# Proposal for an experimentally realizable quantum simulator of symplectic Anderson localization

Zoubair Daouma, Radu Chicireanu, Adam Rançon

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Zoubair Daouma, Radu Chicireanu, Adam Rançon. Proposal for an experimentally realizable quantum simulator of symplectic Anderson localization. 2025. hal-05317276

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# Proposal for an experimentally realizable quantum simulator of symplectic Anderson localization

Zoubair Daouma,¹ Radu Chicireanu,¹ and Adam Rançon¹,²

¹Univ. Lille, CNRS, UMR 8523 - PhLAM - Laboratoire de Physique des Lasers Atomes et Molécules, F-59000 Lille, France

²Institut Universitaire de France

We propose an experimentally realizable dynamical spin-orbit coupling model for studying Anderson localization in the symplectic symmetry class, using a one-dimensional spin-1/2 double-kicked rotor (SDKR), which displays dynamical localization in momentum space. By tuning the model's parameters, the SDKR can realize all three Dyson-Wigner symmetry classes: orthogonal, unitary, and symplectic. This allows for direct comparison of their respective localization properties within a single framework, as shown by the spectral statistics of the associated Floquet operator, which match random matrix theory predictions. Focusing on the symplectic class, we identify clear signatures of weak anti-localization in momentum space, corresponding to an enhanced return to the origin in momentum, including an anti-coherent backscattering (anti-CBS) dip and a coherent forward scattering (CFS) peak. These structures serve as direct probes of destructive quantum interference and the onset of localization. We further characterize the system's transport using the one-parameter scaling function $\beta(g)$, confirming its universality across disorder realizations and models. These results establish the SDKR as a minimal yet versatile platform for experimentally exploring localization beyond the standard symmetry classes.

## I. INTRODUCTION

Anderson localization, the suppression of transport due to interference in disordered systems [1], is a paradigmatic phenomenon in condensed matter physics and is a universal phenomenon of classical and quantum waves [2–18]. Its occurrence and critical properties depend crucially on the symmetries of the system, particularly time-reversal symmetry (TRS). Time reversal is a fundamental transformation in quantum mechanics that inverts the time coordinate and reverses dynamical variables such as momentum and spin [19]. A system is said to be time-reversal invariant if its dynamics remain unchanged under this transformation, which strongly influences interference effects and transport in disordered media.

In spinless systems, weak localization arises in the presence of TRS, as in disordered metals, due to coherent backscattering (CBS). The constructive interference of time-reversed trajectories enhances the backscattering amplitude, leading to a reduction of conductivity at low temperatures [1, 20–22]. The same quantum interferences also produce an enhanced return-to-origin (ERO) effect, i.e., an increased probability of finding a particle at its initial position [23–25]. These effects, observed in both real and momentum space and collectively referred to as CBS effects, serve as signatures of the weak localization regime. Spinless systems with and without TRS belong to the orthogonal and unitary universality classes, respectively.

However, the spin of the particle can have a strong influence on the dynamics relevant for instance, for materials with strong spin-orbit coupling (SOC). In the context of Anderson physics, these systems then belong to a distinct universality class characterized by TRS with breaking of spin rotational symmetry, known as the symplectic symmetry class. In this class, weak anti-localization effects are expected to be observable [26], particularly in 1D systems. For instance, quantum interferences lead in this case to a decreased return to the origin (or anti-ERO), and to an enhancement of diffusive transport at short times [27–33]. These aspects, and their dependences on the universality class, are encapsulated in the so-called beta function $\beta(g)$, which describes the universal behavior of the conductance $g$ as the system size is increased [21].

Floquet systems are particularly versatile quantum simulators of spatially disordered materials, to explore Anderson localization in a highly controllable setting. A paradigmatic example is the quantum kicked rotor (QKR), which describes a spinless particle periodically driven by a 1D position-dependent sinusoidal potential applied in short bursts (kicks) and evolving freely between kicks. The QKR has been extensively studied in the context of quantum chaos with cold atoms [34, 35] and provides a highly controlled platform to investigate interference effects analogous to those in spatially disordered systems [36]. In the QKR, localization, called dynamical localization, occurs in momentum space, and experimental studies with ultracold atoms have observed phenomena equivalent to Anderson localization in one and two dimensions [37–39], as well as the metal-insulator transition and its critical properties in three [40–44] and four dimensions [45]. The QKR Hamiltonian is invariant under a generalized time-reversal operator squaring to one, placing the system in the orthogonal symmetry class. This symmetry can be broken, e.g., by breaking spatial invariance [46], applying a strong magnetic field [47], or using phase- or amplitude-modulated QKR variants [48, 49], to realize the unitary class. Coherent return-to-origin (ERO) and related interference effects have been theoretically
predicted [50] and experimentally observed in cold-atom QKR setups [48, 51], providing a direct probe of the system's symmetry properties in both orthogonal and unitary universality classes, see also [52].

Spin-1/2 versions of the kicked rotor have attracted intense theoretical and numerical interest, as they allow access to the symplectic universality class and support the realization of topological phases [53–61]. These models typically employ spin-dependent and position-dependent potentials, involving at least two Pauli matrices. Such potentials generate spin-dependent kicks and position-dependent population transfers, which must be applied briefly and with sufficient strength. These requirements present significant experimental challenges, explaining why the exploration of this physics in atomic Floquet systems remains limited.

In this paper, we introduce a spin-1/2 double-kicked rotor (SDKR) that realizes the symplectic symmetry class, as well as the orthogonal and unitary classes, while significantly alleviating these experimental challenges. Unlike earlier theoretical models, the SDKR requires only spin-dependent kick potentials, with population transfer occurring during free evolution, and time-reversal symmetry is restored by reversing the sign of the transfer pulse every two kicks. This design makes the SDKR readily implementable in current cold-atom experiments. In the absence of disorder, the model maps onto a lattice system with spin-orbit coupling, albeit with a non-standard dispersion relation.

The manuscript is organized as follows. In Section II, we present the model, analyze its symmetries, and discuss its experimental relevance. In Section III, we examine the quasi-energy level statistics of the Floquet operator and compare them with the predictions of random matrix theory (RMT), confirming that the SDKR allows for studying the three universality classes. In Section IV, we compute the scaling function $\beta(g)$ capturing the universal aspects of transport. In Section V, we study the dynamics of wavepackets and the signatures of the ERO, both at short times, in the regime of weak (anti-)localization, and in the strong localization regime. We give our conclusions in Section VI.

## II. SPIN-1/2 DOUBLE-KICKED ROTOR MODEL

### A. Definition of the disordered model with effective spin-orbit coupling

Our SDKR model is designed to simulate Anderson localization in various symmetry classes, including the symplectic class. This model generalizes the well-known quantum kicked rotor by incorporating spin-dependent and spin-independent kicks. It is loosely inspired by a model studied in [62]. The time-dependent Hamiltonian reads:

$$ \hat{H}(t) = \frac{\hat{p}^2}{2} + (-1)^{|t|} k \Omega \hat{\sigma}_x + \hat{H}_- \sum_n \delta(t-2n) + \hat{H}_+ \sum_n \delta(t-2n-1), \quad (1) $$

where

$$ \hat{H}_{\pm} = K(1 \pm \alpha) [\cos(\hat{x}) + \varepsilon \sin(\hat{x})\hat{\sigma}_z]. \quad (2) $$

All quantities are dimensionless units, with time in units of the period $\tau$ of the kicks and length in units of the inverse of the kick-potential wavenumber $k_K$. Momenta are normalized such that $[\hat{x}, \hat{p}] = ik$, with $k = \hbar k_K^2 \tau/M$ is the effective Planck constant and $M$ is the mass of the particle. The periodicity in space of the kicking potential enforces quantization of momentum: $p = (m + q)k$, with $m \in \mathbb{Z}$ and $q \in [-1/2, 1/2]$ the conserved quasi-momentum. The eigenstates of $\hat{\sigma}_z$ and $\hat{\sigma}_x$ are denoted as $|\uparrow\downarrow\rangle$ and $|\pm_x\rangle$, respectively.

The key ingredients of the SDKR are the alternating spin rotations $\pm\Omega\hat{\sigma}_x$ between kicks, and the sequence of spin-dependent kicks that alternate between $\hat{H}_-$ and $\hat{H}_+$. These alternating terms are crucial: the sign flip of $\Omega$ every unit time ensures a breaking of spin-rotation symmetry while preserving time-reversal symmetry, see below. The parameter $\alpha$ controls the asymmetry between the two kicks, which is also needed, while $\varepsilon$ sets the strength of the spin-dependent potential.

The Hamiltonian is periodic with period $T = 2$, and we define the Floquet operator over one period

$$ \hat{F} = \hat{U}_{K_{-}}^{1/2} \hat{U}_{P_{-}} \hat{U}_{K_{+}} \hat{U}_{P_{+}} \hat{U}_{K_{-}}^{1/2}, \quad (3) $$

with

$$ \begin{aligned} \hat{U}_{K_{\pm}} &= e^{-i\hat{H}_{\pm}/k}, \\ \hat{U}_{P_{\pm}} &= e^{-i\left[\frac{\hat{p}^2}{2k} \pm \Omega\hat{\sigma}_x\right]}. \end{aligned} \quad (4) $$

Note that we have symmetrized the operator by implementing half a kick sequence with $\hat{U}_{K_{-}}$ on each side for later convenience.
<figure><img src="image_3.png" /><figcaption>FIG. 1. Functions $f_0(x)$, $f_1(x)$ and $f_3(x)$ describing the effective Hamiltonian of the SDKR in the absence of the kinetic term. Parameters: $K = 4$, $\alpha = 0.8$, $\epsilon = 2$ and $\Omega = \sqrt{5}\pi$.</figcaption></figure>

During the free evolution induced by

$$ \hat{H}_0 = \frac{\hat{p}^2}{2} + (-1)^{|t|} k \Omega \hat{\sigma}_x, \qquad (5) $$

its eigenstates $|p, \pm_x\rangle$ accumulate the phases

$$ \varphi_{\mp}(p) = \frac{p^2}{2k} \mp (-1)^{|t|} \Omega. \qquad (6) $$

When $k$ and $\Omega$ are incommensurate with $2\pi$, the accumulated phase becomes pseudo-random in momentum space, effectively introducing disorder. Meanwhile, the kicks change the particles' momentum, effectively generating a hopping in momentum space (at fixed quasi-momentum), which is spin dependent thanks to $\epsilon$. As a result of these two competing effects, we expect localization in momentum space, with its detailed features determined by the system's time-reversal symmetry.

The dynamics induced by Eq. (1) can be mapped onto that of a lattice model with spin-orbit coupling. This is most easily shown in the absence of “disorder”, i.e. if we neglect the kinetic energy term. This is possible for the state with $q = 0$ at a quantum resonance, $k = 4\pi$, for which the $\frac{p^2}{2k} = 2\pi$. In this case, the Floquet operator is diagonal in the real space (which is the equivalent of the reciprocal space of a standard lattice model), and can be written as

$$ \hat{F} = e^{-iH_{eff}(\hat{x})/k}, \qquad (7) $$

with effective Hamiltonian

$$ H_{eff}(\hat{x}) = f_0(\hat{x}) + f_1(\hat{x})\hat{\sigma}_x + f_3(\hat{x})\hat{\sigma}_z. \qquad (8) $$

The analytical expressions of the $2\pi$-periodic functions $f_i(x)$ are rather cumbersome and uninformative and are thus not given, they are shown in Fig. 1 for some generic values of the parameters. One finds that $f_0(x) = 2K \cos(x)$ is even, while the two other functions are odd. This gives rise to a spin-dependent “dispersion” with the correct symmetries under time-reversal. In the presence of disorder induced by the kinetic energy term, this will give rise to Anderson physics with spin-orbit coupling.

Note that our approach is the reverse of that used to study the integer quantum Hall effect with the QKR [56–58]. There, a simple lattice model was mapped onto a complicated spin-dependent QKR model, while we started from a simple spin-dependent QKR and showed that it can be mapped on an a priori complicated lattice model. See also the model introduced in Section II D for comparison.
## B. Symmetry properties

The spinless kicked rotor possesses a generalized time-reversal $\hat{T}_0 = \hat{P}\hat{T}$ symmetries, where $\hat{P}$ is parity operator, and $\hat{T}$ is the usual time-reversal operator. Their action on coordinates reads:

$$ 
\begin{aligned}
\hat{T} & : t \to -t, \quad \hat{p} \to -\hat{p}, \quad \hat{x} \to \hat{x}, \\
\hat{P} & : t \to t, \quad \hat{p} \to -\hat{p}, \quad \hat{x} \to -\hat{x}, \\
\hat{T}_0 & : t \to -t, \quad \hat{p} \to \hat{p}, \quad \hat{x} \to -\hat{x}.
\end{aligned}
\qquad (9) $$

The operator $\hat{T}$ is implemented through the anti-linear operator $\hat{K}$, the complex conjugation (leaving the position basis invariant) [19]. These symmetries can be generalized to families of operations $\hat{T}_\tau$ and $\hat{P}_{x_0}$ that shift time or reflect about an arbitrary time $\tau$ or point $x_0$. For the spinless QKR, the relevant operator is $\hat{T}_0$, since it preserves quasi-momentum [19].

In the presence of spin, the natural extension of $\hat{T}_0$ is the antiunitary operator

$$ \hat{T} = i\hat{\sigma}_y \hat{P} \hat{K} : \begin{cases} t \to -t, \\ \hat{x} \to -\hat{x}, \\ \hat{p} \to \hat{p}, \\ \hat{\sigma}_j \to -\hat{\sigma}_j, \quad j = \{x, y, z\}, \end{cases} \qquad (10) $$

which is such that $\hat{T}^2 = -1$.

One can easily check that

$$ 
\begin{aligned}
\hat{T}\hat{U}_{K_{\pm}}\hat{T}^{-1} &= \hat{U}_{K_{\pm}}^{\dagger}, \\
\hat{T}\hat{U}_{P_{\pm}}\hat{T}^{-1} &= \hat{U}_{P_{\pm}}^{\dagger}
\end{aligned}
\qquad (11) $$

which implies that $\hat{T}\hat{F}\hat{T}^{-1} = \hat{F}^{\dagger}$ is time-reversal invariant. Since $\hat{T}^2 = -1$, the spectrum is doubly degenerated, with eigenstates coming in (Kramers) pairs. Furthermore, the system will generically be in the symplectic class, unless additional unitary symmetries are present [19]. This happens, for instance, if $\Omega$ or $\varepsilon$ vanish. The Hamiltonian can then be diagonalized in the $|\uparrow\downarrow\rangle$ basis, giving rise to two independent copies of the QKR belonging to the orthogonal class.

The unitary class is realized easily by fully breaking time-reversal symmetry. This can be done by using two spin rotations $\Omega_1$ and $\Omega_2 \neq -\Omega_1$ every two kicks, such that $\hat{T}e^{-i\Omega_2\hat{\sigma}_x}\hat{T}^{-1} \neq (e^{-i\Omega_1\hat{\sigma}_x})^{\dagger}$. In this case, there is no Kramers degeneracy. We show in Appendix A that in the case $\alpha = 0$ (and $\Omega_1 = \Omega = -\Omega_2$, such that $\hat{T}$ is a symmetry), the dynamics belong to (two copies of) the unitary class, even though we were not able to identify any additional unitary symmetry. This is the reason why two different kick strengths are needed in Eq. (2).

The main difference between the SDKR and previously studied models [53–61] is the fact that in our model, there is only one spin component ($\hat{\sigma}_z$) coupled to a space-dependent potential. The second spin component ($\hat{\sigma}_x$) is coupled to a homogeneous field, which, as we discuss below, makes this model easier to realize experimentally. Time-reversal invariance is recovered by changing the sign of the field after every kick.

## C. Numerical implementation and the random SDKR

To numerically study the localization properties, we define a random-phase version of the model [63], the random SDKR. Since the kinetic energy phases of the kicked rotor are quasi-random (but deterministic) in momentum space, disorder averaging may be hindered by parasitic effects, such as correlations between kicks. In the random-phase variant, the kinetic energy operator $\hat{p}^2/2$ is replaced with $\sum_{p \in \mathbb{Z}} e^{-i\phi_p}|p\rangle\langle p|$, where $\phi_p \in [0, 2\pi]$ are i.i.d. random phases. This construction destroys correlations of the “disorder” in momentum, mimicking a fully disordered medium [64]. We show in Appendix B that our conclusions are unaffected by this choice, and that the random SDKR shows the same localization properties as the original, deterministic version of the SDKR. Unless stated otherwise, all numerical simulations presented in the following are performed using the random-phase model, with $\bar{k} = 1$.

The wavefunction of the system reads

$$ |\psi(t)\rangle = \sum_{p \in \mathbb{Z}} (\psi_{\uparrow}(p, t)|p, \uparrow\rangle + \psi_{\downarrow}(p, t)|p, \downarrow\rangle), \qquad (12) $$
and we choose the initial state $|\psi(t=0)\rangle = |p=0, \uparrow\rangle$. The total density at momentum $p$ is noted $|\psi(p,t)|^2 = |\psi_\uparrow(p,t)|^2 + |\psi_\downarrow(p,t)|^2$, where the overline means averaging over disorder (i.e. over the random phases in the case of the random SDKR, or random sampling of quasi-momenta for the deterministic SDKR).

While the physical Hilbert space is infinite, the numerical simulations are performed using a truncated momentum basis of dimension $N$, leading to a total Hilbert space size $2N$, due to spin. Care is taken to ensure that the dynamics is not significantly affected by this truncation. It is solved using Fast Fourier Transform to change from position to momentum basis for applying the kicks or free propagations.

In practice, we do not use the Floquet operator Eq. (3), but we always perform the sequence $\hat{U}_{K_-} \rightarrow \hat{U}_{P_+} \rightarrow \hat{U}_{K_+} \rightarrow \hat{U}_{P_-} \rightarrow \hat{U}_{K_-} \rightarrow \dots$

## D. Thaha-Blümel-Smilansky model

To benchmark our model, we will compare our results against the Thaha-Blümel-Smilansky (TBS) model [46], with time-periodic Hamiltonian

$$ \hat{H}_{TBS}(t) = \frac{\hat{p}^2}{2} + K \left[ \cos(\hat{x}) + \frac{\mu}{2} \sin(2\hat{x}) \hat{\sigma}_x + \epsilon \sin(\hat{x}) \hat{\sigma}_z \right] \sum_n \delta(t-n). \quad (13) $$

This model has been shown to belong to the three universality classes depending on the parameters, in particular by comparing its level spacing to that of random matrix theory. This model is time-reversal invariant under $\hat{T}$. For generic values of $K$, $\mu$, and $\epsilon$, the model belongs to the symplectic class.

When $\epsilon = 0$, the existence of a so-called geometric symmetry $\hat{R}_x = i\hat{\sigma}_x$ which commutes with $\hat{T}$ and squares to $-1$ implies that the system now belongs to the unitary class (with a doubly degenerated spectrum for each Kramers pair) [19]. Note that while $\hat{T}' = \hat{R}_x\hat{T}$, such that $\hat{T}'^2 = 1$, is a time-reversal symmetry of the model in this case, it does not imply the orthogonal class, as $\hat{T}'$ does not constrain the Floquet operator more than $\hat{R}_x$ and $\hat{T}$ do already.

When $\mu = 0$, the potential term can be written as $K \cos(\hat{x} - \varphi\hat{\sigma}_z)/\cos(\varphi)$, where $\varphi = \arctan \epsilon$. There are now two geometric symmetries, $\hat{R}_z = i\hat{\sigma}_z$ and $\hat{P}_z = \hat{R}_x e^{2i\varphi\hat{p}\hat{\sigma}_z}$. Here $\hat{P}_z$ is a spin-dependent translation of $\hat{x}$ followed by a spin-flip, which leaves $\cos(\hat{x} - \varphi\sigma_z)$ constant. Note that $[\hat{P}_z, \hat{T}] = 0 = [\hat{R}_z, \hat{T}]$, $\hat{P}_z^2 = -1 = \hat{R}_z^2$ and $\{\hat{P}_z, \hat{R}_z\} = 0$. Thus, one can diagonalize $\hat{R}_z$ and $\hat{H}_{TBS}(t)$, with $\hat{P}_z$ still off-diagonal. One then shows that $\hat{T}'' = \hat{P}_z\hat{T}$ is a new time-reversal symmetry squaring to 1 that implies that the system belongs to the orthogonal class [19].

In practice, our numerical simulations of the TBS model are done using the random version of the model, where we replace the kinetic term with random phases.

## E. Experimental relevance

The main interest of Hamiltonian in Eq. (1) lies in the fact that it can be easily implemented in ultracold atom experiments, by encoding the (pseudo)spin in two internal hyperfine or Zeeman states chosen to have magnetic moments of opposite sign. The spin-dependent potential can be obtained by using the vector component of the atomic polarizability, in a standing wave configuration produced with linearly polarized beams having a non-zero relative angle (the so-called ‘lin-angle–lin’ configuration [61, 65]). This configuration effectively creates a spin-dependent phase shift of the kick potential. Note that previous schemes [53, 56–58] (including the TBS model [46]) require that the spin coupling be applied during the very short duration of the kicks and also have a spatial modulation of this coupling on the scale of the optical wave length. A key practical advantage in our model lies in the fact that the coupling between the spin states in Eq. (1) occurs during the longer time of free evolution between the kicks, and is homogeneous in space. In practice, the coupling can easily be realized using microwave or Raman transitions, with an additional periodic sign flip, implemented by adding $\pi$ phase jumps in the driving field after each kick. Note that this scheme is also robust with respect to decoherence sources related, e.g. to spontaneous photon scattering, which limit coherence and lifetime in more traditional SOC implementation based on near-resonant Raman dressing fields.

## III. SPECTRAL FLUCTUATIONS SIGNATURES

A fundamental principle of quantum chaos predicts that the spectral properties of chaotic systems should follow the predictions of RMT [19, 66–71], with Floquet systems being described by the circular ensembles. The QKR has been a pioneering model for studying quantum chaos, where it was observed that the fluctuations of quasi-energies indeed
<table><caption>TABLE I. Average values of the consecutive level spacing ratio $\langle\tilde{r}_n\rangle$ obtained from simulations for the three symmetry classes (CSE, CUE, COE), compared with the theoretical predictions from reference [71]. Same parameters as in Fig. 2.</caption><thead><tr><th>Class</th><th>⟨r̃<sub>n</sub>⟩ (SDKR)</th><th>⟨r̃<sub>n</sub>⟩ (RMT) [71]</th></tr></thead><tbody><tr><td>CSE</td><td>0.6746(1)</td><td>0.6744(1)</td></tr><tr><td>CUE</td><td>0.5993(1)</td><td>0.5996(1)</td></tr><tr><td>COE</td><td>0.5306(1)</td><td>0.5307(1)</td></tr></tbody></table>

correspond to the predictions of RMT [72, 73]. In particular, when the QKR preserves parity-time reversal invariance $\hat{P}\hat{T}$, the spectral statistics are described by circular orthogonal ensembles (COE). Conversely, if this symmetry is broken, the spectral statistics align with those of circular unitary ensembles (CUE). Finally, spin-dependent QKR models with $\hat{T}^2 = -1$ belong to the symplectic orthogonal ensemble (CSE).

We follow an approach similar to that used in [54, 74] to characterize the properties under time-reversal of the random SDKR as a function of the parameters. Specifically, we numerically study the spectral fluctuations of the Floquet operator depending on the underlying symmetries.

To this end, we construct the Floquet operator $\hat{F}$ by discretizing both position and momentum spaces using $N = 128$ grid points for each spin component, yielding a total Hilbert space dimension of $2N = 256$. The momentum basis is truncated at $m = -N/2, \dots, N/2-1$. It is crucial to note that RMT statistics apply only when the basis size $N$ remains smaller than the localization length $l$, i.e., $N < l$. In this regime, Floquet eigenstates exhibit significant overlap, which leads to level repulsion consistent with universal RMT behavior. Conversely, when $N \gg l$, the eigenstates become localized at centers separated by distances larger than $l$, and the eigenvalues become uncorrelated. In such a regime, level statistics are expected to follow a Poisson distribution, independent of the symmetry class under consideration [73, 74].

For each realization of the random phases $\phi_m$ (see Section II C), we compute the free propagation operators and the potential operators and construct the Floquet operator $\hat{F}$. We average over $10^5$ independent disorder realization. For each realization, we diagonalize the corresponding Floquet operator to obtain its eigenvalues $e^{-i\epsilon_n}$, where the quasi-energies $\epsilon_n$ lie within the interval $[0, 2\pi[$. The quasi-energies are sorted in ascending order, and we compute the normalized level spacings as $s_n = (\epsilon_{n+1} - \epsilon_n)/\delta$, with $\delta$ denoting the mean level spacing.

When analyzing the statistics of quasi-energy level spacings, an accumulation of zero-spacings is expected due to Kramers degeneracy in presence of time-reversal invariance with $\hat{T}^2 = -1$. To avoid counting each degenerate pair twice, which would artificially create a peak at zero, we retain only every second eigenvalue. The distribution $P(s)$ of normalized level spacings is obtained by constructing a histogram of all $s_n$ collected over the ensemble, as illustrated in Fig. 2. The resulting $P(s)$ are consistent with those of the Dyson-Wigner circular ensembles [19, 75],

$$ 
\begin{align} 
P_{\text{CSE}}(s) &= \frac{2^{18}}{3^6 \pi^3} s^4 e^{-\frac{64}{9\pi} s^2}, \notag \\ 
P_{\text{CUE}}(s) &= \frac{32}{\pi^2} s^2 e^{-\frac{4}{\pi} s^2}, \tag{14} \\ 
P_{\text{COE}}(s) &= \frac{\pi}{2} s e^{-\frac{\pi}{4} s^2}. \notag 
\end{align} 
$$

The ratio of consecutive level spacings [71, 76],

$$ \tilde{r}_n = \frac{\min(s_n, s_{n-1})}{\max(s_n, s_{n-1})} = \min\left(\frac{s_n}{s_{n-1}}, \frac{s_{n-1}}{s_n}\right), \quad (15) $$

is another marker of the universality class. The average value $\langle\tilde{r}_n\rangle$ is estimated using a bootstrap procedure, which allows us to assess its statistical uncertainty. The results presented in Table I are in excellent agreement with those expected for the different RMT classes. This clearly confirms that the model we have introduced belongs to the corresponding universality class, depending on the parameters.

## IV. ONE-PARAMETER SCALING LAW OF THE CONDUCTANCE

Transport in disordered systems is governed by the principle of one-parameter scaling, encapsulated by a function $\beta(g) = \frac{\partial \ln g}{\partial \ln L}$, with $g$ the dimensionless conductance and $L$ the system size [21]. Alternatively, transport can be analyzed through the evolution of a wave packet in a disordered potential, with localization characterized by the saturation of the width of the wave packet.
<figure><img src="image_4.png" /><figcaption>FIG. 2. Histograms of normalized level spacings for the three universality classes compared with the theoretical distributions given in Eq. (14): a) symplectic; b) unitary; c) orthogonal. The distributions are averaged over $10^5$ random phase realizations. In c), the choice of $\varepsilon = \tan(\Phi)$ is important when discretizing the bases to ensure proper invariance under time reversal. We use $\Phi = 2\pi/N$.</figcaption></figure>

In the context of the QKR, suitable dynamical definitions of $L$ and $g$ have been introduced in

$$L = \frac{\sqrt{\langle p^2 \rangle_t}}{k}, \\ g = \frac{\sqrt{\langle p^2 \rangle_t}}{kt/N}, \qquad (16)$$

where $\langle p^2 \rangle_t = \sum_p p^2 |\psi(p, t)|^2$, $t$ is the number of kicks, and $N = T(2s+1)/d$ with $T = 2$ the period, $2s+1$ the number of spin degrees of freedom, where $s$ is the spin ($s = 1/2$ here) and $d = 2$ in the presence of Kramers degeneracy and $d = 1$ if not. Defined in this way, $L$ represents the size explored by the system at a given time, and $g$ is proportional to the diffusion coefficient at short time (where $\langle p^2 \rangle_t \propto Dt$ with $D$ the diffusion coefficient and thus $g \propto D/L$) while it goes to zero at long time (where $\langle p^2 \rangle_t$ is constant). The scaling function $\beta(g)$ thus exhibits two distinct regimes. In the weak-localization regime, i.e. at short times, one finds $\beta(g) = -1$, characteristic of metallic behavior governed by Ohm's law. In the strong localization regime, i.e. at times larger than the localization time, the system transitions to a fully localized regime where the conductance $g$ vanishes and $\beta(g)$ diverges to $-\infty$. The factor $N$ has been introduced to normalize the time scale on which the dynamics occurs: it increases with the period of the kicking sequence, which can be interpreted as introducing $T$ effective “transverse channels” [48], as does the number of spin states. Finally, Kramers degeneracy reduces the density of eigenenergies of the Floquet eigenstates in each Kramers subspace, and thus accelerates the dynamics compared to when there is not such degeneracy.<fnref n="1" />

The analytic form of the $\beta$-function at small $1/g$ can be obtained from the short-time behavior of the kinetic energy. The detailed analysis of the main corrections to $\langle p^2 \rangle_t$ due to loop effects is given in [77],<fnref n="2" />

$$\begin{align}
\beta_O(g) &= -1 - \frac{4\sqrt{2}}{3\sqrt{\pi}g} + \dots, \nonumber \\
\beta_U(g) &= -1 - \frac{1}{2g^2} + \dots, \tag{17} \\
\beta_S(g) &= -1 + \frac{2\sqrt{2}}{3\sqrt{\pi}g} + \dots \nonumber
\end{align}$$

The sign of the first correction to diffusion indicates the effect of quantum interferences. The minus sign (orthogonal and unitary cases) indicates a decrease of transport, and thus weak localization, while the plus sign signals an increase of transport at short time and thus anti-weak localization – which is characteristic to the symplectic symmetry class.

The scaling functions $\beta(g)$ corresponding to the three Dyson-Wigner ensembles (symplectic, unitary, and orthogonal) can be obtained from the random SDKR, depending on the chosen parameters, as illustrated in Fig. 3, panel a). As shown in Fig. 3-b), the scaling functions $\beta(g)$ obtained from the TBS model Eq. (13) perfectly match them, for all three symmetry classes. In particular, the short time (small $1/g$) regime is well described by Eq. (17). Despite differences in microscopic dynamics, both systems display the same universal behavior, in agreement with the predictions of one-parameter scaling theory.

<footnote n="1">¹ We have checked the factor $2s + 1$ by performing numerical simulations of our model for spin 3/2 and 5/2.</footnote>

<footnote n="2">² The first correction to the symplectic case can be obtained from that of the orthogonal case by multiplying it by $-1/2$ [78].</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 3. a) One-parameter scaling function $\beta(g)$ for the three symmetry classes—symplectic, unitary, and orthogonal—computed using the random SDKR model. Results are averaged over $10^4$ random phase realizations. Dashed lines show the corresponding small $1/g$ predictions Eq. (17). Same parameters as in Fig. 2. b) Scaling function $\beta(g)$ computed for the random TBS model Eq. (13) in three different symmetry classes, averaged over $10^4$ random phase realizations. Parameters are: $(K, \mu, \epsilon) = (49.229, 0.295, 0.125)$ for the symplectic class, $(25, 0.8, 0)$ for the unitary class, and $(39.345, 0, 1)$ for the orthogonal class.</figcaption></figure>

## V. WAVEPACKET DYNAMICS

In the present section, we study the short-time and long-time dynamics of a wavepacket with initial condition $|\psi(t = 0)\rangle = |p = 0, \uparrow\rangle$. We show that at short time weak (anti-)localization effects allow for distinguishing the different symmetry classes through the CBS ERO, whereas at long time all three universality classes show an ERO associated with more complex localization effects.

### A. Weak anti-localization

The weak localization (WL) and weak anti-localization (WAL) phenomena are manifestations of quantum interference in disordered systems [79]. These effects appear as corrections to classical electrical conductance, caused by the interference between time-reversed particle trajectories. Coherent backscattering (CBS) enhances the return probability due to constructive interference, while its counterpart in systems with strong spin-orbit coupling—WAL—leads to destructive interference and reduced return probability. These phenomena are well-established and have been extensively discussed in the literature [80].

In this section, we investigate these interference effects in the case of the SDKR. In kicked rotors, Anderson localization manifests in momentum space. The localization, as well as its weak localization (CBS) or anti-localization (anti-CBS) precursors, arise from interference between time-reversed paths. In momentum space, these correspond to the enhanced (ERO) or suppressed (anti-ERO) return to the origin. In real space, the corresponding phenomena are CBS and anti-CBS [50]. For brevity, we use CBS/anti-CBS to refer to both momentum and real space phenomena.

In the symplectic class, Fig. 4-d) shows a clear anti-CBS dip at $p = 0$ for an odd number of kicks, reflecting destructive interference. In contrast, for an even number of kicks Fig. 4-a), the anti-CBS dip disappears. To understand this dependence on the parity of the number of kicks, we analyze the symmetry structure of the time-evolution operators at given times $t$. The evolution operator at an odd number of kicks, $t = 2n + 1$ reads:

$$ \hat{U}(t = 2n + 1) = \hat{U}_{K-} \left( \hat{U}_{P-} \hat{U}_{K+} \hat{U}_{P+} \hat{U}_{K-} \right)^n, \qquad (18) $$

which is time-reversal symmetric:

$$ \hat{T}^{-1} \hat{U}(t = 2n + 1) \hat{T} = \hat{U}^{\dagger}(t = 2n + 1). \qquad (19) $$

By contrast, the even number of kicks, $t = 2n$, evolution operator reads:

$$ \hat{U}(t = 2n) = \hat{U}_{K+} \hat{U}_{P+} \hat{U}_{K-} \left( \hat{U}_{P-} \hat{U}_{K+} \hat{U}_{P+} \hat{U}_{K-} \right)^{n-1}, \qquad (20) $$

ending with an unpaired $\hat{U}_{\text{kick2}}$, which breaks time-reversal symmetry:

$$ \hat{T}^{-1} \hat{U}(t = 2n) \hat{T} \neq \hat{U}^{\dagger}(t = 2n). \qquad (21) $$
<figure><img src="image_6.png" /><figcaption>FIG. 4. Short-time momentum distribution of the random SDKR model in the three symmetry classes: first column: symplectic; second column: orthogonal; third column: unitary class. Top and bottom rows represent even and respectively odd kick numbers. Same parameters as in Fig. 2.</figcaption></figure>

This symmetry breaking explains the disappearance of anti-CBS for an even number of kicks. A similar effect has previously been observed in spinless kicked rotor systems with a period-two phase modulations of the kick sequence [51], where a CBS peak only appears for an even number of kicks.

As shown in the previous sections, choosing the parameters of the SDKR model also allows us to realize other symmetry classes, which directly affects the interference WL signatures. In the orthogonal class ($\Omega = 0$), constructive interferences between time-reversed paths lead to a positive CBS signal, present once again only at odd number of kicks, as shown in Fig. 4-b) and d). On the other hand, in the unitary class ($\Omega_1 \neq \Omega_2$), where time reversal symmetry is broken at all kick numbers, no interference peaks are present at odd and even number of kicks, see Fig. 4-c) and f). We refer to [51] for a detailed discussion of WL in the unitary and orthogonal classes in the QKR.

## B. Strong localization regime

In the presence of (strong) Anderson localization, a related phenomenon to the CBS, known as coherent forward scattering (CFS), can also emerge [81, 82]. It appears even in the absence of TRS and is indicative of the onset of strong localization [83] and can serve as a marker of the Anderson transition in higher dimensions [84]. For the QKR, in momentum space, the CFS corresponds to an additional ERO effect, also robust to TRS breaking [48].

Figure 5 displays the disordered average probability distribution $|\psi(p)|^2$ in the strong localization regime for odd and even numbers of kicks in the symplectic class, panels a) and d), orthogonal class, panels b) and e), and unitary class, panels c) and f). Strong localization can be deduced by the exponentially-localized shape of the probability distributions (except near $p = 0$). In this regime, a second type of interference peak appears at the origin $p = 0$, with distinct features compared to the CBS peaks in Fig. 4. The CFS peaks appear at even numbers of kicks, and their presence is independent of the symmetry of the system - as can be seen in Fig. 5-a), b) and c). Moreover, these peaks are absent at short times (see corresponding panels in Fig. 4), and their amplitude is always positive and gradually increases in time, until saturation at long times.

Although its presence is independent of the symmetry class, the dynamics of the CFS peak does depend on symmetry. A feature of the SDKR model lies in the presence of the period-two modulations, which allows for a time separation of the CBS and CFS contributions to the return probability, and thus studies the dynamics of the CFS contribution
<figure><img src="image_7.png" /><figcaption>FIG. 5. Top and middle rows: asymptotic momentum-space probability density $|\psi(p)|^2$ at long times, in the Anderson localization regime in the three symmetry classes (first column: symplectic; second column: orthogonal; third column: unitary class) for even (top row) and odd (middle row) kick numbers. Bottom row: corresponding contrasts of the return probability, at even and odd numbers of kicks. The parameters are the same as in Fig. 2.</figcaption></figure>

[48]. See Appendix C for a discussion of the CBS and CFS peaks for the TBS model, which does not have this time separation of the two contributions to the ERO. Thus, the parity of the number of kicks naturally leads to two contrast measures:

$$C_{\text{CBS}} = \frac{|\psi|_{0,\text{odd}}^2 - \langle |\psi|_{\text{neigh.}}^2 \rangle_{\text{odd}}}{\langle |\psi|_{\text{neigh.}}^2 \rangle_{\text{odd}}}, \qquad (22)$$

$$C_{\text{CFS}} = \frac{|\psi|_{0,\text{even}}^2 - \langle |\psi|_{\text{neigh.}}^2 \rangle_{\text{even}}}{\langle |\psi|_{\text{neigh.}}^2 \rangle_{\text{even}}},$$

where $|\psi|_0^2$ is the value of $|\psi(p = 0, t)|^2$, and $\langle |\psi|_{\text{neigh.}}^2 \rangle$ is its average in a small momentum window around and excluding $p = 0$. The index odd/even corresponds to considering only odd or even times. Figure 5-g), h) and i) show these CBS and CFS contrasts over time, in the three symmetry classes. While the (anti-)CBS contrast quickly stabilizes to a constant value from the first few kicks, the CFS contrast grows slowly and saturates in the localized regime, with a saturation time that depends on the symmetry class.

We leave a detailed discussion of the dynamics of the CBS and CFS peaks, especially how they depend on the spin components, for future work.
# VI. DISCUSSION

We have introduced a spinful kicked rotor model that realizes the symplectic, unitary, or orthogonal symmetry class depending on the choice of parameters. We have shown that it can be interpreted, in the absence of disorder, as a lattice model with spin-orbit coupling. This classification was confirmed through the quasienergy statistics of the Floquet operator, as well as experimentally accessible observables, the dynamics of the momentum distribution and the scaling function $\beta(g)$.

Our results provide a concrete and experimentally viable platform to explore both weak and strong localization phenomena in systems with symplectic symmetry. Earlier proposals for simulating the symplectic class and other topological phases, such as the TBS model Eq. (13), are challenging to implement experimentally, as they require coupling both spin components to spatially varying potentials with different spatial periods. In contrast, our model requires only one spin component to be coupled to a spatially modulated potential, while the other interacts with a homogeneous field during free evolution. Time-reversal symmetry is enforced by reversing this field every two kicks. This setup is readily compatible with current cold atom experiments: the $\Omega\hat{\delta}_x$ coupling can be realized via RF coupling between hyperfine states, and the spin-dependent potential can be engineered as proposed in [61]. The generalization to other universality classes with non-trivial topological invariants using these simplifications is an exciting perspective that we are currently exploring.

# ACKNOWLEDGMENT

We thank C. Tian, J.-C. Garreau, C. Hainaut and G. Usaj for useful discussions. This work received financial support from the projects ANR-24-CE47-4949 UniQ-RingS and ANR-23-PETQ-0001 Dyn1D at the title of France 2030. ZD was supported by the Région Hauts-de-France and Labex CEMPI (GrantNo. ANR-11-LABX-0007-01). AR has benefited from the financial support of the Grant No. ANR-24-CE30-6695 FUSIoN.

## Appendix A: Special case: $\alpha = 0$

In this appendix, we discuss the special case $\alpha = 0$ and show that it belongs to the unitary class with Kramers pairs. First, we note that the discussion on symmetries in Section II B does not depend on the value of $\alpha$. In particular, $\hat{T}$ is a symmetry of the Floquet operator for $\alpha = 0$, i.e. $\hat{T}\hat{F}\hat{T}^{-1} = \hat{F}^\dagger$. This implies in particular that there are Kramers pairs with equal eigenvalues also for $\alpha = 0$.

To better understand the properties of the system in this case, it is convenient to define a different Floquet operator

$$ \hat{F}' = \hat{U}_{P-}^{1/2} \hat{U}_K \hat{U}_{P+} \hat{U}_K \hat{U}_{P-}^{1/2}, \quad (A1) $$

which is unitarily related to $\hat{F}$ (we call $\hat{U}_{K\pm} = \hat{U}_K$ since $\alpha = 0$). Note that for this Floquet operator, the correct time-reversal operator that squares to $-1$ is $\hat{T}\hat{U}_{P+}^{1/2}\hat{U}_K\hat{U}_{P-}^{1/2}$.

Interestingly, we find that $\hat{T}' = \hat{R}_z\hat{T}$ ($\hat{R}_z = i\hat{\delta}_z$) is a time-reversal operator that squares to 1 such that $\hat{T}'\hat{F}'\hat{T}^{-1} = \hat{F}'^\dagger$. This is similar to what happens in the case of the TBS model when it belongs to the unitary class, see Section II D. However, in the present case, $\hat{R}_z$ is not a geometric symmetry of $\hat{F}'$ and thus the discussion of that section cannot be used to explain why the SDKR with $\alpha = 0$ belongs to the unitary class. In fact, we were not able to find a geometric symmetry, squaring to $-1$ and commuting with the time-reversal operator. We leave this problem for future work.

Instead, we give numerical evidence that this case indeed belongs to the unitary class in Fig. 6. Panel a) shows the distribution of level spacing and panel b) shows the beta function, both agreeing with the unitary universal shape. Concerning the CBS and CFS peaks, the situation is quite different from the unitary case discussed in the main text, where time-reversal symmetry is fully broken.

Interestingly, the existence of the two operators $\hat{T}$ and $\hat{T}'$, squaring to $-1$ and 1 respectively, allows for the presence of an anti-CBS peak at even times and the presence of a CBS peak at odd times, respectively. The contrast of those two peaks is $\mp 1/2$, as they appear only on one component of the spins (we leave the detailed study of the spin dependence of these peaks to future work). At longer time, the CFS peak grows at even kicks only, leading to a contrast $+3/2$ for even kicks.
<figure><img src="image_8.png" /><figcaption>FIG. 6. Random SDKR with $\alpha = 0$. a) level statistics b) one-parameter scaling function $\beta(g)$ c) return probability contrast d-e) probability density at short times f-g) at long times. Parameters are given in panel a).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 7. Histograms of normalized level spacings for the three symmetry classes (symplectic, unitary, orthogonal) of the deterministic SDKR, compared with the theoretical RMT distributions Eq. (14). Figures a), b), and c) correspond to the symplectic, unitary and orthogonal classes, respectively.</figcaption></figure>

## Appendix B: Comparison with the deterministic SDKR model

In this appendix, we show that the conclusions obtained with random SDKR are unchanged for the deterministic SDKR, Eq. (1), with the correct kinetic energy term. The parameters used in the simulations are the same as that used for the random SDKR (with $k = 1$). In this case, the averaging is performed over $10^5$ randomly sampled quasi-momenta $q \in [-1/2, 1/2[$.

Figure 7 shows the histograms of the level spacings for the three universality classes (see Section III), which are in perfect agreement with those of the circular ensembles.

Figure 8 shows the $\beta(g)$ of the deterministic SDKR for the same parameters as in Fig. 3. We observe the same universal behavior in the three symmetry classes. At short time (corresponding to large conductance), we observe increased fluctuations that we associate to the temporal correlations induced by the quasi-randomness of the disorder [64]. The oscillations are reduced using the random version of the SDKR (see Fig. 3).
<figure><img src="image_10.png" /><figcaption>FIG. 8. One-parameter scaling function $\beta(g)$ of the deterministic SDKR model, for the three symmetry classes: symplectic, unitary, and orthogonal (the color code and parameters are the same as in Fig. 3). The inset is a zoom on the behavior at large conductance (short time).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 9. a) and b): momentum-space probability density $|\psi(p)|^2$ at short ($t \ll t_{\text{loc}}$) and long ($t \gg t_{\text{loc}}$) times respectively, using the random TBS model in the symplectic symmetry class. c) Time evolution of the contrast of the return probability. The parameters are $K = 20$; $\varepsilon = 0.2$; $\mu = 0.5$.</figcaption></figure>

## Appendix C: CBS and CFS peaks of the TBS model

In this section, we show the CBS and CFS return probability peaks are also observed in numerical simulations using the TBS model. Contrary to the SDKR model, the period of the TBS model is equal to one kick. This means that the two components of the return probability are not separated in time, as in the case the SDKR, and manifest at each kick. However, the respective temporal dynamics are different and can be used to distinguish between the two. Fig. 9 a-b) show the momentum-space probability distribution in the symplectic class at short and long times, respectively. In the diffusive regime, we observe an anti-CBS dip in the return probability with a contrast of $-1/2$, similar to the one observed in the SDKR model (see Fig. 4-d). On the other hand, in the localized regime we observe an increase in return probability, which manifests by a contrast $+1$ CFS peak superposed to the anti-CBS one - leading to a total contrast of $+1/2$. The corresponding dynamics of the *total* return probability is shown in Fig. 9-c), with a slow evolution between the two asymptotic values $-1/2$ at short times and $+1/2$ determined by the onset of strong dynamical localization in the symplectic TBS model.

## Appendix D: Universality of the one-parameter scaling function

In this appendix, we show the universality of the one-parameter scaling function $\beta(g)$. To this end, we have plotted $\beta(g)$ for the random SDKR model for several sets of parameters corresponding to the three fundamental symmetry classes: orthogonal, unitary, and symplectic, as shown in Fig. (10). Despite the different microscopic parameters used
in each case, all data collapse onto the same universal curves characteristic of their respective symmetry class. The specific values of the parameters employed for each simulation are summarized in Table D.

<figure><img src="image_12.png" /><figcaption>FIG. 10. Universality of the one-parameter scaling function $\beta(g)$ for the random SDKR model, shown for the three symmetry classes (orthogonal, unitary, and symplectic).</figcaption></figure>

<table><caption>TABLE II. Parameters used for each curve in Fig. 10. The symbol and color next to each “Set” column correspond exactly to the markers in the main figure.</caption><thead><tr><th>Class</th><th>Parameter</th><th>Set 1 ●</th><th>Set 2 ■</th><th>Set 3 ▲</th></tr></thead><tbody><tr><td rowspan="4">Symplectic</td><td>K</td><td>20</td><td>15</td><td>22</td></tr><tr><td>Ω</td><td>√2π</td><td>√7π</td><td>√5π</td></tr><tr><td>ε</td><td>0.7</td><td>0.6</td><td>0.9</td></tr><tr><td>α</td><td>0.7</td><td>0.3</td><td>0.5</td></tr><tr><td rowspan="6">Unitary</td><td></td><td>Set 1 ●</td><td>Set 2 ■</td><td>Set 3 ▲</td></tr><tr><td>K</td><td>20</td><td>17</td><td>15</td></tr><tr><td>Ω<sub>1</sub></td><td>√2π</td><td>√5π</td><td>√7π</td></tr><tr><td>Ω<sub>2</sub></td><td>√3π</td><td>√7π</td><td>√9π</td></tr><tr><td>ε</td><td>0.5</td><td>0.6</td><td>0.9</td></tr><tr><td>α</td><td>0.3</td><td>0.3</td><td>0.1</td></tr><tr><td rowspan="4">Orthogonal</td><td></td><td>Set 1 ●</td><td>Set 2 ■</td><td>Set 3 ▲</td></tr><tr><td>K</td><td>20</td><td>14</td><td>22</td></tr><tr><td>Ω</td><td>0</td><td>0</td><td>0</td></tr><tr><td>ε</td><td>0.3</td><td>0.4</td><td>0.1</td></tr><tr><td></td><td>α</td><td>0.2</td><td>0.1</td><td>0.5</td></tr></tbody></table>

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