[Paper Review] Curved spacetimes in the lab

Nikodem Szpak|arXiv (Cornell University)|Oct 6, 2014

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper proposes using ultra-cold atoms in engineered optical lattices to simulate quantum fields in curved spacetimes, leveraging tunable lattice potentials to realize effective geometries with positive, negative, and dynamical curvature. By mapping the discrete hopping Hamiltonian of atoms to the Laplace-Beltrami operator in curved space, the authors demonstrate that realistic optical lattices can emulate compact, curved spacetimes—enabling laboratory studies of relativistic quantum field theory effects such as particle creation and Hawking radiation.

ABSTRACT

We present some new ideas on how to design analogue models of quantum fields living in curved spacetimes using ultra-cold atoms in optical lattices. We discuss various types of static and dynamical curved spacetimes achievable by simple manipulations of the optical setup. Examples presented here contain two-dimensional spaces of positive and negative curvature as well as homogeneous cosmological models and metric waves. Most of them are extendable to three spatial dimensions. We mention some interesting phenomena of quantum field theory in curved spacetimes which might be simulated in such optical lattices loaded with bosonic or fermionic ultra-cold atoms. We also argue that methods of differential geometry can be used, as an alternative mathematical approach, for dealing with realistic inhomogeneous optical lattices.

Motivation & Objective

• To develop a scalable, experimentally feasible method for simulating quantum fields in curved spacetimes using ultra-cold atoms in optical lattices.
• To extend analogue gravity models beyond static metrics to include time-dependent and cosmological spacetimes.
• To demonstrate that realistic, inhomogeneous optical lattices can give rise to effective curved geometries via spatial modulation of tunneling amplitudes.
• To enable the simulation of relativistic quantum fields, including fermionic and bosonic systems, in effective curved spacetime backgrounds.
• To provide a framework for studying strong-field quantum gravity effects such as particle creation and horizons in a controlled laboratory environment.

Proposed method

• Use Wannier functions as a discrete basis to map the continuous Schrödinger equation onto a tight-binding Hamiltonian with site-dependent hopping parameters.
• Engineer spatially varying optical potentials with Gaussian intensity profiles to create position-dependent tunneling amplitudes, mimicking curved spacetime geometry.
• Map the discrete Laplacian in the lattice to the Laplace-Beltrami operator in curved space via a transformation of the hopping terms.
• Utilize bichromatic laser potentials to create supercells with multi-component band structures, enabling effective relativistic dispersion relations (e.g., Dirac-like) for fermions.
• Apply differential geometry to model inhomogeneous lattices as effective Riemannian manifolds, allowing curvature to emerge from lattice inhomogeneities.
• Include trapping potentials to confine atoms and stabilize the system, while allowing finite-size effects to simulate compact, finite-geodesic-distance spaces.

Experimental results

Research questions

• RQ1 Can realistic, inhomogeneous optical lattices with spatially varying potential depth produce effective curved spacetime geometries?
• RQ2 How can the discrete hopping Hamiltonian of ultra-cold atoms be mapped to the wave operator in a curved spacetime?
• RQ3 What types of curvature—positive, negative, or dynamical—can be realized in 2D and 3D optical lattices using simple laser configurations?
• RQ4 Can relativistic quantum fields, including fermions described by the Dirac equation, be simulated in such lattices?
• RQ5 To what extent do finite-size and edge effects in optical lattices lead to compact, finite-geodesic-distance spacetimes?

Key findings

• Spatially modulated optical potentials with Gaussian intensity profiles lead to anisotropic, position-dependent tunneling amplitudes that effectively generate curved spacetime geometries.
• Finite 2D optical lattices with smoothly vanishing potential depth at the edges map to compact surfaces with positive curvature or curvature transitions, as confirmed by numerical simulations.
• The effective metric in the lattice system is derived from the hopping parameters, with the discrete Laplacian mapping to the Laplace-Beltrami operator in curved space.
• Bichromatic 2D potentials with three minima per supercell produce a relativistic-like dispersion relation $ E(\vec{p}) \approx \pm \sqrt{\vec{p}^2 c^2 + m^2 c^4} $, enabling simulation of pseudo-relativistic fields.
• The system supports the emergence of compact, finite-geodesic-distance spaces even in finite lattices due to the absence of barriers at the edges, allowing atoms to tunnel to infinity in finite time.
• The framework enables the simulation of quantum field theory effects such as particle creation and Hawking radiation in a controlled, experimentally accessible setup.