Competition and interplay between topology and quasi-periodic disorder in Thouless pumping of ultracold atoms

Correspondence and requests for materials should be addressed to S. Nakajima.

We thank H. Aoki, Y. Hatsugai, and K. Imura for valuable discussions and A. Sawada for experimental assistance. This work was supported by the Grant-in-Aid for Scientific Research of JSPS (Nos. JP25220711, JP26247064, JP16H00990, JP16H01053, JP17H06138, JP18H05405, JP18H05228, and JP18K13480), the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) program, JST CREST (No. JPMJCR1673), and MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant No. JPMXS0118069021. Y. K. acknowledges the support of the Grant-in-Aid for JSPS Fellows (No.17J00486). P. M. is supported by the Japan Science and Technology Agency (JST) of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), JST CREST Grant. No. JPMJCR19T2 and by the (MEXT)-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006) and by JSPS Grant-in-Aid for Early-Career Scientists (Grant No. 20K14375).

S. N. and N. T. contributed equally to this work. S. N., N. T. and K. S. carried out experiments and the data analysis. Y. K. and P. M. carried out the theoretical calculation. Y. T. conducted the whole experiment. All the authors contributed to the writing of the manuscript.

The authors declare no competing interests.

In order to estimate the qualitative properties of our experimental system modeled by the cRM model, we consider an approximated tight-binding picture. Our experimental system is based on the following dynamical superlattice potential in a clean limit:

Hereafter, since in our experiments only the first band is almost occupied, and the band gap between the second and third bands are fairly large, we can focus on only the first and second bands created by our experimental superlattice potential. Then, we can describe the system with an effective tight-binding model, namely, the tRM model in a clean limit, which reads

In the topological charge pumping (TCP), $\delta$ and $\Delta$ are adiabatic dynamical parameters, which can be approximately determined by the band structure induced by the superlattice potential $V_{\mathrm{clean}}(x,t)$. Here, we approximate these dynamical sequence for $\delta$ and $\Delta$ as a circular trajectory, $\delta=\delta_{0}\sin\theta$, $\Delta(t)=\Delta_{0}\cos\theta$, where $\theta=2\phi(t)$. Although the actual dynamical sequence for $\delta$ and $\Delta$ in our experimental system is subtly different from the circular one, this approximation facilitates the ideal theoretical treatment of the TCP phenomena, especially for calculating the Chern number, and can gives insight for qualitative properties of the TCP described by the cRM in our experiment. The physical values of $J$, $\delta_{0}$ and $\Delta_{0}$ can be directly determined by the band structure of the superlattice potential $V_{\mathrm{clean}}(x,t)$. The values of $J$ and $\delta_{0}$ are determined by the band width in the double-well lattice case ($\theta=\pi/2$, $3\pi/2$), while the value of $\Delta_{0}$ by the band gap in the staggered lattice case ($\theta=0$, $\pi$) as discussed in the previous study Nakajima . For the band structure determined by our experimental setup, the tRM parameters are approximately determined. The tRM model is expected to capture the qualitative behavior for our experimental system, since in the clean limit, the circular parameter sequence for $\delta$ and $\Delta$ is topologically connected to our actual experimental sequence without gap closing. Regarding the basic topological aspect of the tRM model, for $\Delta=0$ (at inversion symmetric point, $\theta=\pi/2$, $3\pi/2$), the model reduces to the celebrated Su-Schrieffer-Heeger (SSH) model, classified in the Altland-Zirnbauer class BDI of the periodic classification of topological statesSchnyder ; Kitaev . The SSH model exhibits a topological phase and zero energy topological edge modes Asboth . However, a finite $\Delta$ breaks the chiral symmetry without gap closing, i.e., ${\hat{\sigma}}_{z}{\hat{H}}_{\rm RM}(k){\hat{\sigma}}_{z}\neq-{\hat{H}}_{\rm RM}(k)$ where ${\hat{H}}_{\rm RM}(k)$ is the bulk momentum Hamiltonian of Eq. (S2). Accordingly, the tRM model is not necessarily in the BDI class.

In our experiment, a quasi-periodic disorder lattice created by an additional laser with wavelength $\lambda_{D}$ (= 798 nm) applied by tilting 45^∘ for the one-dimensional superlattice axis, is given by

(see Fig. 1a of the main text). This quasi-periodic disorder potential $V_{\rm Dis}(x)$ can be effectively implemented in the tRM model as an on-site potential given by

where $\beta=d/(2d_{D})$. The potential amplitude $\Delta_{D}$ in $\hat{V}^{\prime}_{\rm Dis}$ does not strictly correspond to the value of the continuous system potential $V_{D}$, but gives a good approximation. The quasi-periodic disorder potential $\hat{V}^{\prime}_{\rm Dis}$ is diagonal and breaks the chiral symmetry even for $\theta=\pi/2$ or $2\pi/3$ SSH case. In general, this type of the disorder strongly affects the topological system and leads to the breakdown of the topological phase, compared to a chiral symmetric disorder employed in a recent experiment Meier . On the other hand, in this tight-binding picture we ignore modulated effects of $V_{\rm Dis}(x)$ to the tRM parameters, $J$, $\delta_{0}$ and $\Delta_{0}$. Thus, we consider ${\hat{H}}_{RM}+\hat{V}^{\prime}_{\rm Dis}$ as an approximated tight-binding model for our experimental system.

For the sake of simplicity, we consider the effect of quasi-periodic disorder only on the on-site potential and ignore the effects of the trapping potential. However, the resulting model is nevertheless able to capture the essence of the physical behavior of the cRM model and of the experimental system. In particular, for small $V_{D}$, the tRM model reasonably approximates the cRM model. However, for large $V_{D}$, deviations between the tRM model and cRM models become noticeable. Since $\hat{V}^{\prime}_{\rm Dis}$ is diagonal, the total tight-binding model has no chiral symmetry even for inversion symmetric point, therefore $\hat{V}^{\prime}_{\rm Dis}$ is expected to have strong influence to the TCP.

To estimate the Anderson localization point $V_{AL}$ in our experimental system, we calculate effects of quasi-periodic disorder created in a disordered SSH model, corresponding to the $\Delta=0$ case in the tRM model of Eq. (S2) including the $\hat{V}^{\prime}_{\rm Dis}$ of Eq. (S4) as an effect of the disordered lattice potential. Here, we consider a quasi-periodic disorder lattice with a wavelength of $798$ nm. In general, since we consider quasi-periodic disorder, we expect an Anderson localization transition at a finite disorder strength even in the one dimensional system. In conventional Anderson localization, single atoms are localized at each single lattice site. To estimate the localization transition in our considering system, we employ the inverse participation ratio (IPR). The IPR for each eigenstate is defined as

where $|j\rangle$ is a $j$-site localized single particle state and $|\psi_{\ell}\rangle$ is $\ell$-th eigenstate in the SSH model with quasi-periodic disorder potential. Large value of the IPR is a signature of the localized tendency of the single particle wave function. We perform the IPR calculation for four different parameter sets of ($J$,$\delta$) corresponding to those in the four different pump trajectories in our experiments. The results are shown in Fig. S1, where we plot the IPR as a function of $\Delta_{D}$ and of the eigenstate number in $L=200$ lattice site system with periodic boundary conditions. The sets $C_{1}$ and $C_{2}$ correspond to the parameter sets A and B in Fig. 2a in the main text, and the set $C_{3}$ and $C_{4}$ correspond to the trajectories $C_{\mathrm{inner}}$ and $C_{\mathrm{outer}}$ in Fig. 5b, respectively. The order of the eigenstate numbers corresponds to the eigenenergy in ascending order. Due to the quasi-periodic disorder, all cases in Fig. S1 exhibit mobility edges, i.e., different eigenstates localize for different disorder strengths. Here we take the condition $({\rm IPR})_{\ell}\gtrsim 0.1$ as a localization criterion, while the transition point is determined by the case where all eigenstates fulfill this condition. For all the cases in Fig. S1, all the eigenstates already tend to be localized for $\Delta_{D}/2\Delta_{0}<0.04$. We define the Anderson localization transition point $V_{AL}$ in the main text, using eigenstates with lower eigenenergies. These states are localized at $\Delta_{D}/2\Delta_{0}\sim 0.02$ and $\Delta_{D}/2\Delta_{0}\sim 0.004$ for the pump trajectories $C_{1}$ and $C_{2}$ (or the set A and B in the main text), respectively. These values correspond to $V_{AL}^{A}\sim 0.3E_{R}$ and $V_{AL}^{B}\sim 0.07E_{R}$. Even though from the numerical calculation our experimental system (cRM model) is also expected to be localized around the SSH parameters points, this fact does not mean gap closing directly. Even when the system exhibits localization and fulfills our criteria, the pumping does not necessarily break down, as shown numerically in Ref. Wauters .

The tRM model can be mapped into the Harper–Hofstadter–Hatsugai (HHH) model Hatsugai through a dimensional extension Kraus ; Lohse . By regarding the pumping parameter $\theta$ in ${\hat{H}}_{\rm RM}$ of Eq. (S2) as $y$-direction momentum $k_{y}$ and applying the Fourier transformation for $k_{y}$, we can obtain the HHH model from the tRM model:

Here $\hat{c}^{(\dagger)}_{m,n}$ is the annihilation (creation) operator at a two dimensional lattice site $(m,n)$, where $m=1,2,\cdots,L_{x}$ and $n=1,2,\cdots,L_{y}$, i.e., $L_{x}\times L_{y}$ lattice system. The HHH model is known to have two topological bands with a non-zero Chern number Hatsugai ; Kraus . This is a peculiar property compared to a conventional quantum Hall system on lattice described by the Hofstadter model Hofstadter , since the Hofstadter model does not exhibit two separated bands with non-trivial topology. The two topological bands with a non-zero Chern number in the HHH model is the origin of the TCP in the tRM model. The value of the Chern number corresponds to the total pumped current per one pumping cycle in the tRM model Asboth . For the HHH model, we can further implement the effect of the quasi-periodic disorder $\hat{V}^{\prime}_{dis}$ of the tRM model as $\hat{V}^{\prime\prime}_{\rm Dis}=\frac{\Delta_{D}}{2}\sin(2\pi\beta m+\alpha){\hat{c}}^{\dagger}_{n,m}{\hat{c}}_{n,m}$. Here, it should be noted that $\hat{V}^{\prime\prime}_{\rm Dis}$ depends only on the $x$-component site $m$ and is uniform along $y$-direction.

In our experimental system, atoms almost occupy the lowest band and the occupation of atom in higher bands is fairly suppressed. We assume that the lower band of the tRM model is fully-occupied approximately. Accordingly, in the numerical calculation of the HHH model the half-filling case is considered. Then, to make comparison with the experimental results shown in the main text, we set the HHH model parameters ($J$, $\delta_{0}$, $\Delta_{0}$) to four experimental parameter sets: (I) ($V_{L}$,$V_{S})=(20,14)E_{R}$, ($J$,$\delta_{0}$,$\Delta_{0})=(0.861,0.851,6.45)E_{R}$, (II) ($V_{L}$,$V_{S})=(30,20)E_{R}$, ($J$,$\delta_{0}$,$\Delta_{0})=(0.860,0.857,8.54)E_{R}$, (III) ($V_{L}$,$V_{S})=(15,10)E_{R}$, ($J$,$\delta_{0}$,$\Delta_{0})=(0.914,0.898,4.86)E_{R}$, and (IV) ($V_{L}$,$V_{S})=(36,24)E_{R}$, ($J$,$\delta_{0}$,$\Delta_{0})=(0.827,0.825,9.605)E_{R}$. These approximated circular pumping trajectories determined by the above four parameter sets are denoted by $C_{1}$, $C_{2}$, $C_{3}$ and $C_{4}$. These parameter sets ($J$, $\delta_{0}$, $\Delta_{0}$) are determined by comparing to the energy spectra of the cRM model as in the previous paper Nakajima . The $C_{1}$ and $C_{2}$ trajectories are the approximated version of the set A and B in the experimental pumping sequences in Fig. 2a in the main text, respectively. The $C_{3}$ and $C_{4}$ trajectories correspond to the experimental pumping sequences $C_{\mathrm{inner}}$ and $C_{\mathrm{outer}}$ in Fig. 5b, respectively. In what follows, We numerically calculated the Chern number for $C_{1}$, $C_{3}$ and $C_{4}$ trajectories and use $L_{x}=40$ (20 unit cells). This $x$-direction system size is close to our experimental system.

We calculate the Chern number for the model of $\hat{H}_{2D}+\hat{V}^{\prime\prime}_{\rm Dis}$ in the following manner. Since discrete translational invariance is broken due to the term $\hat{V}^{\prime\prime}_{\rm Dis}$, we employ a calculation method to obtain the Chern number from the real space Hamiltonian, namely, the so-called coupling matrix method YFZhang ; Castro ; Sriluckshmy ; Kuno . In this method, we impose twisted periodic boundary conditions with two twisted phase $\theta_{x}$ and $\theta_{y}$ for each spatial direction. Their twisted phases play a role of the momentum $k_{x}$ and $k_{y}$ (corresponding to $\theta$) in the TKNN formula Niu85 . Since this twisted phase boundary conditions can be employed even for a system without translational invariant such as a disordered system, we can calculate the Chern number for the model of $\hat{H}_{2D}+\hat{V}^{\prime\prime}_{\rm Dis}$.

Figure S2 displays the calculated Chern number in the presence of quasi-periodic disorder and its dependence of the artificial $y$-spatial system size $L_{y}$ with fixed $L_{x}=40$ (20 unit cells system from the view point of the tRM model). The result indicates that the system-size dependence is fairly small, and the topological phase transition point does not depend much on the value of $L_{y}$. Therefore, the system size used in the numerical calculation is large enough to capture the behavior of our experimental system. $L_{y}=24$ data in Fig. S2 is displayed in Fig. 2b in the main text.

Next, we employ this numerical method to study the disorder-induced charge pump (DICP). The schematic pumping trajectory plotted in the $\delta-\Delta$ plane is shown in Fig. S3a. This trajectory is a combination of circular $C_{3}$ and $C_{4}$ trajectories. Our experiment is conducted with a trajectory similar to this schematic trajectory. In particular, the experimental trajectory can be connected by continuous deformation, and it is therefore topologically equivalent, to the trajectory shown in Fig. S3a. The DICP trajectory does not wrap the origin $(\Delta_{0},\delta_{0})=(0,0)$ corresponding to the gap closing point in the tRM model in the clean limit. In this sense, in the clean limit, the tight-binding RM model of Eq. (S2) exhibits no TCP, i.e., total pumped current is zero. However, once the quasi-periodic disorder $\hat{V}^{\prime}_{\rm Dis}$ is switched on, the disordered model has possibility to exhibit a non-trivial pumped current that could not occur at the clean limit. This corresponds to the fact that the disordered HHH model also has possibility to exhibit a topological non-trivial phase with a non-vanishing Chern number, $C_{N}\neq 0$.

To study the DICP, we split the DICP trajectory into two circles, $C_{3}$ and $C_{4}$. For each circular trajectory, we calculate the Chern number by using the disordered HHH model. Here, various system sizes for $L_{y}$ are calculated to check the finite size effects. Moreover, for all data we average over different samples corresponding to different values of $\alpha$ (60 samples). The numerical results for each trajectory are shown in Fig. S3b and c. The data display $\Delta_{D}$-dependence (not $2\Delta_{0}$ scale) of the Chern number. Furthermore, Fig. S3d shows the sum of the Chern number between $C_{3}$ and $C_{4}$ trajectories. The result clearly captures the presence of a non-trivial pump between $10E_{\rm R}\lesssim\Delta_{D}\lesssim 30E_{\rm R}$, quantified by the imperfect cancellation between the Chern numbers for $C_{3}$ and $C_{4}$ trajectories. All results in Fig. S3b-d indicate that the finite system size ($L_{y}$) dependence is fairly small. $L_{y}=24$ data in Fig. S3 are used in Fig. 5b and c in the main text.

In addition, we show that this type of pump protocol in Fig. S3a theoretically exhibits a topological DICP. Here, we consider that the inner and outer trajectories are farther apart compared with the experimental ones, i.e., we set the inner circle (C3) with $(J,\delta_{0},\Delta_{0})=(1,0.9,5)E_{R}$ and the outer circle (C4) with $(J,\delta_{0},\Delta_{0})=(6,5,45)E_{R}$. The numerical results are shown in Fig. S3e-g. As seen in Fig.S3g, there exists a quantized plateau within the errors or fluctuations due to numerical instability inevitable for disordered systems for moderate disorder strength.

To obtain the band structure in the continuous system, we use a continuous model with a lattice potential given by

and we solve the time-independent Schrödinger equation

using space discretization. Using the recoil energy as the energy unit and the lattice constant of the long lattice as the length unit one has $E_{R}={h^{2}}/{(8m)^{2}}=1$.

Since $d_{\mathrm{D}}/d$ is an irrational number, the quasi-periodic disorder lattice is incommensurate with respect to the short and the long lattice. As a consequence, the translational symmetry of the lattice is broken, and thus the unit cell of superlattice is not well-defined. Hence, one cannot meaningfully define the periodic boundary conditions for the system. Therefore, we will approach the irrational value $d_{\mathrm{D}}/d={3}/{2\sqrt{2}}$ by taking a succession of rational approximations of the irrational number, obtained in terms of continued fraction representation continued_fractions_hardy . Every irrational number $R\in\mathbb{R}-\mathbb{Q}$ can be written uniquely as an infinite continued fraction as

with $N_{i}\in\mathbb{Z}$ integers. The successive approximations $R_{n}$ are obtained by truncating the continued fraction representation $[N_{0};N_{1},N_{2},\,\ldots,N_{n}]$. Since $R_{n}$ are rational numbers one can write as

where $P_{n},Q_{n}$ are coprimes. These represent successive rational approximations of the irrational number $R$.

In our specific case one has

For $V_{\mathrm{D}}=0$ the unit cell of the superlattice has length $d$. For $V_{\mathrm{D}}>0$ and $R=d/d_{\mathrm{D}}$ being a rational number $R=P/Q$ with $P,Q$ coprimes, the total unit cell of the superlattice has length $Qd$ as one can see by direct substituting $x\to x+Q$ Eq. (S7). The energy levels are then obtained by solving the Schrödinger equation with periodic boundary conditions over a system of length $L=N_{\mathrm{c}}Qd$ where $N_{\mathrm{c}}$ is the total number of unit cells. We take $P/Q=35/33$ and a total length of $198d$.

Figure S4 shows the bulk density of states (DoS) as a function of $V_{\mathrm{D}}$ and the band structure in the clean and in the strong disorder regimes, for different values of the lattice depths $V_{\mathrm{L}}$ and $V_{\mathrm{S}}$. The DoS and the band structure are calculated using closed (periodic) boundary conditions. In the shallower lattice cases (a and b) we can clearly see from the DoS that the global gap closes and reopens at $V_{\mathrm{D}}\approx 5$. For the case (c) , a very small gap closes and reopens at $V_{\mathrm{D}}\approx 7$. In the case (d) , which is the same as Fig. 3d of the main text, the numerical calculations show the gap closing at $V_{\mathrm{D}}\approx 5$ but do not show such re-opening of the gap.

Figure S5 shows the band structure calculated in the case of open boundary conditions in the clean $V_{\mathrm{D}}=0$ and strong $V_{\mathrm{D}}=15$ disorder regimes, for the parameter set $V_{\mathrm{L}},V_{\mathrm{S}}=5,3.5$ (as in Fig. S4a). In the clean limit, one can clearly see an edge state which connects the lowest energy band with the first excited band. Therefore the lowest energy gap is topologically non-trivial. In the strong disorder case there is an intraband state in the lowest energy gap. This intraband state do not connect the two bands and it is therefore topologically trivial. Non-trivial edge states are also present in the lowest energy gap in the clean limit $V_{\mathrm{D}}=0$ for $V_{\mathrm{L}},V_{\mathrm{S}}=7.5,5$, $V_{\mathrm{L}},V_{\mathrm{S}}=10,7$, and $V_{\mathrm{L}},V_{\mathrm{S}}=20,14$. However, for these choices of the lattice depths, the energy bands dispersion becomes so broad that the lowest energy gaps become very narrow, and the presence and identification of intraband edge states cannot be resolved unambiguously.

In Fig. 3a of the main text, we investigate the gap closing between the first and second bands by measuring the fraction of atoms excited to the second band after three cycles of cRM pumping under quasi-periodic disorder with the lattice parameter $(V_{L},V_{S})=(20,14)E_{R}$. The band structure is defined for the long lattice, spanned by the quasi-momentum $q$, since we adiabatically turn off the quasi-periodic disorder lattice from the whole lattice setup in 130 ms. We then utilize a band-mapping technique, where an adiabatic turn-off of the remaining optical lattices maps quasi-momentum $q$ of $n$-th band to free-particle momentum $p$ of $n$-th Brillouin zone Greiner2001 ; Kohl2005 . By taking absorption images after time-of-flight of 10 ms, we evaluate atom fractions in the first and second band from the momentum distribution.

In the case of a clean limit with $\phi$ fixed at 0, it is expected that only the first band is occupied. Figure S6 shows the measured momentum distribution corresponding to this case. An almost homogeneous distribution is obtained within the first Brillouin zone. However, the homogenous distribution was smoothed around $p=\pm\hbar\pi/d$ owing to experimental imperfection such as non-adiabaticity in the band mapping. We approximate this smoothing by the following function $f_{1}(p)$:

here $a$ and $s$ are fitting parameters and the fitted $s$ value $s_{\mathrm{fit}}$ denotes the smoothness. We assume that this fitted function represents the momentum distribution when the first band is homogeneously occupied.

During the pumping under the quasi-periodic disorder, atoms can be excited to the second band. Suppose that the atom distribution in the second band is also homogeneous, and the smoothness of the value $s_{\mathrm{fit}}$ obtained above is common for both the first and second bands. Then we fit the measured data for the pumping under the quasi-periodic disorder using the following function $f_{t}(p)$:

here

and $b$ is a fitting parameter. The fitted functions $f_{1}(p)$ and $f_{2}(p)$ show the momentum distributions in the first and second bands, respectively. Figure S7 shows one of the fitted results at the disorder $V_{D}=20E_{R}$ in Fig. 3a in the main text. After normalizing the integral of the fitted $f_{t}(p)$, we evaluated the second-band fraction by calculating $\int dpf_{2}(p)$. The mean and error bar representing the standard deviation in Fig. 3a of the main text are obtained for ten independent measurements at each disorder strength.

We can understand intuitively why the pump is suppressed at $V_{D}\sim V_{L}$ in the following way. As long as the lowest band is occupied, Thouless pumping with a cRM lattice is topologically equivalent to that only with a sliding long lattice Nakajima . Therefore we can capture the essence of the results of the pump suppression by considering a sliding lattice superimposed with a quasi-periodic disorder lattice, namely the second and third terms of Eq (1) in the main text.

Figure S8 shows the trapping potential produced by these two lattices as a function of position $x$. In addition, the position change of the local minimum point of the trap during three pump cycles is depicted by the red line. The trap depth of the long lattice is fixed to $V_{L}=20E_{R}$, whereas the disorder strength $V_{D}$ is varied from $0E_{R}$ to $23E_{R}$ (from top to bottom). When $V_{D}\leq 21E_{R}$, the local minimum point moves across several lattice sites during the pump cycle in accordance with the sliding lattice. This means that the whole lattice is sliding with the phase $\phi$ swept. For $V_{D}\geq 22E_{R}$, however, the minimum point stops moving and wanders around the particular trap minimum. Accordingly the whole lattice does not slide. This transition happens around $V_{D}\sim V_{L}$. Therefore it is expected that the pump is suppressed at $V_{D}\sim V_{L}$. Because this discussion does not depend largely on the wavelength of the quasi-periodic disorder lattices, it is reasonable that we observe no clear difference among three measurements in Fig. 4 in the main text.

Since the dominant lattice is that of the static quasi-periodic disorder lattice in the case of $V_{D}>V_{L}$, the above-mentioned situation also indicates that the trivial band gap should be open. Although the re-opening of the gap is not clearly visible in the numerical calculations in Fig. 3 in the main text, this does not necessarily mean that the state is metallic. In fact, the energy difference between the ground and first-excited vibrational levels within individual lattice sites becomes large in the strong disorder as shown in Fig. S9. This large local energy gap should suppress the excitations to higher levels as observed in Fig. 3a in the main text.