Single-qubit quantum memory exceeding $10$-minute coherence time

2 MB·

by  Ye Wang

Single-qubit quantum memory exceeding 10-minute coherence time
Ye Wang1, Mark Um1, Junhua Zhang1, Shuoming An1, Ming Lyu1,
Jing -Ning Zhang1, L.-M. Duan1,2, Dahyun Yum1∗ & Kihwan Kim1∗
1 Center for Quantum Information, Institute for Interdisciplinary Information Sciences,
Tsinghua University, Beijing 100084, P. R. China
2 Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
(Dated: January 17, 2017)
A long-time quantum memory capable of storing and measuring quantum information at the
single-qubitlevelisanessentialingredientforpracticalquantumcomputationandcom-munication[1,
2]. Recently,therehavebeenremarkableprogressesofincreasingcoherencetimeforensemble-based
quantum memories of trapped ions[3, 4], nuclear spins of ionized donors[5] or nuclear spins in a
solid[6]. Until now, however, the record of coherence time of a single qubit is on the order of a few
tens of seconds demonstrated in trapped ion systems[7–9]. The qubit coherence time in a trapped
7
ionismainlylimitedbytheincreasingmagneticfieldfluctuationandthedecreasingstate-detection
1
efficiency associated with the motional heating of the ion without laser cooling[10, 11]. Here we
0
2 report the coherence time of a single qubit over 10 minutes in the hyperfine states of a 171Yb+ ion
sympathetically cooled by a 138Ba+ ion in the same Paul trap, which eliminates the heating of
n
the qubit ion even at room temperature. To reach such coherence time, we apply a few thousands
a
of dynamical decoupling pulses to suppress the field fluctuation noise[5, 6, 12–16]. A long-time
J
quantum memory demonstrated in this experiment makes an important step for construction of
6 thememoryzoneinscalablequantumcomputerarchitectures[17,18]orforion-trap-basedquantum
1
networks[2,19,20]. Withfurtherimprovementofthecoherencetimebytechniquessuchasmagnetic

fieldshieldingandincreaseofthenumberofqubitsinthequantummemory,ourdemonstrationalso
]
h makes a basis for other applications including quantum money[21, 22].
p
-
t The trapped ion system constitutes one of the lead- state of the qubit. Due to the motional heating, the
n
a ing candidates for the realization of large-scale quantum wavepacket of the ion is expanding with time, which is
u computers[1]. It also provides a competitive platform subject to influence of nonuniform magnetic field. In ad-
q
for the realization of quantum networks which combines dition,themotionalheatingreducesthecountoffluores-
[
long-distance quantum communication with local quan- cence photons, which makes the qubit detection increas-

1 tum computation[2]. One scalable architecture for ion- ingly inefficient[10, 11]. Here, we completely eliminate
v trap quantum computer is to divide the system into op- the ion heating during long-time storage by the sympa-
5
eration and memory zones and to connect them through thetic cooling of a different species of atomic ions.
9
ionshuttling[17,18]. Forthisarchitecture,thebasicunit
1 In the experiment, we use two species of ions,
4 of operation zone has been demonstrated[23, 24]. As the 171Yb+ as the qubit ion and 138Ba+ as the cooling
0 size of the system scales up, the needed storage time of
ion, confined together in a standard Paul trap as shown
1. the qubits in the memory zone will correspondingly in- in Fig. 1a. We choose 138Ba+ ion as the refrigera-
0 crease. To keep the qubit error rates below a certain tor since it has similar mass to 171Yb+ , which makes
7 threshold for fault-tolerant computation, it is crucial to
the sympathetic cooling efficient. The Doppler cool-
1
extend the coherence time of qubits. For the quantum ing laser and the repumping laser for 138Ba+ are de-
:
v network based on probabilistic ion-photon mapping[25],
tuned by more than 200 THz from the corresponding
Xi the basic units of ion-photon and ion-ion entanglement transitions of 171Yb+ ion, therefore the coherence of
r have been demonstrated[26–28]. The required coherence 171Yb+ qubitisnotaffectedbythecoolingof138Ba+ ion
a time of qubits increases in this approach as the size (see Methods). The two hyperfine levels of 171Yb+ ion
of the system grows. A long-time quantum memory is in the 2S manifold are used as a qubit represented
1/2
therefore important for both quantum computation and
by |↓(cid:105) ≡ |F =0,m =0(cid:105) and |↑(cid:105) ≡ |F =1,m =0(cid:105),
F F
communication[2, 29]. which is separated by 12642812118+310.8B2 Hz, where
B ismagneticfieldinGauss. Inexperiment,weinitialize
For trapped ion qubits, the main noise is not relax-
ation with time T but instead dephasing with time T∗ the qubit state to |↓(cid:105) by the standard optical pumping
1 2
method and discriminate the state by the florescence de-
induced by fluctuation of magnetic fields. The current
tection scheme. We perform the coherent manipulation
records of single-qubit coherence time in trapped ion
ofqubitbyapplyingtheresonantmicrowaveasshownin
systems are around tens of seconds, achieved by using
Fig. 1b.
magneticfieldinsensitivequbits[7,9]ordecoherence-free-
subspace qubits[8]. The coherence time is mainly lim- With the help of sympathetic cooling, the remaining
ited by motional excitations without cooling laser beam, dominantfactorofdecoherenceisthemagneticfieldfluc-
as the latter would immediately destroys the quantum tuation, which leads to phase randomization[13]. We
2
A standard technique to preserve the qubit coher-
ence against random phase noise is dynamical decou-
pling, which expands Hahn spin echo into a multi-pulse
a
sequence[5, 6, 12–16]. Since the performance of the se-
quence is sensitive to the characteristics of noise envi-
ronment, we study the noise spectrum of our system,
which guides us to choose the proper dynamical decou-
pling pulses. In the experiment, we probe the specific
171Yb+ 138Ba+
P frequency part of the noise spectrum by monitoring the
1/2
F=1 P response of the qubit under the specific sequence[30].
1/2
The initial state of qubit |ψ(0)(cid:105) under a dynamical
F=0
461.31 THz
decoupling sequence in noisy environment evolves ac-
811.25 THz 607.43 THz D3/2 cording to |ψ(T)(cid:105) = eiFN(T)σz|ψ(0)(cid:105), where FN(T) =
(cid:80)N (−1)i+1(cid:82)τi+1dtβ(t), T is the total evolution time,
FS=11/2 S1/2 τi iis=0thetimestτaimpofthei-thπpulse. Weobtainthein-
1ω =12.642812118GHz
HF formationofF (T)bymeasuringthecontrastofRamsey
F=0 N
fringe as (cid:104)cos(2F (T))(cid:105), which we define as the coher-
N
b 0 Microwave PMT ence of the qubit (see Methods). We set magnetic field
Microwave Horn strength to 8.8 G and apply CPMG (Carr, Purcell, Mei-
Switch
Oscillator
Mixer boomandGill)[16]sequenceshowninFig. 2atomeasure
12.4 GHz
the noise spectrum of the environment.
Switch In our system, the dominant components of noise are
493 nm
& at 50 Hz and its harmonics coming from the power line,
Fast 650 nm which is modeled as the sum of discrete noises β˜(ω) =
Control 20D0D MSHz AOMs (cid:80)dk=1βkδ(ω−ωk). Here βk is the strength of the noise.
Trigger The Ramsey contrast of the final state becomes[14]
Slow EO Pulse
Control Picker d
(cid:89)
Mechanical (cid:104)cos(2F (T))(cid:105)= J (|β y˜(ω ,T)|), (1)
Shutter N 0 k k
369 nm Laser k=1
where J is the 0th of Bessel function, y˜(ω,T) =
0
ω1 (cid:80)Nj=0(−1)j(cid:2)eiωτj −eiωτj+1(cid:3), τ0 = 0, τN+1 = T. With
31 pulses, the resultant coherence as a function of the
pulse interval is shown in Fig. 2b. By fitting, we ob-
FIG. 1. Experimental setup a, Schematic diagram of a
tain the discrete noise spectrum as B = 18.3 µG,
trapped-ionsystemwithtwospecies. Wesimultaneouslytrap 50Hz
B = 57.5 µG with no significant components at
171Yb+ and 138Ba+ with distance of 10 µm in a linear Paul 150Hz
trap. The laser beams for 171Yb+ and 138Ba+ cover both other frequencies.
ions so that initialization and detection of 171Yb+ is not af- We further study the rest of the noise spectrum by
fected by the change of the ion position due to hopping in using the continuous noise model. Given an arbitrary
around every 5 minutes. We do not observe any difference noise spectrum S (ω), the Ramsey contrast is given by
β
in the strength of magnetic field and π time of a single qubit e−χ(T), where χ(T) is written as[13]
gate for both positions. b, The microwave and laser control
system. Microwave is generated by mixing a fixed 12.442812 2 (cid:90) ∞
GHzsignalfromanAgilentmicrowaveoscillatorandthesig- χ(T)= S (ω)|y˜(ω,T)|2dω. (2)
π β
nalaround200MHzsignalfromaDDSwiththecapabilityof 0
changing phase within 100 ns, which is controlled by FPGA.
Herethefunction|y˜(ω,T)|2 canbeviewedasabandpass
AllsourcesarereferencedtoRbclock. Weapplyathree-stage
ofswitchsystemsforthe369nmlaserbeam,whichareAOMs, filter with the center frequency of 1 and the width pro-
2τ
EO pulse picker and mechanical shutter (see Methods). portional to 1 , where τ is the pulses interval shown in
2πT
Fig. 2a. As shown in Fig. 2c, we observe the Ramsey
contrasts depending on the total evolution time T with
can write the Hamiltonian of the qubit system as H = various total number of pulses as N = 2, 10, 100, 500,
(cid:126)(ω +β(t))σ ,whereω isthesplittingofthequbit,β(t) 400, 6000 and 10000. Applying Eq. (2) to the results of
2 0 z 0
is the random phase noise proportional to magnetic field Fig. 2c, we obtain S (ω) as shown in Fig. 2d. We ob-
β
fluctuation and σ is the Pauli operator. The accumula- serve sharp increase of noise strength below (2π) 2 Hz,
z
tion of the random phase causes dephasing of the qubit whichisconsistentwiththeresultoffluxgaugemeasure-
state. ment. We also observe slow increase of the noise above
3
FIG. 2. Measurement of noise spectrum of the system a, Diagram of the CPMG sequence. All the π pulses have
the same phase of π/2. Interval between every π pulses is τ. b, The fringe contrast (cid:104)cos(2F (T))(cid:105) as a function of the pulse
N
interval for 31 CPMG pulses. By fitting the data with Eq. (1), we obtain B = 18.3µ G, B = 57.5µ G. To suppress
50Hz 150Hz
50 Hz and 150 Hz noises with KDD , we choose τ = 200 ms instead of 250 ms, since the values of |y˜|2 near 50 Hz and 150
xy
Hz has around 10 orders of differences as shown in the inset, respectively. c, Ramsey contrasts (cid:104)cos(2F (T))(cid:105) depending on
N
the total evolution time for various numbers of pulses as N = 2, 10, 100, 500, 4000, 6000 and 10000. The data are normalized
to the contrast of continuous application of the same number of pulses. Error bars are the standard deviations of 50 to 200
repetitions. d,AnalyzednoisespectrafromthecoherencedecayshownincbyusingEq.(2). Thesmallestnoiselevelislocated
at the range of (2π) 2∼100 Hz.
(2π) 100 Hz, which could come from our ground line for sequence (see Methods). Our single qubit gate fidelity is
current sources generating the magnetic field. measured to be 99.994±0.002% by randomized bench-
In order to extend coherence time with the CPMG marking method[31] as shown in Fig. 3b. We test
type of sequence under our current noise environment, the robustness of KDDxy by continuously applying it to
we carefully choose the interval of the pulse sequence. |↑(cid:105) state. We have more than 85% population back to
Wecanlocatethebandpassfrequencybetween2Hzand the initial |↑(cid:105) state after the application of 20000 pulses
100Hz,wherethenoisespectrumhaslowestvalue,which as shown in Fig. 3c.
means 5 < τ < 250 ms. In this range, a larger τ is For the measurement of single-qubit coherence time,
preferred, which reduces the number of pulses and leads we set the magnetic field to 3.5 G. We measure the co-
to smaller accumulation of gate errors. We choose the herence time of single-qubit memory with 6 different ini-
√ √
pulse interval τ properly so that the noise components tial states as |↑(cid:105), |↓(cid:105), 1/ 2(|↑(cid:105)+|↓(cid:105)), 1/ 2(|↑(cid:105)+i|↓(cid:105)),
√ √
at 50 Hz and 150 Hz are suppressed to a negligible level. 1/ 2(|↑(cid:105)−|↓(cid:105)), 1/ 2(|↑(cid:105)−i|↓(cid:105)). In Fig. 4a, we see no
Considering all the factors, the optimal τ =200 ms. significant relaxation for the initial states of |↑(cid:105) and |↓(cid:105)
In experiment, we use the KDD [6, 15] sequence to underKDD sequence. Thereductionoftheircontrasts
xy xy
store arbitrary quantum states of single qubit. KDD mainly comes from the accumulation of gate errors. The
xy
is a robust dynamical decoupling sequence insensitive other four initial states have a similar decoherence rate,
to imperfections of gate operations including flip-angle whichcorrespondstothecoherencetimeof666.9±16.6s
errors and off-resonance errors. As shown in Fig. 3a, fromtheexponentialdecayfitting. Thedemonstratedco-
KDD can be considered as an extension of the CPMG herence time is primarily limited by non-vanishing noise
xy
4
a ×N/10 a e0
π π π π π π π π π π
π π
2 τ τ 2
2 τ τ τ τ τ τ τ τ τ 2 st e-0.5
120° 90° 180° 90° 120°KDD21x0y° 180° 270° 180° 210° ntra ● |↑〉+|↓〉
Co ● |↑〉-|↓〉
b 1 nge e-1 ● |↑〉+i|↓〉
Fri ● |↑〉-i|↓〉 DecoherenceThreshold
● |↑〉
0.98 ● |↓〉
e-1.5
y
Fidelit 0.96 Fidelity 0.91◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲◆◆◆◆◆◆◆◆■■■■■■■■●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲■■■■■■■■ ◆◆◆◆◆◆◆◆●●●●●●●●▲▲▲▲▲▲▲▲■■■■■■■■ 0 100 200 T301(.0s0)c 400 500 600
0.94 0.80 100 200 300 400 500 0.8
GateNumber
0.6
0 100 200 300 400 500
GateNumber
0.4
c 1
0.2
0
n
o
ulati 0.9 FIG. 4. Coherence time measurement and quantum
p process tomography a,Thecoherencetimeofsixdifferent
o
P initial states. For the state |↑(cid:105) and |↓(cid:105), the coherence time
is 4744.72±1762.33 s. For the other four initial states, the
coherence time is 666.9±16.6 s. b, The result of quantum
process tomography for the duration of 8 minutes. Here the
0.8
tomographyisobtainedfromtheentireprocessincludingini-
2000 10000 20000
GateNumber tialization, storage and detection. Identity is the dominant
diagonal part with χ = 0.699±0.058. c, The transforma-
II
FIG. 3. KDD sequence and gate fidelity a, Diagram tion from the initial states lying on the meshed surface to
xy
ofKDD sequence. Knillpulseconsistsoffiveπ pulseswith the final states lying on the solid surface after 8 minutes of
xy
angles φ+π/6,φ,φ+π/2,φ,φ+π/6. The KDD consists storage.
xy
of two Knill pulses equally spaced with φ = π/2 and φ = π.
We apply N/10 sets of the KDD , which leads to the total
xy
numberofN pulses. NotethatN shouldbeamultipleof20to
formanidentityoperation[15]. b,Randomizedbenchmarking
befurtherincreasedbydecreasingS (ω)atthebandpass
test of the single qubit gates. We use 32 settings of random β
frequency,whichcanbeachievedbyinstallingamagnetic
sequencesfollowedbyRef.[31]. Ateachsettingwerepeat500
times. Theinsetshowstheresultsof32differentsettings. The field shield or by using a magnetic field insensitive qubit
blackdotsinbrepresenttheaveragevaluesof32settingswith at zero crossing regime. The number of memory qubits
theerrorbarofstandarddeviation. Byfitting,weobtainthe can be increased with sympathetic cooling for quantum
gatefidelityas99.994±0.002%. c,Wecontinuouslyapplythe cryptographicapplicationsincludingquantummoney[21,
KDD pulsesontheinitial|↑(cid:105)statetotestitsperformance.
xy 22].
We observe more than 85% population back to the initial
|↑(cid:105) state after the application of 20000 π pulses.
strength at the bandpass filter frequency of 2.5 Hz. We
also perform the process tomography on our dynamical
decoupling process for the duration of 8 minutes. The
Acknowledgment
results shown in Fig. 4b and 4c demonstrate that the
process is approximated as an identity operator with the
main error corresponding to pure dephasing, in agree- This work was supported by the National Key Re-
ment with our knowledge about the hyperfine qubit of searchandDevelopmentProgramofChinaunderGrants
the trapped ion system. No. 2016YFA0301900 (No. 2016YFA0301901), the Na-
We note that our measured coherence time over 600 tional Natural Science Foundation of China 11374178,
s is not fundamentally limited. The coherence time can 11504197 and 11574002.
5
METHOD Magnetic field fluctuation
Control System The second order Zeeman effect of the 171Yb+ qubit
transition[35] is described by ∆f = K(cid:104)B2(cid:105) Hz, K =
2OZ
310.8 Hz/G2. In the Hamiltonian H = (cid:126)(ω +β(t))σ ,
We use two FPGA boards to separately control the 2 0 z
where ω and β(t) are given by
laser system and the microwave system. Two parts are 0
synchronized in less than 10 ps of time jitter. The leak-
ω =ω +K(B2+B2+B2), (4)
age of 369 nm laser beams greatly limits the coherence 0 HF x y z
time of the qubit of 171Yb+ ion. To block the 369 nm β(t)=K(2Bx(cid:104)bx(t)(cid:105)+2By(cid:104)by(t)(cid:105)+2Bz(cid:104)bz(t)(cid:105)
laser beams for the Doppler cooling, initializing and de- +(cid:104)bx(t)2(cid:105)+(cid:104)by(t)2(cid:105)+(cid:104)bz(t)2(cid:105)), (5)
tecting the quantum state of 171Yb+ ion, we use three
and ω is the hyperfine splitting, B , B , B represent
stages of switches: AOMs (Acousto-Optic Modulator), HF x y z
the average values of the magnetic field in three direc-
EO (Electro-Optic) pulse picker and mechanical shutter.
tions and b (t), b (t), b (t) describe fluctuations in the
The AOM switches leave a few nW of laser beam leak- x y z
corresponding directions. The fluctuations are believed
age, the EO pulse picker provides 200 times of attenu-
tocomefromtheenvironmentandthepowersourcesgen-
ation for the leakage and the mechanical shutter blocks
erating the magnetic field. Since b (t) is much smaller
the laser beam completely with about 50 ms of response x
than b (t) and b (t) according to the flux gauge mea-
time. With only the AOM switches, we observe a max- y z
surement, we put the magnetic field along x-axis. In the
imum of 50 ms coherence time. The coherence time is
experiment, we set B = 0, B = 0 and B = 3.5 G,
increased to 10 s by applying EO pulse picker and dy- z y x
whichisthesmallestmagneticfieldstrengthtomaintain
namical decoupling sequence. Therefore it is necessary
detection efficiency in our system. To study noise spec-
to use the mechanical shutter to completely block the
trumofoursystem,weusethesettingofB =0,B =0
laser beam. The resonant microwave with the frequency z y
and B =8.8 G.
of 12.642812 GHz is generated by mixing the signal of a x
microwaveoscillatorandthesignalofaround200MHzof
a DDS (Direct Digital Synthesizer). The DDS is used to
change the phase of the microwave within 100 ns, which
Dynamical decoupling
is controlled by the FPGA. All sources are referenced to
a Rb clock.
Intheexperiment,CPMGisusedforthestudyofnoise
environment and KDD is used for storage. Here we
xy
show the filter function of KDD is the same as that of
xy
Limitation of coherence time due to the cooling CPMG.Sincealltherotationaxesweuseareinxy-plane,
laser beams for 138Ba+ we write the rotation as
D (γ)=D (φ)D (γ)D (−φ)
We calculate the scattering rate between 2S man- φ z x z
1/2
ifold and 2P , 2P manifold of 171Yb+ due to the =e−2iσzφe−2iσxγe2iσzφ, (6)
1/2 3/2
coolinglaserbeams,493nmand650nm,for138Ba+ ion.
whereφistheanglebetweentherotationaxisandxaxis
The scattering rate is calculated by the model of Raman
and γ is the rotation angle along the axis. In dynamical
scattering[32–34], which is given as
decoupling sequences, γ is always π, therefore, D (π) =
x
g2Γ(cid:18) 1 2 (cid:19) cos(π/2)+iσxsin(π/2)=iσx. Below we will only use σx
Γscat = 6 ∆2D1 + (∆D1+∆fs)2 , (3) because the factor i is an irrelevant global phase. √
We start from initial state |ψ(0)(cid:105) = (|↓(cid:105)+i|↑(cid:105))/ 2,
(cid:113)
where Γ ≈ 2π × 20 MHz, g = Γ I , ∆ ≈ 2π × the final state is shown as
2 2Isat fs
100 THz. For the 493 nm laser, the parameters are as |ψ(T)(cid:105)=R˜(T)|ψ(0)(cid:105), (7)
follows: laser power P = 40 µw, beam waist ω = 22
µm, I493 = 49.6Isat and ∆D1= 203.8 THz, which yields R˜(T)=e−iσz(cid:82)ττNN+1β(t)dtDφN(π)···
tahsecaptatrearminegterarsteaoref 3a.s8×fol1lo0w−s8:Hlaz.seFroprotwheer6P50=nm10laµsewr,, Dφ2(π)e−iσz(cid:82)ττ12β(t)dtDφ1(π)e−iσz(cid:82)ττ01β(t)dt, (8)
beam waist ω = 20.7 µm, I = 14.0I and ∆ = whereR˜(T)istheevolutionduringtotaltimeT,N isthe
650 sat D1
349.9 THz, which yields a scattering rate of 1.5×10−8 total number of π pulses, τ is the time stamp of the ith
i
Hz. Therefore, 493 nm and 650 nm laser beams have π pulse, τ =0, τ =T and φ is the phase of the ith
0 n+1 i
no effect on the coherence of 171Yb+ ion hyperfine qubit π pulse. Since N is even for our dynamical decoupling
during hours of storage. sequence and only σ operations in R˜(T) flip the spin,
x
6
we can obtain 76, 033411 (2007).
[11] J. Wesenberg, R. Epstein, D. Leibfried, R. Blakestad,
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C. Langer, et al., Physical Review A 76, 053416 (2007).
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xy
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