Sideband cooling while preserving coherences in the nuclear spin state in group-II-like atoms

0.45 MB·English

by  Iris Reichenbach

Sideband cooling while preserving coherences in the nuclear spin state in
Group-II-like atoms
Iris Reichenbach∗ and Ivan H. Deutsch
Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA
We propose a method for laser cooling group-II-like atoms without changing the quantum state
of their nuclear spins, thus preserving coherences that are usually destroyed by optical pumping
in the cooling process. As group-II-like atoms have a 1S0 closed-shell ground state, nuclear spin
andelectronicangularmomentumaredecoupled,allowingfortheirindependentmanipulation. The
hyperfine interaction that couples these degrees of freedom in excited states can be suppressed
throughtheapplicationofexternalmagneticfields. Ourprotocolemploysresolved-sidebandcooling
on the forbidden clock transition, 1S0 →3P0, with quenching via coupling to the rapidly decaying
8 1P1 state, deep in the Paschen-Back regime. This makes it possible to laser-cool neutral atomic
qubits without destroying the quantum information stored in their nuclear spins as shown in two
0
examples, 171Yband 87Sr.
0
2

n Theabilitytocoherentlycontrolatomshasallowedfor regime, as in optical lattices currently under consider-
a
majoradvancesinprecisionmeasurementfortestsoffun- ation for next-generation atomic clocks based on opti-
J
damentalphysics[1]andnewproposedapplicationssuch cal frequency standards [7]. We will explain our cooling
9
as quantum simulators and universal quantum comput- scheme in detail using two example species, 171Yb and
2
ers[2]. Thoughtraditionallyalkali-metal(group-I)atoms 87Sr,bothofwhichareactivelystudiedinthelaboratory

4 are used in many of these applications given our ability for application to optical clocks [8, 9]. The I = 1/2
v to laser cool and trap them with relative ease, alkaline- nucleus of 171Yb is a natural qubit for storing quan-
0 earth-metal (group-II) atoms are also being considered tum information. In the case of 87Sr, whose nucleus has
2
given long-lived metastable states, accessible lasers for spin I = 9/2, we will choose two sublevels in the 10-
1
2 cooling and trapping, and a rich (but tractable) inter- dimensional manifold to encode a qubit. Exquisite opti-
0 nal structure. One distinguishing feature of group-II vs. cal control of the nuclear spin of 87Sr has recently been
7 group-Ielementsisthatthegroundstateisaclosed-shell demonstrated by Boyd et al. [10].
0 1S with zero electron angular momentum, and thus no In order to preserve the quantum state of a qubit en-
/ 0
h hyperfinecouplingtothenuclearspin. Thiswillallowus coded in a nuclear spin while laser cooling, the spin co-
p to independently manipulate nuclear and electronic de- herencesmustbetransferredinbothexcitationandspon-
-
t greesoffreedominwaysthatarenoteasilyaccomplished taneous decay. Optical fields interacting with atoms via
n
with group-I atoms. theelectricdipolecoupledirectlytotheelectrons,affect-
a
u From the perspective of storing and manipulating ing nuclear spin states only indirectly via the hyperfine
q quantum information, nuclear spins in atoms are attrac- interaction. A key requirement of our protocol is, there-
: tive given their long coherence times and the mature fore, to excite states with negligible hyperfine coupling,
v
i techniques of NMR [3]. Recently, we have studied a and/ordecouple electronsfrom nuclear spin throughthe
X
protocol in which nuclear spins can mediate quantum application of a sufficiently strong magnetic field that
r logic solely throughquantum statistics via an“exchange the Zeeman effect on the electrons dominates over the
a
blockade” [4]. This allows us to isolate nuclear spins hyperfine interaction (Paschen-Backregime [11]). More-
from noisy perturbing forces while simultaneously pro- over,weseektorecoolatomsintrapstoneartheground
vidingamechanismbywhichstrongqubit-qubitinterac- vibrationalstatewhichrequiresresolvablemotionalside-
tion can be implemented via cold collisions. As is typi- bands.
calin atomic quantum logicprotocols,heating of atomic The dual requirement of very narrow linewidths and
motion degrades performance. In atoms with group-I- small hyperfine coupling leads us to consider cooling on
likeelectronicstructures,lasercoolingcannotbe usedto the weakly allowed intercombination “clock” transition
re-initialize atomic vibration in the course of quantum 1S 3P in a “magic wavelength” optical lattice [7].
0 0
→
evolution because this is accompanied by optical pump- Cooling to the ground vibrational state can proceed by
ingthaterasesthe qubitstoredinthe internaldegreesof coherentexcitationofaπ pulseonthefirstredsideband,
freedom. One mustthereforeresortto sympathetic cool- 1S ,n 3P ,n 1 , where n is the vibrational quan-
0 0
| i→| − i
ing with another species [5]. In this Letter we propose tum number, followed by recycling to the ground state
andanalyzeanewprotocolinwhichonecancontinuously with Lamb-Dicke suppression of recoil. Relatively large
refrigerateatomicmotionwhilesimultaneouslymaintain- vibrational frequencies of ω /2π =90(260) kHz have al-
v
ing quantumcoherencestoredin nuclearspins. Our pro- readybeen realizedfor Yb (Sr) in suchan opticallattice
posal is based on resolved-sideband laser cooling [6] of in 1D [12, 13]. In principle, given these tight confine-
group-II-like atoms, tightly trapped in the Lamb-Dicke ments and the tiny linewidth γ of the clock transition
2
(few millihertz), the motionalsidebandsareverywellre-
solved, and extremely cold temperatures can be reached
with a steady state mean vibrational excitation on the
order of n γ2/(2ω )2 =10−15(10−18) [6]. Of course,
v
h i≈
the true achievable temperature will depend strongly on
the suppression of the different heating mechanisms.
Ourconcernistocarryoutthiscoolingwhilemaintain-
ing nuclear spin coherence. To good approximation, the
3P state has total electron angular momentum J = 0,
0
andunderthis condition,thereisnointeractionbetween
electrons and nuclear spin, as in the ground state, 1S .
0
| i
However, the clock transition is J = 0 J = 0, and
laser excitation is only allowed because→in the excited FIG. 1: Schematic cooling and readout level-diagram for
171Yb (analogous for 87Sr). Sideband cooling in a trap oc-
state, the hyperfine interaction leads to a small admix-
curs via excitation on the 1S0 → 3P0 clock transition (vi-
ture of the higher-lying P states [10, 14]. The nuclear brational levelsnot shown),quenchedbycouplingtothe 1P1
spin projection mI is thus not an exact quantum num- state. An external magnetic field splits the 1S0 and 3P0 nu-
ber for the magnetic sublevels in 3P0 ; a very small ad- clear sublevels (spin-up and spin-down shown) and ensures
| i
mixture of electronic angular momentum renders mF a preservation of the nuclear spin during resonant excitation
good quantum number. This leads to two effects which and repumping. The dashed arrows demonstrate a possible
we must address in the context of transferring the nu- readout schemeby resonance fluorescence.
clear spin state. While the clock transition is only al-
loweddue to the hyperfine interaction, it is still the case
that the sublevels of 3P are almost pure m , and a π-
0 I
| i
polarized pulse will preserve nuclear spin projections in the nuclear spin and electronic degrees of freedom of an
excitationontheclocktransition[10]. Moreimportantly, excitedstatearedecoupled(no hyperfineinteraction),in
even a small admixture of electron angular momentum a product state with zero projection of electron angular
can strongly affect the magnetic moment, so that the g- momentum, |ei≈|I,m′Ii⊗|J′,m′J =0i, all decay chan-
factorofthe excitedstatediffersfromthatoftheground nels to the ground state are indistinguishable. To see
state [10]. This implies that in a bias magnetic field, the this note that the selection rules dictate that the elec-
π-transitions|1S0,ni⊗|mIi→|3P0,n−1i⊗|mF =mIi tric dipole matrix element satisfies, he|dq|gi=hJ′,m′J =
will have different resonant frequencies for different val- 0|d0|J =0,mJ =0iδq,0δmI,m′I. Onlyπ-transitionsareal-
ues of m . The protocol for the transfer of nuclear spin lowed and all decay channels have the same polarization
I
coherence from the ground state to the clock state will without change of nuclear spin projection. Moreover, in
thus depend on the details of the experimental condi- the m′J = 0 manifold, the g-factor is solely nuclear, gI,
tions, i.e., the relative size of the differential Zeeman equivalenttothatinthegroundstate. Thus,theZeeman
splitting in the given bias field when compared to the splitting in excited and ground states will be equal and
vibrational spacing in the trap. Under any operating decay channels will not be distinguished by frequency.
condition, these coherences can be transfered due to the Deviations of the excited state from a nuclear-electron
extremely narrow linewidth of the clock transition, be it productstate (due to residualhyperfine interaction)will
throughsequentialaddressingofeachsublevelinaseries leadtodecoherencewhentherearenuclearspinchanging
of narrow band pulses [10], or in a short pulse that does decays (polarization distinguishable) and/or when the
notresolvethe differentialZeemanshift but does resolve different channels are frequency resolvable (i.e. have a
the sidebands. Finally, in this step, one must also take frequency difference that is not negligible compared to
into account that the differential g-factor leads to dif- the decay linewidth).
ferential relative phases between the sublevels; this is a In the case of the clock state 3P , spontaneous decay
0
unitary transformation which can be reversed. is completely due to the admixture of higher P states
Oncetheatomistransferredtotheclockstate3P ,the andthedecaychannelswithdifferentpolarizationsoccur
0
cooling cycle must be completed through a spontaneous withaprobabilityproportionaltotherespectiveClebsch
eventthatreturnstheatomtoitsgroundelectronicstate Gordan coefficients. Therefore, the nuclear spin is not
1S , with Lamb-Dicke suppression of recoil. Of impor- preservedduring spontaneous decay. Moreover,the long
0
tance in our protocol is that this occurs without deco- lifetime makes the cooling cycle too long for practical
herence of the nuclear spin state. The key requirement purposes. We thus considerquenchingthe clockstateby
for spontaneous transfer of coherences is that the decay pumpingtothe shortlived1P state(seeFig.1)[15,16].
1
paths of the different sublevels are indistinguishable, i.e. Thisleveldecayswithveryhighprobabilitytotheground
the decay channels cannot be distinguished by their po- state 1S and has a very broad linewidth, Γ/(2π) > 10
0
larization or frequency. Under circumstances in which MHz. Direct excitation 3P 1P , is weakly magnetic-
0 1
→
3
dipole-allowed, and has been considered in the context
of electromagnetically induced transparency [17]. Alter-
natively,we canquenchthe clock state via a two-photon
3P 3S 1P (m = 0) transition, off resonance
0 1 1 J
→ →
fromtheintermediatestate. Thishastheadvantagethat,
thoughonelegisanintercombinationline,alltransitions
areelectric-dipoleallowed,leadingtolargerdepopulation
rates for the same intensity.
In order to obtain the product states of nuclear-
electron spin degrees of freedom in 1P as required for
1
preserving nuclear spin coherence, we employ a suffi-
ciently strong magnetic field. The 1P subspace is gov-
1
erned by the Hamiltonian including Zeeman interaction,
magnetic spin coupling and quadrupole effects,
3(ˆI Jˆ)2+3/2ˆI Jˆ I(I +1)J(J +1)
Hˆ =AˆI Jˆ+Q · · −
· 2IJ(2I 1)(2J 1)
− −
+g µ Jˆ B g µ ˆI B, (1)
J B I N
· − ·
where g and g are the relevant electron and nuclear
J I
g-factors. The magnetic hyperfine constants are A/h =
216 Mhz and A/h = 3.4 MHz for 171Yb and 87Sr
− −
respectively [10, 18]. For 171Yb, because I = 1/2, the
quadrupoleconstantQ=0andwecananalyticallysolve FIG.2: Zeemandiagramofthe1P1manifold. (a)In171Ybthe
Breit-Rabiformulaapplies. Thehyperfinesublevelsdecouple
fortheenergyspectrumandeigenstatesusingamodified
toproductstatesofelectronandnuclearspin(Paschen-Back)
Breit-Rabi formula,
with modified linear Zeeman shift. The qubit is encoded the
states |↑i and |↓i in the mJ = 0, subspace. (b) 87Sr Zee-
E
E = HF +g µ Bm man diagram for the 2I +1 = 10 sublevels that asymptote
mJ,mI −2(2J +1) J B F to the mJ = 0 subspace in the Paschen-Back regime (other
subspacesnotshown). Becauseofthelargequadrupoleeffect,
E 4m x
HF 1 F +x2, (2) pairswith±mI arecloselyspacedformagneticfieldsbetween
± 2 r − 2J +1
50and120mT.Qubitscanbeencodedinthesepairswithout
loosing coherence in spontaneous emission.
where x = (g µ +g µ )B/E , m = m +m and
I B J N HF F I J
E = A(J +1/2). The resulting Zeeman diagram can
HF
be seen in Fig. 2. The eigenstates are specified by
|mFi= qcq(mF)|mI =mF −qi|mJ =qi, q =0,1,−1, with equal |mI| are paired and very close in energy (less
with fielPd dependent expansion coefficients cq. For high than 2 MHz), whereas the energy splitting between the
magnetic fields, in the subspace of interest, c0 1, and pairsisontheorderoftensofmegahertz,nearlyasbigas
→
the states are in a product state mI mJ = 0 . De- thelinewidth. Giventheneardegeneracyofthesublevels
| i⊗| i
viations from this limit lead to the residual differential m , we will consider preserving nuclear spin coherence
I
±
g-factor. in a qubit encoded in one of these two-dimensional sub-
Since 87Sr has a large quadrupole constant Q/h = 39 spaces, thereby ensuring that the frequencies of the dif-
MHz [19], the Breit-Rabi formula does not apply. Diag- ferent decay channels are not resolvable.
onalizing Eq. (1) numerically gives the Zeeman diagram Inthefollowingwequantitativelystudythetransferof
shown in Fig. 2 (for an analytic form, see [10]). Each nuclear spin-coherence via spontaneous emission. Con-
of the three subspaces for mJ = 1,0, 1 consists of 10 sider a qubit in the excited state αe, +β e, which
sublevelswhichasymptotetothe2I+1−projectionsasso- decays to the ground state g . W| e↑iwish |to↓tiransfer
| i
ciatedwiththe nuclearspin ofthis isotope,I =9/2. For the qubit into the superposition αg, +β g, . Here
| ↑i | ↓i
fields of order 10 mT or greater the Zeeman effect dom- e, ( ) = 1P ,m = m and g, ( ) = 1S , m .
1 F I 0 I
inates over the hyperfine coupling and the sublevels ap- |Th↑e e↓violuti|on of the at±om cian be|des↑cr↓ibied b|y a m±asteir
proachproductstatesofelectronandnuclearspin. How- equation in Lindblad form,
ever,the residualquadrupole interactionleads to a com-
i 1
plex spectrum that is not described by a linear Zeeman ρ˙ = [H,ρ] (L†L ρ+ρL†L 2L ρL†), (3)
shift with an effective g-factor. For B-fields between 50 −~ − 2Xq q q q q− q q
and120mT,thesubspacewithm =0isnearlyflat. Ad-
J
ditionally, due to the quadrupole symmetry, the states where L = √Γ c (m )g,m = m q e,m
q mF q F | I F − ih F|
P
4
lows for final readout of the nuclear spin state via res-
onance fluorescence. Consider first 171Yb. According
to Eq. (1), the splitting between the m ,m = 1/2
J I
| ± i
states is given by ∆E = g µ B+Am , where A is the
I N J
hyperfine constant for the 1P state. Given 171Yb in a
1
magnetic field of 1 T, the splitting between the m =
J
| 6
0,m = 1/2 states is on the order of 200 MHz. The
I
± i
splitting between the and in the 1S groundstate
0
|↑i |↓i
arises solely due to the Zeeman interaction with the nu-
clear spin, equal to 7 MHz for the applied field (g 1).
I
≈
This makes it possible to selectively drive the transition
1S ,m = +1/2 1P ,m = 1,m = +1/2 as
0 I 1 J I
| i → | − i
shown in Fig. 1 for readout.
For 87Sr in a magnetic field of few millitesla, the split-
ting between neighboring substates in m = 1, domi-
J
±
nated by the hyperfine interaction, is on the orderof the
linewidth and therefore not well resolved. Thus we can-
notselectivelyexciteagivenm levelinthesamewayas
I
wedescribedfor171Yb. Wecan,however,takeadvantage
FIG.3: (Coloronline)Fidelityfortransferofcoherence,F =
Γ′2/(Γ2+δ2) (see text), as a function of magnetic field. (a) ofthedifferenceintheg-factorsofthe1S0 and3P0 states
The spin-1/2 qubit of 171Yb. (b) Different choices of qubit inSrinordertomanipulate populationinthe individual
encoding as ±mI in 87Sr. magnetic sublevels. Consider a robustcontrolpulse that
transfers all population from 1S to 3P level. The dif-
0 0
ference in Zeeman splitting implies that a narrow-band
π-pulse can selectively return population in a chosenm
(with c defined above) are the “jump operators” for I
q
leveltothegroundstate,leavingtheremainingsublevels
the spontaneous emission of a photon with polarization
π,σ+,σ−, (q = 0,+1, 1). Spin coherences, described shelved in the metastable clock state. The occupation
− of level m can then be probed via fluorescence on the
by off-diagonal matrix elements, satisfy I
1S 1P transition. The procedure of shelving and
0 1
→
ρ˙(↑e,↓) =(−i∆e−Γ)ρ(↑e,↓) (4a) qacuteinvtaitailnlyg mtheeasstuarteepoofpiunltaetrieosntimnIe,atchhussuabllleovwesl.us to se-
ρ˙(g) = i∆ ρ(g) +Γ′ρ(e) (4b)
↑,↓ − g ↑,↓ ↑,↓ We thank Chad Hoyt, Leo Hollberg, and Jun Ye
where ∆ arethe Zeemansplittings ofthe excited and for helpful discussions. This work was partly sup-
e(g)
ground qubits and Γ′ = Γc (m )c ( m ). Solving for ported by Grants No. ARDA DAAD19-01-1-0648 and
0 F 0 F
−
the ground state coherences in the limit t Γ, when all No. ONR N00014-03-1-0508
≫
population and coherence resides in the ground states,
Γ′
ρ(g)(t)=ρ(e)(0) ei∆gt, (5)
↑,↓ ↑,↓ Γ iδ
− ∗
Electronic address: [email protected]
where δ = ∆g ∆e is the differential energy shift. [1] For a review see Nature(London) 416, 1 (2002).
−
The parameters δ and Γ′ Γ are functions of applied [2] For a review see D. Jacksch Contemp. Phys. 45, 367
−
magnetic field, both approaching zero in the perfect (2004).
Paschen-Back limit. For finite fields, imperfect decou- [3] L.M.K.VandersypenandI.L.Chuang,Rev.Mod.Phys.
pling of electron and nuclear spin results in imperfect 76, 1037 (2005).
[4] D. Hayes and P. S. Julienne and I. H. Deutsch, Phys.
transfer of coherences, characterized by the “fidelity”
Rev. Lett.98, 070501 (2007).
= ρ(g) 2/ρ(e)(0)2 = Γ′2/(Γ2 + δ2). The resulting
F | ↑,↓| | ↑,↓ | [5] A. J. Daley, P. O.Fedichev, and P. Zoller, Phys.Rev. A
transfer of coherence as a function of the magnetic field 69, 022306 (2004); M. D. Barrett et al., Phys. Rev. A
can be seen in Fig. 3. For 171Yb, a magnetic field of 1 68, 042302 (2003).
T is required to reach a fidelity of 99%, whereas for the [6] F. Diedrich et al.,Phys. Rev.Lett. 62, 403 (1989).
m qubit in 87Sr a magnetics field of order 10 mT is [7] M. Takamoto et al. Nature (London) 435, 321 (2005).
± I [8] A.D.Ludlowet al.,Phys.Rev.Lett.96,033003 (2006).
sufficient to obtain this threshold.
[9] C.W. Hoyt et al.,Phys. Rev.Lett. 95, 083003 (2005).
Although we have argued throughout this Letter that
[10] M.M.Boydetal.,Science314,1430(2006);M.M.Boyd
one needs to decouple nuclear and electronic degrees of
et al.,arXiv:0704.0912v1.
freedom in order to preservenuclear spin coherence dur- [11] C.J.Foot,AtomicPhysics(OxfordUniversityPress,Ox-
ing laser cooling, we still require an interaction that al- ford, 2004).
5
[12] AndersBruschetal.,Phys.Rev.Lett.96,103003(2006). 20, 977 (2003).
[13] Z.W.Barber et al.,Phys.Rev.Lett.96, 083002 (2006). [17] R. Santraet al.,Phys.Rev. Lett.94, 173002 (2005).
[14] S. G. Porsev, A. Derevianko, and E. N. Fortson, Phys. [18] R.W.BerendsandL.Maleki,J.Opt.Soc.Am.B9,332
Rev.A 69, 021403(R) (2004). (1992).
[15] U.Sterr et al., C. R.Physique 5, 845 (2004). [19] H.-J. Kluge and HattoSauter Z.Phys. 270, 295 (1974).
[16] A.Curtis,C.OatesandL.Hollberg,J.Opt.Soc.Am.B