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arXiv:1906.10867

Conclusive nonlinear phase sensitivity limit for a Mach-Zehnder interferometer with single-mode non-vacuum inputs

Jiandong Zhang, Zijing Zhang, Hu, Jun-Yan, Longzhu Cen, Yifei Sun, Chen-Fei Jin, Yuan Zhe Zhao

Published Jun 26, 2019• 1 citation
arXivHTMLPDF

Abstract

Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of $1/N$ with $N$ being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.

DOI
10.48550/arxiv.1906.10867

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Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of 1/N1/N1/N with NNN being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
We theoretically study the phase estimation based on a Mach-Zehnder interferometer (MZI) with a two-mode squeezed coherent state. By maximizing the quantum Fisher information, we find that the quantum Cramér-Rao bounds (QCRB) can reach sub-Heisenberg limit under the phase-matched condition. The optimal phase sensitivity can reach the sub-shot noise limit (SNL) and approach the QCRB by employing the intensity difference detection. Meanwhile, compared with the MZI fed with a coherent plus a single-mode squeezed vacuum state, this scheme can have better performance by adjusting the squeezing parameter and the mean photon number. With the same parameter, our scheme shows more sensitive phase measurement than the SU(1,1) interferometer with a coherent plus a vacuum state. We also show that the phase sensitivity of our proposal can still reach the SNL when the loss of the photon is 36%. This scheme can provide potential applications in optical sensors.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.
Phase-shifting interferometry (PSI) promises excellent performance for quantitative phase measurement. However, the PSI is commonly realized by utilizing the Mach-Zehnder interferometer (MZI) fed with classical resources, and hence its phase sensitivity is hindered by the photon shot noise. In this paper, we propose to introduce specific phase-shifting algorithms into the well-established quantum-enhanced MZI, seeded by coherent and squeezed vacuum states and followed by single-mode intensity detection and difference intensity detection, to achieve quantitative phase measurement with sensitivity beyond that achieved by the corresponding Mach-Zehnder-type PSI with classical input. We show that when the average photon number of the coherent state is dominant, the optimal phase sensitivity of the Mach-Zehnder-type PSI with coherent plus squeezed vacuum input is better than the phase sensitivity of the Mach-Zehnder-type PSI with single coherent input under the same phase-sensing intensity condition in both of detection setups. We also show that for the typical MZI seeded with a high-power coherent state and a finitely squeezed vacuum state, the combination of intensity detection and phase-shifting algorithm can still improve the sensitivity of quantitative phase measurement. Furthermore, we derive the phase sensitivity of the fixed MZI as well as the quantum Cramér-Rao bound for two detection schemes. Finally, we explore the impacts of photon losses on the phase sensitivity enhancements of the Mach-Zehnder-type PSI and the fixed MZI in the high-intensity regime for each detection setup. The robustness to losses is compared between the Mach-Zehnder-type PSI and the fixed MZI. Our results may be helpful to promote the exploration of quantum-enhanced quantitative phase sensing.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
We investigate the phase sensitivity of the SU(1,1) interfereometer [SU(1,1)I] and the modified Mach–Zehnder interferometer (MMZI) with the entangled coherent states (ECS) as inputs. We consider the ideal case and the situations in which the photon losses are taken into account. We find that, under ideal conditions, the phase sensitivity of both the MMZI and the SU(1,1)I can beat the shot-noise limit (SNL) and approach the Heisenberg limit (HL). In the presence of photon losses, the ECS can beat the coherent and squeezed states as inputs in the SU(1,1)I, and the MMZI is more robust against internal photon losses than the SU(1,1)I.