Distributed Quantum Metrology with a Single Squeezed-Vacuum Source

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 relys upon a single squeezed state source, an antisqueezing operation at the interferometer output, and on-off photodetectors.

A novel scheme was recently proposed to tackle distributed quantum metrology with Heisenberg-limited sensitivity [32]. However, its main limitation is the fact that it relies on two Fock states with a large number of photons as probes in order to achieve Heisenberg-limited sensitivity. Schemes to make high-photon-number Fock states do not currently exist. Furthermore, it requires a number of auxiliary interferometric channels up to the number M of unknown distributed phases, and photon-number-resolving detectors. Therefore, devising measurement schemes which can exhibit supersensitivity while making use of probe states which are simple to produce in the laboratory with current technology is a matter of great interest.
We overcome these limitations by introducing an interferometric scheme (Fig. 1) which employs only a single squeezed source and on-off photodetectors. Indeed, squeezed states of light are a natural candidate for Heisenberg-limited probing [33][34][35], on account of their experimental availability with a high mean photon number and their nonclassical character. While such states have been largely used to yield supersensitivity in the estimation of a single unknown parameter, their quantum metrological advantage in the case of multiple distributed parameters has not yet been fully explored [13,14]. Here, we demonstrate how a simple M-channel linear optical interferometer with only a single squeezed-vacuum source and on-off photodetectors can achieve Heisenberg-limited sensitivity in distributed quantum metrology with M unknown phase delays. Remarkably, such a scheme can be implemented experimentally with present quantum optical technologies.
The optical interferometer. We describe here in details an interferometric setup ( Fig. 1) able to estimate the combination of M unknown phases ϕ j ( j = 1, . . . , M) for any given set of non-negative weights {w j } M j=1 [36]. Without loss of generality 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 prepared in the squeezed-vacuum state where | = |0 1 · · · |0 M is the vacuum state,Ŝ 1 (z) = e 1 2 (z * â2 1 −zâ †2 1 ) is the squeezing operator,â 1 is the photonic annihilation operator of the first mode, and z is the squeezing parameter. The squeezing parameter z fixes the mean number The probe travels through the first linear optical transformation, described by the unitary operatorÛ through the equationÛ †â where U is an M × M unitary matrix associated with the transition amplitudes from the channel j to the channel i, with i, j = 1, . . . , M. We set these amplitudes to where j = 1, . . . , M and the w i 's are the weights of Eq. (1). This can always be achieved with an appropriate combination of beam splitters [37]. More importantly, this step enables the estimation of ϕ by making the output measurement explicitly dependent on it [see Eq. (12) later on], as well as creating useful entanglement [32], distributed across all channels containing the phase delays ϕ 1 , . . . , ϕ M . After the linear optical transformationÛ the probe undergoes phase shifts ϕ 1 , . . . , ϕ M through the respective channels, and finally evolves through the inverse linear optical transfor-mationÛ † . Reversing the linear optical transformation will allow us to effectively project the output state onto the input state (see below). Thus, given the generator of the phase shifts, the state at the output of our interferometer is Heisenberg-limited estimation. We now demonstrate Heisenberg-limited sensitivity [in Eq. (16)] by means of the observableÔ 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 value ofÔ is the measurement ofÔ is equivalent to projecting onto the vacuum | after the action of an antisqueezing operation on the first channel, described byŜ † 1 (z). This can be experimentally achieved, for instance, by retroreflecting the down-converted photons onto the crystal generating the original squeezed light [38][39][40][41][42], and then using on-off photodetectors.
Since Ô out = | in | out | 2 is the probability of the output state to coincide with the input state, if the phases are small, the total interferometric operator should be close to the identity, and therefore Ô out should be close to one. More precisely, since −|ϕ| maxN Ĝ |ϕ| maxN , with |ϕ| max = max i |ϕ i |, and the interferometer preserves the total number of photons, if we can perform an expansion of Ô out in powers ofĜ. By using the notation Ĝ m U for the expectation value of the operatorĜ m with m = 1, 2 taken at the state | U = U | in , and G 2 U = Ĝ 2 U − Ĝ 2 U , for the variance ofĜ, we obtain up to fourth-order terms (see the first section of the Supplemental Material [43]). By using Eq. (6) and the canonical commutation relations (see the second section of the Supplemental Material [43]), we obtain that the exact expression for the variance ofĜ depends on ϕ in Eq. (1), and on ϕ 2 = j w j ϕ 2 j as where N 2 = in |N 2 | in . The variance ofĜ is made of a contribution from number fluctuations and a contribution from the fluctuations of the phases ϕ j with respect to the weights w j . This result is valid for any (not necessarily Gaussian) Mboson state | in with all modes but the first in the vacuum. In our case, since the first mode is in a squeezed-vacuum state, its photon-number statistics is super-Poissonian [44], with the mean photon number N =N given by (3) and a variance N 2 − N 2 = 2N (N + 1), (13) which scales asN 2 . This scaling is unlike a coherent state which has a Poissonian photon-number statistics with variance equal to the mean N . As we will see, this is an essential ingredient for obtaining a Heisenberg-limited sensitivity. For largeN one gets G 2 U 2N 2 ϕ 2 , from which the expectation value of our observable (11) reads Estimation of the linear combination two unknown phases ϕ 1 , ϕ 2 with weights w 1 , w 2 . The interferometric scheme in Fig. 1 reduces to a Mach-Zehnder interferometer, with unbalanced beam splitters with R/T = w 1 /w 2 , where R is the reflectivity and T is the transmittivity. and differs from 1 by a small quantity, as expected. Indeed, we are in the regime of largeN and small ϕ, such that |ϕ|N |ϕ| maxN is small, in accordance with Eq. (10).
The sensitivity in the estimation of ϕ is obtained by the error propagation formula [7] By using the fact thatÔ =Ô 2 is a projection, and by virtue of Eq. (14), we easily get i.e., the sensitivity scales at the Heisenberg limit.
Example. Let us consider the M = 2 case, i.e., we wish to estimate in a two-mode interferometer, for assigned weights w 1 , w 2 0, w 1 + w 2 = 1. One possible choice for U which satisfies Eq. (5) is This is just the matrix describing a beam splitter of reflectivity R = w 1 and transmittivity T = w 2 , therefore the interferometric setup is simply that of a Mach-Zehnder interferometer (see Fig. 2). Remarkably, here both phases ϕ 1 and ϕ 2 are unknown, differently from previous proposals where only one parameter is unknown [1,45]. Even more interestingly, the scheme in Fig. 2 is sensitive to the sum, rather than the difference, of the phases ϕ 1 and ϕ 2 with positive weights w 1 and w 2 , respectively. To see how this is possible, let us set w 1 = w 2 = 1/2 for simplicity, and let us consider the optical unitary transformation describing the balanced Mach-Zehnder, whereĴ y = − i 2 (â † 1â 2 −â 1â † 2 ) [2]. As we can see, the output state | out =Û MZ | in does depend, in general, on both ϕ 1 and ϕ 2 , however, the information on the sum of the phases can be "washed out" by the choice of the measurement protocol. Indeed, if | in is an eigenstate of the number operatorN, | out depends only on the relative phase ϕ 1 − ϕ 2 , because the second exponential in Eq. (19) gives rise to a global complex phase, and the information on ϕ = (ϕ 1 + ϕ 2 )/2 is completely lost. Furthermore, if one measures an observablê O which commutes withN, then Ô out , as well as all the higher moments Ô k out [2], will again depend on ϕ 1 − ϕ 2 only, sinceÛ † MZÔÛ MZ = e − i 2 (ϕ 1 −ϕ 2 )Ĵ yÔ e i 2 (ϕ 1 −ϕ 2 )Ĵ y . Discussion. We have shown how squeezed light can be used to estimate an arbitrary superposition of phases with nonnegative weights. Our protocol can overcome the limitations of Ref. [32], most notably we can achieve the Heisenberg limit with a single squeezed state rather than two Fock states. Futhermore, our protocol does not necessitate the use of auxiliary channels nor photon-number-resolving detectors. The interferometric setup is easily realizable for any set of weights by using only beam splitters and phase shifters [37]. The initially separable input state acquires entanglement across the various channels where the phase shifts are distributed owing to the linear optical network. The squeezed source can be produced in a number of ways, including spontaneous parametric down-conversion and four-wave mixing. Antisqueezing has already been achieved with high efficiency [38][39][40][41][42] by retroreflecting the down-converted photons and the pump back onto the crystal. We would like to mention that it is also possible to have, instead of the antisqueezer S † (z) =Ŝ(−z), an output squeezerŜ(z ) where |z | = |z|, but z and z have opposite complex phases. Analysis of this sort of interferometer protocol is beyond the scope of this Rapid Communication. Remarkably, given the parameter of the first squeezer, increasing the parameter of the second one can compensate for detection losses [33]. A final comment is in order. We have shown that by a simply implementable setup, with a single-mode squeezed state and on-off detectors, one can attain Heisenberg-limited sensitivity δϕ ∝ 1/N in the presence of an arbitrary number of unknown phases. A detailed analysis of the quantum Fisher information matrix and of the quantum Cramér-Rao bound-that will be deferred to a more technical publication in order not to obscure the main point of the work-can reveal what is the optimal prefactor and whether it is attained already by our simple setup. In conclusion, our protocol can achieve Heisenberg-limited sensitivity for distributed quantum metrology while being well within the realm of current quantum optical technologies.