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

We realize experimentally a cold-atom system, the quasiperiodic kicked rotor, equivalent to the three-dimensional Anderson model of disordered solids where the anisotropy between the x direction and the y–z plane can be controlled by adjusting an experimentally accessible parameter. This allows us to study experimentally the disorder versus anisotropy phase diagram of the Anderson metal–insulator transition. Numerical and experimental data compare very well with each other and a theoretical analysis based on the self-consistent theory of localization correctly describes the observed behavior, illustrating the flexibility of cold-atom experiments for the study of transport phenomena in complex quantum systems.

making this system an almost ideal environment to study Anderson-type quantum phase transitions 5 .
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][18][19][20][21][22] for previous theoretical studies), as well as an analytical description of these properties based on the self-consistent theory of Anderson localization [23][24][25], which adds another important brick to our detailed knowledge on the Anderson metal-insulator transition.

Controlled disorder and anisotropy within a cold-atom system
The QpKR [8,26,27] is described by the one-dimensional (1D) time-dependent Hamiltonian H qpkr = p 2 2 + K cos x (1 + ε cos ω 2 t cos ω 3 t) n δ(t − n). (1) Experimentally, it is realized by placing laser-cooled atoms (of mass M) in a standing wave (formed by counterpropagating beams of intensity I 0 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 T 1 ) 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 ω 2 and ω 3 , proportionally to 1 + ε cos ω 2 t cos ω 3 t. Lengths are measured in units of (2k L ) −1 , time in units of T 1 and momenta in units of M/2k L T 1 ; note that [x, p] = ik withk = 4hk 2 L T 1 /M playing the role of a reduced Planck constant. The pulse amplitude is K =kτ 2 /8 L , where is the resonance Rabi frequency between the atom and the laser light and L the laser-atom detuning. Fixed parameters used throughout the present work arek = 2.89, ω 2 = 2π √ 5 and ω 3 = 2π √ 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: 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 √ t. Quantum mechanically, this system displays the phenomenon of dynamical localization, which corresponds to an asymptotic saturation of the average square momentum p 2 [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][32][33].
If 2π/T 1 , ω 2 , ω 3 ,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 with an initial condition taken as a planar source in momentum space (completely delocalized along the transverse directions p 2 and p 3 ). 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 |p = | p =kp , where the p i are integers 7 . In this basis, the evolution operator over one temporal period reads as the product U = J V of an on-site operator, V (p) = e −iφ p with phases φ p = kp 2 1 2 + ω 2 p 2 + ω 3 p 3 , and of a hopping operator J such that The phases V (p) are different on each site of the momentum lattice, and, although perfectly deterministic, constitute a pseudo-random sequence 8 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 /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]. The color plot corresponds to the growth rate α of p 2 (t) ∝ t α at long times, estimated from numerical simulations. Blue represents localization (α = 0) and red represents diffusive dynamics (α = 1). Intermediate colors (cyan, yellow, etc) appear because the numerical calculation is done at a finite time t = 1000 kicks, where, close to the transition, the asymptotic regime is not yet entirely attained. The black line corresponds to α = 2/3, that is the critical line of the Anderson transition. Paths (white dashed lines) form the grid used for the determination of K c (ε). 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 (p; t) ∼ exp (− |p| /p loc ) (with p loc the localization length) if K is smaller than a critical value K c (ε) and Gaussian diffusive (p; t) ∼ exp −p 2 /4Dt (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 following the anomalous diffusion at criticality: p 2 = c (ε)t 2/3 (here α = 3 1/6 (2/3) −1/2 ) (see also [27,39]). The fundamental quantities characterizing the threshold of the transition are therefore K c (ε) and c (ε), and we will consider in the following their dependence as a function of the anisotropy parameter ε (see figure 1).

Experimental determination of the anisotropy phase diagram
In our experiment, we measure the population of the zero velocity class (0; t) using Raman velocimetry [41][42][43]. This quantity is proportional to p 2 (t) , with a proportionality factor that depends on the specific shape of (p). This factor varies smoothly across the Anderson transition, so that the transition can be studied using either p 2 (t) or (0; t). The scaling theory of localization [12,44] predicts that p 2 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] Asymptotically, (t) ∝ t −2/3 , t 0 , t 1/3 in the localized, critical and diffusive regimes, respectively, so that ln (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 transition 9 . 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 can be written as a one-parameter scaling function 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 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 p 2 , and hence c . We have measured the value of the critical parameters K c (ε) and 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 (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 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 , 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. Experimental results on the determination of the critical point of the metal-insulator Anderson transition, for various values of the parameter ε of the QpKR. The second column indicates the range of K where data have been taken. The experimentally measured values of both K c and c are compared with the numerically calculated values. The uncertainties on the experimental data are rather small. The numerical data (with times up to 1000 kicks) have similar uncertainties, but also incorporate systematic shifts of (K c , 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 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 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 c (marked deviation), an effect that is not present in the numerical data.

Self-consistent theory of the anisotropy phase diagram
We shall now try to describe theoretically the observed anisotropy dependences of the two critical parameters K c (ε) and c (ε). The approach we shall follow is based on the selfconsistent 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 selfconsistent 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 p i to a site p f in N steps, It consists of propagations mediated by the hopping amplitudes p n+1 |J |p n and by collisions on the disorder represented by V (p) = e −iφ p . Two important points are the following: (i) one can consider in a first approximation the φ p as completely random phases [31,49,50] and we will consider quantities averaged over those phases, for example P(p i , p f , t = N ) where the line over the quantity represents this The numerical data (with times up to 1000 kicks) have similar uncertainties, but also incorporate systematic deviations of (K c , c ) when estimated over various temporal ranges. These systematic deviations cannot be measured in the experiment (limited to 120 kicks). In plot (a) one observes a very good agreement between numerical and experimental results. The agreement is fair in plot (b): the deviations in the value of K c between experimental and numerical data are converted into significant deviations of c because of the rapid variation of versus K . Experimental and numerical values deviate at low epsilon because of decoherence that is expected to have a significant impact at t = 120 kicks and to decrease K c and c . In the region of large ε, the deviation is due to the finite variance of the initial momentum distribution that has only a small increasing effect on K c but significantly increases c at 120 kicks.
averaging; (ii) p n+1 |J |p n 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 p 1 , this is just a Bessel function that decreases exponentially fast for |p n+1 − p n | K /k and one can thus see K /k as the analogue of the mean free path, with the limit of small disorder corresponding to K /k 1. One can attack the problem of the calculation of 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 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 (p, t)) as [24,25,34] Here, the diffusive tensor D-computed in [38] for large K -is anisotropic, but diagonal, with 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 j k j 1, with k j the mean free path along direction j which is such that D j j = k 2 j /4. Equation (7) means that in the regime of long times and large distances (in momentum space), we should have diffusive transport with p 2 j = 2D j j 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 paths 10 , 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 P takes then the approximate form P(q, ω) = 1 −iω+ j D j j (ω)q 2 j , where D is the frequency-dependent diffusion constant obeying the self-consistent equation [23,24,34] Quite remarkably, this approach is able to account for a certain number of observed features 11 : it predicts a transition between a metallic phase of diffusive transport for with p loci the localization length along direction i. At the threshold, the self-consistent theory 10 The QpKR or the equivalent periodic 3D pseudo-rotor is not invariant by time reversal. However, the Hamiltonian, equation (2) is invariant by the product of time reversal and parity. The existence of such an antiunitary symmetry is sufficient for the system to belong to the orthogonal universality class, and consequently for the existence of the weak localization correction. 11 Note that the self-consistent theory wrongly predicts a critical exponent ν = 1, in disagreement with the numerical data ν ≈ 1.58 [40,47]. However, incorporating in this theory an ad hoc momentum dependence for the diffusion coefficient at the threshold allows for a better prediction ν = 1.5 [52].
predicts an anomalous diffusion with D ii (ω) ∼ (−iω) 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 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] with C 1 a numerical constant of the order of one. We make the following change of variables, Y j = D j j (ω) −iω q j , and define the rescaled cutoff: (9) it is clear that the ratio D j j (ω)/D j j is isotropic; thus (ω) is isotropic). One obtains The threshold K c (ε) of the Anderson transition is then approached from the diffusive regime, which is characterized by D ii (ω) From expressions (8) for the diffusion tensor, one deduces the following dependence of the threshold versus anisotropy: The self-consistent theory allows also for a determination of c (ε). In fact, at finite but sufficiently small ω (i.e. at sufficiently large times), (ω) is large and one can evaluate the righthand side of (10) at the lowest order in 1/ (ω), which gives We know from the study of the critical state of the Anderson transition [13] that D 11 (ω) (−iω) 1/3 = Using the diffusion tensor relations (8) Upper panel: numerical data for ln = ln p 2 t 2/3 versus K at different times ranging from t = 36 to 956. The threshold corresponds to the crossing of these lines where we have the critical anomalous diffusion p 2 ∼ t 2/3 , whereas for K < K c (ε) (t) decreases at long time (localized regime) and for K > K c (ε) (t) increases at long time (metallic regime). The uncertainty region (between the orange and violet dash-dotted lines) corresponds to the region where the evolution of (t) is not monotonous. Lower panel: the different degrees of approximation for 16π 4 D 11 D 22 D 33 . According to equation (11) (with C 1 = 1/2, see text), this quantity should be equal to unity at the critical point. The black line corresponds to the simple analytic prediction (8) for the diffusion tensor. The red line shows the theoretical prediction incorporating oscillating corrections for the diffusion tensor (see equation (16) and text). The green line with points shows numerical data for the diffusion tensor D of the 3D kicked rotor (2), at short time. The parameters arek = 2.89, ω 2 = 2π √ 5, ω 3 = 2π √ 13 and ε = 0.036.
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 c (ε) diverge when ε tends to zero as K c (ε) ∼ ε −2/3 and 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 .

Experimental and numerical tests of the self-consistent theory of the anisotropy phase diagram
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 . The anisotropy dependence of K c (ε) with the associated uncertainty is determined from numerical simulations of the dynamics of the QpKR and is represented by the white filled region between the blue (localized) and pink (metallic) filled regions. The three degrees of approximation of the selfconsistent theory prediction (11) (with C 1 = 1/2) are shown: (i) the black line corresponds to the simple analytic prediction (12), (ii) the red line incorporates oscillating corrections for the diffusion tensor (see equation (16) and text) while (iii) the green line with points corresponds to numerical data for the diffusion tensor D of the 3D kicked rotor (2), at short time. The blue points represent the experimental data. The parameters arek = 2.89, ω 2 = 2π √ 5 and ω 3 = 2π √ 13.
K c (ε) and c (ε) from the crossing of the curves ln = ln p 2 t 2/3 versus K at different times (see figure 4). At the critical point, (t) is a constant (t) = c (ε) corresponding to the critical anomalous diffusion p 2 ∼ t 2/3 and the crossing of the curves in figure 4, whereas for K < K c (ε) (K > K c (ε), respectively), (t) decreases (increases, respectively) as time increases. We have evaluated the uncertainty of the parameters (K c , c ) by the region where the evolution of (t) is not monotonous (due to systematic shifts of (K c , c ) as a function of time [40]). The results are represented in figure 5 for K c (ε) and in figure 6 for 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.
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 c (ε) are well accounted for by these simple predictions; however, they fail to reproduce the oscillating corrections observed in the numerical data.  (14) of the self-consistent theory is shown with the three different degrees of approximation considered: (i) the simple analytic prediction (15) is shown as a black line, (ii) the red line is based on the analytic prediction equation (16) for the diffusion tensor incorporating oscillating corrections in K andk and (iii) numerical data for the diffusion tensor D of the 3D kicked rotor at short time (2) give the green line with points. The blue points represent the experimental data. The parameters arek = 2.89, ω 2 = 2π √ 5 and ω 3 = 2π √ 13.
• 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 andk. 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: withK ≡ K sink/2 k/2 and J 2 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 p 2 i = 2D 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 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.

Conclusion
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.