Simultaneous optical trapping and detection of atoms by microdisk resonators

0.31 MB·English

by  Michael Rosenblit

Simultaneous optical trapping and detection of atoms by microdisk resonators
Michael Rosenblit,1 Yonathan Japha,1 Peter Horak,2 and Ron Folman1
1Department of Physics and Ilse Katz Center for Meso- and Nanoscale Science and Technology,
Ben Gurion University of the Negev, P.O.Box 653, Be’er Sheva 84105, Israel
2Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom
(Dated: February 1, 2008)
We propose a scheme for simultaneously trapping and detecting single atoms near the surface
of a substrate using whispering gallery modes of a microdisk resonator. For efficient atom-mode
coupling the atom should be placed within approximately 150 nm from the disk. We show that a
combination of red and blue detuned modes can form an optical trap at such distances while the
back-action of the atom on the field modes can simultaneously be used for atom detection. We
investigatethesetrappingpotentialsincludingvan-der-WaalsandCasimir-Polderforcesanddiscuss
6 correspondingatomdetectionefficiencies,dependingonavarietyofsystemparameters. Finally,we
0 analyze thefeasibility of non-destructivedetection.
0
2 PACSnumbers: 42.50.Ct,42.82.-m,32.80.Qk

n
a
J I. INTRODUCTION analysisoftheeffectsofdetectionontheatomicexternal
3 degreesoffreedom. Inparticular,weshowthatthesame
2 Opticalmicro-resonatorsarecurrently attractinga lot optical modes which are used for atom detection can be
ofinterestinavarietyoffieldsrangingfromtelecommuni- exploited to create a trapping potential for the atom.
1 cation[1]tobiological/chemicalsensors[2]. Inparticular, The parameterscan be adjusted to provide a sufficiently
v theadvancementofmicrodiskresonatorsmayleadtothe deeppotentialminimumatanappropriatedistancefrom
4
5 developmentofcompactandintegrableoptical-electronic thedisksurfacetoholdtheatomsecurelyinplaceduring
1 devices. Recently, significant progress in increasing the the detection process.
1 finesse of such resonators has been reported [3], which This workis organizedas follows. First,we reviewthe
0 makesthese devices interestingfor future applicationsin system under consideration and the principles of optical
6 the emerging field of quantum technology [4, 5, 6, 7, 8]. single-atom detection in such a device in Sec. II. Next,
0 inSec.III,wediscuss the differentoptical,magneticand
Combininghigh-Qmicro-resonatorswithminiaturized
/
h magnetictrapsforcold,neutralatomsaboveasubstrate, surface forces and potentials operating on the atom. In
p so-called atom chips [9], may lead to integrated devices Sec. IV we investigate a trapping scheme which simul-
- taneously allows for optical atom detection. In Sec. V
t which allow for a high degree of control over light-atom
n interaction. Such systems may have a significant impact we present results on an optimized set of parameters for
a simultaneoustrappinganddetectionanddiscusstheper-
incontextssuchascavityQED[10],singlephotonsources
u
formance of the detector and the dynamics of the atom
q [4], memory and purifiers for quantum communication
during the detection. Finally, we discuss the experimen-
: [5], manipulation of matter wavesin interferometric sen-
v tal feasibility and conclude in Sec. VI.
sors [6], atomic clocks [7], and the quantum computer
i
X [5, 8]. Such an integrated photonics device for quantum
r technologywouldnotonly improvetechnicalcapabilities
a suchasenhancedrobustnessandaccuracywhilereducing II. SYSTEM DESIGN AND OPERATION
SCHEME
size, cost and power consumption; it may also give rise
to complex new functionalities such as non-destructive
atom-lightinteractionandhighsignal-to-noisedetection, A. System design
high-fidelity qubit transfer and entanglement for quan-
tum communication, and scalability for, e.g., the quan- The basic system under consideration and its optical
tum computer. properties have been discussed in detail elsewhere [13].
Several different realizations of micro-resonators are Here we will therefore only give a brief summary.
currently under investigation in the context of their in- Weconsideranatomchipconsistingofamagnetictrap
tegration on atom chips, e.g., Fabry-Perot fiber cavities forcoldatomsandanopticalresonatorforatomtrapping
[11],photonicbandgapstructures[12],andmicrodiskres- and detection as shown in Fig. 1.
onators [13]. The latter possibility is attractive since it The optical resonator is a microdisk or a toroid made
combines the high optical quality of the much studied of a dielectric material [3], which supports high-finesse
microsphere[14, 15,16]withadvancedmicro-fabrication whisperinggallerymodesnearthewavelengthsoftheD1
and integration technology. andD2lines ofrubidiumatoms,i.e.,around795nmand
In this paper we follow up on our recent proposal of 780nm,respectively. Lightiscoupledintoandoutofthe
using the whispering gallery modes (WGM) of a toroid cavitythroughevanescent-fieldcouplingacrossanairgap
microcavityforsingle-atomdetection[13],withadetailed byataperedlinearwaveguidewhichismode-matchedto
2
D (µm) l λ (nm) Q1/106 Q2/108 g0 (MHz) α (1/µm)
30 168 774.2 3.2 1.49 100.5 7.49
30 167 778.73 3.0 1.47 102.6 7.7
30 166 783.27 2.79 1.46 103.2 7.19
30 163 797.2 2.27 1.39 105.0 7.3
15 82 771.3 1.56 0.77 202.5 7.06
15 79 799.2 1.02 0.70 209.8 6.75
TABLE I: Optical properties of selected WGMs. Q1 (Q2)
is the quality factor for a waveguide-disk gap size of 0.5µm
(0.9µm), l isthelongitudinalmodeindex(theradial indexis
q = 1), g0 is the single-photon Rabi frequency for an atom
at the disk boundary, and α is the decay constant of the
evanescent field.
FIG. 1: The structure under consideration. The evanescent uide is mixed with a strong localoscillatorfield in a bal-
wave from the slab waveguide (1) is coupled into the disk anceddetector. Thepresenceoftheatomistheninferred
(2) and back through a small gap between them. The adia- fromachangeintheintensitydifferencebetweenthetwo
baticwaveguidetapers(3)serveforcouplinglightfromoptical arms of the balanced detector.
fibers(notshown)intoandoutofthewaveguide. Coldatoms We have shown previously that, in the limit of far de-
(4)canbebroughttothedisksideviathemagneticfieldofa
tuning of the light field from the atomic transition and
Z-shapedcurrent-carryingwire(5)[9]. However,asexplained
forlowatomicsaturation,thesignal-to-noiseratioofthis
in thiswork,such amagnetic trapis notsuitable tohold the
atom detection scheme is given by
atoms during detection, and so other means of trapping are
described for this operation.
κ g2
S =4√τ A T , (1)
| in|∆κ2
achieve best coupling. whereτ isthemeasurementtime,A istheamplitudeof
in
Atoms are initially loaded into a magnetic trap [17] the pump light in the linear waveguide normalized such
formed by a Z-shaped current carrying wire, as shown that A 2 is the power in units of photons per second,
in
in Fig. 1, with its center located at about 50 - 250 nm κ is|the|cavity decay rate due to microdisk-waveguide
T
distance from the side wall of the disk. Typically the coupling,κisthetotalcavitydecayrateincludinglosses,
wire has a rectangular shape with width and height of ∆ is the atom-light detuning, and g is the single photon
1 µm, embedded 0.5 µm below the surface of the chip. Rabi frequency. The corresponding photon number in
Assuming wirecurrentsof 100mAandahomogeneous the cavity is given by
∼
bias field of 100 G, a magnetic trap is formed approx-
κ
imately 2 µm above the surface. However, as will be N =2A 2 T. (2)
discussed later, the magnetic trap is in general too weak | in| κ2
to provide atom confinement during the optical detec-
The optical properties of selected WGMs near reso-
tion process. The atoms are therefore transferred into nance with either the D1 or D2 transition lines of 87Rb
an optical trap formed by the evanescent waves of two
(at795nm and780nm, respectively)aresummarizedin
microdisk modes of opposite detuning. The atom-light
Table I for disk diameters 15 µm and 30 µm. Q (Q ) is
1 2
coupling also changes the optical properties of the disk
the total quality factor defined by Q = ω/(2κ) [13]. As
modes, which can subsequently be measured in order to
discussed below (Sec. IV) only modes that are blue de-
infer the presence of the atom.
tuned with respect to the D2 line and red-detuned with
Theresultspresentedinthisworkarebasedonasemi-
respect to the D1 line are eventually proposed for de-
analyticcoupledmodetheory[18,19]oftheopticalprop-
tection and trapping. We assume a root mean square
erties of the system and a standard Jaynes-Cummings
surface roughness of σ =1 nm and a surface correlation
type model of the atom-light interaction.
length of L =5 nm throughout this paper [13].
c
Let us now derive an estimate for the light field inten-
sity requiredfor single-atomdetection. Assume we want
B. Single-atom detection to detect a Rb atom using the interaction of the D2 line
(780nmwavelength)withthe (l =167,q=1)mode ofa
Aswasdescribedindetailin[13]ourdetectionscheme 30 µm diameter disk, see Table I for details. The atom
works as follows. Light that is resonant with one mi- is assumed to be located 100 nm off the surface of the
crodisk mode is coupled into the linear waveguide from disk for a time τ = 10 µs. In order to detect the atom
oneside. The outputfieldatthe otherendofthe waveg- with a signal-to-noiseratioof S =10,the requiredinput
3
poweris A 2 0.12 1014 photonsper second(3µW). fixed at a given point x have the steady-state solution
in
| | ≈ ×
The corresponding number of photons in the cavity is
N 2.4 105 for a gap size of 0.6 µm. 1 Ω(x)2
≈ × ρ11(x,t) = 2 Ω(x)2|+2∆|2+2Γ2 (6)
| |
Ω(x)(Γ i∆)
III. FORCES AND POTENTIALS ρ01(x,t) = − Ω(x)2+2−∆2+2Γ2e−iωt, (7)
| |
In this section we describe the forces on an atom situ- where 2Γ is the decay rate of the excited state popula-
atednearthesurfaceofthemicrodisk: opticalforcesdue tion by spontaneous emission and ∆ = ω ω is the
01
−
totheinteractionwiththe lightfields nearthedisk,van- detuning of the mode frequency from the atomic transi-
der-Waalsforcesduetotheinteractionwiththedielectric tion frequency. The light force on the atom is given by
surface,andoptionallymagneticforcesgeneratedbycur-
rent carrying metal wires on the surface of the chip. In
general,the combinedeffectoftheseinteractionsisquite F= Tr ρ(x) Hlight . (8)
− { ∇ }
complex. In order to simplify the discussion, we will in
the following assume the limit of low atomic saturation, Iftheatomicmotionisslowenoughsuchthatthechange
wherethe combinedpotentialisa simple sumofthe var- in the light intensity at the atomic location is small dur-
ious contributions, ing the decaytime 1/(2Γ),anadiabaticatomicpotential
may be defined which is the spatial integral of the force
V =V +V +V . (3) (8) using the steady-state density matrix (6), (7). This
light AS mag
yields
Here V is the optical potential, V is the atom-
light AS
surface potential, and V is the magnetic potential of ¯h∆ Ω2 ¯h∆
mag V = log 1+ log[1 2ρ ].
the wire trap. light 2 2Γ2+2∆2 ≈− 2 − 11
(cid:20) (cid:21)
(9)
For Ω,Γ ∆ the potential may be approximated by
≪
A. Atom-light interaction
¯hΩ(x)2
V (x) | | , (10)
light
In this section we will assume fixed light intensities ≈ 4∆
in the cavity and only discuss the effects of the light on
which is positive (repulsive) for ∆ > 0 (”blue detuned”
the atom. The Hamiltonian of the interaction of a sin-
light)andnegative(attractive)for∆<0(”red-detuned”
gle atom with several optical modes in the dipole and
light). UsingthevaluesoftheparameterestimateofSec.
rotating-waveapproximations is given by
IIB,wefindthattheforceonanatom100nmawayfrom
the disk surface is about 70 µK/nm.
i¯h
Hlight = [Ωj,mn(x)eiωjt m n Forlateruseinthispaperwenowdiscussthecombined
2 | ih |−
j m>n optical potential of two light fields interacting simulta-
X X
Ω∗ (x)e−iωjt n m]. (4) neously with the atom, where one field is blue detuned
j,mn | ih | and the other is red detuned. Two possibilities can be
Here m and n are different atomic electronic levels, considered: (i) where the two fields couple the atomic
| i | i
and Ω is the interaction amplitude between the disk ground state to two different excited states (three-level
j,mn
modej withfrequencyω andthe atomtransition n situation), or (ii) where both light fields operate on the
j
| i→
m , such that level m has a higher energy than n . same atomic transition (two-level situation).
| i | i | i
Note that different modes may be frequency degenerate In the three-level situation, e.g., when the two light
(e.g.,twocounter-propagatingmodes)andmaycoupleto fields operate on the D1 and D2 lines of rubidium, re-
the same atomic transition. In the following we assume spectively,anexactanalyticsolutionoftheopticalBloch
coherent states for the cavity modes and therefore [13] equationsfortheatomicdensitymatrixcanbefound. In
the limit of low atomic saturation, the steady-state 3x3
Ω (x)=2g (x) N (5) densitymatrixmaybeapproximatedbyadirectproduct
j,mn j j
of the 2x2 matrices for the two atomic transitions inde-
where N is the number of cavitypphotons, g (x) is the pendently. Theopticalpotentialisthenfoundasthesum
j j
single-photon Rabi frequency of mode j for the relevant of two terms given by Eq. (10) corresponding to the two
transition in an atom at position x in the evanescent fields.
field of the disk mode. The maximum values of g at The situation is slightly more complicated in the two-
j
the disksurfaceforseveralselectedmodesandthe decay level case. Because of interference between the two light
constants of their evanescent fields are given in Table I. fields oscillating at different frequencies, no steady-state
Inthesimplecaseofalightfieldwithasinglefrequency solution exists in this case. Instead, the atomic density
ω the Bloch equations for the density matrix of the two- matrix, and hence the optical force and the dipole po-
level atom (with ground state 0 and excited state 1 ) tential, oscillatewith the difference frequency ∆ ∆ .
1 2
| i | i | − |
4
However, in the limit of low atomic saturation it can be
0
shown that the potential oscillates around a mean value
which is the sum of the steady-state potentials expected )−0.2
m
from each of the light fields alone. Moreover, because
n
of the large detuning envisaged here between microdisk K/−0.4
µ
modes and atomic transitions of the order of THz, the (
e −0.6
oscillation frequency ∆1 ∆2 is much larger than the c
| − | r
typical kinetic energy of the atom. Therefore, the atom o
F−0.8
will move under an effective potential which is the time
average of the oscillating potential. −1
Wethereforefindthatinbothconfigurations,the two- 100 120 140 160 180
Distance from disk (nm)
level and the three-level situations, the optical potential
canbeapproximatedbythesumoftwotermsoftheform
(10), FIG. 2: Atom-surface asymptotic forms of the force versus
atom distance from the micro disk: Casimir-Polder potential
V (x) V (x)+V (x) (solid line) and van-der-Waalspotential (dashed line).
light light,1 light,2
≈
¯hΩ (x)2 ¯hΩ (x)2
1 2
| | + | | . (11)
≈ 4∆ 4∆ We assume an isotropic atomic dipole moment and use
1 2
the identity
j d i 2
B. Atom-surface interaction
α = |h | | i| (14)
0
E
ji
j
X
An atom near a dielectric or conducting surface ex-
whereα isthestaticpolarizabilityoftheatom. Wethen
periences an effective potential due to the interaction of 0
obtain
the atomic dipole with the dipole moments created in
the material. At very short distances this potential is α ¯hc
V (x)= 0 (2ck+c⊥). (15)
a van-der-Waals potential due to static dipole-dipole in- CP −2π2ǫ (2x)4 4 4
0
teraction, while at larger distances of the order of one
wavelength of the atomic transition, the retardation ef- The form of the van-der-Waals potential given by
fect changes the nature of the potential and it is then Eq.(12)is validforveryshortdistances(ofthe ordersof
called the Casimir-Polder potential [20]. For a ground- few nanometers), while the forms of the Casimir-Polder
stateatomtheatom-surfacepotentialsareusuallyattrac- potential, Eq. (13) and (15), hold for large distances (of
tive. The asymptotic form of the atom-surface potential the order of one micron or more). In the intermediate
at very short distances from a non-magnetic and non- regimetheexactpotentialinvolvescumbersomeintegrals
dissipative dielectric material with refractive index n is which may be performed numerically. For simplicity we
make the following approximation:
x
V (x)= (n2−1)hd2ki+2hd2⊥i (12) VAS(x)≈ dx′max{FvdW(x′),FCP(x′)}, (16)
vdW − n2+1 8πǫ (2x)3 Z
0
where F and F are the derivatives of the corre-
vdW CP
where dk,d⊥ are the components of the atomic dipole sponding potentials in the radialdirection. At short dis-
moments parallel and perpendicular to the surface, re- tances we hence approximate the atom-surface potential
spectively. For an isotropic atom d2 +2 d2 = 4e2 r2 by V , while at large distances we use V =V .
h ki h ⊥i 3 h i vdW AS CP
whereeistheelectronchargeand r2 istheexpectation Figure 2 shows the two asymptotic forms of the force
h i
value of the square of the atomic radius. (derivativeofthepotentials)plottedoverthewholerange
Theatom-surfacepotentialintheretardedregime,the ofdistancesforarubidiumatomnearasilicasurface(re-
Casimir-Polder potential, takes the form fractive index n=1.454). The transition point between
these two asymptotic forms is found at a distance of
V (x)= ¯hc ck4hd2ki+c⊥4hd2⊥i, (13) aWbaoaults1fo3r0cenmatftrhoemdtihsteanmciecroofd5is0knsmurfiascaeb.oTuhte3vµaKn-/dnemr-
CP −2π2ǫ (2x)4 E
0 j ji and at 100 nm it is about 1 µK/nm. These values are
X
therefore much smaller than the repulsive force created
where Eji is the energy difference between atomic levels by the blue-detuned detection light as estimated above.
i and j, dk,d⊥ are matrix elements of the atomic dipole NotethattheCasimir-Polderinteractionchangeswith
between the two levels at directions parallelandperpen- temperature [21]. However, the temperature dependent
diculartothesurface,andthecoefficientsck,c⊥ arefunc- correction factor is mainly important for long atom-
4 4
tions of the refractive index n (see Ref. [20]) ranging be- surface distances of the order of a few microns and here
tween0forn=1and1forn (aperfectconductor). we neglect these temperature effects.
→∞
5
C. Magnetic trap potential implies a maximum current of 0.1 A. It follows that for
87Rb atoms in the state F = 2,m = 2 a typical har-
F
| i
ThemagneticfieldBcreatedbycurrentcarryingwires monic oscillator frequency of a magnetic trap can reach
on the chip interacts with the magnetic moment µ of up to ωho = 2π 35 kHz at trap heights of z0 = 3 µm
×
the atom through the Hamiltonian H = µB. A above the wire center (2.5 µm above the top surface of
mag
hyperfine atomic level with F >0 splits in the−magnetic the wire). The energy splitting between magnetic states
field into its Zeeman sub-levels with magnetic moments with different mF (∆mF =1) is given by
followingthedirectionofthemagneticfieldwhilemoving
adiabatically in the regionof the field. The potential for ∆E ∆mFgFµB Boffset . (22)
∼ | |
a specific Zeeman level with quantum number m is
F
For the strong magnetic trapata heightof 3 µm the en-
V (x)=m g µ B(x), (17)
mF F F B| | ergysplitting ofa Z-trapis approximately50MHz. The
maximal force that can be applied to the atom by the
where µ is the Bohr magneton and g is the Land´e
B F
magnetic field is given by the magnetic potential gradi-
factor corresponding to the hyperfine level F. Atomic
ent, F 2 10−3 µ I/z2, which is of the order of
levels with mF >0 are attracted to the minimum of the | mag| ≈ × B 0
1.5 µK/nm. This value is much smaller than the force
field, the m = 0 level feels no potential, and m < 0
F F
applied by the evanescent light fields of the disk. This
levels are repelled from the minimum of the field. A
implies that a magnetic field by itself cannot hold the
two-dimensional confinement of levels with m > 0 is
F
atom against the repulsive force of the detection light.
achievedabovea straightwire with the help of anexter-
nalbiasfield. Threedimensionalconfinementisachieved Magnetic fields may still be considered for atom trap-
by the combination of several wires or by a single wire ping in the angular direction (along the perimeter of the
with a U or Z shape. disk) where no light forces exist. However,this option is
A magnetic trap above an atomic chip is usually also problematic because of light-induced Raman transi-
formed by a combination of magnetic field components tions into untrapped Zeeman levels. Raman transitions
from 3 sources: (a) A wire in the y direction on the chip betweendifferentmagneticatomiclevelsoccurbecauseof
surface carrying a current I generates a magnetic field stimulated photon absorption and re-emission processes
B 2 10−3I(zxˆ xˆz)/(x2+z2) above the wire (I is in without significantly populating a higher electronic level
Am≈per·e,B inGau−ss,andx, z inmeter). (b)Abiasfield of the atom. In principle, such transitions can be sup-
B xˆ which cancels the magnetic field generated by the pressed by controlling the polarizations of the optical
0
−wire at height z = 2I/B from the center of the wire fields. However, in the case of an evanescent field near
0 0
at x=0. (c) An offset field B yˆ, which prevents the thesurfaceofamicrodiskthefieldpolarizationcannotbe
offset
magnetic field at the center of the trap to be zero. The controlled. Raman transitions will therefore occur with
potential near the center of the trap at x = (0,0,z ) is an effective oscillation frequency
0 0
then given by
Ng(x)2
B (z z )∂Bx(x0),B ,x∂Bz(x0) (18) Ωeff ∼ ∆ . (23)
0 offset
≈ − ∂z ∂x
(cid:18) (cid:19)
If the value of Ω between two magnetic levels is larger
where ∂B /∂z = ∂B /∂x 2 10−3I/z2. In the range eff
x z ≈ · 0 than the frequency splitting δ between the two levels,
where (z z )∂B /∂z, x∂B /∂x B the potential
− 0 x z ≪ offset significant oscillations will take place between the levels
is approximately harmonic,
with flipping times t =π/2Ω . On the other hand if
flip eff
Vmag(x)≈mFgFµB{Boffset+ z24IB210−3[x2+(z−z0)2]}, tδh>e oΩtehffe,romnlayganfertaicctsiotantΩe.2effI/n(Ωou2effr+syδs2t)eimst,rtaynpsifcearlrevdailnuteos
0 offset ofthe effective Rabifrequency areof the orderof tens of
(19)
MHz. Thisis ofthe sameorderasthe magneticsplitting
with oscillation frequency
of50MHz estimatedabove,andsignificantRamantran-
2 10−3I m g µ sitions into different magnetic levels are expected. Note
F F B
ω = · . (20)
ho z2 mB thataccordingto Eq.(22)the magneticlevelsplittingat
0 r offset the trap center can be increased, and thus the Raman
For a Z-shaped trap usually B B 10−32I/z . effect reduced, by increasing the offset magnetic field,
offset 0 0
≈ ≈
Thepotentialisthenharmonicintherange x, z z < e.g., with the help of additional current-carrying wires
0
| | | − |
z and the trapping frequency becomes on the chip. However, this comes at the cost of smaller
0
trapdepths and oscillationfrequencies,see Eq.(20). We
1 2 10−3Im g µ therefore conclude that magnetic trapping in the pres-
F F B
ω · . (21)
ho ≈ z0s mz0 ence of the evanescent light fields from the disk would
be difficult to achieve in practice, even in the direction
Conventional wires can carry a current density of up along the perimeter of the disk where no optical dipole
to 107 A/cm2. For a wire of cross section 1 µm2 this forces exist.
6
crodisk. The light intensities create a trapping potential
15
a) of about 1.9 mK at a distance of 115 nm from the sur-
10 face. The surface potential reduces the potential depth
K) byloweringthepotentialbarriertowardsthedisksurface
m 5
ntial ( 0 anBdeaflosroesilnigvhetsltyigsahtiifntgs tthraepceonptteirmpizoastiitoionn.numerically in
ote −5 moredetail,wewillfirstdiscussafewtrappropertiesina
P simpleanalyticapproximation. Forthis,weapproximate
−10
the evanescent fields by decaying exponentials, i.e., the
−15 blue detuned potential is written as Vb(r) = Vb0e−αbr,
100 200 300 400
where V > 0 and r is the distance from the disk sur-
Distance (nm) b0
face, and the red detuned potential is Vr(r) = Vr0e−αrr
withV <0. Thedecaycoefficientsα aregiveninTa-
b) r0 r,b
0 ble I. A minimum of the combined effective potential is
K) formed when the two corresponding forces cancel. Close
m−0.5 to the surfacethe repulsiveforcemustdominate,thatis,
ntial ( −1 αbVb0 >αr|Vr0|. The minimum is formed at a distance
e
ot−1.5 1 Vb0αb
P r = log . (24)
min
α α V α
−2 b r r0 r
− | |
The potential at this point is given by
100 200 300 400
Distance (nm)
α α
V(r )= V e−αrmin b− r (25)
min b0
FIG.3: Theradialpotentialnearthesurfaceofa30µmdisk. − αr
(a)Potentialformedbyblue-detunedlight(dash-dottedline)
which is typically a few percent of the single-frequency
and red-detunedlight (dotted),surface potential (solid), and
sumpotential(dashed). (b)Thesumoftheopticalpotentials potentials. At the minimum, the potential is quadratic
(solid), and thesum of theoptical potentials and thesurface with oscillator frequency
potential (dashed). In this example, the number of blue de-
tuned cavity photons is N = 2.4×105, and the number of ω = α α V(r )/m. (26)
b ho r b min
red detuned photonsis N =3.68×105.
r
p
Note that the potential is very sensitive to the field
intensities: if V changes by δV , the corresponding
r0 r0
| |
IV. ATOM TRAPPING change in the location of the minimum is
1 δV
A. Trapping potential δr = r0. (27)
min
α α V
b r r0
− | |
Thediscussionintheprevioussectionshowsthatmag- Because of the small difference between the two coeffi-
neticandvan-der-Waalsforceswillingeneralbetooweak cients α and α (see Table I), a small change of the
b r
toformtrappingpotentialsforatomsnearthe microdisk magnitude of the potentials will significantly shift the
surfaceinthe presenceofthe detectionlightfield. Inthe minimum of the combined potential. Due to this high
following we will therefore investigate atom trapping us- sensitivity to light intensities, a more accurate form of
ing two lightfields: a blue detuned detectionlightwhich the optical potential must be examined. For the follow-
also provides a repulsive potential to keep atoms away ing calculations, three effects are taken into account:
from the surface against the atom-surface force, and an (i) The exact form of the potential is given by Han-
additional red detuned field to provide a long-range at- kel functions [13] instead of exponentials. Furthermore,
tractivepotentialwhichwillkeeptheatomscloseenough atomic saturation effects render the optical potential
to the interaction region with the detection light. nonlinear in the light intensities, see Eq. (9).
Two objectivesmust be achievedby the choiceof trap (ii) The combined potential formed by the two modes
parameters. (i) The trap center must be within 50-200 is slightly different than the sum of the two potentials
nm of the disk surface to allow for atom detection with from each mode alone, cf. the discussion in Sec. IIIA.
sufficient signal-to-noise ratio, e.g., S > 10. (ii) The (iii) While the van-der-Waals potential is significantly
depth has to be large enough to keep the atom trapped smallerthanthepotentialofeachlightfieldindividually,
for the detection time, taking into account any heating it may still have anappreciable effect on the sum poten-
effects. tial,astheredandbluedetunedfieldslargelycanceleach
In Fig. 3 we show sample potentials for a rubidium other.
atom coupled to the (l =167,q=1) blue detuned mode The sensitivity of the optical trap to the intensity of
andthe(l =163,q=1)reddetunedmodeofa30µmmi- the red detuned light for a fixed intensity of the other
7
a)
al depth (mK)468 N00000b...../0235710946666 6× (10)b000...468 0.40.81.21.60.282.124..8213.6.232.6422..483.21.622.4 1.1.62 10..28 0.8 0.4 0.4
enti2 N 0.4 1.2 0.8 0.4
ot 0.8
P 0.2 0.4
0 0.4
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
50 100 150 200 250 300
N/106 r (nm)
r min
b)
400
FIG. 5: Contour plot of the potential depth (in mK) vs. po-
sition of the trap minimum r and photon number N in
300 min b
thebluedetuned mode for a disk diameter of 15 µm.
m)
n
(n200
rmi10000 0.2 0.4 0N.6r/1006.8 1.0 1.2 1.4 6×N ( 10)b00000.....345670.20.40.60.20.811.401.04.2.61.60.81.82.42.62.2112.411..623.63.4823..843.23144..28.424..482.65.252.42.556..46356.6..2863.446.83.4.44.485.4.223.22.5834.62.2.14.68433..682.23.43.222.832.26.41.16.412..8212.21 1.4 1.20.180.6 0.40.6
fFimeIrGuenm.t4v:vsa(.laup)ehsPootootfenNnntiu,amlthdbeeeprptNhhoratinonndt(nhbue)mrpebdoesrditeiintountnhoeefdtbmhleuoedtredafepotrumndieinfd-- 00..12 0.20.600.8.421.41.161.2 0.0.68 0.4 0.2
b 50 100 150 200 250
mode. The disk diameter is 30 µm. r (nm)
min
FIG. 6: Contour plot of the potential depth (in mK) vs. po-
mode, as motivated by Eq. (27), is discussed in Fig. 4.
sition of the trap minimum r and photon number N in
min b
The figure shows the potentialdepth, i.e, the energy dif- thebluedetuned mode for a disk diameter of 30 µm.
ference between the minimum of the potential and the
maximumtowardsthe disk surfaceas seenin Fig.3, and
the position of its minimum versusthe reddetuned pho- B. 3D trapping
ton number N . For too small values of N , no trapping
r r
potential exists and atoms are either attracted to the So far, we have only discussed trapping in the radial
disk surface by the atom-surface interaction or repelled direction from the disk surface. However, we would like
towards infinity by the blue detuned light. For too large to ensure full three-dimensional trapping. First, this
values of Nr, the attractive optical force always domi- wouldenableustouseanatomagainafterdetectionand
natesovertherepulsiveforceandallatomsareattracted furthermore, tight 3D confinement would reduce atomic
to the disk surface. Trapping occurs for a small range heating as discussed in Section IVC.
of intermediate red-detuned photon numbers. The max- The optical potential that is formed by the evanes-
imum potential depth for the chosen parameters is 7 cent field of the WGMs in the disk also depends on
∼
mK andthe correspondingcenter of the trapis at 120 the z coordinate along the height of the disk. The
∼
nm. field intensity along this axis can be approximated by
We summarize the dependence of the potential depth V(z) V(H/2)cos2[π(1 2z/H)/2], where H is the
≈ −
on the intensity of the detection (blue detuned) light Nb height of the disk [22]. Thus, the intensity vanishes at
andonthe distancermin fordisk sizes15µmand30µm the edges z =0 and z =H of the disk. In the center, at
in Figs. 5 and 6, respectively. For each pair of values Nb z =H/2, a potential minimum is formed with harmonic
andrmin inthesefigures,theintensityofthered-detuned oscillator frequency
light N is chosen to yield a trap center at this position
r π
rmin. ωz,ho = Vmax /m. (28)
Notethatbyincreasingtheintensityofboth lightfields H | |
p
arbitrarilydeepopticalpotentialscanbecreatedinprin- This frequency is typically a few kHz, which is an order
ciple. However, this will also change the detection effi- ofmagnitudesmallerthanthe radialtrappingfrequency.
ciencyandincreasethebackactionontheatomitself,as Confinement in the angulardirectionmay be achieved
will be discussed later. if we create a red-detuned standing wave by inputting
8
red-detunedlight fromthe two sides ofthe linear waveg- to remain in the ground state of motion during the de-
uide equally to create a superposition of clockwise and tection time.
counterclockwise propagation around the disk. In this Letusassumethatthepotentialmaybeapproximated
case the red-detuned and blue-detuned light potential as a harmonic potential V = 1m(ω2x2 +ω2y2 +ω2z2)
2 x y z
will be given by with groundstate sizes x , y , z in the three directions,
0 0 0
respectively. The probability that an atom in the har-
Vred(x,y,z) = Vred(0,0,H/2)e−αrx (29) monicgroundstate ψ0 returnsintothesamestateafter
| i
cos2(π(z H/2)/H)cos2(l y/R) the emissionof a photon with wave vector k is given by
r
× −
V (x,y,z) = V (0,0,H/2)e−αbx (30)
blue blue P (k)= ψ ei(kxx+kyy+kzz) ψ 2. (33)
0 0 0
cos2(π(z H/2)/H) |h | | i|
× −
Averaging over all possible directions of k yields
wherel isthewindingnumberofthered-detunedWGM
r
and R = D/2 is the microdisk radius. The harmonic √πerf(kr )
0
P = , (34)
oscillator frequency of the trapping potential in the y 0 2 kr
0
direction is then
where
l
ω = r Vmax /m. (31)
y,ho R | red | ¯h 1 1 1
q r = x2+y2+z2 = + + . (35)
where Vrmedax is the red-detunedpotentialatthe centerof 0 q 0 0 0 s2m(cid:18)ωx ωy ωz(cid:19)
the trap.
Finally, in order to ensure trapping we need to ensure For kr0 1 this may be approximated by
≪
that the tunneling probability to the disk is negligible,
k2r2 ¯hk2 1 1 1
Fortypicalvaluesofbarrierheightof1-2mKandbarrier P 1 0 =1 + + . (36)
0
widthof30-60nmweexpectthatnosignificanttunneling ≈ − 3 − 6m (cid:18)ωx ωy ωz(cid:19)
will occur in a time of less than few seconds. In order to
AfterM spontaneousemissionstheprobabilityofstay-
keepthesevaluesofbarrierheightandwidththetrapping
ing in the ground state is PM and thus the probability
distanceoftheatomfromthesurfacemustbelargerthen 0
of scattering into excited states of motion is given by
80 nm.
∼
¯hk2 1 1 1
P (τ)=1 PM 2Γρ τ + + .
C. Heating of a trapped atom by spontaneous other − 0 ≈ 11 6m (cid:18)ωx ωy ωz(cid:19)
emission (37)
If this probability is much smaller than unity, the atom
willstayinthegroundstateduringthedetectionprocess
Interaction of the atom with the trapping and detec-
with high probability.
tion light fields will in general also induce heating of the
atom. Two heating mechanisms can be distinguished:
fluctuations of the dipole forces due to, for example, me-
D. Trapping stability
chanicalvibrationsorlaserinstability,andrecoilheating
afterspontaneousemissionevents[23]. Herewewillfocus
Bi-chromatic atom trapping has already been used in
on the latter heating mechanism.
severalapplications,beginning fromthe workofOvchin-
Inashallowpotentialwell,eachspontaneouslyemitted
nikov et al. [24], where two colors with different evanes-
photonwillonaverageaddonephotonrecoilenergyE =
r
¯h2k2/(2m) to the kinetic energy of the photon, where k cent decay lengths have been proposed in order to trap
atoms at a distance λ from a prism surface. The two
is the wave-vector of the photon. Thus, if we define
color scheme has been considered also for a dielectric
M =2Γτρ (32) microsphere, a free-standing channel waveguide and an
11
integrated optical waveguide [23, 25, 26, 27, 28]. The
asthenumberofspontaneousemissioneventsduringthe back-action of an atom on the bi-chromatic light field
atomdetectiontimeτ,thetotalheatingisgivenbyME . in the strong coupling regime was considered for atom
r
We therefore require potential depths V > ME in or- trapping and cooling in Ref. [29].
r
der to hold an initially ultra-cold atom in the trapping The main disadvantage of bi-chromatic trapping, as
potential during the interaction with the light fields. was pointed out by Burke et al. [27], is that trapping
Inasteeppotentialwell,ontheotherhand,atomtrap- is achieved by a fine balance between two optical fields,
ping is aided by the Lamb-Dicke effect: an atom in the such that even a small intensity fluctuation may dras-
oscillatory ground state is much more likely to return to tically change the trapping conditions or even destroy
the same state after a spontaneous emission than to be the trap. Such spatial intensity fluctuations usually ex-
excited to a higher oscillation state. In the following we istin realwaveguidesdue to imperfections thatgenerate
will derive an expression for the probability of the atom backscatterredwaveswhichinterfere with the mainlight
9
wave. It was shown [27] that even a backscatteredwave, a)
whose intensity is only 0.001 of the propagating mode 11 9 7 5
intensity,maydecreasethe potentialdepthbyahalfand 0.6 10 8 6 3
consequently destroy the trap. 0.5 4
Here we analyze the intensity spatial fluctuations due 6
ttohabtabckascckastctaetrtienrginignmthaeymbeicrdoedcirsekassetdrutcotuarel.evWelewshheorwe N/10b0.498 7 6 5 3
0.3 4 2
such fluctuations will not severely change the trapping
conditions. This is achievable by increasing the coupling 0.2 6 5 3
rate between the linear waveguide and the microdisk. 4 2
This increase mayaffect the detection signal-to-noisera- 100 120 140 160 180 200
r (nm)
tio andconsequentlythe integrationtime willneedtobe min
increased. b)
tchoemAsptwlweoxasccoodueeffinsctcreiireb-npetdroǫip,nawg[1ah3tii]cnhtghiems lorinedleeaastreidscotduoepsltcihnriegbeibndettrwbinyeseinac 00..56 5.52.544.4.644.8 4.4233..483.63.2 32.28.6 2.42.2 2 1.8 1.61.41.2
lloossss rraatteeoofftthheeddiisskkκκinitsdauseutmo iomfptherisfeicnttiorinnss.icTlhoesstoatnadl 6/10b0.4 44.3.234.8 3.63.2 322..86 2.42.2 2 1.8 1.611..42 1
the loss κT due to waveguide - disk coupling [13], such N 0.3 322..68 2.42.2 2 1.8 1.61.4 1 0.8
athreatgiκviennt =byκc−omκpTl.exInnutmhibsemrsoαde+l tahnedmαo−d,ereasmpepclittiuvedleys, 0.2 2 1.8 1.16.4 1 1.2 0.8 0.6
stwucohmthoadtes|α. ±If|2t=heNd±iskariestphuemnpuemdbweristhofraptheotηonthseinn tthhee 10101.2 1200.8 1400.6 160 108.40 200
r (nm)
steady state values of α are given by min
±
α− = (ǫ/κ)α+ (38) FIG.7: (a) Signal tonoise ratio and (b)total (red and blue)
η scatteredphotonnumbervs. distancefromthediskandblue
α = (39)
+ κ(1+ǫ2/κ2) detuned photon number for a disk diameter of 15 µm. The
integration time is 125 µs.
so that the backscattered to propagating ratio is given
by
ǫ κ κ ǫ κ disk sizes are investigated: a 30 µm disk using the (l =
(I−/I+)1/2 = − T = 1 T (40) 167,q=1)modeasthe blue detuneddetectionlightand
κ κ κ κ − κ
− T int (cid:16) (cid:17) the (l = 163,q = 1) mode as the red detuned attractive
Withhigh-qualitymicrodiskresonators,ǫ/κ 1can trapping light, anda 15µm disk with the (l =82,q=1)
int
≈
be achieved [3]. Using this value we find that if we wish mode as the blue-detuned detection light and the (l =
to keep the trap depth fluctuations smaller than 2% 79,q = 1) mode as the red-detuned attractive trapping
(meaning (I /I )1/2 < 1% [30]), we require κ /±κ > light. The optical properties of these modes are given
− + T
0.99. This value may be achieved by decreasing the gap in Table I. For both configurations we assume surface
between waveguide and microdisk to 0.46µm in both quality parameters of σ = 1 nm and L = 5 nm, and a
c
∼
disks of diameters D = 30µm and D = 15µm, with cor- gap size of 0.5 µm.
respondingQ values ofQ 1.7 106 andQ=0.9 106, We start∼our parameter optimization procedure by in-
respectively. In this case,I≈ /I ×=10−4, i.e., anor×derof vestigatingthesingle-atomdetectionefficiencyasafunc-
− +
magnitude better than that considered by Burke et al., tion of light intensities. To this end we show in in Figs.
whichleadstoacceptabletrappinginstabilities. Thuswe 7(a)and8(a),contourplotsofthesignal-to-noiseratioS,
believe the concerns of Burke et al. describedearlier,re- versus blue detuned photon number in the disk and ver-
gardingthe destructionofthe trapduetobackscattered sus the position r ofthe trapminimum fromthe disk
min
light, are not warrantedfor state-of-the-art micro-disks. surface (this is shown for disk diameters 15 and 30 µm
with observation times τ = 125 and 75 µs respectively).
As expected, S increases with increasing light intensity,
V. DETECTION OF TRAPPED ATOMS becauseofthebetterphotonstatistics,andwithdecreas-
ing atom-disk distance, because of increasing coupling
Havingdiscussedtheprinciplesofatomdetection(Sec. constant g(x).
II) andtrapping (Secs.III andIV), we willnow combine Next, we calculate the intensity of the red-detuned
all the components and find an optimized set of param- light required to achieve a trapping potential minimum
eters for efficient simultaneous atom detection and trap- at given values of r . From this we obtain the total
min
ping. numberofphotonsM,definedineq.(32),spontaneously
We consider the detection and trapping of an atom scattered out of the two light fields by the atom during
in a three-level configuration with two light fields. Two the detection process. The results show that, in general,
10
a)
00..45111567 14131211 10987 65 4 3 2 1 60) 00..56 Hp5%reoa bitnian b1gi2li5tµys S=5
610 1413 109 6 3 2 × 1 0.4 Tpuronbnaebliinligty
N/b0.3 12 87 5 4 1 N (b0.3 2% in 125µs
11
0.2
9 6 3 2 0.2
87
0.1 5 4 1 80 100 120 140 160 180 200
100 150 200 250 r (nm)
r (nm) min
min
b)
FIG. 9: Performance characteristics of a disk with diameter
0.5 9.3 89.8.78.417.7.872.56.69.665.7 4.844..523.93.63.3 32.7 2.1 Dblu=e-d1e5tuµnemdapnhdotofonrnaum12b5eµrssainntdegartaotmiondisttimanec,esa.tTdhiffeearerenat
6/10b00..349.899.8.877..471.29.78.656.6.9656.7 6.5.344.84.4.235.5.913.63.3 23.7 22..41 1.8 1.5 1.2 0.9 ec<toen5rfi%rneeagdniodbnytuwtnhhneeerleliinsngteaspbrrloeebpaartbeosimleintyttsroaSfpp<>in2g%5,acnhodenatdtaeiintnegscttpihoreonbpcaaabrnialimbtye-
N 0.254.6.4..483342..953.53..613 2.73 2.21.4 1.8 1.5 1.2 0.9 0.6 oatofclhe±ire2av%necdeinwshietoihtwhhseirtghhtahtceofobnrlfiudteheenoscreer.oepdAt-nidmeatiznueandleydpsiaslrigaohmfttehitneetrepsnaasriacthmyaewntgeilerl
0.1 2.4 1.8 1.5 1.2 0.90.6 0.3 srteislelnletsadthteocahnaontgheerinpotrinatpwpiotshiitniotnheifttrhiaenignltee.nTsihtyeoafrrtohwerreepd--
100 150 200 250
r (nm) or blue-detunedlight is varied by ±2%.
min
FIG.8: (a)Signaltonoiseratioand(b)totalscatteredphoton
0.5
numbersvs. distancefrom thediskandbluedetunedphoton
Heating
number for a disk diameter of 30 µm. The integration time probability
is 75 µs. 0.4 7% in 75µs
6
0
1 S=5
/ b0.3
largervaluesofS implylargernumbersofscatteredpho- N
tons M as depicted in Figs. 7(b) and 8(b). However, 0.2 Tunneling
probability
we note that the parameter dependence of S and M is 2% in 75µs
different, so that optimization is possible.
0.1
100 120 140 160 180 200
InFig.9wecombineforaD =15µmdiskthecontour r (nm)
lines foranS =5detection, aheatingprobabilityof5%, min
Eq. (37), and a tunneling probability of 2%, calculated
FIG. 10: Performance characteristics of a disk with diame-
usingthestandardexpressionbasedontheWKBapprox-
ter D = 30 µm and for a 75µs integration time, at different
imation. The optimal point for atom detection within
blue-detunedphoton numbersand atom distances. The area
the parameter region of Fig. 9 is found in the middle of
confined by the lines of S = 5, heating probability 7% and
the triangle for Nb = 6 105 and a trap at distance of curverepresentstunnelingprobabilityof2%containsthepa-
×
r = 115 nm from the disk. In order to achieve atomic rameterregionwherestableatomtrappinganddetectioncan
trappingatthispointweneed2.5 105red-detunedpho- be achieved with high confidence. The arrow represents the
×
tons in the disk. At this point we obtain S = 8. The change in trap center position if the intensity of the red- or
potential depth is 2.6 mK and the harmonic oscillator blue-detunedlight is varied by ±2%.
frequency is (ω ,ω ,ω ) = 2π (1.5,4,0.14) MHz. The
x y z
×
ground state energy of a single atom in this trap is ap-
proximately160µKandthegroundstatedimensionsare through tunneling to the disk surface.
(∆x,∆y,∆z) = (15,6.5,40) nm. The heating probabil- ThesameanalysiswiththecontourlinesforS =5and
ity is 4%. An analysis of the parameter tolerance shows heating probability of 7% for the disk diameter D = 30
that for these optimized parameters a change of 2% µm gives the optimal point for atom detection within
in either the blue or red-detuned light intensity will±still the parameter regionof Fig. 10 for N =2.4 105 and a
b
×
lead to another point within the triangle of parameters trap at distance r = 120 nm from the disk. In order to
indicated inFig.9. A better tolerancecouldbe achieved achieve atomic trapping at this point we need 3.6 105
×
if light intensities are increased so that the gap between red-detunedphotonsinthe disk. Atthis pointweobtain
the two lines of signal-to-noise and heating probability S = 7, the potential depth is 1.6 mK and the harmonic
in Fig. 9 broadens. However, in this case the potential oscillatorfrequencyis(ω ,ω ,ω )=2π (0.92,4.2,0.11)
x y z
×
depth would be decreased, with increased loss of atoms MHz. The ground state energy is approximately 126