NOON states with ultracold bosonic atoms via resonance- and chaos-assisted tunneling

G. Vanhaele and P. Schlagheck

Phys. Rev. A 103, 013315 – Published 19 January 2021

Abstract

We investigate theoretically the generation of microscopic atomic NOON states, corresponding to the coherent $| N, 0 〉 + | 0, N 〉$ superposition with $N \sim 5$ particles, via collective tunneling of interacting ultracold bosonic atoms within a symmetric double-well potential in the self-trapping regime. We show that a periodic driving of the double well with suitably tuned amplitude and frequency parameters allows one to substantially boost this tunneling process without altering its collective character. The timescale to generate the NOON superposition, which corresponds to half the tunneling time and would be prohibitively large in the undriven double well for the atomic populations considered, can thereby be drastically reduced, which renders the realization of NOON states through this protocol experimentally feasible. Resonance- and chaos-assisted tunneling are identified as key mechanisms in this context. A quantitative semiclassical evaluation of their impact on the collective tunneling process allows one to determine the optimal choice for the driving parameters in order to generate those NOON states as fast as possible.

Received 27 August 2020
Revised 11 December 2020
Accepted 17 December 2020

DOI:https://doi.org/10.1103/PhysRevA.103.013315

Physics Subject Headings (PhySH)

Bose gases Cold gases in optical lattices Dynamical tunneling

Nonlinear DynamicsAtomic, Molecular & Optical

Authors & Affiliations

G. Vanhaele and P. Schlagheck

CESAM Research Unit, University of Liege, 4000 Liège, Belgium

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Vol. 103, Iss. 1 — January 2021

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Images

Figure 1
(a) Sequential and [(b) and (c)] collective tunneling of ultracold bosonic atoms in a double-well potential, in the [(a) and (b)] absence and (c) presence of a periodic driving. (a)–(c) A measurement process of the number of atoms within individual wells is numerically simulated for various evolved times, based on the numerical time evolution generated by the Hamiltonian (7). (d)–(f) Corresponding phase spaces in the mean-field approximation [see Eqs. (4) and (5) or the Hamiltonian (10)]. The population imbalance and the phase difference between sites 1 and 2 are $z = (N_{1} - N_{2}) / 2$ and $ϕ = θ_{1} - θ_{2}$ , respectively. The red trajectories are the classical counterparts of the upper quantum dynamics. [(a) and (d)] In the absence of interaction, tunneling occurs one by one, leading to Josephson oscillations. [(b) and (e)] In the presence of interaction ( $U = 20 J$ ), collective tunneling takes place, but on a very long timescale $τ = 6.0 \times 10^{5} ℏ / J$ . The phase space displays two qualitatively different dynamics, namely, Josephson oscillation for low population imbalances and self-trapping for high population imbalance. [(c) and (f)] In the presence of a periodic shaking characterized by an amplitude $δ / J = 19.5$ and a frequency $ℏ ω / J = 20$ , collective tunneling occurs on a much shorter timescale $τ = 1.9 \times 10^{2} ℏ / J$ . The presence of a central chaotic layer and a 1:4 resonance roughly situated at $z = \pm 2$ is able to explain the decrease of the NOON time.
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Figure 2
NOON doublet in the Floquet spectrum, as a function of the driving amplitude: (a) a block of the periodic Floquet spectrum (with even-parity levels being marked in black and odd-parity levels in red), (b) a close-up of the block in (a), and (c) the NOON time calculated from (b) and the relation (3). The minimal $τ$ appears roughly in $δ / J ≃ 19.5$ , which was used to produce Figs. 1 and 1. This result is robust in the sense that a range between $δ / J ≃ 12$ and 22 can be chosen to observe the reduction by several orders of magnitude of the NOON time. The parameters are $N_{p} = 5, U / J = 20$ , and $ℏ ω / J = 20$ .
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Figure 3
Overlap between the quasimodes $| n, N_{p} - n 〉$ ( $n = 0, 1, 2, 3, 4, 5$ ) and the Floquet eigenstate according to the relation (15). Even for the perturbed system in (b), the two-level approximation can still be justified [see Eq. (2)] as $M_{0} = M_{N_{p}} \approx 0.5$ . [(c) and (d)] Detection probabilities knowing that the system is prepared in $| 0, 5 〉$ . While (a) and (c) show the unperturbed case, (b) and (d) display the situation with a frequency $ℏ ω / J = 20$ that produces a chaotic layer and a 1:4 resonance [see Fig. 1].
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Figure 4
Evolution of the phase space with $N_{p}$ . The nonlinear parameter $(N_{p} + 1) U / J$ and $ℏ ω / J$ are kept constant in order to preserve the phase space for $δ = 0$ . Here $δ / J$ is chosen such that a minimum of the NOON time is reached for $N_{p} = 5, 6, 7$ . We choose $δ / J = 19.5, 50, 44$ , respectively. Note that for $N_{p} = 6$ and 7 the NOON states (highlighted by red shading on the phase space) are separated from the chaotic sea by a partial barrier.
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Figure 5
NOON states in a near-integrable regime for $U / J = 20$ : [(a) and (b)] stroboscopic sections of the phase space, [(c) and (d)] admixtures of quasimodes, and [(e) and (f)] time evolution of the detection probabilities. The left column represents the case where a 2:1 resonance situated in $z_{2 : 1} = 1.5$ couples the quasimodes $| 5, 0 〉$ and $| 3, 2 〉$ ( $δ / J = 75$ and $ℏ ω / J = 120$ ), while the right column represents the case where the 1:1 resonance located in $z_{1 : 1} ≃ - 2$ couples the quasimodes $| 0, 5 〉$ and $| 1, 4 〉$ ( $δ / J = 12$ and $ℏ ω / J = 80$ ). For the latter, the two-level approximation does not hold and the dynamic takes place on two timescales [see the relation (16) for (d) and (f)]. For the 2:1 resonance case, the NOON time is given by $τ = 2.4 \times 10^{3} ℏ / J$ , with an excellent purity $p = 0.999$ , while $τ_{s}$ is equal to 112.3 $ℏ / J$ in the 1:1 case.
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Figure 6
Close-up of $τ_{s}$ of the detection probabilities for the 1:1 resonance case [see Fig. 5]. Even though this configuration is not able to produce a perfectly unbiased NOON state ( $P_{| 0, 5 〉} = P_{| 5, 0 〉} = 0.5$ ), a slightly biased NOON state with almost no impurity, i.e., $P_{| 0, 5 〉} + P_{| 5, 0 〉} ≃ 1$ and $P_{| 0, 5 〉} - P_{| 5, 0 〉} = 8.5 \times 10^{- 3}$ , can be realized by choosing the measurement time slightly larger than $τ_{s} = 112.3 ℏ / J$ , namely, $t_{m} = 113.25 ℏ / J$ .
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