Quantum-Enhanced Sensing Based on Time Reversal of Nonlinear Dynamics

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Quantum-Enhanced Sensing Based on Time Reversal of Nonlinear Dynamics

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler

Phys. Rev. Lett. 117, 013001 – Published 28 June 2016

Abstract

We experimentally demonstrate a nonlinear detection scheme exploiting time-reversal dynamics that disentangles continuous variable entangled states for feasible readout. Spin-exchange dynamics of Bose-Einstein condensates is used as the nonlinear mechanism which not only generates entangled states but can also be time reversed by controlled phase imprinting. For demonstration of a quantum-enhanced measurement we construct an active atom SU(1,1) interferometer, where entangled state preparation and nonlinear readout both consist of parametric amplification. This scheme is capable of exhausting the quantum resource by detecting solely mean atom numbers. Controlled nonlinear transformations widen the spectrum of useful entangled states for applied quantum technologies.

Received 24 February 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.013001

Physics Subject Headings (PhySH)

Atom interferometry Cold atoms & matter waves Quantum entanglement

Bose-Einstein condensates

Atomic, Molecular & Optical

Authors & Affiliations

D. Linnemann^1,*, H. Strobel¹, W. Muessel¹, J. Schulz¹, R. J. Lewis-Swan², K. V. Kheruntsyan², and M. K. Oberthaler¹

¹Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
²The University of Queensland, School of Mathematics and Physics, Brisbane, Queensland 4072, Australia

^*timereversal@matterwave.de

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Vol. 117, Iss. 1 — 1 July 2016

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Images

Figure 1
Disentangling with nonlinear time reversal. (a) Our nonlinear readout scheme exploits a time reversal sequence. For this, the Hamiltonian $H$ used for entangled state generation is inverted and reapplied for readout. Interrogation takes place in between both periods of nonlinear dynamics. (b) Time trace of the characteristic variance of $N_{+} = N_{↑} + N_{↓}$ during entangled state generation, interrogation, and time reversal. The initial drastic increase in variance is revoked by nonlinear evolution under the time inverted generation process. A pronounced minimum is found close to matched times, $t_{1} = t_{2}$ . (c) Spin-changing collisions in a Bose-Einstein condensate are used as the nonlinear process. Atom numbers are detected by high resolution absorption imaging after Stern-Gerlach separation. A typical absorption image with counting regions indicated by ellipses is shown. (d) The side-mode population exhibits characteristic thermal-like fluctuations, approaching the variance of the entangled two-mode squeezed vacuum state (diagonal). Results of a truncated Wigner simulation (dashed) and the expected variance of a coherent state (gray) are shown for comparison. (e) Variance of the side-mode population before (red diamonds) and after (blue) time reversal sequence for the matched case of two equal durations of nonlinear dynamics. We find reversion to the initial vacuum state for a wide range of effective evolution times. The red line is a fit to the expected behavior in undepleted pump approximation.
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Figure 2
Interferometry based on (dis-) entangling nonlinear dynamics. (a) Schematic representation of an optical SU(1,1) interferometer and its realization in atom optics. This scheme takes advantage of the entanglement-enabled deamplification of fluctuations by time inversion of parametric amplification (PA). (b) Typical experimental population histograms of $N_{+}$ (black lines are fits to a thermal distribution convolved with our detection noise) for different spinor phase shifts $φ$ applied within the active interferometer. The blue histograms are recorded at the output, while the red one is obtained by omitting the final spin-mixing period. The dark colored bins depict the corresponding mean values, which are plotted in the lower panel (zoom-in into the gray shaded area), revealing the interferometry fringe. The horizontal dark red line denotes the average probe atom number of $⟨ N_{+}^{inside} ⟩ = 2.8 \pm 0.2$ inside the interferometer.
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Figure 3
Quantum-enhanced phase sensitivity with nonlinear readout. (a) The phase sensitivity is experimentally extracted by Gaussian error propagation on $⟨ N_{+} ⟩$ . The standard quantum limit (gray horizontal bar) is surpassed in close vicinity of phase $φ = π$ . At phase $π$ the sensitivity diverges due to the vanishing slope of the signal. The undepleted-pump theory (dashed) additionally taking into account the nonperfect reversibility is shown as a solid line. The observed phase sensitivity agrees with the theoretical model of an active SU(1,1) interferometer over 2 orders of magnitude (inset). (b) Mean side-mode population $⟨ N_{+} ⟩$ in vicinity of the fringe minimum. The signal’s derivative is determined by the sinusoidal fit. (c) Variance of $N_{+}$ at the interferometer output. Our detection noise of $33.5 \pm 1.3$ is indicated by the horizontal dotted line and subtracted for determining the phase sensitivity. Good agreement to the undepleted-pump theory is found when considering the nonperfect reversibility by including an offset (black line).
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