Arbitrary Rotation of a Single Spinwave Qubit in an Atomic-Ensemble Quantum Memory

3.4 MB·English

by  Jun Rui

Arbitrary Rotation of a Single Spinwave Qubit in an Atomic-Ensemble Quantum
Memory
Jun Rui,1,2 Yan Jiang,1,2 Bo Zhao,1,2 Xiao-Hui Bao,1,2 and Jian-Wei Pan1,2
1Hefei National Laboratory for Physical Sciences at Microscale and Departmentof Modern Physics,
University of Science and Technology of China,Hefei,Anhui 230026, China
2CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
(Dated: January 29, 2015)
We report the first experimental realization of single-qubit manipulation for single spinwaves
stored in an atomic ensemble quantum memory. In order to have high-fidelity gate operations, we
make use of stimulated Raman transition and controlled Lamor precession jointly. We characterize
the gate performances with quantum state tomography and quantum process tomography, both
5 of which imply that high-fidelity operations have been achieved. Our work complements the ex-
1 perimental toolbox of atomic-ensemble quantum memories by adding the capability of single-qubit
0 manipulation, thus may have important applications in future scalable quantum networks.
2
PACSnumbers: 42.50.Dv
n
a
J
Manyphysicalsystemshavebeenexperimentallystud- a single spinwave lies in how to manipulate a single col-
8
ied for quantum information applications [1]. Among lective excitation mode precisely and coherently without
2
them,atomicensembles[2]aremainlyfamousfortheca- influencing (depopulating or exciting) the rest majority

pability of long-term storage of quantum states and the atoms. Evenasingleindeliberateexcitationfromthema-
]
h capability of efficient interaction with single-photons. In jority of atoms could possibly ruin a stored single spin-
p recentyears,significantachievementsonmemorylifetime wave. The manipulation process has also to preserve the
-
t [3, 4] and efficiency [5, 6] have been made. Neverthe- spinwave wave-vector since its amplitude and direction
n
less, direct manipulation of single qubits in an atomic- determines the mode direction for the retrieval photons.
a
ensemble quantum memory has not been realized so Preliminarystudyonmanipulationofclassicalspinwaves
u
q far, which limits further applications of atomic-ensemble has been performed previously by H. Wang et al. Either
[ quantum memories. For instance, in order to teleport a by manipulating the majority atoms [18] or the excited

qubit from one ensemble to another [6] or entangle re- atoms[19], theyobservedthefloppingbetweentwospin-
1
mote atomic ensembles [7] through interfering indepen- wave modes. Nevertheless, their experiments still lie in
v
dentphotons, randomsingle-qubitmanipulationsonone the classical regime, and number of excitation is on the
7
6 ensemblearerequiredinordertorestorethetargetstates order of ∼ 106 which is far more larger than a single
0 conditioned on different Bell state measurement results. spinwave.
7 However, due to the incapability of qubit manipulation, In this paper, we report the first experiment of single-
0 in all previous experiments [7–9], one has to convert the qubit manipulation for single spinwaves in an atomic
.
1 spinwave states into single-photon states and apply the ensemble quantum memory. High-fidelity operation is
0 qubit rotations on photons instead. Inefficiency is the achieved by making use of stimulated Raman transition
5 main drawback for this alternative way. For large-scale and controlled Larmor procession jointly. An arbitrary
1
applications, the overall efficiency can be extremely low. single spinwave state is heraldedly prepared through the
:
v If the spinwave states can be directly manipulated and process of spontaneous Raman scattering and making
i detected in the atomic ensemble quantum memories, the projective measurement on the scattered single photons.
X
efficiency will be largely enhanced. Fortherealizationofanarbitrarysingle-qubitoperation,
r
a Arbitrary single-qubit rotation is an essential element the capabilities of rotating an arbitrary angle along two
in quantum information science, and has been success- orthogonal axes in the Bloch sphere are required. For
fully implemented in single emitter systems, such as sin- the rotation along the axis in the direction of poles, we
gle neutral atoms [10], single ions [11] and single quan- make use of controlled Larmor procession. For the rota-
tum dot [12, 13] and single NV center [14, 15]. However, tion along an axis in the equatorial plane, we make use
coherently manipulate the spinwave qubit in atomic en- of stimulated Raman transitions. Experimental results
sembles remains challenging. For single-emitter systems, are characterized with quantum state tomography and
single-qubit manipulation can be easily realized by ad- quantum process tomography.
dressingthesingleemitterusingmicrowavesorradiofre- Our experimental scheme is shown in Fig. 1. An en-
quencyfields. Nevertheless,qubitmanipulationforasin- semble of 87Rb atoms are captured using a standard
glespinwavequbitrequirescoherentlymanipulateallthe magneto-optical trap (MOT). With 2 ms polarization
atoms simultaneously, since a single spinwave is a collec- gradientcooling,theatomicensembleiscooledtoatem-
tive superposition of all the atoms with only one excita- perature of about 10 µK. All the atoms are initially
tion [16, 17]. The main difficulty of manipulating such pumped to the state of |g(cid:105) ≡ |F = 1,m = 0(cid:105). To
f
2
create a single spinwave exication [16], as shown in Fig. a Write Rotation Read
1(a), a σ− polarized write pulse is applied to couple the e ∆
transition of |g(cid:105) → |e(cid:105) with |e(cid:105) ≡ |F(cid:48) = 2,m = −1(cid:105)
weakly. Raman scattered σ+ and σ− idler phoFtons are + (cid:60) idler k+ k- signal
selected through spacial and frequency filtering. A σ+
idler photon heralds the creation of a single excitation g
in | ↓(cid:105) ≡ |F = 2,mf = 2(cid:105), while a σ− idler photon b z z
heralds the creation of a different single excitation in
| ↑(cid:105) ≡ |F = 2,m = 0(cid:105). For each write pulse, the Write Read
f
heralding probability is 3×10−3, which guarantees that
y Lamor y
probabilityforcreatingmultiple(≥2)excitationsisneg-
x Raman x
ligibly small. Interference between these two channels
gives rise to the entanglement between an idler photon
and an atomic spinwave [20, 21] in the form of Lamor Rotation Raman Rotation
|Ψ (cid:105)=(cid:112)2/5|s (cid:105)|σ+(cid:105)−(cid:112)3/5|s (cid:105)|σ−(cid:105), c D1
AP ↓ ↑ Raman idler
Bias field
where the amplitude and relative phase are determined
Write
by the atomic transitions and the collective spin states
are defined as |s↓(cid:105) = Σjei(cid:126)ks·(cid:126)rj|g1... ↓j ...gN(cid:105) and |s↑(cid:105) = D2a MOT Read
Σjei(cid:126)ks·(cid:126)rj|g1... ↑j ...gN(cid:105), with (cid:126)ks = (cid:126)kw −(cid:126)ki. Once the signal FP Cavity PBS SPDC
idlerphotonismeasuredunderspecificpolarizations,the D2b
Atom Filter HWP QWP
spinwave will be projected to the corresponding states.
In order to prepare an arbitrary single spinwave state
FIG. 1: Experiment scheme for arbitrary rotations of spin-
of |ψ (cid:105) = cosθ|s (cid:105) + sinθeiφ|s (cid:105), one just needs to
s ↓ ↑ wave qubit. a, Energy levels: The write process projects the
project the idler photon onto a corresponding state of
|ψ (cid:105)=((cid:112)3/5cosθ−(cid:112)2/5sinθe−iφ)|H(cid:105)+i((cid:112)3/5cosθ+ spinwave onto a superposition state between |s↓(cid:105) and|s↑(cid:105), by
i measuringtheidlerphotonalongaspecificpolarizationstate.
(cid:112)
2/5sinθe−iφ)|V(cid:105), where a normalizing factor is omit- Thenaseriesofrotationsbetweentwobasisstatesarecarried
ted. The projection of idler photon can be done through out. At last, the spinwave qubit state is verified by convert-
standard linear-optics method [22]. Before performing ing it back to a signal photon during the read process. The
qubit rotations on this spinwave state, we first charac- levels chosen here are designed to provide almost identical
tize the state preparation fidelities. The spinwave state retrieval efficiencies for both spinwave states, since the tran-
sition strengths of |F(cid:48) = 2,m = −1/+1(cid:105) → |g(cid:105) are the
is measured by applying a strong read pulse which con- f
same. b, Two elemental rotations in the Bloch sphere. One
verts |s (cid:105) to a σ− polarized signal photon and converts
↓ is the rotation along z axis, which is due to different Larmor
|s (cid:105) to σ+ polarized signal photons. Arbitrary single-
↑ precession frequencies of these two spin states. The other is
qubitmeasurementisrealizedthroughmeasuringthesig-
alonganaxisinthex-yplane,wheretheorientationangleϕis
nal photon in arbitrary polarization bases, which is also determinedbytherelativephasebetweentwoRamanbeams.
one of the highlights of our system. In our experiment, c, Experimental Layout. The write beam has a waist of 150
we select six spinwave states to prepare: |s (cid:105), |s (cid:105), |s (cid:105), µm,apowerof1.5µWandadetuningof-10MHz. Theidler
√↓ ↑ D
|s (cid:105), |s (cid:105) and |s (cid:105), with |s (cid:105) = 1/ 2(|s (cid:105) ± |s (cid:105)) photon is collected in the direction with an angle of 1.5◦ rel-
A R L√ D/A ↓ ↑
ative to the write beam, and has a mode waist of 100 µm. It
and |s (cid:105) = 1/ 2(|s (cid:105) ± i|s (cid:105)). We make use of
R/L ↓ ↑ isfurtherfrequency-filteredwithaFabry-Perotcavity. While
quantum state tomography and the maximum likelihood
the signal photon is filtered by a Rubidium vapor filter to
method[23] to characterize the prepared states, and cal-
avoidbackreflectionsintotheidlerchannalduringwritepro-
culate the fidelities in comparison with the target states. cess. The polarization of a single Raman beam is prepared
Results are shown in Tab.I, and an average fidelity of to provide both helicities for stimulated Raman transitions
97.2%isachieved. Standarderrorsforeachstatearecal- between |s (cid:105) .
↓\↑
culated by poissonian sampling of the measured counts
for 200 times.
Inordertorealizearbitrarysingle-qubitmanipulation, MHz/Guass. Thus the relative phase between |s /s (cid:105)
↓ ↑
the ability of rotation along two orthogonal axes in the will evolve as φ(t) = ω t, which effectively is a rotation
L
Blochspherearerequired. AsshowninFig. 1(b),wefirst around z-axis in Bloch sphere. A specific rotation angle
make use of the Larmor procession process to realize the can be realized by changing the duration of the Larmor
rotationofR (φ)whichcorrespondstorotationalongan procession. Withthebiasfieldapplied, wemeasuredthe
z
axis in the direction of poles. As the magnetic moment Larmorfrequencytobeω =2π×180kHz. Therotation
L
of the |s (cid:105) state is zero, the Larmor phase of this state is matrix of Larmor operation is,
↑
constantduringtheexperiment. Whileforthe|s (cid:105)state,
↓
(cid:20) (cid:21)
itsmagneticmomentprecessesaroundthebiasmagnetic eiΩLtL 0
R (t )= ,
field, with an angluar frequency of ω = 2πB × 1.4 z L 0 1
L 0
3
Idler State Fidelity(%) Spinwave State Fidelity(%) frequency, the maximum population leakage onto |saux(cid:105)
s during Rabi flopping is estimated to be ∼ 1%. Thus
H 98.3(1.6) 97.6(1.0)
we can restrict the population mainly flops between our
s
V 97.4(1.2) 99.0(0.6) target spinwave states of |s↑(cid:105) and |s↓(cid:105).
s Then we can successfully create a pure rotation along
D 96.2(1.2) D 95.1(1.3)
a definite axis in the x-y plane of the Bloch sphere when
s
A 97.6(1.1) A 96.3(1.3) a Raman pulse is applied. Note the angle ϕ between the
R 97.4(1.1) sR 97.5(1.3) Ramanrotationaxisandxaxisisdeterminedbytherel-
ative phase ϕ between k+ and k− Raman components.
s R
L 99.1(0.5) L 97.8(1.2) An additional merit of our scheme is that this phase can
be precisely controlled and it’s stable in itself, excluding
TABLEI:Fidelitiesofstatepreparations. (Left)Fidelitiesbe- other electronic phase stabilizing controls. The rotation
tween measured and expected signal photon states for differ- matrix of Raman operation can be written as,
entidlerphotonpolarizations. Fidelitiesarerelativelyhigher
due to the polarization needed is more accurate to prepare.
(tRonigshtta)teFsidfoelritdieiffsebreetnwtesepninmweaavseursetadtaens.dexpectedsignalpho- R(cid:126)n(tR)=(cid:20)−ie−cioϕsRφs(itnRφ)(tR) −ieciϕoRsφsi(ntRφ)(tR)(cid:21),
whereφ(t )=Ω t /2,andt isthedurationofapplied
R R R R
where tL is the duration of Larmor operation. Ramanpulse. BychoosingϕR =0,rotationalongx-axis
We make use of stimulated Raman transition to re- can be realized. While with ϕR = −π/2, the rotation is
alize the rotation along an axis in the equatorial plane. along y-axis, as shown in Fig. 1(b). With Raman beam
The detailed scheme and experimental arrangement are paramters stated above, the measured Raman Rabi fre-
shown in Fig. 1. First, we notice that if both Raman quencyisabout190kHz. Thedeviationwiththeoretical
beams are red (or blue) detuned relative to the |F(cid:48) =2(cid:105) predicationsmaybeduetotheimperfectinterferencebe-
and |F(cid:48) =1(cid:105) hyperfine states of the D1 line, destructive tween two Raman transition channels.
interference between these two Raman transition chan- We first evaluate the gate performances with quan-
nelswilloccur. Toavoidthis,wesetthefrequencyofthe tum state tomography [23] by measuring the quantum
Raman beams to be at the middle of these two states, state of the spinwave after a single-qubit operation and
as shown in Fig. 1(a), where the detuning is ∆ ≈ 408 compare it with the ideal target state using F(ρ ,ρ ) ≡
√ √ 1 2
MHz. Thus these two Raman transitions constructively {tr[( ρ1ρ2 ρ1)12]}2. Rotations of Rx,Ry,Rz, are car-
interfere with each other, and lead to faster rotation op- ried out for a series of rotation angles as shown in Fig.
erationswithlimitedlaserpower. Second,it’sfoundthat 2(a). For Raman R ,R rotations, the prepared ini-
x y
whentheRamanbeamsareapplied,theycouplenotonly tial spin state is |s (cid:105). And for Larmor R rotation, the
↓ z
thetransitionof|s↑(cid:105)↔|s↓(cid:105)butalsoanunexpectedtran- initial spin state is projected by choosing the state of
sitionof|s↑(cid:105)↔|saux(cid:105)≡|F =2,mF =+2(cid:105). Asthestate idler photon to be |H(cid:105). In Fig. 2(c ∼ e), Stokes pa-
|saux(cid:105) cannot be converted to signal photons in the read rameters are presented for each rotation, and are com-
process, we should minimize the population leakage into pared with theoretical curves which fit well. The state
thisstate, andrestricttheRabifloppingtooccurmainly fidelities of R ,R ,R operations averaged for all rota-
x y z
between |s↓(cid:105) and |s↑(cid:105). tionanglesare98.6(0.6)%,98.8(0.5)%and99.1(0.3)%re-
The Raman beam in our setup has a total power spectively. Note here the expected state is generated
of about 7 mW, with a waist of 1.9 mm. The po- by applying ideal operations on the measured density
larization state of the beam is adjusted to be |ψ (cid:105) = matrix of initial states, which takes account of imper-
R
(cid:112)1/7|σ+(cid:105) + eiϕR(cid:112)6/7|σ−(cid:105), where σ+/σ− terms serve fections of state preparations and diagnose the rotation
as the k+/k− Raman beams in Fig. 1(a) respectively, operations exclusively. To show how well the quantum
and ϕ is the relative phase between these two terms. nature is preserved after operations, non-classical corre-
R
With such a polarization, AC Starks shifts due to the lations between idler and signal photons are shown in
Raman beams of |s (cid:105), |s (cid:105) and |s (cid:105) states are calcu- Fig. 2(g) and(h), where the visibilities are both well
↓ ↑ aux
lated to be about +40 kHz, -140 kHz and +240 kHz, above 6. Moreover an arbitrary rotation can be decom-
respectively. Thus the frequency splitting between |s↑(cid:105) posed as R(cid:126)n(φ) = Rz(α)Ry(β)Rz(γ)eiδ [24] by combin-
and |s↓(cid:105) induced by AC Starts effect is -180 kHz, which ing Ry and Rz operations. As an example, we choose to
just cancels the Zeeman splitting (+180 kHz) between realize R(cid:126)n(φ) = exp(−i(cid:126)n·(cid:126)σφ) with (cid:126)n = (√13,√13,√13),
these two states. However, the overall frequency split- which is illustrated in Fig. 2(b). To make the rotation
tingbetween|s (cid:105)and|s (cid:105)areshiftedtoanevenlarger plane meet with zero point, we prepare the initial spin
aux ↑
value of 560 kHz. The theoretical two-photon Rabi fre- state on |ψ (cid:105) = cos3π|s (cid:105)+sin3π|s (cid:105), with a state fi-
s 8 ↓ 8 ↑
quencies of |s (cid:105) ↔ |s (cid:105) and |s (cid:105) ↔ |s (cid:105) are both 240 delityof96.6(1.7)%. Thenaseriesofrotationoperations
↑ ↓ ↑ aux
kHz. As the two-photon detuning of the |s (cid:105) ↔ |s (cid:105) are performed on the initial state, with roation angles
↑ aux
transition is significantly larger than the Raman Rabi φ = 0, 1 π,...,11π. For each rotation, it’s decomposed
12 12
4
z s s likelihood method [25]. In our experiment quantum pro-
a b z
cesstomographyismeasuredforPaulioperations(σ ,σ
x y
and σ ) and the Hadamard gate, with the results shown
ii i z
y
s inTab.II.Theprocessfidelityaveragedforthesefourop-
R
erations is calculated to be 94.7(7)%. With the process
y
x iii sR iv sDx fiofdaelpitrioesc,esosn,ewchainchesistidmeafitneedthaesatvheeraagveersatgaeteoffisdtealtietyfiFdealvie-
s ties between input and output states, where input states
D
s are randomly selected from the Bloch sphere. It’s shown
-x -y -z
that,theaveragefidelityofaprocess,canbesimplifiedto
1 1
c ϕR=0 d ϕR=−π/2 evaluate the mean state fidelities of six pure input states
0 0 locating at the cardinal points [26]. The average fidelity

i ii is related with the process fidelity by, F = dFproc+1,
-1 -1 ave d+1
0 1 2 3 4 5 0 1 2 3 4 5 where d is the dimensionality of the system and d = 2
Raman (µs) Raman (µs) in our case[27]. Both measured average fidelity F and
e 1 iii f 1 iv average fidelity Fatvhe derived from Fproc are also laisvteed in
Tab.II. We note that F is slightly higher than Fth ,
0 0 ave ave
which is mainly due to the fact that the prepared ini-
-1 -1 tial states are not pure and thus a different definition
0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8
Lamor (µs) Pha se/π of F(ρ1,ρ2) is adopted. Besides, we notice that, the
g h40 constraint of completely positive mapping on the fitted
40 ϕ=0
R R H process matrixes, also reduces the calculated process fi-
g220 L g220 V delities, thus further lowering down the Fth values in
ave
0 0 Tab.II.
1 2 3 4 5 1 2 3 4 5
Raman (µs) Lamor (µs) Traget Operation Fpro（c %） Fav（e %） Fa vthe（%）
FIG. 2: Arbitrary rotations of the Spinwave qubit. a-b, Il- x R= 2, tR=2.6(cid:43)s 95.3(1.1) 97.6(1.6) 96.9
lustratons of state rotations in the Bloch sphere. a, rota-
tions along x-, y-, z-axis. b, special rotation along axis (cid:126)n = y R=0, tR=2.6(cid:43)s 93.5(1.8) 97.7(1.5) 95.7
{√13,√13,√13}, and the rotation plane crosses zero point. c-f, z tL=2.7(cid:43)s 95.3(1.1) 95.3(3.1) 96.9
Expreimentalresultsofrotations,dottedpointsaremeasured
t =2.7(cid:43)s
values of each axis, and compared with theoretical curves. c, H L 94.8(1.8) 98.6(1.3) 96.6
Rx rotations realized by Raman operations with ϕR =0. d, R= 2, tR=1.3(cid:43)s
R rotations by Raman operations with ϕ = −π/2. e, R
y R z
rotations by Larmor operations. f, R(cid:126)n rotations by combina- TABLE II: Process Fidelity of Pauli and Hadamard Opera-
tions of Rz and Rx rotations. g-h, non-classical correlations tions. Pauli operations and Hadamard gate are realized by
betweenidlerandsignalphotonsafteroperationsfordifferent choosing appropriate Raman and Larmor parameters. The
states of signal photons. g, the initial spin state is on |s↓(cid:105), measured process matrices by maximum likelihood method
Raman rotations with ϕR =0 are applied. h, the initial spin arecomparedwithmatricesofidealoperations,todetermine
stateisapproximatelyon|sA(cid:105)withtheidlerphotonprojected theprocessfidelities. Averagestatefidelitiesofeachoperation
along |H(cid:105) state. from measured results are also shown here.
To summarize, by making use of stimulated Raman
into values of α,β,γ, with the global phase term δ ne- transitionandcontrolledLarmorprocession,wehaveim-
glected. TheresultisshowninFig. 2(f),withanaverage plementedsingle-qubitoperationsforsinglespinwavesin
state fidelity of 98.3(0.8)%. an atomic ensemble quantum memory for the first time.
Inordertogivemorecompletecharacterizationforour Wehavemadeuseofquantumstatetomographytomea-
single-qubit gate, we make use of quantum process to- sure the target states for arbitrary rotations. Average
mography[24]. Anarbitrarysingle-qubitoperationonan state fidelity for R , R and R rotations is measured to
x y z
inputstateρin canbefullydescribedbyaprocessmatrix be 98.8(3)%. We have also adopted the method of quan-
χ, which is defined as ρ = (cid:80)3 χ σ ρ σ†, where tum process tomography to characterize the Pauli oper-
out i,j=0 i,j i in j
σ arePaulimatriceswithσ =I,σ =σ ,σ =σ ,σ = ations and the Hadamard gate. Average process fidelity
i 0 1 x 2 y 3
σ . Thedistancebetweenparacticalandidealoperations is measured to be 94.7(7)%. By making use of Raman
z
can be characterized by process fidelity, F =tr[χ χ ]. beams with better intensity homogeneity and actively
proc 1 2
Inordertomeasuretheprocessmatrixforasingle-qubit controllingtheambientmagneticfield,evenhigherfideli-
operation, we select six different spinwave states, which tiescanbegot. Ourworkenrichestheexperimentaltool-
arelistedinTab.I,asinputstatesandcarryoutquantum boxofharnessingatomicensemblesforhigh-performance
state tomography for each output state. In order to get quantum memories, thus could possibly have lots of ap-
a physical process matrix we make use of the maximal plications in future scalable quantum networks [28].
5
This work was supported by the National Natu- 2011CB921300), and the Chinese Academy of Sciences.
ral Science Foundation of China, National Fundamen- B.Z. and X.-H.B. acknowledge support from the Youth
tal Research Program of China (under Grant No. Qianren Program.
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