Trapped Atoms in One-Dimensional Photonic Crystals

3 MB·English

by  C. -L. Hung

Trapped Atoms in One-Dimensional Photonic Crystals
C.-L. Hung∗,1,3 S. M. Meenehan∗,2,3 D. E. Chang,4 O. Painter,2,3 and H. J. Kimble1,3
1 Norman Bridge Laboratory of Physics 12-33
2 Thomas J. Watson, Sr., Laboratory of Applied Physics 128-95
3 Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA and
4 ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
(Dated: January 23, 2013)
We describe one-dimensional photonic crystals that support a guided mode suitable for atom
trapping within a unit cell, as well as a second probe mode with strong atom-photon interactions.
AnewhybridtrapisanalyzedthatcombinesopticalandCasimir-Polderforcestoformstabletraps
forneutralatomsindielectricnanostructures. Bysuitabledesignofthebandstructure,theatomic
3 spontaneous emission rate into the probe mode can exceed the rate into all other modes by more
1 than tenfold. The unprecedented single-atom reflectivity r (cid:38)0.9 for the guided probe field should
0
0 enable diverse investigations of photon-mediated interactions for 1D atom chains and cavity QED.
2
PACSnumbers: 42.50.Ct,37.10.Gh,37.10.Jk,42.50.Ex
n
a
J
New opportunities in Atomic, Molecular, and Optical tions are exploitedto close the trap perpendicular to the
2
Physics and Quantum Information Science emerge from plane of structure, which would otherwise be unstable
2
the capability to achieve strong radiative interactions with either the FORT or the CP potential alone [37].

] between single atoms and the fields of nanoscopic op- In addition to the waveguide trapping proper-
s
c tical waveguides and resonators [1]. For example, strong ties, strong near-resonance atom-photon interactions of
i atom-photoninteractionsinlithographicstructures[2–6] trapped atoms are found to arise for waveguides with
t
p could be used to create quantum optical circuits with properly tailored band structure [38–42]. For practically
o long-range atom-atom interactions mediated by single realizable structures, we find γ /γ(cid:48) (cid:38) 10, where γ
. 1D 1D
s photons [7, 8]. Moreover, linear arrays of atoms ra- is the atomic decay rate into the (guided) probe mode
c
diatively coupled to nanophotonic waveguides exhibit a and γ(cid:48) the rate into all other modes. One atom trapped
i
s wide range of remarkable phenomena, including coher- within the structure could thereby attenuate a resonant
y
h ent transport of atomic emission [9–11], guided superra- probe with transmission |1−r0|2 (cid:46)10−2 [9, 16].
p dianceandpolaritons[12–14],aswellashighlyreflecting AsillustratedinFig. 1,wefocusontwoofthesimplest
[ atomicmirrors[15,16]. Theinterplayofatomicemission quasi-1D photonic crystal geometries. The first waveg-

into the waveguide and photon-mediated forces can lead uideconsistsofasinglesilicon-nitridenanobeam(refrac-
1
v to self-organization of atoms into exotic spatial configu- tive index n=2) with a 1D array of filleted rectangular
2 rations along the waveguide [17, 18]. holesalongthepropagationdirection;atomsaretrapped
5 A long-standing obstacle to this scientific frontier is inthecentersoftheholes(Fig. 1(a)). Thesecondwaveg-
2
the challenge of trapping atoms in vacuum near dielec- uide consists of two parallel silicon nitride nanobeams,
5
tricsurfaces(∼100nm)whileatthesametimeachieving each with a periodic array of circular holes, with atoms
.
1
stronginteractionsbetweenoneatomandphoton. Afar- trapped in the gap between the beams (Fig. 1(e)).
0
off resonance dipole-force trap (FORT) [19] can provide The design of a 1D-photonic crystal waveguide with
3
1 atomic localization by using modes of the dielectric for distinct modes for optical trapping and strong atom-
: optical trapping [20–22] and has been used to trap cold photon interactions is constrained by the region of the
v
atoms within hollow-core optical fibers [23–26] and ex- optical band structure containing a continuum of un-
i
X ternal to fiber-taper waveguides [27–29]. guided optical modes (i.e., the light cone indicated in
r Motivated by these advances, in this manuscript we gray in Fig. 1(b, f)). Modes within the light cone can
a
present principles for the design of optical traps and stillhavelargeamplitudeinthestructurebutradiateen-
strongatom-photoninteractionsinone-dimensional(1D) ergy into the surrounding vacuum leading to unaccept-
photonic crystal waveguides. We analyze the potential able loss. The top of the vacuum light line is at the
U (r) due to light-shifts from a FORT [30–32] together Brillouin zone boundary (X-point, where k a = π), so
tot x
with Casimir-Polder (CP) interactions with the dielec- thelatticeconstantaisconstrainedbya<λ/2, whereλ
tric [33–36] (Figs. 1(a, e)). Despite the proximity of the is the smaller of the (vacuum) wavelengths for trapping
surfaces, stable potentials U (r) are achieved for mod- and probe fields. Here, k is the Bloch wavevector along
tot x
est optical intensities (∼ 5mW/µm2) for blue-detuned the waveguide axis x.
FORTs operated at a ‘magic’ wavelength for the D2 line Onceaisfixed,additionalguidedmodescanbe‘pulled’
of atomic Cesium [19]. A new possibility for trapping is belowthelightlinebyincreasingthewidthandthickness
also identified for which vacuum forces from CP interac- ofthestructure. Withappropriatemodesbelowthelight
2
(a) hy a (b) 400 otheratomictransitionsbywayofthescaleinvarianceof
Maxwell’s equations [44].
z)380 The photonic crystals are assumed to be suspended in
H
z hx t ν (T360 vcuaclautuemd uasnindgcothmepMosIeTd pofhoStioNni.cBbaannddsstsroufctwtuarreespaareckcaagle-
y 340 [45]. Field profiles for guided modes are calculated using
w
x
the finite-element-method (FEM) simulations [46]. Re-
(c) (d) 0.4 0.42 0.44 0.46 0.48 0.5
sults for the single and double nanobeam structures are
k (2π/a)
x
presented in Fig. 1.
With suitable guided modes for trapping in hand, we
have developed numerical tools for evaluating the FORT
(e) (f) 440 and CP potentials inside the waveguide, and hence the
g a 400 total potential Utot = UFORT+UCP. The adiabatic po-
z y d wt ν (THz)360 tfi[3ee0nld–ti3ad2l]i.sUtHrFiOebrRueTtiu(orn)(oisrf)rteihsaedthitlreyappcearmlicoouddlaiect,eBdElo(urcs)hin=wgautvhekexf(uernl)ecectitkiroxinxc
x 320 kx
at propagation constant k .
x
280 The surface potential U (r ) is determined from the
(g) (h) 0.34 0.38 0.42 0.46 0.5 CP a
k (2π/a) formalism in Ref. [35] for the imaginary component
x
of the scattering Green’s tensor G (r ,r ,ω), which is
sc a a
the Green’s tensor from Maxwell’s equations for a point
dipole at the atomic location r with the vacuum con-
a
tribution (i.e., no dielectric structure) subtracted. We
FIG.1: a)Schematicforthesinglenanobeamstructurewith
evaluate G (r ,r ,ω) numerically by adapting the pro-
sc a a
dimensions (a,w,t,h ,h ) = (367,845,825,246,745)nm. b)
x y cedures from Ref. [36], as described in [47].
Band diagram for the single nanobeam in a) showing only
Figures 2 and 3 display numerical results for U (r),
bands with even vector symmetry about the y and z sym- CP
metry planes. The trapping and probing bands are shown as UFORT(r), and Utot(r) for the single and double
thicker lines, with the trap ω /2π (probe ω /2π) frequency nanobeams for k below the X-point. The calculations
T A x
as a blue (red) dashed line. c) Field intensity of the blue are for the 6S ,F =4 hyperfine ground state of Cs for
1/2
trapping mode and d) field amplitude of the probe mode in theFORTmodesindicatedinFig. 1(c,g)[48]. Forthese
the center plane z =0 for the single nanobeam in a). Green
initial calculations, we make the reasonable assumption
spheres mark the locations of minima of the trapping poten-
for SiN that the dielectric constant (cid:15) is frequency inde-
tial. e) Schematic for the double nanobeam structure with
pendent, (cid:15)(r,ω)→(cid:15)(r).
(a,w,t,d,g) = (335,335,200,116,250)nm. f) Band diagram
forthedoublenanobeamine)displayingonlymodesofeven Utot(r) for the single nanobeam in Fig. 2 reveals that
vector symmetry in z. The proximity of the two nanobeams modest optical intensity is sufficient to overcome the at-
resultsinabandstructurecomposedofeven(green)andodd tractive CP interactions and create a stable potential
(magenta) superpositions of single nanobeam modes. We fo- minimum in the center of the vacuum space at r =0
min
cus on the even parity supermodes due to their large field
within a unit cell. An atom would be localized at dis-
amplitude in the gap. g) Field intensity of the blue trapping
tances (d ,d ) = (123,373)nm from the walls of the di-
modeandh)fieldamplitudeoftheprobemodeinthecenter x y
electric. The trap oscillation frequencies for a Cs atom
plane z = 0 of the double nanobeam in e). The black dia-
monds in b) [f)] mark resonances for finite structures of 81 would be (fx,fy,fz)(cid:39)(612,180,484)kHz.
unit cells from Figs. 4 [5]. For the double nanobeams [49], the FORT alone is
insufficient to trap the atom, as the mode has a (weak)
lineforprobingandtrapping,thespacingofthesemodes localintensitymaximumalongthezdirectionthatrepels
at the X-point can be tuned by altering the size of the an atom. However, the CP potential U (r) along z
CP
holes, whichenablestheprobemodetoberesonantwith provides the force necessary to overcome the repulsive
the frequency ω of the atomic transition while simul- optical force and to form a stable trap. The result is a
A
taneously matching the optical frequency ω of the trap hybrid optical-vacuum trap that circumvents the ‘no-go’
T
mode to a ‘magic’ frequency for the atom [19]. theorem for vacuum trapping alone [37].
Within this general context, here we consider only Potentials for our hybrid trap are illustrated in Fig.
blue-detunedFORTsforwhichthetrappingmodehasan 3. At the trap minimum r = 0, an atom would be
min
intensity minimum at the trapping site [43]. Our analy- localized at distance d =125nm from adjacent surfaces
y
sisisfortheD2lineofatomicCswithprobewavelength of the dielectric beams. Oscillation frequencies for a Cs
near the atomic resonance λ =852nm and with a blue- atom would be (f ,f ,f )(cid:39)(1013,390,57)kHz.
A x y z
detuned FORT at the magic wavelength λ = 793nm As concerns strong radiative interactions, our struc-
T
[19]. Note that our results are readily transcribed to tures trap an atom in a region of large amplitude for the
3
(a) (c) (a) (c)
1.2 4
U (mK)CP--00-5..480 -0.04 U (mK)000...048 U (K)μCP-2000 0 U (mK)20
m) 0 -100 0 100 -50 -100 0 100
x (n 50 -200 y (n0m) 200 -1.0 (dmK))12..00 x (nm) y (nm) 050 -200 0z (nm20)0 -250 (d)K)1 x (nm)
(b) U ( (b) U (m
0
0.0
U (mK)tot1-05..550 2.0 (e)2.0 -200 y (n0m) 200 U (K)μtot1000 120 (e) 0-100 y (n0m) 100
x (nm)500 200 -0.04 U (mK)1.0 y (n-m50) 0 0 200 -35 U (K)μ--2100
0
-200 y (nm) 0.0 50 -200 z (nm) -30
-300 0 300
-200 0 200
z (nm)
z (nm)
FIG.2: Trappingpotentialsforthesinglenanobeamstructure FIG. 3: Trapping potentials for the double nanobeam struc-
in Fig. 1(a) for Cs 6S1/2,F = 4 level and λT = 793nm. ture in Fig. 1(e) for Cs 6S1/2,F =4 level and λT =793nm.
(a) Casimir-Polder potential UCP(r) and (b) total potential (a) Casimir-Polder potential UCP(r) and (b) total potential
Utot(r)=UCP(r)+UFORT(r) in the central z =0 plane. (c- Utot(r) = UCP(r)+UFORT(r) in the transverse x = 0 plane.
e) show line cuts of UCP (red solid), UFORT (blue dashed), (c-e)showlinecutsofUCP (redsolid),UFORT (bluedashed),
and Utot (blue solid) along the (c) x-, (d) y-, and (e) z-axis. and Utot (blue solid) along the (c) x-, (d) y-, and (e) z-axis.
Average trap intensity for a unit cell is 4.9mW/µm2. Average trap intensity for a unit cell is 3.5mW/µm2.
modes. IntheregionneartheCsD2line(i.e.,ν (cid:39)ν =
probe field, leading to small mode volume per unit cell d A
ω /2π =352THz),wefindpeaksinIm[G ]forthesin-
[47]. It is well known that atom-photon interactions can A xx
gle nanobeam and in Im[G ] for the double nanobeam.
befurtherenhancednearabandedge[38–42], wherethe yy
These peaks are due to emission into our designated
density of states diverges due to a van Hove singular-
probe modes for the respective structures, where for the
ity. Toquantifytheradiativecoupling,wedeterminethe
single (double) nanobeam(s), the probe mode is princi-
decay rate γ for a point dipole located at r = 0 for
tot a
pally polarized along the x-(y-) axis. Each peak is from
a structure with N unit cells [50]. FDTD calculations
are performed to evaluate the classical Green’s tensor a discrete set of propagation constants kx(n) (cid:39) πn/aN
G(r ,r ,ω) and thence γ following Refs. [34, 35, 51], imposed by the boundary conditions for the finite struc-
a a tot
as described in [47]. tures. Here, N is the total number of cells in the single
Figures4(a)and5(a)displaythediagonalcomponents (double) beam, and n≤N is an even (odd) integer. We
Im[G (ν )] of the Green’s tensor as functions of dipole findexcellentagreementbetweenthefrequenciesofthese
ii d
frequency ν = ω /2π and relate to the emission rate of resonancesandthebanddiagramoftheprobemode(‘di-
d d
resonant point dipoles polarized along the i = x, y, or amonds’inFigs.1(b)and(f))forvariousvaluesof(n,N).
z-axis for the single and double nanobeams. Firstly, in The peaks become larger and narrower as kx(n) ap-
Fig.4(a),Im[G ]isenhancedalongthex-(periodic)di- proaches the X-point, owing to the diminishing group
xx
rectionacrossabroadfrequencyrange,andissuppressed velocity [40, 42, 47]. Beyond the X-point, the probe
inthey-andz-directions,ascanbeexplainedbytheori- resonances disappear, leaving a broad background cor-
entation of the induced array of image dipoles along the responding to coupling into lossy (radiation) modes.
single nanobeam. When the source dipole is polarized On an expanded frequency scale around ν (cid:39) ν ,
d A
along the x-axis, the image dipoles line up head-to-tail Figs. 4(b) and 5(b) show calculated atomic decay rates
and, just below the X-point, constructively interfere to γ for the 6P ,F(cid:48) = 5 → 6S ,F = 4 transition
tot 3/2 1/2
enhancedipoleemission;y,z orientationsrenderdestruc- in atomic Cs [47]. When the atomic dipole is aligned
tive interference and suppressed emission. Comparable along the principal polarization of the designated probe
suppression is not apparent for the double nanobeams in mode(xˆ forthesinglebeamandyˆforthedoublebeam),
Fig. 5(a) since no such array of image dipoles is formed. the emission rate γ into the probe mode is strongly
1D
Secondly, Figs. 4(a) and 5(a) display a series of res- enhanced at frequencies corresponding to k(n) near the
x
onant peaks due to strong emission into various guided X-point. Specifically, for ν = ν large enhancements
d A
4

(a) (a)
] ]
G010 x G010 x
m[ y z m[ y z
G]/Iii 1 G]/Iii 1
m[ m[
I I
200 250 300 350 400 450 200 250 300 350 400 450
νd (THz) νd (THz)
(b) (c) 8 (b) (c) 2
15
m’ = 0 m’ = 5 20 m’ = 0 m’ = 5
F F F F
γ0 10 γ0 γ0 γ0
γ/tot 5 y x γ/tot 4 γ/tot10 y x γ/tot1
0 0 0 0
344 348 352 356 340 350 360 344 348 352 356 340 350 360
νd (THz) νd (THz) νd (THz) νd (THz)
FIG.4: Green’stensorandtotalatomicdecayrateγ versus FIG. 5: As in Fig. 4 now for the double nanobeam at r=0.
tot
source dipole frequency ν for the single nanobeam at r=0. (a) shows the diagonal components of the Green’s tensor,
d
(a) shows the diagonal components of the Green’s tensor, Im[G ] (solid black), Im[G ] (red), and Im[G ] (blue),
xx yy zz
Im[G ] (solid black), Im[G ] (red), and Im[G ] (blue), normalized to Im[G ] for free space (dashed line). N = 81
xx yy zz 0
normalizedtothefreespacevalueIm[G ](dashedline). The unit cells. Cesium D2-line frequency ν = 352 THz is cen-
0 A
number of unit cells is N = 81. Cesium D2-line frequency tered in the shaded area. The vertical dotted line marks the
ν = 352 THz is centered in the shaded area. The vertical light line, beyond which all decay channels are lossy. (b, c)
A
dottedlinemarksthelightline,beyondwhichalldecaychan- show γ (red curves), normalized to the free space value γ
tot 0
nels are lossy. (b, c) show γ (black curves), normalized to (dotted line), in the frequency range marked by the shaded
tot
the free space value γ (dotted line), in the frequency range areain(a). Thesolid(dashed)curveisevaluatedusing81(61)
0
markedbytheshadedareain(a). Thesolid(dashed)curveis unit cells. The atomic spin is aligned to the y-axis, with the
evaluated using 81(61) unit cells. The atomic spin is aligned spinprojectionquantumnumber(b)m(cid:48) =0and(c)m(cid:48) =5.
F F
to the x-axis, with the spin projection quantum number (b)
m(cid:48)F =0 and (c) m(cid:48)F =5. leads to r0 (cid:39) 0.95. For a cavity QED system with
one ‘impurity’ atom surrounded by N ‘mirror’ atoms
A
in γ occur for the initial excited state 6P ,F(cid:48) =
1D 3/2 alonga1D-lattice[16],theratioofthecoherentcoupling
√
5,m(cid:48)F = 0, while γ1D is suppressed for the initial state rate g1 = NAγ1D/2 to the effective dissipative rate γ(cid:48)
m(cid:48) =5. This is because the probe mode predominantly
F would exceed unity even for NA = 1 atom. For conven-
supports π-polarization and hence ∆m = 0. Coupling
F tionalcavityQED,weestimateavacuumRabifrequency
between states with ∆m (cid:54)= 0 is small. Of course, ad-
F ∼ 2π × 6GHz for an atom trapped in the 1D-photon
ditional guided modes can contribute to γ , as is evi-
tot crystal waveguides studied here, making the reasonable
denced for the m(cid:48) = 5 state due to field polarizations
F assumption of a cavity formed from N ∼10 unit cells.
perpendicular to the atomic spin, such as zˆ (xˆ) for the
Certainly there are challenges to the implementation
single (double) beam(s) in Fig. 4(a) (5(a)).
of our designs for trapping atoms in 1D photonic crys-
Fromγtot(νd)andananalyticmodelofcouplingtothe talwaveguideswithstrongsingle-photoninteractions,in-
guided-mode near the X-point, we estimate the contri- cludingatomloadingintothesmalltrapvolumeandlight
butions of γ1D and γ(cid:48) to γtot =γ1D+γ(cid:48) near the largest scattering from device imperfections. We are working
resonances in Figs. 4(b) and 5(b) [47]. For m(cid:48) = 0 and to address these issues by numerical simulation, device
F
N = 81, we find that γ1D/γ0 (cid:39) 15 and γ(cid:48)/γ0 (cid:39) 1.2 for fabrication, and cold-atom experiments with nanoscopic
thesinglenanobeam,whileγ1D/γ0 (cid:39)21andγ(cid:48)/γ0 (cid:39)1.0 structures. Our efforts are motivated by the predic-
for the double nanobeams [52]. Here, γ0/2π = 5.2MHz, tion γ1D/γtot (cid:38) 0.9 in Figs. 4, 5, which is unprece-
the free-space Cs decay rate. dentedinAMOphysicsandwhichcouldcreatenewscien-
The ratios γ /γ and γ /γ(cid:48) serve as metrics for tific opportunities (e.g., quantum many-body physics for
1D tot 1D
thestrengthofatom-photoninteractionsforour1Dpho- 1D atom chains with photon-mediated interactions, and
tonic crystals. For example, the resonant reflectivity r high-precision studies of vacuum forces). Moreover, our
0
of a trapped atom for the probe field should scale as double nanobeam structure provides proof-of-principle
r = γ /γ [9, 16], which for the double nanobeams for a promising new concept that combines optical and
0 1D tot
5
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[31] Fam Le Kien, P. Schneeweiss, and A. Rauschenbeutel,
arXiv:1211.2673v1 (2012).
[32] D. Ding, A. Goban, K. S. Choi, and H. J. Kimble,
6
SUPPLEMENTAL MATERIAL CALCULATION OF γ
tot
Todeterminethetotalspontaneousdecayrateγ for
tot
an atom in our structures, we also solve for the classical
CALCULATION OF CASIMIR-POLDER
POTENTIALS Green’s tensors and evaluate γtot via [2, 6, 7]
 
The Casimir-Polder potential UCP(ra) in Ref. [1] is γ = 2µ0ωj2Im(cid:88)Tr[D ·G(r ,r ,ω )], (3)
calculated from the following integral [2]: tot (cid:126) j a a j
 
{0}
UCP(ra)= where Dj =(cid:104){0}|d†|j(cid:105)(cid:104)j|d|{0}(cid:105) is the dipole matrix ele-
(cid:126)µ (cid:26)(cid:90) ∞ (cid:27) ment between the ground state manifold and the excited
− 2π0Im dωω2Tr[α0(ω)·Gsc(ra,ra,ω)] , (1) state j, and ωj is the transition frequency. The total de-
0 cay rate γ =γ +γ(cid:48) includes the decay rate γ to a
tot 1D 1D
guidedmodeofinterestaswellastherateγ(cid:48) toallother
where Tr[.] denotes the trace, α0 is the dynamic po-
modes of the structure, including lossy modes. As dis-
larizability tensor of ground-state Cesium atom, and
cussed below, the contributions of γ ,γ(cid:48) to γ can be
1D tot
G (r ,r ,ω) = G(r ,r ,ω)−G (r ,r ,ω) is the scat-
sc a a a a 0 a a estimated from the global frequencydependence γ (ω).
tot
tering Green’s tensor, that is, the Green’s tensor G
To obtain γ (ω), we evaluate the Green’s tensor for
tot
subtracted by the vacuum contribution G evaluated
0 therealdielectricfunction(cid:15)(r)usingtheFDTDmethod,
at atomic location r ; 2π(cid:126) is Planck’s constant, and
a followed by a discrete Fourier analysis.
µ is the vacuum permeability. The Green’s tensor
0
(cid:104)
is the solution to the Maxwell equation ∇ × ∇ ×
VALIDATION
(cid:105)
−ω2(cid:15)(r,ω) G(r,r(cid:48),ω) = Iδ(3)(r−r(cid:48)), corresponding to
c2
theelectricfieldresponsetoapointdipolecurrentsource. To validate our numerical procedures, we have per-
(cid:15)(r,ω)isthedielectricfunction,andIistheunitytensor. formed calculations of UCP for several geometries where
analyticalsolutionsareavailable,includinganatomnear
We employ finite-difference-time-domain (FDTD) cal-
an infinite dielectric or metal half-space [8] and an atom
culations [3] to solve numerically for the Green’s tensors
locatedaboveaninfinitedielectricgrating[9], andfound
of our structures. The integral of Eq. (1) is evaluated by
excellent agreement between our simulations and the ex-
adaptingaprocedureestablishedinRef. [4]andbyusing
(cid:112) act results.
adeformedcontourω(ξ)=ξ 1+iσ/ξ intheupperhalf
Wehavevalidatedourcalculationsofγ forthecases
of the complex frequency plane, parametrized by a real tot
of an (atomic) dipole near an infinite dielectric, metallic
numberξ ≥0andaconstantσ >0. AsexplainedinRef.
parallel plates, a nanofiber [10], and 2D-photonic band-
[4],thisisequivalenttosolvingtheGreen’stensoratreal
gap microcavities [11].
frequenciesξ withafictitiousglobalconductivityapplied
to the dielectric function (cid:15)(cid:48)(r,ξ)=(1+iσ/ξ)(cid:15)(r,ξ). The
integration can then be performed in the time domain GUIDED MODE RESONANCES
(via the convolution theorem) and converges quickly due
to fast decay from σ. For our structure with an infinite number of unit
Specifically, Eq. (1) is numerically evaluated using cells, a guided mode (denoted by λ) contribution to
the imaginary part of the Green’s tensor can be calcu-
(cid:126) (cid:90) ∞ lated as [7, 12], Im(cid:2)Gλ1D(ra,ra,ω)(cid:3) = ac2uλ(ra;kx) ⊗
UCP(ra)= 2π dtIm[gµν(−t)]xˆµ·Esc,ν(ra,t), (2) u∗λ(ra;kx)/2ωvλ,whenthefrequencyωintersectsthefre-
0 quency band ω at a propagation constant k below the
λ x
light line. Here, u (r;k ) is the orthonormal mode func-
λ x
where E (r ,t) is the (real) electric field generated
sc,ν a tion, and v is the group velocity, both available via nu-
λ
by a point dipole current source J = δ(t)xˆ (xˆ =
ν ν mericalcalculations[13,14]. Aswescanthefrequencyω,
xˆ,yˆ,zˆ) located at the position ra and scattered by a Im(cid:2)Gλ (cid:3)increasesmonotonicallyanddivergesask ap-
structure with a dielectric function (cid:15)(cid:48)(r,ξ) [5]. Here, 1D x
proaches the X-point, where v → 0. The guided mode
λ
the indices µ and ν are repeated for summation con-
Green’stensorvanisheswhenωliesbeyondthefrequency
vention, g (t) is the Fourier transform of g (ξ) =
µν µν of the band edge.
(cid:113) (cid:16) (cid:17)
−iξ 1+ iσ 1+ iσ Θ(ξ)α0 (ω(ξ)), and Θ(ξ) is the Based on this analysis, we can evaluate the decay rate
ξ 2ξ µν
Heaviside step function. For the initial calculations in γ(∞)(ω) into the designated probe mode for an infinite
1D
Ref.[1], wetakethedielectricconstant(cid:15)tobefrequency structure, and compare it with the heights of resonant
independent, (cid:15)(r,ω)→(cid:15)(r). features in γ (ω) for finite structures with different
tot
7
numbers of unit cells, as shown in Figs. 4(b) and 5(b) plitude for the probe field. One measure of the strength
of Ref. [1]. Indeed, the actual γ of a finite-size struc- of the atom-field coupling is the effective mode volume
1D
turemustdeviatefromγ(∞) duetoboundaryconditions Vm per unit cell, where
1D
that transform a continuous spectrum into a discrete set
(cid:90)
of resonant peaks [15], as shown in Figs. 4(b) and 5(b) V = (cid:15)(r)|E(r)|2d3r/(cid:15)(r )|E(r )|2. (4)
m min min
[1]. When the number of unit cells is increased in our
calculation of γ over the range N =11 to N =81, we
1D
Here the integration is carried out over the volume of a
find that the frequencies ω(n) of the resonant peaks shift
unit cell. That is, the integration domain along propa-
inpositionandthepeakschangeheight. Asdocumented
gation direction x extends over the distance a (i.e., the
by the black diamonds in Figs. 1(b), 1(f) of Ref. [1], the
lattice constant), while in the transverse y,z directions,
ω(n) arise from the discrete set of propagation constants
the integration domain is from −∞ to +∞.
k(n) (cid:39) πn/aN imposed by the boundary conditions for
x
For the single nanobeam, the probe mode has a global
the finite structures with n either even or odd.
maximum at r = 0 and an effective mode volume
The peaks in γ at the set of frequencies ω(n) build min
tot V ∼ 0.13 µm3. For the double nanobeams, the probe
upontopofafairlyconstantbackgroundwithinthefre- m
mode has a saddle-like intensity distribution around
quency range displayed in Figs. 4(b) and 5(b) of Ref.
r =0, resulting in V ∼0.11 µm3 for a unit cell.
[1]. We assume that this background represents the con- min m
tribution of γ(cid:48) to γ , and subtract the background to
tot ∗These authors contributed equally to this work.
estimateγ . Theresultingformforγ (ω)consistsofa
1D 1D
setofresonantpeakswhoseheightsatdiscreteω(n) qual-
itatively map out γ(∞) calculated for the infinite struc-
1D
ture, with the maximum peak height for γ occurring
1D
for the peak closest to the band edge. Moving further [1] C.-L. Hung, S. Meenehan, D. E. Chang, O. J. Painter,
away from the band edge, we find that our numerical es- and H. J. Kimble, Trapped Atoms in One-Dimensional
timate of γ (ω)−γ(cid:48)(ω) asymptotes to the calculated Photonic Crystals (2013).
tot [2] S. Y. Buhmann et al., Phys. Rev. A 70, 052117 (2004).
value of γ(∞)(ω) reasonably well.
1D [3] A. F. Oskooi et al., Comput. Phys. Commun. 181, 687
(2010).
Estimation of γ and γ(cid:48) [4] A. W. Rodriguezet al, PRA 80, 012115 (2009).
1D
[5] To obtain the CP potential for a single atom inside a
From the previous discussions, we identify that the periodic structure, we impose Bloch-periodic boundary
decay rate into other modes γ(cid:48) can be read off from conditionsonasingleunit-cellintheFDTDcalculations.
the broad background in γ . Specifically, we estimate WethensumoverallfieldsEsc,ν(ra,t;kx)withtheBloch
tot
wavevector k across the first Brillouin zone to obtain
γ(cid:48) =γtot(ω(cid:48)) at a frequency ω(cid:48) just across the band edge E (r ,t)=xa (cid:82)π/a dk E (r ,t;k ),whichismath-
and away from any resonant peak for a guided mode. sc,ν a 2π −π/a x sc,ν a x
ematically equivalent to solving for the fields of a single
The decay rate into the probe mode γ can then be
1D pointdipoleinaninfiniteperiodicstructure.Theintegral
estimated using γ1D =γtot−γ(cid:48). is approximated using a 10 point Gaussian quadrature.
For the single-beam structure and the atomic spin ori- [6] G. S. Agarwal, PRA 12 1475 (1975).
entationshowninFig.4(b)[1],wefindapeaktotaldecay [7] T. Søndergaard and B. Tromborg, Phys. Rev. A. 64
rate γ /γ ≈ 15 and a background level γ(cid:48)/γ ≈ 1.2 033812 (2004).
tot 0 0
near the Cesium D2-line frequency ν = 352 THz. [8] I. E. Dzayloshinkii, E. M. Lifshitz, and L. P. Pitaevskii,
A
Sov. Phys. Usp. 4, 153 (1961).
We estimate the coupling to the resonant probe mode
[9] A. M. Contreras-Reyes et al., Phys. Rev. A 82, 052517
γ = γ − γ(cid:48) ≈ 14γ . For the double-beam struc-
1D tot 0 (2010).
ture and spin orientation shown in Fig. 5(b), we find [10] F. L. Kien, S. D. Gupta, V. I. Balykin, and K. Hakuta,
γtot/γ0 ≈ 22, γ(cid:48)/γ0 ≈ 1, and, therefore, γ1D/γ0 ≈ 21. Phys. Rev. A 72, 032509 (2005).
Here, γ /2π =5.2MHz is the free-space (vacuum) decay [11] J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, Phys. Rev. B
0
rate for the D2 line. 60, 46884695 (1999).
[12] P. Yao, V. S. C. Manga Rao and S. Hughes Laser &
Photon. Rev. 4: 499516 (2010).
EFFECTIVE AREA AND MODE VOLUME FOR [13] S. G. Johnson and J. D. Joannopolous, Opt. Express 8,
PROBE 173 (2001).
[14] COMSOL Multiphysics http://www.comsol.com/
Both the single and double nanobeam structures in [15] V.S.C.MangaRaoandS.Hughes,Phys. Rev. Lett.99,
Ref. [1]leadtoatomlocalizationinaregionoflargeam- 139901 (2007).