Phonon-mediated decay of an atom in a surface-induced potential
Fam Le Kien,1,∗ S. Dutta Gupta,1,2 and K. Hakuta1
Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
School of Physics, University of Hyderabad, Hyderabad, India
(Dated: August 4, 2021)
We study phonon-mediated transitions between translational levels of an atom in a surface-induced
potential. We present a general master equation governing the dynamics of the translational states
of the atom. In the framework of the Debye model, we derive compact expressions for the rates
for both upward and downward transitions. Numerical calculations for the transition rates are
performed for a deep silica-induced potential allowing for a large number of bound levels as well
as free states of a cesium atom. The total absorption rate is shown to be determined mainly by
the bound-to-bound transitions for deep bound levels and by bound-to-free transitions for shallow
bound levels. Moreover, the phonon emission and absorption processes can be orders of magnitude
larger for deep bound levels as compared to the shallow bound ones. We also study various types of
transitions from free states. We show that, for thermal atomic cesium with temperature in the range
from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption
(free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay)
rate, while the cooling (free-to-free downward decay) rate is negligible.
PACS numbers: 34.50.Dy,33.70.Ca
I. INTRODUCTION
Over the past few years, tight confinement of cold
atoms has drawn considerable attention. The interest in
this area is motivated not only by the fundamental na-
ture of the problem, but also by its potential applications
in atom optics and quantum information. A method
for microscopic trapping and guiding of individual atoms
along a nanofiber has been proposed [1]. Surface–atom
quantum electrodynamic effects have constituted another
interesting area, where a great deal of work has been
carried out. Modification of spontaneous emission of
an atom [2] and radiative exchange between two distant
atoms [3] mediated by a nanofiber have been investigated.
Surface-induced deep potentials have played a major role
and have received due attention in recent years. Oria
et al. have studied various theoretical schemes to load
atoms into such potentials [4, 5]. A rigorous theory of
spontaneous decay of an atom in a surface-induced po-
tential invoking the density-matrix formalism has been
developed [6]. The role of interference between the emit-
ted and reflected fields and also the role of transmission
into the evanescent modes were identified. Further cal-
culations on the excitation spectrum have been carried
out [7]. Bound-to-bound transitions were shown to lead
to significant effects like a large red tail of the excita-
tion spectrum as compared to the weak consequences of
free-to-bound transitions. A crucial step in this direction
was the experimental observation of the excitation spec-
trum and the channeling of the fluorescent photons along
the nanofiber [8], opening up avenues for novel quantum
information devices.
In most of the problems involving surface–atom inter-
action, the macroscopic surface is usually kept at room
temperature. Thus the pertinent question that can be
asked is what would be the effect of heating on the cold
atoms. It is understood that transfer of heat to the
trapped atoms will lead to a change in the occupation
probability of the vibrational levels as well as their co-
herence. Phonon-induced changes in the populations of
the vibrational levels have been studied by several groups
[5, 9, 10]. In a nice and compact treatment based on the
dyadic Green function and the Fermi golden rule, Henkel
et al. showed that the effects can be very different de-
pending on the nature of the atomic/molecular species
[9]. The time scales for various species were estimated.
It should be stressed that the trap considered by Henkel
et al. was not necessarily a surface trap and misses out
on many of the aspects of the surface–atom interaction
[9]. Based on the assumption that the surface–atom in-
teraction can be represented by a Morse potential, the
phonon-mediated decay was estimated by Oria et al. [5].
Their estimate was based on the formalism developed by
Gortel et al. [10]. However, all the previous theories
focus on only the transition rates and thus are not gen-
eral enough. In this paper, we present a general density-
matrix formalism to calculate the phonon-mediated de-
cay of populations as well as the changes in coherence.
We derive the relevant master equation for the density
matrix of the atom. We emphasize that our density-
matrix equation describes the full dynamics of the cou-
pling between trapped atoms and phonons and does not
assume any particular form of the trapping potential.
Under the Debye approximation, we derive compact ex-
pressions for the phonon-mediated decay rates. Numer-
ical calculations are carried out assuming the potential
model considered in [4]. In contrast to the previous work,
we include a large number of vibrational levels due to the
deep surface–atom potential. We show that there can be
significant differences in the decay rates when the initial
level is chosen as one of the shallow or deep bound levels.
We also calculate and analyze the decay rates for various
http://arxiv.org/abs/0704.0340v1
types of transitions from free states.
The paper is organized as follows. In Sec. II we de-
scribe the model. In Sec. III we derive the basic dynam-
ical equations for the phonon-mediated decay processes.
In Sec. IV we present the results of numerical calcula-
tions. Our conclusions are given in Sec. V.
II. DESCRIPTION OF THE MODEL SYSTEM
We assume the whole space to be divided into two re-
gions, namely, the half-space x < 0, occupied by a nondis-
persive nonabsorbing dielectric medium (medium 1), and
the half-space x > 0, occupied by vacuum (medium 2).
We examine a single atom moving in the empty half-
space x > 0. We assume that the atom is in a fixed
internal state |i〉 with energy h̄ωi. Without loss of gen-
erality, we assume that the energy of the internal state
|i〉 is zero, i.e. ωi = 0. We describe the interaction be-
tween the atom and the surface. We first consider the
surface-induced interaction potential and then add the
atom-phonon interaction.
A. Surface-induced interaction potential
In this subsection, we describe the interaction between
the atom and the surface in the case where thermal vi-
brations of the surface are absent. The potential en-
ergy of the surface–atom interaction is a combination of
a long-range van der Waals attraction and a short-range
repulsion [11]. Despite a large volume of research on the
surface–atom interaction, due to the complexity of sur-
face physics and the lack of data, the actual form of the
potential is yet to be ascertained [11]. For the purpose
of numerical demonstration of our formalism, we choose
the following model for the potential [4, 11]:
U(x) = Ae−αx − C3
. (1)
Here, C3 is the van der Waals coefficient, while A and
α determine the height and range, respectively, of the
surface repulsion. The potential parameters C3, A, and
α depend on the nature of the dielectric and the atom.
In numerical calculations, we use the parameters of fused
silica, for the dielectric, and the parameters of ground-
state atomic cesium, for the atom. The parameters for
the interaction between silica and ground-state atomic
cesium are theoretically estimated to be C3 = 1.56 kHz
µm3, A = 1.6× 1018 Hz, and α = 53 nm−1 [6].
We introduce the notation ϕν(x) for the eigenfunc-
tions of the center-of-mass motion of the atom in the
potential U(x). They are determined by the stationary
Schrödinger equation
+ U(x)
ϕν(x) = Eνϕν(x). (2)
Here m is the mass of the atom. In the numerical ex-
ample with atomic cesium, we have m = 132.9 a.u.
= 2.21 × 10−25 kg. The eigenvalues Eν are the center-
of-mass energies of the translational levels of the atom.
These eigenvalues are the shifts of the energies of the
translational levels from the energy of the internal state
|i〉. Without loss of generality, we assume that the
center-of-mass eigenfunctions ϕν(x) are real functions,
i.e. ϕ∗ν(x) = ϕν(x).
In Fig. 1, we show the potential U(x) and the wave
functions ϕν(x) of a number of bound levels with en-
ergies in the range from −1 GHz to −5 MHz. We also
plot the wave function of a free state with energy of about
4.25 MHz. In order to have some estimate about the spa-
tial extent of a wave function ϕν(x), we define a crossing
point xcross, which corresponds to the rightmost solution
of the equation U(x) = Eν . Note that, for shallow lev-
els, the wave function generally peaks close to the point
xcross. We plot the eigenvalue modulus |Eν | and the cross-
ing point xcross in Figs. 2(a) and 2(b), respectively. It is
clear from the figure that, for ν in the range from 0 to
300, the eigenvalue varies dramatically from about 158
THz to about 322 kHz, while the wave function extends
only up to 170 nm.
FIG. 1: Energies and wave functions of the center-of-mass
motion of an atom in a surface-induced potential. The pa-
rameters of the potential are C3 = 1.56 kHz µm
3, A =
1.6 × 1018 Hz, and α = 53 nm−1. The mass of the atom
is m = 2.21 × 10−25 kg. We plot bound levels with energies
in the range from −1 GHz to −5 MHz and also a free state
with energy of about 4.25 MHz.
FIG. 2: Eigenvalue modulus |Eν | (a) and crossing point xcross
(b) as functions of the vibrational quantum number ν. The
parameters used are as in Fig. 1.
We introduce the notation |ν〉 = |ϕν〉 and ων = Eν/h̄
for the state vectors and frequencies of translational lev-
els. Then, the Hamiltonian of the atom in the surface-
induced potential can be represented in the diagonal form
h̄ωνσνν . (3)
Here, σνν = |ν〉〈ν| is the population operator for the
translational level ν. We emphasize that the summation
over ν includes both the discrete (Eν < 0) and continuous
(Eν > 0) spectra. The levels ν with Eν < 0 are called
the bound (or vibrational) levels. In such a state, the
atom is bound to the surface. It is vibrating, or more
exactly, moving back and forth between the walls formed
by the van der Waals part and the repulsive part of the
potential. The levels ν with Eν > 0 are called the free (or
continuum) levels. The center-of-mass wave functions of
the bound states are normalized to unity. The center-of-
mass wave functions of the free states are normalized to
the delta function of energy.
B. Atom–phonon interaction
In this subsection, we incorporate the thermal vibra-
tions of the solid into the model. Due to the thermal
effects, the surface of the dielectric vibrates. The surface-
induced potential for the atom is then U(x− xs), where
xs is the displacement of the surface from the mean po-
sition 〈xs〉 = 0. We approximate the vibrating potential
U(x− xs) by expanding it to the first order in xs,
U(x− xs) = U(x) − U ′(x)xs. (4)
The first term, U(x), when combined with the kinetic
energy p2/2m, yields the Hamiltonian HA [see Eq. (3)],
which leads to the formation of translational levels of
the atom. The second term, −U ′(x)xs, accounts for the
thermal effects in the interaction of the atom with the
solid. Note that the quantity F = −U ′(x) is the force
of the surface upon the atom. Hence, the force of the
atom upon the surface is −F = U ′(x) and, consequently,
U ′(x)xs is the work required to displace the surface for
a small distance xs.
It is well known that, for a smooth surface, the gas
atom interacts only with the phonons polarized along
the x direction [10]. In the harmonic approximation, we
2MNωq
iqR + b†qe
−iqR). (5)
Here, M is the mass of a particle of the solid, N is
the particle number density, ωq and q are the frequency
and wave vector of the x-polarized acoustic phonons, re-
spectively, R = (0, y, z) is the lateral component of the
position vector (x, y, z) of the atom, and bq and b
q are
the annihilation and creation phonon operators, respec-
tively. Without loss of generality, we choose R = 0.
Meanwhile, the operator U ′ can be decomposed as U ′ =
νν′ σνν′ 〈ν|U ′|ν′〉, where σνν′ = |ν〉〈ν′| is the operator
for the translational transition ν ↔ ν′. Hence, the en-
ergy term −U ′(x)xs leads to the atom–phonon interac-
tion Hamiltonian [10]
HI = h̄
S(bq + b
q), (6)
gνν′σνν′ . (7)
Here we have introduced the atom–phonon coupling co-
efficients
gνν′ =
Fνν′√
2MNh̄
, (8)
Fνν′ = −
ϕν(x)U
′(x)ϕν′ (x)dx (9)
being the matrix elements for the force of the surface
upon the atom. We note that Fνν′ = −mω2νν′xνν′ , where
xνν′ = 〈ν|x|ν′〉 and ωνν′ = ων −ων′ are the surface–atom
dipole matrix element and the translational transition
frequency, respectively. Hence, the coupling coefficient
gνν′ depends on the dipole matrix element xνν′ and the
transition frequency ωνν′ . Since ωνν = 0, we have gνν =
We note that the Hamiltonian of the x-polarized acous-
tic phonons is given by
h̄ωqb
qbq. (10)
The total Hamiltonian of the atom–phonon system is
H = HA +HI +HB. (11)
We use the above Hamiltonian to study the phonon-
mediated decay of the atom.
III. DYNAMICS OF THE ATOM
In this section, we present the basic equations for the
phonon-mediated decay processes. We derive a general
master equation for the reduced density operator of the
atom in subsection IIIA, obtain analytical expressions
for the relaxation rates and frequency shifts in subsec-
tion III B, and calculate the rates and the shifts in the
framework of the Debye model in subsection III C.
A. Master equation
In the Heisenberg picture, the equation for the phonon
operator bq(t) is
ḃq(t) = −iωqbq(t)−
S(t), (12)
which has a solution of the form
bq(t) = bq(t0)e
−iωq(t−t0) − iWq(t). (13)
Here, t0 is the initial time and Wq is given by
Wq(t) =
e−iωq(t−τ)S(τ) dτ. (14)
Consider an arbitrary atomic operator O which acts only
on the atomic states but not on the phonon states. The
time evolution of this operator is governed by the Heisen-
berg equation
∂O(t)
[HA(t) +HI(t),O(t)], (15)
which, with account of Eqs. (6) and (13), yields
∂O(t)
[HA(t),O(t)]
[S(t),O(t)][bq(t0)e−iωq(t−t0) − iWq(t)]
[b†q(t0)e
iωq(t−t0) + iW †q(t)][O(t), S(t)].
We assume the initial density of the atom–phonon sys-
tem to be the direct product state
ρΣ(t0) = ρ(t0)ρB(t0), (17)
with the atom in an arbitrary state ρ(t0) and the phonons
in a thermal state
ρB(t0) = Z
−1 exp[−HB(t0)/kBT ]. (18)
Here, Z is the normalization constant and T is the tem-
perature of the phonon bath. For the initial condition
(17), the Bogolubov’s lemma [12], applied to an arbitrary
operator Θ(t), asserts the following:
〈Θ(t)bq(t0)〉 = n̄q〈[bq(t0),Θ(t)]〉, (19)
where the mean number of phonons in the mode q is
given by
n̄q =
exp(h̄ωq/kBT )− 1
. (20)
Let Θ be an atomic operator. We then have the commu-
tation relation [bq(t),Θ(t)] = 0, which yields
[bq(t0),Θ(t)] = ie
iωq(t−t0)[Wq(t),Θ(t)]. (21)
Combining Eq. (19) with Eq. (21) leads to
〈Θ(t)bq(t0)〉 = ieiωq(t−t0)n̄q〈[Wq(t),Θ(t)]〉. (22)
We perform the quantum mechanical averaging for ex-
pression (16) and use Eq. (22) to eliminate the phonon
operators bq(t0) and b
q(t0). The resulting equation can
be written as
∂〈O(t)〉
〈[HA(t),O(t)]〉
n̄q + 1√
〈[S(t),O(t)]Wq(t) +W †q(t)[O(t), S(t)]〉
〈Wq(t)[O(t), S(t)] + [S(t),O(t)]W †q(t)〉.
We note that Eq. (23) is exact. It does not contain
phonon operators explicitly. The dependence on the
phonon operators is hidden in the time shift of the oper-
ator S(τ) in expression (14) for the operator Wq(t).
We now show how the dependence of the operator
Wq(t) on the phonon operators can be approximately
eliminated. We assume that the atom–phonon coupling
coefficients gνν′ are small. The use of the zeroth-order
approximation σνν′(τ) = σνν′ (t)e
iωνν′(τ−t) in the expres-
sion for S(τ) [see Eq. (7)] yields
S(τ) =
gνν′σνν′ (t)e
iωνν′(τ−t), (24)
which is accurate to first order in the coupling coeffi-
cients. Inserting Eq. (24) into Eq. (14) gives
Wq(t) =
gνν′σνν′(t)δ−(ων′ν − ωq), (25)
where
δ−(ω) = lim
e−i(ω+iǫ)τ dτ
δ(ω). (26)
Here, in order to take into account the effect of adiabatic
turn-on of interaction, we have added a small positive
parameter ǫ to the integral and have used the limit t0 →
−∞. Introducing the notation
gνν′σνν′δ−(ων′ν − ωq), (27)
we can rewrite Eq. (23) in the form
∂〈O(t)〉
〈[HA(t),O(t)]〉
(n̄q + 1)〈[S(t),O(t)]Kq(t) +K†q(t)[O(t), S(t)]〉
n̄q〈Kq(t)[O(t), S(t)] + [S(t),O(t)]K†q(t)〉. (28)
In order to examine the time evolution of the reduced
density operator ρ(t) of the atom in the Schrödinger
picture, we use the relation 〈O(t)〉 = Tr[O(t)ρ(0)] =
Tr[O(0)ρ(t)], transform to arrange the operator O(0) at
the first position in each operator product, and eliminate
O(0). Then, we obtain the Liouville master equation
∂ρ(t)
= − i
[HA, ρ(t)]
(n̄q + 1){[Kqρ(t), S] + [S, ρ(t)K†q]}
n̄q{[S, ρ(t)Kq] + [K†qρ(t), S]}. (29)
Equations (28) and (29) are valid to second order in
the coupling coefficients. These equations allow us to
study the time evolution and dynamical characteristics of
the atom interacting with the thermal phonon bath. We
note that Eq. (29) is a particular form of the Zwanzig’s
generalized master equation, which can be obtained by
the projection operator method [13].
B. Relaxation rates and frequency shifts
We use Eq. (29) to derive an equation for the matrix
elements ρjj′ ≡ 〈j|ρ|j′〉 of the reduced density operator
of the atom. The result is
∂ρjj′
= −iωjj′ρjj′ +
(γejj′νν′ + γ
jj′νν′)ρνν′
[(γejν + γ
jν)ρνj′ + (γ
j′ν + γ
j′ν)ρjν ], (30)
where the coefficients
γejj′νν′ = 2π
n̄q + 1
gjνgj′ν′ [δ−(ωνj − ωq)
+ δ+(ων′j′ − ωq)],
γejν = 2π
n̄q + 1
gjµgνµδ−(ωνµ − ωq) (31)
γajj′νν′ = 2π
gjνgj′ν′ [δ−(ωj′ν′ − ωq)
+ δ+(ωjν − ωq)],
γajν = 2π
gjµgνµδ+(ωµν − ωq) (32)
are the decay parameters associated with the phonon
emission and absorption, respectively. Here, the nota-
tion δ+(ω) = δ
−(ω) has been used.
Equation (30) describes phonon-induced variations in
the populations and coherences of the translational levels
of the atom. We analyze the characteristics of the relax-
ation processes. For simplicity of mathematical treat-
ment, we first consider only transitions from discrete lev-
els. The equation for the diagonal matrix element ρjj for
a discrete level j can be written in the form
(γejjνν + γ
jjνν )ρνν
− (γejj + γajj + c.c.)ρjj
+ off-diagonal terms. (33)
When the off-diagonal terms are neglected, Eq. (33) re-
duces to a simple rate equation. It is clear from Eq. (33)
that the rate for the downward transition from an upper
level l to a lower level k (k < l) is
Rekl = γ
kkll = 2π
n̄q + 1
g2lkδ(ωlk − ωq), (34)
while the rate for the upward transition from a lower level
k to an upper level l (l > k) is
Ralk = γ
llkk = 2π
g2lkδ(ωlk − ωq). (35)
Equations (34) and (35) are in agreement with the re-
sults of Gortel et al. [10], obtained by using the Fermi
golden rule. We note that Rekl and R
lk with l ≤ k are
mathematically equal to zero because they have no phys-
ical meaning. For convenience, we introduce the notation
Rlk = R
lk, R
lk, or 0 for l < k, l > k, or l = k, respec-
tively. It is clear that the off-diagonal coefficients Rlk
with l 6= k are the rates of transitions. However, the di-
agonal coefficients Rkk have no physical meaning and are
mathematically equal to zero.
As seen from Eq. (33), the phonon-mediated depletion
rate of a level k is Γkk = 2Re(γ
kk + γ
kk). The explicit
expression for this rate is
Γkk = 2π
n̄q + 1
g2kµδ(ωkµ − ωq)
g2µkδ(ωµk − ωq). (36)
We note that Γkk =
µk +R
µk) =
µ Rµk. We can
write Γkk = Γ
kk + Γ
kk, where
Γekk =
Reµk (37)
Γakk =
Raµk (38)
are the contributions due to downward transitions
(phonon emission) and upward transitions (phonon ab-
sorption), respectively. In the above equations, the sum-
mation over µ can be extended to cover not only the
discrete levels but also the continuum levels.
Meanwhile, the equation for the off-diagonal matrix
element ρlk for a pair of discrete levels l and k can be
written in the form ∂ρlk/∂t = −(iωlk + γell + γall + γe∗kk +
γa∗kk)ρlk + . . . , or, equivalently,
= −i(ωlk +∆lk − iΓlk)ρlk + . . . . (39)
Here the frequency shift ∆lk is given by
∆lk =
n̄q + 1
ωlµ − ωq
ωµk + ωq
ωlµ + ωq
ωµk − ωq
, (40)
while the coherence decay rate Γlk is expressed as
Γlk = π
n̄q + 1
g2lµδ(ωlµ − ωq) + g2kµδ(ωkµ − ωq)
g2µlδ(ωµl − ωq) + g2µkδ(ωµk − ωq)
When we set l = k in Eq. (40), we find ∆kk = 0.
When we set l = k in Eq. (41), we recover Eq. (36).
We note that Γlk =
µl + R
µk + R
µl + R
µk)/2 =
µ(Rµl + Rµk)/2. Comparison between Eqs. (41) and
(36) yields the relation Γlk = (Γll +Γkk)/2. We can also
write Γlk = Γ
lk + Γ
lk, where Γ
µl + R
µk)/2
and Γalk =
µl + R
µk)/2 are the contributions due
to downward transitions (phonon emission) and upward
transitions (phonon absorption), respectively. In the
above equations, the summation over µ can be extended
to cover not only the discrete levels but also the contin-
uum levels.
We now discuss phonon-mediated transitions from con-
tinuum (free) levels. We start by considering free-to-
bound transitions. For a continuum level f with energy
Ef > 0, the center-of-mass wave function ϕf (x) is nor-
malized per unit energy. In this case, the quantity Rνf
becomes the density of the transition rate. A free level f
can be approximated by a level of a quasicontinuum [14].
A discretization of the continuum can be realized by using
a large box of length L with reflecting boundary condi-
tions [15]. We label En the energies of the eigenstates
in the box and φn(x) the corresponding wave functions.
Note that such states are standing-wave states [14, 15].
The relation between a quasicontinuum-state wave func-
tion φnf (x), normalized to unity in the box, and the cor-
responding continuum-state wave function ϕf (x), nor-
malized per unit energy, with equal energies Enf = Ef ,
is [15]
ϕf (x) ∼=
]−1/2
φnf (x)
)1/2 (
φnf (x). (42)
Consequently, for a single atom initially prepared in the
quasicontinuum standing-wave state |nf 〉 = |φnf 〉, the
rate for the transition to an arbitrary bound state |ν〉 is
approximately given by
Gνf =
vfRνf , (43)
where vf = (2Ef/m)1/2 is the velocity of the atom in the
initial standing-wave state |f〉. The phonon-mediated
free-to-bound decay rate (adsorption rate) is then given
Gνf , (44)
where the summation includes only bound levels. It is
clear from Eq. (43) that, in the continuum limit L → ∞,
the rate Gνf tends to zero. This is because a free atom
can be anywhere in free space and therefore the effect of
phonons on a single free atom is negligible.
In order to get deeper insight into the free-to-bound
transition rate density Rνf , we consider a macroscopic
atomic ensemble in the thermodynamic limit [14]. Sup-
pose that there are N0 atoms in a volume with a large
length L and a transverse cross section area S0. Assume
that all the atoms are in the same quasicontinuum state
|nf 〉 and interact with the dielectric independently. The
rate for the transitions of the atoms from the quasicon-
tinuum state |nf〉 to an arbitrary bound state |ν〉, defined
as the time derivative of the number of atoms in the state
|ν〉, is Dνf = N0Gνf . In order to get the rate for the con-
tinuum state |f〉, we need to take the thermodynamical
limit, where L → ∞ and N0 → ∞ but N0/L remains
constant. Then, the rate for the transitions of the atoms
from the continuum state |f〉 to an arbitrary bound
state |ν〉 is given by Dνf = πh̄ρ0S0vfRνf = 2πh̄NfRνf .
Here, ρ0 = N0/LS0 is the atomic number density and
Nf = ρ0S0vf/2 is the number of atoms incident into the
dielectric surface per unit time. It is clear that the tran-
sition rate Dνf is proportional to the incidence rate Nf
as well as the transition rate density Rνf . We emphasize
that Dνf is a characteristics for a macroscopic atomic en-
semble in the thermodynamic limit while Gνf is a mea-
sure for a single atom. When the length of the box, L,
and the number of atoms, N0, are finite, the dynamics of
the atoms cannot be described by the free-to-bound rate
Dνf directly. Instead, we must use the transition rate
per atom Gνf = Dνf/N0, which depends on the length
L of the box that contains the free atoms [see Eq. (43)].
In a thermal gas, the atoms have different velocities
and, therefore, different energies. For a thermal Maxwell-
Boltzmann gas with temperature T0, the distribution of
the kinetic energy Ef of the atomic center-of-mass motion
along the x direction is
P (Ef ) =
πkBT0
e−Ef/kBT0
. (45)
The transition rate to an arbitrary bound state |ν〉 is
then given by GνT0 =
GνfP (Ef ) dEf , i.e.
GνT0 =
e−Ef/kBT0RνfdEf , (46)
where λD = (2πh̄
2/mkBT0)
1/2 is the thermal de Broglie
wavelength. The phonon-mediated free-to-bound decay
rate (adsorption rate) is given by
GT0 =
GνT0 =
GfP (Ef ) dEf . (47)
In the above equation, the summation over ν includes
only bound levels. Note that Eq. (46) is in qualitative
agreement with the results of Refs. [5, 14].
It is easy to extend the above results to the case of
free-to-free transitions. Indeed, it can be shown that the
density of the rate for the transition from a quasicontin-
uum state |nf 〉, which corresponds to a free state |f〉, to
a different free state |f ′〉 is given by
Qf ′f =
vfRf ′f . (48)
For convenience, we introduce the notation Qef ′f = Qf ′f
or 0 for Ef ′ < Ef or Ef ′ ≥ Ef , respectively, and Qaf ′f =
Qf ′f or 0 for Ef ′ > Ef or Ef ′ ≤ Ef , respectively. Then, we
have Qf ′f = Q
f ′f , 0, or Q
f ′f for Ef ′ < Ef , Ef ′ = Ef , or
Ef ′ > Ef , respectively. The downward (phonon-emission)
and upward (phonon-absorption) free-to-free decay rates
for the free state |f〉 are given by
Qef =
Qef ′fdEf ′ (49)
Qaf =
Qaf ′fdEf ′ , (50)
respectively. The total free-to-free decay rate for the free
state |f〉 is Qf = Qef +Qaf =
Qf ′fdEf ′ .
For a thermal gas, we need to replace the transition
rate density Qf ′f and the decay rate Qf by Qf ′T0 =
Qf ′fP (Ef ) dEf and QT0 =
QfP (Ef ) dEf , respec-
tively, which are the averages of Qf ′f and Qf , respec-
tively, with respect to the energy distribution P (Ef )
of the initial state. Like in the other cases, we have
Qf ′T0 = Q
f ′T0
+Qaf ′T0 and QT0 = Q
+QaT0 , where
Qef ′T0 =
Qef ′fP (Ef ) dEf ,
Qaf ′T0 =
∫ Ef′
Qaf ′fP (Ef ) dEf (51)
are the downward and upward transition rate densities
QeT0 =
QefP (Ef ) dEf ,
QaT0 =
QafP (Ef ) dEf (52)
are the downward and upward decay rates. The thermal
decay ratesQeT0 andQ
describe the cooling and heating
processes, respectively. It can be easily shown thatQeT0 <
QaT0 , Q
> QaT0 , and Q
= QaT0 when T0 < T , T0 >
T , and T0 = T , respectively. The relation Q
< QaT0
(QeT0 > Q
), obtained for T0 < T (T0 > T ), indicates
the dominance of heating (cooling) of free atoms by the
surface.
C. Relaxation rates and frequency shifts in the
framework of the Debye model
In order to get insight into the relaxation rates and
frequency shifts, we approximate them using the Debye
model for phonons. In this model, the phonon frequency
ωq is related to the phonon wave number q as ωq = vq,
where v is the sound velocity. Furthermore, the summa-
tion over the first Brillouin zone is replaced by an integral
over a sphere of radius qD = (6π
2N/V )1/3, where V is the
volume of the solid. The Debye frequency and the Debye
temperature are given by ωD = vqD and TD = h̄ωD/kB,
respectively. For fused silica, we have v = 5.96 km/s,
NM/V = 2.2 g/cm3, and M = 9.98× 10−26 kg [16]. Us-
ing these parameters, we find qD = 109.29 × 106 cm−1,
ωD = 10.4 THz, and TD = 498 K. In order to perform
the summation over phonon states in the framework of
the Debye model, we invoke the thermodynamic limit,
i.e., replace
· · · = V
|q|≤qD
. . . dq =
. . . ω2qdωq. (53)
Then, for transitions between an upper level l and a lower
level k, where 0 < ωlk < ωD, Eqs. (34) and (35) yield
Rekl =
Mh̄ω3D
(n̄lk + 1)ωlkF
lk (54)
Ralk =
Mh̄ω3D
n̄lkωlkF
lk. (55)
Here, n̄lk is given by Eq. (20) with ωq replaced by ωlk.
We emphasize that, according to Eqs. (54) and (55),
the phonon-emission rate Rekl and the phonon-absorption
rate Ralk depend not only on the matrix element Flk of
the force but also on the translational transition fre-
quency ωlk. The frequency dependences of the transi-
tion rates are comprised of the frequency dependences
of the mean phonon number n̄lk, the phonon mode den-
sity 3Nω2lk/ω
D, and the matrix element Flk = −U ′lk =
−mω2lkxlk of the force. An additional factor comes from
the presence of the phonon frequency in Eq. (5) for
the surface displacement and, consequently, in the atom–
phonon interaction Hamiltonian (6). It is clear that an
increase in the phonon frequency leads to a decrease in
the mean phonon number and an increase in the phonon
mode density. The matrix element of the force usu-
ally first increases and then decreases with increasing
phonon frequency. Due to the existence of several com-
peting factors, the frequency dependences of the tran-
sition rates are rather complicated. They usually first
increase and then decrease with increasing phonon fre-
quency. We note that, for transitions with ωlk > ωD, we
have Rekl = R
lk = 0.
We conclude this section by noting that the use of Eq.
(53) in Eq. (40) yields the frequency shift
∆lk = ∆
lk +∆
lk , (56)
where
2Mh̄ω3D
F 2lµ
ωlµ − ω
F 2µk
ωµk + ω
ωdω (57)
Mh̄ω3D
ω2lµ − ω2
ω2µk − ω2
n̄ωωdω
are the zero- and finite-temperature contributions, re-
spectively. In Eq. (58), n̄ω is given by Eq. (20) with ωq
replaced by ω.
IV. NUMERICAL RESULTS AND
DISCUSSIONS
In this section, we present the numerical results based
on the analytical expressions derived in the previous
section for the phonon-mediated relaxation rates of the
translational levels of the atom. In particular, we use
Eqs. (54) and (55), obtained in the framework of the De-
bye model, for our numerical calculations. We consider
transitions from bound states as well as free states. The
transitions from bound states to other translational lev-
els occur in the case where the atom is initially already
adsorbed or trapped near the surface. The transitions
from free states to other translational levels occur in the
processes of adsorbing, heating, and cooling of free atoms
by the surface. Due to the difference in physics of the ini-
tial situations, we study the transitions from bound and
free states separately.
A. Transitions from bound states
FIG. 3: Phonon-emission rates Reν′ν from the vibrational lev-
els (a) ν = 280 and (b) ν = 120 to other levels ν′ as functions
of the lower-level energy Eν′ . The arrows mark the initial
states. The parameters of the solid are M = 9.98 × 10−26 kg
and ωD = 10.4 THz. The temperature of the phonon bath is
T = 300 K. Other parameters are as in Fig. 1.
FIG. 4: Phonon-absorption rates Raν′ν from the vibrational
levels (a) ν = 280 and (b) ν = 120 to other levels ν′ as func-
tions of the upper-level energy Eν′ . The left (right) panel
in each row corresponds to bound-to-bound (bound-to-free)
transitions. The arrows mark the initial states. The param-
eters used are as in Fig. 3. The temperature of the phonon
bath is T = 300 K.
We start from a given bound level and calculate the
rates of phonon-mediated atomic transitions, both down-
ward and upward. The profiles of the phonon-emission
(downward-transition) rate Reν′ν [see Eq. (54)] and the
phonon-absorption (upward-transition) rateRaν′ν [see Eq.
(55)] are shown in Figs. 3 and 4, respectively. The upper
(lower) part of each of these figures corresponds to the
case of the initial level ν = 280 (ν = 120), with energy
Eν = −156 MHz (Eν = −8.4 THz). The left (right) panel
of Fig. 4 corresponds to bound-to-bound (bound-to-free)
upward transitions. The temperature of the surface is
assumed to be T = 300 K. As seen from Figs. 3 and 4,
the transition rates have pronounced localized profiles.
Due to the competing effects of the mean phonon num-
ber, the phonon mode density, and the matrix element
of the force, the transition rates usually first increase
and then decrease with increasing phonon frequency. It
is clear from a comparison of Figs. 3(a) and 3(b) and
also a comparison of Figs. 4(a) and 4(b) that transitions
from shallow levels have probabilities orders of magni-
tude lower than those from deeper levels. The main rea-
son is that the wave functions of the shallow states are
spread further away from the surface than those for the
deep states. Due to this difference, the effects of the sur-
face vibrations are weaker for the shallow levels than for
the deep levels. Another pertinent feature that should
be noted from the figure is the following: Since transi-
tion frequencies involved are large, they may overshoot
the Debye frequency ωD = 10.4 THz, leading to a cutoff
on the lower (higher) side of the frequency axis for the
emission (absorption) curve.
In order to see the overall effect of the individual tran-
sition rates shown above, we add them up. First we ex-
amine the phonon-absorption rates of bound levels. The
total phonon-absorption rate Γaνν of a bound level ν is the
sum of the individual absorption rates Raµν over all the
upper levels µ, both bound and free [see Eq. (38)]. We
plot in Fig. 5 the contributions to Γaνν from two types
of transitions, bound-to-bound and bound-to-free (des-
orption) transitions. The solid curve of the figure shows
that the bound-to-bound phonon-absorption rate is large
(above 1010 s−1) for deep and intermediate levels. How-
ever, it reduces dramatically with increasing ν in the
region of large ν and becomes very small (below 10−5
s−1) for shallow levels. Meanwhile, the dashed curve of
Fig. 5 shows that the bound-to-free phonon-absorption
rate (i.e., the desorption rate) is zero for deep levels, since
the energy required for the transition is greater than the
Debye energy [5]. However, the desorption rate is sub-
stantial (above 105 s−1) for intermediate and shallow lev-
els. Thus, the total phonon-absorption rate Γaνν is mainly
determined by the bound-to-bound transitions in the case
of deep levels and by the bound-to-free transitions in the
case of shallow levels. One of the reasons for the dramatic
reduction of the bound-to-bound phonon-absorption rate
in the region of shallow levels is that the number of up-
per bound levels µ becomes small. The second reason is
that the frequency of each individual transition becomes
small, leading to a decrease of the phonon mode den-
sity. The third reason is that the center-of-mass wave
functions of shallow levels are spread far away from the
surface, leading to a reduction of the effect of phonons
on the atom.
Unlike the bound-to-bound phonon-absorption rate,
the bound-to-free phonon-absorption rate is substantial
in the region of shallow levels. This is because the free-
state spectrum is continuous and the range of the bound-
to-free transition frequency can be large (up to the De-
bye frequency ωD = 10.4 THz). The gradual reduction of
the bound-to-free phonon-absorption rate in the region of
shallow levels is mainly due to the reduction of the time
that the atom spends in the proximity of the surface.
FIG. 5: Contributions of bound-to-bound (solid curve) and
bound-to-free (dashed curve) transitions to the total phonon-
absorption rate Γaνν versus the vibrational quantum number
ν of the initial level. The parameters used are as in Fig. 3.
The temperature of the phonon bath is T = 300 K.
The total phonon-emission rate Γeνν [see Eq. (37)] and
the total phonon-absorption rate Γaνν [see Eq. (38)] are
shown in Fig. 6 by the solid and dashed curves, respec-
tively. It is clear from the figure that emission is com-
parable to but slightly stronger than absorption. Such a
dominance is due to the fact that phonon emission moves
the atom to a center-of-mass state closer to the surface
while phonon absorption changes the atomic state in the
opposite direction (see Figs. 1 and 2). Our results for the
rates are in good qualitative agreement with the results
of Oria et al., albeit with the Morse potential [5]. We
stress that we include a large number of vibrational lev-
els as a consequence of the deep silica–cesium potential.
Note that the earlier work on this theme involved much
fewer levels [5].
FIG. 6: Phonon-emission decay rate Γeνν (solid lines) and
phonon-absorption decay rate Γaνν (dashed lines) of a bound
level as functions of the vibrational quantum number ν. The
inset shows the rates in the linear scale to highlight the dif-
ferences in the dissociation limit. The parameters used are as
in Fig. 3. The temperature of the phonon bath is T = 300 K.
FIG. 7: Same as in Fig. 6 except that T = 30 K.
We next study the effect of temperature on the decay
rates. The results for the phonon-mediated decay rates
for T = 30 K are shown in Fig. 7. In contrast to Fig. 6,
the absorption rate is now much smaller than the corre-
sponding emission rate for both shallow and deep levels.
Thus, while it is difficult to distinguish the two log-scale
curves for deep and shallow levels at room temperature
(see Fig. 6), they are well resolved at low temperature.
B. Transitions from free states
We now calculate the rates for transitions from free
states to other levels. We first examine free-to-bound
transitions, which correspond to the adsorption process.
According to Eq. (43), the free-to-bound (more exactly,
quasicontinuum-to-bound) transition rate Gνf depends
not only on the continuum-to-bound transition rate den-
sity Rνf but also on the length L of the free-atom quan-
tization box. To be specific, we use in our numerical
calculations the value L = 1 mm, which is a typical size
of atomic clouds in magneto-optical traps [17].
FIG. 8: Free-to-bound transition rates Gνf for transitions
from the free plane-wave states with energies (a) Ef = 2 MHz
and (b) Ef = 3.1 THz to bound levels ν as functions of the
bound-level energy Eν . The arrows mark the energies of the
initial free states. The insets show Gνf on the log scale versus
Eν in the range from −200 MHz to −0.2 MHz to highlight
the rates to shallow bound levels. The length of the free-
atom quantization box is L = 1 mm. The temperature of the
phonon bath is T = 300 K. Other parameters are as in Fig. 3.
We plot in Fig. 8 the free-to-bound transition rate Gνf
[see Eq. (43)] as a function of the vibrational quantum
number ν. The upper (lower) part of the figure corre-
sponds to the case of the initial-state energy Ef = 2
MHz (Ef = 3.1 THz), which is close to the average ki-
netic energy per atom in an ideal gas with temperature
T0 = 200 µK (T0 = 300 K). We observe that the free-to-
bound transition rate first increases and then decreases
with increasing transition frequency ωfν = (Ef − Eν)/h̄.
Such behavior results from the competing effects of the
mean phonon number, the phonon mode density, and the
matrix element of the force, like in the case of bound-
to-bound transitions (see Fig. 3). We also see a cut-
off of the transition frequency, which is associated with
the Debye frequency. Comparison of Figs. 8(a) and 8(b)
shows that the transitions from low-energy free states
have probabilities orders of magnitude smaller than those
from high-energy free states. One of the reasons is that
the transition rate Gνf is proportional to the velocity
vf = (2Ef/m)1/2 [see Eq. (43)]. The dependence of
the transition rate density Rνf on the transition fre-
quency ωfν also plays an important role. Because of this,
the rates for the transitions from low-energy free states
to shallow bound levels are very small [see the inset of
Fig. 8(a)].
FIG. 9: Free-to-bound decay rate Gf as a function of the
free-state energy Ef . The inset highlights the magnitude and
profile of the decay rate for Ef in the range from 0 to 20
MHz. The temperature of the phonon bath is T = 300 K.
Other parameters are as in Fig. 8.
We show in Fig. 9 the free-to-bound decay rate Gf
[see Eq. (44)], which is a characteristic of the adsorp-
tion process, as a function of the free-state energy Ef .
We see that Gf first increases and then decreases with
increasing Ef . The increase of Gf with increasing Ef in
the region of small Ef (see the inset) is mainly due to
the increase in the atomic incidence velocity vf . In this
region, we have Gf ∝ vf ∝
Ef [see Eqs. (43) and
(44)]. For Ef in the range from 0 to 20 MHz, which is
typical for atoms in magneto-optical traps, the maximum
value of Gf is on the order of 10
4 s−1 (see the inset of
Fig. 9). Such free-to-bound (adsorption) rates are sev-
eral orders of magnitude smaller than the bound-to-free
(desorption) rates (see the dashed curve in Fig. 5). The
decrease of Gf with increasing Ef in the region of large
Ef is mainly due to the reduction of the atom–phonon
coupling coefficients.
FIG. 10: Free-to-bound transition rates GνT0 for transitions
from the thermal states with temperatures (a) T0 = 200 µK
and (b) T0 = 300 K to bound levels ν as functions of the
bound-level energy Eν . The insets show GνT0 on the log scale
versus Eν in the range from −200 MHz to −0.2 MHz to high-
light the rates to shallow bound levels. The temperature of
the phonon bath is T = 300 K. Other parameters are as in
Fig. 8.
FIG. 11: Free-to-bound decay rate GT0 as a function of the
atomic temperature T0 in the ranges (a) from 100 µK to 400
µK and (b) from 50 K to 350 K. The temperature of the
phonon bath is T = 300 K. Other parameters are as in Fig. 8.
In a thermal gas, the adsorption process is charac-
terized by the transition rate GνT0 [see Eq. (46)] and
the decay rate GT0 [see Eq. (47)], which are the av-
erages of the free-to-bound transition rate Gνf and the
free-to-bound decay rate Gf , respectively, over the free-
state energy distribution (45). We plot the free-to-bound
transition rate GνT0 and the free-to-bound decay rate
GT0 in Figs. 10 and 11, respectively. Comparison be-
tween Figs. 10(a) and 9(a) shows that the transition rates
from low-temperature thermal states and low-energy free
states look quite similar to each other. The reason is that
the spread of the energy distribution is not substantial
in the case of low temperatures. The spread of the en-
ergy distribution is however substantial in the case of
high temperatures, leading to the softening of the cut-
off frequency effect [compare Fig. 10(b) with Fig. 9(b)].
Figure 11 shows that the free-to-bound decay rate GT0
first increases and then reduces with increasing atomic
temperature T0. For T0 in the range from 100 µK to 400
µK, which is typical for atoms in magneto-optical traps,
the maximum value of GT0 is on the order of 10
4 s−1 [see
Fig. 11(a)]. Such free-to-bound (adsorption) rates are
several orders of magnitude smaller than the bound-to-
free (desorption) rates (see the dashed curve in Fig. 5).
Figure 11(a) shows that, in the region of low atomic tem-
perature T0, one has GT0 ∝
T0, in agreement with the
asymptotic behavior of Eqs. (46) and (47).
FIG. 12: Free-to-free transition rate densities Qf ′f for the
upward (solid lines) and downward (dashed lines) transitions
from the free states |f〉 with energies (a) Ef = 2 MHz and (b)
Ef = 3.1 THz to other free states |f
′〉 as functions of the final-
level energy Ef ′ . The arrows mark the energies of the initial
free states. The inset in part (a) shows Qf ′f versus Ef ′ in
the range from 0 to 4 MHz to highlight the small magnitude
of the rate density for downward transitions (dashed line).
The temperature of the phonon bath is T = 300 K. Other
parameters are as in Fig. 8.
We now examine free-to-free transitions, both upward
and downward, which corresponding to the heating and
cooling processes of free atoms by the surface. We plot in
Fig. 12 the free-to-free transition rate density Qf ′f [see
Eq. (48)] as a function of the final-level energy Ef ′ . The
upper (lower) part of the figure corresponds to the case
of the initial-state energy Ef = 2 MHz (Ef = 3.1 THz),
which is close to the average kinetic energy per atom in
an ideal gas with temperature T0 = 200 µK (T0 = 300
K). The rate densities are shown for the upward (phonon-
absorption) and downward (phonon-emission) transitions
by the solid and dashed lines, respectively. The fig-
ure shows that the free-to-free transition rate density in-
creases or decreases with increasing transition frequency
if the latter is not too large or is large enough, respec-
tively. We also observe a signature of the Debye cutoff
of the phonon frequency. Comparison of Figs. 12(a) and
12(b) shows that transitions from low-energy free states
have probabilities orders of magnitude smaller than those
from high-energy free states. Figure 12(a) and its inset
show that, when the energy of the free state is low, the
free-to-free downward (cooling) transition rate is very
small as compared to the free-to-free upward (heating)
transition rate.
FIG. 13: Free-to-free upward and downward decay rates Qaf
(solid lines) and Qef (dashed lines) as functions of the energy
Ef of the initial free state. The insets highlight the magni-
tudes and profiles of the decay rates for Ef in the range from
0 to 20 MHz. The temperature of the phonon bath is T = 300
K. Other parameters are as in Fig. 8.
We show in Fig. 13 the free-to-free upward (phonon-
absorption) and downward (phonon-emission) decay
rates Qaf [see Eq. (50)] and Q
f [see Eq. (49)] as functions
of the free-state energy Ef . We observe that Qaf and Qef
increase with increasing Ef in the range from 0 to 8 THz.
The increase of Qaf with increasing Ef in the region of
small Ef (see the left inset) is mainly due to the increase
in the atomic incidence velocity vf . In this region, we
have Qaf ∝ vf ∝
Ef [see Eqs. (48) and (50)]. The
increase of Qef with increasing Ef in the region of small
Ef (see the right inset) is due to not only the increase in
the atomic incidence velocity vf [see Eq. (48)] but also
the increase of the transition rate density Qef ′f and the
increase of the integration interval (0, Ef) [see Eq. (49)].
In this region, the dependence of Qef on the energy Ef
is of higher order than E3/2f . The left inset of Fig. 13
shows that, for Ef in the range from 0 to 20 MHz, the
maximum value of Qaf is on the order of 10
4 s−1. Such
free-to-free upward (heating) decay rates are comparable
to but about two times smaller than the corresponding
free-to-bound (adsorption) decay rates (see the inset of
Fig. 9). Meanwhile, the right inset of Fig. 13 shows that,
in the region of small Ef , the free-to-free downward (cool-
ing) decay rate Qef is very small.
FIG. 14: Free-to-free transition rate densities QafT0 for up-
ward transitions (solid lines) and QefT0 for downward tran-
sitions (dashed lines) from the thermal states with tempera-
tures (a) T0 = 200 µK and (b) T0 = 300 K to free levels f
as functions of the free-level energy Ef . The inset in part (a)
shows the rate densities versus Ef in the range from 0 to 8
MHz to highlight the small magnitude of QefT0 (dashed line).
The temperature of the phonon bath is T = 300 K. Other
parameters are as in Fig. 8.
FIG. 15: Free-to-free decay rates QaT0 (solid lines) and
(dashed lines) for upward and downward transitions, re-
spectively, as functions of the atomic temperature T0 in the
ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350
K. For comparison, the free-to-bound decay rate GT0 is re-
plotted from Fig. 11 by the dotted lines. The temperature of
the phonon bath is T = 300 K. Other parameters are as in
Fig. 8.
In the case of a thermal gas, the phonon-mediated heat
transfer between the gas and the surface is characterized
by the free-to-free transition rate densitiesQafT0 andQ
[see Eqs. (51)] and the free-to-free decay rates QaT0and
QeT0 [see Eqs. (52)]. We plot the free-to-free transition
rate densities QafT0 and Q
in Fig. 14. Comparison
between Figs. 14(a) and 12(a) shows that the transition
rate densities from low-temperature thermal states and
low-energy free states are quite similar to each other.
The spread of the initial-state energy distribution is not
substantial in this case. However, the energy spread of
the initial state is substantial in the case of high tem-
peratures, concealing the cutoff frequency effect [com-
pare Fig. 14(b) with Fig. 12(b)]. We display the free-
to-free decay rates QaT0 and Q
in Fig. 15. The solid
and dashed lines correspond to the upward (heating) and
downward (cooling) transitions, respectively. For com-
parison, the free-to-bound decay rate (adsorption rate)
GT0 is re-plotted from Fig. 11 by the dotted lines. We
observe that, for T0 in the range from 100 µK to 400 µK
[see Fig. 15(a)], the adsorption rate GT0 (dotted line) is
about two times larger than the heating rate QaT0 (solid
line), while the cooling rate QeT0 (dashed line) is negligi-
ble. Figure 15(a) shows that, in the region of low atomic
temperatures, one has QT0
∼= QaT0 ∝
T0, in agreement
with the asymptotic behavior of expressions (52). The
figure also shows that QeT0 quickly increases with increas-
ing atomic temperature T0. The relation Q
< QaT0 ,
obtained for T0 < T , indicates the dominance of heating
of cold free atoms by the surface. The substantial mag-
nitude of the free-to-bound transition rate GT0 (dotted
line) indicates that a significant number of atoms can be
adsorbed by the surface. According to Fig. 15(b), the
free-to-free downward transition rate QeT0 (dashed line)
crosses the upward transition rate QaT0 (solid line) when
T0 = T = 300 K, and then becomes the dominant decay
rate. The relation QeT0 > Q
, obtained for T0 > T , indi-
cates the dominance of cooling of hot free atoms by the
surface.
V. CONCLUSIONS
In conclusion, we have studied the phonon-mediated
transitions of an atom in a surface-induced potential.
We developed a general formalism, which is applicable
for any surface–atom potential. A systematic derivation
of the corresponding density-matrix equation enables us
to investigate the dynamics of both diagonal and off-
diagonal elements. We included a large number of vi-
brational levels originating from the deep silica–cesium
potential. We calculated the transition and decay rates
from both bound and free levels. We found that the
rates of phonon-mediated transitions between transla-
tional levels depend on the mean phonon number, the
phonon mode density, and the matrix element of the force
from the surface upon the atom. Due to the effects of
these competing factors, the transition rates usually first
increase and then reduce with increasing transition fre-
quency. We focused on the transitions from bound states.
Two specific examples, namely, when the initial level is
a shallow level also when it can be one of the deep levels
have been worked out. We have shown that there can be
marked differences in the absorption and emission behav-
ior in the two cases. For example, both the absorption
and emission rates from the deep bound levels can be sev-
eral orders (in our case, six orders) of magnitude larger
than the corresponding rates from the shallow bound lev-
els. We also analyzed various types of transitions from
free states. We have shown that, for thermal atomic ce-
sium with temperature in the range from 100 µK to 400
µK in the vicinity of a silica surface with temperature of
300 K, the adsorption (free-to-bound decay) rate is about
two times larger than the heating (free-to-free upward de-
cay) rate, while the cooling (free-to-free downward decay)
rate is negligible.
Acknowledgments
We thank M. Chevrollier for fruitful discussions. This
work was carried out under the 21st Century COE pro-
gram on “Coherent Optical Science.”
[∗] Also at Institute of Physics and Electronics, Vietnamese
Academy of Science and Technology, Hanoi, Vietnam.
[1] V. I. Balykin, K. Hakuta, Fam Le Kien, J. Q. Liang, and
M. Morinaga, Phys. Rev. A 70, 011401(R) (2004); Fam
Le Kien, V. I. Balykin, and K. Hakuta, Phys. Rev. A 70,
063403 (2004).
[2] Fam Le Kien, S. Dutta Gupta, V. I. Balykin, and K.
Hakuta, Phys. Rev. A 72, 032509 (2005).
[3] Fam Le Kien, S. Dutta Gupta, K. P. Nayak, and K.
Hakuta, Phys. Rev. A 72, 063815 (2005).
[4] E. G. Lima, M. Chevrollier, O. Di Lorenzo, P. C. Se-
gundo, and M. Oriá, Phys. Rev. A 62, 013410 (2000).
[5] T. Passerat de Silans, B. Farias, M. Oriá, and M.
Chevrollier, Appl. Phys. B 82, 367 (2006).
[6] Fam Le Kien and K. Hakuta, Phys. Rev. A 75, 013423
(2007).
[7] Fam Le Kien, S. Dutta Gupta, and K. Hakuta, e-print
quant-ph/0610067.
[8] K. P. Nayak, P. N. Melentiev, M. Morinaga, Fam Le Kien,
V. I. Balykin, and K. Hakuta, e-print quant-ph/0610136.
[9] C. Henkel and M. Wilkens, Europhys. Lett. 47, 414
(1999).
[10] Z. W. Gortel, H. J. Kreuzer, and R. Teshima, Phys. Rev.
B 22, 5655 (1980).
[11] H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980).
[12] N. N. Bogolubov, Commun. of JINR, E17-11822, Dubna
(1978); N. N. Bogolubov and N. N. Bogolubov Jr., Ele-
mentary Particles and Nuclei (USSR) 11, 245 (1980).
[13] R. Zwanzig, Lectures in Theoretical Physics, eds. W. E.
Brittin, B. W. Downs, and J. Downs (Interscience, New
York, 1961) Vol. 3, p. 106; G. S. Agarwal, Progress in
Optics, ed. E. Wolf (North-Holland, Amsterdam, 1973)
Vol. 11, p. 3; L. Mandel and E. Wolf, Optical Coherence
and Quantum Optics (Cambridge, New York, 1995) p.
[14] J. Javanainen and M. Mackie, Phys. Rev. A 58, R789
(1998); M. Mackie and J. Javanainen, ibid. 60, 3174
(1999).
[15] E. Luc-Koenig, M. Vatasescu, and F. Masnou-Seeuws,
Eur. Phys. J. D 31, 239 (2004).
[16] See, for example, G. P. Agrawal, Nonlinear Fiber Optics
(Academic, New York, 2001).
[17] H. J. Metcalf and P. van der Straten, Laser Cooling and
Trapping (Springer, New York, 1999).
http://arxiv.org/abs/quant-ph/0610067
http://arxiv.org/abs/quant-ph/0610136

  We study phonon-mediated transitions between translational levels of an atom
in a surface-induced potential. We present a general master equation governing
the dynamics of the translational states of the atom. In the framework of the
Debye model, we derive compact expressions for the rates for both upward and
downward transitions. Numerical calculations for the transition rates are
performed for a deep silica-induced potential allowing for a large number of
bound levels as well as free states of a cesium atom. The total absorption rate
is shown to be determined mainly by the bound-to-bound transitions for deep
bound levels and by bound-to-free transitions for shallow bound levels.
Moreover, the phonon emission and absorption processes can be orders of
magnitude larger for deep bound levels as compared to the shallow bound ones.
We also study various types of transitions from free states. We show that, for
thermal atomic cesium with temperature in the range from 100 $\mu$K to 400
$\mu$K in the vicinity of a silica surface with temperature of 300 K, the
adsorption (free-to-bound decay) rate is about two times larger than the
heating (free-to-free upward decay) rate, while the cooling (free-to-free
downward decay) rate is negligible.

Phonon-mediated decay of an atom in a surface-induced potential
Fam Le Kien,1,∗ S. Dutta Gupta,1,2 and K. Hakuta1
Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
School of Physics, University of Hyderabad, Hyderabad, India
(Dated: August 4, 2021)
We study phonon-mediated transitions between translational levels of an atom in a surface-induced
potential. We present a general master equation governing the dynamics of the translational states
of the atom. In the framework of the Debye model, we derive compact expressions for the rates
for both upward and downward transitions. Numerical calculations for the transition rates are
performed for a deep silica-induced potential allowing for a large number of bound levels as well
as free states of a cesium atom. The total absorption rate is shown to be determined mainly by
the bound-to-bound transitions for deep bound levels and by bound-to-free transitions for shallow
bound levels. Moreover, the phonon emission and absorption processes can be orders of magnitude
larger for deep bound levels as compared to the shallow bound ones. We also study various types of
transitions from free states. We show that, for thermal atomic cesium with temperature in the range
from 100 µK to 400 µK in the vicinity of a silica surface with temperature of 300 K, the adsorption
(free-to-bound decay) rate is about two times larger than the heating (free-to-free upward decay)
rate, while the cooling (free-to-free downward decay) rate is negligible.
PACS numbers: 34.50.Dy,33.70.Ca
I. INTRODUCTION
Over the past few years, tight confinement of cold
atoms has drawn considerable attention. The interest in
this area is motivated not only by the fundamental na-
ture of the problem, but also by its potential applications
in atom optics and quantum information. A method
for microscopic trapping and guiding of individual atoms
along a nanofiber has been proposed [1]. Surface–atom
quantum electrodynamic effects have constituted another
interesting area, where a great deal of work has been
carried out. Modification of spontaneous emission of
an atom [2] and radiative exchange between two distant
atoms [3] mediated by a nanofiber have been investigated.
Surface-induced deep potentials have played a major role
and have received due attention in recent years. Oria
et al. have studied various theoretical schemes to load
atoms into such potentials [4, 5]. A rigorous theory of
spontaneous decay of an atom in a surface-induced po-
tential invoking the density-matrix formalism has been
developed [6]. The role of interference between the emit-
ted and reflected fields and also the role of transmission
into the evanescent modes were identified. Further cal-
culations on the excitation spectrum have been carried
out [7]. Bound-to-bound transitions were shown to lead
to significant effects like a large red tail of the excita-
tion spectrum as compared to the weak consequences of
free-to-bound transitions. A crucial step in this direction
was the experimental observation of the excitation spec-
trum and the channeling of the fluorescent photons along
the nanofiber [8], opening up avenues for novel quantum
information devices.
In most of the problems involving surface–atom inter-
action, the macroscopic surface is usually kept at room
temperature. Thus the pertinent question that can be
asked is what would be the effect of heating on the cold
atoms. It is understood that transfer of heat to the
trapped atoms will lead to a change in the occupation
probability of the vibrational levels as well as their co-
herence. Phonon-induced changes in the populations of
the vibrational levels have been studied by several groups
[5, 9, 10]. In a nice and compact treatment based on the
dyadic Green function and the Fermi golden rule, Henkel
et al. showed that the effects can be very different de-
pending on the nature of the atomic/molecular species
[9]. The time scales for various species were estimated.
It should be stressed that the trap considered by Henkel
et al. was not necessarily a surface trap and misses out
on many of the aspects of the surface–atom interaction
[9]. Based on the assumption that the surface–atom in-
teraction can be represented by a Morse potential, the
phonon-mediated decay was estimated by Oria et al. [5].
Their estimate was based on the formalism developed by
Gortel et al. [10]. However, all the previous theories
focus on only the transition rates and thus are not gen-
eral enough. In this paper, we present a general density-
matrix formalism to calculate the phonon-mediated de-
cay of populations as well as the changes in coherence.
We derive the relevant master equation for the density
matrix of the atom. We emphasize that our density-
matrix equation describes the full dynamics of the cou-
pling between trapped atoms and phonons and does not
assume any particular form of the trapping potential.
Under the Debye approximation, we derive compact ex-
pressions for the phonon-mediated decay rates. Numer-
ical calculations are carried out assuming the potential
model considered in [4]. In contrast to the previous work,
we include a large number of vibrational levels due to the
deep surface–atom potential. We show that there can be
significant differences in the decay rates when the initial
level is chosen as one of the shallow or deep bound levels.
We also calculate and analyze the decay rates for various
http://arxiv.org/abs/0704.0340v1
types of transitions from free states.
The paper is organized as follows. In Sec. II we de-
scribe the model. In Sec. III we derive the basic dynam-
ical equations for the phonon-mediated decay processes.
In Sec. IV we present the results of numerical calcula-
tions. Our conclusions are given in Sec. V.
II. DESCRIPTION OF THE MODEL SYSTEM
We assume the whole space to be divided into two re-
gions, namely, the half-space x < 0, occupied by a nondis-
persive nonabsorbing dielectric medium (medium 1), and
the half-space x > 0, occupied by vacuum (medium 2).
We examine a single atom moving in the empty half-
space x > 0. We assume that the atom is in a fixed
internal state |i〉 with energy h̄ωi. Without loss of gen-
erality, we assume that the energy of the internal state
|i〉 is zero, i.e. ωi = 0. We describe the interaction be-
tween the atom and the surface. We first consider the
surface-induced interaction potential and then add the
atom-phonon interaction.
A. Surface-induced interaction potential
In this subsection, we describe the interaction between
the atom and the surface in the case where thermal vi-
brations of the surface are absent. The potential en-
ergy of the surface–atom interaction is a combination of
a long-range van der Waals attraction and a short-range
repulsion [11]. Despite a large volume of research on the
surface–atom interaction, due to the complexity of sur-
face physics and the lack of data, the actual form of the
potential is yet to be ascertained [11]. For the purpose
of numerical demonstration of our formalism, we choose
the following model for the potential [4, 11]:
U(x) = Ae−αx − C3
. (1)
Here, C3 is the van der Waals coefficient, while A and
α determine the height and range, respectively, of the
surface repulsion. The potential parameters C3, A, and
α depend on the nature of the dielectric and the atom.
In numerical calculations, we use the parameters of fused
silica, for the dielectric, and the parameters of ground-
state atomic cesium, for the atom. The parameters for
the interaction between silica and ground-state atomic
cesium are theoretically estimated to be C3 = 1.56 kHz
µm3, A = 1.6× 1018 Hz, and α = 53 nm−1 [6].
We introduce the notation ϕν(x) for the eigenfunc-
tions of the center-of-mass motion of the atom in the
potential U(x). They are determined by the stationary
Schrödinger equation
+ U(x)
ϕν(x) = Eνϕν(x). (2)
Here m is the mass of the atom. In the numerical ex-
ample with atomic cesium, we have m = 132.9 a.u.
= 2.21 × 10−25 kg. The eigenvalues Eν are the center-
of-mass energies of the translational levels of the atom.
These eigenvalues are the shifts of the energies of the
translational levels from the energy of the internal state
|i〉. Without loss of generality, we assume that the
center-of-mass eigenfunctions ϕν(x) are real functions,
i.e. ϕ∗ν(x) = ϕν(x).
In Fig. 1, we show the potential U(x) and the wave
functions ϕν(x) of a number of bound levels with en-
ergies in the range from −1 GHz to −5 MHz. We also
plot the wave function of a free state with energy of about
4.25 MHz. In order to have some estimate about the spa-
tial extent of a wave function ϕν(x), we define a crossing
point xcross, which corresponds to the rightmost solution
of the equation U(x) = Eν . Note that, for shallow lev-
els, the wave function generally peaks close to the point
xcross. We plot the eigenvalue modulus |Eν | and the cross-
ing point xcross in Figs. 2(a) and 2(b), respectively. It is
clear from the figure that, for ν in the range from 0 to
300, the eigenvalue varies dramatically from about 158
THz to about 322 kHz, while the wave function extends
only up to 170 nm.
FIG. 1: Energies and wave functions of the center-of-mass
motion of an atom in a surface-induced potential. The pa-
rameters of the potential are C3 = 1.56 kHz µm
3, A =
1.6 × 1018 Hz, and α = 53 nm−1. The mass of the atom
is m = 2.21 × 10−25 kg. We plot bound levels with energies
in the range from −1 GHz to −5 MHz and also a free state
with energy of about 4.25 MHz.
FIG. 2: Eigenvalue modulus |Eν | (a) and crossing point xcross
(b) as functions of the vibrational quantum number ν. The
parameters used are as in Fig. 1.
We introduce the notation |ν〉 = |ϕν〉 and ων = Eν/h̄
for the state vectors and frequencies of translational lev-
els. Then, the Hamiltonian of the atom in the surface-
induced potential can be represented in the diagonal form
h̄ωνσνν . (3)
Here, σνν = |ν〉〈ν| is the population operator for the
translational level ν. We emphasize that the summation
over ν includes both the discrete (Eν < 0) and continuous
(Eν > 0) spectra. The levels ν with Eν < 0 are called
the bound (or vibrational) levels. In such a state, the
atom is bound to the surface. It is vibrating, or more
exactly, moving back and forth between the walls formed
by the van der Waals part and the repulsive part of the
potential. The levels ν with Eν > 0 are called the free (or
continuum) levels. The center-of-mass wave functions of
the bound states are normalized to unity. The center-of-
mass wave functions of the free states are normalized to
the delta function of energy.
B. Atom–phonon interaction
In this subsection, we incorporate the thermal vibra-
tions of the solid into the model. Due to the thermal
effects, the surface of the dielectric vibrates. The surface-
induced potential for the atom is then U(x− xs), where
xs is the displacement of the surface from the mean po-
sition 〈xs〉 = 0. We approximate the vibrating potential
U(x− xs) by expanding it to the first order in xs,
U(x− xs) = U(x) − U ′(x)xs. (4)
The first term, U(x), when combined with the kinetic
energy p2/2m, yields the Hamiltonian HA [see Eq. (3)],
which leads to the formation of translational levels of
the atom. The second term, −U ′(x)xs, accounts for the
thermal effects in the interaction of the atom with the
solid. Note that the quantity F = −U ′(x) is the force
of the surface upon the atom. Hence, the force of the
atom upon the surface is −F = U ′(x) and, consequently,
U ′(x)xs is the work required to displace the surface for
a small distance xs.
It is well known that, for a smooth surface, the gas
atom interacts only with the phonons polarized along
the x direction [10]. In the harmonic approximation, we
2MNωq
iqR + b†qe
−iqR). (5)
Here, M is the mass of a particle of the solid, N is
the particle number density, ωq and q are the frequency
and wave vector of the x-polarized acoustic phonons, re-
spectively, R = (0, y, z) is the lateral component of the
position vector (x, y, z) of the atom, and bq and b
q are
the annihilation and creation phonon operators, respec-
tively. Without loss of generality, we choose R = 0.
Meanwhile, the operator U ′ can be decomposed as U ′ =
νν′ σνν′ 〈ν|U ′|ν′〉, where σνν′ = |ν〉〈ν′| is the operator
for the translational transition ν ↔ ν′. Hence, the en-
ergy term −U ′(x)xs leads to the atom–phonon interac-
tion Hamiltonian [10]
HI = h̄
S(bq + b
q), (6)
gνν′σνν′ . (7)
Here we have introduced the atom–phonon coupling co-
efficients
gνν′ =
Fνν′√
2MNh̄
, (8)
Fνν′ = −
ϕν(x)U
′(x)ϕν′ (x)dx (9)
being the matrix elements for the force of the surface
upon the atom. We note that Fνν′ = −mω2νν′xνν′ , where
xνν′ = 〈ν|x|ν′〉 and ωνν′ = ων −ων′ are the surface–atom
dipole matrix element and the translational transition
frequency, respectively. Hence, the coupling coefficient
gνν′ depends on the dipole matrix element xνν′ and the
transition frequency ωνν′ . Since ωνν = 0, we have gνν =
We note that the Hamiltonian of the x-polarized acous-
tic phonons is given by
h̄ωqb
qbq. (10)
The total Hamiltonian of the atom–phonon system is
H = HA +HI +HB. (11)
We use the above Hamiltonian to study the phonon-
mediated decay of the atom.
III. DYNAMICS OF THE ATOM
In this section, we present the basic equations for the
phonon-mediated decay processes. We derive a general
master equation for the reduced density operator of the
atom in subsection IIIA, obtain analytical expressions
for the relaxation rates and frequency shifts in subsec-
tion III B, and calculate the rates and the shifts in the
framework of the Debye model in subsection III C.
A. Master equation
In the Heisenberg picture, the equation for the phonon
operator bq(t) is
ḃq(t) = −iωqbq(t)−
S(t), (12)
which has a solution of the form
bq(t) = bq(t0)e
−iωq(t−t0) − iWq(t). (13)
Here, t0 is the initial time and Wq is given by
Wq(t) =
e−iωq(t−τ)S(τ) dτ. (14)
Consider an arbitrary atomic operator O which acts only
on the atomic states but not on the phonon states. The
time evolution of this operator is governed by the Heisen-
berg equation
∂O(t)
[HA(t) +HI(t),O(t)], (15)
which, with account of Eqs. (6) and (13), yields
∂O(t)
[HA(t),O(t)]
[S(t),O(t)][bq(t0)e−iωq(t−t0) − iWq(t)]
[b†q(t0)e
iωq(t−t0) + iW †q(t)][O(t), S(t)].
We assume the initial density of the atom–phonon sys-
tem to be the direct product state
ρΣ(t0) = ρ(t0)ρB(t0), (17)
with the atom in an arbitrary state ρ(t0) and the phonons
in a thermal state
ρB(t0) = Z
−1 exp[−HB(t0)/kBT ]. (18)
Here, Z is the normalization constant and T is the tem-
perature of the phonon bath. For the initial condition
(17), the Bogolubov’s lemma [12], applied to an arbitrary
operator Θ(t), asserts the following:
〈Θ(t)bq(t0)〉 = n̄q〈[bq(t0),Θ(t)]〉, (19)
where the mean number of phonons in the mode q is
given by
n̄q =
exp(h̄ωq/kBT )− 1
. (20)
Let Θ be an atomic operator. We then have the commu-
tation relation [bq(t),Θ(t)] = 0, which yields
[bq(t0),Θ(t)] = ie
iωq(t−t0)[Wq(t),Θ(t)]. (21)
Combining Eq. (19) with Eq. (21) leads to
〈Θ(t)bq(t0)〉 = ieiωq(t−t0)n̄q〈[Wq(t),Θ(t)]〉. (22)
We perform the quantum mechanical averaging for ex-
pression (16) and use Eq. (22) to eliminate the phonon
operators bq(t0) and b
q(t0). The resulting equation can
be written as
∂〈O(t)〉
〈[HA(t),O(t)]〉
n̄q + 1√
〈[S(t),O(t)]Wq(t) +W †q(t)[O(t), S(t)]〉
〈Wq(t)[O(t), S(t)] + [S(t),O(t)]W †q(t)〉.
We note that Eq. (23) is exact. It does not contain
phonon operators explicitly. The dependence on the
phonon operators is hidden in the time shift of the oper-
ator S(τ) in expression (14) for the operator Wq(t).
We now show how the dependence of the operator
Wq(t) on the phonon operators can be approximately
eliminated. We assume that the atom–phonon coupling
coefficients gνν′ are small. The use of the zeroth-order
approximation σνν′(τ) = σνν′ (t)e
iωνν′(τ−t) in the expres-
sion for S(τ) [see Eq. (7)] yields
S(τ) =
gνν′σνν′ (t)e
iωνν′(τ−t), (24)
which is accurate to first order in the coupling coeffi-
cients. Inserting Eq. (24) into Eq. (14) gives
Wq(t) =
gνν′σνν′(t)δ−(ων′ν − ωq), (25)
where
δ−(ω) = lim
e−i(ω+iǫ)τ dτ
δ(ω). (26)
Here, in order to take into account the effect of adiabatic
turn-on of interaction, we have added a small positive
parameter ǫ to the integral and have used the limit t0 →
−∞. Introducing the notation
gνν′σνν′δ−(ων′ν − ωq), (27)
we can rewrite Eq. (23) in the form
∂〈O(t)〉
〈[HA(t),O(t)]〉
(n̄q + 1)〈[S(t),O(t)]Kq(t) +K†q(t)[O(t), S(t)]〉
n̄q〈Kq(t)[O(t), S(t)] + [S(t),O(t)]K†q(t)〉. (28)
In order to examine the time evolution of the reduced
density operator ρ(t) of the atom in the Schrödinger
picture, we use the relation 〈O(t)〉 = Tr[O(t)ρ(0)] =
Tr[O(0)ρ(t)], transform to arrange the operator O(0) at
the first position in each operator product, and eliminate
O(0). Then, we obtain the Liouville master equation
∂ρ(t)
= − i
[HA, ρ(t)]
(n̄q + 1){[Kqρ(t), S] + [S, ρ(t)K†q]}
n̄q{[S, ρ(t)Kq] + [K†qρ(t), S]}. (29)
Equations (28) and (29) are valid to second order in
the coupling coefficients. These equations allow us to
study the time evolution and dynamical characteristics of
the atom interacting with the thermal phonon bath. We
note that Eq. (29) is a particular form of the Zwanzig’s
generalized master equation, which can be obtained by
the projection operator method [13].
B. Relaxation rates and frequency shifts
We use Eq. (29) to derive an equation for the matrix
elements ρjj′ ≡ 〈j|ρ|j′〉 of the reduced density operator
of the atom. The result is
∂ρjj′
= −iωjj′ρjj′ +
(γejj′νν′ + γ
jj′νν′)ρνν′
[(γejν + γ
jν)ρνj′ + (γ
j′ν + γ
j′ν)ρjν ], (30)
where the coefficients
γejj′νν′ = 2π
n̄q + 1
gjνgj′ν′ [δ−(ωνj − ωq)
+ δ+(ων′j′ − ωq)],
γejν = 2π
n̄q + 1
gjµgνµδ−(ωνµ − ωq) (31)
γajj′νν′ = 2π
gjνgj′ν′ [δ−(ωj′ν′ − ωq)
+ δ+(ωjν − ωq)],
γajν = 2π
gjµgνµδ+(ωµν − ωq) (32)
are the decay parameters associated with the phonon
emission and absorption, respectively. Here, the nota-
tion δ+(ω) = δ
−(ω) has been used.
Equation (30) describes phonon-induced variations in
the populations and coherences of the translational levels
of the atom. We analyze the characteristics of the relax-
ation processes. For simplicity of mathematical treat-
ment, we first consider only transitions from discrete lev-
els. The equation for the diagonal matrix element ρjj for
a discrete level j can be written in the form
(γejjνν + γ
jjνν )ρνν
− (γejj + γajj + c.c.)ρjj
+ off-diagonal terms. (33)
When the off-diagonal terms are neglected, Eq. (33) re-
duces to a simple rate equation. It is clear from Eq. (33)
that the rate for the downward transition from an upper
level l to a lower level k (k < l) is
Rekl = γ
kkll = 2π
n̄q + 1
g2lkδ(ωlk − ωq), (34)
while the rate for the upward transition from a lower level
k to an upper level l (l > k) is
Ralk = γ
llkk = 2π
g2lkδ(ωlk − ωq). (35)
Equations (34) and (35) are in agreement with the re-
sults of Gortel et al. [10], obtained by using the Fermi
golden rule. We note that Rekl and R
lk with l ≤ k are
mathematically equal to zero because they have no phys-
ical meaning. For convenience, we introduce the notation
Rlk = R
lk, R
lk, or 0 for l < k, l > k, or l = k, respec-
tively. It is clear that the off-diagonal coefficients Rlk
with l 6= k are the rates of transitions. However, the di-
agonal coefficients Rkk have no physical meaning and are
mathematically equal to zero.
As seen from Eq. (33), the phonon-mediated depletion
rate of a level k is Γkk = 2Re(γ
kk + γ
kk). The explicit
expression for this rate is
Γkk = 2π
n̄q + 1
g2kµδ(ωkµ − ωq)
g2µkδ(ωµk − ωq). (36)
We note that Γkk =
µk +R
µk) =
µ Rµk. We can
write Γkk = Γ
kk + Γ
kk, where
Γekk =
Reµk (37)
Γakk =
Raµk (38)
are the contributions due to downward transitions
(phonon emission) and upward transitions (phonon ab-
sorption), respectively. In the above equations, the sum-
mation over µ can be extended to cover not only the
discrete levels but also the continuum levels.
Meanwhile, the equation for the off-diagonal matrix
element ρlk for a pair of discrete levels l and k can be
written in the form ∂ρlk/∂t = −(iωlk + γell + γall + γe∗kk +
γa∗kk)ρlk + . . . , or, equivalently,
= −i(ωlk +∆lk − iΓlk)ρlk + . . . . (39)
Here the frequency shift ∆lk is given by
∆lk =
n̄q + 1
ωlµ − ωq
ωµk + ωq
ωlµ + ωq
ωµk − ωq
, (40)
while the coherence decay rate Γlk is expressed as
Γlk = π
n̄q + 1
g2lµδ(ωlµ − ωq) + g2kµδ(ωkµ − ωq)
g2µlδ(ωµl − ωq) + g2µkδ(ωµk − ωq)
When we set l = k in Eq. (40), we find ∆kk = 0.
When we set l = k in Eq. (41), we recover Eq. (36).
We note that Γlk =
µl + R
µk + R
µl + R
µk)/2 =
µ(Rµl + Rµk)/2. Comparison between Eqs. (41) and
(36) yields the relation Γlk = (Γll +Γkk)/2. We can also
write Γlk = Γ
lk + Γ
lk, where Γ
µl + R
µk)/2
and Γalk =
µl + R
µk)/2 are the contributions due
to downward transitions (phonon emission) and upward
transitions (phonon absorption), respectively. In the
above equations, the summation over µ can be extended
to cover not only the discrete levels but also the contin-
uum levels.
We now discuss phonon-mediated transitions from con-
tinuum (free) levels. We start by considering free-to-
bound transitions. For a continuum level f with energy
Ef > 0, the center-of-mass wave function ϕf (x) is nor-
malized per unit energy. In this case, the quantity Rνf
becomes the density of the transition rate. A free level f
can be approximated by a level of a quasicontinuum [14].
A discretization of the continuum can be realized by using
a large box of length L with reflecting boundary condi-
tions [15]. We label En the energies of the eigenstates
in the box and φn(x) the corresponding wave functions.
Note that such states are standing-wave states [14, 15].
The relation between a quasicontinuum-state wave func-
tion φnf (x), normalized to unity in the box, and the cor-
responding continuum-state wave function ϕf (x), nor-
malized per unit energy, with equal energies Enf = Ef ,
is [15]
ϕf (x) ∼=
]−1/2
φnf (x)
)1/2 (
φnf (x). (42)
Consequently, for a single atom initially prepared in the
quasicontinuum standing-wave state |nf 〉 = |φnf 〉, the
rate for the transition to an arbitrary bound state |ν〉 is
approximately given by
Gνf =
vfRνf , (43)
where vf = (2Ef/m)1/2 is the velocity of the atom in the
initial standing-wave state |f〉. The phonon-mediated
free-to-bound decay rate (adsorption rate) is then given
Gνf , (44)
where the summation includes only bound levels. It is
clear from Eq. (43) that, in the continuum limit L → ∞,
the rate Gνf tends to zero. This is because a free atom
can be anywhere in free space and therefore the effect of
phonons on a single free atom is negligible.
In order to get deeper insight into the free-to-bound
transition rate density Rνf , we consider a macroscopic
atomic ensemble in the thermodynamic limit [14]. Sup-
pose that there are N0 atoms in a volume with a large
length L and a transverse cross section area S0. Assume
that all the atoms are in the same quasicontinuum state
|nf 〉 and interact with the dielectric independently. The
rate for the transitions of the atoms from the quasicon-
tinuum state |nf〉 to an arbitrary bound state |ν〉, defined
as the time derivative of the number of atoms in the state
|ν〉, is Dνf = N0Gνf . In order to get the rate for the con-
tinuum state |f〉, we need to take the thermodynamical
limit, where L → ∞ and N0 → ∞ but N0/L remains
constant. Then, the rate for the transitions of the atoms
from the continuum state |f〉 to an arbitrary bound
state |ν〉 is given by Dνf = πh̄ρ0S0vfRνf = 2πh̄NfRνf .
Here, ρ0 = N0/LS0 is the atomic number density and
Nf = ρ0S0vf/2 is the number of atoms incident into the
dielectric surface per unit time. It is clear that the tran-
sition rate Dνf is proportional to the incidence rate Nf
as well as the transition rate density Rνf . We emphasize
that Dνf is a characteristics for a macroscopic atomic en-
semble in the thermodynamic limit while Gνf is a mea-
sure for a single atom. When the length of the box, L,
and the number of atoms, N0, are finite, the dynamics of
the atoms cannot be described by the free-to-bound rate
Dνf directly. Instead, we must use the transition rate
per atom Gνf = Dνf/N0, which depends on the length
L of the box that contains the free atoms [see Eq. (43)].
In a thermal gas, the atoms have different velocities
and, therefore, different energies. For a thermal Maxwell-
Boltzmann gas with temperature T0, the distribution of
the kinetic energy Ef of the atomic center-of-mass motion
along the x direction is
P (Ef ) =
πkBT0
e−Ef/kBT0
. (45)
The transition rate to an arbitrary bound state |ν〉 is
then given by GνT0 =
GνfP (Ef ) dEf , i.e.
GνT0 =
e−Ef/kBT0RνfdEf , (46)
where λD = (2πh̄
2/mkBT0)
1/2 is the thermal de Broglie
wavelength. The phonon-mediated free-to-bound decay
rate (adsorption rate) is given by
GT0 =
GνT0 =
GfP (Ef ) dEf . (47)
In the above equation, the summation over ν includes
only bound levels. Note that Eq. (46) is in qualitative
agreement with the results of Refs. [5, 14].
It is easy to extend the above results to the case of
free-to-free transitions. Indeed, it can be shown that the
density of the rate for the transition from a quasicontin-
uum state |nf 〉, which corresponds to a free state |f〉, to
a different free state |f ′〉 is given by
Qf ′f =
vfRf ′f . (48)
For convenience, we introduce the notation Qef ′f = Qf ′f
or 0 for Ef ′ < Ef or Ef ′ ≥ Ef , respectively, and Qaf ′f =
Qf ′f or 0 for Ef ′ > Ef or Ef ′ ≤ Ef , respectively. Then, we
have Qf ′f = Q
f ′f , 0, or Q
f ′f for Ef ′ < Ef , Ef ′ = Ef , or
Ef ′ > Ef , respectively. The downward (phonon-emission)
and upward (phonon-absorption) free-to-free decay rates
for the free state |f〉 are given by
Qef =
Qef ′fdEf ′ (49)
Qaf =
Qaf ′fdEf ′ , (50)
respectively. The total free-to-free decay rate for the free
state |f〉 is Qf = Qef +Qaf =
Qf ′fdEf ′ .
For a thermal gas, we need to replace the transition
rate density Qf ′f and the decay rate Qf by Qf ′T0 =
Qf ′fP (Ef ) dEf and QT0 =
QfP (Ef ) dEf , respec-
tively, which are the averages of Qf ′f and Qf , respec-
tively, with respect to the energy distribution P (Ef )
of the initial state. Like in the other cases, we have
Qf ′T0 = Q
f ′T0
+Qaf ′T0 and QT0 = Q
+QaT0 , where
Qef ′T0 =
Qef ′fP (Ef ) dEf ,
Qaf ′T0 =
∫ Ef′
Qaf ′fP (Ef ) dEf (51)
are the downward and upward transition rate densities
QeT0 =
QefP (Ef ) dEf ,
QaT0 =
QafP (Ef ) dEf (52)
are the downward and upward decay rates. The thermal
decay ratesQeT0 andQ
describe the cooling and heating
processes, respectively. It can be easily shown thatQeT0 <
QaT0 , Q
> QaT0 , and Q
= QaT0 when T0 < T , T0 >
T , and T0 = T , respectively. The relation Q
< QaT0
(QeT0 > Q
), obtained for T0 < T (T0 > T ), indicates
the dominance of heating (cooling) of free atoms by the
surface.
C. Relaxation rates and frequency shifts in the
framework of the Debye model
In order to get insight into the relaxation rates and
frequency shifts, we approximate them using the Debye
model for phonons. In this model, the phonon frequency
ωq is related to the phonon wave number q as ωq = vq,
where v is the sound velocity. Furthermore, the summa-
tion over the first Brillouin zone is replaced by an integral
over a sphere of radius qD = (6π
2N/V )1/3, where V is the
volume of the solid. The Debye frequency and the Debye
temperature are given by ωD = vqD and TD = h̄ωD/kB,
respectively. For fused silica, we have v = 5.96 km/s,
NM/V = 2.2 g/cm3, and M = 9.98× 10−26 kg [16]. Us-
ing these parameters, we find qD = 109.29 × 106 cm−1,
ωD = 10.4 THz, and TD = 498 K. In order to perform
the summation over phonon states in the framework of
the Debye model, we invoke the thermodynamic limit,
i.e., replace
· · · = V
|q|≤qD
. . . dq =
. . . ω2qdωq. (53)
Then, for transitions between an upper level l and a lower
level k, where 0 < ωlk < ωD, Eqs. (34) and (35) yield
Rekl =
Mh̄ω3D
(n̄lk + 1)ωlkF
lk (54)
Ralk =
Mh̄ω3D
n̄lkωlkF
lk. (55)
Here, n̄lk is given by Eq. (20) with ωq replaced by ωlk.
We emphasize that, according to Eqs. (54) and (55),
the phonon-emission rate Rekl and the phonon-absorption
rate Ralk depend not only on the matrix element Flk of
the force but also on the translational transition fre-
quency ωlk. The frequency dependences of the transi-
tion rates are comprised of the frequency dependences
of the mean phonon number n̄lk, the phonon mode den-
sity 3Nω2lk/ω
D, and the matrix element Flk = −U ′lk =
−mω2lkxlk of the force. An additional factor comes from
the presence of the phonon frequency in Eq. (5) for
the surface displacement and, consequently, in the atom–
phonon interaction Hamiltonian (6). It is clear that an
increase in the phonon frequency leads to a decrease in
the mean phonon number and an increase in the phonon
mode density. The matrix element of the force usu-
ally first increases and then decreases with increasing
phonon frequency. Due to the existence of several com-
peting factors, the frequency dependences of the tran-
sition rates are rather complicated. They usually first
increase and then decrease with increasing phonon fre-
quency. We note that, for transitions with ωlk > ωD, we
have Rekl = R
lk = 0.
We conclude this section by noting that the use of Eq.
(53) in Eq. (40) yields the frequency shift
∆lk = ∆
lk +∆
lk , (56)
where
2Mh̄ω3D
F 2lµ
ωlµ − ω
F 2µk
ωµk + ω
ωdω (57)
Mh̄ω3D
ω2lµ − ω2
ω2µk − ω2
n̄ωωdω
are the zero- and finite-temperature contributions, re-
spectively. In Eq. (58), n̄ω is given by Eq. (20) with ωq
replaced by ω.
IV. NUMERICAL RESULTS AND
DISCUSSIONS
In this section, we present the numerical results based
on the analytical expressions derived in the previous
section for the phonon-mediated relaxation rates of the
translational levels of the atom. In particular, we use
Eqs. (54) and (55), obtained in the framework of the De-
bye model, for our numerical calculations. We consider
transitions from bound states as well as free states. The
transitions from bound states to other translational lev-
els occur in the case where the atom is initially already
adsorbed or trapped near the surface. The transitions
from free states to other translational levels occur in the
processes of adsorbing, heating, and cooling of free atoms
by the surface. Due to the difference in physics of the ini-
tial situations, we study the transitions from bound and
free states separately.
A. Transitions from bound states
FIG. 3: Phonon-emission rates Reν′ν from the vibrational lev-
els (a) ν = 280 and (b) ν = 120 to other levels ν′ as functions
of the lower-level energy Eν′ . The arrows mark the initial
states. The parameters of the solid are M = 9.98 × 10−26 kg
and ωD = 10.4 THz. The temperature of the phonon bath is
T = 300 K. Other parameters are as in Fig. 1.
FIG. 4: Phonon-absorption rates Raν′ν from the vibrational
levels (a) ν = 280 and (b) ν = 120 to other levels ν′ as func-
tions of the upper-level energy Eν′ . The left (right) panel
in each row corresponds to bound-to-bound (bound-to-free)
transitions. The arrows mark the initial states. The param-
eters used are as in Fig. 3. The temperature of the phonon
bath is T = 300 K.
We start from a given bound level and calculate the
rates of phonon-mediated atomic transitions, both down-
ward and upward. The profiles of the phonon-emission
(downward-transition) rate Reν′ν [see Eq. (54)] and the
phonon-absorption (upward-transition) rateRaν′ν [see Eq.
(55)] are shown in Figs. 3 and 4, respectively. The upper
(lower) part of each of these figures corresponds to the
case of the initial level ν = 280 (ν = 120), with energy
Eν = −156 MHz (Eν = −8.4 THz). The left (right) panel
of Fig. 4 corresponds to bound-to-bound (bound-to-free)
upward transitions. The temperature of the surface is
assumed to be T = 300 K. As seen from Figs. 3 and 4,
the transition rates have pronounced localized profiles.
Due to the competing effects of the mean phonon num-
ber, the phonon mode density, and the matrix element
of the force, the transition rates usually first increase
and then decrease with increasing phonon frequency. It
is clear from a comparison of Figs. 3(a) and 3(b) and
also a comparison of Figs. 4(a) and 4(b) that transitions
from shallow levels have probabilities orders of magni-
tude lower than those from deeper levels. The main rea-
son is that the wave functions of the shallow states are
spread further away from the surface than those for the
deep states. Due to this difference, the effects of the sur-
face vibrations are weaker for the shallow levels than for
the deep levels. Another pertinent feature that should
be noted from the figure is the following: Since transi-
tion frequencies involved are large, they may overshoot
the Debye frequency ωD = 10.4 THz, leading to a cutoff
on the lower (higher) side of the frequency axis for the
emission (absorption) curve.
In order to see the overall effect of the individual tran-
sition rates shown above, we add them up. First we ex-
amine the phonon-absorption rates of bound levels. The
total phonon-absorption rate Γaνν of a bound level ν is the
sum of the individual absorption rates Raµν over all the
upper levels µ, both bound and free [see Eq. (38)]. We
plot in Fig. 5 the contributions to Γaνν from two types
of transitions, bound-to-bound and bound-to-free (des-
orption) transitions. The solid curve of the figure shows
that the bound-to-bound phonon-absorption rate is large
(above 1010 s−1) for deep and intermediate levels. How-
ever, it reduces dramatically with increasing ν in the
region of large ν and becomes very small (below 10−5
s−1) for shallow levels. Meanwhile, the dashed curve of
Fig. 5 shows that the bound-to-free phonon-absorption
rate (i.e., the desorption rate) is zero for deep levels, since
the energy required for the transition is greater than the
Debye energy [5]. However, the desorption rate is sub-
stantial (above 105 s−1) for intermediate and shallow lev-
els. Thus, the total phonon-absorption rate Γaνν is mainly
determined by the bound-to-bound transitions in the case
of deep levels and by the bound-to-free transitions in the
case of shallow levels. One of the reasons for the dramatic
reduction of the bound-to-bound phonon-absorption rate
in the region of shallow levels is that the number of up-
per bound levels µ becomes small. The second reason is
that the frequency of each individual transition becomes
small, leading to a decrease of the phonon mode den-
sity. The third reason is that the center-of-mass wave
functions of shallow levels are spread far away from the
surface, leading to a reduction of the effect of phonons
on the atom.
Unlike the bound-to-bound phonon-absorption rate,
the bound-to-free phonon-absorption rate is substantial
in the region of shallow levels. This is because the free-
state spectrum is continuous and the range of the bound-
to-free transition frequency can be large (up to the De-
bye frequency ωD = 10.4 THz). The gradual reduction of
the bound-to-free phonon-absorption rate in the region of
shallow levels is mainly due to the reduction of the time
that the atom spends in the proximity of the surface.
FIG. 5: Contributions of bound-to-bound (solid curve) and
bound-to-free (dashed curve) transitions to the total phonon-
absorption rate Γaνν versus the vibrational quantum number
ν of the initial level. The parameters used are as in Fig. 3.
The temperature of the phonon bath is T = 300 K.
The total phonon-emission rate Γeνν [see Eq. (37)] and
the total phonon-absorption rate Γaνν [see Eq. (38)] are
shown in Fig. 6 by the solid and dashed curves, respec-
tively. It is clear from the figure that emission is com-
parable to but slightly stronger than absorption. Such a
dominance is due to the fact that phonon emission moves
the atom to a center-of-mass state closer to the surface
while phonon absorption changes the atomic state in the
opposite direction (see Figs. 1 and 2). Our results for the
rates are in good qualitative agreement with the results
of Oria et al., albeit with the Morse potential [5]. We
stress that we include a large number of vibrational lev-
els as a consequence of the deep silica–cesium potential.
Note that the earlier work on this theme involved much
fewer levels [5].
FIG. 6: Phonon-emission decay rate Γeνν (solid lines) and
phonon-absorption decay rate Γaνν (dashed lines) of a bound
level as functions of the vibrational quantum number ν. The
inset shows the rates in the linear scale to highlight the dif-
ferences in the dissociation limit. The parameters used are as
in Fig. 3. The temperature of the phonon bath is T = 300 K.
FIG. 7: Same as in Fig. 6 except that T = 30 K.
We next study the effect of temperature on the decay
rates. The results for the phonon-mediated decay rates
for T = 30 K are shown in Fig. 7. In contrast to Fig. 6,
the absorption rate is now much smaller than the corre-
sponding emission rate for both shallow and deep levels.
Thus, while it is difficult to distinguish the two log-scale
curves for deep and shallow levels at room temperature
(see Fig. 6), they are well resolved at low temperature.
B. Transitions from free states
We now calculate the rates for transitions from free
states to other levels. We first examine free-to-bound
transitions, which correspond to the adsorption process.
According to Eq. (43), the free-to-bound (more exactly,
quasicontinuum-to-bound) transition rate Gνf depends
not only on the continuum-to-bound transition rate den-
sity Rνf but also on the length L of the free-atom quan-
tization box. To be specific, we use in our numerical
calculations the value L = 1 mm, which is a typical size
of atomic clouds in magneto-optical traps [17].
FIG. 8: Free-to-bound transition rates Gνf for transitions
from the free plane-wave states with energies (a) Ef = 2 MHz
and (b) Ef = 3.1 THz to bound levels ν as functions of the
bound-level energy Eν . The arrows mark the energies of the
initial free states. The insets show Gνf on the log scale versus
Eν in the range from −200 MHz to −0.2 MHz to highlight
the rates to shallow bound levels. The length of the free-
atom quantization box is L = 1 mm. The temperature of the
phonon bath is T = 300 K. Other parameters are as in Fig. 3.
We plot in Fig. 8 the free-to-bound transition rate Gνf
[see Eq. (43)] as a function of the vibrational quantum
number ν. The upper (lower) part of the figure corre-
sponds to the case of the initial-state energy Ef = 2
MHz (Ef = 3.1 THz), which is close to the average ki-
netic energy per atom in an ideal gas with temperature
T0 = 200 µK (T0 = 300 K). We observe that the free-to-
bound transition rate first increases and then decreases
with increasing transition frequency ωfν = (Ef − Eν)/h̄.
Such behavior results from the competing effects of the
mean phonon number, the phonon mode density, and the
matrix element of the force, like in the case of bound-
to-bound transitions (see Fig. 3). We also see a cut-
off of the transition frequency, which is associated with
the Debye frequency. Comparison of Figs. 8(a) and 8(b)
shows that the transitions from low-energy free states
have probabilities orders of magnitude smaller than those
from high-energy free states. One of the reasons is that
the transition rate Gνf is proportional to the velocity
vf = (2Ef/m)1/2 [see Eq. (43)]. The dependence of
the transition rate density Rνf on the transition fre-
quency ωfν also plays an important role. Because of this,
the rates for the transitions from low-energy free states
to shallow bound levels are very small [see the inset of
Fig. 8(a)].
FIG. 9: Free-to-bound decay rate Gf as a function of the
free-state energy Ef . The inset highlights the magnitude and
profile of the decay rate for Ef in the range from 0 to 20
MHz. The temperature of the phonon bath is T = 300 K.
Other parameters are as in Fig. 8.
We show in Fig. 9 the free-to-bound decay rate Gf
[see Eq. (44)], which is a characteristic of the adsorp-
tion process, as a function of the free-state energy Ef .
We see that Gf first increases and then decreases with
increasing Ef . The increase of Gf with increasing Ef in
the region of small Ef (see the inset) is mainly due to
the increase in the atomic incidence velocity vf . In this
region, we have Gf ∝ vf ∝
Ef [see Eqs. (43) and
(44)]. For Ef in the range from 0 to 20 MHz, which is
typical for atoms in magneto-optical traps, the maximum
value of Gf is on the order of 10
4 s−1 (see the inset of
Fig. 9). Such free-to-bound (adsorption) rates are sev-
eral orders of magnitude smaller than the bound-to-free
(desorption) rates (see the dashed curve in Fig. 5). The
decrease of Gf with increasing Ef in the region of large
Ef is mainly due to the reduction of the atom–phonon
coupling coefficients.
FIG. 10: Free-to-bound transition rates GνT0 for transitions
from the thermal states with temperatures (a) T0 = 200 µK
and (b) T0 = 300 K to bound levels ν as functions of the
bound-level energy Eν . The insets show GνT0 on the log scale
versus Eν in the range from −200 MHz to −0.2 MHz to high-
light the rates to shallow bound levels. The temperature of
the phonon bath is T = 300 K. Other parameters are as in
Fig. 8.
FIG. 11: Free-to-bound decay rate GT0 as a function of the
atomic temperature T0 in the ranges (a) from 100 µK to 400
µK and (b) from 50 K to 350 K. The temperature of the
phonon bath is T = 300 K. Other parameters are as in Fig. 8.
In a thermal gas, the adsorption process is charac-
terized by the transition rate GνT0 [see Eq. (46)] and
the decay rate GT0 [see Eq. (47)], which are the av-
erages of the free-to-bound transition rate Gνf and the
free-to-bound decay rate Gf , respectively, over the free-
state energy distribution (45). We plot the free-to-bound
transition rate GνT0 and the free-to-bound decay rate
GT0 in Figs. 10 and 11, respectively. Comparison be-
tween Figs. 10(a) and 9(a) shows that the transition rates
from low-temperature thermal states and low-energy free
states look quite similar to each other. The reason is that
the spread of the energy distribution is not substantial
in the case of low temperatures. The spread of the en-
ergy distribution is however substantial in the case of
high temperatures, leading to the softening of the cut-
off frequency effect [compare Fig. 10(b) with Fig. 9(b)].
Figure 11 shows that the free-to-bound decay rate GT0
first increases and then reduces with increasing atomic
temperature T0. For T0 in the range from 100 µK to 400
µK, which is typical for atoms in magneto-optical traps,
the maximum value of GT0 is on the order of 10
4 s−1 [see
Fig. 11(a)]. Such free-to-bound (adsorption) rates are
several orders of magnitude smaller than the bound-to-
free (desorption) rates (see the dashed curve in Fig. 5).
Figure 11(a) shows that, in the region of low atomic tem-
perature T0, one has GT0 ∝
T0, in agreement with the
asymptotic behavior of Eqs. (46) and (47).
FIG. 12: Free-to-free transition rate densities Qf ′f for the
upward (solid lines) and downward (dashed lines) transitions
from the free states |f〉 with energies (a) Ef = 2 MHz and (b)
Ef = 3.1 THz to other free states |f
′〉 as functions of the final-
level energy Ef ′ . The arrows mark the energies of the initial
free states. The inset in part (a) shows Qf ′f versus Ef ′ in
the range from 0 to 4 MHz to highlight the small magnitude
of the rate density for downward transitions (dashed line).
The temperature of the phonon bath is T = 300 K. Other
parameters are as in Fig. 8.
We now examine free-to-free transitions, both upward
and downward, which corresponding to the heating and
cooling processes of free atoms by the surface. We plot in
Fig. 12 the free-to-free transition rate density Qf ′f [see
Eq. (48)] as a function of the final-level energy Ef ′ . The
upper (lower) part of the figure corresponds to the case
of the initial-state energy Ef = 2 MHz (Ef = 3.1 THz),
which is close to the average kinetic energy per atom in
an ideal gas with temperature T0 = 200 µK (T0 = 300
K). The rate densities are shown for the upward (phonon-
absorption) and downward (phonon-emission) transitions
by the solid and dashed lines, respectively. The fig-
ure shows that the free-to-free transition rate density in-
creases or decreases with increasing transition frequency
if the latter is not too large or is large enough, respec-
tively. We also observe a signature of the Debye cutoff
of the phonon frequency. Comparison of Figs. 12(a) and
12(b) shows that transitions from low-energy free states
have probabilities orders of magnitude smaller than those
from high-energy free states. Figure 12(a) and its inset
show that, when the energy of the free state is low, the
free-to-free downward (cooling) transition rate is very
small as compared to the free-to-free upward (heating)
transition rate.
FIG. 13: Free-to-free upward and downward decay rates Qaf
(solid lines) and Qef (dashed lines) as functions of the energy
Ef of the initial free state. The insets highlight the magni-
tudes and profiles of the decay rates for Ef in the range from
0 to 20 MHz. The temperature of the phonon bath is T = 300
K. Other parameters are as in Fig. 8.
We show in Fig. 13 the free-to-free upward (phonon-
absorption) and downward (phonon-emission) decay
rates Qaf [see Eq. (50)] and Q
f [see Eq. (49)] as functions
of the free-state energy Ef . We observe that Qaf and Qef
increase with increasing Ef in the range from 0 to 8 THz.
The increase of Qaf with increasing Ef in the region of
small Ef (see the left inset) is mainly due to the increase
in the atomic incidence velocity vf . In this region, we
have Qaf ∝ vf ∝
Ef [see Eqs. (48) and (50)]. The
increase of Qef with increasing Ef in the region of small
Ef (see the right inset) is due to not only the increase in
the atomic incidence velocity vf [see Eq. (48)] but also
the increase of the transition rate density Qef ′f and the
increase of the integration interval (0, Ef) [see Eq. (49)].
In this region, the dependence of Qef on the energy Ef
is of higher order than E3/2f . The left inset of Fig. 13
shows that, for Ef in the range from 0 to 20 MHz, the
maximum value of Qaf is on the order of 10
4 s−1. Such
free-to-free upward (heating) decay rates are comparable
to but about two times smaller than the corresponding
free-to-bound (adsorption) decay rates (see the inset of
Fig. 9). Meanwhile, the right inset of Fig. 13 shows that,
in the region of small Ef , the free-to-free downward (cool-
ing) decay rate Qef is very small.
FIG. 14: Free-to-free transition rate densities QafT0 for up-
ward transitions (solid lines) and QefT0 for downward tran-
sitions (dashed lines) from the thermal states with tempera-
tures (a) T0 = 200 µK and (b) T0 = 300 K to free levels f
as functions of the free-level energy Ef . The inset in part (a)
shows the rate densities versus Ef in the range from 0 to 8
MHz to highlight the small magnitude of QefT0 (dashed line).
The temperature of the phonon bath is T = 300 K. Other
parameters are as in Fig. 8.
FIG. 15: Free-to-free decay rates QaT0 (solid lines) and
(dashed lines) for upward and downward transitions, re-
spectively, as functions of the atomic temperature T0 in the
ranges (a) from 100 µK to 400 µK and (b) from 50 K to 350
K. For comparison, the free-to-bound decay rate GT0 is re-
plotted from Fig. 11 by the dotted lines. The temperature of
the phonon bath is T = 300 K. Other parameters are as in
Fig. 8.
In the case of a thermal gas, the phonon-mediated heat
transfer between the gas and the surface is characterized
by the free-to-free transition rate densitiesQafT0 andQ
[see Eqs. (51)] and the free-to-free decay rates QaT0and
QeT0 [see Eqs. (52)]. We plot the free-to-free transition
rate densities QafT0 and Q
in Fig. 14. Comparison
between Figs. 14(a) and 12(a) shows that the transition
rate densities from low-temperature thermal states and
low-energy free states are quite similar to each other.
The spread of the initial-state energy distribution is not
substantial in this case. However, the energy spread of
the initial state is substantial in the case of high tem-
peratures, concealing the cutoff frequency effect [com-
pare Fig. 14(b) with Fig. 12(b)]. We display the free-
to-free decay rates QaT0 and Q
in Fig. 15. The solid
and dashed lines correspond to the upward (heating) and
downward (cooling) transitions, respectively. For com-
parison, the free-to-bound decay rate (adsorption rate)
GT0 is re-plotted from Fig. 11 by the dotted lines. We
observe that, for T0 in the range from 100 µK to 400 µK
[see Fig. 15(a)], the adsorption rate GT0 (dotted line) is
about two times larger than the heating rate QaT0 (solid
line), while the cooling rate QeT0 (dashed line) is negligi-
ble. Figure 15(a) shows that, in the region of low atomic
temperatures, one has QT0
∼= QaT0 ∝
T0, in agreement
with the asymptotic behavior of expressions (52). The
figure also shows that QeT0 quickly increases with increas-
ing atomic temperature T0. The relation Q
< QaT0 ,
obtained for T0 < T , indicates the dominance of heating
of cold free atoms by the surface. The substantial mag-
nitude of the free-to-bound transition rate GT0 (dotted
line) indicates that a significant number of atoms can be
adsorbed by the surface. According to Fig. 15(b), the
free-to-free downward transition rate QeT0 (dashed line)
crosses the upward transition rate QaT0 (solid line) when
T0 = T = 300 K, and then becomes the dominant decay
rate. The relation QeT0 > Q
, obtained for T0 > T , indi-
cates the dominance of cooling of hot free atoms by the
surface.
V. CONCLUSIONS
In conclusion, we have studied the phonon-mediated
transitions of an atom in a surface-induced potential.
We developed a general formalism, which is applicable
for any surface–atom potential. A systematic derivation
of the corresponding density-matrix equation enables us
to investigate the dynamics of both diagonal and off-
diagonal elements. We included a large number of vi-
brational levels originating from the deep silica–cesium
potential. We calculated the transition and decay rates
from both bound and free levels. We found that the
rates of phonon-mediated transitions between transla-
tional levels depend on the mean phonon number, the
phonon mode density, and the matrix element of the force
from the surface upon the atom. Due to the effects of
these competing factors, the transition rates usually first
increase and then reduce with increasing transition fre-
quency. We focused on the transitions from bound states.
Two specific examples, namely, when the initial level is
a shallow level also when it can be one of the deep levels
have been worked out. We have shown that there can be
marked differences in the absorption and emission behav-
ior in the two cases. For example, both the absorption
and emission rates from the deep bound levels can be sev-
eral orders (in our case, six orders) of magnitude larger
than the corresponding rates from the shallow bound lev-
els. We also analyzed various types of transitions from
free states. We have shown that, for thermal atomic ce-
sium with temperature in the range from 100 µK to 400
µK in the vicinity of a silica surface with temperature of
300 K, the adsorption (free-to-bound decay) rate is about
two times larger than the heating (free-to-free upward de-
cay) rate, while the cooling (free-to-free downward decay)
rate is negligible.
Acknowledgments
We thank M. Chevrollier for fruitful discussions. This
work was carried out under the 21st Century COE pro-
gram on “Coherent Optical Science.”
[∗] Also at Institute of Physics and Electronics, Vietnamese
Academy of Science and Technology, Hanoi, Vietnam.
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http://arxiv.org/abs/quant-ph/0610067
http://arxiv.org/abs/quant-ph/0610136
