Institute for Experimental Physics University of Innsbruck€¦ · Innsbruck, MPQ" ! ! we assume an...

Prospects for ion-cavity optomechanics

Tracy Northup Institute for Experimental Physics

University of Innsbruck

Optomechanics without a cavity •  (anti-)Jaynes-Cummings Hamiltonian •  cooling •  heating & lasing •  coupled oscillators

What role can cavities play? •  cavity cooling

State of the art ion-cavity systems

•  parameters, tools at hand, “challenges”

Ion traps in a nutshell

•! state initialization •! coherent rotations & high-fidelity gates •! state readout via electron shelving •! cooling to ground state

...!

•! provided early insight into coherent dynamics of ion-trap systems

...!

Motional interactions are analogous to cavity QED

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! Jaynes-Cummings Hamiltonian

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Driving the red sideband: ground-state cooling

VOLUME 83, NUMBER 23 P HY S I CA L R EV I EW LE T T ER S 6 DECEMBER 1999

ion from the D5!2 level to the electronic ground state viathe P3!2 level.Calcium ions are loaded into the trap from a thermal

beam by crossing it with an electron beam inside the trapvolume. To make sure the ion rests at the node of thequadrupolar rf field, we compensate for stray dc fieldsby applying small bias voltages to the end caps and twonearby auxiliary electrodes.Preparation of the motional ground state is accom-

plished by a two-stage cooling process. First the ion iscooled to the Doppler limit by driving the S1!2 to P1!2dipole transition. In the second stage, a resolved-sidebandcooling scheme similar to Ref. [2] is applied on the nar-row S1!2 to D5!2 quadrupole transition: The laser is reddetuned from the line center by the trap frequency v (redsideband), thus removing one phonon with each electronicexcitation. The cooling cycle is closed by a spontaneousdecay to the ground state which conserves the phononnumber. When the vibrational ground state is reached theion decouples from the laser. The weak coupling betweenlight and atom on a bare quadrupole transition would ne-cessitate long cooling times. However, the cooling rateis greatly enhanced by (i) strongly saturating the transi-tion and (ii) shortening the lifetime of the excited state viacoupling to a dipole-allowed transition.For coherent spectroscopic investigation and state engi-

neering on the S1!2 $ D5!2 transition at 729 nm we use apulsed technique which consists of five consecutive steps.(i) Laser light at 397, 866, and 854 nm is used to pumpthe ion to the S1!2 ground state and prepare the vibra-tional state at the Doppler limit E ! hG!2 [7]. This cor-responds to a mean vibrational quantum number "n# ! 10for v ! $2p% 1 MHz. (ii) The S1!2$m ! 11!2% sub-state is prepared by optical pumping with s1 radiation at397 nm. A magnetic field of 4 G at 70± to the k-vector di-rection of the light at 729 nm provides a quantization axisand splits the ten Zeeman components of the S1!2 $ D5!2transition in frequency. (iii) Sideband cooling step: TheS1!2$m ! 1!2% $ D5!2$m ! 5!2% transition is excited onone of the red sidebands at approximately 1 mW laserpower focused to a waist size of 30 mm. The laser at854 nm is switched on to broaden the D5!2 level at apower level which is set for optimum cooling. Opti-cal pumping to the S1!2$m ! 21!2% level is preventedby interspersing short laser pulses of s1-polarized lightat 397 nm. The duration of those pulses is kept at aminimum to prevent unwanted heating. (iv) State engi-neering step: We then excite the S1!2$m ! 11!2% $D5!2$m ! 15!2% transition at 729 nm with laser pulsesof well-controlled frequency, power, and timing. Theseparameters are chosen according to the desired state ma-nipulation. (v) Final state analysis: We collect the ion’sfluorescence under excitation with laser light at 397 and866 nm and detect whether a transition to the shelvinglevel D5!2 has been previously induced.Sequence (i)–(v) is repeated typically 100 times

to measure the D5!2 state occupation PD after the

state engineering step. We study the dependence of PDon the experimental parameters such as the detuning dvof the light at 729 nm with respect to the ionic transitionor the length of one of the excitation pulses in step (iv).The duration of a single sequence is typically 20 ms, sowe can synchronize the sequence with the ac power lineat 50 Hz to reduce ac-magnetic field fluctuations.For the quantitative determination of the vibrational

ground state occupation probability p0 after sidebandcooling we compare PD for excitation at dv ! 2vand dv ! 1v, i.e., on the red and the blue side-bands. If the vibrational ground state is reached with100% probability, PD$2vtrap% vanishes completely. Fora thermal phonon probability distribution, the ratio ofexcitation on the red and blue sidebands is given byPred!Pblue ! "n#!$1 1 "n#%.The ground state occupation after sideband cooling

is determined by probing sideband absorption immedi-ately after the cooling pulse. Figure 2 shows PD$v%for frequencies centered around the red and blue vzsideband. Comparison of the sideband heights yieldsa 99.9% ground state occupation for the axial modewhen vz ! $2p% 4.51 MHz. By cooling the radial modewith vy ! $2p% 2 MHz, we transfer 95% of the popu-lation to the motional ground state. Ground state cool-ing is also possible at lower trap frequencies, howeverslightly less efficient. At trap frequencies of vz ! $2p%2 MHz, vy ! $2p% 0.92 MHz, we achieve "nz# ! 0.95and "ny# ! 0.85, respectively. The x direction is left un-cooled because it is nearly perpendicular to the coolingbeam. We also succeeded in simultaneously cooling allthree vibrational modes by using a second cooling beamand alternating the tuning of the cooling beams betweenthe different red sidebands repeatedly.The best cooling results of 99.9% ground state occu-

pation were achieved with a cooling pulse duration oftcool ! 6.4 ms. The power of the cooling laser was set

FIG. 2. Sideband absorption spectrum on the S1!2$m !11!2% $ D5!2$m ! 15!2% transition after sideband cooling(full circles). The frequency is centered around (a) red and(b) blue sidebands at vz ! 4.51 MHz. Open circles in (a)show the red sideband after Doppler cooling. Each data pointrepresents 400 individual measurements.

4714

Ch. Roos et al., Phys. Rev. Lett. 83,4713 (1999)

First experiments at Boulder, Innsbruck, MPQ"

! !

we assume an F ! 4 atom is present; if N > 0:75Ne, weassume an F ! 4 atom is not present; otherwise, the mea-surement is inconclusive (<2% of the time) and we ignorethe result. Whenever we detect the atomic state, we per-form two such measurements: the first with EP to find out ifan F ! 4 atom is present, the second with EP together with!3 as a repumper [26], to detect an atom, regardless of itsinternal state.

We measure the Raman transfer probability P4 for agiven !R by preparing an atom in F ! 3, applying aRaman pulse, and then detecting the atomic state usingthe above scheme (with Ne " 22). For each measurementcycle (or trial), we first Raman-sideband cool the atom foran interval "tc. Next, we pump it into F ! 3 by alternating1 "s pulses of !4 lattice light with 1 "s pulses of !04linearly polarized resonant F ! 4! F0 ! 4 light from theside of the cavity (10 pulses of each). After the atom ispumped to F ! 3, we apply a "tR ! 500 "s Raman pulse,which sometimes transfers it toF ! 4. Finally, we measurethe atomic state and check if the atom is still present. Foreach atom, we fix the absolute value of the Raman detuningj!Rj and alternate trials at #j!Rj with trials at $j!Rj(299 trials each). By combining data from atoms withdifferent values of j!Rj, we map out a Raman spectrum.Note that, because the initial Zeeman state of the atom israndom, all allowed F ! 3! F ! 4 Zeeman transitionscontribute to these spectra.

Two example Raman spectra are plotted in Fig. 3. Forthe curve in Fig. 3(a), we cool for "tc ! 250 "s, for thecurve in Fig. 3(b) for "tc ! 5 ms. These scans are per-formed after nulling the magnetic field to within"40 mG;the widths of the peaks are set by the splitting of differentZeeman levels due to the residual magnetic field. For thecurve in Fig. 3(a), we see peaks at the carrier (!R ! 0), aswell as at the blue/red sidebands (!R=2# ’ %530 kHz !%!a). Already we note a sideband asymmetry, indicatingthat a significant fraction of the population is in the n ! 0vibrational state. For the data in Fig. 3(b), the red sidebandat !R=2# ’ $530 kHz is suppressed such that it cannot bedistinguished from the background and contribution fromoff-resonant excitation of the carrier.

The ratio r of transfer probabilities for the red and bluesidebands gives information about the temperature of theatom. For a two-state atom in a thermal state, this ratio r0 atj!Rj ! !a is related to the mean vibrational quantumnumber #n by r0 ! #n=& #n# 1' [23]. In Fig. 3(c), we plot ras a function of j!Rj for the "tc ! 5 ms data. As shown inFig. 3(b), we fit a Lorentzian curve to the carrier, thensubtract its contribution from both the red and the bluesideband data, as shown in Fig. 3(c). We find r0 ’ #n !0:01% 0:05, and the ground state population P0 ! 1=& #n#1' ! 0:99% 0:05, where the error bars reflect fluctuationsin the data around j!Rj ! !a. If, instead, we subtract theconstant background of PB4 ! 0:024 but not the carrier’sLorentzian tail, we find r0 ’ #n ! 0:05% 0:04 and P0 !0:95% 0:04. Finally, if we use the raw data from Fig. 3(b)

with no subtractions, we obtain r0 ! 0:10% 0:03, #n !0:12% 0:04, and P0 ! 0:89% 0:03. Because the atom isnot a two-state system and the motional state is not knownto be thermal, these estimates are approximate.

The axial cooling rate and asymptotic value of #n dependon !R, on the !R1;R2 Rabi frequencies, and on the powerand detuning of !4. We have performed computer simu-lations to help us choose optimal values for these parame-ters. A common feature of both our theoretical andexperimental investigations is the robustness of #n undervariations of the cooling parameters. As an example [30],in Fig. 4 we plot the measured sideband ratio r0 at!R=2# ! $500 kHz ’ $!a as a function of (a) the de-tuning !R used for sideband cooling and (b) the recyclingintensity I4. The sideband asymmetry is maintained over arange of at least 200 kHz in detuning and of 2 orders ofmagnitude in the intensity I4 of the !4 beams. The insetsgive results from a 2-state calculation of r0, displayingsimilar insensitivity to the exact values of !R and I4.

We use two methods for estimating the mean energy Erfor radial motion. The first method involves adiabaticallylowering the FORT depth to zero, so that only the UR

0.4

0.3

0.2

0.1

0

r, P

' 4

700600500400|!R|/2" [kHz]

(c)

#a

0.50

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0.30

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0.10

0

P4

(a)

0.50

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0.30

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P4

-1.0 -0.5 0 0.5 1.0!R/2" [MHz]

(b)

FIG. 3 (color online). Population P4 in the F ! 4 state versusRaman detuning !R=2#. The data in (a) are taken with "tc !250 "s of cooling and an !4 total 4-beam intensity I4 ! 5Isat

4 ;those in (b) with "tc ! 5 ms, I4 ! 0:5Isat

4 (on average,"33 atoms per data point). The arrow marks the detuningused for sideband cooling. (c) Zoom-in on the two sidebandregions for the data in (b), with detuning axis folded around!R ! 0. The red (!) and blue (") sidebands, and their ratio r(#), are shown after subtracting a Lorentzian fit to the carrier[superimposed in (b)].

PRL 97, 083602 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending25 AUGUST 2006

083602-3

A. D. Boozer et al., Phys. Rev. Lett. 97, 083602 (2006)

…in cavity QED, access to center-of-mass motion

atom-cavity coupling constant g. To determine this value,we fit the normal modes with a theory curve (solid red line)with g and the atomic detuning as the only free parameters.From this fit, we find g=2! ! "6:7# 0:1$ MHz, close tothe theoretical value of g0=2! ! 8 MHz (dashed red line).This again proves that we are able to accurately localize theatom at the center of the cavity field and that the system isin the single-atom strong coupling regime of CQED.

After demonstrating the good control achieved over theposition of the atom, we now turn to its motion. In order tomeasure the temperature, we use Raman sideband spec-troscopy. For an atom at low temperature, the sinusoidallattice can be approximated by a harmonic potential, lead-ing to a quantization of the atomic vibrational energyEfx;y;zg ! "nfx;y;zg % 1=2$h"fx;y;zg for each lattice axis [23].Here, nfx;y;zg is an integer and "fx;y;zg denotes the trapfrequency that depends on the intensity and wavelengthof the lattice light along the fx; y; zg direction. We drivetransitions between the different motional states usingRaman beams. One of the beams is polarized orthogonallyto the cavity axis and impinges at an angle of 45& with the xand y axes (Fig. 1). Another beam, polarized along thecavity axis, is counterpropagating to the first one, and athird, also linearly polarized beam is applied along thecavity axis. The latter two are detuned by the hyperfinesplitting of 6.8 GHz from the first, while all of them arered-detuned by 0.3 THz from the D1 line at 795 nm.Because of this large detuning, the Raman beams lead toan effective coupling of the two hyperfine ground stateswithout populating the excited state. The linewidth ofthis coupling can be much smaller than the natural line-width of the D1 transition. Thus, the sidebands can be

spectroscopically resolved and addressed individuallywhen the intensity of the lasers is low enough and theduration of the Raman pulse is long enough.To obtain a sideband spectrum, the atom is optically

pumped to the F ! 1 hyperfine state. Subsequently, weapply the Raman lasers for 200 #s. In order to measure thepopulation transfer to F ! 2, we perform cavity-basedhyperfine-state detection [28,33]. We can thus determinewithin 30 #s whether the atomic population has beentransferred to F ! 2. The transfer probability after thepreviously described intracavity Sisyphus cooling as afunction of the detuning between the Raman beams isshown in the green (grey) curve of Fig. 4(a), where zero

FIG. 3 (color online). Normal-mode spectroscopy of the atom-cavity system with the atom trapped in the 3D optical lattice. Thetransmission of the cavity is a Lorentzian curve when the atom isnot coupled (black squares and black fit curve), while a resonantatom leads to a normal-mode splitting (red dots and solid red fitcurve). The slight asymmetry is caused by a small residualdetuning between atom and cavity. The error bars are statistical.The dashed curve shows the spectrum expected for g0=2! !8 MHz, the value calculated from our cavity parameters.

(a)

(b)

FIG. 4 (color online). (a) Sideband spectrum after intracavitySisyphus [green (grey) line] and after sideband cooling (blackline). The statistical standard error of the data is given by thethickness of the lines. The three peaks at positive detuningscorrespond to a transition on the blue sideband for each axis ofthe 3D lattice potential (right to left: x, y, and z axes). The carrierpeak at the center (dashed blue Lorentzian fit curve) is saturated.Transitions on the red sideband (negative detunings) are stillobserved after Sisyphus cooling [green (grey) line] but nearlyvanish after 5 ms of sideband cooling (black line). (b) Transferprobability on the red (red squares) and blue (blue dots) side-bands after Raman sideband cooling. The solid curves arenumerical fits of the sum of three Lorentzian curves, with theshaded areas indicating the 66% confidence interval. The atomictemperature after sideband cooling is determined from these fits.

PRL 110, 223003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending31 MAY 2013

223003-3

A. Reiserer et al., Phys. Rev. Lett. 110, 223003 (2013)

12.5 13 13.5 14 14.5 15 15.5 16 16.50

1

2

3

4

5

Raman detuning (MHz / 2/)

APD

cou

nt ra

te (k

Hz)

awithout sideband coolingwith sideband cooling

neutral atoms in 1D: …and 3D:

ions in 3D: A. Stute et al., Appl. Phys. B 107, 1145 (2012)

Driving the blue sideband: heating " and lasing

thermal to coherent transition

NATURE PHYSICS DOI: 10.1038/NPHYS1367ARTICLES

100 m m

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Figure 1 | Ion-trap illustration, observation of coherent motion andphysical origin of stimulated phonon emission for the single-ion phononlaser. a, An ion in a harmonic trap interacting with a cooling beam(frequency !c) and an amplification beam (frequency !a). The ion withposition x and mass m moves in a trap potential v(x). The optical transitionhas frequency !0 and linewidth � . b, Ion-luminescence series showingtime-averaged motion. The amplification beam intensity is increasinglystepped from the lowest (no amplification) to upper images. The coolingbeam intensity is constant. A threshold that marks a transition fromthermal to coherent motion is apparent. c, At the quantum level,amplification results from net production of phonons through stimulatedemission (rate R

+) and absorption (rate R

�) of centre-of-mass phonons.The corresponding, phonon-assisted transitions to the upper level areshown. Phonon emission and phonon absorptive transitions inducepolarization at frequencies corresponding to levels 2+ and 2� in thesediagrams. These lie within the distribution of transition frequencies givenby the lineshape function. Analogies to stimulated Raman amplification arediscussed in the text.

v�2sat is merely the second-order Taylor coefficient. The Van der Poloscillator is commonly used to understand lasers and electronicoscillators18,19. It has a small-signal and large-signal (saturated)regime of operation; these are studied separately below.

Setting v to zero in the damping and amplification terms (small-signal regime), consider letting ia increase from zerowhile holding icconstant (fixed cooling). As ia increases, ultimately, g (0)ia = (0)ic(the threshold condition) at an intensity ia = iT ⌘ ((0)/g (0))ic. Foria > iT, the overall damping becomes negative in equation (1) (thatis, g (0)ia > (0)ic), and ion motion grows, at first, exponentiallyin time; before finally stabilizing in the large-signal regime. Beforeconsidering this large-signal, saturation process, experimentalobservations of threshold are presented. A single, magnesium ionin a linear radiofrequency trap was studied. The experimentalarrangement is nearly identical to that detailed in ref. 31. TheMg+D2 transition at 279.6 nm has a natural linewidth of 41.8MHzand the axial secular frequency of the trap was set to 71 kHz inall measurements; micro-motion was negligible. The amplificationbeam was directed along the trap’s long axis to excite only axialoscillations. The cooling beam was applied slightly off axis fromthe long axis so as to project onto all three axes of motion.Cooling to approximately 1mK was obtained. Figure 1b shows aseries of time-averaged images of the ion motion, for increasinglevels of amplification beam intensity (ia = 0 in the lowest image).A threshold at which the amplified, Brownian, thermal motionof the ion transitions to a double-lobed pattern (produced by

Figure 2 |A comparison of measured and predicted ion motion as afunction of pumping that illustrates threshold behaviour and increasingcoherent motion for pumping above threshold. Measured, square of ion-velocity amplitude (yellow, circles) plotted versus ia/ic (ic held constant).Amplification/cooling beams are detuned by +12 MHz/�74 MHz.The solid curve is the steady-state solution of equation (3). To matchthe ab initio calculation, the data-set origin was shifted by (�0.05,�0.72 m2 s�2). Inset: The role of amplification saturation in establishingthe operating point is illustrated by graphical solution of G(v0)ia =K(v0)ic(steady-state solution of equation (2)). This excludes spontaneousemission since the Langevin force averages to zero. G(v0)ia is plotted in redfor beam intensities ia = 0.5,1,2,4⇥ iT, whereas K(v0)ic is plotted in blueat a single cooling intensity (both are normalized to their threshold values,set equal to unity). For ia = iT,G(0)iT =K(0)ic so that, trivially, v0 = 0.However, with increasing amplification beam intensity, the point v0 = 0 isno longer stable (that is, G(0)ia >K(0)ic for ia > iT). In the terminology oflaser oscillators, G(0)ia for ia > iT is the unsaturated amplification.Saturation of the mechanical amplification restores balance ofamplification with damping by increase of v0 as illustrated. Thecorresponding operating points are diamonds, and are superimposed onthe dashed curve (solution of G(v0)ia =K(v0)ic) in the main panel. Theoperating points are dynamically stable with respect to fluctuations as canbe verified using the inset (perturbations create slight changes in netamplification/damping so as to restore the operating point).

the oscillatory motion of the ion) is apparent in the data.From these images, the velocity amplitude is determined usingv0 = ⌦0x0; and the threshold behaviour is further apparent inFig. 2 where the squared-velocity amplitude (to emphasize thethreshold) is plotted versus ia normalized by ic. The ratio ia/icis experimentally accessible, and, as discussed below, physicallyappropriate if, as was the case in this experiment, saturation ofthe atomic transition can be neglected. It was determined bymeasuring separately the ion luminescence induced by each beamat a reference, negative detuning. The amplification (cooling) beamdetuning is set to +12MHz (�74MHz) in these measurements;and the solid curve in Fig. 2 is the theoretical operating pointcurve as discussed below.

The oscillator operating point (that is, amplitude of motionat a given pumping level) results from a saturation process,similar to amplification saturation in a Van der Pol oscillator (orlaser oscillator). (v) and g (v), through their imbalance in thesmall-signal regime (that is, g (0)ia > (0)ic), cause growth in theamplitude of oscillation. Increases in motional amplitude inducesaturation that restores the balance of amplification and damping.As in a Van der Pol oscillator and laser oscillator this balance occursin a time-averaged sense over one cycle of motion. To analyse thesaturation process, the velocity is expressed as v = v0cos(⌦0t +�),where v0 and � are a slowly varying amplitude and phase (thatis, v0 ⌧ ⌦v0 and � ⌧ ⌦0). On substitution of this expression into

NATURE PHYSICS | VOL 5 | SEPTEMBER 2009 | www.nature.com/naturephysics 683

K. Vahala et al., Nature Phys. 5, 682 (2009)

K. R. Brown et al., Nature 471, 196 (2011)

Coupled oscillators exchange motional states

54 +m

0.0

1.01.52.02.5

0.5

0

Pote

ntia

l (m

eV)

–40 –20 4020Axial position ( +m)

pendula connected by a (massless) spring. If one pendulum initiallyoscillates while the other is at rest, the motion is periodically exchangedbetween them. The first pendulum comes to a complete stop after acharacteristic time, Tswap, which, for small coupling, is inversely pro-portional to the associated spring constant. A quantum-mechanicaldescription may be given using the vibrational quanta, ni, oftenlabelled ‘phonons’, in the individual wells. Under the dipole–dipoleinteraction, an initial motional state jn1, n2æ becomes entangled with allother motional states n’1,n’2j i for which n’1zn’2~n1zn2. Only atodd (even) multiples of Tswap is the swapped (original) basis staterecovered. Notably, the initial state j0, 1æ evolves into the Bell state

0,1j iz 1,0j i! "! """

2p

after time Tswap/2, yielding a maximally entangledstate of motion. This motional entanglement can be mapped on to theinternal electronic state of the ions18.

In the experiment presented here, the coherent energy exchange isdemonstrated between singly charged 40Ca1 ions. They are held in anion trap with gold-on-alumina electrodes arranged in a two-layer geo-metry (Fig. 1) similar to the one described in ref. 19. Applying d.c.voltages of up to 110 V to seven adjacent electrode pairs, a double-wellpotential with a trap separation of 54mm and an axial trap frequency,f0 5 537 Hz, is created (Methods). The ions are Doppler-cooled on theS1/2–P1/2 transition at 397 nm using a single, elliptically shaped laserbeam, and detected by collecting the fluorescence light on an electron-multiplied CCD (charge-coupled device) camera and on a photo-multiplier tube. Two 729-nm laser beams, individually focused on thetwo trapping sites, are used to perform sideband cooling on the S1/2–D5/2transition and to map out the sideband spectrum20. During the 4-msperiod of sideband cooling, the trapping sites are alternately illuminatedwith 729-nm light. This alternation is carried out at the approximaterate of the energy exchange to achieve an imbalance of phonon popu-lation between the traps. By comparing Rabi-oscillations on the redand blue sideband21, the mean phonon number in the first well iscalculated. Figure 2 shows the oscillatory behaviour of the phononnumber in this well as a function of waiting time. The theoretical fitto the data (Methods) indicates an exchange within Tswap 5 222(10)msand an initial phonon population of Æn2æ 5 9(1) in the second well; allnumbers in parentheses denote 1s standard deviations with respect tothe last digit. Currently, the ions experience a heating rate, _nh i, of 1.3(7)quanta per millisecond, which limits the efficiency of the sideband

cooling and is comparable to other room-temperature traps of similarsize22.

Ideally, the exchange rate would be significantly larger than theaverage heating rate. To enhance the exchange rate, strings of severalions can be used in the trapping wells. Equation (4) suggests that thecoupling strength scales in proportion to the number of ions. Thisassumes the ion strings to be point-like objects. In practice, theextended nature of the strings can cause a significantly faster increasein the coupling strength.

This increase is demonstrated by mapping out the ions’ excitationspectrum as the trapping frequencies of the individual sites arescanned through the resonance condition. The frequency scan isachieved by applying a control voltage Uax to an outer trap-electrodepair (Methods). The dipole–dipole coupling manifests itself as anavoided crossing, separating the mode frequencies by Vc/2p. Closeto resonance, the motion of the ion strings is strongly coupled, andthe oscillation can be excited with 729-nm light on either of the twotrapping sites20. Figure 3a shows an example of this avoided crossingmeasured with two ions in each well, while Fig. 3b represents anindividual sideband spectrum. The motional spectra of five ion con-figurations have been analysed, using up to three ions in each well. Theconfigurations and corresponding mode spectra are displayed in Fig. 4.The curves present calculations using a common set of fit parametersfor the potential and for the action of the control voltage Uax (Methods),and explicitly include the extended nature of the ion strings. For theconfiguration with one ion per well, the observed splitting of 1.9 kHzagrees within one standard deviation with the energy-exchange ratefrom Fig. 2, which was measured with the same trap parameters. Thedata further show that the splitting is increased from 1.9(3) kHz to14(1) kHz by using up to three ions in each well, without any modifica-tion of the external potential. The sevenfold increase of the coupling isbeyond the factor of three that is expected from a simple point-chargemodel and is due to the anharmonicity of the individual potential wells:as the outer potential walls are steeper than the inner ones, the ion-strings’ centres of mass get closer as more ions are added. At the sametime, the average oscillation frequency is reduced. Both of these effectslead to an increase in the coupling strength (equation (4)). The extent ofthe ion strings provides a comparable increase, as it is non-negligiblewith respect to the inter-well distance.

The demonstrated coupling can be used in diverse schemes to createentanglement or to perform gates. (See also independent experimentsat NIST Boulder with trapped 9Be1 ions23.) The creation of Bell states

0 0.2 0.4 0.6 0.8 10

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⟨n1⟩

Figure 2 | Energy exchange between two trapped ions over a distance of54 mm. The data show the average number of phonons in the first trappingwell, Æn1æ, after sideband cooling and a variable waiting time, t. The sequencefor sideband cooling has been arranged to yield different phonon numbers inthe two wells, revealing the state exchange as an oscillation of phonon numbersat the level of a few quanta. The observed time for a complete exchange isTswap 5 222(10)ms, indicating a mode splitting of Vc < 2p3 2.25(4) kHz. Adamping constant tdamp 5 3(2) ms and a constant background heating, _nh i, of1.3(7) quanta per millisecond are inferred from the fit to the data. The error barsindicate one standard deviation as inferred from Monte Carlo simulations.Lateral deviations of the data from the fit are attributed to small drifts in theresonance condition which modify the exchange rate.

!50 !25 0 25 50440

450

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z)

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Figure 3 | Demonstration of avoided crossing derived from sidebandspectra. a, Oscillation frequencies of four trapped ions (two in each well) as afunction of the axial control voltage Uax, yielding a mode splitting of 5.5(3) kHz.The data points (filled circles) correspond to peaks in individual sidebandspectra taken on the S1/2–D5/2 transition. The error bars are smaller than thefilled circles. b, Example of an individual sideband spectrum. The resonantfrequencies are marked with a dot-dashed line, and the corresponding datapoints in a are coloured red.

LETTER RESEARCH

1 0 M A R C H 2 0 1 1 | V O L 4 7 1 | N A T U R E | 2 0 1

Macmillan Publishers Limited. All rights reserved©2011

!n1"

time (ms)

M. Harlander et al., Nature 471, 200 (2011)

Udipole-dipole

=~⌦c

2(a

1

a†2

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We can already cool to the ground state. Why is cavity cooling useful?

1.  quantitative understanding of cooling dynamics

2.  “nondestructive” method that preserves coherence

Leibrandt et al., Phys. Rev. Lett. 103, 103001 (2009)

88Sr+ in linear Paul trap •  measured heating & cooling rates •  found agreement with rate equation model 3

FIG. 3: (color online) Photon scattering rate into the cavityas a function of laser-cavity detuning !lc with the ion locatedat a cavity standing-wave antinode (a), halfway between anode and an antinode (b), and at a node (c). Each data pointis measured by preparing the ion at fixed temperature byDoppler cooling for 200 µs, then measuring the rate of scat-tering photons from the cavity cooling beam into the cavityfor 50 µs with !ci/(2") = !60 MHz and !sc = 1.2" 107 pho-tons/s. The solid, dashed, and dotted vertical lines are atthe carrier, first-order motional sideband, and second-ordermotional sideband transition frequencies, respectively. Thecurves are Lorentzian fits with linewidths consistent with thecombined linewidth of the cavity and laser.

motional sideband transitions. The best cooling alongthe z-direction as investigated here is achieved via the!nz = !1 transition when the atom is located at a nodeof the cavity field.

We investigate one-dimensional cavity cooling alongthe z direction by measuring separately the recoil heat-ing rate, as well as the cavity cooling and heating ratesfor pumping on the cavity red (!lc = !"z) and blue(!lc = +"z) motional sidebands, respectively. To real-

0 1 2 3 40

10

20

30

40

50

60

Cavity cooling pulse length, t [ms]

Axia

l occ

upat

ion

num

ber,

<nz>

δlc = 0δlc = −ωzδlc = +ωz

FIG. 4: (color online) Cavity cooling dynamics. The ionis sideband cooled to the three-dimensional motional groundstate; a cavity cooling pulse with detuning !lc = 0 (carrier),!lc = !#z (red axial sideband), or !lc = +#z (blue axial side-band) is applied; and the mean number of motional quanta inthe z mode is measured. The three lines are a simultaneousfit to the model described in the Supplementary Informationwith fit parameters n0 = 0.30(6), !sc = 2.87(2) " 106 pho-tons/s, and $ = 0.0148(2). The reduced %2 of the fit is 1.7.

ize a situation that allows simple quantitative compari-son with the theoretical model for cavity cooling [8, 9],we prepare the ion in its motional ground state by stan-dard sideband cooling on the narrow S1/2, m = !1/2 "D5/2, m = !5/2 transition. We then apply the cavitycooling laser for a variable time t with detunings !lc = 0or !lc = ±"z and !ci/(2#) = !10 MHz. Finally, the meanvibrational quantum number #nz$ is determined by mea-suring the Rabi frequencies of the red and blue motionalsidebands of the S1/2, m = !1/2 " D5/2, m = !5/2transition [5]. This cavity-ion detuning is near the opti-mum value for conventional Doppler cooling, but the ge-ometry of the setup dictates that the cavity cooling laserDoppler cools the x and y motional modes to maintainthem at #nx,y$%< 10 but does not Doppler cool the z mo-tional mode. The ion position is locked to a node of thecavity standing wave for this measurement by applyingDC compensation voltages to the trap electrodes. The re-coil heating rate is the slope d#nz$/dt for !lc = 0 (Fig. 4green line), and the cavity cooling and heating rates arethe di"erences of the slopes d#nz$/dt for !lc = ±"z (Fig. 4red and blue lines) and for the recoil heating rate. Thesignature of cavity cooling is that the temperature afterpumping on the cavity red sideband is smaller than thetemperature after pumping on the cavity carrier, which issmaller than the temperature after pumping on the cavityblue sideband. Cavity cooling (!lc = !"z) counteractsrecoil heating by free-space scattering, and results in a fi-nite steady-state vibrational quantum number n! & 20.

We fit the cavity cooling dynamics to a rate equa-

Ions enable quantitative study of cavity cooling

blue SB

red SB

carrier

•  limited by small cooperativity to n = 22.5

4

tion model parameterized by the initial mean occupationnumber n0, the free space scattering rate !sc, and thee"ective cooperativity ! [9] (see the Supplementary In-formation for a description of the model). The data inFig. 4 fits the model with fit parameters n0 = 0.30(6),!sc = 2.87(2) ! 106 s!1, and ! = 0.0148(2). These val-ues of the fit parameters are consistent with those de-rived from independent direct measurements, so that therate equation model is consistent with our data withoutany free parameters. The steady-state mean occupationnumber due to cavity cooling is limited by the relativelysmall cooperativity of the cavity to n" = 22.5(3), con-sistent with the value calculated from the model for ourparameters.

In conclusion, we have demonstrated one-dimensionalcavity cooling of a single 88Sr+ ion in the resolved side-band regime. While our small e"ective cooperativity pre-vents us from cooling to the motional ground state, wehave observed clearly resolved motional sidebands in thecavity emission spectrum, and we have measured the cav-ity cooling and heating rates. Our results validate therate equation model proposed by Vuletic et al. [9], whichpredicts that it is possible to cavity cool atoms or ionsto the motional ground state without decohering the in-ternal state. This would require a large detuning of theincident laser compared to the atomic fine structure, acriterion which is easier to meet with light ions such asBe+ [12].

Resolved sideband cavity cooling might also be usedto cool large molecular ions to the motional ground state[22, 23]. While some species of molecular ions have beenpreviously cooled using sympathetic cooling [24, 25, 26],large mass ratios between the atomic cooling ions andthe molecular ions prevent e#cient sympathetic coolingof the molecular ions at temperatures near the motionalground state. Resolved sideband cavity cooling couldenable exciting new studies of large molecular ions in thequantized motional regime.

We thank S. Urabe for providing the linear RF Paultrap used in this work and J. Simon, M. Cetina, andA. Grier for advice. This work is supported in part bythe Japan Science and Technology Agency, the NSF, andthe NSF Center for Ultracold Atoms.

SUPPLEMENTARY INFORMATION

For cavity cooling in the weak coupling regime ! " 1,coherences [27, 28] decay rapidly, and rate equations aresu#cient to describe the cooling [9]. For one-dimensionalcooling along the cavity axis z, the rate of transitionsfrom motional state |n# to |n $ 1# is

!sc!!2LDn

1 + 4 ("lc + #)2 /$2% R!n, (2)

and the rate of transitions from motional state |n# to|n + 1# is

!sc!!2

LD(n+1)

1+4("lc!#)2/$2+ !scC!2

LD + next % R+(n + 1) + N+,

(3)where !sc is the photon scattering rate into free space,the number C is defined such that C!2

LDh# is the av-erage recoil heating along the z direction per free spacescattering event, and next is the heating rate along the zdirection due to environmental electric field fluctuationsin quanta per second. Here, !2

LD = Erec/(h#) is theLamb-Dicke parameter, as determined by the ratio of re-coil energy Erec and trap vibration frequency #. Notethat these transition rates are only valid in the Lamb-Dicke regime !2

LD &n# " 1, which limits the applicabil-ity of this model to &n# " 70 for our experimental pa-rameters. The expectation value of the mean vibrationalquantum number &n#t evolves according to

&n#t = n0e!Wt + n"

!

1 $ e!Wt"

(4)

for "lc = $# (cooling),

&n#t = n0 +!

R+ + N+"

t (5)

for "lc = 0, and

&n#t = (n0 + n" + 1) eWt $ (n" + 1) (6)

for "lc = +# (heating), where n0 % &n#t=0 and n" %&n#t#" are the initial and steady-state value of &n#t, re-spectively. The cavity cooling rate constant W is givenby

W =!sc!!2

LD

1 + $2/ (2#)2, (7)

and the steady state average occupation number n" isgiven by

n" =# $

4#

$2+

%

C

!+

next

!sc!!2LD

& %

1 +# $

4#

$2&

. (8)

For cavity cooling of the z motional mode in our ex-periment, we calculate C = 1/3 (photons are scatteredisotropically for a J = 1/2 ' J $ = 1/2 transition), andmeasure independently next = 17(2) s!1. Thus, for ourexperimental parameters, the heating due to environmen-tal field fluctuations is negligible (next " !sc!!2

LD) andthe expression for the steady-state occupation numberreduces to Eq. (1).

! [email protected][1] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J.

Wineland, Phys. Rev. Lett. 62, 403 (1989).

12.5 13 13.5 14 14.5 15 15.5 16 16.50

1

2

3

4

5

Raman detuning (MHz / 2/)

APD

cou

nt ra

te (k

Hz)

awithout sideband coolingwith sideband cooling

Sidebands are not enough"

theory: C. Genes, K. Hammerer, P. Rabl, H. Ritsch 1.! write down effective master equation for " 2.! "and for "m, motional component.

13.5 14 14.5Raman detuning (MHz / 2

drive system on red sideband" "but no evidence of cooling.

P3/2!

S1/2!

D5/2!

Master-equation analysis of cavity cooling Obtained expressions for

cooling rate "cool mean occupation nsteady-state

as functions of •! excited-state population, •! power spectrum of dipole moment correlation,

evaluated at red and blue sidebands power spectrum of dipole moment correlation,

asymmetry needed!

!1.5 !1 !0.5 0 0.5 1 1.53.5

4

4.5

5

5.5

6

6.5

7

7.5

8x 10!5

detuning of cavity from Raman resonance [MHz]

S(!) [1/MHz]

Simulations: No asymmetry for our parameters. Maybe in the new experiment"

!30 !20 !10 0 10 20 302

4

6

8

10

12

14x 10!6 S(!) [1/MHz]

detuning of cavity from Raman resonance [MHz]

) [1/MHz]





In current experiments"

What cavity parameters are accessible?

Two comments: 1.! In general, not two-level systems. 2.! In a Raman system, we can work with effective parameters.

secular trap frequencies are !x ! !z ! 2!" 1:5 MHzand !y ! 2!" 2:5 MHz. We have measured the heatingrate of the trapped ion using the Doppler-recooling method[26] and find #1:0$ 0:1% K=s, in line with previous mea-surements of similar trap geometries [23]. The trap depthaccording to simulations is 550 K, and we find ion lifetimesof 30 min with laser cooling, probably limited by decayinto the F7=2 dark state, and 100 s without laser cooling.

The fiber cavity is aligned in the plane orthogonal to thesymmetry axis of the ion trap. We mount the fiber ends instainless steel sleeves with an outer diameter of 300 "m inorder to provide mechanical stability, electrical grounding,and shielding of the fibers from ultraviolet stray light,which would charge up the dielectric material. Duringloading and initial laser cooling of the ion, the fiber cavityis located 1.8 mm away from the center of the ion trap inorder to prevent contamination of the mirrors with neutralYb atoms from the atomic beam. Subsequently, the cavitymode is overlapped with the trapped ion by translating thecavity assembly towards the ion trap using a nanoposition-ing stage with a speed of 1 mm=s, during which we main-tain laser cooling of the ion. Experimentally, we find thatthe cold ion remains trapped during the overlap procedure

with the cavity in>90% of the attempts and that the cavityalignment is unaffected by the transport. We emphasizethat our methodology provides an alternative to the conceptof having spatially separated loading and processingregions in ion traps [9]. When the cavity is in the finalposition [see Fig. 1(c)], the length of the optical cavity isactively stabilized by a piezoelectric transducer to asecondary light field near 780 nm using the Pound-Drever-Hall technique in transmission. The presence ofthe rf-grounded fiber sleeves increases the geometric quad-rupole efficiency of the trap from 10% to approximately19%. Accordingly, we reduce the rf power and reachsecular trap frequencies of !x=2! ! 2:1 MHz, !y=2! !2:7 MHz, and !z=2! ! 1:3 MHz. Previously, the heatingof trapped ions due to fluctuating potentials on nearbydielectric surfaces has been identified as a critical limita-tion for integrating ion traps and small optical cavities [16].We find that for our case with &110 "m spacing betweenion and cavity mirror the ion heating rate doubles to#2:3$ 0:4% K=s, which is significant but not prohibitivelylarge. The ion lifetimes are 30 min with laser cooling,unchanged from the case without cavity, and 6 s withoutlaser cooling.The optical fiber cavity interacts with the Yb' ion on

the D3=2 (D)3=2*1=2 transition near # ! 935 nm [seeFig. 1(d)]. The fiber cavity is designed to provide a

peak single-ion–single-photon coupling strength of g !d

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4c=#@$0#w2L%

p! 2!" 6 MHz, where c is the speed

of light and $0 is the vacuum permittivity. The cavity fieldhas a (in situ) measured decay rate of % ! 2!" 320 MHz,corresponding to a finesse of F ! 1000$ 50, which islimited by scattering and absorption losses of the mirrors.Taking into account the branching of the decay of theexcited state, the total decay rate of the dipole momentamounts to ! ! 2!" 2 MHz and hence C ! 0:05.We demonstrate control over the ion-cavity coupling by

displacing the cavity with respect to the ion [9]. To thisend, we mechanically translate the cavity structure usingthe nanopositioning stage and, simultaneously, measurethe optical pump rate 1=&D out of the D3=2 state. Afterpreparation of the ion in the D3=2 state, we employ laserlight on the cavity mode to optically pump into the S1=2ground state, which we detect by fluorescence on theS1=2 ( P1=2 transition [see Figs. 2(a) and 2(b)]. All mea-surements are performed at a magnetic offset field of 2 G.In Fig. 2(c) we display a map of the cross section of thecavity mode together with Gaussian fits along the y andthe z axes [Figs. 2(d) and 2(e)]. We find waists of wy !#7:6$ 0:9% "m and wz ! #6:6$ 0:9% "m, in agreementwith the measured radii of curvature of the mirrors. Owingto the small radius of curvature available with the opticalfiber cavity technology, these mode waists are significantlysmaller than in previous cavities used with ion traps andemploying conventional mirrors [9–13,15]. In Fig. 2(f) weshow the standing-wave pattern as the cavity assembly is

(a)

xy

z

2S1/2

Coolinglaser369 nm

Couplinglaser297 nm

Cavitymode935 nm

2P1/2

2D3/2

3D[3/2]1/2(d)

Cavity-stimulatedtwo-photon transition

1 mm

(b)

100 m m

(c)

FIG. 1 (color online). Experimental setup and relevant energylevels of Yb'. (a) The ion trap is formed by end cap electrodes(gray, vertical), which are surrounded by compensation elec-trodes (yellow, arranged on a square) and the optical cavity(oriented horizontally). For loading and initial cooling of theion, the cavity is retracted using a nanopositioning stage.(b) Composite image of the end cap electrodes (shadow image)and the trapped ion (fluorescence image, inset). (c) Compositeimage with the cavity in position. (d) Cooling and detection ofthe ion is performed on the S1=2 ( P1=2 transition, the cavityoperates on the D3=2 (D)3=2*1=2 transition at 935 nm, and the" transition is driven by a coupling laser at 297 nm and thevacuum cavity mode. The branching ratio from the D)3=2*1=2state is 55:1 in favor of decay into the S1=2 state.

PRL 110, 043003 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

25 JANUARY 2013

043003-2

M. Steiner et al., Phys. Rev. Lett. 110, 043003 (2013)

P3/2!Ca+ !

S1/2!

D5/2!

Yb+!

Single-ion strong coupling has not yet been reached.

NATURE PHYSICS DOI: 10.1038/NPHYS1302LETTERS

b

a

d

c

100 µm

100 µm

Ion trap

CM

LC

APDAPD

PZT

LCPP, RB

CCD

x

y

z

OP

RP

PP RBCM

+3/2¬3/2

¬1/2 +1/2

¬1/2 +1/2

¬1/2 +1/2

Optical pumping

P1/2

+3/2¬3/2

¬1/2 +1/2

¬1/2 +1/2

¬1/2 +1/2

Cooling

+3/2¬3/2

¬1/2 +1/2

-1/2 +1/2

¬1/2 +1/2

Probing

LC

RP OP PP

LC

D3/2S1/2

P1/2 P1/2

D3/2S1/2D3/2S1/2

Figure 1 | Schematic diagram and ion Coulomb crystal images. a, Schematic diagram of the experimental set-up with the linear radiofrequency ion trapincorporating an optical cavity along its radiofrequency-field-free axis (z axis). The incoupling mirror of the cavity is mounted on a piezoelectric transducer(PZT) allowing for tuning of the cavity around the atomic resonance. LC: laser cooling beam, RP: repumping beam, OP: optical pumping beam, PP: probepulse, RB: reference beam, CM: cavity mirrors, PZT: piezoelectric transducer, CCD: CCD camera, APD: avalanche photodiode. b, Energy levels of 40Ca+

including the relevant transitions for the three main parts of the experimental sequence. First, the ions are Doppler laser cooled by driving the4s2S1/2–4p2P1/2 transition using 397 nm laser beams and repumping on the 3d2

D3/2–4p2P1/2 transition by light at 866 nm. Next, the ions are opticallypumped into the m

j

=+3/2 Zeeman substate of the 3d2D3/2 level using the optical pumping beam in combination with the laser cooling beams. Finally, thecoupling of the Coulomb crystals to the cavity field is measured by injecting a weak probe pulse of 866 nm light into the cavity and detecting the reflectionsignal by an avalanche photodiode. c,d, Projection images of a 1-mm-long ion Coulomb crystal recorded with the CCD camera by collecting 397 nmfluorescence light emitted during laser cooling: the whole crystal (c) and ions within the cavity mode volume (d) (see the text for details).

3d2D3/2-level of 40Ca+ (lifetime: ⇠1s) using a beam making anangle of 45� with respect to the z axis and having a proper ellipticalpolarization (Fig. 1a,b).

To investigate the collective coupling of the ions in theCoulomb crystal to the TEM00 mode of the cavity, a 1.4-µs-long and 99%-mode-matched probe pulse is injected. The fieldstrength of this pulse corresponds to an average intra-cavityphoton number of about one or less at any time. The frequencyof the probe light can be tuned around the 3d2D3/2–4p2P1/2transition, and it is left-handed circularly polarized to obtain thestrongest coupling with the optically pumped ions (Fig. 1b). Aweak off-resonant continuous-wave laser beam at 894 nm is also

injected into the cavity as a reference of the cavity fluctuationsand drifts. Photons in the probe pulse and the reference beam aredetected in reflection and transmission, respectively, by avalanchephotodiodes with an overall efficiency of 16% after spectral andspatial filtering (Fig. 1a).

An accurate way to quantify the strength of the collectivecoupling is, for a series of detunings of the probe frequency !pwith respect to the 3d2D3/2–4p2P1/2 transition frequency !a, tomeasure the width and resonance phase shift of the reflected probesignal when scanning the cavity resonance frequency !c around!p. Figure 2a, b shows results of such measurements for a crystalsimilar to the one shown in Fig. 1. According to the two-level atom

NATURE PHYSICS | VOL 5 | JULY 2009 | www.nature.com/naturephysics 495

Aarhus Ca+, ~540 ions g = 2# · 0.5 MHz $ = 2# · 2.2 MHz % = 2# · 11.2 MHz

Innsbruck

Ca+

g = 2# · 1.4 MHz $ = 2# · 0.05 MHz % = 2# · 11.2 MHz

P. F. Herskind et al., Nature Phys. 5, 494 (2009)

Single-ion strong coupling has not yet been reached.

Aarhus Ca+, ~540 ions g = 2𝜋 · 0.5 MHz 𝜅 = 2𝜋 · 2.2 MHz 𝛾 = 2𝜋 · 11.2 MHz

Innsbruck

Ca+

g = 2𝜋 · 1.4 MHz 𝜅 = 2𝜋 · 0.05 MHz 𝛾 = 2𝜋 · 11.2 MHz

5

Figure 1. (a) A picture of the experimental set-up, with the microfabricatedplanar-electrode ion trap, the 22 mm-long optical cavity, and the Yb oven. (b)Layout of the microfabricated trap chip. Outer RF electrodes are shown in green,and the inner RF electrode is split into three periodic electrodes shown in blueand red. The 24 large rectangular DC electrodes are shown in white. (c) A portionof the inner periodic electrodes generating the electrostatic periodic potential. (d)Periodic potential at the position of the ions 138 µm above the trap surface fora �1 V DC voltage applied to the inner periodic electrode (blue) and +0.9 Vto the outer periodic electrodes (red). (e) Pseudopotential produced in the radialdirections by the RF voltage applied to the outer RF electrodes. The equipotentialcontours are spaced by 5 mV. (f) Image of five array sites containing small ionchains where single ions can be resolved. The ions are illuminated by light in thecavity mode, which is overlapped with the array.

New Journal of Physics 15 (2013) 053001 (http://www.njp.org/)

MIT Ca+ g = 2𝜋 · 0.8 MHz 𝜅 = 2𝜋 · 2.7 MHz 𝛾 = 2𝜋 · 11.2 MHz

Cambridge/Bonn

Yb+ g = 2𝜋 · 6 MHz 𝜅 = 2𝜋 · 320 MHz 𝛾 = 2𝜋 · 1 MHz

M. Cetina et al., New J. Phys. 15, 053001 (2013)






!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4c=#@$0#w2L%




(a)

xy

z

2S1/2

Coolinglaser369 nm

Couplinglaser297 nm

Cavitymode935 nm

2P1/2

2D3/2

3D[3/2]1/2(d)


1 mm

(b)

100 µm

(c)



25 JANUARY 2013

043003-2



Ions coupled to the cavity… How many?

Spacing? Tunable?

large Coulomb crystals: this morning’s talk by A. Dantan


Ions coupled to the cavity" How many?

Spacing? Tunable?

4° Trap axis

Ion 1

Ion 2

Cavity shift (a.u.)yCavity shift (a.u.)

Inte

nsity

(a.u

.)

Inte

nsity

(a.u

.)

B. Casabone et al., Phys. Rev. Lett. 111, 100505 (2013)

Can we align the cavity and trap axes?

5



1.! Paul trap: mirrors in endcaps e.g., Keller group, Sussex

2.! segmented trap with multiple zones e.g., Chuang/Vuletic, MIT

5



5



M. Cetina et al., New J. Phys. 15, 053001 (2013)

In fiber cavities, a tradeoff: coupling vs. ion number. •! strong coupling expected,

but just one ion in cavity

Other designs: max. 1 ion trapped






!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!4c=#@$0#w2L%




(a)

xy

z

2S1/2

Coolinglaser369 nm

Couplinglaser297 nm

Cavitymode935 nm

2P1/2

2D3/2

3D[3/2]1/2(d)


1 mm

(b)

100 m m

(c)



25 JANUARY 2013

043003-2


2

a) b)

FIG. 1. (color online). (a) Endcap trap with integrated op-tical fibers. Half of the upper electrode structure is cut awayto reveal the upper fiber. (b) Cross-section of the pseudopo-tential of the trap with fiber, obtained from a finite elementcalculation by averaging the potential over one rf-cycle.

aperture of the fiber, a capture e�ciency of 12.3% canbe achieved at an ion-fiber separation of 183 µm.

Calcium ions are loaded from an e↵usive oven mountedto the side of the trap. To reduce coating of the trapelectrodes with calcium, the atomic beam is collimatedby means of a tube of inner diameter 250 µm, 2.45 mmaway from the center of the trap. The atoms are pho-toionized using a resonant laser at 423 nm and a secondstage at 375 nm for the actual ionization [13]. Once cap-tured in the trapping potential, a single ion is stored forseveral hours. Compared to an endcap trap with solidcentral electrodes, tubular electrodes generate a pseu-dopotential with a 25% lower trap depth [Fig 1(b)]. Foran rf-amplitude of 200 V, the calculated potential bar-riers are 2.8 eV in the radial and 2.1 eV in the axialdirection. The secular frequencies were measured to be!r=(2⇡)1.9 MHz and !z=(2⇡)3.8 MHz for a drive fre-quency of 14.9 MHz.

With a view to applications in spectroscopy, quantuminformation processing and cavity-QED, the localizationof the ion in the trap is of particular importance. Asa first step, the ion is laser-cooled on the S1/2 ! P1/2

transition with a wavelength of �=397 nm. Lasers with apower of several µW are injected from the side under dif-ferent angles, red-detuned by roughly half the linewidth�=(2⇡)22.3 MHz. To avoid optical pumping to the D3/2

and the D5/2 level, we apply lasers at 850 nm and and854 nm, returning the ion to the ground state via theP3/2-level [Fig 2].

The ion may also undergo a periodic oscillation, drivenby the rf-trapping field (micromotion), if dc electric strayfields push the ion o↵ the central node of the rf-field.In order to minimize micromotion, we compensate strayfields by applying voltages to a set of compensation elec-trodes. The atomic beam collimator serves as the elec-trode compensating dc stray fields in the direction of theoven. Horizontal dc-fields orthogonal to the oven arecompensated via a small wire mounted to the side, whilea vertical dc o↵set-field can be applied via the rf-ground

FIG. 2. (color online). Scheme of the relevant transitions in40Ca+. We probe the quantum properties of light scattered onthe resonance transition at 397 nm (bold arrows). Populationtrapping due to decay to the D-states is avoided by repumpingvia the upper P-state (thin arrows).

electrodes. With the help of these electrodes, we positionthe ion at the rf-center of the trap, where its micromotionis minimal. The required dc-voltages are determined byprobing the rf-modulation of the fluorescence intensityusing UV pump-beams in three non-collinear directions.

The ion’s localization that can be achieved depends onthe sensitivity with which we detect micromotion. Thesensitivity to Doppler modulation in our experiment is0.016�/

pHz. For an acquisition time of 4 s, we can detect

an axial displacement of 0.006��/!z ⇡ 0.04�. Therefore,by compensating stray fields and nulling the micromo-tion, we localize the ion in a region smaller than thewavelength (Lamb-Dicke regime), a fact which is essen-tial for applications in cavity-QED.

The presence of the fiber facets close to the ion mightpotentially lead to stray electric fields at the position ofthe ion due to the accumulation of charges on dielectricsurfaces [14, 15]. If left uncompensated, the ion wouldbe pushed o↵ the rf-node and hence become subject tomicromotion, resulting in line broadening and reducedcoupling to light. Charges might be created by directlaser-illumination of surfaces or in the photoionization ofatomic calcium. We minimize these e↵ects by reducingthe beam waists and recessing the end of the fibers as de-scribed above. Another source of stray fields are contactpotentials in sections of the electrodes partially coatedwith calcium.

As a sensitive probe for the presence of stray fields,we have utilized the trapped ion itself. Automaticallynulling the micromotion at intervals of 2 minutes, wetracked the compensation voltages over time after load-ing the trap. Using a model of the trap fields obtainedfrom a 3D finite element calculation, the compensationvoltages are converted to an electric field at the cen-ter of the trap which must be equal and diametricallyopposed to the instantaneous stray field. The sensitiv-ity of the measurement was 60 mV/(cm

pHz). Our data

showed stationary conditions during normal operation ofthe trap. Immediately after loading, stray fields on theorder of 1 V/cm appear, but decay exponentially at arate of 5 ⇥ 10�4s�1 on average. Since we measure all

H. Takahashi et al., New J. Phys. 15, 053011 (2013)

RF

RF GND

GND GND

endcap

RF

Ions as mechanical oscillators: prepare & control interesting quantum states.

Cavity cooling of ions would require the right parameter regime.

But we know how to simulate it now!

Interesting properties for optomechanics experiments: •  fast cavity decay •  tunable cavity interactions •  ion number: depends on trap geometry

Cavity team:

Birgit Brandstätter Bernardo Casabone

Konstantin Friebe Klemens Schüppert

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