Casimir forces in the time domain: I. Theory
Alejandro W. Rodriguez,1Alexander P. McCauley,1John D. Joannopoulos,1and Steven G. Johnson2
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
We introduce a method to compute Casimir forces in arbitrary geometries and for arbitrary
materials based on the finite-difference time-domain (FDTD) scheme. The method involves the
time-evolution of electric and magnetic fields in response to a set of current sources, in a modified
medium with frequency-independent conductivity. The advantage of this approach is that it allows
one to exploit existing FDTD software, without modification, to compute Casimir forces. In this
manuscript, part I, we focus on the derivation, implementation choices, and essential properties of
the time-domain algorithm, both considered analytically and illustrated in the simplest parallel-plate
geometry. Part II presents results for more complex two- and three-dimensional geometries.
In recent years, Casimir forces arising from quantum
vacuum fluctuations of the electromagnetic field [1, 2, 3]
have become the focus of intense theoretical and ex-
perimental effort [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21]. This effect has been veri-
fied via many experiments [22, 23, 24, 25], most com-
monly in simple, one-dimensional geometries involving
parallel plates or approximations thereof, with some ex-
ceptions . A particular topic of interest is the ge-
ometry and material dependence of the force, a sub-
ject that has only recently begun to be addressed in ex-
periments  and by promising new theoretical meth-
ods [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].
For example, recent works have shown that it is possi-
ble to find unusual effects arising from many-body in-
teractions or from systems exhibiting strongly coupled
material and geometric dispersion [39, 40, 41, 42, 43].
These numerical studies have been mainly focused in two-
dimensional [13, 44, 45, 46] or simple three-dimensional
constant-cross-section geometries [33, 40, 47] for which
numerical calculations are tractable.
In this manuscript, we present a simple and general
method to compute Casimir forces in arbitrary geome-
tries and for arbitrary materials that is based on a
finite-difference time-domain (FDTD) scheme in which
Maxwell’s equations are evolved in time . A time-
domain approach offers a number of advantages over
previous methods. First, and foremost, it enables re-
searchers to exploit powerful free and commercial FDTD
software with no modification. The generality of many
available FDTD solvers provides yet another means to
explore the material and geometry dependence of the
force, including calculations involving anisotropic di-
electrics  and/or three-dimensional problems. Sec-
ond, this formulation also offers a fundamentally different
viewpoint on Casimir phenomena, and thus new oppor-
tunities for the theoretical and numerical understanding
of the force in complex geometries.
Our time-domain method is based on a standard for-
mulation in which the Casimir force is expressed as
a contour integral of the frequency-domain stress ten-
sor . Like most other methods for Casimir calculations,
the stress tensor method typically involves evaluation at
imaginary frequencies, which we show to be unsuitable
for FDTD. We overcome this difficulty by exploiting a
recently-developed exact equivalence between the system
for which we wish to compute the Casimir force and a
transformed problem in which all material properties are
modified to include dissipation . To illustrate this
approach, we consider a simple choice of contour, corre-
sponding to a conductive medium, that leads to a simple
and efficient time-domain implementation. Finally, using
a free, widely-available FDTD code , we compute the
force between two vacuum-separated perfectly-metallic
plates, a geometry that is amenable to analytical calcu-
lations and which we use to analyze various important
features of our method. An illustration of the power and
flexibility of this method will be provided in a subse-
quent article , currently in preparation, in which we
will demonstrate computations of the force in a number
of non-trivial (dispersive, three-dimensional) geometries
as well as further refinements to the method.
In what follows, we derive a numerical method to
compute the Casimir force on a body using the FDTD
method. The basic steps involved in computing the force
(1) Map the problem exactly onto a new problem with
dissipation given by a frequency-independent con-
(2) Measure the electric E and magnetic H fields in
response to current pulses placed separately at each
point along a surface enclosing the body of interest.
(3) Integrate these fields in space over the enclosing
surface and then integrate this result, multiplied by
a known function g(−t), over time t, via Eq. (29).
The result of this process is the exact Casimir force
(in the limit of sufficient computational resolution), ex-
pressed via Eq. (29) and requiring only the time-evolution
of Eqs. (15–16).
arXiv:0904.0267v2 [quant-ph] 21 Apr 2009
In this section, we describe the mathematical develop-
ment of our time-domain computational method, start-
ing from a standard formulation in which the Casimir
force is expressed as a contour integral of the frequency-
domain stress tensor.We consider the frequency do-
main for derivation purposes only, since the final tech-
nique outlined above resides entirely in the time domain.
In this framework, computing the Casimir force involves
the repeated evaluation of the photon Green’s function
Gij over a surface S surrounding the object of interest.
Our goal is then to compute Gijvia the FDTD method.
The straightforward way to achieve this involves com-
puting the Fourier transform of the electric field in re-
sponse to a short pulse. However, in most methods a
crucial step for evaluating the resulting frequency inte-
gral is the passage to imaginary frequencies, correspond-
ing to imaginary time. We show that, in the FDTD
this, gives rise to exponentially growing solutions and is
therefore unsuitable. Instead, we describe an alternative
formulation of the problem that exploits a recently pro-
posed equivalence in which contour deformations in the
complex frequency-domain ω(ξ) correspond to introduc-
ing an effective dispersive, dissipative medium at a real
“frequency” ξ. From this perspective, it becomes simple
to modify the FDTD Maxwell’s equations for the pur-
pose of obtaining well-behaved stress tensor frequency
integrands. We illustrate our approach by considering
a contour corresponding to a medium with frequency-
independent conductivity σ. This contour has the ad-
vantage of being easily implemented in the FDTD, and
in fact is already incorporated in most FDTD solvers.
Finally, we show that it is possible to abandon the fre-
quency domain entirely in favor of evaluating the force
integral directly in the time domain, which offers several
conceptual and numerical advantages.
A. Stress Tensor Formulation
The Casimir force on a body can be expressed  as an
integral over any closed surface S (enclosing the body) of
the mean electromagnetic stress tensor ?Tij(r,ω)?. Here
r denotes spatial position and ω frequency. In particular,
the force in the ith direction is given by:
The stress tensor is expressed in terms of correlation
functions of the the field operators ?Ei(r,ω)Ej(r?,ω)?
where both the electric and magnetic field correlation
functions can be written as derivatives of a vector poten-
tial operator AE(r,ω):
Ei(r,ω) = −iωAE
µHi(r,ω) = (∇×)ijAE
We explicitly place a superscript on the vector poten-
tial in order to refer to our choice of gauge [Eqs. (3–
4)], in which E is obtained as a time-derivative of A.
The fluctuation-dissipation theorem relates the corre-
lation function of AEto the photon Green’s function
j(r?,ω)? = −?
electric dipole current J along the ˆ ejdirection:
Eq. (5) to express the field correlation functions at points
r and r?in terms of the photon Green’s function:
ijis the vector potential AE
i in response to an
µ(r,ω)∇ × − ω2ε(r,ω)
j(ω;r,r?) = δ(r−r?)ˆ ej,
ij, one can use Eqs. (3–4) in conjunction with
?Hi(r,ω)Hj(r?,ω)? = −?
In order to find the force via Eq. (1), we must first
integration S, and for every ω .
be solved numerically in a number of ways, such as by
a finite-difference discretization : this involves dis-
cretizing space and solving the resulting matrix eigen-
value equation using standard numerical linear algebra
techniques [53, 54]. We note that finite spatial discretiza-
tion automatically regularizes the singularity in GE
r = r?, making GE
ij(r,r?= r,ω) at every r on the surface of
Equation (6) can
ijfinite everywhere .
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