Scheme for teleporting an arbitrary superposition of atomic Dicke states via multi-fold coincidence detection

We propose a novel scheme for teleporting an arbitrary superposition of two-atom Dicke states with atoms trapped in optical cavities. The scheme requires a multi-fold coincidence detection and is insensitive to the imperfection of the photon detectors, and the fidelity is unity for any superposition of Dicke states. Further, we also point out that scheme can be extended to teleport an arbitrary superposition of N-atom Dicke states with unit fidelity.


Introduction
Quantum teleportation is not only a fundamental phenomenon of the quantum world, but also one of the key procedures in the area of quantum information processing [1]. By teleportation, an unknown quantum state can be transported from one place to another without moving through the intervening space. Since the pioneering contribution of Bennett et al [1], quantum teleportation was first demonstrated experimentally by using spontaneous parametric down-conversions [2]. maximally entangled state between atoms and two-mode cavity fields. In the detection stage, Alice uses photon detectors to measure the photons leaking out from two cavities. If each of the four detectors registers a single click during this stage, the scheme is successful, superposition of Dicke states is transferred from cavity A to B with unit fidelity. In the scheme, imperfection of the photon detectors decreases the success probability, but has no influence on the fidelity of the teleportation.
We now analyse the scheme in detail. The level structure of atoms is shown in figure  2, which has four ground states and three excited states. For concreteness, we consider a possible implementation using 87 Rb, whose usefulness in the quantum information context has been demonstrated in recent experiments [8]. The ground states |g L , |g 0 , |g R correspond to |F = 1, m = −1 , |F = 1, m = 0 and |F = 1, m = 1 of 5 2 S 1/2 , respectively, and one could use |F = 2, m = 0 of 5 2 S 1/2 as the ground state |g a . The excited states |e L , |e 0 and |e R correspond to |F = 1, m = −1 , |F = 1, m = 0 and |F = 1, m = 1 of 5 2 P 3/2 , respectively. The lifetimes of the atomic levels |g L , |g R , |g 0 , |g a are comparatively long so that spontaneous decay of these states can be neglected. We encode the ground states |g L and |g R as logic zero and one states, i.e. |g L = |0 and |g R = |1 . The transitions |e 0 ⇐⇒ |g R and |e L ⇐⇒ |g 0 (|e 0 ⇐⇒ |g L and |e R ⇐⇒ |g 0 ) are coupled to cavity mode a L (a R ) with the left-circular (right-circular) polarization. On Alice's side, the transition |e L ⇐⇒ |g L and |e R ⇐⇒ |g R are driven by π-polarized classical fields with the Rabi frequency A (t). The transition between |e 0 and |g 0 is electric dipole forbidden [20]. On Bob's side, one use π-polarized classical fields to drive the transition |e 0 ⇐⇒ |g a with the Rabi frequency B (t).  Alice's qubit is encoded in the two Zeeman sublevels |g L and |g R . For the initial state equation (2), only four transitions are included. |e L A → |g 0 A (|e R A → |g 0 A ) is coupled to the left-circularly (right-circularly) polarized mode of the cavity. |e L A ⇐⇒ |g L A and |e R A ⇐⇒ |g R A are driven by π-polarized classical fields, and the transition between |e 0 A and |g 0 A is electric dipole forbidden. (b) The involved atomic levels and transitions for each atom of Bob. Bob's qubit is also encoded in the two Zeeman sublevels |g L and |g R . For initial state |g a , g a |0, 0 , only three transitions are included. |e 0 B → |g R B (|e 0 B → |g L B ) is coupled to the left-circularly (right-circularly) polarized mode of the cavity. |e 0 B ⇐⇒ |g a B is driven by π-polarized classical fields.
Initially the atomic state in cavity A is prepared in a Dicke-state superposition of the form where coefficients C i are arbitrary and satisfy |C 1 | 2 + |C 2 | 2 + |C 3 | 2 = 1. We also assume that two cavity modes are in the vacuum state |0, 0 A . Thus, the initial state of the system is

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT For such an initial state, only four transitions are involved as depicted in figure 2(a), and the Hamiltonian describing the system is given by where a † L and a † R denote the creation operators for the corresponding polarized mode of the cavity. g A is the atom-cavity coupling, and the time dependence of the A (t) can be controlled by the external laser field. The Hamiltonian (3) has the following orthogonal dark states Based on the dark states equations (4)-(6), one can map the two-atom Dicke states equation (1) into the two-photon cavity states. Initially, we choose the parameters to satisfy A g A . In the limit of A (t) g A , the dark states |D LL , |D LR and |D RR coincide with the states |g L , g L |0, 0 A , (|g L , g R + |g R , g L )|0, 0 A / √ 2 and |g R , g R |0, 0 A , respectively. Therefore, by adiabatically increasing the coupling A (t), the initial state (2) of the system evolves in the dark space into the following state at the time t 6 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT For Bob, two atoms are initially prepared in the ground states |g a , g a and cavity fields are in the vacuum states. For such an initial state, only three transitions are involved as depicted in figure 2(b), and the Hamiltonian describing the system is given by By adiabatically tuning B (t) to go from B g B , the initial state |g a , g a |0, 0 B evolves into the following dark state at the time t In order to consider the effect of the cavity decay and photon observation on the state evolution in the physical model, it is convenient to follow a quantum trajectory description [21]. The evolution of the system's wave function is governed by a non-Hermitian Hamiltonian as long as no photon decays from the cavity. In this case, the state of the atom-cavity system j(j = A, B) at the time t evolves into Here we assume that two optical cavities have the same loss rate κ for the all modes, and H A and H B correspond to the equations (3) and (8). Following [19,21], a direct calculation gives the following results for Alice's state at the time t with the success probability . (3) and (8) are applied to the atom-cavity system A and B simultaneously, so that the preparation of the atom-cavity states | (t) j (j = A, B) ends at the same time. The joint state of two atom-cavity systems A and B is given by

If we assume that the interaction Hamiltonian
This implements the preparation stage of the protocol. The success probability is given by P suc = P A P B , i.e. the probability that no photon decays from either atom-cavity system during the preparation. If the conditions j e −κt g j are satisfied, the last terms in equations (12) and (13) are much larger than other terms, then | (t) A and | (t) B are reduced into the forms and which demonstrates that Alice maps the unknown Dicke-state superposition equation (1) into the two-mode state (15) and Bob generates a maximally entangled state between atoms and two-mode cavity fields (16). Now we consider the detection stage, in which we make a photon number measurement with four photon detectors D j (j = AH, AV, BH, BV) on the output modes of the set-up. We assume that photons are detected at the time τ. This assumption is posed to calculate the system's time evolution during this time interval in a consistent way with the 'no-photon-emission-Hamiltonian' (10). The detection of one photon with the detector D j (j = AH, AV, BH, BV) can 8 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT be formulated with the operator b j on the joint state | (τ) A | (τ) B . As shown in figure 1, the photons leaking out from the cavities A and B are mixed at the PBS, whose action is to transmit the horizontal polarization and reflect vertical polarization. Since the photons coming from cavities are circularly polarized, two QWPs are inserted before PBS to map the left-circular photons (right-circular photons) into the horizontal polarization photons (vertical polarization photons). After leaving the PBS, photon polarizations are rotated by polarization-rotations, whose actions are given by transformation a H → cos θa H + sin θa V and a V → cos θa V − sin θa H , where θ is rotation angle and will be determined later. Thus the operators of the four detectors have the following forms If each of the four detectors detects one photon, the state of the total system is projected into If we choose parameter θ = π/8, equation (18) becomes Based on the four-photon coincidence, Bob performs local unitary operations 1 to his atoms to transform state (19) into equation (1), the teleportation is thus finished. The success probability to achieving four-photon coincidence is P succ = (1 − e −2κτ ) 4 /12. We now give a brief discussion on the influence of the quantum noise on the scheme. Firstly, it is evident that the scheme is inherently robust to photon loss, which includes the contribution from channel attenuation, and the inefficiency of the photon detectors. All these kinds of noise can be considered by an overall photon loss probability η [4]. It is noticed that the present scheme is based on the four-photon coincidence detection. If one photon is lost, a click from each of the detectors is never recorded. In this case, the scheme fails to teleport superposition of Dickestates. Therefore the imperfection of photon detectors only decreases the success probability P succ by a factor of (1 − η) 4 , but have no influence on the fidelity of the expected operation. Next we show that the scheme is insensitive to the phase accumulated by the photons on their way from the ions to the place where they are detected. The phases ϕ 1 = kL 1 and ϕ 2 = kL 2 , where k is the wavenumber and L j are the optical lengths which photons travel from the jth cavity towards the photon detectors, lead only to a multiplicative factor e i2(ϕ 1 +ϕ 2 ) in equation (19). This result demonstrates that phase accumulated by the photons has no effect on the conditional implementation of the quantum operation. Thirdly, since the scheme is based on adiabatic passage technology, atomic excited states can be decoupled from the evolution, and the influence of atomic spontaneous emission on the teleportation could be suppressed.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Teleportation of N-atom Dicke states
In this section, we turn to the problem of teleportation of N-atom Dicke states from one cavity to another. We use the notation |g ⊗m L g ⊗N−m R to denote a normalized Dicke state where m atoms are in the level g L and N a − m atoms are in the level g R . The atomic state in cavity A is initially prepared in Dicke-state superposition of the form and cavity fields are in the vacuum states. For such an initial state, the Hamiltonian describing the system is still given by equation (3), which has the following dark states Thus, by adiabatically adjusting the coupling A (t) from A g A to A g A , based on the dark states, one can map the N-atom Dicke superposition states into the two-mode N-photon states For Bob, N atoms are initially prepared in the state |g a and cavity fields are in the vacuum states. For such initial state, the Hamiltonian describing the system is given by equation (8). In the adiabatic limit, the initial state |g a ⊗N |0, 0 B evolves into the following dark state By adiabatically tuning B (t) to go from B g B to B g B , achieves the maximally entangled states of N-atom Dicke states and N-photon polarization states In order to realize teleportation, photons which originate from the state | A and | B are injected into the set-up shown in the state of the total system is projected into Based on the measurement result, Bob performs local operation to transform equation (27) to equation (20). This demonstrates that the proposed set-up, which is shown in figure 3, definitely implements the teleportation of N-atom Dick-state superposition with unit fidelity. The success probability of conditional realization is P meas = N 1−2N N i=1 cos θ 4 i , which decreases exponentially with increasing N. The quantum noise has no influence on the fidelity of the teleportation, but decreases the success probability (1 − η) 2 N.
In summary, we have presented schemes to teleport an arbitrary superposition of two-atom Dicke states from one cavity to another. In contrast to the scheme [19], our scheme requires 2 The action of measurement device M is to project on the two-mode N-photon entangled states | N = N m=0 |mH, (N − m)V a . In terms of creation operators, such an N-photon state can be written in the form | N = (a † H cos θ N + a † V sin θ N e −iϕ N ) · · · (a † H cos θ 1 + a † V sin θ 1 e −iϕ 1 )|0, 0 a , where parameters tan θ 1 e −iϕ 1 , . . . , tan θ N e −iϕ N are chosen to be the N complex roots of equation (26).
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT multi-photon coincidence detection, and is insensitive to quantum noise, which has no influence on the fidelity of the teleportation, but decreases the success probability. The fidelity is unity for any superposition of Dicke states. Finally, it is briefly pointed out that scheme can be modified to teleport a superposition of N-atom Dicke states.