Distributed Quantum Metrology with a Single Squeezed-Vacuum Source

PHYSICAL REVIEW RESEARCH 1, 032024(R) (2019)

Rapid Communications

Distributed quantum metrology with a single squeezed-vacuum source

Dario Gatto ,1,* Paolo Facchi ,2,3 Frank A. Narducci ,4 and Vincenzo Tamma 1,5,† 1School of Mathematics and Physics, University of Portsmouth, Portsmouth PO1 3QL, United Kingdom 2Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy 3INFN, Sezione di Bari, I-70126 Bari, Italy 4Department of Physics, Naval Postgraduate School, Monterey, California 93943, USA 5Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom

(Received 29 June 2019; published 25 November 2019)

We propose an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number M > 1 of unknown phase delays, distributed across an M-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number-resolving detectors, or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relies upon a single squeezed-state source, an antisqueezing operation at the interferometer output, and on-off photodetectors.

DOI: 10.1103/PhysRevResearch.1.032024

Introduction and motivations. Quantum metrology aims at produce in the laboratory with current technology is a matter harnessing inherently quantum features such as entanglement, of great interest. multiphoton interference, and squeezing, to develop novel We overcome these limitations by introducing an interfero- quantum enhanced technologies for sensing and imaging be- metric scheme (Fig. 1) which employs only a single squeezed yond any classical capabilities [1–8]. More recently, a great source and on-off photodetectors. Indeed, squeezed states of deal of attention has been devoted to distributed quantum light are a natural candidate for Heisenberg-limited probing metrology, particularly on the problem of measuring a linear [33–35], on account of their experimental availability with a combination of several unknown phase shifts distributed over high mean photon number and their nonclassical character. a linear optical network [9–14]. More explicitly, we will While such states have been largely used to yield super- be interested in measuring a linear combination of M > 1 sensitivity in the estimation of a single unknown parameter, unknown distributed phases. This problem is of interest in a their quantum metrological advantage in the case of multiple variety of settings: from the mapping of inhomogeneous mag- distributed parameters has not yet been fully explored [13,14]. netic fields [15–19], phase imaging [20–25], and quantum- Here, we demonstrate how a simple M-channel linear optical enhanced nanoscale nuclear magnetic resonance imaging interferometer with only a single squeezed-vacuum source [9,26,27], to applications in precision clocks [28], geodesy, and on-off photodetectors can achieve Heisenberg-limited and geophysics [29–31]. sensitivity in distributed quantum metrology with M unknown A novel scheme was recently proposed to tackle distributed phase delays. Remarkably, such a scheme can be implemented quantum metrology with Heisenberg-limited sensitivity [32]. experimentally with present quantum optical technologies. However, its main limitation is the fact that it relies on two The optical interferometer. We describe here in details an Fock states with a large number of photons as probes in interferometric setup (Fig. 1) able to estimate the combination order to achieve Heisenberg-limited sensitivity. Schemes to M make high-photon-number Fock states do not currently exist. ϕ = w jϕ j, (1) Furthermore, it requires a number of auxiliary interferomet- j=1 ric channels up to the number M of unknown distributed ϕ = ,..., phases, and photon-number-resolving detectors. Therefore, of M unknown phases j ( j 1 M) for any given set of { }M devising measurement schemes which can exhibit supersen- non-negative weights w j j=1 [36]. Without loss of generality = sitivity while making use of probe states which are simple to we will assume in the following the normalization j w j 1, so that the w j’s are probability weights. The general situation will differ just by an immaterial factor. The probe light at the input of our interferometer is pre- *[email protected] pared in the squeezed-vacuum state † [email protected] |in=Sˆ1(z)|, (2) |=| ···| ˆ = Published by the American Physical Society under the terms of the where 0 1 0 M is the vacuum state, S1(z) 1 (z∗aˆ2−zaˆ†2 ) Creative Commons Attribution 4.0 International license. Further e 2 1 1 is the squeezing operator,a ˆ1 is the photonic distribution of this work must maintain attribution to the author(s) annihilation operator of the first mode, and z is the squeezing and the published article’s title, journal citation, and DOI. parameter. The squeezing parameter z fixes the mean number

2643-1564/2019/1(3)/032024(5) 032024-1 Published by the American Physical Society GATTO, FACCHI, NARDUCCI, AND TAMMA PHYSICAL REVIEW RESEARCH 1, 032024(R) (2019)

+ |0 S(z) ϕ1 S (z) |0 0| associated with the projection of the output state over the input state, i.e., to the probability that the probe leaves the interferometer with its state unaltered. Since the expectation |0 ϕ2 |0 0| value of Oˆ is

+ U U Oôut =out|Oˆ|out |0 . . . |0 0| † −iGˆ 2 . . . =|in|Uˆ e Uˆ |in| =|| ˆ† | |2, S1 (z) out (9) |0 ϕM |0 0| the measurement of Oˆ is equivalent to projecting onto the vacuum | after the action of an antisqueezing operation on the FIG. 1. Interferometric setup with only a single squeezed- ˆ† first channel, described by S1 (z). This can be experimentally vacuum source for the Heisenberg-limited estimation of the linear achieved, for instance, by retroreflecting the down-converted ϕ = ,..., combination of unknown phases i, with i 1 M,asinEq.(1). photons onto the crystal generating the original squeezed light ˆ , ˆ † The linear optical networks represented by U U are set up in [38–42], and then using on-off photodetectors. such a way as to satisfy Eq. (5). The operators Sˆ(z), Sˆ†(z)represent Since Oˆ =| | |2 is the probability of the out- squeezing and antisqueezing operations, respectively. out in out put state to coincide with the input state, if the phases are small, the total interferometric operator should be close † ˆ of input photons Nˆ =in|Nˆ |in, with Nˆ = aˆ aˆ j,by to the identity, and therefore O out should be close to j j −|ϕ| ˆ ˆ |ϕ| ˆ the relation Nˆ =N¯ , where one. More precisely, since maxN G maxN, with |ϕ|max = maxi |ϕi|, and the interferometer preserves the total N¯ = sinh2(|z|). (3) number of photons, if The probe travels through the first linear optical trans- |ϕ|max Nˆ 1, (10) ˆ formation, described by the unitary operator U through the ˆ ˆ we can perform an expansion of O out in powers of G. equation ˆ m By using the notation G U for the expectation value M m of the operator Gˆ with m = 1, 2 taken at the state |U = † 2 Uˆ aîUˆ = Uijaˆ j, (4) ˆ | 2 =ˆ 2 −ˆ ˆ U in , and GU G U G U , for the variance of G,we j=1 obtain U M × M −iGˆ 2 1 2 2 where is an unitary matrix associated with the Oôut =|e U | 1 − iGˆ − Gˆ transition amplitudes from the channel j to the channel i, with 2 U − 2 , i, j = 1,...,M. We set these amplitudes to 1 GU (11) √ up to fourth-order terms (see the first section of the Supple- U j1 = w j, (5) mental Material [43]). where j = 1,...,M and the wi’s are the weights of Eq. (1). By using Eq. (6) and the canonical commutation relations This can always be achieved with an appropriate combination (see the second section of the Supplemental Material [43]), we ˆ of beam splitters [37]. More importantly, this step enables the obtain that the exact expression for the variance of G depends estimation of ϕ by making the output measurement explicitly ϕ ϕ2 = ϕ2 on in Eq. (1), and on j w j j as dependent on it [see Eq. (12) later on], as well as creating 2 = ϕ2 ˆ 2−ˆ 2 + ϕ2 − ϕ2 ˆ , useful entanglement [32], distributed across all channels con- GU ( N N ) ( ) N (12) taining the phase delays ϕ ,...,ϕ . 1 M where Nˆ 2= |Nˆ 2| . The variance of Gˆ is made of After the linear optical transformation Uˆ the probe under- in in a contribution from number fluctuations and a contribution goes phase shifts ϕ ,...,ϕ through the respective channels, 1 M from the fluctuations of the phases ϕ with respect to the and finally evolves through the inverse linear optical transfor- j weights w . mation Uˆ †. Reversing the linear optical transformation will j This result is valid for any (not necessarily Gaussian) M- allow us to effectively project the output state onto the input boson state | with all modes but the first in the vacuum. In state (see below). Thus, given the generator of the phase shifts, in our case, since the first mode is in a squeezed-vacuum state, M its photon-number statistics is super-Poissonian [44], with the ˆ = ϕ † , = G j aˆ j aˆ j (6) mean photon number Nˆ N¯ given by (3) and a variance j=1 Nˆ 2−Nˆ 2 = 2N¯ (N¯ + 1), (13) the state at the output of our interferometer is which scales as N¯ 2. This scaling is unlike a coherent state † −iGˆ which has a Poissonian photon-number statistics with vari- |out=Uˆ e Uˆ |in. (7) ance equal to the mean Nˆ . As we will see, this is an essential Heisenberg-limited estimation. We now demonstrate ingredient for obtaining a Heisenberg-limited sensitivity. ¯ 2 ¯ 2ϕ2 Heisenberg-limited sensitivity [in Eq. (16)] by means of the For large N one gets GU 2N , from which the observable expectation value of our observable (11) reads 2 2 Oˆ =|inin|, (8) Oôut 1 − 2N¯ ϕ , (14)

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+ |0 S(z) ϕ1 S (z) |0 0| be “washed out” by the choice of the measurement protocol. Indeed, if |in is an eigenstate of the number operator Nˆ , | ϕ − ϕ w1/w2 w1/w2 out depends only on the relative phase 1 2, because |0 ϕ2 |0 0| the second exponential in Eq. (19) gives rise to a global complex phase, and the information on ϕ = (ϕ1 + ϕ2 )/2is FIG. 2. Estimation of the linear combination two unknown completely lost. Furthermore, if one measures an observable ˆ ˆ ˆ phases ϕ1,ϕ2 with weights w1, w2. The interferometric scheme in O which commutes with N, then O out,aswellasallthe ˆ k ϕ − ϕ Fig. 1 reduces to a Mach-Zehnder interferometer, with unbalanced higher moments O out [2], will again depend on 1 2 † − i (ϕ −ϕ )Jˆ i (ϕ −ϕ )Jˆ R/T = w /w R T ˆ ˆ ˆ = 2 1 2 y ˆ 2 1 2 y beam splitters with 1 2,where is the reflectivity and only, since UMZ OUMZ e Oe . is the transmittivity. Discussion. We have shown how squeezed light can be used to estimate an arbitrary superposition of phases with non- and differs from 1 by a small quantity, as expected. Indeed, negative weights. Our protocol can overcome the limitations we are in the regime of large N¯ and small ϕ, such that |ϕ|N¯ of Ref. [32], most notably we can achieve the Heisenberg limit with a single squeezed state rather than two Fock states. |ϕ|maxN¯ is small, in accordance with Eq. (10). The sensitivity in the estimation of ϕ is obtained by the Futhermore, our protocol does not necessitate the use of aux- error propagation formula [7] iliary channels nor photon-number-resolving detectors. The interferometric setup is easily realizable for any set of weights 2 2 Oˆ out −Oˆ by using only beam splitters and phase shifters [37]. The δϕ2 = out . (15) d Oˆ 2 initially separable input state acquires entanglement across dq out the various channels where the phase shifts are distributed By using the fact that Oˆ = Oˆ 2 is a projection, and by virtue of owing to the linear optical network. The squeezed source Eq. (14), we easily get can be produced in a number of ways, including sponta- neous parametric down-conversion and four-wave mixing. 1 δϕ2 , (16) Antisqueezing has already been achieved with high efficiency 8N¯ 2 [38–42] by retroreflecting the down-converted photons and i.e., the sensitivity scales at the Heisenberg limit. the pump back onto the crystal. We would like to mention Example. Let us consider the M = 2 case, i.e., we wish to that it is also possible to have, instead of the antisqueezer † estimate Sˆ (z) = Sˆ(−z), an output squeezer Sˆ(z ) where |z | =|z|,but z and z have opposite complex phases. Analysis of this sort ϕ = ϕ + ϕ , w1 1 w2 2 (17) of interferometer protocol is beyond the scope of this Rapid in a two-mode interferometer, for assigned weights w , Communication. Remarkably, given the parameter of the first 1 squeezer, increasing the parameter of the second one can w2 0, w1 + w2 = 1. One possible choice for U which sat- isfies Eq. (5)is compensate for detection losses [33]. A final comment is √ √ in order. We have shown that by a simply implementable w w setup, with a single-mode squeezed state and on-off detectors, U = √ 1 √ 2 . (18) w2 − w1 one can attain Heisenberg-limited sensitivity δϕ ∝ 1/N¯ in the presence of an arbitrary number of unknown phases. A This is just the matrix describing a beam splitter of reflectivity detailed analysis of the quantum Fisher information matrix = = R w1 and transmittivity T w2, therefore the interfero- and of the quantum Cramér-Rao bound—that will be deferred metric setup is simply that of a Mach-Zehnder interferometer to a more technical publication in order not to obscure the ϕ ϕ (see Fig. 2). Remarkably, here both phases 1 and 2 are main point of the work—can reveal what is the optimal pref- unknown, differently from previous proposals where only one actor and whether it is attained already by our simple setup. parameter is unknown [1,45]. In conclusion, our protocol can achieve Heisenberg-limited Even more interestingly, the scheme in Fig. 2 is sensitive sensitivity for distributed quantum metrology while being ϕ to the sum, rather than the difference, of the phases 1 and well within the realm of current quantum optical technologies. ϕ with positive weights w and w , respectively. To see how 2 1 2 Acknowledgments. This project was partially supported this is possible, let us set w = w = 1/2 for simplicity, and 1 2 by the Office of Naval Research (ONR) Global (Award No. let us consider the optical unitary transformation describing N62909-18-1-2153). D.G. is supported by the University of the balanced Mach-Zehnder, Portsmouth. P.F. is partially supported by Istituto Nazionale i ϕ −ϕ ˆ − i ϕ +ϕ ˆ ( 1 2 )Jy ( 1 2 )N di Fisica Nucleare (INFN) through the project “QUANTUM,” UˆMZ = e 2 e 2 , (19) and by the Italian National Group of Mathematical Physics ˆ =−i † − † where Jy 2 (â1aˆ2 aˆ1aˆ2 )[2]. As we can see, the output of Istituto Nazionale di Alta Matematica (GNFM-INdAM). state |out=UˆMZ|in does depend, in general, on both ϕ1 F.N. is supported by the Naval Postgraduate School Research and ϕ2, however, the information on the sum of the phases can Initiation Program.

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