Anderson localization and quantum chaos maps

• Shmuel Fishman, Department of Physics, Technion, Haifa, Israel

Quantum interference suppresses transport in some deterministic systems in the same way as in disordered systems, namely by the mechanism of Anderson Localization.

Introduction

The classical diffusion of particles in disordered solids is suppressed by quantum interference. This is the phenomenon of Anderson localization. Anderson localization has a dynamical counterpart whereby the chaotic classical diffusion of momentum or energy found in such deterministic classical systems as the Chirikov standard map (sometimes called the Chirikov-Taylor map), is suppressed when the same Hamiltonian is quantized. This phenomenon was realized experimentally for cold atoms. Implications for other maps will be discussed as well. The subject is reviewed in great detail in [1] and discussed in books [2,3].

Anderson Localization

Quantum mechanical interference may result in localization of particles by a random potential. Localization may take place even if the energy is higher than all potential barriers. It is most remarkable in one dimension. The transmission of a wave through a one dimensional strip where the potential is random decays exponentially with the length of the strip. This phenomenon is known as Anderson Localization following the 1958 seminal paper by P.W. Anderson [4]. (For a review see for example [5]). In his original paper, Anderson considered the electronic motion in disordered solids and predicted that in some situations transport for such solids will vanish. He proposed a model for the exploration of this phenomenon, known as the Anderson Model. It is a tight-binding model which can be represented in one dimension by the time independent Schrödinger equation: \[\tag{1} Eu_n=\epsilon_n u_n +u_{n+1}+u_{n-1}, \]

where the \( \epsilon_n \) are independent random variables, \( E \) is the energy eigenvalue and \(\{u_n\}\) is the corresponding eigenstate. For this (one dimensional model) all the states are exponentially localized with a characteristic length, called the localization length \( \xi\). Consequently if the system is initially in a state localized in space, only few of the expansion coefficients will be of substantial magnitude. Therefore it will remain localized for arbitrarily long time. Also in two dimensions all the states are localized while in three dimensions states are localized in some energy range and are extended in another energy range. A critical energy separates these regimes. Anderson Localization is a wave effect which is found, for example in classical optics [6].

Figure 1: Typical eigenstates of the standard kicked rotor (open and full circles represent numerical results, dashed lines are introduced to guide the eye) (from [9]).

The Standard Map and the Kicked Rotor

Maps are dynamical systems evolving in discrete time steps. For example the standard map (or Chirikov-Taylor) (see [7,3], or any standard book on chaos) is \[\tag{2} p_{m+1}=p_m+K\sin\theta_m, ~~~~~~~~~~ \theta_{m+1}=\theta_m+p_{m+1} \]

which is defined on the cylinder \( -\infty<p<\infty;~~~0\leq\theta<2\pi\). The map defines a transformation of a point \((\theta_m,p_m)\) where the system is at time \(m\) to the point \((\theta_{m+1},p_{m+1})\) where is at time \(m+1\). The phase space of the system is mixed. In some regions the motion is chaotic and in other regions it is regular. The map is controlled by the stochasticity parameter \(K\). As it increases the area of the chaotic regions increases while the area of the regular ones shrinks. For large \(K\) the motion is similar to diffusion in momentum space \[\tag{3} <(p_N-p_0)^2>=DN \]

for large \(N\), where the average \(<....>\) is an average over the initial conditions and \( D\) is the diffusion coefficient in momentum. The quantum version of the standard map is generated by the Hamiltonian of the kicked-rotor \[\tag{4} H=\frac{\tau}{2}\hat{n}^2+\frac{K}{\tau}\cos\theta\sum_m\delta(t-m) \]

where \(\hat{n}=-i\frac{\partial}{\partial \theta}\) is the angular momentum operator. The units were chosen so that the time between the kicks is unity. The parameter \(\tau\) is proportional to the physical time between the kicks and to Planck's constant \(\hbar\). The Hamilton equations generated by (4) form the standard map (2) with \(p=n\tau\). A quantum map is obtained from the Schrödinger equation with the Hamiltonian generating the classical map, \(H\) of (4) in our case. Since the Hamiltonian is time periodic the motion is determined by the one step evolution operator \(\hat{U}\), defined by \[\tag{5} \hat{U}\psi_t=\psi_{t+1}, \]

where \(\psi_t\) is the wave function of the system at time \( t.\) The eigenvalues of \(\hat{U}\) are \(e^{-i\omega}\) with real \(\omega\) that is called the quasienergy. The corresponding eigenstate is the quasienergy state. The wave functions can be expanded in terms of quasienergy states. Therefore these states play the role of the eigenstates of the Hamiltonian, when it is time independent.

Figure 2: Typical eigenstates of a modified kicked rotor (open and full circles represent numerical results, dashed lines the analytical results for the localization length, if the phases are random) (from [9].

Anderson Localization for the Kicked Rotor

The dynamics are determined by the eigenstates of the evolution operator \(\hat{U}\). Its matrix elements \(U_{n,n'}\) are negligibly small for \(|n-n'|\gg K/\tau\) , while the phases are \(\varphi_n=\frac{1}{2}\tau n^2\). If the phases can be considered random \(\hat{U}\) is a matrix banded around the diagonal and the model is similar to (1) resulting in exponentially localized states, as argued by Prange, Grempel and Fishman [8,9]. Why can \(\varphi_n\) be considered random? These are phases, therefore the functions in question actually depend on \[\tag{6} \alpha_n\equiv\varphi_n (mod2\pi)=\frac{1}{2}\tau n^2 (mod2\pi) \]

that for large \(n\) is a small fraction of a large number (\(\varphi (mod 2\pi)\) is the fraction of \(\varphi\) found by subtracting the largest multiple of \(2\pi\) which is smaller than \(\varphi\) ). The sequence of the \(\alpha_n\) is deterministic but shares some properties with random sequences. Such a sequence is called pseudo-random. It was argued by Prange, Grempel and Fishman [8,9] that the degree of randomness of (6) is sufficient for localization. Moreover they introduced a mapping of (4) to a system like (1). A somewhat different mapping was introduced by Shepelansky [10]. The eigenstates are indeed exponentially localized, typical eigenstates are presented in Fig. 1 (Fig. 10 of [9]). There is a model where \(\cos\theta\) of \(H\) is replaced by another function so that a model of the type of (1) where the localization length \(\xi\) is known analytically, if the sequence \(\alpha_n\) is truly random. In Fig. 2 (Fig. 4 of [9]) eigenstates of such a model are presented. The approximation of truly random \(\alpha_n\) is excellent.

Figure 3: The intensity of atoms as a function of momentum and time (from [11]).

The kicked-rotor was realized experimentally by the group of Raizen [11]. Cold Sodium atoms were periodically kicked and their center of mass momentum was measured. The center of mass motion is modelled by \(H\), where the linear momentum is \(p=\beta+\gamma n\) ( \(\beta\) and \(\gamma\) are constants), the coordinate is \(x=\theta+2\pi n_x\) (\(n_x\) is a constant) and \(K\) is proportional to the intensity of the laser. In spite of the kicking, the spreading in momentum (and the resulting growth of the kinetic energy) stops. This is demonstrated in Fig. 3 (Figure 3 of [11]). The intensity (number of atoms) is plotted as a function of momentum (in units of \(\hbar k_L\) where \(k_L\) is the wave-number of the laser) and the time \(N\)(number of kicks). The distribution saturates at an exponential profile characteristic of Anderson localization. This saturation is obvious from Fig. 4 (Fig. 4 of [11] ). The solid line shows the diffusive growth (3) one finds for the corresponding classical problem . The dashed line is the asymptotic (in time) result of the localization theory with the localization length \(\xi=8.3\) corresponding to the parameters of the experiment. The asymptotic (in time) profile is presented in the inset. The experimental results, shown by the dots, exhibit a clear crossover from the initial classical diffusive behavior to asymptotic quantum behavior predicted by the localization theory. It is a consequence of the exponential decay of the eigenstates presented in Figs. 1 and 2.

Figure 4: The kinetic energy as a function of time (experimental results are represented by dots, classical diffusion by solid line, quantum asymptotic behavior by dashed line). The inset shows the asymptotic profile (from [11]).

Pseudo-randomness

The sequence \(\{\alpha_n\}\) is deterministic but it has some properties of random sequences. Such sequences are called pseudo-random. A natural question is how close a pseudo-random sequence should be to a truly random one so that their localization properties are identical. The sequence \(\{\alpha_n\}\) leads to localization as truly random sequences do, but with some differences [10,12]. In particular for the kicked rotor, in the semiclassical limit, the localization length is given in terms of the parameters of (4) by [10]: \[\tag{7} \xi=\frac{D(K)}{2\tau^2} \]

where \(D\) is the diffusion coefficient (3). The assumption that \(\{\alpha_n\}\) is random leads to different results [10,12]. It turns out that localization requires a very low degree of pseudo-randomness as found from the exploration of several models [13,14,15].

So far it was assumed that \(\tau/\pi\) is a typical irrational number. For rational \(\tau/\pi\) the sequence {\(\{\alpha_n\}\)} is periodic, therefore in this case the quantum motion is typically ballistic [16], as in a periodic solid.

Perturbation theory methods which are used in the analysis of Anderson localization for disordered solids, were introduced in the analysis of (4) in [17]. Here the small parameter is \(\tau/K\). In particular the growth of the kinetic energy with time was computed in [18]. In this calculation some properties of \(\tau\), resulting in deviations of the sequence \(\{\alpha_n\}\) from a truly random one, are exhibited, in agreement with other types of analysis [19].

Anderson Localization for Various Quantum Maps

The most natural extension of the kicked-rotor (4) and the resulting standard map (2) to higher dimensions is by increasing the number of angle variables (and their conjugate momenta). In this case one can explore how the classical diffusion is suppressed by Anderson Localization [20]. It was found indeed that in two dimensions diffusion is suppressed and the localization length depends on parameters as predicted by localization theory.

A more sophisticated extension to a three a dimensional model was introduced by Casati, Guarneri and Shepelyansky [21]. They replace the constant \(K\) of the Hamiltonian (4) by \[\tag{8} K_1(t)=K[1+K'(\cos\omega_2t+\cos\omega_3t)] \]

where \(K\) and \(K'\) are constants, while \(\omega_2\) and \(\omega_3\) are incommensurate frequencies which are incommensurate with \(2\pi\) the frequency of the \(\delta\) kicking. It was demonstrated [21] that the motion of the system driven with the frequencies \((2\pi,\omega_2, \omega_3)\) is similar to one on a three dimensional lattice. An Anderson transition between a regime where the states are localized and one where these are extended was predicted. It is the analog of a metal insulator-transition in solid state physics. Following this proposal, such a transition was observed experimentally for kicked cold Cesium atoms [22]. The model can be extended to an arbitrary dimension varying the number of frequencies in (8).

It was found that ionization of Hydrogen atoms driven by a microwave field is suppressed by quantum interference. With the help of a map which was developed for this exploration (the Kepler map) it was demonstrated that this suppression is a manifestation of Anderson Localization [23].

A complicated interplay between localization and diffusion was found for the Kicked Harper Model [24].

Another interesting extension is the inclusion of noise. Quantum mechanically it has been shown [25] in the context of the original Anderson problem of spatial diffusion on a disordered lattice that even rather small finite temperature leads to decoherence and a consequent suppression of the quantum localization, resulting in restoration of diffusion. The analogous process also occurs in the dynamical context. In particular, it has been shown by Ott, Antonsen and Hanson [26] that small noise ( e.g., created by addition of a small time dependent fluctuation to the kicking strength \(K\)) can lead to restoration of the classical chaos induced energy or momentum diffusion [26,27]. This large effect on the quantum result can occur even for noise levels that are classically inconsequential.

The Maryland Model

If for the Hamiltonian (4) the term \( \tau\hat{n}^2/2\) is replaced by \(\tau\hat{n}\) , the corresponding phase \(\varphi\) is linear in \(n\) and the classical dynamics is integrable [28]. The resulting model is exactly solvable [29]. For typical values of \(\tau\) all the eigenstates are localized. The corresponding tight-binding model of the form (1) is a model where the potential is \(\epsilon_n=\tan(\tau n/2+const)\). Typically its period is incommensurate with the chain. The model is of importance since it is an exactly solvable example of the family of incommensurate models. It helped to improve the mathematical insight for these models (see detailed discussion by Simon [30]). The model can be defined for arbitrary dimension and typically all its eigenstates are localized [31,32,33] in any dimension.

History

Quantization of the standard map (2) leading to the kicked-rotor (4), was first proposed by Casati, Chirikov, Izrailev and Ford [34]. Some indication for localization was provided in this work and it was numerically established later by Chirikov, Izrailev and Shepelyansky [35]. Their preprint reached Edward Ott at the University of Maryland. He presented the result in a very clear way at a seminar during the winter of 1981-82 and considered it a puzzle. At that time the chaos community was not aware of the deep ideas introduced by P.W. Anderson in 1958 [4] which were followed and further developed by Mott, Thouless and other leaders of the condensed matter community. During the seminar of Edward Ott, Richard Prange and Shmuel Fishman realized immediately, thanks to the clear presentation, that Chirikov, Izrailev and Shepelyansky found Anderson localization in their system. In a lunch discussion with Daniel Grempel that followed, it became completely clear that this suppression of transport is indeed Anderson localization. In a short time [8,9,29] as well as other related papers were submitted for publication.

The Maryland model was discovered and solved by Grempel, Fishman and Prange at the University of Maryland [29]. It was dubbed "Maryland Model" by Simon [30].

Comments

• For the quantum mechanical dynamics generated by the Hamiltonian (4), Bourgain proved [36] that for small \(K/\tau\) there is a set of large measure of values of \(\tau\) where localization in momentum takes place.

• Pseudo-randomness of rotational phases \(\varphi_n\) in Eq.(6) is of primary importance. For the Maryland model, where the phases change with level number in a linear way, the exponential localization of eigenstates had been proven for any dimension in [28-33].

• In contrast to Anderson localization in disordered systems with a random potential, the quantum localization of dynamical chaos takes place in deterministic systems without any randomness. Due to that it is often called dynamical localization or Chirikov localization (see e.g. [37]).

References

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12. R. Blumel, S. Fishman, M. Griniasty and U.Smilansky, Localization in the Quantum Description of the Periodically Perturbed Rotor, in"Quantum Chaos and Statistical Nuclear Physics", Proc.of the 2nd International Conference on Quantum Chaos, Curnevaca, Mexico, Edited by T.H.Seligman and H. Nishioka, (Springer-Verlag, Heidelberg, 1986).
13. M. Griniasty and S. Fishman, Localization by Pseudorandom Potentials in One Dimension, Phys. Rev. Lett. 60, 1335 (1988).
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16. F.M.Izrailev and D.L.Shepelyansky, Quantum Resonance for a Rotator in a Nonlinear Periodic Field, Theor. Math. Phys. 43, 553 (1980).
17. C. Tian, A. Kamenev and A. Larkin, Weak Dynamical Localization in Periodically Kicked Cold Atomic Gases, Phys. Rev. Lett. 93, 124101 (2004); Ehrenfest Time in the Weak Dynamical Localization, Phys. Rev. B 72, 045108 (2005).
18. C. Tian and A. Altland, Theory of Localization and Resonance Phenomena in the Quantum Kicked Rotor, New J. Phys. 12 043043 (2010).
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24. R.Artuso, G.Casati, D.L.Shepelyansky, Fractal Spectrum and Anomalous Diffusion in the Kicked Harper Model, Phys. Rev. Lett. 68, 3826 (1992)
25. N.F. Mott and E.A. Davis, "Electronic Processes in Non-Crystaline Materials", (Clarendon Press,Oxford,1971)
26. E. Ott, T.M. Antonsen, Jr. and J.D. Hanson, Effect of Noise on Time-Dependent Quantum Chaos, Phys. Rev. Lett. 53, 2187 (1984).
27. D. Cohen, Quantum Chaos, Dynaical Correlations and the Effect of Noise on Localization, Phys. Rev. A 44, 2292 (1991).
28. M.V. Berry, Incommensurability in an Exactly-Soluble Quantal and Classical Model for a Kicked Rotator, Physica, 10D, 369 (1984).
29. D.R. Grempel, S.Fishman and R.E. Prange, Localization in an Incommensurate Potential: An Exactly Solvable Model, Phys. Rev. Lett. 49, 833 (1982).
30. B.Simon, Almost Periodic Schrödinger Operators IV. The Maryland Model 159, 157 (1985).
31. J. Bellissard, R. Lima and E.Scoppola, Localization in \(\nu\)-Dimensional Incommensurate Structures, Comm. Math. Phys. 88, 465 (1983).
32. A.L.Figotin and L.A.Pastur, An Exactly Solvable model of a Multidimensional Incommensurate Structure, Comm. Math. Phys. 95, 401 (1984);JETP lett. 37, 686 (1983).
33. S. Fishman, D.R. Grempel and R.E. Prange, Localization in a d-Dimensional Incommensurate Structure, Phys. Rev. B 29, 4272 (1984)
34. G. Casati, B.V. Chirikov, F.M. Izrailev and J. Ford, Stochastic Behavior of a Quantum Pendulum Under Periodic Perturbation, in "Stochastic Behaviour in Classical and Quantum Hamiltonian Systems", Vol 93 of lecture Notes in Physics, edited by G. Casati and J.Ford (Springer,Berlin 1979), p 334.
35. B.V. Chirikov, F.M. Izrailev and D.L. Shepelyansky, Dynamical Stochasticity in Classical and Quantum Mechanics, Sov. Sci. Rev. Sec. C2, 209 (1981).
36. J. Bourgain, Estimates on Green's functions, Localization and the Quantum Kicked Rotor Model, Ann. Math. 156, 249 (2002).
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Internal references

• Dima Shepelyansky (2008) Boris Valerianovich Chirikov. Scholarpedia, 3(10):6628.

• Boris Chirikov and Dima Shepelyansky (2008) Chirikov standard map. Scholarpedia, 3(3):3550.

• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Recommended Reading

1. Eric Akkermans and Gilles Montambaux, Mesoscopic Physics of Electrons and Photons, (Cambridge University Press,2007).
2. Yoseph Imry, Introduction to Mesoscopic Physics, (Oxford University Press, 2008).
3. Il'ya M. Lifshits, Sergyi A. Gredeskul and, Leonid A. Pastur, Introduction to the Theory of Disordered Systems, (Wiley-Interscience, 1988).
4. Michael V. Berry in Topics in Nonlinear Dynamics, AIP Conference Proceedings No. 46 (AIP NY 1978).
5. Michael Tabor,"Chaos and Integrability in Nonlinear Dynamics" (Wiley, NY 1989).
6. A.M. Ozorio de Almeida, "Hamiltonian Systems,Chaos and Quantization" (Cambridge Press, Cambridge 1988)

External links

• Workshop "Mathematics and Physics of Anderson localization: 50 Years After", Newton Institute, Cambridge (2008)
• Author web page

See also internal links

Anderson localization, Chirikov standard map, Kicked top, Cold atom experiments in quantum chaos

Sponsored by: Dima Shepelyansky, Laboratoire de Physique Théorique, CNRS, Université Paul Sabatier, Toulouse

Reviewed by: Anonymous

Accepted on: 2010-08-04 12:24:28 GMT