[Paper Review] Integrable atomtronic interferometry

D. S. Grun, L. H. Ymai+4 more|arXiv (Cornell University)|Apr 24, 2020

Neural Networks and Reservoir Computing4 citations

TL;DR

This paper proposes an integrable four-site Bose-Hubbard model for atomtronic interferometry, leveraging exact solvability via the algebraic Bethe Ansatz to derive analytic expressions for quantum dynamics. It demonstrates that the system functions as both a NOON state producer and interferometric identifier, with time evolution equivalent to a controlled-phase gate on two hybrid qudits, enabling Heisenberg-limited sensitivity and robustness against small integrability-breaking perturbations.

ABSTRACT

High sensitivity quantum interferometry requires more than just access to entangled states. It is achieved through deep understanding of quantum correlations in a system. Integrable models offer the framework to develop this understanding. We communicate the design of interferometric protocols for an integrable model that describes the interaction of bosons in a four-site configuration. Analytic formulae for the quantum dynamics of certain observables are computed. These expose the system's functionality as both an interferometric identifier, and producer, of NOON states. Being equivalent to a controlled-phase gate acting on two hybrid qudits, this system also highlights an equivalence between Heisenberg-limited interferometry and quantum information. These results are expected to open new avenues for integrability-enhanced atomtronic technologies.

Motivation & Objective

• To develop a theoretically tractable, integrable model for atomtronic interferometry with high quantum sensitivity.
• To identify how quantum correlations and conserved operators in an integrable system can be harnessed for interferometric state preparation and measurement.
• To establish a direct link between integrable quantum dynamics and quantum information operations, such as controlled-phase gates.
• To analyze the robustness of interferometric performance under small deviations from integrability, modeling realistic experimental noise.

Proposed method

• The study employs the integrable four-site Bose-Hubbard Hamiltonian with long-range dipolar interactions and closed boundary conditions, derived via the Quantum Inverse Scattering Method.
• It identifies two additional conserved operators, $ Q_1 $ and $ Q_2 $, beyond total particle number, ensuring integrability and enabling exact solution via the algebraic Bethe Ansatz.
• In the resonant tunneling regime, an effective Hamiltonian $ H_{\text{eff}} = (N+1)\Omega(Q_1 + Q_2) - 2\Omega Q_1 Q_2 $ is derived using second-order perturbation theory in $ J/U $.
• Analytic formulae for the time evolution of observables, including particle number distributions and fractional imbalances, are computed using Bernstein polynomials and binomial distributions.
• The system's operation is mapped to a two-qudit quantum information processor, with time evolution equivalent to a controlled-phase gate acting on hybrid qudits.
• Robustness is quantified via fidelity between time-evolved states under the ideal Hamiltonian and a perturbed one with $ \epsilon \nu(N_1N_3 + N_2N_4) $, modeling experimental noise.

Experimental results

Research questions

• RQ1 Can an integrable many-body system be used to design high-fidelity quantum interferometric protocols for state identification and preparation?
• RQ2 How do conserved operators in an integrable system enable exact analytic control over quantum dynamics in a four-boson system?
• RQ3 To what extent is the interferometric functionality preserved under small deviations from integrability, such as parameter fluctuations?
• RQ4 What is the quantum information-theoretic interpretation of the system’s dynamics, particularly in terms of entanglement generation and gate equivalence?

Key findings

• The system’s time evolution produces a NOON state with maximal fidelity when the swap operator expectation value reaches $ \pm 1 $, indicating maximal symmetry or antisymmetry in the output state.
• The fractional imbalance correlation $ \mathcal{I}_{\Phi(\phi)}(t) $ exhibits a cosine dependence on phase $ \phi $, with contrast bounded by the swapped fractional imbalance correlation (SFIC), which achieves $ |C_{AB}^{\Psi_0}| = 1 $ at the optimal time $ t = t_m $.
• For even $ N $, the particle number distribution at site 3 follows a binomial distribution with maximal support; for odd $ N $, it collapses to a double-delta function, indicating strong quantum localization.
• The fidelity of the time-evolved state remains above 0.95 for $ \epsilon \leq 0.3 $ in the robustness analysis, indicating resilience to small integrability-breaking perturbations.
• The interferometer is operationally equivalent to a controlled-phase gate acting on two hybrid qudits, establishing a direct bridge between Heisenberg-limited interferometry and quantum information processing.