Quantum gate using qubit states separated by terahertz

0.14 MB·English

by  K. Toyoda

Quantum gate using qubit states separated by terahertz
K. Toyoda,1,2 S. Haze,1 R. Yamazaki,2 and S. Urabe1,2
1 Graduate School of Engineering Science, Osaka University,
1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
2 JST-CREST, 4-1-8 Honmachi, Kawaguchi, Saitama 331-0012, Japan
Atwo-qubitquantumgateisrealizedusingelectronicexcitedstatesinasingleionwithanenergy
separation on the order of a terahertz times the Planck constant as a qubit. Two phase locked
lasers are used to excite a stimulated Raman transition between two metastable states D and
3/2
D separated by 1.82 THz in a single trapped 40Ca+ ion to construct a qubit, which is used as
5/2
thetarget bit for the Cirac-Zoller two-qubit controlled NOT gate. Quantum dynamicsconditioned
on a motional qubit is clearly observed as a fringe reversal in Ramsey interferometry.
0
1
PACSnumbers: 03.67.Lx,32.80.Qk,37.10.Ty
0
2

n Atomic systems including trapped ions and neutral simplified form using internal states and a motional de-
a atoms are considered attractive for quantum informa- gree of freedomin a single 9Be ion[15]. A full implemen-
J tion processing (QIP) since they can be made to be well tation of the scheme in a scalable manner using a 40Ca+

6 isolated from the environment and hence enable con- ion string with the technique of individual addressing is
2 struction of qubits with small decoherence/dephasing. reported in 2003[16].

Among experimental approaches toward QIP using dif- It has been shown that all unitary operations on arbi-
]
h ferentphysicalsystems,some ofthe mostadvancedhave trarymany qubits canbe decomposedinto two-bitgates
p been experiments using trapped ions[1, 2], which are and one-bit gates[17]. One example of such decomposi-
- basedonqubitlevelswithseparationintherf/microwave tion of unitary operations uses controlledNOT (CNOT)
t
n region and the optical region. Recent advances in opti- gates and rotation operations on single qubits[18, 19].
a calcombgenerationandopticalfrequencymetrology[3–5] Analogously to a classical exclusive-OR (XOR) gate, a
u
offers much flexibility in choosing qubit states, including CNOT quantum gate realizes the following operation:
q
[ atomic states with frequency separations that have not ǫ1 ǫ2 ǫ1 ǫ1 ǫ2 withǫ1,2 =1,2and representing
been explored before. a|nia|ddiit→ion| mi|odu⊕lo 2i. ⊕
1
In view of recent progress in experiments of ultracold In the Cirac-Zoller (CZ) proposal[14], for implement-
v
0 molecules transferred to the ground state of both inter- ing this CNOT operation,a red-sidebandpulse, whichis
0 nal and external degrees of freedom, molecular systems detuned to the lower side of the resonance of the qubit
6 whichhaverichinternalstructuresarealsoconsideredat- transitionbythefrequencyofacollectivemotionalmode,
4 tractive for application to QIP. In the recentworks[6–9], isappliedbetweenonebasisstateofthetargetqubitand
.
1 byperformingstimulatedRamanadiabaticpassageusing an auxiliary state. When the collective motional state
0 twolaserswithhighrelativecoherence,weaklyboundul- has one quantum, the red-sideband pulse applied for a
0 tracold Feshbach molecules are transferred to their rovi- durationcorrespondingtoa2π rotationcausesaπ phase
1 bronic ground state. In addition, there are proposals to
:
v encode qubits in molecular states with small dipole mo-
Xi mentsandtransferthesetostateswithlargerdipole mo- P3/2 ∆
ments, thereby realizing switchable interaction between
r molecular qubits[10–13]. The required transfer can be P1/2
a
performed by applying two phase locked lasers through Stimulated Raman (850, 854nm)
stimulated Raman process. D5/2 mJ=-5/2 mJ=1/2
↑
Inthisarticle,wepresenttheresultofaquantumgate
1.82THz
experiment using phase-locked lasers to excite a stim- ↓
D
ulated Raman transition. Two metastable states D 3/2
3/2 mJ=1/2
and D5/2 in 40Ca+ separated by 1.82 THz are used as Sideband cooling Preparation (729nm)
the target bit to perform the Cirac-Zoller gate[14]. This (729nm)
is the first attempt to use phase locked lasers to bridge
an energy separation larger than a terahertz and realize S1/2 mJ=-g1/2
aquantumgate,andisanimportantsteptowardobtain-
ingawiderchoiceofqubitlevelsincludinginternallevels FIG. 1: (Color online) Level scheme for 40Ca+ and tran-
of atoms and molecules. sitions relevant for implementing the CZ gate. Two sub-
CiracandZollerproposedin1995arealisticschemefor levels each from D3/2 and D5/2 metastable states, |↑i ≡
scalablequantumcomputationusingastringofionsina (cid:12)(cid:12)D5/2(mJ =1/2)(cid:11) and |↓i ≡ (cid:12)(cid:12)D3/2(mJ =1/2)(cid:11), are used as
linear trap[14]. It was experimentally demonstrated in a thequbit states here.
2
ICCD Magnification: 854 nm phase locked by using a passive-type optical
x50 Copropagating Raman beams
866nm, 854nm(resonant) (850, 854 nm) comb[5] in combination with an acousto-optic modula-
Linear Paul trap
tor (AOM) and an electro-optic modulator[21] are used
ωz/2π=0.7 MHz
to excite the stimulated Raman transition. For excita-
397nm, B To Ti-subl. and ion pump tion of the quadrupole transition, a titanium sapphire
423nm, 375nm Pressure ~ 6(cid:7587)10-9 Pa laser at 729 nm stabilized to a high-finesse low-thermal-
expansion cavity having a linewidth of < 400 Hz and a
Magnetic shield
root-mean-squareintensitynoiseof0.3%isused. Control
397nm(σ-) 729nm beam of optical frequency/phase/amplitude is done by AOM
PMT
andrffieldsusedforthemaregeneratedbydirect-digital
synthesis (DDS) boards which are controlled by a field-
FIG. 2: (Color online) Experimental setup for the terahertz-
programmable gate array (FPGA).
qubit quantum-gateexperiment. See text for details.
See Fig. 2 for the details of the beam configuration.
For realizing gate experiments, all the motional degrees
shift between two basis states of the target qubit states. of the ion are cooled to near the ground states using
Ontheotherhand,whenthecollectivemotionalstatehas Doppler cooling (with 397 and 866 nm lasers) and side-
no quantum, such rotation does not occur and no phase band cooling (SBC). For SBC, the S1/2(mJ = 1/2)–
−
shift is given to the target qubit. This corresponds to a D5/2(mJ′ = 5/2) transition at 729 nm is used, and an
−
unitary operation conditioned on the motional quantum additional quenching laser resonant to D5/2–P3/2 at 854
number, thereby implementing a controlled phase gate, nm is also applied. All the three dimensions are cooled
andaCNOT gate is realizedwhenthis is combinedwith for 2 ms each and then this is repeated for 20 times.
−
certain single qubit operations. Optical pumping is performed using 397 nm σ tran-
To realize the CZ gate using the metastable states sition in S1/2(mJ = +1/2)–P1/2(mJ′ = 1/2) before,
−
in 40Ca+ and its motional states, we adopt an excita- every 6ms during, and after SBC, each with duration of
6 µs. The final quantum numbers obtained after SBC
tion scheme using the stimulated Raman transition be-
are (n ,n ,n ) (1,1,0.02).
tweenD andD alongwitha quadrupoletransition x y z
3/2 5/2 ∼
that connects the ground state S with D (see Fig. Forpreparationtothe state,acarrier/BSBπ pulse
1/2 5/2 |↑i
1). The stimulated Raman transition is used for sin- on g – is applied. This is a ∆mJ = 1 transition
| i |↑i | |
glequbitoperationonthe metastablestatesqubit, while whichrequiresapolarizationdifferentfromthatusedfor
thequadrupoletransitionisusedforrealizingconditional sideband cooling transition for which ∆mJ =2. In our
| |
phase shift required for a CZ gate, as well as sideband case the former is parallel with and the latter is perpen-
coolingandstate preparation. Asthe targetqubitstates dicular to the surface of the optical table on which the
D (m = 1/2) and D (m = 1/2) are trap chamber is placed (see Fig. 2). In order to perform
5/2 J 3/2 J
c|↑hios≡en,whileasthecontrolqu|↓biit≡thelow-lyingtwostates both in one configuration,the polarizationof the 729nm
of the axial motion initialized to the ground state are light is chosen to be linear and rotatedfrom the perpen-
used: 0 (1 ) n =0 (n =1 ), where n denotes dicular direction by 45 degree.
z z z
theaxia|lim|otiion≡al|quantumi |numberi. Aconditionalphase The target qubit states are discriminated by shining
shift is implemented by applying a blue sideband (BSB) the cooling lasers at 397 and 866 nm for 7 ms and ob-
2π pulse between and g S (m = 1/2) state, serving fluorescence photons by a photomultiplier.
1/2 J
|↑i | i ≡ −
which gives a π phase shift to the 1 . The coherence times of the Raman transition have
| i|↑i
We use a single 40Ca+ ion trapped in a vacuum pres- been measured to be 5.1 ms (1.6 ms) with (without)
sureof6 10−9Pa. Thetrapusedhereisaconventional spin echo in a setup w∼ithout a magnetic shield[22]. The
×
linear trap with an operating frequency of 23 MHz and coherencetimeforthequadrupoletransitionwithamag-
secular frequencies of (ω ,ω ,ω )/2π =(1.91,1.68,0.72) netic shield, which is deduced from decay of Rabi oscil-
x y z
MHz. Amagneticfieldof 3.1 10−4Tisappliedtolift lation signals, is about 0.8 ms.
∼ ×
the degeneracyofZeemanstates and to define the quan- Figure 3 shows Rabioscillationsignalson the relevant
tization axis for optical pumping. To reduce the effects transitions including the carrier/BSB on g – and the
| i |↑i
of the ambient ac magnetic field, the vacuum chamber carrier on the stimulated Raman transition. Based on
is enclosed in a magnetic shield. Loading of single ions theseresults,wecanexpectnearlyunitfidelityforcarrier
is performed by using two-step photoionization from the pulses on g – while less fidelity for BSB Rabi pulses
4s1S ground state of Ca via 4p1P1 with correspond- and carrie|r ipu|l↑sies on the stimulated Raman transition.
0
ing wavelengths of 423 nm and 375 nm for the first and The figure also shows results of numerical simulation,
second step of the photoionization, respectively. the details of which are given later. By comparing the
Aboutthephaselockedlasersusedforexcitationofthe simulation with the experiment, we can quantitatively
stimulated Raman transition, the setup has been mod- characterize the fidelity limiting factors, and this helps
ified from the one described in our previous article[20] estimation of possible fidelity of Bell state generation as
in order to improve the noise in the difference frequency described later.
of the two lasers. Two Ti-sapphire lasers at 850 and Figure 4(a) shows the pulse sequence for the Cirac-
3
Fidelitylimitingfactorsexceptforthatfromaxialmo-
(a) 1 tional state distribution, which include laser phase fluc-
P↑ 0.5 tuation and magnetic field fluctuation, are incorporated
into the equation as exponential decay of off-diagonal
0
0 5 10 15 20 25 30 35 density matrix elements for the internal degrees of free-
Time [µs]
dom. For the axial motional state, the initial distribu-
(b) 1
tion is assumed to be a thermal distribution based on
P↑ 0.5 the experimentalresultsofsidebandcooling(nz 0.02).
∼
Heating during the gate operation is neglected, which
0
0 200 400 600 800 1000 is reasonable since our measured heating rate is 0.005
Time [µs] ∼
(c) 1 quanta/ms for the axial motion and the typical gate se-
quences are shorter than 1 ms.
P↑ 0.5 The parameters for the exponential dephasing are ex-
tracted from experimental results by manually fitting
0
0 100 200 300 400 500 600 simulation results for simple one-pulse sequences to the
Time [µs]
experimental Rabi oscillation results. Dotted curves
in Fig. 3 represent such manually fitted simulation re-
FIG. 3: (Color online) Rabi oscillation signals in the tran-
sults. Representing exponential decay of off-diagonal
sitions relevant to the terahertz qubit scheme. (a) Carrier
density matrix elements using a proportionality factor
(BSB) Rabi oscillation and (b) BSB Rabi oscillation in |gi–
exp[ (γ/2)t],thevaluesofγ areextractedtobe2π 400
|↑i. (c) Carrier Rabi oscillation in the stimulated Raman
− ×
transition |↑i–|↓i. Thedotted curvesareresults of numerical Hz for g – and 2π 300 Hz for – .
| i |↑i × |↑i |↓i
simulation. Based on the above-mentioned assumptions, the CZ
gateexperimentis simulatedwith4pulses assumed,and
the result is shown in the Fig. 4(b) as curves. It well
Zoller gate experiment. The first pulse (preparation reproducesthe reductionof fringe contrastsand also the
pulse) is applied either on the carrier or BSB on g – negativeoffsetinthe caseof 1 preparationwithoutany
| i | i
to prepare the motional state 0 or 1 respectively. fitting parameters. It is presumable that the negative
|↑i | i | i
Then the first stimulated-Raman π/2 pulse is applied, offsetiscausedbythe infidelityintheBSBexcitationon
which is followed by a BSB 2π pulse on g – and the
| i |↑i
secondstimulated-Ramanπ/2pulse. For 0 preparation
| i
the BSB 2π pulse cause no effect since there is no mo- (a) Raman π/2 Raman π/2
φ=0 φ=φ1
tional state to reach in g , while for 1 preparation the
| i | i
BSB2π pulsecause2π rotationandgivesaπ phaseshift
to the originalstate. This conditionalphase flip (π rota- Quadrupole carrier/BSB Quadrupole BSB
tion around the z axis in the Bloch sphere) is converted π pulse 2π pulse
(b) 1
intoaconditionalbitflip(π rotationaroundahorizontal
axis in the Bloch sphere) by the two π/2 pulses. 0.9
Figure 4(b) shows the result of the Cirac-Zoller gate 0.8
experiment. Here the phase of the second pulse is ro- 0.7
stautreedd.frComros0sestore4pπreasenndttthhee p0opuplraetpioanratinion|↑ciaisse,maenad- bility0.6
| i a0.5
filled circles the 1 preparation case. The interference ob
fringes for the tw|oicases clearly show a π phase differ- Pr0.4
ence to each other, which is an evidence of a conditional 0.3
dynamics caused by the BSB 2π pulse. The contrasts 0.2
of the fringes are limited to 0.4 0.6, and for the BSB
∼ 0.1
preparation case there is a negative offset 0.05, which
∼ 0
is consistently observed in similar measurements. These 0 0.5 1 1.5 2
Phase[rad]/2p
imperfections are explained later.
Numerical simulation is performed to quantitatively
FIG.4: (Coloronline)(a)PulsesequencefortheCirac-Zoller
analyze the CZ gate result and to estimate possible fi-
gate experiment. (b) Result of a Cirac-Zoller gate experi-
delity for Bell state generation. A Liouville equation
ment. Ramsey interference signals are plotted against the
with exponential decay is solved for three internal lev-
phase of the second pulse. Crosses (filled circles) represent
els (g , and ) and 5 external levels representing the |0i (|1i) preparation case. The solid (dashed) curve rep-
| i |↑i |↓i
the axial motional states truncated at nz =4. Coupling resents numerical simulation for |0i (|1i) preparation. The
between the internal and externalstates is considered to π phasedifferencebetween interference fringes indicatesthat
the second order of the Lamb-Dicke factor η for carrier a controlled dynamics characterizing the Cirac-Zoller gate is
excitation and to the first order for sideband excitation. realized.
4
g – for the first and third pulse. The infidelity result LossoffidelityintheBellstategenerationprocesscan
| i |↑i
in populationleft in g , which is consideredas a leakage be also estimated by simulation. Infidelity in excitation
| i
out of the computational subspace spanned by , ofthestimulatedRamantransition – ,whichinclude
|↑i |↓i |↑i |↓i
leading to a signal decrease. phasenoise betweenthe lasersandmagneticfield fluctu-
The conditional phase shift confirmed above is an es- ation,contributes12-14%. Infidelity inexcitationof g –
| i
sential result characterizing the CZ gate scheme, while ,whichinclude729-nmlaserfrequencynoiseandmag-
|↑i
the whole functionality of the gate operationmay be ex- netic field fluctuation, contributes 5-7%. Axial quantum
amined more thoroughly through constructing truth ta- number distribution contributes 6-7%. Intensity fluctu-
bles,performingquantumstate/processtomography[23], ation is estimated to contribute by as low as 0.1%, and
and estimating fidelity in entangled state generation us- the effect of spontaneous emission is 0.1%.
∼
ing the gate. Here we numerically simulate Bell state Inconclusion,aquantumgateisdemonstratedwithan
generation using the CZ gate and estimate the fidelity. atomic qubit consisting of electronic excited states with
A Bell state Ψ = (1/√2)(0 + 1 ) can be a separation on the order of a terahertz. A conditional
B
| i | i|↑i | i|↓i
producedfromaninitialstate g byfirstapplyingaπ/2 dynamics is clearlyobservedas a fringereversalin Ram-
| i
carrier pulse and a π BSB pulse on g – to prepare sey interferometry. Fidelity for Bell-state generation is
| i |↑i
(1/√2)(0 + 1 ) and then performing a controlled estimated to be 0.74, and decoherence factors are anal-
| i | i |↑i
NOT operation that flips the internal qubit conditioned ysed. The excitation scheme using stimulated Raman
on the motional state. Using exactly the same param- transitions with phase-locked lasers offers much flexibil-
eters as used for the simulation in Fig. 4(b), the time ity, and is eventually used for atomic transitions which
dependence of the density matrix in the process of the arenotexploredbeforeasqubittransitionsaswellasfor
generation of the Bell state Ψ is simulated. The fi- molecular transitions.
B
| i
delity of the final state F Ψ ρ Ψ is obtained to We thank I. Chuang and his group for their coopera-
B B
≡ h | | i
be 0.74, which well exceed the value 0.5 expected for tion in the development of the pulse programming sys-
product states[24]. tem.
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