Measuring acceleration using the Purcell effect
Kacper Ko
˙
zdo
´
n
1
, Ian T. Durham
2
, and Andrzej Dragan
1
1
Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
2
Department of Physics, Saint Anselm College, Manchester, NH 03102, USA
August 14, 2018
We show that a two-level atom resonantly cou-
pled to one of the modes of a cavity field can be
used as a sensitive tool to measure the proper
acceleration of a combined atom-cavity system.
To achieve it we investigate the relation between
the transition probability of a two-level atom
placed within an ideal cavity and study how it
is affected by the acceleration of the whole. We
indicate how to choose the position of the atom
as well as its characteristic frequency in order to
maximize the sensitivity to acceleration.
The spontaneous emission of a photon from an atom
is a property of the atom-vacuum system, rather than of
the atom itself. The irreversibility of such systems arises
from the fact that an infinite number of vacuum states
is typically available to the radiated photon. Modifica-
tions to the vacuum states can thus be used to either
inhibit or enhance the spontaneous emission. One such
modification involves placing an excited atom between
mirrors in an optical cavity. In the weak coupling limit
between a two-level atom and a single-mode cavity, the
resulting change to the transition rate of the atom is
known as the Purcell effect [15, 19].
Additional modifications to the vacuum states oc-
cur as a result of acceleration. The Unruh effect, for
instance, implies that an observer in an accelerated
reference frame should detect the presence of a back-
ground thermal bath of photons where an inertial ob-
server would detect none [8, 13, 21]. Thus, for an ac-
celerated atom, there are no longer an infinite number
of vacuum states available to a radiated photon which
makes the atom theoretically capable of detecting the
non-inertial background bath. Unfortunately, this ef-
fect is very weak and has not yet been detected, despite
continuing efforts.
In this work we investigate another effect of acceler-
ation on the mode structure of the vacuum state that
can significantly modify the atom-field interaction. We
consider an ideal optical cavity in uniformly acceler-
ated motion with an atom placed inside that is co-
accelerating with the cavity. We show how to position
the atom and tune its interaction with the cavity field
Ian T. Durham: idurham@anselm.edu
in order to maximize the sensitivity of the transition
rate of such a system to the acceleration.
Specifically we consider an Unruh-DeWitt detector
which is a simplified model of an atom moving along
an arbitrary classical trajectory [10, 21]. Accelerated
Unruh-DeWitt detectors in stationary cavities have
been analyzed, but the observed effects can be at-
tributed to the relative acceleration that exists between
the detectors and the cavity [1–3, 5, 11, 20]. Solutions
have also been found for scalar fields in cavities with
a time-varying size, though these results only apply to
the scalar field modes of the cavities themselves and
lack any coupling to detectors [6, 7, 16, 17].
It is worth asking, then, what the effect of the ac-
celerated motion of the combined atom-cavity system
is on the dynamics of the quantum state of this sys-
tem. In the weak coupling limit, the atomic transition
is irreversible. In this limit the presence of the cavity
necessarily modifies the vacuum states (via the Purcell
effect), and thus any changes to the dynamics of the
system will manifest as changes to the Purcell effect if
the entire system, consisting of detector and cavity, is
accelerated as one. In the strong coupling limit, the
atomic transition is reversible meaning an emitted pho-
ton can be reabsorbed by the atom before escaping the
cavity. In this article we confine ourselves to discussion
of the weak coupling limit.
We begin by considering the probability for a detec-
tor to transition between an excited state |ei and a
ground state |gi. Such a transition necessarily involves
the emission of a photon whose probability amplitude
is modeled by the Klein-Gordon equation. We consider
both massive and massless 1+1 dimensional real scalar
field models (~ = c = 1):
( + m
2
)
ˆ
φ(x, t) = 0 (1)
with the canonical scalar product
ˆ
ψ
1
,
ˆ
ψ
2
= −i
Z
dx
ˆ
ψ
∗
1
←→
∂
τ
ˆ
ψ
2
. (2)
When considered in the interior of a stationary cavity
of proper-length L, the walls of the cavity introduce
Accepted in Quantum 2018-08-06, click title to verify 1
arXiv:1708.05489v2 [quant-ph] 12 Aug 2018
Dirichlet boundary conditions and the solution becomes
ˆ
φ(t, x) =
∞
X
k=1
F
k
(x)
e
−iω
k
t
ˆa
k
+ e
iω
k
t
ˆa
†
k
, (3)
where the F
k
(x) describe eigenmodes of the cavity with
the corresponding quantized frequencies ω
k
. For a cav-
ity at rest, the spatial part of the solution is given by:
F
k
(x) =
1
√
ω
k
L
sin
kπ
x+
L
2
L
ω
k
=
q
kπ
L
2
+ m
2
.
(4)
Let us now consider a single Unruh-DeWitt detector
governed by the interaction Hamiltonian:
ˆ
H
I
(τ) ∝ (τ)
ˆ
φ(x(τ))
e
−iωτ
ˆ
d + e
iωτ
ˆ
d
†
, (5)
with τ being the proper time along the detector’s tra-
jectory x(τ ) and with (τ ) representing the coupling
strength. The annihilation operator and characteristic
frequency of the detector are
ˆ
d and ω respectively. We
can then position the detector in the center of the cavity
such that it always lies on the cavity’s reference trajec-
tory and is thus always at rest with respect to it. Ini-
tially, the detector is in its excited state |ei and we cal-
culate the amplitude of a transition to its ground state
|gi in the lowest order in . We will assume that the
cavity is initially in the vacuum state of all the modes
in its co-moving reference frame. Due to the interaction
between the detector and the cavity field, the transition
from |ei to |gi has a non-zero probability, which we will
minimize by tuning the detector frequency to match the
frequency of the second eigenmode of the cavity. In this
case, the atom couples most strongly to the second cav-
ity mode which vanishes at the position of the detector
when the cavity is at rest. In such a situation, only
the off-resonant interaction with the remaining modes
contributes to the transition probability and even then
it will become negligible for sufficiently long interaction
times. As a result, such an atom in the middle of the
cavity will remain in its excited state provided the cav-
ity remains at rest.
To show this we compute the first-order probabil-
ity amplitude for the atom to make a transition to the
ground state with the field going from the initial vac-
uum state to some arbitrary final state |ψi:
A
ψ
= −i
∞
Z
−∞
dτ hg|hψ|
ˆ
H
I
(τ) |ei|0i. (6)
The total probability for the detector ending up in the
ground state is given by:
P =
X
ψ
|A
ψ
|
2
. (7)
Using the decomposition (3) one finds that the prob-
ability of the decay is given by the following sum of
integral expressions:
P =
∞
X
k=1
Z
dτ(τ)F
k
(x(τ)) e
−i(ωτ−ω
k
t(τ ))
2
.
(8)
For a resting detector switched on for a time τ this
yields:
P
rest
=
τ
√
ω
2
L
sin
2π
x
0
+
L
2
L
!
2
+
X
k6=2
ω
k
− ω
2
1
√
ω
k
L
sin
kπ
x
0
+
L
2
L
!
e
−i(ω
2
−ω
k
)τ
− 1
2
, (9)
where x
0
= 0 corresponds to the position of the detec-
tor inside the cavity. As we can see, the first (resonant)
term vanishes and, for finite interaction times, the re-
maining off-resonant, oscillating terms fall off quickly as
|ω
k
− ω
2
| increases, making the total transition proba-
bility approximately zero.
Let us now consider the situation in which the detec-
tor and the cavity comove with a uniform proper accel-
eration along the cavity’s length. In this case the cavity
will relativistically contract [4, 9] in which case the cav-
ity’s second eigenmode will not vanish at the position of
the detector thus making the transition to the ground
state possible. As a consequence, a transition from the
excited state to the ground state can serve as an in-
dication that the system is accelerating. As such, the
system consisting of the detector and cavity together
functions as an accelerometer.
In order to find stationary solutions for a uniformly
accelerated cavity, we introduce Rindler coordinates
(τ, χ) for the accelerated frame,
τ =
1
a
atanh
t
x
χ =
√
x
2
− t
2
,
(10)
where a is a parameter corresponding to the proper ac-
celeration of an arbitrarily chosen accelerated reference
trajectory. As mentioned above, we will choose to have
this reference trajectory lie in the very center of the
Accepted in Quantum 2018-08-06, click title to verify 2
cavity, so that when the length L of the cavity is much
shorter than a
−1
, the parameter a can be treated as
the proper-acceleration of the entire cavity. In this case
τ, being the proper time along the reference trajectory,
then becomes the proper time of the entire cavity. The
eigenmodes of the accelerated cavity are then the solu-
tions of
1
a
2
χ
2
∂
2
τ
− ∂
2
χ
−
1
χ
∂
χ
+ m
2
ˆ
φ = 0 (11)
which is the massive Klein-Gordon equation (1) in
Rindler coordinates. General solutions to this equa-
tion are of the form given in (3) where the eigen-
modes can now be expressed in terms of modified Bessel
functions of the first kind with purely imaginary order
I
±iν
(mχ) [11, 12]:
F
Ω
k
(χ) = N
k
I
i
Ω
k
a
(mχ) I
−i
Ω
k
a
(mχ
2
)
−I
−i
Ω
k
a
(mχ) I
i
Ω
k
a
(mχ
2
)
, (12)
where Ω
k
is the corresponding eigenfrequency and N
k
is a normalization constant. We of course choose the
boundary conditions such that the eigenmodes vanish
at the positions of the cavity mirrors, χ
1
=
1
a
−
L
2
,
χ
2
=
1
a
+
L
2
, which necessarily quantizes Ω
k
.
When the cavity and detector are subjected to the
same uniform acceleration, (8) becomes:
P
acc
=
X
k
Ω
k
− ω
2
N
k
I
i
Ω
k
a
m
1
a
I
−i
Ω
k
a
(mχ
2
) − I
−i
Ω
k
a
m
1
a
I
i
Ω
k
a
(mχ
2
)
e
i(Ω
k
−ω
2
)τ
− 1
2
.
(13)
0.0
0.1
0.2
0.3
0.4
0.5
a
P
acc
Hauxiliary unitsL
Figure 1: Decay probability as a function of the proper accel-
eration a of the cavity for arbitrary m and .
We note that, due to the semi-arbitariness of the cou-
pling constant (we are only restricting ourselves to
the weak coupling regime), the normalization constant
is also arbitrary. As such we are working in arbitrary
units. For an equally arbitrary, non-zero mass, we have
plotted (13) as a function of the proper acceleration of
the cavity for the second field mode in Fig. 1. We see
that the probability rises rapidly for small accelerations
before beginning to oscillate. At the same time, the
local maxima decrease with a.
If we were to choose to couple the detector to a higher
field mode, we would have multiple nodes at which we
could choose to place the detector. For the resonant
detector’s frequency ω
n
this corresponds to placing the
detector at positions
1
a
−
L
2
+
kL
n
, k = 1 . . . n − 1 at
which there would be no decay in a resting cavity. We
find that the effect is not uniform across the nodes as
shown in Figs. 2-4. The effect is greatest in the node
in the middle of the cavity, and if there is no node in
the middle, then the node closest to the middle in the
direction of acceleration (see inset, Fig. 2).
The situation is quite different for massless fields. In
that case, the spatial part of the solution to the Klein-
Gordon equation takes the form:
F
Ω
k
, m=0
(χ) =
1
√
kπ
sin (Ω
k
(ξ − ξ
l
))
Ω
k
=
kπ
ξ
2
−ξ
1
ξ =
1
a
log(aχ)
ξ
i
=
1
a
log(aχ
i
), i = 1, 2
(14)
which is just (4) with x → ξ(a) and ω
k
→ Ω
k
. The
decay rate is then
P
acc
(a)
Ω
k
6=ω
2
=
X
k
1
kπ
sin
2
kπ
−log(1 −
aL
2
)
L
0
a
!
×
1
(ω
2
− Ω
k
)
2
e
−i(ω
2
−Ω
k
)τ
− 1
2
L
0
=
1
a
log(
1+
aL
2
1−
aL
2
).
(15)
a plot of which is shown in Fig. 5. Notice that there
is no oscillatory behavior for the massless scalar field
in the considered range, even though we observe such a
behavior in massive case. In addition, the decay prob-
ability increases steadily with increasing a whereas for
Accepted in Quantum 2018-08-06, click title to verify 3
0.0
0.2
0.4
0.6
0.8
a
P
acc
Hauxiliary unitsL
rear mirror
front mirror
• •
direction of acceleration
A
Figure 2: The inset shows the probability amplitude for the
third field mode as a function of location in a simple 1D cavity
at rest. It also shows the direction that this initially resting
cavity is accelerated. The plot shows the probability amplitude
as a function of a for the accelerated cavity. The green line
corresponds to a detector positioned at the green node in the
inset while the red line corresponds to a detector positioned at
the red node in the inset.
0.0
0.1
0.2
0.3
0.4
0.5
a
P
acc
Hauxiliary unitsL
Figure 3: Dependence of the decay probability on acceleration
for the detector coupled to the fourth field mode. The prob-
ability for each node is given from the rearmost node forward
by the red, green, and blue lines respectively.
0.0
0.1
0.2
0.3
0.4
0.5
a
P
acc
Hauxiliary unitsL
Figure 4: Dependence of the decay probability on acceleration
for the detector coupled to the fifth field mode. The probability
for each node is given from the rearmost node forward by the
red, green, blue, and orange lines respectively.
0.00
0.05
0.10
0.15
0.20
a
P
acc
Hauxiliary unitsL
Figure 5: Decay probability as a function of the proper accel-
eration a of the cavity for a massless scalar field.
Accepted in Quantum 2018-08-06, click title to verify 4
the massive field, the local maxima for the decay prob-
ability decrease with increasing a, eventually dying out
entirely.
In our simplified detector model we have ignored the
relativistic effects of acceleration on the atom itself.
Such effects are described in [18]. In other words, in our
model we are only considering changes to the cavity it-
self whereas accelerated atoms will experience changes
to their energy levels independent of the presence of
the cavity. In addition, realistic models would be three-
dimensional. For a single-mode cavity, the transition
rate can be computed from the density of states and
Fermi’s Golden rule and is a function of several factors
that can be affected by relativistic acceleration. Thus
additional work would need to be done to fully under-
stand the relativistic effects on a more realistic model.
That said, experimental realization of such a model
might be difficult. A condensed matter analog does po-
tentially exist, though. Work has been done on frus-
trated spontaneous emission in metamaterial photonic
band gaps. For example, work by Ginzberg, et. al.
has demonstrated an analog Purcell effect in a nonlocal
metamaterial that is based on a plasmonic nanorod as-
sembly [14]. Observing the changes to such a metama-
terial under relativistic acceleration would potentially
serve as an experimental test bed for our model, though
additional work would need to be done to ensure that
the analogy holds.
Acknowledgements
We thank Jorma Louko and Keith Schwab for useful dis-
cussions and Mario Serna for alerting us to the work on
metamaterials. We also thank an anonymous reviewer
for helpful comments. ITD acknowledges FQXi and the
Silicon Valley Community Foundation for funding. AD
acknowledges support from the National Science Cen-
tre, Sonata BIS Grant No. 2012/07/E/ST2/01402.
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