Transport of Quantum States of Periodically Driven Systems H.P

Transport of quantum states of periodically driven systems H.P. Breuer, K. Dietz, M. Holthaus

To cite this version:

H.P. Breuer, K. Dietz, M. Holthaus. Transport of quantum states of periodically driven systems. Journal de Physique, 1990, 51 (8), pp.709-722. 10.1051/jphys:01990005108070900. jpa-00212403

HAL Id: jpa-00212403 https://hal.archives-ouvertes.fr/jpa-00212403 Submitted on 1 Jan 1990

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Classification Physics Abstracts 03.65 - 32.80 - 42.50

Transport of quantum states of periodically driven systems

H. P. Breuer, K. Dietz and M. Holthaus

Physikalisches Institut, Universität Bonn, Nussallee 12, D-5300 Bonn 1, F.R.G.

(Reçu le 17 août 1989, accepté sous forme définitive le 23 novembre 1989)

Résumé. 2014 On traite des holonomies quantiques sur des surfaces de quasi-énergie de systèmes soumis à une excitation périodique, et on établit leur topologie globale non triviale. On montre que cette dernière est causée par des transitions diabatiques entre niveaux au passage d’anti- croisements serrés. On donne brièvement quelques conséquences expérimentales concernant le transport adiabatique et les transitions de Landau-Zener entre états de Floquet.

Abstract. 2014 We discuss the transport of quantum states on quasi-energy surfaces of periodically driven systems and establish their non-trivial structure. The latter is shown to be caused by diabatic transitions at lines of narrow avoided crossings. Some experimental consequences pertaining to adiabatic transport and Landau-Zener transitions among Floquet states are briefly sketched.

1. Introduction.

Periodically driven quantum systems are of very broad physical interest and are experimentally readily realised by studying laser/maser - matter interactions. Among the many facets of related phenomena we shall address theoretical problems and their experimental repercussions which specifically pertain to the regime of strong laser/maser - fields ; strong field - matter interaction cannot be described by perturbation theory, non-perturbative methods have to be applied. The particular problem we shall treat in this note is the transport of quantum states [1] by variation of external parameters, i. e. their adiabatic - diabatic motion. In order to get a clear-cut physical picture let us consider a strong single - mode laser field interacting with a quantum system (atom, molecule or, generally speaking, any mesoscopic system) which is described by the following set of Hamiltonians

where

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005108070900 710

we explicitly indicated the parameter dependence of H : À --+- strength of the laser/maser field to laser/maser frequency e1 cp = arg - phase controlling the transition from linearly to circularly polarised plane -e2 wave fields ;

Ho describes the static system, the dipole approximation is not assumed at this state of argument, D is the matter current density. Introducing now the région A in which the external parameters À, (J), cp can be varied (in a given experiment) and

we can formulate our problem as follows : a time-variation in an interval of length Ta of external parameters is represented as a curve r in parameter space

the problem of quantum transport is now to study the behaviour of states, i.e. of solutions of the Schrôdinger equation

under motions on the curve r. A particularly transparent description emerges if we use Floquet states [2, 3] as a basis [4]. We write the general solution of the Schrôdinger equation as a superposition of Floquet states

where the « stationary » states Ua (t) obey

Of course, ua and the quasi-energies Ea do depend on the parameters R

adiabatic motion of the state gi is characterised by the requirement that the probabilities 1 aa 12 in (1.5) remain constant for a sufficiently slow change of parameters along T and are thus independent of R. Speaking in terms of experiments, parameter variations on curves T are well known : the simplest example is a laser/maser pulse where 711 curves necessarily embedded in higher dimensional parameter regions appear if in addition to a varying field strength the frequency cv and/or the phase cp are modulated. Closed curves T

are of particular interest if at least two of the three parameters vary in time such that a non- vanishing surface is enclosed. The ad libitum realisation of such curves is doubtlessly an experimental challenge, we believe, however, and hope to convince the reader that new physics can be uncovered in experiments of this type. The requirement of periodicity for the « stationary » states ua in the eigenvalue problem (1.6) entails energy conservation modulo W , i. e.

and the quasi-energy spectrum is arranged in Brillouin zones. We assume a discrete spectrum of the static Hamiltonian

and anticipate a choice for {e j} such that

we then have

Nonetheless, the quasi-energy spectrum {Ea} is, in general, dense. For our context this fact has as a consequence that the adiabatic theorem cannot be proven in its original form for periodically driven quantum systems : an essential part in its proof is the requirement that the state to be transported adiabatically be separated in energy from all other states of the system [5]. On the other hand, simple model calculations [6, 7] have clearly demonstrated the validity of adiabatic concepts ; more realistic calculations [8, 9] pertaining to the problem of microwave ionisation of excited hydrogen atoms [10] have shown the importance of adiabatic evolution for the interpretation of experimentally observed features. Numerical evidence of this kind and new experimental support [11] for the concept of diabatic - adiabatic transitions at avoided level crossings may be taken as sound guide - lines in the following argumentation. We start by dividing the quasi-energy spectrum into sets of specific symmetry of states, sets which we denote by { ê a } sym The point is here that levels belonging to the same symmetry class do not cross [12] in the generic case, avoided crossings (AC’s) appear as characteristic patterns. A typical feature of periodically driven systems is that AC’s emerge, in general, densely distributed : the set (s/) sym 8> JL contains dense patterns of AC’s. Not all of these AC’s are dynamically relevant for the description of the parameter motion (1.3). In a recent publication [13] we have described a simple selection procedure : given an experimental set-up and, hence, a time-scale Ta we determine all the AC’s for which 712

with

5 e = level distance in a Brillouin zone OT = time which the system spends in the vicinity of the AC (more precisely speaking OT = 15R 1/1 Ji1 where 1 5R1 is the parameter distance within which the level distance changes by a factor of J2). The levels connected by these AC’s determine a finite dimensional of states in a subspace {ua } eff which, very good approximation, effectively suffices to describe the diabatic-adiabatic motion of quantum states in experiments performed with the very set-up considered :

where the aa do depend, in general, on time during the motion (1.3) on the parameter curve r. We are thus led to the discussiôn of a finite set of smooth effective quasi-energy surfaces

It so happens that quasi-energy surfaces in this set get very close : they almost touch along curves of, in general, finite length. A cross section of this approach is a very narrow AC, very narrow in the scale set by Ta defined in (1.3) and used for the construction of these narrow AC’s are while on T in the sense that the { Ua } eff’ Dynamically, ignored moving system jumps across from one quasi-energy surface to the adjacent one with probability very close to one. Geometrically speaking, on the other hand, this line of narrow AC’s plays the role of a twist, the two surfaces should be visualised as manifolds cut along the line of narrow AC’s with cross-wise identified borders, the cross-wise structure being due to the purely diabatic nature of transitions at these narrow AC’s. To get this point unequivocally clear : having set the time-scale Ta by specifying experimental conditions we are led to a coarse- graining [13] of the quasi-energy spectrum and thus to the construction of an effective set of states {ua} cff’ narrow AC’s imply diabatic transitions and, hence, cross-identification of quasi-energy surfaces the geometric purport of which we express by taking the two surfaces diabatically connected as two branches of one geometrical object connected in the twisted manner just described. Needless to say, this is again an effective, however, consequent construction translating into geometry the dynamic notions of diabatic-adiabatic and purely adiabatic transitions in periodically driven quantum systems. for a M the set in the Clearly, sufficiently large parameter region {ua} cff will, generic case, appear as one geometrical entity in which all the states ua are connected to all the other states by the above described cut ; the assignment of quantum numbers for its various sheets u,,,, is a matter of convention fixed by the choice of parameters R. In the following, we shall describe the precise construction of such an « universal covering » in some detail and discuss, in particular, connections for adiabatic motion and their implications for generalised Berry phases [14]. Before going into the details we should state very clearly the logic underlying our construction : starting with the heuristically determined effective set of dynamically relevant states {ua} cff which suffices to describe the dynamics of our system the more acurately the clearer narrow and effective AC’s (cf. (1.14)) are separated on the set scale Ta, we proceed to investigate, mainly in mathematical terms, the geometrical 713

structures responsible for diabatic - adiabatic motion. It should become clear to the reader that even basic experiments on this type of questions need such a framework for an adequate theoretical explanation. The following two sections will first introduce the necessary mathematical concepts and then discuss their physical implications.

2. Formal considérations on diabatic-adiabatic motion.

Motion on a curve r in parameter space

of a system driven by a single mode laser/maser field is govemed by two time-scales : the laser/maser period

which is, of course, in general, T dependent, and the time Ta required to cover the curve T. We treat these scales separately by first defining the eigenvalue problem (1.6) explicitly as

a on e.g. for non-relativistic n-electron system, 3C R acts functions ua (rl, ..., rn, t). The « slow » motion on r is integrated [8] by means of

w (t ) : = e (t, t ) is then a solution of the Schrôdinger equation (1.4) including the parameter variation on r. For every the set of effective spans a finite dimensional vector R (T ) states {ua} cffz complex space

The fundamental mathematical structure [15] underlying the diabatic - adiabatic motion of our system is the N-dimensional (N = dim FReff)’ complex vector bundle B over the base space M

The projection

defines fibres

Because of the very construction of { ua} eff’ to a high degree of accuracy, motion on r with the fixed scale Ta never drives the state of the system out of 9 ; motion from, say, R ( T 1 ) E T to R ( T 2) E r simply maps the corresponding fibres onto each other : 714

We write a solution et> a (T ) of (2.4) (we do not indicate the t-dependence explicitly) obeying the « initial » condition

and stipulate that and

Inserting into (2.4)

and demanding that the projection of the right-hand side onto the fibre F,,R4’) vanish we obtain the equation for the transport matrix Uf3a (T) thus defining the gauge field A on the bundle t which effectuates parallel transport of quantum states for parameter motion on r. Explicitly we find

where

is the scalar product in the Hilbert space underlying the eigenvalue problem (2.3), a(J) and dots indicate differentiation with respect to (J) and T respectively which, when acting on periodic functions with period T = 2 TI’ / (J) (T), acts only on the Fourier components, the shift of the Fourier basis being already explicitly taken into account in (2.13). Introducing the scaled variable s = T /Ta we see that the first two terms in (2.13) are proportional to Ta whereas the last term is independent of the scale Ta (if the curve r is kept fixed) and defines a hermitian connection 1-form

hermiticity is immediate because of « uo 1 uy > = S f3’Y ; .ae can be taken as a Lie-algebra valued 1-form :

where TM is the tangent bundle on M, C(U(N)) is the Lie-algebra of U(N). 715

A concurrent gauge-structure now emerges [16]. Changing the basis in the fibre Feff

induces a transformation of the connection 1-form

The corresponding curvature 2-form reads explicitly

We now consider closed curves T, i.e.

and write the formal solution of (2.12) as

where we decomposed (2.13) and used the symbol 3’ to indicate the T ordering to define the exponential integrals in (2.21). The holonomies U(l’) ; i.e. the images of all closed curves in parameter space mapped into the set of solutions of our transport equation (2.12)

or, rather, the holonomy group

’ closed curve through : contain information on the topological structure of the bundle t ; the non-triviality of g, for instance, is related to its curvature. From (2.21) we have

Assuming the existence of a surface S on which A is smoothly defined such that j’ = aS we find from Stokes’ theorem for the geometric phase factor

since, obviously, tr dA = tr Y by (2.18). 716

1 To recall a familiar example, let r = S and S± denote the upper and lower half-sphere connected to it. We then have

and, hence,

The quantisation [15] of the first Chern class

again appears as a consistency condition on the holonomy U(T ). We now come to the essential point of our discussion, namely to establish a relation between the time-scale of the motion on T and the form of energy-surfaces in the corresponding parametrisation R = (À, (J), cp) (1.2). The situation qualitatively shown in figure 1 is relatively uninteresting. The quasi-energy surfaces are separated by finite, approximately constant gaps which are bounded from below such that for sufficiently large Ta the adiabatic limit is attained. The non-diagonal elements of the gauge-field A are exponentially small such that the equations for the transport matrix Ul3a (T) decouple :

- surfaces as functions of Fig. 1. A sketch of quasi-energy tat’ ..., taN plotted parameters R E M in a case in which purely adiabatic motion is expected for sufficiently large time-scale à 717 and

where

Initial states propagate independently with their « own » Berry phase [14], the holonomy group is simply the direct sum of N copies of U ( 1 )

A richer case is qualitatively shown in figure 2. The two quasi-energy surfaces can be visualised as being constructed by moving a double cone along some line of finite length and then cutting out a small region containing this line, smoothing the cut, of course. The typical feature is now that our system e.g. moved along the closed loop F depicted in figure 2

Fig. 2. - Quasi-energy surfaces Eal, Ea2 plotted as functions of k, co showing narrow (S1 ) and broad (S2 ) avoided crossings in cross-sections Si and S2. The line separating the two « roofs » has to be visualised as separating them by a line of narrow AC’s as indicated in the cross-section Sl. For time scales such that transitions induced by e.g. moving the system in Si are (almost) completely diabatic, opposite surfaces of these roofs are identified and a square-root like Riemannian surface appears giving an effective description of our system. A closed parameter curve r passing through a narrow and a broad AC is indicated. 718

encounters two very different AC’s : a very narrow and a broad one, the corresponding cross- sections Si and S2 are likewise shown in figure 2. We now demonstrate the relation between the time Ta needed to run along r and the geometry of the bundle 6. i) Certainly, there is a scale Ta such that the system passes Si as well as S2 adiabatically and the geometry of 6 is characterised by (2.30), the situation being identical to the one shown in figure 1. ii) If we now increase the speed in traversing r, i.e. decrease Ta, we shall reach the point where diabatic-adiabatic (Landau-Zener) transitions occur at S1. The transport matrix obtains non-diagonal elements ; the holonomy group has a non-abelian structure. iii) A further decrease of Ta leads to (almost) pure diabatic transitions at Si and adiabatic transitions at S2. In the latter case the system, starting on one quasi energy surface, jumps on the other surface and remains there until the starting point on the loop T is reached again : in traversing the loop T the system has changed its quantum state

The energy surfaces Eal and Ea2 are thus melted, under the premiss that time-scales be such that the line of narrow AC’s along the double-rim displayed in figure 2 cannot be resolved and purely diabatic transitions take place, into one surface which has, topologically speaking, the structure of a square-root Riemannian surface, the outermost point in figure 2 being the branch-point. Hence, the diabolical point of a double-cone structure can turn into a square- root branch pÓÎ11t when the system under consideration is periodically driven with a sufficiently large amplitude. The integration of e.g. the form (2.15), therefore, does not encounter any singularity and is well defined. Geometrically speaking, iii) means that states corresponding to parameters on the line in u1t tracing out the narrow AC’s have to be cross-wise identified in order to model the diabatic jumps thus introducing non-trivial topological structure into the bundle 6. To see this in some detail, let us identify the fibre F,,R40) with C2

The cases (i) and (üi) with (2.32) then correspond to the holonomies

and

respectively. Diabatic jumps reflect themselves in the geometry of 6 locally as topological structures of square-root type Riemannian surfaces. In general terms, the point to be noted here is that it is the effective adiabatic theorem which introduces non-trivial structures in the set of holonomies. We see that the bundle 6 in our example has, roughly speaking, the structure of a Môbius band : to construct the identity element the appropriate loop r has to be followed twice. The observation that the notion of an effective adiabatic theorem contains a time scale, either of intrinsic nature or given by the experimental apparatus, leads to the important physically observable fact that the connectivity of quasi-energy surfaces, tested by holonomies U( ), is not an a priori notion but rather influenced and determined by experimentally imposed scales. 719

The physical realisation and experimental implications of such geometrical features of 8 will be discussed in the following section.

3. Conclusions.

The non-perturbative character of strong laser/maser - matter interaction is most effectively taken into account by treating its dynamics in a space of Floquet states ; in order to illustrate their importance for quantum transport phenomena we discuss a simple example in a separate appendix. Considered as a function of external parameters the quasi-energy spectrum has a complicated structure even if the static spectrum is discrete : avoided crossings are densely distributed over any region M of parameters which we took as field strength, frequency and polarisation angle in the case of single-mode laser/maser interaction. The observation that avoided crossings appear in very distinct scales, extracted from model calculations (see, e.g. figure 1 in [13]), led us to the construction of effective sets of Floquet which describe the of an molecule etc. for a motion states { ua } eff dynamics atom, parameter on curves F in it. Physical examples are interactions with pulsed laser/maser beams, switch- on processes, atomic and molecular beams in cavities etc. which introduce a time-scale Ta. This scale set, states in {ua} eff are, at a fixed laser/maser frequency, connected only via avoided crossings producing Landau-Zener transitions. Seen in the 3-dimensional region f1, states in also connected lines of narrow avoided where our {ua} eff are, however, by crossings system jumps diabatically between adjacent states on the time-scale Ta. The set of states {ua} eff’ hence, seems to be connected at square-root type branch-points the square-root character resulting from the cross-wise identification of states at narrow avoided crossings. The typical situation is depicted in figure 2 where the diabatic transition in Si leads to the cross-identification ; starting at parameters R(0) with a state ua1R(0) a modulated laser/maser pulse of length Ta will lead us along the curve r to the state UR(T.)with R ( Ta ) = R(0) : we do not end up in the same state, the Ua, a E Xeff are multivalued functions of the extemal parameters, it is only an even number of pulses that will lead us back into the same state. We took care of this non-uniqueness of Floquet states by constructing the bundle t defined in (2.6) to (2.9) where the assignment of quantum numbers to the states u a is a matter of choice of coordinates, quantum numbers being a path-dependent concept. The set of effective states is thus assembled into one the bundle 6 which a {ua} eff structure, represents « universal covering » of the parameter space M. On such a bundle, adiabatic motion is a well-defined procedure, multi-valuedness is translated into non-trivial holonomies on 8. On the other hand, this bundle construction is also necessary for the discussion of adiabatic and Landau-Zener dynamics : quasi-energy surfaces separated by avoided crossings leading to Landau-Zener transitions in one small parameter region will be connected by narrow avoided crossings in a parameter region sufficiently enlarged, the latter leading to purely diabatic transitions and, hence, introducing non-trivial geometry. There is a simple line of reasoning which helps to visualise the occurence of structures like the one shown in figure 2 : choose w such that for given integer n and static eigenvalues E J O) ’ E k (0)

and, hence, and a Brillouin zone are exactly degenerate at representatives £a.J £ak ion À = 0. For non-vanishing À these levels will repel each other and the resulting AC will be characterised by a small 6 e for small À to begin with ; twist identification up to a time-scale 720

Ta is the corresponding dynamical mechanism. Increasing the field strength À leads to an increase of 5 e, for 6 E dT = h Landau-Zener transitions become important and the « branch- point » in figure 2 is reached. We checked this scenario in model calculations similar to the ones reported in [6, 7]. In particular, we considered a one-dimensional, periodically driven electron in a box and reproduced quantitatively the structure which we displayed in figure 2 as a qualitative, paradigmatic example relevant for the construction of a non-trivial holonomy (see (2.33)). We repeatedly emphasised that the position of this « branch-point » depends on Ta : in the vicinity of each such point the wave function splits into a superposition of the two wave functions of the AC-levels. Thus, purely diabatic and adiabatic motions are occurring in parameter regions M with holes around « branch points », the holes being chosen such that parameters leading to Landau-Zener transitions are removed from M. An experimental verification of elemental structures as the one depicted schematically in figure 2 has to cope with the following boundary conditions : first of all, the time scale Ta faces an upper limit which is either due to spontaneous emission or to the continuous part in atomic and molecular spectra leading to a finite life-time of Floquet states. The question at stake, is, of course, whether or not the time-scale Ta can be chosen large enough for effective adiabatic motion to be dominant. Furthermore, one has to search for systems where technically feasible loops r in parameter space do not enclose too many branch points and AC’s in order not to obscure the basic square-root type connectedness in {ua} eff and the corresponding ramifications in the effective space of states caused by Landau-Zener transitions. This does not mean that the construct of a bundle 6 of physical states is meaningless for the description of, say, microwave ionisation of hydrogen atoms where, in certain parameter regions 4t, the reaction is dominated by a large number of close, overlapping AC’s. On the contrary, it is our belief that even a statistical treatment of such agglomerations of AC’s has to take into account the non-trivial topological structure of 8 : an understanding of such experiments in more detail will require a deciphering of these global geometrical properties of quasi-energy surfaces. In preceeding publications [8, 9] we have shown that a simple purely quantum mechanical frame for the description of microwave ionisation of excited hydrogen atoms and related phenomena is constituted by invoking Landau-Zener transitions in an effective set of Floquet states : typical scales for the parameters of the corresponding AC’s and their distributions over parameter regions M are responsible for typical features of, for instance, ionisation curves. In this paper, we extended these notions pointing out non-trivial geometrical structures caused by diabatic transitions which prevail in significant sets of external parameters. It is doubtlessly an interesting challenge to hunt for experimental verification. On the theoretical side, an understanding of such a rich but elementally quantum mechanical scenario in terms of a semi-classical approach to classically complicated, chaotic phenomena is an outstanding problem. Remains the question of the continuum. As we have remarked in (1.10) to (1.13), a discrete spectrum of Ho leads, in the generic case, to a dense quasi-energy spectrum ; a continuous part of the spectrum of Ho to a quasi-energy spectrum comprising the whole real axis. The provocative question arises of how any computer simulation of the dynamics of periodically driven quantum systems should ever distinguish the two cases ! It is again a question of time- scales and coarse-graining and leaving out dynamically irrelevant states which can be applied in both cases ; we hope to come back to this point. 721

Acknowledgment.

This work was done in part at the Harvard-Smithsonian Observatory, one of us (K. D.) would like to thank Alex Dalgarno, Kate Kirby and George Victor for their kind hospitality. Fruitful discussions with B. Zygelman are equally acknowledged.

Appendix. To illustrate the fact that Floquet states provide the appropriate basis for a description of adiabatic transport in periodically driven systems we give an explicit example for an abelian phase factor. Let us consider a quantum system interacting with a linearily polarized radiation mode described by the Floquet Hamiltonian

with variable phase 5. It is easy to see that the quasi-energies Sa are independent of 8 and that the general solution of is given by

with 8-independent Fourier modes uam. We now vary 6 from 0 to 2 ir and obtain from (2.29) the Berry-phase

Since

we finally arrive at

As an observable quantity this phase factor does, for £a - c + mw, not depend on m, i.e. on the second index in (1.13). Furthermore, the role of Floquet states for adiabatic transport is reflected in the explicit occurence of the Floquet index E a . It is instructive to compare the expression (A6) with the result of Bamett et al. [17]. Considering a two level system with level spacing Wo and using the rotating wave approximation these authors obtain the following expression for the phase

where detuning 722

To compare (A7) with out formula (A6) we have to calculate the Floquet indices for the same system and obtain

Therefore we have

and we see (A7) to be a special case of the general formula (A6) which is neither restricted to a two level system nor to the rotating wave approximation. We remark that (A6) could also be of experimental interest in order to distinguish an avoided crossing in the E - w-plane from a real crossing by means of a pure phase variation since at an avoided crossing we have 2 ’TT dEa/dW = 0 mod 2 ?r.

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