Spinor dynamics of quantum accelerator modes near higher-order resonances
Laura Rebuzzini,1,2,
*
Italo Guarneri,1,2,3and Roberto Artuso1,3,41Dipartimento di Fisica e Matematica and Center for Nonlinear and Complex Systems, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy
2Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Via Ugo Bassi 6, 27100 Pavia, Italy
3CNISM, Unità di Como, Via Valleggio 11, 22100 Como, Italy
4Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy 共Received 28 November 2008; published 13 March 2009兲
Quantum accelerator modes were discovered in experiments with kicked cold atoms in the presence of gravity. They were shown to be tightly related to resonances of the quantum kicked rotor. In this paper a spinor formalism is developed for the analysis of modes associated with resonances of arbitrary order qⱖ1. Decou- pling of spin variables from the orbital ones is achieved by means of an ansatz of the Born-Oppenheimer type that generates q independent band dynamics. Each of these is described, in classical terms, by a map, and the stable periodic orbits of this map give rise to quantum accelerator modes, which are potentially observable in experiments. The arithmetic organization of such periodic orbits is briefly discussed.
DOI:10.1103/PhysRevA.79.033614 PACS number共s兲: 03.75.Kk, 05.45.Mt, 03.75.⫺b, 37.10.Vz
I. INTRODUCTION
A kicked system is a Hamiltonian system that is periodi- cally driven by pulses of infinitesimal duration. More than 30 years after the invention of the paradigmatic model of such systems, namely the kicked rotor 共KR兲 关1兴, kicked quantum dynamics is still the focus of active research for a twofold reason. On one hand, it has given birth to an ever increasing list of variants of the basic original prototypes, which have provided formally simple models for the investigation of quantum-classical correspondence and of some general prop- erties of quantum transport. These include dynamical local- ization 关2兴, anomalous diffusion 关3–6兴, decay from stable phase-space islands 关7–9兴, electronic conduction in mesos- copic devices关10–12兴, nondispersive wave-packet dynamics 关13兴, effects of dissipation on quantum dynamics 关14兴, and lately directed transport关15–17兴. On the other hand, renewed interest on the physical side has been stimulated by experi- mental realizations 关18–21兴, which are now possible, under excellent control conditions, thanks to the science and tech- nology of cold and ultracold atoms. Unexpected advances of the theory have been prompted by such experiments. For instance, the so-called quantum accelerator modes 共QAMs兲 were discovered in experiments with cold atoms in periodi- cally pulsed optical lattices关22–25兴. Their underlying theo- retical model is a variant of the kicked rotor model, where the difference is that in between kicks, atoms are subject to gravity. When the kicking period is close to a half-integer multiple of the Talbot time关26兴, which is a natural time scale for the system, a fraction of atoms steadily accelerates away from the bulk of the atomic cloud at a rate and in a direction which depend on various parameter values. Though QAMs are a somewhat particular phenomenon, their theory关27–30兴 is a vast repertory of classic items of classical and quantum mechanics. QAMs are rooted in subtle aspects of the Bloch theory and have a relation to the Wannier-Stark resonances
of solid-state physics 关9兴. They are a purely quantal effect, and yet they are explained in terms of trajectories of certain classical dynamical systems by means of a “pseudo-quasi- classical” approximation, where the role of the Planck con- stant is played by a parameter
⑀
, which measures the detun- ing of the kicking period from a half-integer multiple of the Talbot time. This theory hinges on existence of a“pseudoclassical limit” for
⑀
→0. That means, for kicking periods close to half-integer multiples of the Talbot time, the quantum dynamics may formally be obtained from quantiza- tion of a classical dynamical system using⑀
as the Planck constant. This system is totally unrelated from the classical system that is obtained in the proper classical limit ប→0.Experimental and theoretical investigations on QAMs are currently focused on novel research lines: the observation of QAMs in a Bose-Einstein condensate关31–33兴, which allows a precise control on the initial momentum distribution, the analysis of QAMs for special values of the physical param- eters 关34,35兴, and in particular, when the kicking period is close to a rational multiple of the Talbot time 关36,37兴. The- oretical aspects concerning the latter problem are considered in the present paper.
QAMs are connected with an important feature of the KR model, namely, the KR resonances关38兴, which occur when- ever the kicking period is rationally related to the internal frequencies of the free rotor. The dynamics of the rotor at a quantum resonance 共QR兲 is invariant under momentum translations by multiples of an integer number. The least positive integer q, such that translation invariance in momen- tum space holds, is of the “order” of the resonance共Sec.II兲.
The half-Talbot time in atom-optics experiments is the period of the KR resonances of order q = 1 共i.e., “principal” reso- nances兲, so the originally observed QAMs are related to KR resonances of order 1.
In this paper we consider quantum motion in the vicinity of a higher-order KR resonance 共q⬎1兲 in the presence of gravity. Numerical共see Fig.1兲 and heuristic indications 关37兴 suggest that higher-order KR resonances, too, may give rise to QAMs. This has been substantiated by a theory关36兴 based on a nontrivial reformulation of the original pseudoclassical
*laura.rebuzzini@uninsubria.it
1050-2947/2009/79共3兲/033614共13兲 033614-1 ©2009 The American Physical Society
approximation. It has been remarked that in the case of higher-order resonances, no pseudoclassical limit exists and similarity to the case of quasiclassical analysis for particles with spin was noted but not explored. About the latter gen- eral problem 关39,40兴, it is known that although no single well-defined classical limit exists, and so no global quasi- classical phase-space approximation in terms of a unique classical Hamiltonian flow is possible, local quasiclassical approximations are nevertheless still possible, as provided by bundles of trajectories which belong to a number of different Hamiltonian systems.
In this paper we develop a formulation of the problem of QAMs near higher-order resonances in spinor terms. The quantum evolution at exact resonance is described by a mul- ticomponent wave function, namely a spinor of rank q given by the order of the resonance关38,41兴, and is generated by a time-independent spinor Hamiltonian关42,43兴. We show that the small-
⑀
analysis of quantum dynamics is formally equivalent to semiclassical approximation for a particle with spin-orbit coupling. Thus QAMs near higher-order reso- nances constitute a particular, though experimentally rel- evant, model system, in which this crucial theoretical issue can be explored. The semiclassical theory in关40兴 is not di- rectly applicable here because the dynamics is not specified by a self-adjoint spinor Hamiltonian but by a spinor unitary propagator instead. We therefore resort to an “adiabatic” an- satz, which allows decoupling spin dynamics from orbital motion. In this way we obtain q distinct and independent orbital one-period propagators. Each of them may be viewed as the quantization of a formally classical dynamical system, given by a map; however, the “pseudo-Planck-constant”⑀
explicitly appears in such maps in a form that precludes ex- istence of a⑀
→0 limit for the maps themselves, except for the q = 1 case, in which the pseudoclassical theory of Refs.关27,28兴 is recovered.
QAMs, detected by numerical simulations of the exact quantum dynamics near higher-order resonances, tightly cor- respond to stable periodic orbits of the maps. The accelera- tion of the modes is expressed in terms of the winding num- bers of the corresponding orbits and of the order of the resonance. Moreover, we derive some theoretical results, which generalize those obtained in 关28,29兴 for the principal resonances: a formula for the special values of quasimo- menta, which dominate the mode, and a classification of de- tectable modes by a Farey tree construction 关44,45兴 as a function of the gravity acceleration.
This paper is organized as follows. In Sec.IIthe Floquet operator, describing the one-step evolution of a kicked atom in a free-falling frame, is recalled and the resonant spinor dynamics in kicked particle共KP兲 model is briefly reviewed;
in Sec.III, the quantum motion in the vicinity of a resonance of arbitrary order is related to the problem of a particle with spin-orbit coupling. In Sec. IV, a “formally” classical de- scription of the orbital dynamics, associated to the QAMs, is achieved. Finally, in Sec.Vconnections between the theoret- ical results and possible experimental findings are discussed.
II. BACKGROUND
A. Floquet operator in the temporal gauge
In the laboratory frame, the quantum dynamics of the at- oms moving under the joint action of gravity and of the kicking potential is ruled by the time-dependent Hamiltonian 共expressed in dimensionless units兲
Hˆ
L共t兲 =pˆ2 2 −
xˆ + kV共xˆ兲兺n=−⬁
+⬁
␦
共t − n
兲, 共1兲where pˆ and xˆ are the momentum and position operators. The potential V共x兲 is a smooth periodic function of spatial period FIG. 1.共Color online兲 Momentum distributions in the time-dependent gauge, after t=100 kicks, for different values of the kicking period near the resonanceres= 3. Red color 共near p=0兲 corresponds to highest probability. The vertical dashed line corresponds to the resonant value. White full lines show the theoretical curves 关Eq. 共32兲兴, with 共left兲 q=2, 共r,s兲=共1,1兲 and 共right兲 q=7, 共r,s兲=共4,1兲 close to the higher-order resonance res/2=p/q=11/7. The initial quantum distribution is a Gaussian wave packet, reproducing the experimental conditions. The other parameters are k = 1 and g = 0.0386. All our numerical simulations refer to the choice V共兲=k cos . All plotted quantities are in dimensionless units.
2
. By denoting M, T, K, g, and 2
/G the atomic mass, the temporal period of the kicking, the kicking strength, the gravity acceleration, and the spatial period of the kicks, re- spectively, the momentum, position, and mass of the atom in Eq. 共1兲 are rescaled in units of បG, G−1, and M; then time t and energy come in units of M/共បG2兲 and ប2G2/M. The three dimensionless parameters k,
, and
in Eq.共1兲, which fully characterize the dynamics, are expressed in terms of physical quantities by k = K/ប,
= MgT/共បG兲, and
=បTG2/M =4
T/TB. TB= 4
M/共បG2兲 is the Talbot time 关26兴 and g=
/
is the rescaled gravity acceleration.Throughout the followingប=1 is understood.
For
= 0, Hamiltonian 共1兲 reduces to that of the KP model, which is a well-known variant of the KR model. The KP differs from the KR because the eigenvalues of particle momentum are continuous while those of the angular mo- mentum of the rotor are discrete. Due to Bloch theorem, the invariance of the KP Hamiltonian, under space translations by 2
, implies conservation of the quasimomentum
, which, in the chosen units, is the fractional part of the mo- mentum. The particle momentum is decomposed as p = N +
with N苸Z and 0ⱕ
⬍1. Conservation of quasimomen- tum enables a Bloch-Wannier fibration of the particle dynam- ics; the particle wave function is obtained by a superposition of Bloch waves, describing the states of independently evolving kicked rotors with different values of the quasimo- mentum共called
rotors兲.A remarkable feature of Hamiltonian 共1兲 is that, unless rescaled gravity g =
/
assumes exceptional commensurate values, the linear potential term breaks invariance under 2
space translations. Such an invariance may be recovered by going to a temporal gauge, where momentum is measured with respect to free fall. This transformation gets rid of the linear term and the new Hamiltonian reads关28兴Hˆ
g共t兲 =1
2
冉
Nˆ +
+
t冊
2+ kV共
ˆ 兲n=−兺⬁ +⬁␦
共t − n
兲, 共2兲where
= x mod共2
兲, Nˆ=−i
with periodic boundary condi- tions.The quantum motion of a

rotor in the “temporal gauge”共that is, “in the falling frame”兲 is described by the following Floquet operator on L2共T兲 共T denotes the 1–torus, param- etrized by
苸关−
,
关兲:Uˆ
共n兲 = e−ikV共ˆ兲exp兵− i共2兲关Nˆ +

+
n +共2兲兴2其, 共3兲 where n苸Z denotes the number of kicks. Operator 共3兲 de- scribes evolution from time t = n
to time t =共n+1兲
.B. Quantum resonances
We consider the problem of quantum accelerator modes in the vicinity of a generic resonance of the

rotor; the concept of quantum resonance is reviewed in this subsection. A QR occurs whenever quantum evolution commutes with a non- trivial group of momentum translations. A momentum trans- lation Nˆ →Nˆ+ᐉ 共recall ប=1兲 with ᐉ苸Z is described by the operator Tˆᐉ= eiᐉˆ. Throughout this section, we assume
= 0and then operator共3兲 is time independent. It commutes with Tˆᐉ if and only if 关46兴 共i兲
/2
= p/q with p and q coprime integers,共ii兲 ᐉ=rq with r苸N, and 共iii兲
=
/rp+rq/2 共mod 1兲 with
苸Z.In this paper we restrict to “primary” resonances, i.e., to resonances with r = 1 and ᐉ=q; in this case, q defines the order of the resonance. QRs of order 1 are called “principal resonances.”
A theory for QAMs in the vicinity of principal resonances was proposed in 关27,28兴. In this paper we consider quantum resonances of arbitrary order qⱖ1. The resonant values of the kicking period 共i兲, expressed in physical units, coincide with rational multiples of half of the Talbot time. We generi- cally denote Uˆ
res operator共3兲 at resonance, and

0the reso- nant values of quasimomentum, given by the above condi- tion 共iii兲, i.e.,
0=
/p+q/2 with
= 0 , . . . , p − 1.C. Bloch theory and spinors
Translation invariance under Tˆq enforces conservation of the Bloch phase
⬅
mod共2
/q兲, taking values in the Bril- louin zone B =关−
/q,
/q关. Loosely speaking, this means that
only changes by multiples of 2
/q, so
has the mean- ing of “quasiposition.” As we show below, at a QR, a Bloch- Wannier fibration of the rotor dynamics holds with respect to the quasiposition
.We use a rescaled quasiposition
⬅q
and accordingly resize the Brillouin zone to 关−
,
关. In all representations where quasiposition is diagonal, the state兩
典 of the rotor is described by a q-spinor
, specified by q complex functions
l共
兲=具
, l兩
典 共l=1, ... ,q兲. We shall use a representation where the spinor
共
兲, which corresponds to a given rotor wave function
共
兲=具
兩
典, is defined by
l共
兲 = 1冑2
兺m苸Z
ˆ 共l + mq兲eim,共4兲 l = 0, . . . ,q − 1,
where
ˆ 共n兲 共n苸Z兲 are the Fourier coefficients of
共
兲.Equation 共4兲 defines a unitary map a of L2共T兲 onto L2共T兲
丢Cq. Under this map, the共angular兲 momentum operator Nˆ is transformed to
Nˆ = − i
→ a共Nˆ兲a−1= − iq
丢Iˆ + Iˆ丢Sˆ, 共5兲 where Iˆ and Iˆ are the identity operators in L2共T兲 and in Cq, respectively, and Sˆ is the spin operator inCq,Sˆ =兺
l=0 q−1
l兩l典具l兩, 共6兲
where兩l典, l=0, ... ,q−1, is the canonical basis in Cq. Thus, in spinor representation, the momentum operator is the sum of the orbital operator −iq
ˆ丢Iˆ and the spin operator Iˆ丢Sˆ. In this picture, the rotor is characterized by “orbital” observ- ables 共
, −i
兲 and by the spin observable. Bold symbols denote vectors inCqand q⫻q matrices.D. Resonant spin dynamics
At resonance, quasiposition is conserved under the discrete-time evolution defined by Eq. 共3兲, so whenever it has a definite value
, no orbital motion occurs, and spin alone changes in time. Therefore the evolution is described by a unitary q⫻q matrix Aˆ共
兲 such that as
共
兲 evolves into Uˆres
共
兲, the corresponding spinor
共
兲 evolves into the spinor Aˆ 共
兲
共
兲. The explicit form of the spin propagator Aˆ 共
兲 is easily computed by using Eq. 共3兲 under resonance conditions. With the specific choice V共
兲=cos共
兲, one finds 共details can be found in Appendix A兲Aˆ 共
兲 = e−ikVˆ 共兲e−iGˆ, 共7兲Gˆ ⬅ Gˆp,q,0=
pq共Sˆ +

0Iˆ兲2, 共8兲Vˆ 共
兲 =12
再
兺q−2l=0共兩l典具l + 1兩 + 兩l + 1典具l兩兲 + 兩0典具q − 1兩ei+兩q − 1典具0兩e−i
冎
. 共9兲Operator 共7兲 may be written in terms of a “resonant Hamil- tonian.”
E. Bands
The resonant Hamiltonian Hˆres共
兲 is a Hermitian matrix of rank q such thatAˆ 共
兲 = e−iHˆres共兲. 共10兲 It is uniquely defined under the condition that its eigenvalues 关i.e., the eigenphases of Aˆ共
兲兴 lie in 关0,2
关. Explicit calcu- lation of eigenvalues and eigenvectors of Aˆ 共
兲, hence of the resonant Hamiltonian, is trivial for q = 1 and is easily per- formed for q = 2 in terms of Pauli matrices关43兴 共such a case is reviewed in Appendix B兲. However, for q⬎2 analytical calculation is prohibitive.Eigenphases of Aˆ 共
兲 are smooth periodic function of qua- siposition
. As
varies in关−
,
关, they sweep bands in the quasienergy spectrum of the resonant evolution described by Uˆres 关47兴. They also depend on the kicking strength k and will be denoted by
l=
l共
, k兲 in the following 共l= 0 , . . . , q − 1兲. In the case q=1,
0共
, k兲=k cos共
兲. For q⬎1 the eigenvalues are nontrivial functions of the kick strength k. For fixed q⬎2 bandwidths tend to increase with k, eventually giving rise to complex patterns of avoided crossings. Examples of
and k dependence of eigenphases are shown in Fig. 2for 共a兲 q=2 and 共b兲 q=7. For q⬎2 the bandwidths depend also on l. In the resonant representation 关i.e., in the representation in which the resonant propagators in Eq.共10兲 are diagonal兴, the spinor components in Eq. 共4兲 evolve independently.III. NEAR-RESONANT DYNAMICS AND SPIN-ORBITAL DECOUPLING
We are interested in quantum motion, described by Eq.
共3兲, in the vicinity of a QR, namely, when the kicking period is
= 2
p/q+⑀
, where the detuning⑀
of the period from the resonant value
res= 2
p/q is assumed to be small. The one- step evolution operator 共3兲 may be factorized as 共apart an irrelevant phase factor兲Uˆ
共n兲 = UˆresUˆ
nr共n兲, 共11兲
Uˆ
nr共n兲 = exp关− i共12
⑀
Nˆ2+ DnNˆ 兲兴, 共12兲 where Dn=
共
+
n +
/2兲−2
p
0/q.A. Adiabatic decoupling of spinor and orbital motions Translation invariance 共in momentum兲 is now broken by Uˆ
nr, so quasiposition is not conserved any more. The evolu- tion of a spinor
苸L2共T兲丢Cq is ruled by the time- dependent Schrödinger equationi
⑀
t
= Hˆ 共t兲
, 共13兲Hˆ 共t兲 =
⑀
Hˆres兺n=−⬁ +⬁
␦
共t − n兲 + Hˆ0共t兲, 共14兲Hˆ
0共t兲 =1
2
⑀
2共− iq
丢Iˆ + Iˆ丢Sˆ兲2+⑀
D关t兴共− iq
丢Iˆ + Iˆ丢Sˆ兲, 共15兲 where 关t兴 denotes the integer part of t and 共Hˆres
兲共
兲= Hˆres共
兲
共
兲. Note that Hˆ0共t兲 is constant in between kicks.Both sides of the Schrödinger equation have been multiplied by
⑀
to make it apparent that the detuning⑀
plays the role of an effective Planck constant in what concerns the motion between the␦
-kicks.The Hamiltonian operators Hˆres and Hˆ
0 are not simulta- neously diagonal on the same basis. In the resonant represen-
0 2 4 6
-2 0 2
υ ω
j(υ ,k)
(a)
-2 0 2
υ
(b)
FIG. 2. 共Color online兲 Eigenvalues of the resonant Hamiltonian Hˆres共兲 for different values of the kicking constant k=1 共red兲, 3 共green兲, and 5 共blue兲 and 共a兲 q=2, p=3 and 共b兲 q=7, p=11. Wider bands in 共a兲 correspond to higher values of k; narrower bands to small k.
tation, the spinor components are mixed by Eq.共15兲 during the evolution; we use an ansatz of the Born-Oppenheimer type in order to decouple orbital 共slow兲 motion from spin 共fast兲 motion.
The detuning
⑀
controls as well the separation between the different time scales of the system. At exact resonance 共i.e.,⑀
= 0兲 the decoupling is exact because motion is re- stricted to the eigenspaces of the resonant propagator 共10兲.These subspaces are defined by the spectral decomposition of the resonant Hamiltonian, which we write in the form
Hˆres共
兲 =兺j=0 q−1
j共
,k兲Pˆj共
兲,Pˆ
j共
兲 = 兩
j共
兲典具
j共
兲兩, 共16兲 where
j共
兲 is the normalized eigenvector of Hˆres共
兲 which corresponds to the eigenvalue
j共
, k兲. For each value of
, the operators Pˆj共
兲 in Eq. 共16兲 are projectors in Cq. We de- note by Pˆj the projectors in the full Hilbert space L2共T兲
丢Cq, which act on spinors according to 共Pˆj
兲共
兲= Pˆ
j共
兲
共
兲. The subspaces Hj whereupon the Pˆj project are the “band subspaces” and are not invariant for the full Hamiltonian 共14兲. By using the ansatz that band subspaces are almost invariant for small
⑀
, we next decouple the共as- sumedly “fast”兲 spin variables from the orbital 共“slow”兲 ones. We assume that the decoupled evolution inside the band subspaces provides a good description of the exact evo- lution when⑀
is small because the leading error terms are linear in⑀
.Our approximation consists in replacing the exact dynam- ics, ruled by the Hamiltonian in Eqs. 共14兲 and 共15兲, by an adiabatic evolution generated by the Hamiltonian
Hˆdiag共t兲 =兺
j=0 q−1
PˆjHˆ 共t兲Pˆj=
⑀
Hˆresn=−⬁兺+⬁␦
共t − n兲 +兺j=0 q−1
PˆjHˆ
0共t兲Pˆj. 共17兲 In the case of time-independent Hamiltonians, such pro- jection on band subspaces, aimed at separating fast and slow time scales, is basically a Born-Oppenheimer approximation 关48兴. In the case of kicked dynamics this projection should be performed on the “effective,” time-independent Hamil- tonian Hˆ
eff, which generates over a unit time the same evo- lution as does the kicked Hamiltonian. The effective Hamil- tonian is not known in closed form, although it can be expressed by a sum of infinite terms, ordered in powers of
⑀
关43,49兴. Our ansatz is somehow related to a rough approxi- mation Hˆeff⯝
⑀
Hˆres+ Hˆ0. We assume that this is valid in some restricted parameter regimes 共see further comments in Sec.IV兲.
A spinor in Hj has the form
共
兲
j共
兲 with
苸L2共T兲 and may thus be described by a scalar wave function
共
兲 共the amplitude of the spinor on the jth resonant eigenstate兲.Evolution inside the band subspaceHjis ruled by the “band
Hamiltonian” Hˆ
j共t兲= PˆjHˆ 共t兲Pˆjand direct calculation by using Eqs. 共14兲 and 共15兲, shows that band Hamiltonians have the following form:
Hˆ
j共t兲 =
⑀
j共
,k兲n=−⬁兺+⬁␦
共t − n兲 + Hˆ0共j兲共t兲, 共18兲 whereHˆ
0共j兲共t兲 = −1
2
⑀
2q2
2−共⑀
2q2具
j兩
˙j典 + i⑀
2qSj+ i⑀
qD关t兴兲
+12
⑀
2共Sj⬙
− q2具
j兩
¨j典 − i2qSl⬘
兲+
⑀
D关t兴共Sj− iq具
j兩
˙j典兲. 共19兲 The dots denote derivatives with respect to
, andSj共
兲 = 具
j共
兲兩Sˆ兩
j共
兲典,S
⬘
j共
兲 = 具
j共
兲兩Sˆ兩
˙j共
兲典,S
⬙
j共
兲 = 具
j共
兲兩Sˆ2兩
j共
兲典. 共20兲B. Band Hamiltonians
We now note that the problem can be formulated as the evolution of a particle in a fictitious magnetic field, which takes into account the average effects of spin degree of free- dom on the orbital motion. We derive a simpler form for the band Hamiltonians 关Eq. 共26兲兴. By the introduction of mag- netic vector and scalar potentials, operator共19兲 may be writ- ten in the form
Hˆ
0共j兲共t兲 =1
2
⑀
2q2关− i
−Aj共
兲兴2 +⑀
qD关t兴关− i
−Aj共
兲兴 +12
⑀
2Bj共
兲. 共21兲 The “geometric” vector potentialAj共
兲 and the scalar poten- tial Bj共
兲 are determined by the structure of the resonant eigenvectors
j共
兲 via the following relations:Aj共
兲 = i具
j共
兲兩
˙j共
兲典 −1qSj共
兲, 共22兲 Bj共
兲 = Sj⬙
共
兲 + 2qISj⬘
共
兲 − q2A2j共
兲 + q2具
˙j共
兲兩
˙j共
兲典.共23兲 Reality of such potentials follows from Eqs. 共20兲 and from the fact that 具
j共
兲兩
˙j共
兲典 is purely imaginary thanks to normalization. The vector potential is gauge dependent;eigenvectors
j共
兲 are determined up to arbitrary
-dependent phase factors and so operator共21兲 may be fur- ther simplified by a gauge transformation,
j共
兲→
j共
兲eij共兲. Under such a transformation,Aj共
兲 changes to A˜j共
兲=Aj共
兲−˙j共
兲 and Bj共
兲 does not change. The transformation may be chosen so thatA˜j共
兲 = const = −␥
j,q− ⬅␣
j, 共24兲 where␥
j,q= 12
i冕
− d
具
j共
兲兩
˙j共
兲典, = 1
2
q冕
− d
Sj共
兲.This immediately follows from Eq. 共22兲 and from the re- quirement that eigenvectors be single valued. Note that 2
␥
j,qis the geometric共Berry’s兲 phase 关50,51兴. We thus as- sumeA˜j=
␣
j; this choice corresponds to the Coulomb gauge.In conclusion, in the jth band subspace, the band dynam- ics is described by the following Schrödinger equation:
i
⑀
t
共
,t兲 = Hˆj共t兲
共
,t兲, 共25兲Hˆ
j共t兲 =
⑀
j共
,k兲兺n=−⬁
⬁
␦
共t − n兲 +1 2⑀
2Bj共
兲 +12
⑀
2q2共− i
−␣
j兲2+⑀
qD关t兴共− i
−␣
j兲. 共26兲 The multicomponent Schrödinger Eq.共13兲, for the q-spinor wave function
共
, t兲, is then reduced to q scalar Schrödinger equations 共25兲, each of which determines the independent evolution of a rotor wave function
共
, t兲.IV. PSEUDOCLASSICAL DESCRIPTION OF ORBITAL MOTION
We now derive a description of the dynamics of the or- bital observables共
, −i
兲, restricted inside each of the band subspacesHj, by formally classical equations of motion. We introduce a “pseudoclassical” momentum operator Iˆ, defined as follows:Iˆ = − i
⑀
, 共27兲which differs from the orbital momentum because of the re- placement of the Planck constant共=1兲 by
⑀
. If the same role is granted to⑀
in Eq.共25兲, then in classical terms, the effec- tive band dynamics in the jth band subspace looks like a rotor dynamics, with angle coordinate
and conjugate mo- mentum I, ruled by the kicked HamiltonianHj共
,I,t兲 =⑀
j共
,k兲n=−⬁兺+⬁␦
共t − n兲 + Fj共
,I,t兲,Fj共
,I,t兲 =12q2I2+ D关t兴qI −
⑀
q2␣
jI +12
⑀
2Bj共
兲. 共28兲 Terms, independent of I and
, have been neglected. This Hamiltonian describes a classical kicked dynamics. By drop- ping terms beyond first order in⑀
, the map from immediatelyafter the nth kick to immediately after the共n+1兲th kick is
n+1=
n+ q2In+ 2
⍀qn + mod共2
兲,In+1= In−
⑀
˙j共
n+1,k兲, 共29兲 where ⍀=
/共2
兲 and =q共−⑀
q␣
j+
⍀+
− 2
p
0/q兲.The meaning of the pseudoclassical map 共29兲, as a descrip- tion of the nearly resonant quantum dynamics, will be dis- cussed in Sec.IV C.
A. Pseudoclassical maps and quantum accelerator modes We now describe how quantum accelerator modes appear in the present framework. The explicit dependence on time of map共29兲 is removed by changing the momentum variable to Jn= q2In+ 2
⍀qn+. In the variables 共J,
兲 the map is 2
periodic in J and so it may be written as a map on the 2–torus,
n+1=
n+ Jn mod共2
兲,Jn+1= Jn−
⑀
q2
˙j共
,k兲 + 2
⍀q mod共2
兲. 共30兲 In the case of q = 1, this map reduces to one which was in- troduced in关27兴 in order to explain the QAMs that had been experimentally observed near principal resonances. For q⬎1, it has q different versions, labeled by j=0, ... ,q−1.
Similar to the case q = 1, the stable periodic orbits of each of these versions are expected to give rise to QAMs. Indeed, each stable periodic orbit of map共30兲 corresponds to a stable accelerating orbit of map 共29兲 because the difference be- tween momentum In and momentum Jn linearly increases with time. More precisely, let 共
0, J0兲 be initial conditions for a periodic orbit of period s and winding number r/s. The increment of J after time ns 共measured in the number of kicks兲 is 2
rn; therefore, the increment of the original mo- mentum variable isIsn− I0= aIsn, aI=2
q
冉
qsr −⍀冊
, 共31兲with I0=共J0− 兲/q2. This formula共31兲 yields the acceleration of a stable orbit of the pseudoclassical dynamics共29兲, and it is precisely this orbit that may give rise to QAMs in the vicinity of resonances of arbitrary order. As a matter of fact, numerical simulations reveal QAMs near higher-order reso- nances, in correspondence with periodic orbits of maps in Eq.共30兲. In Sec.V, we explain how the analysis of the stable periodic orbits of maps in Eq. 共30兲 may help to resolve the complex pattern of QAMs presented in Fig.1.
Thanks to Eqs.共5兲 and 共27兲, the physical momentum N is related to I by N = qI/
⑀
+ j; therefore, the physical accelera- tion is given bya =2
⑀ 冉
qsr −⍀冊
. 共32兲Although the analytical derivation of the maps is based on the resonant Hamiltonian, which is known in closed form only for q = 1 , 2, the practical use of Eqs. 共30兲 only requires the resonant eigenvalues, which can be easily computed by a numerical diagonalization of a q⫻q matrix.
In Fig.3, examples of phase space of maps in Eq.共30兲 are shown for 共a兲 and 共b兲 q=2, 共c兲 q=7, and 共d兲 q=15, in pa- rameter regimes in which QAMs are present. The plotted periodic orbits correspond to some of the modes shown in Fig.1. For instance, in Fig.3共a兲the stability island of a fixed point of one of the maps in Eq. 共37兲 for q=2 is plotted for
/2
= 1.455; this fixed point corresponds to the huge mode on the left side of Fig.1. As shown in Fig.4, a distribution of phase-space points, which initially fall inside the stability island, describes an ensemble of atoms generating the QAM.Of the q maps in Eqs.共30兲, the one, which crucially con- tributes in determining the observed QAMs, is generally that with the widest bandwidth; indeed this map presents, in most cases, classical structures, such as stability islands, with an area greater than the effective Planck constant
⑀
and thus supporting many quantum states 关9兴. However, also islands with a size comparable or smaller than⑀
, and scars, related to unstable periodic orbits, may affect the quantum system, giv- ing rise to minor structures in the atomic momentum distri- bution关28兴.B. Special values of quasimomenta
A QAM arises when the initial wave packet is centered in momentum N0, related to I0 by
N0= qI0
⑀
+ j = 1q
⑀
共J0+ 2
n兲 + q␣
j− 1
⑀
共
⍀ +
− 2
p
0/q兲 + j, 共33兲 with n苸Z. As in the case for main resonances, we expectthat the modes will be especially pronounced when quasimo- mentum is fine tuned; in view of Eq.共33兲 such optimal val- ues of

are determined by the condition
= −⑀
共N0− j − q␣
j+
0兲−J0+ 2
m q
+ −
2 +

0 mod共1兲, 共34兲 with
0=p+q2 and
= 0 , 1 , . . . , p − 1. A wave packet initially localized in N0+
will be mostly captured inside a QAM;indeed in this case, the overlap between the stability island and the initial wave packet is maximal. Formula 共34兲 is a generalization of the result derived for q = 1 in关27兴 and ex- perimentally verified in 关31兴; it reduces to the expression in 关27兴 for
␣
0= 0关see Appendix B兴.This picture is confirmed by Fig.4, in which the quantum phase-space evolution of a

rotor, with a quasimomentum given by Eq.共34兲, and the pseudoclassical motion are com- pared. The initial state of the rotor is a coherent wave packet centered in the共r,s兲=共1,1兲 fixed point, plotted in Fig.3共a兲, corresponding to the⑀
-classical accelerator mode on the left part of Fig. 1, in the vicinity of q = 2 resonance. The mode moves with an acceleration equal to 0.2988 according to Eq.共32兲.
C. Validity of the pseudoclassical description
We now come back to the meaning of the pseudoclassical description, as pseudoclassical dynamics共29兲 still explicitly retains the “Planck constant”
⑀
. In the case when q = 1, there is a single resonant eigenvalue, given by
0共
, k兲=k cos共
兲, so the pseudoclassical dynamics共29兲 has a well-defined limit for⑀
→0, k→⬁, and k⑀
→k˜ with 兩k˜兩⬍⬁. This limit dynam- ics was discovered and analyzed in关27,28兴. This is no longer true when q⬎1, and then the relation between the band dy- -20 2
-2 0 2
-2 0 2
J
(a) (b)
υ
J
υ
J
(c)
υ
(d)
-2 0 2
FIG. 3. 共Color online兲 Phase portraits of the two-torus maps with k = 1共k˜=⑀兲 and g=0.0386. 共a兲 and 共b兲 refer to map 共37兲 with mp,= 1;共c兲 and 共d兲 refer to Eq. 共30兲. The periodic orbit in 共d兲 has period s = 132 and is associated to 132 stability islands; r and s are not coprime共r=1056, s=132兲. One of the small islands of the chain is magnified in the inset. The values of p/q;/2; 共⑀兲; 2⍀; 共r,s兲;
and j are respectively:共a兲 3/2, 1.445 共-0.2827兲, 3.2261, 共1,1兲, 1; 共b兲 3/2, 1.5025共0.0157兲, 3.4401, 共23,21兲, 0; 共c兲 11/7, 1.5375 共0.2132兲, 3.6023,共4,1兲, 3; 共d兲 22/15, 1.485 共0.1152兲, 3.3605, 共8,1兲, 15.
0 10 20 30
-2 0 2
θ
p
(a)
26 28 30 32 34
-2 0 2
(b)
θ
FIG. 4. 共Color online兲 共a兲 Contour plot at time t=100 of the Husimi distribution of the wave packet of a  rotor with 
= 0.1672, given by Eq.共39兲 with N0= J0= 0, p = 3, =0, and n=1.
The rotor is initially prepared in a coherent state centered in the 共r,s兲=共1,1兲 fixed point of Fig.3共a兲.共b兲 Magnification of 共a兲. The black spots in the centers of the contours are an ensemble of clas- sical phase points, initially distributed in a circle of area ⬃⑀ cen- tered at the mode. They evolve according to the⑀-classical dynam- ics 共38兲 with =−0.5709. The other parameter values are k=1,
= 1.455⫻2共⑀=−0.2827兲, and g=0.0386.
namics and the pseudoclassical dynamics 共29兲 is less trans- parent. The quantum band dynamics is still, formally, the quantization of the classical kicked dynamics共29兲 using
⑀
as the Planck constant. Nevertheless, the latter dynamics con- tains the Planck constant⑀
in crucial ways, which preclude existence of a limit for⑀
→0. To see this, note that
j共
, k兲 depends on its arguments only through the real variables u= k sin共
/q兲 and v=k cos共
/q兲 关cf. the form of the resonant evolution 共A3兲 in Appendix A兴, that is,
j共
, k兲=G共u,v兲, where G is a smooth oscillatory function independent of k.Hence,
⑀
˙j共
,k兲 =⑀
kq兵cos共
/q兲
uG − sin共
/q兲
vG其. 共35兲 Existence of a limit demands⑀
k→k˜; but then, except in the trivial case k˜ =0, the arguments of the G functions in Eq. 共35兲 diverge and so the second term of Eq.共35兲 appears to oscil- late faster and faster as⑀
→0 without a well-defined limit.Nonexistence of a pseudoclassical limit for the quantum dynamics was established in 关36兴, by a stationary phase ap- proach, with no recourse to the band formalism. It was none- theless pointed out that despite absence of such a limit, QAMs may be associated to certain rays, which correspond to trajectories of some formally classical maps. The meaning of the latter maps is, at most, that of providing local phase- space descriptions near QAMs. Similar remarks apply in the case of the pseudoclassical maps in Eq.共29兲.
It is worth recalling that maps in Eq. 共29兲 were derived from an ansatz, which would be optimally justified if the effective Hamiltonian of kicked dynamics could be replaced by the sum of the free and of the kicking Hamiltonians共Sec.
III兲. This approximation is obviously invalid in a global sense, yet in spinless cases, it is known to work remarkably well near stable fixed points 关9兴; indeed, in the KR case it yields a pendulum Hamiltonian, which provides a good de- scription of the motion near the stable fixed point of the standard map. This may be seen as a qualitative justification for the use of maps in Eq. 共29兲 if restricted to the search of QAMs.
D. Case q = 2
While the expressions in Sec.IV Aare quite general, we may accomplish a detailed analysis when q = 2 and V共
兲= k cos共
兲. We recall that the resonant values of quasimo- menta are
0=
/p with
= 0 , . . . , p − 1 according to共iii兲 in Sec. II B.In such a case the eigenvalues
j共
, k兲 共j=0,1兲 of the resonant Hamiltonian can be written down explicitly 共see Appendix B兲,
j=
2
2p − mp,
冋
4 +共− 1兲j arccos冉
cos冑2v冊 册
, 共36兲withv = k cos共
/2兲 and mp,=共−1兲关共p+1兲/2兴+.Therefore, our theory produces two maps 关Eqs. 共30兲兴, which take the form
t+1=
t+ Jt mod共2
兲,Jt+1= Jt+ 4
⍀ + 2k˜hj共
t+1兲 mod共2
兲, 共37兲 withhj共
兲 = − mp,共− 1兲j sin冉
2冊
冑1 + sinsin关k cos共2关k cos共2兲兴2兲兴. Going back to the time-dependent form, the maps are written as
t+1=
t+ 4It+ 4
⍀t + mod共2
兲,It+1= It+˜k
2hj共
t+1兲, 共38兲 with = 2共⑀␦
j,1+
⍀+
−
兲 and where we have used␣
j= −12
␦
j,1关see Appendix B兴.We remark that in the case q = 2, there aren’t any avoided crossings between the eigenvalues关Eq. 共36兲兴, for an arbitrary value of k. We may also check the selection criterion for quasimomenta that in the present case assumes the form

= −⑀
冉
N0+
p冊
+J0+ 2q
n−
2 +
p mod共1兲, 共39兲 with
= 0 , . . . , p − 1. A scan over possible
values reveals that indeed QAMs are greatly enhanced around the values predicted by Eq.共39兲; this is confirmed by Fig. 5, in which the momentum probability transferred to the mode is shown as a function of
.V. MODE SPECTROSCOPY AND CONNECTIONS WITH COLD ATOM EXPERIMENTS
A. Farey ordering of QAMs near a fixed resonance We now elucidate how our findings apply to inspection of density plots similar to the one illustrated in Fig.1. We point out that such a picture is of direct physical significance since typical experimental protocols maintain k and g fixed while
0 0.2 0.4 0.6
0 0.2 0.4 0.6 0.8 1
β P
L(β )
FIG. 5. 共Color online兲 Probability inside a box of extension equal to L = 6⯝⌬Jq/兩⑀兩 in momentum, moving according to Eq.
共32兲 for p=3, q=2, 共r,s兲=共1,1兲, and ⑀=−0.2828 as a function of quasimomentum of the  rotor. ⌬J is the size of the island in J, plotted in Fig.3共a兲. Dashed vertical lines refer to special values of quasimomenta, given by formula 共39兲 with =0,1,2 and N0= J0
= 0, n = 1, and = g=0.3529. The probability is shown at time t
= 100共red, or higher curve兲 and t=200 共blue curve兲. The parameter values are the same as in Fig.1.
performing a scan on the pulse period
. Such a scan, in the present context, has to be carried out around a resonant value, namely,
共⑀
兲=2
p/q+⑀
. Density plots of momentum distribution disclose the presence of QAMs, since after a fixed number of kicks their momentum is linearly related to acceleration共32兲; a depends on⑀
through the “bare” winding number q⍀,q⍀共
⑀
兲 = q2
g冉
2
pq+⑀ 冊
2, 共40兲and on the “dressed” winding number of the pseudoclassical map r/s, which individuates the mode. We denote by ⍀ⴱthe resonant 共
⑀
= 0兲 value of the bare winding number 共notice that for⑀
= 0 the maps correspond to pure rotation in J兲,⍀ⴱ⬅ q⍀共0兲 = 2
p2qg, 共41兲
which is independent of the mapping index j. Formula共41兲 is a generalization of the analogous result found for q = 1 关29,52兴.
As analyzed in 关29,52兴 for principal resonances, the pa- rameter space of map共30兲 is characterized by the presence of regions共Arnold tongues兲, in which stable periodic orbits ex- ist. Close to resonances we expect that mode-locking struc- ture of the pseudoclassical maps singles out modes whose winding number provides rational approximants to⍀ⴱ; at the same time fat tongues are associated to small s values so the corresponding modes should be more clearly detectable. This is the physical motivation underlying Farey organization of observed modes; whenever we observe two modes labeled by winding numbers r1/s1and r2/s2共r1/s1⬍⍀ⴱ⬍r2/s2兲, the fraction with the smallest denominator bracketed by the winding pair is the Farey mediant共r1+ r2兲/共s1+ s2兲.
We can now analyze in more detail共Fig.1兲 which repre- sents a numerical simulation of experimental momentum dis- tribution after t = 100 vs
, which assumes values around a second-order resonance, namely,
res/2
= p/q=3/2. All pa- rameters are chosen to be accessible to experiments and the initial atomic distribution reproduces that employed in 关22–25兴; g=0.0386, k=1, and the initial state is a mixture of 100 plane waves sampled from a Gaussian distribution of momenta with full width at half maximum 共FWHM兲 ⬃9.Full lines in the figure delineate momentum profiles consis- tent with acceleration 共32兲, with 共r,s兲 given by winding number r/s of corresponding stable periodic orbits of maps 关Eq. 共30兲兴.
The value of ⍀ⴱ and the first few rational approximants 共obtained upon successive truncation of the continued frac- tion expansion兲, corresponding to detectable modes, are
⍀ⴱ⯝ 1.091 389 3 = 1 + 关10,1,16,3,3, ...兴,
r
s= 1, 11 10, 12
11, . . . 共42兲
The first one共r,s兲=共1,1兲 is shown with the white line on the left of Fig. 1 and the stability island of the corresponding fixed point is shown in Fig.3共a兲. The second and third are marked by full yellow lines in Fig. 6, which is an enlarge-
ment of Fig. 1 in the region 1.49ⱕ
/2
ⱕ1.506 25, calcu- lated for time t = 200. Farey organization is exemplified by the appearance of the共23, 21兲 QAM, whose winding number is the Farey composition of the 共11, 10兲 and the 共12, 11兲 modes; the correspondent stable periodic orbit is plotted in Fig.3共b兲. Through Farey composition law we may also iden- tify observed modes to the left of
res, as shown in Fig.6.B. Visibility of resonances of different order
The complexity of mode spectroscopy is further enhanced by the fact that within some interval in
, arbitrarily many different resonant values occur. As a matter of fact it is pos- sible to recognize in Fig. 1 the modes coming from a wide set of resonances; besides q = 2 also q = 7 , 15, 17, 21, 36, 40 contribute QAMs in the selected range; this is shown in Fig.1 for q = 7 and in Fig. 7for the other resonances. No QAM with q = 13 could be resolved in the range of Fig.1.
Farey composition is still of some use in the identification of the resonances to which modes they belong; for instance, the very large mode on the right of the figure belongs to a QR between p/q=3/2 and p/q=2/1; applying Farey com- position successively, we get the sequence p/q=5/3,8/5 共outside the plotted range in
兲 and then 11/7, to which the mode belongs to. The accumulation point of the resonance p/q=11/7 is ⍀ⴱ⯝4.192 320 7=4+关5,...兴. The mode shown in Fig.1corresponds to the first principal convergent of⍀ⴱ, i.e., to the fixed point共r,s兲=共4,1兲, shown in Fig.3共c兲. The same occurs for the modes near resonances of higher q, shown in Fig. 7.We remark that a hierarchy in resonant fractions looks more cumbersome than the one considered for winding num- bers, as for instance there does not seem to be any straight- forward dependence on the size of q. Numerical data how- ever suggest that detectable modes appear in the vicinity of resonances leading to almost integer⍀ⴱ, i.e., when the frac- FIG. 6. 共Color online兲 Enlargement of Fig. 1 in the region 1.49ⱕ/2ⱕ1.506 25 around the resonance res= 3 共p/q=3/2兲.
Momentum distributions are calculated after t = 200 kicks. Full lines show the theoretical curves 关Eq. 共32兲兴; the yellow ones refer to principal convergents of ⍀ⴱ, listed in Eq. 共42兲. Starting from the left, the modes correspond to the stable periodic orbits of maps in Eq. 共37兲 with 共r,s兲=共14,13兲, 共25, 23兲, 共12, 11兲, 共23, 21兲, and 共11, 10兲.
