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Abstract
Ultracold atoms in quasiperiodic lattices offer a versatile platform for studying many intriguingly disorder-driven phenomena. Here we study the quantum transport of noninteracting ultracold 133Cs atoms in a one-dimensional quasiperiodic momentum lattice, which is formed by using the laser-coupled momentum states as synthetic dimensions. Through measurements of the site-resolved density distribution of atoms in the synthetic quasiperiodic lattice, we directly observe the transition from the atomic diffusive transport to Anderson localization in the initial injection site with increasing strength of the quasiperiodic modulation. We show the dependence of momentum width characterizing the atomic expansion degree on the quasiperiodic modulation strength, and our observation is in good agreement with the theoretical calculation. Our work provides significant insights for understanding the quantum spreading in disordered systems.
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1. Introduction
Anderson localization (AL) is an ubiquitous phenomenon in disordered systems, in which both the wave propagation and the transport of quantum particles are strongly affected by the uncorrelated disorders [1,2]. AL has been actively pursued in many areas of physics, including condensed matter physics, light waves, microwaves, acoustic waves and atomic matter waves [3–16]. Experiments with ultracold atoms afford the control over disorder properties that are not easily realized in conventional mediums, and allow to precisely study the critical properties involved in the transition of AL. The exponential localization of atomic wave functions has been observed in three-dimensional (3D) random disordered potentials, which are formed by using optical speckle [12–14]. The 3D AL occurs at a finite disorder strength, which sets a critical energy, i.e., mobility edge, that separates localized from extended states. For the 1D and 2D systems, the arbitrarily weak random disorder can make all single-particle eigenstates localized [17].
Different from random disorders, the localization phenomena of ultracold atoms have also been extensively studied in the low-dimensional quasiperiodic lattice systems with incommensurate modulations on the on-site energy, such as those described by the Aubry-André (AA) model [18–22]. The experimental realization of quasi-periodic AA model in a bichromatic incommensurate optical lattice has enabled the observations of both the AL of a noninteracting atomic Bose-Einstein condensate (BEC) and the many-body localization of interacting Fermi gases [23,24]. Recently, momentum lattices synthesized through the laser-coupled discrete atomic momentum states further facilitate the experimental study of the AA model in synthetic dimensions, where the on-site energy is precisely tunable in a site-resolved manner [25–29]. Moreover, the tunneling dynamics of atoms in momentum lattices can be directly observed with single-site resolution [25,26,30,31]. However, the direct observation for the effect of quasiperiodic modulation strength on the atomic density distribution has been not reported largely.
In this work, we experimentally realize a quasiperiodic lattice based on a series of laser-coupled discrete momentum states of condensed $^{133}$Cs atoms, where the detunings in the Bragg transitions between those momentum states are individually engineered for the incommensurate modulation of the on-site energy in the synthetic 1D momentum lattice. We measure the quantum transport of atoms in the 1D quasiperiodic lattice through the evolution of atomic density distribution in a site-resolved manner, and observe the transition from the diffusion to Anderson localization with increasing strength of quasiperiodic modulation in the noninteracting regime. The atomic momentum width obtained from the condensate displacement in the momentum-space lattice is used for characterizing the atomic expansion degree after an evolution time, and the measured dependence of momentum width on the quasiperiodic modulation strength shows good agreement with the theoretical calculation based on the Gross-Pitaevskii equation (GPE).
2. Implementing quasiperiodic lattice
Our experiment starts with a BEC of $\thicksim$ $4\times 10^{4}$ $^{133}$Cs atoms in the hyperfine ground state $6S_{1/2}, |F=3, m_{F}=3\rangle$ in a cigar optical trap with the trap frequency of $(\omega _{x}, \omega _{y}, \omega _{z})$ = $2\pi$ $\times$ $(125, 96, 13)$Hz [32,33]. A broad Feshbach resonance is used to tune the atomic $s$-wave scattering length into the noninteracting regime [34]. The trap laser beam that provides the strong radial confinement (with the wavelength of $\lambda$ = 1064nm) is retro-reflected to form a pair of counter-propagating laser beams along the $z$-direction to illuminate the weakly trapped BEC. Here two acoustic optical modulators driven by the programmable radio-frequency signals are used in the reflected laser beam to generate multi-frequency components $\omega _{n}$ with $n\in \{-10, 9\}$, in contrast the incident laser beam with the single frequency $\omega$ as shown in Fig. 1(a). These laser fields drive a series of two-photon Bragg transitions for coupling 21 discrete atomic momentum states with the increment of $2\hbar k$, where $\hbar$ is the reduced Planck’s constant and $k = 2\pi /\lambda$ [35,36]. The frequency difference between the incident and retro-reflected laser beams is engineered according to the quadratic dispersion of free atoms, as $\Delta \omega _{n}$ = $\omega - \omega _{n}$ = $(2n+1)4E_{R}/\hbar$, and $E_{R}$ is the atomic recoil energy. As a result, we synthesize the 1D momentum lattice by using these laser-coupled momentum states, and the tunneling energy is characterized by $J$.
The on-site energy in momentum lattices can be individually controlled through the distinct frequency difference $\Delta \omega _{n}$ in each Bragg transition, where the detuning of multi-frequency components in the retro-reflected laser beam can be precisely tuned. A quasiperiodic lattice potential in the momentum space of ultracold atoms can be realized by incommensurately modulating the on-site energy with $\varepsilon _{n}$ = $\Delta \cos (2\beta \pi n+\phi )$, as shown in Fig. 1(b), where $\Delta$ and $\phi$ are the modulation strength and phase, respectively, as well as $\beta$ = $(\sqrt {5}-1)/2$ [27,28]. As predicted by the AA model, all eigenstates of the system are extended for $\Delta /J<2$ and localized for $\Delta /J>2$ in the noninteracting limit.
3. Observing Anderson localization transition
Before switching on all the Bragg laser beams, the atoms are prepared in the momentum-lattice site $n$ = 0. Then the quantum transport of atoms in the quasiperiodic lattice are probed by measuring the atomic density distribution in a site-resolved manner, where each site is correspondingly mapped to the unique momentum state. After an evolution time $t$ = 4$\hbar /J$, we turn off all the laser beams, and measure the density distribution of atoms in different momentum states after a 22-ms time-of-flight via the standard absorption imaging technique. In Fig. 2, we show the atomic population distribution in momentum space for different ratios of the quasiperiodic modulation strength to the tunneling energy $\Delta /J$. As expected from the AA model, the condensed atoms spread over several momentum-lattice sites for $\Delta /J<2$, but become localized sharply near the initial injection site $n$ = 0 for $\Delta /J>2$ [18]. Specifically, we observe that a large number of atoms are localized for the critical quasiperiodic modulation strength with $\Delta /J=2$, while the number of atoms populating outside the site $n$ = 0 is very small.
Fig. 1. Construction of quasiperiodic lattice in momentum space of ultracold atoms. (a) A $^{133}$Cs BEC is illuminated by two counterpropagating, far-detuned laser beams, one has the single frequency $\omega$ and the other contains multifrequency components $\omega _{n}$ ($n$ = −10,…,9). These laser fields drive a series of two-photon Bragg transitions to couple the discrete atomic momentum states $p_{n}$ = $2n\hbar k$ for synthesizing a 1D momentum lattice with 21 sites in the noninteracting regime. (b) The momentum-lattice site energies are engineered by individually controlling the distinct frequency detuning of multi-frequency components in (a) involved in the Bragg transitions. A quasiperiodic lattice potential is synthesized by incommensurately modulating the on-site energy in the form of $\varepsilon _{n}$ = $\Delta \cos (2\beta \pi n+\phi )$ with the periodicity $\beta$ = $(\sqrt {5}-1)/2$, amplitude $\Delta$ and phase $\phi$.
Fig. 2. Direct observation of the transition from the diffusion of atoms in quasiperiodic lattices to Anderson localization. The population distribution of atoms in the synthetic quasiperiodic lattice for different quasiperiodic modulation strengths with $\Delta /J$ = 0, 1, 2 and 3. The optical density profile of condensate in each momentum-lattice site that corresponds to the distinct momentum state is extracted from the images in the insets, where the standard absorption images are implemented for obtaining the atomic density distribution after the evolution time $t$ = 4$\hbar /J$ and 22 ms time-of-flight. The atoms become localized in the site $n$ = 0 with increasing $\Delta /J$. In all panels, the tunneling energies are $J/\hbar$ = 2$\pi$ $\times$ 500 Hz.
To quantitatively characterize the atomic expansion degree in the quasiperiodic lattice, we define the momentum width as
where $P_{n}(t)$ is the fraction of atoms populating at the momentum state $|2n\hbar k\rangle$ at the time $t$. In Fig. 3, we plot the measured momentum width $\langle d\rangle$ as a function of $\Delta /J$ for three different evolution times $t$ = 2$\hbar /J$, 3$\hbar /J$ and 4$\hbar /J$. Our measurement shows that $\langle d\rangle$ decreases with increasing $\Delta /J$ within the range of $\Delta /J<2$, and the applied quasiperiodic disorders strongly reduce the diffusion of atoms in momentum lattices. For $\Delta /J>2$, $\langle d\rangle$ $\thicksim$ 0 represents the localization of atoms in the site $n$ = 0.Fig. 3. The quantitative description of Anderson localization transition in the transport dynamics of atoms in the quasiperiodic lattice. The momentum width $\langle d\rangle$ obtained from the atomic population distribution in momentum lattices at different evolution time $t$ = 2$\hbar /J$, 3$\hbar /J$ and 4$\hbar /J$ as a function of the quasiperiodic modulation strength $\Delta /J$. The blue solid lines represent the theoretical calculations based on Eq. (2), and the numerical results are in good agreement with the experimental data (red circles). The error bars denote the standard errors. In all panels, the tunneling energies are $J/\hbar$ = 2$\pi$ $\times$ 500 Hz.
4. Theoretical calculation based on the GPE
In theory, for the far-off-resonant Bragg laser beams, we adiabatically eliminate the excited states, and the interactions of the Bragg laser beams with atoms lead to the time-dependent lattice potentials on the ground state $V_\mathrm {Bragg}(z, t)$ = $\Sigma _{n}2\hbar J\cos (2kz-\Delta \omega _{n})$ [37–39]. For a BEC in the combined presence of a harmonic trap $V_{\mathrm {trap}}$ and the trap of the Bragg lasers $V_{\mathrm {Bragg}}$, the time evolution of condensate wavefunction $\psi$ follows the real-space GPE:
Based on Eq. (2), we prepare the initial BEC ground state through an imaginary-time evolution with a time-split-operator numerical method, and then implement the quench dynamics for different $\Delta /J$. A Fourier transformation is performed for the numerical solution, i.e., $\psi (\textbf {r}, t)$, to yield the momentum-space $\psi (\textbf {k}, t)$. Then, the density distribution along the $k_{z}$-direction is given by
Considering the narrow population distribution of atoms in the discrete momentum states, the atomic population $P_{n}$ is further calculated by integrating the momentum-space density $n_{k_{z}}$ with the range of $[(2n-1)k, (2n+1)k]$. We perform the theoretical calculation based on Eq. (2), and the numerical results are in good agreement with the experimental data in Fig. 3. For the case of $\Delta /J>2$, the momentum width $\langle d(t)\rangle$ is not zero mainly due to presence of harmonic trapping potential $V_{\mathrm {trap}}$.5. Conclusion
We have synthesized a 1D momentum lattice by using the laser-coupled discrete atomic momentum states, and studied the quantum transport of atoms in a quasiperiodic momentum lattice with the incommensurate modulation on the on-site energy. We have measured the evolution of atomic density distribution in a site-resolved manner, and presented the direct observation of the transition from the diffusion to Anderson localization. Our observations show good agreement with the theoretical calculations based on the GPE, which contains the effect of optical trap for ultracold atoms. The advantage of both the site-resolved measurement and the controllable quasiperiodic modulation strength featured in our experiment provides new opportunities for engineering the quantum transport and quantum phase transitions in disordered systems.
Funding
National Key Research and Development Program of China (2022YFA1404201); National Natural Science Foundation of China (11974331, 12074234, 62020106014, 62175140, 92165106); Applied Basic Research Project of Shanxi Province, China (202203021224001).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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