Bichromatic control of dynamical tunneling: influence of the irregular Floquet states
BBichromatic control of dynamical tunneling: influence of the irregular Floquet states
Archana Shukla and Srihari Keshavamurthy
Department of Chemistry, Indian Institute of Technology, Kanpur U. P. 208 016, India (Dated: October 4, 2018)Bichromatic control, in terms of the amplitude and relative phase of the second field as controlknobs, is an useful approach for controlling a variety of quantum processes. In this context, un-derstanding the features of the control landscape is important to assess the extent and efficiency ofthe control process. A key question is whether, for a given quantum process, one can have regionswherein there is a complete lack of control. In this work we show that such regions do exist and canbe explained on the basis of the phase space nature of the quantum Floquet states. Specifically, weshow that robust regions of no control arise due to the phenomenon of chaos-assisted tunneling. Wealso comment on the possible influence of such regions on the phenomenon of directed transport inquantum Hamiltonian ratchets.
PACS numbers: 05.45.Mt, 05.60.Gg, 32.80.Qk, 05.60.-k
I. INTRODUCTION
Coherent control of quantum tunneling is currently anarea of active research. In many instances one is facedwith the task of controlling dynamical tunneling - a pro-cess wherein the barriers are in the phase space and thesystem tunnels between two symmetrically related regu-lar regions in the phase space[1, 2]. Dynamical tunnel-ing is rather ubiquitous in nature and manifests in sys-tems ranging from the dynamics of trapped cold atoms[3–5] to the flow of vibrational energy in molecules[6–8].Over the last decade, considerable advances[9] have beenmade in our understanding of the mechanism of dy-namical tunneling. It is now well established that dy-namical tunneling is extremely sensitive to the classi-cal phase space structures. In particular, for systemsin the mixed regular-chaotic regimes, the role of the non-linear resonances resulting in resonance-assisted tunnel-ing (RAT)[10–12] and the influence of the chaotic sealeading to the phenomenon of chaos-assisted tunneling(CAT)[13, 14] have been studied in great detail. Clearly,any attempt to control dynamical tunneling must there-fore take into cognizance the interplay of RAT and CAToccurring in the system.An attractive possibility, given that the tunneling hap-pens between symmetry related regions, is to break thesymmetry and arrest the tunneling dynamics. A naturalapproach when dealing with periodically driven systemsis the so called laser induced symmetry breaking[15–17].Thus, consider a Hamiltonian periodically driven by afield with amplitude E and frequency ωH ( q, p, t ) = H ( q, p ) − µ ( q ) E ( t ) (1) ≡ H ( q, p ) − E µ ( q ) cos( ωt ) . As usual, it is advantageous to characterize the solu-tions Ψ α ( q, t ) = e − i(cid:15) α t/ ¯ h Φ α ( q, t ) to the time-dependentSchr¨odinger equation in terms of the time periodic Flo-quet eigenstates Φ α ( q, t ) = Φ α ( q, t + 2 π/ω ) and the as-sociated quasienergies (cid:15) α . If the above Hamiltonian pos- sesses the dynamical symmetry[16, 18] H (cid:16) − q, − p, t + πω (cid:17) = H ( q, p, t ) (2)then the Floquet states of eq. 2 come as even-odd sym-metric doublets Φ ± α ( q, t ). The quasienergy splitting ofthe doublets ∆ (cid:15) α = | (cid:15) + α − (cid:15) − α | relates to the timescaleof dynamical tunneling. In order to break the symmetryand control the tunneling process one introduces a second(control) field to the Hamiltonian in eq. 2 as follows H ( q, p, t ) = H ( q, p, t ) − E µ ( q ) cos(2 ωt + θ ) , (3)with E and θ being the control field amplitude and rel-ative phase respectively. For E , E (cid:54) = 0 the symme-try of eq. 2 is violated. Moreover, for the relative phase θ (cid:54) = 0 , ± π an additional symmetry ( q, p, t ) → ( q, − p. − t )is also violated. Incidentally, note that the violationof the two symmetries is a prerequisite for observingdirected transport i.e., ratcheting in driven Hamilto-nian systems[18–21]. Bichromatic control Hamiltoni-ans as in eq. 3 have proved to be useful in many con-texts including reaction dynamics[22, 23], ionization ofatoms[24–27] and oriented molecules[28, 29], control ofpopulation imbalance of trapped BEC[30], high harmonicgeneration[31, 32], and orientation of rotationally coldmolecules[33–36].From the above arguments it appears that controllingdynamical tunneling by bichromatic fields as in eq. 3should be a relatively easy task. However, there areindications[37, 38] that this viewpoint is too simplistic insituations when the tunneling dynamics involves, apartfrom the doublets, additional Floquet states. For in-stance, an earlier study[39] showed that the driven doublewell system[40, 41] could be controlled using bichromaticfields. In contrast, a subsequent study[38] of the bichro-matic control landscape of the driven double well estab-lished that involvement of Floquet states delocalized inthe stochastic region of the phase space leads to a lackof control of the tunneling dynamics. Nevertheless, thereare important questions that still need to be answered. a r X i v : . [ n li n . C D ] N ov For instance, is there a specific class of delocalized Flo-quet states that can lead to a complete loss of bichromaticcontrol? What happens to the nature of the bichromaticcontrol landscape upon varying the effective value of thePlanck’s constant? In this work we address these ques-tions in detail for a simple yet paradigmatic model of thedriven pendulum. We show that the control knobs ( E , θ )are ineffective in certain regimes, further strengtheningthe notion that symmetry breaking is not always effectivein controlling processes dominated by CAT. II. MODEL HAMILTONIAN AND THEPROCESS OF INTEREST
The Hamiltonian of interest[20] is a periodically drivenpendulum and can be expressed as H ( q, p, t ) = 12 p + (1 + cos q ) − q E ( t ) , (4)where E ( t ) ≡ E cos( ωt )+ E cos(2 ωt + θ ) is a bichromaticexternal field. The driven pendulum is a paradigmaticmodel system that appears in various contexts. Here E is the amplitude of the driving field and E and θ are theamplitude and the relative phase of the controlling field.Due to the periodicity of the Hamiltonian in eq. 4, it isadvantageous to study the time evolution in terms of theFloquet states | Φ α (cid:105) and their associated quasienergies (cid:15) α with − π < (cid:15) α < π .As discussed in the introduction, our interest in thispaper is to use the 2 ω -field to control a dynamical tun-neling process occurring in the system for E = 0. Westart with a brief overview of the nature of the classicalphase space associated with eq. 4 and then specify thequantum process of interest.The nature of the phase space for eq. 4, a 1 . E values of interest with E = 0 and the driving field frequency ω = 2. Note thatthe main features of the phase space are nearly the samethroughout the entire range. In order to identify the var-ious nonlinear resonances one writes eq. 4 in terms of theaction-angle variables ( J, φ ) of the unperturbed pendu-lum Hamiltonian and considering E ( t ) = E cos ωt yields H ( J, φ, t ) = H ( J ) + E ∞ (cid:88) n = −∞ V n ( J ) e i ( nφ + mωt ) , (5)with m = ± E < E >
2) orbits. Nevertheless, both casescan be written in the above form with different Fourieramplitudes V n ( J ). In addition, only odd values of n ap-pear for the trapped case. Thus, all field-matter nonlin-ear resonances are of the form n ˙ φ ≡ n Ω( J ) = ω and aredenoted in this work as n : 1. The nonlinear frequencies FIG. 1: The right panels show the stroboscopic surface ofsection for the driven pendulum in eq. 4 for E = 0 . E = 1 . ω = 2 and E = 0. The left panel shows the tunneling(shown as arrows in the phase space) time between the sym-metry related Ω : ω = 1 : 1 islands for a coherent state placedat the center of the left island. The three data shown in theleft panel correspond to ¯ h = 0 . h = 0 . h = 0 .
05 (blue line). can be determined to beΩ( J ) = (cid:40) π K ( k t ) trapped πk u K ( k u ) untrapped , (6)with K ( k ) being the complete elliptic integral of the firstkind with moduli k t = (cid:112) E / k u = (cid:112) /E . InFig. 1 a prominent Ω : ω = 1 : 1 nonlinear resonanceis present. It can be shown that this corresponds toa resonance between an untrapped (rotor) state of thependulum and the ω -field. For the values of the effectivePlanck constant used in this work, this 1 : 1 resonanceisland can support several Floquet states. In addition, aΩ : ω = 3 : 1 resonance involving a trapped state of thependulum and the ω -field can be observed for E = 0 . E = 1 . ω = 1 :1 resonance islands since the 1 : 1 islands play a key rolein enhancing the directed current in the system[19, 20].Classically, trajectories in the left island are trapped for-ever. However, quantum mechanically, it is well knownthat dynamical tunneling destroys the classical localiza-tion. Thus, a coherent state initially localized in the leftisland of Fig. 1 can tunnel to the symmetry related rightisland. In Fig. 1 the tunneling time for an initial state ψ ( q ; q c , p c ) = (cid:104) q | ψ (cid:105) = 1(2 πσ ) / exp (cid:20) − ( q c − q ) σ + i p c q ¯ h (cid:21) (7)localized about the island center ( q c , p c ) are shown as afunction of E for various values of ¯ h with σ = ¯ h/ | ψ (0) (cid:105) = (cid:88) α C α (0) | Φ α (cid:105) (8)and obtaining the time evolved state | ψ ( kT f ) (cid:105) at integermultiples of the field period T f = 2 π/ω using the Floquetoperator. The Floquet states of eq. 4 are determinedusing an efficient method used in an earlier work[20]. Thefluctuations, increasing in number and amplitude withdecreasing ¯ h , of the tunneling times seen in Fig. 1 areparadigmatic of the RAT and CAT phenomena. A ratherdetailed understanding of these fluctuations in terms ofthe various phase space structures is possible[8, 12, 13,42]. However, in the present work we are interested incontrolling the dynamical tunneling process of Fig. 1 byswitching on the second field in eq. 4 and mapping outthe bichromatic control landscape. III. BICHROMATIC CONTROL LANDSCAPE
We now switch on the 2 ω -field with θ (cid:54) = 0 , ± π andstudy the dynamics of the initial state centered on theleft 1 : 1 island in Fig. 1. As discussed in the introduc-tion, such a symmetry breaking field should destroy thedynamical tunneling process and essentially localize theinitial state. In the context of Hamiltonian ratchets, theinitial state is centered about a transporting island andsymmetry breaking is expected to result in directed mo-tion. In order to obtain a comprehensive view of the con-trol process it is useful to construct a control landscape.Such a control landscape should unambiguously show re-gions of control or lack of control. Several measures canbe used to construct the control landscape. For instance,the decay time of the survival probability associated withthe initial state S ( τ ) ≡ |(cid:104) ψ (0) | ψ ( τ ) (cid:105)| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) α e − i(cid:15) α τ/ ¯ h (cid:104) ψ (0) | Φ α (0) (cid:105)(cid:104) Φ α (0) | ψ (0) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) α,β p α p β e − i ( (cid:15) α − (cid:15) β ) τ/ ¯ h (9)can be used as a measure for the control landscape. Inthe above, p α ≡ |(cid:104) ψ (0) | Φ α (0) (cid:105)| is the overlap intensity.Note that the tunneling times shown in Fig. 1 are de-termined by such an approach. It is also possible to usethe time averaged version of eq. 9 as a measure for thecontrol landscape[38]. However, such approaches can be-come unwieldy for various reasons and in this work weuse a simpler measure to map out the landscape. Thismeasure is the long time limit of S ( τ ) given bylim τ →∞ S ( τ ) = (cid:88) α p α ≡ L ( E , E , θ ) (10) FIG. 2: Control landscape L ( E , E , θ ) as in eq. 10 for ¯ h = 0 . h = 0 . θ = π/ ω =2. Lowest values signal multiple states participating in thedynamics of the initial coherent state and lack of bichromaticcontrol. Landscapes computed on an equally spaced 101 × and represents the number of Floquet states that par-ticipate in the dynamics of | ψ (0) (cid:105) . In eq. 10 we havedenoted L ( E , E , θ ) as the landscape ‘function’. It isalso possible to have the driving frequency ω as anotherparameter for the landscape function. However, in caseof the present work, this is of limited utility since the na-ture and location of the 1 : 1 island changes with varying ω . The measure in eq. 10 is particularly useful from thedynamical tunneling perspective since CAT and RAT areexpected to be multistate processes.In Fig. 2 we summarize our results for the control land-scapes L ( E , E , π/
2) computed using eq. 10, and theinitial state (cf. eq. 7) being localized about the left1 : 1 resonance island in the phase space. In Fig. 2 (top)we show L ( E , E , π/
2) for ¯ h = 0 . L ( E , E , π/ ω -field isunable to control the dynamical tunneling process. Thecalculation for ¯ h = 0 . h = 0 .
2. In addition, new regions, for in-stance around E ≈ .
7, emerge in the control landscape.Insights into the origins of such regions of no controlseen in Fig. 2 comes from inspecting the variation ofthe quasienergy spectrum with E for E = 0 shown inFig. 3. Clearly, regions of no control in Fig. 2 correlatewell with the avoided crossings involving the tunnelingFloquet doublets with other Floquet states. Althoughthe observation that such avoided crossings can lead tothe loss of bichromatic control has been made earlier[37],a comprehensive study of the landscape as a function ofthe key control knobs ( E , θ ) is still lacking. To this end,in Fig. 3 we also show L ( E , E , θ ) for fixed values of E corresponding to three of the avoided crossings. Theresults show that not all avoided crossings are equallyeffective in disrupting bichromatic control. While thereare regions that exhibit a complete loss of control as inFig. 3(III), there are regions as in Fig. 3(II) where con-trol is restored with small variations in the the relativephase θ . One may argue, and correctly so, that the differ-ences come from the extent of sharpness of the avoidedcrossings and hence the effective coupling between theFloquet states. Nevertheless, in the following section weshow that deeper insights come from investigating thephase space nature of the participating states near theavoided crossings.We note that similar correlations between avoidedcrossings and landscape features, including the mecha-nisms and arguments presented below, are seen in the¯ h = 0 . h = 0 . IV. CLASSICAL-QUANTUMCORRESPONDENCE FOR BICHROMATICCONTROL
In order to study the quantum-classical correspon-dence for the results shown in Fig. 2 and Fig. 3 it isimportant to establish the phase space nature of the Flo-quet states that play an important role in the dynamicsof the initial state of interest. Consequently, we computethe Husimi distribution[43] ρ H ( q c , p c ) = 12 π |(cid:104) Φ α | ψ (cid:105)| , (11)given by the overlap of coherent states ψ ( q ; q c , p c ) local-ized at various phase space points ( q c , p c ) (cf. eq. 7 ) withthe Floquet states of interest | Φ α (cid:105) . Note that a scalingfactor involving the Planck constant in the denominatorof eq. 11 has been ignored. This is of no consequence tothe present study where only the qualitative features ofthe Husimi distributions are of interest. The phase spacenature of the various Floquet states thus obtained areexpected to be important in understanding the featuresof the control landscapes. A. Role of the irregular Floquet states
In Fig. 4(a) the Husimi representation of two of thekey Floquet states, corresponding to the case I of Fig. 3with E = 0, are shown along with the overlap inten-sity spectrum. Clearly, a state influenced by the 1 : 3 FIG. 3: Left panel shows a portion of the quasienergy spec-trum as a function of the ω -field amplitude E . The otherfixed parameters are E = 0, ¯ h = 0 . ω = 2. Three re-gions I, II, and III highlight different types of avoided cross-ings. Right column shows the ( E , θ ) control landscapes withfixed E = 0 .
919 (bottom, case I), E = 1 .
129 (middle, caseII), and E = 1 .
399 (top, case III). All landscapes in this plotare computed over an equally spaced 101 ×
101 grid of therelevant variables. nonlinear resonance is involved in the avoided crossingobserved in Fig. 3 and this scenario, also highlighted inan earlier work[37], is typical of the RAT mechanism.The 1 : 3 state continues to influence the initial statedynamics for increasing amplitude of the 2 ω -field. Thus,the lack of bichromatic control seen in Fig. 3 (landscape,middle panel) arises due to the phenomenon of RAT andconfirms earlier predictions made in the context of localcontrol of the dissociation of a driven Morse oscillator[44].Results for the comparatively more complex case IIIare shown in Fig. 4(b) and it is immediately clear thatthe dynamics involves at least four Floquet states withtwo of them being significantly delocalized in the phasespace. Interestingly, of the two most delocalized Flo-quet states shown in Fig. 4(b), one of them resemblesa “Janus” state[12] which has its Husimi density local-ized around the border between the central stable islandand the chaotic sea. Such states are expected[12] to beinvolved in CAT and are less sensitive to the symmetrybreaking induced by the 2 ω -field. Indeed, our computa-tions show that the delocalized states persist upon intro-ducing the symmetry breaking field and continue to playan important role in the dynamics of the initial state.Thus, the “wall of no control” seen in the L ( E , E , π/ FIG. 4: Husimi distributions of delocalized Floquet statesparticipating in the dynamics of a coherent state centered onthe left 1 : 1 resonance island for E = 0. (a) Floquet statesfor E ≈ .
919 influenced by the 1 : 3 resonance, correspond-ing to case I in Fig. 1 and involved in an avoided crossing.(b) Two of the Floquet states for E ≈ .
399 delocalized inthe chaotic region, corresponding to case
III in Fig. 1. Axisrange of the phase space are identical in all the plots andmaximum Husimi density is in green. The overlap intensityspectrum in each case is also shown. from Fig. 5 where snapshots of the time evolving Husimidistributions are shown. It is clear that the survival prob-ability oscillates with a time period of ∼ τ correspond-ing to significant revivals of the initial coherent state.Furthermore, the snapshots at specific times indicate theinvolvement of the delocalized Floquet states shown inFig. 4(b) and hence the lack of control observed in thelandscape.In contrast to case I and III discussed above, Fig. 3(middle) shows that in case II it is possible to control thedynamics. The key difference here is that the observedavoided crossing in Fig. 3 involves the regular tunnelingdoublet and a regular rotor doublet. Note that here tooone observes lack of control in the L ( E , E , π/
2) land-scape in isolated regions around E ≈ . E ≈ . ω -field but the initial co-herent state does not localize. Instead, as shown in Fig. 6, FIG. 5: Survival probability S ( τ ) for a symmetry broken casewith E = 1 . E = 0 .
3, ¯ h = 0 .
2, and θ = π/
2. Initialstate is centered around the left island in the classical phasespace (shown in the inset) and time τ is in units of the fieldperiod. Dynamical tunneling is observed as shown by thesnapshots of time evolving Husimi distributions in the leftcolumn. The Husimis are shown at select times, with the topmost panel being at τ = 0 and the bottom most panel at τ = 20, highlighted by red circles in the survival probabilityplot. a two state sharp avoided crossing induced by the sym-metry breaking field results in the delocalization of theinitial state. Our computations show that very little tono amplitude is built up in the right 1 : 1 island over longtimes and hence quite different from the results shown inFig. 5 . Thus, the mechanism for the lack of controlis distinct from cases I and III. This is also reflected inFig. 3 (middle) wherein small variations in the controlknobs E or θ disentangles the Floquet states and leadto control. B. Influence on directed transport
It has been argued that symmetry breaking leadsto directed transport classically as well as quantummechanically. Classically, a non-zero dc-current J ch comes about[19] due to the desymmetrization of thechaotic layer and perturbative arguments lead to J ch ∼ E E sin θ . Quantum mechanically, however, significantenhancements in the current can occur when the relevantFloquet states are involved in an avoided crossing[20]. Inparticular, Denisov et al showed that avoided crossingsbetween Floquet states localized on the 1 : 1 transportingisland and Floquet states delocalized in the chaotic layerplay an important role. Moreover, tuning the ( E , θ ) con-trol knobs allows one to vary the extent of the enhanced -0.64-0.63-0.62 (cid:161) (cid:95) A B AB
FIG. 6: Variation of the quasienergies with E for fixed E ≈ . h = 0 .
2, and θ = π/
2. Note that one of theFloquet states is involved in a sharp avoided crossings withdesymmetrized rotor states at E ≈ . .
5, labeled asA and B respectively. Right column shows the snapshots ofHusimi representation for the time evolved initial state at τ = 1146 and τ = 1454 corresponding to A and B respec-tively. In both cases, the times (in units of field period) arechosen when the survival probability is a minimum. current.From the results in the previous section, as exempli-fied by Fig. 4(b) and Fig. 5, it is evident that Floquetstates delocalized in the chaotic layer are also responsi-ble for the loss of bichromatic control due to CAT. Aquestion then arises - to what extent will the failure ofsymmetry breaking influence the directed transport? Isit possible for CAT to partially suppress the magnitudeof the asymptotic current? To this end we compute thequantum asymptotic current J ( t ) = (cid:88) α | C α ( t ) | (cid:104) p (cid:105) α (12)for an initial state which is an eigenstate of the momen-tum with eigenvalue zero. Note that the study of Denisovet al.[20] utilized such an initial state to highlight the roleof the delocalized Floquet states in enhancing currents atresonance. In the above equation, (cid:104) p (cid:105) α denotes the av-erage momentum in the Floquet state | Φ α (cid:105) and C α ( t )represents the expansion coefficient of the initial state inthe Floquet basis, with t being the initial time.In Fig. 7 the computed asymptotic current for maxi-mum desymmetrization and initial time t = 0 is shown.Note that, quantum mechanically and in contrast tothe classical case, the asymptotic current is expected tobe dependent on the initial time. However, as arguedearlier[20], the strong enhancements in J (0) due to reso-nant interaction between Floquet states are independentof the initial time. Several such enhancements can beseen in Fig. 7 and indeed do correspond to avoided cross-ings between Floquet states (localized and delocalized)in the central region of the phase space (cf. Fig. 1) with FIG. 7: Magnitude of the quantum asymptotic current givenby eq. 12 as a function of E with E = 0 . θ = π/
2, andinitial time t = 0. The initial state is a plane wave statewith zero wavevector. Inset shows the magnification of theregion enclosed by the dashed rectangle. In the inset data for θ = π/ , π/
4, and π/ θ = π/
2. Top,middle and bottom panels correspond to A, B (inset) and Cin the current plot. Axes, their range and color code are sameas in the previous figures of Husimi representations. the Floquet states localized in and around the 1 : 1 trans-porting islands. Nevertheless, two key observations canbe made. Firstly, | J (0) | is significantly smaller in therange E ∈ (1 . , . V. CONCLUDING REMARKS
In this work we have shown that bichromatic control ofdynamical tunneling can be compromised due to the pres-ence of delocalized Floquet states in the classical phasespace. In essence, participation of the Floquet statesdelocalized in the chaotic regions of the phase space re-sults in chaos-assisted tunneling which is robust despitebreaking of the symmetries upon addition of a secondfield. Since the present work deals with the control ofdynamical tunneling, a quantum process with no classi-cal limit, the results do not necessarily invalidate the ideaof laser induced symmetry breaking approach to control.Indeed, our landscape computations for decreasing val-ues of the Planck constant do show that the “pillars” ofno bichromatic control are reduced to “dust” in the deepsemiclassical limit. Nevertheless, it remains to be seen asto whether chaos-assisted tunneling can reduce the effi-ciency or significantly modify the mechanism of controlbased on the principle of interference of optically inducedpathways[17].Moreover, our analysis suggests that such a loss ofbichromatic control can potentially influence current rec-tification in driven systems. An earlier study[45] by Gongand Brumer has also hinted at the possibility of reduceddirected currents in Hamiltonian ratchets due to dynami-cal tunneling between a chaotic sea and and transportingislands embedded in the chaotic sea. However, furtherdetailed studies are required in order to clearly establish the role of chaos-assisted tunneling in ratcheting systemswith[46] and without dissipation. In the latter case, therole of hierarchical states[47, 48] in both control of dy-namical tunneling and generation of directed transportneeds to be explored further. Note that one possible ex-planation for the “pillars” of no control being reducedto “dust” in the ¯ h → Acknowledgments
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