SUPPLEMENTARY INFORMATION
doi: 10.1038/nPHYS1837
nature PHYSicS
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1
Supplementary Information -
Strong Interactions of Single Atoms and Photons near a Dielectric Boundary
D. J. Alton,
1
N. P. Stern,
1
Takao Aoki,
2
H. Lee,
3
E. Ostby,
3
K. J. Vahala,
3
and H. J. Kimble
1
1
Norman Bridge Laboratory of Physics, California Institute of Technology 12-33, Pasadena, California 91125, USA
2
Department of Physics, Kyoto University, Kyoto, Japan
3
T. J. Watson Laboratory of Applied Physics, California Institute of Technology 128-95, Pasadena, California 91125, USA
(Dated: October 22, 2010)
Supporting documentation of the experimental meth-
ods and theoretical modeling is provided for Ref. 1.
I. THEORETICAL DESCRIPTION OF
MICROTOROID CAVITY QED
Here we outline a cQED model for an atom coupled
to a cylindrically symmetric resonator as originally pre-
sented in the supplementary material of Refs. 2 and 3 and
shown schematically in Fig. S1. A microtoroidal cavity
supports two degenerate counter-propagating whispering
gallery modes at resonance frequency
ω
c
with annihila-
tion operators
a
and
b
, which are coupled via scattering at
a rate
h
[4]. Each travelling-wave mode has an intrinsic
loss rate,
κ
i
, due to absorption, scattering, and radiation.
A tapered fiber carries input fields
{
a
in
,
b
in
}
at frequency
ω
p
which couple to the cavity modes with an extrinsic
coupling rate
κ
ex
. The output fields of the fiber taper can
be written in terms of the input fields as
{
a
out
,
b
out
}
=
−{
a
in
,
b
in
}
+
√
2
κ
ex
{
a
,
b
}
[2, 3]. For single-sided excita-
tion,
b
in
=
0 and
a
in
drives the
a
mode with strength
ε
p
=
i
√
2
κ
ex
a
in
. The transmitted and reflected photon
fluxes,
P
T
=
a
†
out
a
out
and
P
R
=
b
†
out
b
out
, are calcu-
lated from the input flux
P
in
=
a
†
in
a
in
, with the transmis-
sion and reflection coe
ffi
cients defined as
T
=
P
T
/
P
in
and
R
=
P
R
/
P
in
, respectively.
We consider a two-level atom with transition fre-
quency
ω
a
at location
r
(
ρ,φ,
z
)
(in standard cylindrical
coordinates) coupled to the travelling wave modes
{
a
,
b
}
with single-photon coupling rate
g
tw
(
r
)
=
g
max
tw
f
(
ρ,
z
)e
±
i
θ
,
where
f
(
ρ,
z
) is a function determined by the cavity mode,
θ
=
k
ρφ
, and
k
is the wavevector of the circulating mode.
The atomic frequency
ω
a
may in general be shifted from
the free-space value
ω
(0)
a
by frequency
δ
a
from the vacuum
frequency due to interactions with the dielectric resonator.
An approximate form for the function
f
(
ρ,
z
) for the low-
est order toroid mode in the evanescent region can be writ-
ten as
f
(
ρ,
z
)
∼
e
−
d
/
e
−
(
ψ/ψ
0
)
2
where
d
=
d
(
ρ,
z
) is the
closest distance to the toroid surface,
ψ
(
ρ,
z
) is the angle
around the
ρ
−
z
circular cross-section of the toroid (
ψ
=
0
at
z
=
0),
ψ
0
is a characteristic angle, and
0
≡
λ
0
/
2
π
where
λ
0
is the free-space wavelength.
The Hamiltonian in a frame rotating at
ω
p
is given by
[2–4]:
H
/
=∆
ap
σ
+
σ
−
+∆
cp
a
†
a
+
b
†
b
+
h
a
†
b
+
b
†
a
+
ε
∗
p
a
+
ε
p
a
†
(1)
+
g
∗
tw
a
†
σ
−
+
g
tw
σ
+
a
+
g
tw
b
†
σ
−
+
g
∗
tw
σ
+
b
where
σ
±
are the atomic raising and lowering operators,
∆
ap
=
ω
a
−
ω
p
and
∆
cp
=
ω
c
−
ω
p
. Dissipation is treated
using the master equation for the density operator of the
system
ρ
:
̇
ρ
=
−
i
[
H
,ρ
]
+
κ
2
a
ρ
a
†
−
a
†
a
ρ
−
ρ
a
†
a
+
κ
2
b
ρ
b
†
−
b
†
b
ρ
−
ρ
b
†
b
(2)
+
γ
�
2
σ
−
ρσ
+
−
σ
+
σ
−
ρ
−
ρσ
+
σ
−
Here,
κ
=
κ
i
+
κ
ex
is the total field decay rate of each cavity
mode, and 2
γ
(
r
) is the atomic spontaneous emission rate,
which is orientation dependent near a dielectric surface
(Sec. III B 2). The Hamiltonian (Eq. 1) can be rewritten in
a standing wave basis using normal modes
A
=
(
a
+
b
)
/
√
2
and
B
=
(
a
−
b
)
/
√
2,
H
/
=∆
ap
σ
+
σ
−
+
(
∆
cp
+
h
)
A
†
A
+
(
∆
cp
−
h
)
B
†
B
+
ε
∗
p
A
+
ε
p
A
†
/
√
2
+
ε
∗
p
B
+
ε
p
B
†
/
√
2 (3)
+
g
A
A
†
σ
−
+
σ
+
A
−
ig
B
B
†
σ
−
−
σ
+
B
where
g
A
(
r
)
=
g
max
f
(
ρ,
z
) cos
θ
,
g
B
(
r
)
=
g
max
f
(
ρ,
z
) sin
θ
,
and
g
max
=
√
2
g
max
tw
. Depending on the azimuthal coordi-
nate
θ
, coupling may occur predominantly, or even exclu-
sively, to one of the two normal modes. For such
θ
, the
system can be interpreted as an atom coupled to one nor-
mal mode in a traditional Jaynes-Cummings model with
dressed-state splitting given by the single-photon Rabi
frequency
Ω
(1)
=
2
g
≡
2
g
max
f
(
ρ,
z
), along with a sec-
ond complementary cavity mode not coupled to the atom.
For a fixed phase of
h
set by the scattering in the toroid,
this decomposition is not possible for arbitrary atomic co-
ordinate
θ
; for non-zero
h
the atom in general couples to
both normal modes as
θ
is varied [2].