Quantum diffusion with disorder, noise and interaction

The experiment is based on a Bose–Einstein condensate of ³⁹K atoms in the fundamental energy band of a quasi-periodic potential, which is generated by perturbing a strong primary optical lattice with a weak, incommensurate, secondary one (figure 1(a)). The site-to-site tunneling energy J of the main lattice and the quasi-disorder strength Δ characterize the corresponding Hamiltonian [42]. In the non-interacting case the system shows an Anderson-like localization transition for a finite value of the disorder Δ = 2J. Above this threshold all eigenstates are exponentially localized, with a localization length ξ ≈ d/ln(Δ/2J) [43], where d is the spacing of the main lattice.

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The noise is introduced by an amplitude modulation of the secondary lattice, with controllable strength A and with frequency ω_m that is randomly varied in a proper interval (see appendix A). This corresponds to a broadband spectrum with a controllable width of the same order as the energy bandwidth of the quasi-periodic potential, i.e. W ≈ 2Δ + 4J. We note that our noise is non-dissipative, and in general, we have an out-of-equilibrium situation in which the fluctuation–dissipation relation does not hold. However, the finite bandwidth of the lattice sets a limit of the order of 4J to the maximum kinetic energy that can be pumped into the system by the noise source. Numerical simulations we performed indicate that after a typical time t = 0.1–1 s our system has reached this limit, and most of the long-time dynamics happens in a quasi-equilibrium regime.

A magnetic Feshbach resonance allows one to control the contact interaction [44], by tuning the s-wave scattering length a and in turn the mean interaction energy per particle $E_{\mathrm { int}}\approx 2\pi \hbar ^2 a \bar {n}/m$, where $\bar {n}$ is the mean density and m is the particle mass.

To study the dynamics, we initially prepare the condensate close to the ground state of the lattice, also in presence of a tight axial trap. We then switch off the trap and study the expansion dynamics along the lattice, thanks to an additional radial confinement. To do so, we detect the axial density profile n(x,t) with destructive absorption imaging at increasing times t, up to t = 10 s. The measurements we present are typically referred to as the root mean square (rms) width σ of the sample along the lattice.

A typical time evolution of the width of the system for Δ > 2J in the presence of noise and interaction is shown in figure 1. The disorder-induced localization is broken by interaction or noise alone and also by their combination. In all cases, we observe a short-time transient that evolves into asymptotic behavior, which is different in the three cases. To analyze the expansion, we start by recalling that a solution of the diffusion equation

$$\begin{equation} \frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\frac{\partial}{\partial x}\Big(D\frac{\partial n(x,t)}{\partial x}\Big) \end{equation} \tag{ 1 }$$

is a Gaussian distribution $n(x,t)\approx \exp {[-x^2/2\sigma (t)^2]}$, with dσ²(t)/dt = D. This would imply a time dependence of the distribution width as

$$\begin{equation} \sigma (t) = \sigma_{0} (1+t/t_{0})^{\alpha} \end{equation} \tag{ 2 }$$

with the time exponent α = 0.5. For our experimental distributions, we find that a solution of a generalized diffusion equation of the form of equation (2) with α as a free parameter can be used to fit the rms width of n(x,t) in all cases. Here σ₀ is the initial width, t₀ is a free parameter that represents the crossover time from the short-time dynamics to the asymptotic regime and the exponent α characterizes the expansion at long times. In our data, we typically see about one decade (in time) of asymptotic expansion. In the presence of noise alone we typically observe normal diffusion, i.e. α = 0.5, as expected in the case of white noise. The dynamics is instead sub-diffusive, i.e. α < 0.5, in the presence of a repulsive interaction; as we will discuss later, this is essentially due to a reduction of the interaction coupling as the system expands. In the presence of both noise and interaction, we typically observe a non-trivial anomalous diffusion, with expansion exponent α depending on the relative value of interaction energy E_int and noise amplitude A. As we discussed above, we are not aware of any theory for this combined problem. We therefore model the expansion with a generalized diffusion equation for σ²(t) where the instantaneous diffusion coefficient is the sum of the two coefficients of interaction and noise alone, i.e.

$$\begin{equation} \frac{\mathrm{d} \sigma^2 (t)}{\mathrm{d} t} = D_{\rm noise} + D_{\rm int}(t). \end{equation} \tag{ 3 }$$

Here D_noise = const is the diffusion coefficient due to noise alone, whereas D_int(t) is the time-dependent diffusion coefficient for interaction alone. A main result of this work is that such a general diffusion equation is valid in a wide range of parameters, as we will show in detail in the following through an analysis of the individual diffusion mechanisms and their combination. A similar generalized diffusion equation has been theoretically predicted for Brownian motion of classical interacting particles [45].

Let us start by exploring in detail the effect of noise on the dynamics of the linear system, which we realize by tuning the scattering length a close to zero. As shown in figure 1(b), we observe that a finite noise amplitude A ≠ 0 results in a slow expansion of the initially localized sample. The shape of n(x) keeps being Gaussian at all times. We fit σ(t) to equation (2); from these and other data we measure α = 0.45(5), which is therefore consistent with normal diffusion, and we extract a diffusion coefficient D = σ²₀/t₀.

We have performed an extensive investigation of the diffusion dynamics, by exploring different values of the noise amplitude A and the disorder strength Δ/J, i.e. different localization lengths. The measured diffusion coefficients D, shown in figure 2, are in good agreement with numerical simulations in terms of a generalized Aubry–André model [30, 46], including a dynamical disorder analogous to the experimental one (appendix B).

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We can interpret the observed diffusion as incoherent hopping between localized states driven by the broadband noise. A perturbative approach suggests that D∝Γξ² and Γ∝A², where the localization length ξ represents the natural length scale of the hopping and Γ is the perturbative transition rate [19]. For the Aubry–André model, in the limit of a noise bandwidth equal to the lattice bandwidth, we calculate a diffusion coefficient

$$\begin{equation} D \propto \frac{A^{2} J}{\hbar} \frac{(\xi+d)^{2}}{1+\mathrm{e}^{d/\xi}}\; , \end{equation} \tag{ 4 }$$

which is in good agreement with both the experiment and numerical simulations (appendix B). As shown in figure 2, we observe the linear dependence of D on A² and also an increase of D with ξ.

The described behavior persists in most of the range of values of A and ξ we have been able to explore in the experiment. The simulations, however, give a clear indication that large values of A and/or ξ would bring the system into a different regime, where the dependence on the localization length becomes weaker. This is expected since, in the presence of sufficiently strong noise, the perturbative approach based on localized states must fail. Interestingly, the experimental data, the numerical ones and the perturbative model indicate that the crossover to this second regime happens at strong noise amplitudes, A ≈ 1, for our range of rather short localization lengths, i.e. ξ ≈ d.

We now discuss the effect of the interaction, which is introduced in our system by changing the scattering length a to a finite value, hence introducing a finite E_int. The effect of such nonlinearity on the dynamics on a lattice with static disorder has already been studied in theory [26–32] and experiments [25, 37]. Basically, the finite interaction energy breaks the orthogonality of the single-particle localized states, weakening the localization. In this case one can describe perturbatively the resulting dynamics as an interaction-assisted coherent hopping between localized states, with a coupling strength that decreases as the sample expands, since the density and hence E_int decrease [37]. This results in subdiffusive behavior, i.e. in diffusion with a decreasing instantaneous diffusion coefficient

$$\begin{equation} D_{\rm int}(t)=2\alpha_{\rm int} \frac{\sigma_0^{\alpha_{\rm int}^{-1}}}{t_0} \sigma(t)^{2-\alpha_{\rm int}^{-1}}, \end{equation} \tag{ 5 }$$

where α_int is the time exponent. From dσ(t)²/dt = D_int(t), one gets indeed an evolution of the form σ(t) = σ₀(1 + t/t₀)^α_int. The precise value of such an exponent depends on the details of the interaction strength and of the spatial correlations of the disorder: for an uncorrelated random disorder and E_int ≈ Δ, a perturbative approach predicts α_int = 0.25. In our experiment, we find exponents in the range 0.2 < α_int < 0.35. A detailed description of the various regimes achievable in a quasi-periodic lattice can be found in [47].

When we introduce noise and interaction at the same time, we observe an expansion that is globally faster than that for noise or interaction alone. Note that the solution of equation (3) will result in a time-dependent exponent α(t) such that equation (2) does not hold anymore. However, given the experimentally limited time window, it is still possible to fit well the observed behavior by (2) with a transient exponent 0.3 < α' < 0.5, which is intermediate between the two previous ones. This indicates that both delocalization mechanisms are playing a role in the expansion. One example over many of these observations is shown in figure 3, which reports three characteristic expansion curves for noise or interaction alone, or both. In the presence of both noise and interaction, we also find that the distribution does not manifestly deviate from a Gaussian during the expansion. To analyze the combined dynamics, we use the generalized diffusion equation (3), where D_noise and D_int(t) are separately extracted by fitting, respectively, the case with noise alone and interaction alone to equation (2). The numerical solution of the differential equation (3) is actually in good agreement with the experimental data and also with the numerical results of the theoretical model, as shown in figure 3.

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These examples of the experimental observations and numerical simulations strongly support the hypothesis of additivity of the two delocalization mechanisms, at least at first order. In the experimental data (see figures 3(a) and (b)), we typically observe a slightly faster diffusion with noise and interaction. This difference is due to the axial excitation in the presence of noise, which in turn excites the radial degrees of freedom in the presence of interaction [37] and produces the increased expansion we observe in figure 3(b). In contrast, in numerical simulations, which neglect the axial-to-radial coupling, there are much smaller deviations from the solution of equation (3), of the order of a few lattice constants (see figure 3(d)).

The good agreement between the experimental data and the prediction of equation (3) persists for the whole range of A and E_int that are accessible in the experiment, with A ranging from 0.4 to 1 and E_int adjustable up to ≈J. Furthermore, as we described above (figure 2), in numerical simulations we can explore the region of large A and/or ξ, where the perturbative description of the noise effect fails. Interestingly enough, although this regime is not perturbative, we find the additivity of noise and interaction mechanisms also in this case.

An investigation over a broad range of parameters in theory indicates that the system should behave in a fully symmetric way for attractive and repulsive interactions, provided that also the energy distribution is reversed when changing the sign of the interaction. For example, the diffusion dynamics for a repulsive system initially prepared in a potential minimum is equivalent to that of an attractive system initially prepared in a potential maximum. We have experimentally tested this expectation with a sample prepared with attractive interaction (a ≈ −100a₀ with a₀ being the Bohr radius), where the behavior is fully analogous to that in figures 3(a) and (b), i.e. the two sources of delocalization simply add up. Note that systems prepared with a < 0 do not have a stable state at low energy, but typically occupy the whole energy band in a way similar to the a > 0 case.

Therefore, the large region of validity of equation (3) supports the idea that the observed anomalous diffusion is driven by the additivity of the two delocalization mechanisms, which indeed act independently. Note that the interaction-assisted diffusion tends to vanish as the sample expands, so that one should expect a long-time crossover to a regime where interaction effects are negligible, and the system diffuses normally due to noise alone. While we can clearly observe this crossover in the simulations, its experimental characterization is prevented by the limited observation time.

The one-dimensional quasi-periodic potential is created by a primary optical lattice combined with a weaker incommensurate one:

$$\begin{equation} V(x)=V_1\sin^2(k_1x+\varphi_1)+V_2\gamma^2\sin^2(\gamma k_1x+\varphi_2). \end{equation} \tag{ A.1 }$$

Here k_i = 2/λ_i are the wavevectors of the lattices (λ₁ = 1064.4 nm and λ₂ = 859.6 nm) and $\gamma =\frac {\lambda _1}{\lambda _2}$ measures the commensurability of the lattices. The relevant parameters are the spacing d = λ₁/2, the tunneling energy J (typically 150 Hz) of the primary lattice and the disorder strength Δ, which scales linearly with V₂ [46]. Non-interacting particles in the fundamental band of this lattice are described by the Aubry–André model [43] which shows a metal–insulator transition for a finite value of the disorder Δ = 2J [46].

The scattering length a is changed by means of a broad Feshbach resonance to values ranging from a ≈ 0.1a₀ to about a = 300a₀ [44]. We can define a mean interaction energy per atom $E_{\mathrm { int}}=2\pi \hbar ^2 a \bar {n}/m$, where $\bar {n}$ is the mean on-site density. The atomic sample is radially trapped with a frequency ω_r = 2π × 50 Hz. The presence of the radial degrees of freedom limits E_int to values that are not much larger than the kinetic energy J or the disorder energy Δ.

In order to have a controllable amount of noise, we introduce a time dependence on the amplitude of the secondary lattice potential, which produces a time variation of the on-site energies. In the experiment, the noise must be broadband enough to couple to as many states as possible within the lattice bandwidth (W/ℏ ≈ (2Δ + 4J)/ℏ), but at the same time one must avoid excitation of the radial modes (ω_r = 2π × 50 Hz) as well of the second band of the lattice (ΔE_gap/h ≈ 3–5 kHz). This is achieved with a sinusoidal amplitude modulation, i.e.

$$\begin{equation} V_2(t)=V_0(1+A\,\sin(\omega_m t+\phi_m)) \end{equation} \tag{ A.2 }$$

with the frequency ω_m that is randomly varied in a finite interval ω_m∈[ω₀ − δω, ω₀ + δω] with a time step T, while the phase ϕ_m is adjusted to preserve the continuity of the modulation and the sign of its first derivative and to avoid frequency components outside the chosen band. The values used in the experiment, whose typical spectrum is shown in figure A.1, are ω₀/2π = 250 Hz, δω₀/2π = 50 Hz and T = 5 ms. At long times, t ≫ T the resulting noise spectrum decays exponentially outside the band [ω₀ − δω, ω₀ + δω], on a frequency scale ≈1/T. Note that at short times, t ≈ T, this spectrum contains only a few components, which might result in an inefficient excitation of the localized states to provide diffusion [56]. In the experiment, however, we do not see such an effect. We believe that this is due to an additional broadband background noise in the relative phase of the two lattices, which is due to fluctuations of the position of the retroreflecting mirror at acoustic frequencies. We measured this phase noise via a Michelson interferometer scheme, and we estimate it to be equivalent to a broadband amplitude noise on the secondary lattice, about 25 dB below the main noise when A = 1. The estimated value is consistent with the small diffusion we have for A = 0 and with the diffusion we observe for A > 0 and a single frequency modulation, i.e. δω = 0.

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In the numerical simulations, we find, however, a reduced expansion until t ≫ T. For this reason, we have numerically tested also a second noise scheme corresponding to a larger width of the power spectrum already at short times and is able to better reproduce the experimental combination of the external frequency noise and the uncontrollable phase one. In this scheme the frequency ω_m is kept fixed, while the phase ϕ_m is randomly varied every time step T. Here, the power spectrum is essentially a sinc function with width δω = 2π/T as soon as t > T.

We have checked with extensive simulations that both types of noise lead to normal diffusion at long times. In the second case, we observe that the diffusion coefficient D grows linearly with the bandwidth of the noise as long as such a bandwidth is smaller than the lattice bandwidth W/ℏ. Larger bandwidths, i.e. very small T, actually lead to a reduction of D, since not all the frequencies can effectively produce hopping between the localized states.

The numerical simulations presented in the paper are based on a generalized version of the Aubry–André model [43], including a mean-field interaction term [30, 46]:

$$\begin{eqnarray} &&\mathrm{i}\dot{\psi}_j(t) = -(\psi_{j+1}(t)+\psi_{j-1}(t))+V_{j}|\psi_j(t)|^2+\beta|\psi_j|^2\psi_j,\; \ \ \ \ \ \end{eqnarray} \tag{ B.1 }$$

where V_j = Δ/J sin(2πγj), β is the interaction strength and ψ_j(t) are the coefficients of the wave function in the Wannier basis, normalized in such a way that their squared modulus corresponds to the atom density on the jth site of the lattice. All energies are measured in units of the next-neighbor tunneling energy J, while the natural units for time are ℏ/J. Note that this model can describe only the lowest energy band of the real system. Excited bands are, however, not populated in the experiment. The interaction parameter β in the model can be connected to the mean interaction energy per particle in the real system, which is defined as

$$\begin{equation} E_{\rm int}=\frac{2\pi\hbar^2}{m} a \frac{\int{n({\bf r})^2 \,\mathrm{d}^3r}}{\int{n({\bf r}) \,\mathrm{d}^3r}}. \end{equation} \tag{ B.2 }$$

Here n(r) is the mean on-site density distribution, i.e. the solution of the interacting Gross–Pitaevskii problem in a single well, normalized to an atom number $N/\bar {n}_{\mathrm { s}}$, where $\bar {n}_{\mathrm { s}}$ is the mean number of sites occupied by the atomic distribution. The relation between E_int and the interaction strength in the model is approximately $E_{\mathrm { int}}\approx 2 J\beta /\bar {n}_{\mathrm { s}}$ [30, 46].

To quantify the localization, we consider two quantities: the width of the wavepacket measured as the square root of the second moment of the spatial distribution |ψ_j(t)|²,

$$\begin{equation} \sigma (t) = \sqrt{m_2(t)} = \left[\sum_j(j-\langle j \rangle)^2 \left|\psi_j(t) \right|^2\right]^{1/2}\; \end{equation} \tag{ B.3 }$$

and the participation ratio PR,

$$\begin{equation} {\rm PR}(t)=\frac{1}{\sum_j {\left|\psi_j(t)\right|^4}}\; \end{equation} \tag{ B.4 }$$

measuring the number of significantly occupied lattice sites. The quantity 〈j〉 represents the average over the spatial distribution, defined as $\langle j \rangle = \sum _j j\left |\psi _j \right |^2$.

In this section, we consider a perturbative approach to describe the diffusion induced by noise. The dynamics of the disordered system in the absence of temporal noise is described by equation (B.1). In the regime of small noise amplitude (A ≪ 1), we can assume that the only effect of noise is to induce hopping between different localized states of the unperturbed system. The coupling is driven by the frequency component of the noise spectrum that is resonant with the energy difference between the states. In the regime of ξ ≈ d investigated in this work, we can restrict our analysis just to the coupling between neighboring states. The hopping rate depends on the noise strength and on the overlap between the noise spectrum s(ν) and the unperturbed system energy distribution P(ν):

$$\begin{equation} \Gamma = \frac{A^{2}J}{\hbar} \eta \left(\frac{\Delta}{J}\right)^{2}|\langle i|\sin (2 \pi \gamma j)|f\rangle|^{2}\; , \end{equation} \tag{ B.5 }$$

where $\eta =\frac {1}{A^2}\int\!s(\nu ) P(\nu )\,\mathrm { d} \nu$ is ≲1. To obtain an analytic prediction of the effect of the noise, we approximate the energy difference between states as a flat spectrum with the same width as the lattice band, W/h ≈ (2Δ + 4J)/h. We approximate the overlap |〈i|sin(2πγj)|f〉|² with |〈i|f〉|², where the states |i〉 and |f〉 are assumed to be exponentially localized over a distance ξ. It is also reasonable to consider Δ/J ∼ 2 e^d/ξ. Finally, when the noise spectrum is a sinc function with T equal to the inverse of the bandwidth W/h, we obtain an analytical estimation of the diffusion coefficient

$$\begin{equation} D = \xi^{2}\Gamma \approx \frac{A^{2}J}{3\hbar} \frac{(\xi+d)^{2}}{1+\mathrm{e}^{d/\xi}}\; . \end{equation} \tag{ B.6 }$$

This expression is evaluated for the optimal case of T = h/W, which determines the specific numerical prefactor 1/3. In the general case, we find a more complex dependence of D on ξ, and a coefficient D that decreases with the width of the noise spectrum δν = 1/T, i.e. the overlap between s(ν) and P(ν) decreases. As shown in figure B.1, the diffusion coefficient obtained with this heuristic model almost perfectly captures the evolution of D with both A and ξ. Nevertheless, we numerically observe that the perturbation approach does not hold for large values of A and ξ, i.e. ξ ⩾ 2d. In these regimes, in fact, the ξ dependence of D becomes weaker. The good agreement between the perturbative model and the observed evolution of D confirms the picture that this diffusion is induced by the hopping between localized states, in analogy with other disordered noisy systems, such as the kicked rotor with noise [18].

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We can estimate that the perturbative approach fails when the energy associated with the perturbation rate becomes comparable with the mean separation energy between states in a localization volume, i.e. $\hbar \Gamma \approx \Delta \frac {d}{3 \xi }$. This corresponds to a critical noise amplitude

$$\begin{equation} A_{\rm c} \approx \Big(2 \mathrm{\,e}^{d/\xi} \frac{1+\mathrm{e}^{d/\xi}}{(1+\frac{d}{\xi})^{2}}\frac{d}{3 \xi}\Big)^{1/2}\; . \end{equation} \tag{ B.7 }$$

As shown in figure 2, A_c and the relative D, calculated using respectively equations (B.6) and (B.7), correctly capture the order of magnitude for the transition from the perturbative regime into a new regime in which the ξ dependence of D is weaker. On top of this, the fact that A_c increases as ξ decreases, i.e. when the disorder is stronger, is also reproduced by our simulations.

There is a further regime of strong disorder that can, in principle, be conceived in which the localization length is smaller than the lattice spacing and d becomes therefore the relevant length scale for the diffusion [18]. This has so far not been accessible in the experiment, because the resulting diffusion is too slow to be characterized accurately.