# Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum

- Bryan T Gard
^{1}Email authorView ORCID ID profile, - Chenglong You
^{1}, - Devendra K Mishra
^{1, 2}, - Robinjeet Singh
^{1}, - Hwang Lee
^{1}, - Thomas R Corbitt
^{1}and - Jonathan P Dowling
^{1}

**Received: **6 January 2017

**Accepted: **30 March 2017

**Published: **7 April 2017

## Abstract

The use of an interferometer to perform an ultra-precise parameter estimation under noisy conditions is a challenging task. Here we discuss nearly optimal measurement schemes for a well known, sensitive input state, squeezed vacuum and coherent light. We find that a single mode intensity measurement, while the simplest and able to beat the shot-noise limit, is outperformed by other measurement schemes in the low-power regime. However, at high powers, intensity measurement is only outperformed by a small factor. Specifically, we confirm, that an optimal measurement choice under lossless conditions is the parity measurement. In addition, we also discuss the performance of several other common measurement schemes when considering photon loss, detector efficiency, phase drift, and thermal photon noise. We conclude that, with noise considerations, homodyne remains near optimal in both the low and high power regimes. Surprisingly, some of the remaining investigated measurement schemes, including the previous optimal parity measurement, do not remain even near optimal when noise is introduced.

## Keywords

## PACS Codes

## 1 Introduction

Typical parameter estimation with the use of interferometric schemes aims to estimate some unknown parameter which is probed with the input quantum states of light. In principle, the sensitivity of these measurements depends on the chosen input states of light, the interferometric scheme, the noise encountered and the detection scheme performed at the output. For a real-world example, perhaps the most sensitive of these types of interferometers are the large scale interferometers used as gravitational wave sensors [1–7]. In general, if classical states of light are used, then the most sensitive measurement is limited to a classical bound, the shot-noise limit (SNL) [8–10]. Despite the remarkable precision possible with classical states, improvements are still possible. Here we discuss nearly optimal measurements achievable when one considers input states of coherent and squeezed vacuum [11, 12], under many common noisy conditions and in realistic power regimes which are applicable to general interferometry.

It is of practical interest to consider the difficulty with implementing any particular measurement scheme as every additional optical element introduces further loss into the interferometer. It has been previously shown that the parity measurement is one example of an optimal measurement for coherent and squeezed vacuum input states under lossless conditions [13]. It was also previously shown that a more involved detection scheme is optimal under photon loss [14]. However, here we will discuss various common detection schemes, which are easier to implement in practice and perform nearly optimal. Discussion of a lossy MZI for Fock state inputs is also discussed in previous works [15, 16].

While there are many technical challenges in using squeezed states of light, we show here that some of the measurement techniques commonly used in a classical setup are no longer near optimal. In addition, some measurements exhibit problems with effects such as phase drift and thermal photon noise. With the goal of choosing a simple, yet well-performing measurement, we investigate homodyne [17], parity measurements [18–22] and compare them to a standard intensity measurements. These measurements form a set that are either simple to implement, or are known to be optimal in the lossless case. Specifically, we confirm that, under lossless conditions, the parity measurement achieves the smallest phase variance. However, under noisy conditions, surprisingly the parity measurement suffers greatly, while the homodyne measurement continues to give a nearly optimal phase measurement. The parity measurement under losses was briefly discussed in the context of entangled coherent states by Joo et al. [23]. For the lossless case, we divide our results into two regimes, the low power regime (\(|\alpha|^{2}<500\), e.g. small scale sensors), in which different detection schemes can lead to significantly different phase variances, and the high power regime (\(|\alpha|^{2}>10^{5}\), e.g. large scale, devoted interferometry), where all detection schemes are nearly optimal. While our scheme may hint at applications for setups like LIGO, a much more focused analysis, outside the scope of our investigation, would be required before drawing conclusions about LIGO’s performance.

## 2 Method

*N*is the mean number of photons entering the MZI [11].

*α*,

*θ*are the coherent amplitude and phase, respectively while

*r*,

*δ*denote the squeezing parameter and phase. As the input state we consider is a product state, it can be written in terms of the product [25],

*ϕ*represents the unknown phase difference between the two arms of our MZI. We have chosen to use a symmetric phase model in order to simplify calculations as well as agree with previous results [27, 28]. Our goal then will be minimizing our uncertainty in the estimation of the unknown parameter

*ϕ*. Using these transforms, the total transform for the phase space variables is given by,

We can also consider photon loss in the model by way of two mechanisms, photon loss to the environment inside the interferometer and photon loss at the detectors, due to inefficient detectors. Both of these can be modeled by placing a fictitious beam splitter in the interferometer with vacuum and a interferometer arm as input and tracing over one of the output modes, to mimic loss of photons to the environment [29]. This linear photon loss mechanism can be modeled with the use of a relatively simple transform, since these states are all of Gaussian form. Specifically this amounts to a transform of the covariance matrix according to \(\sigma_{L}=(1-L)\mathbb{I} \cdot\sigma +L\mathbb{I}\), \(0\leq L\leq1\) is the combined photon loss and \(\mathbb {I}\) is the 4×4 identity matrix. Similarly the mean vector is transformed according to \(\mathbf{d}_{L}=\sqrt{(1-L)}\mathbb{I}\cdot \mathbf{d}\) [14, 30].

## 3 Results and discussion

### 3.1 Quantum Cramér-Rao bound

We consider an optimal measurement scheme with the meaning of saturating the quantum Cramér-Rao bound (QCRB) [31, 32], which gives the best phase sensitivity possible for a chosen interferometer setup and input states. This optimality is independent of measurement scheme and it remains a separate task to show which measurement scheme achieves this optimal bound [12]. In what we call the classical version of this setup, a coherent state and vacuum state are used as input. With these two input states, the best sensitivity one can achieve is bounded by the SNL, which is achievable with many different detection schemes. Many interferometer models mainly focus on analytical analysis of Fisher information [33, 34] when there is loss and phase drift. While this analysis is useful in that it demonstrates a ‘best case scenario’, it is unknown whether the optimal detection scheme is hard to realize in an actual experimental setup. Thus, in our analysis, we are more focused on Fisher information *and* how it compares with specific detection schemes, under noisy conditions.

### 3.2 Specific measurements under lossless conditions

Now that we have a bound on the best possible sensitivity, we now seek to show how various choices of measurement compare to this bound. We consider some standard measurement choices including single-mode intensity, intensity difference, homodyne, and parity. While each of these measurements would require a significant reconfiguration of any interferometer, it is worthwhile to show how each choice impacts the resulting phase sensitivity measurement. We utilize the bosonic creation and annihilation operators (\(\hat{a}^{\dagger}\), *â*), which obey the commutation relation, \([\hat{a},\hat{a}^{\dagger}]=1\). We also utilize the quadrature operators (*x̂*, *p̂*) which are related to the creation and annihilation operators by the transform \(\hat {a}_{j}=\frac{1}{\sqrt{2}}(\hat{x}_{j}+i \hat{p}_{j})\). These quadrature operators obey a similar commutator, \([\hat{x},\hat{p}]=i\).

In terms of our output Wigner function, \(\langle\hat{O}_{\mathrm {sym}}\rangle=\int_{-\infty}^{\infty}O \times W(\mathbf{X}) \, d\mathbf {X}\), where ‘sym’ indicates that this integral calculates the symmetric ordered expectation value of the operator *Ô*. Each measurement operator, \(\langle\hat{O}\rangle\), gives rise to a phase uncertainty by way of \(\varDelta ^{2} \phi=\varDelta ^{2}\hat{O}/|\partial\langle \hat{O}\rangle/\partial\phi|^{2}\).

*x*quadrature). For a balanced homodyne detection scheme, one would impinge one of the outgoing light outputs onto a 50-50 beam splitter, along with a coherent state of the same frequency as the input coherent state (usually this is derived from the same source) and perform intensity difference between the two outputs of this beam splitter. While there exist other implementations of homodyne than we describe here, we choose a standard balanced homodyne scheme, for simplicity. A standard intensity difference is defined as \(\hat{O}= \hat {a}^{\dagger}\hat{a}- \hat{b}^{\dagger}\hat{b}\). This particular measurement choice is also explored in Ref. [39]. Parity detection is defined to be \(\hat{O}=(-1)^{\langle\hat{a}^{\dagger}\hat {a}\rangle}=\pi W(0,0)\equiv\langle\hat{\varPi }\rangle\). Parity detection has been implemented experimentally, though focusing on its ability for super-resolution [40]. While all chosen measurements can surpass the SNL, in the lossless case, to various degrees, in order of improving phase sensitivity, single-mode intensity performs the worst, followed by intensity difference, homodyne, and finally parity. The analytical forms of each detection scheme, at their respective minima, are listed below and we confirm that, under lossless conditions, the parity measurement matches the QCRB [13],

*α*. From these forms then, we can say that in the low-photon-number regime (\(|\alpha |^{2}<500\)), the difference in these detection schemes can be significant, but in the high photon number regime (\(|\alpha |^{2}>10^{5}\)), there is little difference between the various detection schemes.

### 3.3 Lossy inteferometer

Note that this QCRB with loss only considers linear photon loss caused by photon loss inside the interferometer and photon loss due to inefficient detectors. In reality, there may be more specific sources of noise one needs to consider, but our method’s purpose is to show a preliminary case when simple loss models are considered. We note that a measurement scheme proposed by Ono and Hofmann is exactly optimal (thus it is able to achieve the bound given by Eq. (11)) under loss [14], but we wish to explore how simpler measurement schemes perform when compared with this bound.

*r*and the amplitude of the coherent state \(|\alpha |\). Therefore, fluctuations in the source will actually affect the optimal phase setting and in general degrade the phase measurement in this measurement scheme. Note that, in practice, typical experiments use an offset to remain near these optimum values, but purposely remain slightly away from the minimum, due to noise considerations. At this point we can note, that current technological limits enforce \(|\alpha |^{2}\gg\sinh^{2}(r)\), as generally it is relatively easier to increase laser power, than to increase squeezing power. Just as it was in the lossless case, under lossy conditions then, Figure 2 shows that homodyne remains nearly optimal in the low power regime. In contrast, the previously optimal measurement, parity, is now not able to even reach sub-SNL.

*r*, the trend of homodyne achieving near optimal measurement generalizes to other parameter choices as well.

### 3.4 Phase drift

Returning to Figure 2, it is clear at which value of phase the various measurements attain their lowest value. It is this value of phase that one attempts to always take measurements at with the use of a control phase inside the interferometer. The width of each of curve then can be interpreted as the chosen measurement scheme’s resistance to phase drift. The mechanism of phase drift comes about due to the limited ability to set control phases in the interferometer with infinite precision. In general, the control phase value will vary around the optimal phase setting. For this reason we aim to show this phase drift in a more rigorous way. We therefore will use the analytical forms of the various measurement phase variances, as a function of unknown phase *ϕ*. We simulate phase drift by computing a running average of the phase variance, with a pseudo-randomly chosen phase, near the optimal phase, for each measurement. This pseudo-random choice is made from a Gaussian distribution, whose mean is fixed at each measurements respective optimal phase choice and has a chosen variance of \(\sigma=0.15\). As predicted in the previous discussion, this gives a clearer picture of each measurement’s behavior under phase drift. For simplicity, we focus on the lossless case for this treatment of phase drift.

### 3.5 Thermal photon noise

We recommend that a homodyne measurement is the simplest, nearly optimal measurement choice for a setup as discussed here. Homodyne is a typical measurement choice in interferometer experiments, as well as being a single mode measurement, likely resistant to photon loss, detector efficiency, and phase drift. It shows its main benefits in the low power regime, but performs nearly optimal in both the low and high power regimes.

## 4 Conclusion

In this paper, we have seen the performance of many common interferometric measurement schemes. While all are able to achieve a sub-SNL phase variance measurement in the lossless case, for the choice of a coherent and squeezed vacuum input state, all are outperformed by a homodyne measurement when loss is introduced. While these measurements each come with their own challenges in implementing, we have shown that each measurement’s performance can vary significantly under different noise models. We have also shown that in the high-photon regime, with loss, most measurement schemes approach the QCRB except for parity which suffers significantly. Our results may imply that simpler measurement schemes are overall appealing when using large powers. The behavior of each measurement scheme under phase drift and thermal photon noise is also discussed, and we find that homodyne and intensity difference measurement behave best within these models. This should be expected as both homodyne and intensity difference measurements operate in a similar way, subtracting intensities between two modes, removing common noise sources. Therefore, when considering ease of implementation as well as near optimal detection, we conclude that homodyne is nearly optimal under loss, phase drift, and thermal photon noise, for the specific choice of input states of coherent and squeezed vacuum and in both power regimes.

## Declarations

### Acknowledgements

BTG would like to acknowledge support from the National Physical Science Consortium & National Institute of Standards and Technology graduate fellowship program as well as helpful discussions with Dr. Emanuel Knill at NIST-Boulder. CY would like to acknowledge support from an Economic Development Assistantship from the National Science Foundation and the Louisiana State University System Board of Regents. DKM would like to acknowledge support from University Grants Commission, New Delhi, India for Raman Fellowship. TRC would like to acknowledge support from the National Science Foundation grants PHY-1150531. This document has been assigned the LIGO document number LIGO-P1600084. JPD would like to acknowledge support from the Air Force Office of Scientific Research, the Army Research Office, the Boeing Corporation, the National Science Foundation, and the Northrop Grumman Corporation. We would all also like to thank Dr. Haixing Miao and the MQM LIGO group for helpful discussions. All authors contributed equally to this work.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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