Quantum chaology: The photoeffect and beyond

PRAMANA © Printed in India Vol. 48, No. 2, --journal of February. 1997

physics pp. 469-486

Quantum chaology: The photoeffect and beyond

BALA SUNDARAM Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA Present address: Department of Mathematics, CSI-CUNY, 2800, Victory Boulevard, Staten Island, NY 10314, USA

Abstract. I will show how aspects of quantum chaology are relevant even in a seemingly well understood quantum phenomenon like the photoelectric effect. This example together with recent experiments in atom optics are used to define and discuss the larger questions, recent progress made by us in resolving some of these issues and future directions.

Keywords. Quantum chaos; photoelectric effect; atom optics; standard map.

PACS Nos 03.65; 05.45; 32.80

1. Introduction

In his 1987 Bakerian lecture to the Royal Society in London, M V Berry defined quantum chaology as "the study of semiclassical, but nonclassical, behaviour characteristic of systems whose classical motion exhibits chaos" [1]. For more than two decades prior to this, work on systems which exhibited such behaviour had been ongoing under the nominal definition of 'quantum chaos' [2]. Unfortunately, the very nature of quantum mechanics precludes many of the defining characteristics of classical chaos, making this latter definition burdensome. In fact it has even led some researchers to ignore the unquestionable fact that there are quantifiable changes in the quantum dynamics when the limiting classical dynamics exhibits chaos, as compared with a nonchaotic counterpart.

In this article, I shall illustrate that aspects of these cbanges are readily observed even in a simple quantum paradigm - the photoelectric effect. The fact that this phenomenon embodies wave-particle duality makes it ideal to introduce and discuss quantum-classical correspondence in the larger context of non-integrable systems. I will then consider another new experimental system [3] in atom optics which facilitates the realization of theoretical paradigms like the quantum 6-kicked rotor and provides an example of a different manifestation of quantum chaology. Proceeding from these illustrations, I shall define the larger picture in the study of quantum dynamics with classically chaotic limit and briefly discuss our work, in collaboration with Rainer Scharf, in integrating all the relevant issues within the framework of a single model system. Concluding remarks address ongoing directions with a view on future experiments in atom optics.

Most of us remember the photoelectric effect as the emission of electrons when a surface is irradiated with light. The threshold is defined by the condition hv = W where v is the frequency of the light and W a characteristic binding energy for the electron. It was

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Bala Sundaram

soon realized that a generalized multiphoton threshold Nthhv = W was also possible though the intensity of the light had to be considerably higher with increasing Ntn. To put this principle to practice required the advent of the laser. Atoms and molecules became the targets of choice, W was taken to be the ionization or dissociation energy and the age of experimental multiphoton physics was born. Since that time, laser power has rapidly gone through the kilo-, mega-, giga- and tera- prefixes and now borders on the peta-Watt regime. This increase in intensity has brought with it new and manifestly non-perturbative phenomena such as above-threshold ionization (where the electron absorbs more photons than it needs just for the ionization threshold) and harmonic generation (where the irradiated system behaves as a driven anharrnonic oscillator and scatters photons at frequencies which are large multiples of the incident frequency). These have led to a revision of the traditional analysis of multiphoton physics [4]. Most importantly from our perspective, it has brought dynamics to the foreground.

Despite these changes in our understanding of the photoeffect, the one feature which had not been challenged was the fact that the probability of freeing the electron was still expected to increase with increasing intensity of the light. In other words, the lifetime of the quantum state was expected to decrease with growing intensity.

The phenomenon of 'stabilization' was reported in numerical experiments [5] which considered a model Hamiltonian, for a 1-D atom, of the form

H(x,p, t) = p2/2 - l/V/1 + x 2 + xFcos('.~t + ~) , (a.u.) (1)

where F, ~v, and ~b are the field strength, frequency, and phase of the oscillating electric field. This 1-D model potential asymptotes to the Coulombic potential for large x, but eliminates the singular behaviour at the origin. On solving the corresponding time- dependent Schrrdinger equation, it was found that on increasing F, the probability for freeing the electron decreased dramatically, for fixed interaction time. The wavefunction at this time also exhibited peaks which were consistent with an effective double well potential [5]. Both 'stabilization' and the distinct 'dichotomous' form of the residual wavefunction were soon confirmed in more realistic 3-D simulations [6]. We shall show how stabilization provides an illustration of effects in 'quantum chaology' which can then be used to predict the extent of stabilization and the shape of the wavefunction [7].

It is true that advances in classical nonlinear dynamics and chaos have been successfully applied to strongly coupled quantum systems like the photo-absorption spectrum of Rydberg atoms in strong magnetic fields [8], microwave ionization of highly excited hydrogen atoms [9], and the excitation of doubly excited states of helium atoms [10]. However, the fact that the atoms are initially in the ground state, far from the classical-quantum correspondence limit, makes the application of these ideas to the remarkable phenomenon of 'stabilization' in super-intense laser fields still more counter- intuitive [11].

2. Stabilization and scarring of wavefunctions

In the high-field limit (F > 1 atomic unit meaning that it is greater than the binding potential) the smoothed Coulomb potential in (1) can be treated as a perturbation on the regular, classical motion of a free electron in an oscillating field. So, let us first consider

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Special issue on "Nonlinearity & Chaos in the Physical Sciences"


the Hamiltonian for the one-dimensional motion of a free electron in the oscillating electric field E -- F cos(cot(9),

H0(x, v, t) -- v2/2 - xF cos(wt + ¢) . (2)

The classical equations of motion can be integrated exactly and the solution for the position of the electron as a function of time

x ( t )=xo ~ [ c o s ( w t + ( 9 ) - c o s ( 9 ] + v o - z s i n ( 9 J t , (3)

describes a particle that oscillates back and forth in the electric field with frequency u = w/2~r and amplitude c~ = F / J , and drifts away from its initial position, x0, with velocity

F V d = IJ 0 - - - - sin (9. (4)

CO

(Note that even if the initial velocity, v0, is zero, a nonzero initial phase, (9 of the oscillating field can lead to a large drift velocity, Vd =-(F/CO) sin(9.) The classical motion is considerably simplified if we consider variables in the oscillating frame

q = x + a[cos (COt + (9) - cos (9]

p = v _ [Fsin(cot+(9)]. (5)

In the absence of any other forces p = vd is constant and the oscillation center drifts at the constant velocity q(t) = x(O) + Vdt. In particular, if Vd = 0 (eg., v0 = 0 and (9 = 0 or 70, then q(t) = xo is constant. On adding a potential V(x), we have in the transformed frame

H(q,p, t) = p2/2 + V(q - c~[cos (COt + (9) - cos (9]), (6)

which can be conveniently expanded in a discrete Fourier series

H(q,p, t) = p2/2 + ~ Vn(q)e i'~t, (7) r l = - - O G

as the perturbation is periodic with period T = 27r/CO. The Fourier coefficient Vo(q) is simply the time-averaged potential which is all that is required in the high-frequency approximation. Away from this limit, the other coefficients V,(q) contain valuable dynamical information. The ansatz that V,(q) ~ Vo(q) for a large number of n provides the other extreme. In this case, the Hamiltonian in the oscillating frame reduces to

H(q,p, t) ~ p2/2 + Vo(q) ~ e in~' e in~(~ n = - - o c ~

= p2/2 + Vo(q) ~ 6(t - nT + (gT/27r), (8) n = - - O C

using the Poisson sum rule. The phase (9 can be set to zero with no loss of generality and integrating the classical equations of motion over one period T leads to the nonlinear,

P r a m a n a - J. P h y s . , Vol. 48, No. 2, February 1997 (Part H)

Special issue on "Nonlinearity & Chaos in the Physical Sciences" 471

Bala Sundaram

-O.1

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\

[ I

/ / /

i i /

I I

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I [ -60 -40 -80 20

q

Figure 1. Kick potential as a function of q for F = 5.0 (a.u.) and w = 0.52 (a.u.), for both exact and approximate (evaluated only in the vicinity of the classical turning points) kicking terms. The curves clearly show that a simple constant shift is the only difference. Note that F = 5 means that the external field is five times larger than the binding field.

area-preserving map,

qn+l = qn + Tpn+l, Pn+l = Pn + G(qn), (9)

where qn and pn are the positions and momenta of the electron in the oscillating frame, evaluated once every period T = 27r/w of the field. The total impulse of each kick G(q,) is the time-integral of the force over one period and can be simply expressed in terms of the space derivative of the time-integral of the oscillating potential,

d a(qn) = --~q TVo(q) lq=q . (10)

A more physical motivation for the map as well as the explicit conditions for its validity are discussed in ref. [7]. Let us now consider the smoothed Coulomb potential

V(x) = - Z / v / a 2 + x 2, (11)

which in the moving frame is

V(q,t) = - Z / v / a 2 + (q - ol[cos wt - 1]) 2, (12)

P r a m a n a - J. Phys . , Vol. 48, No. 2, February 1997 (Part H)

472 Special issue on "Nonlinearity & Chaos in the Physical Sciences"


z (b o.e (d x ~ x

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q F i g u r e 2. Comparison of the impulse generated over a single period from the differential dynamics (crosses) and the single valued function G for the map. The parameter a is (a) 0.37, (b) 3.7, (e) 18.5 and (d) 295.9. (d) displays only a small portion of the total range so as to emphasize the maximum discrepancy.

where Z is the nuclear charge and a eliminates the x -1 singularity at the origin and also defines a characteristic length scale. The time-average of V(q, t) can be expressed exactly in terms of elliptic integrals though for our purposes it is enough to note that, as shown in figure 1, it is a double-welled potential with minima near the classical turning points at 0 and - 2 a for the free electron in the oscillating field. The map approximates the impulse only at these turning points and results in G(q,) = kF(q,) where k : V/8/F and F(x) involves complete elliptic integrals of the first and second kinds. The effective potential in the map approximation, also shown in figure 1, is simply shifted relative to the exact potential. The product n = kT is the stochasticity parameter and determines the nature (locally/globally/not chaotic) of the phase space of classical solutions. Note that both the limits of high field F and high frequency ov lead to small ~ which corresponds to 'linear' or integrable dynamics.

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S p e c i a l i s sue on " N o n l i n e a r i t y & C h a o s in the P h y s i c a l S c i e n c e s " 473

Bala Sundaram

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F i g u r e 3. Poincar6 section for F = 5.0 and w = 0.52 which is typical for large values of the stochasticity parameter• As discussed in the text, no stable (island) structures are expected and the classical dynamics is dominated by rapidly ionizing orbits.

The validity of the map approximation can be checked a posteriori by comparing the magnitude of the impulses experienced by the electron after each period of the oscillating field with those predicted by the single-valued function in the map. Figure 2 contrasts the impulse computed from the differential dynamics with G for a range of c~. It is clear that the map becomes an effective description of the true dynamics with increasing c~.

Having established the map as a good approximation to the true classical dynamics, we can now use a general stability analysis of trajectories to predict the classical parameter bounds for stabilization. This is also readily seen from the Poincar6 surfaces of section which provide an overview of the classical dynamics. For example, figure 3 considers field parameters for which almost all the classical electron orbits are chaotic and ionizing. By contrast figure 4 shows a case where many initial conditions starting out near the nucleus are regular and do not ionize, tracing out regular island structures in the classical phase space. Thus, classical prediction of stabilization is determined merely by the existence or otherwise of stable regions in phase space.

The clearest representation of the parameter values for which regular regions exist in the classical phase space is a stability diagram in the space of F -~v, shown in figure 5. The dotted line divides the overall parameter space into two regions where a < 1 (on the right side) and c~ > 1 (on the left side). When c~ << 1 the map potential is

474

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S p e c i a l i s s u e o n " N o n l i n e a r i t y & C h a o s in t h e P h y s i c a l S c i e n c e s "

Quantum chaology." The photoeffect and beyond

0 . 5

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Figure 4. Poincar6 section for parameters w = 1.34,F = 5.0 of showing the presence of stable classical islands. The occurrence of these stable regions is typical for smaller values of the stochasticity parameter•

a single well centered on a fixed point near -c~. Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below an analytically predicted stability border (solid line) after which no classical mechanism for stabilization exists. The predicted boundary is consistent with the existence of stable structures in both map dynamics (indicated by the dotted region) and the full differential dynamics (indicated by the open circles), as seen from figure 5. Note the region of low F and ~v where map and differential dynamics disagree. In this regime, the relevant time scale is the internal period rather than the external field period we use. This results in a different stroboscopic approximation called the Kepler map [12].

We consider three parameter sets corresponding to the points A, B and C in the stability diagram. Purely classical stability arguments would suggest no stabilization for A and a larger fraction for C as compared with B. C is also closer to the line a = I which means a near single-well effective potential. As I shall now show, the quantum determination of stability is considerably more complicated. For the quantum dynamics, we simply integrate the time-dependent Schr6dinger equation (in atomic units)

O~P(x,t) [ 1 0 z ] i -~ - ~ ~x 2 V(x) - xE(t) k~(x, t), (13)

on a space-time grid using the standard two-sweep algorithm [13]. The initial condition is the ground state of the undriven potential. Given our picture of the stabilization

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Bala Sundaram

o

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Figure 5. The classical stability of the map (dots) and the full differential dynamics (circles) was assessed by advancing the equations of motion 200 periods of the perturbation. If the trajectory returned to the vicinity of the nucleus a point was plotted in the (w - F) parameter plane. The dividing line a = 1.0 is indicated by the dotted line while the stability boundary is shown by the solid line.

mechanism as resulting from a free-particle interacting periodically with trapping centers (in this case a double well potential), a simple measure of ionization is the fraction of wavefunction that is outside the "interaction volume". A smaller ionized fraction means increased stabilization. As the maximum spacing of the double-wells (averaging over an external period) is 4a, a suitable choice for this interaction volume (in one dimension) is - 4 a < x < 4a. Thus, a definition of ionization as

PI(t) = fx I~(x't)lZdx (14) l>4a

is adequate to establish when stabilization is significant. The results of the quantum simulations for cases A, B and C are shown in the lower

two panels in figure 6. The corresponding classical phase portraits shown reinforce our inferences from the stability diagram; no stabilization for A while larger islands exist for C as compared with B. However, the ionized fraction as calculated from the quantum evolution supports the contrary result that there is more stabilization for A as compared with B. Case C is the most stable which is at least consistent with the classical prediction. What is the origin of this discrepancy?

476



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Figure 6. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, oJ = 0.52; (B) F = 20, a; = 1.04; and (C) F = l0 and ~ = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potential (see text).

The resolution comes from the fact that the classical predictions were based entirely on the stability analysis of low-order fixed points. Though a fixed point may be unstable, there remains an important dynamically invariant, but still unstable, classical structure - the homoclinic tangle. Homoclinic tangles corresponding to cases A and B are shown in figures 7 and 8 respectively. The tangle emanates from unstable fixed points and its complicated appearance is simply a consequence of the area-preserving constraint of Hamiltonian evolution. The unstable tangle also provides support for quantum wavefunctions - an example of a phenomenon referred to as 'scarring' [14]. A heuristic estimate of the 'support ' provided by the tangle for quantization is obtained by considering how the size (area) of the tangle compares with Planck's constant. This measure clearly shows that despite larger stable regions, the overall support stable



I I

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Bala Sundaram

-1 I I I -60 -40 -20 0 20

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Figure 7. Homoclinic tangle associated with the fixed point at ( - a , 0) for case A. Near the fixed point, the solid line gives the unstable direction while the dashed line is the stable direction. The size of Planck's constant h is shown to illustrate that several states can be supported by the single structure. An estimate of the number of states is given by the number of h boxes needed to cover the structure.

and unstable, is considerably less in case B as compared with case A. This is the reason for the reduced stabilization. However, the wavefunction at the end of the interaction time in case B exhibits three distinct peaks which clearly reflect the stable regions in the phase space, unlike A. Case C is one where the phase space is dominated by a large stable region and classical and quantum intuition agree. Thus, the discrepancy in cases A/B is a direct consequence of what was stated to be quantum chaology - "..nonclassical behaviour characteristic of systems whose classical motion exhibits chaos".

These considerations can be extended to the full 3-D system where a four-dimensional map and considerably more involved analysis is necessary [15, 16], though scarfing is once again a relevant feature. For those concerned by the seemingly unphysically large field strengths used in the illustrative cases, it should be noted that these results can be scaled to other frequency regimes and also to excited state initial conditions, where an experimental realization is more likely [17]. The mobility of the light component of the two-component plasmas considered in wakefield accelerators is another potential experimental system.


478 Special issue on "Nonlinearity & Chaos in the Physical Sciences"

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Figure 8. Same as figure 7 but for case B. Note that the classical excursion a = F / w 2 is the same in the two cases.

3. Atom optics and dynamical localization

Our earlier discussion makes clear that quantum dynamics of systems with chaotic classical limit are usually much more stable in their evolution than their classical counterparts. However, contrary to expectations, the mechanisms that lead to quantum stabilization are not solely associated with classically stabilizing phase space structures like the regions around elliptic periodic orbits, KAM-tori, and separatices. Even structures connected with unstable classical motion such as cantori, hyperbolic periodic orbits, homoclinic and heteroclinic tangles can support quantum mechanical wavefunctions. Even though these 'scarred' wavefunctions are ubiquitous in both autonomous and non-autonomous strongly coupled quantum systems, they form a small part of the bigger picture of quantum dynamics with chaotic classical limit.

For example there is also a genuinely quantum dynamical stabilization mechanism. 'Dynamical localization' refers to the fact that eigenstates of quantum dynamics can be localized in an action variable like momentum despite the fact that the classical limit exhibits (deterministic) diffusion [18]. This effect is analogous to Anderson localization in tight-binding models [19] and provides a global mechanism (does not distinguish details of the classical phase space) for the quantum suppression of classical chaos. In one-dimension, the localized wavefunction has a characteristic exponential form as in the case of Anderson localization.



Bala Sundaram

This is in contrast with the phenomenon of 'scarring' where the quantum wavefunctions show enhanced probability around specific classically unstable invariant structures. Thus, the excitation of scarred wavefunctions provides a 'local' mechanism of suppression of the classically chaotic dynamics.

As the emphasis in this article is on the usefulness of these ideas in unexpected contexts, I will now consider a recent example of dynamical localization in atom optics. Instead of the smoothed Coulomb potential, let us take V(q) = k cos q. In the presence of the time-drive and in the moving frame, we have

H = p2/2 + V(q - ~ sin (cvt))

= p2/2 + kcos (q - a sin (wt))

= p2/2 + k Z Jn (c~) cos (q - nwt), (15) /1

which describes a modulated pendulum where the Fourier weights J , are Bessel functions of order n. For large a, the asymptotic form for J , can be taken where the amplitude varies slowly with n (as required by the ansatz of the earlier example). The resulting map is a textbook paradigm for Hamiltonian dynamics and chaos - the standard or Chirikov- Taylor map. More significantly, the quantum standard map or kicked rotor is the paradigm for dynamical localization.

A novel realization of this driven system was reported recently by considering the momentum transfer from a modulated standing wave of light to a sample of independent, ultra-cold atoms [3,20]. The thermal (cold) sample of approximately 105 sodium atoms provide a Gaussian initial condition in momentum. The relevance of the internal structure of the atom to the dynamics can be eliminated by tuning the frequency of the standing wave far from resonance. The remaining dynamics is that of a 'structureless' atom, described by the above Hamiltonian, which undergoes a momentum change of 2likE (kL is the wavenumber of the light wave) when the velocities of the atom and standing wave are matched. These 'resonant kicks' occur twice every standing wave period though they are not equally spaced in time. The experiment measures the final momentum distribution of the atoms as the modulation amplitude of the standing wave is varied.

The classical dynamics generated by (15) is governed by resonances at p = / / = mo with approximate widths 4 ~ in momentum. These resonances overlap as c~ is increased after which the classical particle can diffuse in momentum. It should be noted that there is a maximum momentum value pmax = o~ beyond which the interaction turns off. This is easily seen as the velocities of the atom and standing wave have to match (stationary phase condition) for the momentum transfer to occur. Thus, the classical diffusion is over a bounded region in momentum space, the extent of which can be controlled by changing ~.

The first column shown in Figure 9 corresponds to zero modulation amplitude, which is just the case of the simple pendulum. The classical, experimentally measured and quantum distribution of final momenta of the atomic ensemble agree once again because the phase space is entirely regular. On increasing ~, the phase space becomes increasingly chaotic and the stable regions shrink as shown in column two. The dominant classical mechanism is now diffusion in momentum. The

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Figure 9. Phase portraits (upper panel), classical momentum distributions (middle panel), and experimentally measured momentum distributions compared with quantum theory (bottom panel, theory marked by lines). The first column is for a = 0 and is just the case of the simple pendulum while a ~ 0 in column two. The initial condition corresponds to a Gaussian distribution in momentum and fully extended (plane waves) in position. The vertical scales for the distributions are logarithmic and are marked in decades.

Pramana - J. Phys., Vol. 48, No. 2, February 1997 (Part II)


Bala Sundaram

Quantum Dynamics with

Semielassieal Quantization for Nonintegrable Dynamics

Periodic orbit th. Tunneling (Chost orbits)

Classical Chaotic Limit

', ]3

t

Quantum Chaology - Signatures of Chaos?

i,'t , /

Eigenvalue Statistics; ", Eigenvector Composition; '.

Phase-space Transport etc. '.

Scarring

Dynamical Localization

Figure 10. Schematic representation of issues which arise when considering the quantization of classically chaotic dynamics.

final classical momentum distribution is nearly uniform over the allowed region, except for a small peak corresponding to the remnant stable region• This is in sharp contrast with the quantum and experimental distributions which both exhibit an exponential distribution in momentum (note the logarithmic y-axis). This is an example of dynamical localization.

Anderson localization describes the motion of electrons on a disordered lattice where the randomness is externally imposed• In dynamical localization randomness or, more correctly, pseudorandomness is generated by the dynamics leading to localization in an action variable, rather than in space• Dynamical equivalents of higher dimensional Anderson models [21] also exist which should be realized soon by the atom optics experiments• It should be mentioned that these experiments have already confirmed many of the details of the theory' of dynamical localization [22] as well as the mechanism of overlapping resonances to achieve global chaos [23].

4, The bigger picture

Scarring and dynamical localization are merely two aspects of a much larger view of quantum dynamics with classical chaotic limit• As shown by the schematic in figure 10, this picture originates from two simple questions which still demarcate current approaches. (a) How does semiclassical quantization work for nonintegrable dynamics? and (b) What are the semiclassical signatures of chaos?

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The first question resulted in periodic orbit quantization, an idea which dates back to Van Vleck in 1928 and which was extensively pursued by Gutzwiller [24] The seminal idea is the use of classical trajectories to calculate the quantum propagator of a mechanical system within the semiclassical approximation. The trace of this propagator, which gives information about the spectrum of the dynamics, is therefore linked to the periodic orbits of the classical dynamics. This approach works not only for integrable dynamics but also for strongly chaotic dynamics as well. It has been applied successfully to describe the quantum properties of model systems like the anisotropic Kepler problem [25, 26] and the motion of a particle on a surface of constant negative curvature, where the trace-formula is exact (see for example [27]); as well as to the spectral analysis of a hydrogen atom in a strong magnetic field [8] and the doubly excited spectrum of the helium atom [10]. A related and yet outstanding problem is periodic orbit quantization for dynamics in a mixed phase space, where regular and chaotic structures coexist.

A recent twist to the periodic-orbit analysis came with the observation that in systems where new orbits are created through bifurcation, the impact of these orbits is felt in the quantum spectrum before they appear as real orbits of the classical dynamics (parametrically speaking) [28, 29, 30]. These so-called "ghost" orbit contributions further modify the semi-classical analysis.

The second question leads to "quantum chaology". It includes the study of the statistical features of the eigenvalue spectra, as well as of the eigenvector components in a fixed representation. Using this approach, a variety of interesting results have surfaced (see for example [31, 32]). An example of consequence to experiments is the relationship between the statistical features of the spectra of most strongly coupled systems to certain matrix ensembles devised to understand nuclear spectra [33]. However, this statistical description results in randomized eigenfunctions and has to be modified as soon as a dynamically localized or scarred wavefunction is observed.

The interplay of the different mechanisms had not been addressed till very recently. In large part this is due to the fact that the demands made on model systems by the two branches are quite distinct. For example, most aspects of quantum chaology can be studied within the &kicked rotor or standard map paradigm. However. semicJassical or periodic orbit quantization is greatly facilitated by the existence of a symbolic coding (alphabet) and pruning rules (grammar) for the classical periodic orbits, as in the 3- and 4-disk scattering systems. It is not easy to calculate classical invariant structures in the standard map. As such even the competition between local (scarring) and global (dynamical localization) suppression mechanisms has not been explored. Scamng [34] and dynamical localization [35] have only recently been reconciled with semiclassical theory, though even there open questions still remain.

Piecewise continuous potentials have been used to address a number of issues related to classical transport [36]. Scharf and I [37] showed that a piecewise linear standard map [38] facilitated the calculation of classical invariant structures for scarfing and provided a simple coding for periodic orbits. We used this modified dynamics to explore the strength of scarring [39] as well as the construction of local effective Hamiltonian flows based on classical periodic orbits [40[. This local effective description becomes necessary in mixed phase spaces though the crucial issue of matching solutions then becomes the open question. Current work on relating these local effective flows to Frobenius-Perron [41] operator dynamics appears very promising.



Bala Sundaram

Experimentally, a strong focus of the next generation of experiments in atom optics will be mixed phase space dynamics and the effects of noise [42]. Direct excitation of scarred states and an assessment of their coherence with an atom interferometer may also be possible. Advances in fabrication technology have opened up another new testing ground in the area of layered heterostructures. The role of the ideas we have discussed to the transport or magnetic properties of driven quantum wells and a variety of mesoscopic systems is currently under intense scrutiny with the anticipa- tion that these might impact on device physics as well. In particular, experiments measuring changes in the magnetoconductance as the shape of the scattering domain, for a two-dimensional electron gas, is changed from integrable (circle or square) to nonintegrable (stadium) [43, 44] provide interesting deviations from the theoretical paradigms. We had suggested that ghost orbit contributions are important to the dynamics once the Lorentz force is considered in computing classical trajectories [30]. Recently experimental evidence also indicates deviations from random matrix theory predictions [45] as well as the role of scarring [46]. Inelastic scattering and decoherence are also issues which invoke the role of parametric or systemic noise on any stabilized quantum structure [47]. It is my view that just as turbulence provides strong motivation in classical dynamical analyses, these inherently 'messy', truly many- body systems pose an analogous challenge for nonintegrable quantum dynamical systems.

Acknowledgements

The author would like to acknowledge fruitful collaborations with R V Jensen and the experimental group of M G Raizen. The 'big picture' developed over many years of working with his friend and 'sounding board' for ideas, Rainer Scharf. This work was supported in part by the U.S. National Science Foundation.

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Bala Sundaram

.1 A Folk, S R Patel, S F Godijn, A G Huibers, S M Cronenwett, C M Marcus, K Campman and A C Gossard, Phys. Rev. Lett. 76, 1699 (1996)

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Quantum chaology: The photoeffect and beyond