arXiv:1107.3726v1 [quant-ph] 19 Jul 2011 Quantum interferometry for noisy detectors Nicol` o Spagnolo, 1,2 Chiara Vitelli, 1 Vito Giovanni Lucivero, 1 Vittorio Giovannetti, 3 Lorenzo Maccone, 4 and Fabio Sciarrino 1,5, ∗ 1 Dipartimento di Fisica, Sapienza Universit` a di Roma, piazzale Aldo Moro 5, I-00185 Roma, Italy 2 Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, piazzale Aldo Moro 5, I-00185 Roma, Italy 3 NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 4 Dip. Fisica “A. Volta”, INFN Sez. Pavia, Universit` a di Pavia, via Bassi 6, I-27100 Pavia, Italy 5 Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (INO-CNR), largo Fermi 6, I-50125 Firenze, Italy † The sensitivity in optical interferometry is strongly affected by losses during the signal propagation or at the detection stage. The optimal quantum states of the probing signals in the presence of loss were recently found. However, in many cases of practical interest, their associated accuracy is worse than the one obtainable without employing quantum resources (e.g. entanglement and squeezing) but neglecting the detector’s loss. Here we detail an experiment that can reach the latter even in the presence of imperfect detectors: it employs a phase-sensitive amplification of the signals after the phase sensing, before the detection. Since our method uses coherent states as input signals, it is a practical technique that can be used for high-sensitivity interferometry and, in contrast to the optimal strategies, does not require one to have an exact characterization of the loss beforehand. From gravitational wave measurements to the realization of optical gyroscopes, the estimation of an optical phase φ through interferometric experiments is an ubiquitous tech- nique. For each input state of the probe, the maximum accu- racy of the process, optimized over all possible measurement strategies, is provided by the quantum Fisher information H φ through the Quantum Cram´ er-Rao (QCR) bound [1, 2]. The QCR sets an asymptotically achievable lower bound on the mean square error of the estimation δ 2 φ ≥ (MH φ ) −1 , where M is the number of repeated experiments. In the ab- sence of noise and when no quantum effects (like entangle- ment or squeezing) are exploited in the probe preparation, the QCR bound scales as the inverse of the mean photon number, the Standard Quantum Limit (SQL). Better performances are known to be achievable when using non classical input signals [3–9]. However, the difficulty involved in implementing them implies that all experiments up to now have been performed using post-selection, and cannot claim a sub-SQL sensitivity when all resources used in the phase sensing are accounted for [10]. Additionally, in the presence of loss, the SQL can be asymptotically beaten only by a constant factor [11–13], so that sophisticated sub-SQL strategies [6, 14–17] (imple- mented up to now only for few photons) may not be worth the effort. This implies that, for practical high-sensitivity inter- ferometry, the best resource exploitation (or, equivalently, the minimally invasive scenarios) currently entail strategies based on the use of a coherent state |α〉, i.e. a classical signal. Its QCR bound takes the form δφ 2 SQL  (2Mηξ |α| 2 ) −1 , where we consider separately the loss 1 − ξ in the sensing stage and the loss 1 − η in the detectors (η being their quantum effi- ciency). Here we present the experimental realization of a robust phase estimation protocol that improves the above ac- curacy up to ∼ (2Mξ |α| 2 ) −1 , while still using coherent sig- nals as input. It achieves the SQL of a system only affected by the propagation loss 1 − ξ , and not by the detectors’ 1 − η. Our scheme employs a simple, conventional interferometric phase sensing stage that uses coherent-state probes. These are amplified with an optical parametric amplifier (OPA) after the interaction with the sample, but before the lossy detectors. No post-selection is employed to filter [4, 10] the output signal. The OPA (an optimal phase-covariant quantum cloning ma- chine [18]) transfers the properties of the injected state into a field with a larger number of particles, robust under losses and decoherence [19]. Our approach is suitable for the analysis of fragile samples (weak regime of the interaction), since the am- plification acts after the interaction of the probe state with the sample. A similar scheme was used to perform interferometry with single-photon probes [20], but the small intensity of its probing signal could only achieve limited accuracy. Theory - The probe is a H -polarized coherent state |α〉 H |0〉 V , with α = |α|e ıθ . The interaction with the sample induces a phase shift φ between the polarization components and the sample loss 1 − ξ reduces the state amplitude to β = √ ξα: the state evolves to |Ψ β φ 〉 = |e −iφ/2 β cos(φ/2)〉 H |ıe −ıφ/2 β sin(φ/2)〉 V . The QCR bound for this state is δ 2 φ  1/(M 2|β| 2 ). In the absence of ampli- fication, the detection losses 1 − η would increase the QCR to δ 2 φ SQL  1/(M 2η|β| 2 ). To prevent this and to attain the previous bound, we implemented the operations shown in Fig.1a: a λ/4 wave-plate and the OPA, described by the uni- tary U OPA = exp[g(a † 2 H − a † 2 V )/2+h.c.], where g = |g|e iλ is the amplifier gain, and a H and a V are the annihilation op- erators of the two modes. After these operations, the state |Ψ β,g φ 〉 = U OPA |Ψ β φ 〉 is measured by lossy detectors which measure the photon number difference D = n H −n V between the two modes, with n x ≡ a † x a x . The detector loss 1 − η can be described through a loss map L η acting on |Ψ β,g φ 〉, which outputs the mixed state ρ β,g,η φ . The uncertainty on the phase φ can be evaluated [1] as δφ = σ(〈 ˆ D〉)| ∂〈 ˆ D〉 ∂φ | −1 , where 〈D〉 is the expectation value of D on ρ β,g,η φ . A calculation of the sensitivity S = δφ −1 of the whole procedure (see Supplementary Material) shows that S depends on the value of the phase φ to be estimated.