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periodically - kicked quantum systems , such as the @xmath0-kicked particle ( @xmath0-kp ) , have long played a central role in studies of quantum chaos and the correspondence between quantum behavior and the underlying classical dynamics @xcite . advances in the manipulation of cold atoms have permitted the experimental realization of these systems in pulsed optical lattices @xcite . experiments with sodium and cesium atoms have demonstrated the phenomenon of `` dynamical localization '' ( dl )  the quantum suppression of classical chaotic diffusion  and established the suitability of these systems as an arena for the study of effects arising from quantum chaos . when treating conservative quantum systems it is frequently useful to study the system s energy - eigenstates , and for periodically driven systems the appropriate generalization of these states is given by the floquet states . this approach has provided extensive insight into the properties of the standard qkp , and has shown , for example , that dl arises directly from the exponential localization of the system s floquet states @xcite . observed momentum oscillations associated with chaos - assisted tunneling , in experiments using periodically - driven cold atoms @xcite and becs @xcite have also been analysed with floquet theory ; it was found that the oscillation period is determined by the splittings of the floquet phases of a pair of symmetry - related eigenstates . the statistics of qkp floquet quasi - energy spectrum have been studied extensively see e.g. @xcite and compared with the predictions of random matrix theory . notably , though , the @xmath0-qkp has poissonian short - range statistics ( which are typically associated with integrable dynamics ) even for very large values of @xmath1 , where the dynamics is fully chaotic . this has been shown to be a further consequence of dl @xcite .    however , a series of recent theoretical @xcite and experimental @xcite studies of cold atom systems subjected to repeating cycles of unequally spaced kicks revealed dynamics rather different from that found in the corresponding standard qkp systems . two types of unequally @xmath0-kicked systems were investigated . the first @xcite consists of a @xmath0-kp with small perturbations made to the kick - period . we term it the perturbed - period kp . in the second system , @xcite the system is periodically subjected to _ pairs _ of closely - spaced kicks . this is referred to as the double @xmath0-kp or 2@xmath0-kp .    in a theoretical study , the perturbed - period kp was found to yield a quantum momentum current even in the chaotic regime @xcite . this was unexpected in a chaotic hamiltonian system , since to date only mixed phase - space ratchet mechanisms had been investigated @xcite . a simple definition of a ratchet is a spatially periodic device which produces a current without net bias ( ie the time and space - averaged forces are zero ) . most of the extensive ratchet literature deals with dissipative or brownian ratchets @xcite and comparatively little theoretical work has been undertaken on hamiltonian ratchets , which are dissipation and noise - free . in @xcite , a momentum distribution with a non - zero average ( constant in time ) was obtained experimentally from an atomic cloud with initial zero average momentum . we are unaware of any other experimental studies of hamiltonian quantum ratchets : all implementations to date have been of dissipative / brownian ratchets . hence the results from @xcite and also reproduced here , represent the only implementation of a type of hamiltonian quantum ratchet , whether chaotic or mixed phase - space . in ref . @xcite it was also proposed that the chaotic diffusive properties of the perturbed - period kp could be exploited to filter cold atoms , according to their momenta , by controlling the dynamical localization .    for the second system , the 2@xmath0-kp , a theoretical and experimental study @xcite revealed that the diffusion is dominated by long - ranged correlations which control escape from well - defined momentum trapping regions . this combination of strong chaotic diffusion and long - ranged correlations is novel : strong chaos is generally associated with rapidly decaying correlations . it is clear that floquet theory is central to the analysis of chaotic , time - periodic quantum systems . the need to understand further the chaotic hamiltonian ratchet as well as the 2@xmath0-kp motivated this study of the floquet states of these systems . the paper is organized as follows . in the next section we review the well - known @xmath0-kp , then introduce the perturbed @xmath0-kp and the double @xmath0-kp systems . in section iii we give a brief review of the floquet approach . in section iv we compare the results with recently obtained experiments on these systems . finally in section v we give our conclusions . the hamiltonian for the usual @xmath0-kp can be written as @xmath2 where @xmath1 is the kick strength and @xmath3 is the time interval between successive kicks . consider its effect on an ensemble of particles with a gaussian momentum distribution centered on @xmath4 , @xmath5 $ ] . the classical dynamics depends only on the parameter @xmath1 , and for values larger than @xmath6 , the chaotic diffusion is not bounded by classical barriers . in this regime the ensemble will diffuse in momentum space , its average energy growing linearly with time as @xmath7 , where , to lowest order , the diffusion rate is given by @xmath8 . the distribution will thus remain gaussian , although its width will increase with time as @xmath9 . in contrast , the quantum system only follows this behavior up to a timescale @xmath10 @xcite , after which diffusion in momentum space is suppressed  dynamical localization ( dl ) . such a system will asymptotically evolve towards a characteristic exponential momentum distribution , @xmath11 $ ] , with constant width @xmath12 , which thus acts as an experimental fingerprint for dl @xcite . as dl is a wave - coherent effect , the quantum system must preserve coherence over at least the timescale @xmath13 for this effect to be observable . indeed , it has been verified experimentally @xcite that the dl profile does not survive the presence of noise or dissipation , and that with decoherence a more gaussian profile for @xmath14 will be produced .    in both classical and quantum cases the behavior of the standard @xmath0-kp is essentially independent of @xmath4 since , even for modest values of @xmath1 , the effects of small fluctuations in the structure of phase - space are on negligible scales relative to @xmath15 . even if there are small stable islands , they are of size @xmath16 so have little effect on the general form of @xmath14 , since typically @xmath17 . the classical dynamics of the @xmath0-kp is obtained by iterating the well - known `` standard map '' . for the perturbed - period and double @xmath0- kicked systems , on the other hand , the dynamics is given by a 2-kick map : @xmath18 clearly , setting @xmath19 and @xmath20 , we recover the standard map . for the perturbed - period kp , the lengths of the two kicking periods are @xmath21 and @xmath22 , where @xmath23 . the perturbation thus consists of slightly altering the kicking period about its mean . for the double @xmath0-kp we take @xmath24 , @xmath25 , although we shall also show the effect of interchanging the two kick periods . it should be noted that these systems are _ time - periodic _ , with period @xmath26 , and are thus quite distinct from the recent interesting study of two independent kicking sequences , which can be non - periodic and hence non - localizing @xcite .    as in the standard map , we consider a sinusoidal potential @xmath27 . however , to obtain a ratchet current in the case of the perturbed - period kp , we need to break the spatio - temporal symmetries , and so we add an additional `` rocking '' linear potential of strength @xmath28 . in this case the form of the potential is @xmath29 $ ] , where @xmath30 is the kick number . in experimental implementations of this system , the rocking linear term was obtained by means of an accelerated lattice @xcite . we first consider in general terms how the introduction of the second timescale @xmath31 modifies the classical behavior of the standard map . if we neglect all correlations , the standard map has a constant momentum diffusion rate , @xmath32  this is what one would expect if the momenta at consecutive kicks are uncorrelated and so evolve as a random walk . however , unless @xmath1 is exceedingly large , the time - evolution of the standard map will contain some short - range ( 2-kick and 3-kick ) correlations . including these corrections yields a modified diffusion constant @xmath33 $ ] . of particular interest is the @xmath34 term , representing correlations @xmath35 between nearest - but - one kicks ( the 2-kick correlation ) .    for the modified systems it is also possible to obtain analytically the important correlations @xcite . for instance , for the perturbed - period system the corrected diffusion is @xmath36 $ ] . we see that we have a modified 2-kick correlation which _ oscillates _ with the initial momentum @xmath4 . this effect is clearly most significant for values of @xmath1 such that @xmath37 . the key point is that perturbing the kick spacings @xmath3 by a small amount can result in large scale ( relative to @xmath38 ) variations in the classical momentum diffusion , and that these are present even in fully chaotic regimes ( we take this to mean the absence of visible stable structures on the poincar surface of section ) . for the analysis of experiments , one must now consider a _ local _ diffusion rate @xmath39 , which depends on the initial relative momentum between the atoms and the optical lattice .    in @xcite , the perturbed - period system was implemented experimentally with a cloud of cesium atoms for the case @xmath40 . it was verified that the energy absorbed by the cloud after dynamical localization @xmath41 as expected . however , @xmath40 corresponds to a symmetric potential . the case @xmath42 in the perturbed - period system is particularly interesting since then the momentum diffusion is asymmetric about @xmath43 . this implies that atoms with positive momenta will absorb kinetic energy at different rates from those with momenta of the same magnitude but moving in the opposite direction . this asymmetric momentum diffusion represents a type of fully chaotic _ momentum _ ratchet : in other words , roughly equal numbers of particles will diffuse to the left or to the right , but those diffusing to the right , on average , move faster .    in a further experimental study of the perturbed - period kp @xcite , the @xmath42 potential was implemented by means of an accelerated lattice . it was found that an atomic cloud prepared initially with a gaussian momentum distribution centered on @xmath44 evolved into a distribution with non - zero , but constant , @xmath45 which persisted even beyond the break - time @xmath13 , as expected from the theory @xcite . this type of chaotic directed motion was first identified in a slightly different system : a kicked asymmetric double - well potential @xcite . however , the latter potential gives rise to a rather more complicated diffusive behavior , and also has proved much harder to implement experimentally . for these reasons here we do not consider the case of the asymmetric double - well ratchet , but note that our floquet analysis of the perturbed - period kp can be carried - over to the system investigated in @xcite . the second system we consider explicitly in this work , the 2@xmath0-kp , has diffusive behavior which is qualitatively different to both the standard map and perturbed - period kp . while for these other kicked systems we can analyze the diffusion as an uncorrelated term , @xmath46 , corrected by short - ranged correlations ( typically only 2 or 3-kick correlations for @xmath47 ) , for the double @xmath0-kp , we find that the diffusion at long times is dominated by families @xcite of long - ranged `` global '' correlations ( `` global '' in the sense that they correlate all kicks up to the time under consideration ) . at short times , the diffusion is dominated by a 1-kick correlation not present in other kick systems . at longer times , the global diffusion terms , though weak , accumulate and eventually become dominant . the method of correlations provides a generic and accurate way of interpreting experimental data for this system @xcite . there is also a simple physical picture . for particles subjected to kicks of form @xmath48 , consecutive kicks will be out of phase and will hence cancel if @xmath49 where @xmath50 . in other words , an impulse @xmath51 will be immediately followed by another which cancels it , since @xmath52 . this cancellation means that particles become trapped at these momenta . in contrast , particles for which @xmath53 , will experience enhanced diffusion . it was shown in @xcite that the new types of global families of correlations control the escape from , and through , these `` trapping regions '' . an unexpected feature of the classical calculations ( and also seen in experiment ) was the observation that particles initially prepared in the trapping regions will eventually gain more energy than those initially prepared in regions of enhanced diffusion , after a timescale @xmath54@xcite . if a hamiltonian has a @xmath3-periodic time dependence , @xmath55 , then the floquet theorem implies that solutions to the time - dependent schrdinger equation can be written in the form @xmath56 \phi(t)\ ] ] where @xmath57 is a @xmath3-periodic function called a floquet state , and @xmath31 is a quantity with dimensions of energy , termed a quasienergy . this type of relation is familiar in the context of solid - state physics , where a hamiltonian s invariance under discrete shifts of _ spatial _ position ( typically arising from a lattice structure ) allow solutions to be written analogously in terms of bloch states and quasimomenta . the floquet states provide a complete basis , and thus the time - evolution of a general quantum state under periodic driving can be expressed as @xmath58 \phi_n(t ) , \label{floquet}\ ] ] where @xmath59 are _ time - independent _ expansion coefficients . it is clear from this expression that the floquet states and quasienergies play a similar role for periodically - driven systems to that of energy eigenstates and eigenvalues in the time - independent case . the time - evolution operator @xmath60 may be used to evolve a quantum state from time @xmath61 to a time @xmath62 . for a time - periodic system , the single - period propagator @xmath63 allows a quantum system to be evolved `` stroboscopically '' at intervals separated by the period @xmath3 with great efficiency , by defining the quantum map @xmath64^n \psi(0)$ ] . in terms of floquet states , it is straightforward to show that the time - evolution operator is given by @xmath65 \label{define } \\ { } & = & \sum_n e^{-i \epsilon_n t / \hbar }    @xmath66 is the time - ordering operator , and thus it can be seen that the quasienergies and floquet states can be conveniently obtained by simply diagonalising the one - period propagator . the eigenvectors of this operator are the floquet states , while its eigenvalues are related to the quasienergies via @xmath67 $ ] . obtaining an explicit form for this propagator is normally a complicated procedure , as in general the driving field does not commute with the static hamiltonian . for the case of @xmath0-kicking , however , the problem is simplified considerably which allows an analytic form for the propagator to be written . for the qkp ( eq.[ham_kp ] ) the propagator is given by @xmath68 \exp\left [ -i ( k / \hbar ) \sin x \right ] .\ ] ] using a basis of plane - wave states , the matrix - elements @xmath69 of this operator can easily be shown to be @xmath70   j_{m - n}(k / \hbar ) \delta ( q - q ' ) , \label{prop}\ ] ] where @xmath71 is the quasimomentum ( @xmath72 ) and @xmath73 is the @xmath74-th bessel function of the first kind . for practical purposes it is useful to note that @xmath75 decreases extremely rapidly with increasing @xmath76 , thus giving @xmath77 an effectively banded - structure . the single - period propagators for the unequally - kicked systems can now be expressed as the product of two matrices of this form , @xmath78 , where @xmath79 and @xmath80 are the two kick - periods . it should be noted that although the single - kick propagators do not conserve quasimomentum , their product does . as a consequence the evolution of an ensemble of non - interacting particles which can be modelled by the evolution of a superposition of states with different quasimomenta , represents a computationally efficient procedure : we can consider each quasi - momentum component independently . in our study of floquet states in effect this means we can diagonalise the smaller matrix given by [ prop ] , which is block - diagonal in @xmath71 . having obtained the floquet states by diagonalising ( [ prop ] ) , it is useful to analyse their structure , particularly their spread in momentum space . we do this by evaluating a localization length , @xmath81 of each floquet state at @xmath82 . we note that in this case we can not assume that the floquet states have the usual exponential momentum distribution @xmath83 $ ] , of the usual @xmath0-kp . hence we take @xmath81 to be simply the root mean square deviation from the mean , @xmath84 . in this expression @xmath85 is the mean momentum of the @xmath74-th floquet state at @xmath82 , @xmath86 , and @xmath87 . if the floquet states do not have a strong time - dependence , such as for the standard , or even the perturbed - period kp , this is adequate to quantify the degree of spreading in momentum space . we shall see , however , that for the case of the 2@xmath0 kicked system it is not sufficient to measure the localization at a single time , due to the extremely strong time - dependence of the floquet states . fig.[fig1 ] shows a plot of the experimental ratchet current obtained in @xcite . a series of momentum distributions @xmath88 as a function of @xmath4 for a cloud of cold cesium atoms in an optical lattice pulsed with unequal periods . the momentum distribution is essentially unchanged after about 60 kicks ; the plotted values correspond to about @xmath89 kicks , hence well after dynamical localization . full details are given in @xcite , but by employing an accelerated lattice , the experiment simulated an effective rocking potential with @xmath90 . the first moment of each localized distribution @xmath91 was then calculated and plotted as a function of @xmath4 . in particular , a distribution centered at @xmath44 initially , and with zero initial momentum current @xmath92 at @xmath82 , yielded a finite and constant momentum current @xmath93 at long times . for non - zero initial momenta @xmath94 , an oscillation @xmath95 , was observed and is seen in fig.[fig1 ] . this may be qualitatively understood from the form of the classical two - kick momentum - diffusion correction introduced in sec.ii , @xmath96 . if we consider a very small momentum displacement @xmath97 , differential absorption of energy for particles moving to the left @xmath98 or right @xmath99 is proportional to the gradient @xmath100 . an accurate analytical form for the classical current was derived in @xcite . of course , it follows that a finite and persistent constant momentum current is also obtained classically . it was found in @xcite that asymmetric diffusion persists only on a timescale @xmath101 and for this ( unbounded ) chaotic system , the acquired momentum asymmetry is never lost . for a bounded ( `` compact phase - space '' ) system , such asymmetries would vanish on a long time - scale , since the distribution of a fully chaotic system would eventually become uniform . for this reason , until recently , it was argued that a fully chaotic system could not generate directed motion . so , although as shown in @xcite , the fully chaotic classical system can keep a constant current for long times , practical implementation is ultimately less interesting since the average kinetic energy of the ensemble grows linearly with time and without limit . hence , this type of chaotic ratchet is of most interest as a _ quantum _ rather than a _ classical _ ratchet since in the quantum case dl halts the diffusion and `` freezes - in '' the asymmetry , without the need for classical barriers like tori . , for the perturbed - period kp , obtained with cold cesium atoms in a pulsed optical lattice for @xmath47 , @xmath102 . the solid line is a best - fit to the data , showing that the current oscillates sinusoidally as @xmath103 for @xmath42.,width=288 ]     for the perturbed - period kp obtained with cold cesium atoms in a pulsed optical lattice for @xmath47 , @xmath102 . the distributions have localized , and hence remain essentially constant with time . the results show clearly that the origin of the net non - zero value of @xmath45 obtained at long times is in the asymmetry of the dl profiles . as expected , the asymmetry is reversed by changing the sign of @xmath28 , the amplitude of the rocking potential.,height=288 ]    in fig.[fig2 ] we reproduce two experimental momentum distributions for @xmath47 obtained with cesium atoms in ref.@xcite , for @xmath104 . we clearly see that the origin of the non - zero momentum current is the asymmetric momentum distribution . as expected , fig.[fig2 ] shows that changing the sign of @xmath28 reverses the asymmetry . at this stage it may be unclear to the reader what the significance of altering the sign of @xmath28 in the experiment might be , since after all , the rocking potential involves alternating impulses @xmath105 . in fact the distinction ( as may be ascertained from the form of the classical diffusion ) is between the case where an impulse @xmath106 precedes free evolution for a time interval @xmath107 ( obviously followed by an impulse @xmath108 and interval @xmath109 ) and the separate experimental case where an impulse @xmath108 precedes free evolution for a time - interval @xmath107 and so forth ( which corresponds to a reversed current ) . note that the experimental range of @xmath110 does correspond to a classical surface of section with some islands . however , we note that classical quantities such as the average energy are very accurately given by diffusion rates ( with 2 and 3-kick corrections ) . the essential mechanism is asymmetric chaotic diffusion : similar behavior was found at larger @xmath1 in @xcite in regimes where there are no visible classical islands ( but for which experiments are not available ) ; hence , in the analysis of this type of ratchet , the presence ( or otherwise ) of small stable islands is immaterial . what is important , though , is that since the asymmetric diffusion term is @xmath111 , we need @xmath112 . thus the much - studied ( for the standard map ) parameter value @xmath113 does _ not _ produce asymmetry , since @xmath114 . values of @xmath115 , @xmath116 , on the other hand , turned out to be experimentally convenient and produced the strongest asymmetries .     of floquet states as a function of average momentum @xmath117 ( i.e. @xmath45 ) for @xmath118 , @xmath119=1 . results are shown ( a ) for the standard qkp case , i.e. @xmath120 ( a non - zero @xmath28 was used to break spatial symmetry ) , @xmath121 , ( b ) for @xmath122 , @xmath120 and ( c ) @xmath122 , @xmath42 . the graph shows that for the standard kicked rotor the @xmath81 are distributed within a narrow range in comparison with other two below . for the rocking case , @xmath81 oscillates with @xmath117 as expected from the 2-kick correction @xmath123 ; the oscillations of the two lower graphs are shifted relative to each other by a phase @xmath124 . the density of eigenstates corresponding to average momentum range is roughly the same in all three cases.,width=288,height=288 ]    we now examine the form of the underlying floquet states . in fig.[fig3 ] we compare the localization lengths for the standard qkp , with those of the perturbed - period kp for @xmath118 , @xmath122 . the difference is quite striking ; while the standard qkp eigenstates are quite uniform across all regions of phase - space , the perturbed - period localization lengths oscillate sinusoidally with @xmath117 , with a period of @xmath125 . introducing the additional rocking potential with the accelerated lattice ( @xmath42 ) clearly leads to a @xmath124 shift in the oscillations . inspecting fig.[fig3](c ) for @xmath126 , we see that for positive momenta the localization lengths are increasing , while for negative momenta , the localization lengths decrease . note the nearly regular row of states for the standard qkp case with @xmath127 . these correspond to states localized on a series of stable islands separated by @xmath128 , due to the momentum periodicity of phase - space in that case . we have chosen a parameter range for which @xmath129 : that is , the localization length of each state is much smaller than the oscillation in @xmath117 . hence individual floquet states really do sample `` local '' diffusion rates . we found that if we move towards a regime where @xmath130 , the conclusions remain valid , but the amplitude of the oscillations is considerably damped . similarly , if the sign of @xmath131 changes , so does the sign of the sinusoidal oscillation . we now consider the shape of the floquet states in detail . in fig.[fig4 ] we show the momentum distributions @xmath132 for floquet states of the standard qkp . the distributions ( with @xmath14 on a logarithmic scale ) all show the well - known triangular form @xcite  the hallmark of dynamical localization . it may be clearly seen that the localization lengths vary little from state to state .    in fig.[fig5 ] , by contrast , the localization lengths of the floquet states of the perturbed period @xmath0-kp display a strong dependence on the mean momentum of the states . in addition , the figure shows that states localized close to @xmath133 are markedly asymmetric . the states are considerably extended towards positive momentum , but are strongly localized towards negative @xmath134 . this behavior neatly accounts for the form of the experimental momentum distribution shown in fig.1 , which for @xmath42 were also more extended towards positive @xmath134 . the states localized near @xmath135 and @xmath136 correspond to , respectively , minima and maxima of the classical diffusion . they are roughly symmetrical ( typically ) but vary by up to a factor of @xmath137 in @xmath81 . in ref.@xcite it was proposed that the observed variation in the energy absorption rates between atoms prepared with an initial drift momentum @xmath138 ( which absorb very little energy ) and those with @xmath139 might be exploited to filter traffic of atoms through an optical lattice . the form of the underlying floquet states explains this differential rate of energy absorption . subsequently , it was found experimentally that the double @xmath0-kp in fact shows much more pronounced differential absorption rates , without requiring the application of a rocking field @xmath28 . we next report a study of the floquet states of this system . , @xmath140 . as expected , all the states are exponentially localized , giving the characteristic triangular shape of @xmath14 when plotted on a logarithmic scale . they all have approximately similar localization lengths.,width=288,height=144 ]    , @xmath122 , @xmath42 and @xmath119=1 . here we plot @xmath132 as a function of @xmath134 . ( a ) states with @xmath141 . this corresponds to a minimum of the 2-kick correction @xmath142 . the states are narrow , but in general , roughly symmetric . ( b ) states with @xmath143 . the typical state here is asymmetric ( c ) states with @xmath144 . this corresponds to a maximum of the 2-kick correction . states here are generally symmetrical , but broad and flat - topped.,title="fig:",width=288 ] +      -kp realization with cesium atoms ( see @xcite for details ) . each data point ( star ) shows the energy absorbed ( after 100 kicks , @xmath145 , @xmath140 ) by a cloud of atoms with average momentum @xmath146 ( relative to the optical lattice ) at initial time , @xmath82 . with increasing @xmath31 , we see the minima ( maxima ) in the energy flip into maxima ( minima ) as a long - ranged family of classical correlations gradually overtakes the 1-kick classical correlation . the dashed lines represent a classical simulation using 100,000 particles , all with momenta @xmath4 at @xmath82 , and @xmath1 within the range @xmath147 . * ( a ) * @xmath148 . regime where a one - kick correlation is the dominant correction to the classical diffusion . here , atoms prepared near the trapping regions @xmath149 remain trapped . results follow closely the formula @xmath150 . * ( b)*@xmath151 . regime showing the inverted peaks of the poisson correlation terms analyzed in @xcite , which determine the momentum trapping very close to the resonant condition @xmath152 . * ( c ) * @xmath153 . regime dominated by correlation family @xmath154 , but sharp inverted peaks due to the poisson correlations are still visible.,title="fig:",width=288 ] +    a study of the experimental and classical behavior of the double @xmath0-kp was carried out in @xcite . the classical dynamics is very different to that of the perturbed period kp . at very short times , the chaotic diffusion comprises an uncorrelated diffusion term @xmath46 and one dominant 1-kick correction . it was found in @xcite that one can approximate the growth in the mean energy with time @xmath155 , by the simple expression @xmath156 $ ] . in fig.[fig6](a ) we show experimental results for cesium atoms which localized in this regime . the experiment measured the energy of a series of clouds of @xmath157 atoms moving through the pulsed optical lattice with varying average drift momenta @xmath4 . for fig.[fig6](a ) , the simple expression given above gives an excellent fit to the experiment , if we take @xmath158 , where @xmath13 is the break time . this regime corresponds to @xmath159 . however , a more detailed study of the classical correlations showed that for later times , a new type of correction appeared . families of long - ranged , or `` global '' , correlations which coupled _ all _ kicks appeared . these corrections are individually very weak , but accumulate to eventually dominate the diffusive process . one family ( termed the `` poisson family '' in @xcite ) was shown to lead to well - localized , inverted peaks in the energy absorption at values of @xmath160 , where @xmath161 . these values of @xmath4 correspond to trapping regions in phase - space ( at low values of @xmath1 , structures corresponding to islands and broken phase - space barriers are evident ) . however there is no need to investigate detailed transport through this complex mixed phase - space structure , as the correlations give us a generic and quantitative handle on the energy diffusion with time . in this intermediate regime , dominated by the poisson correlations , atoms prepared outside the trapping regions rapidly diffuse across the regions between them . particles prepared in the trapping regions remain there . this regime occurs for @xmath162 and corresponds approximately to the experimental results shown in fig.[fig6 ] . finally , at the longest timescales , there is the @xmath154 correction investigated in @xcite , which is a long - ranged global - correlation family . @xmath154 results in an oscillation of the form @xmath163 and becomes dominant at the longest timescales . the oscillation is of the same period as the 1-kick correlation , but is of opposite sign . this means that at the longest timescales , the minima in energy absorption shown in fig.[fig6](a ) become maxima in energy absorption ; and vice - versa : the maxima become minima . fig.6(c ) shows experiments tending towards this regime . the inverted peaks of the poisson family are still in evidence , but a @xmath164 oscillation is clearly superposed . this is a somewhat counter - intuitive result since it implies that atoms initially prepared in the momentum trapping regions are the ones which at long times , for @xmath165 , will absorb the most energy ( there are no further reversals of this behavior at even longer times ) . -kp ( @xmath166 , @xmath167 ) corresponding to the three classical diffusion regimes investigated in the experiments in @xcite.,width=288 ]    the classical analysis thus reveals that there are three distinct classical diffusive regimes occurring at three timescales . we can expect that the corresponding _ quantum _ behavior will depend on which regime is dominant when dynamical localization arrests the quantum momentum diffusion . to investigate this , we now investigate how the form of the floquet states varies in these different regimes .    in fig.7(a ) we show the localization lengths of the floquet states for a weak kicking - strength . it can clearly be seen that the localization varies periodically as a function of momentum , staying within the range @xmath168 for the majority of points , with the exception of a series of sharp cusp - like features at which the localization dramatically falls . these location of these cusps exactly corresponds to the the `` trapping momenta '' , @xmath169 , predicted from classical arguments . the floquet states centered in the trapping regions have widths of @xmath170 , much narrower than states localized on stable islands , which are also visible in this figure as regular strings of points at @xmath171 . at the experimental values of @xmath47 and @xmath172 a similar behavior is produced , with the broadest floquet states having localization lengths of @xmath173 , while the narrowest have widths of @xmath174 , over one thousand times narrower . the floquet states at the tips of the cusps have such low localization lengths that they are effectively pure plane - wave states ( this can be further corroborated by evaluating the inverse participation ratio for these states , which indeed takes a value of almost unity ) . it is thus unsurprising that the presence of these states corresponds to the classical trapping effect , as a quantum system prepared in such a state will have a vanishingly small overlap with any other state and so will remain frozen ( or trapped ) in its initial state . it is important to note , however , that this quantum trapping effect depends critically on the _ order _ of the two kick- periods  that is , whether the system is driven with a short - long kick - sequence or the inverse long - short ordering . this may appear surprising at first , since the floquet states are periodic , with the same period @xmath175 as the driving , and this period is not altered by interchanging the order of the kicks . although it is frequently neglected , however , it is important to recall that the floquet states do have an explicit time - dependence within each period , and this _ is _ able to produce substantially different behavior @xcite when the phase of the driving field is altered . to illustrate this , we show in fig.[floq ] the time - evolution of one of the localized floquet states , which experiences @xmath0-kicks at times @xmath176 and @xmath177 . as can be seen , the state has only a trivial time - evolution during the first time - interval ( @xmath178 ) , since it is almost a plane wave and is thus approximately an eigenstate of the free hamiltonian . the first kick at @xmath79 causes the wave - packet to spread considerably in momentum space , before the second kick restores this broadened state to its original narrow form . thus in this brief window of time between the two kicks , even the most localized floquet states have a considerable spread in momentum . as a consequence , if the phase of the kicking field is shifted so that the system experiences the short - long kick - sequence , none of the floquet states are sharply localized in momentum at @xmath82 . we show in fig.[evolve ] the time - evolution of the system s kinetic energy when it is prepared in a momentum eigenstate in a trapping region . for the long - short kick - sequence this state projects onto essentially a single floquet state at @xmath82 , and so its time evolution is trivial and its energy remains constant . for the case of the short - long kick - sequence , however , the initial state projects onto a number of floquet states ( eq.[floquet ] ) , giving rise to a complicated quasi - periodic behavior arising from beating between the different quasienergies .    , for physical parameters : @xmath179 , @xmath180 and @xmath167 . initially the floquet state is sharply peaked at @xmath181 , in the center of a trapping region . the first kick at @xmath182 causes the state to spread across a much broader range of momentum , until the second kick at @xmath183 restores the localized state.,width=288 ]    -kp , for the same physical parameters as in fig.7a . the solid line shows the evolution of the system under the long - short kicking sequence , and shows little features . the dotted line shows the result of the short - long sequence , and exhibits a complicated quasi - periodic behavior.,width=288 ]    in fig.7(b ) , we see the effect of increasing the kick - strength . in this regime we find there is an almost constant localization length for momenta in between the trapping regions , which are again signalled by sharp cusp - like structures . this indicates that the floquet states are confined between the classical broken phase barriers in the trapping regions . early studies indicate that the level statistics of the corresponding quasienergies are not pure poisson in this regime , as would be the case for the standard qkp .    in fig 7(c ) , we see an inversion of the broad momentum modulation in fig 7(a ) , similar to the reversal seen in the experiment . in this regime , the eigenstates localized in the trapping regions near @xmath184 are typically _ broader _ than those localized in between . we have presented a study of the floquet states of @xmath0-kicked particles pulsed with unequal periods , and used them to analyze experimental data on these systems . we conclude that the chaotic ratchet effect proposed in @xcite and observed experimentally in @xcite is associated with asymmetric floquet states localized around @xmath133 . we conclude also that the behavior of the localization lengths of the floquet states for the double @xmath0-kicked rotor broadly accompany the change over between the three distinct classical diffusion regimes investigated experimentally in @xcite .
recently , cesium atoms in optical lattices subjected to cycles of unequally - spaced pulses have been found to show interesting behavior : they represent the first experimental demonstration of a hamiltonian ratchet mechanism , and they show strong variability of the dynamical localization lengths as a function of initial momentum . 
 the behavior differs qualitatively from corresponding atomic systems pulsed with equal periods , which are a textbook implementation of a well - studied quantum chaos paradigm , the quantum @xmath0-kicked particle ( @xmath0-qkp ) . 
 we investigate here the properties of the corresponding eigenstates ( floquet states ) in the parameter regime of the new experiments and compare them with those of the eigenstates of the @xmath0-qkp at similar kicking strengths . 
 we show that , with the properties of the floquet states , we can shed light on the form of the observed ratchet current as well as variations in the dynamical localization length .