Abstract
Ultracold atoms in optical lattices offer a unique platform for investigating disorderdriven phenomena. While static disordered site potentials have been explored in a number of experiments, a more general, dynamical control over siteenergy and offdiagonal tunnelling disorder has been lacking. The use of atomic quantum states as synthetic dimensions has introduced the spectroscopic, siteresolved control necessary to engineer more tailored realisations of disorder. Here, we present explorations of dynamical and tunneling disorder in an atomic system by controlling laserdriven dynamics of atomic population in a momentumspace lattice. By applying static tunnelling phase disorder to a onedimensional lattice, we observe ballistic quantum spreading. When the applied disorder fluctuates on time scales comparable to intersite tunnelling, we instead observe diffusive atomic transport, signalling a crossover from quantum to classical expansion dynamics. We compare these observations to the case of static siteenergy disorder, where we directly observe quantum localisation.
Introduction
Over the past two decades, dilute atomic gases have become a fertile testing ground for the study of localisation phenomena in disordered quantum systems^{1}. They have allowed for some of the earliest and most comprehensive studies of Anderson localisation of quantum particles^{2,3,4,5,6,7,8}, strongly interacting disordered matter^{9,10,11,12,13,14} and manybody localisation^{15,16,17,18}. Still, the emulation of many types of disorder relevant to real systems—e.g., crystal strain and dislocation, site vacancies, interstitial and substitutional defects, magnetic disorder and thermal phonons—will require types of control that go beyond traditional methods based on static disorder potentials^{10}.
The recent advent of using atomic quantum states as synthetic dimensions has broadened the cold atom toolkit with the spectroscopic, siteresolved control of fielddriven transitions^{19,20,21,22,23,24}. This technique has aided the study of synthetic gauge fields^{19,20,21, 24,25,26,27}, and its spatial and dynamical control offers a prime way to implement specifically tailored, dynamical realisations of disorder that would otherwise be difficult to study. However, current studies based on internal states^{20, 21, 25,26,27}, have been limited to a small number of sites along the synthetic dimension, inhibiting the study of quantum localisation in the presence of disorder.
Here, we employ our recently developed technique of momentumspace lattices^{22, 28}, to engineer tailored and dynamical disorder in synthetic dimensions. Our approach introduces several key advances to cold atom studies of disorder: the achievement of pure offdiagonal tunnelling disorder, the dynamical variation of disorder, and siteresolved detection of populations in a disordered system. For the case of tunnelling disorder, we examine the scenario in which only the phase of tunnelling is disordered. As expected for a onedimensional (1D) system with only nearestneighbour tunnelling, these random tunnelling phases are of zero consequence when applied in a static manner. When this phase disorder fluctuates on time scales comparable to intersite tunnelling, however, we observe a crossover from ballistic to diffusive transport^{29}. We compare to the case of static siteenergy disorder, observing Anderson localisation at the siteresolved level.
Results
Implementation
Our bottomup approach^{22, 28}, to Hamiltonian engineering is based on the coherent coupling of atomic momentum states to form an effective synthetic lattice of sites in momentum space (see Fig. 1). This approach may be viewed as studying transport in an artificial dimension^{19}, of discrete spatial eigenstates^{30} (as opposed to a bounded set of atomic internal states^{20, 21}) through resonant or nearresonant fielddriven transitions.
Starting with ^{87}Rb BoseEinstein condensates of ~5 × 10^{4} atoms, we initiate dynamics between 21 discrete momentum states by applying sets of counterpropagating fardetuned laser fields (wavelength λ = 1064 nm, wavevector k = 2π/λ), specifically detuned to address multiple twophoton Bragg transitions, as depicted in Fig. 1a, b. Our spectrally resolved control of the individual Bragg transitions permits a local control of the system parameters, similar to that found in photonic simulators^{31,32,33,34,35}. We can tune the strength, phase and detuning from Bragg resonance of each frequency component to control the tunnelling amplitude, tunnelling phase and site energy of each lattice link/site, respectively. This control is enabled by creating a multifrequency beam (Laser 2 in Fig. 1a) with tailored spectral components. This is achieved by passing a single frequency laser through an acoustooptic modulator (AOM) that is driven by a tailored rf spectrum. Unique to our implementation is the direct and arbitrary control of tunnelling phases^{22}, and the realised tightbinding model is depicted in Fig. 1c. Here, we use this capability to explore the dynamics of cold atoms subject to disordered and dynamical arrangements of tunnelling elements.
Specifically, we explore disorder arising purely in the phase of nearestneighbour tunnelling elements. In higher dimensions, such disordered tunnelling phases would give rise to random flux patterns that mimic the physics of charged particles in a random magnetic field^{36,37,38}. In 1D, however, the absence of closed tunnelling paths renders any static arrangement of tunnelling phases inconsequential to the dynamical and equilibrium properties of the particle density. Timevarying phases, however, can have a nontrivial influence on the system’s dynamical evolution.
Diffusive transport under annealed disorder
We engineer annealed, or dynamically varying, disorder^{39,40,41} of the tunnelling phases and study its influence through the atoms’ nonequilibrium dynamics following a tunnelling quench. Our experiments begin with all population restricted to a single momentum state (site). We suddenly turn on the Bragg laser fields, quenching on the (in general) timedependent effective Hamiltonian
where τ is the time variable, t is the (homogeneous) tunnelling energy, and \({\hat c_n}\) \(\left( {\hat c_n^\dag } \right)\) is the annihilation (creation) operator for the momentum state with index n (momentum p _{ n } = 2nħk). The tunnelling phases φ _{ n } and site energies ε _{ n } are controlled through the phases and detunings of the twophoton momentum Bragg transitions, respectively. After a variable duration of laserdriven dynamics, we perform direct absorption imaging of the final distribution of momentum states, which naturally separate during 18 ms time of flight. Analysis of these distributions, including determination of site populations through a multiGaussian fit, is as described in ref. ^{22}.
As a control, we first examine the case of no disorder, with all site energies set to zero and uniform, static tunnelling phases φ _{ n }(τ) = φ. Figure 2a shows the evolution of the 1D momentum distribution, obtained from timeofflight images integrated along the axis normal to the imparted momentum, displaying ballistic expansion characteristic of a continuoustime quantum walk. For times before the atoms reflect from the open boundaries of the 21site lattice, we find good qualitative agreement between the observed momentum distributions and the expected form P _{ n } = J _{ n }(ϑ)^{2}, where J _{ n } is the Bessel function of order n and ϑ = 2τt/ħ. Figure 2e shows the (symmetrised) momentum profile at time τ = 2.96(2)ħ/t along with the Bessel function distribution for ϑ = 5.4. The discrepancy between the measured evolution time and the argument of the Bessel distribution stems from the uncertainty in the measured tunnelling time ħ/t, which is dependent on local laser intensity and prone to variations.
In comparison, Fig. 2b shows the case of zero site energies and static, random tunnelling phases φ _{ n }∈[0,2π). The dynamics are nearly identical to the case of uniform tunnelling phases. This is consistent with the expectation that any pattern of static tunnelling phases in 1D is irrelevant for the dynamics of the effective tightbinding model realised by our controlled laser coupling, since these phases can be gauged away with local transformations. For this case, Fig. 2f shows the (symmetrised) momentum profile at τ = 2.52(2)ħ/t along with the Bessel function distribution for ϑ = 5.35.
While static phase disorder has little impact on the quantum random walk dynamics, we may generally expect that controlled random phase jumps or even pseudorandom variations of the phases should inhibit coherent transport, mimicking random phase shifts induced through interaction with a thermal environment. To probe such behaviour, we implement dynamical phase disorder by composing each tunnelling phase φ _{ n } from a broad spectrum of oscillatory terms with randomly defined phases θ _{ n,i } but welldefined frequencies ω _{ i }, the weights of which are derived from an ohmic bath distribution. Specifically, the dynamical tunnelling phases take the form
where S(ω) = (ħω/k _{B} T)exp[−(ħω/k _{B} T)], the θ _{ n,i } are randomly chosen from [0, 2π), and T is an artificial temperature scale that sets the range of the frequency distribution. In this discrete formulation of φ _{ n }(τ), we include N = 50 frequencies ranging between zero and 8k _{B} T/ħ. The frequency spectrum and dynamics for one tunnelling phase φ _{ n }(τ) are shown in Fig. 2i for the case of k _{B} T/t = 1.
Figure 2c displays the population dynamics in the presence of this dynamical disorder, characterised by an effective temperature k _{B} T/t = 0.66(1) and averaged over three independent realisations of the disorder using different phase distributions θ _{ n,i }. We note that the population spreads asymmetrically because we do not average over a large range of θ _{ n,i } distributions. The dynamics no longer feature ballistically separating wavepackets, instead displaying a broad, slowly spreading distribution peaked near zero momentum. A clear deviation of the (symmetrised) momentum distribution from the form P _{ n } = J _{ n }(ϑ)^{2} describing the previous quantum walk dynamics can be seen in Fig. 2g. Instead, this more diffusive behaviour is better described by a Gaussian distribution characterised by a width \({\sigma _n} = \sqrt {2\tau t{\rm{/}}\hbar } \). We find excellent agreement with a Gaussian distribution at our measured evolution time of τ = 3.80(3)ħ/t, consistent with spreading governed by an effectively classical or thermal random walk.
Localisation under siteenergy disorder
Lastly, while no influence of static tunnelling phase disorder is expected in 1D, the effect of static siteenergy disorder is dramatically different. Here, with homogeneous static tunnelling terms, we explore the influence of pseudorandom variations of the site energies governed by the AubryAndré model^{4, 9, 12, 16}. With an irrational periodicity \(b = \left( {\sqrt 5  1} \right){\rm{/}}2\), the site energies \({\varepsilon _n} = \Delta \,{\rm{cos}}(2\pi bn + \phi )\) do not repeat, and are governed by a pseudorandom distribution. For an infinite system, this AubryAndré model with diagonal disorder features a metalinsulator transition at the critical disorder strength Δ_{c} = 2t. The expansion dynamics for the strong disorder case Δ/t = 5.9(1) are shown in Fig. 2d, with population largely restricted to the initial, central momentum order. The exponentially localised distribution of site populations (symmetrised and averaged over all profiles in the range τ = 5.1(1) to 6.4(1)ħ/t) is shown in Fig. 2h, along with an exponential distribution with decay length ξ = 0.6 lattice sites. The theoretically predicted localisation length can be described by \(1{\rm{/}}\xi = {\rm{ln}}(\Delta {\rm{/}}2t)\) ^{42}, giving a value of ξ = 0.9 lattice sites that deviates from the value we fit from the shorttime dynamics. Analogous population distributions (again symmetrised and averaged over the same time range) are shown for the cases of weaker disorder [Δ/t = 0.98(1), 1.96(3), 3.05(4), 4.02(9)] in Fig. 2j. Because atoms in different lattice sites (momentum states) eventually separate spatially, we have a limited experimental timescale to observe localisation. Close to the critical point, we cannot accurately describe the population distributions with localisation lengths, though they still exhibit an apparent transition to exponential localisation for Δ/t ≳ 2.
Comparison of expansion dynamics
For all of the explored cases, we study these expansion dynamics in greater detail in Fig. 3. Figure 3a examines the momentumwidth (σ _{ p }) dynamics of the atomic distributions for the cases of static and dynamic random phase disorder. For static phase disorder, we observe a roughly linear increase of σ _{ p } until population reflects from the open system boundaries, while dynamical phase disorder leads to subballistic expansion. In particular, for time τ measured in units of ħ/t and momentumwidth σ _{ p } in units of the site separation 2ħk, these two cases agree well with the displayed theory curves for ballistic and diffusive expansion, having the forms \({\sigma _p} = \sqrt 2 \tau \) and \({\sigma _p} = \sqrt {2\tau } \), respectively (with the latter curve shifted by 0.35ħ/t). To explore these two different expansions more quantitatively, we fit the momentum variance \({V_p} \equiv \sigma _p^2\) to a powerlaw \({V_p}(\tau)=\alpha\tau^\gamma\) ^{43}, performing a linear fit to variance dynamics on a double logarithmic scale as shown in Fig. 3c. The fitdetermined expansion exponents γ for the cases of static and dynamically disordered tunnelling phases are 2.05(2) and 1.27(2), respectively. These values are roughly consistent with a coherent, quantum random walk for the case of static tunnelling phases (γ = 2) and an incoherent, nearly diffusive random walk for the case of dynamical phase disorder (γ = 1).
The observed transport dynamics cross over from ballistic to diffusive as the effective thermal energy scale k _{B} T approaches the coherent tunnelling energy t, matching our expectation that randomly varying tunnelling phases can mimic the random dephasing induced by a thermal environment. We note that similar classical random walk behaviour has been seen previously for both atoms and photons, due to irreversible decoherence^{44,45,46,47} and dissipation^{48, 49}, and thermal excitations^{50}. However, this observation is based on reversible engineered noise of a Hamiltonian parameter. These observations of a thermal random walk suggest that annealed disorder may provide a means of mimicking thermal fluctuations and studying thermodynamical properties^{39} of simulated models using atomic momentumspace lattices, and by extension other nonequilibrium experimental platforms such as photonic simulators.
We also analyse the full expansion dynamics for the case of static site energy disorder in Fig. 3b, d. For homogeneous static tunnellings and thus zero disorder (Δ/t = 0), we observe momentumwidth dynamics similar to the case of static random tunnelling phases, but with one distinct difference: while σ _{ p } features a linear increase for random static phases, it increases in a stepwise fashion for uniform tunnelling phases^{22}. Because our underlying implementation applies a comb of 20 discrete, equallyspaced frequency teeth to the atoms (see Fig. 1), each Bragg transition is addressed not only by an onresonant frequency tooth, but also by 19 other frequencies in an offresonant fashion. These offresonant couplings add up constructively to generate jumps in the dynamics with a frequency that exactly matches the spacing between frequency teeth. By introducing random tunnelling phases onto the teeth, this constructive behaviour is suppressed, resulting in smoother dynamics. We note that the expected smooth behaviour emerges in the limit where the tunnelling is far smaller than the spacing between frequency teeth, though due to dephasing concerns we cannot work at such low tunnelling rates.
Evolution of the momentumwidth (σ _{ p }) for the siteenergy disorder cases of Δ/t = 0.98(1), 2.47(3), 5.9(1) are also shown in Fig. 3b. We observe the reduction of expansion dynamics with increasing disorder, with nearly arrested dynamics in the strong disorder limit. More quantitatively, fits of the variance dynamics as shown in Fig. 3d reveal subballistic, nearly diffusive expansion for intermediate disorder [γ = 1.00(2) for Δ/t = 0.98(1)], giving way to a nearly vanishing expansion exponent for strong disorder [γ = 0.12(6) for Δ/t = 5.9(1)].
The extracted expansion exponents for all of the explored cases are summarised in Fig. 3e. For static siteenergy disorder (red circles), while longer expansion times than those explored (τ ≲ 6.3ħ/t) would better distinguish insulating behaviour from subballistic and subdiffusive expansion, a clear trend towards arrested transport (γ ~ 0) is found for \(\Delta {\rm{/}}t \gg 1\). Numerical simulation (red curve) verifies this qualitative trend, but reaches a finite value of γ due to our fits taking into account transient dynamics at short times (compared to the localisation time). The deviation from this simulation curve can possibly be attributed to the same offresonant tunnelling terms that give rise to the steplike behaviour in Fig. 3b. Combined with the observation of exponential localisation of the site populations in Fig. 2h, j, these observations are consistent with a crossover in our 21site system from metallic behaviour to quantum localisation for Δ/t ≳ 2.
Our observations of a crossover from ballistic expansion (γ ~ 2) to nearly diffusive transport (γ ~ 1) for randomly fluctuating tunnelling phase disorder are also summarised in Fig. 3e. In the experimentally accessible regime of low to moderate effective thermal energies (k _{B} T/t ≲ 1), our experimental data points (blue squares) match up well with numerical simulation (open black circles). For the magnitude of tunnelling energy used in these experiments, we are restricted from exploring higher effective temperatures (k _{B} T/t ≳ 1), as rapid variations of the tunnelling phases introduce spurious spectral components of the Bragg laser fields that could drive undesired transitions. Simulations in this hightemperature regime suggest that the expansion exponent should rise back up for increasing temperatures, saturating to a value γ ~ 2. This results from the fact that the timeaveraged phase effectively vanishes when the time scale of pseudorandom phase variations is much shorter than the tunnelling time.
Discussion
The demonstrated levels of local and timedependent control over tunnelling elements and site energies in our synthetic momentumspace lattice have allowed us to perform explorations of annealed disorder in an atomic system. Such an approach based on synthetic dimensions should enable myriad future explorations of engineered Floquet dynamics^{51,52,53,54} and unconventional disordered lattices^{55, 56}. Furthermore, the realisation of designer disorder in a system that supports nonlinear atomic interactions^{57, 58} should permit us to explore aspects of manybody localisation^{59}.
Methods
Experimental setup
As described in ref. ^{22}, our experiment starts with the preparation of a ^{87}Rb BoseEinstein condensate containing ~5 × 10^{4} atoms through alloptical evaporation in a trap comprised of several optical dipole beams. The condensate is then transferred to a trap formed mainly from one of these beams (wavelength λ = 1064 nm), which we use as our lattice beam. To apply a desired Hamiltonian for the atoms and initiate dynamics, we use AOMs to imprint on the retro lattice beam multiple frequency sidebands ω _{ j,j+1}, which address Bragg transitions between atomic momentum states with momenta 2jħk and 2(j + 1)ħk. By addressing transitions between many adjacent momentum states, we create an effective lattice of sites in a synthetic dimension. We control the detunings from Bragg resonances as well as the amplitudes and phases of each frequency component so as to tune the site energies, tunnelling amplitudes, and tunnelling phases of each element in our lattice, respectively. We use this local parameter control to generate the many realisations of disorder presented in this work.
In this work we create 21site lattices, but in general we can reach lattice sizes of over 50 sites. However, we cannot populate all of these sites in the experimental timeframe, due to eventual decoherence from the spatial separation of atoms in different momentum orders.
Mean field interactions in this system cause shifts in the Bragg resonance frequencies from the singleparticle resonances. By directly measuring this shift to be 2π × 430(40) Hz^{58}, we find a peak meanfield energy of μ _{0} = gn _{0} = ħ × 2π × 760(70) Hz, relating to the peak atomic density n _{0} ≈ 10^{14} cm^{−3} at the center of our harmonic trap^{60}. Here, g = 4πħ ^{2} a/M _{Rb} for M _{Rb} the mass of Rubidium and a the scattering length.
Calibrated tunnelling times
The tunnelling times for all data were calibrated using twosite Rabi oscillations. These times are: ħ/t = 111.6(7) μs for the clean, nondisordered data (Fig. 2a, e), ħ/t = 115.3(9) μs for the random static tunnelling phases data (Fig. 2b, f), ħ/t = 126.4(9) μs for the annealed disorder data (Figs. 2c, g and 3a, c) and ħ/t = 158(7) μs averaged over all of the AubryAndré model data (Figs. 2d, h and 3b, d).
Data availability
All data sets presented here are available from the corresponding author upon request.
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Acknowledgements
We thank J. Ang’ong’a for helpful discussions and careful reading of the manuscript.
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F.A.A. performed the experiments with assistance from E.J.M. F.A.A. and B.G. analysed the data. All authors contributed to the preparation of the manuscript. B.G. supervised the project.
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An, F.A., Meier, E.J. & Gadway, B. Diffusive and arrested transport of atoms under tailored disorder. Nat Commun 8, 325 (2017). https://doi.org/10.1038/s4146701700387w
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DOI: https://doi.org/10.1038/s4146701700387w
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