In our recent paper [1], we reported observations of photon blockade by one
atom strongly coupled to an optical cavity. In support of these measurements,
here we provide an expanded discussion of the general phenomenology of photon
blockade as well as of the theoretical model and results that were presented in
Ref. [1]. We describe the general condition for photon blockade in terms of the
transmission coefficients for photon number states. For the atom-cavity system
of Ref. [1], we present the model Hamiltonian and examine the relationship of
the eigenvalues to the predicted intensity correlation function. We explore the
effect of different driving mechanisms on the photon statistics. We also
present additional corrections to the model to describe cavity birefringence
and ac-Stark shifts. [1] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T.
E. Northup, and H. J. Kimble, Nature 436, 87 (2005).Comment: 10 pages, 6 figure

A Boca

A Imamoḡlu

A. Boca

A. D. Boozer

ET Jaynes

F Diedrich

H. J. Kimble

HJ Carmichael

HJ Kimble

II Smolyaninov

J Kim

J McKeever

K. M. Birnbaum

KK Likharev

L Mandel

L Tian

MA Kastner

MJ Werner

P Grangier

R. Miller

RJ Brecha

S Rebić

T. E. Northup

TA Fulton

English

arXiv

Crossref

Photon blockade in an optical cavity with one trapped atom

At low temperatures, sufficiently small metallic and semiconductor devices exhibit the 'Coulomb blockade' effect, in which charge transport through the device occurs on an electron-by-electron basis. For example, a single electron on a metallic island can block the flow of another electron if the charging energy of the island greatly exceeds the thermal energy. The analogous effect of 'photon blockade' has been proposed for the transport of light through an optical system; this involves photon–photon interactions in a nonlinear optical cavity. Here we report observations of photon blockade for the light transmitted by an optical cavity containing one trapped atom, in the regime of strong atom–cavity coupling. Excitation of the atom–cavity system by a first photon blocks the transmission of a second photon, thereby converting an incident poissonian stream of photons into a sub-poissonian, anti-bunched stream. This is confirmed by measurements of the photon statistics of the transmitted field. Our observations of photon blockade represent an advance over traditional nonlinear optics and laser physics, into a regime with dynamical processes involving atoms and photons taken one-by-one

Birnbaum, K. M.

Boca, A.

Miller, R.

Boozer, A. D.

Northup, T. E.

Kimble, H. J.

Caltech Authors - Main

Supplemental Information –
Photon Blockade in an Optical Cavity with One Trapped Atom
K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble
Norman Bridge Laboratory of Physics 12-33
California Institute of Technology, Pasadena, CA 91125
In our Letter [1], we employ a model of a two mode cavity coupled to an atom with multiple internal states. In
this Supplement we make explicit the coupling used in the model Hamiltonian to determine the eigenvalues displayed
in Figure 1(b) of Ref. [1]. This Hamiltonian was also incorporated into the master equation for the damped, driven
system used to compute the theoretical results in Fig. 2(b) of Ref. [1]. We furthermore present an extension to this
model which includes the effect of cavity birefringence and FORT induced ac-Stark shifts in the atomic states. The
modified cavity transmission and intensity correlation function are presented for comparison.
We approximate the atom-cavity coupling to be a dipole interaction. We define the atomic dipole transition
operators for the 6S1/2, F = 4 → 6P3/2, F ′ = 5′ transition in atomic Cæsium as
Dq =
4∑
mF =−4
|F = 4,mF 〉〈F = 4,mF |µq|F ′ = 5′,mF + q〉〈F ′ = 5′,mF + q| , (1)
where q = {−1, 0, 1} and µq is the dipole operator for {σ−, π, σ+}-polarization, respectively, normalized such that
for the cycling transition 〈F = 4,mF = 4|µ1|F ′ = 5′,mF = 5〉 = 1. The matrix element of the dipole operator
〈F = 4,mF |µq|F ′ = 5′,m′F 〉 is equivalent to the Clebsch-Gordan coefficient for adding spin 1 to spin 4 to reach total
spin 5, namely 〈j1 = 4, j2 = 1; m1 = mF ,m2 = q|jtotal = 5;mtotal = m′F 〉.
The Hamiltonian of a single atom coupled to a cavity with two degenerate orthogonal linear modes is
H4→5′ = ~ωA
5∑
m′F =−5
|F ′ = 5′,m′F 〉〈F ′ = 5′,m′F |+ ~ωC1(a†a + b†b) (2)
+~g0(a†D0 + D†0a + b
†Dy + D†yb) ,
where Dy = i√2 (D−1 + D+1) is the dipole operator for linear polarization along the y-axis. We are using coordinates
where the cavity supports ŷ and ẑ polarizations and x̂ is along the cavity axis. The annihilation operator for the ẑ
(ŷ) polarized cavity mode is a (b).
Assuming ωA = ωC1 ≡ ω0, we find that the lowest eigenvalues of H4→5′ have a relatively simple structure. In the
manifold of zero excitations all nine eigenvalues are zero. In manifolds with n excitations, the eigenvalues are of the
form En,k = n~ω0 + ~g0ε(n)k , where ε
(n)
k is a numerical factor and k is an index for distinct eigenvalues. There are 29
states in the n = 1 manifold, but due to degeneracy k has only 13 distinct values, k ∈ {−6, . . . 6}; in the n = 2 manifold
there are 49 states but k ∈ {−11, . . . 11}. The number of states in any manifold can be understood by considering how
the excitations can be distributed among the atom and the two cavity modes. For example, in the n = 1 manifold,
the atom can be in one of its 9 ground states (mF ∈ {−4, . . . 4}) and either cavity mode ly or mode lz can have one
photon (giving 18 possible states), or the atom can be in one of its 11 excited states (m′F ∈ {−5, . . . 5}) while both
cavity modes are in the vacuum state, yielding a total of 29 states. Table I lists numerical values for ε(1,2)k as well
as their respective degeneracies η(1,2)k . The numerical factors and degeneracies have the symmetries ε
(n)
−k = −ε(n)k and
η
(n)
−k = η
(n)
k . The resulting eigenvalues En,k for n = {0, 1, 2} are displayed in Fig. 1(b) of Ref. [1].
Although these eigenvalues are certainly not sufficient for understanding the complex dynamics associated with
the full master equation, they do provide some insight into some structural aspects of the atom-cavity system. For
example, the eigenvalues ε(1)±6 = ±1 correspond to the vacuum-Rabi splitting for the states |1,±〉 for a two-state
atom coupled to a single cavity mode [cf., Fig. 1(a) in Ref. [1]]. The one-photon detunings for transitions from the
n = 1 → n′ = 2 manifold are largest for the eigenstates associated with ε(1)±6. Indeed, just as for the two-state atom
with one cavity mode, transitions from the eigenstates at ±g0 have frequency detunings ±(2−
√
2)g0 relative to the
nearest states in the n′ = 2 manifold (at ε(2)±11 = ±
√
2, respectively). Hence, as a function of probe frequency ωp,
the eigenvalue structure in Table I suggests that the ratio of two-photon to one-photon excitation would exhibit a
minimum around ωp = ω0 ± g0, resulting in reduced values g(2)(0) < 1, which the full calculation verifies in Fig. 2(b)
of Ref. [1].
2
k ε
(1)
k η
(1)
k ε
(2)
k η
(2)
k
0 0 7 0 5
1 0.667 1 0.516 1
2 0.683 2 0.556 2
3 0.730 2 0.662 2
4 0.803 2 0.805 2
5 0.894 2 0.966 3
6 1 2 0.978 2
7 1.014 2
8 1.073 2
9 1.155 2
10 1.265 2
11 1.414 2
TABLE I: Numerical factors ε
(n)
k for the eigenvalues of the Hamiltonian H4→5′ in Eq. 2, together with their degeneracies η
(n)
k .
10
-3
10
0
10
3
10
6
g z
z(
2
) (
0
),
g y
z(
2
) (
0
)
60x10
-3
40
20
0
T
zz
,
T
yz
-1.0 0.0 1.0
( p- 0)/g0
 zz
 yz
FIG. 1: Tzz and g
(2)
zz (0) (dashed), and Tyz and g
(2)
yz (0) (red) versus normalized probe detuning (see Ref. [1] for definition of
variables). We consider an F = 4 → F ′ = 5′ transition driven by linearly polarized light in a cavity containing two modes of
orthogonal polarization that are frequency degenerate. Parameters are (g0, κ, γ)/2π = (50, 1, 1) MHz. The probe strength is
such that the intracavity photon number on resonance without an atom is 0.05. The blue dotted line indicates g(2)(0) = 1 for
Poissonian statistics.
For excitation to the other eigenstates in the n = 1 manifold, such blockade is not evidenced in Fig. 2(b) [1]. A
contributing factor suggested by the structure of eigenvalues in Table I is interference of one and two-photon excitation
processes. For example, excitation at ωp ' ω0 ± g0/4 results in two-photon resonance for the eigenstates associated
with ε(2)±1 ' ±0.5, and to photon bunching with g(2)(0) À 1 as confirmed by our full calculation of photon statistics.
Figure 1 provides a global perspective of these various effects. Here, we calculate transmission spectra and intensity
correlation functions analogous to those shown in Fig. 2(b) of Ref. [1], but now with coherent coupling g0 much larger
than the dissipative rates (κ, γ) and well beyond what we have achieved in our experiments, g0/κ = g0/γ = 50. At
ωp = ω0 ± g0, g(2)yz (0) ' 0.002 in evidence of the previously discussed photon blockade suggested by the eigenvalue
structure in Table I. As anticipated, large photon bunching results near ωp ' ω0±g0/4 associated with the two-photon
3
10
-2
10
0
10
2
10
4
g z
z(
2
) (
0
),
g y
z(
2
) (
0
)
0.2
0.1
0.0
T
zz
,
T
yz
-1.0 -0.5 0.0 0.5 1.0
( p- 0)/g0
 zz
 yz
FIG. 2: Tzz and g
(2)
zz (0) (dashed), and Tyz and g
(2)
yz (0) (red) versus normalized probe detuning (see Ref. [1] for definition
of variables). We consider an F = 4 → F ′ = 5′ transition driven by linearly polarized light in a cavity containing two
nondegenerate modes of orthogonal polarization. Parameters are (g0, κ, γ, ∆ωC1 , U0)/2π = (33.9, 4.1, 2.6, 4.4,−43) MHz, and
ωCz1 = ωA ≡ ω0. The probe strength is such that the intracavity photon number on resonance without an atom is 0.05. The
blue dotted line indicates g(2)(0) = 1 for Poissonian statistics.
resonance to reach the eigenstates with ε(2)±1 ' ±0.5. Between these two extremes for the eigenvalues with the largest
and smallest nonzero magnitudes (ω0 ± g0 ≤ ωp ≤ ω0 ± g0/4), g(2)yz (0) displays a complex structure involving multiple
excitation pathways through states in the n = 1 manifold to reach states in the n′ = 2 manifold. The extremely large
peak at ωp = ω0 is discussed in Refs. [2, 3].
We now consider the effects of cavity birefringence and m′F -dependent ac-Stark shifts. These modify the Hamiltonian
H4→5′ in Eq. 2 to
Hfull =
5∑
m′F =−5
~ωm′F |F ′ = 5′,m′F 〉〈F ′ = 5′, m′F |+ ~ωCz1 a†a + ~ωCy1 b†b (3)
+~g0(a†D0 + D†0a + b
†Dy + D†yb)
The birefringent splitting ∆ωC1 is the difference of the resonant frequencies of the two polarization modes, ∆ωC1 =
ωCz1 −ωCy1 . The atomic excited state frequencies are given by ωm′F = ωA+U0βm′F , where ωA is the unshifted frequency
of the F = 4 → F ′ = 5′ transition in free space, U0 is the FORT potential, and βm′F for the FORT wavelength of the
experiment is given by {m′F , βm′F } = {±5, 0.18}, {±4, 0.06}, {±3,−0.03}, {±2,−0.10}, {±1,−0.14}, {0,−0.15} [4].
The effect of these corrections to the Hamiltonian on the transmitted field from the steady-state solutions to the
master equation are displayed in Fig. 2. The heights and shapes of the multiplets in Tyz,zz are modified, but the
basic structure is unaffected relative to Fig. 2(b) of Ref. [1]. The structure of g(2)yz,zz(0) is also qualitatively unchanged.
The value of g(2)yz (0) for ωp = ω0 − g0 is 0.02 (ignoring the above corrections yields g(2)yz (0) ' 0.03). These values are
consistent with the experimental result g(2)yz (0) = 0.13± 0.11 [1].
[1] Birnbaum, K. M., Boca, A., Miller, R., Boozer, A. D., Northup, T. E., and Kimble, H. J. Photon Blockade in an Optical
Cavity with One Trapped Atom (2005).
4
[2] Carmichael, H. J, Brecha, R. J., & Rice, P. R. Quantum interference and collapse of the wavefunction in cavity QED. Opt.
Commun. 82, 73-79 (1991).
[3] Brecha, R. J., Rice, P. R., and Xiao, M. N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon
scattering for weak incident fields. Phys. Rev. A 59, 2392-2417 (1999).
[4] McKeever, J., et al. State-Insensitive Cooling and Trapping of Single Atoms in an Optical Cavity. Phys. Rev. Lett. 90 ,
133602 (2003).


We have observed photon blockade, as evidenced by the photon statistics for light transmitted by an optical cavity containing one trapped atom. The measurements also reveal the energy distribution for atomic motion in the trap

Photon Blockade in an Optical Cavity
with One Trapped Atom
K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup,
and H. J. Kimble
Norman Bridge Laboratory of Physics 12-33,
California Institute of Technology, Pasadena, CA 91125
andreea@caltech.edu
Abstract: We have observed photon blockade, as evidenced by the photon statistics for
light transmitted by an optical cavity containing one trapped atom. The measurements also
reveal the energy distribution for atomic motion in the trap.
c©2006 Optical Society of America
OCIS codes: (020.5580) Quantum Electrodynamics; (270.5290) Photon Statistics
1. Photon blockade in cavity QED
The phenomenon of photon blockade, first proposed in Ref. [1] in analogy with Coulomb blockade for elec-
trons, occurs when the absorption of a first input photon by an optical device blocks the transmission of a
second one, thereby leading to nonclassical output photon statistics. One of the settings for which photon
blockade has been predicted is that of a single atom in cavity quantum electrodynamics [2–4], in which
the blockade is due to the anharmonicity of the Jaynes-Cummings ladder of eigenstates [5]. If an incoming
photon resonantly excites the atom-cavity system from its ground state to |1,±〉 (where |n,+(−)〉 denotes
the n-excitation dressed state with higher (lower) energy), then a second photon at the same frequency will
be detuned from either of the next steps up the ladder, i.e. from states |2,±〉. In the strong coupling regime
[6], for which the coherent rate of evolution g exceeds the dissipative rates κ and γ, this detuning will be
much larger than the excited-state linewidths, so that the two-excitation manifold will rarely be populated.
This in turn leads to the ordered flow of photons in the transmitted field, which emerge from the cavity one
at a time.
2.0
1.5
1.0
0.5
0.0
g y
z(2
)  (τ
)
-1.0 -0.5 0.0 0.5 1.0
τ (µs)
FIG. 1: Second order intensity correlation function of light emerging one photon at a time from the atom-cavity system.
We have recently observed photon blockade in the light transmitted by a high-finesse optical cavity
containing one trapped Cesium atom strongly coupled to the cavity field [7]. The coherent excitation at
the cavity input is near-resonant with one of the two sidebands in the previously-determined vacuum-Rabi
spectrum for our system [8]. The resulting output displays both antibunching and sub-Poissonian statistics,
as shown in Figure 1. The measured second order intensity correlation function demonstrates the manifestly
       a1068_1.pdf        
       JTuD1.pdf
    
1-55752-813-6/06/$25.00 ©2006 IEEE
2nonclassical character of the output field, with g(2)(0) = (0.13± 0.11) < 1 and g(2)(0) < g(2)(τ).
2. Motional energy distribution
The atom is localized within the cavity mode by the anharmonic potential of a red-detuned far-off-resonant
trap. The axial and radial motion-induced modulation on the atom-cavity coupling g can be observed on the
cavity transmission and hence on its correlation function. Figure 2 shows a Fourier transform of the second
order intensity correlation function, with a narrow peak just below the calculated maximum oscillation
frequency in the axial direction, corresponding to the lowest-lying vibrational level. We use the shape of this
peak to estimate that the atoms are distributed among only the lowest ten levels, with maximum energy for
axial motion E/kB ∼ 250 µK.
30x106
25
20
15
10
5
0
 
g(
2) (
f) 
8006004002000
 f(kHz)
~
fmin f0
FIG. 2: Fourier transform of g(2)(τ), with the dotted lines indicating the predicted maximum and minimum frequencies allowed
for harmonic motion along the cavity axis.
[1] A. Imamog¯lu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev.
Lett. 79, 1467-1470 (1997).
[2] S. Rebic´, A. S. Parkins, and S. M. Tan, “Photon statistics of a single-atom intracavity system involving electromagnetically
induced transparency,” Phys. Rev. A 65, 063804 (2002).
[3] L. Tian and H. J. Carmichael, “Quantum trajectory simulations of two-state behavior in an optical cavity containing one
atom,” Phys. Rev. A 46, R6801-R6804 (1992).
[4] R. J. Brecha, P. R. Rice, and M. Xiao, “N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon
scattering for weak incident fields,” Phys. Rev. A 59, 2392-2417 (1999).
[5] E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the
beam maser,” Proc. IEEE 51, 89 (1963).
[6] H. J. Kimble, “Strong Interactions of Single Atoms and Photons in Cavity QED,” Physica Scripta T76, 127 (1998).
[7] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity
with one trapped atom,” Nature 436, 87-90 (2005).
[8] A. Boca, R. Miller, K. M. Birnbaum, A. D. Boozer, J. McKeever, and H. J. Kimble, “Observation of the Vacuum Rabi
Spectrum for One Trapped Atom,” Phys. Rev. Lett. 93, 233603 (2004).
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       JTuD1.pdf
    


Supplemental Information –
Photon Blockade in an Optical Cavity with One Trapped Atom
K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble
Norman Bridge Laboratory of Physics 12-33
California Institute of Technology, Pasadena, CA 91125
In our Letter [1], we employ a model of a two mode cavity coupled to an atom with multiple internal states. In
this Supplement we make explicit the coupling used in the model Hamiltonian to determine the eigenvalues displayed
in Figure 1(b) of Ref. [1]. This Hamiltonian was also incorporated into the master equation for the damped, driven
system used to compute the theoretical results in Fig. 2(b) of Ref. [1]. We furthermore present an extension to this
model which includes the effect of cavity birefringence and FORT induced ac-Stark shifts in the atomic states. The
modified cavity transmission and intensity correlation function are presented for comparison.
We approximate the atom-cavity coupling to be a dipole interaction. We define the atomic dipole transition
operators for the 6S1/2, F = 4→ 6P3/2, F ′ = 5′ transition in atomic Cæsium as
Dq =
4∑
mF=−4
|F = 4,mF 〉〈F = 4,mF |µq|F ′ = 5′,mF + q〉〈F ′ = 5′,mF + q| , (1)
where q = {−1, 0, 1} and µq is the dipole operator for {σ−, pi, σ+}-polarization, respectively, normalized such that
for the cycling transition 〈F = 4,mF = 4|µ1|F ′ = 5′,mF = 5〉 = 1. The matrix element of the dipole operator
〈F = 4,mF |µq|F ′ = 5′,m′F 〉 is equivalent to the Clebsch-Gordan coefficient for adding spin 1 to spin 4 to reach total
spin 5, namely 〈j1 = 4, j2 = 1;m1 = mF ,m2 = q|jtotal = 5;mtotal = m′F 〉.
The Hamiltonian of a single atom coupled to a cavity with two degenerate orthogonal linear modes is
H4→5′ = ~ωA
5∑
m′F=−5
|F ′ = 5′,m′F 〉〈F ′ = 5′,m′F |+ ~ωC1(a†a+ b†b) (2)
+~g0(a†D0 +D†0a+ b
†Dy +D†yb) ,
where Dy = i√2 (D−1 +D+1) is the dipole operator for linear polarization along the y-axis. We are using coordinates
where the cavity supports yˆ and zˆ polarizations and xˆ is along the cavity axis. The annihilation operator for the zˆ
(yˆ) polarized cavity mode is a (b).
Assuming ωA = ωC1 ≡ ω0, we find that the lowest eigenvalues of H4→5′ have a relatively simple structure. In the
manifold of zero excitations all nine eigenvalues are zero. In manifolds with n excitations, the eigenvalues are of the
form En,k = n~ω0 + ~g0ε(n)k , where ε
(n)
k is a numerical factor and k is an index for distinct eigenvalues. There are 29
states in the n = 1 manifold, but due to degeneracy k has only 13 distinct values, k ∈ {−6, . . . 6}; in the n = 2 manifold
there are 49 states but k ∈ {−11, . . . 11}. The number of states in any manifold can be understood by considering how
the excitations can be distributed among the atom and the two cavity modes. For example, in the n = 1 manifold,
the atom can be in one of its 9 ground states (mF ∈ {−4, . . . 4}) and either cavity mode ly or mode lz can have one
photon (giving 18 possible states), or the atom can be in one of its 11 excited states (m′F ∈ {−5, . . . 5}) while both
cavity modes are in the vacuum state, yielding a total of 29 states. Table I lists numerical values for ε(1,2)k as well
as their respective degeneracies η(1,2)k . The numerical factors and degeneracies have the symmetries ε
(n)
−k = −ε(n)k and
η
(n)
−k = η
(n)
k . The resulting eigenvalues En,k for n = {0, 1, 2} are displayed in Fig. 1(b) of Ref. [1].
Although these eigenvalues are certainly not sufficient for understanding the complex dynamics associated with
the full master equation, they do provide some insight into some structural aspects of the atom-cavity system. For
example, the eigenvalues ε(1)±6 = ±1 correspond to the vacuum-Rabi splitting for the states |1,±〉 for a two-state
atom coupled to a single cavity mode [cf., Fig. 1(a) in Ref. [1]]. The one-photon detunings for transitions from the
n = 1 → n′ = 2 manifold are largest for the eigenstates associated with ε(1)±6. Indeed, just as for the two-state atom
with one cavity mode, transitions from the eigenstates at ±g0 have frequency detunings ±(2−
√
2)g0 relative to the
nearest states in the n′ = 2 manifold (at ε(2)±11 = ±
√
2, respectively). Hence, as a function of probe frequency ωp,
the eigenvalue structure in Table I suggests that the ratio of two-photon to one-photon excitation would exhibit a
minimum around ωp = ω0 ± g0, resulting in reduced values g(2)(0) < 1, which the full calculation verifies in Fig. 2(b)
of Ref. [1].
2k ε
(1)
k η
(1)
k ε
(2)
k η
(2)
k
0 0 7 0 5
1 0.667 1 0.516 1
2 0.683 2 0.556 2
3 0.730 2 0.662 2
4 0.803 2 0.805 2
5 0.894 2 0.966 3
6 1 2 0.978 2
7 1.014 2
8 1.073 2
9 1.155 2
10 1.265 2
11 1.414 2
TABLE I: Numerical factors ε
(n)
k for the eigenvalues of the Hamiltonian H4→5′ in Eq. 2, together with their degeneracies η
(n)
k .
10
-3
10
0
10
3
10
6
g z
z(
2
) (
0
),
g y
z(
2
) (
0
)
60x10
-3
40
20
0
T
zz
,
T
yz
-1.0 0.0 1.0
(?p-?0)/g0
 zz
 yz
FIG. 1: Tzz and g
(2)
zz (0) (dashed), and Tyz and g
(2)
yz (0) (red) versus normalized probe detuning (see Ref. [1] for definition of
variables). We consider an F = 4 → F ′ = 5′ transition driven by linearly polarized light in a cavity containing two modes of
orthogonal polarization that are frequency degenerate. Parameters are (g0, κ, γ)/2pi = (50, 1, 1) MHz. The probe strength is
such that the intracavity photon number on resonance without an atom is 0.05. The blue dotted line indicates g(2)(0) = 1 for
Poissonian statistics.
For excitation to the other eigenstates in the n = 1 manifold, such blockade is not evidenced in Fig. 2(b) [1]. A
contributing factor suggested by the structure of eigenvalues in Table I is interference of one and two-photon excitation
processes. For example, excitation at ωp ' ω0 ± g0/4 results in two-photon resonance for the eigenstates associated
with ε(2)±1 ' ±0.5, and to photon bunching with g(2)(0)À 1 as confirmed by our full calculation of photon statistics.
Figure 1 provides a global perspective of these various effects. Here, we calculate transmission spectra and intensity
correlation functions analogous to those shown in Fig. 2(b) of Ref. [1], but now with coherent coupling g0 much larger
than the dissipative rates (κ, γ) and well beyond what we have achieved in our experiments, g0/κ = g0/γ = 50. At
ωp = ω0 ± g0, g(2)yz (0) ' 0.002 in evidence of the previously discussed photon blockade suggested by the eigenvalue
structure in Table I. As anticipated, large photon bunching results near ωp ' ω0±g0/4 associated with the two-photon
310
-2
10
0
10
2
10
4
g z
z(
2
) (
0
),
g y
z(
2
) (
0
)
0.2
0.1
0.0
T
zz
,
T
yz
-1.0 -0.5 0.0 0.5 1.0
(?p-?0)/g0
 zz
 yz
FIG. 2: Tzz and g
(2)
zz (0) (dashed), and Tyz and g
(2)
yz (0) (red) versus normalized probe detuning (see Ref. [1] for definition
of variables). We consider an F = 4 → F ′ = 5′ transition driven by linearly polarized light in a cavity containing two
nondegenerate modes of orthogonal polarization. Parameters are (g0, κ, γ,∆ωC1 , U0)/2pi = (33.9, 4.1, 2.6, 4.4,−43) MHz, and
ωCz1 = ωA ≡ ω0. The probe strength is such that the intracavity photon number on resonance without an atom is 0.05. The
blue dotted line indicates g(2)(0) = 1 for Poissonian statistics.
resonance to reach the eigenstates with ε(2)±1 ' ±0.5. Between these two extremes for the eigenvalues with the largest
and smallest nonzero magnitudes (ω0 ± g0 ≤ ωp ≤ ω0 ± g0/4), g(2)yz (0) displays a complex structure involving multiple
excitation pathways through states in the n = 1 manifold to reach states in the n′ = 2 manifold. The extremely large
peak at ωp = ω0 is discussed in Refs. [2, 3].
We now consider the effects of cavity birefringence andm′F -dependent ac-Stark shifts. These modify the Hamiltonian
H4→5′ in Eq. 2 to
Hfull =
5∑
m′F=−5
~ωm′F |F ′ = 5′,m′F 〉〈F ′ = 5′,m′F |+ ~ωCz1 a†a+ ~ωCy1 b†b (3)
+~g0(a†D0 +D†0a+ b
†Dy +D†yb)
The birefringent splitting ∆ωC1 is the difference of the resonant frequencies of the two polarization modes, ∆ωC1 =
ωCz1 −ωCy1 . The atomic excited state frequencies are given by ωm′F = ωA+U0βm′F , where ωA is the unshifted frequency
of the F = 4→ F ′ = 5′ transition in free space, U0 is the FORT potential, and βm′F for the FORT wavelength of the
experiment is given by {m′F , βm′F } = {±5, 0.18}, {±4, 0.06}, {±3,−0.03}, {±2,−0.10}, {±1,−0.14}, {0,−0.15} [4].
The effect of these corrections to the Hamiltonian on the transmitted field from the steady-state solutions to the
master equation are displayed in Fig. 2. The heights and shapes of the multiplets in Tyz,zz are modified, but the
basic structure is unaffected relative to Fig. 2(b) of Ref. [1]. The structure of g(2)yz,zz(0) is also qualitatively unchanged.
The value of g(2)yz (0) for ωp = ω0 − g0 is 0.02 (ignoring the above corrections yields g(2)yz (0) ' 0.03). These values are
consistent with the experimental result g(2)yz (0) = 0.13± 0.11 [1].
[1] Birnbaum, K. M., Boca, A., Miller, R., Boozer, A. D., Northup, T. E., and Kimble, H. J. Photon Blockade in an Optical
Cavity with One Trapped Atom (2005).
4[2] Carmichael, H. J, Brecha, R. J., & Rice, P. R. Quantum interference and collapse of the wavefunction in cavity QED. Opt.
Commun. 82, 73-79 (1991).
[3] Brecha, R. J., Rice, P. R., and Xiao, M. N two-level atoms in a driven optical cavity: Quantum dynamics of forward photon
scattering for weak incident fields. Phys. Rev. A 59, 2392-2417 (1999).
[4] McKeever, J., et al. State-Insensitive Cooling and Trapping of Single Atoms in an Optical Cavity. Phys. Rev. Lett. 90 ,
133602 (2003).


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