Three-dimensional strong localization of matter waves by scattering from atoms in a lattice with a confinement-induced resonance

The possibility of using ultracold atoms to observe strong localization of matter waves is now a subject of great interest, as undesirable decoherence and interactions can be made negligible in these systems. It was proposed that a static-disordered potential can be realized by trapping atoms of a given species in randomly chosen sites of a deep three-dimensional (3D) optical lattice with no multiple occupation. We analyze in detail the prospects of this scheme for observing localized states in 3D for a matter wave of a different atomic species that interacts with the trapped particles and that is sufficiently far detuned from the optical lattice to be insensitive to it. We demonstrate that at low energy a large number of 3D strongly localized states can be produced for the matter wave, if the effective scattering length describing the interaction of the matter wave with a trapped atom is of the order of the mean distance between the trapped particles. Such high values of the effective scattering length can be obtained by using a Feshbach resonance to adjust the free-space interspecies scattering length and by taking advantage of confinement-induced resonances induced by the trapping of the scatterers in the lattice.

Figure 1

(Color online) Sketch of the model of disorder considered here: a matter wave ($A$, blue) scatters on randomly distributed $B$ particles (red), each occupying the vibrational ground state of a node in a 3D optical lattice (here the average occupancy $p$ is 0.1). The lattice does not act on the matter wave $A$.

Figure 2

Schematic view of the behavior of the wave function $\varphi \left(x\right)=\mathrm{Im}⟨x\mid \mathcal{G}\mid 0⟩$ in presence of a 1D scattering medium of length $L$ much larger than the localization length $\xi$, so that the modulus of the transmission coefficient for each half of the medium $\mid t_{+}\mid ,\mid t_{-}\mid \sim \mid t\mid \propto e^{-L∕2\xi }⪡1$. (a) For a generic positive energy: the wave function decreases exponentially inside the medium, being of modulus $\sim \mid t\mid$ out of the medium and $\sim {\mid t\mid }^2$ in the center $x=0$ of the medium. (b) For specific values of the energy, the wave function is “localized” inside the medium: its modulus decreases from $\sim 1∕{\mid t\mid }^2$ in $x=0$ to $\sim 1∕\mid t\mid$ for $\mid x\mid >L∕2$.

Figure 3

(Color online) Ratio of the values of ${\varphi }^2$ in the center and outside the scattering medium (see text) as a function of $k={\left(2m_{A}E\right)}^{1∕2}∕\hslash$, with $a_{\mathrm{eff}}=d$, for a given realization of the disorder with $N\approx 900$ scatterers that occupy the nodes of a cubic lattice with 21 sites per side, with a filling factor $p=0.1$. The value of ${\varphi }_{\text{out}}^2$ is calculated as explained in [36] (red solid curve) or with the extrapolation from the far field behavior, Eq. (20) with $r=R=20d$ (green dashed curve). The energy intervals where the matter wave significantly penetrates the scattering medium correspond to the narrow peaks in this figure. We have checked that the wave function is actually spatially localized in such an energy interval. The vertical lines mark the locations of the resonances obtained by the spectral method: dashed lines for the very-long-lived resonances $(\Gamma <6\times {10}^{-5}{E}_r^A∕\hslash )$ and dotted lines for the broader resonances. The temporal decay rate $\Gamma$ is obtained from Eq. (23).

Figure 4

Plot of $\varphi {\left(\mathbf{r}\right)}^2$ along the straight line passing through ${\mathbf{r}}_0$ and parallel to $z$ axis, for two values of $k$, (a) a generic value $k=0.3∕d$ and (b) a value $k=0.350134274724∕d$ corresponding to a peak for ${\varphi }_{\text{in}}^2∕{\varphi }_{\text{out}}^2$ as a function of $k$ (in the spirit of Fig. 3). Note the similarity with the 1D case sketched in Fig. 2. In (a) the position ${\mathbf{r}}_0$ is close to the center of the scattering medium, ${\mathbf{r}}_0=(d∕2,-d∕2,-d∕2)$; in (b) it is close to the “center” of the localized state: ${\mathbf{r}}_0=(15d∕2,-3d∕2,-11d∕2)$. The effective scattering length is given by $d∕a_{\mathrm{eff}}=1.20530122302$. To get a clear evidence of the exponential decay of $\varphi$, we used a larger scattering medium than in Fig. 3: $N\simeq 3400$ atoms on the lattice within a sphere of radius $20d$, with an occupation probability $p=0.1$.

Figure 5

(Color online) Analytic properties of the resolvent $\mathcal{G}\left(z\right)=1∕(z-\mathcal{H})$ in the complex plane $z$. The resolvent has poles on the negative side of the real axis, corresponding to bound states, and a branch cut on the positive side of the real axis, corresponding to the continuum. The analytic continuation of the resolvent across the branch cut from $\mathrm{Im}z>0$ to $\mathrm{Im}z<0$ may present a discrete set of poles in the fourth quadrant: the associated states are the resonances of the system.

Figure 6

(Color online) Representation in the complex plane of the eigenvalues $m^{\infty }$ of $M^{\infty }$ for different values of the wave number $k$ of the matter wave. The eigenvalues of $M$ are then simply deduced from these eigenvalues by a shift of $1∕a_{\mathrm{eff}}$ along the real axis. Left column: cubic lattice with 21 sites/side, $p=0.1$ and 872 scatterers; the inverse mean distance between scatterers is $p^{1∕3}∕d\simeq 0.46∕d$ (same realization of disorder as in Fig. 3). Right column: cubic lattice with 209 sites/side, $p={10}^{-4}$ and 878 scatterers; the inverse mean distance between scatterers is $p^{1∕3}∕d\simeq 0.046∕d$. The green dashed lines mark the values $\mathrm{Re}\left(m^{\infty }\right)=-2p^{1∕3}∕d$. The real axis is in units of $p^{1∕3}∕d$, and common values of $kd∕p^{1∕3}$ are taken in both columns, so as to reveal a possible universality in the low $p$ limit. The imaginary axis is in units of $k$, as justified by Eq. (22).

Figure 7

(Color online) For a fixed positive energy, corresponding to $k=0.35∕d$, ratio of the values of ${\varphi }^2$ in the center and outside the scattering medium as a function of the effective scattering length $a_{\mathrm{eff}}$ [36]. The dashed line gives the result for $a_{\mathrm{eff}}=0$. The same realization of disorder is used as in Fig. 3. The inset is a magnification of the region $a_{\mathrm{eff}}\sim d$.

Figure 8

(Color online) For a single realization of disorder, histogram giving the number of imaginary eigenvalues of $M$ per class of inverse lifetimes ${\Gamma }^{-1}$. The filling factor of the lattice is $p=0.1$ and the effective scattering length is $a_{\mathrm{eff}}=d$. Black dashed histogram: $N=872$ scatterers within a cube with 21 sites per side. Red solid line histogram: $N=2985$ scatterers within a cube with 31 sites per side. These histograms were constructed by a dichotomy search of the values $k_0$ of the momentum $k∊[0,1∕d]$ such that the matrix $M$ has a purely imaginary eigenvalue; the associated lifetime was then calculated with Eq. (23). Eigenvalues with negative values of $\Gamma$ are of course not included (see [41]). The decay rate $\Gamma$ is given in units of the recoil angular frequency $E_r^A∕\hslash =\hslash k_L^2∕2m_A$ of the species $A$. In practice, only the eigenvalues with small enough values of $\Gamma$ are expected to produce an observable resonance in ${\varphi }_{\text{in}}^2∕{\varphi }_{\text{out}}^2$ as a function of $k$, see Fig. 3. If the matter wave is made of atoms of ${}^6\mathrm{Li}$, a duration of the experiment of $0.3\mathrm{s}$ corresponds to a minimal observable inverse lifetime of $\Gamma \sim {10}^{-5}{E}_r^A∕\hslash$ for an optical lattice laser wavelength of $779\mathrm{nm}$.

Figure 9

(Color online) Effective scattering length $a_{\mathrm{eff}}$ (solid line) as a function of $a^{-1}$ for $m_B∕m_A=0.15$ (trapped ${}^6\mathrm{Li}$ and free ${}^{40}\mathrm{K}$). The vertical lines (red) mark the positions of the resonances (a step of $a_{\mathrm{ho}}∕a=0.01$ is used to sample the curves, and some of the resonances are too narrow to be seen on the graph).

Figure 10

(Color online) Same as Fig. 9 for two particles of equal mass. The blue dashed line is the effective range $r_e$, and the magenta dotted line is $r_{e,\text{Born}}=-(\mu ∕m_A)a_{\mathrm{ho}}^2∕a$, obtained in the Born approximation by replacing ${\psi }_{\mathrm{reg}}\left(\rho \right)$ with ${\psi }_0(\rho ,\rho )$ in Eq. (37). A green dot at $a=-1.6a_{\mathrm{ho}}$ marks the position where $a_{\mathrm{eff}}\approx d$ if $V_0^B=50E_r^B$.

Figure 11

(Color online) Same as Fig. 9 for $m_B∕m_A=6.67$ (trapped ${}^{40}\mathrm{K}$ and free ${}^6\mathrm{Li}$). The blue dashed line is the Born approximation $a_{\mathrm{eff},\mathrm{Born}}=a{m}_A∕\mu$, obtained by replacing ${\psi }_{\mathrm{reg}}^{k=0}\left(\rho \right)$ by ${\varphi }_0\left(\rho \right)$ in Eq. (38).

Figure 12

(Color online) Energy dependence of the scattering amplitude for two particles of equal mass at $a=-1.6a_{\mathrm{ho}}$ (green marker in Fig. 10). The linear fit at low energy, $\mathrm{Re}(1∕f_k)=-1∕a_{\mathrm{eff}}+r_e{k}^2∕2$, yields $a_{\mathrm{eff}}=8.3a_{\mathrm{ho}}$ and $r_e=1.135a_{\mathrm{ho}}$ (units of $a_{\mathrm{ho}}^{-2}$ and $a_{\mathrm{ho}}^{-1}$ on the horizontal and vertical axis, respectively).

Figure 13

(Color online) Position of the broadest resonances for positive (upper graph) and negative (lower graph) $a$ as a function of $m_B∕m_A$. The dashed lines are the theoretical predictions: for $a>0$ they are given by Eq. (44); for $a<0$ and $m_B<m_A$ they are given by the Hamiltonian in Eq. (48); for $a<0$ and $m_B>m_A$ it is given by Eq. (51). In the upper graph, from top to bottom symbols correspond to $n=4,3,2,1$ in Eq. (44). The dotted vertical line indicates the value of the mass ratio $m_B∕m_A$ when $A={}^6\mathrm{Li}$ and $B={}^{87}\mathrm{Rb}$.

Figure 14

(Color online) Decay of the off-diagonal real space matrix elements of the resolvent as a function of the distance from the center of the cloud of scatterers (in units of the lattice spacing $d$), averaged over 100 different realizations of the random potential. The $\approx 4600$ scatterers $(a_{\mathrm{eff}}=0.3d)$ are distributed in a cubic box of side $21d$, occupying each site of the lattice with $p=0.5$. The $y$ axis is in units of ${({\hslash }^2∕2m_A)}^{-2}$.