {\displaystyle 1/{\sqrt {N}}} , where N {\displaystyle N} is the mean number of particles (photons for electromagnetic waves) entering the input port of the interferometer. The shot noise limit can be overcome by using light that utilizes quantum properties such as quantum entanglement (e.g. squeezed states, NOON states), at the unused input port. In principle, this can achieve the Heisenberg limit of sensitivity which scales as 1 / N {\displaystyle 1/N} with the change in the mean number of photons entering the input port. SU(1,1) interferometers were first proposed by Yurke et al. in which the beam splitters in conventional interferometers were replaced by optical parametric amplifiers. The advantage that comes with the parametric amplifiers is that the input fields can be coherently split and interfere that would be fundamentally quantum in nature. This is attributed to the nonlinear processes in parametric amplifiers such as four-wave mixing. Theoretically, SU(1,1) interferometers can achieve the Heisenberg limit of sensitivity with fewer optical elements than conventional interferometers.
Theory
To briefly understand the benefit of using a parametric amplifier, a balanced SU(1,1) interferometer can be considered. Treating the input fields as quantum fields described by operators a 1 i n {\displaystyle a_{1}^{in}} , a 2 i n {\displaystyle a_{2}^{in}} , the output quantum fields from a parametric amplifier can be written as:
a 1 o = G a 1 i n + g a 2 i n {\displaystyle a_{1}^{o}=Ga_{1}^{in}+ga_{2}^{in}}
a 2 o = g a 1 i n + G a 2 i n {\displaystyle a_{2}^{o}=ga_{1}^{in}+Ga_{2}^{in}}
where G {\displaystyle G} is the amplitude gain and | G | 2 − | g | 2 = 1 {\displaystyle \left\vert G\right\vert ^{2}-\left\vert g\right\vert ^{2}=1} . For a coherent state input | α ⟩ {\displaystyle |\alpha \rangle } at the first parametric amplifier with initial input intensity I 0 = | α | 2 {\displaystyle I_{0}=\left\vert \alpha \right\vert ^{2}} , the output intensities will be:
… excerpt ends here. Continue reading the full article.
