Multi-particle entanglement of hot trapped ions

4 Pages·0.13 MB·English

by  Klaus Molmer

Multi-particle entanglement of hot trapped ions
Klaus Mølmer and Anders Sørensen
∗ †
Instituteof Physics and Astronomy,University of Aarhus,
DK-8000 ˚ArhusC, Denmark
Wepropose anefficient method toproducemulti-particleentangled states ofionsin an ion trap
forwhichawiderangeofinterestingeffectsandapplicationshavebeensuggested. Ourpreparation
scheme exploits the collective vibrational motion of the ions, but it works in such a way that this
motion need not be fully controlled in the experiment. The ions may, e.g., be in thermal motion
and exchange mechanical energy with a surrounding heat bath without detrimental effects on the
internal state preparation. Ourscheme does not require access to theindividualions in thetrap.
PACS:32.80.Pj,03.65 Bz,75.10 Jm
9
9
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WedescribeinthisLetteramethodtoconstructstates the J operator, J J,M = h¯M J,M , M = J,...,J,
1 z z| i | i −
of the form where ¯h is Planck’s constant. J ”counts” the number
z
n 1 of excited ions, so that a state, invariant under permu-
a Ψ = [eiφg gg...g +eiφe ee...e ], (1) tation of the ions, with a certain number N of excited
J | i √2 | i | i e
ions is described as J = N/2,M = Ne N/2 , and
6 | − i
where gg...e and ee...e (N terms) are product states thestate(1)isasuperpositionofthestates N/2, N/2
2 | i | i | − i
describing N ions which are all in the (same) internal and N/2,N/2 . An increase of the quantum number M
2 state g or e. The unique correlation between different by u|nity corresiponds to a single ionic excitation, and it
v particles in such an entangled state is a significant non- is described by the angular momentum ladder operator
0 classicalfeatureofquantummechanics. Entangledstates J+ J,M = h¯p(J +M +1)(J M)J,M + 1 , where
4 with two particles have been employed to test Bell’s in- the|normialization factor account−s prec|isely for tihe com-
0 equalityandtoruleoutlocalrealisticdescriptionsofNa- binatorialaspectsofthis excitationofthe system. As we
0
ture[1]. Stateslike(1)withthreeparticles,the so-called shalldetailbelow,itispossibletoapplyclassicallaserir-
1
GHZ states [2], and with more particles allow a closer radiation which cannot be absorbed by a single ion, but
8
9 scrutiny of the role of quantum correlations, and there which requires two ions to undergo the g e transi-
| i↔ | i
/ is a large current interest in carrying out experiments tion,simultaneously. Whenthisfieldisappliedtoallions
h
on such states. It has also been shown that the states inthe trap,the effectiveHamiltoniandescribingthe evo-
p
(1) are ideal for spectroscopic investigations [3], and the lution of the state of the ions in the spin representation
-
t studyofentangledstatesofmanyparticlesisinlinewith can be written,
n
a the rapidly growing interest in quantum encoding and
χ χ
u quantum information processing [4]. Together with an H = (J2 +J2 +J J +J J )=4 J2 (2)
q analysis of their applicability within high precision fre- ¯h + − + − − + ¯h x
v: quency measurements, a method to produce such states
i inaniontrapwasproposedin[3]. Thatmethodrequires where J is the hermitian conjugate of J+ and where
X experimental control of two collective vibrational modes Jx =(J+−+J )/2 denotes the x-component of the effec-
r ofthe ions,whichareinvolvedinstepwisemanipulations tive spin. −
a
of the internal states of the ions. Timeevolutionisaccommodatedbyapplicationofthe
Inordertodescribeourmorestraightforwardproposal, operator U(t) = exp( i4χJ2t/¯h2) to the initial state
− x
we shall show that the desired state (1) can be obtained N/2, N/2 and the population becomes distributed on
| − i
byapplicationofonesimpleinteractionHamiltonian. We all J,M states,withM differingfrom N/2byaneven
| i −
shallthenshowthatthisHamiltonianisrealizedinanion number. If N is even, population is transferred all the
trapby simplyilluminating allthe ionswithlaserbeams way to the state N/2,N/2 , which is the second com-
| i
with two different colours, and we shall present numeri- ponent of (1). Moreover, at the instant t = π/(8χ), the
cal results that display the performance of our proposal state is precisely of the form (1), with φg = π/4 and
−
under quite relaxed assumptions for the functioning of φe =π/4+Nπ/2. WeshowinFig. 1. thetimeevolution
the ion trap. ofthepopulationsofthetwoextremalstatesfordifferent
We begin with all ions in the groundstate g , and we (even) numbers of ions. At t = π/(8χ) and at later odd
| i
assume that our external perturbation interacts in the multiples of this value, both populations equal one half.
samewaywithallions,sothatthejointinternalstatere- This result can be understood from the ro-
mainssymmetric withrespecttointerchangeofthe ions. tation properties of angular momenta: The ini-
Insuchasituation,itisconvenienttoapplythespinrep- tial state can be expanded on eigenstates of J :
x
resentation of atomic states, where a collective state of N/2, N/2 = c N/2,M , and from the prop-
| − i PM M| ix
allionsis representedby J,M whichisaneigenstateof erties of the Wigner rotation functions [5], it follows
| i
1
that N/2,N/2 = c ( 1)N/2 M N/2,M . In- and we assume η = η = η. Numerical solutions of
| i PM M − − | ix 1 2
serting the values for φ and φ mentioned above, we the Schr¨odingerequation with the Hamiltonian givenby
g e
can therefore write the state (1) as c 1 (e iπ/4+ Eqs.(3,4), are readily obtained within the basis of 2N
PM M√2 −
internal states and a sufficiently large set of vibrational
( 1)Meiπ/4)N/2,M . The net factors multiplying the
− | ix states f f ...f n , f = e or g. First, however, we
initial amplitudes cM are unity for M even and −i for discuss{|a1pe2rturNbia|tivi}e anialysis pointing to appropriate
M odd. The action of U(t) in the J basis amounts to a
x choicesofparameterswhichwilleffectivelyyieldthespin
multiplication of each amplitude c by exp( 4iχM2t),
M − Hamiltonian (2).
and for t = π/(8χ) this factor just attains the desired
value of unity for M even and i for M odd. The energy levels of the ion string constitute a lad-
−
Note that the state (1) is not only a multi-particle en- derofequidistantvibrationallevelsaddedtotheinternal
tangled state; it is a superposition of two “mesoscopi- ionicenergies. Thisladderstructurehasbeenextensively
cally distinguishable” states. In Quantum Optics, such used in laser manipulation of ions in traps, where laser
states have been studied experimentally for single quan- frequenciesdifferingfromtheatomictransitionfrequency
tum systems, suchas a harmonicallytrapped ion[6] and by a multiple of the trap frequency may be used to con-
asinglemode ofthe quantizedradiationfield[7],excited troltheinternalandexternaldynamics[6],andexcitation
into superpositions of states of “mesoscopic” separation. of collective vibrational modes for several ions has been
Mathematically,ouranalysisaboveresemblestheonefor proposed as a means for implementing quantum logic in
production of such superpositions by propagation of an an ion trap quantum computer [11].
opticalfieldthroughanamplitudedispersivemedium[8].
Wenowturntoa“microscopic”analysisofacollection Here, we make use of the idea we put forward in [12]
of N two-level ions in a linear ion trap. The choice of a toapplylaserfieldswithtwodifferentfrequenciessothat
lineartrapeliminatesthe so-calledmicro-motionforions the two-photon process, exciting any pair of ions in the
on the axis, and by choosing the transverse confinement trap gg ee is resonant, i.e., ω1 +ω2 = 2ωeg, but
| i → | i
much stronger than the longitudinal one, we can neglect neither of the frequencies are resonant with single ex-
coupling to transverse vibrations. Due to the Coulomb citations of the ions. By choosing ω1 = ωeg δ and
−
repulsionamongthe ions,thevariouslongitudinalvibra- ω2 =ωeg+δ, where δ is close to, but not resonant with,
tional modes have different frequencies and by choosing the center-of-mass vibration frequency we may limit our
the laserdetuning closetothe center-of-massvibrational perturbative analysis to involve only the intermediate
frequencywemayexcludeothermodesfromouranalysis states with one excited ion and a vibrational quantum
[9]. number raisedor loweredby unity. Note that the lackof
The quantized energy of the total system is composed degeneracyofvibrationalmodesiscrucialbecauseitpre-
of the center-of-massvibrationalenergy of the ion string ventsanumberofmodestointerfereandobstructthecol-
and the internal electronic energy of the ions. The vi- lective transitionsinthe multi-particle system. We show
brational energy is represented by a harmonic oscillator in Fig. 2 the effective transition paths between states
with ladder operatorsa anda and frequency ν, andthe gg n and ee n and between eg n and ge n me-
† | i| i | i| i | i| i | i| i
internalenergyisdescribedbyPaulimatricesσ andthe diated by the interactionwith the laser fields within any
zi
energy difference ¯hω , pair of ions. Although they serve as intermediate states
eg
in the process, the intermediate states are only virtually
¯hωeg i.e., negligibly, populated. The interaction strengths of
H0 =h¯ν(a†a+1/2)+ Xσzi. (3)
2 the different one-photon transitions in Fig. 2 depend on
i
thevalueofthevibrationalquantumnumbernandinthe
The ions interact with two laser fields with frequencies limitofη√n+1<<1,termsbetweenstatesnandn+1
ωj (j =1,2). We assume that both fields have the same areproportionalto √n+1 whereasterms between n 1
−
intensity and we also assume that the ion-laser interac- and n are proportional to √n. When we add the con-
tion strength characterized by the Rabi frequency Ω is tributions fromthe four different transitionpaths shown
the same for all ions. Pauli matrices σ i describe inter- in either panel of Fig. 2, a “miracle” occurs: The paths
nal atomic transitions and the interac±tion Hamiltonian (virtually)involvingintermediatestateswithn+1vibra-
can be written tionquantayield(√n+1Ωη)2/(δ ν)andtheoneswith
n 1quantayield(√nΩη)2/(ν δ−). Thesignsofthede-
Hint =X¯hΩ(σ+iei(ηj(a+a†)−ωjt)+h.c.). (4) no−minatorscauseadestructivei−nterferencewhichexactly
2
i,j removes the n-dependence from the total effective Rabi
frequencyofthetwo-photontransition. Thisimpliesthat
(h.c. in (4) denotes the hermitian conjugate.) The Dicke
theinternalstateevolvesinamannerindependentofthe
parameters η are products of the projection of the k-
j external state, which may be in a mixed (e.g., thermal)
vector of the j’th laser field along the direction of the
state and which may even evolve with time.
string of ions and the rms excursion of the ionic center-
of-massalongthisdirection[10]. Inpracticalrealizations When the summation over all ions is carried out, the
the k-vectors can be nearly identical for the two fields, couplings lead to the spin Hamiltonian (2) with
2
η2Ω2ν mal motion. Our analysis adds the surprising possibil-
χ= , (5)
2(ν2 δ2) ity of coherently controlling the mutual internal state of
−
several particles by explicit use of the uncontrolled in-
and to the extent that our perturbative treatment is coherently excited degrees of freedom. We believe that
valid,wethereforeexpecttorealizetheresults,presented much wider possibilities for quantum state control be-
inFig. 1. InFig. 3. weshownumericalsimulationsofthe come feasible with the introduction of the crucial ele-
dynamicsgovernedbythecorrectHamiltonian(3,4)for4 mentsofourmethod: resonanceconditionsallowingonly
ions,i.e. beyondperturbationtheoryandtoallordersin pairwise transitions, virtual excitations of un-controlled
the Dicke parameter η. We assume a distribution on vi- degreesoffreedom,andpartiallydestructiveinterference
brational levels with quantum numbers ranging between betweentransitionpaths. Wehaverecentlyappliedthese
0 and40, and we assume that the center-of-massmotion ideas to quantum computation in an ion trap [12] and
exchanges energy with a surrounding thermal reservoir we are currently investigating their possible use in other
represented by relaxation operators c1 = pΓ(nth+1)a physical systems.
and c = √Γn a . Our calculation for a reservoir with
2 th †
a mean excitation of n = 5 vibrational quanta at the
th
trap frequency and a coupling strength Γ = 0.0001 may
represent a heating mechanism towards higher temper-
atures counteracted by laser cooling on one ion in the
string,reservedforthis purpose. Theactionofthereser-
voirisdescribedbyquantumjumpsandanon-Hermitian ∗ Email: [email protected]
Hamiltonian in a Monte Carlo wave function simulation † Email: [email protected]
[13]. Part (a) of the figure represents the result of a sin- [1] A. Aspect et al,Phys. Rev.Lett. 47, 1804 (1982).
gle Monte Carlo realisation with quantum jumps in the [2] D. M. Greenberger et al, Am.J. Phys.58, 1131 (1990).
vibrationalstatesatinstantsoftime,indicatedbyarrows [3] J. J. Bollinger et al, Phys.Rev.A 54, R4649 (1996).
[4] Physics World 11, 33-57, March 1998. Special issue on
in the figure. Part(b) is an averageover10 suchrealisa-
Quantum Information.
tions. The full lines present the populations and coher-
[5] D. M. Brink and G. R. Satchler, Angular Momentum,
encesoftheinternalstateoftheions,obtainedbytracing
Clarendon Press, Oxford (1975).
out the vibrational degrees of freedom. For comparison
[6] C. Monroe et al, Science272, 1131, (1996).
we also show the result of a calculation with the cor-
[7] M. Bruneet al, Phys. Rev.Lett. 77, 4887 (1996).
responding spin Hamiltonian (2) (dashed curve). With
[8] B. Yurkeand D. Stoler, Phys. Rev.Lett. 57, 13 (1986).
the realistic parameterschosen, the state at νt 1500is
∼ [9] Numericaltestsincludinganadditionalvibrationalmode
very close to the desired state in (1). Small oscillatory
prove that it is sufficient to deal with the near-resonant
deviations from the smooth results of Fig. 1. are seen.
center-of-mass mode of the ions.
Themagnitudeoftheoscillationsisgivenbythephysical [10] D. M. Meekhof et al, Phys. Rev. Lett. 76, 1796 (1996).
parameters, so that in the limit of small η and Ω, they This Letter shows how the two-level formalism also de-
can be made arbitrarily small [12]. scribes two stable states coupled by Raman transitions.
The pairwise interaction with ions does not produce The k-vector in the definition of the Dicke parameter is
a coherent coupling of the two components in (1) if N then a k-vector difference, and η can be varied more or
is odd. In this case, however, one can verify within the less freely in an experiment.
spin model, that it suffices to apply a linear coupling [11] J.ICiracandP.Zoller,Phys.Rev.Lett.74,4091(1995).
H = 4ξJ for the duration t = π/(8ξ) in addition to [12] AndersSørensenandKlausMølmer, Quantum computa-
1 x
application of our quadratic term (2). Since J and J2 tionwithionsinthermalmotion,toappearinPhys.Rev.
x x
commute, the linear Hamiltonian may actually be ap- Lett., quant-ph/9810039.
plied before, after or simultaneously with the quadratic [13] J. Dalibard, Y.Castin and K.Mølmer, Phys. Rev.Lett.
one. This additional operation is easy to implement in 68, 580 (1992).
the ion trap, where one only needs to drive the ions on
the atomic resonance frequency ω .
eg
Insummary,wehavepresenteda methodto preparea
multi-particle entangled state of trapped ions. Our sug-
gestion aims at experimental application, and it is rel-
evant both for experiments with limited control of the
vibrational motion, where the ions heat up for different
technical reasons, or where there is not access to indi-
vidualions. The possibilityto exercisecontroloversome
quantumdegreesoffreedomofasystemwhile othersare
uncontrolledis commonplace in coherentoptical excita-
tion experiments, where the internal states of atoms in
a gas may be controlled regardless of the external ther-
3
1 1
(a)
0.8
0.8
0.6
0.6
0.4
0.4 0.2
0
0.2
-0.2
0 0 1000 2000 3000 4000
0 0.2 0.4 0.6 0.8 1 νt
χt 1
(b)
FIG.1. Time evolutionofthepopulation ofthejointionic
0.8
ground state |gg...gi = |N/2,−N/2i (curves starting from
the value of unity at t=0), and the joint ionic excited state
0.6
|ee...ei=|N/2,N/2i. Results are presented for different val-
ues of the number of ions: N=2 (solid curves), N=4 (dashed 0.4
curves) and N=8 (dotted curves). At t = π/(8χ) the states
are in a 50-50 superposition and thestate (1) is obtained. 0.2
0
-0.2
|een+1>
ω |een> ω2 ||eeeenn->1> 0 1000 2ν0t00 3000 4000
1
ω ω
|egn+1> 2 |gen+1> 1
|egn> |gen> |egn> |gen> FIG.3. ResultsofnumericalintegrationoftheSchr¨odinger
|egn-1> ω |gen-1> ω
2 1 equation for 4 ions. The physical parameters are δ = 0.9ν,
ω ω
1 2 |ggn+1> Ω = 0.1ν, η = 0.1, Γ = 0.0001ν and nth = 5. (a) Single
|ggn> |ggn> Monte Carlo realisation with quantum jumps in the vibra-
|ggn-1>
tional states at instants of time, indicated by arrows in the
FIG.2. Levelschemeforapairofionssharinganoscillator figure. (b) Average over 10 realisations. The first full curve
degreeoffreedom. Left: byapplicationoflaserlightwithfre- (counted from above at νt < 1000) is the population of the
quenciesωeg±δ,whereδissomewhatsmallerthanthevibra- joint ground state, the second one is the imaginary part of
tional frequency ν, we identify four transition paths between theinternaldensitymatrixelementbetweenthejointground
thestates |ggi|ni and |eei|ni, which interfere as described in andthejoint excitedstate, thethirdoneisthepopulation of
the text. Right: four similar transition paths are identified the joint excited state, and the fourth one is the real part of
betweenstates|egi|niand|gei|ni,yieldingthesameeffective the off-diagonal density matrix element. The dashed curves
coupling among thesestates as between thestates in the left are equivalent to the N = 4 curves of Fig. 1, obtained by
panel. application of thespin Hamiltonian (2) with χ= 2(ην22Ω−2δν2).
4