Double-port homodyne detection in a squeezed-state interferometry with a binaryoutcome data processing

Likun Zhou ,Pan Liu and Guang-Ri Jin,?

1 Key Laboratory of Optical Field Manipulation of Zhejiang Province and Physics Department of Zhejiang Sci-Tech University,Hangzhou 310018,China

2 China Academy of Electronics and Information Technology,Beijing 100041,China

Abstract Performing homodyne detection at a single output port of a squeezed-state light interferometer and then separating the measurement quadrature into two intervals can realize super-resolving and super-sensitive phase measurements,which is equivalent to a binary-outcome measurement.Obviously,the single-port homodyne detection may lose almost part of the phase information,reducing the estimation precision.Here,we propose a data-processing technique over the doubleport homodyne detection,where the two-dimensional measurement quadrature (p1,p2) has been divided into two regions.With such a binary-outcome measurement,we estimate the phase shift accumulated in the interferometer by inverting the output signal.By analyzing the full width at half maximum of the signal and the phase sensitivity,we show that both the resolution and the achievable sensitivity are better than that of the previous binary-outcome scheme.

Keywords: quantum-enhancement phase estimation,squeezed-state light interferometer,homodyne detection

1.Introduction

Quantum metrology is of importance for multiple areas of scientific research,including gravitational wave detection [1,2],biological sensing [3,4],atomic clocks [5,6],and so on.One of the main tasks of quantum metrology can be attributed to a phase estimation problem,where the estimation precision of an unknown phase shift θ depends on the probes,the phase accumulation,and the measurement scheme [7–14].For instance,performing the intensity measurements at the output ports of a coherent-state light interferometer,the output signalwhere θ is an unknown phase shift.The signal exhibits the full width at half maximum(FWHM)=π,corresponding to the Rayleigh resolution limit[15].On the other hand,the achievable phase sensitivity is subject to the shot-noise limit(SNL)[16,17],whereis the mean number of photons injected into the interferometer.

To beat the above classical limits,one can use nonclassical resources such as squeezed state [18–21] andNphoton entangled state[22–28].In 1981,Caves[18]proposed a squeezed-state interferometer,where a coherent state and a squeezed vacuum state are injected into the two input ports,as depicted by figure 1(a).It has been shown that the achievable sensitivity can beat the SNL.Furthermore,the so-called Heisenberg-limit (HL) can be reached by using theN-photon entangled states [23–25],or squeezed states [19–21].However,the entangled states are difficult to prepare and are extremely fragile.On the other hand,the measurement schemes based on a coherent state and a squeezed vacuum state [19–21] with coincidence photon counting or parity detection are not highly efficient and are dependent on the number resolvable counters.Indeed,the achievable sensitivity is subject to the finite number resolution of the photon counters [29,30].Recently,Sch?fermeieret al[31] have realized deterministic super-resolution and super-sensitivity simultaneously in the squeezed-state interferometer with a single-port homodyne detection,where the measurement data is divided intop1?[-a,a] andp1?[-a,a],whereais a parameter that can be adjusted artificially.This is equivalent to a binary-outcome measurement.

Figure 1.(a) Homodyne detection (i.e.measuring the quadrature operators) at two ports of the interferometer that is fed by a coherent state|α0〉and a squeezed vacuum|ξ0〉.(b)The achievable phase sensitivity for the double-port homodyne detection(blue solid line),is better than that of the single-port homodyne detection (red dashed line).Vertical lines in (b): the SNL and the best sensitivity The parameters: e-r=0.6 and ρ=0.5.

Obviously,the above single-port homodyne detection neglects almost half of the total phase information at another output port,leading to a reduced phase estimation precision.In this paper,we propose a double-port homodyne detection scheme with proper data processing.Specifically,we divide the two-dimensional measurement quadrature(p1,p2)into two regions,which realize a similar binary-outcome measurement with that of Sch?fermeieret al[31].We calculate the full width at half maximum (FWHM) of the output signal and the phase estimation precision using the inversion estimator.Our results show that a 1.8-fold improvement in the sensitivity and a 30-fold improvement in the resolution beyond their associated classical limits can be realized with about 400 input photons,which is slightly better than that of [31].Finally,it should be mentioned that there are many kinds of data processing in the two-dimensional measurement quadrature (p1,p2).Here we show the simplest one to realize a further improvement in the resolution and in the sensitivity beyond that of [31].

2.Double-port homodyne detection without data processing

IntegratingWout(α,β) overx1andx2,we obtain the conditional probability for detecting a measurement outcome:

where we introduced

The achievable phase sensitivity is given by

whereF(θ) is the classical Fisher information (CFI):

In figure 1(b),we show the achievable phase sensitivity of double-port homodyne detection (blue solid line) as a function of the phase θ.One can find that the phase sensitivity can beat SNL around the optimal phase point θ=0,±π,which gives the maximal CFI

and the best sensitivity

3.Single-port homodyne detection without data processing

If we do not take into account the quadraturep2,the scheme is reduced to the single-port homodyne detection,as demonstrated in [31].IntegratingP(p1,p2|θ) overp2,we can obtain the conditional probability of the single-port homodyne detection,i.e.

With the single-port homodyne detection,one can easily obtain the output signal

which exhibits FWHM=2π/3,and hence the Rayleigh limit in fringe resolution [31,33].On the other hand,the achievable CFI reads [34]:

and hence the achievable sensitivity

When the coherent-state component dominates over the squeezed vacuum(i.e.),the best sensitivity occurs at the optimal working point θ=0.Using equation (13) and taking θ=0,we obtain the maximal CFI

and hence the best sensitivity approximates to[34],which is in agreement with the light intensity-difference measurement as[18,19].When the squeeze factorr=0,the phase sensitivity is limited to the SNL,as shown in[32,33].To beat the SNL,it is necessary to take the squeeze factorr>0.

Comparing equations (9) and (15),the maximal CFI of double-port homodyne detection is greater than that of the single-port homodyne detection,which means that we can attain a better phase sensitivity by the double-port homodyne detection.This is because,the single-port homodyne detection only takes into account the contribution ofp1,which lost some information of the unknown phase θ.In figure 1(b),we show the difference in achievable phase sensitivity between single-port homodyne detection and double-port homodyne detection.One can find that the double-port homodyne detection is better than the singleport homodyne detection in the whole phase interval θ ?[-π,π],and can beat the SNL around θ=0,±π.In order to improve the resolution determined by the FWHM,we can adopt suitable data processing over the measurement outcomes.In the next section,we will discuss binary-outcome measurement data processing.

4.Binary-outcome homodyne detection

Figure 2.(a)?P(p1|θ)/?θ as a function of the quadrature p1.(b)Density plot of ?P(p1,p2|θ)/?θ as a function of the quadratures p1 and p2,where we take θ=0.(c) and (d) The data processing schemes of binary-outcome measurement corresponding to single-port homodyne detection(c)and double-port homodyne detection(d),where the region width of the two schemes are 2a.(e)and(f)The output signal and the sensitivity of the binary-outcome measurement with single-port homodyne detection(red dashed lines)and double-port homodyne detection(blue lines).The black circles are simulated by M=500 repeats of N=300independent measurements.Parameters:=42,e-r=0.47,

The phase resolution can be improved by suitable data processing over the measurement outcomes,at the cost of reduced phase sensitivity.For the case of the single-port homodyne detection,Sch?fermeieret al[31] have demonstrated super-sensitive and super-resolving phase measurement in the squeezed-state interferometer.The dataprocessing method they adopted is to separate the measurement quadraturep1?(-∞,∞) into two bins:p1?[-a,a]andp1?[-a,a] (shown in figure 2 (c)),whereais a controllable parameter.This is equivalent to a binary-outcome measurement[35,36],with the observablewhere μ0andμ?are arbitrary numbers.Furthermore,the projection operators are given byUsing the relationwe obtain the probabilities of each outcome:

where Erf [x,y]≡erf[y]-erf [x]denotes a generalized error function,and

with η-being defined by equation (6).With such a kind of binary-outcome measurement,one can construct the output signal

which can quantify the performance of the estimator θinv.According to the error-propagation formula [37],the phase sensitivity is

Next,we consider the double-port homodyne detection with a binary-outcome measurement data processing,i.e.separate the measurement quadraturep1andp2into two regions.The data processing method we use is inspired by figure 2(b),in which we show the density plot of ?P(p1,p2|θ)/?θ as a function of the measurement quadraturesp1andp2with θ=0.One can find that it is symmetric with respect to the linewhich inspires us to treat the regionas an outcome ‘0’ and other regions as an outcome‘?’(shown in figure 2(d)),where

and the region width 2ais a controllable parameter.The corresponding conditional probabilities are:

where

We construct the output signal by takingμ?=0,and replacingP0(θ) byin equation (19).In figure 2(c),we show the output signals with the parametera=0.5.The resolution is determined by the FWHM of the signal.One can find that the blue line (double-port detection) shows a better phase resolution than that of the single-port detection adopted by Sch?fermeieret al[31].

The phase sensitivity can be also given by equation (22)withP0(θ) replaced byIn figure 2(e),we show the phase sensitivity with parametera=0.5,which is almost saturated bywhere σ is simulated byM=500 repeats of N=300in dependent measurements.For the double-port detection (the blue line),the achievable sensitivityδθminappears at the optimal point 0.11=and is better than the single-port detection (red dashed line).

The value ofais a trade-off parameter that controls the achievable sensitivity and the resolution.In figures 3(a) and(b),we show a density plot of the ratiosas functions of the average photon numberand the parametera,whereδθminis the best sensitivity.Using the parameters similar to Sch?fermeieret al[31],one can find that the resolution is optimal asa→0,while the best sensitivity appears ata～0.42.When the average photon numberand the parametera～0.4,we can obtain a 1.8-fold improvement in the sensitivity and a 30-fold improvement in the resolution,which is better than the experimental result in[31],where a 1.7-fold improvement in the sensitivity and a 22-fold improvement in the resolution with430=has been demonstrated.In figures 3(c) and (d),we show the log–log plot of the best sensitivityδθminand the FWHM against the photon number.One can find that both the best sensitivity and the FWHM of the double-port detection are better than that of the single-port detection.

5.Conclusion

In summary,we have proposed a double-port homodyne detection in the interferometer with proper data processing,where the input states are the coherent state and squeezed vacuum state.Performing the double-port homodyne detection without any data processing,we show that the best sensitivity isbetter than that of the single-port homodyne detectionTo improve the resolution,we separate the two-dimensional measurement quadratures into two regions (as shown in figure 2(d)),equivalent to a binary-outcome measurement.We find that a 30-fold improvement in the resolution and a 1.8-fold in the sensitivity can be realized simultaneously,slightly better than that of [31].The data processing technique,as the most simple one with respect to the two-dimensional measurement quadratures,can be generalized to a multi-outcome phase measurement [38,39].

Figure 3.(a)and(b)Density plots of the ratiosas functions of the average photon number and the parameter a,with the parameters e-r=0.47 and ρ=0.5 [31].(c) and (d) Log–log plot of the achievable sensitivityδθmin and the FWHM against the photon number.The green solid line in (c) is given by equation (10).

Acknowledgments

Project supported by the Science Foundation of Zhejiang Sci-Tech University,grant number 18 062 145-Y,and the National Natural Science Foundation of China (NSFC) grant number 12 075 209.

Appendix.Quantum Fisher information

The phase accumulation can be described by a unitary operator:

which is independent of the phase shift θ.For the input state|ψin〉=|α0〉?|ξ0〉,we have