PHYSICAL REVIEW A 89, 033854 (2014)

Photon propagation in a one-dimensional optomechanical lattice
Wei Chen and Aashish A. Clerk*
Department of Physics, McGill University, Montreal, Canada H3A 2T8(Received 13 January 2014; published 27 March 2014)
We consider a one-dimensional optomechanical lattice where each site is strongly driven by a control laser toenhance the basic optomechanical interaction. We then study the propagation of photons injected by an additional
probe laser beam; this is the lattice generalization of the well-known optomechanically induced transparency(OMIT) effect in a single optomechanical cavity. We find an interesting interplay between OMIT-type physics
and geometric, Fabry-Perot-type resonances. In particular, phononlike polaritons can give rise to high-amplitudetransmission resonances which are much narrower than the scale set by internal photon losses. We also find thatthe local photon density of states in the lattice exhibits OMIT-style interference features. It is thus far richer thanwould be expected by just looking at the band structure of the dissipation-free coherent system.
DOI: 10.1103/PhysRevA.89.033854

PACS number(s): 42.50.Wk, 42.65.Sf, 42.25.Bs

I. INTRODUCTION

The rapidly growing field of quantum optomechanics
involves studying the coupling of mechanical motion to light,
with the prototypical structure being an optomechanical cavity:
Photons in a single mode of an electromagnetic cavity interact
via radiation pressure with the position of a mechanicalresonator (see Ref. [1] for a recent review). Among the many
recent breakthroughs in this area are the laser cooling of
mechanical motion to the ground state [2,3], the generationof squeezed light via the optomechanical interaction [4–6],
and demonstration of entanglement between microwaves andmechanical motion [7].
A particularly exciting direction for this field is the development of multimode structures composed of many individual
optomechanical cavities which are coupled via photonic and/or
phononic tunneling. Such multisite optomechanical arrays canbe realized in so-called optomechanical crystals, fabricatedusing a planar photonic bandgap material, and using defects
to localize both mechanical and optical modes [8,9]. Recent
theoretical studies have investigated synchronization andphase-transition physics in these systems [10–13], applications
to quantum information processing [14], as well as the effective
band structure describing excitations when the system isdriven with a red-detuned laser [15]. Arrays based on multiple
membranes in a single cavity have also been studied [16–18].
In this work, we consider how a prototypical optomechanical effect—optomechanically induced transparency (OMIT)[19–22]—is modified in a lattice setting. OMIT in a single
cavity is an effect whereby even a weak optomechanicalinteraction can, in the presence of a large-amplitude control
laser, lead to a strong modification of the cavity photon densityof states. This modification can in turn be probed using either
the reflection or transmission of a weak auxiliary probe field.
It is analogous to the well-known electromagnetically induced
transparency effect (EIT) [23–25] involving the interaction ofa three-level atom with light, where destructive interference
prevents optical absorption by the atom. We consider aone-dimensional (1D) optomechanical array where photons
can tunnel between adjacent sites, and consider what happens*

clerk@physics.mcgill.ca

1050-2947/2014/89(3)/033854(14)when, similar to a standard OMIT experiment, the entire
system is driven by a large-amplitude control laser (detuned
in such a manner to avoid any instability). We work inthe standard regime where the single-photon optomechanicalcoupling is negligible, and only the drive-enhanced manyphoton coupling plays a role. Unlike the recent work from
Schmidt et al. [15], we restrict attention to one dimension, but
do more than simply characterize the band structure describing
excitations in this system: We also provide a full descriptionof transport phenomena, where additional probe photons are
injected at one site of the lattice, and extracted at some otherpoint in the lattice (see Fig. 1). Our work also strongly differsfrom the previous study by Chang et al. [26], who studied analternate geometry where a regular 1D array of optomechanicalcavities is side coupled to a one-dimensional waveguide. In thatsystem there is no direct tunneling between optomechanicalcavities, whereas in our system, probe photons can only betransported via such a tunnel coupling. As we discuss, thisgives rise to markedly different behaviors.
Our key results can be summarized as follows. After
introducing our system in Sec. II, we first look at the local
photon density of states of the lattice in Sec. III. In the presenceof a many-photon optomechanical coupling but ignoring
dissipation, one will generically obtain an energy gap between
the polariton bands of the coupled system. Similar to OMIT,
we find that this gap can persist even if it is much smaller than
the intrinsic photon damping rate κ0 . Unlike OMIT, the sizeof the spectral gap in the coupled system can be identical insize to that expected in the coherent system. We show how
this density-of-states phenomenon could be directly measured
in an OMIT-style experiment involving the local reflection ofprobe light from a given site of the lattice.
We next consider in Sec. IV the question of transport in oursystem, where probe light is injected at site j = 1 and extracted
at site j = N + 1 of the lattice. We find that polariton modes
that are almost entirely phononic can contribute strongly tophoton transport. We also study the interplay between these
OMIT-style effects and the presence of geometric resonances(caused either by reflections resulting from the coupling to theexternal probe waveguides, or from the finite size of the lattice).
We again find a striking result: Interference effects involving
phononlike polaritons can yield transmission resonances thathave extremely high maximum transmissions, and which have033854-1©2014 American Physical Society

WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

a frequency width much narrower than the intrinsic photon
loss rate of the lattice κ0 . Note that the interplay between
conventional atomic EIT and geometric optical resonances waspreviously studied by Lukin et al. [27]; as we discuss further
in Sec. IV B, there are marked differences from our work, as
the analogy between OMIT in our system and propagation in
an atomic EIT medium is not complete.
We stress that the effects we describe in this work,
like OMIT itself, are not simply the result of the coherent
hybridization of light and matter—the effects we describe
persist even in a regime where this coherent coupling is muchsmaller than the photonic dissipation. Instead, these effects rely
crucially on the dissipative physics of the system, in particular
the markedly different intrinsic damping rates of photons andphonons, and the fact that the dissipation is not “diagonal”
in the basis of energy eigenstates of the coherent system (thepolariton basis). In the case of EIT, this crucial difference has
been emphasized in a recent work by Anisimov et al. [28].

the phonon energy. OMIT Ĥdis describes both dissipation ofphoton and phonons. We assume that phonons and photons oneach site are coupled to their own (independent) Markovian,Gaussian dissipative baths, and treat these as is done instandard input-output theory [29,30]. This gives rise to adamping rate γ for each phonon mode, and a damping rate
κ0 for each photon mode. κ0 includes both dissipation due tointernal losses, as well as the coupling to port used to drive eachsite of the lattice. We will focus throughout on the resolved
sideband regime here, where κ0  M . Note that while we
focus on an infinite system, the key effects we describe also
have direct relevance to finite-sized systems (see Appendix C).
As mentioned in the introduction, the model describedabove could most easily be realized experimentally using
so-called optomechanical crystals, where an array of optomechanical cells is formed by creating a superlattice of
defects in a photonic bandgap material (typically a patternedsilicon membrane); each defect localizes an optical and aphononic mode, providing an effective optomechanical cavity.
Fabrication and characterization of one- and two-dimensional
optomechanical crystals have been achieved in several recent
experiments [31,32], where typical parameters have the values
ωc /2π  200 THz, M /2π = 10 GHz, g/2π = 250 kHz, andthe intrinsic optical and mechanical damping rates are in therange 200 kHz–2 GHz and 10–200 kHz, respectively. The photon hopping J depends on the distance between neighboringsites and typically has the same order of magnitude as thephoton frequency ωc in experiments. As discussed recently
in Ref. [15], for such realistic optomechanical crystals, one
expects the phonon hopping to be smaller than the photon
hopping J by a large factor ∼ 104 ; hence the neglect of phonon
hopping in our model is well motivated.
Note that for simplicity, in this work we do not consider theeffect of disorder arising, e.g., from site-to-site fluctuations of
the photon and/or phonon frequencies. Such disorder effects
would generically give rise to localization effects in a 1D
system. However, as discussed in Ref. [15], for realistic finitesized cavity arrays the disorder can be weak enough such thatthe localization length is much larger than the size of the array(see, e.g., Ref. [33]).

II. MODEL AND POLARITON BAND STRUCTURE

B. Many-photon optomechanical coupling

A. Optomechanical lattice Hamiltonian

In standard single-cavity optomechanics experiments, thesingle-photon coupling is extremely weak (i.e., g  κ0 ), andhence the cavity is strongly driven to enhance the interaction.
We assume a similar situation in our lattice, now taking eachsite to be strongly driven by a control laser at frequency ωL ; wetake the laser amplitude to be the same on all sites. Working
in a rotating frame at the laser frequency, the laser will inducea classical, time-independent mean amplitude for each photon
mode āj and each mechanical mode b̄j . These stationary
amplitudes are readily found from the classical equations of
motion for our system:κ0āj − ig āj (b̄j + b̄j∗ )0 = i −2√+ iJ (āj −1 + āj +1 ) − κ0 āin ,
(2a)




γ
(2b)
0 = − iM −b̄j − ig āj2 .
2FIG. 1. (Color online) Schematic picture of the 1D optomechanical array studied in this work. On each site, the blue (red) circle
represents a localized mechanical (photonic) mode; they interact
locally via a standard radiation pressure coupling. Photons can hop
between neighboring site with hopping strength J . A uniform control
laser is applied to each site at frequency ωL = ωc + , where  is thedetuning between the laser frequency and cavity photon frequency
ωc . For transport, external waveguides are attached to sites 1 andN + 1; these are used to send and extract probe light at frequency ωp .

We first consider an infinite one-dimensional lattice where
each site j contains a localized photonic and phononic mode
(lowering operators âj , b̂j , respectively), coupled locally via
the standard dispersive optomechanical interaction (strengthg). We allow for photon hopping between neighboring sites
via standard tight-binding hopping terms (hopping strength
J ); for simplicity, we ignore mechanical hopping, although
this could easily be included in the model. The HamiltonianĤ of the lattice thus takes the form Ĥ = Ĥcoh + Ĥdiss where †Ĥj − Jâi âj ,(1a)Ĥcoh =ji,j ††† †Ĥj = ωc âj âj + M b̂j b̂j + g(b̂j + b̂j )âj âj .

(1b)Here Ĥj is the standard Hamiltonian of a single optomechanical cell [1], with ωc the photon mode energy and M

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PHYSICAL REVIEW A 89, 033854 (2014)

Here, āin is the amplitude of the drive laser on each site,
and the laser detuning  ≡ ωL − ωc . The presence of hoppingchanges these equations from their single-cavity counterpart.
We look for a solution to these equations that does not break
the translational invariance of the Hamiltonian, i.e., āj = āand b̄j = b̄; the equations then reduce to the single-site casewith a shifted detuning  →  + 2J . We assume that thedrive is small enough so the equations do not exhibit bistability[1,20,34,35], and hence yield a single solution for ā,b̄; without
loss of generality, we take ā to be real and positive.
With these classical solutions in hand, we next makestandard canonical displacement transformations of photon
and phonon operators. We simply shift each b̂j by its meanvalue, and define d̂j = âj − ā (again, in a rotating framewith respect to the control laser frequency). The resulting
Hamiltonian has an optomechanical interaction enhanced bya factor of ā. Assuming this factor to be large, we makethe standard linearization, dropping interaction terms that are
independent of ā. We are assuming the standard regime where
the single-photon optomechanical coupling g is so weak thatit can only play a role when enhanced by the amplitudeinduced by the strong control laser. The resulting coherent(i.e., dissipation-free) Hamiltonian has the form,Ĥcoh =††††[− d̂j d̂j + M b̂j b̂j + G(d̂j + d̂j )(b̂j + b̂j )]
j

−J



†

(3)

d̂i d̂j ,i,j 

where G = āj g is the many-photon optomechanical coupling.
We stress that G can be tuned by changing the control laser
intensity. The effective laser detuning in the displaced frame =  − g(b̄ + b̄∗ ); we just denote this  in what follows.
Note that our primary interest in this work is on the lattice
analog of OMIT and not on strong-coupling effects associatedwith the many-photon optomechanical coupling G. As such,
we will focus on regimes where G  κ0 , G  M .
Given its translational symmetry, the Hamiltonian has asimpler structure in momentum space. Setting the lattice
constant a = 1, we haveĤcoh =

 π
−π

†

†

†

† †

dk[ωk d̂k d̂k + M b̂k b̂k + G(d̂k b̂k + d̂k b̂−k + H.c.)].
(4)Here, d̂k and b̂k are the momentum-space photon and mechanical fields (−π  k < π ), and have a delta-function†
normalization (e.g., [d̂k ,d̂k ] = δ(k − k )). We useωk = − − 2J cos k

Note that if one drops terms in Ĥcoh which do notconserve excitation number, then the Hamiltonian for eachinvariant k subspace is equivalent to an effective single-site
optomechanical cavity in the same approximation, with aneffective laser detuning eff = −ωk . In contrast, including theexcitation number nonconserving terms ruins the correspondence to the single site problem, as now a given photon modewith momentum k couples both to phonons with momentum
k and −k.
C. Polariton band structure

The coherent Hamiltonian Ĥcoh in Eq. (4) is readily
diagonalized in terms of eigenstates that are mixtures ofphonons and photons; we term these excitations polaritons.
1. Parametric instabilities

An immediate issue is that, similar to a singlecavity optomechanical system, the excitation nonconserving“parametric-amplifier” terms in Eq. (4) can give rise to
instabilities (see also Ref. [15]). Note that in the interaction
picture, the bottom of the bare photon band is at an energyωk=0 = − − 2J ; it can be negative even when  < 0 (i.e.,the control laser is detuned to the red of the middle of the barephoton band). The existence of negative energy photon states
can make the parametric-amplifier terms in Eq. (4) resonant,making the system especially susceptible to instability. In
particular, for a detuning −  2J − M , we will have sucha resonance for some value of k. Including dissipation, aparametric instability will occur for such detunings when
the cooperativity 4C ≡ G2 /γ κ0 > 1 [1]. Our focus in thiswork will be on regimes free of such instabilities. As we areinterested in weak optomechanical couplings G  M , wewill thus restrict attention to laser detunings  that position thebare photon band to lie above ω = 0 in the interaction picture.
In such cases, the parametric amplifier terms are off resonantby a minimum amount M , and parametric instabilities cannot
occur for optomechanical couplings G  M .
2. Rotating-wave approximation

Given the choice of detunings and parameters discussed
above, we will be in a regime where the effects of the particle
nonconserving terms in Eq. (4) are extremely weak. As such,
we can make a rotating-wave approximation (RWA) and dropthem. The resulting Hamiltonian is diagonalized as
 π
†
†
†
Ĥ =
dk[ωk d̂k d̂k + M b̂k b̂k + G(d̂k b̂k + H.c.)]
−π

=

(5)

to denote the dispersion relation of the bare photon band (i.e., in
the absence of optomechanical interaction). The corresponding
bare phonon band has no dispersion, consistent with ourneglect of phonon hopping. In the interaction picture, theposition of the bare phonon band [described by Eq. (5)] relative
to the flat photon band at ω = M can be controlled by thecontrol laser detuning .

 π
σ =± −π†dk Eσ,k ψ̂σ,k ψ̂σ,k .

(6)Here, σ = +,− denotes the two polaritons (normal modes) ateach momentum k (energy Eσ,k , annihilation operator ψ̂σ,k ).
One finds that the energies are

 2
δk
δk
± G2 +
E±,k = M +
,
(7)
2
2

033854-3

WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

a

c

b
1.0

2

2

cos2 θk

M

J

1

1
0.6
0
0.4

Ek

0

Ek

M

J

0.8

1

1

2

2

0.2
sin 2 θk

0
2

0

2

2

k

0

2

2

k

0

2

k

FIG. 2. (Color online) (a) Band structure of the bare photons (red dashed curve) and mechanical modes (purple solid curve) for red sideband
detuning  = −M and G → 0. (b) Band structure of the polaritons, same parameters in (a) but now G = 0.1J . The upper (dashed) curve
represents the + polariton band and the lower (solid) curve represents the − polariton band. (c) Variation of the factors sin2 θk , cos2 θk which
describe the photon and phonon parts of the polariton wave functions [c.f. Eq. (9)].

whereδk ≡ ωk − M = (− − 2J cos k) − M(8)is the energy detuning between the bare photon and phononbands at wave vector k. Similarly, the polariton mode operatorsareψ̂+,k = cos θk · d̂k + sin θk · b̂k ,(9a)ψ̂−,k = − sin θk · d̂k + cos θk · b̂k ,(9b)

Similar to the single-site case, this is not the case: The fact thatthe mechanical damping rate γ is vastly different in magnitude
than the photonic damping rate κ0 leads to features and physics
that go well beyond a simple energy broadening.
A. Calculation of photon density of states

The local photon DOS on site j of the lattice, ρc (ω,j )measures how hard it is to add a photon of energy ω to site j .
It is formally defined aswhere the mixing angle θk is given bytan 2θk = 2G/δk .

ρc (ω,j ) = −2ImDcR (ω; j,j ),(10)

Figure 2 shows a typical polariton band structure (both withand without the optomechanical interaction) in the simple casewhere the bare photon band is bisected by the bare mechanicalband (i.e.,  = −M ). While the RWA Hamiltonian in Eq. (6)describes a simple two-band bosonic tight-binding model, it
does not tell the complete story. As we will see in what follows,
the dissipative nature of our system will play a crucial rolein determining the spectral and transport properties we areinterested in (namely the fact that photonic and phononic
damping rates are vastly different from one another). This
makes our system markedly different from standard two-bandmodels studied in the condensed matter literature.

(GF) of cavity photons, defined in the standard manner:†

DcR (j,t; j ,t ) = −iθ (t − t )[d̂j (t),d̂j (t )],DcR (ω; j,j ) = ∞−∞dt DcR (j,t ; j ,0)eiωt .

(12a)
(12b)

As we are dealing with a translationally invariant system,ρc (ω,j ) will be independent of j .
The retarded GF of the cavity photons in k space iseasily obtained from the Heisenberg-Langevin equations of
motion corresponding to our RWA Hamiltonian Eq. (6) anda standard input-output treatment of the dissipative baths(see Appendix A),

III. CAVITY PHOTON DENSITY OF STATES AND OMITEFFECT IN AN OPTOMECHANICAL ARRAY

Having discussed the simple properties of the dissipationfree, linearized optomechanical lattice, we now turn to themore interesting and subtle features that arise when dissipation
is included. We first will examine the local photon density ofstates (DOS) of the lattice, a quantity that can be directly
measured via OMIT-style experiments where one looks at
the local reflection of an auxiliary probe laser; we explicitly
calculate this reflection coefficient in Sec. III D. One might
naively expect that dissipation would simply broaden thedensity-of-states features present in the coherent system.

(11)

where DcR (ω; j,j ) is the retarded real-space Green function

DcR (ω,k) =12

ω +  + 2J cos k + i κ20 − ω−GM +i γ

. (13)

2

For each k mode, this has exactly the same form as the photon
GF for a single optomechanical cavity (also treated within theRWA) with an effective k-dependent laser detuning.
Transforming to real space yields
 π1R
dk eik(N−1) DcR (ω,k)
Dc (ω; N,1) =2π −π033854-4=

eiq[ω](N−1).
2iJ sin q[ω](14)PHOTON PROPAGATION IN A ONE-DIMENSIONAL . . .

PHYSICAL REVIEW A 89, 033854 (2014)

3
2

CJ 0.1
CJ 1
CJ 10
κ0
0

b
1.0

G2 J 10 3 κ0

0.9
Runits of 1 J

a

1

0.7

ρph
0

0.8

0.6

1.5 1.0 0.5

0 0.5 1.0 1.5
units
of G2 JM

1.5 1.0 0.5 0 0.5 1.0 1.52
M units of G J

FIG. 3. (Color online) (a) The local cavity photon DOS in the optomechanical array [c.f. Eq. (17)] for frequencies near the energy gap in
the coherent band structure; the detuning − = M and optomechanical coupling G = 0.01J for all the curves (see Fig. 2 for a representative
coherent band structure). κ0 = 0.1J for all the curves except the purple curve (dot-dashed) which corresponds to κ0 = γ = 0. The different
cooperativities in the figure are achieved by tuning the mechanical decay rate γ . One sees that if the effective cooperativity CJ is larger than
∼ 1, the gap in the coherent-system density of states persist, even if its magnitude is smaller than κ0 . (b) The local reflection probability
R[ω] = |r[ω]|2 [Eq. (23)] at frequencies near M (center of the coherent-system gap) for a single weak probe on site 1 with κex,1 = 0.1J .
Curves correspond to the legend in panel (a). R[ω] directly mirrors the behavior of the DOS in (a).

Here q[ω] is complex and satisfies
ω++i

G2
κ0
+ 2J cos q[ω] −= 0. (15)2ω − M + i γ2

As usual, the relevant solution here is the one with Im q > 0(which thus yields behavior consistent with causality). It isuseful to decompose q[ω] into its real and imaginary parts,
hence we writeq[ω] ≡ k̃[ω] + iα[ω]/2.

(16)k̃[ω] represents the effective propagation wave vector at
frequency ω, while α[ω] describes the spatial attenuation of awave at this frequency.
Using Eq. (14), we thus find that the local photon DOS isgiven by2ρc (ω,j ) = Re 
22
4J − ω +  + i κ20 − ω−GM +i γ

2

.

2

(17)
Without dissipation (i.e., κ0 ,γ → 0), the local density ofstates will exhibit the expected features corresponding to thecoherent band structure: There will be a gap for frequencies
between the two polariton bands, and there will be van Hove
singularities at frequencies corresponding to the edges of eachband. One might expect that dissipation simply smears out
these features: We now show explicitly that this is not the case.
In particular, we will see that even if the photon dissipation κ0is larger than the gap in the coherent band structure, a gap in
the density of states can survive.
B. Resilience of the gap against photonic dissipation

We consider for simplicity the symmetric situation where
the laser detuning is  = −M , which results in the barephoton band being bisected by the bare mechanical band (asdepicted in Fig. 2). As discussed, we focus on the realistic

situation where the optomechanical coupling is weak: G < κ0 .
We also consider the photon hopping to be much larger than
the internal photonic loss rate, but keep it small enough that
our RWA treatment remains valid for our choice of detuning:κ0 < J  M .
Recall first what happens in this regime in the absenceof dissipation. The optomechanical interaction leads to theformation of two polariton bands, separated by an energy gap.
The gap is centered on ω = M in the interaction picture, andhas a width  G2 /J [see Fig. 3(a), dashed-dotted curve]. The
polariton states with energies at the edges of this gap region(i.e., top of − band, bottom of + band) are almost entirely
phononlike: The photon component of these polaritons issmaller by a factor ∼ G/J compared to the phonon component[c.f. Eq. (9) and Figs. 2(c) and 2(d)].
Introducing dissipation, we would expect that if the photon
damping rate κ0 is larger than the size of the coherent-system
gap G2 /J , the gap will be smeared away. Figure 3 shows that
this is not necessarily true: The gap can persist in the photon
DOS even for relatively large values of κ0 .
While at first glance this behavior is surprising, a simple
explanation is possible. In the presence of dissipation, thephononlike polariton modes with energies at the edges of the
gap will have a total damping rate given bygap-edge  γ + opt ,(18a)
opt ∼ (G/J )2 κ0 .

(18b)

opt is an effective optical damping of these phononlike modes:
It is given simply by the intrinsic cavity damping rate κ0 , times
the small but nonzero photon weight of these polariton modes,
cos2 θk ∼ (G/J )2 . We see that these modes see only weakly
the photon loss rate κ0 . The approximate condition that now
ensures that dissipation does not smear out the gap is that theenergy broadening of a gap-edge polariton gap-edge needs to bemuch smaller than the size of the gap in the coherent system,
G2 /J . This can be recast as the conditions,

033854-5

κ0  J, CJ

1,

(19)

WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

where we have introduced the effective lattice cooperativity,
CJ ≡ G2 /J γ .

(20)

We thus see that the gap in the photon density of states canremain pronounced even though it is much smaller than κ0 .
We stress that the structure of the dissipation in our system
plays a key role in obtaining this result (namely that there
are two very different damping rates κ0 and γ , and that thedissipation is not diagonal in the eigenbasis of the coherent
Hamiltonian). The dissipative physics in our system cannot
be accurately described by introducing a simple homogeneousenergy broadening into our system. If one has κ = γ , thenthere is no gap feature left, one gets essentially the red (dotted)
curves in Figs. 3(a) and 3(b).

additional waveguide is coupled to a site of the lattice (without
loss of generality, we take this to be the site j = 1). The
coupling to this waveguide is again treated using standard
input-output theory [29]; it increases the damping rate on site
1 from κ0 to κ0 + κ1,ex ≡ κ1 , where κ1,ex is the coupling rate
to the probe waveguide. Site 1 is now driven by an additional
input field d̂1,p,in (t) describing the probe waveguide. This field
has a nonzero average value describing the applied probetone:d̂1,p,in (t) = d̄1,p,in e−iωp t .

(21)

Input-output theory now tells us the corresponding average
output field emanating back into the probe waveguide:d̂1,p,out = d̂1,p,in +√κex,1 d̂1 .

(22)

C. Comparison with single-cavity OMITThe fact that the gap in the photon density of states persists
even when it is much smaller than the photon loss rate κ0is analogous to the situation in standard, single-cavity OMIT
[19,20]. We begin by summarizing the essential features of thiseffect, which involves driving a single-cavity optomechanicalcavity with a strong control laser near the red mechanicalsideband  = −M . One obtains a standard beam-splitterinteraction which coherently converts photons to phonons andvice versa. In the absence of dissipation, the optomechanicalcavity would thus have two normal modes (polaritons) split
by an energy 2G. As a result, in the rotating frame, thecavity density of states would be zero for frequencies between
ω − M = −G and ω − M = G. Introducing dissipation,
one would naively think that this density-of-states suppression
would be completely lost if κ0
G. Instead, one finds that
there is a density-of-states suppression in this regime atω = 0 as long as the standard optomechanical cooperativity
C = 4G2 /κ0 γ is larger than one. Furthermore, the width of
the DOS suppression in this regime is set by the effective
mechanical linewidth (which includes an optical damping
contribution); this width is much smaller than G. This is
the standard single-cavity OMIT effect; it can be interpreted
as arising from interference between different pathways foradding a photon to the cavity, and can be measured bylooking at the transmission or reflection of an auxiliary probe
laser.
Returning now to our lattice system, we find a similar
situation to the single cell case described above (again forG  κ0 ), but with some important differences. First, it is
now the lattice cooperativity CJ which plays a crucial rolein determining whether the gap persists. Second, the spectral
width of the DOS suppression is about the same as in thecoherent system, ∼G2 /J ; this is in contrast to single-site
OMIT, where the width of the DOS suppression is muchsmaller than the “gap” in the coherent system 2G.
D. Measuring OMIT in an optomechanical array

Similar to the situation with single-cavity OMIT, one canmeasure the DOS suppression described above by applying
an additional weak probe laser with frequency ωp to a siteof the array, and measuring its reflection as ωp is varied.
To model an OMIT-style experiment, we assume that an

The average value of this output field will give us the desiredreflection coefficient of the probe field. As our system is linear,
this quantity can be related directly to the retarded photon
Green function (e.g., by solving the averaged Heisenberg
equations of motion, or by using the Kubo formula). Onefindsr[ωp ] = d̂1,p,out /d̂1,p,in  = 1 − iκex,1 D̃cR (ωp ; 1,1). (23)The GF D̃cR (ωp ; 1,1) is defined as per Eq. (12a), except that it
now includes the effects of the coupling to the probe waveguide(i.e., the additional photon damping on site j = 1). Notethat this coupling breaks the translational invariance of themodel.
To calculate the retarded GF in the presence of the probewaveguide, we need to include the additional photon damping
due to the probe waveguide. This is equivalent to introducing
an extra non-Hermitian potential for photons on site 1,
V̂ = −iκex,1 †d̂ d̂1 .
2 1(24)

Solving the Dyson equation corresponding to this extra
potential allows us to calculate the “dressed” GF D̃ R (seeAppendix B). We findD̃cR (ω; 1,1) =1,
2iJ sin q[ω] + i κex,1
2(25)

where the complex function q[ω] is again defined via
Eq. (15).
For the case of a weak probe waveguide coupling (i.e.,κex,1  J ), the “dressed” GF D̃cR is similar to the intrinsic GF
DcR of the probe-free system. Thus, as expected, the reflection
coefficient as a function of probe frequency r[ωp ] directlyreflects the structure in the density of states discussed earlier.
This is demonstrated by comparing Figs. 3(b) and 3(c). On aheuristic level, for a weak coupling, the only way for a probe
photon to not be reflected is if there is a state in the cavity for it
to enter. If it enters such a state, the much stronger coupling to
the rest of the lattice means there is essentially no chance thatthe photon will be reflected back into the coupling waveguide.
Hence, for weak coupling, large reflection is associated withsmall photon density of states.

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PHYSICAL REVIEW A 89, 033854 (2014)

IV. TRANSPORT OF PROBE LASER PHOTONS IN THE
PRESENCE OF OPTOMECHANICAL COUPLING

In the previous section, we examined how even a weak
many-photon optomechanical interaction could lead to a
suppression of the photon DOS, and how this could manifest
itself in the local reflection of a probe laser beam. In thissection, we now address the more general and rich question of
the transport of photons injected by a probe laser beam: How
does the probe light injected at site 1 propagate through thelattice to site N + 1?
We will again focus exclusively on the experimentally
relevant parameter regime where the optomechanical coupling
G is small: G  J,M . In addition, to avoid any possibilityof parametric instability [and hence again justify the use of the
RWA Hamiltonian in Eq. (6)], we will focus on control laser
detunings  < −2J ; for this regime, the bare photon band [asgiven by Eq (5)] always has positive energy in our interaction
picture.
A. Calculation of transmission amplitude

For a probe-light transmission experiment, we imagine
coupling auxiliary waveguides to sites 1 and N + 1 of ourD̃cR (ω; N + 1,1) =t˜i [ωp ] (i = 1,N + 1) describe the transmission of photons
between the lattice and the two external waveguides, whiler̃i [ωp ] describes the back-reflection of photons in the lattice atsites 1, N + 1 due to the effective impedance mismatch causedby the external waveguide couplings. One finds
κex,i;
κex,i + 4J sin q[ω]t˜i =d̂N+1,out (t)=d̄1,in e−iωp t
d̂1,in (t)√= −i κex,1 κex,N+1 D̃cR (ωp ; N + 1,1).

t[N,ωp ] =d̂N+1,out (t)8J κex,i sin q[ω].
κex,i + 4J sin q[ω](29)Note that both the optomechanical interaction G andthe presence of internal dissipation in the lattice enter
completely through the effective complex wave vector q[ω][c.f. Eq. (15)].
B. Differences from transport in atomic EIT systems

Before proceeding with our analysis, it is worth pausing toemphasize a key difference between probe photon transport in(26)

Here, D̃cR (ωp ; N + 1,1) is the retarded photon Green function
calculated in the presence of the couplings to the probewaveguides (which break translational invariance).
We can again calculate D̃cR (ωp ; N + 1,1) by introducing an
imaginary potential for cavity photons on sites 1 and N + 1,†d̂ d̂ , and then solve the resulting
i.e., V̂ = −i i=1,N+1 κex,i
2 i i
Dyson equation for the full GF. The GF is then found to be(see Appendix B for details)

8iJ sin q[ω] eiq[ω]N,κex,1 κN+1,ex (ei2Nq[ω] − 1) − 4J sin q[ω](κex,1 + κN+1,ex ) − (4J sin q[ω])2

where the complex propagation wave vector q[ω] ≡ k̃[ω] +iα[ω]/2 is independent of the waveguide couplings, and isagain defined via Eq. (15).
For more intuition, note that the presence of two coupling
waveguides makes our system somewhat analogous to a FabryPerot interferometer, as these couplings can induce multiple
local reflections within the lattice. It is thus useful to rewrite thetransmission obtained from Eqs. (26) and (27) in the standardform:t˜1 [ωp ]t˜N+1 [ωp ]eiq[ωp ]Nt[N,ωp ] =.
(28)
1 − r̃1 [ωp ]r̃N+1 [ωp ]e2iq[ωp ]Nr̃i =lattice, as shown in Fig. 1(b). These additional waveguide
couplings are treated analogously to what was done inSec. III D; they increase the photon damping rate on sites
1 and N + 1 to κ0 + κex,1 ≡ κ1 and κ0 + κN+1,ex ≡ κN+1 ,respectively. We assume that a weak probe tone is incident
on site 1 at frequency ωp , and then ask what the induced
amplitude of the outgoing light is leaving site N + 1 throughthe probe waveguide. Using standard input-output relations
combined with linear response (analogously to Sec. III D), we
obtain the transmission coefficient:(27)

our optomechanical lattice, and light propagation in standard
atomic EIT systems. Recall that atomic EIT involves threelevel “lambda” -system atoms, with two lower levels |g1,|g2
and an upper excited level |e. In an atomic EIT medium, thephysics can be best understood in terms of dark-state polaritons[36,37]: Light propagates as a coherent superposition of aphotonic excitation and an atomic excitation |g1 → |g2, insuch a way that destructive interference prevents the uppermost
atomic level |e from being populated. In the analogy to OMIT
[19,20], the role of the |g1 → |g2 excitation is played bya mechanical phonon excitation, while the role of the upperatomic excitation |e is played by having a photon in the cavity.
Thus, the destructive interference that prevents excitation to|e in atomic EIT now prevents a probe photon from entering
the cavity in OMIT.
Turning to propagation physics, in our system probe
photons can only be transported through the lattice by entering
the cavity. As such, this cannot be mapped onto an atomic
EIT dark state polariton, as having a photon enter the cavity in
our system is analogous to exciting an atom to the upper |estate in the atomic system. For this basic reason, the physics
we describe here is not equivalent to light propagation in an
atomic EIT medium [27]. We note that the alternative system
studied by Chang et al. [26], where optomechanical cavities
are periodically side coupled to a waveguide, is more closely
analogous to this situation. In that system, one can discussprobe photon propagation in terms of a dark-state polariton
involving a probe photon in the waveguide, and a phonon in
the mechanical mode; having a photon in the cavity is notnecessary.

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WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

C. Transmission for a weak probe waveguide-lattice coupling

For very weak probe couplings, the reflection amplitudesr̃1 ,r̃N+1 are negligible, and the transmission is well approximated by the numerator of Eq. (28). Equivalently, one caninterpret the transmission as being a probe of the intrinsic
photon GF DcR (ω; N), similar to the local reflection coefficientin Eq. (23) in this limit. However, in contrast to the reflection
coefficient, one is now probing the intrinsic GF at nonzero
propagation distance N .
The expectation would thus be that t[N,ωp ] is simply
a weighted measure of the photon density of states, where
the contribution of each polariton is weighted both by how
photonic it is [i.e., the appropriate factor of sin θk or cos θk ; c.f.
Eq. (10)], and by a factor describing attenuation. Attenuation
due to propagation is described by the factor α[ω] given in
Eqs. (15) and (16). In the absence of any optomechanicalinteraction, and for κ0  J , it is simply given byα[ω] =

κ0κ0=,
2J sin k[ω]vg [ω](30)

where vg [ω] = 2J sin k[ω] is the group velocity of the barephoton band, and k[ω] is the photon wave vector corresponding
to the band energy ω.
Based on the above, one might conclude that polaritonsthat are almost entirely phononlike will make a negligible
contribution to the transmission of probe photons. This isincorrect, as is shown in Fig. 4. Here, we pick parameterssuch that the bare phonon and photon bands only touch at
k = 0. As such, the lower polariton band is almost entirely
phononlike (except for a narrow range of k vectors near k = 0,i.e., near the bottom of the − band). The figure shows thetransmission probability of probe light from site 1 to 11, in a
regime where the internal cavity loss is relatively large. We see
that, surprisingly, the phononlike lower polariton band makes
an appreciable contribution to the transmission over its fullaD. Transport for strong waveguide coupling: interplayof Fabry-Perot interference and OMIT

We now consider probe photon transmission in the regime
where the couplings to the probe waveguides are large,
κex > J . Unlike the weak-coupling regime considered in theprevious subsection, transmission probabilities of order unity
are now possible. From Eqs. (28) and (29), we see that a
new aspect of the physics in this regime will be geometric,

c
5.0

0.4
sin2 θk

3.0
2.0

4.0
105 t 2 N 10,

4.0
105 t 2 N 10,

G2
κ0 .
(31)
J2
If this induced optical damping dominates the intrinsic
damping γ of the phononlike modes in the − band, thenthese modes can be used for probe photon transport with atransmission on par with the purely photonic modes in the +polariton band.
The transmission profile associated with the photonlike +polariton band in Fig. 4 reflects the shape of the correspondingDOS, exhibiting van-Hove singularities at the band edges. In
contrast, the transmission profile of the phononlike − polariton
band is more asymmetric; this reflects the variation of themixing angle θk [c.f. Eq. (10)] through the Brillouin zone.
opt ∼

b

0.5

5.0

width: It yields maximum transmissions that are on par withthose associated with the photonlike upper polariton band.
The explanation is straightforward: even though most −band polaritons are largely phononlike, their dissipation (forthese parameters) is dominated by photonic losses (κ0 ,κex )associated with the small photon part of their wave function.
As a result, they are effectively as well coupled to the external
probe waveguides as states in the more photonlike + polariton
band. Focusing on k vectors away from k = 0, we see from
Eq. (9) that the photonic amplitude associated with a −polariton scales as sin θk ∼ G/J  1. As such, the “optical
damping” of such a mode will be

0.3
0.2
0.1

1.0
0
10 2 10 3 10 4 10 5
M J

0
10 2

3.0
2.0
1.0
0

10 3

10 4
M J

10 5

0

1

2
M

3
J

4

FIG. 4. (Color online) Transmission probability of probe light from sites 1 to 11 as a function of probe frequency, for the case of a relatively
weak coupling to probe waveguides (κex,1 = κN,ex = 0.01J ), and for relatively large internal cavity loss (κ0 = 0.1J ). We have picked parameterssuch that the bare photon and phonon bands just touch at k = 0 (i.e., δ0 = − − M − 2J = 0), and taken G = 0.1κ0 ,γ = 10−6 κ0 . (a) Shows
the contribution from the lower, phononlike polariton band (using a logarithmic ω axis); (b) shows on the same scale the variation of thephotonic component of the − polariton as a function of frequency; (c) shows the transmission probability from the photonlike upper polariton
band. Surprisingly, both polariton bands can make comparable contributions to the photon transmission in this limit; this is due to the fact thatthe optical damping of the lower polariton band opt is much larger than γ .

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PHYSICAL REVIEW A 89, 033854 (2014)

FIG. 5. (Color online) Transmission probability for probe light from sites j = 1 to j = 21, as a function of the probe beam frequency ω, inthe regime of a strong coupling to the external probe waveguides: κex = 5J . We also take a control laser detuning yielding δ0 = 0 [c.f. Eq. (8)],implying that the bare photon band just touches the bare phonon band at k = 0. (a) (Blue curve) Transmission spectrum without optomechanicalcoupling, G = 0, and with κ0 = 0.01J . One sees well-developed Fabry-Perot style resonances associated with the bare photon band. (Redcurve) Same, but now with G = κ0 /2 and γ = 10−5 κ0 . The main effect of increasing the optomechanical coupling is the emergence of an
extremely sharp transmission feature near ω ∼ M , associated with the phononlike − polariton band. (b) and (c) Zoom-in of the − polariton
band features in (a). One again sees Fabry-Perot style resonances, leading to features in the transmission that are much narrower than κ0 .

Fabry-Perot style resonances caused by constructive interference of photons in the lattice reflecting between sites 1
and N + 1. Understanding how these geometric resonances
are modified by the optomechanical coupling and the physics
underlying OMIT will be a key goal of this subsection. Notethat the same interplay of OMIT physics and Fabry-Perot
interference can happen in a finite lattice with probe couplingat the end sites 1 and N + 1, resulting in similar transmission
properties; see Appendix C for details.
We again focus on the simple case where bare photon
and phonon bands only touch at k = 0, and where G M ,J ; for such a system, the − polariton band is mostly
phononlike, whereas the + band is mostly photonlike. Shownin Figs. 5(a)–5(c) is an example of the transmission spectrum
in this regime, both with and without the optomechanicalcoupling G. Without the coupling [Fig. 5(a)], one sees the
expected pattern of Fabry-Perot resonances associated withthe photon band; resonances occur whenmk[ω] = ± π − π ≡ ±km , m = 0,1,...,N. (32)N
Turning on the optomechanical interaction but keeping it
weak (0 < G < κ0 ), we see that the main features present for
G = 0 persist; these are now resonances associated with the+ polariton band, which is mostly photonic. However, despite
the weak value of G, one obtains a pronounced sharp featurenear ω = M associated with the phononlike − polariton
band. Figures 5(b) and 5(c) show zoom-ins of this feature.
One sees well-defined Fabry-Perot resonances associated withthis extremely narrow − band. Remarkably, we see that theseresonant features are much sharper in frequency than the
intrinsic photon loss rate κ0 . We thus have one of our key
results: Phononlike polariton modes can yield extremely sharp(width ω  κ0 ), large-magnitude transmission resonances,even if the optomechanical interaction is weak.
For weak internal dissipation, the energies in the −polariton band giving resonances can be easily found bycombining Eqs. (7) and (32). For small G, we can simply
expand in G to describe all resonances except those at the verybottom of the band. The spacing of the resonances is then wellapproximated asE−,m ≡ |E−,m − E−,m+1 |π
2G2
J sin km2N,(δ0 + 2J (1 − cos km ))(33)

where δ0 is the detuning between the bare photon and phononband at k = 0; c.f. Eq. (8). Thus, for resonances in the middle ofthe band, we have E−,m ∼ G2 /(J N ), where for resonancesnear the edge of the band, we have an even smaller scale,E−,m ∼ mG2 /(J N 2 ).
Given the extremely small frequency separation of theseresonances, one might think that even a tiny amount of intrinsic
photon damping κ0 would be enough to completely suppress
them. As seen dramatically in Figs. 5(b) and 5(c), this is
not the case: The resonances remain sharp even though theirseparation is much smaller than κ0 . The explanation is simply
that the majority of the resonances are due to polaritons which
are mostly phononlike, and hence only weakly see the photon
loss rate κ0 .

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WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

To make this more quantitative, note that the dominant
effect of intrinsic dissipation is to make the effective wave
vector q[ω] in Eq. (28) complex. From the denominatorof Eq. (28), this means that perfect interference between
subsequent reflections will be impossible. By insisting that thisdeviation from perfect constructive interference in Eq. (28) is
1, we can derive a quantitative condition for how weak
the internal dissipation has to be to see the Fabry-Perot
resonances in the − polariton band. Consider the relevant caseκex,jJ | sin q|, where the transmission coefficients from theprobe waveguides to the lattice t˜j in Eq. (28) have a small
magnitude, and the reflection coefficients r̃ have magnitude
near 1. In this case, the condition becomes




eff
J | sin(km )| |t˜|2 = √.
(34)
Nα = Nvg
κex,1 κex,N+1

V. CONCLUSION

Here eff is the effective damping rate of the resonant polariton,and vg is its group velocity.
We can further simplify this condition by focusing on
resonances away from the bottom of the band, where a small-G
expansion of the − polariton dispersion relation is possible.
For such resonances, the group velocity vg ∼ (G2 /J ) sin km ,while the effective damping rate eff is just the sum of themechanical damping rate γ and the optical damping rate
(G2 /J 2 )κ0 . Taking κex,1 = κex,N+1 = κex for simplicity, ourcondition becomes 2  m2 2
GG2
if m  N ;π
N3
max γ , 2 κ0 
J
κex
1/N
if m ∼ N/2 .

We have presented a study of OMIT-type effects in a
one-dimensional optomechanical lattice with photon hopping
between adjacent sites, and with a strong control laser applied
to each site to enhance the optomechanical interaction. Similar
to single-cavity OMIT, we find that the photon densityof states can be strongly suppressed by the many-photon
optomechanical interaction G, even in regimes where it isextremely weak compared to the intrinsic photon loss rate κ0 ;we discussed how these effects could be measured via the local
reflection of probe photons.
We also studied the transport of a beam of probe photons
through the lattice, showing both that phononlike polariton
modes can be surprisingly effective in transport, and that
one can have an interesting interplay between OMIT-styleeffects and geometric resonances in the lattice. One finds theinteresting result that phononlike polaritons can give rise tohigh-magnitude transmission resonances whose spectral widthis much narrower than the intrinsic photon loss rate κ0 .
The general physics we describe could play an importantrole when one now tries to understand the effects of disorder in
the OMIT lattice, as disorder could also give rise to geometricresonances. Our work also demonstrates the importance ofviewing an optomechanical lattice as a driven, dissipative
system; as we have stressed, the OMIT-style interference
effects that arise are a direct consequence of the structured
dissipation in our system, and cannot be described by simply
energy broadening the results expected for the coherent(dissipation-free) system.

(35)
ACKNOWLEDGMENTS

2

Figure 5 corresponds to the case γ < G
κ . One can easily
J2 0
2
κ
,
the
resonance peaks
verify that in the opposite case γ > G
J2 0with width much narrower than κ0 can also be resolved only
if Eq. (35) is satisfied. We thus see that while κ0 needs to be
sufficiently small to get well-defined resonances, the relevant
condition is in general not as stringent as requiring that κ0 be
smaller than the spacing of the resonances.
We note that the interplay of OMIT and Fabry-Perot
interference discussed here occurs in a similar fashion in a
finite optomechanical lattice. One could imagine two distinct
situations. The first is where the probe waveguides are coupled
to sites in the interior of the lattice. In this case, the results forthe infinite lattice are recovered as long as κex is sufficiently
large; the additional reflections possible at the ends of thelattice play a minor role, as the probability of being transmitted
past the sites coupled to probe waveguides is weak.
The second situation is where the probe waveguides arecoupled to the end sites of the lattice. This case is treated indetail in Appendix C. As shown in that section, one obtains aqualitatively similar transmission spectrum to that in Fig. 5, but
now for a finite array (keeping other parameters the same). On a
qualitative level, the important physics here is the possibility ofpolariton reflections at sites 1 and N . In the infinite lattice case,these are only due to the impedance mismatch caused by the
coupling to the probe waveguide; in the finite lattice, one also
has reflections from the end of the lattice. This difference does
not play a large role in the qualitative nature of the interplay
we discuss.

We would like to thank Marc-Antoine Lemonde for useful
discussions. This work is supported by NSERC and the
DARPA ORCHID program under a grant from the AFOSR.
W.C. would like to thank the Institute of Advanced Study atTsinghua University for their hospitality during the writing
stage of this paper.
APPENDIX A: HEISENBERG-LANGEVIN EQUATIONSIn this Appendix, we have a brief derivation of the equationsof motion for the photon and phonon fields in momentum space
under the rotating wave approximation. The retarded GF in
Eq. (13) of cavity photons in k space then follows from theequations of motion (EOM) of the photon and phonon fields.
The RWA Hamiltonian in k space without coupling to thebath is shown in Eq. (6). The coupling to the bath is treated asin the standard input-output theory. The equations of motion of
the photon and phonon fields after this treatment of the baths
are


∂ d̂k
κ0
√
= ωk − id̂k + Gb̂k − i κ0 ξ̂k ,
i
∂t
2
(A1)


∂ b̂k
γ
√
= m − ib̂k + Gd̂k − i γ η̂k ,i∂t2where ξ̂k and η̂k are the noise operators of the baths which thephoton and phonon modes couple to, respectively.

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PHYSICAL REVIEW A 89, 033854 (2014)

Denoting φ̂k = ( b̂d̂kk ), the retarded GF in k spacefrom Eq. (14) to be†

DR (k; t,t ) ≡ −iθ (t − t )[φ̂k (t),φ̂k (t )] satisfies(i∂t − Mk )G (k; t,t ) = δ(t − t ),Rκ0where Mk = (ωk −Gi 2R(ω; 1,1) =
D0,c

(A2)

G
m − i γ2 ) is obtained from the EOM in

Eq. (A1).
Fourier transforming Eq. (A2) to the frequency domain,
one gets the retarded GF in the frequency domain as
DR (k,ω) = (ω1 − Mk )−1 .

APPENDIX B: CALCULATION OF PHOTON GREEN
FUNCTION IN THE PRESENCE OF EXTERNALWAVEGUIDES

In this Appendix, we present the calculation to obtain thefull retarded Green’s function of cavity photons in Eqs. (25)and (27) taking into account the effect of coupling to the probewaveguide.
We first get the local retarded Green function of cavity
photons in Eq. (25) dressed by probe coupling.
The coupling to probe waveguide on site 1 for reflection
experiments introduces an effective imaginary potential tosite 1, which in the position representation is(B1)

and vanishes on all other sites. The bare retarded Green’s
Rfunction of cavity photons in real space D0,c
(ω; 1,1) is obtained

DcR (ω; 1,1)

DcR (ω; 1,N + 1)

(B2)

From Dyson’s equation, the full local retarded Green’s
function of cavity photons can be obtained asRR(ω; 1,1) +D0,c(ω; 1,1)V (1,1)DcR (ω; 1,1).
DcR (ω; 1,1) = D0,c

(B3)

(A3)

The retarded GF DR (k,ω) obtained above is a 2 × 2 matrix.
The (1,1) component gives the retarded GF for the photon
component DcR (ω,k) in Eq. (13).

V (1,1) = −i 12 κex,1 ,1.
2iJ sin q[ω]The full retarded Green’s function DcR (ω; 1,1) is thenDcR (ω; 1,1) =1−1
D0,R c (ω; 1,1)
− V (1,1),(B4)

as obtained in Eq. (25).
For the transmission experiments, two probe waveguides
are introduced on site 1 and site N + 1, respectively. The
couplings to the waveguides introduce an effective imaginary
potential to site 1, i.e., V (1,1) = −i 12 κex,1 , and site N + 1, i.e.,
V (N + 1,N + 1) = −i 12 κex,N+1 , respectively. These potentials
can be written as a matrix in the space spanned by states |1
and |N + 1 asV (1,1)
V (1,N + 1)
V=
V (N + 1,1) V (N + 1,N + 1)0−i 12 κex,1
.
(B5)
=
0
−i 12 κex,N+1
The full retarded Green’s functions of cavity photons satisfy
the Dyson’s equation,

DcR (ω; N + 1,1) DcR (ω; N + 1,N + 1)


R
R
D0,c
(ω; 1,1)
D0,c(ω; 1,N + 1)=RR
(ω; N + 1,1) D0,c
(ω; N + 1,N + 1)
D0,c

 
R
R
D0,c
(ω; 1,1)
D0,c(ω; 1,N + 1)
DcR (ω; 1,1)+V
RR
(ω; N + 1,1) D0,c
(ω; N + 1,N + 1)
D0,cDcR (ω; N + 1,1)

The bare retarded Green’s function of cavity photonsR(ω; N,1) is given by Eq. (14). Combined with Eq. (B6), theD0,cfull retarded Green’s function of cavity photons DcR (ω; N +1,1) can be obtained as shown in Eq. (27).
APPENDIX C: PHOTON GREEN’S FUNCTION IN A
FINITE OPTOMECHANICAL ARRAY

In this Appendix, we present the retarded Green’s functionof cavity photons in a finite optomechanical array with N +1 cells. We will show that the Fabry-Perot interference also

DcR (ω; 1,N + 1)
DcR (ω; N + 1,N + 1)


.

(B6)

occurs in a finite optomechanical array of N + 1 cells withprobe coupling on the two end sites. In the limit κex,1 ,κex, N+1
J the full retarded Green’s functions DcR (ω; N + 1,1) for theinfinite array and finite array reduce to the same form so thetransmission properties are the same in the two systems in thislimit.
The retarded Green’s function of cavity photons in a finite
optomechanical array can be obtained from the bare retarded
Green’s function in an infinite array following the sameprocedure in Appendix A.

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WEI CHEN AND AASHISH A. CLERKPHYSICAL REVIEW A 89, 033854 (2014)

We first obtain the retarded Green’s function in a semi-infinite array by introducing an effective potential to cut the
hopping between site 0 and site 1 in an infinite array. In the position representation spanned by the position states |0 and |1, theeffective potential can be written asV (0,0)V=
V (1,0) 0V (0,1)=
V (1,1)JJ.
−i 12 κex,1(C1)Here V (1,1) = −i 12 κex,1 takes into account the external decay rate due to coupling with the probe waveguide on site 1.
R
The retarded Green’s function Ds,c
(ω; n,n ) in a semi-infinite optomechanical array can be obtained from the Dyson’s equations
below: 
T (0,0) T (0,1)
V (0,0) V (0,1)=
T (1,0) T (1,1)
V (1,0) V (1,1)



 R
R
(ω; 0,1)
D0,c (ω; 0,0) D0,c
T (0,0) T (0,1)
V (0,0) V (0,1);+
R
R
T (1,0) T (1,1)
V (1,0) V (1,1)
D0,c(ω; 1,0) D0,c
(ω; 1,1)
(C2)


 
R
R
R
R
(ω; n,n) D0,c
(ω; n,n )D0,c(ω; n,n) Ds,c
(ω; n,n )
Ds,c
=
R
R
R
R
D0,c
(ω; n ,n) D0,c
(ω; n ,n )(ω; n ,n) Ds,c
(ω; n ,n )
Ds,c

+

R
D0,c
(ω; n,0)RD0,c(ω; n ,0)T (0,0)R
T (1,0)D0,c (ω; n ,1)RD0,c(ω; n,1)

T (0,1)
T (1,1)R(ω; 0,n)
D0,c

R
D0,c
(ω; 0,n )RD0,c(ω; 1,n)RD0,c(ω; 1,n ).

We get the retarded Green’s function for a semi-infinite optomechanical array asR(ω; n,n ) =Ds,c1(eiq[ω]|n−n | − eiq[ω]|n+n | ζ1 ),2iJ sin q[ω](C3)

where q[ω] is the same as defined in Eq. (15) and ζ1 is definedaswhere Z ≡ eiq[ω] and ζN+1 is defined as

1 − i κex,1e−iq[ω]2
ζ1 =
κex,1 iq[ω] .
1−i 2 eζN+1 =(C4)

To get the retarded Green’s function for the finite array with
N + 1 cells, we follow the same idea as above and introduce
an effective potential to cut the hopping between site N + 1
and site N + 2 as
V (N + 1,N + 1) V (N + 1,N + 2)
V=
V (N + 2,N + 1) V (N + 2,N + 2)

 1
−i 2 κex, N+1 J
.
=
J
0

(C5)

Here V (N + 1,N + 1) = −i 12 κex, N+1 takes into account theexternal decay rate due to coupling with the probe waveguideon site N + 1.
Following the same procedure in Eq. (C2), we get theretarded Green’s function of cavity photons in a finite array of
N + 1 cells as

1 − i κex,2N+1 e−iq[ω].
1 − i κex,2N+1 eiq[ω](C7)

One can easily verify that Eqs. (C6) and (27) share the samedenominator of Eq. (27). Since the Fabry-Perot interference ismanifested by the same parameter Z 2N − 1 in the denominator,
the two systems show very similar properties of Fabry-Perot
interference, i.e., the resonance frequencies and width of thetransmission spectrum in the two systems are the same, though
the peak heights of the two systems are different due to thedifferent numerators of the Green’s functions in Eqs. (C6) and(27). For comparison, the transmission spectrum from the firstsite to the last site of a finite array of N + 1 = 21 cells isshown in Fig. 6, with the same parameters as in Fig. 5.
In the limit κex,1 ,κex, N+1
J , one can Taylor expand
Eqs. (C6) and (27) treating J /κex,1 and J /κex, N+1 as small
parameters and the retarded Green’s functions of the twosystems reduce to the same form,
DcR (ω; N + 1,1) =8J Z N sin q[ω].
√(Z 2N − 1) κex,1 κN+1,ex(C8)R(ω; N + 1,1)Dc,f(−1 + ζ1 Z 2 )(−ζN+1 + 1/Z 2 )1=,2i sin q[ω] ζ1 ζN+1 Z N+2 − 1/Z N+2(C6)

In this limit, the two systems are equivalent and the transmission properties from site 1 to site N + 1 are the same.

033854-12

PHOTON PROPAGATION IN A ONE-DIMENSIONAL . . .

PHYSICAL REVIEW A 89, 033854 (2014)ab
T N 21,Ω

0.8

0.4
0.2
0
0.25

0.2

0.15
M

0.1
Κ0

0.05

M

0.001
Κ0

0

0.4
c
0.2

T N 21,Ω

T N 21,Ω

0.6

0.6

G 0
G 0.005J
1

2
M

3

0.4
0.2
0
0.002

0.0
0

0.6

4

J

0.00150.0005

FIG. 6. (Color online) Transmission probability for probe light from sites j = 1 to j = 21 for a finite optomechanical array of 21 cells,
as a function of the probe beam frequency ω. (a) (Black curve) Transmission spectrum without optomechanical coupling, G = 0. (Red curve)
Transmission spectrum with G = κ0 /2. All the other parameters are the same as in Fig. 5. (b) and (c) Zoom-in of the − polariton band featuresin (a). One again sees Fabry-Perot style resonances. The location and width of the Fabry-Perot resonances are almost the same as in the infinite
array in Fig. 5, whereas the resonance heights are different.

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