Multistep Two-Copy Distillation of Squeezed States via Two-Photon Subtraction

Stephan Grebien, Julian Göttsch, Boris Hage, Jaromír Fiurášek, and Roman Schnabel

Phys. Rev. Lett. 129, 273604 – Published 29 December 2022

Abstract

Squeezed states are nonclassical resources of quantum cryptography and photonic quantum computing. The higher the squeeze factor, the greater the quantum advantage. Limitations are set by the effective nonlinearity of the pumped medium and energy loss on the squeezed states produced. Here, we experimentally analyze for the first time the multistep distillation of squeezed states that in the ideal case can approach an infinite squeeze factor. Heralded by the probabilistic subtraction of two photons, the first step increased our squeezing from 2.4 to 2.8 dB. The second step was a two-copy Gaussification, which we emulated. For this, we simultaneously measured orthogonal quadratures of the distilled state and found by probabilistic postprocessing an enhancement from 2.8 to 3.4 dB. Our new approach is able to increase the squeeze factor beyond the limit set by the effective nonlinearity of the pumped medium.

Received 15 June 2022
Accepted 5 December 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.273604

Physics Subject Headings (PhySH)

Quantum optics

Atomic, Molecular & Optical

Authors & Affiliations

Stephan Grebien¹, Julian Göttsch¹, Boris Hage², Jaromír Fiurášek ³, and Roman Schnabel ^1,*

¹Institut für Laserphysik & Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
²Institut für Physik, Universität Rostock, 18051 Rostock, Germany
³Department of Optics, Faculty of Science, Palacký University, 17. listopadu 12, 77900 Olomouc, Czech Republic

^*roman.schnabel@physnet.uni-hamburg.de

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Issue

Vol. 129, Iss. 27 — 30 December 2022

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Images

Figure 1
Potential of our distillation protocol. Variances of squeezed quadrature uncertainties $Δ^{2} \hat{Y}$ as a function of the squeeze parameter $r_{in}$ before distillation. A: Initial, pure Gaussian squeezed vacuum state. B: Two-photon subtracted non-Gaussian squeezed vacuum state. C: Gaussian squeezed vacuum state in the asymptotic limit of multistep distillation Gaussification. For an initially pure 3 dB squeezed state ( $r_{in} = r_{th} \approx 0.3466$ ), $Δ^{2} \hat{Y} \to 0$ . The Gaussification converges only for $r < r_{th}$ , where curve $C$ ends. Note, our experiment (red arrows) started from a slightly mixed Gaussian state.
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Figure 2
Optical setup. Resonator-enhanced parametric down-conversion (PDC) produced a beam of subsequent modes in identical squeezed vacuum states. 10% of the states’ energy was tapped and distributed onto two superconducting nanowire single-photon detectors (SNSPD1,2). 90% of the optical energy was also split and absorbed by two balanced homodyne detectors (BHD1,2) that simultaneously measured the quadratures ${\hat{X}}^{Q}$ and ${\hat{Y}}^{Q}$ . The subscript “ $Q$ ” indicates data taken on halves of the beam. An interference filter (IF) and two optical filter cavities of different lengths (FC1,2) rejected the optical spectrum outside the BHD bandwidth. LO: continuous-wave local oscillator (1064 nm); PS: phase shifter; SH: second-harmonic pump field (532 nm).
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Figure 3
Variances after two subtracted photons. Shown are the results of two independent measurement runs with slightly different initial squeeze factors. The quadrature variances $Δ^{2} {\hat{X}}^{Q} (t)$ (top, from BHD1) and $Δ^{2} {\hat{Y}}^{Q} (t)$ (bottom, from BHD2) include the time when both SNSPDs clicked, to which both $x$ axes are referenced to ( $t = 0$ ). The traces are calculated from $10^{6}$ individual measurements on halves of the beam and represent data from which the Husimi $Q$ function [36] can be calculated. Around $t = 0$ , the antisqueezing as well the squeezing are enhanced, which represents the distillation success of the first step of our protocol. Note that the data include frequencies outside the bandwidth of the squeezing resonator. This dilutes the actual squeeze factor from $Δ^{2} {\hat{Y}}^{Q} (t) \approx 0.79$ (2.4 dB) to about 0.866 in #2.
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Figure 4
Reconstructed Wigner functions. (a) The initial 2.4 dB-squeezed vacuum state. (b) The initial state distilled by the subtraction of two photons yielding 2.8 dB squeezing. (c) Example of the subsequently two-copy-two-step-distilled and Gaussified squeezed vacuum state for $\bar{n} = 1.3$ having 3.14 dB squeezing, see Fig. 5. The Gaussification step reduced the ensemble size to the fraction $P_{svv} = 0.246$ . Additional two-copy distillation steps are possible in principle if the amount of samples is sufficiently high. The “mode” is defined by the temporal shape $f (t)$ and its Fourier transform limited spectrum.
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Figure 5
Distilled squeezed variance as in Fig. 4 versus success probability of the Gaussification step $P_{svv}^{G}$ . Our best result of about 0.453 ( $\approx - 3.4 dB$ ) occurs when the ensemble size decreases most.
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