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Photon creation in a resonant cavity with a nonstationary plasma mirror
and its detection with Rydberg atoms
Toru Kawakubo and Katsuji Yamamoto
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
(Dated: November 11, 2010)
We investigate the dynamical Casimir effect and its detection with Rydberg atoms. The photons
are created in a resonant cavity with a plasma mirror of a semiconductor slab which is irradiated
by periodic laser pulses. The canonical Hamiltonian is derived for the creation and annihilation
operators showing the explicit time-variation in the couplings, which originates from the external
configuration such as a nonstationary plasma mirror. The number of created photons is evaluated as
squeezing from the Heisenberg equations with the Hamiltonian. Then, the detection of the photons
as the atomic excitations is examined through the atom-field interaction. Some consideration is
made for a feasible experimental realization with a semiconductor plasma mirror.
PACS numbers: 42.50.Pq,42.50.Lc,42.50.Ct,32.80.Ee
I. INTRODUCTION
The quantum nature of the vacuum provides a vari-
ety of physically interesting phenomena, including the
Casimir effect [1]. The so-called dynamical (nonstation-
ary) Casimir effect (DCE), as well as the static force,
has been investigated extensively [2–23] (and references
therein), where photons are created from the vacuum
fluctuation due to nonadiabatic change of the system
such as vibration of a cavity or expansion of the uni-
verse. It is, however, difficult experimentally to realize
the mechanical vibration of the cavity with a sufficient
magnitude at the resonant frequency ∼ GHz which is
required to create a significant number of photons for
detection. As a feasible alternative, it has been proposed
recently that the oscillating wall can be simulated by a
plasma mirror of a semiconductor slab which is irradiated
by periodic laser pulses [15] (see also Refs. [16, 17]).
In this paper, we investigate quantum mechanically
the photon creation via the DCE and its detection with
Rydberg atoms. We particularly intend to examine the
experimental realization of DCE with a plasma mirror of
a semiconductor slab [15, 23]. In Sec. II, the canonical
Hamiltonian for DCE is derived in terms of the creation
and annihilation operators, where the field operators are
expanded simply with the initial modes. Then, in Sec.
III the time-varying frequencies and squeezing couplings
of the Hamiltonian are calculated in an effective 1+1 di-
mensional scalar field model with a plasma mirror. They
exhibit the enhancement of effective wall oscillation for
the DCE which is simulated by the nonstationary plasma
mirror. In Sec. IV, the number of photons created via
the DCE is evaluated as squeezing from the Heisenberg
equations for the creation and annihilation operators.
The results appear to agree essentially with those ob-
tained by the usual instantaneous-mode approach. In
Sec. V, we investigate the excitation process of Rydberg
atoms through the atom-field interaction, which is uti-
lized to detect the created photons. Some conditions on
the physical parameters are clarified for the efficient pho-
ton detection. In Sec. VI, the experimental realization of
DCE with a semiconductor plasma mirror is discussed.
Section VII is devoted to a summary.
II. CANONICAL HAMILTONIAN
We consider a scalar field in 3+1 space-time dimensions
as an effective description of the electromagnetic field in
a resonant cavity. The Lagrangian is given by
L = 12 ǫ(x, t)(φ˙)
2 − 12(∇φ)
2 − 12m
2(x, t)φ2 (1)
(~ = c = 1) [7, 9, 10, 18, 19]. Here, ǫ(x, t) and m2(x, t)
represent the dielectric permittivity and conductivity (ef-
fective “mass” term), respectively, in the matter region
such as a semiconductor slab. As specified later, they are
space-time dependent, simulating the boundary oscilla-
tion. Conventionally, the instantaneous modes f¯α(x, t)
(real, orthonormal and complete) at each time t with
time-varying frequencies ω¯α(t) are adopted according to
the boundary oscillation:
[−∇2 + m2(x, t)]f¯α(x, t) = ǫ(x, t)ω¯2α(t)f¯α(x, t) (2)
with the orthonormalization
∫
V
ǫ(x, t)f¯α(x, t)f¯β(x, t)d3x = δαβ/[2ω¯α(t)]. (3)
Instead, we here specify the particle representation sim-
ply in terms of the initial modes
f0α(x) = f¯α(x, t = 0), ω0α = ω¯α(t = 0). (4)
The canonical field operators in the Heisenberg picture
are expanded with the creation and annihilation opera-
tors a†α(t) and aα(t) as
φ(x, t) =
∑
α
[aα(t) + a†α(t)]f0α(x), (5)
Π(x, t) = ǫ(x, 0)
∑
α
iω0α[−aα(t) + a†α(t)]f0α(x), (6)

2where Π(x, t) = ∂L/∂φ˙ = ǫ(x, t)φ˙(x, t). Then, the
canonical Hamiltonian is presented by the usual proce-
dure as
HF(t) =
∫
V
1
2
{ Π2
ǫ(x, t) + φ[−∇
2 +m2(x, t)]φ
}
d3x
=
∑
α
ωα(t)
(
a†αaα +
1
2
)
+
∑
α6=β
µαβ(t)a†αaβ
+
∑
α,β
i
[
gαβ(t)a†αa†β − g∗αβ(t)aβaα
]
, (7)
where the space-integral is taken over the whole region
V which is fixed suitably (not time-dependent) accord-
ing to the physical setup, as illustrated later for the case
of a cavity with a nonstationary plasma mirror. [The
usual oscillating boundary may also be described as a
periodic shift of the region of a high potential wall repre-
sented by m2(x, t).] The explicit time-dependence of the
Hamiltonian HF(t) in Eq. (7) represents the variation
of the couplings which originates from the nonstationary
behavior of the c-number external quantities ǫ(x, t) and
m2(x, t). The second-order field equation (Klein-Gordon
equation) is derived from the Heisenberg equations for
φ(x, t) and Π(x, t).
The mode frequencies ωα(t), intermode couplings
µαβ(t) and squeezing terms gαβ(t) are calculated by con-
sidering the orthonormality of f0α(x) which obey the wave
equation with ǫ(x, 0) and m2(x, 0):
ωα(t) = ω0α + µαα(t) ≡ ω0α + δωα(t), (8)
µαβ(t) = 2Gǫαβ(t) + 2Gmαβ(t), (9)
gαβ(t) = −i[−Gǫαβ(t) +Gmαβ(t)], (10)
Gǫαβ(t) =
1
2ω
0
αω0β
∫
δV (t)
ǫ2(x, 0)
ǫ∆(x, t)
f0α(x)f0β(x)d3x, (11)
Gmαβ(t) =
1
2
∫
δV (t)
m2∆(x, t)f0α(x)f0β(x)d3x. (12)
The space-integrals for Gǫ,mαβ (t) are evaluated actually
in the subregion δV (t) (⊆ V ), possibly time-dependent
when a moving boundary is considered, where ǫ(x, t) and
m2(x, t) vary in time as
ǫ−1∆ (x, t) ≡ ǫ−1(x, t) − ǫ−1(x, 0), (13)
m2∆(x, t) ≡ m2(x, t)−m2(x, 0). (14)
Here, Gǫ,mαβ (0) = 0 with ǫ−1∆ (x, 0) = 0 and m2∆(x, 0) = 0
at t = 0, as the Hamiltonian HF(0) is diagonalized in
terms of the initial modes f0α(x).
Similar formulas are presented for the effective Hamil-
tonian with the instantaneous modes [9, 10]. This ef-
fective Hamiltonian involves even the time-derivatives
of the mode functions since the quantum time evolu-
tion is traced along the instantaneous modes. On the
other hand, in the present approach the time evolution
is viewed on the initial modes according to the Heisen-
berg equations. The canonical Hamiltonian is calculated
without the time-derivatives of the mode functions, and
applicable readily for various physical setups, e.g., the
case of a plasma mirror, clarifying its dependence on
the experimental parameters. There may be some claim
concerning the ambiguity on the particle representation
and photon number since the basis modes are changing
during the DCE. This ambiguity is, however, spurious
physically (but might be essential for the case of the ex-
panding universe, which is beyond the present scope).
In fact, the instantaneous modes return to the initial
modes at each period of the oscillation, where the pho-
ton number operators of the respective descriptions co-
incide with each other by definition. We can check ex-
plicitly that when the mode functions are not deformed
largely in time, as usually considered, this canonical
treatment provides essentially the same result for the
DCE as the instantaneous-mode approach. The effects
of the intermode couplings will be less significant in the
instantaneous-mode approach, where the Hamiltonian is
diagonalized at each time. Anyway, the intermode cou-
plings are usually off resonant, providing subleading con-
tributions to the DCE.
III. VIBRATION WITH A PLASMA MIRROR
We next calculate the time-varying frequencies and
squeezing couplings of the Hamiltonian for DCE in an
effective 1+1 dimensional scalar field model with a non-
stationary plasma mirror which is realized with a semi-
conductor slab irradiated by periodic laser pulses [15].
The dielectric response of the plasma is given by
ǫ(ω) = ǫ1[1 − (ω2p/ω2)] with the plasma frequency ωp =
(nee2/ǫ1m∗)1/2 in terms of the effective electron mass m∗
and the conduction electron number density ne propor-
tional to the laser power Wlaser/pulse. This response for
the dispersion relation, ǫ(ω)ω2 = ǫ1ω2 − (nee2/m∗), can
be taken into account in the slab region [l, l + δ] around
x = l with a thickness δ(≪ L) as
ǫ(x, t) = ǫ1(t),m2(x, t) = m2p(t) ≡ ne(t)e2/m∗, (15)
where m2p(0) = 0 for Wlaser(0) = 0. (The spatial distri-
bution of the conduction electrons along the x direction
may also be considered readily.) The instantaneous mode
functions are given as
f¯k(x, t) =



D sin kx [0, l)
Beik′x + Ce−ik′x [l, l + δ] : slab
A sin k[x− δ + ξ(t)] (l + δ, L]
(16)
with the dispersion relations
ω¯2k = (k2 + k2⊥)/ǫ0 = (k′
2 + k2⊥ +m2p)/ǫ1 (17)
(k′ = i|k′| for k′2 < 0 with large m2p), where k⊥ is the
momentum in the orthogonal spatial two dimensions (not
shown explicitly) [12, 13, 21]. The Dirichlet boundary
condition is adopted at x = 0, L with sink[L−δ+ξ(t)] =
0, corresponding to the case of TE modes. The case of

3TM modes can be treated in a similar way by adopting
m2(x, t) = [(∂ne/∂x)e2/(k2⊥m∗)] [23].
The diagonal couplings δωk(t) and gkk(t) are specifi-
cally calculated in Eqs. (8)–(12) with Eq. (16) for f0k (x)
at t = 0 as
δωk(t) = ω0k[δǫ(t) + δm(t)]/L, (18)
gkk(t) = −(i/2)ω0k[−δǫ(t) + δm(t)]/L. (19)
Here, the effective wall oscillation is enhanced as
δǫ(t)/δ ≃ −[ǫ1(0)/ǫ0][1− ǫ1(0)/ǫ1(t)] sin2 kl, (20)
δm(t)/δ ≃ [ne(t)e2/m∗ǫ0(ω0k)2] sin2 kl. (21)
This effect is almost proportional to the square
of the mode function around the slab, [f0k (l)]2 ∝
sin2 kl, since
∫ l+δ
l [f0k (x)]2dx ≃ [f0k (l)]2δ for k′δ ∼
[ǫ1(0)/ǫ0]1/2(δ/L) ≪ 1 at t = 0. If the slab is placed
at the boundary x = l = 0, sin2 kl is replaced with
(kδ)2/3 ∼ (δ/L)2 ≪ 1, as observed in Ref. [19] claiming
that the DCE is suppressed in the TE mode. The signifi-
cant photon creation, however, can take place even in the
TE mode if the slab is placed apart from the boundaries
x = 0, L which are the nodes of f0k (x) [18, 23].
The shift ξ(t) in the instantaneous mode of Eq. (16)
is determined mainly proportional to δ to give the fre-
quency modulation δω¯k(t). The diagonal squeezing cou-
pling g¯kk(t) is then calculated with the formulas for the
effective Hamiltonian [9, 10]. After some calculations we
find the relations
δω¯k(t) ≃ δωk(t), g¯kk(t) ≃ [i/2ω¯k(t)]g˙kk(t), (22)
where the change of dielectric is assumed to be small,
|ǫ1(t) − ǫ1(0)| ≪ ǫ1(0) as usual [19]. These relations in
Eq. (22) ensure almost the same result for the DCE in
the canonical and instantaneous-mode approaches (ex-
cept for the small contribution of the off-resonant inter-
mode couplings). This will be checked numerically in the
next section.
The above calculations of δωk(t) and gkk(t) are valid
up to |δωk(t)|/ω0k = |δǫ(t) + δm(t)|/L ∼ 0.1, which is
still a significant enhancement of the effective displace-
ment |δǫ,m| ≫ δ for the DCE. The present approach on
the fixed basis, however, does not work effectively in an
extreme situation where the mode functions are largely
deformed in time with |δωk(t)| ∼ ω0k. In such a case the
instantaneous-mode approach is rather suitable though
the deformation of the mode functions cannot be treated
perturbatively [23]. Anyway, as seen in the following, a
reasonable deformation to induce |δωk(t)|/ω0k ∼ 0.01−0.1
is sufficient to create a significant number of photons for
detection with atoms.
IV. PHOTON CREATION AS SQUEEZING
Once the Hamiltonian is presented in terms of the
creation and annihilation operators, the time evolution
for the DCE is determined by the Heisenberg equations
a˙α(t) = i[HF(t), aα(t)] and a˙†α(t) = i[HF(t), a†α(t)]. It is
described as the Bogoliubov transformation,
aα(t) = Aαβ(t)aβ +B∗αβ(t)a†β , (23)
a†α(t) = A∗αβ(t)a†β +Bαβ(t)aβ . (24)
The master equations for the Bogoliubov transformation
are derived from the Heisenberg equations as
A˙αβ = −iωα(t)Aαβ − iµαγ(t)Aγβ + 2gαγBγβ, (25)
B˙αβ = iωα(t)Bαβ + iµ∗αγ(t)Bγβ + 2g∗αγAγβ, (26)
where the intermode couplings are renamed suitably as
µαγ(1− δαγ) → µαγ with µαα ≡ 0.
In the following, we illustrate the characteristic fea-
tures of DCE by concentrating on a single resonant
mode with time-varying frequency ω(t) = ω0+ δω(t) and
squeezing coupling g(t) (the mode index “k” omitted).
The intermode couplings will not provide significant con-
tributions since they are fairly off resonant generally for
the nonequidistant frequency differences [11, 13, 21]. The
master equations read
A˙ = −iω(t)A+ 2g(t)B, B˙ = iω(t)B + 2g∗(t)A (27)
for the Bogoliubov transformation,
a(t) = A(t)a +B∗(t)a†, a†(t) = A∗(t)a† +B(t)a.(28)
The solution is expressed as squeezing and phase rotation
[2],
A(t) = cosh r(t)eiφA(t), B(t) = sinh r(t)eiφB (t), (29)
with the initial condition A(0) = 1, B(0) = 0, ensuring
|A(t)|2 − |B(t)|2 = 1.
An analytic solution for A(t) and B(t) is obtained in
the RWA (rotating-wave approximation) by replacing
ω(t) → ω0 + 〈δω〉(average), (30)
g(t) → 〈g〉Ωe−iΩt(Fourier component), (31)
where ω0 = ω(0). By noting the time-evolution of the
number operator a†(t)a(t) = |B(t)|2aa† + . . ., we obtain
the photon creation via DCE (vacuum squeezing) as
nγ(t) = 〈0|a†(t)a(t)|0〉 = |B(t)|2
≃
∣
∣
∣
∣
2〈g〉Ω
χ
∣
∣
∣
∣
2
×



sinh2 χt (|∆| < |2〈g〉Ω|)
|χ|2t2 (|∆| = |2〈g〉Ω|)
sin2 |χ|t (|∆| > |2〈g〉Ω|)
(32)
with the effective squeezing rate
χ =
√
|2〈g〉Ω|2 −∆2. (33)
Here, the detuning ∆ is introduced for the frequency Ω
of laser pulses [12, 13] as
Ω = 2(ω0 + 〈δω〉+ ∆). (34)
The resonance condition for DCE is then given by
Ω(resonance) = 2(ω0 + 〈δω〉), (35)

4involving the average shift of the frequency 〈δω〉 [18, 23],
rather than the naive condition Ω = 2ω0. If Ω = 2ω0
is taken with ∆ = −〈δω〉, the squeezing rate χ is signif-
icantly reduced, even possibly becomes imaginary with
nγ(t) . 1 oscillating as sin2 |χ|t. The photon damping
with the factor e−Γt due to the cavity loss should further
be taken into account, where
Γ = ω0/Q (36)
with the cavity quality factor Q. Hence, the threshold
condition for the squeezing by DCE is placed as
χ > Γ/2, (37)
which is readily satisfied with a large enough Q.
We have solved numerically the master equations in
Eq. (27) without the RWA. The time-varying couplings
are taken typically as ω(t) = ω0 + 〈δω〉(1 − cosΩt) and
g(t) = 2〈g〉Ω(1 − cosΩt), where |2〈g〉Ω| ∼ |〈δω〉|/2 as
indicated in Eqs. (18) and (19) for the plasma mir-
ror. The instantaneous-mode solution has also been ob-
tained by considering the relations δω¯(t) = δω(t) and
g¯(t) = [i/2ω¯(t)]g˙(t) in Eq. (22). In Fig. 1, the pho-
ton creation nγ(t) in the early stage of DCE is plotted
for Npulse = t(Ω/2π) ≤ 30 (the number of periodic laser
pulses). The results of the canonical and instantaneous-
mode approaches are shown with the solid and dotted
curves, respectively. Here, the parameters are taken typ-
ically as 〈δω〉 = 0.02ω0, 2〈g〉Ω = i0.01ω0, and ∆ = 0
(upper curves), −〈δω〉 (lower curves) for Ω in Eq. (34).
We can see that nγ(t) increases rapidly via the DCE on
the resonance with Ω = 2(ω0 + 〈δω〉) (∆ = 0), while
nγ(t) does not grow for Ω = 2ω0 (∆ = −〈δω〉) due to the
effective detuning brought by the average shift 〈δω〉. In
Fig. 2, the photon creation nγ(t) is plotted through the
DCE period for Npulse = t(Ω/2π) ≤ 300. The squeezing
rate is determined from this plot to be χ ≃ 0.01ω0, as
indicated in Eq. (33) with ∆ = 0. This result confirms
that a large number of photons can be created via the
DCE with a reasonable squeezing rate χ ∼ 0.01ω0 when
the laser pulses are applied many times. It is also found
that the canonical and instantaneous-mode approaches
provide almost the same result (except for the small con-
tribution of the off-resonant intermode couplings). The
analytic solution under the RWA in Eq. (32) overlaps
almost with the instantaneous-mode result though it is
not plotted explicitly in Figs. 1 and 2.
We briefly discuss the effect of the intermode couplings.
Specifically, the coupling µ12a†1a2 + µ∗12a†2a1 between the
modes 1 and 2 becomes resonant under a condition ω02 =
3ω01 → ω02 − ω01 = 2ω01 ≈ Ω for the case of the TE111
and TE115 modes in a cubic cavity due to the relation
(12 + 12 + 52)1/2 = 3(12 + 12 + 12)1/2. Then, through
this resonant intermode coupling the significant photon
creation occurs in both the modes 1 and 2 as nγ1(t) ∼
nγ2(t), increasing the total of photon numbers [13, 21,
23]. The photons of the mode 2 are, however, fairly off
resonant with the Rydberg atoms tuned to detect the
photons of the mode 1. Hence, they cannot be detected
efficiently.
0 10 20 30
Npulse=t(Ω/2pi)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
n
γ
canonical
instantaneous
∆=0
∆= –<δω>
FIG. 1. Photon creation nγ(t) (linear plot) in the early stage
of DCE for Npulse = t(Ω/2pi) ≤ 30 (the number of periodic
laser pulses). The results of the canonical and instantaneous-
mode approaches are shown with the solid and dotted curves,
respectively. The parameters are taken typically as 〈δω〉 =
0.02ω0, 2〈g〉Ω = i0.01ω0, and ∆ = 0 (on-resonance: upper
curves), −〈δω〉 (off-resonance: lower curves) for Ω.
0 100 200 300
Npulse=t(Ω/2pi)
10–3
10–2
10–1
100
101
102
103
104
105
106
107
108
n
γ
canonical
instantaneous
∆=0
FIG. 2. Photon creation nγ(t) (log plot) through the DCE pe-
riod for Npulse = t(Ω/2pi) ≤ 300. The results of the canonical
and instantaneous-mode approaches are shown with the solid
and dotted curves, respectively. The parameters are taken
typically as 〈δω〉 = 0.02ω0, 2〈g〉Ω = i0.01ω0, and Ω = 2.04ω0
(∆ = 0).
V. DETECTION WITH RYDBERG ATOMS
The photons created via the DCE are detected suit-
ably by Rydberg atoms with principal quantum number
n ≈ 100 and transition frequency ∼ GHz [11, 23]. Ryd-
berg atoms may be treated as a two-level system with a
transition frequency ωe for the resonant photon absorp-
tion with ωe ≈ ω0. They are initially prepared in the

5lower level |g〉, and injected into the cavity. A part of
these atoms are excited to the upper level |e〉 by absorb-
ing the photons, and detected outside the cavity as the
signal of photons. Recently, a high-sensitivity measure-
ment of blackbody radiation has been performed at a fre-
quency 2.527 GHz and low temperatures 67 mK – 1 K by
employing a Rydberg-atom cavity detector with a newly
developed selective field ionization scheme for n ≈ 100
(the atoms excited by absorbing photons are selectively
ionized by applying an electric field) [24]. Here, we note
that in order to observe purely the vacuum squeezing via
DCE, the cavity should be cooled well below 100 mK
to suppress the thermal photons as nγ(thermal) ≪ 1.
In fact, if photons are present initially with an expecta-
tion value 〈a†a〉, they are also amplified by the DCE as
(1 + 2|B(t)|2)〈a†a〉.
Consider that NRyd Rydberg atoms (actually NRyd ∼
100 − 1000 [24]), which are all prepared at the lower
level |g〉, are injected into the cavity to detect the cre-
ated photons after the period of DCE, for simplicity of
argument. (The following features for the photon de-
tection are essentially valid even if the atomic beam is
injected continuously during and after the DCE, as dis-
cussed later.) The nγ photons and NRyd atoms (all lo-
cated at the same position for simplicity) are coupled
with the Jaynes-Cummings Hamiltonian under the RWA
as
HAF = κ
√
NRyd(aD+ + a†D−). (38)
(The effect of the counter-rotating terms is negligible
near the resonance.) Here, the collective atomic spin-like
operators are defined (in the Schro¨dinger picture) [25] by
D+ ≡
NRyd
∑
i=1
|e〉〈g|(i)/
√
NRyd, (39)
D− ≡
NRyd
∑
i=1
|g〉〈e|(i)/
√
NRyd, (40)
and the complex phase for κ is absorbed in the atomic
levels. The single atom-photon coupling κ is explicitly
given by
κ = d
√
ω0/2ǫ0V (|f0(x1)|/|f0(x0)|) (41)
in terms of the magnitude of the electric dipole transition
matrix element d, the cavity volume V and the mode
function f0(x), where x1 and x0 represent the atomic
position and the antinode, respectively. The collective
atom-photon coupling is suitably defined by
κ¯ = κ
√
NRyd. (42)
The single atom-field coupling is typically κ ∼ 3×103s−1
at the antinode for the Rydberg atom of principal quan-
tum number n ≈ 100 with ωe ≈ ω0 ∼ 1.5 × 1010s−1
(2.4GHz× 2π) and V ∼ (0.1m)3 [24, 25]. Then, the col-
lective coupling amounts to κ¯ ∼ 105s−1 ∼ 10−5ω0 for
NRyd ∼ 103, which is still much smaller than the reso-
nant frequency ωe ≈ ω0.
The commutation relations among the collective oper-
ators are given by
[D+, D−] = Dz
≡
NRyd
∑
i=1
[|e〉〈e|(i) − |g〉〈g|(i)]/NRyd, (43)
[Dz, D±] = ±(2/NRyd)D±. (44)
The operators Nˆe and Nˆg to represent the populations of
the upper and lower levels |e〉 and |g〉, respectively, are
given by
Nˆe =
NRyd
∑
i=1
|e〉〈e|(i) = (NRyd/2)(1 +Dz), (45)
Nˆg =
NRyd
∑
i=1
|g〉〈g|(i) = (NRyd/2)(1−Dz), (46)
satisfying the completeness
Nˆe + Nˆg =
NRyd
∑
i=1
[|e〉〈e|(i) + |g〉〈g|(i)] ≡ NRyd. (47)
The created photons are detected by counting the num-
ber of excited atoms which is represented by Nˆe with
eigenvalues 0, 1, . . . , NRyd. The initial atomic state is
prepared as
|0e〉 = |g1, g2, . . . , gNRyd〉, (48)
which is an eigenstate of Nˆe with zero atomic excitation
satisfying D−|0e〉 = 0. The one-excitation state is gener-
ated as
|1e〉 = D+|0e〉
= 1√
NRyd
NRyd
∑
i=1
|g1, . . . , ei, gi+1, . . . , gNRyd〉, (49)
and so on for the multi-excitation states.
The Heisenberg equations are derived by taking
the total Hamiltonian HA + HAF + HF with HA =
(NRyd/2)ωeDz for the free atomic system:
a˙ = −iω0a− iκ¯D−, (50)
D˙− = −iωeD− + iκ¯aDz, (51)
D˙z = −i(2/NRyd)κ¯(aD+ − a†D−). (52)
We solve these equations perturbatively to see the evolu-
tion of the atomic excitation Ne(t) = 〈Nˆe(t)〉. First, Eqs.
(50) and (51) for a(t) and D−(t) = D†+(t) are integrated
up to the first order of κ¯ with the initial atomic operators
D±(t1) in Eqs. (39) and (40) and the photon operator
a(t1) at t = t1 after the DCE with one sequence of Npulse
laser pulses for the duration
t1 = Npulse(2π/Ω). (53)

6Then, the results are applied to Eq. (52) to obtain Dz(t)
up to the second order of κ¯. This determines the atomic
excitation as
Ne(t) = 〈Nˆe(t)〉 = (NRyd/2)[1 + 〈Dz(t)〉]
≃ nγ(2κ¯/∆e)2 sin2[∆e(t− t1)/2], (54)
where the atomic detuning is given by
∆e = ωe − ω0. (55)
In these calculations, the following relations are con-
sidered: {a, a†}Dz + {D+, D−} = 2(a†aDz + D+D−),
[a, a†]Dz − [D+, D−] = 0, 〈0e|D±(t1)|0e〉 = 0,
〈0e|D+(t1)D−(t1)|0e〉 = 0, 〈0e|Dz(t1)|0e〉 = −1, and
〈0|a†(t1)a(t1)|0〉 = nγ (the photons created via the
DCE). Note here that Ne(t) ≪ NRyd with 〈Dz〉 ≈ −1 in
the early epoch of photon detection (the linear regime).
Although it is difficult in practice to trace exactly the
time evolution beyond the linear regime for the system
of the many atoms interacting with the resonant cavity
mode, we may survey the essential features for the atomic
excitation to detect the photons as follows.
Suppose that nγ ≫ NRyd, namely the photons are cre-
ated much more than the Rydberg atoms, as desired and
feasible experimentally. Then, the atomic excitation is
eventually saturated as Ne(t) ∼ (κ¯t)2nγ ∼ NRyd for
t ∼ 1/(κ√nγ), which is expected by extrapolating Eq.
(54) roughly up to κ¯t ∼
√
NRyd/nγ ≪ 1 near the reso-
nance ∆e ≈ 0 (henceforth t − t1 → t). This excitation
process may be viewed as the onset of Rabi oscillation
between |g〉 and |e〉 at a rate
Ωe ∼ κ
√nγ , (56)
which takes place almost independently for the NRyd
atoms in the presence of the large field (many photons
with nγ ≫ NRyd).
On the other hand, if nγ < NRyd though less inter-
esting experimentally, the excitation is exchanged be-
tween the atoms and field as Ne(t) ∼ nγ/2 on average
for κ¯t ∼ 1. This may be understood from the fact that
the interaction Hamiltonian HAF in Eq. (38) describes
the oscillation with a rate Ωe ∼ κ¯ = κ
√
NRyd between
the atomic and field operators in the linear regime. The
collective atomic excitation can be treated as a quantum
oscillator, satisfying approximately the bosonic commu-
tation relation [D−, D+] ≈ −〈Dz〉 ≈ 1 with nγ ≪ NRyd
in Eq. (43), that is D+ and D− act as the creation and
annihilation operators, respectively [25].
The cavity loss eventually becomes significant for t &
1/Γ. Then, the atomic excitation is also relaxed with a
rate
Γe ∼
{
4(κ¯/Γ)2Γ (κ¯ < Γ/4)
Γ/2 (κ¯ ≥ Γ/4) (57)
through the transition |e〉 → |g〉 + γ and the loss of the
emitted photon in the cavity [25]. We also note that the
atom-field interaction terminates when the atoms transit
through the cavity. The atomic transit time is given by
ttr = L/v ≡ Γ−1tr , (58)
where v and L are the atomic velocity and the cavity
length, respectively. We have typically
Γtr ∼
300m/s
0.1m = 3× 10
3s−1, (59)
which is comparable to the single atom-field coupling κ.
By considering these damping effects, we realize that the
created photons are detected efficiently with the atoms
under the conditions,
Ωe & Γ,Γtr, (60)
Γtr & Γe. (61)
That is, the atomic excitation should take place for t ∼
Ω−1e before the significant loss of the created photons due
to the cavity damping (Γ ≥ 2Γe), and the actual cutoff of
the atom-field interaction by the atomic transit (Γtr). It
is also required that the excitation damping (Γe) induced
by the cavity loss does not become significant before the
atoms transit through the cavity (Γtr).
As investigated so far, if the photons are created copi-
ously via the DCE with nγ ≫ NRyd, they are detected
by the atomic excitation as
Ne(ttr) ∼ NRyd/2. (62)
Here, the condition Ωe & Γtr is less restrictive, requiring
merely nγ & (Γtr/κ)2 ∼ 1 for Γtr ∼ κ. The atomic
detuning may be suppressed readily as ∆e < Ωe, e.g., for
Ωe ∼ 3× 106s−1 with κ ∼ 3× 103s−1 and nγ ∼ 106. The
conditions Ωe & Γ and Γtr & Γe ∼ 2(κ¯/Γ)2Γ (κ¯ < Γ/4)
imply lower and upper bounds, respectively, on the cavity
quality factor,
(ω0/κ)/
√nγ . Q . (ω0/κ)(Γtr/κ)/NRyd, (63)
where ω0/κ ∼ 5 × 106. These bounds are combined as a
requirement for the number of created photons,
nγ & (κ/Γtr)2N2Ryd ≫ NRyd. (64)
For example, we estimate Q ∼ 5 × 103 and nγ ∼ 106
for Γtr ∼ κ and NRyd ∼ 103. This range of Q meets
consistently the condition κ¯ < Γ/4 for Γe.
On the other hand, if Γe = Γ/2 (κ¯ ≥ Γ/4) the condi-
tion Γtr & Γe places a significant bound
Q & ω0/Γtr ∼ 5× 106. (65)
This range of Q meets consistently the condition κ¯ ≥ Γ/4
for Γe. We also note that Ne(ttr) ∼ nγ/2 for nγ < NRyd.
In this case with Ωe ∼ κ¯, the condition Ωe & Γ implies
κ¯ ≥ Γ/4. Hence, the above range of Q in Eq. (65) is
effective either for nγ & NRyd or nγ < NRyd.
The atomic beam may be injected continuously
through the period of DCE. Then, we can show that
the atomic excitation is squeezed together as Ne(t) ∼

7(κ¯/ω0)2nγ(t) during the DCE. This atomic excitation
is usually smaller than NRyd ∼ 100 − 1000, e.g., for
κ¯/ω0 ∼ 10−5 and nγ < 1010. Anyway, the created
photons are detected with the atoms efficiently after the
DCE.
VI. EXPERIMENTAL REALIZATION
We now discuss a feasible experimental realization of
DCE with a semiconductor plasma mirror [15, 23]. Based
on the analyses presented so far for the DCE and pho-
ton detection, we can find desired values for the physical
parameters.
The photons are created as
nγ ∼
1
4e
2χt1 ∼ 14e
2π(χ/ω0)Npulse (66)
with the squeezing rate χ for the resonant mode, where
t1 = Npulse(2π/Ω) and Ω ≃ 2ω0 (see also Fig. 2). Hence,
the desired number nγ of created photons places a re-
quirement for the squeezing rate as
χ/ω0 ∼
ln(4nγ)
2πNpulse
. (67)
Typically, χ ∼ 0.01ω0 to obtain nγ ∼ 106 − 108 with
Npulse = 300 laser pulses, where the threshold condition
χ > Γ/2 for the DCE is also satisfied sufficiently with
Q & 103.
The effective displacement in Eq. (21) is achieved by
applying a laser power Wlaser/pulse for the period T =
2π/Ω ∼ 0.2ns:
δm/L ∼ (nse2/ǫ0m∗)L/π2, (68)
where sin2 kl = 1 for definiteness (the slab is placed in
the middle of cavity l = L/2), ω0L ∼ π, and ns = neδ
(∝ Wlaser) is the surface number density of electrons. We
may readily obtain (nse2/ǫ0m∗)L ∼ 1 with a reasonable
laser power Wlaser/pulse ∼ 0.01µJ/pulse [23], achieving
a significant displacement δm ∼ 0.1L. In this case, the
conductivity effect δm in Eq. (21) dominates over the
dielectric effect δǫ in Eq. (20) for ǫ1(0) ∼ 1 − 10 and
ǫ1(0) ≤ |ǫ1(t)| [the photon damping by the complex ǫ1(t)
does not exceed the squeezing by the DCE mainly with
δm]. We estimate the variation of the mode frequency as
δω ≃ (δm/L)ω0 ∼ 0.1ω0(Wlaser/0.01µJ). (69)
By noting the relation |δω| ≃ |2g|, the desired squeezing
rate for the DCE can be obtained in Eq. (33) with ∆ = 0
as
χ = |2〈g〉Ω| ∼ 0.01ω0(rΩ/0.1)(Wlaser/0.01µJ). (70)
Here, the factor rΩ represents the Fourier component
〈g〉Ωe−iΩt of g(t), which may be optimized by suitably
designing the time-profile Wlaser(t) of laser pulse. As
seen in Eqs. (34) and (35), the tuning of Ω is required
for the resonance by taking into account the average shift
〈δω〉/ω0 ∼ 0.01− 0.1.
As for the photon detection, the analyses in Sec. V
indicate that roughly NRyd/2 ∼ 100 atomic excitations
are detected per mean atomic transit time ttr ∼ 0.1ms
for the creation of nγ ∼ 106 − 108 photons via the DCE.
The quality factor of cavity should be chosen suitably
to ensure the efficient atomic excitation and detection.
Specifically, Q ∼ 5 × 103 in Eq. (63) or Q & 5 × 106
in Eq. (65). We note that even if an excessive amount
of photons (nγ ≫ 108) are created, their detection is
actually limited by the number of Rydberg atomsNRyd ∼
100−1000. After the detection, the photons remaining in
the cavity are relaxed finally as nγ → 0 for t & 10ms ≫
Γ−1, ttr; namely the field returns to the vacuum. Then,
the subsequent rounds of photon creation and detection
are performed repeatedly.
VII. SUMMARY
We have investigated quantum mechanically the pho-
ton creation via DCE and its detection with Rydberg
atoms, specifically considering the experimental realiza-
tion in a resonant cavity with a plasma mirror of a semi-
conductor slab irradiated by laser pulses. The canoni-
cal Hamiltonian for the DCE is derived in terms of the
creation and annihilation operators showing the explicit
time-variation which originates from the external config-
uration such as the nonstationary plasma mirror. Then,
the photon creation is evaluated as squeezing from the
Heisenberg equations. This confirms that a sufficiently
large number of photons can be created via the DCE
with a reasonable squeezing rate when the laser pulses
are applied many times. The atomic excitation process
to detect the photons is described with the atom-field in-
teraction, which clarifies the conditions for the efficient
detection. Based on these analyses, desired values of the
physical parameters are considered for a feasible exper-
iment for DCE and its detection with a plasma mirror
and Rydberg atoms.
ACKNOWLEDGMENTS
The authors appreciate valuable discussions with S.
Matsuki, Y. Kido, T. Nishimura, W. Naylor and the Rit-
sumeikan University group.

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