We consider a strongly repulsive two-component Fermi gas in a one-dimensional (1D) optical lattice described in terms of a Hubbard Hamiltonian. We analyze the response of the system to a periodic modulation of the hopping amplitude in presence of large two body interaction. By (essentially) exact simulations of the time evolution, we find a non-trivial double occupancy frequency dependence. We show how the dependence relates to the spectral features of the system given by the Bethe ansatz. The discrete nature of the spectrum is clearly reflected in the double occupancy after long enough modulation time. We also discuss the implications of the 1D results to experiments in higher dimensional systems.
Deep Dive into Hopping modulation in a one-dimensional Fermi-Hubbard Hamiltonian.
We consider a strongly repulsive two-component Fermi gas in a one-dimensional (1D) optical lattice described in terms of a Hubbard Hamiltonian. We analyze the response of the system to a periodic modulation of the hopping amplitude in presence of large two body interaction. By (essentially) exact simulations of the time evolution, we find a non-trivial double occupancy frequency dependence. We show how the dependence relates to the spectral features of the system given by the Bethe ansatz. The discrete nature of the spectrum is clearly reflected in the double occupancy after long enough modulation time. We also discuss the implications of the 1D results to experiments in higher dimensional systems.
Ultracold atomic gases systems couple weakly with the surrounding environment and are highly controllable [1,2,3,4], therefore they offer excellent possibilities to investigate the dynamics of strongly correlated quantum many-body systems. Much attention has been recently devoted to the study of the dynamical properties both from the experimental [5,6] and theoretical [7,8,9,10,11] point of view. Especially, one-dimensional (1D) systems, accessible by experiments and theoretically exactly solvable in some cases, can be used to obtain thorough understanding of the many-body ground state and the dynamics. In this letter, we present an (essentially) exact time-evolving block decimation TEBD simulations of the dynamics of a repulsively interacting 1D system and reveal a non-trivial time-dependence which we explain using the Bethe ansatz (BA). We extend the analysis also to the harmonically trapped case essential for ultracold gases experiments. In higher dimensions, the relevance of Mott and antiferromagnet (AF) physics in connection to high-T c superconductivity (see e.g. [12]), suggests that the investigation of the equivalent systems in the framework of ultracold gases, especially in two dimensions, may shed new light on high-T c superconductor physics. Our results are relevant for such experiments by showing -with analysis that does not assume mean-field approximation nor linear response -how the discrete nature of the spectrum is reflected in the dynamics of a lattice modulation experiment.
We examine the dynamical properties of a two-species ultracold atomic gas loaded in a 1D optical lattice both for open boundary conditions (obc) and in presence of parabolic confinement. In particular we perform an (essentially) exact numerical simulation of this system when a periodic lattice modulation is applied. To this end we consider the 1D Hubbard Hamiltonian, in presence of an external parabolic confining potential
where
J is the hopping amplitude, and U the on-site interaction. In the limit U/J ≫ 1, the only effect of the lattice modulation is a modulation of the hopping amplitude J, the effect on U being negligible (see [11]).
We focus on the double occupancy (d.o.) expectation value < n i ↑ n i ↓ >, when a (small) periodic modulation δJ sin(ωt) of the hopping amplitude J is applied for a given time to the ground state of the Hamiltonian given in Eq. ( 1) for two different situations: U/J = 60, L = 20, particle number N p = 12 and U/J = 20, L = 40, N p = 24 respectively. Our simulation is performed with a TEBD algorithm [13,14], both for the ground-state calculation (imaginary-time evolution) and the real-time evolution. The numerical results show that the ground state is constituted by a central Mott region with one atom per site, surrounded by two small metallic (Luttinger liquid) regions where the filling is less than one. In order to avoid finite-size effects, we have considered a lattice size exceeding the actual extent of the atomic cloud by a few lattice sites. Heuristically, the Luttinger liquid phase corresponds to the regions where < n 2 i > -< n i > 2 = const, see [15]. Moreover, the static structure factor S(q) in the ground state of the finite systems here considered exhibits the same qualitative features of the 1D Heisenberg AF chain (i.e. a slow decaying peak centered around q = π), as expected in the U ≫ J limit of the Hubbard Hamiltonian [16].
For an infinite chain at half filling, if hopping and parabolic confinement are suppressed, the (highly degenerate) first excited state is represented by a site with an empty site and a doubly occupied one (henceforth particle-hole excitation). The energy gap between this state and the ground state is equal to U . To investigate the spectral properties of the system when U ≫ J, it seems then natural to choose ω ≃ U in δJ sin(ωt), with δJ/J = 0.1 throughout the paper.
In Fig. 1 (inset) we show the d.o. as a function of frequency for short times, from which one notices that a broad peak appears as in correspondence of the value Ω = U , consistently with the value of the gap of the particle-hole excitation. However, for larger times a richer structure appears (Fig. 1). From Fig. 2, it is clear how it is possible to distinguish between short-(t < t * ) 0.9 0.95
as a function of frequency for long (t = 10 main panel) and short (t = 0.5 inset) times. A single broad peak appears for ω/U ≃ 1 (L = 20, U/J = 60, Ω/J = 0.1). and long-(t > t * ) time behavior, the explicit value of the threshold t * will be derived below.
The key idea in understanding the time dependence of < n i ↑ n i ↓ > is that the resolution of degeneracy depends on the modulation timescale. For t < t * it is possible to consider the particle-hole excited states as quasi-degenerate, hence the transition probability between the ground state and the quasi-continuum (centered around U ) of excited states is given by P (t) ∝ δJ 2 sin 2 [(U -ω) t/ 2 ] (U -ω) 2 . The threshold time is then c
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