Dynamical instability in kicked Bose-Einstein condensates
Bose-Einstein condensates subject to short pulses (‘kicks’) from standing waves of light represent a nonlinear analogue of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (ie exponential proliferation of non-condensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics.
The production of Bose-Einstein condensates (BECs) in dilute atomic gases has opened up a new domain for research in quantum dynamics, since BECs are intrinsically phase-coherent and can be controlled experimentally to an extremely high degree of precision Stringari (). An increasingly interesting aspect of the dynamics of BECs is that they represent a new arena for investigation of the interaction between nonlinearity and quantum dynamics, including quantum chaos Shep (); Gardiner (); Garreau (); Duffy (); Zhang (); Zhang2 (); Wimberger (); Adams ().
A BEC subject to periodic short pulses, or kicks, from standing waves of light represents a nonlinear generalization of the well-known chaos paradigm, the quantum kicked rotor (QKR). The QKR has been realized using (non-condensed) cold atoms, permitting experimental investigation of a range of interesting chaos phenomena Raizen (). The regime where the kick-period is a rational multiple of has also proved of particular interest: several studies have investigated the dynamics here with or without linearity Darcy (); Fish (); Wimberger (); Rebuzzini (). A number of experimental studies have also investigated kicked BECs Sadgrove (). Ensuring dynamical stability of the condensate is thus very important in studies of its coherent dynamics: if the condensate is dynamically unstable, numbers of non-condensate particles grow exponentially. If it is stable, they grow more slowly (polynomially). More broadly, the study of diffferent types of instability in static Wu () and driven BECs Dalfovo () is of much current interest.
Previous work on kicked systemsGardiner (); Zhang (); Zhang2 () considered the onset of dynamical instability and investigated the relation with classical chaos. In Gardiner (), the possibility that instability is related to chaos in the one-body limit was investigated for the Kicked Harmonic Oscillator. In Zhang (); Zhang2 () the correlation between chaos in the mean-field dynamics, rather, and the onset of dynamical instability, was investigated. An “instability border”, determined by the kick strength and the nonlinearity was mapped out; it was then found Zhang2 () that the parameter ranges for this border corresponds closely to a transition from regular to chaotic motion, of an effective classical Hamiltonian derived from the mean-field dynamics. Hence, present understanding of onset of dynamical instability in kicked BECs suggests that it may somehow be related to a transition to chaos.
In this work, we conclude that a quite different mechanism is primarily responsible for dynamical instability in the QKR-BEC. Our key finding is that it is the strong resonant driving of certain condensate modes by the kicking, which triggers loss of stability of the condensate. This mechanism is unrelated to the transition to chaos, but is rather an example of parametric resonance. In another context, the relationship between parametric resonance and dynamical instability of a BEC in a time-modulated trap is a topic of much current theoretical PRTHEO (); Dalfovo () and experimental interest PREXPT (). But to date, “Bogoliubov spectroscopy” in the analogous time-dependent system, the -kicked BEC, has not been investigated. Our study shows that the temporally kicked BECs open up many new possibilities in this arena.
We find that in general, for the kicked-BEC, there is no single stability border: typically, for moderate , the condensate restabilizes just above the stability border. For small and the number of non-condensed atoms grows exponentially only very close to a few, isolated resonance peaks. With increasing and , the number of resonances which can be strongly excited by the kicking proliferates and overlaps. Our calculations show this is associated with generalized exponential instability; however this regime is, to a large degree, beyond the scope of our methods. For lower and , though, we introduce a simple perturbative model which provides the approximate position and width of the important resonances for both rational and irrational .
A key finding is that, for the integer values of (where is integer) values, the focus of the study in Zhang (), the onset of instability can occur at nonlinearities much lower than those required to resonantly excite even the very lowest collective mode – a key reason why the mechanism of parametric resonance may so far been overlooked in respect of destabilization of kicked BECs. Our model demonstrates that for this case, resonant excitation involves two excited modes in addition to the initial ground state mode. Hence we can explain the position of the critical stability border found in Zhang (); Poletti ().
We investigate both the usual QKR-BEC as well as a simple modification, obtained by applying a series of pairs of closely-spaced opposing kicks (the QKR2-BEC). This modifies substantially the relative strengths of the resonances, and provides the added novelty that the lowest modes are excited by an effective imaginary kick-strength. It is closely related to the double-kicked quantum rotor, investigated in cold atoms experiments and theory Jones (). We introduce a simple analytical model based on the properties of the unperturbed condensate, which gives the distinctive properties and scaling behavior of the condensate oscillations on and off resonance.
In Section II we introduce briefly the kicked and double-kicked BEC systems. In Section III we introduce the time-dependent Bogoliubov method proposed by Castin and Dum and present numerics for the growth of non-condensate atoms. In Section IV we introduce a simple perturbative model, based on the one period time evolution operator for a kicked BEC. In Section V we show that the simple model and the time-dependent Bogoliubov numerics give excellent agreement in the limit of weak kicks. In Section VI we consider the case with both numerics and the perturbative model and show that the instability border found in Zhang (); Zhang2 () is due to a novel type of compound Bogoliubov resonance.
Ii Kicked BEC systems
As in Zhang (), we consider a BEC confined in a ring-shaped trap of radius . We assume that the lateral dimension of the trap is much smaller than its circumference, and thus we are dealing with an effectively 1D system rescalg (). The dynamics of the condensate wavefunction at temperatures well below the transition temperature are then governed by the 1D Gross-Pitaevskii (GP) Hamiltonian with an additional kicking potential:
The short-range interactions between the atoms in the condensate are described by a mean-field term with strength , where is the s-wave scattering length, and is the total number of atoms. For the QKR-BEC system, , while for the QKR2-BEC,
where is the total period of the driving;
and thus the second kick nearly cancels the first.
Experimental and theoretical studies of the double-kicked rotor Jones () have shown that its quantum behavior is largely determined by an effective kick strength , provided . Here we take . Hence, while for the QKR-BEC, the value represents a relatively large impulse for a kicked BEC, for a double kicked BEC, in the numerics below corresponds to , and represents only a very weak impulse. The reason for this is the near cancellation of consecutive kicks in each pair.
This mechanism has certain analogies with the so-called “quantum antiresonance” investigated in Zhang (): for QKRs kicked at , consecutive kicks effectively cancel. This means that even large values of and represent only weak driving; for example, the instability border was found by Zhang () to occur at and .
Iii time-dependent Bogoliubov method
The number of non-condensed atoms were calculated by making the usual Bogoliubov approximation, and following the formalism of Castin and Dum castin (). This adaptation of the Bogoliubov linearization for time-dependent potentials has been used in all studies to date of the dynamical stability of kicked condensates Gardiner (); Zhang (); Poletti (); Rebuzzini (). The mean number of non-condensed atoms at zero temperature is given by , where the amplitudes of the Bogoliubov quasiparticle operators are governed by the coupled equations
In this expression, are projection operators that orthogonalize the quasiparticle modes with respect to the condensate castin (). We assume that at time , we have a homogeneous condensate . Further discussion of the theory is given in gard ().
The regime of validity of the method is discussed in castin (). The method is valid in the weakly interacting limit where is the density. A limit is identified where this condition is satisfied, if one works with a constant ; thus the limit corresponds to . A further requirement is that condensate depletion remains negligible. This condition fails after a few kicks in exponentially unstable regions. Here the method is employed only to identify the the parameter range for the onset of instability. We cut-off our calculations for (a reasonable threshold for small depletion in a condensate with ).
In Figs.1 (a) and (b) we show the number of non-condensed atoms,
, calculated from the Bogoliubov
equations (4) after .
For small , , a single resonance is seen
For small , resonances occur whenever the resonance
condition Dalfovo ()
where is an integer and is the eigenfrequency
of the collective mode.
For larger , the figure shows that
resonances are extremely dense and overlap with each
other (and we show the behavior in this regime
for ). For overlapping resonances, unambiguous
identification of each resonance is no longer possible.
The key point here, however, is that in the stable regions outside the resonances,
remains very small even after prolonged kicking.
shows oscillations of , as a function of time, for weak , ,
close to the isolated resonance at . The three possible
regimes of: (non-resonant) weak quasi-periodic oscillations in time;
(near-resonant) slower, large periodic oscillations;
and (resonant) exponential growth are illustrated.
The condensate energy,
after kicks, obtained from the GPE itself, is also shown, for
comparison, in the inset: at resonance, large oscillations are also seen.
Fig.2 shows the corresponding behavior for the double-kicked BEC, but now as a function of , keeping , constant and or (hence or ). The curve corresponds to weak impulses and shows two isolated Bogoliubov resonances. While values of are large compared with current experimental values of (see discussion of experimental in Rebuzzini ()), resonances at small more suitable for experimental spectroscopy can be excited by considering larger . The curve is in the overlapping resonance regime, so produces generalised instability.
In order to understand the behavior at the resonances, we introduce in Section II a model for the time evolution of perturbations from the kicked condensate, based on the usual linearization with respect to small perturbations.
Iv II: Kicked condensate model
The time-evolution of small perturbations of the condensate itself are described by an equation similar to Eq.4, see castin (). We write the condensate wavefunction in the form , where is the unperturbed condensate and represent the excited components. Inserting this form in the GPE and linearizing with respect to , we can write:
The analysis of condensate stability for a time-periodic system Dalfovo () reduces to the analysis of the operator over one period . In general, for systems like BECs in modulated optical lattices, inter-mode coupling requires a detailed analysis of the instantaneous evolution. The nature of the -kicked potential permits considerable simplification.
The effect of reduces to the free-ringing of the eigenmodes of the unperturbed condensate for period , interspersed by instantaneous impulses which mix the modes. Even for an experiment (where the kicks are approximated pulses of very short, but finite duration) numerical time-propagation is avoided: intermode coupling occurs over a very short time-scale, during which eigenmode phases remain essentially constant.
Excluding the kick term for the moment, we recall that the time propagation under can be analyzed in terms of the eigenmodes and eigenvalues of of the matrix on the right hand side of Eq.6. Setting , the matrix can be diagonalized and there are well-known analytical expressions for the unperturbed eigenmodes Stringari ()
where , and .
In order to understand the behavior at the resonances, we introduce below a simple model using the eigenmodes Eq.7 as a basis. Writing the small perturbation in this basis:
Neglecting the kick, evolving the modes from some initial time , each eigenmode simply acquires a phase ie:
After a time interval , a kick is applied which couples the eigenmodes. Its effect is obtained by expressing the perturbation in a momentum basis, where , and we can restrict ourselves to the symmetric subspace of the initial condensate (parity is conserved in our system). Then, we can see by inspection that
Note that for this system. Conversely, the corresponding amplitude in each eigenmode is given trivially from Eq.8 using orthonormality of the momentum states and the relation , yielding
If the evolving condensate is given in the momentum basis, the effect of a kick operator is well-known. The matrix elements:
The are Bessel functions.
The amplitudes are given by
where denotes the amplitude in state just after/before the kick.
We can now define a “time-evolution” operator
, where denotes free ringing of the eigenmodes, is the transformation from momentum basis to Bogoliubov basis and is the action of the kick. A usual procedure for stability analysis of a driven condensate is to examine the eigenvalues of to ascertain whether there is one (or more eigenvalues) which have a real, positive component Dalfovo (), ie whether they produce exponential growth in the amplitudes .
However, to compare with GPE numerics, we simply evolve the mode amplitudes in time over a few kicks and examine the overall condensate response to the kicking (in the limit of very weak kicking). Hence we can evolve the amplitudes of the condensate perturbation from period to period :
using only the simple analytical coefficients in Eq.13 and Eq.9, provided we use the simple transformations in Eqs.10 and Eq.11 to switch between the Bogoliubov mode basis and the momentum basis. is non-unitary, but the method is quantitative in the perturbative limit provided , ie we assume .
We calculate the average energy over the first few kicks, . Slow, large amplitude oscillations in yield a large and indicate a resonance. Fig.3(a) shows the QKR-BEC behavior, for equivalent parameters to Fig.1(a). For low , there is the same single resonance at as in Fig.1(a). For higher the method is far from quantitative: the model Eq.14 is only a valid means of time-evolving the perturbation over a few kicks for small since it assumes the perturbed component is negligible; nevertheless, for it illustrates the regime of dense, overlapping resonances.
In Fig.4(b) we compare the perturbative Eq.(14) results with full GPE numerics
for the first 20 kick pairs of the QKR2 in the limit of weak kicks.
It shows remarkably good agreement.
Moreover the scaling of the resonances with is well described.
The QKR2 resonant Bogoliubov spectrum differs appreciably from the
Fig.4(b) shows that for QKR2, even for low
and low , both and resonances are strongly excited.
The QKR2-BEC resonance intensity depends strongly on
: the resonances scale as , while the scaling is closer to
. In the full GPE numerics, the position of the maxima depends slightly
on and , but remains within a few percent of the unperturbed value,
even for longer kicking times if remains small.
In the limit of weak driving, one can obtain explicit expressions for the condensate wavefunction as a function of time. We assume that . Then Eq.13 can be approximated by . From Eq.11 and Eq.9 we see that the amplitude accumulated over a single period in each eigenmode is
Summing all contributions iteratively from , taking , we obtain
and so for all the contributions add in phase, analogously to the well-known (but unrelated) resonances of the non-interacting limit Darcy ().
We can write where the function is:
We thus expect oscillations in each set of momentum components of amplitude
Off-resonance there will be quasi-periodic oscillations (in e.g. the condensate energy) from the superposition of contributions characterized by different eigenfrequencies . Close to resonance, a single component dominates; if the mode is resonant we can write where is the de-phasing from resonance. Then
and there are slow, periodic oscillations of large amplitude , at a frequency which is not related to any eigenmode frequency, but given rather by the de-phasing from resonance.
The QKR2 resonant excitation spectrum is rather different from the QKR, and is analysed further in the next section.
V III: Resonances of the QKR2-BEC
In the limit , we can obtain analytical expressions for the BEC wavefunction of the double-kicked system. Firstly note that when , the non-linearity has little effect during the short time-interval . Using the relation,
the time evolution can be given as a ‘one-kick’ operator
In the limit , one can split the operators in Eq.21 and neglect a term to obtain the approximation
leaving an effective single-kick quantum rotor with a kicking potential
The second term, curiously, appears as kicking potential with an imaginary, and dependent, kick strength . It is of purely quantum origin as it arises from the non-commutativity of and , i.e.
Nevertheless, as seen below, it is important for weak driving as it controls the amplitude of the first excited mode .
The matrix elements of the modified kick , like those in Eq.13, are Bessel functions. Specifically, the effect of on the condensate amplitudes is given by
where , and indicates momentum amplitudes before(-) and after(+) the kick, as in Eq.13. Since and , only Bessel functions of low order ( or ) will be non-negligible, and we can use the small-argument approximations for them, namely , .
Then, if the condensate is relatively unperturbed, the main effect of the kick will be to simply excite a small amount of and from the state
We obtain a similar equation to the QKR-BEC for the mode amplitudes, i.e.
But if only the lowest excited modes are significant,
then, in particular,
. For all the contributions add in phase and we will have a resonance of either the or modes, the regime illustrated in Fig2(b).
Similarly as for the QKR-BEC, we can sum all the contributions to obtain an approximate analytical expression for the evolving condensate wavefunction including excited modes and ,
Eq.28 shows that the amplitudes and scale as and respectively, as seen in the numerics in Fig.4(b). Fig.5(a) shows that Eq.28 gives excellent agreement with GPE numerics, giving accurately the non-resonant quasi-periodic condensate oscillations. Near the resonance of Fig.2, Fig.5(b) confirms the QKR2 condensate oscillations (obtained from the GPE) scale quite accurately as as expected from Eq.19 and Eq.28.
Vi IV: Bogoliubov resonances for
The kick period , in a non-interacting system of cold atoms
(i.e. ) corresponds to a so-called “quantum anti-resonance”
where the cold atom cloud exhibits periodic (period-2) oscillations.
Hence the isolated Bogoliubov resonance regime to higher
than would be expected for generic .
The effect of a non-zero for was investigated
in Zhang (). An instability border occurring at a critical
value of nonlinearity, e.g. for at ,
was identified where the growth on non-condensate particles with time
In Fig.5(a) we investigate the behavior near critical , for . We see that if a wider range of is considered, the stability border is also a resonance: the condensate rapidly recovers stability after the instability border is passed. The condensate is exponentially unstable for , but is quite stable for both and , as shown. Fig.5(b) shows oscillations in the condensate energy, as a function of time; a smoothed plot is also shown. For and (off-resonance) the smoothed plots are flat; for and (near-resonant), slow deep oscillations are apparent.
The behavior is analogous to that of generic ; however, the analysis of the condensate resonances for is less straightforward: the strongest resonances, even for low , do not in fact occur for , where .
A significant difference between generic and is that, for the generic case, if we write
we see that for arbitrary generic , the distance from the nearest resonance, for the different modes, depends on . In contrast, for , for large (i.e. ) we find ; in other words, the de-phasing from the nearest resonance (and hence the period of the mode oscillations) is similar (either or ) for for all modes. So all mode oscillations for high are approximately in phase with each other.
For , only low modes are significantly populated. These low modes ( and ) are only in phase with each other at certain precise values of . For these parameters, the model of Eq.14 predicts large resonances whenever the condition is satisfied. In particular, for the resonance near , we find that for the mode, while for the mode , with .
These results suggest that “two-mode resonances”, i.e. synchronized oscillations of pairs of the lowest excited modes are the dominant mechanism for (NB this could be viewed as a “three-mode” resonance, if we include the lowest, initial mode, but for our system). They account for the shifting position of the critical instability border found by Zhang () in the case. For example, for slightly higher kick strengths, such as , a resonance appears for corresponding to , which accounts for the displacement of the instability border to lower values of . Note that the resonance positions in the full numerics are -dependent, whereas in the perturbative model of Eq.14 this dependence is neglected; the model is only valid for very small .
Vii V: Conclusion
In conclusion, we have shown that exponential instability in kicked BECs is related to parametric resonances, ie driving of low-lying collective modes at their natural frequencies, rather than to chaos in the underlying mean-field dynamics gard2 ().
The signature of this process is in the onset of slow, large amplitude periodic oscillations in the condensate energy as well as the number of non-condensate atoms calculated from the time-dependent Bogoliubov formalism, as a resonance is approached. The resonances proliferate and overlap for large kick-strengths , leading to instability over wider ranges of and . The time-dependent Bogoliubov approximation used here and in all other previous studies is only valid in regimes where the condensate depletion is negligible; for realistic condensates analysis of the dynamics in the narrow (for weak driving) windows of parametric instability, would require other approaches beyond Bogoliubov. However, away from these windows, the kicked condensate remains stable and relatively unperturbed, even after prolongued kicking.
JR acknowledges funding from an EPSRC-DHPA scholarship. The authors would like to thank Chuanwei Zhang for valuable advice. This research was supported by the EPSRC.
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