Quantum Simulation of an Extra Dimension

Abstract

We present a general strategy to simulate a D + 1-dimensional quantum system using a D-dimensional one. We analyze in detail a feasible implementation of our scheme using optical lattice technology. The simplest nontrivial realization of a fourth dimension corresponds to the creation of a bivolume geometry. We also propose single-and many-particle experimental signatures to detect the effects of the extra dimension.Introduction. There is long-standing interest in Physics for the possible existence and effects of extra dimensions. This interest was brought by the seminal papers by Kaluza and Klein [1] aimed at unifying interactions using the presence of a fourth spacial dimension. Later on, extra dimensions became a sine qua non element for the construction of string theory [2]. Separately, enormous progress has been made in recent years to achieve real quantum simulations, that is, to simulate quantum mechanical models using other well controlled quantum systems [3]. It is now reasonable to investigate to what extent present technology can be used to faithfully simulate a quantum theory living in extra dimensions.Let us briefly recall recent progress on quantum simulation of condensed matter models using cold atoms [4,5]. By confining atoms to an optical lattice, the Hubbard model may be realized [6], and the superfluid-to-Mott-insulator transition observed [7]. Furthermore, several schemes to couple neutral cold atoms to artificial Abelian [8] and non-Abelian [9, 10] magnetic and electric fields have been put forth [11][12][13][14]. This opens the door to creating strongly correlated quantum-Hall states with cold atoms [15]. Although cold atoms are non-relativistic, it is possible to simulate relativistic effects by looking at the low-energy behaviour of some special lattice models -lattice models where the band structure presents Dirac cones, e.g. honeycomb lattices [16,17] or lattices dressed with internal degrees of freedom [18]. Hence, it is not far-fetched that that cold atoms may provide some insight into particle physics models that are not completely understood, such as quantum chromodynamics (see for instance [19]) or, as presented here, in the analysis of extra dimensions.General strategy. The basic idea to achieve a quantum simulation of an extra dimension consists in engineering the connectivity of the system partly on real dimensions, and partly on the use of different species for the degrees of freedom (for an alternative approach cf. [20]). Let us illustrate this construction in the case of the simplest quantum mechanical model of a free particle on a hypercubic D + 1 spacial lattice. The Hamiltonian for this system is

Synthetic Dimensions and Spin-Orbit Coupling with an Optical Clock Transition

“…This interaction was first synthesized in cold atomic gases by using twophoton Raman transitions [11] coupling two hyperfine spin states with a transfer of momentum. The coupling between spin states also enables a new powerful tool for engineering topological states of matter, which relies on the "synthetic dimension" (SD) concept [12,13]. According to this approach, the internal states of an atom are treated as effective sites along a synthetic lattice dimension, and coherent coupling between them is interpreted in terms of an effective tunneling.…”

Correlated Dynamics in a Synthetic Lattice of Momentum States

“…There has been recent interest in extending quantum simulation studies from real-space potentials to synthetic lattice systems composed of discrete internal [3,4] or external [5] states. These synthetic dimensions enable many unique capabilities for quantum simulation, including new approaches to engineering nontrivial topology [4,6], access to higher dimensions [3], and potential insensitivity to finite motional temperature.The recent development of momentum-space lattices (MSLs), based on the use of discrete momentum states as effective sites, has introduced a fully synthetic approach to simulating lattice dynamics [7][8][9][10][11]. As compared to partially synthetic systems [12,13], fully synthetic lattices offer complete microscopic control of system parameters.…”

Quantum magnetism of ultracold atoms with a dynamical pseudospin degree of freedom

“…We calculated the ground state and low excitations by Lanczos diagonalization of H eff , and evaluated observables like ground state magnetization, defined in Eq. (16), and spin correlations. The latter we use, in particular, to identify antiferromagnetic order, which can be tested by an anticorrelation order parameter…”

“…In the extradimension picture of Ref. [16], the internal degree of freedom becomes equivalent to a compactified spatial dimension. Assuming SU(N ) symmetry, we show in Sec.…”

“…Recently, it has been pointed out in Ref. [16] that the internal degrees of freedom can also be used for simulating an "extradimension", once the internal states are properly coupled to provide the hopping between the synthetic sites given by the atomic species.…”