We demonstrate how spontaneous emission in a cavity can be controlled by the
application of a dc field. The method is specially suitable for Rydberg atoms.
We present a simple argument for the control of emission.Comment: 3-pages, 2figure. accepted in Phys. Rev. 

E. M. Purcell

G. S. Agarwal

P. K. Pathak

T. W. Mossberg

English

arXiv

We demonstrate how spontaneous emission in a cavity can be controlled by the application of a dc field. The method is especially suitable for Rydberg atoms. We present a simple argument based on Stark shifts for the control of emission

Agarwal, G. S.

Pathak, P. K.

Indian Academy of Sciences

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DC field induced enhancement and inhibition of spontaneous emission in a cavity
G. S. Agarwal and P. K. Pathak
Physical Research Laboratory, Navrangpura, Ahmedabad-380009, India
(Dated: February 1, 2008)
We demonstrate how spontaneous emission in a cavity can be controlled by the application of a
dc field. The method is specially suitable for Rydberg atoms. We present a simple argument based
on Stark shifts for the control of emission.
PACS numbers: 42.50.Lc, 42.50.Ct, 42.50.Hz
The manipulation of spontaneous emission has been
extensively studied. Methods involving either external
fields [1, 2, 3, 4, 5, 6, 7, 8] or appropriate cavities [9, 10]
have been suggested. Use of external fields enables one to
control spontaneous emission via quantum interference
effects [5, 6]. Purcell [11] recognized how the emission
rate in a single mode cavity can be much higher than in
free space. Several demonstrations of the cavity enhanced
spontaneous rates exists in the literature [9, 10]. Klepp-
ner [12] discovered that the radiation rate in a cavity can
be inhibited by choosing the transition frequency such
that the density of states at this frequency is insignif-
icant. Quantum interference between various channels
[3, 4, 6, 7] could also result in the inhibition of emission.
Further very interesting experiments [1, 2, 6, 7, 8] on
the field induced inhibition of emission in a cavity were
reported. In these experiments the applied fields were
resonant with atomic transitions. In this paper we show
how a possible control of spontaneous emission can be
obtained by using dc fields. We treat the case of atoms
in a cavity and explain in rather simple terms the origin
of the control produced by the dc fields. To be precise
we are considering only the effect of dc field on the part
of decay which is due to the emission in the cavity mode.
The decay of the atom depends on the detuning between
the atomic frequency and the cavity frequency. The ap-
plication of the dc field makes the detuning dependent
on the field (Stark effect) and thus the dc field provides
a control of the spontaneous emission. The dc field in-
duced modification of spontaneous emission in free space
is treated in reference [3].
We next describe how to calculate the dc field induced
modification of the decay characteristics in a cavity. For
our purpose we consider a two level atom placed in a
cavity and a dc field (or low frequency field) is injected
inside the cavity . The Hamiltonian of the system can be
written as,
H = ~ω0S
z + ~ωca
†a+ ~g
(
aS+ + S−a†
)
+~E cosΩt
(
S+ + S−
)
, (1)
where ω0 is atomic transition frequency, ωc is cavity mode
frequency and g is the atom cavity coupling constant.
The term E cosΩt corresponds to a low frequency field if
Ω is chosen to be very small. Note that E has dimensions
of frequency. The cavity field has been expressed in terms
of annihilation and creation operators a, a† and S+, S−,
Sz are usual atomic spin operators. We perform master
equation calculation for atom-cavity system. The density
matrix of the system ρ will evolve as,
ρ̇ = −
i
~
[H, ρ] − κ
(
a†aρ− 2aρa† + ρa†a
)
, (2)
where 2κ gives the leakage of photons. It is related to
the cavity Q via κ = ωc/2Q. We will work in a frame
rotating with atomic frequency ω0 The density matrix in
this frame is given by
ρ̃ = eiω0(S
z+a†a)t/~ρe−iω0(S
z+a†a)t/~. (3)
Using Eq(2) and (3) we obtain the equation for ρ̃
˙̃ρ = −
i
~
[Ha, ρ̃] − κ
(
a†aρ̃− 2aρ̃a† + ρ̃a†a
)
−
i
~
[Hd, ρ̃], (4)
where
Ha = −~∆a
†a+ ~g
(
aS+ + S−a†
)
,
Hd = ~
E
2
{
S+
(
ei(ω0+Ω)t + ei(ω0−Ω)t
)
+S−
(
e−i(ω0+Ω)t + e−i(ω0−Ω)t
)}
, (5)
and ∆ = ω0 − ωc is the detuning. We first note that the
experiments of Lange and Walther correspond to using a
microwave field and thus Ω ∼ ω0. The results of Purcell
and Kleppner also follow from the master equation (4).
For E = 0 and g << κ, we can derive an equation for the
atomic density matrix ρ̃a
ρ̃a = Trcρ̃, (6)
where Trc is trace over cavity field, by adiabatically elim-
inating cavity variables. This leads to
˙̃ρa = −i[δ0S
z, ρ̃a] − Γ0
(
S+S−ρ̃a − 2S
−ρ̃aS
+ + ρ̃aS
+S−
)
,(7)
where
Γ0 =
g2κ
κ2 + ∆2
, δ0 =
g2∆
κ2 + ∆2
. (8)
For resonant cavity ωc = ω0, δ0 = 0 and the decay rate
Γ0 = g
2/κ. There is cavity induced enhancement if g2/κ
2
is greater than the free space decay rate. Note that as the
cavity is detuned (∆ 6= 0) Γ0 decreases which is Klepp-
ner’s result for a single mode cavity. The first experimen-
tal observation of the Purcell effect was made by Goy et
al [9]. Next we investigate the effect of the applied dc or
low frequency field. Note that the last term in the master
equation (4) is highly oscillating. We do time averaging
for this as such terms oscillating at the cavity frequency
would not be normally observed. The time averaging is
well justified here as all other relevant time scales g−1,
κ−1, ∆−1 are much larger than (ω0±Ω)
−1. The inequal-
ity ω0 >> g, κ, ∆ enables us to do the time averaging in
a much simpler fashion i.e. we can essentially ignore the
terms having Ha and κ in (4). We relegate the details of
time averaging to the appendix. The calculation leads to
the following time averaged master equation
˙̃ρ = i
[
∆ea
†a, ρ̃
]
− ig
[(
aS+ + S−a†
)
, ρ̃
]
−κ
(
a†aρ̃− 2aρ̃a† + ρ̃a†a
)
, (9)
where
∆e = ∆ + 2ω0E
2/(ω20 − Ω
2). (10)
We note that the dc field contributes to the Stark shift
of the two levels in question. We further note that these
two atomic levels can also be shifted because of the in-
teraction of the dc field with other levels. These can be
accounted for by introducing the polarizabilities αe and
αg of the levels |e〉 and |g〉 [13, 14], we can rewrite Eq(10)
as
∆e = ∆ + α0E
2
d ; α0 = αe − αg ; (11)
where Ed is now the dc field in esu. The formulation
of the appendix can also be used to produce the well
known expressions for the α,s. The value of α0 is known
for many low lying as well as Rydberg transitions. The
values of α0 have been calculated in the literature by
converting infinite sums into the solution of differential
equations.
The Eq(9) can be solved assuming that the atom is
initially excited and the cavity field is in vacuum state.
The Eq(9) can be converted into a set of coupled equa-
tions in terms of the states |e, 0〉 , |g, 1〉 and |g, 0〉 . The
results of the numerical integration are shown in the Fig1
for different values of the parameter ∆e. Clearly there
is inhibition as ∆e increases. The effective detuning ∆e
changes due to the applied dc field . For a fixed cavity
detuning ∆ the dc field can make ∆e larger or smaller de-
pending on the sign of ∆. The results can be understood
by deriving analytical results in the bad cavity limit [15]
g << κ (and more precisely g2 << κ2 + ∆2e). In this
limit we can obtain a simpler equation for the atomic
density matrix ρ̃a defined by Eq(6). The final result for
the atomic system is
˙̃ρa = −i[δeS
z, ρ̃a] − Γe
(
S+S−ρ̃a − 2S
−ρ̃aS
+ + ρ̃aS
+S−
)
,(12)
where
Γe =
g2κ
κ2 + ∆2e
, δe =
g2∆e
κ2 + ∆2e
. (13)
Here Γe is the dc field modified decay parameter and δe
is the net frequency shift. The ratio η of the decays in
the presence and absence of dc field is given by
η =
Γe
Γ0
=
κ2 + ∆2
κ2 + ∆2e
. (14)
Clearly the dc field modifies the decay rate which depends
on the detuning. For the cavity resonant to the atomic
transition (∆ = 0), using Eq(10), η reduces to
η =
κ2
κ2 + α20E
4
d
≈
(
1 +
4E4
κ2ω20
)−1
, for Ω = 0. (15)
It is clear from the Eq(15) that dc field inhibits the decay
rate. Note that the inhibition starts becoming significant
for
α0E
2
d ∼ κ. (16)
Let us estimate the condition (16) forNa Rydberg transi-
tion 23S1/2 → 22P3/2 whose frequency is 340GHz . For
the sake of argument we also assume α0E
2
d ∼ 2E
2/ω0.
This transition has a dipole moment d ∼ 10−15esu. The
atom is placed in the cavity having one mode resonant
to the atomic transition. Let us choose the cavity decay
rate κ = 1MHz. The condition (16) then leads to a Rabi
frequency E of the order 400MHz which in turn requires
a dc field of the order of 10−2esu. We note that the re-
quired dc field is small enough so that the perturbative
results for the Stark shift hold. We further note that the
scalar and tensor polarizabilities are available for some
S and P levels of Na [13, 14] though the absolute val-
ues for both 23S1/2 and 22P3/2 level are not available in
Fabre et al [13]. However the reported polarizabilities for
say 23P level are of the order of few MHz/(V olt/cm)2.
0 5 10 15 20
gt/π
0
0.2
0.4
0.6
0.8
1
ρ e
e(
t)
α0εd
2
=0
α0εd
2
=5 g
α0εd
2
=10 g
α0εd
2
=15 g
FIG. 1: The probability of the atom remaining in its excited
state, ρee ≡ 〈e, 0|ρ|e, 0〉 vs time, for κ = 5g, ∆ = 0, Ω = 0,
and for the different values of the dc field Ed.
3
Thus the condition (16) is realistic and our finding that
the dc field can be used to control spontaneous emission,
can be implemented by the appropriate choice of the Ry-
dberg transitions, [cf the condition (16)]. We emphasize
that we are discussing the inhibition or enhancement of
spontaneous emission on a given transition which is reso-
nant with the cavity. This, for example, is the transition
23S → 22P in the experiments of Goy et. al [9]. The
authors of ref. [9] emphasize this as well and it is in the
spirit of the original suggestion of Purcell [11]. It must
be noted that the field ionization techniques enable one
to study transitions selectively [16].
In the case of cavities detuned from the atomic tran-
sition, spontaneous decay is smaller and the decay rate
is given by Γ = g2κ/
(
κ2 + ∆2
)
. Further inhibition of
decay rate is possible by applying dc field. When cavity
is tuned below the atomic transition frequency (∆ is pos-
itive) then there is significant inhibition of spontaneous
decay, which increases further as the applied dc field is in-
creased. On the other hand when cavity is tuned above
the atomic frequency (∆ is negative) there is enhance-
ment in the atomic decay i.e. on increasing the value
of applied dc field the atom decays faster. In Fig2 we
−15 −5 5 15
∆/κ
0
5
10
η=
Γ e
/Γ
0
ε=0
ε=300 κ
ε=700κ
FIG. 2: The ratio (η) of the decays in the presence and the
absence of dc field vs ∆/κ . The parameters are ω0 = 3.4 ×
105κ and Ω = 0.
show the behavior of the factor η as a function of ∆ for
different values of the dc field. The enhancement as well
as inhibition of spontaneous decay occurs depending on
whether the cavity is tuned above or below the atomic
frequency. The results shown in the Fig2 are consistent
with the results obtained by direct solution of the Eq(9).
In conclusion we find that in presence of dc field spon-
taneous emission can be inhibited significantly in the
case of cavities resonant to atomic transition. In the
case of cavities having negligible mode density around
atomic frequency spontaneous emission itself is smaller
and the presence of dc field shows significant inhibition
or enhancement depending on cavity is tuned below the
atomic transition frequency or above the transition fre-
quency.
One of us (GSA) thanks G. Rempe and H. Walther for
discussions on this subject.
APPENDIX A
We outline how the time averaging is to be done. Let
us consider schrodinger equation
∂
∂t
|ψ(t)〉 = −
i
~
V (t)|ψ(t)〉, (A1)
where V (t) consists of rapidly oscillating terms only so
that the time average of V (t) is zero. Let |ψ〉 be written
as
|ψ〉 = |ψ̄〉 + |φ〉, (A2)
where |ψ̄〉 is time averaged part and |φ〉 is the rapidly
oscillating part. On substituting (A2) in Eq (A1) we
find that to the lowest order in V (t)
|φ〉 = −
i
~
∫ t
0
V (τ)dτ |ψ̄〉, (A3)
and
∂
∂t
|ψ̄(t)〉 = −
i
~
V̄ (t)|ψ̄〉, (A4)
where
V̄ (t) = −
i
~
V (t)
∫ t
0
V (τ)dτ . (A5)
The field induced shift term in (9) is obtained by using
Eq(A5).
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