Quantum walk and Anderson localization of rotational excitations in disordered ensembles of polar molecules
QQuantum walk and Anderson localization of rotational excitationsin disordered ensembles of polar molecules
T. Xu and R. V. Krems Department of Chemistry, University of BritishColumbia, Vancouver, BC V6T 1Z1, Canada (Dated: October 10, 2018)
Abstract
We consider the dynamics of rotational excitations placed on a single molecule in spatially dis-ordered 1D, 2D and 3D ensembles of ultracold molecules trapped in optical lattices. The disorderarises from incomplete populations of optical lattices with molecules. This leads to a model cor-responding to a quantum particle with long-range tunnelling amplitudes moving on a lattice withthe same on-site energy but with forbidden access to random sites (vacancies). We examine thetime and length scales of Anderson localization for this type of disorder with realistic experimen-tal parameters in the Hamiltonian. We show that for an experimentally realized system of KRbmolecules on an optical lattice this type of disorder leads to disorder-induced localization in 1Dand 2D systems on a time scale t ∼ a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y . INTRODUCTION Scattering of electrons by impurities in disordered crystals leads to Anderson localization[1, 2], which determines the conductor - insulator transitions in metals [3], quasi-crystals [4]and granular metal films [5], and is associated with many interesting phenomena, such as thequantized phases of the integer Hall effect [2, 6]. If written in a second-quantized form, theHamiltonian of an electron in a disordered lattice can be mapped onto the Hamiltonians for avariety of systems [7]. This mapping can be used to study transport properties of electronsin disordered lattices by examining the properties of other quantum particles and quasi-particles. For example, disordered-induced localization has been studied with microwavesin a tubular waveguide [8, 9], optical photons in opaque media [10, 11] and ultracold atomsin optical lattices [12–15]. Despite these studies, there are still many open questions aboutAnderson localization. For example, the scaling hypothesis providing the relation betweenquantum localization in systems of different dimensionality [16] still awaits experimentalconfirmation. Equally important and widely debated are the effects of inter-particle inter-actions [17] and dissipative forces [18] on localization in far-from-equilibrium systems orthe role of long-range tunnelling amplitudes [19–21] allowing particles to transition betweendistant lattice sites.The development of experimental methods for producing molecules at ultracold temper-atures has created new possibilities for studying disorder-induced localization. As demon-strated in recent experiments [22–26], an ensemble of ultracold KRb diatoms, produced byphotoassociation of ultracold K and Rb atoms and trapped in an optical lattice, forms a ran-dom spatial distribution of KRb molecules in the ro-vibrational ground state. If the trappingfield of the optical lattice has high intensity, the molecular motion is strongly confined andthe spatial distribution is quasi-stationary, eliminating molecule - molecule collisions. Theexcitation of molecules from the rotational ground state to the lowest rotationally excitedstate leads to the formation of delocalized excitations [25, 26], representing the wave packetsof rotational Frenkel excitons [27, 28]. The purpose of the present work is to explore if thesewave packets can be used as probe particles to study dynamics of localization induced bythe disorder potential stemming from the randomness of the molecular positions.The rotational excitations in an ensemble of polar molecules randomly distributed on anoptical lattice possess several unique properties:2
The rotationally excited states of molecules have long radiative lifetimes ( > − • The rotational excitations can be placed on molecules in well defined lattice sites. Thiscan be achieved by applying a spatial gradient of an electric field and a microwavefield pulse resonant with rotational transitions for molecules in the desired part of thelattice [29]. The rotational excitations thus produced form spatially localized wavepackets [28]. Once the gradient of the field is removed, these wave packets can travelthroughout lattice due to resonant energy transfer between molecules, undergoingquantum walk [30, 31] and scattering by empty lattice sites. By shaping the gradientof the field, it may be possible to prepare the excitation wave packets representingsuperpositions of excitations in spatially separated lattice sites, thus enabling thecontrolled study of the role of initial conditions and entanglement between spatiallyseparated particles on the dynamics of quantum walk. • The interactions between the rotational excitations can be tuned by applying a DCelectric field [32]. • The transfer of rotational excitations between molecules is enabled by long-range dipole- dipole interactions, leading to long-range transfer of excitations between distantlattice sites. This can be used to study the effects of long-range tunnelling on disorder-induced localization. • The translational motion of molecules can be controlled by varying the strength ofthe optical lattice potential. As shown in Ref. [33, 34], the translational motion ofmolecules can be coupled to the rotational excitations. This can be used to studydisorder-induced localization and diffusion of quantum particles in the presence ofcontrolled coupling to a dissipative environment. • 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, effectivelyconfining the rotational excitations to one-dimensional (1D) or two-dimensional (2D)disordered lattices [35].These properties make the rotational excitations in spatially disordered ensembles of ultra-cold molecules a unique new platform for the study of quantum walk and disorder-induced3ocalization. In particular, the ability to change the dimensionality of the optical latticepotential opens up a unique possibility to verify the universality of Anderson localization indifferent dimensionalities [16] and the ability to tune the interactions between the rotationalexcitations [27] suggests a new way to study the effects of particle interactions on Andersonlocalization. However, the microscopic Hamiltonian describing rotational excitations in anoptical lattice leads to a specific model corresponding to a quantum particle moving on alattice with the same on-site energy but with forbidden access to random sites, with thetunnelling amplitudes decaying as the inverse cube of the hopping distance. Whether or notsuch a model can be used to observe Anderson localization in finite, or even infinite [36, 37],2D systems and whether or not such a model allows Anderson localization in 3D systems ofany size [38] is – to the best of our knowledge – unknown [39].The specific goal of this work is to consider the following questions:(i) While quantum particles placed in a 2D disordered potential are generally expectedto undergo Anderson localization [16, 36, 37], the localization length is sensitive tothe microscopic details of the particle - disorder interactions and can be very large.Given that the experiments with optical lattices employ finite ensembles of molecules,typically with less than 100 sites per dimension, is the disorder potential for rota-tional excitations in a partially populated optical lattice strong enough to allow forobservations of Anderson localization in 2D systems on the length scale of 100 latticesites?(ii) If the disorder potential is strong enough to induce Anderson localization in finite 2Darrays, what is the minimum concentration of lattice vacancies leading to localizationlengths less than 100 lattice sites?(iii) Quantum particles placed in a 3D disordered potential may or may not undergo An-derson localization [16], depending on the microscopic details of the particle - disorderinteractions. Is the disorder potential for rotational excitations in 3D disordered en-sembles strong enough to induce Anderson localization?(iv) What is the effect of the long-range tunnelling amplitudes on the dynamics in thedisordered system of this specific kind?4v) What is the time scale of the disorder-induced localization, if any, for excitationsplaced in specific lattice sites?If the disorder potential is too weak, the localization dynamics may not be observ-able over experimentally feasible time scales. If the disorder potential is too strong,localized rotational wave packets may remain pinned to their original location and notexhibit any dynamics of quantum walk over experimentally feasible time scales.Questions (i) - (iv) are general for the lattice model with tunnelling amplitudes randomlyomitted and the results pertaining to question (v) can be renormalized for a particularsystem by re-scaling the time dependence of observables. To answer these questions, weconsider the dynamics of rotational excitations placed in individual sites of an optical lat-tice partially populated with KRb molecules. To simulate an experiment with destructivemeasurements, we average the dynamics of rotational energy transfer between molecules inthe disordered ensembles over multiple disorder orientations and examine the time scales ofthe formation of disorder-induced distributions of rotational excitations in lattices of variousdimensionality. We examine the effects of the long-range tunnelling amplitudes in lattices ofvarious dimensionality and discuss the effect of disorder on diffusion of rotational excitationsin lattices of various dimensionality.
II. METHODOLOGY
The Hamiltonian for an ensemble of molecules on an optical lattice of arbitrary dimen-sionality can be written as ˆ H = (cid:88) n ˆ H n + (cid:88) n (cid:88) n (cid:48) ˆ V n , n (cid:48) , (1)where ˆ H n is the Hamiltonian of an isolated molecule placed in lattice site n positionedat r = n a , a is the lattice constant, and ˆ V n , n (cid:48) is the dipole - dipole interaction betweenmolecules in sites n and n (cid:48) . The vector index is n = n x for 1D lattices, n = ( n x , n y ) for 2Dlattices 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 = 2 N + 1 is the number of lattice sites per dimension. We assume thatthe translational motion of molecules can be neglected. This approximation is valid for thetrapping field potential with the trapping frequency exceeding 100 kHz [33].5ollowing the experimental work in Ref. [25], we assume that each molecule is initiallyin the lowest eigenstate of ˆ H n , corresponding to zero rotational angular momentum j = 0 ofthe molecule. Some of the molecules are subsequently excited to an isolated hyperfine statewithin 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 lowestenergy state and this specific excited state for the molecule in site n are hereafter denotedas | g n (cid:105) and | e n (cid:105) . The states | g (cid:105) and | e (cid:105) have the opposite parity.Since the dipole - dipole interaction operator can be written as a sum over productsof rank-1 spherical tensors acting on the subspace of the individual molecules, the matrixelements (cid:104) g n |(cid:104) g n (cid:48) | ˆ V n , n (cid:48) | g n (cid:105)| g n (cid:48) (cid:105) vanish so the molecules are non-interacting when in theground state. However, the dipole - dipole interaction has non-zero matrix elements betweenthe product states | g n (cid:105)| e n (cid:48) (cid:105) and | e n (cid:105)| g n (cid:48) (cid:105) . The matrix elements t n , n (cid:48) = (cid:104) g n |(cid:104) e n (cid:48) | ˆ V n , n (cid:48) | e n (cid:105)| g n (cid:48) (cid:105) (2)stimulate resonant energy transfer of the | g (cid:105) → | e (cid:105) excitation between molecules in sites n and n (cid:48) . For the specific states | g (cid:105) = | j = 0 (cid:105) and | e (cid:105) = | j = 1 , m j = − (cid:105) , the matrix elements(2) have the following form: t n , n (cid:48) = 14 π(cid:15) · d n d n (cid:48) a | n − n (cid:48) | (3 cos θ nn (cid:48) −
1) (3)where d n = d n (cid:48) = 0 .
57 Debye is the permanent dipole moment of KRb molecules [25], θ nn (cid:48) isthe angle between the intermolecular axis joining the molecules in lattice sites n and n (cid:48) andthe z -axis. We assume that θ nn (cid:48) = π/ a = 532 nm [25]. For the specific system considered here,the value of these matrix elements is t n , n (cid:48) = α × (3 cos θ nn (cid:48) − | n − n (cid:48) | . (4)with α = 52 .
12 Hz.The matrix elements (2) also lead to the delocalized character of the rotational excitations.6n general, the | g (cid:105) → | e (cid:105) excitation generates the following many-body excited state: | ψ exc (cid:105) = (cid:88) n C n | e n (cid:105) (cid:89) i (cid:54) = n | g i (cid:105) . (5)In an ideal lattice, the coefficients C n = e ia p · n / √N and | ψ exc (cid:105) ⇒ | ψ exc ( p ) (cid:105) represents arotational Frenkel exciton with wave vector p [40] , i.e. a rotational excitation completelydelocalized over the entire lattice. If the lattice is disordered, the excited state is a localizedcoherent superposition of the Frenkel excitons with different p [28].When the lattice is partially filled with molecules, the excitation energy of the molecularensemble depends on the location of molecules in the excited state. The excited states of themany-body system of molecules in a disordered lattice are the eigenstates of the followingHamiltonian: ˆ H = (cid:88) n w n c † n c n + (cid:88) n (cid:88) n (cid:48) t n , n (cid:48) c † n c n (cid:48) , (6)where the operator c n removes an excitation from site n , w n is the excitation energy of amolecule in site n , if the site is populated, and zero otherwise. The transition amplitudes t n , n (cid:48) are given by Eq. (2) if both of the sites n and n (cid:48) are populated by molecules and zerootherwise. It is this disorder in the transition amplitudes t n , n (cid:48) that localizes the delocalizedexcitations.In order to compute the time-evolution of the rotational excitations, we diagonalize thematrix of the Hamiltonian (6) in the site-representation basis | n (cid:105) ≡ c † n | (cid:105) , where the vacuumstate is | (cid:105) = (cid:89) m | g m (cid:105) , (7)with both m and n running only over the lattice sites populated with molecules.The eigenstates of the full Hamiltonian (6), | λ (cid:105) = (cid:88) n C λ n | n (cid:105) , (8)7re then used to compute the time evolution of the excitation wave packet as follows | ψ ( t ) (cid:105) = (cid:88) λ C λ e − iE λ t/ (cid:126) | λ (cid:105) = (cid:88) n f n ( t ) | n (cid:105) , (9)where E λ are the eigenvalues of the Hamiltonian (6), the coefficients f n ( t ) = (cid:88) λ C λ n C λ e − iE λ t/ (cid:126) , (10)and the coefficients C λ are the projections of the excitation wave packet at t = 0 onto theeigenstates of the Hamiltonian, C λ = (cid:104) λ | ψ ( t = 0) (cid:105) . (11)The quantities | f n ( t ) | are averaged over multiple calculations with different realizations ofdisorder (i.e. different random distributions of empty lattice sites).As will be demonstrated in the following section, the rotational excitations scattered bythe empty lattice sites form peaked distributions. In this article, we will use two measuresof distributions: the distribution width L and the standard deviation σ r . The distributionwidth L is defined as the length of the lattice (for a 1D lattice), the radius of a circle (fora 2D lattice) or the radius of a sphere (for a 3D lattice) containing 90 % of the excitationprobability. The standard deviation is defined as σ r = (cid:112) (cid:104) r (cid:105) − (cid:104) r (cid:105) , where r = x fora 1D lattice, r = x + y for a 2D lattice and r = x + y + z for a 3D lattice. Thedistribution width characterizes the physical spread of the rotational excitation over themolecular ensemble. However, this quantity may be significantly affected by fluctuationsdue to specific disorder realizations. The standard deviation depends on the shape of thedistribution but is less affected by the fluctuations. We will use the standard deviation inorder to characterize non-ambiguously the dynamical properties such as the time dependenceof the distributions. III. QUANTUM WALK OF ROTATIONAL EXCITATIONS
As originally proposed by DeMille [29], ultracold molecules trapped in an optical latticecan be selectively excited by applying a gradient of a DC electric field that separates therotational energy levels of molecules in different lattice sites by a different magnitude. This8ethod can, in principle, be used to excite rotationally a single molecule in a specific latticesite. If the gradient of the field is removed, the rotational excitation thus generated maybe transferred to molecules in other lattice sites by resonant energy transfer. The dynamicsof this energy transfer can be probed by applying another gradient of the DC electric fieldat time t and detecting either the excited molecules or molecules remaining in the groundstate using resonantly-enhanced multiphoton ionization (REMPI). The REMPI detectionis a destructive measurement. If the optical lattice is partially populated with randomlydistributed molecules, repeating the experiment multiple times to determine the dependenceof rotational energy transfer on t is equivalent to averaging over different realizations ofdisorder (i.e. different distributions of empty lattice sites).In order to simulate the outcome of such measurements, we consider an optical latticepartially populated with randomly distributed KRb molecules. We compute the time evolu-tion of a rotational excitation placed in a single lattice site at t = 0 and average the resultsat each t over multiple random distributions of empty lattice sites with a fixed concentra-tion of vacancies. Figures 1, 2 and 3 illustrate the characteristic distributions of rotationalexcitations as a result of quantum walk of a single rotational excitation initially placed inthe centre of a 1D, 2D (square) or 3D (cubic) lattice, respectively. In the following sec-tions, we explore the detailed dynamics of how these distributions are formed and the roleof long-range tunnelling effects in determining the quantum walk dynamics. A. Localization dynamics in 1D lattice
A quantum particle placed in a 1D disordered lattice must undergo Anderson localization[1]. 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 distribu-tions shown in Figure 1. These distributions characterize the probability of molecules inthe corresponding lattice sites to be in the rotationally excited state. Figure 1 illustratesthat rotational excitations in an optical lattice partially populated with molecules behave asquantum particles in a disordered lattice. It is important to note that the distributions pre-sented in Figure 1 are time-independent. Although the rotational excitation initially placedin a specific lattice site represents a wave packet that, given sufficient time, can potentiallyexplore the entire lattice and exhibit revivals, the averaging over disorder realizations leads9o stationary distributions in the limit of long time. In order to illustrate this, we presentin 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 forthe parameters representing rotational excitations in an ensemble of KRb molecules, theaveraged excitation probability distributions change most significantly at t between zeroand 100 - 300 ms. The distribution widths exhibit a temporary peak at short times. Thisoscillation survives the averaging over disorder realizations and is present in the results fordifferent concentrations of vacancies. Since the distribution width presented in Figure 4 ismuch smaller than the lattice size (1001 × a ), this short-time oscillation cannot be due toreflection from the lattice boundaries and is likely the result of constructive interference dueto 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 stationarydistributions is determined by the strength of the disorder potential, i.e. the concentrationof lattice vacancies. This is illustrated more clearly in the upper panel of Figure 5 showingthe time dependence of the standard deviations σ r for different concentrations of vacancies.Figure 5 demonstrates that the rotational excitations in a 1D lattice of KRb moleculesreach the time-independent distributions in less than one second, provided the vacancyconcentration is > τ requiredfor a rotational excitation initially placed in the center of a lattice to form a time-independentdistribution as a function of vacancy concentration in lattices of different dimensionality.Interestingly, Figure 6 reveals that the dependence of τ on the vacancy concentration isqualitatively different for lattices of different dimensionality. In 1D lattices, τ is a monoton-ically decaying function of the vacancy concentration. This illustrates that increasing thedisorder of 1D lattices impedes the dynamics of quantum walk leading to narrower distribu-tions of the rotational excitation (cf., Figure 1) and decreases the time required to reach thestationary distributions (less time since the quantum walker has less space to explore). Thecalculations shown in Figure 6 have two implications: (i) the time evolution of the rotationalexcitations in 1D disordered lattices can be used as a probe of the disorder strength; (ii)the mechanisms for the formation of the distributions shown in Figures 1 - 3 are different10or lattices of different dimensionality. In the following section, we correlate the rising orfalling dependence of τ on the concentration of vacancies with the regime of classical diffu-sion or sub-diffusion, arguing that the sign of the gradient of τ can be used as a signatureof sub-diffusion stimulated by Anderson localization. B. Localization and diffusion in 2D and 3D lattices
As can be seen from Figures 2 and 3 and the comparison of the different curves in Figures5 and 6, the dynamics of rotational energy transfer between molecules trapped in 2D and3D lattices is quantitatively and qualitatively different from the localization dynamics in 1Dlattices. It is apparent that the formation of localized distributions requires higher concen-trations of vacancies and the approach to time-independent distributions takes longer timein lattices of higher dimensionality. In particular, Figure 6 reveals that the dependence ofthe time τ required to reach a stationary distribution on the vacancy concentration is quali-tatively different for lattices of different dimensionality. In 1D lattices, τ is a monotonicallydecaying function of the vacancy concentration. In 2D lattices, τ exhibits a maximum as afunction of the vacancy concentration. In 3D lattices, τ increases monotonically with thedisorder strength.In order to elucidate the dynamics of quantum walk in higher dimensions and the impli-cations of Figure 6, we present in Figure 7 the time dependence of the distribution widths L computed for a rotational excitation in 2D and 3D lattices of varying size with varyingconcentrations of vacant sites. The results for the 2D lattices illustrate that at low con-centration of vacancies (20 %) corresponding to the range, where the dependence of τ onthe disorder strength rises, the rotational excitation width approaches the size of the latticein the limit of long time. This shows that the localization length of the excitation for thisconcentration of vacancies is greater than the size of the lattice. When the concentrationof empty sites is 70 %, the rotational excitation in a 2D lattice is well localized and thedifferent curves shown in Figure 7 converge to the same value in the limit of long time. Thislocalization regime corresponds to the range in Figure 6, where τ decreases with the disorderstrength.Since the experiments with ultracold molecules on optical lattices employ finite samplesof molecules, typically ∼
100 per dimension [25], it is important to find the threshold11oncentration of vacancies that leads to localized states in 2D lattices with dimension < × L ( t → ∞ ) < τ on the vacancyconcentration exhibits a maximum at the concentration of vacancies between 40% and 50%.We conclude that the rotational excitations can be localized in 2D disordered lattices ofmolecules with 100 ×
100 lattice sites, provided the concentration of vacancies is > τ rises monotonically for 3D lattices. This indicates that therotational excitations do not exhibit Anderson localization on the length scale considered.To prove this, we compare the dynamics of the spatial expansion of the rotational excitationsin 3D disordered lattices with classical diffusion.The diffusion of quantum particles in disordered lattices can be examined for an alterna-tive signature of Anderson localization. A quantum particle with cosine dispersion placedin an individual site of an ideal lattice expands ballistically to form a distribution with thestandard deviation σ r = (cid:104) r (cid:105) − (cid:104) r (cid:105) ∝ t [41, 42]. In contrast, the diffusion of a classi-cal particle in Brownian motion is characterized by σ r ∝ t . In the presence of Andersonlocalization, the diffusive dynamics must be suppressed.The time-dependence of σ r can thus be used as a quantitative measure of quantuminterference, constructive or destructive, on particle propagation in disordered lattices.The upper panels of Figure 8 (2D lattices) and Figure 9 (3D lattices) show the timedependence of σ r computed for 2D and 3D lattices with various concentrations of emptylattice sites. As expected, we find that σ r ∝ t for ideal lattices, both in 2D and 3D.However, the time-dependence of σ r is dramatically modified by the presence of vacancies.In particular, Figure 8 illustrates that the rotational excitation in disordered lattices expandsballistically at short times with σ r ∝ t and exhibits the dependence σ r ∝ t , characteristic ofthe diffusive regime, at later times. That σ r becomes a linear function of time illustrates thatthe presence of vacancies destroys interferences accelerating diffusion of quantum particles1243–46]. At even later times ( t >
300 ms), the dependence of σ r on time departs fromlinearity and the propagation of the rotational excitations enters a sub-diffusive regime( σ r ∝ t γ with γ < σ r from linearity is due to disorder or due to the finite size effects. In order todo this, we first consider the ballistic expansion of the rotational excitations in ideal latticesof varying size (upper right panels of Figures 8 and 9). We observe that, for each lattice size, σ r increases as a quadratic function of time at t < t , until the reflection from the boundariesslows down the ballistic expansion. We denote by t the time, at which the dependence of σ r on time in an ideal lattice departs from the quadratic function due to the reflection fromthe boundaries. The insets of Figures 8 and 9 show the dependence of t on the latticesize. In the presence of disorder, the expansion is significantly slower and the reflection fromthe boundaries must occur at times later than t . Therefore, we can use the time t as thelower limit for the onset of the boundary effects and conclude that the dynamics, whetherin disordered or ideal lattices, is not affected by the boundary effects at t < t .The lower panel of Figure 8 shows that the dynamics of rotational excitations in 2Dlattices with the disorder concentration >
50% enters the sub-diffusive regime at times t < t . This proves that the transition to the sub-diffusive regimes at these concentrationof vacancies is due to disorder-induced localization, and not due to the boundary effects.For the disorder concentration 20%, we observe that the departure of σ r from linearityoccurs at t > t . These observations are consistent with the results of Figures 6 and 7.We thus conclude that the increase of the time required for the formation of the stationarydistributions shown in Figure 6 reflects that increasing the disorder strength slows down thediffusion of rotational excitations to the lattice boundaries, hence the dramatically differentdependence of τ on the disorder strength for 2D (at low vacancy concentrations) and 3Dlattices from that in 1D lattices.For 3D lattices, the transition to the sub-diffusive regimes always occurs at t > t ,indicating (but not proving) that the departure of σ r from linearity is due to the finite sizeof the system, and not the scattering by disorder. The lower panel of Figure 9 shows theresults for an extreme case of a 3D lattice with 90 % of the lattice sites vacant. To prove13hat the disorder does not lead to the sub-diffusive regime in 3D lattices, we perform thecalculations illustrated in the lower panel of Figure 9 for different lattice sizes. For eachlattice size, we determine the time t , at which the expansion of σ r undergoes the transitionto the sub-diffusive regime. We find that the dependence of t on the lattice size is a linearfunction of the lattice size for extended 3D lattices (31 – 55 sites), as expected for a particlein the diffusive regime. This proves that there is no Anderson localization in 3D lattices onthe length scale of 55 lattice sites for the disordered system considered here. C. Effect of long-range tunnelling
Most models of quantum particles in disordered lattices are based on the nearest neigh-bour (tight-binding) approximation assuming that the tunnelling amplitudes t n , n (cid:48) in Eq. (6)are non-zero only when n and n (cid:48) are adjacent lattice sites. However, the tunnelling ampli-tudes (2) responsible for rotational energy transfer between molecules decay as ∝ / | n − n (cid:48) | functions of the lattice site separation. This makes the rotational excitations in an ensembleof polar molecules a new platform for the experimental investigation of long-range tunnellingeffects on the formation of localized states. The 1 / | n − n (cid:48) | dependence of the tunnellingamplitudes is particularly interesting [19]. Beyond allowing for direct transitions betweendistant lattice sites, it leads to non-analyticity of particle dispersion at low particle velocities[20]. It may also lead to interesting long-range correlations in the localization dynamics ofmultiple interacting particles [47, 48].The effect of the long-range character of the tunnelling amplitudes t n , n (cid:48) can be – inprinciple – investigated by comparing the dynamics of rotational excitations in ensemblesof polar molecules with those in ensembles of homonuclear diatomic molecules. Ultracoldhomonuclear molecules have been produced in multiple experiments worldwide [49]. Therotational excitations in an ensemble of non-polar molecules can be resonantly transferredbetween molecules due to quadrupole - quadrupole interactions, the leading term in themultipole expansion of the interaction between non-polar molecules separated by a largedistance. The quadrupole - quadrupole interaction decays as a ∝ /R function of themolecule - molecule separation R . Therefore, the rotational excitations in an ensemble ofhomonuclear molecules trapped in an optical lattice should be expected to undergo quantumwalks governed by the Hamiltonian (6) with t n , n (cid:48) ∝ / | n − n (cid:48) | .14n this section, we compare the localization dynamics of particles with the tunnellingamplitudes t n , n (cid:48) ∝ / | n − n (cid:48) | and t n , n (cid:48) ∝ / | n − n (cid:48) | . To remove ambiguity associated withthe different magnitudes of the dipole and quadrupole moments, we compare the dynamicsof rotational excitations stimulated by the tunnelling amplitudes as given in Eq. (2) andthe tunnelling amplitudes ˜ t n , n (cid:48) = b/ | n − n (cid:48) | , where the constant b is chosen such that˜ t n , n (cid:48) = t n , n (cid:48) for | n − n (cid:48) | = 1. In other words, we enhance the quadrupole - quadrupoleinteraction to be the same as the dipole - dipole interaction between molecules in adjacentlattice sites so that the difference between the calculations with ˜ t n , n (cid:48) and t n , n (cid:48) arises solelyfrom the different radial dependence of the tunnelling amplitudes.The results presented in Figure 10 illustrate that the long-range matrix elements t n , n (cid:48) leadto less localized distributions in all dimensions. The effect of the long-range matrix elementsappears to depend both on the strength of the disorder potential and the dimensionality.To elucidate these dependencies, we present in Figure 11 the ratio of the distribution widthscomputed in the limit of long time with the long-range amplitudes t n , n (cid:48) ∝ / | n − n (cid:48) | andthe short-range amplitudes ˜ t n , n (cid:48) ∝ / | n − n (cid:48) | . The long-range matrix elements appear tohave little effect at low concentrations of empty sites, but increase the distribution widthsby a factor of 2 or more at large concentrations of vacancies.The results presented in Figure 11 indicate that the long-range tunnelling amplitudes t n , n (cid:48) play little role at low disorder strengths in lattices of higher dimensionality, i.e. whenthe dynamics is expected to be more dominated by diffusion. At the same time, Figure11 shows that the long-range matrix elements become exceedingly important in the regimewhere disorder-induced localization is expected to dominate. In order to illustrate this moreclearly, we show in Figure 12 the distribution widths L computed as functions of time for 2Dlattices with two concentrations of vacancies: 20 % corresponding to the regime dominatedby diffusion (cf. Figures 6 and 8) and 70 % corresponding to the regime, where the rotationalexcitations in 2D lattices are well localized. The results show that the long-range tunnellingamplitudes insignificantly accelerate the expansion dynamics of the rotational excitation inthe diffusive regime, enhancing the distribution widths at short times by about 20 % or less(upper panel of Figure 12). In stark contrast, the distribution widths obtained with the twotypes of tunnelling amplitudes are dramatically different in the regime of strong localization(lower panel of Figure 12). 15 V. CONCLUSION
In this work, we study the dynamics of quantum walk of rotational excitations in finitedisordered 1D, 2D and 3D ensembles of ultracold molecules trapped in optical lattices. Thedisorder arises from incomplete populations of optical lattices with molecules. Thus, therotational excitations travelling between molecules in a partially populated optical latticeare described by the Hamiltonian (6), which can be obtained from the tight-binding latticemodel by randomly omitting some of the tunnelling amplitudes and extending the remainingtunnelling amplitudes to decay as ∝ α/ | n − n (cid:48) | . While the dynamics of particles described bysuch a Hamiltonian should be expected to exhibit the general features of quantum particlesin a disordered potential, whether or not such a model can be used to observe Andersonlocalization in finite 2D systems and whether or not such a model can be used to observeAnderson localization in 3D systems (finite or infinite), depends on the microscopic detailsof particle - disorder interactions. We note that there are only two free parameters in thisHamiltonian: the magnitude of the constant α in Eq. (4) and the concentration of emptylattice sites. While the magnitude of α determines the time scales of quantum walk andlocalization dynamics, the localization length in 1D and 2D systems and the presence orabsence of Anderson localization in 3D systems is entirely determined by the concentrationof vacancies.In order to guide the ongoing experiments with ultracold molecules, we chose the value of α to correspond to a specific experimentally realized system of KRb molecules on an opticallattice with lattice constant a = 532 nm. We have confirmed that increasing the value of α accelerates the time evolution of the dynamics but leads to the same results in the limit oflong time. The qualitative features of the time-dependent results and the results in the limitof long time are, therefore, general for any system described by the Hamiltonian consideredhere. The time-dependent results can be renormalized for other systems by changing thevalue of the coupling constant in Eq. (2), which controls the time-evolution of the system.The main results of this work can be summarized as following: • For the specific system of 1D arrays of KRb molecules on an optical lattice with latticeconstant a = 532 nm, the rotational excitations placed in individual lattice sites formlocalized distributions within t ∼ < a . This indicates that Anderson localiza-16ion can be studied with rotational excitations in 1D arrays of molecules, providingcoherence can be preserved for longer than 1 second. • In 2D, this particular type of disorder requires concentrations of empty lattice sites >
50% in order to localize the rotational excitations within 100 ×
100 sites. • For 3D disordered arrays with 55 sites in each dimension, we observe no localizationat vacancy concentrations ≤
90 %, i.e. the rotational excitations generated in themiddle of the lattice diffuse to the edges of the lattice as classical particles. Thevacancy concentration 90 % is near the percolation threshold for a 3D network of sites[25]. We note that the results of Ref. [38] indicate that the model for the particle withnearest-neighbour hopping on a lattice with random on-site energies and substitutionaldisorder does allow quantum localization in 3D. Figure 5 of Ref. [38] suggests thatquantum localization is present in a system with substitutional disorder even in theabsence of disorder in the on-site energies. Our results thus indicate (but not yet prove)that the interval of quantum localization in Figure 5 of Ref. [38] between the regimes ofdiffusion and classical localization may be reduced to zero due to long-range tunnelling.This raises a general question whether quantum particles with long-range tunnellingcan undergo quantum localization in 3D lattices with substitutional disorder. • Our results show that the long-range character of the tunnelling amplitudes in theHamiltonian (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 significantlythe localization dynamics in lattices with large concentrations of vacancies, enhancingthe width of the localized particle distributions in 2D lattices by a factor of 2. • Our results show that the diffusive vs localization regime can be identified by measuringthe dependence of the minimum time τ (cf. Figure 6) required for a particle placed ina specific lattice site to form a time-independent averaged distribution as a functionof vacancy concentration. • Our calculations illustrate that the diffusion of rotational excitations placed in 2Dlattices with disordered tunnelling amplitudes has three distinct time scales, whichcan be varied by changing the concentration of empty lattice sites. At short times,the dynamics is characterized by the quadratic time dependence of σ r = (cid:104) r (cid:105) − (cid:104) r (cid:105) ,17haracteristic of quantum particles with cosine dispersion. The rise of σ r is suppressedby the disorder at later times. This suppression leads to an extended time interval,where σ r exhibits a linear dependence on time, characteristic of classical diffusion. Thisinterval of time increases with decreasing disorder strength and can be as long as ∼ σ r on time is sub-diffusive, i.e. slower than linear. This suggeststhat rotational excitations in 2D disordered ensembles of polar molecules can be usedto study the crossover from ballistic expansion to classical diffusion, where quantuminterferences enhancing diffusion are suppressed. At longer times, the dynamics ofrotational excitations in 2D lattices can be used to study the crossover from classicaldiffusion to a sub-diffusive regime, where quantum interferences leading to localizationbecome important.In our previous work [33], we showed that the rotational excitations of molecules onan optical lattice can be coupled to the translational motion of molecules, leading to theformation of polarons with a wide range of tunable properties [34]. We also showed that,if the molecules on an optical lattice are subjected to DC electric fields, the rotationalexcitations exhibit strong interactions that may – depending on the electric field strength –lead to the formation of quasi-particle bound pairs [32]. In combination with this previouswork, the present results suggest an exciting research avenue for the study of quantumdiffusion through disordered lattices in the presence of dissipation and the role of inter-particle interactions on Anderson localization. Acknowledgment
We acknowledge useful discussions with Joshua Cantin, Marina Litinskaya, EvgenyShapiro, John Sous and Ping Xiang. The work was supported by NSERC of Canada.18 − − − E xc i t a t i onp r obab ili t y (a) (b) (c) − −
250 0 250 500
Site index − − − E xc i t a t i onp r obab ili t y (d) − −
250 0 250 500
Site index(e) − −
250 0 250 500
Site index(f)
FIG. 1: Averaged rotational excitation probability distributions formed at t = 2 s by a singlerotational excitation placed at t = 0 in site n = 0 of a 1D lattice of KRb molecules with randomlydistributed empty sites. The concentration of vacancies is zero (a), 1 % (b), 10 % (c), 20% (d),50% (e) and 70% (f). The results are averaged over 100 realizations of disorder. − S i t e i nde x (a) × (f) × (b) × (c) × − − S i t e i nde x (d) × (e) × − − − E xc i t a t i onp r obab ili t y − −
10 0 10 20
Site index 85%10% 20% − −
10 0 10 20
Site index − − − E xc i t a t i onp r obab ili t y − −
10 0 10 20
Site index 70%
FIG. 2: Averaged rotational excitation probability distributions formed at t = 5 s by a singlerotational excitation placed at t = 0 in site n x = 0 , n y = 0 of a 2D lattice of KRb with a totalof 51 ×
51 sites containing a random distribution of empty sites. The concentration of vacanciesis zero (a), 10 % (b), 20 % (c), 50% (d), 70% (e) and 85% (f). The results are averaged over100 realizations of disorder. For better visualization, the probability values in the two-dimensionalplots are magnified by the indicated factor. − − − x direction − − − y direction − − − E xc i t a t i onp r obab ili t y − − z direction − − − − − − − x directiony direction − − z direction − − − − x directiony direction − − z direction FIG. 3: Averaged rotational excitation probability distributions formed at t = 5 s by a singlerotational excitation placed at t = 0 in site n x = 0 , n y = 0 , n z = 0 of a 3D lattice of KRb moleculeswith a total of 31 × ×
31 sites containing a random distribution of empty sites. The concentrationof vacancies is 50% (left panels), 70% (middle panels) and 85% (right panels). The results areaveraged over 300 realizations of disorder. The upper panels show the part of the distributions for x ≤ y ≥
0. For better visualization, the probability values in the three-dimensional plots aremagnified by a factor of 27 (left), 10 (middle), and 2.5 (right). −
250 0 250 500 site − − − E xc i t a t i onp r obab ili t y . s . s . s . s . . . . . time (s) L ( a ) FIG. 4: Upper panel: Time dependence of the rotational excitation probability distributionsformed from one rotational excitation placed at t = 0 in the lattice site n = 0 of a 1D latticeof KRb molecules with 20 % of vacancies. Lower panel: Time dependence of the distributionwidth L (in units of the lattice constant a ) defined as the range of the lattice containing 90 %of the rotational excitation probability: squares – vacancy concentration 10 %; circles – vacancyconcentration 20 %. The results at each time are averaged over 1000 random realizations of thevacancy distributions. time (s) σ r ( a ) time (s) σ r ( a ) time (s) σ r ( a ) FIG. 5: Time dependence of the standard deviations σ r of the rotational excitation probabilitydistributions formed from one rotational excitation placed at t = 0 in the centre of a lattice partiallypopulated with molecules. Upper panel: 1D lattice with 1001 sites; middle panel: 2D square latticewith 51 ×
51 lattice sites; lower panel: 3D cubic lattice with 31 × ×
31 lattice sites. The resultsat each time are averaged over >
100 random realizations of disorder. The different sets of datacorrespond to different concentrations of empty lattice sites. vacancy concentration (%) t i m e ( s )
1D 2D3D
FIG. 6: Minimum time τ required for a single rotational excitation to form a time-independentaveraged distribution as a function of vacancy concentration in 1D (circles), 2D (squares) and 3D(triangles) disordered lattices as a function of vacancy concentration. The results at each vacancyconcentration are averaged over >
100 realizations of disorder. time (s) L ( a ) × × × time (s) L ( a ) × × × time (s) L ( a ) × × × × × × FIG. 7: Time dependence of the width L (in units of the lattice constant a ) of the averagedrotational excitation distributions formed by one rotational excitation placed at t = 0 in the centreof 2D and 3D lattices with different size. The upper and the lower panels show that the excitationsdiffuse to the edges of the lattice, while the middle plane illustrates that the excitation is localizedwithin the lattice and is not affected by the lattice boundaries. . . . . . time (s) σ r ( a ) . . . . . . time (s) σ r ( a )
51 101 151 201 size (n) t i m e ( m s ) − − time (s) σ r ( a ) t FIG. 8: Time dependence of σ r = (cid:104) r (cid:105) − (cid:104) r (cid:105) for a rotational excitation initially placed in themiddle of a 2D lattice. Upper left panel: circles – ideal lattice; squares – lattice with 20 % ofvacancies; triangles – lattice with 50 % of vacancies. Upper right panel: ideal lattice with size 51 ×
51 (diamonds); 101 ×
101 (circles); 151 ×
151 (triangles); 201 ×
201 (squares). Inset: Latticesize dependence of time t marking the deviation of the wave packet dynamics from that in anideal infinite lattice due to the boundary effects. Lower panel: 2D lattice with 20 % of vacancies(squares); 50 % of vacancies (circles); 70 % of vacancies (triangles). The symbols represent thenumerical calculations; the full curves are the analytical fits σ r = Dt ; the dashed curves are thelinear fits σ r = bt + c . The size of the lattice for the results in the upper left panel and the lowerpanel is 101 × . . . . time (s) σ r ( a ) .
00 0 .
01 0 .
02 0 .
03 0 .
04 0 . time (s) σ r ( a )
21 25 31 35 size (n) . . . . t i m e ( m s ) − − time (s) σ r ( a ) t t FIG. 9: Time dependence of σ r = (cid:104) r (cid:105) − (cid:104) r (cid:105) for a rotational excitation initially placed in themiddle of a 3D lattice. Upper left panel: 3D ideal lattice (circles), disordered lattice with 50 %of vacancies (squares) and disordered lattice with 85 % of vacancies with size 31 × ×
31 sites.Upper right panel: 3D ideal lattice with size 21 × ×
21 (squares); 25 × ×
25 (triangles); 31 × ×
31 (circles); 35 × ×
35 (diamonds). Inset: Lattice size dependence of time t marking thedeviation of the wave packet dynamics from that in an ideal infinitely lattice due to the boundaryeffects. Lower panel: 3D disordered lattice of size 55 × ×
55 with 90 % of vacancies. The symbolsrepresent the numerical calculations; the full curves are the analytical fits σ r = Dt ; the dashedcurves are the linear fits σ r = bt + c . −
20 0 20 40
Site index − − − − E xc i t a t i onp r obab ili t y − −
20 0 20 40
Site index1D: 50% − −
13 0 13 25
Site index − − − E xc i t a t i onp r obab ili t y − −
13 0 13 25
Site index2D: 70% − − − Site index − − − E xc i t a t i onp r obab ili t y − − − Site index3D: 70%
FIG. 10: Averaged rotational excitation probability distributions formed at t = 2 s (for 1Dlattices) and t = 5 s (for 2D and 3D lattices) by a single rotational excitation placed at t = 0 inthe center of the lattices. The solid curves are the results of the calculation with the tunnellingamplitudes t n , n (cid:48) as defined in Eq. (2). The dashed curves are the result of the calculation with thetunnelling amplitudes ˜ t n , n (cid:48) ∝ / | n − n (cid:48) | . The middle and lower panels show the cross sectionsof the 2D and 3D distributions along the x -axis. The vertical dotted lines show the distributionwidths containing 99 % of the rotational excitation. The dimensions of the lattices are 1001 sitesfor 1D; 51 ×
51 sites for 2D; and 31 × ×
31 sites for 3D. vacancy concentration (%) . . . . . . L S / L L FIG. 11: The ratio of the widths of the rotational excitation distributions L S /L L . The distri-bution widths L L are computed with the long-range amplitudes t n , n (cid:48) as defined in Eq. (2). Thedistribution widths L S are computed with the short-range amplitudes ˜ t n , n (cid:48) ∝ / | n − n (cid:48) | . Theresults at each vacancy concentration are averaged over >
100 realizations of disorder. . . . . . time (s) L ( a ) Long rangeShort range time (s) L ( a ) Long rangeShort range
FIG. 12: Time dependence of the width L (in units of the lattice constant a ) of the averagedrotational excitation distributions formed by one rotational excitation placed at t = 0 in the centreof 2D lattices with different concentrations of empty lattice sites: circles – the results computedwith the long-range amplitudes t n , n (cid:48) as defined in Eqs. (2) and (3); squares – the results computedwith the short-range amplitudes ˜ t n , n (cid:48) ∝ / | n − n (cid:48) | .
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