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Entanglement generation by adiabatic navigation in the space of symmetric multi-particle states

We propose a technique for robust and efficient navigation in the Hilbert space of entangled symmetric states of a multiparticle system with externally controllable linear and nonlinear collective interactions. A linearly changing external field applied along the quantization axis creates a network of well separated level crossings in the energy diagram of the collective states. One or more transverse pulsed fields applied at the times of specific level crossings induce adiabatic passage between these states. By choosing the timing of the pulsed field appropriately, one can transfer an initial product state of all N spins into (i) any symmetric state with n spin excitations and (ii) the N-particle analog of the Greenberger-Horne-Zeilinger state. This technique, unlike techniques using pulses of specific area, does not require precise knowledge of the number of particles and is robust against variations in the interaction parameters. We discuss potential applications in two-component Bose condensates and ion-trap systems.

We propose a technique for robust and efficient navigation in the Hilbert space of entangled symmetric states of a multiparticle system with externally controllable linear and nonlinear collective interactions. A linearly changing external field applied along the quantization axis creates a network of well separated level crossings in the energy diagram of the collective states. One or more transverse pulsed fields applied at the times of specific level crossings induce adiabatic passage between these states. By choosing the timing of the pulsed field appropriately, one can transfer an initial product state of all N spins into (i) any symmetric state with n spin excitations and (ii) the N -particle analog of the Greenberger-Horne-Zeilinger state. This technique, unlike techniques using pulses of specific area, does not require precise knowledge of the number of particles and is robust against variations in the interaction parameters. We discuss potential applications in two-component Bose condensates and ion-trap systems.

I. INTRODUCTION
Entanglement is a unique quantum feature which has enjoyed considerable attention in the last few years. It plays a crucial role in many rapidly developing areas of contemporary quantum physics, such as quantum information [1] and fundamental tests of quantum mechanics [2]. Various quantum systems have been suggested for controlled creation of entanglement, e.g. trapped ions [3], spins in magnetic field [4], quantum dots [5], cavityquantum-electrodynamics systems [6], crystal lattices [7], Josephson junctions [8], and others.
In order to entangle N spin-1 2 particles interaction between the spins is required and external control of this interaction is necessary to generate specific many-particle states. For the latter purpose one can use sequences of resonant external pulses of precise area, e.g. π-pulses. While this technique is conceptually simple, it is very sensitive to variations in the pulse area and resonance mismatch, which can be caused by temporal and spatial fluctuations of the external field and may lead to significant errors. Hence an important practical challenge is to design robust and efficient methods for a controlled navigation in the multi-particle Hilbert space. This is of particular importance in mesoscopic systems, where only limited control over the interaction parameters and the number of particles is possible.
The simplest interaction that can lead to entanglement in a collection of spins involves either pairwise nearestneighbor interactions or a collective coupling between all particles. An example for the first case is the Ising model [9], while the collective coupling of ions to a phonon mode in an ion trap [3] or the self-interaction in a Bose-Einstein condensate (BEC) [10] is an example for the second.
We here analyze the second type of systems and propose a method for controlled, efficient and robust navigation in the space of symmetric multi-particle entangled states. No precise knowledge of interaction parameters or particle number is needed and only certain adiabaticity criteria have to be fulfilled. The proposed technique is a multi-particle generalization of our earlier proposal [11] for creation of entanglement in a pair of two-state systems by using adiabatic passage induced by a suitably crafted external field. The multi-particle problem adds some new challenges as it involves in general multi-step as well as direct transitions; this opens a variety of paths between any pair of multi-particle states. We use this to advantage and demonstrate that certain paths are insensitive to the number of particles N , a property that is particularly significant for systems with large N , for which N is not usually known exactly.
The scheme proposed in the present paper allows to create robustly and efficiently maximum entanglement starting from a product state. In particular, it can be used to create various special entangled states, such as of the Greenberger-Horne-Zeilinger (GHZ) type or the socalled W -states. Moreover arbitrary transitions among the collective states can be realized. After introducing the concept of the method we discuss two specific implementations: an ion-trap scheme similar to that of Mølmer and Sørenson [12] and a coupled two-component BEC in the two-mode approximation [13].

A. Controlled collective spin interaction
We consider a collection of N identical spin-1 2 particles with a total angular momentum operatorĴ. The simplest collective interaction that allows to entangle individual spins is quadratic in one of the components ofĴ, viz. (in unitsh = 1)Ĥ where ξ is the spin-spin interaction constant and B(t) is some time-dependent external field. Special cases of this type of Hamiltonian have been discussed by several authors (see, e.g., [14] and references therein).
Since the Hamiltonian (1) contains only collective spin operators, only multi-particle states with the same symmetry upon particle exchange are coupled. In particular, if the system is initially in the product state of all spins in the spin-down state |↓↓ . . . ↓ , only symmetric collective states with the same (maximum) angular momentum J = 1 2 N will interact. Thus the 2 N -dimensional Hilbert space reduces to the (N + 1)-dimensional subspace of symmetric multiparticle states. Because each of these states is characterized by a definite number of excitations n = 0, 1, . . . , N (which corresponds to an angular momentum projection m = −J + n ∈ {−J, J}), we shall use n to label the states. They are given by . . .
whereσ + i is the Pauli spin-flip operator, which inverts the spin of the ith particle. In particular, the singleexcitation symmetric state |1 is an N -particle analog of the W -state [15]: This state is maximally robust against disposal of any of its qubits [15]. Obviously, state |N − 1 , which has N − 1 spins up and one spin down, is also of the W -type. We assume, similarly to [11], that the particle-field coupling consists of a linearly changing component along the z-direction (chosen as the quantization axis) and a pulsed component in the x-direction. Then the Hamiltonian (1) takes the form where A is assumed real and positive. Because the opera-torĴ x connects only adjacent states, the linkage pattern is chainwise. The direct coupling between each pair of adjacent states |n and |n + 1 is given by i.e. all couplings have the time dependence of the external pulse, but different magnitudes. The energy of each state |n changes linearly in time with a slope proportional to m = −J + n, This creates a web of level crossings in the energy diagram of the collective states as shown in Fig. 1 for the case of N = 4 particles. These level crossings can be used to design various navigation routes between the collective states.
As follows from Eq. (5), states |n and |k cross at time These crossings are equidistant and separated by a time interval τ = ξ/A, which does not depend on the number of particles N , but only on the interaction parameters. Pairs of states with the same total number of excitations n + k cross at the same time. However, due to the presence of the nonlinear interaction term m 2 ξ, all level crossings are well separated in energy.
The positions of the crossings may or may not depend on N . For example, the crossing between states |0 and |1 is situated at t 01 = −(N − 1)ξ/A. Hence this time will be known only if N is known exactly; we will see below that this restricts the possible scenarios of adiabatic transfer. In contrast, the crossing between the two product states |0 and |N does not depend on N , because it is situated at t 0N = 0, and the time t = 0 is determined by the zero value of the longitudinal field, B z (0) = 0. The crossing between states |N and |1 , which is situated at t 1N = ξ/A, does not depend on N either.

Navigation routes
Once a network of level crossings is created, one can design in principle any navigation route in the Hilbert space by choosing properly the timing of the pulsed external field. It is most natural to assume that the multiparticle system is prepared initially in one of the product  (5) of the collective states. States |0 and |4 are product states, all others are entangled symmetric states. Application of an external coupling pulse at time t nk leads to an avoided crossing and adiabatic population transfer between states |n and |k . The four frames apply to the cases when an external coupling pulse is applied at the crossings (a) t01, (b) t02, (c) t03, (d) t04. As a result, four navigation routes are created connecting the initial state |0 to states |1 , |2 , |3 , and |4 , respectively.
states, for example in |0 . If the system evolves along this state and the designed route requires that it must make a transition to the entangled state |n , one should apply a sufficiently strong (adiabatic) external pulse at the time of diabatic crossing t 0n between |0 and |n . The interaction will open an avoided crossing between the energies of the corresponding adiabatic states and will force the system to make an adiabatic transition from |0 to |n [16]. On the contrary, if the route requires the system to remain in the same collective state, one should ensure that there is negligible interaction between this state and the other collective states. With the leeway in choosing the time dependence and the intensity of the pulsed external field, one can link any initial state to any final state by using one or more suitably timed pulses.
In Fig. 2 we show the numerically calculated populations of the five symmetric collective states |n in a four-particle system plotted against the center T 0 of a Gaussian coupling pulse. Depending on the timing of this pulse, the population is transferred from the initial state |0 to different collective states. In agreement with the above discussion, the maximum transfer efficiency for each collective state |n is achieved when T 0 is near the crossing t 0n between |0 and |n . One also recognizes that the range of times over which complete transfer occurs decreases as the number of excitations associated with the transfer increases.
The present technique allows also to transfer entanglement. Indeed, if the many-particle system is initially in the entangled state |n , it can be transferred adiabatically to another entangled state |k by applying an adiabatic pulse at the crossing time t nk of these states. An entangled state |n can also be transferred into one of the unentangled states |0 or |N ; this can be used for measurement of entanglement.

Conditions
For such a state engineering to be successful, the level crossings must be well separated, i.e. the width T of the external pulsed field must be small in comparison with the time separation between the crossings: T ≪ ξ/A.
On the other hand, the condition for Landau-Zener population transfer around the crossing at t nk can be shown to lead to the following conditions where Ω nk (t nk ) is the effective coupling between states |n and |k , estimated at the crossing time t nk . For adjacent states, which are connected directly, this coupling is given by Eq. (4). For example, the above conditions suffice to estimate the feasibility of the direct transition |0 → |1 . For states that are coupled via one or more intermediate states, Ω nk (t) is an effective multi-quantum coupling. This coupling can be estimated perturbatively when Ω ≪ N ξ [17]   Alternative routes for the transition between the product states |0 → |4 in a four-particle system: a single narrow pulse applied at the crossing t04 between |0 and |4 (top), and a wide pulse covering all crossings (bottom).
Thus for Ω ≪ N ξ the effective coupling between |n and |k is very small and cannot induce adiabatic evolution. Thus we have to consider the case Ω > ∼ N ξ, which is however accessible only numerically.

C. Choice of navigation path
As the energy diagram in Fig. 1 suggests, there are multiple paths linking each pair of collective states. Each of these paths has certain advantages and disadvantages, depending on the particular experimental situation.

Transition between the product states
The two product states |0 and |N can be linked in several different ways. First, one can apply a single adiabatic pulse at their level crossing at time t 0N = 0, as shown in Fig. 3 (upper frame). This approach is independent of the number of particles N because the crossing time t 0N = 0 is well defined by the zero of the linearly increasing z-field. In other words, even if we do not know the exact N , we can find the exact location of the crossing between the completely unexcited state |0 and the completely excited state |N . However, using this crossing between |0 and |N requires much stronger field because these states are not coupled directly, but only via an N -quanta transition.
Alternatively, one can use a train of pulses centered at each crossing, so that the population will flow from |0 through all intermediate states to reach |N at the end: |0 → |1 → |2 → . . . → |N . Because this navigation route (the lowest solid curve in Fig. 3) passes only through crossings of directly coupled adjacent levels, much less field intensity is needed to satisfy the adiabatic condition. However, the first crossing at t 01 = −(N − 1)ξ/A depends on the number of particles N , i.e. this approach is only applicable if N is known exactly.
A third possibility, applicable only to the transition |0 → |N , is to apply a sufficiently wide single pulse, covering all crossings, as shown in Fig. 3 (lower frame). In this case, only an approximate knowledge of N is required, because there are no stringent restrictions on the pulse width and timing.

Creation of W -states
One faces similar choices for the transitions from one of the product states to any entangled state |n . For example, the N -particle W -state can be created using the transition |0 → |1 by applying a single adiabatic pulse at time t 01 = −(N − 1)ξ/A, as shown in Fig. 4 (upper plot). Because this is a transition between adjacent collective states, only a moderately strong field is required. However, this approach is only applicable if the number of particles is known precisely because the crossing time t 01 depends on N : an error in N , even as small as ∆N = 1, is inadmissible since then the pulse will be applied at a wrong crossing.
Alternatively, state |1 can be populated using the transition |N → |1 , as shown in Fig. 4 (lower plot). For this path, the crossing time t N 1 = ξ/A does not depend on N . However, because this is an (N − 1)-quanta transition, the coupled states differ by a large number of spin excitations. Then a much stronger field may be needed to widen the much narrower avoided crossing and induce adiabatic evolution.
Similar conclusions apply to the other W -state |N − 1 and to any other entangled state |n .

Feasibility of the N -invariant scenario
The N -invariance of the latter approach, which uses the (multi-quanta) crossings near t = 0 rather than the (single-quanta) outside crossings, is a very attractive feature because it allows to use this technique without knowing the precise number of particles N . The only problem is that for multi-quanta transitions a stronger external field is needed to induce adiabatic passage. Therefore we have performed numerical simulations to estimate the minimal pulse area A min needed to achieve 90% transfer efficiency for the transition |N → |1 from the product state |N to the W -state |1 . This area is plotted in Fig. 5 as a function of the particle number N .
As the figure demonstrates, A min increases nearly quadratically for small N and approaches a linear dependence for large N . This (slow) linear increase implies that the conditions on the required resourses for application of this technique to many-particle systems are not very strong. The linear behavior can be understood qualitatively by noting that, as the numerical simulations show, the coupling termĴ x in Eq. (3) (which is ∝ Ω 0 ∝ A min ) is of the same order of magnitude as the vertical energy splitting (which is ∝ N ξ). Hence, the transition probability must scale with the parameter Ω 0 /(N ξ). FIG. 5. Numerically calculated minimal pulse area Amin (defined as the pulse area for which 90% transfer efficiency is achieved) for the transition from the product state |N to the W -state |1 as a function of the particle number N . We have used a Gaussian pulse,

D. Generation GHZ states
Another interesting application of the present scheme, after a slight modification, is the creation of the so-called GHZ state [18], Starting from the product state |0 one has to create first an equal superposition of this state and the W -state |1 , for example, by a single π/2-pulse. Then the present technique can be used to transfer the population of |1 to the other product state |N by applying a single adiabatic pulse at the crossing t 1N ,

III. IMPLEMENTATIONS
In the following we briefly discuss two implementations of the technique discussed above. The first one involves two-photon transitions in trapped ions, the second a coherently coupled two-component Bose-Einstein condensate.

A. Ion-trap system
Recently Sørenson and Mølmer suggested a realization of a nonlinear Hamiltonian in a two-level ion trap system [12], displayed in Fig. 6. In their scheme, two laser fields (in the Lamb-Dicke limit of light coupling) are applied with frequencies ω 1 and ω 2 that are symmetrically detuned from the single-photon resonance by a detuning δ. Then there is a two-photon resonance between states  6. Ion-trap system for realization of an effective nonlinear spin Hamiltonian. The application of a symmetrically detuned bichromatic field to ions with ground state |g and excited state |e leads to two-photon resonance between the collective states |gg and |ee independent of the trap oscillation quantum number n.
with two ions in the ground level |gg and two ions in the excited level |ee . If the detuning is larger than the Rabi frequencies of the two lasers (δ ≫ Ω 1 = Ω 2 ≡ Ω) there is no single-photon excitation.
In a coarse-grained time-averaged picture an effective Hamiltonian proportional to ξĴ 2 z emerges [12], with ξ = 2η 2 Ω 2 ν/(ν 2 − δ 2 ), η being the Lamb-Dicke parameter, and ν the phonon frequency. If, in addition to the detuned bichromatic fields, a resonant laser coupling the z component and one coupling the x component are applied, the Hamiltonian (3) is obtained. Since ξ is proportional to the Lamb-Dicke parameter η ∼ 1/ √ N , the crossing times (6) depend on the number of particles. However, this is not a significant problem because N is usually precisely known for trapped ions.

B. Two-component BEC
Another example for our model Hamiltonian (3) is the trapped atomic BEC in two different internal states a and b [10,13], demonstrated in recent experiments with the hyperfine states |F = 1, M = ±1 in sodium [19]. The external coupling field B(t) can be provided by a chirped radio-frequency pulse.
The Hamiltonian of a two-component BEC with swave scattering and coherent coupling between the components in the two-mode approximation is given bŷ