Theory of localization and resonance phenomena in the quantum kicked rotor
We present an analytic theory of quantum interference and Anderson localization in the quantum kicked rotor (QKR). The behavior of the system is known to depend sensitively on the value of its effective Planck’s constant . We here show that for rational values of , it bears similarity to a disordered metallic ring of circumference and threaded by an Aharonov-Bohm flux. Building on that correspondence, we obtain quantitative results for the time–dependent behavior of the QKR kinetic energy, (this is an observable which sensitively probes the system’s localization properties). For values of smaller than the localization length , we obtain scaling , where is the quasi–energy level spacing on the ring. This scaling is indicative of a long time dynamics that is neither localized nor diffusive. For larger values , the functions saturates (up to exponentially small corrections ), thus reflecting essentially localized behavior.
pacs:05.45.Mt, 71.23.-k, 71.55.Jv
The kicked rotor, or standard map Chirikov59 () is one of the most prominent model systems of nonlinear dynamics. In spite of its very simple construction – the rotor Hamiltonian depends on just a single parameter, the dimensionless kicking strength, – it shows complex physical behavior. Specifically, the classical rotor undergoes a transition from integrable to chaotic dynamics as the kicking strength is increased. Quantization in the chaotic regime leads to a quantum system that bears strong similarity to a disordered multi-channel quantum wire. Much like in these systems, the dynamics of the quantum kicked rotor (QKR) is governed by mechanisms of quantum interference and localization Casati79 (); Chirikov81 (). The applied interest in researching these structures has increased since the mid nineties Raizen00 () when concrete realizations of the rotor in atom optics settings became an option Raizen95 ().
The analogies to condensed matter systems notwithstanding, there are also a number of important differences. The QKR is not a genuinely disordered system. Rather, its effective stochasticity roots in mechanisms of incommensurability. The differences from generic disorder manifest themselves in anomalies showing as Planck’s constant assumes peculiar “resonant” values Casati79 (); Izrailev79 (); Chang85 (); Izrailev90 (); Casati00 (). (In this paper, we will not discuss the role of classical anomalies, such as accelerator mode formation Zaslavsky97 () at special values of .) At these values, the system becomes effectively periodic (in the space of angular momentum quantum numbers) and generally escapes localization. Various aspects of QKR resonances have been discussed before in terms of general observations, phenomenological reasoning and direct diagonalization (see Ref. Izrailev90 () for review.) However, a microscopic theory of the interplay of resonances and localization has not yet been formulated. The construction of such a theory is the subject of the present paper.
Below, we will introduce an analytic and parametrically controlled theory of localization phenomena in the QKR both on and off resonant values of Planck’s constant, . It turns out that at resonant values, , with coprime numbers, localization phenomena in the QKR can be described in terms of a field theory defined on a ring subject to an Aharonov-Bohm flux (the latter implementing the Bloch phases pervasive in the physics of periodically extended systems.) We will apply perturbative and non–perturbative methods to analyze this field theory in terms of different observables. Emphasis will be put on the discussion of the time dependent increase in the rotor’s kinetic energy, an observable that carries immediate information on the transport characteristics of the system.
Before turning to a qualitative discussion of results, it is worthwhile to put the present theory into the context of earlier work. The connections to disordered systems were first discussed in Ref. Fishman83 (). That work introduced a mapping of the QKR Hamiltonian onto an effectively disordered tight binding system. However, the long-ranged correlated ’disorder’ inherent to that model made it difficult to treat analytically. A different approach was introduced in Ref. Altland93 (), where a diagrammatic approach conceptually similar to the impurity diagram language of disordered systems was introduced. In this work, perturbative quantum interference processes (“weak localization”) were discussed in terms of low energy fluctuations in angular momentum space (for a refined implementation of the diagrammatic approach, see Ref. Tian05 ().) These results inspired a subsequent field theoretical study Zirnbauer96a () in which the large scale dynamics of the quantum kicked rotor was mapped onto a one-dimensional supersymmetric nonlinear –model (otherwise known as the effective field theory of Anderson localization in disordered quasi-one-dimensional wires Efetov97 ().)
None of these works payed attention to the resonance phenomena at rational , a lack of resolution that has prompted some criticism Casati98 (); Casati00 (). Specifically, it has been argued that the insensitivity of earlier work to the algebraic characteristics of might be indicative of ad hoc assumptions hidden in the construction. One of the motivations behind the present work is to show that these objections have been ill founded. Rather, we will see that the field theory below – by and large a refined variant of the earlier theory Zirnbauer96a () – is capable of describing the system from large scales down to the fine structure of individual quasi-energy levels.
Nonetheless, it is worthwhile to point out that our theory is far from ’mathematically rigorous’. Rather, it is based on a mapping of QKR correlation functions onto a functional integral (exact) followed by an identification of field configurations of least action, plus an integration over fluctuations. The last two steps are approximate, if parametrically controlled (by a small parameter .) In this sense, our theory below falls into the general framework of semiclassical approaches to nonlinear systems. The semiclassical limit enables us to explicitly describe system observables which more rigorous lines of reasoning Jitomirskaya03 () can only characterize in implicit terms. We also emphasize that no averaging procedures other than a global average over the quasi-energy spectrum will be performed throughout.
The rest of this paper is organized as follows: we start with a qualitative discussion of the system and a summary of our main results (Section II). In Section III we introduce the functional integral approach to the problem, which will be reduced to an effective semiclassical action in Section IV. The low energy physics of this action will then be discussed in Sections V (off-resonance) and VI (on resonance). We summarize in Section VII. Some technical details are deferred to Appendices A-C.
Ii Qualitative discussions
The QKR is a periodically driven (“kicked”) quantum particle moving on a circle. In units where the particle mass and the circular radius are set to unity, its Hamiltonian reads
where and are the period and the kicking strength, respectively. The angular operator, , and the angular momentum operator, obey the canonical quantization condition where . In rescaled dimensionless time, , the dynamics of the system is determined by two parameters: the effective Planck’s constant and the classical nonlinear parameter . (In the chosen units, both parameters are dimensionless.)
At integer times , the solution of the Schrödinger equation can be expressed as , where is the Floquet operator, i.e. the unitary operator governing the time evolution during one elementary time step:
where we have introduced as a rescaled angular momentum operator with integer–valued spectrum quantum.
For Planck’s constants commensurable with , , with coprime , the Floquet operator is invariant under the shift , or
where is the translation operator by steps, on eigenstates . This discrete translational symmetry implies that the (quasi)energy states of the Floquet operator can be expanded in a basis of Bloch states,
where and . The infinite extension of Bloch states in angular momentum space means the absence of localization at these “resonant values” of . Casati79 (); Izrailev90 (); Chang85 (); Izrailev79 (); Casati00 () (However, for periodicity intervals much larger than the intrinsic localization length of the system, the wave function amplitudes at the boundaries of the “unit cell” are exponentially small, which means that the system behaves effectively localized.) Below, we will explore the interplay of localization and Bloch periodicity in some detail. Our discussion will include the case of irrational as a limit of co–prime values with diverging . We will assume that , which means that the angular momentum unit cell is larger than the length scales at which diffusive dynamics begins to form.
In dirty metals, the manifestations of localization depend sensitively on the presence or absence of time reversal, (where and are coordinate and momentum respectively.) In the context of the kicked rotor, momentum and coordinate space change their roles, which is why time reversal followed by space inversion, becomes a relevant symmetry. It has been shown Altland93 (); Tian05 (); Smilansky92 () that plays a role analogous to time reversal in metals. The Hamiltonian (1) is –invariant, and in our analysis below, we need to take this symmetry into account. (–invariance may be broken e.g. by a shift However, in this paper, we focus on the invariant system.) In a basis of angular momentum eigenstates, , , the symmetry acts by matrix transposition (much like conventional time reversal acts by transposition in a real space basis.)
ii.2 Off resonance
Let us temporarily assume that is irrational (or rational with , so that the periodicity of the system plays no role.) Throughout this paper, we consider the system prepared in a pure initial state with zero angular momentum, i.e., . Under the Floquet dynamics it will evolve into . Its (squared) deviation from the point of departure is measured by the expectation value . Comparison with the Hamiltonian (1) shows that this quantity may be interpreted in terms of the system’s change in kinetic energy. In the classical limit, value grows indefinitely in time, a phenomenon that may be interpreted as a mechanism of ’heating’. More precisely,
where has the status of a diffusion constant in angular momentum space. Notice that the energy expectation value can be represented as , where is the density-density correlation function in angular momentum space (for a precise definition, see Section III below.) The linear growth in time is indicative of diffusive behavior , where is Fourier conjugate to . Empirically, it has long been known Chirikov81 () that in the quantum model, the linear growth in energy comes to an end at times . Simple scaling arguments show that this saturation implies a crossover at frequencies . The scaling is a manifestation of quantum localization Efetov97 ().
A microscopic quantum theory of the system must be capable of establishing diffusive dynamics at intermediate time scales, and to describe the quantum interference processes rendering the diffusion constant frequency dependent.
ii.3 On resonance
We now discuss what happens for rational values . The commutativity (3) means that we are considering a variant of a -periodic quantum system. For the convenience of the reader, a few basic structures of periodically extended quantum systems are recapitulated in Fig. 1. The Bloch function basis can be understood as a basis whose elements exhibit a definite phase change across a single unit cell (top panel.) For a given , one may then consider the system within the reduced scheme of just a single unit cell, subject to twisted boundary conditions (middle left). Equivalently, one may interpret the system in terms of an Aharonov-Bohm ring (middle right) subject to a gauge flux . The energy levels of this system (bottom) form a periodic family, each defines a Bloch band.
Pioneering work on the physics of the kicked rotor at these “resonant” values of Planck’s constant has been done by Izrailev and Shepelyansky (IS) Izrailev79 () (see Ref. Izrailev90 () and Section VI.1 below for review.) In this work, it has been argued that at resonance, the energy stored in the system increases as
(designation of constants and function taken from the review Ref. Izrailev90 ()). Let us briefly discuss the meaning of this expansion, as discussed in IS. Eq. (4) has the status of a large time asymptotic. For times large enough such that is smaller than the level spacing of the unit cell, , we are in a “deep quantum regime”, where transitions between individual Bloch bands can be neglected. In this regime, the energy increase is governed by expressions such as (symbolic notation)
Thus, the contribution is a measure of the Bloch–phase auto–correlation of individual levels. The auto–correlation of levels also yields a quantum correction linear in , and a residual contribution of indefinite time dependence. The coefficients of both terms, and , reflect the dispersion of Bloch bands, and vanish if bands are flat. Finally, in regimes with localization, we expect the increase in to come to an end at large times . At larger times, will saturate. (With exponentially small corrections proportional to the wave function overlap across the ring.)
The construction of IS that led to (4) has been formal in that it relied on explicit knowledge of eigenfunctions and -values of the Floquet operator; apart from a few exceptional values of , no concrete results for the coefficients of the expansion have been available. Equally important, at times shorter than mechanisms of energy diffusion different from those discussed in IS play a role. Specifically, for small values of time, the finite extension of the unit cell is not yet felt, and angular momentum will diffuse as in an infinite system. This generates a (leading) dependence in , which is purely classical. (Within the framework of a quantum approach, diffusion has to do with inter–band transitions.) At short times, the profile of will, thus, be different from (4).
Below, we will analytically derive the function for periodicity volumes obeying . As a result, we obtain a universal scaling function
where is the inverse of the classical diffusion time through the periodicity volume. Referring for a more substantial discussion to Section VI.2, we here merely note that Eq. (5) predicts diffusive scaling for times . For larger times, we obtain a quadratic increase, , governed by the universal coefficient . For the discussion of the meaning of this coefficient and of the correction terms in (5), we refer to section VI.2.
In Section VI.3, we will briefly discuss the regime of large resonance volumes . Our results are in line with the general expectations formulated above.
ii.4 Close to fundamental resonance
The sensitivity of the rotor to the algebraic properties of manifests itself not only on resonances, but also in the close vicinity of low–order resonant values. To be specific, consider values , close to the fundamental resonance . Straightforward substitution into (2) shows that the Floquet operator defined for the pair , is identical to one with , where Fishman03 ()
This duality phenomenon has been dubbed “-classics”: a rotor with large value of Planck’s constant becomes equivalent to one with small . Faithfulness to this symmetry is a benchmark criterion which the present approach obeys. In practice, this means that system characteristics such as the localization length will depend on and through certain invariant combinations (that have been conjectured phenomenologically before Shepelyansky87a ().)
We now turn to the quantitative formulation of our theory.
Iii Functional integral formulation
As in condensed matter physics, localization properties of the QKR can be conveniently probed in terms of “two particle Green functions”. In this paper, emphasis will be put on the density correlation function
where the frequency arguments, , with understood as , and we have introduced an average over the quasienergy spectrum, . Summation over obtains
and the () superscript designates the retarded (advanced) Green function. Physically, the function describes the probability of propagation in time . Exponential decay of at signals the onset of localization. The density correlation function also contains information on the time dependent expectation value of the energy. Comparing with the definition given in the beginning of Section II.2, it is straightforward to show
In the present formalism, the consequences of time reversal symmetry are best exposed by “taking the square root” of the Floquet operator,
We next introduce the matrix Green function
We will refer to the newly introduced two-component structure as “time reversal space”, or “T”–space and label it by indices . The matrix is related to the original Green function as , i.e.
As a first step towards the construction of a field integral formulation, we introduce a superfield
where component structure refers to the matrix structure in (13), and the superfields carry complex commuting (anticommuting) components (). The index will be used to discriminate between the retarded () and the advanced () sector of the theory. Finally, the commuting variables are the complex conjugate of while in the anticommuting sector and are independent variables.
We next define the compound Green function
where matrices with subscript “AR” (“BF”) act in the two-dimensional spaces of () indices. Matrices contain zeroes everywhere except for a unity at position and are unit matrices. The correlation function can then be represented as a Gaussian integral
where the pre–exponential factor
The absence of a normalization factor in (16) follows from the fact that the integration over anticommuting variables normalizes the integral to unity,
We next reformulate the field integral in a way that will later enable us to make the consequences of the symmetries of the problem manifest. Beginning with time reversal, , we use the symmetry of the Green function, to write
where we defined
and the Pauli matrix accounts for the anti–commutativity of the variables , i.e. , and the same for . The newly defined integration variables exhibit the “reality condition” (the representative of symmetry in the present construction)
where the matrix is defined through
and matrices with subscript “T” act in the two–component “time reversal” space introduced in (17). Armed with these definitions, the partition function may now be rewritten as
The next step in the construction of the field theory is the average over the phase . We will perform this average with the help of the “color–flavor” transformation Zirnbauer96 (), an exact integral transform stating that for arbitrary supervectors, ,
where integration domain and algebraic structure of the supermatrices and will be discussed in a moment.
Applied to the present context, the transformation obtains
where the supermatrices are subject to the (convergence generating) condition . The integration measure is given by
where “sdet” and “str” are superdeterminant and supertrace, respectively sdetdef (). Notice that these operations include the angular momentum indices of the theory. Finally, the integration domain in the boson–boson sector is restricted by the constraint (indices in the BF–sector.)
Using Eq. (18), the Gaussian integration over variables may be performed to obtain
Both the integration measure, and the action of this functional integral are invariant under transformations
We may use this freedom to transform the Floquet operator by to the form
which will turn out to be convenient in the following. To keep the notation simple, we denote the unitarily transformed operator again by .
So far, all operations have been exact. In the rest of the paper, we aim to reduce the functional integral to a more manageable form describing long–ranged correlations in the system. To this end, we need to identify field configurations of low action. Field configurations of this type necessarily have to obey the constraint
The reason is that any violation of this symmetry leads to huge mismatch between the three terms in the exponent of Eq. (21) and thereby to large values of the action.
Let us briefly discuss the conceptual status of the symmetry relation (22). The symmetry essentially involves the T–degrees of freedom of the theory. Retracing their origin, we notice that the T–indices have been introduced to account for the symmetry of the operator (2) under transposition – the physical –symmetry. The ensuing doubling of indices of fields, and the corresponding symmetry (22) lead to the appearance of two distinct low-energy field configurations in the theory. Formally, these are the field components diagonal and off-diagonal in T–space, respectively. Physically, these fields have a status analogous to the diffusion and Cooperon modes in disordered metals.
The partition function reduced to field configurations obeying (22) is given by
We next turn to the discussion of the second fundamental symmetry of the problem, the symmetry under –translation in angular momentum space, . The presence of this symmetry suggests to use a basis of Bloch functions as a preferential basis. A wave function in angular momentum space would thus be expanded as
where and the states are –periodic, . This representation defines an analogous expansion of the fields :
where is a container index comprising BF- and T-index, and the fields are -periodic,
It may be convenient to think of the fields (we suppress indices in the notation whenever possible) as matrices in a tensor space , where is –dimensional reduced angular momentum space (the ’unit cell’), and is spanned by Bloch angular wave functions , also with periodic boundary conditions . Substituting the above expansion into the action (23) and using Eq. (3), it is straightforward to verify that the action assumes the form
where the supertrace now extends over the Bloch quantum numbers,
and the residual “str” traces over the internal indices of . Further, is an effective Floquet operator, diagonal in , with matrix elements
It is instructive to compare this with the matrix elements of the original Floquet operator (defined in unrestricted -space):
The difference is that the compactification of the theory to a unit cell in –space implies a discretization of the angular integration, . Second, the quantized “momenta” get shifted by the Bloch momentum . One may think of as an effective Floquet operator on a ring (in –space) of circumference which is threaded by an Aharonov–Bohm flux . (Of course, this flux is purely fictitious and does not break time reversal invariance.) It is straightforward to verify the unitarity of the effective Floquet operator,
where angular momentum sums are now all meant to run over the unit cell. Finally, notice that the action (28) for the field basis introduced in (26) defines a faithful representation of the theory (23); there are no approximations involved.
Let us close this section with a brief discussion of the general structure of the theory. To this end, we define the matrix
With the formal representation , the zero frequency action can be represented as (cf. Appendix A)
This action is manifestly invariant under transformations provided that (or, equivalently, ). The ’s fulfilling this condition are given by
where are unit matrices in angular momentum space. The zero-mode configurations (30) define the field space of the nonlinear –model (of orthogonal symmetry) Efetov97 (). The global invariance under the “zero-mode” fluctuations spanning this manifold bears important consequences for the formulation of the fluctuation expansion. It implies that we may organize our later analysis of general fluctuations (fluctuations non-commutative with ) around any zero mode reference configuration . Below, we will expand around , or , corresponding to . In the end of the calculation, the extension to generic –field configurations may then be obtained by generalization of the Gaussian fluctuation action to an action displaying the full rotation invariance.
For later reference, we finally note (cf. Appendix B) that the -matrix representation of the density correlation function is given by
where , the average is over the functional with action (28) and the representation (30) is implied. Note that the fields , by definition, are also -periodic, i.e., , and in Eq. (33) may be larger than .
Iv Effective action
In this section we reduce the functional integral formulation (23) (or, equivalently, (28)) to a more manageable field theory of localization in angular momentum space. In doing so, we need to pay attention to fluctuations inhomogeneous in angular momentum space. We will begin by looking at the zero frequency action, , which describes the physics of infinitely long–lived correlations.
Conceptually, the field describes the dynamical evolution of states . (For example, the action (23) essentially measures the overlap of the one–step evolved state, with the un-evolved configuration.) Configurations of low dynamical action are to be expected if (a) is close to (in which case, evolves close to a “classical trajectory” through angular momentum space, and (b) variations in are shallow, such that the classical action of the reference trajectory is low. Indeed, in the limiting case and independent of , the operators and in (23) cancel out and the action vanishes. Our goal is to compute the action associated to soft fluctuations around this zero mode limit.
iv.1 Fluctuation action
As discussed in the end of the foregoing section, it will be sufficient to analyze angular momentum space fluctuations around the reference point . The structure of the full action then follows from its invariance under uniform rotations . We thus start by considering the quadratic expansion of the action (28) around and at . Denoting the ensuing quadratic action by , we have
This is an exact representation of the quadratic fluctuation action. We now need to identify those configurations whose action is asymptotically vanishing. All other, “massive” field configurations can then be integrated out by perturbation theory. Following our discussion in the end of the previous section, we split the fields as
into “massless” configurations and “massive” fluctuations . The massless sector will be parameterized as
where is a shorthand for a summation over “mode indices” smaller than a certain cutoff index to be discussed in a moment. The fields describe fluctuations diagonal in angular momentum space, . To understand the meaning of this limitation, recall the interpretation of as the representative of a bilinear . Turning to a Wigner representation (symbolic notation), , we notice that the degree of off-diagonality encodes information about the direction of propagation in angular momentum space (as measured by the angular variable of the rotating particle.) On physical grounds, we expect that sense of direction to decay rapidly if the kicking strength is sufficiently large (for fixed ). Indeed we will see in Appendix C that the effect of off–diagonal fluctuations amounts to largely innocent corrections to the theory. The essential information is carried by the modes describing fluctuations uniform in angular space. The Fourier mode decomposition in terms of an index states that we will concentrate on fluctuations at scales . Deferring the self-consistent determination of to Appendix C we here state that , i.e. fields of low action will fluctuate on scales , where the scale plays a role analogous to that of the elastic mean free path in disordered systems. It is important that the cutoff index is invariant under the transformation (7). (Note also that the condition implies , as we assume above. Otherwise, the massless sector will contain only the constant mode )
For the moment we ignore the contribution of the massive modes and substitute , with parameterized as in (35) into the action. This leads to
where the integral kernel is given by
Substituting (29), it is a straightforward calculation to reduce this expression to
In the last expression, we have approximated the discrete –sum by an integral. (For the small values of under consideration, this is an innocent operation.) Substituting this result into the action and switching back to a momentum space representation , we obtain
where we have passed from sum to integration because is interpreted as a smooth function of the variable . Here, we have introduced the (bare) diffusion constant
and the commutators act as
The action above was obtained by quadratic expansion around the reference configuration (cf. discussion in the end of Section III.) To obtain its rotationally invariant generalization, we rewrite the action as
where and and describe infinitesimally small deviations off . The generalization to an arbitrary configuration is then given by and , where need no longer be close to unity. We thus conclude that action (37) defined for arbitrary
is the unique rotationally invariant generalization of the quadratic action above. (Starting from (28), the same action can be obtained by the straightforward if more tedious fluctuation expansion around arbitrary .)
To conclude the derivation of Eq. (37), we need to explore its stability with respect to the “massive mode” fluctuations that were so far neglected. This analysis is conceptually straightforward yet tedious in practice. Referring to Appendix C for the actual formulation of the massive mode integration, we here summarize the main conclusions:
The soft mode action is stable at length scales .
Massive mode fluctuations are damped out at scales .
Their principal feedback into the soft mode action is a weak renormalization of the diffusion constant,