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									Development of quantum gates and quantum state manipulation with Ca+ ions

        Graduate School of Engineering Science, Osaka University
              S.Urabe, U.Tanaka, K.Toyoda, R.Yamasaki

1.Cooling to the motional ground state
     Single ions in a spherical Paul trap, Two ions in a linear trap
2.Qubit using stimulated Raman transitions between matastable states of 40Ca+
     High resolution Raman spectroscopy,
     Coherence time measurement of matastable Raman qubits
3.Trapping of a rare odd isotope 43Ca+
4.Surface traps for large-scale integration (research collaboration with MIT)

       Staffs S.Urabe, U.Tanaka, K.Toyoda
       CREST Researcher R. Yamazaki
       Graduate school students
       D3    H.Sawamura
       M2    K.Kanda,H.Shiibara, H.Iwata, R.Naka, R.Kitaike
       M1 H.Inoue, T.Ujimaru, K.Nagano, S.Haze, Y.Hata
             Cooling to the motional ground state

Initialization of bus bits for ion trap quantum
computing
     (Cirac-Zoller gate, Molmer&Sorensen gate etc)


Laser cooling
 Doppler pre-cooling ・・・ ~1mK
 (ion strings, confinement in Lamb-Dicke region)
    elimination of excess micromotion
 Sideband cooling・・・cooling to the motional ground
                state (n~0)
    electric quadrupole transitions, Highly stable laser


Evaluation of the final quantum number
   Measurement of height of motional sidebands
   in electric quadrupole transitions
Experimental setup for laser cooling of single 40Ca+ ions




                 Spherical rf trap
                    size: r0=0.6mm
                   rf excitation: 20MHz
                   amplitude~200V
                    Doppler pre-cooling
Motional sideband spectrum (729nm) of a single ion in the Lamb-Dicke regime




          ωr      ωr
     ωz                ωz



          T ≈ 4mK                              Near Doppler Limit
                                 Ground-state cooling
                                         2P
  Steps toward motional                    3/2                       854nm
  ground state                           2D                              729nm (red sideband)
                                           5/2
                                                                                    ω0-ωv

       Doppler cooling                                                     l3>
                                         2S
                                                             l2>
                                           1/2
                                                       l1>
       with S1/2 - P1/2 - D3/2                   l0>




   Zeeman pumping of
   ground state populations


Sideband cooling
 with quenching laser at 854nm
 using closed cycle

  2S           2
  1/2(m=±1/2)→ D5/2(m=±5/2)
                                                                   300mW at trap
                                                                   Beam radius ω0~30µm
 →2P3/2(m=±3/2)→2S1/2(m=±1/2)                                      →carrier Rabi freq. Ω0 ~ 2π×1.9 MHz
                                                                     sideband Rabi freq. > 2π×100 kHz

                                                                   cf. Freq. fluctuation ~ 2kHz
   Results of ground-state cooling of axial motion of single ions
         Red sideband                 Blue sideband
           S-∝<n>                      S+∝ (<n>+1)             Blue points:
                                                               after Doppler cooling
                                                               Red points:
                                                               after sideband cooling


                                                           S- : S+ = <nz> : (<nz>+1)
                                                           ~ 0.003 : 0.33
                                                           ⇒Final quantum number :
                                                           <nz> ~0.01


Optimization of sideband cooling: master equation analysis
  External heating rate, linewidth of cooling laser
     → Optimum value for minimum <n>
            intensity and detuning of 854nm laser (Γeff)
            intensity of 729nm cooling laser (ηΩ20)

         H.Sawamura, K.Kanda, R.Yamazaki, K.Toyoda, S.Urabe, in preparation
                                      Three segmented linear trap

                                            Rf voltage: 400V
                                            Rf frequency: 27MHz
                                            DC voltage: 200V




COM mode                                                     Sideband spectrum of two ions
               2κQU z
 ② ωz =                    ≅ 2π × 900kHz
                mz 0
                       2                                                     carrier
                           QκU 0
                       2
             Q 2V0
 ③ ωr   =
            2m 2 Ω 2 R04
                         −     2
                            mz 0
                                 ≅ 2π × 2.14 MHz

Stretch mode

  ④ ω st = 3ω st ≅ 2π × 1.56 MHz
Rocking mode
                              1
  ⑤ ω rock = (ω r − ω z )             ≅ 2π × 1.94 MHz
                2     2           2
                 Matastable-states qubit of 40Ca+

          Candidates of qubit states in Ca+ ions
            1. groud state (S1/2) - metastable state (D5/2) in 40Ca+
            2. Hyperfine states in 43Ca+

Novel qubit using 40Ca+: metastable states D3/2-D5/2
  qubit transitions: Raman transitions
   Manipulating qubit transitions by using
         phase-locked 850nm and 854nm lasers.

     High two photon Rabi frequency
      Rabi frequency of Raman transitions
      Ω∝Ω1Ω2/2Δ
       Ω1,Ω2 : rabi frequency of E1 transitions
       Δ:detuning
     Light sources
       The effective linewidth is determined by the stability of
            microwave source.
          High resolution stimulated Raman spectroscopy
                       of the D-D transition
                                                              Observation of D-D transition
     Energy   levels (measured from 4s 2S1/2)                 Ba+ ion(12.48μm)
                                                              A.A.Madej et al, Phys. Rev.A,
                3d 2D3/2: 13,650.19 cm-1                      45,1742,1992
                3d 2D5/2: 13,710.88 cm-1
               from NIST atomic spectra database;
                 Edlen, B., and Risberg, P. (1956), Ark. Fys. 10, 553:
                 Obtained using hollow-cathode lamp and spectrometer

                                                             854nm             850nm
Splitting between 2D3/2 and 2D5/2
        Only known from the difference
        with 0.01 cm-1 ~ 300 MHz inaccuracy


    Dn = 1.8194・・ THz
      ~ 6.252 GHz × 291 lines + 1×102 MHz
        fm (mod. freq.     index    fb (the smallest of
        of opt. comb)                the beat frequencies)
Experimental setup of Raman spectroscopy with single ions
   Result of Raman spectroscopy: Copropagating carriers
           Wide scan over the D3/2-D5/2 transition

                                                         mJ=1/2
                                                         ⇒mJ’=1/2




                                     B ~ 2.8 G


R. Yamazaki, T.Iwai, K.Toyoda, S.Urabe, Opt. Lett. Vol.32, No.15, 2085, 2007
     Results of high-resolution Raman spectroscopy




R.Yamazaki, H.Sawamura, K.Toyoda, S.Urabe, to be published in Phys. Rev. A
      Measurement of coherence time of internal states
      for realization of stimulated Raman quantum gate

Ramsey spectroscopy: Identification of the transverse relaxation rate


                   Pulse width
                   ~100µs
                   Pulse separation
                   ~400µs


                                                Decay of the off-diagonal
                                                density matrix element
                    Pulse width
                    ~100µs
                    Pulse separation
                    ~1000µs

                                       │2ρ12│



                                                  Pulse separation [ms]
                Quantum gates using matastable-state qubits of 40Ca+
 Experiments toward realization of quantum gates
      initialization of internal states→Adiabatic Rapid Passage(ARP)
      Laser pulse control, state tomography etc.
                                                                                            Two pairs of Raman beams
 Two qubits gate                                                            r      r         r
   Mølmer & Sørensen gate                                            k z = (k L1 − k L 2 ) ⋅ z
                                                                        z 0 = n(2π / k z )
                                                                                        r              z0                     r
  A..Sørensen & K. Mølmer, Phys. Rev. A,
   62,022311,2000.                                                                      kL2                                   k L1
  P.C.Haljan et al., Phys. Rev. A,
  72, 062316, 2005
                                                                                                                          r
                                                                       ω2 ≡ ωl′1 − ωl′2 = ω0 − δ ′            ω1 ≡ ωl1 − ωl 2 = ω0 + δ ′
                                                                                                                          e
                                                                                       ω0
            ψ (t) = U (t ) ψ (0)
                     ˆ                                                                                                    g
                                                                                            ν
• stretch mode, σ x - diagonal basis ↑ x ↑ x , ↓ x ↓ x , ↑ x ↓ x , ↓ x ↑ x                               e − iΦ ↑ x ↓ x
                                                                                                                              p




U (t ) = ↑ x ↑ x ↑ x ↑ x + ↓ x ↓ x ↓ x ↓ x
 ˆ
                                                                                                                Φ                      Φ
                                                                                                                                                   z
                   + e −iΦ D(α ) ↑ x ↓ x ↑ x ↓ x + e −iΦ D(−α ) ↓ x ↑ x ↓ x ↑ x
                           ˆ                             ˆ

   α (t ,ν − δ ′) = α 0 (1 − e   i (ν −δ ′ ) t
                                                 ), Φ (t) = α {(ν − δ ′)t − sin(ν − δ ′)t}            Ωη                          e − iΦ ↓ x ↑ x
                                                                                                 α0 = 0
                                                              2
                                                              0
                                                                                                     ν −δ ′
             Selective trapping of an odd isotope 43Ca+
                                                                            Mass     Abunda
                                                                           number    nce(%)
    Hyperfine qubit: 2S1/2 (F=4,mF=0)⇔(F=3,mF=0)                             40        96.9
         very long lifetime                                                  42       0.647
         magnetic-field independent transitions
                                                                             43       0.135
                                                                             44        2.09
                                                                             46       0.004

                                Continuum                                    48       0.187
   Photo-ionization
                                        390 nm           Isotope-selective loading using
                          4s4p 1P1                       isotope shifts in the 1S0-1P1
   423 nm        390 nm                  423 nm          transition

        Ca beam           4s2 1S0                    Merit: No charging of insulating parts
        from oven                                            Reduction of unwanted material
                                                               sputtered onto trap electrodes


First step                                Second step (390nm)
(423nm)                                    Incoherent light is available

                                          U. Tanaka, H. Matsunishi, I. Morita, S. Urabe,
                                          Appl. Phys. B 81, 795 (2005)
      Evaluation of isotope selectivity using isotope shifts of
              1S -1P transitions (423nm)
                0   1

The ratio, R, of even isotope population to 43Ca population in the 1P1 state
is estimated when the laser frequency is tuned to resonance of 43Ca.
       Voigt profile is assumed for the evaluation (natural width: 35 MHz)

Dependence on Doppler widths                       Dependence on laser intensity
 (Laser intensity :0.15 mW/mm2)                  (Doppler width :45 MHz)




Even if collimation of atomic beam is improved, it is impossible to select only
 43Ca isotopes completely. Purification is necessary.
                                      Purification of trapped isotopes
                        F
   2P                   3
     1/2                    581MHz
                        4
            397nm
           Cooling     Repumping
                       scan
                            F
                            3
   2S                       3226MHz
     1/2




                                            (1)excitation spectrum(866nm laser: (2)excitation spectrum(866nm laser:
                                              tuned to the resonance of 40Ca+)    tuned to the resonance of 44Ca+)
              (2)    Locked      (1)
                     866nm
             Procedure of purification
             ●Laser heating of 40Ca+ and 44Ca+.
                (One of the 866nm laser is tuned
                to resonance of 40Ca+ or 44Ca+,
                and the power of 397nm laser (4:4)
                is increased.)                               Excitation spectrum
             ●Decreasing the trap potential                     of 43Ca+ after
                                                            eliminating 40Ca+ and
U.Tanaka, I Morita, S.Urabe, Appl.Phys.B, vol.89, 195, 2007         44Ca+.
        Planar trap           Research collaboration with MIT
                              K. R. Brown, R. J. Clark, J. Labaziewicz,
 (for integrated ion traps)   D. R. Leibrandt, I. L. Chuang

                                                             Rf voltage
                                                             ground

             Linear rf trap   Planar trap


Layout of trap electrode
 trap electrode: Copper         Trap on CPGA mount
 insulator: Rogers 4350B
                        Images of trapped ions

Fluorescence from 40Ca+ ion cloud (397 nm)
                                                                  Scattered light




                                                                                    Electrode ~0.5 mm
                                     Electrode ~0.5 mm
                                                                      Single ion

         Image of a single 40Ca+ ion                     Image of two 40Ca+ ions
                           Progress and plan
Progress in last years’ objectives
   1.Lase cooling
       Cooling single and two ions to the motional ground state (n<0.02)
   2.Qubit using stimulated Raman transitions between matastable states of 40Ca+
      Development of 1.8THz phase locked infrared laser system
       Precise measurement of energy level separation of metastable D states
       Coherence time measurement of matastable Raman qubits
      Observation of Zeeman resonance
   3.Trapping of a rare odd isotope 43Ca+
      Isotope selective trapping, purification and cooling of 43Ca+
   4.Surface traps for large-scale integration (research collaboration with MIT)
      Trapping of single and two ions
Research plan for the next 12 months
   1.Quantum gates with metastable-states qubits
        Initialization of internal states (Adiabatic Rapid Passage)
        Development of Mølmer & Sørensen gate
            (Laser pulse control, state tomography etc.)
   2.Surface traps
        Compensation of micromotion and measurement of heating in
        ion movement, development of trap array

								
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