# Deterministic phase measurements exhibiting super-sensitivity and super-resolution

###### Abstract

Phase super-sensitivity is obtained when the sensitivity in a phase measurement goes beyond the quantum shot noise limit, whereas super-resolution is obtained when the interference fringes in an interferometer are narrower than half the input wavelength. Here we show experimentally that these two features can be simultaneously achieved using a relatively simple setup based on Gaussian states and homodyne measurement. Using photons shared between a coherent- and a squeezed vacuum state, we demonstrate a -fold improvement in the phase resolution while we observe a -fold improvement in the sensitivity. In contrast to previous demonstrations of super-resolution and super-sensitivity, this approach is fully deterministic.

© Optica Vol. 5, Issue 1 (2018), pp. 60–64, Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited.

Current address: ]Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, St. Lucia, Queensland 4072, Australia

## I Introduction

Quantum interference of light plays a pivotal role in high-precision quantum sensing ^{1}, optical quantum computation ^{2} and quantum state tomography ^{3}.
It is typically understood as two-beam interference which can be observed, for instance, in a Mach–Zehnder interferometer or a double slit experiment.
At the output, such interferometers create an oscillatory pattern with a periodicity given by half of the wavelength () of the radiation field, which may be referred to, in analogy to the resolution-benchmark in optical imaging, as the “Rayleigh criterion” for phase measurements.
This limit can however be surpassed using different types of states or measurement schemes ^{4; 5; 6; 7; 8; 9; 10}.
In particular, measurement schemes which are based on parity detection ^{11; 9} or approximate parity detection via a phase-space relation ^{8} are utilised to beat this limit with classical states, i.e. they do not require quantum states
^{1}^{1}1A simple approach for increasing the fringe resolution by classical means is a multi-pass interferometer.
As the mentioned techniques can be mapped to multi-pass configurations, we focus on the single-pass configuration..
The arguably best-known quantum approach to observe a fringe narrowing uses NOON states, .
Surpassing the Rayleigh criterion is referred to as super-resolution ^{13; 14} and is studied in the context of, e.g., optical lithography ^{5}, matter-wave interferometry ^{15} and radar ranging ^{16}.

In quantifying the performance for applications in quantum sensing and imaging, it is common to evaluate the Fisher information ^{17} or, equivalently, determine the sensitivity in the interferometric phase measurement.
Using coherent states of light the optimal sensitivity is given by where is the mean number of photons of the state ^{18}.
This sensitivity constitutes the shot noise limit (SNL).
Overcoming the SNL is commonly referred to as super-sensitivity and can be achieved by non-classical states ^{19; 20; 1; 21}.
Super-sensitivity based on squeezed states of light has proven to be a powerful and practical way to enhance the sensitivity of gravitational wave detectors ^{22; 23}.

The effects of super-sensitivity and super-resolution can be obtained simultaneously.
For example, optical NOON states offer a sensitivity with Heisenberg scaling, , and a phase resolution that scales as corresponding to fringes per half-wavelength.
NOON states thus exhibit an equal scaling in the two effects.
In contrast, this work shows how resolution and sensitivity are tuneable and can, in fact, compete with each other.
Due to the high fragility of NOON states, the complexity in their generation and the commonly probabilistic way of generation, super-sensitivity and super-resolution have been only measured in the coincidence basis and in a highly probabilistic setting ^{6; 24; 19; 25}.
It has also been suggested to use two-mode squeezed vacuum states in combination with parity detection to attain the two “super-features” simultaneously ^{26}.
However, possibly due to the complications in implementing a parity detection scheme, it has so-far never been achieved experimentally.
The complexity associated with the two schemes are due to the involved non-Gaussian states (NOON states) or the non-Gaussian measurements (parity detection).
A natural question to ask is whether the same “super-features” can be realised using simple Gaussian operations.
Here we answer this question in the affirmative.

We propose and experimentally demonstrate that, by using Gaussian states of light and Gaussian measurements, it is possible to realise a phase measurement which features super-resolution and super-sensitivity simultaneously. Using displaced squeezed states of light in conjunction with homodyne detection followed by a data-windowing technique, we show that the interferometric fringes can be made arbitrarily narrow while at the same time beating the shot noise limit. In stark contrast to the NOON state scheme which, in any practical setting, is highly probabilistic both in preparation and in detection, our approach provides a deterministic demonstration of super-resolution and super-sensitivity.

## Ii Materials and methods

An illustration of the basic scheme is shown in Fig. 1. A vacuum squeezed state is combined with a coherent state of light at the entrance to a symmetric Mach–Zehnder interferometer. The Wigner function at the input is given by

(1) |

where and are the amplitude- and phase-quadratures, is the amplitude of the coherent state, , where denotes the squeezing parameter, and represents the purity of the squeezed state. In the scheme, an amplitude modulated coherent state and a phase-squeezed vacuum state interfere on the first beam splitter of the interferometer. Then the resulting state acquires a relative phase shift , next interferes on the second beam splitter and finally one of the outputs is measured. As we used weak input signals, a homodyne readout scheme was employed. Fig. 1 illustrates the trajectory of the output state in phase space for different phase shifts.

If the interferometer is operated near a dark fringe, i.e. biasing the phase shift such that most of the light exits the second output of the interferometer, the phase-squeezed vacuum state will be detected.
Thereby, the shot noise around the bias is suppressed and the phase sensitivity improved.
The approach of feeding the commonly unused input mode with a vacuum squeezed state is equal to the proposal by Caves ^{20} to beat the shot noise limit in phase measurements.
However, since the phase response for Caves’ scheme reads , which is an oscillating function with a period equal to , the resolution coincides with the mentioned “Rayleigh criterion” for phase measurements.
In the following we show that by implementing a homodyne windowing scheme, the setup yields super-resolution and super-sensitivity.

The quadrature measurement of the homodyne detector is divided into two bins, set by the ‘bin size’ : If the phase quadrature is measured, we categorise two different results which are associated with the intervals and . We describe such a measurement strategy by the projectors

(2) |

The measurement observable can thus be written as , where and are the eigenvalues associated with the two measurement outcomes. Now the detector response is found by evaluating which, in the idealised case of , i.e. , and a pure squeezed vacuum state, yields

(3) |

The full-width-half-maximum (FWHM) of this function follows for , thereby indicating that the interference fringes become narrower as is increasing and thus demonstrating super-resolution.
It should be stressed that setting is an idealisation, as it means a projection on an infinitely squeezed state, i.e. even number state.
However it points out that the operator is in some sense an approximation of the parity operator ^{11; 9}.
Considering instead a realistic setting where and the squeezed state is not pure (), the response function reads

(4) |

where and

(5) |

The scaling of the FWHM is preserved for a general value , i.e. FWHM . In Fig. 2a we plot the FWHM-improvement as a function of the squeezing parameter and the bin size . It is clear from this plot that the super-resolution feature only depends weakly on the degree of squeezing, and a similar conclusion is found for the purity of the state. The only critical parameter for attaining high resolution is the mean photon number of the input coherent state. More details and a derivation may be found in the Supplemental Document.

We now turn to the investigation of the sensitivity using the above scheme. The sensitivity can be found using the uncertainty propagation formula,

(6) |

where , and for our measurement operator it follows

(7) |

with the notation and . For a specific parameter regime defined by the purity , the bin size and the squeezing parameter , this sensitivity beats the shot noise limit. In Figs. 2b,c we plot the sensitivity relative to the shot noise limited sensitivity as a function of the bin size and the squeezing parameter for two different purities. It is shown in Fig. 2c that it is possible to achieve super-sensitivity in a setting where the squeezed state is impure. In conclusion, both super-sensitivity and super-resolution can be achieved in a practical setup for the parameter space shown in Fig. 2c. Furthermore, sensitivity and resolution features are neither independent nor fixed with respect to each other, but can be varied by the homodyne windowing technique. A discussion of the ultimate sensitivity may be found in the Supplemental Document.

We proceed by discussing the experimental realisation depicted in Fig. 3. A squeezed vacuum state and a coherent state with a controllable photon number is injected into the input ports of a polarisation based Mach–Zehnder interferometer. The polarisation basis ensures high stability and quality of the interference. Furthermore it allows for simple control of the relative phase shift. The phase shift is varied by a half-wave plate mounted on a remote-controlled rotation stage. One output of the interferometer is measured with a high-efficiency homodyne detector exhibiting an overall quantum efficiency of , given by efficiency of the photo diodes and visibility to the local oscillator (LO). The relative phase of the two input beams of the interferometer as well as the phase of the LO is actively stabilised via real-time feedback circuits, thereby recreating the scheme in Fig. 1 and projecting the output on the quadrature. A detailed description may be found in the Supplemental Document.

Squeezed vacuum is generated by parametric downconversion in a long periodically-poled KTP crystal embedded in a long cavity comprising a piezo-actuated curved mirror and a plane mirror integrated with end-facet of the crystal. A Pound–Drever–Hall (PDH) scheme is adopted to stabilise the cavity resonance. The downconversion process is pumped by a continuous-wave laser beam operating at , such that squeezed light is produced at . To stabilise the pump phase, the radio-frequency signal used also for cavity stabilisation is down-mixed with a phase shift of . Using a local oscillator, we observe shot noise suppression at sideband frequency, while the anti-squeezed quadrature is above shot noise. The squeezed state parameters read, on average, and . A complete characterisation of the squeezed light source is presented in the Supplemental Document.

The coherent input state is produced by an electro-optical modulator (EOM) at a sideband-frequency of . The chosen frequency ensures the creation of a coherent state far from low-frequency technical noise and with an amplitude that is conveniently controlled by the modulation depth of the EOM.

To measure the interferometer’s output state at , the electronic output of the homodyne detector is down-mixed at this frequency, subsequently low-pass filtered at and then digitised with resolution. For each phase setting, samples are acquired at a sampling rate of . The data is recorded on a computer for post-processing which includes the dichotomic windowing strategy given by (2) in which we set the bin size . After dividing the data according to , we calculate as well as the standard deviation for each phase setting from the data. Finally, is computed according to (6). The term in (6) is extracted directly from the data. Instead of calculating the derivative of also directly, it is estimated from the theoretical model of fitted to the data. This approach is chosen to increase the confidence in the computation of and a comparison between this and a direct evaluation is shown in the Supplemental Document. In the panels on the right of Fig. 4, is shown in comparison to the theoretical model given by (7).

## Iii Results and discussion

The results for a mean photon number of and , with a mean of photons contained in the squeezed state and an overall efficiency of circa , are shown in Fig. 4 and compared with theoretical predictions. The latter is denoted by solid lines. It is clear from the plots that the scheme exhibits super-resolution as well as super-sensitivity for certain phase intervals. We expect that the resolution and sensitivity improves as the mean photon number is increased. This expectation is confirmed in Fig. 5 where the measurement of these two features for increasing photon numbers in the coherent state is depicted. Specifically, at we obtain a -fold improvement in the phase resolution compared to a standard interferometer and a -fold improvement in the sensitivity relative to the shot noise limit.

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