Dynamical Anderson transition in one-dimensional periodically kicked incommensurate lattices
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Dynamical Anderson transition in one-dimensional periodically kickedincommensurate lattices
Pinquan Qin, Chuanhao Yin, and Shu Chen
1, 2, ∗ Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China
We study the dynamical localization transition in a one-dimensional periodically kicked incom-mensurate lattice, which is created by perturbing a primary optical lattice periodically with a pulsedweaker incommensurate lattice. The diffusion of wave packets in the pulsed optical lattice exhibitseither extended or localized behaviors, which can be well characterized by the mean square displace-ment and the spatial correlation function. We show that the dynamical localization transition isrelevant to both the strength of incommensurate potential and the kicked period, and the transitionpoint can be revealed by the information entropy of eigenfunctions of the Floquet propagator.
PACS numbers: 03.75.Lm, 72.15.Rn, 05.30.Rt
I. INTRODUCTION
As a fundamental phenomenon of quantum systems inthe presence of disorder, Anderson localization has beenfound in a broad range of physical systems beyond thescope of traditional condensed matter physics , includ-ing light waves in photonic lattices and atomic matterwaves in a one-dimensional (1D) disordered or quasi-periodic potential . Particularly, for a Bose-Einsteincondensate (BEC) trapped in a 1D quasi-periodic po-tential, it has been demonstrated that a transition froman extended state to an exponentially localized state ex-ists with the change of the disorder strength . Althoughmost of studies on the Anderson localization focused onstatic disordered systems, the dynamic localization prob-lem, which was originally put forward in the study of pe-riodically kicked quantum rotors , has also attractedmuch attention recently due to experimental realizationsof the quantum kicked rotor in trapped cold atom sys-tems interacting with pulsed standing wave of light andthe observation of Anderson localization in the kickedsystem .As no external random element is introduced, the dy-namic localization in the kicked rotor can be viewed as ananalog of 1D Anderson localization in momentum spaceby mapping the system onto a quasi-random 1D Ander-son model . The effective randomness in the kicked ro-tor is rooted in mechanisms of incommensurability in-duced by the periodic driving, and consequently the lo-calization for the kicked rotor occurs in momentum space,instead of real space as in the usual Anderson model.An interesting issue arose here is to study the interplayof periodic driving and disorder, which is not yet ad-dressed in the previous study of static disorder systemsand kicked rotor systems. To this end, we study thedynamic localization in a 1D optical lattice perturbedby an additional pulsed incommensurate lattice. Differ-ent from previous works , the disorder induced by theapplied incommensurate potential is periodically added,and the system can be described by a periodically kicked Aubry-André (AA) model. For the static AA model ,its eigenstates are either extended or localized and a lo-calization transition occurs by increasing the strength ofincommensurate potential , which has been experi-mentally verified in a bichromatic optical lattice by ob-serving the expansion dynamics of a trapped noninter-acting BEC . While 1D static incommensurate opticallattices have been well studied , less attention hasbeen paid on the pulsed incommensurate optical lattices.In this work, we study the dynamical localization tran-sition in the periodically kicked incommensurate latticeand find the dynamics is not solely determined by thestrength of incommensurate potential, but also relevantto the driven frequency of the kicked potential. The tun-ability of the incommensurate optical lattices makesit feasible to experimentally study the dynamical local-ization transition through the diffusion of wave packetsin the pulsed 1D incommensurate optical lattice. II. MODEL WITH PERIODICALLY DRIVENINCOMMENSURATE POTENTIAL
We consider the model with periodically driven incom-mensurate potentials described by the following Hamil-tonian: H = X i [( − J ˆ c † i ˆ c i +1 + H.c. ) + X n δ ( t − nT ) V i ˆ n i ] , (1)where ˆ n i = ˆ c † i ˆ c i is the particle number operator and ˆ c † i ( ˆ c i ) the creation (annihilation) operator. Here J is thenearest-neighbor hopping amplitude and the incommen-surate potential V i = λ cos(2 πiα ) varies at each lattice site with α being an irrationalnumber and λ the strength of the incommensurate po-tential. In contrast to the AA model described by H = P i [( − J ˆ c † i ˆ c i +1 + H.c. ) + λ AA cos(2 πiα )ˆ n i ] , the on-site incommensurate potential in Eq.(1) is periodicallyadded with a pulsed period T . Because of this resem-blance, we will refer to systems described by Eq.(1) asthe periodically kicked AA model.Experimentally, the AA model can be realized bysuperposing two optical lattices with incommensuratefrequency . Similarly, the periodically kicked AA modelmay be realized by superimposing two optical lattices ofthe form V ( x ) = V ( x ) + V ( x ) X n δ ( t − nT ) (2)with V ( x ) = s E R sin ( k x ) and V ( x ) = s E R T sin ( k x ) , where k i = 2 π/λ i are the latticewave-numbers and s i are the heights of the two latticesin units of their recoil energies E R i = h / (2 mλ i ) . Thepotential V ( x ) is used to create a primary lattice, thatis weakly perturbed by adding V ( x ) periodically whentime equals multiples of the kicked period. We note thatthe periodically kicked AA model is also related to thekicked Harper model and thus our scheme in terms ofperiodically added incommensurate optical lattices alsoprovides a possible physical realization of kicked Harpermodel. III. DYNAMIC EVOLUTION AND DYNAMICANDERSON TRANSITION
The dynamical evolution of the periodicallykicked system is determined by the Floquet uni-tary propagator over one period, which can be writtenas U ( T,
0) = exp ( − iH T ) exp (cid:16) − i P Lj V j ˆ c † j ˆ c j (cid:17) , where H = − P j (cid:16) ˆ c † j ˆ c j +1 + H.c. (cid:17) and L is the lattice size. Forconvenience, we have set ~ = 1 and J = 1 as the unit ofthe energy. Given an initial state | ψ (0) i = P Li =1 C i | i i at t = 0 , the evolution state after one kicked period is givenby | ψ ( T ) i = U ( T, | ψ (0) i , where | i i = ˆ c † i | i representsthe state with a particle located in the i -th site. Toget the distribution function of the evolution state,we need calculate the matrix element of the Floquetpropagator h i | U ( T, | j i . Representing | φ µ i = P i C µi | i i as the µ th eigenvector of H with the single particleeigenenergy E µ , i.e., H | φ µ i = E µ | φ µ i , we can calculatethe matrix element of the Floquet propagator via theexpression of h i | U ( T, | j i = P µ C µi C µ ∗ j e − i ( E µ T + V j ) .By applying the Floquet propagator repeatedly,the state after N periods can be written as | ψ ( N T ) i = [ U ( T )] N | ψ (0) i = P Li =1 C i ( N T ) | i i . Here U ( T ) = U ( T, and we have used the relation U ( T,
0) = U (2 T, T ) = · · · = U ( nT, ( n − T ) .For convenience, we take the initial state as | ψ (0) i = | L/ i , i.e., with the initial state located in the centerof the lattice, and then study the expansion dynamicsof the initial state in the pulsed incommensurate poten-tial. To give a concrete example, in the following cal-culation we take α = ( √ − / and focus our study
100 200 300 400 500 600 700 800 9000.10.20.30.40.5 | C i | T=0.3100 200 300 400 500 600 700 800 9000.20.4 | C i | T=0.490 100 200 300 400 500 600 700 800 90000.010.020.03 | C i | T=0.50 100 200 300 400 500 600 700 800 90000.0050.010.0150.02 | C i | T=0.51100 200 300 400 500 600 700 800 900246 x 10 −3 i | C i | T=0.7
FIG. 1: The probability distribution of the state after N =10 periods in the periodically kicked AA model with λ = 1 .
100 200 300 400 500 600 700 800 9002468 x 10 −3 | C i | λ =0.010 100 200 300 400 500 600 700 800 90000.0050.010.0150.02 | C i | λ =0.19100 200 300 400 500 600 700 800 9000.050.10.150.2 | C i | λ =0.2100 200 300 400 500 600 700 800 9000.10.20.3 | C i | λ =0.21100 200 300 400 500 600 700 800 9000.20.4 i | C i | λ =0.31 FIG. 2: The probability distribution of the state after N =10 periods in the periodically kicked AA model with T = 0 . . on the high-frequency regime with /T > . It is knownthat the expansion dynamics on a static incommensuratelattice is only determined by the strength of incommen-surate potentials, i.e., the evolution of the initial stateexhibits quite different behaviors in the delocalization orlocalization regime . However, for the periodically kickedsystem, the expansion dynamics is determined by boththe strength of incommensurate potentials and the drivenfrequency. To see it clearly, we first consider periodicallykicked systems with the strength of the incommensuratepotential fixed and variable driven frequencies. Fixingthe strength of the incommensurate potential at λ = 1 ,we show distributions of expansion states after N = 10 pulsed periods for systems with different driven periods T = 0 . , . , . , . and . , respectively, in Fig.1. Wecan find that the final evolution state is still localizedaround the initial position when the driven period of theperiodically kicked potential is smaller than a threshold,i.e., T < . . On the other hand, the final state expandsto the whole lattice when the driven period is larger thana threshold.Next we consider systems with the driven period ofthe periodically kicked potential fixed and study theevolution dynamics for systems with different potentialstrengths. In Fig.2, we show distributions of expansionstates after N = 10 pulsed periods with the driven pe-riod fixed at T = 0 . for systems with different potentialstrengths. Our results clearly indicate that the evolu-tion state is localized for λ > . , whereas it is extendedwhen λ < . . Results shown in Fig.1 and Fig.2 indicatethat the dynamic evolution of the periodically kicked sys-tems is relevant to both the strength of incommensuratepotentials and the driven frequency. The dynamic local-ization transition is determined by the ratio of λ and T ,i.e., the evolution state is either localized or extended for λ/T > or λ/T < .To see how the wave packet spreads as a function oftime, we calculate the mean square displacement whichis defined as σ ( t ) ≡ L X i =1 ( i − L/ | C i ( t ) | . In general, during the expansion process, the meansquare displacement increases as the power law of thetime given by σ ( t ) ∼ t γ . The parameter γ takes dif-ferent values for the expansion in different lattices, forexample, γ = 2 in uniform lattices; γ = 0 in disorderedlattices. While γ = 2 and γ = 0 correspond to bal-listic diffusion and localization, respectively, the super-diffusion ( < γ < ) and sub-diffusion ( < γ < ) canoccur in quasi-periodic lattices. In Ref. , the quan-tum hyper-diffusion ( γ > ) was also discovered. For thekicked driven AA model, one can expect that the diffu-sion process is quite different for λ/T > or λ/T < .To see it clearly, we calculate the mean square displace-ment as a function of time in units of the driven periodwith λ fixed for different periods T . In Fig.3, we showdistributions of the mean square displacement σ ( τ ) withthe strength of the periodically kicked potential fixed at λ = 1 . for systems with L = 900 and different drivenperiods T = 0 . , . , . and . , respectively. For con-venience, we have defined τ = t/T . It is clear that thetime-dependent mean square displacement displays dif-ferent behaviors for T > . or T < . . While themean square displacement shows a power-law increasefor T = 0 . and T = 0 . , it oscillates around a givenvalue after some expansion time and has zero power-lawindex for T = 0 . and T = 0 . . As shown in Fig.3, thelong-time power-law increase of σ ( τ ) can be approxi-mately described by σ ( τ ) ∝ τ . for T = 0 . and σ ( τ ) ∝ τ . for T = 0 . , respectively. These power-lawindexes indicate that the dynamical expansion is a super-diffusion process, which is in contrast to the localization −1 τ τ τ σ ( τ ) T=0.8T=0.65T=0.55T=0.4
FIG. 3: Time dependence of σ ( τ ) in the periodically kickedAA model with λ = 1 . and L = 900 . The dash line representsa power-law fitting, which given σ ( τ ) ∼ τ . with T = 0 . , σ ( τ ) ∼ τ . with T = 0 . . −10 −9 −8 −7 −6 −5 −4 −3 x G ( x ) T=0.8T=0.41T=0.39T=0.3100 200 300 40010 −12 −10 −8 −6 −4 x G ( x ) T=0.39T=0.3
FIG. 4: The correlation function as a function of x with λ =0 . , L = 2000 , N = 1 . × and various T . The dash lineindicates a power-law fit. In the inset, the dash line indicatesan exponential-law fit. process with zero power-law index for the expansion with T = 0 . and . . The property of the mean squaredisplacement also indicates the occurrence of dynamicallocalization transition when λ/T exceeds a threshold.The dynamical localization can be also revealedby the correlation function defined as, G ( x, t ) ≡ L − P Li (cid:12)(cid:12)(cid:12)D ψ ( t ) (cid:12)(cid:12)(cid:12) c † i c i + x (cid:12)(cid:12)(cid:12) ψ ( t ) E(cid:12)(cid:12)(cid:12) = L − P Li | C ∗ i ( t ) C i + x ( t ) | ,where t = N T . Fixing the strength of the incommen-surate potential at λ = 0 . , we show distributions ofthe correlation functions after N = 1 . × pulsedperiods for systems with different driven periods T =0 . , . , . , . , respectively, in Fig.4. Our results in-dicate that the correlation function exhibits a power-law decay when the driven period of the periodicallykicked potential is larger than a threshold, for examples, G ( x ) ∝ x − . for T = 0 . and G ( x ) ∝ x − . for T =0 . . On the other hand, the correlation function has anexponential-law decay when the driven period is smallerthan the threshold, for examples, G ( x ) ∝ e − . x for T = 0 . and G ( x ) ∝ e − . x for T = 0 . as shown inthe inset of Fig.4. The exponential-law decay of the spa-tial correlation function is the characteristic of the systemin a dynamical localized state.We have demonstrated that the extended or localizedproperty of the dynamic evolution state can be well char-acterized by the mean square displacement and the spa-tial correlation function of the evolution state. More-over, we find that the eigenfunction of the Floquet uni-tary propagator can be also used to determine the tran-sition point from the dynamical extended state to local-ized state, which is irrelevant to the choice of the ini-tial state. Given that | ψ η i is the eigenstate of the Flo-quet propagator U ( T ) with the Floquet energy E η , i.e., U ( T ) | ψ η i = e − iE η T | ψ η i , in the basis of | i i , we can rep-resent | ψ η i = P Li =1 C i ( E η ) | i i . Then one can introducethe information entropy defined as S infη ≡ − L X i =1 | C i ( E η ) | ln | C i ( E η ) | . The information entropy takes it’s minimum S infη = 0 ,whenever the state is localized in a single site, while ittakes it’s maximum S infη = ln( L ) , when the state is com-pletely extended with the wave function probability am-plitudes given by | C i ( E η ) | = 1 / √ L .Fixing the strength of the incommensurate potentialat λ = 1 , we show the mean information entropy of theFloquet unitary propagator versus the driven period T inFig.5, where the mean information entropy is defined as S inf ≡ L − P Lη =1 S infη . It shows that the mean informa-tion entropy increases from a tiny value to a finite largevalue with the increase of the pulsed period T , which in-dicates the wave function of the periodically kicked AAmodel undergoing a translation from localized state toextended state. In the up inset of Fig.5, we show thederivative of the mean information entropy as a functionof T for systems with different potential strengths. Itturns out that the extremum of the derivative appears at T = 0 . for λ = 0 . , T = 0 . for λ = 1 . and T = 0 . for λ = 1 . . Similarly, in down inset of Fig.5, the deriva-tive of the mean information entropy as a function of λ for systems with different pulsed periods is displayed,with extremum of the derivative located at λ = 0 . for T = 0 . , λ = 0 . for T = 0 . and λ = 1 . for T = 0 . .It is clear that the extremum of the derivative of meaninformation entropy appears at λ/T = 2 for differentsystems, corresponding to the transition point from the T S i n f λ d S i n f / d λ d S i n f / d T FIG. 5: The mean information entropy versus T for the sys-tem with λ = 1 and L = 1500 . The left up inset showsthe derivative of the mean information entropy versus T with λ = 0 . (left plot); λ = 1 . (middle plot); λ = 1 . (right plot).The right down inset shows the derivative of the mean infor-mation entropy versus λ with T = 0 . (left plot); T = 0 . (middle plot); T = 0 . (right plot). dynamical localization to delocalization state.Our numerical results indicate that, in the high-frequency regime, the dynamical localization transitionpoint of the periodically kicked Aubry-André model islocated at λ/T = 2 . To understand this explicitly, we ex-plore the effective Hamiltonian of the periodically kickedAubry-André model in the high-frequency regime. Theeffective Hamiltonian H eff can be obtained from the Flo-quet unitary propagator by the relation U ( T ) = exp ( − iH eff T ) . As displayed in the appendix, the effective Hamiltonianis derived by using the Baker-Campbell-Hausdorff for-mula. In the high frequency and weak disorder limit with /T ≫ and λ ≪ , the effective Hamiltonian takes theform of H eff = − X i (cid:16) ˆ c † i ˆ c i +1 + H.c. (cid:17) + λ/T X i cos(2 πiα )ˆ n i by omitting high-order terms which contain commuta-tors. This effective Hamiltonian is just the static AAmodel with scaled potential strength λ/T , which indi-cates the localization transition point given by λ/T = 2 .However, in the low frequency regime, the high-orderterms can’t be omitted, and one can not expect thatthe dynamical localization transition can be describedin the scheme of effective AA model. To see it clearly, wealso calculate the mean information entropy in a largerparameter region, which is displayed in Fig.6 with themean information entropy as a function of the strengthof incommensurate potential and the kicked period. It’sshown that there is a sharp change across the line of FIG. 6: (Color online) The mean information entropy versusboth λ and T for the system with L = 900 . λ/T = 2 in the high frequency region, similar to the spe-cific case displayed in Fig.5. However, as shown in Fig.6,the mean information entropy displays a more compli-cated distribution pattern when the system deviates thehigh frequency region. Consequently, the dynamic lo-calization transition occurring at λ/T = 2 breaks downwhen /T < .In order to connect to the potential experimental re-alization, we refer our system to the parameter in Ref. in which K in a incommensurate lattices with heightof the primary lattice s = 5 in units of the recoilenergy E R = 4785 h Hz . Thus the hopping magni-tude J = 323 . h Hz . Correspondingly, the period T in unit of ~ /J is of the order of − s . Therefore thepulsed period T ∼ µ s − µ s in typical kicked atomexperiments is in the high-frequency regime dis-cussed in the current work, which makes it possible tostudy the phenomena of the dynamical localization tran-sition. IV. SUMMARY
In summary, we have revealed the dynamical Ander-son localization transition in a 1D periodically kicked in-commensurate optical lattice by studying the diffusion ofwave packets. The dynamical evolution of wave packetsindicates that the dynamical state is either extended orlocalized, depending on both the strength of incommen-surate potential and the kicked period. We characterizethe dynamical transition from various aspects by calcu-lating the mean square displacement, the spatial correla-tion function and the information entropy of eigenfunc-tions of the Floquet propagator. These quantities all in-dicate the dynamical localization transition occurring at λ/T = 2 in the high frequency regime, which can be alsointerpreted from the effective Hamiltonian of the system.Our observations and theoretical analysis should stimu-late experimental studies of the phenomena of dynamicallocalization transition in the pulsed incommensurate op-tical lattices.
Acknowledgment
This work has been supported by National Program forBasic Research of MOST, by NSF of China under GrantsNo. 11374354, No. 11174360, and No. 11121063, and bythe Strategic Priority Research Program of the ChineseAcademy of Sciences under Grant No. XDB07000000.
Appendix A: Derivation of effective Hamiltonian ofperiodically kicked Aubry-André model
In Sec.III, the Floquet unitary propagator of the peri-odically kicked Aubry-André model is written as a prod-uct of two exponential operators. From this propagator,we can derive the effective Hamiltonian of the kicked sys-tem, namely U ( T ) = exp ( − iH T ) exp (cid:16) − iλ ˆ V (cid:17) = exp ( − iH eff T ) (A1)where ˆ V = P Lj cos(2 πiα )ˆ c † j ˆ c j . In the following analysis,we shall use the Baker-Campbell-Hausdorff (BCH) for-mula which determines ˆ Z such that e ˆ A e ˆ B = e ˆ Z in thefollowing way e ˆ A e ˆ B = exp ˆ A + ˆ B + 12 h ˆ A, ˆ B i + 112 h ˆ A, h ˆ A, ˆ B ii + 112 hh ˆ A, ˆ B i , ˆ B i + · · · ! (A2)Using this formula to Eq.(A1), we derive the effectiveHamiltonian as H eff = H + λT ˆ V − i λ h H , ˆ V i − T λ h H , h H , ˆ V ii − λ hh H , ˆ V i , ˆ V i + · · · . (A3)From the above expression, it is obvious that the effectiveHamiltonian can be simplified as H eff = H + λT ˆ V in thelimit of /T ≫ and λ ≪ , which is just the AA modelwith J = 1 and λ AA = λ/T . However, one must considerthe high-order terms when the system is not in the high-frequency regime. ∗ Electronic address: [email protected] P. W. Anderson, Phys. Rev. , 1492 (1958). F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008). E. Abrahams,
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