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Realizing a scalable building block of a U(1) gauge theory with cold atomic mixtures

Alexander Mil, Torsten V. Zache, Apoorva Hegde, Andy Xia, Rohit P. Bhatt, Markus K. Oberthaler, Philipp Hauke, Jürgen Berges, Fred Jendrzejewski

arXiv:1909.07641v1cond-mat.quant-gashep-phphysics.atom-phquant-ph

TL;DR

A scalable implementation of an extended U(1) gauge theory requires localizing U(1) symmetry on building blocks associated with links and neighboring matter fields. The platform combines 23Na–7Li atomic mixtures, inter-species spin-changing collisions, and optical-lattice localization to realize gauge-invariant elementary building blocks. The building-block Hamiltonian describes the experimental data with phenomenological damping, with a fitted decoherence time of 1/γ = 46(12) ms.

Abstract

from arXiv · show

In the fundamental laws of physics, gauge fields mediate the interaction between charged particles. An example is quantum electrodynamics -- the theory of electrons interacting with the electromagnetic field -- based on U(1) gauge symmetry. Solving such gauge theories is in general a hard problem for classical computational techniques. While quantum computers suggest a way forward, it is difficult to build large-scale digital quantum devices required for complex simulations. Here, we propose a fully scalable analog quantum simulator of a U(1) gauge theory in one spatial dimension. To engineer the local gauge symmetry, we employ inter-species spin-changing collisions in an atomic mixture. We demonstrate the experimental realization of the elementary building block as a key step towards a platform for large-scale quantum simulations of continuous gauge theories.

EXPERIMENTAL IMPLEMENTATION

The experiment prepares and probes a 23Na–7Li Bose–Einstein-condensate mixture, using controlled spin exchange and imaging to measure matter-state population transfer. External dynamics and detection limits constrain the observable and observation time.

EXPERIMENTAL IMPLEMENTATION

  • The mixture uses 23Na and 7Li condensates in an optical dipole trap, with gravitational sag reduced to approximately 1.1 µm and atom numbers of about 300 × 10^3 and 50 × 10^3.The trap frequencies are specified separately for sodium and lithium, with lithium frequencies 2.08 times higher.Source: EXPERIMENTAL IMPLEMENTATION · p. 6
  • Heteronuclear spin transfer is enabled near B0, where sodium and lithium transition frequencies approach each other at approximately 1.45 MHz.The relevant sodium and lithium states are |↓⟩, |↑⟩, |v⟩, and |p⟩, and an offset field BA = 2.118(2) G lifts degeneracies.Source: EXPERIMENTAL IMPLEMENTATION · p. 6
  • A two-pulse microwave transition sequence drives sodium toward a desired superposition of |↑⟩ and |↓⟩ while lithium remains spin polarized during the quench, with atoms initially prepared in |↓⟩ and |v⟩.It uses sodium |F = 2, mF = 0⟩ as an intermediate state; the second pulse is fixed at 100 µs and is a π-pulse.Source: EXPERIMENTAL IMPLEMENTATION · p. 6
  • The experiment measures lithium transfer into |p⟩ through Stern–Gerlach separation and absorption imaging after evolution times of up to 100 ms.The spin populations are extracted by integrating the separated density distributions.Source: EXPERIMENTAL IMPLEMENTATION · p. 6
  • The observable is the relative lithium population Np/N, and fringe removal reduces the experimental noise floor below 1%.Data points average at least three measurements in Fig. 3 and ten in Fig. 4, with Fig. 4 binned in 5% intervals of Lz/L.Source: EXPERIMENTAL IMPLEMENTATION · p. 6
  • Sodium population changes from spin-changing collisions are expected to be at most 1%, while external sodium-component dynamics vary the density distribution and bias imaging calibration.These effects currently prevent detection of the sodium population change with the imaging routine.Source: EXPERIMENTAL IMPLEMENTATION · p. 6

MICROSCOPIC HAMILTONIAN

The microscopic model describes two-component 23Na and 7Li Bose gases with free single-particle dynamics and spin-conserving or spin-changing interactions. Atomic masses, trapping potentials, Zeeman shifts, and scattering lengths determine the Hamiltonian parameters.

MICROSCOPIC HAMILTONIAN

  • The total Hamiltonian splits into free terms for sodium and lithium and interaction terms comprising sodium–sodium, lithium–lithium, and sodium–lithium contributions.The model uses two internal states, mF = 0 and 1, from the F_s = 1 hyperfine manifolds.Source: MICROSCOPIC HAMILTONIAN · p. 6
  • Each free-species Hamiltonian contains kinetic energy, trapping potential, and the Zeeman shift E_s,α(B) generated by the external magnetic field.The Zeeman shift is given by the Breit–Rabi formula, and α denotes the internal mF state.Source: MICROSCOPIC HAMILTONIAN · p. 7
  • Spin-conserving intra-species and inter-species interactions are parameterized by interaction constants g_s and corresponding scattering lengths.The supplied Hamiltonian description lists separate sodium, lithium, and mixed-species interaction parameters.Source: MICROSCOPIC HAMILTONIAN · p. 7
  • Heteronuclear spin-changing collisions provide the interaction channel that transfers population between the selected sodium and lithium spin states.Their strength is characterized by the scattering length aSCC = 0.35aB.Source: MICROSCOPIC HAMILTONIAN · p. 7

PROPOSED EXTENDED GAUGE THEORY

The proposal realizes a one-dimensional U(1) lattice gauge theory from localized atomic-mixture building blocks, with gauge fields represented by spins and matter fields on neighboring sites. Laser-assisted tunneling connects these blocks while preserving Gauß’ law.

PROPOSED EXTENDED GAUGE THEORY

  • Each deep-lattice well forms an elementary building block containing both matter species, with negligible inter-well tunneling and interactions.The extended system is assembled by coupling these blocks through matter-field tunneling.Source: PROPOSED EXTENDED GAUGE THEORY · pp. 7–8
  • Gauge fields are encoded through a Schwinger representation of angular-momentum operators, with the spin z-component serving as a discrete electric field.The continuous gauge-field limit is approached using long spins.Source: PROPOSED EXTENDED GAUGE THEORY · p. 7
  • Matter fields occupy lattice sites while gauge fields reside on links, and their charge enters local Gauß’ law operators.The construction associates matter components with neighboring sites and a link-centered building block.Source: PROPOSED EXTENDED GAUGE THEORY · p. 7
  • The building-block construction ensures Gauß’ law and localizes the U(1) symmetry through conservation of total magnetization.This localization links one block to a gauge link and matter components on neighboring sites.Source: PROPOSED EXTENDED GAUGE THEORY · p. 7
  • The proposal identifies a feature as key to its scalability.Source: PROPOSED EXTENDED GAUGE THEORY · p. 8
  • Laser-assisted tunneling connects the building blocks into an extended one-dimensional system while respecting Gauß’ law.With fermions replacing bosons, the corresponding first-line Hamiltonian becomes lattice QED with Wilson fermions.Source: PROPOSED EXTENDED GAUGE THEORY · p. 8

THE BUILDING BLOCK

The experiment isolates a single atomic-mixture building block containing two matter components and one gauge spin. Its fixed conserved quantities yield a gauge-invariant Hamiltonian that commutes with neighboring reduced Gauß’ law operators.

THE BUILDING BLOCK

  • The experimental building block contains two matter components, one gauge spin, and fixed total magnetization and particle numbers.The microscopic Hamiltonian conserves magnetization and the particle numbers of both species.Source: THE BUILDING BLOCK · p. 8
  • The reduced building-block Hamiltonian is gauge-invariant because it commutes with the two Gauß’ law operators associated with neighboring lattice sites.The fixed conserved quantities simplify the Hamiltonian up to additive constants.Source: THE BUILDING BLOCK · p. 8
  • The effective parameters depend on interaction constants, mean-field shifts, single-particle energy differences, and Wannier-function overlap integrals.The precise energy levels and interaction constants depend on the optical lattice and its Wannier functions.Source: THE BUILDING BLOCK · p. 8
  • The experiment tunes magnetization through the initial gauge-spin projection and magnetic field, producing a corresponding decomposition of the matter-component energy difference.The decomposition is used for small magnetic-field changes around a reference field and fixed lithium initial conditions.Source: THE BUILDING BLOCK · p. 8

EFFECTIVE DESCRIPTION OF THE BUILDING BLOCK DYNAMICS

The building-block dynamics are modeled as two coupled effective spins with phenomenological damping and fitted field-dependent parameters. The model reproduces the observed data qualitatively, including resonance shifts, while microscopic estimates have the correct signs but overestimate measured values.

EFFECTIVE DESCRIPTION OF THE BUILDING BLOCK DYNAMICS

  • All experimental data are well described by the mean-field building-block Hamiltonian supplemented with phenomenological damping.The damping accounts for decoherence of the observed oscillations.Source: EFFECTIVE DESCRIPTION OF THE BUILDING… · pp. 8–9
  • 1/γ = 46(12) ms, corresponding to γ/2π = 3.54(94) Hz, characterizes the fitted decoherence time and damping rate.Experimental imperfections and quantum fluctuations contribute to decoherence and parameter renormalization.Source: EFFECTIVE DESCRIPTION OF THE BUILDING… · p. 9
  • The dynamics are solved as two coupled effective spins, with damping-modified equations fitted to particle-production data across initial spin and magnetic-field conditions.The observable is the particle number, and model parameters are obtained from sequential fits to the resonance datasets.Source: EFFECTIVE DESCRIPTION OF THE BUILDING… · p. 9
  • Microscopic estimates exceed measured parameter values by factors of about 2 to 8, but their signs agree and predict the qualitative shifts with magnetic field and initial spin.The comparison indicates systematic uncertainty in the microscopic parameters while retaining the correct order of magnitude.Source: EFFECTIVE DESCRIPTION OF THE BUILDING… · p. 9
  • The resonance condition 2χLz,N ∼ ∆ explains the observed resonance qualitatively, while nonlinearities prevent accurate prediction of the peak position.The condition follows from matching the uncoupled spin-rotation frequencies when Lz,N remains approximately constant.Source: EFFECTIVE DESCRIPTION OF THE BUILDING… · p. 9
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