Quantum resonances of kicked rotor in the position representation
aa r X i v : . [ qu a n t - ph ] A ug Quantum resonances of kicked rotor in the position representation
Kush Mohan Mittal ∗ and M. S. Santhanam † Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India (Dated: August 16, 2019)The study of quantum resonances in the chaotic atom-optics kicked rotor system is of interest fromtwo different perspectives. In quantum chaos, it marks out the regime of resonant quantum dynamicsin which the atomic cloud displays ballistic mean energy growth due to coherent momentum transfer.Secondly, the sharp quantum resonance peaks are useful in the context of measurement of Talbottime, one of the parameter that helps in precise measurement of fine structure constant. Most of theearlier works rely on fidelity based approach and have proposed Talbot time measurement throughexperimental determination of the momentum space probability density of the periodically kickedatomic cloud. Fidelity approach has the disadvantage that phase reversed kicks need to be impartedas well which potentially leads to dephasing. In contrast to this, in this work, it is theoreticallyshown that, without manipulating the kick sequences, the quantum resonances through positionspace density can be measured more accurately and is experimentally feasible as well.
I. INTRODUCTION
Kicked rotor system, a particle that is periodicallykicked by an external sinusoidal field, is a fundamentalmodel of chaotic dynamics in a Hamiltonian system [1, 2].Atom-optics kicked rotor is an experimentally accessibleversion of the quantum kicked rotor model implementedby subjecting cold atomic gases to flashing optical lat-tices [3]. For large kick strengths, classical kicked ro-tor displays diffusive mean energy growth while in thecorresponding quantum regime it is suppressed by quan-tum interference effects leading to dynamical localization[1, 2, 4]. Such distinct dynamical behaviour in the classi-cal and quantum regime is what makes this system usefulto study the quantum to classical transition and decoher-ence effects. Recently, atom-optics kicked rotor has beenemployed to demonstrate non-exponential coherence de-cays and optimal diffusion rates [5]. In general, it hasbeen used widely to study unusual quantum transportscenarios [6], quantum entanglement [7–9] and in quan-tum metrology for acceleration measurements [10].From the point of view of atomic interferometry [11]and metrology applications, the quantum resonances inthe atom-optics kicked rotor [12, 13] are of special inter-est. Such resonances are purely quantum effects withoutany classical analogue. In the kicked rotor system, quan-tum resonances occur if the scaled Planck’s constant isof the form ~ = 4 π ls , where l and s are integers. For s = 1, time interval between successive kicks is called theTalbot time and the energy of the kicked rotor systemgrows quadratically due to coherent momentum transferto the atomic cloud. However, if s = 1, then an initialstate recurs after s kicks, in which the phase added bysuccessive kicks tends to cancel each other. This is calledthe anti-resonances [14]. Measurement of Talbot timethrough resonance effects is important for determining ∗ E-mail: [email protected] † E-mail: [email protected] precise values of fine structure constant and hence a cru-cial ingredient for quantum metrology [15]. The quantumresonances are at the heart of atom interferometers andwere experimentally realised using atom-optics kicked ro-tors [16–18]. One proposal for atom-optics kicked ro-tor based inteferometry relies on manipulating kick se-quences in such a way that the N pulses imparted to theatomic cloud are followed by N pulses whose phases dif-fer by π with respect to the former sequence [22]. Thisis shown to be capable of measuring Talbot time and thelocal gravitational field [22]. It is also known that quan-tum resonances are reasonably robust to small amountsof phase noise [23] and amplitude noise [24, 25] in thekick sequences, thus making these systems attractive forinterferometric studies and measurement of Talbot time.Atom optics kicked rotor system can be used to mea-sure Talbot time and it mainly relies on momentum spacemeasurements. The first order effects of perturbationabout the Talbot time shows up as a change in the phaseof the coefficients in momentum basis. Hence, this effectcannot be detected directly by measuring the momentumdistribution of atoms [19, 22]. To measure these effects,a fidelity measurement has been proposed involving theapplication of phase reversed kicks [19]. In this scheme, N periodic kicked rotor pulse sequences of strength k arefollowed by the last pulse which will have opposite phaseand strength N k . This is shown to display a sharp res-onance peak whose width is proportional to N − [19] incontrast to mean energy resonant profiles which scalesas N − . This scenario was experimentally observed ina fidelity measurement performed on Bose-Einstein con-densate in pulsed external field [20]. Although this wasable to capture the first order changes in the phase, ex-perimentally the kick reversal process led to pronounceddephasing. Hence this scheme was not feasible for largekick numbers. Thus, dephasing became an impedimentto achieving even sharper resonance peaks.In order to overcome this deficiency, in this paper, wepropose position space measurement and this does not re-quire manipulating the standard pulse sequence impartedto the kicked rotor system. This is motivated by the fact Position P r ob a b ilit y D e n s it y -30 -20 -10 0 10 20 30 momentum FIG. 1. Position space density after N = 40 kicks have beenimparted to the initial state evolving under the effect of kickedrotor Hamiltonian. The dotted line corresponds to ε = 0 andsolid line to ε = 10 − . The inset shows the momentum spacedensity for the same values of ε as the main figure. Note thepronounced difference between the cases of these two valuesof ε observed in the position representation, but not in thecase of momentum representation. that if the kick period differs from the Talbot time by ε ≪
1, then to first order, the resulting effect shows updirectly in the spatial distribution of atoms as a narrowerinitial peak about the Talbot time without the require-ment of the kick reversal process. On the other hand,in the momentum space, the effect is not distinct fromthe case with ε = 0. Hence, quantum resonance is betteranalysed in the position representation. This is illus-trated in Fig. 1. In this, the position space density isdisplayed for fixed values of ε = 0 and 10 − after N = 40kicks are imparted. Clearly, in comparison with the caseof ε = 0, for small deviation from Talbot time, positiondensity shows pronounced difference. In the momentumrepresentation (shown in the inset of Fig. 1), the prob-ability densities for ε = 0 and 10 − after N = 40 kicksdo not show any perceptible difference. The analysis pre-sented in this paper is motivated by this observable effectin position space as well as the fact that the experimentsusing optical mask techniques can directly probe positionspace density of cold atomic cloud [21]. Further, we alsoextend the results in Ref. [26] to derive analytical resultfor change in position density around the Talbot time.We compare the analytical results with the simulations. II. QUANTUM RESONANCES IN THE δ -KICKED ROTOR Kicked rotor system is experimentally realized usingultra-cold atoms in optical lattices. In this, the periodickicks are imparted using two far-detuned counter propa-gating laser beams. The pulse are considered to be in theshort pulse or Raman-Nath limit, i.e. , the evolution of the atomic cloud during the pulse duration is negligible.In the ideal limit of delta kicks the system is describedby the dimensionless Hamiltonian [4] given by b H = b P M + K cos b X N − X n =0 δ ( t − n ) . (1)where we have assumed that a total of N kicks are im-parted to the atomic cloud. In this, K is the kickingstrength, k L is the wave number of the standing wave, T is periodicity of the kicks. Both ˆ X and ˆ P are thescaled canonical variables. The corresponding evolutionoperator is b U = exp (cid:18) − i K ~ s cos X (cid:19) exp (cid:18) − i P ~ s (cid:19) , (2)in which ~ s = 4 ~ k L T /M represents the scaled Planck’sconstant. The evolution operator splits into the kick andthe free evolution part due to the δ -kicks. As the kick-ing potential is spatially periodic, the eigenstates have aBloch wave structure. Hence, an initial state | P o i corre-sponding to definite momentum, under the action of theevolution operator, gets mapped to states of the form | P o + m ~ s i , with m being an integer. In the analysispresented below, we shall be considering the case wherethe particles are initially in the zero momentum state, i.e, | P o = 0 i . It must be noted that if the profile of the initialstate has a width due to finite temperature effects, thenthe temporal evolution of the initial state strongly devi-ates from that corresponding to | P o = 0 i for longer times.This time-scale is inversely proportional to the width ofthe initial momentum distribution [27]. In experiments,at longer times (more kicks) the dephasing effects becomesignificant as well. Hence, the analysis presented here isvalid until dephasing becomes pronounced even if initialstates are not ideal. Further, the results of numericalsimulations shown in this were performed for an effectivekick strength of K/ ~ s = 0 .
485 and λ L corresponding toa wavelength of 780 nm.Considering this, we can write the state of the particleat any arbitrary time in the form,Ψ( X ) = 1 √ π ∞ X m = −∞ ψ ( m ) e imX . (3)Quantum resonance is characterized by the value of ~ s =4 πl , l > U given in Eq.(2) becomes exp( − i πm l ), effectivelyan identity operator. For this choice of ~ s , the kick pe-riod T is referred to as the Talbot time T b correspondingto the physical picture that the kicks accumulate phasesin a coherent manner [26]. In the analysis that follows,we compute the change in Ψ( X ) due to small perturba-tion in the Talbot time T B . Unlike the earlier studiesin momentum representation, we study the resonances inposition representation. X | Ψ ( X ) | AnalyticalNumerical
FIG. 2. Position space density after N = 5 kicks have beenimparted to the initial state evolving under the effect of kickedrotor Hamiltonian. The deviation from Talbot time is ε =10 − . The solid curve is the analytical result in Eq. 12 andsymbols are the numerically computed result. III. PERTURBATION ABOUT THE TALBOTTIME
As pointed out earlier, we assume that only N kicks areto be imparted. In the experimental context, dependingon the parameters of the set-up, typically the numberof kicks do not exceed a few hundred in units of kickperiod. Let Ψ( X, t −
1) represent the state at time t − N − th kick isjust applied. Following this, the free evolution operatoris applied to it for time t = T B + ε where ε ≪ X, t − ) = exp (cid:16) − i P κ (cid:17) Ψ( X, t − , (4)in which t − denotes the state just before the N th kickis applied. Now, as discussed earlier, P = m ~ s and itis evident that for small perturbations about the Talbottime the scaled Planck’s constant becomes ~ s = 4 πl (1 + εT B ). Substituting these values for P and ~ s in Eq.(4),we getΨ( X, t − ) = exp (cid:18) − i m πl (1 + εT B ) (cid:19) Ψ( X, t − . (5)The exponential term can be split into two parts; one cor-responding to the ideal Talbot time condition and othercorresponding to the perturbation. For ε ≪
1, the expo-nential is expanded as a Taylor series and terms of order ε and higher are discarded. We consider the primaryresonance by taking l = 1, and this leads toΨ( X, t − ) = e − i πm Ψ( X, t −
1) exp (cid:16) − i m π εT B (cid:17) = Ψ( X, t − − i m π εT B Ψ( X, t −
1) (6) Using Eq.(3), Ψ(
X, t − ) can be rewritten asΨ( X, t − ) =Ψ( X, t − − (7)1 √ π ∞ X m = −∞ ψ ( m, t − e imX (cid:16) i m π εT B (cid:17) To analyze the first order effect of the perturbation, theposition space density is obtained as | Ψ( X, t − ) | = | Ψ( X, t − | − πiε √ πT B ∞ X m = −∞ m (cid:2) Ψ ∗ ( X, t − ψ ( m, t − e imX +Ψ( X, t − ψ ∗ ( m, t − e − imX (cid:3) , (8)in which only the terms first order in ε are retained. Thefirst term on the right hand side corresponds to the prob-ability density at time t − N -th kick. Us-ing Eq.(3) again, the correction terms appearing in Eq.(8)can be written as ∞ X n = −∞ ∞ X m = −∞ im εT B (cid:16) ψ ∗ ( n, t − ψ ( m, t − e i ( m − n ) X + ψ ( n, t − ψ ∗ ( m, t − e i ( n − m ) X (cid:17) . (9)It is evident that the terms for which n = m cancel eachother and the summation is left with terms with n = m .Thus, the corrections can equivalently be written as, ∞ X m = −∞ X n>m Re h π ψ ∗ ( n, t − ψ ( m, t − e i ( m − n ) X (cid:16) i ( n − m ) 2 πεT B (cid:17)i (10)where Re ( . ) represents the real part. Since we are work-ing upto first order in ǫ , the amplitude ψ in Eq.(10) corre-sponds to that when the Talbot condition is met. Hencethey can be written in terms of n -th order Bessel func-tion J n ( . ) as ψ ( n, t −
1) = ( − i ) n J n (( N − φ d ) [28] where φ d = K/ ~ s at time t − N − th kick has beenapplied. This leads to the expression for the correctionterm C N ( ε ) : C N ( ε ) = 1 π ∞ X m = −∞ X n>m Re h e i ( m − n ) X ( n − m ) 2 πiεT B i n J n (( N − φ d )( − i ) m J m (( N − φ d ) i (11)In this, J m ( . ) is the Bessel function of order m with realargument. Thus, the probability density under the firstorder approximation can be finally written as, | Ψ( X, t − ) | = | Ψ( X, t − | + C N ( ε ) . (12) -3e-06 -2e-06 -1e-06 0 1e-06 2e-06 3e-06 ε σ X FIG. 3. The numerically computed standard deviation σ X ofthe probability density function as a function of perturbationabout Talbot time ε for a fixed kick number. It is displayedfor the kick numbers N = 5 (outer most curve) to N = 10(inner most curve). The value of σ X decreases with increasein the kick number. Physically, the effect of kick is to give a phase factorof exp( − i K ~ s cos X ). This does not affect the probabilitydensity in position basis, but leads to occupancy of highermomenta states which in turn affects the perturbationterm. It is clear that Eq.(12) forms a recursive equationconnecting Ψ at successive kicks and complete analyticalthough cumbersome approximation can be found recur-sively. IV. POSITION SPACE ANALYSIS
In Fig. 2, perturbation based analytical result ob-tained in Eq.(12) is compared with the numerical sim-ulations for ε = 10 − and N = 5 kicks. The analyticalresult is calculated recursively starting from momentumeigenstate p = 0 at t = 0. The kicked rotor Hamiltonianis evolved with kick strength φ d = K/ ~ s = 0 . | Ψ( X ) | . This is conveniently measured using thestandard deviation denoted by σ X . It is anticipated thatas the kick period deviates from Talbot time, as quanti-fied by ε , the position space density will evolve from anuniform profile at N = 0 (shown in Fig. 1) to a narrowprofile as N ≫
1. The width σ X of this profile will decaywith increasing kick number. As is evident from the nu-merical results in Fig. 3, as the number of kicks increase,the width does indeed decrease. It takes a power-lawform σ X ∝ N − γ , and the exponent γ is estimated byregression to be 2.10. N) -7-6.5-6-5.5 l og σ X fidelity ( γ = −3.08) position ( γ = −2.10) FIG. 4. The width σ of distribution about the Talbot timeas a function of kick number for fidelity analysis (reported in[19]) and the position space analysis studied in this paper.The symbols are the simulation results and the solid linesrepresent best-fit lines. The slope for fidelity analysis is esti-mated to be γ = − .
08 while for the position space analysisit is γ = 2 . It must be noted that the width in the case of fidelitybased analysis scales with N whose exponent is γ ≈ − . /N whereas for fi-delity analysis it scales as 1 /N . Notice also that in gen-eral the width of position space based method starts fromfar lower width in comparison to the fidelity method. Itturns out that till N = 16 kicks, the position space dis-tribution has lesser σ than that of fidelity approach andhence can potentially lead to better Talbot time mea-surement.However, one significant problem with the fidelity tech-nique is the requirement of kick reversal process, whichcan potentially lead to dephasing. This implies that, fora fair comparison of both these approaches, if positionspace analysis presented here applied M kicks, then fi-delity technique will require 2 M kicks, i.e. , M normalkicks plus M phase reversed kicks, to be applied. If thisis taken into account, then for identical total number ofkicks applied, it is clearly seen that position space densitybased analysis of quantum resonance far outperforms thefidelity based approach. V. CONCLUSIONS