Quantum walk and Anderson localization of rotational excitations in disordered ensembles of polar molecules

We consider the dynamics of rotational excitations placed on a single molecule in spatially disordered 1D, 2D and 3D ensembles of ultracold molecules trapped in optical lattices. The disorder arises from incomplete populations of optical lattices with molecules. This leads to a model corresponding to a quantum particle with long-range tunnelling amplitudes moving on a lattice with the same on-site energy but with forbidden access to random sites (vacancies). We examine the time and length scales of Anderson localization for this type of disorder with realistic experimental parameters in the Hamiltonian. We show that for an experimentally realized system of KRb molecules on an optical lattice this type of disorder leads to disorder-induced localization in 1D and 2D systems on a time scale $t \sim 1$ sec. For 3D lattices with $55$ sites in each dimension and vacancy concentration $ 90~\%$, the rotational excitations diffuse to the edges of the lattice and show no signature of Anderson localization. We examine the role of the long-range tunnelling amplitudes allowing for transfer of rotational excitations between distant lattice sites. Our results show that the long-range tunnelling has little impact on the dynamics in the diffusive regime but affects significantly the localization dynamics in lattices with large concentrations of vacancies, enhancing the width of the localized distributions in 2D lattices by more than a factor of 2. Our results raise a general question whether quantum particles with long-range tunnelling can undergo quantum localization in 3D lattices with substitutional disorder.


I. INTRODUCTION
Scattering-induced localization of electrons in disordered crystals determines the conductor -insulator transitions in metals [1], quasi-crystals [2] and granular metal films [3], and is associated with many interesting phenomena, such as the quantized phases of the integer Hall effect [4]. If written in a second-quantized form, the Hamiltonian of an electron in a disordered lattice can be mapped onto the Hamiltonians for a variety of systems [5]. This mapping can be used to study transport properties of electrons in disordered lattices by examining the properties of other quantum particles and quasi-particles. For example, disordered-induced localization has been studied with microwaves in a tubular waveguide [6,7], optical photons in opaque media [8,9] and ultracold atoms in optical lattices [10][11][12][13]. Despite these studies, there are still many open questions about disorder-induced localization. For example, the scaling hypothesis providing the relation between quantum localization in systems of different dimensionality [14] still awaits experimental confirmation. Equally important and widely debated are the effects of inter-particle interactions [15] and dissipative forces [16] on localization in far-from-equilibrium systems or the role of long-range tunnelling amplitudes [17][18][19] allowing particles to transition between distant lattice sites.
The development of experimental methods for producing molecules at ultracold temperatures has created new possibilities for studying disorder-induced localization. As demonstrated in recent experiments [20][21][22][23][24], an ensemble of ultracold KRb diatoms, produced by photoassociation of ultracold K and Rb atoms and trapped in an optical lattice, forms a random spatial distribution of KRb molecules in the ro-vibrational ground state. If the trapping field of the optical lattice has high intensity, the molecular motion is strongly confined and the spatial distribution is quasi-stationary, eliminating molecule -molecule collisions. The excitation of molecules from the rotational ground state to the lowest rotationally excited state leads to the formation of delocalized excitations [23,24], representing the wave packets of rotational Frenkel excitons [25,26]. The purpose of the present work is to explore if these wave packets can be used as probe particles to study dynamics of localization induced by the disorder potential stemming from the randomness of the molecular positions.
The rotational excitations in an ensemble of polar molecules randomly distributed on an optical lattice possess several unique properties: (i) The rotationally excited states of molecules have long radiative lifetimes (> 1 − 10 sec), allowing one to probe coherent dynamics over long time scales.
(ii) The rotational excitations can be placed on molecules in well defined lattice sites. This can be achieved by applying a spatial gradient of an electric field and a microwave field pulse resonant with rotational transitions for molecules in the desired part of the lattice [27]. The rotational excitations thus produced form spatially localized wave packets [26]. Once the gradient of the field is removed, these wave packets can travel throughout lattice due to resonant energy transfer between molecules, undergoing quantum random walk and scattering by empty lattice sites. By shaping the gradient of the field, it may be possible to prepare the excitation wave packets representing superpositions of excitations in spatially separated lattice sites, thus enabling the controlled study of the role of initial conditions and entanglement between spatially separated particles on the dynamics of quantum walk.
(iii) The transfer of rotational excitations between molecules is enabled by long-range dipole -dipole interactions, leading to long-range transfer of excitations between distant lattice sites. This can be used to study the effects of long-range tunnelling on disorderinduced localization.
(iv) The translational motion of molecules can be controlled by varying the strength of the optical lattice potential. As shown in Ref. [28,29], the translational motion of molecules can be coupled to the rotational excitations. This can be used to study disorder-induced localization and diffusion of quantum particles in the presence of controlled coupling to a dissipative environment.
(v) The optical lattice potential can be designed to produce a three-dimensional (3D) finite lattice of molecules or to separate molecules in one or two dimensions, effectively confining the rotational excitations to one-dimensional (1D) or two-dimensional (2D) disordered lattices [30].
(vi) The probability distribution of rotational excitations in a disordered lattice of molecules can be imaged directly by resonantly enhanced multi-photon ionization [29].
These unique properties make the rotational excitations in spatially disordered ensembles of ultracold molecules a unique new platform for the study of quantum random walk and 3 disorder-induced localization. However, the strength of the disorder potential produced by the random distribution of molecular positions as experienced by rotational excitations is unknown. If the disorder potential is too weak, the localization dynamics may not be observable over experimentally feasible time scales. If the disorder potential is too strong, localized rotational wave packets may remain pinned to their original location and not exhibit any dynamics of quantum walk over experimentally feasible time scales.
The specific goal of this work is to answer the following questions: Is the disorder potential for rotational excitations in a disordered ensemble of ultracold molecules strong enough to stimulate the localization dynamics of rotational excitations in two-dimensional and threedimensional lattices over experimentally feasible time scales? Is the disorder potential weak enough to allow interesting dynamics of quantum walk over experimentally feasible time scales? To answer these questions, we consider the dynamics of rotational excitations placed in specific sites of an optical lattice partially populated with KRb molecules. To simulate an experiment with destructive measurements, we average the dynamics of rotational energy transfer between molecules in the disordered ensembles over multiple disorder orientations and examine the time scales of the formation of disorder-induced distributions of rotational excitations in lattices of various dimensionality. We also examine the effects of the longrange tunnelling amplitudes in lattices of various dimensionality and discuss the effect of disorder on diffusion of rotational excitations in lattices of various dimensionality.

II. METHODOLOGY
The Hamiltonian for an ensemble of molecules on an optical lattice of arbitrary dimensionality can be written asĤ whereĤ n is the Hamiltonian of an isolated molecule placed in lattice site n positioned at r = na, a is the lattice constant, andV n,n is the dipole -dipole interaction between molecules in sites n and n . The vector index is n = n x for 1D lattices, n = (n x , n y ) for 2D lattices and n = (n x , n y , n z ) for 3D lattices, with each of n x , n y and n z running from −N to N , so that N = 2N + 1 is the number of lattice sites per dimension. We assume that 4 the translational motion of molecules can be neglected. This approximation is valid for the trapping field potential with depth exceeding 100 kHz [28].
Following the experimental work in Ref. [23], we assume that each molecule is initially in the lowest eigenstate ofĤ n , corresponding to zero rotational angular momentum j = 0 of the molecule. Some of the molecules are subsequently excited to an isolated hyperfine state within the triplet of the hyperfine states corresponding to the rotational angular momentum j = 1 and the projection of j on the space-fixed quantization axis m j = −1. The lowest energy state and this specific excited state for the molecule in site n are hereafter denoted as |g n and |e n . The states |g and |e have the opposite parity.
Since the dipole -dipole interaction operator can be written as a sum over products of rank-1 spherical tensors acting on the subspace of the individual molecules, the matrix elements g n | g n |V n,n |g n |g n vanish so the molecules are non-interacting when in the ground state. However, the dipole -dipole interaction has non-zero matrix elements between the product states |g n |e n and |e n |g n . The matrix elements t n,n = g n | e n |V n,n |e n |g n (2) stimulate resonant energy transfer of the |g → |e excitation between molecules in sites n and n . For the specific states |g = |j = 0 and |e = |j = 1, m j = −1 , the matrix elements (2) have the following form: where d n = d n = 0.57 Debye is the permanent dipole moment of KRb molecules [23], θ nn is the angle between the intermolecular axis joining the molecules in lattice sites n and n and the z-axis. We assume that θ nn = π/2 for 1D and 2D lattices. The molecules are trapped in a lattice with lattice constant a = 532 nm [23]. For the specific system considered here, the value of these matrix elements is t n,n = 52.12 × (3 cos 2 θ nn − 1) |n − n | 3 Hz.
The matrix elements (2) also lead to the delocalized character of the rotational excitations.
In general, the |g → |e excitation generates the following many-body excited state: In an ideal lattice, the coefficients C n = e iap·n / √ N and |ψ exc ⇒ |ψ exc (p) represents a rotational Frenkel exciton with wave vector p [31] , i.e. a rotational excitation completely delocalized over the entire lattice. If the lattice is disordered, the excited state is a localized coherent superposition of the Frenkel excitons with different p [26].
When the lattice is partially filled with molecules, the excitation energy of the molecular ensemble depends on the location of molecules in the excited state. The excited states of the many-body system of molecules in a disordered lattice are the eigenstates of the following where the operator c n removes an excitation from site n, w n is the excitation energy of a molecule in site n, if the site is populated, and zero otherwise. The transition amplitudes t n,n are given by Eq. (2) if both of the sites n and n are populated by molecules and zero otherwise. It is this disorder in the transition amplitudes t n,n that localizes the delocalized excitations.
In order to compute the time-evolution of the rotational excitations, we diagonalize the matrix of the Hamiltonian (6) in the site-representation basis |n ≡ c † n |0 , where the vacuum state is with both m and n running only over the lattice sites populated with molecules.
The eigenstates of the full Hamiltonian (6), are then used to compute the time evolution of the excitation wave packet as follows where E λ are the eigenvalues of the Hamiltonian (6), the coefficients and the coefficients C λ are the projections of the excitation wave packet at t = 0 onto the eigenstates of the Hamiltonian, The quantities |f n (t)| 2 are averaged over multiple calculations with different realizations of disorder (i.e. different random distributions of empty lattice sites).
As will be demonstrated in the following section, the rotational excitations scattered by the empty lattice sites form peaked distributions. In this article, we will use two measures of distributions: the distribution width L and the standard deviation σ r . The distribution width L is defined as the length of the lattice (for a 1D lattice), the radius of a circle (for a 2D lattice) or the radius of a sphere (for a 3D lattice) containing 90 % of the excitation probability. The standard deviation is defined as σ r = r 2 − r 2 , where r 2 = x 2 for a 1D lattice, r 2 = x 2 + y 2 for a 2D lattice and r 2 = x 2 + y 2 + z 2 for a 3D lattice. The distribution width characterizes the physical spread of the rotational excitation over the molecular ensemble. However, this quantity may be significantly affected by fluctuations due to specific disorder realizations. The standard deviation depends on the shape of the distribution but is less affected by the fluctuations. We will use the standard deviation in order to characterize non-ambiguously the dynamical properties such as the time dependence of the distributions.

III. QUANTUM WALK OF ROTATIONAL EXCITATIONS
As originally proposed by DeMille [27], ultracold molecules trapped in an optical lattice A. Localization dynamics in 1D lattice A quantum particle placed in a 1D disordered lattice must undergo Anderson localization [32]. Mapped onto a rotational excitation travelling in a disordered ensemble of molecules, this well-known result is reflected in the formation of the exponentially localized distributions shown in Figure 1. These distributions characterize the probability of molecules in the corresponding lattice sites to be in the rotationally excited state. Figure 1 illustrates that rotational excitations in an optical lattice partially populated with molecules behave as quantum particles in a disordered lattice. It is important to note that the distributions presented in Figure 1 are time-independent. Although the rotational excitation initially placed in a specific lattice site represents a wave packet that, given sufficient time, can potentially explore the entire lattice and exhibit revivals, the averaging over disorder realizations leads 8 to stationary distributions in the limit of long time. In order to illustrate this, we present in Figure 4 the distributions of the rotational excitation computed at different times.
The lower panel of Figure 4 illustrates the time dependence of the distribution width L in a 1D lattice with the vacancy concentrations 10 % and 20 %. The results show that for the parameters representing rotational excitations in an ensemble of KRb molecules, the averaged excitation probability distributions change most significantly at t between zero and 100 -300 ms. The distribution widths exhibit a temporary peak at short times. This oscillation survives the averaging over disorder realizations and is present in the results for different concentrations of vacancies. Since the distribution width presented in Figure 4 is much smaller than the lattice size (1001 × a), this short-time oscillation cannot be due to reflection from the lattice boundaries and is likely the result of constructive interference due to back-scattering of the rotational excitation from the empty lattice sites.
As indicated by the results in the lower panel of Figure 4, the approach to the stationary distributions is determined by the strength of the disorder potential, i.e. the concentration of lattice vacancies. This is illustrated more clearly in the upper panel of Figure 5 showing the time dependence of the standard deviations σ r for different concentrations of vacancies. It is instructive to examine the dependence of the minimum time required to form the stationary distributions on the concentration of vacancies. Figure 6 shows the time τ required for a rotational excitation initially placed in the center of a lattice to form a time-independent distribution as a function of vacancy concentration in lattices of different dimensionality.
Interestingly, Figure 6 reveals that the dependence of τ on the vacancy concentration is qualitatively different for lattices of different dimensionality. In 1D lattices, τ is a monotonically decaying function of the vacancy concentration. This illustrates that increasing the disorder of 1D lattices impedes the dynamics of quantum walk leading to narrower distributions of the rotational excitation (cf., Figure 1) and decreases the time required to reach the stationary distributions (less time since the quantum walker has less space to explore). The calculations shown in Figure 6  The time-dependence of σ r can be used as a quantitative measure of diffusion in disordered lattices. In particular, the diffusion of classical particles in Brownian motion is characterized by σ 2 r = r 2 − r 2 ∝ t, while the diffusion of quantum particles with cosine dispersion is characterized by σ 2 r ∝ t 2 [33,34]. Figure 8 shows the time dependence of σ 2 r computed for 2D and 3D lattices with various concentrations of empty lattice sites. As expected, we find that σ 2 r ∝ t 2 for ideal lattices, both in 2D and 3D. However, the time-dependence of σ 2 r is dramatically modified by the presence of vacancies. In particular, Figure 8 illustrates that the rotational excitation in disordered lattices diffuses at short times as a quantum particle with σ 2 r ∝ t 2 and exhibits the dependence σ 2 r ∝ t, characteristic of classical particles in Brownian motion, at later times. At even later times (t > 300 ms), the dependence of σ 2 r on time departs from linearity and the diffusion is suppressed.
Qualitatively, these results can be interpreted as follows. If the concentration of vacancies is finite, the lattice can be separated into short pieces of ∆ lattice sites. The lattice is unperturbed on the scale ≤ ∆. At short times, when the rotational excitations are localized to σ r ∼ a, they expand as quantum particles in an ideal lattice, until σ r reaches the size ∼ ∆. When the size of the rotational excitations exceeds ∆, their propagation is perturbed by the vacancies. The fact that σ 2 r becomes a linear function of time illustrates that the presence of vacancies destroys interferences accelerating diffusion of quantum particles. The interferences are destroyed either because of the incoherent averaging over disorder realizations [35] or because the rotational excitations enter a chaotic scattering regime [36][37][38].
Had the rotational excitations been classical particles, σ 2 r would continue to expand as a linear function of time at all subsequent times. However, we observe that the propagation of the rotational excitations enters a sub-diffusive regime (σ 2 r ∝ t α with α < 1) at long times. This shows that the dynamics of rotational excitations in disordered 2D and 3D lattices of molecules is a combination of disorder-induced localization and diffusion, with the latter significantly -but not entirely -blocked by disorder, even at the concentration of empty lattice sites as high as 85 % in a 3D lattice. The increase of the time required for the formation of the stationary distributions shown in Figure 6 indicates that increasing the disorder strength impedes the diffusion of rotational excitations to the lattice boundaries, hence the dramatically different dependence of τ on the disorder strength for 2D (at low vacancy concentrations) and 3D lattices from that in 1D lattices.

C. Effect of long-range tunnelling
Most models of quantum particles in disordered lattices are based on the nearest neighbour (tight-binding) approximation assuming that the tunnelling amplitudes t n,n in Eq. (6) are non-zero only when n and n are adjacent lattice sites. However, the tunnelling amplitudes (2) responsible for rotational energy transfer between molecules decay as ∝ 1/|n−n | 3 functions of the lattice site separation. This makes the rotational excitations in an ensemble of polar molecules a new platform for the experimental investigation of long-range tunnelling effects on the formation of localized states. The 1/|n − n | 3 dependence of the tunnelling amplitudes is particularly interesting [17]. Beyond allowing for direct transitions between distant lattice sites, it leads to non-analyticity of particle dispersion at low particle velocities [18]. It may also lead to interesting long-range correlations in the localization dynamics of multiple interacting particles [39,40].
The effect of the long-range character of the tunnelling amplitudes t n,n can be -in principle -investigated by comparing the dynamics of rotational excitations in ensembles of polar molecules with those in ensembles of homonuclear diatomic molecules. Ultracold homonuclear molecules have been produced in multiple experiments worldwide [41]. The rotational excitations in an ensemble of non-polar molecules can be resonantly transferred between molecules due to quadrupole -quadrupole interactions, the leading term in the multipole expansion of the interaction between non-polar molecules separated by a large distance. The quadrupole -quadrupole interaction decays as a ∝ 1/R 5 function of the molecule -molecule separation R. Therefore, the rotational excitations in an ensemble of homonuclear molecules trapped in an optical lattice should be expected to undergo quantum walks governed by the Hamiltonian (6) with t n,n ∝ 1/|n − n | 5 .
In this section, we compare the localization dynamics of particles with the tunnelling amplitudes t n,n ∝ 1/|n − n | 3 and t n,n ∝ 1/|n − n | 5 . To remove ambiguity associated with the different magnitudes of the dipole and quadrupole moments, we compare the dynamics of rotational excitations stimulated by the tunnelling amplitudes as given in Eq. (2) and the tunnelling amplitudest n,n = b/|n − n | 5 , where the constant b is chosen such that t n,n = t n,n for |n − n | = 1. In other words, we enhance the quadrupole -quadrupole interaction to be the same as the dipole -dipole interaction between molecules in adjacent lattice sites so that the difference between the calculations witht n,n and t n,n arises solely from the different radial dependence of the tunnelling amplitudes.
The results presented in Figure 9 illustrate that the long-range matrix elements t n,n lead to less localized distributions in all dimensions. The effect of the long-range matrix elements appears to depend both on the strength of the disorder potential and the dimensionality.
To elucidate these dependences, we present in Figure 10 the ratio of the distribution widths computed in the limit of long time with the long-range amplitudes t n,n ∝ 1/|n − n | 3 and the short-range amplitudest n,n ∝ 1/|n − n | 5 . The long-range matrix elements appear to have little effect at low concentrations of empty sites, but increase the distribution widths by a factor of 2 or more at large concentrations of vacancies.
As illustrated in Figures 6 and 8, the dynamics of rotational excitations in disordered 3D lattices is a combination of diffusion and disorder-induced localization. The results presented in Figure 10 indicate that the long-range tunnelling amplitudes t n,n play little role at low disorder strengths in lattices of higher dimensionality, i.e. when the dynamics is expected to be more dominated by diffusion. At the same time, Figure 10 shows that the longrange matrix elements become exceedingly important in the regime where disorder-induced localization is expected to dominate. In order to illustrate this more clearly, we show in  Figure 11). In stark contrast, the distribution widths obtained with the two types of tunnelling amplitudes are dramatically different in the regime of strong localization (lower panel of Figure 11).

IV. CONCLUSION
In this work, we study the dynamics of quantum walk of rotational excitations in disordered 1D, 2D and 3D ensembles of ultracold molecules trapped in optical lattices. The disorder arises from incomplete populations of optical lattices with molecules. We consider a specific experimentally realized system of KRb molecules on an optical lattice with lattice constant a = 532 nm and show that the rotational excitations form localized distributions within t ∼ 1 sec. We also demonstrate that this particular type of disorder requires concentrations of empty lattice sites > 40% in order to localize the rotational excitations in 2D lattices to distributions with σ r ≤ 20 a. For 3D lattices with 55 sites in each dimension (the maximum size of the system we can consider), we observe no localization at vacancy concentrations ≤ 90%, i.e. the rotational excitations generated in the middle of the lattice diffuse to the edges of the lattice. Thus, whether the disorder potential produced by a random distribution of empty lattice sites will be strong enough to induce Anderson localization of rotational excitations in 3D lattices remains an open question.
These results indicate that disorder-induced localization can be studied with rotational excitations as probe quasi-particles, providing coherence can be preserved for longer than 1 second and large ensembles of molecules can be trapped in an optical lattice. The study of Anderson localization in 3D systems will likely require 10 5 or more molecules trapped in an optical lattice, yielding disordered lattices with 100 sparsely populated lattice sites per dimension. The study of Anderson localization in 2D systems can be performed with lattices containing less than 50 sites per dimension, or with less than a total of 10 4 molecules, which is currently experimentally feasible [23]. While the results in this work were obtained for the specific system of KRb molecules on an optical lattice with a = 532 nm, they can be renormalized for other molecular systems by changing the value of the coupling constant in Eq. (2), which controls the time-evolution of the system. We chose to present the results for the specific parameters relevant for KRb molecules in order to provide direct guidance for future experiments. We have confirmed that multiplying the matrix elements t n,n in Eq.
(2) by a factor of 2 accelerates the time evolution of the dynamics but leads to the same results in the limit of long time.
Our calculations also reveal several general features in the dynamics of quantum particles in disordered lattices. In particular, we observe that the diffusion of quantum particles placed in 2D and 3D lattices with disordered tunnelling amplitudes has three distinct time scales. At short times, the diffusion is characterized by the quadratic time dependence of σ 2 r = r 2 − r 2 , characteristic of quantum particles with cosine dispersion. The rise of σ 2 r is suppressed by the disorder at later times. This suppression leads to an extended time interval, where σ 2 r exhibits a linear dependence on time, characteristic of classical diffusion. This interval of time increases with decreasing disorder strength and can be as long as 200 ms for the dynamics of the rotational excitations in a 3D lattice with 85 % of empty sites.
At still longer times, the dependence of σ 2 r on time is sub-diffusive, i.e. slower than linear, in both the 2D and 3D lattices. This suggests that rotational excitations in disordered ensembles of polar molecules can be used to study the crossover from ballistic expansion to classical diffusion, where quantum interferences enhancing diffusion are suppressed. At longer times, the dynamics of rotational excitations can be used to study the crossover from classical diffusion to a sub-diffusive regime, where quantum interferences leading to localization become important.
Since the rotational energy transfer between polar molecules is mediated by matrix elements that decay with the separation between the lattice sites as ∝ 1/|n − n | 3 , it is important to understand the effect of the long-range matrix elements allowing transfer of excitations between distant lattice sites. Our results show that the long-range character of the tunnelling amplitudes in the Hamiltonian (6) has little impact on the dynamics of particles in the diffusive regime (i.e. in 2D and 3D lattices with low concentrations of vacancies) but affects significantly the localization dynamics in lattices with large concentrations of vacancies, enhancing the width of the localized particle distributions in 2D lattices by a factor of 2. In future work, it will be interesting to explore the effect of the long-range tunnelling amplitudes on the qualitative features of diffusion discussed above.
In our previous work [28], we showed that the rotational excitations of molecules on an optical lattice can be coupled to the translational motion of molecules, leading to the formation of polarons with a wide range of tunable properties [29]. We also showed that, if the molecules on an optical lattice are subjected to DC electric fields, the rotational excitations exhibit strong interactions that may -depending on the electric field strengthlead to the formation of quasi-particle bound pairs [42]. In combination with this previous work, the present results suggest an exciting research avenue for the study of quantum diffusion through disordered lattices in the presence of dissipation and the role of inter-      (3); squares -the results computed with the short-range amplitudest n,n ∝ 1/|n − n | 5 .