Bose-Hubbard ladder subject to effective magnetic field: Quench dynamics in a harmonic trap

Wladimir Tschischik, Roderich Moessner, and Masudul Haque

Phys. Rev. A 92, 023845 – Published 24 August 2015

Abstract

Motivated by a recent experiment with optical lattices that has realized a ladder geometry with an effective magnetic field [Atala et al., Nat. Phys. 10, 588 (2014)], we study the dynamics of bosons on a tight-binding two-leg ladder with complex hopping amplitudes. This system displays a quantum phase transition even without interactions. We study the nonequilibrium dynamics without and with interactions, in the presence of a harmonic trapping potential. In particular we consider dynamics induced by quenches of the trapping potential and of the magnitude of the rung hopping. We present a striking “slowing down” effect in the collective mode dynamics near the phase transition. This manifestation of a slowing down phenomenon near a quantum phase transition can be visualized unusually directly: The collective mode dynamics can be followed experimentally in real time and real space by imaging the atomic cloud.

Received 15 July 2015

DOI:https://doi.org/10.1103/PhysRevA.92.023845

Authors & Affiliations

Wladimir Tschischik, Roderich Moessner, and Masudul Haque

Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, 01187 Dresden, Germany

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Issue

Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
(a) Sketch of the magnetic tight-binding ladder with complex interleg hoppings. Hopping around a plaquette increases the phase by $ϕ$ . (b) Low-energy excitation spectrum of the trapped magnetic ladder with $ϕ = \frac{π}{2}$ for $N = 1, L = 500, k_{tr} = 10^{- 4}$ . The eigenstates are nearly degenerate for $K < K_{c}$ (vortex phase) and nondegenerate for $K > K_{c}$ (Meissner phase). (c) Density distributions of four lowest eigenstates for $N = 1, L = 120, k_{tr} = 10^{- 5}$ , and $K = {0.5; 1} < K_{c}$ and $K = {1.5; 2} > K_{c}$ on a leg. For $K < K_{c}$ the degenerate eigenstate pairs have identical density distributions. The first degenerate pair have a Gaussian-like shape, and the second pair corresponds to the dipole mode. For $K > K_{c}$ the states are rearranged: The second eigenstate takes the shape of a dipole mode state, and the third eigenstate becomes a breathing mode state. The labels mark the corresponding eigenenergy in panel (b), and the $\oplus, ⊖$ signs indicate parity under pseudoinversion.
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Figure 2
Dipole mode oscillation after a quench (sudden shift) of trap position. (a) Real time dynamics of center-of-mass in both legs (L, R), after a trap shift of $d_{0} = 1$ for a system with $L = 300, k_{tr} = 0.001$ for several $K$ values. The slowing down effect is clearly visible: Oscillation period is largest close to the critical point. (a1) Exact result for $N = 1$ . (a2) The time evolution is calculated using Gutzwiller mean-field approximation for $N = 7, U = 1$ . (b) Increase of dipole mode oscillation period close to the critical point at $K_{c}$ . (b1) Exact results: Noninteracting $(N = 1)$ and interacting $(N = 4)$ systems compared. At small $K$ and large $K$ , the frequencies become identical in trap units (Kohn theorem). (b2) Same effect is seen with mean-field dynamics.
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Figure 3
Excitation energy spectrum of the trapped magnetic ladder with $N = 2$ bosons, $L = 50, k = 0.001$ and (a) $U = 0$ and (b) $U = 1$ . The line thicknesses logarithmically encode overlaps $| 〈 ϕ_{0} | ψ_{l} 〉 |^{2}$ of eigenstates with initial ground state of the Hamiltonian with a trap shift by $d_{0} = 0.5$ , i.e., after dipole mode excitation. The thick, colored lines correspond to states with an overlap larger than $10^{- 4}$ . Dashed lines correspond to states with overlap smaller than $10^{- 4}$ .
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Figure 4
Breathing mode oscillation after a trap quench for various $K$ , shown using the second moment of the density distribution for $L = 300, k_{tr} = 0.00105 \to 0.001$ . (a) Exact result for $N = 1$ . (b) Gutzwiller mean-field approximation for $N = 7, U = 1$ . (c) Ratio of the breathing and dipole mode frequencies for $N = 1, L = 300, k_{tr} = 0.0001$ . The two excitations are obtained using a $d_{0} = 1$ quench (dipole mode) and a $k_{tr} = 0.00011 \to 0.0001$ quench (breathing mode).
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Figure 5
Excitations after a $K$ quench. We show overlaps of the eigenstates with an initial state, encoded in the thickness of the solid lines. The initial state is one state of the ground state manifold with $K_{i} = 0.5, N = 1, L = 200, k_{tr} = 0.001$ with (a) even and (b) odd parity under inversion. Only low-energy states with the same symmetry as the initial state are excited.
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Figure 6
Energy gap between lowest two single-particle eigenstates, obtained using double-precision numerics. (a) Open boundary conditions, various sizes. The gap in the vortex phase is seen to be numerically significant, $≫ 10^{- 15}$ . (b) Trapped system with various $k_{tr}$ . Data is obtained with $L = 4000$ , for which the trapped cloud does not reach the edges up to numerical accuracy. The gap is seen to reach “numerical zero,” $\sim 10^{- 15}$ .
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