Selective control of the symmetric Dicke subspace in trapped ions

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Selective control of the symmetric Dicke subspace in trapped ions

C. E. López,¹J. C. Retamal,¹and E. Solano^2,3

1Departamento de Física, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

2Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany

3Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado 1761, Lima, Peru 共Received 20 May 2007; published 24 September 2007兲

We propose a method of manipulating selectively the symmetric Dicke subspace in the internal degrees of freedom ofN trapped ions. We show that the direct access to ionic-motional subspaces, based on a suitable tuning of motion-dependent ac Stark shifts, induces a two-level dynamics involving previously selected ionic Dicke states. In this manner, it is possible to produce, sequentially and unitarily, ionic Dicke states with increasing excitation number. Moreover, we propose a probabilistic technique to produce directly any ionic Dicke state assuming suitable initial conditions.

DOI:10.1103/PhysRevA.76.033413 PACS number共s兲: 42.50.Vk, 03.67.⫺a, 03.67.Mn

I. INTRODUCTION

Multipartite entangled states play a fundamental role in quantum information, where these states are used for different applications including the improvement of spectroscopy towards the Heisenberg limit关1兴. In this sense, general sequential techniques for building entangled multipartite states have been recently proposed关2兴. In Ref. 关3兴, an experiment is described where the robust one-excitation symmetric Dicke states关4兴, calledW, ofN艋8 ions are prepared in their electronic levels with the aid ofNentangling pulses. Also, a maximally entangled关Greenberger-Horne-Zeilinger 共GHZ兲兴 state with six ions has been experimentally realized 关5兴.

From a theoretical point of view, adiabatic ground-state tran- sitions were proposed for generating GHZ states and symmetric Dicke states withN/ 2 excitations inNions关6兴. More recently, a method for generating multiqubit entangled states via global addressing of an ion chain in the frame of the Tavis-Cummings model has been discussed关7兴. A four-qubit W state with two excitations has already been realized in linear optics关8兴, which may present astonishing multipartite properties 关9兴, and more general proposals may be consid- ered关10兴. It is well-established that a physical system must fulfill several requirements in order to qualify as a potential candidate for quantum computing tasks关11兴. Among them, overcoming decoherence and scalability considerations may require not only efficient single- and two-qubit gates but also the availability of collective multipartite operations in suitable subspaces.

In this paper, we consider a system composed of N trapped ions addressed collectively by two laser fields in a global lambda-type excitation scheme. We will introduce a method for tailoring the Hilbert space in order to restrict the quantum dynamics to the symmetric Dicke subspace. As we show below, this method allows a different and useful way to manipulate selectively the collective ionic-motional system.

In particular, these multipartite selective interactions will permit the generation of ionic Dicke states with any number of excitations in a sequential manner or, through a probabilistic technique, in a single-shot measurement. This method is based on global selective interactions characterized by a proper tuning of collective motion-dependent Stark shifts.

Selective interactions with a single atom have been proposed in the realm of cavity QED 关12兴 and trapped ions 关13,14兴. Furthermore, it has been demonstrated that they also allow the generation of arbitrary harmonic oscillator states关15兴and their measurement via instantaneous interactions关16兴.

II. MODEL

Let us consider a Raman laser excitation ofNthree-level trapped ions as shown in Fig.1. We will make use of these internal levels and the collective center-of-mass motional mode associated with the frequency␯. A traveling-wave field excites the transition between the states兩g_j典↔兩c_j典, with coupling strength ⍀2j=⍀2j共rជj兲 and detuning ⌬共⌬Ⰷ⍀2j兲. Simi- larly, a standing-wave field excites off resonantly the transition between the electronic internal states 兩ej典↔兩cj典, with position-dependent coupling strength ⍀1j=⍀1j共rជj兲 and detuning ⌬+␯Ⰷ⍀1j. This scenario is described, after a first optical rotating-wave approximation共RWA兲, by the Hamil- tonian

1 2 j N-1 N

… …

1j

2j

c_j

e_j g_j

FIG. 1.Nthree-level ions in a linear Paul trap where the energy diagram of thejth ion is displayed.

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Hˆ =ប␯^aˆ^†^aˆ⁺ប␻e

兺

j=1 N

兩ej典具ej兩+ប␻c

兺

j=1 N

兩cj典具cj兩

+ប

冋

^cos共k¹^zˆ兲eⁱ^␻¹^t

^兺

^j=1^N ^⍀^1j^兩e^j^典具c^j^兩

+e⁻ⁱ^共^k²^zˆ−␻²^t^兲

兺

j=1 N

⍀2j兩gj典具cj兩+ H.c.

册

^. ^共1兲

We go then to an interaction picture inside the Lamb-Dicke regime:␩i

冑

n¯Ⰶ1, wheren¯is the average phonon number and

␩i⬅ki

冑

ប/ 2m␯ are the Lamb-Dicke parameters. In this way, we can adiabatically eliminate levels兩cj典, obtaining the blue- sideband second-order effective Hamiltonian

Hˆ

eff= −ប⌬ˆ+ប共aˆ^†˜ˆJ⁺+aˆJ˜ˆ⁻兲, 共2兲 where˜ˆJ⁺=兺^N_j=1⍀_eff^j ␴^ˆj

†, with⍀_eff^j = 2i␩2⍀1j⍀_2j^* /⌬,␴^ˆj

†=兩ej典具gj兩, and

⌬ˆ = 1

⌬

兺

j=1 N

关1 −␩12共2aˆ^†aˆ+ 1兲兴兩⍀1j兩²兩gj典具gj兩+ 1

⌬

兺

j=1 N

兩⍀2j兩²兩ej典具ej兩共3兲 is the motion-dependent ac Stark shift. In this case, we can discard terms involving level 兩c_j典 by assuming no initial population. The phonon-number dependence of the Stark shift 共3兲 is due to the standing-wave Raman laser, which together with the traveling wave produce the dynamics of Eq.共2兲. Note that ac Stark shifts have already been used for experimental realization of two-qubit gates and multipartite entanglement关17兴.

The detuning ⌬ˆ can be corrected by a fixed position- dependent quantity␦0

j via dc Stark shift or retuning of the lasers frequencies. In this manner, Hamiltonian 共2兲 can be written as

Hˆ

eff= −ប

兺

j=1 N

⍀₀^j共nˆ−␦0

j兲兩gj典具gj兩+ប共aˆ^†˜ˆJ⁺+aˆJ˜ˆ⁻兲, 共4兲 where⍀₀^j= 2␩12兩⍀1j兩²/⌬. It will be convenient to rewrite the Hamiltonian of Eq.共4兲in the interaction picture with respect to the first term, where it reads

Hˆ

eff

I =ប

兺

j=1 N

⍀_eff^j aˆ^†␴^ˆj

†e⁻ⁱ^关⍀⁰^j^共^nˆ−^␦⁰^j^兲+兺^k^⫽^j^⍀⁰^k^兩^g^k^典具^g^k^兩兴^t+ H.c. 共5兲

III. SELECTIVE CONTROL IN THE HOMOGENEOUS COUPLING CASE

A. Generalized selectivity

In order to illustrate how selectivity appears in theN-ion case, let us study the special situation of theNions coupled homogeneously to the Raman lasers where ⍀_eff^j ⬅⍀eff

= 2i␩2⍀1⍀₂^*/⌬, ⍀₀^j⬅⍀0= 2␩1

2兩⍀1兩²/⌬, and ␦0

j⬅␦0. In this case, the interaction part in Hamiltonian共4兲corresponds to

an anti-Tavis-Cummings model关18兴, a spinj=N/ 2 generali- zation of the Jaynes-Cummings model 关19兴. In this case,

˜ˆJ+→⍀effJˆ^±, and the new collective terms Jˆ^±can be consid- ered as angular momentum operators, establishing a permu- tation symmetry on the ionic subsystem dynamics. That means that if the system is found at any time inside the symmetric Dicke subspace关4兴, associated with total angular momentum j=N/ 2, it will stay there along its evolution, re- ducing the Hilbert space dimension from 2^NtoN+ 1. Under this plausible assumption, the collective operatorsJˆ^±can be effectively and exclusively rewritten in the symmetric Dicke subspace via the following assignments:

兺

j=1 N

兩gj典具gj兩→

兺

k=0 N−1

共N−k兲兩Dk典具Dk兩,

Jˆ⁺→

兺

k=0 N−1

fk兩Dk+1典具Dk兩. 共6兲 Here,

兩Dk典=

冉

^N^k

冊

^−1/2

^兺

k

Pk共兩g1,g₂, . . . ,g_N−k,e_N−k+1, . . . ,eN典兲共7兲 are the symmetric Dicke states withkexcitations,兵Pk其is the set of all distinct permutations, and f_k=

冑

共k+ 1兲共N−k兲. It is noteworthy to stress that in the assignments of Eq. 共6兲 we have omitted the nonsymmetric components due to the as- sumed initial symmetric conditions. In this case, and under homogeneous driving, we can derive from Eq.共4兲an analog to Eq.共5兲,

H¯ˆ

eff

I =បaˆ^†⍀eff

兺

k=0 N−1

fkeⁱ^⍀⁰共−nˆ+N−1−k+␦0兲t兩Dk+1典具Dk兩+ H.c., 共8兲 a compact expression that will prove useful to study selective interactions inside the symmetric subspace. Let us consider the system prepared in the initial state 兩N0典兩Dk₀典. Then, the suitable choice of laser frequencies␦0=k₀+N₀−N+ 1 yields a selective resonant coupling inside the subspace 兵兩N₀典兩D_k

0典,兩N₀+ 1典兩D_k

0+1典其. Moreover, provided that ⍀0

Ⰷ⍀eff, all other subspaces will remain off resonance obtaining an effective two-level dynamics. That is, by selecting a determined subspace the Hamiltonian共8兲can be written as

H˜ˆ =ប

冑

N₀+ 1⍀efff_k

0共␴^ˆN+₀Jˆ_k

0 + +␴^ˆN−₀Jˆ_k

0

−兲, 共9兲

where Jˆ_k

0

+=兩Dk₀+1典具Dk₀兩 and ␴^ˆN+₀=兩N0+ 1典具N0兩 are effective spin-1 / 2 operators stemming from the reduced Hilbert space of the collective ionic state and the bosonic field, respec- tively. As we will se below, this selective global interaction will allow us to move comfortably inside the symmetric Dicke subspace with high precision关20兴.

Considering experimental parameters of ion experiments at NIST共Boulder, CO兲关21兴, we could achieve an effective

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coupling⍀eff⬃10⁵Hz, which produces population inversion in the subspace 兵兩N₀典兩D_k

0典,兩N₀+ 1典兩D_k

0+1典其 in a time ␶ 艋0.1 ms, shorter than the typical motional decoherence time

␶d⬃10 ms.

B. Applications of generalized selectivity

We discuss now some applications of our method for selectively manipulating the Dicke subspace. Let us consider the initial state 兩⌿共0兲典=兩0典兩g¯g典⬅兩0典兩D0典. Tuning into resonance the subspace transition兵兩0典兩D0典,兩1典兩D1典其, the evolution of this state is given by

兩⌿共t兲典= cos共

冑

N兩⍀eff兩t兲兩0典兩D0典−ieⁱ^␾sin共

冑

N兩⍀eff兩t兲兩1典兩D1典, 共10兲 where⍀eff=兩⍀eff兩e⁻ⁱ^␾. The one-excitation Dicke state兩D1典is also aW state

兩WN典= 1

冑

N共兩eg¯g典+兩geg¯g典+ ¯ +兩g¯ge典兲. 共11兲 ThisN-partite entangled state has great importance in quantum information theory due to its persistent entanglement properties, as long as more operational effort is needed to disentangle this state关22兴. If this interaction is turned on for a time 2

冑

N兩⍀eff兩t=␲^and␾⁼␲/ 2, then Eq.共10兲becomes

兩⌿共t兲典=兩1典兩D1典 ⬅ 兩1典兩WN典, 共12兲

yielding state兩WN典in the metastableNtwo-level ions. If the system evolves for a time such that cos共

冑

N兩⍀eff兩t兲

= 1 /

冑

N+ 1, then

兩⌿共t兲典=兩WN+1典, 共13兲

where the 共N+ 1兲th qubit is the reduced bosonic spin-1 / 2 system.

Once the system is prepared in the state given in Eq.共12兲, and tuning to resonance the red-sideband subspace transition 兵兩1典兩D₁典,兩0典兩D₂典其, a pulse with Rabi angle 2

冑

N兩⍀eff兩t₂=␲ will lead to

兩⌿共t₂兲典=兩0典兩D₂典. 共14兲 In this manner, it is clear that a successive application of collective blue- and red-sideband interactions can produce deterministically and sequentially all symmetric Dicke states 兩Dk典with number of excitationsk.

Another interesting application of multipartite selective interactions is the possibility to discriminate between ionic states with a different number of excitations. Suppose we have an ionic state prepared in a superposition of states with a different number of excitations 兺_k=0^N ck兩Dk典, with 兺_k=0^N 兩ck兩²

= 1. For example, this state can correspond to an atomic co- herent state关23兴 given by exp共i␪^Jˆx兲兩g¯g典. Note that an in- teraction proportional to Jˆxcan be generated by applying a Raman laser field tuned to the carrier transition on theNions collectively and homogeneously. The center-of-mass mode is initialized in the state兩N0典 and we consider an 共additional兲

ancillary qubit in the ground state 兩g典A. We tune then to resonance the collective blue-sideband subspace 兵兩N0典兩Dk₀−1典,兩N0+ 1典兩Dk₀典其, where 兩Dk₀典 is the state with k₀ excitations we want to discriminate. In this way, after a collective␲-pulse on the ions, we obtain a state of the form

兩⌿1典=

冉

^c^k⁰⁻¹^兩N⁰^{+ 1典兩D}^k⁰^典⁺^兩N⁰^典^k^⫽

^兺

^k0−1 N

ck兩Dk典

冊

^兩g典^A^.

共15兲 Now, a ␲-pulse with the laser field tuned to the first red sideband on the ancillary qubit leads to 兩⌿2典

=共ck₀−1兩Dk₀典兩e典A+兺k⫽k₀−1

N ck兩Dk典兩g典A兲兩N0典. Then, if we mea- sure the ancilla in the excited state兩e典A, the collective ionic state will collapse into the Dicke state兩Dk₀典withk₀ excitations. Remark that the projection on ancillary state兩e典A, that should happen with a probability兩ck₀−1兩², can be done with high precision via well-established electron-shelving techniques.

On the other hand, it has been shown that the use of selective interactions in a single trapped ion can lead to de- terministic and universal manipulation of the motional state 关15兴. Along these lines, similar manipulation could be imple- mented here to grant access to arbitrary states inside the symmetric Dicke subspace. In this case, the motional Fock states would be replaced by symmetric states in the internal ionic degrees of freedom with a fixed number of excitations.

IV. SELECTIVE CONTROL IN THE INHOMOGENEOUS COUPLING CASE

In the more general case of ions interacting inhomogeneously with Raman lasers, we cannot discriminate prese- lected symmetric Dicke states. However, multipartite selectivity will still allow us to manipulate ionic number states, that is, ionic states with a determined number of excitations but not necessarily symmetric. For example, if laser fields interact inhomogeneously with initially deexcited trapped ions in a carrierlike excitation of the form U= exp共−i␪^J˜ˆx兲, where J˜ˆx=˜ˆJ⁺+˜ˆJ⁻, this will not lead to a superposition of symmetric Dicke states. On the opposite, this will lead to a superposition of nonsymmetric collective number states aris- ing from the action of the operators˜ˆJ+and˜ˆJ−on the collective ionic states. It is known that to deal with the unitary evolution of high-dimensional inhomogeneously coupled systems is extremely difficult关23–25兴. In this case, instead of writing the Hamiltonian 共4兲 in the basis of the symmetric Dicke states, as in Eq.共8兲, we should write it in the corre- sponding basis of nonsymmetric collective number states 兩D˜

k

ᐉ典withkexcitations. In this way, we may look for conditions to set into resonance a determined subspace. States兩D˜

kᐉ典 appear naturally from successive applications of˜ˆJ⁺and˜ˆJ⁻on a given initial collective state. The indexᐉ accounts for the fact that, depending on the number of ionic excitations, there could exist more than one nonsymmetric collective state with a determined number of excitations. In the same spirit of Eq.

共2兲, we can write the associated Hamiltonian

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H¯ˆ

eff

I = −ប

兺

_k,ᐉ^具^D^˜^k^ᐉ^兩⌬^ˆ^兩^D^˜^k^ᐉ^典兩^D^˜^k^ᐉ^典具^D^˜^k^ᐉ^兩⁺^ប^aˆ^†

兺

_k,ᐉ^⍀^˜^eff^k,ᐉ^兩^D^˜^k+1^ᐉ ^典具^D^˜^k^ᐉ^兩

+ H.c. 共16兲

Here,⍀˜

effᐉ,k is the new effective coupling constant, which in the homogeneous case corresponds to⍀eff. As in the homogeneous case, if ⍀0Ⰷ⍀˜

effᐉ,k, we can tune to resonance a determined subspace, for example, the inhomogeneous blue- sideband doublet 兵兩N0典兩D˜

k₀

ᐉ典,兩N0+ 1典兩D˜

k₀+1

ᐉ 典其. In this case, from Hamiltonian共16兲in the interaction picture, we can derive that the condition to tune to resonance this subspace is 具D˜

k₀+1 ᐉ 兩⌬ˆ

N₀+1兩D˜

k₀+1 ᐉ 典−具D˜

k₀ ᐉ兩⌬ˆ

N₀兩D˜

k₀

ᐉ典= 0. This condition can be fulfilled by compensating the detuning ⌬ˆ through shifts in the lasers frequencies for fixed values of␦0

j, depending on the subspace we want to select. This procedure is similar to the homogeneous case, but now␦0

j will be inhomogenously distributed, that is, different for each ion.

V. CONCLUSIONS

In conclusion, we have introduced a selective technique that allows a collective manipulation of the ionic degrees of freedom inside the symmetric Dicke subspace. We have stud- ied the homogeneous and inhomogeneous cases, showing applications in both cases, mainly related to the generation and control of number states in the ionic external and internal degrees of freedom. We believe that the introduced concepts may inspire similar physics in other quantum-optical setups with diverse applications, and that they might even be help- ful to transfer collective atomic number states to propagating fields.

ACKNOWLEDGMENTS

C.E.L. is financially supported by MECESUP USA0108 and CONICYT, J.C.R. by Fondecyt 1070157 and Milenio ICM P02-049F, and E.S. by DFG SFB 631, EU EuroSQIP projects, and the German Excellence Initiative via the

“Nanosystems Initiative Munich共NIM兲.” C.E.L. also thanks DIGEGRA USACH and Jan von Delft for hospitality at Ludwig-Maximilian University共Munich兲.

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