Superfluid rings as quantum pendulums

The gist

A feasible experimental proposal to realize a non-dispersive quantum pendulum is presented.

System Setup and Analogy

The proposed setup involves an ultracold atomic cloud with attractive interatomic interactions loaded into a tilted ring potential, where the classical and quantum domains are switched on by tuned interactions, and the classical dynamical stabilization of unstable states is driven by quantum phase imprinting The ring geometries are omnipresent in physics as they endow systems with periodic boundary conditions Approaching the dissipationless limit, superconductors and superfluids are capable of making the cyclic transport of charge or particles, if not perpetual at least persistent, a particularly striking demonstration of which is the persistent flow of superconducting gravimeters. The present work is devoted to exploring the validity of this analogy between ring geometries and pendulum dynamics in attractively interacting, localized BECs.

Pendulum Dynamics in Ultracold Gases

The tilted ring potential is defined by the gravitational potential as U(τ, θ) = −m g R sin τ cos θ along the azimuthal coordinate θ ∈ [−π, π] of the ring. This potential can alternatively be written as U(τ, θ) = −mω2R2 cos θ, where ω = p g sin τ /R is the pendulum angular frequency. For non-interacting particles, the time evolution is described by the Schrödinger equation with γ = 0. The resulting solutions admit general forms that can be written as superpositions of Mathieu functions ce2n(θ, q) and se2n+1(θ, q).

Lattice Structure and Nonlinear States

The tilting potential can be conceptualized as a periodic lattice potential V0 sin2(πx/d) of amplitude V0 = 2gmR sin τ = 2mω2R2, where d = 2πR is the circumference of the ring. The natural energy scale in a lattice is given by EL = ħ2/8mR2, referred to as the recoil energy. The dimensionless ratio between lattice amplitude and the recoil energy, denoted as s = (1/2) V0/EL = 8m2ω2R4/ħ2, quantifies the lattice depth and matches in absolute value the Mathieu-function parameter s = q. A shallow lattice is characterized by s ≲ 1 while a deep lattice corresponds to s ≫ 1. In the limit of very large s, the lowest eigenstates are similar to those of a harmonic oscillator. For the interacting gas, the system dynamics obey the time-dependent Gross-Pitaevskii equation iħ∂tψ = −ħ2/2mR2∂2θ − mω2R2cos θ + γψ2, where γ = -2ħ2/ma1D < 0 denotes the coupling constant of the attractive interparticle interaction.

Quantum Pendulum Analogy and Stabilization

The close analogy between this system and the classical pendulum is evidenced by the time evolution of expectation values, as obtained from the Ehrenfest equations. The nonlinear equation of the pendulum is d2⟨θ⟩/dt2 + ω2⟨sin θ⟩ = 0. The lattice viewpoint offers valuable insight into the structure of the spectrum of the nonlinear system, allowing for new stationary states called gap solitons which emerge as localized states that occupy a few lattice sites. Gap solitons emerge as states centered at the potential maximum. Furthermore, the inverted position of the pendulum can be stabilized by inducing fast vertical vibrations (along the gravitational field direction) of the pendulum pivot as experimentally shown and mathematically proved by introducing the method of averaging of the fast variables by Pyotr Kapitza.

Gravity Measurement Potential

The system can be used as a gravimeter, relying on measuring oscillation period, hence length, positions and corresponding times tracked over repeated, small oscillations. A precise determination of the period would allow one to use the tilted ring for performing sensitive measurements of gravity. By initially preparing the BEC at the energy maximum θ = π, where small displacements produce large differences in the period, a sensible mass at the surface horizontally located with respect to the ring axis could induce an angular displacement δθ ≈ tan δθ = δg/g. This translates into pendulum periods [see Appendix A, Eq. (A3)] that can be approximated by Tk(δθ) ≈ (2/π)(3 ln 2 − ln δθ) T.

Experimental Realization and Precision

The confinement of frequency ν = 350 Hz has a typical size of ξ ≈ 0.6 µm, and even within a small ring of radius R = 12 µm, the relevant parameter R/ξ = 20 is large enough to closely reproduce the dynamics of a classical pendulum. The uncertainty in measuring the motional period might be reduced by resorting to similar procedures as for the classical reversible pendulums of Kater and Bessel. A precise determination of the period would also allow one to use the tilted ring for performing sensitive measurements of gravity. By initially preparing the BEC at the energy maximum θ = π, where small displacements produce large differences in the period, a sensible mass at the surface horizontally located with respect to the ring axis such that δg and g could act orthogonally would induce an angular displacement δθ ≈ tan δθ = δg/g. This translates into pendulum periods [see Appendix A, Eq. (A3)] that can be approximated by Tk(δθ) ≈ (2/π)(3 ln 2 − ln δθ) T.

Conclusion on Period Differences

The differences between the predicted period and the unperturbed period are understood by considering the Ehrenfest equation d2⟨θ⟩/dt2 + ω2⟨sin θ⟩ = 0, which exactly matches the functional form of the classical equation, only when the density profile of the soliton wave function becomes a Dirac delta function. For a free soliton state Eq. (2) of width ξ and moving center θ0(t), this difference can be approximated by ∆(˜γ) ≈ −π/26 ˜γ2/sin⟨θ⟩, resulting in an effective classical equation with a slightly reduced angular frequency ω(γ˜) = ωp/(1 − π/2/(6 ˜γ2)). This estimate produces δ(γ˜ = 8) = 1.31 × 10−2, and δ(γ˜ = 16) = 3.23 × 10−3, in good agreement with the measured results previously reported. The azimuthal variations of the ring potential can introduce a significant source of uncertainty in the measured observables.

Final Statement

The paper presents a feasible experimental proposal to realize a non-dispersive quantum pendulum.

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Superfluid rings as quantum pendulums Antonio Mu˜noz Mateo Departamento de F´ısica, Universidad de La Laguna, 38200 Tenerife, Spain Grigory E. Astrakharchik Departament de F´ısica, Universitat Politicna de Catalunya, 08034 Barcelona, Spain Departament de F´ısica Quantica i Astrof´ısica, Universitat de Barcelona Mart´ı Franqu˜es 1, 08028 Barcelona, Spain. and Institut de Ciˆencies del Cosmos (ICCUB), Universitat de Barcelona Mart´ı Franqu˜es, 1, 08028 Barcelona, Spain. Bruno Juli´a-Díaz Departament de F´ısica Quantica i Astrof´ısica, Universitat de Barcelona Mart´ı Franqu˜es 1, 08028 Barcelona, Spain. and Institut de Ciˆencies del Cosmos (ICCUB), Universitat de Barcelona Mart´ı Franqu˜es, 1, 08028 Barcelona, Spain.

1. Improved Quantum Pendulum Simulation for Non-Dispersive Systems: The AI system can accurately simulate time evolution of a quantum pendulum using the time-dependent Gross-Pitaevskii equation, allowing researchers to study the validity of this analogy between quantum and classical pendulums in ultracold atomic systems.

2. Enhanced Gravimeter Development: The system can be used as a gravimeter for high precision measurements by tracking the oscillation period, hence length, positions and corresponding times over repeated oscillations, potentially reaching uncertainties of delta g approximately 10-9 g.

3. Dynamical Stabilization Control: The AI can implement fast vertical vibrations to stabilize the inverted position of the pendulum by applying Kapitza stabilization techniques, allowing for the realization of a non-dispersive quantum pendulum that maintains its state despite perturbations.

4. Nonlinear State Characterization: The system can identify and characterize new stationary states, specifically gap solitons, by analyzing nonlinear lattice systems where gap solitons emerge as states centered at the potential maximum.

5. Precision Period Measurement: The AI can estimate the difference between classical and quantum periods of oscillation using the derived formula Tk(delta theta), which is better approached at higher interactions and provides a method for sensitive gravity measurements.

A feasible experimental proposal to realize a non-dispersive quantum pendulum is presented. The proposed setup consists of an ultracold atomic cloud, featuring attractive interatomic interactions, loaded into a tilted ring potential. The classical and quantum domains are switched on by tuned interactions, and the classical dynamical stabilization of unstable states, i.e. a la Kapitza, is shown to be driven by quantum phase imprinting. The potential use of this system as a gravimeter is discussed.