Optical pumping into many-body entanglement

0.5 MB·English

by  Jaeyoon Cho

Optical pumping into many-body entanglement
Jaeyoon Cho,1 Sougato Bose,2 and M. S. Kim1
1QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK
2Department of Physics and Astronomy, University College London,
Gower St., London WC1E 6BT, United Kingdom
(Dated: January 18, 2011)
We propose a scheme of optical pumpingby which a system of atoms coupled toharmonic oscil-
lators is driven to an entangled steady state through the atomic spontaneous emission. It is shown
thattheopticalpumpingcanbetailoredsothatthemany-bodyatomicstateasymptoticallyreaches
an arbitrary stabilizer state regardless of the initial state. The proposed scheme can be suited to
various physicalsystems. In particular, theion-trap realization is well within current technology.
Generating entanglement is one of the landmarks of atoms, however, seems daunting and somewhat counter-
modern physics, which has provoked a huge body of intuitive at first sight because the spontaneous emission
1 work. Entanglement—quantum correlation—is distin- occurs independently in each atom, destroying both the
1 guished from the classical counterpart by its coherent correlationand coherence between the atoms.
0 nature [1]. While the correlation itself can be naturally
2 built up by letting multiple bodies mutually interact, In this paper, we show that the concept of optical

n the difficulty usually arises in retaining the coherence, pumping can indeed be applied to the cases of many
a since opening aninteractionchannelinevitably brings in atoms when they are coupled to discrete quantum sys-
J
adversarial environmental effects as well. Consequently, tems, exemplified here by harmonic oscillators. To be
6
both the theoretical and experimental efforts have been specific,we showthatasfarasthe geometrypermits,an
1
devoted mostly to finding a way to isolate the system, arbitrary stabilizer state (a broad class of many-body

] whereby the problem reduces to identifying proper time entangled states) of the atoms can be generated as a
h pointsatwhichthe unitarydynamicsgivesausefulform steady state through the atomic spontaneous emission.
p
ofentanglement[2]. Thisrequirespreparationofdefinite This contrasts with all previous works, mostly with two
-
t initialstatesandprecisecontrolofthesystemparameters atoms, which have rather been case studies simply re-
n
a andtiming. Inaddition,theentanglementgenerationcan vealing that competition between a continuous driving
u be aided by measurements [3]. andthe environmentaldecaycanleadto aparticularen-
q tangled(but far frommaximally entangled) steady state
[
A conceptually different approach is to tailor an open [4,5]orhaveshownthatatomsdrivenbyentangledfields

2 quantum system in such a way that its irreversible dy- canhaveasteady-stateentanglement[6,7]. Oursystem-
v namics drives the system into a steady state that is en- atic study, onthe other hand,suggestsa clear-cutmech-
8 tangled [4–9]. This approach has apparent advantages anismtogenerateabroadclassofentangledstateswitha
8
over those relying on the unitary evolutions. Firstly, the high fidelity using classicalpumping fields. Recent stud-
0
generated entangled state is robust against decoherence ies offer formalisms for engineering system-environment
4
. because the intrinsic decoherence channel of the system couplings to obtain a desired many-body entanglement
8
itself is exploited as a resource. Secondly, the steady as a steady state [8, 9]. However, as the resulting Lind-
0
0 state is determined by the relationship (e.g., ratio) be- blad operators are multipartite ones, which require the
1 tween different parameters, not by their exact values or interaction between the system and the bath to be ef-
: temporal profiles, and moreover is independent of the fectively many-body, the formalism does not necessarily
v
i initial state. This largely alleviates the otherwise nec- make its physical realization apparent, except for some
X
essary manipulation and control mentioned above and special instances leading to only a restricted set of en-
r alsoprovidesextrarobustnessagainstfluctuationsofthe tangled states [8] or those aided in large part by uni-
a
parameters (e.g., of laser fields). In some sense, such a tarymanipulations[12]. Ourscheme,onthe otherhand,
possibility is recognizable from the fact that the envi- deals with realistic physical models: we show that our
ronmental effects in ordinary quantum optical systems ideaisdirectlyapplicabletoion-trapsystemsandalsoto
are well accounted for in terms of the zero-temperature manyothersystemsbyallowingonlysingle-atomunitary
bath [10]. This means that such an environment always transformations in addition. Thanks to these exclusive
playsthe roleof anentropysink that rendersthe system features, our scheme would find straightforwardapplica-
perfectly coherent. For single atoms, optical pumping— tions,e.g.,inestablishingarobustentanglementchannel
more specifically, coherent population trapping—indeed betweencomputationalnodesorcontinuouslystabilizing
usesthisenvironmentaldecay,i.e.,thespontaneousemis- an entangled state generated by a different means. As
sion, to prepare an arbitrary pure superposition of two highlighted in recent literatures, our scheme would also
ground states [11]. Generalizing this idea to multiple have implications in the context of quantum many-body
2
harmonicoscillatorinteractswiththeatominsuchaway
thattheyexchangeenergy,whichcanbedescribedinthe
interaction picture by the following Hamiltonian:
N
H = a gL E L +gR E R +H.c., (1)
ah X (cid:16) j | ijh | j | ijh |(cid:17)
j=1
wherethesubscriptj representsthej-thatom. Thecou-
pling rates gL and gR are assumed to be independently
j j
adjustable. Ontopofthis coupling,we alsoapplyclassi-
calpumping fields withRabifrequencies ωL andωR and
j j
detuningsδL andδR,respectively,asinFIG.1. Forcon-
FIG. 1: Involved atomic levels and transitions. gL,R denotes
venience, let us call the former an L-field and the latter
the coupling rate between the atom and the harmonic oscil-
lator,ωL,R theRabifrequencyofthepumpingfield,δL,R the an R-field. This driving can be described in the interac-
detuning,and ΓL,R thespontaneous emission rate. tion picture by the following Hamiltonian:
N
H = ωLeiδLt E L +ωReiδRt E R +H.c. (2)
p X j | ijh | j | ijh |
simulation and quantum computation [8, 9, 12, 13].
j=1
Stabilizer states represent a large set of entangled
Once the atom is excited, spontaneous emission to L
states including error-correcting codes and graph states | i
and R takes place with rates ΓL and ΓR, respectively,
[14, 15]. Let us employ the following convention. Con- | i
whichcanbedescribedbythefollowingmasterequation:
sider an N-qubit system with a 2N-dimensional Hilbert
space H and M stabilizers Sµ ≡NNj=1uµj with 1≤µ≤ dρ=i[ρ,H +H ]+ N x(ρ), (3)
M, where uµj ∈ {σjX,σjY,σjZ,Ij} is one of the Pauli op- dt ah p Xj=1x=XL,RLj
eratorsor the identity operatoracting on the j-th qubit.
tAhse[SHµil,bSeνr]t =spa0cefocraenvebrey dpiaviirdeadndin(tSoµ)22M=suNbsNjp=a1cIejs,, wΓxherxe ρ iEs tρheEdensxity m1atErix oEfthρe sy1sρtemE andELxj.(ρ)=
(cid:0)| ijh | | ijh |− 2| ijh | − 2 | ijh |(cid:1)
each with dimension 2N M, according to the eigenval- To begin with, suppose only two atoms are coupled to
−
ues of the stabilizers: = (s ,...,s ), theharmonicoscillatorandonlythetransition R E
where µ Ψ = sµ ΨHfor LΨs1,...,sM(=s±11,.H..,sM1 ). OMur is coupled, i.e., g1L = g2L = 0 and g1R = g2R = g|. iW↔e c|ani
aim isSto|puimp all|thie pop|uliat∈ionHinto the subspace denote the state by φ,n , where φ denotes the atomic
| i | i
(+1,+1,...,+1). IfM=N,thedimensionofthisHilbert stateand n then-thexcitedstateoftheharmonicoscil-
sHpace is 1, i.e., it represents a single pure state, but we lator. Not|eithat the coupling between the atom and the
can also consider more general cases where M < N, in harmonic oscillator does not change the number of to-
whichcasethesteadystateisacertainmixedstateinan talexcitationsnt ≡a†a+Pj|EijhE|,i.e.,[Hah,nt]=0.
encoded subspace rather than a pure state. Puttingthepumpingfieldsandthespontaneousemission
The stabilizer states that can be prepared in our aside,the Hilbertspaceisthusdividedintomutuallyde-
scheme depend on the geometry of the arrangement of coupled subspaces according to nt. Furthermore, due to
harmonic oscillators and atoms. For simplicity, let us the coupling, the degeneracy in each excited manifold is
consider a case where one harmonic oscillator is coupled lifted. In the manifold of nt = 1, transitions LR,1
| i ↔
to the atoms. Generalizing this setting to other cases LE,0 and RL,1 EL,0 occur with coupling rate
with more harmonic oscillators is straightforward. The |g, wheireas t|ransitiion↔R| R,1 i 1 (ER,0 + RE,0 )
| i ↔ √2 | i | i
underlying idea is to have the atoms collectively coupled occurs with coupling rate √2g as the atoms collectively
to harmonic oscillators, thereby splitting the otherwise interact with the harmonic oscillator. Consequently, as
degenerate energy levels in the excited manifold of the shown in FIG. 2(a), the excited manifold shows energy
system according to, in our case, the parity. We then splittings,whoseamountsaredeterminedbythenumber
resolve the energy splittings so as to optically pump the of atoms in states R and E . Formally, one can define
| i | i
populationwitha particularparity intothe disjointsub- n = R R + E E andthentheenergysplitting
s Pj| ijh | | ijh |
space. Aswillbecomeapparent,suchanopticalpumping is given by √nsg (except for uncoupled states).
±
is suitable for the generation of stabilizer states. Weresolvetheseenergysplittings byadjustingthede-
We use three-level atoms with two ground levels L tunings to achieve selective pumping of the population.
| i
and R and an excited level E , as shown in FIG. 1. As we aim to generate an entangledstate in the ground-
| i | i
The two atomic transitions L E and R E state manifold (i.e., n = 0) and the spontaneous emis-
t
| i ↔ | i | i ↔ | i
are coupled respectively to orthogonal polarizations of sion always projects the population to the lower man-
light. Letus denote by a (a ) the annihilation(creation) ifold, we need to consider only the two lowest mani-
†
operatorfortheharmonicoscillator. Weassumethatthe folds of n = 0 and n = 1. FIG. 2(b) depicts the
t t
3
toeveryatomR-fieldswithdetunings g, √3g, √5g,
− − −
andsoforth. Here,weexploitred(blue)detuningsforL-
fields(R-fields),butthisisnotmandatory. Forageneral
, we can achieve the same kind of optical pumping
µ
S
by rotating the basis of the operationfor eachj-th atom
accordingtou . Ifu =I ,weturnoffallthecouplings
µj µj j
and pumping fields for the j-th atom so that the atom
(a) is excluded from the pumping process. Otherwise, we
adjust the coupling rates so that gL = α g and gR =
j ∗j j
β g, where α L +β R is the +1 eigenstate of u ,
j∗ j| ij j| ij µj
and instead of an R-field with Rabi frequency ωR, we
j
apply an L-field with α ωR and an R-field with β ωR
∗j j j∗ j
[16]. Likewise, instead of an L-field with Rabi frequency
ωL, we apply an L-field with β ωL and an R-field with
j j j
α ωL.
− j j
A general stabilizer state is now generated by per-
forming the optical pumping processes for all ’s one
µ
S
by one as described above. An important question here
is how the performance, namely, the characteristic time
(b) for the system to reach the steady state, scales as the
number of atoms increases. Unfortunately, it turns out
FIG.2: (a)Energysplittinginthemanifoldofoneexcitation
that the time increases exponentially with the num-
for two atoms (normalization is ignored). (b) Optical pump-
ber of stabilizers M. In order to see this, it is con-
ingfromtheodd-paritysubspacetotheeven-paritysubspace.
venient to think of the pumping process as a random
Straightarrows represent classical pumpingfieldsandwiggly
arrows spontaneous emissions. ωr/√2 and ωl/2 are the cor- walk(orhopping)amongthe2M subspaces (s1,...,sM).
H
responding effective Rabifrequencies. For example, suppose a linear cluster state ΨN =
| i
2−N/2NNj=−11(|LijhL|+|RijhR|σjZ+1)NNk=1(|Lik+|Rik)
of four atoms (N = 4), stabilized by = σXσZI I ,
S1 1 2 3 4
pumping process from the subspace with σ1Zσ2Z = −1 S2 =σ1Zσ2Xσ3ZI4, S3 =I1σ2Zσ3Xσ4Z, and S4 =I1I2σ3Zσ4X,
to that with σ1Zσ2Z = +1. Here, the Pauli operators andconsiderthepumpingstepforS1. Theoriginalaimof
are defined over the ground levels of the atoms, e.g., this step is to pump the population in ( 1,s2,s3,s4)
H −
σjZ = |LijhL|−|RijhR|. Note that the L- and R-fields to H(+1,s2,s3,s4). However, as it destroys the coher-
couple the two manifolds in different ways. While the ence of atom 1 and atom 2, the values of 2 and 3,
S S
L-field induces transition in such a way that when the whichcontaina Paulioperatoronatom1 oratom 2,are
state is excited ns is increased by one, the R-field pre- also affected. Consequently, the pumping for 1 in fact
S
servesns. The spontaneousemission, onthe otherhand, occurs from ( 1,s2,s3,s4) to one of (+1, 1, 1,s4)
H − H ± ±
only decreases n by one while either decreasing n by randomly. The performance is then assessed by estimat-
t s
one or preserving n (see FIG. 2(b)). The overall effect ing the average number of steps to reach the subspace
s
of the pumping is thus that the L-field increases n by (+1,...,+1)bysucharandomwalk,alongwiththe ac-
s
H
one while the R-fielddecreasesn by one. Thanksto the tual pumping time for each step. The latter increases
s
energy splittings, one can apply these optical pumpings as the number of Pauli operators in , which we de-
µ
S
selectively to the population with a particular n : in or- note by k , increases because the energy gap to resolve
s µ
dertopumpthepopulationwithn tothesubspacewith g( k k 1) decreases. This scales only polyno-
s µ µ
p −p −
n +1(n 1),weapplyanL-field(R-field)toeveryatom mially and can be bounded as long as k is finite for
s s µ
−
with detuning √ns+1g ( √nsg). all µ, which is usually the case. However, the former,
± ± S
the number of steps to reach the steady state, increases
Itiseasilyseenthatthe abovepumpingprocessdrives
atwo-qubitsystemintothesubspacewith =σZσZ = exponentially as . M(2lm)M, where each pumping step
Sµ 1 2 affects at most l other stabilizers. Our scheme is thus
+1. Generalizing this idea to the caseof anarbitrary m
µ
S more relevant for a moderate number of atoms.
is straightforward. Suppose first that we need to pump
the population with σZσZ σZ = 1 to the subspace FIG.3 showsthe resultsof oursimulationforthe gen-
1 2 ··· N −
with σZσZ σZ = +1. We achieve this by pumping eration of linear cluster states, which is done by numeri-
1 2 ··· N
every population with an odd n , i.e., that with an odd cally integrating the master equation (3). As the pump-
s
(N+σZ+ +σZ)/2, to the subspace with n 1. For ing exploits the energy splitting, both the pumping field
1 ··· N s±
this,weapplytoeveryatomL-fieldswithdetunings√2g, and the spontaneous emission should be weak enough
√4g, √6g, and so forth. In the same fashion, we apply compared to the energy gap so that the energy split-
4
ter each pumping and detune the transition by an ac
StarkshiftforthosecorrespondingtoI . Itiseasilyseen
j
that this modified scheme is well suited to systems of
atoms sitting in a common cavity mode. Another modi-
fication can be made by noting that the harmonic oscil-
lator can in fact be replaced by a two-level atom, where
the interaction between atoms is the spin-exchange in-
teraction. Along with single-qubit manipulations as de-
scribedabove,whicharerelativelyeasytodo,ourscheme
(a) λ=0.05 (b) λ=0.02
can thus be applied to various Heisenberg spin systems
realized with atoms, where it is understood that auxil-
FIG. 3: Optical pumping to linear cluster states for N =
iary levels are needed to represent another ground level
2,3,4whenafullymixedstateistakenastheinitialstatefor
(a) λ = 0.05 and (b) λ = 0.02. λ is a constant determining L and to realize spontaneousemissions effectively if re-
| i
thedegreetowhichtheenergysplittingisresolved(seetext). quired [21, 22].
tings can be resolved. For this, we choose the parame- WethankHoTsangNgandSunKyungLeeforfruitful
ters as Γ = λg( k k 1), ΓL = ΓR = Γ/2, and discussions. We acknowledge the UK EPSRC for finan-
µ µ
p −p −
ωL = ωR = Γ/ k , where λ 1 is a small constant, cial support.
µ
p ≪
and each round of the pumping for one stabilizer is per-
formedforaperiodoftimeπ/Γ. Ourparticularchoiceof
theparametersis,however,notmandatoryasthepump-
ing is by nature insensitive to the exact values of the
[1] R. Horodecki, P. Horodecki, M. Horodecki, and
parameters. Taking a fully mixed state of atoms along
K. Horodecki, Rev.Mod. Phys.81, 865 (2009).
with the ground state of the harmonic oscillator as an [2] H. H¨affneret al., Nature438, 643 (2005).
initial state, FIG. 3 shows the time evolution of the fi- [3] P. Walther et al.,Nature 434, 169 (2005).
delityofthestatetothe desiredlinearclusterstate. The [4] S.SchneiderandG.J.Milburn,Phys.Rev.A65,042107
weaker the pumping field and the spontaneous emission (2002).
are (smaller λ), the higher is the final fidelity, but the [5] S. Clark, A. Peng, M. Gu, and S. Parkins, Phys. Rev.
Lett. 91, 177901 (2003).
longerit takes to reachthe steady state. One couldvary
[6] B. Kraus and J. I. Cirac, Phys. Rev. Lett. 92, 013602
theparametersintimetooptimizetheperformance,e.g.,
(2004).
bytakinglargerλduringthe initialtransientperiodand
[7] M.Paternostro,W.Son,andM.S.Kim,Phys.Rev.Lett.
taking smaller λ afterward to obtain a higher fidelity. 92, 197901 (2004).
It is evident that the present scheme is well suited to [8] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Buchler,
the existing ion-trapsystems used for quantum informa- and P. Zoller, NaturePhys. 4, 878 (2008).
tion processing [17, 18]. For example, Ca+ ions can be [9] F. Verstraete, M. Wolf, and J. I. Cirac, Nature Phys. 5,
633 (2009).
used, where the two hyperfine levels of S represent
1/2 [10] H. J. Carmichael, An Open Systems Approach to Quan-
L and R , respectively,andone ofD represents E
| i | i 5/2 | i tum Optics (Springer,Berlin, 1993).
[19]. The coupling between each ion and the center of
[11] H. R. Gray, R. M. Whitley, and J. C. R. Stroud, Opt.
mass mode of the ions gL,R can be controlled individu- Lett. 3, 218 (1978).
j
ally by focused beams. As D is metastable, the spon- [12] H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and
5/2
taneousemissionrateΓL,Rcanbecontrolledbyadjusting H. Buchler, NaturePhys. 6, 382 (2010).
pumpingfieldsviaP (andrepumpingfieldsviaP to [13] J. Cho, S. Bose, and M. S.Kim, arXiv:0907.2368.
3/2 1/2 [14] D. Gottesman, Phys.Rev. A 54, 1862 (1996).
makethe transitionclosed). As the typicalcouplingrate
[15] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
is g & 100 kHz, the system reaches the steady state in
5188 (2001).
a few hundredmilliseconds for a smallnumber of atoms, [16] Here, the coupling rates and the Rabi frequencies are
as can be inferred from FIG. 3. One can also incorpo- complex numbers.
rate sympathetic cooling into the pumping sequence to [17] F. Schmidt-Kaleret al.,Nature422, 408 (2003).
overcome motional heating [20]. [18] D. Leibfried et al.,Nature422, 412 (2003).
The present scheme with slight modifications can be [19] C. Roos et al., Phys.Rev.Lett. 83, 4713 (1999).
[20] D. J. Larson, J. C. Bergquist, J. J. Bollinger, W. M.
applied to many other physical systems. First, we can
Itano, and D. J. Wineland, Phys. Rev. Lett. 57, 70
consider a case where only one of the two atomic tran-
(1986).
sitions is persistently coupled, i.e., gL = 0 and gR = 0.
[21] A. Micheli, G. K. Brennen, and P. Zoller, Nature Phys.
6
This enables the optical pumping for stabilizer σZ. 2, 341 (2006).
Nj j
For different stabilizers, one performs single-qubit rota- [22] D. G. Angelakis, M. F. Santos, and S. Bose, Phys. Rev.
tions to the atoms corresponding to σX,Y before and af- A 76, 031805(R) (2007).
j