Atom-light interactions in photonic crystals

Abstract

The integration of nanophotonics and atomic physics has been a long-sought goal that would open new frontiers for optical physics, including novel quantum transport and many-body phenomena with photon-mediated atomic interactions. Reaching this goal requires surmounting diverse challenges in nanofabrication and atomic manipulation. Here we report the development of a novel integrated optical circuit with a photonic crystal capable of both localizing and interfacing atoms with guided photons. Optical bands of a photonic crystal waveguide are aligned with selected atomic transitions. From reflection spectra measured with average atom number , we infer that atoms are localized within the waveguide by optical dipole forces. The fraction of single-atom radiative decay into the waveguide is Γ_1D /Γ'≈(0.32±0.08), where Γ_1D is the rate of emission into the guided mode and Γ' is the decay rate into all other channels. Γ_1D /Γ' is unprecedented in all current atom-photon interfaces.

ARTICLE

Received 19 Dec 2013 | Accepted 2 Apr 2014 | Published 8 May 2014

DOI: 10.1038/ncomms4808

Atomlight interactions in photonic crystals

A. Goban1,2,*, C.-L. Hung1,2,*, S.-P. Yu1,2,*, J.D. Hood1,2,*, J.A. Muniz1,2,*, J.H. Lee1,2, M.J. Martin1,2, A.C. McClung1,2, K.S. Choi3, D.E. Chang4, O. Painter2,5 & H.J. Kimble1,2

The integration of nanophotonics and atomic physics has been a long-sought goal that would open new frontiers for optical physics, including novel quantum transport and many-body phenomena with photon-mediated atomic interactions. Reaching this goal requires surmounting diverse challenges in nanofabrication and atomic manipulation. Here we report the development of a novel integrated optical circuit with a photonic crystal capable of both localizing and interfacing atoms with guided photons. Optical bands of a photonic crystal waveguide are aligned with selected atomic transitions. From reection spectra measured with average atom number N 1:1 0:4, we infer that atoms are localized within the

waveguide by optical dipole forces. The fraction of single-atom radiative decay into the waveguide is G1D/G0C(0.320.08), where G1D is the rate of emission into the guided mode and G0 is the decay rate into all other channels. G1D/G0 is unprecedented in all current atomphoton interfaces.

1 Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA. 2 Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA. 3 Spin Convergence Research Center 39-1, Korea Institute of Science and Technology, Seoul 136-791, Korea. 4 ICFOInstitut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain.

5 Thomas J. Watson, Sr., Laboratory of Applied Physics 128-95, California Institute of Technology, Pasadena, California 91125, USA. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to H.J.K. (email: mailto:[email protected]

Web End [email protected] ).

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Localizing arrays of atoms in photonic crystal waveguides (PCW) with strong atomphoton interactions could provide new tools for quantum networks13 and enable explorations

of quantum many-body physics with engineered atomphoton interactions419. Bringing these scientic possibilities to fruition requires creation of an interdisciplinary toolkit from atomic physics, quantum optics and nanophotonics for the control, manipulation and interaction of atoms and photons with a complexity and scalability not currently possible. Important initial advances to integrate atomic systems and photonics have been made within the setting of cavity quantum electrodynamics with atomphoton interactions enhanced in micro- and nanoscopic optical cavities2026 and waveguides2729. At a minimum, the further migration to photonic crystal structures should allow the relevant parameters associated with these paradigms to be pushed to their limits26 and greatly facilitate scaling. For example, modern lithographic processing can create nanoscopic dielectric waveguides and resonators with optical quality factors Q4106 and with efcient coupling among heterogeneous components3035.

A more intriguing possibility that has hardly been explored is the emergence of completely new paradigms beyond the cavity and waveguide models, which exploit the tremendous exibility for modal and dispersion engineering of PCWs. For example, the ability to tune band edges near atomic transition frequencies can give rise to strongly enhanced optical interactions3640. This enables a single atom to exhibit nearly perfect emission into the guided modes (G1DcG0) and to act as a highly reective mirror

(for example, reection |r1|\0.95 and transmission |t1|t0.05 for one atom41). The entanglement of photon transport with internal states of a single atom can form the basis for optical quantum information processing13 with on-chip quantum optical circuits. At the many-body level, the strong interplay between the optical response and large optical forces of many atomic mirrors can give rise to interesting optomechanical behaviour, such as self-organization15.

Even more remarkable phenomena in PCWs arise when atomic frequencies can be tuned into photonic band gaps, including the ability to control the range, strength and functional form of optical interactions between atoms4,1618. For example, atoms trapped near otherwise perfect photonic crystal structures can act as dielectric defects that seed atom-induced cavities18 and thereby allow atomic excitations to be exchanged with proximal atoms16. The atom-induced cavities can be dynamically controlled with external lasers enabling the realization of nearly arbitrary long-range spin Hamiltonians and spatial interactions (such as an effective Coulomb potential mediated by PCW photons)18, providing a novel tool for quantum simulation with cold atoms. Control over PCW dispersion is also expected to facilitate novel atomic traps based upon quantum vacuum forces19,41,42. The prerequisite to all of these possibilities is a designable platform that allows the simultaneous alignment of optical bands for optical trapping and for interaction physics with atoms, which we demonstrate here for the rst time.

Here, we report advances that provide rudimentary capabilities for such a toolkit with atoms coupled to a PCW. As illustrated in Fig. 1, we have fabricated the integrated optical circuit with a photonic crystal whose optical bands are aligned with atomic transitions for both trapping and interfacing atoms with guided photons41,43. The quasi-1D PCW incorporates a novel design that has been fabricated in silicon nitride (SiN)43,44 (Methods) and integrated into an apparatus for delivering cold caesium atoms into the near eld of the SiN structure. From a series of measurements of reection spectra with

N 1:1 0:4 atoms

coupling to the PCW, we infer that the rate of single-atom radiative decay into the waveguide mode is G1DC(0.320.08)G0,

where G1D is the emission rate without enhancement or inhibition due to an external cavity and G0 is the radiative decay rate into all other channels. The corresponding single-atom reectivity is |r1|C0.24, representing an optical attenuation for one atom greater than 40%12,41. For comparison, atoms trapped near the surface of a fused silica nanober exhibit G1DC(0.040.01)G0 (refs 2729), comparable with observations with atoms and molecules with strongly focused light45,46. By comparing with numerical simulations, our measurements suggest that atoms are guided to unit cells of the PCW by optical dipole forces.

ResultsDesign and characterization of a 1D photonic crystal waveguide. Turning to our experiment, we begin with a scanning electron microscope (SEM) image of a small section of our 1D PCW shown in Fig. 1a. The device consists of two parallel nanobeams with sinusoidal modulation at the outer edges (an alligator PCW or APCW). A challenge in the fabrication of PCWs for atomic physics is placement of the band edges near relevant atomic transition frequencies. Our APCW design facilitates this juxtaposition by ne tuning the gap between the parallel nanobeams and the amplitude of sinusoidal modulation in the APCW. Figure 1b shows the band structure of two fundamental transverse electric (TE-like) modes calculated based on the dimensions measured from SEM images as in Fig. 1a. The two blue dashed lines correspond to the transition frequencies of the Cs (D1, D2) lines at (n1 335, n2 351) THz ((895, 852) nm) that

straddle the band edge frequencies (nD,nA) at kx p/a for the

lower (dielectric) and upper (air) TE bands, respectively. To validate these results, we measure the reection spectrum R(n)

versus the input frequency n for the actual device used in the reported experiments; see Fig. 1c. The large reectivity (RB0.35) from nD to nA corresponds to the band gap for the APCW, while the vertical dashed lines mark (n1, n2) just outside the band gap. Absent propagation loss to and from the APCW, we infer RgapC0.99 from measurements and numerical simulation.

Beyond band edge placement, another requirement for realizing strong atomlight interactions in PCWs is to localize atoms in a region of high mode intensity within a unit cell. The use of two bands enables the separate engineering of the modes for trapping (lower band) and control of spontaneous emission (upper band). The blue-detuned E1 mode excited at n1 in Fig. 1e can guide atoms into the centre of the vacuum space near regions of large |E2|2, with then a eld near n2 in Fig. 1d serving as a probe mode.

The efcacy of this strategy is supported by trajectory calculations of free-space atoms surrounding the APCW (Methods). As shown in Fig. 1f, atoms are guided from free space into the region of high |E2|2, resulting in a density B30% of the remote free-space density. For the simulations, the Casimir Polder potential UCPr for the structure in Fig. 1a is computed

numerically following ref. 41. The optical dipole potential is calculated using a guided mode E1r at kx p/a with total power

of 1 mW and 10 GHz blue-detuning from the F 42F0 4

transition frequency of the D1 line.

An overview of the integrated APCW device is presented in Fig. 2, and shows the optical pathways for excitation to and from the APCW, as well as the supporting structures of the SiN device to a silicon substrate. The entire APCW contains 260 unit cells with a lattice constant a 371 nm, and is terminated on each end

by a mode matching section of 40 cells with tapered sinusoidal modulation and a transition section from a double- to a single-nanobeam waveguide. Input to and output from the device is achieved through an optical bre butt-coupled to one of the single-nanobeam waveguides44 (Methods). The one-way

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efciency for propagation from the APCW to the bre mode is TwfC0.6.

To integrate the device into a cold atom apparatus, the silicon chip in the inset of Fig. 2a and its coupling optical bres are

mounted on a vacuum feedthrough with linear translation and rotation stages and inserted into a UHV chamber. Caesium atoms are delivered to the region surrounding the APCW by a three-stage process of transport and cooling (Methods). The resulting

a d

E (a.u.)

2 E (a.u.)

2

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Ein

r1Ein

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z x

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375 Cs D2 2

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335 340 345 350

(THz)

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Figure 1 | Design and characterization of a 1D PCW. (a) SEM image of the APCW made from 200-nm thick (along z-axis) SiN (ref. 43). Arrows indicate radiative processes of an atom (green circle) coupled to an incident electric eld Ein. Scale bar, 500 nm. (b) Calculated band structure of fundamental TE-like modes E1, E2 (red solid lines), with dominant electric eld polarized in the y-direction. The dashed lines mark the frequencies n1 and n2 of the caesium

D1, D2 lines, respectively. (nA, nD) mark near the band edge. The gray solid line marks the light line. (c) Measured reection spectrum with fast fringe removed (Methods) around the band gap shown in (b). (d) |E|2 cross-sections of E2 mode near n2, and (e) E1 mode near n1 within a unit cell calculated with the MPB software package57 (Methods). Both modes are polarized with their primary electric eld component in the in-plane y-direction. (f) Simulated relative density ~rr of atoms in the x 0 plane of (e) with the optimal excitation of the blue-detuned E1 mode at kx p/a (see text).

VII

I VI

III IV

II V

y

z x

a b

a=371 nm

g=250 nm

y

z x

Figure 2 | Overview of the integrated APCW device. (a) SEM image of the silicon chip showing an integrated optical ber (orange box) coupled, via a SiN nanobeam waveguide, to the APCW region (purple and green boxes). The APCW is located within a 1 3 mm through window (black region without

dielectrics) where free-space atoms and cooling lasers are introduced. Inset shows a picture of the chip and the optical bers glued to a vacuum-compatible holder. Scale bar, 0.5 mm. (b) Detailed schematic of the suspended SiN waveguide. Light enters the system via the optical bre (I) butt-coupled44

to the free end of the waveguide (II) which is supported by a tether array (III). Near the centre of the through window, the waveguide transitions into a double nanobeam, followed by tapering (IV) and APCW (V) sections, and again tapers out to terminate into the substrate (VI). Two parallel rails are added symmetrically to support the structure (one rail is illustrated in VII). The insets, corresponding to the purple and green boxes in (a), show SEM images of segments of the tapering (IV) and APCW (V) sections, respectively. Scale bars in IV and V, 2 mm.

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atomic cloud has a peak number density of r0C2 1010 cm 3 at

temperature TC20 mK measured via time-of-ight absorption imaging.

Atomlight coupling in the APCW. We study atomlight interactions in the APCW by rst shutting off the cooling laser, followed by a delay of 0.1 ms and then interrogating the APCW with atoms by sending a guided probe pulse Eprobe of frequency np in the E2 mode, with typical powerC1 pW and measuring a reected pulse of rEprobe; see Fig. 3a and Methods. Reection

spectra R(Dp) are recorded for 1 ms with a single-photon avalanche photodiode as a function of detuning Dp np n2a,

where v2a is the free-space F 42F0 5 transition frequency of

the D2 line. For 10 ms following termination of the probe pulse, the atom cloud disperses, and then reference spectra R0(Dp) are

recorded for a second probe pulse for 1 ms. For all experiments, the guided mode E1 is driven continuously with power C0.6 mW at 10 GHz blue-detuning from the F 42F0 4 transition

of the D1 line.

In the ideal case of a single atom in an innite PCW, an incident probe beam would be reected with amplitude coefcient |r1| G1D/(G1D G0),12 where G1D refers to emission

into the guided mode of the APCW from the F 42F0 5

transition of the Cs D2 line. Strong spontaneous decay into the guided mode (and hence large |r1|) results from the small area over which the guided mode is concentrated together with a reduced group velocity. These two effects are incorporated into an

fibre APCW

M1 M2

a

b

Eprobe

r Eprobe t Eprobe

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0.0 0.1 0.1 0.0

y (m)

y (m)

x (m)

(r)

Figure 3 | Atomlight coupling in the PCW section. (a) Simplied schematic of a ber-coupled PCW for reection measurements with atoms (green dots). Dashed lines, marked as M1 and M2, illustrate the mirrors for the effective low nesse F 2 cavity formed by the tapered matching sections.

(b,c) Measured reection spectra (circles) with (b) an on-resonant cavity and (c) an off-resonant cavity in the APCW. Solid lines are Lorentzian ts with (b) linewidth of 15.21.8 MHz, peak reectivity Ron=Ron0 1:27 0:02 where the reectivity with no atoms is Ron0 and frequency shift D0 2.50.6 MHz,

and (c) linewidth of 11.51.1 MHz, Roff=Roff0 0:75 0:01 where the reectivity with no atoms is Roff0 and D0 3.70.3 MHz. Error bars for the data points

reect 1 s.d. estimated from the statistical uncertainties. (d) Simulated xr ~rr G1Dr=G1D0 in the x 0, (e) x a/2, and (f) y 0 planes, with a guided

potential of mF 0 using the experimentally excited blue-detuned E1 mode at kx 0.99p/a. Due to the deviation from the band edge, a small bump in the optical

potential at the centre of the unit cells leads to atomic localization near the maxima of x r

in (d,f); see text and Methods for details. Masked areas in gray

represent the APCW.

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effective mode area for an atom at location r within the APCW, namely Amr ngsG0=2G1Dr, where ngC2 is the measured

group index at n2, s 1.4 10 9 cm2 is the free-space atom

photon cross-section for an unpolarized atom, and G0 is the free-space rate of decay. For unpolarized atoms located at the centre of a unit cell r 0; 0; 0 in the APCW, we expect

Am(0) 0.24 mm2, and hence |r1|C0.39, where G0C0.9G0 from

numerical calculations41 (Methods).

In the case of our actual device, the nite lengths of the taper sections lead to imperfect mode matching into the APCW near the band edge. As illustrated in Fig. 3a, the matching sections partially reect an incident probe pulse and form a low nesse (F 2) cavity around the APCW. These weak cavity resonances

in the reection spectrum near n2 are shown in Fig. 1c (without atoms) and complicate the spectra taken with atoms relative to the ideal case, as discussed below.

In Fig. 3b,c, we measure distinctive reection spectra with cold atoms under two congurations of the APCW. Figure 3b displays RonDp=Ron0Dp acquired near a resonance for the matching

cavity where the reectivity with no atom Ron0 is very small (Methods). We observe an increased peak reectivity

Ron=Ron0 1:27 0:02. By comparison, with the matching

cavity excited midway between two cavity resonances where the reectivity with no atoms is Roff0, we observe decreased reectivity with a minimum Roff=Roff0 0:75 0:01 in Fig. 3b.

These reection spectra are in accord with those for a low nesse FabryPerot (FP) cavity containing a frequency-dependent intracavity absorber (Fig. 3a). In Fig. 3b with input frequency resonant with the FP cavity and no atoms, the transmission is high, and there is very small reection (Methods). Atoms inside the PCW reduce light build-up by frequency-dependent atomic absorption, resulting in an increased reectivity Ron=Ron041. By contrast, in Fig. 3c without atoms, the device reectivity is larger Roff04Ron0 due to the off-resonant drive of the FP and is reduced by intracavity atomic absorption,

Roff=Roff0o1.

The reection spectra in Fig. 3b,c represent strong evidence for atomic interactions with the guided mode E2 of the APCW.

Although the cavity formed by the matching sections has a low nesse, R(Dp) nevertheless depends on the cavity detuning as predicted, that is, exhibiting approximately Lorentzian proles for increased (decreased) RonoffDp=Ronoff0Dp for n2a

coincident with (midway between) the weak cavity resonances.

Moreover, our numerical simulations as in Fig. 3df suggest that the blue-detuned E1 mode performs three functions: excluding atoms from the exterior of the APCW, as in Fig. 3e; guiding atoms into regions of large E2 probe intensity near the centre of the unit cells, as in Fig. 3d, f; and expelling atoms from the vicinity of other parts of the waveguide, for example, the single-nanobeam regions in Fig. 2b, leaving only the APCW region with signicant atomeld interactions. Together, these considerations enable us to infer quantitatively the single-atom emission rate absent reections from the tapered sections, as we now describe.

Reection measurements and theoretical t. To obtain quantitative information about atomlight coupling in the APCW region, we compare our measurements with a model based on transfer matrix calculations of the optical pathway to and from the APCW as illustrated in Fig. 3a. Details of the optical elements, including the coupling bre, the supporting structures, and the APCW, are described in Methods and ref. 43. Absent atoms, the optical characteristics of these various elements can be deduced from measurements of reection spectra (for example, Fig. 1c and Methods). With atoms, free parameters are the average atom

N within the APCW, the ratio G1D0=G0 for an atom at

the centre of the probe mode E2, and the frequency shift d0 of the line centre n0 relative to free space, d0 n0 n2a. Here atoms are

drawn from a Poisson distribution and placed randomly along the APCW.

Comparisons between measurements and our model for RoffDp=Roff0Dp are given in Fig. 4. For these data, the

weak cavity formed by the matching sections has a small detuning DcC50 GHz from the midpoint between two resonances (free spectral range B600 GHz). With atoms, the cavity detuning results in asymmetric, dispersive-like reection spectra, which is captured by our model. From ts to the measured reection spectra in Fig. 4, we deduce that G1D/G0C0.350.1

and

N0 1:0 0:1 for loading from a free-space cloud of

density r0.

The inferred value of G1D allows us to determine Amreff for

the atomeld interaction in our experiment, namely Amreff

0:44 mm2 for an unpolarized atom. Together with the E2 mode prole, the value of Am suggests that atoms are distributed in narrow regions around reff 0; 0; 130nm, which is con

sistent with our numerical simulations (Fig. 3df).

The large ratio G1D/G0C0.35 implies a single-atom reectivity |r1|C0.26, which is sufcient to give a nonlinear dependence of RoffDp=Roff0Dp on the atom number

number

N observed in the measured spectra. We are thereby able to disambiguate the product

N G1D into separate parameters in tting our model to

measurement.

a

1.05

1.00

0.95

Roff /R 0off

Roff /R 0off

0.90

0.85

0.80

b

1.05

1.00

0.95

0.90

0.85

0.80

20 10 0

10 20

p (MHz)

Figure 4 | Measured reection spectra and theoretical t for the APCW. Measured reection spectra (circles) with free-space atomic cloud densities r/r0 1 (a), and 0.75 (b) where the reectivity with no atoms is

Roff0. The full curves are ts with a model derived from transfer matrix calculations. Error bars for the data points reect 1 s.d. estimated from the statistical uncertainties. We deduce that G1D=G0;

N; d0=G0

0:35 0:1; 1:0 0:1; 0:33 0:06

(a) and (0.360.1,0.76

0.13,0.480.07) (b). Here, r0 2 1010 cm 3; G0/2p 5.2 MHz (decay

rate in free space). The shaded band gives the uncertainty arising from the position of the matching cavity (Methods).

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0.95

0.90

Scaled Roff /R 0

off

0.85

off

1.0

Roff /R 0

0.9

0.80

0.8 0 25 50

P (pW)

60

0 10 20 30 40Scaled P (pW)

50

Figure 5 | Scaled reection minimum as a function of averaged probe power inside the APCW. The saturation spectra (circles) are measured with four free-space densities r/r0 1 (red), 0.81 (blue), 0.75 (green),

0.44 (magenta), and rescaled to a common free-space density r0C2 1010 cm 3; see text and Methods. The inset shows the saturation

data without scaling. An empirical t (solid curve) gives a saturation power of 22.72.2 pW. Error bars for the data points are 1 s.d. estimated from the statistical uncertainties and uncertainties of the t to the model. The gray area shows 95% condence band.

Saturation measurement. To further investigate the nonlinear dependence of the reection spectra on atom number, we measure reection spectra for increasing values of probe power P in the E2 mode for various values of

N. Figure 5 presents results for

RoffDp=Roff0Dpj

min from a series of measurements as in Fig. 4 for increasing P with the plotted points corresponding to the minima of Roff=Roff0 versus Dp for each spectrum. The inset of

Fig. 5 displays four sets of measurements showing that saturation of the atomic response (that is, Roff=Roff0 ! 1 with increasing P)

requires higher power as the density is increased (that is, increasing

N).

The observed saturation behaviour can be scaled into a common curve using the dependence of cooperative atomic emission on atom number

N. We assume that Roff(P) Roff

(P/Psat) and that the saturation power Psat depends on the average total decay rate as Psat / G0

NG1D2 with G1D/G0 0.350.1,

as determined from our measurements in Fig. 4 with P-0. We rescale the probe power (horizontal axis) for each density in the inset of Fig. 5 to a common density r0. Likewise, RoffP=Roff0

(vertical axis) is rescaled using the density dependence derived from our transfer matrix model (Methods), with

N / r. The

approximate convergence of the data to a common curve in Fig. 5 supports our rudimentary understanding of the underlying atom eld interactions in the APCW, including that the observed line shapes for the data in Fig. 5 taken at higher power (not shown) are predominately homogeneously broadened.

To estimate the saturation power in the APCW, we adopt an empirical form for the saturation behaviour RoffP;

N=Roff0

expf g

N=1 P=Psat

N g. From the t in Fig. 5, we deter-

mine Psat

N0 22:7 2:2 pW and g

N0 0:22 0:01

N0 1:0 0:1 as shown in Fig. 4(a). Combining with the measured

effective mode area, we nd the saturation intensity Isat Psat/

AmC5.2 mW cm 2, close to the expected value Is0 G0

N0G1D2=G20 4:0mW cm 2, where Is0 2.7mW cm 2 is the

free-space saturation intensity.

DiscussionWe have realized a novel APCW device for interfacing atoms and photons. The measured coupling rate G1D (quoted absent Purcell

enhancement and inhibition due to an external cavity) is unprecedented in all current atomphoton interfaces, whether for atoms trapped near a nanober2729, one atom in free space45, or a single molecule on a surface46. For example, in ref. 26, a drop in transmission C0.02 is observed for single atoms trapped outside a photonic crystal cavity. In our work without trapping, we observe a dip in reection C0.25 for

N 1 since atoms are

channelled to near the peak of the probe mode in the centre of unit cells with stronger interactions. Further improvements to the APCW include active tuning of the band edge to near an atomic resonance to achieve an increase \50-fold in G1D41,47, although we are mindful of challenges presented by disorder-induced localization48,49. Other opportunities could be tuning to place the atomic resonance within the band gap to induce long-range atomatom interactions4,1618. By optimizing the power and detuning of the E1 trap mode, we should be able to achieve stable atomic trapping and ground state cooling41,50,51. By applying continuous on-site cooling to Nc1 atoms, we expect to create a 1D atomic lattice with single atoms trapped in unit cells along the APCW, thus opening new opportunities for studying novel quantum transport and many-body phenomena518.

Methods

Design principle. An APCW is designed on a chip in order to observe strong atomlight interactions, as illustrated in Supplementary Fig. 1. The APCW interacts with a cloud of caesium atoms trapped in a magneto-optical trap (MOT) that is centred on the photonic crystal. The APCW (see inset V of Supplementary Fig. 1) consists of two parallel nanobeams with sinusoidal corrugations on the outer edges43. The atoms are guided into the centre of the two nanobeams by a scheme that takes advantage of the TE-like modes (y-polarized) near the band edges52. Their highly symmetric mode proles near the dielectric and air bands at frequencies (nD,nA) and the proximity to the resonant frequencies (n1,n2) of the caesium D1, D2 lines allow us to create strong dipole potentials with small optical power (o10 mW) in the E1 mode, while achieving large atomphoton coupling in the E2 mode. See Fig. 1d,e for calculated mode proles.

The corrugations in the APCW are used rather than the more traditional holes because the corrugation amplitude can be patterned more accurately than hole radii, resulting in more accurate alignment of the band gaps and better adiabatic tapers. The outside-corrugated double-beam design used in this work allows superior band edge positioning by placing the modulation of the dielectric away from the strong-eld region in the centre of the waveguide, hence reducing the sensitivity of band edge frequencies to the modulation geometry parameter imprecision. This design also avoids enclosed hole-based geometry, which is difcult to fabricate using available lithography and etching techniques. The design also enables us to build vanishingly small amplitude modulations required for the gradual tapers. The waveguide is made from 200-nm thick stoichiometric SiN with index n 2.0 (ref. 43). SiN is a widely used material in standard silicon-based

fabrication processes, which we have chosen for its low absorption for the Cs D-line transition wavelengths where Si itself is opaque. Its high intrinsic stress and Youngs modulus makes it mechanically robust, hence it was chosen instead of the also low-loss SiO2. The degrees of freedom for the APCW are the gap g, lattice constant a, width w (inner-edge to centre of peaks), and tooth amplitude A. The APCW has 260 cells, gap 250 nm, width 173 nm and tooth amplitude 132 nm. The photonic crystal is tapered on both sides into an unpatterned (translationally invariant) double nanobeam. The length and prole of the tapering determines the reections from the edges of the APCW. The taper used here has 40 cells, and is carefully designed such that the band gap symmetrically opens about 873 nm, which is between the caesium D1 and D2 lines. The prole and the unpatterned double waveguide width are chosen to minimize reections from the APCW edge.

In order to provide optical access for the trapping and cooling laser beams, the SiN waveguides are suspended across a 1 mm long, 3 mm wide through window on the chip, shown in Fig. 2a, and the APCW (see Supplementary Fig. 1) is positioned at the centre. The suspended waveguide extends beyond the window into a V-groove etched into the Si substrate and then reduces in width for efcient coupling to a conventional optical bre44 (sections III of SupplementaryFigs 1 and 2). The silicon anisotropic etch that forms the window also forms the V-groove, which serves to centre the bre to the waveguide. The far end of the waveguide is extended to a fan shape and terminated into the substrate to minimize reection (section VI).

Atomphoton coupling. To characterize the strength of atomphoton coupling, we calculate the effective mode area for an atom at rA

AmrA

R Er0jE2r0j 2 d3r0

aErAjE2rAj

2 ; 1

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where a is a lattice constant of 371 nm and the integration runs over the space occupied by a unit cell. Supplementary Fig. 3 shows AmrA plotted at the central

x 0 plane. For single atoms channelled to the centre of unit cells, we have

Am(0) 0.24 mm2.

To estimate the atomic emission rate into the guided mode, we use G1DrA G0ngs=2AmrA, where G0 is the atomic decay rate in free space, s the

radiative cross-section, and ng the group index at n2. When the band edge frequency nA is placed near n2, we expect ng441 due to the slow light effect41; ng

c2L Dn

2 for the current device where c the speed of light, Dn the distance

between neighboring resonant dips and LPhc the length of APCW (see Supplementary Fig. 1b). Note that due to our small band edge (Do/o0B0.04), we would have to operate much closer to the band gap than in our current experiment in order to observe a signicant slow light effect (for example, ng 410). For atoms guided to the centre of unit cells and driven by the F 4, mF 02F0 5, mF 0

transition, sC5/9 3l2/2p, where l 852 nm is the free-space wavelength of the

Cs D2 line, with then G1D/G0C0.4ng; for unpolarized atoms, we calculate an averaged G1D/G0C0.29ng.

Device characterization. Supplementary Fig. 4a shows a schematic of the setup for device characterization. The reection spectrum near the band gap is measured by sending a broadband light source into the device via the coupled optical bre, and then recording the reection signal with an optical spectrum analyser. The signal is then normalized by the power spectral density of the light source after considering the loss of each optical element. Finer reection spectra around the D1 and D2 lines are measured by scanning the frequency of narrow bandwidth diode lasers. The polarization in the device is aligned to the TE-like mode by observing the polarization-dependent scattering from the rst tether in the coupler, or equivalently by maximizing the reected signal, since the transverse magnetic mode band gap is located at a higher frequency.

Supplementary Fig. 4b shows the measured reection spectra around the band gap. The gray curve shows the original data that has a fast fringe (free spectral range B50 GHz) resulting from the (parasitic) etalon formed from the cleaved bre end-face and the rst matching taper. The red curve shows the smoothed reection spectrum that approximately represents the response of the APCW without the inuence of the reection from the bre end-face.

Device model. Due to imperfection of the adiabatic tapering, the terminal regions of the APCW form a low nesse cavity. When atoms couple to light in the APCW, the reected spectrum depends on the position of the frequency n2 on the cavity fringe and can take on dispersive line shapes. Further complicating the picture is the presence of a fast B50 GHz fringe due to the parasitic etalons formed by the bre end-face, tethers, and the APCW band edges near nD,nA. In order to t the reected atomic signals, a full model that incorporates all of these elements is developed using the transfer matrix method52,53 as shown in Supplementary Fig. 5. A transfer matrix represents each element, and the reection, transmission and loss coefcients are determined by both experiment and FDTD simulations54.

The light is coupled into the device by matching the modes of a 780HP single mode bre (mode eld diameter 5 mm) to a 130 nm wide rectangular SiN waveguide (parts III in Supplementary Figs 1b and 2). The bre end-face reects3.8% power due to the index mismatch. A 90 nm wide tether anchors the coupling waveguide 5 mm from the free (input) end of the waveguide, which has a theoretical transmission of 87% and reection 0.8%. The waveguide width tapers to 200 nm over 300 mm in order to better conne the light to the dielectric, and then the light propagates through the region of the support rails (part III in Supplementary Fig. 1b) to the APCW at the centre of the window. Our numerical simulations show that the taper, support rails and guide should have negligible loss and reection. The loss in these sections is measured to be 22% per mm for a similar device. The details of loss mechanism are currently under investigation. The total loss from the bre face to the waveguide can be estimated experimentally by measuring the reected signal for frequencies within the band gap, assuming the reectivity of the APCW from numerical simulations is B99%. By tting our model to the envelope of the reection spectrum inside the band gap, we obtain the overall transmission efciency from the internal face of the bre to the input of the APCW to be TtC0.60, including propagation losses in the nanobeam.

The tapers of the APCW (represented by the matching mirrors in Supplementary Fig. 5) also reect near the band edge. There is also loss inside the APCW due to fabrication disorder and absorption. Near the Cs D2 line, as shown in the Supplementary Fig. 4c, the tted spectrum (solid gray line) yields the reection and transmission of the matching sections of the APCW, (Rpc 0.28,

Tpc 0.72), the slope of the reection of the APCW dRpc/dl 0.082 nm 1, and

the one-way transmission inside the APCW, TAPCW 0.400.02.

At the bottom of the fast fringe, for the experiment presented in Fig. 3b, measured reection without atoms is Ron0 jEr j 2=jEin j 2 0:3% and

reection in front of M1 estimated from the device model is <on0 9% ; in Fig. 3c,

Roff0 3% and estimated reection in front of M1, <off0 41% .

The role of disorder in the APCW. The presence of fabrication imperfection and disorder do not signicantly alter our analysis, which we discuss in the following section. It is well known that disorder in PCWs can dramatically modify the

optical properties by creating localized modes with high quality factors, resulting in large Purcell enhancement for an emitter within a localization volume48,49. In Figure 1c, we observe variations in the positions and sizes of the cavity resonances near the band edges compared with the ideal case that are likely due to disorder from imperfect device fabrication. However, for our analysis to determine G1D and

N, the only assumption required is that the elds internal to the APCW near the caesium D2 line behave as spatially extended modes over the entire length of the APCW. In this section, we present evidence that this is indeed the case, with effects due to disorder-induced localization playing a minor role.

Many groups have observed non-dispersive regimes in which the localization length LD for light becomes smaller than the length LPCW of the PCW. The non-dispersive regime onsets for relatively large group indices ng Z20 (refs 55,56).

By contrast, our device is operated in a regime where ng E2, much lower than where localization has been observed. In addition to having a low group index, we have determined from SEM images that the disorder on the sidewalls of our SiN devices has s.d. s E2 nm, which is consistent with state-of-the-art fabrication techniques. For our measured low group index and small disorder, multiple analyses in the literature suggest that we are far from the non-dispersive regime.

By performing full 3D FDTD simulations54 of the APCW with and without disorder, we have conrmed that the measured disorder in our devices does not result in localized modes. The disorder is introduced as Gaussian uctuations of s.d. s 2 nm for sidewall uctuations (with a correlation length of 100 nm), as

determined from SEM images. Similar to our measured devices, we observe variations in the positions and linewidths of the cavity resonances formed by the taper sections near the band edges and mode proles extend over the entire length of the APCW, which is consistent with the ideal case. Even for the cavity resonance closest to the band gap, where the group index is ngB10, we observe no signatures in the intracavity intensity that would suggest localization on a scale shorter than the APCW length.

We have also observed evidence that cavity resonances for reected light correspond to intracavity modes that extend over the entire length of the APCW by imaging spatially resolved scattered light along the APCW. For devices similar to the one used in our work, we have observed intensity proles consistent with the cavity resonances and antiresonances near the band edges. For example, at the third cavity resonance from the band gap, we observe three bright antinodes along the photonic crystal. As show in ref. 43, by measuring the scattered light at various positions along the APCW, we can distinguish between even and odd cavity resonances.

Furthermore, if there were to be a localized mode LD o LPCW that has otherwise escaped our detection (for example, with LDooLPCW and a concomitant non-negligible Purcell effect), such a mode would lead to reection spectra that are incompatible with our observations. Roughly speaking, for xed atomic density around the APCW (which we know from shot-to-shot measurements by absorption imaging of the cold atom cloud), a mode of length LD would interact with an atom number f times smaller than is the case for interactions over the entire length LPCW, where f LD/LPCW o1. Hence to create comparable

absorption leading to the observed reection dip, the atomic coupling strength GPurcell would need to be increased beyond G1D for the entire APCW by roughly 1/f

41 for atomic interaction with the localized mode (assuming Purcell enhancement for the localized mode comparable with that for the APCW with a nesse F 2).

Such an increase in coupling by roughly 1/f (from G1D to GPurcell ) and decrease in

average number by Bf (from

N 1 to n

N f ) would be incompatible with our

observations and modeling. For example, the linewidth for the observed reection spectrum would be increased by B1/f441 for LDooLPCW, which is inconsistent

with our measurements.

In summary, the three pieces of evidence given above strongly suggest that cavity resonances in our reection spectrum are not associated with localized modes of spatial extent smaller than our photonic crystal. It is of course possible that other forms of disorder are playing a role in the photonic crystal. An example is disorder from long-range uctuations in the device dimensions that could smooth the observed band edges but that would not induce a localized mode (as we have conrmed by numerical simulation). The consequences of other forms of disorder are currently under investigation. The main point to stress is that our analysis requires only that the observed structure in reection spectra near the caesium D2 line is associated with cavity resonances for which the intracavity intensity extends over the length of the PCW.

Simulations of relative density. To estimate a relative atomic density near the APCW with a guiding potential, we calculate a relative density ~rr rr=r0,

where r0 is a free-space cloud density, with a Monte Carlo simulation of 5 106

trajectories of thermal atoms with a temperature of 20 mK (ref. 57). For the simulations, the CasimirPolder potential UCPr for the APCW is computed

numerically following ref. 41, with an example cut shown in Supplementary Fig. 6a. The dipole potential Udipoler of the blue-detuned guided mode E1r near n1 is

calculated by using the mode function obtained with MIT Photonic Band package58. Trajectories are obtained by numerically solving the equation of motion with force of F =Utotr =UCPr U

dipole

r. The relative density is

inferred from atomic ux crossing each grid. Note that velocity-dependent forces of polarization-gradient cooling are not included in the simulations.

In Fig. 1f and Supplementary Fig. 6b, we use a guided mode E1r at the band

edge kD,x p/a at nD with total power of 1 mW and 10 GHz blue-detuning from

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F 42F0 4 transition frequency of D1 line, which has zero intensity at the

centre of unit cells of the APCW. Thus, atoms are channelled into unit cells by the combination of CasimirPolder and optical dipole force. The relative density at the centre of unit cells is ~r0 0:3.

In Fig. 3df and Supplementary Fig. 6c, we use experimentally excited E1r

with k1,x near n1 (Cs D1 line), which is B1% below the band edge kD,x. Due to the small deviation from the band edge, the intensity of the mode eld E1r near n1

has a small bump at the centre of the unit cells, which for blue detuning prevents atoms from being channelled to the centre. In addition, the intensity of E1r has a

longitudinal component Ex along the propagation direction, which is p/2 out-of-phase with the transverse components. As a consequence, the polarization of E1r

is elliptical everywhere except for the central y 0 plane of unit cells due to TE

symmetry. The resulting guiding potential has vector shifts due to the ellipticity of E1r, which lead to mF-dependent guiding potentials. Since the centre of guiding

channel for |mF| a0 states moves from y 0 to |y| 40 due to a stronger ctitious

magnetic eld (along z) closer to the structure, the guiding efciency from optical dipole forces is reduced as |mF| increases.

Figure 3df displays xr ~rr G1Dr=G1D0, where ~rr is simulated

relative atomic density with a guided potential for mF 0, and G1D(0) is a decay

rate into the APCW at the centre of unit cells. Due to the small deviation from the band edge and the resulting potential bump, atoms are localized around 4100 nm from the centre of unit cells.

We note that, in the case of Supplementary Fig. 6c, adding to Udipole the dipole

potential from a weak red-detuned E2r mode can help overcome the potential

bump in the gap centre and can create a stable trapping condition. This scheme is

currently under investigation and will be presented elsewhere.

Experimental procedure. To prevent caesium contamination of the APCW due to the background vapor pressure, our vacuum system consists of a source chamber and a science chamber, connected via a differential pumping stage. The source chamber runs in a standard MOT loaded from the background Cs vapor. From the source MOT, a pulsed push beam extracts a ux of cold atoms that is slow enough to be recaptured in a science MOT located in the UHV region59. We load a science MOT for 1 s and compress it for a duration of 50 ms (ref. 60). We then obtain a cloud of cold atoms with a peak density of B1 1011 cm 3 and a temperature of

40 mK about 1 cm away from the APCW of a silicon chip.

In order to transport cold atoms near the APCW, we cool atoms in the moving frame towards the APCW by abruptly changing the centre of magnetic quadrupole eld61. After shutting off cooling beams, a cloud of atoms freely propagates towards the APCW. By turning on additional MOT beams at the time atoms y near the APCW, we cool and recapture propagating atoms with an efciency of B40%. After applying polarization-gradient cooling, we obtain a cloud of cold atoms with a peak number density of r0B2 1010 cm 3, spatially overlapped with the APCW.

An overview of reection measurements in our experiment with cold atoms near the APCW is given in Supplementary Fig. 7. A blue-detuned guiding beam with power B0.6 mW and detuning of 10 GHz from D1 (F 42F0 4) at

k1,x 0.99kD,x is sent into the device, which is kept on throughout the experiment.

The probe pulse is combined with the guiding beam by a volume Bragg grating and then couples to the APCW via a bre beam splitter with T 99% and R 1%. The

reected probe signal from the APCW is efciently picked up by the bre beam splitter through the transmission path. An additional volume Bragg grating at the output reects the return probe beam, which allows us to measure the probe pulses with the guiding beam on. A pair of l/2 and l/4 waveplates in each path is used for aligning the polarization to only excite the TE-like mode.

To maximize the signal-to-noise ratio, we perform reection measurements with atoms after aligning the bottom of the fast fringe to the Cs D2 line, where the probe eld is maximized inside the parasitic cavity formed by the bre end-face and the APCW region; see Supplementary Fig. 8a. The alignment of the fast fringe can be tuned by sending an additional few mW of heating beam to heat up the device and adjust the optical path length between the bre end-face and the APCW region. The heating beam runs at frequency n4nD inside the band gap and is

C5 nm detuned from D1 line, thus does not interfere with atomlight interaction in the APCW region.

Model of reection spectrum of atoms. Reection spectra of guided atoms are obtained by including transfer matrices for atoms62 in the device model. Guided atoms inside the APCW are drawn from a Poisson distribution with mean atom number

N and randomly placed at the centre of unit cells along the APCW. Each of the two matching sections that terminate the APCW partially reects light near the band edge, as depicted in Supplementary Fig. 5. Together the matching sections form a cavity around the APCW (denoted by M1, M2 in Supplementary Fig. 5), whose cavity length has a frequency dependence. We incorporate the uncertainty of the location of the matching mirrors and resulting cavity relative to the APCW into our model. This uncertainty gives rise to a variation in the reection spectra from our model, which is given by the thickness of the lines shown in Supplementary Fig. 8.

The wave vector of probe frequency kp,x is C2% from the band edge at kA,x p/

a as shown in Fig. 1b. In the case of more than one atom coupled to the APCW, the mismatch between kA,x and kp,x causes dephasing since the accumulated phase between atoms is not an integer multiple of p. Due to the moderate one-way

transmission of TAPCWC0.40 inside the APCW as described in Section III, the cooperative effect of two atoms is sensitive to how loss occurs inside of the APCW. Here, we consider two limiting cases: (i) uniform absorption along the APCW, and(ii) loss at the rst matching mirror.

In the case of (i), due to uniform loss along the APCW, only nearby atoms

interact equally with the probe eld. Thus, cooperative effects survive despite of the mismatch of kp,x and kA,x. On the other hand, in the case of (ii), all atoms contribute equally and cooperative effects are washed out due to the phase mismatch for propagation with kp,x as compared with kA,x for the unit cell. Given our current limited knowledge of the microscopic details of the loss mechanisms within the APCW, the models (i, ii) provide a means to estimate the uncertainties in our inferences of G1D/G0 and average atomic number N based upon comparisons of our data with the models.

For the cases shown in Supplementary Fig. 8, the model ts lead to the following: (i) G1D/G0 0.310.05 and average atom number

N 1:5 0:2 in

(b), and (ii) G1D/G0 0.410.04 and

N 0:9 0:1 in (c).

From four sets of data as in Supplementary Fig. 8, taken for comparable atomic densities, we make ts based upon the two models (i) and (ii). We average results for parameters determined from the ts to arrive at the values quoted, namely G1D/

G0 0.320.08, N 1:1 0:4 and d0/G0 0.130.27.

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Acknowledgements

We gratefully acknowledge the contributions of D. Alton, J. Cohen, D. Ding, P. Forn-Diaz, S. Meenehan, R. Norte, and M. Pototschnig. Funding is provided by the IQIM, an NSF Physics Frontiers Center with support of the Moore Foundation, the DARPA ORCHID program, the AFOSR QuMPASS MURI, the DoD NSSEFF program (H.J.K.), and NSF PHY-1205729 (H.J.K.). A.G. is supported by the Nakajima Foundation. S.P.Y. and J.A.M. acknowledge support from the International Fulbright Science and Technology Award. The research of K.S.C. is supported by the KIST institutional program. D.E.C. acknowledges funding from Fundaci Privada Cellex Barcelona.

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All authors contributed extensively to the research presented in this paper.

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How to cite this article: Goban, A. et al. Atomlight interactions in photonic crystals. Nat. Commun. 5:3808 doi: 10.1038/ncomms4808 (2014).

NATURE COMMUNICATIONS | 5:3808 | DOI: 10.1038/ncomms4808 | http://www.nature.com/naturecommunications

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