Observation of interaction-induced mobility edge in a disordered atomic wire

Introduction – The concept of mobility edge (ME), a critical energy separating extended and exponentially localized energy eigen-states in the excitation spectrum, is key for understanding Anderson localization Anderson1958 ; Dalichaouch1991 ; Wiersma1997 ; Chabanov2000 ; Strzer2006 ; Schwartz2007 ; Julien2008 ; Lahini2008 ; Billy2008 ; Kondov2011 ; Jendrzejewski2012 ; Isam2015 ; White2020 induced by random disorder in three dimensions (3D) Abrahams1979 . In low dimensions, arbitrarily weak random disorder can make all single-particle eigenstates localize, hence ME is absent. Localization phenomena have also been actively studied in quasi-periodic lattice systems with incommensurate modulations Aubry1980 ; Grempel1982 ; Kohmoto1983a ; Kohmoto1983b ; DasSarma1986 ; Thouless1988 ; Modugno2009 ; Biddle2009 ; Biddle2011 ; Yan2021 , such as those described by Aubry-André (AA) model Aubry1980 .

Realization of the AA model in cold atoms has led to the first observation of the localization transition of a noninteracting Bose-Einstein condensate (BEC) Roati2008 . As the AA model has self-dual symmetry Aubry1980 , the localization transition is energy-independent (i.e., no ME), with all eigenstates being localized across a single critical point. Intriguingly, variants of AA model which have broken self-duality, such as the generalized Aubry-André (GAA) model Ganeshan2015 , can host ME already in one dimension (1D). So far, the existence of ME has been mainly conjectured in noninteracting quasi-periodic lattice systems DasSarma1988 ; Goedeke2007 ; Biddle2010 ; Ganeshan2015 ; Li2017 ; Yao2019 ; Deng2019 ; Li2020 ; Wang2020 ; Roy2021 , and experimentally confirmed with cold atoms in optical lattices Lschen2018 .

Beyond noninteracting systems, the realization and control of ME are of fundamental interests, but are generally challenging. Recently, some atomic experiments in this direction have been carried out, showing how single-particle MEs are affected by weak interactions An2018a ; An2021a , and many-body ME has been discussed in the context of many-body localization Kohlert2019 . In these experiments, however, ME is already expected without interactions. It is thus highly desired to understand MEs based on systems with tunable interactions.

In this work, we demonstrate that ME can be induced and tuned by interaction; the physical picture is shown in Fig. 1(a): weak interaction can have different dressing effects on different energy eigenstates of the AA model, resulting in a suppressed or enhanced localization of an eigenstate, thus a critical energy (i.e., ME) is expected in the excitation spectrum.

Experimentally, we observe signatures of ME based on the momentum-state lattice of quasi-1D ¹³³Cs BEC that simulates a nonlinear AA model [Fig. 1(b)]. Exploiting the tunable scattering length of Cs atoms with Feshbach resonances Weber2003 ; Kraemer2004 ; Chin2004 ; Chin2010 , we realize a disordered wire with a wide range of interaction, and observe the extended-to-localized transition in the excitation spectrum. In particular, the controllability of all the system parameters including interactions allows us to access the highest excited state, which can be viewed as the ground state of the associated negative Hamiltonian. We demonstrate that interactions can enhance the localization of either low- or high-energy eigenstates, depending on the sign of the interactions. We further construct a mobility-edge phase diagram, which agrees well with the theory.

Such interaction-induced ME can be understood through an effective noninteracting GAA model. While the nonlinear AA model and its variants Lahini2009 ; Deissler2010 ; Lucioni2011 ; An2017 have been studied before in optical and atomic setups McKenna1992 ; Pikovsky2008 ; Deissler2010 ; Lucioni2011 ; Lellouch2014 ; Schwartz2007 ; Lahini2008 ; Lahini2009 ; An2017 ; Deissler2010 ; Lucioni2011 , the exploration of ME in the model remains elusive.

ME in the nonlinear AA model – We start by theoretically showing how ME arises in the nonlinear AA model

Here $\hbar$ is the reduced Planck’s constant, ${\varphi}_{j}$ is the wave function at site $j$ in a lattice of size $L$ with $\sum_{j}|\varphi_{j}|^{2}=1$, and $J$ is the nearest-neighbor coupling. The on-site modulation with $\beta=(\sqrt{5}-1)/2$ has an amplitude $\Delta$ and phase $\phi$, which plays the role of disorder. The nonlinear term characterized by $U$ in our subsequent discussion arises from the atomic interaction. When $U=0$, Eq. (1) reduces to the AA model, with all eigenstates extended for $\Delta/J<2$ and localized for $\Delta/J>2$.

We are interested in the weak interaction regime $|U/J|\lesssim 1$ where the self-trapping Smerzi1997 ; Albiez2005 does not occur, and every eigenstate has a correspondence in the noninteracting counterpart Wu2005 . Insights can be obtained by noting that the combination of the nonlinear term and the disorder results in a density-dependent potential $V_{j}^{\textrm{ext}}=\Delta\cos(2\pi\beta j+\phi)-Un_{j}$, with the density $n_{j}=|\varphi_{j}|^{2}$. As the density distribution is shaped by the incommensurate modulation, $V_{j}^{\textrm{ext}}$ contains multiple harmonics of the quasi-periodicity (i.e., $2\pi\beta$), which breaks the self-duality and leads to ME. For $|U/\Delta|\ll 1$, perturbative analysis suggests $V_{j}^{\textrm{ext}}$ is effectively a GAA lattice potential. For instance, when $\phi=0$, by Fourier expanding the density up to the second harmonics of the quasi-periodicity, we have $V_{j}^{\textrm{ext}}\approx(\Delta-Uc_{1})\cos(2\pi\beta j)-Uc_{2}\cos(4\pi\beta j)$ (apart from some constant), with the expansion coefficients $c_{1}$ and $c_{2}$. It approximates the GAA lattice potential Ganeshan2015

with $\alpha^{*}\ll 1$ and $\alpha^{*}\propto-U$, up to the second harmonics. Thus the physics of Eq. (1) may be understood via an effective noninteracting GAA model: $i\hbar\dot{\varphi}_{j}=\!\!J(\varphi_{j+1}+\varphi_{j-1})\!+\!V^{j}_{\textrm{GAA}}\varphi_{j}$. As the GAA model hosts an exact ME Ganeshan2015 , the location $E_{c}$ of ME in a weakly nonlinear AA model is expected to be

where sgn denotes the sign function. Because $\alpha^{*}\propto-U$, we expect the location of ME in the low- or high-energy region is swapped when $U\rightarrow-U$.

The above analysis is supported by numerical calculations. We focus on the ground state (GS) and the highest excited state (ES) of the nonlinear AA model footnote1 , and characterize their degree of localization via the participation ratio

For a localized state, $r\approx 0$; for an extended state, the maximum possible $r$ is $1$. Figure 2(a) and (b) show the participation ratio $r$ as a function of $\Delta/J$ and $U/J$ for GS and ES, respectively. To identify the transition from the extended to the localized, we use the critical value of $r$ at $\Delta/J=2$ in the noninteracting limit (see white curves) footnote2 ; for $L=21$ sites, the critical value is $r=0.103$. We see that while the transition points of GS and ES coincide without interaction, they differ in the presence of interaction, suggesting the transition is energy-dependent. Moreover, adding $U>0$ enhances localization of GS but delocalizes ES, whereas the opposite occurs for $U<0$. The critical points of GS and ES divide the parameter space $(\Delta,U)$ into four regimes [Fig. 2(c)]. In phase II (IV), all states are localized (extended). But in both phases I and III, extended and localized states coexist, signaling the existence of ME. When increasing disorder, ME emerges from the high-energy regime in I ($U<0$), while it emerges from the low-energy regime in III ($U>0$).

We validate the effective GAA model through numerical simulations. We calculate the effective parameter $\alpha^{*}=(\Delta_{c}^{g}-\Delta_{c}^{e})/(E^{c}_{e}-E^{c}_{g})$ based on Eq. (3), where $E^{c}_{g}$ ($E^{c}_{e}$) denotes the energy of GS (ES) at their critical point $\Delta_{c}^{g}$ ($\Delta_{c}^{e}$), representing where ME coincides with the lowest (highest) energy. Figure 2(d) shows $\alpha^{*}$ as a function of $U/J$ (blue curve). It is linear for small interactions, confirming the previous conjecture. In the linear regime, we expect the location of ME to be given by Eq. (3). When the linearity breaks down, the effective model is no longer suitable.

Experimental realization of the model – We experimentally realize the nonlinear AA model using the momentum-state lattice of ¹³³Cs BEC that contains $4\times 10^{4}$ atoms in the hyperfine state $|F=3,m_{F}=3\rangle$ [Fig. 1(b)] Wang2021 . We start with a BEC confined in a quasi-1D optical trap Supp . Two counter-propagating laser beams with the wavelength $\lambda=1064$ nm are applied, one with a frequency $\omega$, while the other containing multi-frequency components $\omega_{j}=\omega-\Delta\omega_{j}$, $j=-10,...,9$. They drive a series of two-photon Bragg transitions to couple $21$ discrete momentum states with the momentum increment of $2\hbar k$ (with $k=2\pi/\lambda$), which simulates AA model of $L=21$ sites with the nearest-neighbor coupling $J$ Meier2016 . The disorder is realized by engineering $\Delta\omega_{j}$ to yield an on-site energy $\Delta\cos(2\beta\pi j+\phi)$ Supp with both $\Delta$ and $\phi$ controllable via Bragg lasers. We employ a broad Feshbach resonance centered at magnetic field $B=-11.7$ G to tune the atomic $s$-wave scattering length Weber2003 ; Kraemer2004 ; Chin2004 ; Chin2010 . According to the mean-field theory of the momentum-lattice system Chen ; An2021a ; An2018 ; An2021b , the atomic interaction leads to the nonlinear term in Eq. (1), with $U=(4\pi\hbar^{2}a/m)\bar{\rho}$, where $m$ is the atomic mass and $\bar{\rho}$ is an effective atomic density Supp . We note that a quasi-1D BEC can be stable for $a<0$ BECBOOK ; Experimentally, we do not observe the collapse of BEC for $a<0$ on the time scale of $2$ ms relevant for our measurements. To avoid significant three-body loss Kraemer2006 , we restrict ourselves to $a>-100a_{0}$ ($a_{0}$ is the Bohr radius).

To confirm the realization of the nonlinear AA model, we first tune $a\approx 0$ and observe the transport of the noninteracting BEC by controlling the disorder strength. Initially, the BEC with zero momentum is prepared at momentum-state lattice site $j=0$, before the Bragg lasers are switched on. After an evolution time $t$, the momentum distribution is measured through the time-of-flight (TOF) technique Supp . TOF images for $t=4\hbar/J=1.28$ ms under different disorder strength are illustrated in Fig. 3(a). As expected from the AA model, atoms spread over several lattice sites for $\Delta/J<2$, but localizing sharply near the initial position for $\Delta/J>2$. Figure 3(b) measures the time-dependent momentum width, $\langle d(t)\rangle=\sum_{j}|j|n_{j}(t)$, where $n_{j}(t)$ is the measured fraction of atomic population at the momentum state $|2j\hbar k\rangle$ at time $t$. We observe the familiar crossover from the ballistic transport to localization with increasing disorder, in agreement with the numerical simulations (solid lines) based on Gross-Pitaevskii (GP) equation Supp .

Next, we tune the scattering length from $a<0$ to $a>0$ using the broad Feshbach resonance and fix the disorder at $\Delta/J=1$. Figure 3(c) shows the measured momentum width $\langle d\rangle$ at $t=2\hbar/J=0.64$ ms under various interactions. We observe $\langle d\rangle$ decreases with the interaction strength, regardless of its sign, suggesting an increased degree of localization. The experiment agrees qualitatively with the GP calculations Supp (red curve).

Construction of ME phase diagram – Important for our study is the preparation of GS and ES under various disorder and interaction. We prepare GS following the adiabatic protocol in Ref. An2021a . It consists of switching off the laser coupling between lattice sites and initializing atoms in the ground state at zero-momentum site $j=0$. Then, by linearly ramping the coupling from $J=0$ to $J/\hbar=2\pi\times 275$ Hz in $1$ ms and holding there for $1$ ms, the initial state is transferred to GS of the lattice system.

A reliable preparation of ES, however, was nontrivial due to non-adiabatic effects An2021a . Here we adopt a strategy which is motivated by the fact that ES of a Hamiltonian $H$ can be viewed as GS of $-H$. Experimentally, in preparing ES of a system with desired $J$, $\Delta$ and $a$ we instead realize an associated “negative Hamiltonian” (i.e., $-H$) with $-J$, $-\Delta$ and $-a$. To realize $-J$, we introduce a relative phase $\pi$ in the two-photon Bragg transition coupling neighboring momentum states [c.f. Fig. 1(b)]. The $-\Delta$ is achieved by tuning $\phi$. Crucially, the conversion from $a$ to $-a$ is uniquely enabled by Feshbach tuning of the interaction of Cs atoms. Then we adiabatically prepare GS of the negative system similarly as before: after switching off the laser coupling and initializing atoms at a site with the lowest energy, we linearly ramp up $J$, thus achieving ES of the original system.

After the state preparation, we measure the population in each momentum mode to obtain the participation ratio $r$ according to Eq. (4). We identify the potential extended-to-localized transition by numerically fitting the experimental data of $r$ as a function of $\Delta/J$ Supp , as shown in Figs. 4(a)-(b). For $U\approx 0$, the transition is at $\Delta/J=2$ for both GS and ES, as expected. Comparing Figs. 4(a) and (b) shows that the localization of GS is enhanced (suppressed) under $U>0$ ($U<0$), whereas the opposite occurs for ES, in agreement with the predictions. Owing to the non-adiabatic effect in the experimental ramp, the participation ratio $r$ is generically smaller than the idealized value expected from Eq. (1) Supp . Moreover, the experimental $r$ deep in the localized phase is slightly higher than $B=1/21$ due to the residual population. Nevertheless, these imperfections and finite-size effect do not qualitatively change the transitions Supp .

Finally, to construct the phase diagram, we collect the extracted transition points of GS and ES under various interactions into Fig. 4(c). The good agreement between the experiment and theory provides strong evidences on the existence of interaction-induced ME. It also confirms that, depending on whether the atomic interaction $U$ is attractive or repulsive, ME emerges from the high- or low-energy region of the excitation spectrum. In the inset of Fig. 4(c), we compare the experimental data with the predictions from the effective GAA model [c.f. Fig. 2(d)]. As shown, the effective model provides a good explanation of the data for $|U/J|\lesssim 0.25$, suggesting the location of ME in this regime is given by Eq. (3).

Conclusion – In this work we have provided experimental evidence that ME can be induced and controlled by interactions. The interaction-induced ME is different from the single-particle ME in noninteracting models and from the many-body ME Abanin2019 ; Deng2017 ; Li2015 . Our observations shed new light on the interplay between disorder and interaction. The widely tunable atomic interaction featured in our experiment enables the first observation of this intriguing phenomena, which presents new opportunities for engineering the quantum transport and quantum phase transitions in disordered systems.

Acknowledgement – We acknowledge helpful discussions with Biao Wu, Xiaopeng Li, Zhihao Xu and Shuai Chen. This research is funded by National Key R&D Program of China (Grant No. 2017YFA0304203), National Natural Science Foundation of China (Grant No. 62020106014, 92165106, 62175140, 12104276, 11874038, 12034012, 12074234). CC acknowledges support by the National Science Foundation (Grant No. PHY-2103542).