From Anderson to anomalous localization in cold atomic gases with effective spin-orbit coupling

We study the dynamics of a one-dimensional spin-orbit coupled Schrodinger particle with two internal components moving in a random potential. We show that this model can be implemented by the interaction of cold atoms with external lasers and additional Zeeman and Stark shifts. By direct numerical simulations a crossover from an exponential Anderson-type localization to an anomalous power-law behavior of the intensity correlation is found when the spin-orbit coupling becomes large. The power-law behavior is connected to a Dyson singularity in the density of states emerging at zero energy when the system approaches the quasi-relativistic limit of the random mass Dirac model. We discuss conditions under which the crossover is observable in an experiment with ultracold atoms and construct explicitly the zero-energy state, thus proving its existence under proper conditions.

Tightly confined ultracold atomic gases [1] provide an ideal arena for studying a broad spectrum of onedimensional (1D) quantum phenomena. Examples range from Hubbard-type models [2,3], pairing phenomena in Fermi gases [4] to spin-orbit (SO) coupling for neutral atoms [5,6] and quasi-relativistic physics [7,8], generated by the additional use of external laser fields. The same experimental techniques can also be used to address the effects of disorder in condensed matter systems. A prominent example is the demonstration of Anderson localization [9] of a Bose-Einstein Condensate (BEC) in a 1D wave-guide [10,11]. One dimensional Anderson localization is caused by destructive interference in a weak, disordered potential (referred to as diagonal disorder) and leads to exponential localization of the particles [12][13][14][15]. This behaviour may change in half-filled disordered metals [16], random spin-Peierls and spin-ladder systems [17], or as recently predicted in photonic systems with electromagnetically induced transparency [18]. Delocalized zero-energy (mid-gap) states can emerge in these systems showing a power-law behavior for the correlations due to a Dyson singularity in the density of states [19]. Such anomalous localization originates in the chiral symmetry of the corresponding 1D Hamiltonian and can be realized in the system with off-diagonal disorder known as a random-mass Dirac model or the fluctuating gap model (FGM) [20][21][22][23].
In cold atom systems disorder is typically induced by a random potential and is thus of diagonal type. Here we show that the combination of a random potential with SO coupling induced by the motion in space dependent laser fields can lead to effective off-diagonal disorder. Lightinduced SO coupling has been shown to lead to an effective Dirac dynamics in [8]. By investigating the density (Color online) Left: Experimental sketch of the system. The condensate is placed in a tight cigar-shaped trap and driven by three lasers with Rabi frequencies Ω1, Ω2 and Ω3. Right: Internal tripod-type linkage pattern of the atoms exhibiting two dark-states without any contribution of the excited state |0 . of states of the corresponding disorder model we derive conditions under which power-law localization can be observed and argue that they are indeed connected to the emergence of a Dyson singularity in the density of states (DOS). We show by simulating the time evolution of an initially localized wave packet that increasing the SO coupling drives a crossover from exponential (Anderson) localization to an anomalous power-law localization. We note that a model similar to ours has been considered in [24], however only for the case of diagonal disorder and neglecting the kinetic energy part. As will be shown the latter can be neglected only under special circumstances and drastically alters the behavior of the system leading to much richer physics.
We consider an ensemble of atoms exposed to three laser fields in a tripod-type linkage pattern [25] as depicted in Fig. (1). The atoms are characterized by a degenerate manifold of the three ground states |1 , |2 , |3 , coupled to a common excited state |0 via corresponding control lasers of equal wave-number κ. We assume two of the control lasers to be counter-propagating along the x-axis with Rabi frequencies Ω 1 = Ω sin θe −iκx / √ 2 and Ω 2 = Ω sin θe iκx / √ 2, whereas the third laser propagates along the y-axis with Ω 3 = Ω cos θe −iκy , and Ω = 3 i=1 |Ω i | 2 denotes the total Rabi frequency. Following [25] we can write the interaction Hamiltonian ( = 1) of the resulting tripod scheme asĤ 0 = . This Hamiltonian has two darkstates, |D 1 = 1 √ 2 e −iκy (e iκx |1 − e −iκx |2 ) and |D 2 = 1 √ 2 e −iκy cos θ(e iκx |1 + e −iκx |2 ) − sin θ|3 , with zero energy and decoupled from the excited state |0 . We express a general state of the system in the dark state manifold according to |χ(r) = 2 i=1 ψ i (r)|D i (r) , where ψ 1 (r) and ψ 2 (r) are the wave functions corresponding to the two degenerate dark states. An effective equation for the center-of-mass amplitudes ψ i (r) is then given by [7] i where p is the momentum operator, m the atomic mass, and Ψ Ψ Ψ(r) = (ψ 1 (r), ψ 2 (r)) T a two-component vector. The gauge potential A, known as the Mead-Berry connection [26,27], arises from the coordinate-dependence of the dark-states and is given by A k,n = i D k (r)|∇|D n (r) . The external potential has matrix elements V k,n = D k (r)|V |D n (r) , where the potential in the bare basis was assumed to be diagonal, i.e., The scalar potential reads Φ k,n = 2 l=1 A k,l A l,n /2m. By applying an additional strong transverse trapping potential to freeze out the transverse degrees of freedom, Eq. (1) reduces to a 1D equation with 2 × 2-matrices y, z}, are the Pauli matrices. Expanding the square in Eq. (1) and choosing the external potentials as with v D = κ/m. Eq. (2) describes a massive particle with SO coupling moving in a scalar potential ∆(x). The corresponding dispersion relation for a constant potential ∆(x) = δ can easily be calculated and is shown in Fig. (2) for the case of δ = 0 (red dashed line) and small δ = 0 (blue solid line). By changing the relative intensity of the control lasers one can externally control the value of cos θ enabling an easy access to tune the effective SO coupling. In case of a smooth potential and large particle-momenta, i.e., p 2 x /2m ≫ v D p x cos θ, the SO coupling becomes negligible and the problem reduces to two uncoupled massive Schrödinger particles moving in an external potential. In the opposite limit, i.e., for FIG. 2: (Color online). Dispersion relation in the Schrödinger limit, with the inset representing the Dirac limit. The red dotted lines correspond to the case of vanishing potential, i.e. δ = 0, whereas the blue solid lines correspond to δ = mv 2 D /8. The mixing angle is given by θ = 0. Dirac equation for a particle with an effective speed of light c * = v D cos θ and a smooth space-dependent mass ∆(x)/c 2 * [7,8]. By applying Zeeman and Stark shifts, using, e.g., speckle potentials [10,28] or incommensurate optical lattices [11], a spatially fluctuating, random detuning, and thus random potential, can be generated. In the following we assume that the random potential is described by local, Gaussian white noise, where the overbar denotes disorder average and all higher order correlation functions factorize. For vanishing SO coupling one expects exponential localization of the particles according to the Anderson scenario [10,11,19,28,29]. On the other hand, any small SO coupling dominates in the region of small kinetic energy. Neglecting the kinetic energy term proportional to p 2 x in Eq. (2), rather than the SO coupling, one obtains an effective Dirac equation for a particle with a spatially random mass. This model, also known as the fluctuating gap model (FGM) [22,23], is characterized by a Dyson singularity in the density of states (DOS) [30], which is a consequence of the chiral symmetry of Eq. (2) without kinetic energy leading to a power-law localization. What determines the crossover from Anderson-like to anomalous localization? In the case of a random potential it is not immediately obvious in which parameter regimes Eq. (2) leads to Schrödinger or Dirac-like dynamics. To investigate this question we introduce dimensionless units ξ = Γx/v D and τ = Γt resulting in where∆(ξ) = ∆/Γ. In the following we analyze the DOS in the system described by Eq. (4). For a free Schrödinger particle the DOS has a singularity at ω = 0. As can be seen from Fig. 2 the presence of the SO coupling shifts the band-edge of the spectrum away from . Density of states per unit length for a SO coupled massive Schrödinger particle in a random potential with a finite disorder correlation length [33] for various values of the band edge mc 2 * . As the band edge is moving closer to ω = 0 the washing out of the Dyson singularity of the free Dirac case (blue line) becomes more prominent, until the almost purely disordered Schrödinger case (magenta line) is reached.
zero to ω edge = −mc 2 * /2 = −mv 2 D cos 2 θ/2. In the Dirac limit mv 2 D → ∞ the band edge moves to infinity. In the pure Schrödinger case weak disorder leads to a smoothing of the band-edge peak [31], making the DOS analytic, which is associated with the emergence of exponentially localized states (Anderson localization) [32]. To investigate the DOS for our SO-coupled system in the presence of disorder we numerically simulate Eq. (4). As will be the case in any experiment we assume for this a discretized model and furthermore a finite disorder-correlation length [33]. (A smaller correlation length is accompanied by slower numerical convergence of the DOS [22].) The mobility edge in momentum space associated with the finite disorder correlation length is however chosen large enough not to affect the results. Fig. 3 shows the DOS for different values of the dimensionless quantity ∆ 0 /mc 2 * , where ∆ 0 characterizes the rms value of the Gaussian disorder amplitude in the discretized model. It is related to the continuum quantity via Γ = ∆ 2 0 /(v D n kinks ), with n kinks being the impurity density. For mc 2 * → ∞, i.e. in the Dirac limit, one recognizes a Dyson-like singularity at ω = 0. The emergence of a Dyson singularity can in general be taken as an indicator for anomalous, i.e. non-exponential, localization properties [19]. In the random-mass Dirac model or FGM the Dyson singularity has been shown to lead to power-law correlations [20]. As mc 2 * ∼ mv 2 D cos 2 θ decreases the band-edge approaches the singularity. As a consequence the singularity is smoothend out. It should be noted at this point that a true singularity is present only in the exact Dirac limit, but as can be seen from Fig.  (3) a pronounced peak survives as long as the SO coupling, i.e. mc 2 * , is large. Thus one expects in this case a localization scenario where power-law correlations dominate for very long time scales. To quantify this we define following [22] the width ∆ω D of the Dyson singularity as the minimum of the DOS, yielding ∆ω D = αΓ, where α = 0.6257.... Hence, to obtain dynamics associated with the Dyson singularity the condition should be satisfied in order to ensure that the band-edge is sufficiently far away from the singularity. The emergence of the Dyson singularity affects the dynamics of the system around ω = 0. Experimentally this is reflected in a drastically different behavior of the density profile after a long-time expansion of an initially localized wave-packet. From Eq.(5) we expect that for cos 2 θ ≫ Γ/mv 2 D the effective Dirac dynamics along with the creation of anomalous power-law correlations dominates the system whereas in the opposite limit we expect Anderson-like localization. The power-law behavior of correlations will also be reflected in the density distribution of an expanding wavepacket, which is experimentally much easier accessible. To see this we note, that in the power-law case there is no intrinsic length scale. Thus we may assume that an initially well localized wave-packet will show the same behavior as a wavepacket with an initial delta-distribution Ψ Ψ Ψ(x, t = 0) = δ(x)χ χ χ with χ χ χ being a constant vector with unity length. The time-evolution of this wave-packet is given by is the stationary state of a particular disorder realization belonging to the state of energy ω n . It immediately follows that the long-time evolution is given by |Ψ Ψ Ψ(x, t → ∞)| 2 = n |φ φ φ n (0)| 2 |φ φ φ n (x)| 2 . The right hand side of this expression corresponds to the localization criterion defined in [19], and it is thus sufficient to investigate the long-time behavior of the density in order to determine localization properties. In the Schrödinger limit all correlations of the type C n (x) = |φ φ φ n (0)| 2 |φ φ φ n (x)| 2 decay exponentially [19], i.e. C n (x) ∼ e −|x|/L loc , where the localization length L loc is at most of the order of the system size. In the FGM the typical localization length of correlation functions scales as L loc ∼ |lnǫ| 2 [21], where ǫ is the distance from the band-center. For small energies this length is much larger than the system size and consequently the correlation functions should be similar to the zero-energy correlation function decaying with a power law scaling as C n (x) ∼ (Γ|x|/v D ) −3/2 at large distances [21]. Accordingly one expects quite different scaling behavior in the two different limits of large and small SO coupling. To confirm the predictions we numerically simulated the time-evolution according to Eq. can clearly see the exponentially localized wings of the density reminiscent of Anderson localization in cold atom systems [10,11,29]. Increasing the SO coupling strength leads to an intermediate regime at cos θ = 0.5 where neither exponential nor power-law localization can be determined. Finally for strong SO coupling, i.e. cos θ = 1 a fit with a power-law reveals that the density correlation scales as the expected 3/2 power-law at large distances in agreement with the predictions of the FGM.
In summary, we have investigated the one-dimensional dynamics of a SO-coupled massive Schrödinger particle subject to a δ-correlated disorder potential. For weak SO coupling the system is equivalent to two independent Schrödinger particles with diagonal disorder. In the opposite limit the system is described by the random-mass Dirac model with off-diagonal disorder. The model can be implemented with current state-of-the-art techniques by using atomic dark-states in cold atom systems. We showed by calculating the systems' DOS and direct numerical simulation of the time evolution of an expanding wave packet that there is a crossover from an exponential (Anderson) localized regime to a power-law regime governed by a Dyson singularity when varying the strength of the SO coupling relative to the disorder strength. The crossover can be observed by expansion of an initially localized wavepacket with an appropriately chosen width of the momentum distribution in a random potential.
Future investigations should include effects of interactions, which will allow for studying models such as the relativistic Thirring model [35] for fermions as well as for bosons with a random mass. Interactions may well accelerate the crossover to non-exponential behavior as they tend to delocalize particles.