Precise wave-function engineering with magnetic resonance

P. B. Wigley, L. M. Starkey, S. S. Szigeti, M. Jasperse, J. J. Hope, L. D. Turner, and R. P. Anderson

Phys. Rev. A 96, 013612 – Published 10 July 2017

Abstract

Controlling quantum fluids at their fundamental length scale will yield superlative quantum simulators, precision sensors, and spintronic devices. This scale is typically below the optical diffraction limit, precluding precise wave-function engineering using optical potentials alone. We present a protocol to rapidly control the phase and density of a quantum fluid down to the healing length scale using strong time-dependent coupling between internal states of the fluid in a magnetic field gradient. We demonstrate this protocol by simulating the creation of a single stationary soliton and double soliton states in a Bose-Einstein condensate with control over the individual soliton positions and trajectories, using experimentally feasible parameters. Such states are yet to be realized experimentally, and are a path towards engineering soliton gases and exotic topological excitations.

Received 6 January 2015

DOI:https://doi.org/10.1103/PhysRevA.96.013612

Physics Subject Headings (PhySH)

Solitons

Bose-Einstein condensates Quantum fluids & solids

Atomic, Molecular & Optical

Authors & Affiliations

P. B. Wigley¹, L. M. Starkey², S. S. Szigeti³, M. Jasperse², J. J. Hope¹, L. D. Turner², and R. P. Anderson^2,*

¹Department of Quantum Science, The Australian National University, Australian Capital Territory 0200, Australia
²School of Physics and Astronomy, Monash University, Victoria 3800, Australia
³Australin Research Council Centre of Excellence for Engineered Quantum Systems, The University of Queensland, Queensland 4072, Australia

^*russell.anderson@monash.edu

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Vol. 96, Iss. 1 — July 2017

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Images

Figure 1
MRC protocol applied to a pseudospin- $1 / 2$ condensate, showing the position-dependent level splitting, spin state, phases $arg [ψ_{↓, ↑} (z)]$ , and densities ${|ψ_{↓, ↑} (z)|}^{2}$ . (a) Initial wave function: The condensate begins in state $|↓〉$ with uniform phase. (b) First pulse: A magnetic gradient is applied, and a HS pulse transfers the left side of the condensate to state $|↑〉$ with slice sharpness $δ z$ . The phase of $|↓〉$ (blue, solid line) acquired during the first pulse is dominated by the gradient, with the shallower phase variation of $|↑〉$ (red, dashed line) owing to the spin superposition during the pulse. (c) Phase accumulation: The magnetic gradient is removed, and a uniform $π$ phase difference between the two spin components is allowed to accumulate. (d) Second pulse: The magnetic gradient is inverted and a second pulse transfers the left side back to $|↓〉$ , imprinting a phase step of $π$ onto this component. The finite sharpness $δ z$ of the transferred slice carves a notch in the density ${|ψ_{↓} (z)|}^{2}$ , filled by residual $|↑〉$ population. MRC enables independent control of the width of the density notch and the height of the phase step.
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Figure 2
A single black soliton is engineered using the pulse parameters $Ω_{0} / 2 π = 3.00 MHz, μ = 3.87, Γ = 3.86$ , and $α = 0.0104$ , resulting in a single pulse duration of $t_{p} = 8.35 μ s$ (cf. $t_{ξ} = 42 μ s$ ). A gradient of $|d B / d z| = 6992.66 G / cm$ was applied during the two pulses. (a) The coupling and (b)–(c) the densities ${|ψ_{↓, ↑} (z)|}^{2}$ , respectively, during the protocol. (d) The density and phase of the condensate at the conclusion of the protocol. A $π$ phase discontinuity was formed by applying a detuning of $Δ_{ϕ} / 2 π = 490.0 kHz$ during the wait time of $t_{ϕ} = 1.0 μ s$ between pulses. The notch in density ${|ψ|}^{2}$ has a FWHM of $\sim 550 nm$ . (e) The total density of the condensate over five trap periods, demonstrating stationary evolution of the black soliton. We note that the first hyperbolic secant pulse forms a spin domain wall [39, 40].
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Figure 3
The MRC protocol can create two solitons by simply decreasing the width of the transferred slice using $Δ_{0}$ . By varying the offset $|Δ_{1}|$ , the position of the solitons can be precisely controlled. (a) Two solitons initially located at $z = - 6.6 μ m$ and $z = - 8.1 μ m$ , formed using $μ = 0.12, Γ = 124.8$ , and an offset of $|Δ_{1}| / (2 π) = 3.60 MHz$ , exhibit in-phase oscillations. (b) If we instead use $μ = 0.659, Γ = 22.69$ , and $|Δ_{1}| / (2 π) = 1.98 MHz$ , the two solitons are formed at $z = 0.0 μ m$ and $z = - 8.1 μ m$ and oscillate asymmetrically. All other parameters are the same as outlined in Fig. 2.
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