Quantum many-body models with cold atoms coupled to photonic crystals

J. S. Douglas,1, ⇤ H. Habibian,1 C.-L. Hung,2, † A. V. Gorshkov,3 H. J. Kimble,2 and D. E. Chang1

1ICFO-Institut de Ciencies Fotoniques, 08860 Castelldefels, Barcelona, Spain2Norman Bridge Laboratory of Physics and Institute for Quantum Information and Matter,

California Institute of Technology, Pasadena, California 91125, USA3Joint Quantum Institute and Joint Center for Quantum Information and Computer Science,

NIST/University of Maryland, College Park, Maryland 20742, USA(Dated: March 3, 2015)

Using cold atoms to simulate strongly interacting quantum systems represents an exciting fron-tier of physics. However, achieving tunable, coherent long-range interactions between atoms is anoutstanding challenge, which currently leaves a large class of models inaccessible to quantum sim-ulation. Here, we propose a solution exploiting the powerful new platform of cold atoms trappednear nano-photonic systems. We show that the dielectric contrast of an atom trapped near a pho-tonic crystal can seed a localized cavity mode around the atomic position. In a dynamic form of“all-atomic” cavity QED, the length of these cavity modes can be tuned, and atoms separated bythe order of the e↵ective cavity length can interact coherently with each other. Considering realisticconditions such as fabrication disorder and photon losses, coherent long-range potentials or spininteractions can be dominant in the system over length scales up to hundreds of wavelengths.

Trapped ultracold atoms are a rich resource for physi-cists. Isolated from the environment and routinely ma-nipulated, they can act as a quantum simulator for awide variety of physical models [1]. However, while short-range interactions between atoms can be adjusted by Fes-hbach resonance, long-range interactions are typically ab-sent from these systems, leaving a large class of modelsinaccessible to quantum simulation. For example, exoticphases such as the supersolid are predicted in systemswith long-range interactions [2], as well as Wigner crys-tallization [3] and topological states [4]. Long-range in-teractions can also lead to the breakdown of conceptssuch as additivity in statistical mechanics [5] and theviolation of speed limits (Lieb-Robinson bounds) for thepropagation of information [6]. Attempts to achieve long-range interactions with neutral atoms have focused oncreating systems where the particles interact via someintrinsic property [7], for example atoms with large mag-netic moment [8, 9], polar molecules [10], or atoms withRydberg excitation [11, 12]. In all of these scenarios,however, the range of the interaction is fixed and falls ofwith distance like 1/r⌘, where ⌘ � 3.

Here we investigate another paradigm, where insteadof relying on the atomic properties, we design the mediumvia which the atoms interact. Our proposal is inspiredby demonstrations of strong coupling between photons innanophotonic systems and individual solid-state emitters[13] or more recently with cold atoms [14–17]. For exam-ple, systems consisting of ⇠ 103 atoms trapped by andcoupled to the evanescent guided modes of nanofibershave been demonstrated [14, 15], along with couplingof single atoms with photonic crystal cavities [16] andwaveguides [17]. While an original motivation of thesee↵orts was to utilize strong, controlled light-matter in-teractions for quantum information processing and net-works [18], we propose that atoms interfaced with pho-

tonic crystals can also have remarkable consequences forthe exploration of quantum many-body physics [19–21].

The optical modes of a photonic crystal di↵er signif-icantly from those in free space or in a cavity, as theperiodicity of the dielectric leads to a photon dispersionwith band structure. These band structures may alsoexhibit band gaps, frequency ranges where no guidedmodes exist [22]. A conventional photonic crystal cav-ity is fabricated by introducing a local dielectric defect,which pulls a discrete localized mode into the gap fromthe continuum. Here, we show that the dielectric con-trast provided by a single atom trapped near an oth-erwise perfect photonic crystal can itself seed a cavitymode. When many atoms are trapped, coherent interac-tions can take place between atoms [23, 24], mediated bythese dynamically induced cavities. As we show, these in-teractions can realistically extend over distances of orderof 100 optical wavelengths and they can be dynamicallymanipulated in strength and form, providing a new toolto realize fully tunable long-range interactions with coldatoms. More generally, our work provides a glimpse intothe exciting new regimes of physics that are possible withatom-photonic crystal interfaces, which fundamentally gobeyond the transfer of paradigms in macroscopic cavityQED to nanoscale platforms.

E↵ective long range interactions

Long-range interactions between particles often oc-cur through the exchange of photons. A simple exam-ple consists of two-level atoms with transition frequency!a

= 2⇡c/� interacting via a single mode of a Fabry-Perot cavity that has resonance frequency !

c

, as shownin Fig. 1a. Momentarily neglecting losses, a single atomin the cavity is described by the Jaynes-Cummings (JC)

arX

iv:1

312.

2435

v2 [

quan

t-ph]

28

Feb

2015

2

�

|ei

|gi

|ei

|gi

atom 1 atom 2

!k

k ⇡a

a

!a

!b

�

a

b

c

d

band gap

FIG. 1. Progression from cavity-QED to atom-inducedcavities in photonic crystals (a) Two atoms are coupledwith strength gc to a single mode of a Fabry-Perot cavity,enabling an excited atom (atom 1) to transfer its excitationto atom 2 and back. The coherence of this process is reducedby the cavity decay (rate ) and atomic spontaneous emissioninto free space (rate �). (b) Photonic crystals are alternatingdielectric materials, shown here as oval air holes in a dielectricwaveguide, with unit cell length a. A defect, such as causedby removing or altering the hole sizes, can lead to a localizedphotonic mode (red). Atoms coupled to such a system maythen interact via this mode in an analogous manner to in(a). (c) A typical band structure of a 1d photonic crystal,illustrating the guided mode frequency !k versus the Blochwavevector k in the first Brillouin zone. We are interested inthe case where atoms coupled to the crystal have resonancefrequency !a close to the band edge frequency !b, with � ⌘!a � !b. (d) An atom near a photonic crystal can act as adefect, inducing its own cavity mode with an exponentiallydecaying envelope (red). A second atom can couple to thismode to produce an interaction similar to in (a) and (b), butwhere the strength now depends on the inter-atomic distance.

model [25], H = ~!a

�ee

+~!c

a†a+~gc

(�eg

a+h.c.), where�µ⌫

= |µih⌫| operate on the internal atomic state and thecavity mode excitation has associated annihilation oper-ator a. The coupling between the atom and the cavitymode g

c

= deg

p!c

/(2~✏0V ) depends on the strength ofthe dipole matrix element d

eg

of the two level transitionand on the cavity mode volume V . For a single excita-tion, the eigenstates are dressed states — superpositionsof the excitation being purely atomic and purely in thecavity mode — given by | 1i = cos ✓|ei|0i + sin ✓|gi|1iand | 2i = � sin ✓|ei|0i+ cos ✓|gi|1i.

When the detuning between the cavity mode and theatomic resonance is large, such that �

c

= !a

� !c

� gc

,the mixing angle becomes ✓ ⇡ g

c

/�c

⌧ 1, and | 1i ispredominantly an atomic excitation with a small pho-tonic component. A second atom introduced into thecavity can then exchange an excitation with the first viathe weakly populated cavity mode, leading to an e↵ective

interaction. For N atoms this gives [25]

HI

=~g2

c

�c

NX

j,l

�j

eg

�l

ge

, (1)

which describes exchange of excitations between atomswith strength that does not diminish with distance, onlybeing bounded by the volume of the physical cavity.These e↵ectively infinite range interactions, while inter-esting in their own right [26–29], remove the spatial com-plexity of the system, and can often be described usingcollective operators or mean-field methods.To realize long-range interactions that decay with dis-

tance we utilize periodic dielectric structures known asphotonic crystals [22]. Key to our proposal is thatthrough constructive interference of light scattering fromthe periodic structure, frequency windows known as bandgaps can be created in which no propagating modes ex-ist. A typical dispersion relation of photon frequency!k

versus Bloch wavevector k with a band gap is shownin Fig. 1c. Conventionally, a localized photonic crystalcavity mode is created by introducing a local dielectricdefect (Fig. 1b) that pulls a discrete mode out into theband gap from the continuous band spectrum [22]. Here,we show that an atom trapped near the photonic crystalis itself a dielectric defect and capable of seeding a cavitymode localized around the atomic position, via which itcan interact with other atoms (Fig. 1d).

Atom induced cavities and long-range interactions

The interaction between atoms and band edges hasbeen discussed in a number of contexts, such as the for-mation of atom-photon bound states [30] and atomic in-teractions [23, 24, 31, 32]. Here, we provide an elegantinterpretation of this physics in terms of atom-inducedcavities and cavity QED. This mapping enables the pow-erful toolbox of cavity QED to be transferred to thesesystems, and enables one to identify key figures of merit(such as mode volume and cooperativity parameter). Weexploit this mapping to demonstrate that the type of spininteraction and the spatial range can be manipulated dy-namically, with tunability that opens the door to simu-lation of a wide range of long-range interacting models.For the first time, we also identify the limits imposed bysystem imperfections (such as losses and disorder), andanalyze in detail a realistic structure wherein this novellong-range physics can be realized.A simple model to illustrate this mechanism consists of

two-level atoms coupled to the photonic crystal modes,and where the atomic resonance is close to one of theband edges of the photonic crystal (Fig. 1c), with de-tuning � = !

a

� !b

. We assume that the detuningto any other band edge is much larger than � so thatthe influence of other bands is negligible. When the

3

atomic resonance is close to the band edge, the atomis dominantly coupled to modes close to the band edgewavevector k0 due to the van Hove singularity in thedensity of states. In this case we can make the e↵ectivemass approximation and take the dispersion relation tobe quadratic !

k

⇡ !b

(1�↵(k�k0)2/k20) about k0, where↵ characterizes the band curvature [30]. Due to the pe-riodicity of the photonic crystal the photonic modes areof Bloch form and the modes with wavevector k ⇠ k0take the form E

k

(z) ⇡ eikzuk0(z), with annihilation op-

erator ak

. Furthermore for these modes, the couplingg = d

eg

p!b

/(4⇡~✏0A) is approximately independent ofk, where A is the mode cross-sectional area [33].

A system with one atom trapped at z = 0 coupled tothe photonic crystal is then described by the Hamiltonian

H = ~!a

�ee

+

Zdk~!

k

a†k

ak

+~gZ

dk(�eg

ak

Ek

(0)+h.c.).

(2)For a single excitation in the system, we find an exacteigenstate of the Schrodinger equation, H| i = ~!| i,of the form |�1i = cos ✓|ei|0i+sin ✓|gi|1i (see Methods).Here |1i = a†|0i is the single photon state associatedwith photon operator a =

Rdkc

k

ak

. The frequency !lies within the gap, with detuning � = ! � !

b

> 0 fromthe band edge, as shown in Fig. 2a-b. The detuning �is the positive real root of (� � �)

p� = 2�3/2 for � =�

⇡g2|uk0(0)|2k0/

p4↵!

b

�2/3. In the limit � � � (i.e.,

the atomic frequency is deep within the band gap), theeigenmode frequency approaches that of the atom, � ⇡�, while the mode frequency approaches the band edgein the opposite limit when the atomic frequency is wellwithin the band (�� � �).

Importantly, the photonic component is localizedaround the atomic position, as illustrated in Fig. 2c, withspatial mode function

�(z) =

Zdkc⇤

k

Ek

(z) =

r2⇡

Le�|z|/LE

k0(z). (3)

The photon decays exponentially with distance z fromthe atomic position with length scale L =

p↵!

b

/(k20�),reflecting the fact that within the band gap at energy!, the field propagation equation has complex solutionswith attenuation length L.

This confined photonic cloud has the same proper-ties as a real cavity mode, enabling a mapping onto theJC model. Specifically, one can associate an e↵ectiveatom-cavity interaction strength g

c

= gp2⇡/L with the

bandgap system, which scales inversely with square rootof the mode volume, expressed here as the e↵ective cavitylength L for fixed mode area A. We can also identify ane↵ective cavity mode frequency that is the average fre-quency of the photon mode,

Rdk|c

k

|2!k

= !b

� �. Thee↵ective atom-cavity detuning is then �

c

= � + �, asshown in Fig. 2a. The state |�1i then maps to the dressedstate | 1i from the JC model, i.e., the mixing angle and

e

!b

!a

e↵ective cavity

bare atom

!c

� �

�

|�1i

�/�

L/a

�/�

�/�

�/�

� increasing

a b

c

dPp Pe

-10 0 10

0

10

-1

-1

1

0 50 100

10

2

10

3

10

4

-10 0 10

0

0.5

1

FIG. 2. E↵ective cavity mode properties (a) Energy leveldiagram for the photonic crystal dressed state |�1i (blue).The dressed state energy ! is detuned by � from the bandedge into the bandgap (band shown in red). The atom is cou-pled to an e↵ective cavity mode with frequency !c = !b � �formed by superposition of modes in the band. (b) The de-tuning � of the bound state resonance approaches 0 when� ⌧ 0 and approaches � when � � 0. (c) The photoniccomponent of the bound state has an exponentially decay-ing envelope around the atomic position. Increasing � de-creases the length scale L of the exponential decay. At thesame time the bound state superposition changes from be-ing mostly photonic to mostly atomic. (d) The atomic ex-cited state population, Pe = cos2(✓) (green), for state |�1iincreases as a function of detuning from the band edge, whilethe population of the photon mode, Pp = sin2(✓) (red), de-creases as the state switches from photonic to atomic. (e) Thelength of the e↵ective cavity decreases with �. Here L is inunits of the lattice constant a, calculated for ↵ = 10.6 and� = 4.75 ⇥ 10�7!b, which is consistent with the “alligator”photonic crystal waveguide [34] described in the implementa-tion section of the main text.

energy are the same for �c

! �c

and gc

! gc

. Themapping to the JC model breaks down when we considerthe second dressed state, which is not an eigenstate inthe photonic crystal model because of the continuum ofpropagating modes for frequencies below !

b

.

The atomic excited state population Pe

= cos2 ✓ for|�1i is plotted in Fig. 2d. For � � 0, most of the ex-citation resides in the atom, while for � ⌧ 0 the statebecomes mostly photonic, a cavity mode dressed by theatom. Physically, although the atomic frequency can liewell within the band, the atom still provides a weak re-fractive index contrast at frequencies within the gap, andin 1D, arbitrarily weak dielectric defects can seed a cavitymode [35]. In this regime (Fig. 2e), the weak dielectriccontrast yields a very long e↵ective cavity length. Inpractice, this length is only limited by the finite size ofthe photonic crystal structure or by disorder in the pho-

4

tonic crystal structure (as discussed below).It is now apparent that two atoms can exchange an ex-

citation via the induced cavity mode provided they areseparated by a distance . L. In the limit where the pho-tonic modes are weakly populated (� � �), this leadsto an e↵ective dipole-dipole interaction between atoms,with positions z

j

, of the form (see Methods)

HI

⇡ ~g2c

�c

NX

j,l

�j

eg

�l

ge

f(zj

, zl

). (4)

The e↵ective atom-cavity detuning is �c

= 2� (since� ⇠ �) and f(z

j

, zl

) = e�|zj�zl|/LEk0(zj)E

⇤k0(z

l

). Com-pared to the case of a conventional cavity (Eq. (1)), themain feature of interactions emerging from atom-inducedcavities is that the spatial function f(z

j

, zl

) is finite-rangeand tunable, both through the e↵ective interaction lengthL and the Bloch functions (e.g., from which changes insign can be engineered). In addition, this dynamic cav-ity mode follows the atomic position rather than beinga static property set by boundary conditions. Our re-sults also generalize to higher dimensions, e.g., in a 2Dphotonic crystal atom-induced cavities lead to a func-tion f(z

j

, zl

) ⇡ Ek0(zj)E⇤k0(z

l

)e�|zj�zl|/L/p|z

j

� zl

|(see Methods).

As in a real cavity, dissipation limits the possible co-herence time of the system. Imperfections in the pho-tonic crystal cause photon loss at rate

p

, and becauseour structures of interest are not full 3D photonic crys-tals, an excited atom can spontaneously emit into freespace at rate � (usually comparable to the vacuum rate�0 [33]). The e↵ect of losses can be revealed, for exam-ple, by studying the exchange of an excitation betweentwo atoms separated by |z1 � z2| . L. From Eq. (4),the exchange time is given by ⌧ ⇠ �

c

/g2c

, while the to-tal loss is given by ⌧(� cos2 ✓ +

p

sin2 ✓). Optimizingthe detuning, we find loss that scales as 1/

pC, where

C = g2c

/(p

�) is the single-atom cooperativity. For astate of the art photonic crystal (Q ⇠ 106) coupled toCesium atoms (�/(2⇡) ⇠ 5MHz), a cavity with volumeV ⇠ �3 (i.e. L = �) could have g

c

/(2⇡) ⇠ 5GHz, giv-ing a feasible cooperativity of C

�

⇠ 104 [16, 33, 36].Assuming that

p

is dominated by local imperfectionsin the photonic crystal, we expect that it is indepen-dent of the length L (as compared to a Fabry-Perot cav-ity in vacuum, where

p

/ 1/L). The cooperativityof the dynamic cavity mode then scales with length asC

L

= �C�

/L, which limits the length for which interac-tions remain coherent. For C

�

⇠ 104, CL

remains greaterthan 100 for lengths of up to 100 wavelengths.

Our approach to achieving long-range interactionspresents a number of interesting analogies with other sys-tems. In cold Rydberg gases [11, 12], for example, atomsare excited to electronic states with high principal quan-tum number n � 1. Large coherent interactions betweenatoms (scaling with distance like 1/r6) are facilitated by

the exchange of virtual photons, and by the long lifetimesof the Rydberg states. In comparison, we are able to real-ize tunable “Rydberg-like” physics using atoms in theirfirst excited state (principal quantum number n = 2),where the desired strengths and lifetimes are achieved bymodifying the dispersion and vacuum properties of theelectromagnetic field itself. It has also been suggestedthat coupled arrays of photonic crystal cavities contain-ing quantum emitters could be used to realize quantumsimulation [19–21]. However, these systems remain unre-alized in practice, in large part due to the extreme pre-cision required to fabricate homogeneous high-Q cavitiesthat can be coupled simultaneously to atoms. In com-parison, the homogeneity of our atom-induced cavitiesnaturally arises from the identical nature of the atoms.

Implementation in an “alligator” photonic crystalwaveguide

Our simple theoretical model should approximate wellactual physical implementations. As a concrete exam-ple, we consider the one-dimensional “alligator” photoniccrystal waveguide (APCW) as experimentally demon-strated in Ref. [17, 34]. The APCW, as shown inFig. 3a, consists of two parallel, periodically corru-gated nanobeams, whose small distance of separationcouples and hybridizes their optical modes. A combi-nation of far o↵-resonant guided modes and Casimir-Polder forces allows atoms to be localized between thebeams at the periodic points indicated in red in Fig. 3a[33]. The band structure of the APCW, calculated usingthe MIT Photonic-Bands software package, is shown inFig. 3a. The band edge of the fundamental transverseelectric (TE-like) mode, located at !

b

/2⇡ = 333 THz,is closely aligned with the D1 transition of atomic ce-sium to produce the desired long-range interactions.We now calculate the coe�cients U

jl

for the e↵ectiveatom-atom interaction H

I

= ~P

j,l

Ujl

�j

eg

�l

ge

associatedwith the APCW, in the case where the atoms are po-sitioned at the minima of the trapping potentials andwhen the atomic transition is polarized along y (thelocal direction of polarization for the fundamental TEmode). The couplings U

jl

are directly proportional tothe real part of the dyadic electromagnetic Green’s func-tion G

yy

(rj

, rl

,!a

) [37]. The Green’s function physicallydescribes the field produced at position r

j

due to a sourceat r

l

and frequency !a

, and is obtained for the APCWby finite-di↵erence time-domain (FDTD) simulations.In the inset of Fig. 3b, we plot the coupling nor-

malized by the free-space emission rate, |Ujl

|/�0 =|ReG

yy

(rj

, rl

,!a

)|/(2 ImGyy,free(0, 0,!a

)), whereGfree isthe free-space Green’s function. Four curves with atomicfrequency detunings from the band edge ranging from�/2⇡ = 400 GHz to 2.8 THz are shown, over atomicseparations r

jl

/a extending up to 55 lattice sites (a =

5

!b

/(2⇡)

a

b

Frequency

(THz)

330

340

0.48 0.50 0.52

x z

y

k(2⇡/a)

rjl

/a

|Ujl

|/�

a = 371nm

250nm

0 50

0

1

2

0 10 20 30 40 50

0

1

2

FIG. 3. Validation of the single-band model with nu-merical calculations. (a) Band structure of fundamentaltransverse-electric (TE) mode of the sample 1D “alligator”photonic crystal waveguide (APCW), designed for couplingto the D1 line of atomic cesium near the photonic bandedgefrequency !b/2⇡ = 333 THz [17, 34]. The calculated bandstructure has a curvature ↵ ⇡ 10.6 near the band edge atk0 = ⇡/a. Inset shows the dielectric profile of the APCW.Red circles denote the location of trapped atoms. (b) Atom-atom coupling strength Uij evaluated using finite-di↵erencetime-domain simulations (solid circles) and the single-bandmodel from Eq. (4) (solid lines). Results are plotted foratomic detunings from the band edge �/2⇡ = 400 (black),800 (red), 1300 (blue), 2800 (magenta) GHz. Contributionfrom all other photonic and free-space modes in the APCW(open circles) has been estimated numerically and subtractedo↵ from all FDTD results; see text. Inset shows numericallycalculated Green’s functions (solid circles) before the subtrac-tion, together with the single-band model (solid lines).

371 nm is the lattice constant of the APCW). To checkthe validity of our theoretical model given by Eq. (4),we use a parameter of ↵ = 10.6 describing the actualcurvature near the band edge for the APCW [34]. Theatom-field coupling strength g

c

is obtained from firstprinciples by numerical quantization of the guided modes

near the band edge (see Methods), yielding gc

/2⇡ =pa/L⇥ 12.2 GHz.The numerical evaluation of Eq. (4) (solid lines in

Fig. 3b) produces good quantitative agreement with theFDTD simulations with no free fitting parameters. Thedeviation between the theoretical and numerical resultsat �/2⇡ = 400 GHz is primarily attributable to finite-size e↵ects, as the interaction length becomes comparableto the simulated structure size (75a between the sourceand either end of the APCW). At short atomic sepa-rations r

jl

/a < 15, numerical results deviate from themodel at all detunings (Fig. 3b inset). We primarily at-tribute this di↵erence to the contributions coming fromother guided bands (see Ref. [17] for full band struc-ture), as well as leaky and free-space modes that arenot included in our single-band theoretical model. Evenbetter agreement with the model can be reached by ap-proximately subtracting out the contributions from theseother modes. To do this, we note that leaky and free-space modes should exhibit negligible frequency depen-dence over the narrow range of detunings � considered.We then estimate the multimode contribution by calcu-lating the di↵erence between the model and numerical|U

jl

/�0| at �/2⇡ = 3 THz. The results, as plotted inFig. 3b, strongly support the validity of our simple model.

Designing interaction properties

The induced long-range interactions given in Eq. (4)depend on the detuning from the band edge � and bandcurvature ↵, which cannot be easily tuned given a physi-cal structure. This is remedied by considering atoms withan internal ⇤-level structure, as shown in Fig. 4a, withan additional metastable state |si introduced. Here the|gi-|ei transition is coupled to the band edge as before,while |si-|ei is assumed to be de-coupled but address-able by an external laser (e.g. illuminating the photoniccrystal from the side) with Rabi amplitude ⌦ and detun-ing �

L

. This situation may be achieved, for example, ifthe photonic crystal modes and external laser have or-thogonal polarizations. Alternatively, guided modes ofthe photonic crystal with orthogonal polarization maybe used.The photons in the system are now Raman scattered

from the laser field with central frequency !a

+�L

, and in-tuitively the photon cloud size will be determined by theattenuation length at this frequency rather than at !

a

asin the case of pure atomic excitation. Specifically, adia-batically eliminating the excited state and the photonicmodes (see Methods), we obtain an interaction withinthe ground-state manifold of the form in Eq. (4),

HI

=~|⌦|2g2

c

2�L

�2L

NX

j,l

�j

sg

�l

gs

f(zj

, zl

), (5)

6

|si

|gi

|ei

|e0i�L

�L

⌦g,!k

⌦

g,!k

!s

a b

0 50

0

0.5

1

z/a

f(z)

FIG. 4. Designing interaction potentials (a) Driven(black) ⇤ and (black and blue) four level system. In the ⇤scheme, transition |gi-|ei couples with strength g to the pho-tonic crystal modes, while |si-|ei is pumped by a laser withdetuning �L and Rabi frequency ⌦. XX interactions can beachieved by adding level |e0i, where the transition |si-|e0i alsocouples to the modes of the photonic crystal, while a secondpump drives |gi-|e0i. (b) Approximate power law interactionsbetween atoms over a finite region can be achieved by sum-ming the di↵erent exponential interactions associated withmultiple drive fields. This is illustrated here over 50 latticesites, where two exponentials are added to yield a ⌘ = 1/4power law, f(z) = w1e

�s1z/a +w2e�s2z/a ⇡ z�1/4 (solid blue

curve). The error f(z)� z�1/4 is given by the dashed curve.Here w1 = 0.5480, w2 = 0.5684, s1 = 0.2916 and s2 = 0.0089could be achieved by detuning one laser from the band edge by1.723⇥10�3!b and the second by 1.612⇥10�6!b for ↵ = 0.2.

where now � = !a

�!b

is replaced by �L

= �L

+!a

�!b

and gc

is replaced by ⌦gc

/�L

. The strength of the in-teraction is reduced by a factor of |⌦|2/�2

L

, leading to anincrease in the time needed for the spin exchange; how-ever, the population of the atomic excited state is alsoreduced by the same factor reducing the rate of sponta-neous emission. The Raman process e↵ectively narrowsthe natural line width of the excited state, and the co-operativity C, which characterizes the optimal fidelity ofexchange, remains constant.

By tuning the frequency and amplitude of thedrive laser, the interaction strength and length L =p↵!

b

/(�L

k20) can now be dynamically altered. It alsobecomes feasible to build interaction scalings other thanexponential, by driving the |si-|ei transition with twoor more fields of di↵erent frequencies. This leads to aninteraction potential for the atoms which is the sum ofthe potentials due to each individual drive field (the adia-batic elimination of drive fields is additive). For example,a power law interaction f(z

j

, zl

) / |zj

� zl

|�⌘ can be ap-proximated over a finite range, as we show in Fig. 4b for⌘ = 1/4 over 50 unit cells using two pump fields (whoseproperties are numerically optimized). Additional drivefields improve the approximation.

Mapping the interactions to the ground state mani-fold for three-or-more level atoms enables simulation ofspin or quantum magnetism models with long range in-teractions. In one extreme we can suppose the atomsare tightly trapped on a lattice, then the interactionin Eq. (5) corresponds to the XY-model,

Pj 6=l

(�j

x

�l

x

+

�j

y

�l

y

)f(zj

, zl

), for spin-1/2 operators (�x

,�y

,�z

) =(�

gs

+�sg

, i(�gs

��sg

),�ss

��gg

)/2. Other level schemes,such as the four level structure in Fig. 4a, enable otherspin models. Here, the ⇤-structure is extended by addinga transition |si-|e0i that also couples to the photonic crys-tal modes, while a second pump with the same ampli-tude and detuning as the first drives the |gi-|e0i transi-tion. Eliminating the excited state manifold (see Meth-ods) now yields an e↵ective Hamiltonian correspondingto the transverse Ising model with long-range XX inter-actions,

H = ~!s

NX

j

�j

z

+2~|⌦|2g2

c

�L

�2L

NX

j 6=l

�j

x

�l

x

f(zj

, zl

). (6)

It is predicted that long-range interactions in this modellead to a breakdown of Lieb-Robinson bounds [6].In the opposite limit where the atoms move freely,

driving the atoms weakly and o↵ resonance allows theinternal degrees of freedom to be eliminated from thedynamics. In this case the interaction between two-level atoms in Eq. (4), after elimination of the excitedstate, yields a purely mechanical potential for the atoms,

U ⇡ ~|⌦|2g2c

2(!L�!b)(!L�!a)2f(z

j

, zl

) [38]. Engineering the in-teraction to be a power law with ⌘ = 1, for example,would enable simulation of charged particles using neu-tral cold atoms.Beyond the photon losses already discussed, imper-

fections in photonic crystal fabrication can yield disor-dered potentials for propagating fields. Disorder resultsin an Anderson localization length over which opticalfields tend to become trapped, limiting the interactionrange between atoms. As discussed in the Methods, thephysics of weak disorder near a band edge leads to a uni-versal scaling for the localization length as a function ofthe level of disorder. This scaling predicts that localiza-tion lengths exceeding 100 wavelengths are possible withstate-of-the-art fabrication and is consistent with sepa-rate experimental characterization of APCW structures,which reveal a localization length much longer the struc-ture length of ⇠ 200 unit cells [34]. On the other hand,disorder may prove to be a feature of the system, provid-ing access to physical models with interactions that haverandomly varying length scales.

Summary and outlook

In conclusion, we have demonstrated the utility ofatom-induced cavities in photonic crystal structures as atoolbox to achieve tunable long-range interactions. Theseinteractions could lead to long-range entanglement ofatoms [32] and provide a powerful method to investigatemany-body physics with long-range interactions and as-sociated phenomena, such as frustration [39], informationpropagation [6, 40] and many-body localization [41].

7

While the discussion above is quite general to atomsinterfaced with photonic crystals, significant progresshas been made to experimentally realize systems wheresuch physics could be observable. For example, individ-ual atoms have recently been trapped and coupled to aphotonic crystal cavity using optical tweezers [16], andsuch techniques could be extended to realize simultane-ous trapping of several atoms. Trapping many atoms inthe evanescent field of guided modes of photonic crys-tals has been suggested [33] and experiments are quicklyprogressing in this direction [17, 34]. A number of de-sign principles have also been suggested, which can facil-itate various functionalities. For example, the proposalof Hung et al. [33] o↵ers two possibilities, the trappingof atoms in the holes of a photonic crystal as shown inFig. 1d, or the trapping of atoms between two photoniccrystal beams as in the APCW in Fig. 3a. The lattercan in principle allow for atoms to move between trap-ping sites, to observe the mechanical e↵ects of long-rangespatial potentials. In these designs, the polarization ofthe relevant photonic crystal modes can be arranged tobe linear at the trapping sites, allowing the atoms to bedriven with light of orthogonal polarization in our multi-level schemes.

We anticipate that using these systems it should alsobe possible to investigate rich phenomena associated withlong-range interactions in higher spin (greater than 1/2)models, or intermediate regimes where both motion andspin interactions couple to one another. To probe theemerging many-body states, techniques that have beenused for quantum gases that allow single atom resolutioncould be applied [42]. Alternatively, by measuring trans-mission and reflection of probe light within the propa-gating band, internal state correlations or density varia-tions may be revealed [33, 38]. Long-range interactions inthis system could also generate non-local non-linearitiesfor photons, leading to photonic molecules and other ex-otic states [43, 44]. Finally, the use of band edges tomanipulate interactions should find wide use beyond thephotonic crystal setting. In the context of atom-photoninteractions, band structure could be engineered using avariety of fabrication techniques [34, 45, 46], or using pe-riodic arrangements of atoms themselves [47]. Couplingof quantum bits via phononic band structure [48] couldbe controlled in an analogous manner.

METHODS

Single excitation bound state

For a two-level atom interacting with the electromag-netic field modes of a photonic crystal, a bound stateexists that is the superposition of an atomic excita-tion and a localized excitation of the photonic modes[24, 30, 49, 50]. The atom-photon bound state is the so-

lution to the Schrodinger equation, H| i = ~!| i, where! lies within the band gap, for the Hamiltonian given inEq. (2) of the main text. We assume the bound statehas frequency near the lower band edge with detuning� = ! � !

b

> 0. The upper band is assumed to be muchfurther detuned from the atomic resonance and have anegligible e↵ect on the bound state (the complementarysolution near the upper band is found in the same man-ner).Making the e↵ective mass and associated approxima-

tions described in the main text, the eigenenergy is foundto be the positive real root of (� � �)

p� = 2�3/2 for

� =�⇡g2|u

k0(0)|2k0/p4↵!

b

�2/3and � = !

a

� !b

. Thesolution can be expressed explicitly as

� =2

3�+ �

⇣�1/3+ + �

1/3�

⌘(7)

where

�± =

1±

s

1� �3

27�3

!2

. (8)

The corresponding eigenstate is expressible in the form|�1i = cos ✓|ei + sin ✓|1i where |1i =

Rdkc⇤

k

a†k

|0i is asingle photon excitation of the modes in the band. Herecos ✓ = (1+(�/�)3/2)�1/2, sin ✓ = (1+(�/�)3/2)�1/2 andck

= (�/�)3/4guk0(0)/(! � !

k

). In the limit � � �, thedetuning � ! � and the population of the excited statecos2 ✓ ! 1.The weights of the photonic modes give us an indi-

cation of the validity of the approximations used aboveto derive the form of the bound state. The distributionck

/ (� + ↵!b

(k � k0)2/k20)�1 is a Lorentzian in k cen-

tered at k0 with half widthp�/(↵!

b

)k0. Our approxima-tions required that the populated modes have wavevec-tors close to k0, that is, provided

p�/(↵!

b

) ⌧ 1.

Dipole-dipole interactions

Two-level atoms

We first consider the case of many two-level atomstrapped near a photonic crystal. The atomic resonancefrequency is taken to be in the band gap near the lowerband edge, such that the detuning satisfies � � �. Inthis case the photonic modes are weakly populated andwe may eliminate them to obtain a description of thesystem in terms of e↵ective dipole-dipole interactions[23, 24, 31, 32] as detailed below.In the interaction picture, the interaction between the

atom and photonic crystal modes is

HI

= ~X

j

Zdkg

k

�j

eg

ak

uk

(zj

)ei�kt+ikzj +H.c., (9)

8

where �k

= (!a

� !k

). In the presence of loss, we candescribe the evolution of the system by the master equa-tion ⇢ = L

int

(t)⇢+L�

⇢+L

⇢. Here Lint

(t)⇢ = � i

~ [HI

, ⇢]describes the coherent evolution and

L�

⇢ =� �

2

X

j

�{�j

ee

, ⇢}� 2�j

ge

⇢�j

eg

�, (10)

L

⇢ =�

2

Zdk⇣{a†

k

ak

, ⇢}� 2ak

⇢a†k

⌘(11)

describe the loss mechanisms resulting from spontaneousemission from the atomic excited state and loss of pho-tons in the photonic crystal, that occur with rates � and respectively. We now proceed to eliminate the pho-tonic modes using the Nakajima-Zwanzig approach in theBorn-Markov approximation [51]. This yields the masterequation for the reduced density operator, ⇢

s

= Trk(⇢)after tracing over the field modes,

⇢s

= � i

~ [Hef

I

, ⇢] + L�

⇢s

+ Lef

⇢s

. (12)

With the field modes eliminated, the interaction is de-scribed by the e↵ective Hamiltonian

eHef

I

=X

j,l

�j

eg

�l

ge

Zdk

g2k

�k

Ek

(zj

)E⇤k

(zl

)

�2k

+ (/2)2. (13)

We can then integrate over the field modes, applyingthe same approximations near the band edge as used toderive the bound state, that is, the band is quadratic andonly makes significant contributions for k ⇠ k0. To firstorder in the /�, the interaction becomes

Hef

I

=g2c

2�

X

j,l

Ek0(zj)E

⇤k0(z

l

) exp[�|zj

� zl

|/L]�j

eg

�l

ge

.

(14)as in Eq. (4) where we have defined L =

p↵!

b

/(�k20)and g

c

=p2⇡/Lg. If we had considered atoms with

resonance near the upper band instead, with ↵ < 0, theabove relation would still hold, however � would now benegative in the band gap and the interaction would hencehave opposite sign. This provides a method to tune thesign of the interactions by placing the atomic resonanceclose to either the lower or upper band edges. To lowestorder, the loss of coherence due to photon loss in thecrystal is described by

Lef

⇢ = � g2c

8�2

X

j,l

Ek0(zj)E

⇤k0(z

l

)�{�j

eg

�l

ge

, ⇢}� 2�j

ge

⇢�l

eg

�.

(15)The leading order loss terms are hence a factor of /(4�)smaller than the coherent interaction.

The interaction in Eq. (14) is obtained by the integra-tion in Eq. (13) over a one dimensional band structure.For two or three dimensional band structure Eq. (13) is

generalized to become

Hef

I

= ~g2X

j,l

�j

eg

�l

ge

Zdk

E⇤k(zl)Ek(zj)

!a

� !k. (16)

For example, for a two-dimensional band edge of the form!k = !

b

(1 � ↵|k � k0|2/k20), the interaction has spatialdependence f(z

j

, zl

) = 2⇡

K0(|zj � zl

|/L)Ek0(zj)E⇤k0(z

l

),where K0(z) is the Bessel function of the second kind.The interaction now decays approximately exponentiallywith an additional 1/

p|zj

� zl

| scaling due to the two-dimensional spread of the photon cloud.

Weakly driven multi-level atoms

We can now examine the e↵ective interactions betweenatoms with more complicated internal level structuresas depicted in Fig. 4a. We assume the transitions notcoupled to the photonic crystal are weakly excited by anorthogonally polarized laser and we denote by ⌦ the Rabidriving of |si-|ei and by ⌦0 the drive of |gi-|e0i, which iszero in the three level case. For concreteness, we assumespontaneous emission occurs with equal rate � from bothexcited states |e0i and |ei, and decay from each stateoccurs with equal probability into either of the groundstates. We take the driving frequency to be detuned by�L

� �,⌦(0) from each atomic resonance, in which casethe atoms are weakly driven and the Raman scatteredfields are predominantly centered around the two-photonresonance frequency. When the laser frequency is in thephotonic band gap we then expect the atoms to formbound states with photons near the laser frequency.

In this regime we can obtain e↵ective dynamics for thelong-lived atomic ground states |si and |gi. We followa similar procedure as above. In particular, in a framerotating at the laser frequency, we first eliminate the pho-tonic modes using the Nakajima-Zwanzig technique, fol-lowed by the atomic excited states. Following this recipewe obtain e↵ective dynamics to first order in /� and�/�

L

described by

Hef

I

=g2c

2�L

|⌦|2�2L

X

j,l

Ek0(zj)E

⇤k0(z

l

) exp[�|zj

�zl

|/L]S†j

Sl

.

(17)where S

j

= ((⌦0/⌦)�j

sg

+ �j

gs

) and we make the replace-ment � ! �

L

= !L

+ !s

� !b

in the expression for L.The replacement of � by �

L

means L now changes withthe laser frequency, leading to the possibility of dynamictuning of the interaction length.

The loss of coherence in the ground state manifold re-sulting from spontaneous emission into free space is de-

9

scribed by

Lef

�

⇢ = �X

j

�

2

�{�j

ss

, ⇢}� �j

ss

⇢�j

ss

� �j

gs

⇢�j

sg

�

+�0

2

�{�j

gg

, ⇢}� �j

gg

⇢�j

gg

� �j

sg

⇢�j

gs

��(18)

where the e↵ect of the Raman transitions is to narrowthe linewidths of the transitions according to �(0) =|⌦(0)|2�/�2

L

. Losses in the photonic crystal induce an-other dissipation mechanism for the atomic degrees offreedom, which is given by

Lef

⇢ = � g2c

8�L

|⌦|2�2L

X

j,l

Ek0(zj)E

⇤k0(z

l

)⇣{S†

j

Sl

, ⇢}� 2Sj

⇢S†l

⌘.

(19)The interaction and losses are all reduced by the samefactor, |⌦|2/�2

L

, and as a result all processes happen on aslower time scale, while the cooperativity of the systemremains constant.

From Eq. (17) we obtain the e↵ective interactions givenin the main text in Eqs. (5)-(6). We obtain the ⇤-system interaction, Eq. (5), when ⌦0 = 0 such thatSj

= �gs

, and the four level system interaction, Eq. (6),when ⌦0 = ⌦ such that S

j

= 2�x

. For ⌦0 = ⌦ei�, theinteraction in Eq. (17) is rotated to be between spinsSj

= 2 cos(�/2)�j

x

� 2 sin(�/2)�j

y

. We can then adjustthe spin-spin interaction temporally or spatially by ad-justing �.

Numerical analysis of APCW

We adapt the technique of Ref. [52] suited to quantiz-ing the electromagnetic field in the presence of dielectricmedia. Given a classical electric field mode E

k

(r,!k

)of Maxwell’s equations, the corresponding quantum elec-tric field operator is given by E(r) = a

k

Ek

(r,!k

) + h.c.Here E

k

is a rescaled version of Ek

satisfying the energynormalization condition ~!

k

/2 =Rdr ✏0✏(r)|Ek

(r,!k

)|2,where ✏(r) is the dimensionless electric permittivity. In-teractions between an atom at r and the field are de-scribed by the electric dipole Hamiltonian, H = d ·E

k

(r) = ~gk

(r)a�eg

+ h.c., where the coupling strength~g

k

(r) ⌘ deg

· Ek

(r,!k

) depends on the normalized fieldand atomic dipole matrix element.

While the description thus far is completely general,we now specialize to the case of interactions with a singleband of a 1D photonic crystal, where k is now understoodto be the Bloch wavevector. By Bloch’s theorem, theguided mode satisfies E

k

(r,!k

) = uk

(r)eikz, where uk

isa function with periodicity given by the lattice constanta. Now, taking the field profile of Eq. (3) for the localizedphotonic state around an atom and performing the quan-tization prescription, we obtain |g

c

| = gcellpa/L, where

~gcell is the atom-field coupling strength of a mode at theband edge (k = k0) based upon the normalization overa single unit cell, ~!

k0/2 =Rcell dr ✏0✏(r)|Ek0(r,!k0)|2.

Taking the guided mode profile obtained by numericalsimulation of the APCW produces a value of gcell/2⇡ =12.2 GHz.

Disorder near the band edge in a photonic crystalmodel

Disorder in a photonic crystal may be introduced dur-ing fabrication, either deliberately or inevitably due tothe finite accuracy of the fabrication method. We modela one-dimensional photonic crystal to investigate howthis disorder a↵ects the structure of photonic modes. Wetake a simple model of a photonic crystal shown in Fig. 5aconsisting of an alternating dielectric stack, with high re-fractive index n

h

and low refractive index nl

, where eachlayer of the stack has the same phase length �

b

at theband edge. We then introduce weak disorder by addinga random phase length shift "

h(l)�b to each layer, whichhave mean h"

h(l)i = 0, variance h("h(l))

2i = "2 and "⌧ 1.This model of the photonic crystal, as discussed below,

has the same band edge properties as the weakly disor-dered Kronig-Penney model, which is given by [53–55]

"� @2

@x2+ U

X

n

(1 + �n

)�(x� an� a↵n

)

# q

= q2 q

.

(20)The Kronig-Penney model describes the propagation ofthe wavefunction

q

with quasimomentum q through alattice of delta functions with spacing a and strength U .The disorder is introduced to the Kronig-Penney modelthrough the random variables ↵

n

and �n

, which adjustthe position and strength of the nth delta function re-spectively, and is assumed to be weak, that is, the vari-ables have variance h↵2

n

i ⌧ 1 and h�2n

i ⌧ 1. This disor-der in the potential leads to localization of waves in thesystem, which is characterized by the localization length⇠. In Ref. [53] the stochastic dynamics of the waves inthe presence of this disorder are studied by solving theassociated Fokker-Planck equation, from which the local-ization length is found to be

⇠

a=

2�(1/6)

61/3p⇡��2/3 (21)

where � is related to the variances of ↵n

= ↵n+1 � ↵

n

and �n

and � is the Gamma function.We now show that the model of the photonic crystal is

related to the Kronig-Penney model near the band edgeand as a result we can model the localization length inphotonic crystals using Eq. (21). While the two mod-els appear physically di↵erent, their respective e↵ects onwave propagation will be the same if the reflection and

10

b

" 10

�4

10

�3

10

�2

0

300

600

"

⇠/anh nl

a

FIG. 5. E↵ect of disorder (a) Model of disorder in a pho-tonic crystal consisting of alternating layers of dielectric withhigh nh and low nl refractive indexes. For no disorder (tophalf), in the band gap a field (red) decays exponentially. In-troducing disorder (lower half), with standard deviation ", inthe phase length of each layer of the crystal may lead to fasterdecay of the field with a characteristic length ⇠ and fluctua-tions in the intensity. (b) Localization length ⇠ (in units ofthe lattice constant a) as the strength of disorder " varies fornh/nl = 2.

transmission of waves by each unit cell is the same. In-deed at the band edge, in the absence of randomness,the models can be tuned to produce the same behavior,i.e., equal phase and magnitude of the transmission coef-ficient t, by adjusting the free parameters. As we intro-duce randomness we then match the models by equatingthe magnitude and phase of t to first order in the randomvariables. This gives the following relation at the bandedge

�n

! �b

(nh

� nl

)

2pnh

nl

"h

, (22)

↵n

! �b

�KP

"l

+�b

nh

nl

�KP

(n2h

� nh

nl

+ n2l

)"h

. (23)

Eqs. (22) and (23) indicate that a random phase shiftin the low refractive index layer of the photonic crystalis equivalent to a change in the delta function separa-tion in the Kronig-Penney model, while a random shiftin the high index layer is equivalent to changing boththe delta function height and separation. By substi-tuting this mapping into the definition of the variance[53] �2 ⌘ b2(�2

KP

h↵2n

i + sin2(�KP

)h�2n

i) and utilizingsin2(�

KP

) = 4nh

nl

/(nh

+ nl

)2, which holds when theband edges are matched between the two models, we ob-tain

�2 = 4�2b

✓2(r2 + 1)(r � 1)2

r(r + 1)2+

r(r � 1)2

(r2 � r + 1)2

◆"2 (24)

where r = nh

/nl

is the ratio of refractive indices. Thelocalization length for the photonic crystal model at theband edge then follows from Eq. (21).

In Fig. 5b we plot the localization length as a func-tion of the disorder strength " for n

h

/nl

= 2, whichcould be achieved with air and SiN layers for example[33]. Here, we find localization lengths of greater than

100 unit cells, provided that the fractional standard de-viation of the layer thickness is of the order of 10�3. Formore complex geometries, simulations might be neededto quantitatively map the e↵ects of various types of dis-order onto �, but even absent such simulations, Eq. (21)serves as a useful guide as to how much disorder can betolerated to reach a given interaction length. We notethat fabrication of photonic crystals with variations onthe order of 10�3 have been reported [36].The authors thank L. Tagliacozzo, P. Hauke,

M. Lewenstein, A. Gonzalez-Tudela, J. I. Cirac, L. Jiang,J. Preskill, O. Painter, M. Lukin, J. Thompson, andS. Gopalakrishnan for insightful discussions. This workwas supported by Fundacio Privada Cellex Barcelona,the MINECO Ramon y Cajal Program, the MarieCurie Career Integration Grant, the IQIM, an NSFPhysics Frontiers Center, the DoD NSSEFF program,DARPA ORCHID, AFOSR QuMPASS MURI, NSFPHY-1205729, NSF PFC at the JQI, NSF PIF, ARO,ARL, and AFOSR MURI on Ultracold Polar Molecules.

⇤ email address: james.douglas@icfo.es† Present address: Department of Physics and Astronomy,Purdue University, West Lafayette, Indiana 47907, USA

[1] I. Bloch, J. Dalibard, and S. Nascimbene, Nat Phys 8,267 (2012).

[2] G. G. Batrouni, R. T. Scalettar, G. T. Zimanyi, andA. P. Kampf, Phys. Rev. Lett. 74, 2527 (1995).

[3] E. Wigner, Phys. Rev. 46, 1002 (1934).[4] A. Micheli, G. K. Brennen, and P. Zoller, Nat Phys 2,

341 (2006).[5] A. Campa, T. Dauxois, and S. Ru↵o, Physics Reports

480, 57 (2009).[6] P. Hauke and L. Tagliacozzo, Phys. Rev. Lett. 111,

207202 (2013).[7] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and

T. Pfau, Reports on Progress in Physics 72, 126401(2009).

[8] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, andT. Pfau, Phys. Rev. Lett. 94, 160401 (2005).

[9] M. Lu, N. Q. Burdick, S. H. Youn, and B. L. Lev, Phys.Rev. Lett. 107, 190401 (2011).

[10] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er,B. Neyenhuis, J. J. Zirbel, S. Kotochigova, P. S. Julienne,D. S. Jin, and J. Ye, Science 322, 231 (2008).

[11] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote,and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).

[12] M. Sa↵man, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. 82, 2313 (2010).

[13] G. Khitrova, H. M. Gibbs, M. Kira, S. W. Koch, andA. Scherer, Nat Phys 2, 81 (2006).

[14] E. Vetsch, D. Reitz, G. Sague, R. Schmidt, S. T.Dawkins, and A. Rauschenbeutel, Phys. Rev. Lett. 104,203603 (2010).

[15] A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute,M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble,Phys. Rev. Lett. 109, 033603 (2012).

[16] J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist,

11

A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletic, andM. D. Lukin, Science 340, 1202 (2013).

[17] A. Goban, C.-L. Hung, S.-P. Yu, J. Hood, J. Muniz,J. Lee, M. Martin, A. McClung, K. Choi, D. Chang,O. Painter, and H. Kimble, Nat Commun 5, 3808 (2014).

[18] H. J. Kimble, Nature 453, 1023 (2008).[19] A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L.

Hollenberg, Nat Phys 2, 856 (2006).[20] M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio,

Nat Phys 2, 849 (2006).[21] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev.

A 76, 031805 (2007).[22] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and

R. D. Meade, Photonic Crystals: Molding the Flow ofLight (Second Edition), 2nd ed. (Princeton UniversityPress, 2008).

[23] G. Kurizki, Phys. Rev. A 42, 2915 (1990).[24] S. John and J. Wang, Phys. Rev. B 43, 12772 (1991).[25] G. Agarwal, S. Gupta, and R. Puri, Fundamentals of

Cavity Quantum Electrodynamics (World Scientific Pub-lishing Company Incorporated, 1995).

[26] M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight,Phys. Rev. A 59, 2468 (1999).

[27] P. Domokos and H. Ritsch, Phys. Rev. Lett. 89, 253003(2002).

[28] A. T. Black, H. W. Chan, and V. Vuletic, Phys. Rev.Lett. 91, 203001 (2003).

[29] K. Baumann, C. Guerlin, F. Brennecke, andT. Esslinger, Nature 464, 1301 (2010).

[30] S. John and J. Wang, Phys. Rev. Lett. 64, 2418 (1990).[31] S. Bay, P. Lambropoulos, and K. Mølmer, Phys. Rev. A

55, 1485 (1997).[32] E. Shahmoon and G. Kurizki, Phys. Rev. A 87, 033831

(2013).[33] C.-L. Hung, S. M. Meenehan, D. E. Chang, O. Painter,

and H. J. Kimble, New Journal of Physics 15, 083026(2013).

[34] S.-P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin,R. Norte, C.-L. Hung, S. M. Meenehan, J. D. Cohen,O. Painter, and H. J. Kimble, Applied Physics Letters104, 111103 (2014).

[35] P. Markos and C. M. Soukoulis,Wave Propagation: FromElectrons to Photonic Crystals and Left-Handed Materi-

als (Princeton University Press, Princeton, 2010).[36] Y. Taguchi, Y. Takahashi, Y. Sato, T. Asano, and

S. Noda, Opt. Express 19, 11916 (2011).[37] H. T. Dung, L. Knoll, and D.-G. Welsch, Phys. Rev. A

66, 063810 (2002).[38] D. E. Chang, J. I. Cirac, and H. J. Kimble, Phys. Rev.

Lett. 110, 113606 (2013).[39] M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo,

Phys. Rev. Lett. 111, 165302 (2013).[40] J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J.

Daley, Phys. Rev. X 3, 031015 (2013).[41] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, eprint

arXiv:cond-mat/0602510 (2006), cond-mat/0602510.[42] W. S. Bakr, J. I. Gillen, A. Peng, S. Folling, and

M. Greiner, Nature 462, 74 (2009).[43] P. Longo, P. Schmitteckert, and K. Busch, Phys. Rev. A

83, 063828 (2011).[44] O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gor-

shkov, M. D. Lukin, and V. Vuletic, Nature 502, 71(2013).

[45] K. P. Nayak and K. Hakuta, Opt. Express 21, 2480

(2013).[46] M. Sadgrove, R. Yalla, K. P. Nayak, and K. Hakuta,

Opt. Lett. 38, 2542 (2013).[47] D. E. Chang, L. Jiang, A. V. Gorshkov, and H. J. Kim-

ble, New Journal of Physics 14, 063003 (2012).[48] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala,

and O. Painter, Nature 462, 78 (2009).[49] A. Kofman, G. Kurizki, and B. Sherman, Journal of

Modern Optics 41, 353 (1994).[50] S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).[51] H. P. Breuer and F. Petruccione, The theory of

open quantum systems (Oxford University Press, GreatClarendon Street, 2002).

[52] N. A. R. Bhat and J. E. Sipe, Phys. Rev. A 73, 063808(2006).

[53] J. C. Hernandez-Herrejon, F. M. Izrailev, and L. Tessieri,Journal of Physics A: Mathematical and Theoretical 43,425004 (2010).

[54] F. Izrailev, A. Krokhin, and N. Makarov, Physics Re-ports 512, 125 (2012).

[55] Derrida, B. and Gardner, E., J. Phys. France 45, 1283(1984).