Phase diagram of the anisotropic Anderson transition with the atomic kicked rotor: theory and experiment

The interplay of disorder and quantum interference has been an important subject in physics for more than 50 years [1]. Quantum interferences, which are at the heart of most quantum effects, rely on precise relative phases between quantum trajectories, which are strongly sensitive to perturbations such as decoherence (i.e. coupling with a large environment) and scattering of the wavefunction in potential wells. This last effect becomes particularly difficult to describe theoretically in a disordered system, in which these scattering processes have random character. Intuitively, one easily understands that such an effect will play an important role, for example, in the low-temperature electric conductance of solids. In fact, Anderson showed in 1958 that the presence of disorder might produce a spatial localization of the wavefunction, which suppresses conductivity [2], hence the name of 'strong' localization.

Laser cooling opened up the possibility of realizing very clean experiments in disordered systems, which generated a burst of interest in the subject. Under adequate conditions, ultracold atoms placed in spatially structured laser beams feel this radiation as a mechanical potential acting on the center-of-mass variables of the atoms. Disordered potentials created in such a way allowed the realization of spatially disordered systems in one dimension [3, 4] and three dimensions [5, 6]. Despite this progress, the Anderson metal–insulator transition (which manifests itself in three or more dimensions) is still very difficult to study in such systems, because Anderson localization requires very strong disorder and—the cold atomic samples being prepared in the absence of disorder—the energy distribution of the atoms unavoidably spreads across the so-called mobility edge, an energy threshold separating localized and extended eigenstates. This in turn implies that the localized fraction, which can be directly measured in cold-atom experiments from the temporal evolution of the spatial probability distributions, remains small. Fortunately, one can find other systems also described by the Anderson localization physics, which are not a direct transposition of the condensed matter system, but rely on the profound analogy between quantum chaotic systems and disordered systems [7]. Using the quasiperiodic kicked rotor (QpKR)—first studied in [8]— an effectively three-dimensional (3D) variant of the paradigmatic system of quantum chaos [9, 10], the Anderson transition has been observed, its critical exponent measured experimentally [11, 12], its critical wavefunction characterized [13], and its class of universality firmly established [14], making this system an almost ideal environment to study Anderson-type quantum phase transitions⁵.

One advantage of this cold-atom chaotic system as compared to other disordered systems is that the disorder can be controlled very precisely: the mean free path and anisotropy are two experimentally tunable parameters. This allows us to present in this work an experimental study of the disorder versus anisotropy phase diagram of the Anderson transition (see [17–22] for previous theoretical studies), as well as an analytical description of these properties based on the self-consistent theory of Anderson localization [23–25], which adds another important brick to our detailed knowledge on the Anderson metal–insulator transition.

The QpKR [8, 26, 27] is described by the one-dimensional (1D) time-dependent Hamiltonian

$$\begin{equation} H_{\mathrm{qpkr}}=\frac{p^{2}}{2}+K\cos x\left(1+\varepsilon\ \cos\omega_{2}t\,\cos\omega_{3}t\right)\sum_{n}\delta(t-n). \end{equation} \tag{ 1 }$$

Experimentally, it is realized by placing laser-cooled atoms (of mass M) in a standing wave (formed by counterpropagating beams of intensity I₀ and wavenumber k_L), which generates an effective sinusoidal mechanical potential—nicknamed 'optical potential'—cos x acting on the center-of-mass position x of the atom. The standing wave is modulated by an acousto-optical modulator in order to form a periodic (of period T₁) train of short square pulses whose duration τ is short enough that, at the time scale of atom center-of-mass dynamics, they can be assimilated to Dirac δ-functions. The amplitude of such pulses is further modulated with frequencies ω₂ and ω₃, proportionally to 1 + ε cos ω₂t cos ω₃t. Lengths are measured in units of $\left (2k_{\mathrm { L}}\right )^{-1}$, time in units of T₁ and momenta in units of M/2k_LT₁; note that $\left [x,p\right ]={\mathrm { i}}\unicode{x02C9}\mkern-9mu k$ with $\unicode{x02C9}\mkern-9mu k=4\hbar k_{\mathrm { L}}^{2}T_{1}/M$ playing the role of a reduced Planck constant. The pulse amplitude is $K=\unicode{x02C9}\mkern-9mu k\tau \Omega ^{2}/8\Delta _{\mathrm { L}}$, where Ω is the resonance Rabi frequency between the atom and the laser light and Δ_L the laser–atom detuning. Fixed parameters used throughout the present work are $\unicode{x02C9}\mkern-9mu k=2.89$, $\omega _{2}=2\pi \sqrt {5}$ and $\omega _{3}=2\pi \sqrt {13}$.⁶

If ε = 0 one obtains the standard kicked rotor, which is known to display fully chaotic classical dynamics for K ⩾ 6 [30]. At long time, the dynamics is a so-called chaotic diffusion in momentum space, which is—although perfectly deterministic—characterized by a diffusive increase of the rms momentum: 〈p(t) − p(t = 0)〉 = 0, 〈[p(t) − p(0)]²〉 ∼ 2Dt (where D is the classical diffusion constant) where the average 〈〉 is performed over an ensemble of trajectories associated with neighboring initial conditions. The statistical distribution of p(t) has the characteristic Gaussian shape of a diffusion process, whose width increases as $\sqrt {t}.$ Quantum mechanically, this system displays the phenomenon of dynamical localization, which corresponds to an asymptotic saturation of the average square momentum $\langle p^{2} \rangle$ [9, 10] at long time, that is localization in momentum space instead of chaotic diffusion, which has been proved to be a direct analogue of the Anderson localization in one dimension [31–33].

If $2\pi/T_{1},\omega_{2},\omega_{3},\unicode{x02C9}\mkern-9mu k$ are incommensurable and ε ≠ 0, one obtains the QpKR, which can be proved to be equivalent to the Anderson model in three dimensions [8, 12, 34, 35]. In a nutshell, the QpKR, which is a 1D system with a time–dependent Hamiltonian depending on three different quasi-periods, can be mapped on a kicked 'pseudo'-rotor, a 3D system with a time periodic Hamiltonian. As shown in detail in [12], both systems share the same temporal dynamics. The Hamiltonian of the pseudo-rotor is

$$\begin{equation} \mathcal{H}=\frac{p_{1}^{2}}{2}+\omega_{2}p_{2}+\omega_{3}p_{3}+K\cos x_{1}\left[1+\varepsilon\cos x_{2}\cos x_{3}\right]\sum_{n}\delta(t-n) \end{equation} \tag{ 2 }$$

with an initial condition taken as a planar source in momentum space (completely delocalized along the transverse directions p₂ and p₃). Note that the kinetic energy has a different dependence on the momentum in each direction: standard (quadratic) in direction 1, but linear in directions 2 and 3; hence, the name pseudo-rotor (see [36] for a theoretical study of the standard 3D kicked rotor).

The Hamiltonian (2) is periodic in configuration space. It can thus be expanded in a discrete momentum basis composed of states $\vert {\boldsymbol {\mathfrak {p}}} \rangle = \vert {\boldsymbol {p}} = \unicode{x02C9}\mkern-9mu k {\boldsymbol {\mathfrak {p}}}\rangle ,$ where the $\mathfrak {p}_{i}$ are integers⁷. In this basis, the evolution operator over one temporal period reads as the product U = JV of an on-site operator, $V(\boldsymbol {\mathfrak {p}})= {\mathrm { e}}^{-{\mathrm { i}} \phi _{\boldsymbol {\mathfrak {p}}}}$ with phases $\phi _{\boldsymbol {\mathfrak {p}}} = \frac {\unicode{x02C9}\mkern-9mu k \mathfrak {p}_{1}^{2}}{2}+\omega _{2}\mathfrak {p}_{2}+\omega _{3}\mathfrak {p}_{3}$, and of a hopping operator J such that

$$\begin{equation} \fl \langle {\boldsymbol{\mathfrak{p}}}_f \vert J \vert {\boldsymbol{\mathfrak{p}}}_i \rangle = \int \frac{{\rm d} \boldsymbol x }{(2 \pi)^3} \exp{\left[-{\rm i} \frac{K\cos x_{1}\left(1+\varepsilon\cos x_{2}\cos x_{3}\right)}{\unicode{x02C9}\mkern-9mu k}\right]} \ \exp{\left[-{\rm i} ({\boldsymbol{\mathfrak{p}}}_i -{\boldsymbol{\mathfrak{p}}}_f) \boldsymbol x\right]}. \end{equation} \tag{ 3 }$$

The phases $V(\boldsymbol {\mathfrak {p}})$ are different on each site of the momentum lattice, and, although perfectly deterministic, constitute a pseudo-random sequence⁸ analogous to the true random on-site energies of the Anderson model. This makes it possible to identify V as the disorder operator for the QpKR. The parameter K controls the hopping amplitudes, that is the transport properties in the absence of disorder. The larger the K, the larger the distance the system propagates in momentum space (with the operator J) before being scattered by the disorder operator V . As shown below, the associated mean free path in momentum space is of the order of $K / \unicode{x02C9}\mkern-9mu k$. Rather counterintuitively, the weak disorder limit then corresponds to the large K limit, that is strong pulses, while the strong disorder limit where Anderson localization is expected corresponds to small K. It should also be stressed that, for very small K (very strong disorder), the system remains frozen close to its initial state, with a trivial on-site Anderson localization. This is not really surprising as the classical dynamics is then regular instead of chaotic and even the classical chaotic diffusion is suppressed.

In the following, we will concentrate on the role of the ε parameter, which drives the anisotropy between the transverse directions and the longitudinal direction, showing the analogy of (2) with a system of weakly coupled disordered chains as considered in [19].

With such a system, we experimentally observed and characterized the Anderson transition [11, 12], which manifests itself by the fact that the momentum distribution is exponentially localized $\Pi (\mathfrak {p};t)\sim \exp \left (-\left |\mathfrak {p}\right |/\mathfrak {p}_{\mathrm {loc}}\right )$ (with $\mathfrak {p}_{\mathrm {loc}}$ the localization length) if K is smaller than a critical value K_c(ε) and Gaussian diffusive $\Pi (\mathfrak {p};t)\sim \exp \left (-\mathfrak {p}^{2}/4 D t\right )$ (where D is the diffusion coefficient) for K > K_c(ε) after a sufficiently long time. At criticality, K = K_c(ε), the localization length diverges, the diffusion constant vanishes and the critical state is found [13] to have a characteristic Airy shape

$$\begin{equation} \Pi(\mathfrak{p};t)\approx \frac{3}{2}\frac{ \alpha }{ \sqrt{\Lambda_{\rm c}(\varepsilon) t^{2/3}}} \mathrm{Ai}\left[\alpha \sqrt{ \frac{\vert \mathfrak{p}\vert^2}{\Lambda_{\rm c}(\varepsilon) t^{2/3}}}\right] \end{equation} \tag{ 4 }$$

following the anomalous diffusion at criticality: $\langle \mathfrak {p}^{2}\rangle = \Lambda _{\mathrm { c}}(\varepsilon ) t^{2/3}$ (here α = 3^1/6Γ(2/3)^−1/2) (see also [27, 39]). The fundamental quantities characterizing the threshold of the transition are therefore K_c(ε) and Λ_c(ε), and we will consider in the following their dependence as a function of the anisotropy parameter ε (see figure 1).

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In our experiment, we measure the population of the zero velocity class Π(0;t) using Raman velocimetry [41–43]. This quantity is proportional to $\sqrt {\langle \mathfrak {p}^{2}(t)\rangle }$, with a proportionality factor that depends on the specific shape of $\Pi (\mathfrak {p}).$ This factor varies smoothly across the Anderson transition, so that the transition can be studied using either $\langle \mathfrak {p}^{2}(t)\rangle$ or Π(0;t). The scaling theory of localization [12, 44] predicts that $\left \langle \mathfrak {p}^{2}\right \rangle$ has characteristic asymptotic behaviors in t^α, with α = 0 in the localized regime, α = 2/3 in the critical regime and α = 1 in the diffusive regime. This prediction has been fully confirmed by the experimental observations [13]. One can then define the scaling variable [11, 12]

$$\begin{equation} \Lambda(t) \equiv \frac{\left\langle \mathfrak{p}^{2}(t)\right\rangle }{t^{2/3}} \propto \frac{1}{\Pi(\mathfrak{p}=0;t)^{2}\,t^{2/3}}. \end{equation} \tag{ 5 }$$

Asymptotically, Λ(t)∝t^−2/3,t⁰,t^1/3 in the localized, critical and diffusive regimes, respectively, so that lnΛ(t) versus ln t^1/3 displays a positive slope 1 in the diffusive regime, a zero slope at the critical point and a negative slope −2 in the localized regime, which allows one to unambiguously identify the critical point. However, experimental limitations prevent us from performing measurements at large enough times (in our experiments typically t_max = 120) to distinguish precisely between the localized and diffusive behaviors in the vicinity of the transition⁹. The main causes of this limitation are the falling (under gravity action) of the cold atoms out of the standing wave and decoherence induced by spontaneous emission.

Fortunately, a technique known as finite size scaling [45, 46] (which is finite time scaling in our case), based on arguments derived from the so-called one-parameter scaling theory of the Anderson transition [44], allows us to overcome this limitation. The application of this technique to our problem has been discussed in detail in previous work [12, 34, 47]; let us just say here that it relies on the verified hypothesis that Λ can be written as a one-parameter scaling function

$$\begin{equation} \Lambda = F\left(\frac{\xi(K)}{t^{1/3}}\right) \end{equation} \tag{ 6 }$$

with the scaling parameter ξ(K) that plays the role of the localization length p_loc in the localized regime and of the inverse of the diffusive constant in the diffusive regime. This method produces a rather precise determination of the critical parameters K_c(ε) and Λ_c(ε) and of the critical exponent of the Anderson transition [14] even from experimental signals limited to a hundred of kicks or so. An example of such a determination is presented in figure 2. The critical point corresponds to the tip at the right of the curve in figures 2(a) and (c), at the intersection of the two clearly defined branches: a diffusive (top) and a localized branch (bottom). By construction, in principle ξ(K) should diverge at the critical point, but the finite duration of the experiment and decoherence effects produce a cutoff; however, it still presents, as shown in figure 2(b) and (d), a marked maximum at the transition, and a careful fitting procedure taking these effects into account [14] allows a precise determination of K_c. Once the value of K_c has been determined according to the above technique, we measure the full momentum distribution, which is found to be in excellent agreement with the predicted Airy shape, equation (4), as shown in [13]. A simple fit of the experimental data by an Airy function allows us to measure $\langle \mathfrak {p} ^2\rangle ,$ and hence Λ_c.

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We have measured the value of the critical parameters K_c(ε) and Λ_c(ε) (equation (5)) for a grid of seven paths at constant ε in the parameter plane (K,ε) (see figure 1). For each path, 50 uniformly spaced values of K are used and the values of Π(0;t) measured for each K value; the initial and final values of K are chosen symmetrically with respect to the critical point. For each value of K, an average of 20 independent measurements is performed; a full path thus corresponds to more than 7 h of data acquisition. Table 1 gives the details of each path and the results obtained.

Figure 3 displays the experimental and numerical results. Plot (a) indicates the position of the critical point K_c(ε) and plot (b) the critical value Λ_c(ε). In both plots, experimental measurements are indicated by red circles and numerical simulation results by black diamonds and are represented along with their error bars. The uncertainty of the numerical data (see the following for a discussion of the numerical method) is evaluated from data up to t = 1000 kicks and thus incorporates systematic shifts of (K_c,Λ_c) as a function of time [40] that cannot be estimated in the experimental data due to the restricted range of observation times t ⩽ 120 kicks. This results in larger numerical error bars than experimental ones. Note also that a small deviation in K_c implies a much larger error in Λ_c due to its rapid variation as a function of K. In plot (a), one observes a very good agreement between numerical and experimental results. In plot (b), the agreement is only fair: the region of low ε corresponds to high values of K and is thus more sensitive to decoherence effects which decrease Λ_c (while K_c remains nearly unaffected). In the region of large ε, the finite variance of the initial momentum distribution tends to increase the experimental K_c (small deviation) and Λ_c (marked deviation), an effect that is not present in the numerical data.

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We shall now try to describe theoretically the observed anisotropy dependences of the two critical parameters K_c(ε) and Λ_c(ε). The approach we shall follow is based on the self-consistent theory of localization [23] that has been used successfully to predict numerous properties of the Anderson transition, and in particular the disorder versus energy [48] and disorder versus anisotropy [19] phase diagrams of the 3D Anderson model. Moreover, the self-consistent theory of localization has been transposed to the case of the kicked rotor [24, 25]. We will use in the following a simple generalization of this latter approach adapted to the case of the 3D anisotropic kicked pseudo-rotor (2) corresponding to the quasiperiodic 1D kicked rotor.

The starting point is to consider the probability of going from a site ${\boldsymbol {\mathfrak {p}}}_i$ to a site ${\boldsymbol {\mathfrak {p}}}_f$ in N steps, $P({\boldsymbol {\mathfrak {p}}}_i , {\boldsymbol {\mathfrak {p}}}_f, t=N) \equiv \vert \langle {\boldsymbol {\mathfrak {p}}}_f \vert U^N \vert {\boldsymbol {\mathfrak {p}}}_i \rangle \vert ^2$. It consists of propagations mediated by the hopping amplitudes $\langle {\boldsymbol {\mathfrak {p}}}_{n+1} \vert J \vert {\boldsymbol {\mathfrak {p}}}_{n} \rangle$ and by collisions on the disorder represented by $V(\boldsymbol {\mathfrak {p}})= {\mathrm { e}}^{-{\mathrm { i}} \phi _{\boldsymbol {\mathfrak {p}}}}$. Two important points are the following: (i) one can consider in a first approximation the $\phi _{\boldsymbol {\mathfrak {p}}}$ as completely random phases [31, 49, 50] and we will consider quantities averaged over those phases, for example $\overline {P}({\boldsymbol {\mathfrak {p}}}_i , {\boldsymbol {\mathfrak {p}}}_f, t=N)$ where the line over the quantity represents this averaging; (ii) $\langle {\boldsymbol {\mathfrak {p}}}_{n+1} \vert J \vert {\boldsymbol {\mathfrak {p}}}_{n} \rangle$ plays the role of the disorder-averaged Green function (in the usual language of diagrammatic theory of transport in disordered systems [51]), that is the propagation between two scattering events. Indeed, when ε = 0 and in the direction p₁, this is just a Bessel function that decreases exponentially fast for $\vert {\boldsymbol {\mathfrak {p}}}_{n+1} - {\boldsymbol {\mathfrak {p}}}_{n} \vert \gg K/ \unicode{x02C9}\mkern-9mu k$ and one can thus see $K / \unicode{x02C9}\mkern-9mu k$ as the analogue of the mean free path, with the limit of small disorder corresponding to $K / {\unicode{x02C9}\mkern-9mu k \gg 1}$.

One can attack the problem of the calculation of $\overline {P}$ by looking for propagation terms—including of course interference patterns—that survive the disorder averaging. At lowest order, the contribution containing no interference term to the probability $\overline {P}$ is called the diffusion [51]. It corresponds to the classical chaotic diffusion and can be shown to have a diffusive kernel expressed in the reciprocal space (q,ω) (conjugated to $({\boldsymbol {\mathfrak {p}}} , t)$) as [24, 25, 34]

$$\begin{equation} \overline{P_{\mathrm{D}}}(\boldsymbol q , \omega) = \frac{1}{-{\rm i} \omega + \sum_j D_{jj} q_j^2}. \end{equation} \tag{ 7 }$$

Here, the diffusive tensor D—computed in [38] for large K—is anisotropic, but diagonal, with

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} D_{11} = \displaystyle\frac{K^2}{4 \unicode{x02C9}\mkern-9mu k^2} \left(1+ \displaystyle\frac{\varepsilon^2}{4}\right),\\ D_{22} = D_{33} =\displaystyle\frac{K^2 \varepsilon^2}{16 \, {\unicode{x02C9}\mkern-9mu k^2}}. \end{array} \end{eqnarray} \tag{ 8 }$$

This anisotropic diffusive kernel is valid at long times and on large scale in momentum space, that is in the so-called hydrodynamic limit ω ≪ 1 and q_jk_j ≪ 1, with k_j the mean free path along direction j which is such that D_jj = k²_j/4. Equation (7) means that in the regime of long times and large distances (in momentum space), we should have diffusive transport with $\langle \mathfrak {p}_j^2 \rangle = 2 D_{jj} t$. This is certainly not the case near the Anderson transition, which implies that we must go beyond the diffusion approximation.

The simplest interferential correction—known as weak localization correction—is due to the constructive interference between pairs of time-reversed paths¹⁰, or equivalently to the maximally crossed diagrams. The net effect of these interferential contributions is to increase the return probability at the initial point and to decrease the diffusion constant. It is possible to compute perturbatively the weak localization correction as an integral (see below) depending on the diffusion constant itself. Contributions from higher orders are extremely complicated and there is no systematic way of summing them all.

The self-consistent theory of localization is a simple attempt at summing approximately the most important contributions: instead of computing the weak localization correction using the raw diffusion constant, one uses the one renormalized because of weak localization. The whole thing must of course be self-consistent, so that the diffusion constant computed taking into account the weak localization correction should be equal to the one input in the calculation of this correction. The price to pay is that one can no longer define a single diffusion constant—or, in the anisotropic case, a single diffusion tensor—but must introduce a frequency-dependent diffusion constant (or diffusion tensor). This is nevertheless quite natural if one wants to describe the transition from a diffusive behavior at short time (large frequency) when interference terms are small to a localized behavior at long time (small frequency). The intensity propagator $\overline {P}$ takes then the approximate form $\overline {P}(\boldsymbol q , \omega ) = \frac {1}{-{\mathrm { i}} \omega + \sum _j {\cal D}_{jj}(\omega ) q_j^2}$, where $\cal {D}$ is the frequency-dependent diffusion constant obeying the self-consistent equation [23, 24, 34]

$$\begin{equation} {\cal D}_{ii}(\omega) = D_{ii} - 2 {\cal D}_{ii}(\omega) \int \frac{{\rm d}^3 \boldsymbol q}{(2 \pi)^3} \frac{1}{-{\rm i} \omega + \sum_j {\cal D}_{jj}(\omega) q_j^2}. \end{equation} \tag{ 9 }$$

Quite remarkably, this approach is able to account for a certain number of observed features¹¹: it predicts a transition between a metallic phase of diffusive transport for K > K_c(ε) where ${\cal D}_{ii}(\omega ) \underset {\omega \rightarrow 0}{\sim } {\cal D}_{ii}(0)>0$, to a localized phase for K < K_c(ε) where ${\cal D}_{ii}(\omega )\underset {\omega \rightarrow 0}{\sim } -{\mathrm { i}} \omega \; {\mathfrak {p}_{\mathrm{loc}}}_i^2$ with ${\mathfrak {p}_{\mathrm{loc}}}_i$ the localization length along direction i. At the threshold, the self-consistent theory predicts an anomalous diffusion with ${\cal D}_{ii}(\omega ) \sim (-{\mathrm { i}} \omega )^{1/3}$ [53] and this implies the Airy shape of the critical state observed experimentally [13]. In the following, we will calculate explicitly the critical parameters K_c(ε) and Λ_c(ε) from (9) and show that they also comply well with the experimental observations.

One shall evaluate the integral on the right-hand side of equation (9). It is important to remark that, although the system is anisotropic and thus three different equations (9) have to be solved simultaneously, they in fact follow exactly the same renormalization scheme: dividing equation (9) by D_ii produces the same equation in all three dimensions. In other words, there is no anomaly in the anisotropic character at the critical point.

It is well known [23] that in dimension d ⩾ 2, the results of the self-consistent theory are cutoff dependent. Indeed, the integral in (9) diverges at large q and must be limited to q_j < q^max_j, where q^max_j is a cutoff of the order of k⁻¹_j, i.e. the inverse of the mean free path. In the following, we will take [19] $q_j^{\mathrm {max}} \equiv C_1 k_j^{-1} = C_1/(2\sqrt {D_{jj}})$ with C₁ a numerical constant of the order of one. We make the following change of variables, $Y_j = \sqrt {\frac {{\cal D}_{jj} (\omega )}{-\mathrm {i} \omega } } q_j$, and define the rescaled cutoff: $\ell (\omega ) \equiv \sqrt {\frac {{\cal D}_{jj} (\omega )}{-\mathrm {i} \omega } } q_j^{\mathrm {max}}$ (from equation (9) it is clear that the ratio ${\cal D}_{jj}(\omega )/D_{jj}$ is isotropic; thus ℓ(ω) is isotropic). One obtains

$$\begin{equation} \frac{{\cal D}_{ii}(\omega)}{D_{ii}} = 1 - \frac{C_1}{2\pi^2} \frac{1}{\sqrt{D_{11}D_{22}D_{33}}} \left[1 - \frac{\tan^{-1}\ell(\omega)}{\ell(\omega)}\right]. \end{equation} \tag{ 10 }$$

The threshold K_c(ε) of the Anderson transition is then approached from the diffusive regime, which is characterized by $\frac {{\cal D}_{ii}(\omega )}{D_{ii}} \underset {\omega \rightarrow 0}{\rightarrow } \frac {{\cal D}_{ii}(0)}{D_{ii}}\underset {K \rightarrow K_{\mathrm { c}}}{\rightarrow } 0$ and $\ell (\omega ) \underset {\omega \rightarrow 0}{\rightarrow } \infty$. Therefore, K_c(ε) is such that

$$\begin{equation} D_{11} D_{22} D_{33} = \frac{{C_1}^2}{4 \pi^4}. \end{equation} \tag{ 11 }$$

From expressions (8) for the diffusion tensor, one deduces the following dependence of the threshold versus anisotropy:

$$\begin{equation} K_{\rm c}(\varepsilon) = \left(\frac{2^4 C_1 }{\pi^{2}}\right)^{1/3} \frac{\unicode{x02C9}\mkern-9mu k}{(\varepsilon^2 \sqrt{1+\varepsilon^2/4})^{1/3}}. \end{equation} \tag{ 12 }$$

The self-consistent theory allows also for a determination of Λ_c(ε). In fact, at finite but sufficiently small ω (i.e. at sufficiently large times), ℓ(ω) is large and one can evaluate the right-hand side of (10) at the lowest order in 1/ℓ(ω), which gives

$$\begin{equation} \frac{{\cal D}_{ii}(\omega)}{(-{\rm i}\omega)^{1/3}} = \frac{1}{(2\pi)^{2/3}} \frac{D_{11}}{(D_{11}D_{22}D_{33})^{1/3}}. \end{equation} \tag{ 13 }$$

We know from the study of the critical state of the Anderson transition [13] that $\frac {{\cal D}_{11}(\omega )}{(-\mathrm {i} \omega )^{1/3}} = \frac {\Gamma (2/3)}{3} \Lambda _{\mathrm { c}}$, which allows us to write

$$\begin{equation} \Lambda_{\rm c}(\varepsilon) = \frac{3}{\Gamma(2/3)} \left(\frac{1}{2 \pi}\right)^{2/3} \left(\frac{D_{11}^2}{D_{22} D_{33}}\right)^{1/3}. \end{equation} \tag{ 14 }$$

Using the diffusion tensor relations (8), one obtains finally

$$\begin{equation} \Lambda_{\rm c}(\varepsilon)=\frac{3}{\Gamma(2/3) } \left(\frac{2}{\pi}\right)^{2/3}\left(\frac{1+\varepsilon^{2}/4}{\varepsilon^{2}}\right)^{2/3} . \end{equation} \tag{ 15 }$$

Equations (12) and (15) are the most important results of this section. They predict that the threshold K_c(ε) and the critical anomalous diffusion parameter Λ_c(ε) diverge when ε tends to zero as K_c(ε) ∼ ε^−2/3 and Λ_c(ε) ∼ ε^−4/3. Therefore, when ε = 0 we recover the case of the 1D periodic kicked rotor, which is always localized whatever the kicking amplitude K.

In order to test the predictions of the self-consistent theory, we have performed numerical simulations of the dynamics of the QpKR (1). We have determined the critical parameters K_c(ε) and Λ_c(ε) from the crossing of the curves $\ln \Lambda = \ln \frac {\langle \mathfrak {p}^2 \rangle }{t^{2/3}}$ versus K at different times (see figure 4). At the critical point, Λ(t) is a constant Λ(t) = Λ_c(ε) corresponding to the critical anomalous diffusion $\langle \mathfrak {p}^2 \rangle \sim t^{2/3}$ and the crossing of the curves in figure 4, whereas for K < K_c(ε) (K > K_c(ε), respectively), Λ(t) decreases (increases, respectively) as time increases. We have evaluated the uncertainty of the parameters (K_c,Λ_c) by the region where the evolution of Λ(t) is not monotonous (due to systematic shifts of (K_c,Λ_c) as a function of time [40]). The results are represented in figure 5 for K_c(ε) and in figure 6 for Λ_c(ε) by the white filled region between the blue filled region where the system is localized and the pink filled region where the dynamics is diffusive. The data seem to follow an algebraic increase as the anisotropy parameter ε decreases (linear dependence in log–log scale as in figures 5 and 6), on top of which oscillations are also clearly seen.

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The self-consistent theory discussed in the previous section predicts that the critical regime of the Anderson transition is defined by the condition equation (11). Various degrees of approximation—leading to slightly different predictions for the position of the critical point—can be used for the values of the components of the diffusion tensor:

• The simplest approximation is to use equation (8), valid asymptotically for large K. This results in the simple analytic predictions (12) and (15), represented by black lines in figures 5 and 6. The algebraic dependences of K_c(ε) and Λ_c(ε) are well accounted for by these simple predictions; however, they fail to reproduce the oscillating corrections observed in the numerical data.
• The theoretical prediction (8) for the diffusion tensor D misses the oscillations of the diffusion tensor of the 3D kicked pseudo-rotor (2) versus K and $\unicode{x02C9}\mkern-9mu k$. Such oscillations are well known in the case of the 1D periodic kicked rotor [49, 54] and are due to subtle temporal correlation effects. In our case, we have checked that they could be described approximately by the known oscillating form [49], but only along direction 1:
  $$\begin{equation} \begin{array}{@{}c@{}} \tilde{D}_{11} \approx D_{11} \times \{1 - 2 J_2(\tilde{K}) [1 - J_2(\tilde{K})]\},\\ \tilde{D}_{22}= \tilde{D}_{33} = D_{22} = D_{33} \end{array} \end{equation} \tag{ 16 }$$
  with $\tilde {K} \equiv K \frac {\sin {\unicode{x02C9}\mkern-9mu k}/2}{{\unicode{x02C9}\mkern-9mu k}/2}$ and J₂ the usual Bessel function. The use of the above equation for the diffusion tensor allows for a better analytical description of the anisotropy phase diagram; see the red lines in figures 5 and 6.
• The third type of approximation relies on a direct numerical calculation of the diffusion tensor D of the 3D kicked rotor (2) at short time by a linear fitting procedure over the first ten kicks of $\langle \mathfrak {p}_i^2 \rangle = 2 D_{ii} t$. This gives a numerical prediction of the self-consistent theory for the anisotropy phase diagram represented by the green lines with points in figures 5 and 6. They clearly show oscillations around the power-law behaviors of K_c(ε) and Λ_c(ε), and are in very good agreement with the numerical data.

Last but not least, we clearly see in figures 5 and 6 that the predictions of the self-consistent theory agree very well with the experimental data represented by blue points. Therefore, the self-consistent theory appears to be a powerful way describing the Anderson transition with the QpKR.

In conclusion, we presented in this work a rather complete experimental and theoretical study of the anisotropy phase diagram of the Anderson transition in the quasiperiodic kicked rotor. Numerical and experimental results were found to be in good agreement with each other, and theoretical expressions based on the self-consistent theory of the Anderson transition correctly describe the shape dependence of these functions. These results add an important additional brick to our understanding of the Anderson transition.

Future studies made possible by this system are equally promising: one can mention the experimental study of the Anderson localization in two dimensions (lower critical dimension of the Anderson transition) or in four dimensions (in order to address the problem of the upper critical dimension that is believed to be infinite), of the effects of interactions and quantum statistics, and of the effects of symmetry and topology. This puts on even firmer ground the status of the QpKR as one of the simplest—if not the simplest—cold-atom systems that can be used for the experimental study of the Anderson localization.

This work was partially supported by Agence Nationale de la Recherche (LAKRIDI grant) and the Labex CEMPI (ANR-11-LABX-0007-01). CPU time on various computers was provided by GENCI.