Three-dimensional localized-delocalized Anderson transition in the time domain
TThree-dimensional localized-delocalized Anderson transition in the time domain
Dominique Delande, Luis Morales-Molina, and Krzysztof Sacha
3, 4 Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS,ENS-PSL Research University, Collège de France, 4 Place Jussieu, 75005 Paris, France Departamento de Física, Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński,ulica Profesora Stanisława Łojasiewicza 11, 30-348 Kraków, Poland Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński,ulica Profesora Stanisława Łojasiewicza 11, 30-348 Kraków, Poland
Systems which can spontaneously reveal periodic evolution are dubbed time crystals. This is inanalogy with space crystals that display periodic behavior in configuration space. While space crys-tals are modeled with the help of space periodic potentials, crystalline phenomena in time can bemodeled by periodically driven systems. Disorder in the periodic driving can lead to Anderson local-ization in time: the probability for detecting a system at a fixed point of configuration space becomesexponentially localized around a certain moment in time. We here show that a three-dimensionalsystem exposed to a properly disordered pseudo-periodic driving may display a localized-delocalizedAnderson transition in the time domain, in strong analogy with the usual three-dimensional An-derson transition in disordered systems. Such a transition could be experimentally observed withultra-cold atomic gases.
PACS numbers: 71.30.+h, 05.30.Rt, 71.23.An, 67.85.-d
The fact that a detector, placed at some position, hasa large probability to click at a certain moment in timewhen a particle is passing nearby is not very surprising.It is more interesting when this localization has universalcharacteristics such as an exponential shape. This is thecase when temporal Anderson localization is induced bya fluctuating driving force.Usual Anderson localization is the configuration spaceexponential localization of eigenstates in the presence ofa spatially disordered potential [1]. It is accompanied bythe inhibition of transport due to destructive interferencebetween different multiple scattering paths. Andersonlocalization may also take place in momentum space –where it is called ”dynamical localization” – e.g. in theso-called kicked rotor, where an effective pseudo-disorderis induced by the classically chaotic dynamics [2–4].The interference between paths scattered by a disorderdepends on their geometrical properties and especiallyon the dimension of a system. According to the scal-ing theory of localization [5], one-dimensional (1D) andtime-reversal invariant spinless 2D systems reveal local-ization regardless how weak the disorder is. In the 3Dcase, the situation is more complicated since the scal-ing theory predicts a second order phase transition —for a fixed disorder strength, all eigenstates of a systemwith energies up to the so-called mobility edge are lo-calized and the other ones are not. How the mobilityedge depends on the disorder has been analyzed in a va-riety of systems [6]. A rotor driven by a quasi-periodicsequence of kicks with d quasi-periods can be mappedon a d -dimensional pseudo-disordered system, allowingfor a simple experimental method for studying Ander-son localization in dimension d. This made it possible toinvestigate theoretically and experimentally the critical behavior in the vicinity of the 3D mobility edge [7, 8].Anderson localization in the time domain can be real-ized in systems that are perturbed by a time fluctuatingforce, provided the latter is repeated periodically with afrequency that is resonant with the unperturbed motionof the system [9, 10]. In classical mechanics, the fluctu-ating force produces a diffusive motion. In the quantumdescription, interference effects cause the system to local-ize. That is, if we put a detector close to any point of thetrajectory, we will observe that the detection probabil-ity is exponentially localized around a certain moment intime. Moreover, this exponential profile comes back ev-ery period of the classical motion. Thus, we deal with asituation analogous to Anderson localization of a particleon a ring (periodic boundary conditions) in the presenceof a disordered time-independent potential. By travelingperiodically around the ring, one observes periodically alocalized density profile.Anderson localization in time belongs to more generalphenomena dubbed time crystals [11, 12]. Time crys-tals are systems that can spontaneously switch to a pe-riodic motion. That is, even if they are prepared in aneigenstate, which possesses continuous time translationalsymmetry, a small perturbation can push them to peri-odic motion. There is a debate in the literature whethercontinuous time translational symmetry can be sponta-neously broken [13–20]. So far it has been shown [21–26],and demonstrated experimentally [27, 28], that sponta-neous breaking of a discrete time translational symmetryto another discrete one is possible. Here, we will notconsider this problem of spontaneous formation of timecrystals. We will model crystalline behavior by period-ically driven systems [9, 10, 29], in analogy with con-densed matter physics where spatially periodic potentials a r X i v : . [ qu a n t - ph ] N ov are used to model space crystals.Time is a single degree of freedom, therefore, we can-not expect multidimensional time crystals. In this Let-ter, we show that time crystal phenomena with proper-ties of multidimensional condensed matter systems canbe observed. More precisely we demonstrate the ana-log in the time domain of the usual three-dimensionallocalized-delocalized Anderson transition [6].Let us consider a particle with a unit mass movingin the 3D space with periodic boundary conditions (3Dtorus) whose position is denoted by three angles: θ , ψ and φ . We assume that the particle is perturbed by atemporally disordered potential, i.e. the Hamiltonian ofthe system reads H = p θ + p ψ + p φ V g ( θ ) g ( ψ ) g ( φ ) f ( t ) f ( t ) f ( t ) , (1)where V is the amplitude of the perturbation. Thetime dependent functions are periodic but between t and t + 2 π/ω i they perform random fluctuations, i.e. f i ( t +2 π/ω i ) = f i ( t ) = (cid:80) k (cid:54) =0 f ( i ) k e ikω i t where f ( i ) k = f ( i ) ∗− k are independent random numbers. We assume that theratios of the frequencies ω i are irrational numbers. Incontrast, g ( x ) is assumed to be a regular function — wechoose g ( x ) = x/π for x ∈ [ − π, π [ , i.e., g ( x ) = (cid:80) n g n e inx where g n = i ( − n πn for n (cid:54) = 0 and g = 0 . Thus, we dealwith a perturbation which behaves regularly in the con-figuration space (for fixed time) but which is disorderedin time. As shown below, in order to observe Ander-son localization in the time domain, it is important thatboth the spatial function g ( x ) and the temporal disorder f i ( t ) contains many Fourier components. For the sake ofsimplicity, we choose the f ( i ) k components so that: | g k f ( i ) k | = 1 √ k π / e − k / (2 k ) (2)with k a free-to-choose parameter and Arg ( f ( i ) k ) (for k > ) are independent random variables chosen uni-formly in the interval [0 , π [ . Such a Gaussian shapemakes the computation of the localization length easyin 1D [10], but any similar shape will lead to a 3D metal-insulator Anderson transition in the time domain. Thisis a robust phenomenon that takes place in 3D as shownbelow. Similarly, any form of g ( x ) with sufficiently many k components can be used.Let us switch to the moving frame where Θ = θ − ω t , Ψ = ψ − ω t and Φ = φ − ω t . In this frame, Θ , Ψ and Φ are slowly varying variables if we choose the conjugatemomenta P Θ = p θ − ω ≈ , P Ψ = p ψ − ω ≈ and P Φ = p φ − ω ≈ . In the secular approximation [30, 31],the dynamics of the slowly varying variables is describedby an effective Hamiltonian obtained by averaging theoriginal Hamiltonian over time [32]: H eff = P + P + P V eff (Θ , Ψ , Φ) , (3) with V eff = V h (Θ) h (Ψ) h (Φ) where h i ( x ) = (cid:88) k (cid:54) =0 g k f ( i ) − k e ikx . (4)In Eq. (3), the constant term ( ω + ω + ω ) / isomitted. To obtain H eff we take advantage of the factthat the ratios of the frequencies ω i are irrational num-bers. The first order of the secular approximation (3)is valid provided the amplitude of the perturbation V is small or ω i ’s are large and fulfill the relations ω > k ( ω + ω ) , i.e. the second order correction terms, V / ( (cid:80) i n i ω i ) e − (cid:80) i n i / k , never suffer from a small de-nominator problem and are negligible [32].The effective potential V eff (Θ , Ψ , Φ) is a product ofthree independent disordered potentials h i along eachdegree of freedom. It can be characterized by its two-point correlation function which is trivially factorized asa product of three times the same correlation functionalong the three directions. In the limit of relatively large k we are interested in, there is a large number (cid:39) k of random contributions in the sum (4); from the cen-tral limit theorem, we deduce that h i ( x ) has a Gaussiandistribution with zero mean. The correlation function iseasily computed and, for large k , it reads h i ( x (cid:48) ) h i ( x (cid:48) + x ) = V exp (cid:18) − x σ (cid:19) , (5)where denotes the averaging over disorder realizations. σ = √ /k is the correlation length of the disorderedpotential.Let us assume, for a moment, that Θ , Φ and Ψ are notlimited to the interval [0 , π [ but extend from minus infin-ity to infinity. V eff (Θ , Ψ , Φ) being a generic 3D randompotential, one expects a localized-delocalized transitionto take place at some value of the energy E c called themobility edge. There are three different energy scales inthe problem: the strength V of the potential, the energy E of the particle and the so-called correlation energy E σ = 1 σ = k . (6) E σ sets the natural energy scale of the problem [34], sothat the ratio E c /E σ depends only on the ratio V /E σ . In 3D, the Anderson transition takes place in theregime of strong disorder, so that no analytic predictionis available and one has to resort to the numerical calcu-lations in order to compute the position of the mobilityedge as well as the localization length below it. We usedthe transfer matrix method described in [35]. To makea long story short, we discretize the configuration spaceon a (sufficiently dense) 3D rectangular grid and recur-sively compute the total transmission of a bar-shapedgrid with length L and square transverse section M × M ,with M (cid:28) L . This system can be viewed as quasi-1Dand is thus Anderson localized: its total transmissiondecays like exp( − L/λ M ) where λ M is the quasi-1D lo-calization length in units of the lattice spacing. In prac-tice, the log of the total transmission is a self-averagingquantity which can be safely computed. λ M depends on M, on the energy and on the disorder strength. Fig-ure 1(a) shows the ratio λ M /M as a function of en-ergy, for various M values, at a fixed disorder strength V = E σ . At low energy, in the localized regime, λ M /M decreases with M and eventually behaves like λ ∞ /M ,with λ ∞ = lim M →∞ λ M the 3D localization length. Incontrast, at high energy, λ M /M increases with M , asignature of the diffusive regime. At the mobility edge, λ M /M is a constant Λ c of order unity, meaning that thequasi-1D localization length is comparable to the trans-verse size of the system, a signature of marginal 3D lo-calization. Thus, the mobility edge can be obtained bylooking at the point where all curves cross in Fig. 1(a),near E c /E σ = 0 . . In order to pinpoint more accuratelythe position of the mobility edge, we use a finite-size scal-ing analysis [35, 36] which gives E c /E σ = 0 . ± . . It also makes it possible to compute the 3D localizationlength below the mobility edge, shown in Fig. 1(b). Infact, this algebraic divergence of the localization lengthnear the critical energy shown in Fig 1(b) is a character-istic feature of the localized-delocalized Anderson transi-tion in 3D. Notice that the diverging localization lengthappears in units of the correlation length of the disor-dered potential. In this regard, since in our model ispossible to decrease the correlation length upon increas-ing k , it makes feasible the observation of the Andersontransition in a system with finite size. That is, any pointin Fig. 1(b) can be realized in our finite system by achoice of sufficiently small σ . Then, regardless how big ξ/σ is, it is always possible to choose such a small corre-lation length of the effective disordered potential that ξ will be smaller than the system size, i.e. ξ (cid:28) π .Coming back to the initial time-dependent drivenHamiltonian (1), observing a temporal dependence withtime crystal properties requires to have a periodic mo-tion localized along the 3 directions, that is a stationaryeigenstate of the effective Hamiltonian (3) with a local-ization length much smaller than the spatial period π. By inspecting the results in Fig.1(b), we chose an ex-emplary value σ = 0 . (corresponding to k = 10 √ . At energy E = − . E σ , the localization length is pre-dicted to be ξ ≈ . σ = 0 . , sufficiently smaller than π to observe good localization properties. We numer-ically diagonalized the Hamiltonian (3) discretized on a100x100x100 grid using the JADAMILU package [37] toobtain few eigenstates with energy close to − . E σ . Thelocalization properties of a typical eigenstate are shown inFigure 2. As expected, they display an overall exponen-tial localization with the expected localization length andwith the large fluctuations typical of eigenstates. Thelower plots show how the probability density for detect-
FIG. 1. Numerical determination of the mobility edge for V = E σ . For a given energy, we compute the localization length λ M of a long bar-shaped grid with square section M × M .(a): Each curve is computed for a single M value from 25 to55 with step 5 (slope increases with M ) and shows λ M /M vs. energy. The various curves cross at the position of themobility edge E c /E σ = 0 . ± . . (b): The localizationlength ξ vs. energy, in the localized regime. It shows analgebraic divergence near the mobility edge, indicated by thered dashed line. ing a particle at a fixed point in the configuration spacechanges as time evolves. It changes periodically withthe maximum value roughly 5 orders of magnitude largerthan the minimum, i.e. it behaves like the probability inthe case of Anderson localization in a space crystal withperiodic boundary conditions. Other eigenstates, eitherat slightly different energy and/or for a different disorderrealization, have similar localization properties.Finally, let us analyze a possible experimental realiza-tion of Anderson localization in the time domain withultra-cold atomic gases. We will focus on the simplestversion when a single frequency driving is applied andonly one spatial degree of freedom is involved. In or-der to realize a system described by the Hamiltonian H = p / m + V g (2 πz/L z ) f ( t ) , which is analogous to(1), we can use a sawtooth shape periodic potential alongthe z direction with spatial period L z [38, 39]. Initially,an ultra-cold atomic cloud should be prepared in a shal-low trap and in the presence of a strong optical latticealong the z axis which has to be periodic with period L z but can have an arbitrary shape. This creates a se-ries of independent slices of the atomic cloud which con-sist of well defined numbers of atoms but do not haveany mutual phase coherence, see Fig. 3(c). Next, theinitial optical lattice and the shallow trapping poten-tials are turned off while the sawtooth potential tem- -2 0 2 Phase ( ω t, ω t, ω t ) -6 -5 -4 -3 -2 -1 P r ob a b ilit y d e n s it y vs. ω t vs. ω t vs. ω t FIG. 2. Spatial probability density for a typical localizedeigenstate of Hamiltonian (3) with correlation length of thedisorder σ = 0 . , for V = E σ and energy E/E σ = − . . The upper color 3D plot shows the disordered yet localizedcharacter of the state below the mobility edge. Lower plotsshow how probability densities at a fixed position in θ , φ or ψ in the laboratory frame (integrated along two remaining direc-tions) evolve in time. The semi-logarithmic scale indicates anapproximate exponential localization. In the rotating frame,the localization length (cid:39) . σ is in good agreement withthe prediction of the transfer matrix calculation, ξ = 4 . σ inFig. 1. It implies that in the laboratory frame the localizationlength in time reads, e.g., σ/ω if the probability density isintegrated over ψ and φ . porally modulated by the f ( t ) function, is turned on.At the same moment, the scattering length of atomsis adjusted to zero by means of a Feshbach resonanceand atoms are kicked so that their average momentumalong the z direction is (cid:104) p (cid:105) = ω mL z / π , see Fig. 3(d).If these conditions are met, the effective Hamiltonian,in the frame moving with the velocity ω L z / π , reads H eff = P / m + V (cid:80) k g k f (1) − k e ik πz/L z and Anderson lo-calization along the z direction can be expected. Thisrequires the localization length ξ ( E ) corresponding to E = (cid:104) p (cid:105) / m to be smaller than L z where (cid:104) p (cid:105) is the θ / 2π -101 g ( θ ) ω t / -10010 f ( t ) (a) (b)(c) (d) z z p z FIG. 3. Experimental proposal. (a) The red dashed line showsthe shape of a sawtooth potential while the black solid lineits approximation built with the first three spatial harmon-ics only. The latter can be created by means of an opticalstanding wave with wavelength λ and its first two harmon-ics. (b) Temporal modulation function f ( t ) that consistsof three harmonics with random phases and with the ampli-tude | g k f (1) k | = √ k for | k | ≤ k = 3 . (c) Schematic plot ofthe initial stage of the experiment: ultra-cold bosonic atomsare prepared in a strong optical lattice and shallow trappingpotentials. We assume that for the amplitude of the latticepotential of the order of E rec (atomic recoil energy), slicesof the atomic cloud are formed that consist of well definednumbers of particles and do not have mutual phase coher-ence. The average kinetic energy along the z direction is (cid:104) p (cid:105) / m ≈ E rec . (d) Final stage of the experiment: for aperturbation amplitude V = 40 E rec , atoms accelerated tothe average momentum (cid:104) p (cid:105) = mω λ/ π will fly over the timemodulated sawtooth potential and do not spread along z dueto the predicted Anderson localization. The predictions arevalid provided ω ≥ E rec / (cid:126) . initial dispersion of momenta of atoms along the z axisin the presence of the strong optical lattice potential atthe beginning of the experiment. An example of exper-imental parameters is given in Fig. 3. It is not neces-sary to create an exact sawtooth periodic potential. Fortime modulation f ( t ) consisting of, e.g., three harmon-ics ( k = 3 ), only the first three spatial harmonics of thesawtooth potential have to be reproduced. The presenceof Anderson localization in time will have remarkable sig-natures in the described experiment. That is, after theturning off the initial optical lattice and trapping poten-tials, atoms expand slowly in the transverse directionsbut the width of the slices along the z direction remainssmaller than L z despite the fact that atoms fly over thetime modulated sawtooth potential. Although this is asimple setup to implement in the lab, our formulation isnot limited to a sawtooth shape for g ( x ) , thus makingpossible the observation of this phenomenon in a moregeneral setting.In conclusion, we have shown that, using a properlydisordered, but pseudo-periodic, temporal driving of a3D system, one can induce a non-trivial Anderson local-ization in the time domain and the localized-delocalizedAnderson transition. This could be observed – in partic-ular, but not exclusively, using cold atoms – through theexistence of periodically evolving localized wavepacketsdisplaying properties similar to those of space crystalswith disorder and with periodic boundary conditions butin the time domain.This work was performed within the Polish-French bi-lateral POLONIUM Grant 33162XA and the FOCUS ac-tion of Faculty of Physics, Astronomy and Applied Com-puter Science of Jagiellonian University. We thank theauthors of the Jadamilu library [37] that we used forlarge scale diagonalizations. This work was granted ac-cess to the HPC resources of TGCC under the allocations2016-057644 and A0020507644 made by GENCI (GrandEquipement National de Calcul Intensif) and to the HPCresources of MesoPSL financed by the Region Ile deFrance and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenirsupervised by the Agence Nationale pour la Recherche.The work was performed with the support of EU viaHorizon2020 FET project QUIC (No. 641122). Sup-port of the National Science Centre, Poland via projectNo.2016/21/B/ST2/01095 (KS) is acknowledged. [1] P. W. Anderson, Phys. Rev. , 1492 (1958).[2] F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sun-daram, and M. G. Raizen, Phys. Rev. Lett. , 4598(1995).[3] S. Fishman, D. R. Grempel, and R. E. Prange, Phys.Rev. Lett. , 509 (1982).[4] G. Casati, I. Guarneri, and D. L. Shepelyansky, Phys.Rev. Lett. , 345 (1989).[5] E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Phys. Rev. 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We here give the details of the derivation of the effec-tive Hamiltonian, Eq. (3) in the Letter, within the firstorder secular approximation and discuss the second ordercontributions [30, 31].We consider the Hamiltonian of a particle in the 3Dspace with periodic boundary conditions with the form H = p θ + p ψ + p φ V g ( θ ) g ( ψ ) g ( φ ) f ( t ) f ( t ) f ( t ) , (7)where V stands for the amplitude of the perturbation,the angles θ , ψ and φ denote the position of the particleon the 3D torus and p θ , p ψ and p φ are the conjugatemomenta. The time dependent functions are periodic, f i ( t + 2 π/ω i ) = f i ( t ) = (cid:88) k f ( i ) k e ikω i t , (8)where f ( i ) k = f ( i ) ∗− k and f ( i )0 = 0 . We assume that theratios of the frequencies ω i are irrational numbers. Thefunction g ( x ) = x/π for x ∈ [ − π, π [ , has the followingFourier expansion: g ( x ) = (cid:88) n g n e inx , (9)where g n = i ( − n πn for n (cid:54) = 0 and g = 0 .We are interested in the resonant motion when θ , ψ and φ are changing with frequencies close to ω , ω and ω , respectively. Let us perform the canonical transfor-mation to the moving frame, Θ = θ − ω t, P Θ = p θ − ω , (10) Ψ = ψ − ω t, P Ψ = p ψ − ω (11) Φ = φ − ω t, P Φ = p φ − ω , (12)that results in H = P + P + P V (cid:88) kmn (cid:88) opr g k g m g n f (1) o f (2) p f (3) r × e i ( k Θ+ m Ψ+ n Φ) e i ( k + o ) ω t e i ( m + p ) ω t e i ( n + r ) ω t , (13) where the constant additional term ( ω + ω + ω ) / hasbeen omitted. The new variables Θ , Ψ and Φ are slowlyvarying quantities if we choose the conjugate momenta P Θ ≈ , P Ψ ≈ and P Φ ≈ , i.e. if we focus on the mo-tion of the particle in the vicinity of a resonant trajectory.Then, the dynamics of the slowly varying variables canbe described by an effective Hamiltonian obtained by av-eraging the original Hamiltonian (13) over time, H eff = P + P + P V (cid:88) kmn g k g m g n f (1) − k f (2) − m f (3) − n e i ( k Θ+ m Ψ+ n Φ) . (14)Equation (14) is identical to Eq. (3) in the Letter. Inthe following we assume that the absolute values of f ( i ) k fulfill | g k f ( i ) − k | = 1 √ k π / e − k / (2 k ) . (15)The effective Hamiltonian (14) is the first order secularapproximation and it constitutes an accurate descriptionof the resonant dynamics of the particle provided thesecond order terms can be neglected [30, 31]. The latterare proportional to V ( kω + mω + nω ) exp (cid:18) − k + m + n k (cid:19) , (16)where k , m and n are non-zero integers. Even if theratios of ω i are irrational numbers, small denominatorscan arise in (16). To avoid it is sufficient to choose: ω > k ( ω + ω ) . (17)Then, the exponential function in (16) kills the secondorder terms whose denominators are small.Thus, when the condition (17) is fulfilled and V /ω goes to zero, the second order contributions become negli-gible and the effective Hamiltonian (14) provides a quan-titative description of the resonant behavior of the sys-tem. Even if one chooses frequencies ω i whose ratiosare rational numbers, the second order terms are stillnegligible if (17) is satisfied, provided one uses f i ( t ) functions with vanishing high order Fourier components f ( i ) k ≡ for | k | > k0