Quantum control of 88 Sr + in a miniature linear Paul trap
Nitzan Akerman, Shlomi Kotler, Yinnon Glickman, Anna Keselman, Roee Ozeri
QQuantum control of Sr + in a miniature linear Paul trap N. Akerman, S. Kotler, Y. Glickman, A. Keselman and R. Ozeri
Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: September 4, 2018)We report on the construction and characterization of an apparatus for quantum informationexperiments using Sr + ions. A miniature linear radio-frequency (rf) Paul trap was designed andbuilt. Trap frequencies above 1 MHz in all directions are obtained with 50 V on the trap end-capsand less than 1 W of rf power. We encode a quantum bit (qubit) in the two spin states of the S / electronic ground-state of the ion. We constructed all the necessary laser sources for laser coolingand full coherent manipulation of the ions’ external and internal states. Oscillating magnetic fieldsare used for coherent spin rotations. High-fidelity readout as well as a coherence time of 2 . n = 0 .
05 and a heating rate of ˙¯ n = 0 .
016 ms − is measured. PACS numbers:
I. INTRODUCTION
Trapped ions have proven to be a valuable resourcein the field of experimental quantum information andprecision measurements. Their tight spatial localiza-tion, isolation from external perturbations and the abilityof experimenters to coherently manipulate their internaland external degrees of freedom with high fidelity, placetrapped ions among the leading technologies of quantuminformation research [1, 2].The realization of scalable quantum computation andsimulation requires the design and fabrication of smalltrap arrays. The smallest traps demonstrated to datewere built using modern micro-fabrication techniques [3–10]. Due to the technical complexity of micro-fabrication,most of the current research in this field is still con-ducted using manually assembled, mm-scale, traps. Herewe report on the manual assembly of a miniature, submm-scale, trap. Our trap has dimensions that approachthose of micro-fabricated traps but its construction re-quired only commercial parts, conventional workshop ca-pabilities and laser machining which is also commerciallyavailable.Various internal states can be used to encode a qubit,examples include optical, hyperfine and Zeeman split-ted levels [11–14]. Here we focus on a Zeeman qubitin a Sr + ion [15, 16]. We have constructed all the fre-quency stabilized lasers, that drive the various transitionsrequired for laser-cooling, state-selective fluorescence de-tection, and coherent control. One convenient propertyof the Strontium ion is that all its relevant optical tran-sition wavelengths are available using diode lasers.Using a narrow line-width laser, we perform electronshelving on a narrow quadrupole transition, and are thusable to demonstrate high fidelity qubit readout. With thesame laser we performed resolved sideband cooling anddemonstrated 95% ground state occupation of the axialvibrational mode. A heating rate of ˙¯ n = 0 .
016 ms − forthis mode is measured.Although a ground-state Zeeman qubit has practicallyan infinite life time, a coherent superposition is hard to maintain in the presence of ambient magnetic field noise.We constructed a servo system that reduces magneticfield noise and which leads to the extension of the qubitcoherence time by an order of magnitude to 2 . II. EXPERIMENTAL APPARATUSA. Trap and vacuum system
Our trap has the generic linear Paul trap configurationof four parallel rods positioned at the corners of a square,and two additional “end-cap” rods that are aligned alongthe square center, symmetrically from each side of thetrap; see Fig.1(a). The four parallel rods carry the rf anddc voltages necessary for radial confinement, whereas thedc voltages on the end-caps provide confinement in theaxial direction. The square sides length is 0 . . . . . .
65 mm respectively.Structural integrity and electric insulation are pro-vided by four laser machined alumina wafers with ad-equate holes for holding the electrodes. These aluminawafers are aligned and held together by two, 5 mm diam-eter, stainless steel rods that provide a skeleton for thetrap structure. The wafers are separated by three tita-nium spacers of 3 , , a r X i v : . [ qu a n t - ph ] M a y been threaded through the four alumina wafers.Two additional electrodes are placed below the trap,at a distance of 1 .
66 mm from the ion. One electrode isused to drive current that produces the oscillating mag-netic field needed for magnetic-dipole coupling betweenthe qubit levels. The second electrode is used for com-pensation of stray electric fields in the direction of the rfelectrodes.The trap is driven at Ω rf / π = 21 MHz with 0 . ω , rad / π = { . , . } MHz.The degeneracy between the two radial directions is liftedby a 0 . ω ax / π = 1 MHz with a voltage of 50 Vapplied to the end-cap electrodes. The deviation of thetrapping potential from a pure harmonic potential hasbeen characterized as well. The next term beyond linearin the force along the axial direction is cubic, F nl = αx .This term is measured by the study of the ion non-linearresponse to a strong, near resonance, drive. The cubicforce term coefficient is found to be α/ (2 π ) = 1 . ± . · − KgHz /m [17] SolidWorks 2006 Personal Edition (a)(b)FIG. 1: (a) A close-up drawing of our linear Paul trap. Thetrap is constructed of six tungsten rods, held in place byfour laser-machined alumina wafers. Two additional rod-electrodes are placed below the trap. One electrode is usedfor stray field compensation while the other is used for drivingan oscillating current to induce magnetic-dipole transitions.(b) An image of the trap mounted inside the vacuum cham-ber. The trap is held in place by a titanium construction.Two Sr ovens, one from each side of the trap are anchored tothe titanium construction. One of these, pointing to the trapcenter, is seen on the right side of (b)
The trap is mounted inside an octagon-shaped vac-uum chamber (Kimball physics MCF600-SphOct-F2C8) with eight 2 .
75 in. view-ports on the octagon faces andtwo 6 in. ports at the top and the bottom of the cham-ber. The side ports are used for laser access, electri-cal feedthrough and connection to the vacuum pumpswhereas the top port is designated for imaging. Threetypes of vacuum pumps are combined to achieve ultra-high vacuum. A 20 l/s ion getter pump and a Titaniumsublimation pump are connected to the main chamberthrough a 2 .
75 in. port. Inside the main chamber weplaced two non-evaporable getter strips (SAES St707)which are thermally (300 ◦ C) activated during bake-out.After bake-out, the vacuum ion-gauge was indicating apressure below its baseline reading of 4 · − mbar. Dur-ing the last two years ions were regularly trapped for pe-riods of up to a week. We never observed chemistry oftrapped-ions with residual background molecules or lossof ions from the trap, as long as they were laser-cooled.A thermal jet of neutral Strontium atoms is directedinto the trap center from either one of two Sr ovens thatare positioned approximately 2 cm from the trap cen-ter. The ovens are made of 1 mm diameter stainless-steeltubes with one end crimped. Small grains of Strontiummetal are placed inside the tubes in a dry nitrogen at-mosphere. We seal the ovens with a thin layer of Indiumbefore we expose them to room atmosphere. Resistiveheating the oven with a current of 2 A, results in the jetof neutral Strontium atoms necessary for trap loading. B. The Sr + energy levels and laser sources A single or few ions are loaded into the trap by photo-ionization. Photo-ionization is performed using a reso-nant two-photon process [18, 19]. A Sr atom is initiallyexcited to the 5 s p P level and then, by a second pho-ton, to the auto-ionizing 5 p D level, at transition wave-lengths of 461 nm and 405 nm respectively. The first isgenerated by doubling a 922 nm external-cavity diodelaser (ECDL) in a single pass through a PPKTP crys-tal (Raicol Crystals Ltd.). The crystal is mounted in athermally stabilized housing. A 100 mW of 922 nm lightgenerates an output power of 100 µ W at 461 nm. Thesecond beam of 20 mW at 405 nm, is produced using afree running laser diode (InGaN). The 461 nm (405 nm)beam is focused, at the trap center, to a waist of 50 µ mwith a typical power of 15 µ W (3 mW).Doppler-cooling and florescence detection are pre-formed on the S / → P / electric-dipole transition at422 nm. See Fig.2(a) for an energy-level diagram. SinceSr + has low laying 4 D levels, the cooling cycle has tobe closed with additional light at 1092 nm which re-pumps population from the long-lived (0 .
37 s) D / level.The cooling light is generated by a 844 nm ECDL whichis frequency doubled, producing 15 mW of light at 422nm (Toptica SHG110). The fundamental diode laser at844 nm is locked, using a Pound-Drever-Hall (PDH) er-ror signal, to a hemispherical Fabry-Perot cavity . Thecavity is built using an Zerodur spacer and has one ofits mirrors mounted on a piezo-electric stack. The cav-ity maintains long-term stability by locking it to the5 s S / → p P / transition in Rb which is 400 MHzred detuned from the cooling transition [20]. The errorsignal is generated by passing 422 nm light through aheated (120 C) Rb vapor cell and recording the deriva-tive of a saturation absorption signal. The repump lightat 1092 nm is generated by a distributed feedback (DFB)diode laser (Toptica DL100 DFB) producing 15 mW. An-other ECDL (Toptica DL100) at 1033 nm with 30 mW isneeded to repump the D / level. Both the 1092 nm andthe 1033 nm lasers are simultaneously locked to an in-vacuum Zerodur cavity with a finesse of 100 using PDHand lock-in techniques respectively. Typical frequencydrifts of the lasers due to thermal fluctuations of the cav-ity are of the order of 1 MHz in few hours.The narrow S / → D / quadrupole transition isdriven by another ECDL (Toptica DL100) emitting 10mW at 674 nm. The cavity of this laser is 10 cm longleading to a fast line-width of approximately 100 kHz in10 µ s, significantly above the expected Schawlow-Towneslimit. The ECDL is then locked to in-vacuum (10 − mbar) Ultra Low Expansion glass (ULE) cavity with afinesse of 100 ,
000 (ATFilms, [21]) using a PDH tech-nique. A laser line-width of less than 80 Hz is measuredby preforming Ramsey experiment on the ion. Thermalfluctuations cause the cavity resonance to drift on a timescale of few hours with peak to peak amplitude of 1 MHz(The lab temperature is stabilized to within 0 . ◦ C), witha typical slope of 10 Hz/s. These slow drifts are automat-ically compensated for by scanning the atomic transitionevery few minutes and continuously updating the laserfrequency by linear extrapolation. Using this technique,the laser detuning from any desired transition, can besmaller than 300 Hz at all times.All light sources pass through acousto-optical modula-tors (AOM’s), in a double-pass configuration, before theyare brought to the trap by polarization maintaining sin-gle mode fibers. The AOMs have two roles. First, theybridge the frequency difference between the frequency ref-erences and the atomic transitions. Second, they allowfor switching and control of the light intensity by con-trolling their driving rf power. The frequency sources forthe AOM’s are based on either voltage controlled oscilla-tors (VCO’s) or direct digital synthesizers (DDS’s) whichare controlled by a field programable gate array (FPGA).The use of DDS’s allows for phase and frequency controlduring a single experimental sequence.The beams are focused at the trap center to a waist of50 µ m. The schematic drawing of the laser beams config-uration is presented in Fig.2b. A magnetic field of 1 −
20G defines a quantization axis. The cooling and detectionbeams at 422 nm are combined with the two repumpingbeams at 1092 nm and 1033 nm on a dichroic beam com-biner before being focused on the ion. The beams areoriented at 45 ◦ to the trap axis and perpendicular to thequantization axis while having a projection on both of thetrap radial principle directions as well. A circularly po- larized beam at 422 nm, aligned parallel to the magneticfield, is used for optical pumping. The 674 nm beam,that drives the S / → D / quadrupole transition, isalso parallel the quantization axis. For this configura-tion, quadrupole transitions with ∆ m = ± η = 0 . (8ns) (8ns) (435ms) (390ms) B Cooling detectionπ excitation(422nm) Shelving SideBand Cooling (674nm)Optical Pumping σ ± (422nm) Photon-ionization beams (461+405)D P repumper(1032nm)D P repumper (1092nm) Trap axis (a) (8ns) (8ns) (435ms) (390ms) B Cooling detectionπ excitation(422nm)
Shelving
SideBand Cooling (674nm)Optical Pumping σ ± (422nm) Photon-ionization beams (461+405)D P repumper(1032nm)D P repumper (1092nm) Trap axis (b)FIG. 2: (a) The Sr + ion energy levels diagram and relevanttransition (in parenthesis are the levels life time). (b) Geo-metric arrangement of the various laser beams with respect tothe trap long axis and the quantization axis(=magnetic fielddirection). C. Imaging system and photons counting
Ion fluorescence at 422 nm is collected with an
N.A. .
31 objective lens (LENS-Optics). This lens wascustom designed, based on [23], to correct for the view-port spherical aberrations. A flipping mirror switchesbetween forming an image of the trapped-ions onto a elec-tron multiplying CCD (EMCCD) camera(Andor Luca),with magnification of 42 and a diffraction limited reso-lution (0 . µ m), or single photon counting by two photo-multiplier tubes (PMT) each on a different side of a po-larizing beam splitter (PBS) cube. Computer-controlledquarter and half wavelength retardation plates are posi-tioned in front of the PBS to allow for full polarizationanalysis of fluorescence photons.We measured the total photon detection efficiency bycounting detection events in the process of single photonscattering. Single photon scattering is achieved by firstapplying a cooling pulse on the S / → P / transition.After an average of 14 scattering events the electron de-cays to the meta-stable D / level. Then, a short pulseof 1092 nm light pumps the electron back to the P / state, from which it decays back to the S / state viasingle photon emission. The total photon detection effi-ciency was thus found to be 2 . · − . When both the422 nm and 1092 nm beams are applied to the ion, themaximal photon detection rate is 70 kHz. In the absenceof an ion we measure a 1 kHz photon detection rate dueto scattering of laser light from trap surfaces.Figure 3(a) shows an EMCCD image of a three ioncrystal with inter-ion separation of 5 . µ m. Fig.3(b)shows fluorescence spectroscopy of the S / → P / tran-sition based on PMT photons counting. The dip on theleft side of the spectrum is a dark resonance resultingfrom transition amplitude interference in the presence ofboth the 422 nm and the 1092 nm light. The depen-dence of the spectrum on the magnetic field is shown inthe inset. As the magnetic field is increased, the darkresonances due to different Zeeman sub-levels separateand can be resolved. III. MICRO-MOTION COMPENSATION
Displacement of the ion from the rf potential mini-mum due to stray electric fields or structural deforma-tions causes excess micro-motion, i.e. oscillation of theion at the rf trap frequency, which deteriorates coolingand detection efficiency. Once the excess micro-motionis probed it can be eliminated by proper voltage biasingof different trap electrodes in order to bring the ion tothe rf minimum. There are various methods for probingmicro-motion[25]. We implement a method that was re-cently demonstrated by the NIST ion storage group [26].We inject a small additional rf voltage at a frequencyΩ drive = Ω rf + ω ax to the trap rf resonator. Due to po-tential nonlinearities, a drive at the trap frequency, ω ax ,is produced with an amplitude that nulls at the rf po-tential minimum. The resonant driving force heats theion, causing a reduction in its fluorescence rate due to its,large, associated Doppler shifts. A typical micro-motiondetection scan is presented in fig. 4. The 2D gray scaleplots are the ion fluorescence rate as a function of the dcelectrode compensation voltage and the drive frequencyfor both increasing (left) and decreasing (right) frequencysweeps. The asymmetry and hysteretic behavior are a re-sult of the nonlinearity of the trapping potential togetherwith large amplitude of the ion motion[17].At the optimal compensation voltage of − . µ m (a) −50 −40 −30 −20 −10 0 10 200102030405060 422nm detuning [MHz] P ho t on c oun t s i n m s M agne t i c f i e l d [ G ] −80 −60 −40 −20 001020 510152025 (b)FIG. 3: (a) An EMCCD image of a three ion crystal. (b)Spectroscopic scan of the S / − P / transition. Single ionfluorescence counts in 1 ms as function of the 422 nm laserdetuning ∆ in the present of the 1092 nm repump. Thered line is a fit based on the solution of an eight-level coupledoptical Bloch equations[24]. The fit parameters are the re-pump laser detuning ∆ = −
14 MHz, the laser intensities
I/I sat = 7,
I/I sat = 0 .
6, the magnetic field on theion, B = 1 .
18 G, and the laser linewidths, γ L = 0 . and the magnetic field. Here the 1092nm repump is detuned by ∆ = −
80 MHz from resonance.As the magnetic field is increased the dark resonances due todifferent Zeeman sub-levels separate and can be resolved inthe fluorescence spectrum. resonance as a function of the compensation voltage isexpected since compensation is performed only on oneelectrode and thus slightly modified the secular frequen-cies. We have observed that the optimal compensationvoltages do not vary significantly in time. This pointsto geometrical imperfections as the source of the ion dis-placement whereas light or thermally induced charging ofdielectric surfaces seems to be negligible [35]. We foundthis method for micro-motion compensation to be experi-mentally simple as compered with the fluorescence modu-lation method (also know as rf-photon correlation) whichwe had previously employed. One drawback is the needto scan the drive frequency as the trap frequency is mod-ified by the compensation voltages and, therefore, mightmake this method more time consuming. However, thisproblem can be mitigated by first calibrating the depen-dence of the trap frequency on the compensation voltageand then adjusting the drive frequency accordantly bycomputer. Ω drive − Ω rf [kHz] C o m pen s a t i on v o l t age [ V ]
435 436 437 438 439−0.3−0.2−0.100.1 Ω drive − Ω rf [kHz]435 436 437 438 439−0.3−0.2−0.100.1 010203040 FIG. 4: Micro-motion compensation scan. Fluorescencecounts as function of compensation voltage on the DC elec-trode pair and drive frequency for both increasing (left) anddecreasing (right) frequency sweeps. The fluorescence drops(shows as dark region) when the ion experience resonancedrive. The ion is brought to the rf minimum at -0.1V forwhich no fluorescence drop is seen. The asymmetry and hys-teretic behavior are a result of the nonlinearity of the trappingpotential.
IV. SINGLE QUBIT MANIPULATION
There are several ways to encode a qubit using theelectronic states of a trapped-ion. In the Sr + ion, whichlacks a nuclear spin but has a low lying meta-stable Dlevel, two options can be used. One is encoding the qubitin the S / and D / levels which are separated by anoptical transition. The second ,which is discussed here,is encoding the qubit in the two Zeeman sub-levels of the S / ground state. A. Qubit initialization and detection
Initialization of the Zeeman qubit is performed by op-tical pumping, in which a circularly polarized laser beamparallel to the magnetic field ( σ + / − ) and in resonancewith the S / → P / transition, excites only one of theZeeman states. A 3 µ s, σ + polarized, pulse initializesthe qubit to the m = +1 / .
99. The initialization fidelity in this casedepends on the polarization purity which at low mag-netic field is limited by magnetic field fluctuations andat a high magnetic field by stress-induced birefringencein the vacuum chamber view-port. To improve on theinitialization fidelity we further preform optical pumpingusing the narrow quadrupole transition, S / → D / , inwhich spectral selectivity rather than polarization allowsfor the excitation of only one of the Zeeman states. Here,errors as small as 10 − were obtained [27].In contrast to initialization, state detection presents amuch harder challenge. Since the two qubit levels are typ-ically separated by an energy difference which is smaller than the P / excited state linewidth, direct state selec-tive fluorescence is impossible. To discriminant betweenthem, one of the qubit states is shelved to the meta-stable D / level prior to detection by resonance fluorescence.The selective shelving of only one spin state is done usingthe narrow linewidth 674 nm laser. High fidelity state de-tection, therefore, imposes stringent requirements on thelaser performance. Robust shelving can be achieved byapplying several consecutive excitation pulses to differ-ent Zeeman levels in the D / state manifold. We havedemonstrated a total initialization and detection fidelityof 0 . B. Qubit rotations and coherence
The transition between the two S / Zeeman states isdriven by resonant oscillating magnetic field. It is pro-duced by a current flowing in one of the rod electrodesthat are positioned 1 .
66 mm away from the ion; see Fig1(a). The electric current source is an amplified DDSwhich is impedance matched to a few Ohms by a Baluntransformer. The oscillating current has an amplitude of100 mA at a frequency range of 1 −
30 MHz, and givesrise to a Rabi frequency around 50 kHz. Rabi oscillationsbetween the two qubit levels are shown Fig.5.The coherence of a Zeeman superposition is quicklylost in the presence of magnetic field noise. To protectour qubit from decoherence due to magnetic field noise,we have designed and built a servo system that stabilizedthe magnetic field on the ion and reduced noise compo-nents, mainly at the power line frequency (50 Hz) and itsharmonics. We were thus able to reduce these magneticfield noise amplitudes to the few µ G level. The coher-ence time of the qubit, as was measured in a Ramseyexperiment, is 2 . < > µ G) magnetic fieldnoise, as suggested by the un-smooth appearance of thefringes at Ramsey times longer than ∼ µ s. By furtherusing dynamic decoupling methods our qubit coherencetime was extended up to 1 . V. COOLING AND HEATING RATE
Cooling the ion to a low mean vibrational quantumnumber, ¯ n , is useful for both quantum information andprecision measurement applications. Doppler laser cool-ing is a simple and efficient mechanism for cooling buttypically yields ¯ n > + , raises additional complications on reaching theDoppler limit. The atomic line profile on which we pre-form Doppler cooling is shown in Fig.3(b). We slightlydetune the 1092 nm repump laser red of resonance to cre-ate a pronounced dark resonance (seen as a deep in the µ s] | ↑ 〉 s t a t e popu l a t i on FIG. 5: Rabi Oscillation between the two state of the Zeemanqubit. The transition is driven by an oscillating magnetic fieldin resonance with the qubit Zeeman splitting. The red line isa fit to a sin with decaying amplitude µ s] | ↑ 〉 s t a t e popu l a t i on FIG. 6: Qubit coherence measured is a Ramsey experiment.A small detuning is responsible for the oscillation in the pop-ulation. A coherence time of 2.5ms is found from an expo-nential fit to the decaying envelope. The disorder in the oscil-lation after 500 µ s is related to low frequency magnetic noisethat does not average in the time needed to take a single pointin the figure. graph). This results in a steeper slope on which we de-tune the cooling laser to reach a lower final temperature.The ion temperature can be measured in several ways[29–31]. The spectrum of the narrow S / → D / tran-sition has resolved sidebands. The ratio of red to bluesideband excitation probability can be used for thermom-etry, when ¯ n ≤
1. However, after only Doppler cooling¯ n (cid:29)
1. Instead, a simple way to estimate the ions’ tem-perature is to measure the decay of the carrier Rabi oscil-lation due to the ion motion. The carrier Rabi frequencydepends on the occupation of the vibrational mode asΩ n ∝ − η n , up to second order in η . Hence, a thermaldistribution leads to different Rabi frequencies which re-sult in oscillation with a decaying envelope. Fig.7 showsRabi oscillations on the carrier transition with an averageRabi frequency of Ω / π = 180 kHz. Fitting the data to a thermal distribution model, shown by the red solid line,yields ¯ n = 12, consistent with the Doppler cooling limit. µ s] D / l e v e l popu l a t i on FIG. 7: Rabi oscillation on the S / → D / carrier transi-tion after Doppler cooling. The red line is a fit to a thermaldistribution model which agree with an average vibration oc-cupation number ¯ n = 12. This value is consistent with theDoppler limit. High fidelity qubit operations require ¯ n below theDoppler limit and close to the ground state. This isachieved by applying sideband cooling on the narrow S / → D / transition. Since the meta-stable D / level is long lived, it needs to be quenched by the 1033nm repump laser. By properly choosing the saturationparameter of the quenching laser, continuous sidebandcooling can be preformed on the, effectively broadened, S / → D / transition. For optimal performance, therepump laser intensity has to be modified as the ion iscooled, or the cooling time has to be increased [32]. Inorder to shorten the total cooling time, we preform suchcontinuous cooling for only 2 ms, after which the ion iscooled to ¯ n (cid:46)
1. Since most of the population that isnot in the ground state is in n = 1, we then apply twodiscrete population transfer pulses on the red sidebandfollowed by a 1033 nm repump pulse. Figure 8 showsspectroscopy of the axial red sideband (RSB) and bluesideband (BSB) after (a) Doppler cooling only. Here thesideband excitation pulse duration is 15 µ s (b) Resolvedsidebands cooling and a pulse duration of 100 µ s. With-out sideband cooling, the population transfer on the twosidebands is similar, while after sideband cooing, the RSBalmost completely disappears. For a thermal distributionmodel the probabilities to excite the RSB and BSB aregiven by, P { r,b } ex = (cid:88) n P th ( n ) sin (Ω { r,b } n t ) . Here P th ( n ) = n +1 ( ¯ n ¯ n +1 ) n is the thermal distributionand Ω { r,b } n = η Ω (cid:112) ( n + { , } ) are the sidebands Rabifrequencies. A fit to the measured data, shown by theblue and red solid lines in Fig.8, indicates an averageoccupation number of ¯ n = 0 .
05. The small backgroundin the measurement is mainly due to carrier excitationresulting from fast frequency noise in the laser spectrum[33]. We find this combined protocol of continuous andpulsed cooling to be efficient and robust without the needto dynamically optimize the quenching laser intensity. −1.05 −1 −0.95 −0.900.20.40.60.8 RSB 0.9 0.95 1 1.05BSB−0.99 −0.98 −0.97 −0.9600.20.40.60.8 Detuning [MHz] D / l e v e l popu l a t i on FIG. 8: Sideband spectroscopy and ground state cooling. (a)just after Doppler cooling and (b) after Doppler and sidebandcooling. After sideband cooling, the imbalance between thered and blue sidebands indicates 95% occupation of groundstate
We determined the trap heating rate by varying thetime between resolved sideband cooling and the measure-ment of ¯ n . Figure 9 shows the excitation probability ofthe RSB and BSB as function of this delay time. Assum-ing a thermal distribution and a constant heating ratewe fit our data. The fit, shown by the red and blue solidlines, indicates a heating rate of ˙¯ n = 0 .
016 ms − . Us-ing this measurement we calculate the electric field noisespectral density, S E = 4 m Sr (cid:126) ω ax ˙¯ n ax /e [34]. Assuming,1 /f noise, a good quantity to compare between traps ofsimilar dimensions is ωS E . We measure ωS E = 1 . · − (V/m) . This value is consistent with heating rate mea-surements in traps of similar size [9]. During more thentwo years of operation we did not observe a significantchange in the heating rate. VI. SUMMARY
In conclusion, we have constructed various basic build-ing blocks necessary for quantum information related ex- periments with trapped ions. A, sub-mm scale, trap wasmanually assembled using conventional and laser machin-ing and commercially available parts. The trap harmonicfrequencies are in the MHz range, allowing for operationin the Lamb-Dicke regime. All the necessary laser sourceswere constructed and frequency locked to stable refer-ences. A qubit was encoded into the Zeeman splittingof the electronic ground-state of a single Sr + ion. Anarrow linewidth laser enables electron shelving on the S / → D / transition prior to state selective fluores- D / l e v e l popu l a t i on FIG. 9: Heating rate measurement. Probability to excite theaxial RSB (circles) and BSB (diamonds), after sideband cool-ing, and as a function of time. The solid lines are a fit to amodel, assuming a thermal distribution and a constant heat-ing rate. The fit also includes a small constant offset dueto the off-resonance excitation of the carrier. The measuredheating rate is ˙¯ n = 0 .
016 ms − . cence detection. Using the red sideband of this transitionwe have also demonstrated ground-state cooling. Coher-ent transitions between the qubit states are performedby oscillating a current through an additional electrodewhich is integrated in the trap structure. By servo sta-bilization of magnetic field noise, we demonstrate a longqubit coherence time. [1] Leibfried, D., Blatt, R., Monroe, C., and Wineland, D. Rev. Mod. Phys. (1), 281–324 Mar (2003).[2] Wineland, D. J. and Leibfried, D. Laser Phys. Lett. ,175.[3] Stick, D., Hensinger, W., Olmschenk, S., Madsen, M.,Schwab, K., and Monroe, C. Nature Physics (1), 36–39(2005). [4] Seidelin, S., Chiaverini, J., Reichle, R., Bollinger, J. J.,Leibfried, D., Britton, J., Wesenberg, J. H., Blakestad,R. B., Epstein, R. J., Hume, D. B., Itano, W. M., Jost,J. D., Langer, C., Ozeri, R., Shiga, N., and Wineland,D. J. Phys. Rev. Lett. (25), 253003 Jun (2006).[5] Leibrandt, D., Labaziewicz, J., Clark, R., Chuang, I.,Epstein, R., Ospelkaus, C., Wesenberg, J., Bollinger, J., Leibfried, D., Wineland, D., et al.
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