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arXiv:2308.03002

The effect of Quantum Statistics on the sensitivity in an SU(1,1) interferometer

Zeng, Jie, Yingxing Ding, Mengyao Zhou, Gao-Feng Jiao, Keye Zhang, L. Q. Chen, Weiping Zhang, Chun-Hua Yuan

Published Aug 6, 2023• 1 citation
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Abstract

We theoretically study the effect of quantum statistics of the light field on the quantum enhancement of parameter estimation based on cat state input the SU(1,1) interferometer. The phase sensitivity is dependent on the relative phase $θ$ between two coherent states of Schrödinger cat states. The optimal sensitivity is achieved when the relative phase is $π$% , i.e., odd coherent states input. For a coherent state input into one port, the phase sensitivity of the odd coherent state into the second input port is inferior to that of the squeezed vacuum state input. However, in the presence of losses the Schrödinger cat states are more resistant to loss than squeezed vacuum states. As the amplitude of Schrödinger cat states increases, the quantum enhancement of phase sensitivity decreases, which shows that the quantum statistics of Schrödinger cat states tends towards Poisson statistics from sub-Poisson statistics or super-Poisson statistics.

DOI
10.48550/arxiv.2308.03002

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We theoretically study the effect of quantum statistics of the light field on the quantum enhancement of parameter estimation based on cat state input the SU(1,1) interferometer. The phase sensitivity is dependent on the relative phase θθθ between two coherent states of Schrödinger cat states. The optimal sensitivity is achieved when the relative phase is πππ% , i.e., odd coherent states input. For a coherent state input into one port, the phase sensitivity of the odd coherent state into the second input port is inferior to that of the squeezed vacuum state input. However, in the presence of losses the Schrödinger cat states are more resistant to loss than squeezed vacuum states. As the amplitude of Schrödinger cat states increases, the quantum enhancement of phase sensitivity decreases, which shows that the quantum statistics of Schrödinger cat states tends towards Poisson statistics from sub-Poisson statistics or super-Poisson statistics.
Abstract We derive a general phase-matching condition (PMC) for enhancement of sensitivity in SU(1,1) interferometers. Under this condition, the quantum Fisher information (QFI) of two-mode SU(1,1) interferometry becomes maximal with respect to the relative phase of two modes, for the case of an arbitrary state in one input port and an even (odd) state in the other port, and the phase sensitivity is enhanced. We also find that optimal parameters can let the QFI in some areas achieve the Heisenberg limit for both pure and mixed initial states. As examples, we consider several input states: coherent and even coherent states, squeezed vacuum and even coherent states, squeezed thermal and even coherent states. Furthermore, in the realistic scenario of the photon loss channel, we investigate the effect of photon losses on QFI with numerical studies. We find the PMC remains unchanged and is not affected by the transmission coefficients for the above input states. Our results suggest that the PMC can exist in various kinds of interferometers and the phase-matching is robust to even strong photon losses.
We study the phase sensitivity of an SU(1,1) interferometer with two input beams in the displaced squeezed vacuum state and the coherent state, respectively. We find that there exists an optimal squeezing fraction of the displaced squeezed vacuum state that optimizes the phase sensitivity. We also examine the effects of some factors, including the loss, mean photon number of the input beams and amplitude gain of the optical parameter amplifiers, on the optimal squeezing fraction so that we can choose the optimal values to enhance the phase sensitivity.
We theoretically derive the lower and upper bounds of quantum Fisher information (QFI) of an SU(1,1) interferometer whatever the input state chosen. According to the QFI, the crucial resource for quantum enhancement is shown to be large intramode correlations indicated by the Mandel QQQ parameter. The subtraction of photons from the squeezed vacuum state has the effect of increasing the average photon number of the new field state, as well as the intramode correlations. For example, for a coherent state $\ensuremath{\bigotimes}$ a squeezed vacuum state with a given fixed input mean number of photons as the input, if ppp photons are subtracted from the squeezed-vacuum state before inputting the SU(1,1) interferometer, the phase sensitivities can be improved due to the intramode-correlation increment.
We theoretically study the phase sensitivity of an SU(1,1) interferometer with a coherent state in one input port and a squeezed-vacuum state in the other input port using the method of homodyne detection. In this interferometer, beam splitting and recombination are generated by the parametric amplifiers instead of the beam splitters. Compared with the traditional Mach–Zehnder interferometer, the phase sensitivity of this interferometer can be improved due to the amplification process of the parametric amplifiers. Combined with the squeezed state input, the sensitivity can be improved further due to the noise reduction. The phase sensitivity of our scheme can approach the Heisenberg limit and the associated optimal condition is analyzed. The scheme can be implemented with current experimental technology.
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.
We study the improvement in the phase sensitivity of a Mach–Zehnder interferometer using the superposition of Schrödinger’s cat-like state with the Fock state (SCFS) and the coherent state as inputs. With this setup, we examine the effect on phase sensitivity of the interferometer using a two-channel detection (TCD) scheme [ Opt. Express 29 , 95 ( 2021 ) OPEXFF 1094-4087 10.1364/OE.413391 ] in which we take the optimal combination of the intensities of both output ports. We find better phase sensitivity under some conditions for this setup as compared to other known combinations of inputs such as squeezed vacuum and coherent states, vacuum and coherent states, etc. Therefore, we expect that the SCFS may be an alternative nonclassical resource for improvement in the phase sensitivity of a Mach–Zehnder interferometer, having potential application in quantum sensing.