rhoRR_rhoee.tex
Coherent control of atomic tunneling
John Martin and Daniel Braun
Laboratoire de Physique Théorique, IRSAMC, UMR 5152 du CNRS, Université Paul Sabatier, Toulouse, FRANCE
We study the tunneling of a two-level atom in a double well potential while the atom is coupled to
a single electromagnetic field mode of a cavity. The coupling between internal and external degrees
of freedom, due to the mechanical effect on the atom from photon emission into the cavity mode,
can dramatically change the tunneling behavior. We predict that in general the tunneling process
becomes quasiperiodic. In a certain regime of parameters a collapse and revival of the tunneling
occurs. Accessing the internal degrees of freedom of the atom with a laser allows to coherently
manipulate the atom position, and in particular to prepare the atom in one of the two wells. The
effects described should be observable with atoms in an optical double well trap.
PACS numbers: 73.40.Gk, 37.30.+i
I. INTRODUCTION
The tunneling effect is considered one of the hallmarks
of quantum mechanical behavior. Historically, tunneling
was first examined for single particles (e.g. α particles [1],
electrons in field emission [2] and later in mesoscopic cir-
cuits [3]), for Cooper pairs [4], and for molecular groups
[5, 6, 7]. Recently the tunneling of atoms has attracted
substantial attention [8, 9, 10, 11]. Dynamical (chaos as-
sisted) tunneling of ultracold atoms between different is-
lands of stability in phase space was analyzed in [12, 13]
and has been observed experimentally [14, 15]. Reso-
nantly enhanced tunneling of atoms between wells of a
tilted optical lattice has also been observed very recently
[16]. In all of these examples, the atoms have been con-
sidered internally as inert, and only the center of mass
coordinate of the atom was of interest. In [17] it was
shown that by taking into account the internal degrees
of freedom of atoms, an atom/optical double well poten-
tial could be created in which tunneling atoms see their
internal and external states correlated (such an effect is
also known from other contexts [18]). Mechanical effects
of light in optical resonators were also investigated in
[19], but no tunneling was considered.
Here we show that the tunneling effect can be drasti-
cally modified if an internal transition of the atom is cou-
pled to a single electromagnetic mode in a cavity, such
that photon emission is a reversible and coherent pro-
cess. The resulting Rabi oscillations between states with
the excitation in the atom and states with a photon in
the cavity modulate the periodic tunneling motion. De-
pending on the frequencies involved, a rich quasi-periodic
behavior can result. If the cavity is fed with a coherent
state, collapse and revival of the tunneling effect can oc-
cur. Moreover, we show that one may profit from access
to the internal degrees of freedom of the atom (e.g. with
a laser) to control the atomic motion in the external po-
tential.
FIG. 1: (Color online) Two-level atom in a double well po-
tential interacting with a standing wave inside a cavity.
II. MODEL
A. Derivation of the Hamiltonian
Consider a trapped two-level atom (with levels |g〉, |e〉
of energy ∓~ω0/2 respectively) interacting with a stand-
ing wave (with wave number k and frequency ω) inside
a cavity as illustrated in Fig. 1. The atom is assumed to
be bound in the y − z plane at the equilibrium position
y = z = 0 and to experience a symmetric double well po-
tential V (x) along the x direction. We denote by ∆ the
tunnel splitting, i.e. the energy spacing between the two
lowest energy states (the symmetric |−〉 and antisym-
metric |+〉 states) of this double well potential. Below
we also allow the trapped atom to interact resonantly
with an external laser. The Hamiltonian of this system
is given by
H = HA +HF +HAF , (1)
where HA = H
A is the Hamiltonian of the trapped
atom, HF is the Hamiltonian of the free field and HAF
is the interaction Hamiltonian describing the atom-field
interaction. We have
HexA =
+ V (x),
H inA =
σinz ,
HF = ~ωa
HAF = −d.E,
http://arxiv.org/abs/0704.0763v2
where d denotes the atomic dipole,
E = Eωε
a+ a†
sin(k(x− x0)) (3)
is the electric field operator, with Eω =
, where
ǫ0 is the permittivity of free space, V the electromag-
netic mode volume, x0 the abscissa at the left cavity
mirror (x0 < 0), and ε the electric field polarization vec-
tor. We have introduced the operators σini (resp. σ
for i = x, y, z as the Pauli spin operators in the basis
{|e〉, |g〉} (resp. {|+〉, |−〉}). The operator x stands for
the center-of-mass position of the atom, px is the conju-
gate momentum along the x axis, m denotes the atomic
mass, and a (a†) the annihilation (creation) operator of
the cavity radiation field.
We adopt the two-level approximation which consists
of taking into account only the two lowest motional en-
ergy states. This requires the Rabi frequency
4g2 + δ2
(with δ = ω − ω0 the detuning between the cavity field
and the atomic transition frequencies) to be much smaller
than the frequency gap ∆̃ between the upper motional
states and the ground state doublet (see Fig. 1). Within
this approximation, Hamiltonian HexA becomes
HexA =
σexz (4)
and the position operator takes the form x = b
σexx with
b/2 = 〈+|x|−〉. We can form states that are mainly con-
centrated in the left/right wells,
|L〉 = (|+〉 − |−〉)/
|R〉 = (|+〉+ |−〉)/
The average position of a particle localized in the right
well is then given by b/2 (see Fig. 1) and σexx = |R〉〈R| −
|L〉〈L|. The interaction Hamiltonian HAF can then be
written
HAF = −~g(a+ a†)
sinχ cosκ σinx − cosχ sinκ σexx σinx
with the atom-field coupling strength g =
−〈e|d|g〉.εEω/~, and
χ = kx0, κ = kb/2. (6)
For long wavelengths (κ ≪ 1), or κ = nπ with inte-
ger n, the left and right sites of the double well are in-
distinguishable to the cavity photon and HAF reduces
to Jaynes-Cummings Hamiltonian without rotating wave
approximation (with a sine varying coupling constant),
−~g sinχ (a + a†)σinx . Note that κ ≪ 1 would normally
be identified with the Lamb-Dicke regime. Here the sit-
uation is more subtle as the level spacings between the
tunneling split ground state doublet and the next excited
states can be very different such that the recoil energy
~ωrecoil satisfies ∆ ≪ ωrecoil ≪ ∆̃. One may thus be in
the Lamb-Dicke regime concerning transitions to higher
vibrational states but have a significant mechanical ef-
fect on the atomic tunneling. Furthermore, since there is
only one photon mode, the recoil energy cannot vary con-
tinuously and exciting higher vibrational levels requires
ωrecoil close to a level spacing. Our numerical calculations
show that even for κ ∼ 1 the two-level approximation can
still work very well (see Fig. 4).
For δ, ∆ ≪ ω, ω0, a rotating wave approximation is
justified, which consists in eliminating the energy non-
conserving terms aσex± σ
− and a
†σex± σ
+ with σ
+ = |e〉〈g|,
σin− = σ
+ and σ
+ = |+〉〈−|, σex− = σ
+ . Within this
approximation, the total Hamiltonian reads
σexz +
σinz + ~ωa
†a (7)
+~g(aσin+ + a
†σin− )
cosχ sinκ σexx − sinχ cosκ 1ex
Thus, depending on the parameters χ and κ, the cavity
photon may induce internal transitions in the atom only
(cosχ sinκ = 0), or induce transitions between internal
and external states at the same time (cosχ sinκ 6= 0)
even for a vanishing detuning (δ = ω − ω0 = 0). This is
in contrast to conventional sideband transitions of har-
monically bound atoms or ions in the Lamb-Dicke regime
which require an appropriate value of the detuning. For
a fixed potential center (and thus fixed χ), κ can be
changed through a modulation of the well-to-well sep-
aration b. We will neglect in the following the effects
of decoherence, which means that not only g but also
∆ should be much larger than the rate of spontaneous
emission Γ, and the cavity decay rate κcav.
We denote the global state of the atom-field system by
|n, i, j〉 ≡ |n〉⊗|i〉⊗|j〉 where |n〉 stands for the cavity field
eigenstates, |i〉 ∈ {|−〉, |+〉} for the external motional
states, and |j〉 ∈ {|g〉, |e〉} for the internal states. The
total excitation number N is given by a†a+ σin+σ
B. Energy levels
The states |0,±, g〉 are eigenstates of H with eigen-
value (−~ω0 ± ~∆)/2, i.e. these states remain uncou-
pled and represent the two lowest energy states in the
regime δ, ∆ ≪ ω, ω0. It is straightforward to ver-
ify that the Hamiltonian (7) only induces transitions
between states with the same number of excitations
N , {|N − 1,+, e〉, |N,+, g〉, |N − 1,−, e〉, |N,−, g〉} ≡
{|1〉, |2〉, |3〉, |4〉}. It is therefore sufficient to solve the
dynamics in this subspace. In doing so, we obtain the
eigenvalues of H ,
λρµ = (N − 1/2)~ω + ρ
, (8)
for ρ, µ ∈ {±}, N = 1, 2, . . ., and with
2Ng2(1− cos(2κ) cos(2χ)) + δ2 +∆2 ± 2Ω2 , (9)
4Ng2 cos2 κ sin2 χ(∆2 + 4Ng2 sin2 κ cos2 χ) + δ2∆2 . (10)
For a vanishing tunnel splitting (∆ = 0), Ω± reduces to
the maximum (minimum) of the two Rabi frequencies of
the Jaynes-Cummings models in the right and left wells.
For cosκ = 1, the decoupling of external and internal
degrees of freedom manifests itself also in the eigenvalues
with Ω± = |
4Ng2 sin2 χ+ δ2 ±∆|.
C. Evolution operator
The whole dynamics of the system can be described
by means of the evolution operator U(t) = e−iHt/~ with
components Uij = 〈i|U(t)|j〉 = Uji, which can be calcu-
lated exactly. In order to simplify the expressions, we
restrict ourselves in the following to χ = −π/4−2nπ (in-
teger n). We find, up to a an overall phase e−i(N−1/2)ωt,
U11 = −
µSµΩ−µ
ξ + µ(∆− δ)Ω2
− iµΩ+Ω−Cµ(δ∆− µΩ2)
U12 =
Ng cosκ√
µSµΩ−µ(∆
2 + 2Ng2 sin2 κ
+ µΩ2) + iµΩ+Ω−∆Cµ
U13 =
−iNg2 sin(2κ)
µδΩµS−µ + iµΩ+Ω−Cµ
U23 =
Ng sinκ√
µΩµS−µ(δ∆+ 2Ng
2 cos2 κ− µΩ2)
ξ = ∆(δ2 + 2Ng2 cos2 κ− δ∆),
Λ = Ω+Ω−Ω
and where all time dependence is in the coefficients
C± = cos(Ω±t/2), S± = sin(Ω±t/2). (16)
The remaining components can be deduced from the
relations U22(δ,∆) = U33(−δ,−∆) = U44(δ,−∆) =
U11(−δ,∆), U24(δ,∆) = U13(−δ,∆), U14(δ,∆) =
U23(δ,−∆) = U23(−δ,∆), and U34(δ,∆) = U12(δ,−∆),
valid for any χ, where we have made explicit the depen-
dence of the Uij on δ and ∆.
III. INTERNAL AND EXTERNAL DYNAMICS
The reduced density matrix ρex for the atomic center-
of-mass motion alone follows from ρ = |ψ(t)〉〈ψ(t)| by
tracing out the field and internal degrees of freedom,
where the total wave function at time t reads |ψ(t)〉 =
i,j=1 Uij〈j|ψ(0)〉 |i〉. The average position of the atom
in the double well potential is then given by
〈x〉 =
Trex(ρ
exσexx ) =
(1− 2ρLL) (17)
with ρLL = 〈L|ρex|L〉. Similarly, we obtain the reduced
density matrix ρin for the internal atomic state by tracing
out the field and external degrees of freedom, and the
probability to find the atom in the excited state as ρee =
〈e|ρin|e〉.
In the following, we first focus on resonant atom-field
interaction (ω = ω0) before moving to the non-resonant
case (ω 6= ω0). We distinguish three regimes according
to the tunnel splitting compared to the Rabi frequency
g : the small tunnel splitting regime (when ∆/g ≪ 1),
the intermediate regime (when ∆/g ∼ 1), and the large
tunnel splitting regime (when ∆/g ≫ 1).
A. Resonant atom-field interaction
For resonant atom-field interaction (δ = 0), the ex-
pressions for Uij can be greatly simplified. If the system
is initially prepared in the state |N − 1, R, e〉 and for
κ = π/4, we have
ρLL =
∆2 +Ng2
Ωtunt
with the tunnel frequency
Ωtun =
(Ω+ +Ω−) , (19)
ρee =
Ω2µ −∆2
cos(Ωµt) + 4∆
2 cos
Ω+−Ω−
8(Ng2 +∆2)
The atom position oscillates with a single frequency
Ωtun given by Eq. (19), whereas ρee evolves with three in
general incommensurable frequencies Ω+, Ω−, and (Ω+−
Ω−)/2 giving rise to a quasi-periodic signal.
For ∆/g ≪ 1, Eq. (18) leads to ρLL ≃ 0 (up to order
(∆/g)2), indicating that tunneling is suppressed. This is
already obvious from (7), as the term responsible for tun-
neling, (~∆/2)σexz = (~∆/2)(|R〉〈L| + |L〉〈R|) becomes
very small compared to the last term, diagonal in |R〉, |L〉
which leads to internal Rabi flopping. Note, however,
that tunneling is suppressed on all time scales, even for
t≫ 1/∆, due to the reduced amplitude in Eq. (18), very
much in contrast to tunneling without internal degrees of
freedom, where only the period of the tunneling motion,
but not the amplitude is affected when ∆ is reduced. For
κ approaching π, the situation changes because the term
g cosχ sinκ σexx of the interaction Hamiltonian inducing
transitions between vibrational states becomes small in
comparison with ∆ thereby allowing tunneling again.
Because internal and external degrees of freedom are
coupled, the tunneling frequency (Eq. (19)) depends on
the number of photons inside the cavity. As an exam-
ple, let us now consider ∆ ∼ g and a cavity field initially
in a coherent state |α〉 = e− 12 |α|2
|n〉 with |α|2
equal to the mean photon number 〈n〉. Figure 2 shows
that the average position of the atom in the double well
as a function of time for a coherent state exhibits col-
lapses and revivals. The oscillation amplitude decreases
with increasing mean photon number 〈n〉 and decreasing
tunnel splitting ∆ (see Eqs. (18,9)). Since the probabil-
ity to find the atom in the excited state oscillates with
three frequencies, no collapses and revivals are observed
for ρee.
The collapse time tc of the tunneling motion can be
estimated from the condition [20] (Ωtun(〈n〉 +
〈n〉) −
Ωtun(〈n〉−
〈n〉)) tc ∼ 1 with Ωtun(m) given by Eq. (19)
for N = m+ 1, which yields, for 〈n〉 ≫ 1,
(∆/g)2 + 3/4
+O(〈n〉−2) (21)
The time interval between two following revivals, tr,
follows from (Ωtun(〈n〉) − Ωtun(〈n〉 − 1)) tr = 2π, and is
given for 〈n〉 ≫ 1 by
(∆/g)2 + 1/2
+O(〈n〉−2)
For the parameters of Fig. 2, Eq. (22) yields gtr ≃ 68.23
for ∆/g = 2 and gtr ≃ 86.70 for ∆/g = 5. Smaller revival
times are possible for smaller values of 〈n〉, but in general
the observation of revivals will be quite challenging, as
they require ∆ ∼ g ≫ κcav.
For large tunnel splitting, ∆/g ≫ 1, Ωtun = ∆ +
Ng2/(2∆) + O((g/∆)3), and Eq. (18) reduces to ρLL ≃
sin2(∆t/2), which is identical to the tunneling of a parti-
cle without internal structure. Equation (20) reduces to
a Rabi oscillation ρee ≃ cos2(
Ngt/2).
B. Non-resonant atom-field interaction
For non-resonant atom-field interaction (δ 6= 0), and
intermediate tunnel splitting [see Fig. 3 for ∆ = δ = g],
∆/g = 5
∆/g = 2
100806040200
FIG. 2: (Color online) Average position of the atom in the
double well as a function of time for ∆/g = 2 (blue, top curve)
and ∆/g = 5 (red), κ = π/4 and a coherent state with α = 5.
100806040200
FIG. 3: (Color online) Average position of the atom in the
double well as a function of time for ∆ = δ = g, κ = π/4 and
N = 1. The blue solid/red dashed curve corresponds to an
excited atom initially located in the left/right well.
〈x(t)〉 involves in general the two non-commensurate fre-
quencies Ω+ and Ω− and varies therefore quasiperiod-
ically as a function of time. Figure 3 also shows that
an atom initially located in one of the two wells remains
mostly confined to that well.
For small tunnel splitting, ∆/g ≪ 1 and large detuning
|δ|/g ≫ 1 (with ∆|δ|/g2 ∼ 1), the matrix elements of U
simplify to
U13 =
i Ng2 sin 2κ
δ2∆2 +N2g4 sin2(2κ)
(23a)
U33 = cos
δ2∆2 +N2g4 sin2(2κ)
(23b)
up to corrections of order O(∆/g) and a phase factor
ei[(Ng
2/δ+δ)−(2N−1)ω]t/2 while the components U12 and
U23 are of order O(∆/g). In this situation, the system
20151050
FIG. 4: (Color online) Density matrix elements ρRR (top)
and ρee (bottom) as a function of the interaction time gt
for an initially excited atom located in the right well and
for the parameters ∆/g ≃ 0.3336, δ/g = 3, κ = π/4, and
N = 1. Numerical results from the propagation of the time
dependent Schrödinger equation with Hamiltonian (1) and
rotating wave approximation are represented by circles and
analytical results by solid curves. The time propagation was
done with (~ = m = 1) g = 0.01 and the double well potential
V (x) = 0.08x4 − x2 yielding a tunnel splitting ∆ ≃ 0.003336
and a ratio ∆̃/
4g2 + δ2 ≃ 44.4 ≫ 1.
oscillates only between the two states |N − 1,+, e〉 and
|N − 1,−, e〉 with a single frequency
δ2∆2 +N2g4 sin2(2κ)
, (24)
just as a three-level atom undergoing a Raman transition
in the far detuned regime behaves as a two-level system.
If the system is initially in the state |N − 1,−, e〉, we
have from Eqs. (23)
ρLL =
− Nδ∆sin(2κ)
2Ω̄2(δ/g)2
1− cos
, (25)
and ρee = 1. For a detuning δ = ±Ng2 sin(2κ)/∆,
ρLL =
1− cos
. (26)
This regime may be suitable for coherently manipu-
lating the atom position through access to its internal
degrees of freedom with a laser. Coherent manipulation
of the position of neutral atoms has been proposed and
demonstrated before, see e.g. [21, 22, 23]. In these exam-
ples, the manipulation is done by modifying the external
potential. The mechanism we propose here is very dif-
ferent, as the potential remains totally unchanged, and
only internal transitions and the tunneling effect are used
to move the atom in a controlled way. As an example,
we show how the atom can be prepared in the left well
starting from the ground state |0,−, g〉 for δ = −g2/∆.
We first apply a π-pulse with an external laser resonant
with the atomic transition. By using a laser with a wave
vector perpendicular to the Ox-direction, only the atomic
internal degree of freedom is affected, resulting in the
transition |0,−, g〉 → −i|0,−, e〉. We assumed that the
laser Rabi frequency ΩR is much larger than the tunnel
frequency ∆.
Now we use the coupling between the internal and ex-
ternal degrees of freedom to create a superposition of
the |0,±, e〉 states, and then apply a second resonant π-
pulse to get back to the uncoupled states |0,±, g〉. For
∆/g ≪ 1, δ = −g2/∆ and κ = π/4, the initial state
transforms according to
|0,−, g〉 −−−−→
ΩRt=π
|0,−, e〉 −−−−−−→
∆t=π/
|0, L, e〉 −−−−→
ΩRt=π
|0, L, g〉
up to a physically irrelevant phase. Other coherent su-
perpositions of |0,+, g〉 and |0,−, g〉 can be obtained by
choosing appropriate interaction times.
In order to verify that the two-level approximation for
the external motion used in the derivation of the Hamilto-
nian is a good approximation, we have numerically solved
the time dependent Schrödinger equation with Hamil-
tonian (1) and rotating wave approximation but with
the exact external potential V (x) (i.e. with a large num-
ber of vibrational states). Figure 4 shows that provided
4g2 + δ2 as stated before, to take only the two
lowest vibrational states into account is indeed a good
approximation.
We finally comment on possible experimental realiza-
tions of our model. Double well potentials with tunable
well-to-well separation have been demonstrated with op-
tical dipole traps e.g. in [21, 24], and on atom chips
e.g. in [25, 26]. For our model, the double well poten-
tial has to be realized inside the cavity. Optical trapping
and even cooling of atoms close to their ground state
inside a cavity has been achieved in several groups by
now [27, 28, 29, 30], but up to our knowledge double
well potentials have not been realized in a cavity so far.
However, some of the cavities developed have a very long
lateral opening (up to 222 µm [31]) and should allow more
complicated trapping potentials (optical lattices inter-
secting a cavity have been realized in Chapman’s group
[31]). We remark that it is not essential for our model
that the double well potential be aligned with the cavity
axes. Any other orientation is possible, and only leads to
modified coefficients cosχ sinκ and sinχ cosκ.
At certain “magical wavelengths”, Cs, Yb, Sr, Mg, and
Ca atoms in optical traps experience the same potential
for ground and excited internal states coupled by a dipole
transition [27, 32, 33, 34]. In a symmetric potential V (z)
the tunneling frequency ∆ is given in WKB approxi-
mation by ∆ ∼ ωosc exp(−1/~
2m(E0 − V (z)) dz)
where E0 is the ground state energy, ωosc the single well
harmonic oscillation frequency, and z = ±a are the corre-
sponding classical turning points delimiting the range of
the barrier. The exponential factor can approach unity
for a barrier that is only slightly higher than the ground
state energy E0, in which case cooling to temperatures
kBT < ~∆ should be possible with state of the art
techniques [27]. In [27] a trap depth V0/~ = 47 MHz
was achieved inside a cavity with 1.2 mW laser power.
In any case, the trap frequency and thus the tunneling
splitting are determined by the laser power and the fo-
cussing (or the wavelength for optical lattices), and can
therefore be controlled independently of Γ, κcav, such
that there should be no fundamental problem achieving
∆ ≫ Γ, κcav. The detection of the tunneling motion
should be possible by optical imaging, i.e. diffusion of
laser light from another transition in the optical regime
with smaller wavelength than the well separation. Al-
ternatively, one might monitor the transmission through
the cavity in the case that it differs for the two locations
of the wells [35]. Another possibility might be using the
atomic spin as a position meter [17].
Acknowledgments
We thank Jacques Vigué for an interesting discussion
and CALMIP (Toulouse) for the use of their comput-
ers. This work was supported by the Agence National de
la Recherche (ANR), project INFOSYSQQ, and the EC
IST-FET project EUROSQIP.
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(2005).

  We study the tunneling of a two-level atom in a double well potential while
the atom is coupled to a single electromagnetic field mode of a cavity. The
coupling between internal and external degrees of freedom, due to the
mechanical effect on the atom from photon emission into the cavity mode, can
dramatically change the tunneling behavior. We predict that in general the
tunneling process becomes quasiperiodic. In a certain regime of parameters a
collapse and revival of the tunneling occurs. Accessing the internal degrees of
freedom of the atom with a laser allows to coherently manipulate the atom
position, and in particular to prepare the atom in one of the two wells. The
effects described should be observable with atoms in an optical double well
trap.

rhoRR_rhoee.tex
Coherent control of atomic tunneling
John Martin and Daniel Braun
Laboratoire de Physique Théorique, IRSAMC, UMR 5152 du CNRS, Université Paul Sabatier, Toulouse, FRANCE
We study the tunneling of a two-level atom in a double well potential while the atom is coupled to
a single electromagnetic field mode of a cavity. The coupling between internal and external degrees
of freedom, due to the mechanical effect on the atom from photon emission into the cavity mode,
can dramatically change the tunneling behavior. We predict that in general the tunneling process
becomes quasiperiodic. In a certain regime of parameters a collapse and revival of the tunneling
occurs. Accessing the internal degrees of freedom of the atom with a laser allows to coherently
manipulate the atom position, and in particular to prepare the atom in one of the two wells. The
effects described should be observable with atoms in an optical double well trap.
PACS numbers: 73.40.Gk, 37.30.+i
I. INTRODUCTION
The tunneling effect is considered one of the hallmarks
of quantum mechanical behavior. Historically, tunneling
was first examined for single particles (e.g. α particles [1],
electrons in field emission [2] and later in mesoscopic cir-
cuits [3]), for Cooper pairs [4], and for molecular groups
[5, 6, 7]. Recently the tunneling of atoms has attracted
substantial attention [8, 9, 10, 11]. Dynamical (chaos as-
sisted) tunneling of ultracold atoms between different is-
lands of stability in phase space was analyzed in [12, 13]
and has been observed experimentally [14, 15]. Reso-
nantly enhanced tunneling of atoms between wells of a
tilted optical lattice has also been observed very recently
[16]. In all of these examples, the atoms have been con-
sidered internally as inert, and only the center of mass
coordinate of the atom was of interest. In [17] it was
shown that by taking into account the internal degrees
of freedom of atoms, an atom/optical double well poten-
tial could be created in which tunneling atoms see their
internal and external states correlated (such an effect is
also known from other contexts [18]). Mechanical effects
of light in optical resonators were also investigated in
[19], but no tunneling was considered.
Here we show that the tunneling effect can be drasti-
cally modified if an internal transition of the atom is cou-
pled to a single electromagnetic mode in a cavity, such
that photon emission is a reversible and coherent pro-
cess. The resulting Rabi oscillations between states with
the excitation in the atom and states with a photon in
the cavity modulate the periodic tunneling motion. De-
pending on the frequencies involved, a rich quasi-periodic
behavior can result. If the cavity is fed with a coherent
state, collapse and revival of the tunneling effect can oc-
cur. Moreover, we show that one may profit from access
to the internal degrees of freedom of the atom (e.g. with
a laser) to control the atomic motion in the external po-
tential.
FIG. 1: (Color online) Two-level atom in a double well po-
tential interacting with a standing wave inside a cavity.
II. MODEL
A. Derivation of the Hamiltonian
Consider a trapped two-level atom (with levels |g〉, |e〉
of energy ∓~ω0/2 respectively) interacting with a stand-
ing wave (with wave number k and frequency ω) inside
a cavity as illustrated in Fig. 1. The atom is assumed to
be bound in the y − z plane at the equilibrium position
y = z = 0 and to experience a symmetric double well po-
tential V (x) along the x direction. We denote by ∆ the
tunnel splitting, i.e. the energy spacing between the two
lowest energy states (the symmetric |−〉 and antisym-
metric |+〉 states) of this double well potential. Below
we also allow the trapped atom to interact resonantly
with an external laser. The Hamiltonian of this system
is given by
H = HA +HF +HAF , (1)
where HA = H
A is the Hamiltonian of the trapped
atom, HF is the Hamiltonian of the free field and HAF
is the interaction Hamiltonian describing the atom-field
interaction. We have
HexA =
+ V (x),
H inA =
σinz ,
HF = ~ωa
HAF = −d.E,
http://arxiv.org/abs/0704.0763v2
where d denotes the atomic dipole,
E = Eωε
a+ a†
sin(k(x− x0)) (3)
is the electric field operator, with Eω =
, where
ǫ0 is the permittivity of free space, V the electromag-
netic mode volume, x0 the abscissa at the left cavity
mirror (x0 < 0), and ε the electric field polarization vec-
tor. We have introduced the operators σini (resp. σ
for i = x, y, z as the Pauli spin operators in the basis
{|e〉, |g〉} (resp. {|+〉, |−〉}). The operator x stands for
the center-of-mass position of the atom, px is the conju-
gate momentum along the x axis, m denotes the atomic
mass, and a (a†) the annihilation (creation) operator of
the cavity radiation field.
We adopt the two-level approximation which consists
of taking into account only the two lowest motional en-
ergy states. This requires the Rabi frequency
4g2 + δ2
(with δ = ω − ω0 the detuning between the cavity field
and the atomic transition frequencies) to be much smaller
than the frequency gap ∆̃ between the upper motional
states and the ground state doublet (see Fig. 1). Within
this approximation, Hamiltonian HexA becomes
HexA =
σexz (4)
and the position operator takes the form x = b
σexx with
b/2 = 〈+|x|−〉. We can form states that are mainly con-
centrated in the left/right wells,
|L〉 = (|+〉 − |−〉)/
|R〉 = (|+〉+ |−〉)/
The average position of a particle localized in the right
well is then given by b/2 (see Fig. 1) and σexx = |R〉〈R| −
|L〉〈L|. The interaction Hamiltonian HAF can then be
written
HAF = −~g(a+ a†)
sinχ cosκ σinx − cosχ sinκ σexx σinx
with the atom-field coupling strength g =
−〈e|d|g〉.εEω/~, and
χ = kx0, κ = kb/2. (6)
For long wavelengths (κ ≪ 1), or κ = nπ with inte-
ger n, the left and right sites of the double well are in-
distinguishable to the cavity photon and HAF reduces
to Jaynes-Cummings Hamiltonian without rotating wave
approximation (with a sine varying coupling constant),
−~g sinχ (a + a†)σinx . Note that κ ≪ 1 would normally
be identified with the Lamb-Dicke regime. Here the sit-
uation is more subtle as the level spacings between the
tunneling split ground state doublet and the next excited
states can be very different such that the recoil energy
~ωrecoil satisfies ∆ ≪ ωrecoil ≪ ∆̃. One may thus be in
the Lamb-Dicke regime concerning transitions to higher
vibrational states but have a significant mechanical ef-
fect on the atomic tunneling. Furthermore, since there is
only one photon mode, the recoil energy cannot vary con-
tinuously and exciting higher vibrational levels requires
ωrecoil close to a level spacing. Our numerical calculations
show that even for κ ∼ 1 the two-level approximation can
still work very well (see Fig. 4).
For δ, ∆ ≪ ω, ω0, a rotating wave approximation is
justified, which consists in eliminating the energy non-
conserving terms aσex± σ
− and a
†σex± σ
+ with σ
+ = |e〉〈g|,
σin− = σ
+ and σ
+ = |+〉〈−|, σex− = σ
+ . Within this
approximation, the total Hamiltonian reads
σexz +
σinz + ~ωa
†a (7)
+~g(aσin+ + a
†σin− )
cosχ sinκ σexx − sinχ cosκ 1ex
Thus, depending on the parameters χ and κ, the cavity
photon may induce internal transitions in the atom only
(cosχ sinκ = 0), or induce transitions between internal
and external states at the same time (cosχ sinκ 6= 0)
even for a vanishing detuning (δ = ω − ω0 = 0). This is
in contrast to conventional sideband transitions of har-
monically bound atoms or ions in the Lamb-Dicke regime
which require an appropriate value of the detuning. For
a fixed potential center (and thus fixed χ), κ can be
changed through a modulation of the well-to-well sep-
aration b. We will neglect in the following the effects
of decoherence, which means that not only g but also
∆ should be much larger than the rate of spontaneous
emission Γ, and the cavity decay rate κcav.
We denote the global state of the atom-field system by
|n, i, j〉 ≡ |n〉⊗|i〉⊗|j〉 where |n〉 stands for the cavity field
eigenstates, |i〉 ∈ {|−〉, |+〉} for the external motional
states, and |j〉 ∈ {|g〉, |e〉} for the internal states. The
total excitation number N is given by a†a+ σin+σ
B. Energy levels
The states |0,±, g〉 are eigenstates of H with eigen-
value (−~ω0 ± ~∆)/2, i.e. these states remain uncou-
pled and represent the two lowest energy states in the
regime δ, ∆ ≪ ω, ω0. It is straightforward to ver-
ify that the Hamiltonian (7) only induces transitions
between states with the same number of excitations
N , {|N − 1,+, e〉, |N,+, g〉, |N − 1,−, e〉, |N,−, g〉} ≡
{|1〉, |2〉, |3〉, |4〉}. It is therefore sufficient to solve the
dynamics in this subspace. In doing so, we obtain the
eigenvalues of H ,
λρµ = (N − 1/2)~ω + ρ
, (8)
for ρ, µ ∈ {±}, N = 1, 2, . . ., and with
2Ng2(1− cos(2κ) cos(2χ)) + δ2 +∆2 ± 2Ω2 , (9)
4Ng2 cos2 κ sin2 χ(∆2 + 4Ng2 sin2 κ cos2 χ) + δ2∆2 . (10)
For a vanishing tunnel splitting (∆ = 0), Ω± reduces to
the maximum (minimum) of the two Rabi frequencies of
the Jaynes-Cummings models in the right and left wells.
For cosκ = 1, the decoupling of external and internal
degrees of freedom manifests itself also in the eigenvalues
with Ω± = |
4Ng2 sin2 χ+ δ2 ±∆|.
C. Evolution operator
The whole dynamics of the system can be described
by means of the evolution operator U(t) = e−iHt/~ with
components Uij = 〈i|U(t)|j〉 = Uji, which can be calcu-
lated exactly. In order to simplify the expressions, we
restrict ourselves in the following to χ = −π/4−2nπ (in-
teger n). We find, up to a an overall phase e−i(N−1/2)ωt,
U11 = −
µSµΩ−µ
ξ + µ(∆− δ)Ω2
− iµΩ+Ω−Cµ(δ∆− µΩ2)
U12 =
Ng cosκ√
µSµΩ−µ(∆
2 + 2Ng2 sin2 κ
+ µΩ2) + iµΩ+Ω−∆Cµ
U13 =
−iNg2 sin(2κ)
µδΩµS−µ + iµΩ+Ω−Cµ
U23 =
Ng sinκ√
µΩµS−µ(δ∆+ 2Ng
2 cos2 κ− µΩ2)
ξ = ∆(δ2 + 2Ng2 cos2 κ− δ∆),
Λ = Ω+Ω−Ω
and where all time dependence is in the coefficients
C± = cos(Ω±t/2), S± = sin(Ω±t/2). (16)
The remaining components can be deduced from the
relations U22(δ,∆) = U33(−δ,−∆) = U44(δ,−∆) =
U11(−δ,∆), U24(δ,∆) = U13(−δ,∆), U14(δ,∆) =
U23(δ,−∆) = U23(−δ,∆), and U34(δ,∆) = U12(δ,−∆),
valid for any χ, where we have made explicit the depen-
dence of the Uij on δ and ∆.
III. INTERNAL AND EXTERNAL DYNAMICS
The reduced density matrix ρex for the atomic center-
of-mass motion alone follows from ρ = |ψ(t)〉〈ψ(t)| by
tracing out the field and internal degrees of freedom,
where the total wave function at time t reads |ψ(t)〉 =
i,j=1 Uij〈j|ψ(0)〉 |i〉. The average position of the atom
in the double well potential is then given by
〈x〉 =
Trex(ρ
exσexx ) =
(1− 2ρLL) (17)
with ρLL = 〈L|ρex|L〉. Similarly, we obtain the reduced
density matrix ρin for the internal atomic state by tracing
out the field and external degrees of freedom, and the
probability to find the atom in the excited state as ρee =
〈e|ρin|e〉.
In the following, we first focus on resonant atom-field
interaction (ω = ω0) before moving to the non-resonant
case (ω 6= ω0). We distinguish three regimes according
to the tunnel splitting compared to the Rabi frequency
g : the small tunnel splitting regime (when ∆/g ≪ 1),
the intermediate regime (when ∆/g ∼ 1), and the large
tunnel splitting regime (when ∆/g ≫ 1).
A. Resonant atom-field interaction
For resonant atom-field interaction (δ = 0), the ex-
pressions for Uij can be greatly simplified. If the system
is initially prepared in the state |N − 1, R, e〉 and for
κ = π/4, we have
ρLL =
∆2 +Ng2
Ωtunt
with the tunnel frequency
Ωtun =
(Ω+ +Ω−) , (19)
ρee =
Ω2µ −∆2
cos(Ωµt) + 4∆
2 cos
Ω+−Ω−
8(Ng2 +∆2)
The atom position oscillates with a single frequency
Ωtun given by Eq. (19), whereas ρee evolves with three in
general incommensurable frequencies Ω+, Ω−, and (Ω+−
Ω−)/2 giving rise to a quasi-periodic signal.
For ∆/g ≪ 1, Eq. (18) leads to ρLL ≃ 0 (up to order
(∆/g)2), indicating that tunneling is suppressed. This is
already obvious from (7), as the term responsible for tun-
neling, (~∆/2)σexz = (~∆/2)(|R〉〈L| + |L〉〈R|) becomes
very small compared to the last term, diagonal in |R〉, |L〉
which leads to internal Rabi flopping. Note, however,
that tunneling is suppressed on all time scales, even for
t≫ 1/∆, due to the reduced amplitude in Eq. (18), very
much in contrast to tunneling without internal degrees of
freedom, where only the period of the tunneling motion,
but not the amplitude is affected when ∆ is reduced. For
κ approaching π, the situation changes because the term
g cosχ sinκ σexx of the interaction Hamiltonian inducing
transitions between vibrational states becomes small in
comparison with ∆ thereby allowing tunneling again.
Because internal and external degrees of freedom are
coupled, the tunneling frequency (Eq. (19)) depends on
the number of photons inside the cavity. As an exam-
ple, let us now consider ∆ ∼ g and a cavity field initially
in a coherent state |α〉 = e− 12 |α|2
|n〉 with |α|2
equal to the mean photon number 〈n〉. Figure 2 shows
that the average position of the atom in the double well
as a function of time for a coherent state exhibits col-
lapses and revivals. The oscillation amplitude decreases
with increasing mean photon number 〈n〉 and decreasing
tunnel splitting ∆ (see Eqs. (18,9)). Since the probabil-
ity to find the atom in the excited state oscillates with
three frequencies, no collapses and revivals are observed
for ρee.
The collapse time tc of the tunneling motion can be
estimated from the condition [20] (Ωtun(〈n〉 +
〈n〉) −
Ωtun(〈n〉−
〈n〉)) tc ∼ 1 with Ωtun(m) given by Eq. (19)
for N = m+ 1, which yields, for 〈n〉 ≫ 1,
(∆/g)2 + 3/4
+O(〈n〉−2) (21)
The time interval between two following revivals, tr,
follows from (Ωtun(〈n〉) − Ωtun(〈n〉 − 1)) tr = 2π, and is
given for 〈n〉 ≫ 1 by
(∆/g)2 + 1/2
+O(〈n〉−2)
For the parameters of Fig. 2, Eq. (22) yields gtr ≃ 68.23
for ∆/g = 2 and gtr ≃ 86.70 for ∆/g = 5. Smaller revival
times are possible for smaller values of 〈n〉, but in general
the observation of revivals will be quite challenging, as
they require ∆ ∼ g ≫ κcav.
For large tunnel splitting, ∆/g ≫ 1, Ωtun = ∆ +
Ng2/(2∆) + O((g/∆)3), and Eq. (18) reduces to ρLL ≃
sin2(∆t/2), which is identical to the tunneling of a parti-
cle without internal structure. Equation (20) reduces to
a Rabi oscillation ρee ≃ cos2(
Ngt/2).
B. Non-resonant atom-field interaction
For non-resonant atom-field interaction (δ 6= 0), and
intermediate tunnel splitting [see Fig. 3 for ∆ = δ = g],
∆/g = 5
∆/g = 2
100806040200
FIG. 2: (Color online) Average position of the atom in the
double well as a function of time for ∆/g = 2 (blue, top curve)
and ∆/g = 5 (red), κ = π/4 and a coherent state with α = 5.
100806040200
FIG. 3: (Color online) Average position of the atom in the
double well as a function of time for ∆ = δ = g, κ = π/4 and
N = 1. The blue solid/red dashed curve corresponds to an
excited atom initially located in the left/right well.
〈x(t)〉 involves in general the two non-commensurate fre-
quencies Ω+ and Ω− and varies therefore quasiperiod-
ically as a function of time. Figure 3 also shows that
an atom initially located in one of the two wells remains
mostly confined to that well.
For small tunnel splitting, ∆/g ≪ 1 and large detuning
|δ|/g ≫ 1 (with ∆|δ|/g2 ∼ 1), the matrix elements of U
simplify to
U13 =
i Ng2 sin 2κ
δ2∆2 +N2g4 sin2(2κ)
(23a)
U33 = cos
δ2∆2 +N2g4 sin2(2κ)
(23b)
up to corrections of order O(∆/g) and a phase factor
ei[(Ng
2/δ+δ)−(2N−1)ω]t/2 while the components U12 and
U23 are of order O(∆/g). In this situation, the system
20151050
FIG. 4: (Color online) Density matrix elements ρRR (top)
and ρee (bottom) as a function of the interaction time gt
for an initially excited atom located in the right well and
for the parameters ∆/g ≃ 0.3336, δ/g = 3, κ = π/4, and
N = 1. Numerical results from the propagation of the time
dependent Schrödinger equation with Hamiltonian (1) and
rotating wave approximation are represented by circles and
analytical results by solid curves. The time propagation was
done with (~ = m = 1) g = 0.01 and the double well potential
V (x) = 0.08x4 − x2 yielding a tunnel splitting ∆ ≃ 0.003336
and a ratio ∆̃/
4g2 + δ2 ≃ 44.4 ≫ 1.
oscillates only between the two states |N − 1,+, e〉 and
|N − 1,−, e〉 with a single frequency
δ2∆2 +N2g4 sin2(2κ)
, (24)
just as a three-level atom undergoing a Raman transition
in the far detuned regime behaves as a two-level system.
If the system is initially in the state |N − 1,−, e〉, we
have from Eqs. (23)
ρLL =
− Nδ∆sin(2κ)
2Ω̄2(δ/g)2
1− cos
, (25)
and ρee = 1. For a detuning δ = ±Ng2 sin(2κ)/∆,
ρLL =
1− cos
. (26)
This regime may be suitable for coherently manipu-
lating the atom position through access to its internal
degrees of freedom with a laser. Coherent manipulation
of the position of neutral atoms has been proposed and
demonstrated before, see e.g. [21, 22, 23]. In these exam-
ples, the manipulation is done by modifying the external
potential. The mechanism we propose here is very dif-
ferent, as the potential remains totally unchanged, and
only internal transitions and the tunneling effect are used
to move the atom in a controlled way. As an example,
we show how the atom can be prepared in the left well
starting from the ground state |0,−, g〉 for δ = −g2/∆.
We first apply a π-pulse with an external laser resonant
with the atomic transition. By using a laser with a wave
vector perpendicular to the Ox-direction, only the atomic
internal degree of freedom is affected, resulting in the
transition |0,−, g〉 → −i|0,−, e〉. We assumed that the
laser Rabi frequency ΩR is much larger than the tunnel
frequency ∆.
Now we use the coupling between the internal and ex-
ternal degrees of freedom to create a superposition of
the |0,±, e〉 states, and then apply a second resonant π-
pulse to get back to the uncoupled states |0,±, g〉. For
∆/g ≪ 1, δ = −g2/∆ and κ = π/4, the initial state
transforms according to
|0,−, g〉 −−−−→
ΩRt=π
|0,−, e〉 −−−−−−→
∆t=π/
|0, L, e〉 −−−−→
ΩRt=π
|0, L, g〉
up to a physically irrelevant phase. Other coherent su-
perpositions of |0,+, g〉 and |0,−, g〉 can be obtained by
choosing appropriate interaction times.
In order to verify that the two-level approximation for
the external motion used in the derivation of the Hamilto-
nian is a good approximation, we have numerically solved
the time dependent Schrödinger equation with Hamil-
tonian (1) and rotating wave approximation but with
the exact external potential V (x) (i.e. with a large num-
ber of vibrational states). Figure 4 shows that provided
4g2 + δ2 as stated before, to take only the two
lowest vibrational states into account is indeed a good
approximation.
We finally comment on possible experimental realiza-
tions of our model. Double well potentials with tunable
well-to-well separation have been demonstrated with op-
tical dipole traps e.g. in [21, 24], and on atom chips
e.g. in [25, 26]. For our model, the double well poten-
tial has to be realized inside the cavity. Optical trapping
and even cooling of atoms close to their ground state
inside a cavity has been achieved in several groups by
now [27, 28, 29, 30], but up to our knowledge double
well potentials have not been realized in a cavity so far.
However, some of the cavities developed have a very long
lateral opening (up to 222 µm [31]) and should allow more
complicated trapping potentials (optical lattices inter-
secting a cavity have been realized in Chapman’s group
[31]). We remark that it is not essential for our model
that the double well potential be aligned with the cavity
axes. Any other orientation is possible, and only leads to
modified coefficients cosχ sinκ and sinχ cosκ.
At certain “magical wavelengths”, Cs, Yb, Sr, Mg, and
Ca atoms in optical traps experience the same potential
for ground and excited internal states coupled by a dipole
transition [27, 32, 33, 34]. In a symmetric potential V (z)
the tunneling frequency ∆ is given in WKB approxi-
mation by ∆ ∼ ωosc exp(−1/~
2m(E0 − V (z)) dz)
where E0 is the ground state energy, ωosc the single well
harmonic oscillation frequency, and z = ±a are the corre-
sponding classical turning points delimiting the range of
the barrier. The exponential factor can approach unity
for a barrier that is only slightly higher than the ground
state energy E0, in which case cooling to temperatures
kBT < ~∆ should be possible with state of the art
techniques [27]. In [27] a trap depth V0/~ = 47 MHz
was achieved inside a cavity with 1.2 mW laser power.
In any case, the trap frequency and thus the tunneling
splitting are determined by the laser power and the fo-
cussing (or the wavelength for optical lattices), and can
therefore be controlled independently of Γ, κcav, such
that there should be no fundamental problem achieving
∆ ≫ Γ, κcav. The detection of the tunneling motion
should be possible by optical imaging, i.e. diffusion of
laser light from another transition in the optical regime
with smaller wavelength than the well separation. Al-
ternatively, one might monitor the transmission through
the cavity in the case that it differs for the two locations
of the wells [35]. Another possibility might be using the
atomic spin as a position meter [17].
Acknowledgments
We thank Jacques Vigué for an interesting discussion
and CALMIP (Toulouse) for the use of their comput-
ers. This work was supported by the Agence National de
la Recherche (ANR), project INFOSYSQQ, and the EC
IST-FET project EUROSQIP.
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