Spinor Bose-Einstein-condensate phase-sensitive amplifier for SU(1,1) interferometry

J. P. Wrubel, A. Schwettmann, D. P. Fahey, Z. Glassman, H. K. Pechkis, P. F. Griffin, R. Barnett, E. Tiesinga, and P. D. Lett

Phys. Rev. A 98, 023620 – Published 16 August 2018

Abstract

The SU(1,1) interferometer was originally conceived as a Mach-Zehnder interferometer with the beam splitters replaced by parametric amplifiers. The parametric amplifiers produce states with correlations that result in enhanced phase sensitivity. $F = 1$ spinor Bose-Einstein condensates (BECs) can serve as the parametric amplifiers for an atomic version of such an interferometer by collisionally producing entangled pairs of $| F = 1, m = \pm 1 〉$ atoms. We simulate the effect of single- and double-sided seeding of the inputs to the amplifier using the truncated-Wigner approximation. We find that single-sided seeding degrades the performance of the interferometer exactly at the phase the unseeded interferometer should operate the best. Double-sided seeding results in a phase-sensitive amplifier, where the maximal sensitivity is a function of the phase relationship between the input states of the amplifier. In both single- and double-sided seeding we find there exists an optimal phase shift that achieves sensitivity beyond the standard quantum limit. Experimentally, we demonstrate a spinor phase-sensitive amplifier using a BEC of ${}^{23}{Na}$ in an optical dipole trap. This configuration could be used as an input to such an interferometer. We are able to control the initial phase of the double-seeded amplifier and demonstrate sensitivity to initial population fractions as small as 0.1%.

Received 4 May 2018

DOI:https://doi.org/10.1103/PhysRevA.98.023620

Physics Subject Headings (PhySH)

Atom interferometry Entanglement in quantum gases Spinor Bose-Einstein condensates

Quantum Monte Carlo

Atomic, Molecular & Optical

Authors & Affiliations

J. P. Wrubel^1,2, A. Schwettmann^2,3, D. P. Fahey², Z. Glassman², H. K. Pechkis^2,4, P. F. Griffin^2,5, R. Barnett^6,7, E. Tiesinga², and P. D. Lett²

¹Department of Physics, Creighton University, 2500 California Plaza, Omaha, Nebraska 68178, USA
²Quantum Measurement Division, National Institute of Standards and Technology, and Joint Quantum Institute, NIST and University of Maryland, 100 Bureau Drive, Gaithersburg, Maryland 20899-8424, USA
³Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA
⁴Department of Physics, California State University, Chico, California 95973, USA
⁵Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom
⁶Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
⁷Department of Physics, Joint Quantum Institute and Condensed Matter Theory Center, University of Maryland, College Park, Maryland 20742, USA

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Vol. 98, Iss. 2 — August 2018

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Images

Figure 1
Configuration of the seeded spinor SU(1,1) interferometer. The initial spinor amplifier (first green box) mixes the pump coherent state ${| α 〉}_{0}$ (dashed black line) with the probe and conjugate coherent seeds ${| α_{\pm} 〉}_{\pm}$ (blue solid lines) depending on the relative spinor phase $θ$ . For a given set of input states, the amplifier gain is determined by the parameters for spinor dynamics $c, q$ , and $t_{SMD}$ (defined in Sec. 2). A shift in the spinor phase within the interferometer $ϕ$ is sensed when a second parametric amplifier (second green box) reverses the dynamics by negating the values of $c$ and $q$ , and evolving the state for the same time $t_{SMD}$ . The output of the interferometer is the number of atoms $N_{+, out} + N_{-, out}$ .
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Figure 2
Lower bound on the scaled Fisher information obtained from TWA simulations versus interferometer phase $ϕ$ for $t_{SMD} = 20 ms$ with (a) increasing percentages of single-sided seeding into $m = - 1$ , and with (b) increasing spinor phase for 0.1% double-sided seeding. The dashed line shows $f (ϕ)$ calculated from Eq. (4) for the unseeded case. The inset to (a) is the amplified fraction after spin-mixing dynamics versus the initial seed fraction $ρ_{seed}$ . The inset to (b) is $f_{<, max}$ versus the initial spinor phase, with colored dots corresponding to the traces shown in the main figure. Vertical dotted lines indicate the phase of the separatrix (sep) $θ_{sep} = \pm 2.86 rad$ that divides the oscillating phase and running phase spinor dynamics [25].
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Figure 3
The lower bound on the scaled Fisher information versus $ρ$ with (a) increasing percentages of single-sided seeding into $m = - 1$ , and with (b) double-sided $0.1 %$ seeding at various initial phases. The solid black line is $f_{Bog,max} = (N + 2)$ , which is the prediction for the unseeded case (open circles). The gray shading indicates the region of scaled Fisher information that does not reach the standard quantum limit $f_{SQL} = 1$ (based on $N$ atoms in the interferometer arms). The dashed line corresponds to $F = N$ (also $f = 1 / ρ$ ), which is the limit achievable using coherent states in a Mach-Zehnder interferometer having a total of $N$ particles. The phases of the separatrix are $θ_{sep} = \pm 2.86 rad$ .
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Figure 4
The atom fraction in $m = \pm 1$ after 14 ms of evolution time versus the phase shift achieved by a microwave pulse. The amplifier is seeded with 1.0% of the atoms in $m = + 1$ (blue squares) or with 1% of the atoms in each of $m = \pm 1$ (black circles) out of $N = 3.6 (5) \times 10^{4}$ atoms. Solid lines are predictions from the single-mode spinor theory. Single-sided seeding represents multiple experiments, and we quote the mean and standard deviation of the mean. Double-sided seeding represents one experiment per phase.
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Figure 5
Experimental atom fraction after amplification for the seeded nonlinear amplifier versus seeded atom fraction in $m = - 1$ , with $m = + 1$ initially populated with a fraction $ρ_{+, seed} = 0.01$ of the atoms. Solid black circles ( $N = 3.4 (3) \times 10^{4}$ , $q / h = - 1$ Hz, $θ = 0.016$ rad, $t_{SMD} = 27 ms$ ) have an initial spinor phase $θ$ causing amplification, while red diamonds ( $N = 7.8 (9) \times 10^{4}$ , $q / h = - 1.4$ Hz, $θ = 2.45$ rad, $t_{SMD} = 25 ms$ ) have an initial phase causing deamplification. The shading indicates the standard deviation of the mean, and the solid black lines are results based on truncated-Wigner approximation simulations. The green dashed line shows the decrease in the standard deviation ( $Δ θ$ ) of the distribution of initial phases from the TWA simulations.
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