Disordered Bose-Einstein Condensates in Quasi-One-Dimensional Magnetic Microtraps

We analyze the effects of a random magnetic potential in a microfabricated waveguide for ultracold atoms. We find that the shape and position fluctuations of a current carrying wire induce a strong Gaussian correlated random potential with a length scale set by the atom-wire separation. The theory is used to explain quantitatively the observed fragmentation of the Bose-Einstein condensates in atomic waveguides. Furthermore, we show that nonlinear dynamics can be used to provide important insights into the nature of the strongly fragmented condensates. We argue that a quantum phase transition from the superfluid to the insulating Bose glass phase may be reached and detected under the realistic experimental conditions.

Figure 1

Typical setup of a magnetic microtrap: A current $I_0$ flowing in a microfabricated copper conductor and a perpendicular bias field $B_{\perp }$ form a magnetic waveguide for atoms. An offset field $B_{\parallel }$ applied parallel to the wire reduces loss processes near the magnetic field minimum.

Figure 2

(a) Correlation function of the random magnetic potential in a microtrap ${\Delta }_k$, assuming white noise fluctuations of the wire position (i.e., $F_k=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$). Inset: ${\Delta }_k$ when the wire fluctuations have an intrinsic length scale: $F_k\propto [e^{-(k-k_1{)}^2{\eta }_1^2}+e^{-(k+k_1{)}^2{\eta }_1^2}]$. We take $2\pi /k_1=200\mu \mathrm{m}$, ${\eta }_1=100\mu \mathrm{m}$, and $⟨f^{+}(z)f^{+}(z){⟩}_{\mathrm{d}\mathrm{i}\mathrm{s}}=(0.1\mu \mathrm{m}{)}^2$. Solid to dash-dotted lines correspond to $d=50$, 100, 150, and $200\mu \mathrm{m}$, respectively. (b) Condensate density profiles with parameters chosen to be close to the values used in the experiments of Ref. 3. Dotted lines are the results without random potential.

Figure 3

(a) Schematic dynamical phase diagram of fragmented Q1D BEC in relation to the shaking experiments. The dashed line indicates the minimum displacement to generate an observable density fluctuations (i.e., $\zeta >{\zeta }_{\mathrm{m}\mathrm{i}\mathrm{n}}$). Three types of responses can be identified from the power spectra of the oscillations: (b) dipole mode regime, (c) chaos, and (d) self-trapping. Power spectra are obtained from simulations of a three well system with $⟨N_j⟩={10}^5$.

Figure 4

The dotted line shows the probability $\mathcal{P}(Q<1)$, which is close to zero if the system is in the superfluid regime and is close to one if in the insulating Bose glass phase. The solid line shows the probability that one can observe fluctuations in the number of atoms between the neighboring wells with ${\omega }_J\ge {\omega }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $\zeta \ge {\zeta }_{\mathrm{m}\mathrm{i}\mathrm{n}}$ (see text). When this probability approaches zero, the system will appear self-trapped in the shaking experiments. $N_{\mathrm{a}\mathrm{v}\mathrm{e}}$ is the average number of atoms in a single well (minicondensate). Disorder strength for (a) is the same as used in Fig. 2b for $d=100\mu \mathrm{m}$ (denoted to be $u_s^0$), while it is 5 times stronger in (b).