Engineering quantum correlations to enhance transport in cold atoms
Mark Sadgrove,1,* Torben Schell,2 Ken’ichi Nakagawa,1 and Sandro Wimberger2
1Institute for Laser Science, The University of Electro Communications, Chofushi, Chofugaoka 1-5-1, Japan2Institut fur Theoretische Physik and Center for Quantum Dynamics, Universitat Heidelberg, 69120 Heidelberg, Germany
(Received 26 May 2012; revised manuscript received 23 August 2012; published 28 January 2013)
We show experimentally that precise phase modulation of an optical potential allows us to control quantumcorrelations for atomic wave packets in a way that greatly enhances momentum transport. Experimentally, thismeans that for the same laser power and pulse frequency, atoms are accelerated to much higher energies. Weexplain our results with a pseudoclassical analysis along with numerical simulations, highlighting the existenceof transporting islands in phase space.
The properties of physical systems are determined bycorrelations between their degrees of freedom, and control overthese correlations allows one to engineer a system’s behavior[1]. There is, for instance, a famous relationship between staticcorrelations (disorder-induced phase transition) and dynamicalcorrelations (induced by the quantum phase evolution). Theseare responsible for Anderson [2] and dynamical localization[3], respectively. As many experiments have shown, cold-atomsystems are a promising setting for the experimental control ofmany-body and dynamical correlations [4]. In particular bothof the aforementioned phenomena have been realized withcold [5] and ultracold atomic gases [6].
Fundamentally, such localization is ubiquitous for wavepropagation in disordered systems. However, for many ap-plications, it is desirable to engineer transport which avoidslocalization and exceeds classical transport rates. Takingthe specific case of the atom optics kicked rotor (AOKR),examples range from fundamental studies, e.g., tests ofthe semiclassical eigenfunction hypothesis [7], and ratchetsystems [8,9] to possible applications, such as quantumrandom walks [10] and precision measurements [9,11]. Thelink between these applications is a need to control local-ization effects and, ideally, the underlying phase space andthereby the transport properties of the system. In the AOKR,however, only pulse periods which are rational multiples of theinverse recoil frequency (i.e., Talbot time) give nonlocalizedtransport. Furthermore, while it is relatively simple to observemotion maximally suppressed (i.e., frozen) with respect toclassical diffusion in experiments [12], observing resonantmotion requires careful preparation of initial conditions [13]and depends on the details of the system (e.g., the atomicspecies and the optical transition line). Other methods requireacceleration of the atoms [14].
Very recently, modifications of the AOKR in which trans-port is enhanced have been considered theoretically. Typically,such modifications involve adding extra kick frequencies[15], but changing the phase of the potential has also beenconsidered [16]. Here, we experimentally explore a similar
*Now at Center for Photonic Innovation, The University ofElectro Communications, Chofushi, Chofugaoka 1-5-1, Japan;[email protected]
approach, applying control over quantum correlations byengineering the phase of the kicking potential directly. Weidentify experimentally and explain theoretically three primaryenhancements made to the dynamics by our phase controlmethod: (i) the quantum break time (QBT) for the atoms canbe greatly extended, (ii) the quantum resonance (QR) peaksof the kicked rotor are stabilized, and most interesting of all,(iii) the phase modulation of the lattice substantially altersthe phase space of the system, leading to the creation of newtransporting islands. Such transporting islands have so far onlybeen found in accelerated atom systems [14], in which formthey provoked a broad range of insights spanning numbertheory [17] to chaos- and resonance-assisted tunneling [18].Transporting islands are also necessary for proposed tests ofthe semiclassical eigenfunction hypothesis [7]. Although allthe effects we study here take place in the deep quantumregime, we are able to make use of a quasiclassical method[19–21] to provide physical insight into the observed behavior.
This paper proceeds by introducing the experiment, show-ing experimental results for extended quantum break timeand stabilization of QR along with simulations which includeexperimental details, before finally offering a physicallyinsightful explanation of the observed behavior in terms ofthe pseudoclassical picture mentioned above.
II. THE EXPERIMENT
Experimentally, we use laser-cooled 87Rb atoms (tempera-ture ∼20 μK) subjected to pulses of period T from an opticalstanding wave detuned 1 GHz to the blue of the D2 line. Thesystem is described by the Hamiltonian [20]
H (t ′) = τp2
2+ k
∑t∈Z
cos(x + φt )δ(t ′ − t) (1)
for time t ′, atomic center-of-mass momentum p (in units oftwo-photon recoils of the standing wave), position x (in unitsof the spatial period 2kL), kick strength k, and integer time (orkick counter) t and kick period τ . Note that τ = (T/TT )4π
with the Talbot time TT ≈ 66.3μs for 87Rb. Note also that inthis equation, the units of the continuous time variable t ′ areτ . The kick pulses had a temporal width of ∼700 ns. After a12-ms free expansion time, a 4-ms flash of near-resonant lightwas applied, and the resultant atomic fluorescence was imagedon a CCD camera. To control the phase from pulse to pulse,
SADGROVE, SCHELL, NAKAGAWA, AND WIMBERGER PHYSICAL REVIEW A 87, 013631 (2013)
we used a custom-made control mechanism, details of whichmay be found elsewhere [22].
As will be discussed further below, due to constraints onlaser power in the experiment, we chose a relatively smallbeam size relative to the atomic cloud compared with otherrecent experiments. We varied the beam size and the durationof the polarization gradient cooling phase of the experiment(and hence the cloud size) and found that the parameters usedhere gave the best experimental signal at QR. Although theexact distribution of kicking strengths is not trivial, to a fairapproximation it may be considered uniform between zeroand the maximum kicking strength. Results from other studieswhere the cloud-to-beam size ratio is not negligible [23] alongwith numerical simulations (which include the distributionof kicking strengths experienced by the atoms) lead us toconclude that this nonideal feature of the experiment has noeffect on our ability to observe the effects of phase modulation.This is also not surprising in light of results that showed themain structure of the QR peaks survives even maximally largefluctuations in the kicking strength [24].
III. EXPERIMENTAL RESULTS
For the standard AOKR, the kick-dependent phase term φt
is zero. We note that in previous experiments an acceleratedlattice was used to simulate gravity [9,25], and elsewhereperiodic driving was used to control the spatial expansionof a cold atomic gas in an optical lattice [26]. Furthermore,controlling the relative phase between two periodic latticesmay also be used to observe directed transport [27]. However,to our knowledge, direct phase control of individual pulses fora single optical lattice has not been implemented as a methodto enhance transport. Here, we demonstrate how to control thetransport in momentum space by a proper choice of φt in (1). Inthe following we focus on the specific period four sequence forφt given by {0,π/2,0, − π/2, . . .}. This sequence was foundto provide an optimal combination of enhanced initial energygrowth relative to the unmodulated case.
Figure 1 shows momentum distributions obtained without[Fig. 1(a)] and with [Fig. 1(b)] phase modulation. It isnoticeable even from this minimally processed data that inthe presence of modulation the distributions obtained arebroader. Because the mean energy is related to the momentumdistribution width by 〈E〉 = 〈p2〉/(2) (in the units of theHamiltonian) the energy measurements presented in Fig. 2give a quantitative comparison of the distribution widths.
A. Extension of QBT
The central results of our experiments are shown in Fig. 2,where we scan τ over a representative quarter cycle. We alsochecked that the pattern recurred at larger τ . The results showclearly that phase modulation raises the overall kinetic energyof the atoms. In particular, the energy in the shaded regionof Fig. 2 is larger than that of the QR at T = 33.15 μs(corresponding to τ = 2π ). One can show analytically thateven in the presence of modulations, the mean energy ofour atomic ensemble at QR is k2t/4 (see the Appendix),which is equal not only to the unmodulated quantum result[20,21] but also to the so-called quasilinear or correlation-free
32 34 36 38 40 42
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T (μs)p
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)
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−50
0
500
0.02
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0.08
T (μs)p
P(p
)
(a)
(b)
FIG. 1. (Color online) Experimentally measured momentum dis-tributions after 30 kicks obtained as T were scanned over a quartercycle between the half Talbot and Talbot times at T = 33.15 μs andT = 66.3 μs, respectively. (a) Without modulation and (b) with themodulation pattern φt ∈ {0,π/2,0, − π/2, . . .}. Notice the broaderdistributions in (b) corresponding to enhanced transport. In both pan-els, the dotted line indicates the height of the first distribution, and thearrows and vertical lines indicate its two-standard-deviation spread.
energy in the classical kicked rotor [28,29]. The numericalsimulations (lines in Fig. 2) show excellent qualitative andgood quantitative agreement with the experimental resultsin the case of modulation. In the case without modulation,qualitative agreement in the basic shape of the data is seen, butmany points do not lie near the predicted curve. We commenton this further below. In the simulations, we take into accountthe nonzero cloud-to-beam size ratio (3.2 here) along with themomentum cutoff at ±100hk created by the noise floor of theCCD camera. The peak value of the kicking strength was es-timated to be 4.1 by optimizing the numerically predicted andmeasured height of the resonance for the data from the phasemodulated experiment. Considering possible drifts of 10% inpower and 5% in the detuning alone, the peak value of k iscalculated to be 3.4 ± 0.5. We also note that the initial thermalmomentum spread of the atoms contributes a trivial offset tothe energy curves but does not affect their qualitative shape.
Our data show that phase modulation can lead to dynamicswhere, rather than localization being the normal behavior, mostatoms experience transport near or above the quasilinear rate.Additionally, the localized regime near QR becomes muchnarrower in τ . The inset in Fig. 2 shows how the energyincreases with kick number, emphasizing that modulationmakes the near-resonant localized regime into a transportingone for much longer times than without the phase modulation.
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10
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60
Kick period (μs)
Ene
rgy
(2−
phot
on r
ecoi
ls)
0 20 40
20
40
60
80
t
E
FIG. 2. (Color online) Numerical and experimental results scan-ning over an approximate quarter cycle after 30 kicks with (redcircles) and without (blue crosses) phase modulation. Error barsshow standard errors over three separate experiments. Solid linesshow quantum simulations for the same parameters with a momentumcutoff window of ±100 × hkL enforced. The dashed horizontal lineshows the level of the QR peak. The shaded gray region indicates thekicking periods for which the measured energy exceeds the QR energyfor the phase modulation. The inset shows the average energy of theatoms at T = 37 μs vs. kick number with (red circles) and withoutmodulation (blue crosses). The case for T = 33.15 μs (QR) in thepresence of modulation is shown for comparison (black squares),lying below the energies for T = 37 μs with modulation (red circles).
B. Stabilization of QR
We now comment on an important side effect of the phasemodulation, the stabilization of the QR peak at T = 33.15 μs.QR peaks are difficult to observe because resolution ofthe low-amplitude wings of the momentum distribution isrequired. As explained above, in our experiment, an effective100hk momentum cutoff window exists. Thus, our data forthe the case without modulation show a broad, low peak at QRwhich is not the maximum of the measured energy, as would beexpected from simple simulations. This effect was commentedon extensively in [23] and [30] and is the reason why theQR peak is typically smaller than expected in cold-atomexperiments for the case when the kick number is �5.
However, as is clearly seen in the data here, phase mod-ulation dramatically increases the visibility of the resonancepeak relative to the no-modulation case. This is all the moresurprising given that a simple analysis of the evolution operatorapplied to a momentum eigenstate at QR gives a lower ratefor the energy growth in the case of modulation by a factorof 4. Specifically, the one kick evolution operator in our sys-tem is defined by Uφ = exp(−iτ p2/2) exp[−ik cos(x + φ)],so over one period of modulation, we have the period 4evolution operator U = U−π/2U0Uπ/2U0. At QR recall thatexp(−iτ p2/2) = 1. Then, brief inspection shows that, at QR,the U±π/2 terms cancel, leading to a factor of 4 less energygrowth compared with the no-modulation case. Given theresult of this simple analysis coupled with the effect of themomentum window, as mentioned above, how can the apparent
5.5 6 2π 6.5 7
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τ=2π+ε
A B C
0 π 2ππ/2
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θ
J
0 π 2ππ/2
3π/2
5π/2B
θ
J
0 π 2ππ/2
3π/2
5π/2C
θ
J
FIG. 3. (Color online) The main panel shows a comparison ofthe results of pseudoclassical simulations (solid line) with quantumsimulations (circles) for k = 3 after t = 30, as in Fig. 2. Simulationsof the map (2) were performed for 10 000 different initial conditions,whereas the quantum simulations were averaged over 5000 initialplane-wave states with randomly chosen initial momenta within theexperimental distribution (which effectively extends over severalphase-space cells of the pseudoclassical map). The phase-spacediagrams from iterations of M4 [see Eq. (3)], A, B, and C, correspondto the respectively labeled values of τ marked by vertical lines in themain panel.
increase in the resonance height relative to the nonresonantcase be explained?
This apparent paradox is resolved as follows: our systemshows four resonance islands (see leftmost phase space inFig. 3) instead of one island representing the QR in the standardAOKR. [Figure 3 and the associated theory are explained fullyin Sec. IV.] Hence, although the resonant atoms gain momen-tum at a smaller rate with respect to the unmodulated case,there are four times more initial conditions (momenta in phasespace) supporting resonant transport. It is precisely this factwhich means that the QR peak structure is less sensitive to theexperimental momentum cutoff than the no-modulation case.Therefore the apparent growth of the QR peak in the presenceof phase modulation is a result of the fact that the modulationleaves the resonance height unchanged, together with the fourtimes smaller rate of momentum gain for resonance atomswhich makes the momentum distribution easier to measureexperimentally. Beyond being an experimental curiosity, thisimproved stability of the QR is a useful tool in the case wherethe resonance is being used to make measurements [11].
IV. PSEUDOCLASSICAL ANALYSIS ANDTRANSPORTING ISLANDS
Our phase modulation gives rise to new quantum-mechanical energy resonances. Following [32], the correla-tions 〈cos xi cos xj 〉 between kicks i and j can be calculatedexactly up to a few kicks [28,33]. We will comment on thesecalculations in the case of phase modulation elsewhere. Here,however, we are more interested in predicting the temporalevolution over many kicks, in which case no simple quantum
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expression for the energy exists. We now show that thepseudoclassical method [20,21], valid for small detunings inthe kick period from the QR (or multiples of half the Talbottime), can be developed further to include a finite sequenceof phase jumps φt . This tool, even if not fully exact, givesus a good qualitative understanding of the new dynamics forall experimentally accessible kick numbers up to rather largewindows around the QR.
The small parameter in the pseudoclassical map, corre-sponding to an effective Planck constant, is the detuning ε
from the QR condition, i.e., τ = 2π� + ε, with integer �.Then the quantum-mechanical kick-to-kick evolution can beapproximately described by the (pseudo)classical map [19–21]
Jt+1 = Jt + k sin(θt + φt ),(2)
θt+1 = θt + Jt+1,
where we introduce k = kε, Jt = εpt + const, and θt =xtmod2π . To avoid sign problems, we assume ε > 0 through-out. Since the phase φt is time dependent, only a finite sequenceof choices allows us to describe the quantum evolution bya Floquet map or the corresponding classical dynamics by aperiodic map. In our case, this period is four single kicks due tothe effective restriction to the values {0,π/2,0, − π/2}. Eachof these four steps may be represented by a nonlinear functionfrom the initial to the final two variables (Jt ,θt ) → (Jt+1,θt+1).We denote Mφ , φ = 0,π/2, − π/2 for the first, second, andfourth steps, respectively. The new time-periodic map is thenjust the concatenation of these functions:
M4 = M−π/2 ◦ M0 ◦ Mπ/2 ◦ M0 . (3)
The classical evolution is visualized by iterating map M4 andplotting the Poincare surface of sections.
We show simulations for the main experimental observable,the mean kinetic energy after a fixed number of kicks, alongwith the Poincare surface of section of the map M4 for threevalues of kε in Fig. 3. The strong localization (low-energyvalues) near QR (ε → 0) is explained by (nontransporting)island structures in phase space which prevent diffusion.On the other hand, the growing energies for larger kε �0.4 correspond to mainly chaotic regions. Figure 3 stressesalso the good correspondence of classical and true quantumsimulations, not only at small values of ε but even for thewhole range ε = 0, . . . ,0.8 up to experimentally relevant kicknumbers. This good qualitative agreement is not expectedfor the simple case of φt ≡ 0 [20,21] and is related to thenew type of temporal correlations induced by our controlledphase. The latter leads to much longer break times at whichthe quantum diffusion stops due to dynamical localization inthe classically chaotic part of phase space. Figure 4 presentsthe classical diffusion rates (the prefactors of linear energygrowth with time) and the the widths of the wave packets(localization length) after the break time as a function of thescaling parameter kε of the pseudoclassical map. This furtherquantifies the good correspondence between the classicalevolution by M4 and the quantum dynamics.
0 0.5 1 1.5 20
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kε
l loc
0 0.5 1 1.5 20
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kε
Dε−
cl
0 π 2ππ/2
3π/2
5π/2
θ
J
(a)
1.4 1.6 1.83
3.5
4
4.5
θ
J
(b)
FIG. 4. (Color online) (top) Localization length of quantum wavefunctions in momentum space and (bottom) classical diffusion ratesextracted from linear fits to the energy growth with time at smalltimes, both averaged over initial conditions corresponding to an entirephase-space cell and shown over the same range as in Figs. 2 and 3for ε > 0 at fixed k = 6. Insets (a) and (b) show transporting islandsbut plotted modulo 2π in the variable J ; otherwise, the orbits wouldjump to the next phase-space cell after one iteration of M4. In inset(a), the large crosses represent the 1:5 resonance islands themselves,which are too small to be visible as finer points [31]. The larger 1:1resonance is shown in (b).
V. DISCUSSION
In our current experiment, the atoms have a broad initialmomentum distribution spreading over the entire cell of ourpseudoclassical phase spaces [20,21], as shown in Fig. 3. Thismeans that all our curves shown in Figs. 2–4 average over allphase-space structures. Nonetheless we observe features of thephase-modulation-induced transport, such as the near-resonant“horn” (around T � 34 μs in Fig. 2) and off-resonant peaks(around T � 36.5 μs). The phase modulation introduces twoclasses of stable transporting islands embedded into a chaoticsurrounding which appear in visible size for 0.65 � kε � 0.75(responsible for the horn) and kε � 1.7 (for the peak structure).These new stable fixed points correspond to ballisticallyaccelerated trajectories whose energy increases quadraticallywith the number of kicks. This is seen for kε = 2.154 inFig. 5, showing the mean energy of an initial quantum stateand a corresponding pseudoclassical orbit overlapping withthe transporting resonance island in phase space. The insetsexplain the creation and destruction of the stable island whenchanging the control parameter kε. The horn instead arisesfrom a tiny 1:5 resonance shown already in inset (a) in Fig. 4.All those structures come in pairs due to a reflection symmetryin phase space around (J,x) = (3π/2,π ). Experiments usingBose-Einstein condensates with more precise control overinitial conditions in phase space [9,34] could directly focus on,e.g., the larger of the transporting modes seen here. Due to theirsimplicity in realization they may have practical applicationsfor accelerating atoms.
We have shown that it is possible to engineer quantumcorrelations for enhanced transport in a kicked atom system
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FIG. 5. (Color online) (top) Rescaled energy of a quantum wavepacket anchored to the transporting island for kε = 2.154 as afunction of iterations of the map M4. E0 is the initial energy ofthe wave packet. Data are shown for increasing k and decreasing ε
accordingly. The smaller ε is, the closer the quantum result is to theexact pseudoclassical evolution at the fixed point corresponding tothe traveling island (stars, topmost line). (bottom) The evolution ofthe transporting island with changing kε.
by precise control of the phase at each kick. This leads totheoretical peak transport rates (as measured by mean energy)several times greater than at QR, even for thermal atoms.We measured significant enhancements in energy even in thepresence of standard experimental limitations on detection.Our theory explains the experimental results predicting, inparticular, a new form of ballistic motion related to stabletransporting islands in the underlying pseudoclassical phasespace.
Any prospective use of the QR peaks for precisionmeasurements is limited by detection of small numbers ofhigh-momentum atoms. With our modulation method, thepeak shape and height is preserved, but the peak is easierto detect, as our measurements clearly show. Additionally,large localization lengths and transporting islands are ingre-dients necessary for some proposed tests of the semiclassicaleigenfunction hypothesis [7] and the experimental analysisof mixed phase-space systems in general [35], both of muchfundamental interest.
ACKNOWLEDGMENTS
We thank I. Guarneri and S. Fishman for helpful discus-sions. S.W. acknowledges financial support from the DFG(Project No. FOR760), the Helmholtz Alliance ProgramEMMI (Grant No. HA-216), the HGSFP (Grant No. GSC129/1), and the Enable Fund of Heidelberg University.
APPENDIX: RESONANCE PEAK HEIGHT WITHMODULATION
The height of the resonance peak in the absence ofmodulation was first calculated analytically in [20]. Weperform a similar calculation here to derive the height of the
resonance peak in the presence of the {0,π/2,0, − π/2, . . .}phase modulation used in this paper.
We start with the zeroth-order expansion in ε of themap (2) [20],
Jt = J0 + |ε|kt−1∑s=0
sin(θ0 + sJ0 + φs). (A1)
The applied phase modulation is φs = {0,π/2,0, − π/2, . . .},so we can rewrite Eq. (A1) as follows:
Now we use trigonometric angle sum identities to split thesine and cosine terms into s ′ dependent and independent parts.Then, using the trigonometric sum identities for arguments inarithmetic progression, we arrive at
We then evaluate the quantity E = (1/2|ε|2)〈(Jt − J0)2〉θ0 ,which requires averaging over the quantities 2
1, 22, and
12. This gives
〈(Jt − J0)2〉θ0 = k2|ε|2 sin2(tJ0/2)
sin2(2J0)[2 + sin J0 + sin 3J0].
(A5)
Finally, we find the mean energy
Et = 〈〈(Jt − J0)2〉θ0〉J0
2|ε|2 = k2
2
2
2π
∫ 2π
0
sin2(tJ0/2)
sin2(2J0)dJ0 = k2
4t,
(A6)
where we used the fact that sin2(tJ0/2)sin2(2J0) is positive and even
but sin J0 and sin 3J0 are odd and the substitution J ′0 = 2J0
to reach the final result. This result shows that the extraterms created by the phase modulation have no effect on theensemble-averaged energy at quantum resonance.
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