Mixtures of ultracold atoms in one-dimensional disordered potentials

We study interacting one-dimensional two-component mixtures of cold atoms in a random potential, and extend the results reported earlier [Phys. Rev. Lett. 105, 115301 (2010)]. We construct the phase diagram of a disordered Bose-Fermi mixture as a function of the strength of the Bose-Bose and Bose-Fermi interactions, and the ratio of the bosonic sound velocity and the Fermi velocity. Performing renormalization group and variational calculations, three phases are identified: (i) a fully delocalized two-component Luttinger liquid with superfluid bosons and fermions, (ii) a fully localized phase with both components pinned by disorder, and (iii) an intermediate phase where fermions are localized but bosons are superfluid. Within the variational approach, each phase corresponds to a different level of replica symmetry breaking. In the fully localized phase we find that the bosonic and fermionic localization lengths can largely differ. We also compute the long-wavelength asymptotic behavior of the momentum distribution as well as that of the structure factor of the atoms (both experimentally accessible), and discuss how the three phases can be experimentally distinguished.

Figure 1

Superfluid exponents for $K_b=1$ and $U_{\mathrm{BF}}/v_0=1$, with $v_0=(v_f+v_b)/2$, as a function of the fraction of fermions. We take equal masses for both species, so that $U_{\mathrm{BF}}/v_0$ is kept a constant for all fillings. Solid blue line, fermions. Red dashed line, bosons.

Figure 2

Density wave exponents for $K_b=1$ and $U_{\mathrm{BF}}/v_0=1$, with $v_0=(v_f+v_b)/2$, as a function of the fraction of fermions. We take equal masses for both species, so that $U_{\mathrm{BF}}/v_0$ is kept a constant for all fillings. Solid blue line, fermions. Red dashed line, bosons.

Figure 3

Schematic representation of the phase $\phi \left(x\right)$ from the variational solution for the pinning of a density wave by a random potential. $x$ is in unit of the inverse density. The system breaks into domains of order $L_0$ separated by domain walls of the same average size. A schematic description of the classical density also appears. On a domain with constant $\phi$ a regular DW appears while on domain walls, an extra particle ($\nabla \phi <0$) or a hole ($\nabla \phi >0$) is pinned by disorder.

Figure 4

Critical lines as obtained from the RG flow (3.18) and (3.19), for a ratio of velocities $t=3$, and $K_f=1$. Here we have chosen to parametrize interactions with $U=U_{\mathrm{BF}}/\sqrt{v_f{v}_b}$. Note that $U$ is related to the dimensionless parameter $g$ of Eq. (2.23) through $g=U\sqrt{K_f{K}_b}/\pi$.

Figure 5

Critical lines as obtained from the RG flow (3.18) and (3.19), for a ratio of velocities $t=1/3$, and $K_f=1$. We have defined $U=U_{\mathrm{BF}}/\sqrt{v_f{v}_b}$.

Figure 6

Phase diagram of the Bose-Fermi mixture in a random potential as a function of $K_b$ and $U=U_{\mathrm{BF}}/\sqrt{v_b{v}_f}$ for $t=3$. BFG stands for Bose-Fermi glass. BFG${}^{*}$ is the same phase, however, we identify a crossover regime for which the localization length of bosons is much larger than the one of fermions.

Figure 7

Crossover behavior of the function ${\mathcal{J}}_b$ as defined in Eq. (3.48), for $t=3$ and $g=0.9$. Here we have plotted $\tilde{\Lambda }{\mathcal{J}}_b\left(\tilde{\Lambda }\right)/\left(2K_b\right)$, with $\tilde{\Lambda }=\Lambda /\sqrt{{\widehat{\Sigma }}_f}$. It goes to 1 at low momenta and saturates at $X_b/\left(2K_b\right)$ as momentum is increased. We identify a crossover region around $\Lambda ={({\widehat{\Sigma }}_f/\sqrt{1-g^2})}^{1/2}$ (vertical dashed line).

Figure 8

Modified RG flow for bosons using the Gaussian variational solution.

Figure 9

Phase diagram obtained by the Gaussian variational method in replica space as a function of $K_b$ and $U=U_{\mathrm{BF}}/\sqrt{v_b{v}_f}$. We took ${\tilde{D}}_f={\tilde{D}}_b=0.005$ and $t=v_f/v_b=3$.

Figure 10

Fermion mass ${\widehat{\Sigma }}_f$ and $\Delta {\widehat{\Sigma }}_f^{\left(2\right)}$ as a function of $U=U_{\mathrm{BF}}/\sqrt{v_f{v}_b}$, for $t=v_f/v_b=3$ and $K_b=1,1.1,1.2,1.3$ (top to bottom). Note that $\Delta {\widehat{\Sigma }}_f^{\left(2\right)}$ goes to zero simultaneously with ${\widehat{\Sigma }}_f$ as $U$ is increased.

Figure 11

The boson mass ${\widehat{\Sigma }}_b$ as a function of $U=U_{\mathrm{BF}}/\sqrt{v_f{v}_b}$, for $t=v_f/v_b=3$ and $K_b=1,1.1,1.2,1.3$ (top to bottom). Note that ${\widehat{\Sigma }}_b$ goes to zero, simultaneously with ${\widehat{\Sigma }}_f$.

Figure 12

Phase diagram obtained by the Gaussian variational method in replica space, as a function of $K_b$ and $U=U_{\mathrm{BF}}/\sqrt{v_b{v}_f}$, for ${\tilde{D}}_f={\tilde{D}}_b=0.005$ and $v_f/v_b=1/3$. In the fully localized phase there is crossover between two regions where the order of replica symmetry breaking is reverted.

Figure 13

Frequency dependence of the structure factor for bosons (red dashed line) and fermions (blue solid line) in the fully localized phase. Momentum is fixed to $q=0.2\Lambda$ with $\Lambda$ the UV cutoff of the theory. The boson velocity $v_b$ is taken to be equal to 1, making the structure factors dimensionless.

Figure 14

Momentum distribution $n_b\left(q\right)$ (Fourier transform of ${\rho }_b{A}_b\left(x\right)$, dimensionless) in the fully localized phase. We have taken $v_f/v_b=9$, $K_b=1.3$, and $g=0.4$. In addition, ${\widehat{\Sigma }}_f=0.01\Lambda$ and ${\widehat{\Sigma }}_b=0.0001\Lambda$. Related momentum scales, $\sqrt{{\widehat{\Sigma }}_f}$ and $\sqrt{{\widehat{\Sigma }}_b}$, are pinpointed on the plot. Momentum $q$ is in units of the UV cutoff $\Lambda$.

Figure 15

Phase diagram of a 1D ${}^{87}$Rb-${}^{40}$K Bose-Fermi mixture, in the weak disorder limit. We consider an array of tubes, created with lasers of wavelength $\lambda$ $=$ 755 nm, which corresponds to a (2D) lattice constant $d=\lambda /2$. We take $a_{bb}=100a_0$ as Bose-Bose scattering length, while the Bose-Fermi scattering length, $a_{\mathrm{BF}}$, is tuned using a Feshbach resonance. The one-dimensional densities are chosen to be ${\rho }_{f}d=0.3$ and ${\rho }_{b}d=0.2$. The recoil energy is $E_{R,b}=h^2{\lambda }^{-2}/\left(2m_{Rb}\right)$, while $V_{\perp b}$ is the transverse confining potential creating the 1D tubes. The Bose-Bose interaction $U_b$ increases with $V_{\perp b}$. Note that despite a superficial similarity between the present figure and, for instance, Fig. 6, the vertical axis is reverted since $K_b$ decreases as $U_b$ increases. Four phases and the region of instability of the Luttinger liquid theory are shown. BFG, Bose-Fermi glass; BFG${}^{*}$, BFG with an extremely large bosonic localization length; ${\mathrm{AG}}_{\mathrm{SFB}}$, Anderson glass $+$ superfluid bosons; LL, Luttinger liquid.

Figure 16

The momentum distribution of bosons in the fully localized phase exhibits a saturation for momenta below $1/L_b$. Above this threshold, we identify a crossover between two algebraic regimes. Bosonic interactions are renormalized by Bose-Fermi interactions on length scales smaller than the fermionic localization length $L_f$. Indeed for $q>1/L_f$ the exponent of the power law is renormalized by BF interactions, while for $1/L_b<q<1/L_f$ the exponent is controlled by bare interactions.

Figure 17

Sketch of the structure factor (arbitrary units)—for bosons and fermions—in the three phases identified in Fig. 15. (Top panel) Fully localized phase, (middle panel) intermediate phase with localized fermions and superfluid bosons, and (bottom panel) Luttinger liquid phase. The presence of two peaks in each phase is characteristic of the strong coupling between the two components of the gas, even in the localized phases. In the latter the widths of the peaks are inversely proportional to the localization lengths.