Bistable optical transmission through arrays of atoms in free space
C. D. Parmee and J. Ruostekoski
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: February 18, 2021)
We determine the transmission of light through a planar atomic array beyond the limit of low light
intensity that displays optical bistability in the mean-field regime. We develop a theory describing
the intrinsic optical bistability, which is supported purely by resonant dipole-dipole interactions in
free space, showing how bistable light amplitudes exhibit both strong cooperative and weak single-
atom responses and how they depend on the underlying low light intensity collective excitation
eigenmodes. Similarities of the theory with optical bistability in cavities are highlighted, while
recurrent light scattering between atoms takes on the role of cavity mirrors. Our numerics and
analytic estimates show a sharp variation in the extinction, reflectivity, and group delays of the
array, with the incident light completely extinguished up to a critical intensity well beyond the low
light intensity limit. Our analysis paves a way for collective nonlinear optics with cooperatively
responding dense atomic ensembles.
I. INTRODUCTION
Optical bistability is a intriguing nonlinear phenomena
for atoms, where two possible states of a system coexist
for the same parameters. It has been studied theoreti-
cally for a long time in the regime where atoms couple
to a single cavity mode and atomic positions and spa-
tial dependence of interactions do not play a role [1–9].
It has also been observed experimentally [10–15], some-
times alongside a rich phenomenology of other optically-
induced phases [16–19]. However, bistability can also be
found less commonly in systems where it is intrinsic, gen-
erated only by interactions within the sample, and has so
far been observed in Yb
3+
ions in solid-state crystals at
cryogenic temperatures [20] and in highly-excited Ryd-
berg atoms in the microwave regime [21]. Intrinsic bista-
bility was thought to be unachievable for atoms in the op-
tical regime, but recent theoretical studies of many-body
systems suggest that interaction-mediated bistability is
more generic and possible in a variety of systems with
short- and long-range interactions [22–24]. In particular,
we recently demonstrated [25] that intrinsic bistability
and optically-induced phases emerge in arrays of atoms at
sufficiently high densities, due to resonant light-mediated
dipole-dipole (DD) interactions and that these could be
identified in coherently and incoherently scattered light.
Several experiments on interactions of light with
atomic ensembles have in recent years achieved such cold
temperatures and high atom densities [26–35] that the
collective optical responses can start deviating [30, 36]
from those of thermal or low-density samples. The rel-
evant density scale is the number of atoms per cubic
wavenumber k of the resonant light, which takes nonneg-
ligible values also for atoms in optical lattices. In par-
ticular, a Mott-insulator state of
87
Rb atoms was now
studied [37] in an optical lattice with near unit filling,
where a subradiant eigenmode with a spatially uniform
phase profile was driven by the incident field in the limit
of low light intensity (LLI), and observed in a narrowed
transmission resonance for light. Collective interaction of
light with closely related arrays of atoms has attracted
considerable theoretical interest [25, 38–76].
Here we analyze transmission of light through a dense
planar array of cold atoms beyond the limit of LLI, when
the atoms respond nonlinearly to light. We formulate a
theory for optical bistability of free-space atomic arrays
that in general depends on the underlying LLI mode, ex-
panding our earlier analysis of Ref. [25]. In some cases,
the theory can be solved even analytically, with suffi-
ciently small atomic separations (ka) < (π/3)
1/2
needed
for bistability of spatially uniform modes. The bistabil-
ity threshold ka ∼ 1 applies even for the case of just two
atoms and equals the separation at which the single-atom
linewidth γ becomes less than the collective line shift,
originating from recurrent scattering where the light is
scattered more than once by the same atom [77–82].
We find that the transmitted light exhibits a
bistable solution of both “cooperative” and “single-
atom” branches, which we obtain approximate analytic
solutions for at high atomic densities. The coopera-
tive branch represents a collective response where atoms
strongly absorb the incident light, with high extinction
and weak incoherent scattering, while the single-atom
branch represents an independent response with atoms
weakly absorbing the incident field, with low extinction
and strong incoherent scattering. In particular, the co-
operative branch can completely extinguish the incident
light up to a large critical intensity, I
c
/I
sat
' 155, well
beyond the LLI limit. Beyond this intensity, a sharp
change in the transmission behavior of the array occurs,
as light begins to transmit through the lattice. By vary-
ing the frequency and intensity of the incident light, we
find hysteresis between the branches can occur, which is
observable by jumps in the extinction, reflectivity and
phase shifts of the light. We also find the loss of one of
the branches at the edge of the bistability region is asso-
ciated with a first order phase transition, resulting in a
divergence of the group delay and critical slowing, where
increasingly long times are needed to reach the steady
state.
The emergence of optical bistability in a collectively re-
sponding system of regularly spaced array of atoms has
2
a surprisingly close analogy with the optical bistability
in cavities. The presence of closely spaced atoms and
strong DD interactions modify the effective Purcell fac-
tors of atoms, due to substantial recurrent light scatter-
ing. This is reminiscent of the effect of a cavity in which
case a cooperative response results from an atom repeat-
edly scattering the same photon that bounces between
the cavity mirrors. The recurrent scattering in cavities
is quantified by a cooperativity parameter C = g
2
/2γκ,
which depends on the cavity linewidth κ and atom-cavity
coupling g. For specific parameter values the analogy
between bistability in the two systems becomes direct,
with the same equations governing the relationship be-
tween the incident and total light field, in which case we
can define a cooperativity parameter for atomic arrays as
C = ˜γ/2γ, for the collective linewidth γ+˜γ. The bistabil-
ity in both systems then emerges for a sufficiently strong
cooperative response when C > 4.
The layout of this paper is as follows. In Sec. II, we
present the model. In Sec. III, we present the theoretical
description of optical bistability in 2D planar arrays, be-
fore studying transmission properties in Sec. IV. Finally,
in Sec. V, we discuss our results and draw conclusions. In
Appendices A and B, we give further details on the com-
parison between bistability in cavities and atom arrays,
and on the transmission results, respectively.
II. MODEL
A. Atoms and light fields
We consider a system of N two-level cold atoms in-
teracting with light and trapped in a two-dimensional
(2D) array with one atom per site. The electrodynam-
ics are expressed in the length-gauge, obtained by the
Power-Zienau-Woolley transformation [83–85], where the
basic dynamical variable for light is the electric displace-
ment vector,
ˆ
D(r) =
ˆ
D
+
(r) +
ˆ
D
−
(r). The positive
frequency component is
ˆ
D
+
(r) =
P
q
ζˆa
q
e
iq·r
ˆ
e
q
, with
ˆ
D
−
(r) = [
ˆ
D
+
(r)]
†
, where ζ =
p
~ω
q
0
/2V , and we have
introduced the mode frequency, ω
q
, polarization
ˆ
e
q
, pho-
ton annihilation operator, ˆa
q
and mode volume, V . The
polarization of the atoms is expressed through the polar-
ization vector with positive frequency component,
ˆ
P
+
(r) =
X
l
δ(r − r
l
)d
ge
ˆσ
−
l
. (1)
The atoms are at fixed lattice sites r
l
, and we have in-
troduced the dipole moment, d
ge
, and the lowering op-
erator ˆσ
−
l
= |gi
ll
he| = (ˆσ
+
l
)
†
, with |ei
l
and |gi
l
de-
noting the excited and ground state of the two-level
atom on site l, respectively. We assume the atoms
are illuminated by a near monochromatic incident field,
ˆ
D
+
F
(r) =
0
E
+
(r) where E
+
(r) = E
0
ˆ
ee
ik·r
, with wavevec-
tor k and frequency ω = c|k| = ck, and express ob-
servables in terms of slowly varying field amplitudes and
atomic variables,
ˆ
D
+
e
iωt
→
ˆ
D
+
and ˆσ
−
l
e
iωt
→ ˆσ
−
l
. The
incident light is expressed through the Rabi frequency,
R
l
= d
∗
ge
· E
+
(r
l
)/~ acting on an atom at lattice site l,
and the incident and saturation intensity,
I
l
I
sat
= 2
|R
l
|
2
γ
2
, I
sat
= ~c
4π
2
γ
3λ
3
.
(2)
Throughout the paper, we assume d
ge
= D
ˆ
e, where D is
the reduced dipole matrix element.
Integrating over all space, the system Hamiltonian
is [81]
ˆ
H =
X
q
~ω
q
ˆa
†
q
ˆa
q
+
X
l
∆
l
ˆσ
ee
l
+
1
2
0
Z
ˆ
P(r) ·
ˆ
P(r)d
3
r −
1
0
Z
ˆ
D(r) ·
ˆ
P(r)d
3
r.
(3)
The first term is the Hamiltonian of the free electromag-
netic field. The second term is the laser frequency de-
tuning from the atomic resonance, ∆
l
= ω − ω
l
eg
, where
ω
l
eg
is the transition frequency of an atom on site l, and
ˆσ
ee
l
= ˆσ
+
l
ˆσ
−
l
. The third term is the self-polarization,
which is zero for nonoverlapping point atoms. The fi-
nal term is the interaction of the electric displacement
field with the atomic polarization. After carrying out
the spatial integration, we also make a rotating wave ap-
proximation in the Hamiltonian, Eq. (3), to remove the
fast co-rotating terms.
The total electric field
ˆ
E
+
(r) = E
+
(r) +
ˆ
E
+
s
(r) is ob-
tained using
ˆ
D(r) =
0
ˆ
E(r) +
ˆ
P(r) from the scattered
field,
0
ˆ
E
+
s
(r) =
X
l
G(r − r
l
)d
ge
ˆσ
−
l
, (4)
where the field satisfies Maxwell’s wave equation with an
atomic polarization source [81] and the dipole radiation
kernel acting on a dipole located at the origin, with r =
|r| and
ˆ
r = r/r, is given by the familiar dipole radiation
pattern [86, 87]
G(r)d = −
dδ(r)
3
+
k
3
4π
(
(
ˆ
r × d) ×
ˆ
r
e
ikr
kr
− [3
ˆ
r (
ˆ
r · d) − d]
i
(kr)
2
−
1
(kr)
3
e
ikr
)
. (5)
B. Mean-field approximation
1. Mean-field equations
We now describe the dynamical evolution of the atomic
coherences and excited level population in the mean-
field regime where quantum fluctuations between differ-
ent atoms [73] are ignored. The dynamics of the sys-
tem are obtained by solving the Heisenberg equations of
3
Figure 1. Transmission through an array of atoms. An inci-
dent field drives a large collective response at low intensities
and a weak single-atom response at high intensities. For in-
termediate intensities, bistability between the two responses
is possible.
motion for the atomic operators ˆσ
ee
l
and ˆσ
−
l
, assuming
a Born-Markov approximation to eliminate the electric
field operators, ˆa
q
. A Gutzwiller mean-field approxima-
tion is then implemented, corresponding to the factoriza-
tion of internal level correlations,
hˆσ
α
i
ˆσ
β
j
i ≈ hˆσ
α
i
ihˆσ
β
j
i, i 6= j
(6)
where, because atoms are at fixed positions with no po-
sition fluctuations, there are no light-induced correla-
tions [73, 88] between the atoms after the factorization.
The system dynamics are then described by the following
nonlinear equations
˙ρ
(l)
ge
= (i∆
l
− γ) ρ
(l)
ge
− i(2ρ
(l)
ee
− 1)
R
l
+
X
j6=l
(Ω
jl
+ iγ
jl
)ρ
(j)
ge
,
(7a)
˙ρ
(l)
ee
= − 2γρ
(l)
ee
+ 2Im[R
∗
l
ρ
(l)
ge
]
+ 2Im
X
j6=l
(Ω
jl
− iγ
jl
)ρ
(l)
ge
(ρ
(j)
ge
)
∗
,
(7b)
where we define ρ
(l)
ge
= hˆσ
−
l
i and ρ
(l)
ee
= hˆσ
ee
l
i. The sum-
mation terms in Eqs. (7) describe light-mediated inter-
actions between a discrete atoms at fixed points l and j
in a lattice. The DD interaction terms Ω
jl
and γ
jl
de-
pend on the relative positions between the atoms in the
lattice, and are given by the real and imaginary part of
the dipole radiation kernel, Eq. (5),
Ω
jl
=
1
~
0
Re
d
∗
ge
· G(r
j
− r
l
)d
ge
,
γ
jl
=
1
~
0
Im
d
∗
ge
· G(r
j
− r
l
)d
ge
,
(8)
where γ
jj
= γ = D
2
k
3
/(6π
0
~) is the single-atom
linewidth. DD interactions result in recurrent and cor-
related light scattering between the atoms with a strong
collective response from the array. The contact term of
the scattered light field in Eq. (5) is inconsequential in the
atomic interaction coefficients of Eqs. (7) and is the ori-
gin of the local field shift of light inside the medium [89].
In the absence of DD interactions, Eqs. (7) reduce to the
usual independent-atom optical Bloch equations. Mean-
field equations based on related principles as those in
Eqs. (7) have also been used to describe systems with and
without spatial fluctuations [34, 88, 90–92]. Other tran-
sitions can be included, such as the m = ±1 states of the
|J = 0, m = 0i → |J = 1, m = 0, ±1i transition, and the
corresponding general form of Eqs. (7) with arbitrary in-
ternal atomic levels is given in Ref. [88], with the specific
case of three-level system simulated in Ref. [34]. How-
ever, this quickly increases the complexity of the system.
By application of a magnetic field, the m = ±1 states
can be tuned far off-resonance so an effective two-level
system can be obtained with the m = 0 states.
2. Low light intensity
In the limit of LLI, atoms occupy the ground state,
with changes to the coherence, ρ
ge
, linearly proportional
to the incident light field amplitude, E
+
. The LLI
limit [81, 88] then constitutes deriving the equations first
order in light field amplitude by keeping the terms that
include at most one of the amplitudes ρ
ge
or E
+
, and no
ρ
ee
. Equations (7) then reduce to a linear set of equations
describing N dipole-coupled oscillators
˙ρ
(l)
ge
= i
X
j
H
jl
ρ
(j)
ge
+ iR
l
,
H
jl
= (∆ + iγ)δ
jl
+ (Ω
jl
+ iγ
jl
)(1 − δ
jl
).
(9)
The LLI collective excitation eigenmodes are given by
the eigenmodes of H, which satisfy biorthogonality re-
lations, with complex eigenvalues δ
α
+ iυ
α
, where δ
α
and υ
α
describe the collective line shift (from the res-
onance of the isolated atom) and linewidth, respec-
tively [40, 42, 44, 88, 93–96]. Modes with υ
α
> γ
(υ
α
< γ) are termed superradiant (subradiant). For
an infinite lattice, the eigenmodes of the system are de-
scribed by plane waves with wavevector q. The eigenval-
ues are now
˜
Ω(q) + i[˜γ(q) + γ] where
˜
Ω(q) =
X
j6=l
Ω
jl
e
iq·r
j
, ˜γ(q) =
X
j6=l
γ
jl
e
iq·r
j
,
(10)
are the Fourier transforms of the real and imaginary parts
of the dipole kernel, Eq. (8), respectively (excluding the
self-interaction j = l). The plane waves are given by
v
(+)
q
(r
l
) = A
q
cos(q · r
l
), (11)
v
(−)
q
(r
l
) = A
q
sin(q · r
l
), (12)
where A
q
=
p
2/N except for A
q=0,(π/a,π/a)
= 1/
√
N.
For the infinite system, the two-level transition resonance
wavelength defines the light cone, ka/
√
2, where any
mode with |q| > k/
√
2 results in ˜γ(q) = −γ and the
corresponding mode is then completely dark.
4
3. Solutions to the mean-field equations
We now solve the dynamics of Eqs. (7) to obtain the
steady-state solutions by considering uniform level shifts,
∆
l
= ∆. For the incident plane with the wave vector
k, a phase varying Rabi frequency, R
l
= Re
iq·r
l
, can
be obtained by tilting the angle of incidence, such that
q = k − (
ˆ
z · k)
ˆ
z. A general solution to Eqs. (7) is then
given by ρ
(l)
ee
= ρ
ee
and ρ
(l)
ge
= ρ
ge
e
iq·r
l
, with
ρ
ge
=
iRZ
i
h
∆ − Z
˜
Ω(q)
i
− [γ −Z˜γ(q)]
,
(13)
where
Z = 2ρ
ee
− 1. (14)
However, modes with q lying near to or outside the light
cone cannot directly be excited by incident light due to
the rapid phase variation required, and instead must be
driven by applying symmetry-breaking fields to the lat-
tice [25]. Such a symmetry-breaking level shifts could be
generated, for example, by ac Stark shifts [97] of lasers.
The population difference, Z, obeys the cubic equation
p(Z) =
˜γ(q)
2
+
˜
Ω(q)
2
Z
3
+ (∆
2
+ γ
2
)
+
˜γ(q)
2
+
˜
Ω(q)
2
− 2∆
˜
Ω(q) − 2γ˜γ(q)
Z
2
+
∆
2
+ γ
2
+ 2|R|
2
− 2∆
˜
Ω(q) − 2γ˜γ(q)
Z
= 0.
(15)
When |R|
2
≈ 0, Eq. (15) admits the solution ρ
ee
= 0,
with the coherence, Eq. (13), now describing the LLI
eigenmode of the infinite system with a wave vector q.
However, for nonzero incident fields, Eq. (15) can have up
to three real solutions, of which two are dynamically sta-
ble, resulting in optical bistability. Cases where solutions
become unstable can result in the emergence of a typi-
cally rich phase diagram of different solutions, exhibiting
a dependence on the intensity and laser frequency [25].
III. BISTABILITY IN ARRAYS OF ATOMS
A. General formalism
We now establish the formalism used to determine the
parameter ranges where bistability is possible for an ar-
ray of atoms. To do this, we substitute ρ
(l)
ee
= ρ
ee
and
ρ
(l)
ge
= ρ
ge
e
iq·r
l
into Eqs. (7), and rewrite as
˙ρ
ge
= (i∆ − γ) ρ
ge
− i(2ρ
ee
− 1)R
eff
, (16a)
˙ρ
ee
= − 2γρ
ee
+ 2Im [R
∗
eff
ρ
ge
] , (16b)
where we have defined
R
eff
= R + [
˜
Ω(q) + i˜γ(q)]ρ
ge
, (17)
which is the total external electric field (incident plus
scattered field from all the other atoms, given in terms
of the Rabi frequency) driving an arbitrary atom l in
the ensemble. Solving Eqs. (16) gives the coherence and
excited level population in terms of R
eff
[25],
ρ
ge
= R
eff
−∆ + iγ
∆
2
+ γ
2
+ 2|R
eff
|
2
, (18a)
ρ
ee
=
|R
eff
|
2
∆
2
+ γ
2
+ 2|R
eff
|
2
. (18b)
These solutions have a similar form to the solutions of
the optical Bloch equations, but now with the Rabi fre-
quency, R, replaced by R
eff
. Using Eq. (18a) to eliminate
ρ
ge
from Eq. (17) gives
R = R
eff
+ R
eff
2C(∆
2
+ γ
2
)
∆
2
+ γ
2
+ 2|R
eff
|
2
,
(19)
where we have defined the cooperativity parameter [25],
C =
1
2
˜
Ω(q) + i˜γ(q)
∆ + iγ
,
(20)
which is a measure of the collective behavior in the ar-
ray and plays an important role in describing bistabil-
ity. Finally, by taking the absolute value of both sides of
Eq. (19), we obtain
I
I
sat
=
2|R
eff
|
2
γ
2
1
(η
2
+ 2|R
eff
|
2
)
2
4η
4
Im[C]
2
+
η
2
(1 + 2Re[C]) + 2|R
eff
|
2
2
,
(21)
where
η
2
= ∆
2
+ γ
2
,
(22)
and I/I
sat
is given by Eq. (2) (where the intensity is now
the same for all sites, l). Equation (21) is a cubic equa-
tion in |R
eff
|
2
, with either one or two dynamically stable
real solutions, and the bistability region found when the
discriminant is zero as a function of I/I
sat
and ∆. Equa-
tion (15) can also be used to determine bistability in
the system. However, introduction of the effective field
and cooperativity parameter in Eq. (21) recasts the equa-
tions in the same notation used for bistability in cavity
systems [3], making the two systems easier to compare.
For large enough lattice spacings, there is only a single
solution to Eq. (21), and hence no bistability for any
intensity or detuning. To determine the minimal lattice
spacing for the array to support bistability, we consider
I/I
sat
as a function of |R
eff
|
2
in Eq. (21), and find the
lattice spacing where two minima develop, which involves
solving dI/d|R
eff
|
2
= 0, explicitly given by
4|R
eff
|
2
η
2
+ 2|R
eff
|
2
η
2
+ 2|R
eff
|
2
+ 2η
2
Re[C]
+
η
2
− 2|R
eff
|
2
h
η
2
+ 2|R
eff
|
2
+ 2η
2
Re[C]
2
i
+ 4η
4
Im[C]
2
η
2
− 2|R
eff
|
2
= 0.
(23)
5
B. Analytic bistable solutions
For closely-packed arrays where
˜
Ω(q), ˜γ(q) ∆, γ,
Eq. (21) has two well-separated minima and the bistable
solutions can be approximated. For low intensities
[
˜
Ω(q)/γ, ˜γ(q)/γ I/I
sat
], we obtain |R
eff
| from Eq. (21)
using
(η
2
+ 2|R
eff
|
2
+ 2η
2
Re[C])
2
≈ 4η
4
Im[C]
2
+ 4|R
eff
|
2
η
2
(2Re[C] + 1) + (1 + 2Re[C])
2
η
4
,
(24)
and use Eq. (19) to obtain the phase. For high intensi-
ties [I/I
sat
˜
Ω(q)/γ, ˜γ(q)/γ, ∆
2
/γ
2
], R
eff
is found from
Eq. (19) by ignoring the ∆
2
+γ
2
term in the denominator.
The approximate solutions for low and high intensities
are then, respectively, given by
R
coop
eff
=
√
2R
2C + 1
1 −
4|R|
2
η
2
|2C + 1|
2
+
s
1 + (1 − |2C|
2
)
8|R|
2
η
2
|2C + 1|
4
−1/2
,
(25a)
R
SA
eff
=
R
2
1 −
2iη
2
Im[C]
|R|
2
+
s
1 −
4η
2
Re[C]
|R|
2
−
2η
2
Im[C]
|R|
2
2
,
(25b)
where we have labeled the solutions as the “cooperative”
and “single-atom” due to their very different responses to
the incident light
1
, in an analogy with a similar terminol-
ogy in optical cavities [3]. For the cooperative solution,
Eq. (25a), the atoms behave collectively, creating a field
that counteracts the incident light and resulting in the
atoms absorbing strongly, especially at higher atom den-
sities. This is demonstrated most clearly in the LLI limit,
where R
coop
eff
≈ R/(2C + 1), where we can see how the
effective field scales inversely with C, with strongly col-
lective behavior resulting in a small R
eff
. For the single-
atom solution, Eq. (25b), the atoms now saturate and
absorption is weak, with the medium becoming trans-
parent as the atoms react to the incident light almost
independently. The effective field scales linearly with the
incident field for high intensities, where R
SA
eff
≈ R and
there is no dependence on C as collective behavior be-
tween the atoms is lost.
The cooperative and single-atom solutions only de-
scribe the system response for the intensity ranges
I
I
sat
<
η
2
4γ
2
|2C + 1|
4
|2C|
2
− 1
, (26a)
4η
2
γ
2
(Re[C] + |C|) <
I
I
sat
, (26b)
1
Note the cooperative solution presented here is more accu-
rate approximation than the solution presented in our previous
work [25].
respectively, and serve as approximate lower and upper
intensity bound of the bistability region. However, a
more accurate analytic approximation for the upper in-
tensity bound can be found using Eq. (23) in the limit
that
˜
Ω(q), ˜γ(q) ∆, γ by expanding the cubic solutions
to Eq. (23) about small η
2
that yields
4η
2
γ
2
(Re[C] + |C|) <
I
I
sat
<
η
2
γ
2
|C + 1|
2
.
(27)
C. Analytic thresholds
Finding the thresholds for bistability by solving
Eq. (23) can usually only be done numerically. How-
ever, there are two cases where analytic solutions can be
easily obtained [25]. The first case is for real C when
∆/γ =
˜
Ω(q)/˜γ(q), where Eq. (23) gives a threshold of
˜γ(q) > 8γ. (28)
The second case is for imaginary C when ∆/γ =
−˜γ(q)/
˜
Ω(q), where Eq. (23) forms a cubic equation
where two positive real solutions and bistability are pos-
sible when
[
˜
Ω(q)]
2
> 27γ
2
.
(29)
1. Two atom bistability
The analytic results reveal high density thresholds for
the emergence of optical bistability. This can already be
seen in the simplest possible case of two closely spaced
atoms within the mean-field approximation [73] and un-
der uniform illumination. The condition ˜γ = γ
12
> 8γ
cannot be met as γ
12
→ γ in the limit of zero separation.
However, the threshold
˜
Ω = Ω
12
>
√
27γ can be satis-
fied, and a simple dimensional analysis for Ω
12
∼ 1/(ka)
3
yields the threshold ka . 1. This value also equals the
separation required for the collective shift to exceed the
single-atom linewidth Ω
12
& γ; a condition at which cor-
relations due to light-mediated interactions lead to sub-
stantial deviations from standard continuous medium op-
tics [36]. A more accurate calculation gives the bistability
threshold ka . 0.94 (corresponding to a lattice spacing
of a . 0.15λ) and ka . 0.63 (a . 0.10λ) for atoms po-
larized parallel or perpendicular to the separation axis,
respectively.
2. Arrays of atoms
Analytic expressions for the optical bistability can be
obtained for planar arrays for ∆/γ =
˜
Ω(q)/˜γ(q), when
the solutions no longer depend on
˜
Ω(q). The collective
radiative linewidth for the uniform LLI eigenmode with
![Figure 6. Group delay and critical slowing of a pulse through an array of atoms with lattice spacing a = 0.1λ, ê = (1, 0, 0) and ∆/γ = −3.5. (a) Group delay of the spatially uniform mode for the cooperative (blue circles) and single-atom (orange squares) solutions with analytic estimates Eq. (49) (black-dashed) and Eq. (50) (blue-dot-dashed). The group delay divergences at the bistability region edge. The inset shows hysteresis in the group delay from a negative (left arrow) and positive (right arrow) detuning sweep. (b) Longest decay time, τ = −1/Re[λ], for the dynamics to settle to the steady state for the cooperative and single-atom solutions. Critical slowing occurs at the edge of the bistability region, with a diverging decay time.](/figures/figure-6-group-delay-and-critical-slowing-of-a-pulse-through-3pmyzqfd.webp)
![Figure 4. Loss of optical bistability with lattice spacing. (a) Modes that lose bistability (blue dots) as a function of lattice spacing for q = q(1, 1), which are divided into two groups by the light cone (black dashed line). (b) Collective radiative couplings Ω̃(q) and γ̃(q) [Eqs. (10)] for a lattice with ka = 0.2π. For q = 0, Eq. (30) gives γ̃(0) ' 22.9γ.](/figures/figure-4-loss-of-optical-bistability-with-lattice-spacing-a-2xmnc5ub.webp)
![Figure 3. Optical bistability in a planar array of atoms. (a) Region of bistability from Eq. (21) (solid blue line) for q = 0, a = 0.1λ and ê = (1, 1, 0)/ √ 2. (b) |Reff/γ|2 for ∆/γ = Ω̃(0)/γ̃(0) [blue dashed line in (a)] with approximate cooperative (lower black dashed line) and single-atom (upper black dashed line) solutions, Eqs. (25). (c) Bistability region and (d) |Reff/γ|2 for ∆/γ = −γ̃(q)/Ω̃(q) [blue dashed line in (c)] with approximate cooperative and single-atom solutions for the mode q = (π/a, π/a). (e) Bistability region crossing the light cone for the mode q = (π/4a, π/4a) at the lattice spacings a = 0.17λ and (f) a = 0.18λ. In all plots, the intensity thresholds Eqs. (27) (red-dotted and orange-dot-dashed lines) are shown. In (b,d), the cooperative solution intensity threshold Eq. (26a) (gray dashed line) is also shown.](/figures/figure-3-optical-bistability-in-a-planar-array-of-atoms-a-anhdv604.webp)
