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A standard Hamiltonian formulation for the dynamical Casimir effect
Toru Kawakubo and Katsuji Yamamoto
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
(Dated: August 27, 2008)
We present a quantum description of photon creation via dynamical Casimir effect based on the
standard Hamiltonian formulation. The particle representation is constructed in the expansion of
field operators fixed with the initial modes. The Hamiltonian is presented in terms of the creation and
annihilation operators with the time-varying couplings which originate from the external properties
such as an oscillating boundary or a plasma mirror of a semiconductor slab. Some consideration is
also made for the experimental realization with a semiconductor plasma mirror.
PACS numbers: 42.50.Lc,03.70.+k,42.50.Nn,42.50.Dv
Introduction.–The quantum nature of vacuum provides
a variety of physically interesting phenomena, including
the Casimir effect [1]. The so-called dynamical (non-
stationary) Casimir effect (DCE), as well as the static
force, has been investigated extensively [2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] (also
references therein), where photons are created from the
vacuum fluctuation in non-adiabatic change of the sys-
tem driven by vibration of a cavity or expansion of the
universe. Most of the theoretical approaches are based on
the field expansion in terms of the instantaneous modes.
Since there is no unitary map among the instantaneous
modes with different boundary conditions, the Hamilto-
nian formulation to describe the time-evolution does not
exist in the standard sense [3]. Even in this case, the
time-evolution of the instantaneous-mode operators may
be described by means of an effective Hamiltonian [9, 10].
On the other hand, a suitable transformation may be
made to move to a specific coordinate system with fixed
boundaries for canonical quantization [5, 6, 7, 22].
Experimentally, it is difficult to realize a sufficient mag-
nitude of mechanical vibration at a resonant frequency
∼ GHz to create a significant number of photons for de-
tection. As a feasible alternative, it has been proposed
recently that the oscillating wall can be simulated by a
plasma mirror of a semiconductor slab irradiated by pe-
riodic laser pulses [15]. (See also Refs. [16, 17].)
In this paper, we investigate a quantum description
of DCE, presenting the standard Hamiltonian to gov-
ern the unitary time-evolution. We are particularly con-
cerned with the experimental realization of DCE with a
semiconductor plasma mirror. It is indeed important to
develop the Hamiltonian formulation to investigate the
quantum properties of the system, including the detec-
tion of created photons through interaction with suitable
probe such as atoms. This standard description has sev-
eral advantages: (1) The particle representation is con-
structed in the expansion of field operators fixed with
the initial modes. It is neither necessary to trace the
mode change in time, nor to seek a specific coordinate
system for quantization. Simple formulas are presented
to calculate the couplings among the creation and anni-
hilation operators in the Hamiltonian as space-integrals
involving the initial mode functions. Similar formulas are
obtained for the instantaneous-mode effective Hamilto-
nian where the time-derivatives of the mode functions are
further involved [9, 10]. (2) The time-variation of the cre-
ation and annihilation operators in the standard formula-
tion precisely represents the unitary quantum evolution
all the time. On the other hand, the time-variation of
the instantaneous-mode operators and that of the mode
functions together provide the quantum evolution. The
instantaneous-mode operators and mode functions coin-
cide with those of the standard formulation just at each
period of the oscillation. (3) The present formulation is
applicable to various physical setups, including the oscil-
lating wall and the semiconductor plasma mirror, as in-
vestigated later. We can check that under the situations
where the mode functions do not change largely in time,
as usually considered, this standard description provides
essentially the same result for DCE as the instantaneous-
mode description. Furthermore, the Hamiltonian is read-
ily calculated even for the large time-variation of exter-
nal properties, clarifying the dependence on experimental
parameters specifically for the plasma mirror case.
Standard Hamiltonian formulation.–We consider a
scalar field generally in 3+1 space-time dimensions. The
Lagrangian is given as
L = 1
2
ǫ(x, t)(φ˙)2 − 1
2
(∇φ)2 − 1
2
m2(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) with
∫
V ǫ(x, t)f¯α(x, t)f¯β(x, t)d3x =
δαβ/[2ω¯α(t)]. Instead, we here construct the particle rep-
resentation in terms of the initial modes
f0α(x) = f¯α(x, t = 0), ω0α = ω¯α(t = 0). (2)

2The 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), (3)
Π(x, t) = ǫ(x, 0)
∑
α
iω0α[−aα(t) + a†α(t)]f0α(x), (4)
where Π(x, t) = ∂L/∂φ˙ = ǫ(x, t)φ˙(x, t). Then, the
Hamiltonian (Schro¨dinger picture) is presented by the
usual procedure as
H(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
]
, (5)
where the space-integral is taken over the whole region V
which is “fixed” (not time-dependent) according to the
physical setup, as explicitly shown later for typical cases.
The Heisenberg and Schro¨dinger pictures are related by
the unitary transformation U(t) of time-evolution gener-
ated by this Hamiltonian H(t) as aα(t) = U †(t)aαU(t).
Hence, the time-dependence of the couplings in Eq. (5),
which originates from the c-number external quantities
ǫ(x, t) and m2(x, t), is common in any pictures related by
unitary transformations. The Hamiltonian in the Heisen-
berg picture HH(t) = U †(t)H(t)U(t) is obtained simply
by aα → aα(t), a†α → a†α(t), φ → φ(x, t), Π → Π(x, t) in
H(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), (6)
µαβ(t) = 2Gǫαβ(t) + 2Gmαβ(t), (7)
gαβ(t) = −i[−Gǫαβ(t) +Gmαβ(t)], (8)
Gǫαβ(t) =
1
2
ω0αω0β
∫
δV (t)
ǫ2(x, 0)
ǫ∆(x, t)
f0α(x)f0β(x)d3x, (9)
Gmαβ(t) =
1
2
∫
δV (t)
m2∆(x, t)f0α(x)f0β(x)d3x. (10)
The integrals for Gǫ,mαβ (t) are evaluated in practice in the
subregion δV (t) (⊆ V ), possibly time-dependent, where
ǫ(x, t) and m2(x, t) vary in time as ǫ−1∆ (x, t) ≡ ǫ−1(x, t)−
ǫ−1(x, 0) and m2∆(x, t) ≡ m2(x, t)−m2(x, 0). [G
ǫ,m
αβ (0) =
0 at t = 0 with H(0) diagonalized in terms of f0α(x).] In
order to demonstrate the relevance of this Hamiltonian
formulation for DCE, we investigate two typical cases in
the effective 1+1 dimensions (1) oscillating wall and (2)
plasma mirror of a semiconductor slab.
Oscillating wall.–The boundary walls may be repre-
sented by high potential barriers of matter extending in-
finitely (or finite and long) outside the cavity as
m2(x, t) = m2[−∞ < x < δ(t), L < x < ∞]. (11)
Here, the right side is fixed at x = L, while the left side
varies in time around x = 0 as δ(0) = 0 ≤ δ(t) ≤ δ1 ≪ L.
The dielectric is taken uniformly as ǫ(x, t) = ǫ1 in the
matter region. This setup with potential barriers, rather
than the rigid boundary conditions, may be similar to
Ref. [14] where the matter-field interaction is treated
dynamically. The mode functions are given as
f¯k(x, t) =



Ce|k′|(x−δ(t)) (−∞, δ(t)) : wall
A sin k[x− δ(t) + ξ] [δ(t), L]
Be−|k′|(x−L) (L,+∞) : wall
(12)
with the dispersion relations ω¯2k = (k2 + k2⊥)/ǫ0 = (k′
2 +
k2⊥ +m2)/ǫ1, where k′
2 ≃ −m2 < 0 and m ≃ |k′| ≫ k ∼
1/L for the large m2, and k⊥ (∼ k) is the momentum in
the orthogonal spatial 2 dimensions [12, 13, 21].
For the large potential barrier m2, the frequency mod-
ulation δω¯k(t) and the diagonal squeezing coupling g¯kk(t)
in the instantaneous-mode Hamiltonian [9, 10] are calcu-
lated by noting sin k[L − δ(t) + ξ] ≃ 0 at x = L with
ξ ≃ 1/m≪ L and |B/A|, |C/A| ∼ kξ ≪ 1 as
δω¯k(t) ≃ ω0k[δ(t)/L]rk, g¯kk(t) ≃ δ ˙¯ωk(t)/[4ω¯k(t)], (13)
where rk = k2/ǫ0(ω0k)2. (The dielectric contribution is
suppressed significantly by ǫ1ω¯2k/m2 ≪ 1.) This leading
result is independent of the large m2. By taking formally
the limit m → ∞ (ξ → 0), the usual moving boundary
conditions f¯k(δ(t), t) = f¯k(L, t) = 0 are reproduced.
On the other hand, in the present standard description,
the frequency modulation and the squeezing term are
calculated in Eqs. (6)–(10) with ǫ0(ω0k)2/m2 ≪ 1 as
δωk(t) ≃ 2igkk(t) ≃ m2
∫ δ(t)
0
A2 sin2 k(x+ ξ)dx
≃
{
δω¯k(t) [mδ(t) ≪ 1]
δω¯k(t)[mδ(t)]2/3 [mδ(t) ≫ 1] , (14)
where f0k (x) = A sin k(x + ξ) in 0 ≤ x ≤ L with
δ(0) = 0, ξ ≃ 1/m, A ≃ (Lǫ0ω0k)−1/2 (normalization),
and m2∆(x, t) = m2 in δV (t) = (0, δ(x)). In the limit
m → ∞ the standard δωk(t) ∝ [mδ(t)]2 diverges except
at t = 0 with δ(0) = 0, while the instantaneous-mode
δω¯k(t) remains finite as Eq. (13). This corresponds to the
claim that the Hamiltonian does not exist in the moving
boundary problem [3]. The squeezing couplings g¯kk(t) in
Eq. (13) and gkk(t) in Eq. (14) appear to be different
even with δω¯k(t) ≃ δωk(t). It, however, will be shown
that they provide essentially the same result for the pho-
ton creation at the resonance.

3Plasma mirror.–We next consider the case of plasma
mirror which is realized with a semiconductor slab irra-
diated by periodic laser pulses [15]. The dielectric re-
sponse of 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 elec-
tron number density ne proportional to the laser power
Wlaser. The dispersion relation in plasma k2 = 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 distribu-
tion of the conduction electrons may also be considered
readily.) The 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)
(k′ = i|k′| for k′2 < 0 with large m2p). The Dirichlet
boundary condition (corresponding to the TE mode) is
adopted at x = 0, L with sink[L− δ + ξ(t)] = 0.
The standard δωk(t) and gkk(t) are calculated in Eqs.
(6)–(10) with Eq. (16) for f0k (x) at t = 0 as
δωk(t) = ω0k[δǫ(t) + δm(t)]/L, (17)
gkk(t) = (i/2)ω0k[−δǫ(t) + δm(t)]/L. (18)
Here, the effective wall oscillation is enhanced as
δǫ(t)/δ ≃ −[ǫ1(0)/ǫ0][1− ǫ1(0)/ǫ1(t)] sin2 kl, (19)
δm(t)/δ ≃ [m2p(t)/ǫ0(ω0k)2] sin2 kl. (20)
This effect is almost proportional to the square
of 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 DCE is suppressed in the TE mode. The signifi-
cant photon creation, however, will take place even in
the TE mode if the slab is placed apart from the bound-
aries x = 0, L which are the nodes of f0k (x) [18, 23].
The shift ξ(t) in the instantaneous modes of Eq.
(16) is determined mainly proportional to δ to give the
frequency modulation δω¯k(t). The squeezing coupling
g¯kk(t) is then calculated with the formulas for the ef-
fective Hamiltonian [9, 10]. After some calculations we
find again the relations δω¯k(t) ≃ δωk(t) and g¯kk(t) ≃
[i/2ω¯k(t)]g˙kk(t), as seen in Eqs. (13) and (14) for the os-
cillating wall, where the change of dielectric is assumed
to be small, |ǫ1(t) − ǫ1(0)| ≪ ǫ1(0), as usual [19]. This
ensures the same result on DCE in both the descriptions
for the case of small oscillation, as shown later.
The above calculations of δωk(t) and gkk(t) are valid
even for the large ǫ1(t) andm2p(t) to provide the enhanced
displacement |δǫ,m(t)| ≫ δ, which will be plausible exper-
imentally. It is not necessary here to consider the large
deformation of the mode functions in time which inval-
idates the usual perturbative calculation assuming the
small change of the instantaneous modes.
Photon creation as squeezing.–Once the Hamiltonian
is presented in terms of the creation and annihilation
operators, the quantum properties of the system are in-
vestigated readily by using the methods of quantum op-
tics. We here consider the quantum evolution for DCE,
restricted to a single resonant mode with time-varying
frequency ω(t) = ω0+ δω(t) and squeezing coupling g(t),
omitting the mode index “k”. The intermode couplings
will not provide significant contributions [11, 13, 21],
since generally due to the non-equidistant frequency dif-
ferences they are highly oscillating in the rotating-wave
frame (interaction picture) where the term 〈ω〉a†a is elim-
inated for the average frequency 〈ω〉 = ω0+〈δω〉 over the
period T = 2π/Ω of the laser pulse.
The Heisenberg equation ia˙(t) = [a(t), HH(t)] is de-
scribed as the master equation,
A˙ = −iω(t)A+ 2g(t)B, B˙ = iω(t)B + 2g∗(t)A, (21)
in terms of the Bogoliubov transformation,
a(t) = A(t)a +B∗(t)a†, a†(t) = A∗(t)a† +B(t)a. (22)
The solution is expressed as A(t) = cosh r(t)eiφA(t),
B(t) = sinh r(t)eiφB (t), ensuring |A(t)|2 − |B(t)|2 = 1
with A(0) = 1, B(0) = 0. The unitary time-evolution is
then given as a phase rotation and squeezing,
U(t) = eiK(t)e−[λ∗(t)aa−λ(t)a†a†]/2eiφA(t)a†a (23)
with λ(t) = r(t)ei[φA(t)−φB(t)] [2]. The phase factor eiK(t)
withK(t) = φA(t)+
∫ t
0 ω(t′)/2dt′ is included to reproduce
the zero-point energy of H(t) in iU˙(t) = H(t)U(t).
An analytic solution for A(t) and B(t) is obtained in
the rotating-wave approximation by replacing ω(t) →
ω0 + 〈δω〉 (average), g(t) → 〈g〉Ωe−iΩt (Fourier compo-
nent). By noting the time-evolution of the number oper-
ator 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〉 ≃ (|2〈g〉Ω|/χ)2 sinh2 χt (24)
with the effective squeezing
χ =
√
|2〈g〉Ω|2 −∆2, (25)
allowing for the detuning ∆ of the laser pulse [12, 13] as
Ω = 2(ω0 + 〈δω〉+∆). (26)
The resonance for DCE is given precisely by Ω =
2(ω0 + 〈δω〉) rather than Ω = 2ω0, as considered in the

4instantaneous-mode approach [18]. If Ω = 2ω0 is taken
naively with ∆ = −〈δω〉, the effective squeezing χ is sig-
nificantly reduced, even possibly becomes imaginary with
Nγ(t) . 1 oscillating as sin2 |χ|t.
We have solved numerically the master equation typi-
cally with δω(t) = 〈δω〉(1− cosΩt) and g(t) = −iδω(t)/2
to confirm that the rotating-wave approximation is fairly
good for |δω(t)| ≪ ω0. The instantaneous-mode solu-
tion is also obtained with δω¯(t) = δω(t) and g¯(t) =
[i/2ω¯(t)]g˙(t), as seen so far. It almost reproduces the
rotating-wave approximation, smoothing the actual small
oscillation of Nγ(t) due to that of δω(t). The rela-
tions δω¯(t) = δω(t) and g¯(t) = [i/2ω¯(t)]g˙(t) really im-
ply |2〈g〉Ω| ≃ |2〈g¯〉Ω| (Fourier components) around the
resonance Ω = 2(ω0 + 〈δω〉) for |δω(t)| ≪ ω0, giving es-
sentially the same Nγ(t) in Eqs. (24) and (25).
We now discuss the experimental realization of DCE
with the semiconductor plasma mirror. It will be fea-
sible with the sufficient maximal laser power Wmaxlaser
to achieve the enhanced displacement as δmaxm ≃
[(nmaxe e2/ǫ0m∗)/ω20 ]δ ∼ 102δ or larger with sin2 kl = 1
(the slab placed in the middle of cavity l = L/2). In
this case, the conductivity effect δm in Eq. (20) dom-
inates over the dielectric effect δǫ in Eq. (19) with
ǫ1(0) ∼ 1 − 10 and ǫ1(0) ≤ |ǫ1(t)| [even for the complex
ǫ1(t)]. Then, we estimate roughly χ(∆ = 0) = |2〈g〉Ω| ∼
ω0(δmaxm /L) ∼ 10−2ω0 for δ ∼ 10µm and L ∼ 0.1m. This
requires Npulse & 100 repetitions of laser pulse to cre-
ate Nγ & 10 photons with χ(NpulseT ) & 1. The cavity
Q value is reasonable as Q > ω0/χ ∼ 102. The tuning
of Ω for the resonance should be made with the average
shift 〈δω〉 ∼ χ ∼ 10−2ω0. The time-profile of Wlaser(t)
should also be chosen suitably to optimize the Fourier
component 〈g〉Ωe−iΩt in g(t). A detailed analysis will be
made elsewhere based on the present formulation. The
time-varying dielectric function ǫ1(t) (complex) and con-
ductivitym2p(t) are actually given depending on the laser-
power profile Wlaser(t). By using these ǫ1(t) and m2p(t),
the frequency shift δω(t) and squeezing coupling g(t) are
determined in Eqs. (17) and (18). Then, the master
equation is solved to obtain the photon number Nγ(t).
Detection.–The photons created via DCE can be de-
tected suitably by Rydberg atoms with principal quan-
tum number n ≈ 100 and transition frequency ∼ GHz
[11, 23]. Rydberg atoms as two-level system are initially
prepared in the lower level, and injected into the cavity.
Some of these atoms are excited to the upper level by
absorbing the photons, and detected outside the cavity
as the signal of photons. Recently, high-sensitivity mea-
surement of blackbody radiation has been performed at
a frequency 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]. It
exceeds the standard quantum limit, detecting less than
one photon on average in the cavity. Hence, the single-
photon detection with Rydgerg atoms is really capable of
observing even a small number of DCE photons. When
NRyd atoms are injected in the cavity, the number of pho-
tons detected by atoms is limited roughly as Nγ . NRyd
(acutally NRyd ∼ 100 [24]). We also 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 N thermalγ ≪ 1.
The authors appreciate valuable discussions with S.
Matsuki, Y. Kido, T. Nishimura, W. Naylor and the Rit-
sumeikan University group. This work was supported by
KAKENHI (20340060).
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