Typeset with jpsj2.cls <ver.1.2.2b> Letter Coherence of Spin-Polarized Fermions Interacting with a Clock Laser in a Stark-Shift-Free Optical Lattice Masao Takamoto1,2 and Hidetoshi Katori1,2∗ 9 1Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 0 0 Bunkyo-ku, 113-8656 Tokyo, Japan 2 2CREST, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, 332-0012 Saitama, Japan n a J We investigated the coherence of spin-polarized 87Sr atoms trapped in a light-shift-free 2 1 one-dimensional optical lattice during their interaction with a clock laser on the 1S −3P 0 0 transition. Collapses and revivals appeared for more than 50 Rabi cycles, attributed to the ] h thermal distribution of discrete vibrational states in the lattice potential. The population p - oscillation in the clock states lasted more than 1 s, demonstrating high immunity from m decoherence.ThislongatomiccoherencesuggeststhefeasibilityofPauliblockingofcollisions o t in optical clock excitation. a . s KEYWORDS: atomic coherence, spin polarization, Fermi statistics, cold collisions, Rabi oscil- c i lation, collapses and revivals, optical lattice, atomic clock s y h p [ Unprecedented controllability over system parameters makes ultracold atoms in optical 1 lattices apowerfultool for investigating quantum computations1) and quantumsimulations of v 6 stronglycorrelatedsystems.2,3) Particularattentionhasbeenpaidtofermionicatomsconfined 2 5 inoptical lattices, asthey areexpected tosimulate theHubbardHamiltonian thatconjectures 1 high-temperature superconductivity in cuprates.4) Moreover, the noninteracting nature of . 1 0 fermionsoriginatingfromthePauliprincipleprovidesthesystemwithpotentialapplicationsin 9 qubitregisters5,6) and a variety of precision measurement, such as interferometers7) and high- 0 v: performanceatomic clocks.8) Allof theseapplications requireaminimumatomicdecoherence, i X preferably including atom-laser interaction processes to allow the optical manipulation of r a atomic states. In this Letter, we report the coherence properties of spin-polarized 87Sr atoms interacting with resonant light. Atoms were trapped in a one-dimensional (1D) optical lattice operated at the “magic wavelength,” where the light shift perturbation is canceled out in the relevant transition.9) The Lamb-Dicke confinement provided by the lattice potential allowed the sys- tem to be free of atomic-motion-induced decoherence. We investigated Rabi oscillations on the 1S (F = 9/2)−3P (F = 9/2) clock transition at 429 THz. Collapses and revivals remi- 0 0 niscent of those described by the Jaynes-Cummings model10,11) were observed up to 50 Rabi cycles. Furthermore, the oscillations of the atomic population lasted more than 1 s, demon- ∗ E-mail: [email protected] 1/10 J. Phys. Soc. Jpn. Letter (a) z Optical y pumping B : bias field x 0 O E E E L L p Probe Lattice g ¢ (b) mg 3 g ¢ 2 n=3 g ¢ 1 n=2 g ¢ U 0 n=1 0 q ℏn = mgl( /2)sqin n=0 B L Fig. 1. (Color online) (a) Ultracold 87Sr atoms with a temperature of a few µK were loaded into a one-dimensional (1D) optical lattice operated at the magic wavelength of λL = 813.42 nm. A circularly polarized laser optically pumped atoms into the 1S0(F = mF = 9/2) stretched state. (b) The energies of atoms in a 1D lattice show the Bloch bands. This bandwidth can be reduced bytiltingthelatticeθ withrespecttothehorizontalplanetointroducethegravitationalpotential difference ~νB between adjacent sites. strating a remarkably long coherence of the atom-laser system. The origins of decoherence mechanisms were investigated experimentally and numerically. The observed long atomic co- herence suggests that the Pauli blocking of collisions is feasible for 1D optical lattice clocks with spin-polarized fermions.12) Inprecisionspectroscopy, bettersignal-to-noise ratios areobtained byincreasingthenum- ber of atoms, hence atomic density. However, this is often encountered with correspondingly larger atomic interactions that result in problematic systematic uncertainties. Thus, some state-of-the-art atomic clocks such as Cs fountain clocks13) and ballistic Ca clocks,14) which use ultracold bosonic atoms, suffer from collisional frequency shifts. An ultracold fermionic system is expected to be advantageous in such applications, since the mean field energy due to s-wave interaction disappears15) as a consequence of the Pauli principle, and p-wave in- teractions are suppressed at ultralow temperatures. The application of fermionic 134Cs was suggested to eliminate the cold collision shift of Cs microwave clocks.8) The disappearance of the mean field energy was demonstrated experimentally in an RF transition with quantum degenerate 6Li atoms held in a dipole trap.16) 2/10 J. Phys. Soc. Jpn. Letter To apply the Pauli blocking of collisions in atomic clocks, the fermionic system needs to be uniformly excited so as to preserve fermionic identity. At an RF wavelength, as the spatial spread ∆x of laser-cooled atoms can be much smaller than the relevant transition wavelength λ , the Lamb-Dicke condition ∆x ≪ λ is well satisfied; therefore, the system is uniformly 0 0 excited without being significantly affected by Doppler shifts. However, its application to optical transitions remains technically challenging,17,18) because the much shorter transition wavelength λ of less than a µm makes the Lamb-Dicke condition far more severe. We show 0 thatoptical lattices operated at themagic wavelength offer anideal tool forinvestigating such possibilities. We consider a system where spin-polarized fermionic atoms are trapped in an array of disk-shaped potentials (1D optical lattice) formed by a standing wave of a Gaussian beam, as shown in Fig. 1. Atoms are excited on the |Si ⊗ |ni → |Pi ⊗ |ni electronic-vibrational transition by a probe laser with a wave vector k = (k ,k ,k ). For an atom at a position p x y z r= (x,y,z) in the vibrational state n = (n ,n ,n ) with trapping frequencies (ν ,ν ,ν ), the x y z x y z effective Rabi frequency for the carrier (n → n) component is given by19) Ωeff = Ωel Y |hnj|exp(ikjj)|nji|, j=x,y,z where Ω is the Rabi frequency for the electronic transition. When k is nearly parallel to el p k (k xˆ) of a lattice laser so that k y ≈ k z ≈ 0 holds, the effective Rabi frequency Ω is L y z eff determined solely by the x-motional state n= n as x Ω ≈ Ω exp(−η2/2)L (η2). (1) n el x n x Here, η = k h/(2mν )/(2π) is the Lamb-Dicke parameter with a mass m and a Planck x xp x constanth.L (x)isthen-thorderLaguerrepolynomial.Asaconsequenceof then-dependent n vibrational coupling strength, the thermal occupation over n states leads to collapses and revivals of Rabi oscillations. The experimental setup is similar to that described elsewhere.17) Ultracold fermionic 87Sr atoms with a temperature of a few µK20) were loaded into a one-dimensional (1D) optical lattice operated at a magic wavelength λ = 813.428 nm with a wave vector ±k (k xˆ) and L L an electric field E (k zˆ). Typically, 104 atoms were trapped in about 500 lattice sites spread L over 200µm. By focusing the lattice laser with an intensity of 100 mW into an e−2 beam radius of 30 µm, the trap frequencies and lattice depth were ν ≈ 47 kHz and ν ≈ 300 Hz, x y,z and U ≈ 45E , respectively, with E = (h/λ )2/2m the recoil energy. The lattice laser 0 R R L was directed at an angle 0 ≤ θ ≤ 20 mrad with respect to the horizontal plane. This gave a gravitational potential difference ~ν = mg(λ /2)sinθ between adjacent sites, which reduced B L ′ the Bloch bandwidth γ of the optical lattice to γ . n n A circularly polarized (σ+) laser resonant to the 1S (F = 9/2)−3P (F = 9/2) transition 0 1 3/10 J. Phys. Soc. Jpn. Letter was applied along a bias magnetic field B (k zˆ) = 10 mG to optically pump atoms into the 0 outermost 1S (F = m = 9/2) ≡ |Si state. Rabi oscillations were observed by exciting atoms 0 F to the long-lived 3P (F = m = 9/2) ≡ |Pi state with a lifetime of 150 s. The probe laser 0 F was linearly polarized E (k B ) with a linewidth of about 10 Hz at λ = 698 nm. Its wave p 0 p vector k was parallel to k within 2.5 mrad21) to reduce the Lamb-Dicke parameter for p L the radial direction η < 1× 10−2. Consequently, the effective Rabi frequency Ω for the y,z n |Si⊗|ni→ |Pi⊗|ni transition is approximated using eq. (1) with the Lamb-Dicke parameter η ≈ 0.31. Hereafter, |ni denotes the vibrational states of the x-motion. x BeforeobservingRabioscillations, weappliedastatepurificationprocesstoachievehigher spinpolarization,as wellastodepopulateatoms inthehighern(≥ 2)vibrational states.First, atoms were excited on the |Si → |Pi clock transition by a 1-ms-long probe pulsewith an area of π. The remaining atoms in the |Si state, which occupied unwanted Zeeman substates m 6= 9/2 or higher vibrational states, were heated out of the lattices by shining a resonant F laser on the 1S (F = 9/2) − 1P (F = 11/2) transition. Then, atoms in the |Pi state were 0 1 pumped back to the |Si ground state by the same probe pulse as above. Rabi oscillations were measured by determining the atom excitation probability as P(τ ) = N /(N +N ), after applying the resonant probe laser for a duration of τ , where R P S P R N andN werethenumberofatomsinthe|Siand|Pistates,respectively.Thelaser-induced S P fluorescence (LIF) intensity I = κN , where κ gave photon detection efficiency per atom, S S was measured by shining a laser nearly resonant to the 1S −1P transition. After throwing 0 1 the atoms in the |Si state away, another π probe pulse was applied to deexcite the atoms in the |Pi state to the |Si state, where the LIF intensity I (= κN ) was measured in the same P P manner as before. Figure 2(a) shows P(τ ) as a function of the duration of the probe laser with Rabi fre- R quency Ω /2π ≈ 500 Hz for atoms in the n = 0 state. During the Rabi measurements that 0 took about 1 s for each data point, probe laser frequency was stabilized to within a few Hz by alternately referencing the |Si → |Pi transition17) every 2 s and its intensity was stabilized to within 1 % to keep Rabi frequency constant. Collapses and revivals of Rabi oscillations occurred at ∼ 9×(2j−1) ms and ∼ 9×2j ms with j = 1,2,3,..., respectively. The envelope oftheRabioscillations was given bythebeatnote|Ω −Ω |of atoms inthen = 0and1vibra- 0 1 tional states. Figure 2(b) shows a close-up view of the initial oscillations at Ω /2π ≈ 250 Hz. 0 In the first π-pulse excitation at τ = 2 ms, more than 98 % of the atoms were excited on the R clock transition, demonstrating that the purification process produced a high degree of spin polarization. Although the Rabi oscillations started to jitter for τ ≥ 100 ms, which was roughly our R laser coherence time, the population oscillation lasted more than 1 s, as shown in Fig. 3(a). This is because an ensemble of atoms that constituted each data point accumulated the 4/10 J. Phys. Soc. Jpn. Letter 1.0 (a) (b) yy 0.8 tt ii ll ii bb aa bb 0.6 oo rr pp n n oo 0.4 ii tt aa tt ii cc 0.2 xx EE 0.0 0 50 100 150 0 5 10 15 Time (ms) Fig. 2. (Color online) Rabi oscillation of atoms on the 1S0(F = mF = 9/2)−3P0(F = mF = 9/2) clock transition. (a) Collapses and revivals were observed up to 100 ms. (b) For the first Rabi excitation at 2 ms, more than 98 % of the atoms were excited, demonstrating a high degree of spin polarization. The filled circles indicate measured points; each point required a measurement time of about 1 s. same probe laser phase, even if it were jittery, as long as no inhomogeneous perturbations came in. This long population oscillation demonstrated magic lattices’ high immunity from decoherence. Here, to realize nearly single-component Rabi oscillation with an n = 0 state, the atom population at the n ≥ 1 vibrational levels was reduced by gradually lowering the lattice depth down to U ≈ 30E .22) Truncated by the gravitational potential,23) the radial 0 R potential depth was as small as 10E , which was crucial for selecting atoms with low radial R velocities to moderate dephasing due to radial motion. To demonstrate the dephasing by inhomogeneous perturbations, the probe laser beam was misaligned with respect to the lattice laser by about 5 mrad so as to be sensitive to the radial motion. We observed a rapidly decaying Rabi oscillation, as shown in Fig. 3(b). Here, the Lamb-Dicke parameter for the radial direction increased to η ≈ 2 × 10−2 and y the approximation in eq. (1) was no longer applicable, as the vibrational coupling strength exp(−η2/2)L (η2) severely affected each atom’s phase evolution. A much longer waiting time y n y beforerevival would beexpected, since radial vibrational states k T/hν ∼102, which are far B y more than that discussed in Fig. 2, were contributed to the dephasing of the Rabi oscillation. 5/10 J. Phys. Soc. Jpn. Letter 1.00 (a) 0.75 yy tt lili 0.50 ii bb aa 0.25 bb oo rr 0.00 pp 00 2255 5500 7755 110000 11000000 11002255 11005500 nn 1.00 (b) oo ii tt aa 0.75 tt ii cc 0.50 xx EE 0.25 0.00 0 25 50 75 100 125 150 175 Time (ms) Fig. 3. (Color online) Amplitude decay of Rabi oscillations. (a) Although the phase of the Rabi oscillationstartedtojitter afteralasercoherencetime ofabout100ms,thepopulationoscillation continued for more than 1 s. (b) By introducing inhomogeneity in atom-laser interactions, the population oscillation decayed rapidly and approached P ≈0.5. The observed population decay may also be affected by a finite trap lifetime, as well as lattice photon scattering time. At a vacuum pressure of 5×10−9 Torr, the atom trapping lifetimesforthe|Siand|Pistatesweremeasuredtobeτ = 1.4sandτ = 0.5s,respectively. S P Thesetraplifetimes wouldbelimited byglancing collisions withbackground gases.Acollision loss rate nearly three times larger for the 3P state than for the 1S state may be responsible 0 0 for extra collision channels, such as energy pooling and fine state changing collisions in the 3P manifolds.24) Lattice photon scattering lifetimes were estimated to be τ = 12 s and J LS τ = 1.1 s for a lattice with U = 45E . LP 0 R Wesimulatedatomicpopulationdynamicsusingadensitymatrixρ(n)foranatomtrapped in the n-th vibrational state of the lattice potential. In optical Bloch equations (OBEs), we applied measured trap decay rates of τ−1 and τ−1 as the longitudinal decay rates for S P ρ (≡ |SihS|) and ρ (≡ |PihP|), respectively. The transverse decay rate β was given by SS PP n β = γ +1.9γ′ + 1 τ−1+τ−1 , n (cid:0) L n(cid:1) 2 (cid:0) S P (cid:1) where γ = τ−1+τ−1 is the photon scattering rate for the lattice and γ′ is the Bloch band- L LS LP n ′ width for the n-th vibrational state. The effective bandwidth of 1.9γ was derived assuming n that a uniformly populated Bloch band was excited by a probe laser momentum of ~k .25) p 6/10 J. Phys. Soc. Jpn. Letter 1.0 yy tt ii ll 0.8 ii bb aa bb oo 0.6 rr pp nn ê oo 0.4 ii tt aa tt ii cc 0.2 xx EE 0.0 0 50 100 150 200 H L Time (ms) Fig. 4. (Color online) Excitation probability P (t) was calculated for atoms trapped in a lattice OBE with a depth of U0 = 45ER and thermally distributed in n = 0, 1 states as measured in the plot in Fig. 2(a). Figure 4 shows P (τ ) = hρi /(hρi + hρi ) obtained for the experimental pa- OBE R PP SS PP rameters used in Fig. 2. The Bloch bandwidth for the 45E deep lattice was calculated to R be γ /2π = 0.75 Hz and γ /2π = 32 Hz. As the tilt angle θ ≈ 20 mrad used in the cor- 0 1 responding measurement gave marginal values between the Bloch state (ν = 0) and the B ′ Wannier-Stark state (ν > γ ), we assumed an asymptotic bandwidth γ derived experimen- B n n tally26) as γ′ ≈ γ / 1+(ν /0.12γ )2 . We calculated the thermal average of the signal by n n (cid:2) B n (cid:3) nm hρi = p (n)ρ(n), where p (n) was the thermal occupation for the n-th vibrational state P th th n=0 for the atomic temperature of 2 µK and n = 1 was the truncation of vibrational states m provided by the state purification process. The result showed that the contrast of collapses ′ and revivals decreased with the Bloch bandwidth γ . Finally, Rabi oscillations with n = 0 1 atoms survived. In this simulation, in order to fit the slow decay of the Rabi envelope, we includedtheinfluenceof radialmotion by setting η = 5×10−3,whichreasonably agreed with y our experiment. A slight reduction in oscillation amplitude in Fig. 2(a) may be attributed to other decoherence mechanisms, such as interstate atomic collisions. In summary, we investigated the atomic coherence of spin-polarized fermions trapped in a 1D magic wavelength optical lattice during their interactions with a clock laser. The thermal population distribution among a few axial vibrational states showed collapses and revivals for more than 50 Rabi cycles. More interestingly, by carefully removing inhomogeneous atom- 7/10 J. Phys. Soc. Jpn. Letter laser interactions, we observed oscillations of the atomic population for more than 1 s, which was beyond our laser coherence time. This long atomic coherence, which was limited by residual gas collisions and lattice photon scattering, suggested the feasibility of applying the Pauli blocking of collisions in spin-polarized 1D optical lattice clocks with fermionic atoms, e.g., 87Sr,17) where relevant p-wave centrifugal barriers of 96 µK (74 µK) for the 1S (3P ) 0 0 state17,27) are significantly higher than the atomic temperature of a few µK. For a longer π-pulse up to 1 s, time axes of our results such as those shown in Fig. 2, can be extended by simply reducing Ω in Eq. (1), which will allow nearly unit excitation for the first π pulse. el To meet the necessary transverse relaxation time, a vacuum pressure of 10−11 Torr and a shallow lattice of 5E oriented vertically25) should be used. 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