The effect of Landau-Zener tunnelings on the nonlinear dynamics of cold atoms in a modulated laser field

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by  Victor Argonov

The effect of Landau-Zener tunnelings on the nonlinear dynamics of cold atoms in a
modulated laser field
V. Yu. Argonov
Laboratory of Geophysical Hydrodynamics, Pacific Oceanological Institute of the Russian Academy of Sciences,
43, Baltiiskaya Street, Vladivostok, Russia 690041
Westudythemotionofacoldatominafrequency-modulatedstandinglaserwave. Ifthedetuning
betweentheatomicelectronictransitionandthefieldislarge,theatommovesinamodulatedoptical
potential demonstrating known classical nonlinear effects such as chaos and nonlinear resonances.
If the atom-field detuning is small, then two optical potentials emerge in the system, and the atom
performs Landau-Zener (LZ) tunnelings between them. It is a radically non-classical behavior.
However,weshowthatclassicalnonlinearstructuresinsystem’sphasespace(KAM-toriandchaotic
4
layers) survive. Quantum effect of LZ tunnelings only induces small random jumps of trajectories
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between these structures (dynamical tunneling).
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2
PACSnumbers: 05.45.-a,37.10.Vz,05.45.Mt

n
a
J I. INTRODUCTION of these results was not clear. In [12, 13] more rigorous

9 quantum analysis of atomic motion was performed with-
out semiclassical approximation: atoms were considered
Cold atoms in a laser field is a popular system for the
]
as wave packets, not dot-like particles. This new model
h studyofquantumchaos. Since1970s,alotofauthorsre-
p portedvariousnonlineareffectsinsemiclassicalmodelsof was used to test our old results. In a ”chaotic” range
- of parameters of semiclassical models [9, 11], new model
m interactionbetweentwo-levelatomsandlaserwaves(con-
demonstrated random atomic Landau-Zener (LZ) tran-
stant, modulated, and kicked): dynamical chaos [1, 2],
o sitions between two optical potentials. In [12], it was
dynamical localization [3–5], random walking of atoms
t shown that LZ transitions (in quantum model) and dy-
a [6], dynamical tunneling [7, 8], fractals [9], strange at-
. namical chaos (in semiclassical model) produce similar
s tractors[10],nonlinearresonances[11]etc. Someofthese
c studieshavenotonlytheoreticalbutalsomethodological statistical effect on atomic mechanical motion. However,
i subtle nonlinear effects in electronic transitions seem to
s aspect of quantum-classical correspondence: they com-
y pareidealizednonlinear(inparticular,chaotic)dynamics disappear in quantum model.
h
ofasemiclassicalmodelwithrealdynamicsofaquantum In this paper we are trying to overcome limitations of
p
system and analyze the difference. both above-mentioned approaches and to study the non-
[
As a rule, in order to observe nonlinear dynamical ef- lineardynamicsofanatominamodulatedstandingwave
1 fects, the model must be simple enough. There are two near optical resonance. We use a stochastic trajectory
v
popular approaches to observe clear nonlinear dynamics model combining semiclassical dynamics (regular atomic
6
andchaosinatom-fieldmodels: (1)neglectthedynamics motionfarfromstanding-wavenodes)andrandomjumps
5
1 of atomic electronic transitions and study only mechani- (LZ transitions near the nodes). We study the dynamics
2 calmotionofanatominamodulatedlaserfield(detuned by the method of Poincar´e mapping (Poincar´e section)
. farfromopticalresonance), (2)studytheinteractionbe- and report the existence of prominent nonlinear struc-
1
0 tweenatomicelectronicalandmechanicaldegreesoffree- tures (KAM-tori and chaotic areas) in systems’s phase
4 dom in a stationary field (near optical resonance). spaceeveninpresenceofLZtunnelings. Thesizeofsome
1 Firstapproachislessfundamental,becausethesystem nonlinear structures is large enough to overcome Heisen-
: berg limitations. Therefore, they seem to be not just
v is non-autonomous. However, it requires comparatively
artifacts of our model but a real property of the system
i simple experimental setup. Since 1990s, several impor-
X
tant theoretical results on atomic chaos in a modulated We report not only the existence on nonlinear struc-
ar field were directly tested in experiments. In particular, tures but also LZ-induced dynamical tunneling of atom
notable experiments were performed in order to observe betweenthem. ApairofLZtunnelings(fromonepoten-
dynamical localization [4] and dynamical tunneling [7]. tialtoanotherandback)maycauseajumpofatomictra-
The phenomenon of dynamical tunneling is very impor- jectory between two classically isolated nonlinear struc-
tant in the framework of this paper, and we will discuss tures(e.g. twoKAM-tori). Suchquantum-inducedjump
it in details in further paragraphs. is classically prohibited not by energy but by another
Secondapproachismorefundamentalbutmethodolog- constant of motion, and it is called dynamical tunnel-
icallymoreproblematicbecausesemiclassicalapproxima- ing. Thephenomenonofdynamicaltunnelingwasdeeply
tionisnotverygoodforthedescriptionofelectronictran- studied in [7, 8] in the regime of large atom-field detun-
sitions. We applied second approach in our early papers ings without LZ transitions. We report similar effect in
[9–11] using idealized semiclassical models and reported another physical situation: atom is close to optical reso-
a lot of nonlinear effects. However, physical correctness nance, and its dynamical tunnelings are directly caused
2
by pairs of successive LZ tunnelings.
II. EQUATIONS OF MOTION
In this paper we study the same system as in [15, 16].
A two-level atom with transition frequency ω and mass
a
m moves in a standing laser wave with modulated fre-
a
quency ω [t]. We neglect spontaneous emission (excited
f
state has a long lifetime, or some methods are used to
suppress decoherence). In [15, 16], we started the analy-
sis with Jaynes-Cummings Hamiltonian obtaining quan-
tumequationsofatomicmotion. Inthispaper, however,
we use a simplified stochastic trajectory model obtained
in[16]. Themodelisconsistofsemiclassicalandstochas-
tic parts: atom moves as a dot-like particle in a classical
potential, but sometimes it performs random jumps.
Semiclassical part. Semiclassical approximation is
valid when an atom moves between standing-wave nodes
(k X (cid:54)=π/2+πn, cos[k X](cid:54)=0). This motion is regular
f f
and may be described by the Hamiltonian [13–16]
(cid:114)
P2 (ω −ω [t])2
H = +(cid:126)ΩU∓, U∓ =∓ cos2[k X]+ a f .
2m f 4Ω2
a
(1)
Here Ω is a Rabi frequency (field intensity) and k is a
f
wavevector. ThesignofpotentialU∓ isconserved,ifthe
atomdoesnotcrossstanding-wavenodes. Letususethe
following normalized quantities: momentum p≡P/(cid:126)k ,
f
timeτ ≡Ωt,positionx≡k X,massm≡m Ω/(cid:126)k2 and
f a f
detuning ∆[τ] ≡ (ω [τ]−ω )/Ω. Energy and potential
f a
(Fig. 1a, dashed lines) take simple forms
(cid:114)
p2 ∆2[τ]
E = +U∓, U∓ =∓ cos2[x]+ , (2)
2m 4
FIG. 1: Optical potentials and their phase portraits: (a)
and we obtain semiclassical equations of motion dashedline: potentialsU±,solidline: standingwave−cos[x],
(b) phase portrait for U−, (c) phase portrait for U+
p sin[x]cos[x]
x˙ = , p˙ =−gradU∓ = . (3)
m U∓
These equations were originally obtained in [14] for con-
stant field (U∓ = U∓[x]), but they also stay correct for IfLZtunnelingoccurs, thepotentialU∓ changesitssign
modulated field (U∓ = U∓[x,τ]), if the modulation is and the atom continues its motion.
slow [16]. Let us use the harmonic modulation As a summary, a dot-like atom moves according to
the equations (3), but sometimes (when cos[x] = 0) the
∆[τ]=∆ +∆ cos[χ[τ]], χ[τ]≡ζτ +φ,
0 1 (4) potential U∓ changes its sign with the probability (5).
ζ (cid:28)1, ∆ ∼∆ (cid:28)1.
0 1
Stochastic part. When an atom crosses wave nodes
(x=π/2+πn, cos[x]=0), the distance between poten- III. PHASE SPACE STRUCTURE IN LIMIT
tials is minimal (Fig. 1a), and Landau-Zener (LZ) tun- CASES OF STATIONARY FIELD AND LARGE
nelings between them occur with the probability [12] DETUNINGS
−∆2mπ
WLZ (cid:39)exp 4|pnode|, (5) AtIonmamstoavteiosnianray2fiDeldph(∆as1e=sp0a)c,ea:tiotmssitcamteoitsiocnomispsrimehpelne-.
sively given by its position and momentum. In Fig. 1b
where p is an atomic momentum at the moment of
node
and c, phase portraits for potentials U∓ are shown. In
nodecrossing. Forexample,for|∆|(cid:28)1,x[0]=0,U[0]=
U−, it may be approximately estimated as [14] order to compute them, we simulated a series of tra-
jectories with different initial positions (x[0] = 0,±π)
(cid:112)
p (cid:39)± p2[0]−2m. (6) and momenta (a lot of values) and draw them in x,p
node
3
planes. All phase trajectories are periodic: slow atoms In Fig. 2, we put ∆ =−0.3, ∆ =−0.1, ζ =0.02 (∆
0 1
oscillate in potential wells (closed trajectories) and fast oscillatesinarange−0.4≤∆≤−0.2,sotheprobability
atomsmoveballisticallywithoscillatingmomentum. Far ofLZtransitionsisneglible(W (cid:46)exp[−10]))andcom-
LZ
fromresonance(|∆|(cid:38)1)thesignofpotentialisconstant, pute Poincar´e sections for a set of trajectories with dif-
so an atom moves along the same trajectory during the ferent initial atomic momenta and positions in potential
evolution. Near resonance (|∆| (cid:28) 1) LZ transitions be- U−. Most of them look similar to 2D-trajectories shown
tween U− and U+ become possible, so the structure of in Fig. 1b, but some others have sophisticated form.
phase space may switch spontaneously between Fig. 1b This picture is rather typical with chaotic Hamiltonian
and Fig. 1c. This may produce random walk of an atom systems [17, 18]. We see various regular Kolmogorov-
(intermsofstochastictrajectorymodel)andsplittingsof Arnold-Moser(KAM)invariantcurves(producedbynon-
atomic wave packets (in quantum terms) [12–14]. How- linearresonancesofdifferentorder)andchaotic(stochas-
ever, this does not produce any important nonlinear ef- tic) layers. In this figure, the chaotic layers are narrow
fects such as dynamical chaos. and the majority of trajectories are regular. In Fig. 2b
we magnify a small area of Fig. 2a showing a saddle-like
region of chaotic motion between regular curves.
Letusnotethatinnon-autonomoussystem(3),energy
(2) is not conserved. However, for some values of initial
conditions, system contains another integral of motion
(havingnogeneralanalyticalform),sotheeffectiveavail-
able phase space for particle’s motion is a sophisticated
2D surface named KAM-tor (KAM-curves are the sec-
tions of these tori by the plane χ=0). A particle moves
periodicallyorquasiperiodicallyononeofthesesurfaces.
On the other hand, for some other initial conditions no
integrals of motion exist, so the effective available phase
space is 3D and particle’s motion is chaotic. The tran-
sition of particle from one KAM-tor to another (or to
chaotic layer) is prohibited by the integral of motion.
IV. PHASE SPACE STRUCTURE IN
PRESENCE OF MODULATION AND
LANDAU-ZENER TRANSITIONS
ItisnotsurprisingtoseeKAM-toriinaphasespaceof
simple perturbed nonlinear pendulum. More interesting
picture is observed for smaller detunings in presence of
LZ transitions. In Fig. 3 we built Poincar´e sections for
system (3) with the parameters ∆ = −0.2, ∆ = −0.1,
0 1
ζ =0.02. Now ∆ oscillates in a range −0.3≤∆≤−0.1,
FIG. 2: Poincar´e mappings in x,p plane for large detuning so W (cid:46)exp[−2], and LZ tunnelings occur many times
LZ
(∆0 = −0.3): (a) general view, (b) magnified fragment with during the evolution. The surprising fact is that KAM-
chaotic layer and regular curves structures still exist. They are not destroyed by random
events of LZ tunnelings. However, the dynamics has two
new important features.
Presence of field modilation (4) (∆ (cid:54)= 0) radically Firstfeatureisthatduringtheevolution,thepotential
1
changesthedynamics. System’sphasespacebecomes3D switchsbetweenU+andU−. Eachpotentialcorresponds
because the phase of modulation χ[τ] may be regarded toitsownstructureofphasespace. Therefore,inFig.3a
as a third variable. Far from resonance (|∆| (cid:38) 1) atom we observe two sets of overlapping structures. First set
demonstrates typical dynamics of nonlinear pendulum (for U−) is similar to Fig. 2a (although chaotic regions
with periodical modulation of parameters. Similar pen- are larger), while the second one is a perturbed version
dulumregimeswerestudiedincountlessphysicalsystems of Fig. 1c. For any single trajectory in Fig. 3a, the mo-
[17]. In order to demonstrate the structure phase space, tion in U+ is comparatively slow (|p| (cid:46) p ) while the
node
let us use the method of Poincar´e mapping on the plane motion in U− is comparativley fast (|p| (cid:38) p ). How-
node
x,p. The idea of Poincar´e mapping (Poincar´e section) ever, different trajectories correspond to different values
is to draw not the whole trajectory but only its isolated of p (6). Therefore, it is better to study separate
node
points taken periodically after each cycle of modulation. pictures for each trajectory. In Fig. 3b and c we present
To be concrete, let us take points when χ=0. Poincar´e mappings for two particular trajectories from
4
Fig. 3a. Here, it is easy to see structures corresponding tunnelingsbetweentwoopticalpotentials,andthisisnot
to U+ and U− without significant overlapping. a classical pendulum behavior. In this paper, however,
Second feature is that a single atomic trajectory may wehaveshownthatLZtunnelings,beingaquantumran-
visit many separate KAM-curves and chaotic regions. In dom effect, do not destroy classical nonlinear structures
absenseofLZtransitionsrichstructuresshowninFig.3b, in phase space. They affect only particular realizations
canddcanbeproducedonlybyensemblesoftrajectories of atomic motion producing a phenomenon of dynamical
with different initial momenta. In particular, in Fig. 3d tunneling [7, 8]. Large atom-field detuning (used in fa-
weseeafinestructureofchaoticlayerandregularcurves. mous works [2–5, 7, 8]) is not necessary for observation
In a classical system the transition between them is pro- of nonlinear dynamics of atoms in a modulated field.
hibited. However, in presense of LZ transitions they are
These results were obtained with the use of stochastic
all produced by a single atomic trajectory. This is be-
trajectory model simplifying real quantum dynamics of
cause time periods of atomic motion between successive
atoms. However, itseemstobecorrectintheframework
nodecrossingsinU+andU−potentialsaredifferent,and
of our study. We avoided to analyze phase-space struc-
none of them (in general case) is synchronized with field
tures prohibited by Heisenberg relation. All coordinate
modulation. Therefore, after two successive LZ transi-
scaleswereortheorderofopticalwavelenghth(hundreds
tions atom returns to the same potential, but with an-
of nanometers), and all atomic momentums were much
other (in fact, random) values of χ and ∆. If some inte-
larger than the photon momentum. In [16] we presented
gral of motion existed for this trajectory, it changes its
a quantitative comparison between this stochastic tra-
value, and the motion continues along new KAM-curve.
jectorymodelandpurelyquantummodel,andthecorre-
In other words, a pair of LZ tunnelings produces a dy-
spondence was good (the values of variables and param-
namical tunneling [7, 8] of an atomic trajectory between
eters was similar to those used in this paper).
classically isolated areas of phase space
Theperspectiveofexperimentaltestofourresultisnot
a simple issue. A huge number of precise experiments
V. DISCUSSION AND CONCLUSION with single atoms is needed. Each atom must be pre-
paredinappropriateinitialstateanditsmomentumand
positionmustbemeasuredafterexactintegernumberof
Two-level atom in a far-detuned modulated laser field
field modulation periods. In future studies we are going
is a popular physical system for observation of nonlinear
tosimulatesuchexperimentnumerically(usingquantum
effects such as chaos and nonlinear resonance. Its phase
equations),butthiswilltakeahugecomputationaltime.
space contains KAM-tori and chaotic layers typical for
perturbed pendulum. On the other hand, if an atom is This work has been supported by the Grant of the
closetoopticalresonance,itperformsLandau-Zener(LZ) Russian Foundation for Basic Research 12-02-31161.
[1] P.I.Belobrov,G.M.Zaslavskii,G.Kh.Tartakovskii,Zh. [9] V. Yu. Argonov and S. V. Prants, J. Exp. Theor. Phys.
Eksp. Teor. Phys., 71 (1976) 1799. 96 (2003) 832.
[2] W.K.Hensinger,A.G.Truscott,B.Upcroft,N.R.Heck- [10] V. Yu. Argonov and S. V. Prants, Phys. Rev. A, 71
enberg, H. J. Rubinsztein-Dunlop, J. Opt. B: Quantum (2005) 053408.
Semiclass. (2000) 2 659. [11] V. Yu. Argonov and S. V. Prants, J. Russian Laser Re-
[3] R. Graham, M. Schlautmann, and P. Zoller, Phys. Rev. search 27 (2006) 360.
A 45 (1992) R19. [12] V. Yu. Argonov, J. Exp. Theor. Phys. Lett. 90 (2009)
[4] F. L. Moore, J. C. Robinson, C. Bharucha, Bala Sun- 739.
daram, and M. G. Raizen, Phys. Rev. Lett. 73 (1994) [13] S. V. Prants, J. Exp. Theor. Phys. 109 (2009) 751.
2974. [14] V. Yu. Argonov, Phys. Lett. A 375 (2011) 1116.
[5] R. Graham, S. Miyazaki. Phys. Rev. A, 53 (1996) 2683. [15] V. Yu. Argonov, J. Exp. Theor. Phys. Lett. 98 (2013)
[6] S. V. Prants and L. E. Kon’kov, J Exp. Theor. Phys. 655.
Lett., 73 (2001) 180. [16] V. Yu. Argonov, arXiv:1311.6808 (2013).
[7] W. K. Hensinger, H. H¨affner, A. Browaeys, N. R. Heck- [17] R.Z.Sagdeev,D.A.Usikov,G.M.Zaslavsky,Nonlinear
enberg, K. Helmerson, C. McKenzie, G. J. Milburn, W. Physics: From the Pendulum to Turbulence and Chaos.
D. Phillips, S. L. Rolston, H. Rubinsztein-Dunlop, B. New-York: Harwood Academic Publishers (1988).
Upcroft, Nature, 412 (2001) 52. [18] G.M.Zaslavsky,HamiltonianChaosandFractionalDy-
[8] W. K. Hensinger, N. R. Heckenberg, G. J. Milburn, H. namics, Oxford University Press, Oxford (2005).
J. Rubinsztein-Dunlop, J. Opt. B: Quantum Semiclass.
5 (2003) R83.
5
FIG. 3: Poincar´e mappings in x,p plane for small detuning
(∆ =−0.1): (a)generalview(manytrajectories),(b)single
0
chaotic trajectory with p[0]=660, (c) single trajectory with
regular and slightly chaotic parts with p[0] = 460, (d) mag-
nified fragment of (c) showing chaotic layer and neighbour
regular curves visited by a single trajectory