Model

Let us consider particles subjected to a lattice potential. For the description of Bloch oscillations, the starting point is the Gross-Pitaevskii equation

i~∂

∂tΨ =

− ~²

2m∇² +V(r)

Ψ(r) +g|Ψ(r)|²Ψ(r), (6.1) [subsection 2.2.2, equation (2.16)]. Here, we do not include the chemical potential because we work at fixed particle number. The setting differs from the Bogoliubov problem of the extended condensate ground state with the disorder imprint in part I. Here, the condensate is created in a harmonic trap and then transferred into an optical lattice, and the trap is switched off (figure 6.1). Obviously, this is not the ground-state configuration, and the condensate will have the tendency to spread, both due to repulsive interaction and due to the usual linear dispersion of matter waves.

The phenomenon of Bloch oscillation takes place in one direction. Thus, we consider a setup in which the transverse degrees of freedom are frozen.

The transverse harmonic-oscillator ground state is integrated out, leading to a renormalized interaction parameter in the remaining dimension

g1D = mω_⊥

2π~ g3D. (6.2)

Here, ω_⊥ is the transverse oscillator frequency (or the geometric mean in anisotropic configurations). The usual three-dimensional interaction

pa-z

|Ψ(z)|²

Figure 6.4: Typical initial state for Bloch oscillations. This density profile was obtained as the ground state of the continuous Gross-Pitaevskii equation (6.1) with V1D = V0cos²(πz/d) + ¹₂mω_z²z². The trap ωz is then switched off.

This state is well described with a tight-binding ansatz.

rameter g3D = 4π~²as/m is proportional to the s-wave scattering length as. In the one-dimensional Gross-Pitaevskii equation

i~ ∂

∂tΨ(z, t) =

− ~²

2m∇² +V1D(z)

Ψ(z, t) +g1D|Ψ(z, t)|²Ψ(z, t), (6.3) the potential V1D(z) is given as a deep lattice potential V0cos²(πx/d) with spacing d. Later-on, the lattice is accelerated in order to observe Bloch oscillations. This is done by tilting the lattice out of the horizontal plane, or by accelerating the lattice by optical means.

6.2.1. Tight binding approximation

In sufficiently deep lattice potentials, only the local harmonic-oscillator ground state in each lattice site is populated. This regime is called the tight-binding regime (figure 6.4). The condensate is represented by a sin-gle complex number Ψn(t) at each lattice site [139]. Neighboring sites are weakly coupled by tunneling under the separating barrier with tunneling amplitude

J ≈ 4

√π (V0/E_r)³⁴ exp(−2p

V0/E_r)E_r, (6.4) whereE = ~²π²/(2md²) is the recoil energy [136]. Here, changes of the local oscillator function due to the interaction have been neglected. This assump-tion holds in deep lattices with ~ω_k gTB|Ψn|² (see below). Otherwise, the tunneling amplitude (6.4) gets modified already for slight deformations of the Wannier functions [140].

The tight-binding equation of motion thus reads

i~Ψ˙n = −J(Ψn+1+ Ψn−1) +F dnΨn +gTB(t)|Ψn|²Ψn. (6.5) Here, we have added a constant force F. The tight-binding interaction parameter gTB is obtained from the one-dimensional interaction parameter g1D by integrating out also the harmonic-oscillator ground state in the lon-gitudinal direction, gTB = Np

mω_k/2π~g1D. The factor N comes from the convention that the discrete wave function Ψn is normalized to one instead of the particle number N.

Using an appropriate Feshbach resonance, the interaction parameter gTB(t) can be controlled by external magnetic fields. The validity of (6.5) is limited by the transverse trapping potential |gTB| ~ω_k/n0, with n0 = maxn|Ψn|².

The dispersion relation of this single-band model reads (k) =−2Jcos(kd).

Its curvature or inverse mass m⁻¹ = 2Jb²cos(kd)/~² determines the dynam-ics of smooth wave-packets. The equation of motion (6.5) can be derived as iΨ˙n = ∂H/∂Ψ^∗_n from the nonlinear tight-binding Hamiltonian

H = X

n

−J(Ψn+1Ψ^∗_n+ Ψ^∗_n+1Ψn) +F nd|Ψn|² + gTB(t) 2 |Ψn|⁴

. (6.6)

6.2.2. Smooth-envelope approximation

Experimentally, the initial state is prepared by loading a Bose-Einstein con-densate from an optical dipole trap into an optical lattice created by two counter-propagating laser beams [40]. If this is done adiabatically, the low-est oscillator states of the lattice are populated according to the profile of the condensate in the trap (figure 6.4).

In the following, we useJ anddas units of energy and length, respectively.

Furthermore, we set ~ = 1 and omit the subscript TB of the interaction parameter. We tackle Eq. (6.5) by separating the rapidly varying Bloch phase e^ip(t)n from a smooth envelope A(z, t) comoving with the center of mass x(t):

Ψn(t) = e^ip(t)nA(n−x(t), t)e^iφ(t). (6.7) With p(t) = −F t, x(t) = x0 + 2 cos(F t)/F, and φ = φ0 + 2 sin(F t)/F, the envelope is found to obey the nonlinear Schr¨odinger equation

i∂tA= − 1

2m(t)∂_z²A+g(t)|A|²A , (6.8)

with 1/m(t) = 2 cosp(t). Higher spatial derivatives of A have been ne-glected. Note that we choose an immobile wave packet with p(0) = 0 as initial condition, which fixes the phase for the subsequent Bloch oscillations.

Let us consider solutions of (6.8) in two simple limits (a) and (b).

(a) Linear Bloch oscillation

In absence of the nonlinear term, a Schr¨odinger equation with a time-dependent mass is to be solved. This can be done easily in Fourier space withAk(t) ∝ e^{−ik2sin(F t)/F}. For an initial state of Gaussian shape with width σ0, this results in a breathing wave packet

A(z, t) = (2π)⁻¹⁴ The time-dependent complex width σ(t) implies a breathing of the width of the wave packet, as well as a gradient in the phase.

In the first quarter of the Bloch cycle, the mass is positive and the wave packet spreads, as expected for a free-particle dispersion. When the mass changes sign, the time evolution is reversed, and the wave packet recovers its original shape at the edge of the Brillouin zone. Thus, the wave packet shows perfectly periodic breathing on top of the Bloch oscillation. This behavior is independent of the particular initial shape of the wave packet.

(b) Rigid soliton

Let us consider the mass m and the interaction parameter g as constant for the moment. If both have opposite signs, then equation (6.8) admits a soliton solution

A(z, t) = 1

√2ξ

1

cosh (z/ξ) e^−iωt, (6.10) with the quasistatic width ξ = −2/(gm) > 0. If the force F changes the effective mass as function of time, the interaction parameter can be tuned in such a way that the quasistatic width still exists or even is constant.

A perfectly rigid soliton of width ξ0 can be obtained by modulating the interaction like

gr(t) = −2/[ξ0m(t)] = −|gr|cos(F t) with gr = −4/ξ0 < 0. (6.11) More extensive studies based on this idea have been put forward in [141, 142]. If the quasistatic width ξ(t) =−2/[g(t)m(t)] exists for all times but is not constant, then the soliton must be able to follow this width adiabatically in order not to decay. Otherwise its breathing mode will be driven, and other excitations may be created.