Entangling Pairs of Nano-Cantilevers, Cooper-Pair Boxes and Mesoscopic Teleportation

We propose two schemes to establish entanglement between two mesoscopic quantum systems through a third mesoscopic quantum system. The first scheme entangles two nano-mechanical oscillators in a non-Gaussian entangled state through a Cooper pair box. Entanglement detection of the nano-mechanical oscillators is equivalent to a teleportation experiment in a mesoscopic setting. The second scheme can entangle two Cooper pair box qubits through a nano-mechanical oscillator in a thermal state without using measurements in the presence of arbitrarily strong decoherence.

We propose two schemes to establish entanglement between two mesoscopic quantum systems through a third mesoscopic quantum system. The first scheme entangles two nano-mechanical oscillators in a non-Gaussian entangled state through a Cooper pair box. Entanglement detection of the nano-mechanical oscillators is equivalent to a teleportation experiment in a mesoscopic setting. The second scheme can entangle two Cooper pair box qubits through a nano-mechanical oscillator in a thermal state without using measurements in the presence of arbitrarily strong decoherence.

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Probing quantum superpositions and entanglement with mesoscopic mechanical systems has recently developed into an area of substantial interest [1,2,3,4,5,6,7,8,9]. The most striking experimental demonstrations are the interferometry of mesoscopic free particles (molecules) [1] and the entangling of mesoscopic atomic ensembles [2]. Proposals for the generation of entanglement between Bose-Einstein condensates [3] and coherence between states of mesoscopic atomic ensembles have been made [4]. Some early proposals involving harmonically bound mesoscopic systems were based on optomechanical effects where schemes for observing coherent superpositions of states of the movable mirror [5] and entanglement between two such mirrors [6] were proposed. Soon, however, a canonical system of a Cooper-pair box coupled to a mesoscopic cantilever was introduced [7]. It offered an optics-free, fully nano-technological alternative, with switchable couplings for such schemes. Accordingly, a scheme to observe coherent superpositions between states of a mesoscopic cantilever, as well as its entanglement with a Cooper pair box was proposed [7]. Recently, interferometric proposals to probe superpositions of states of movable mirrors have also been proposed [8]. Very recently, a proposal to entangle two well separated nano-electromechanical oscillators through a harmonic chain has also been made [9]. A host of other quantum effects are expected to be seen in mesoscopic mechanical systems [10,11,12,13,14,15,16] including quantum computation [17]. These theoretical proposals are fuelled by the rapid technological progress in the fabrication of nano-mechanical systems and experiments approaching the quantum regime [18,19].
The Hamiltonian which generates entanglement between a Cooper pair box and a cantilever in Ref. [7] offers many more exciting entangling possibilities even with minimal additions to the number of systems, such as just one extra Cooper pair box or just one extra cantilever. In this letter we show that with the above minimal addition, one can entangle two mesoscopic systems of the same dimension: two discrete variable systems (two Cooper pair boxes) or two continuous variable systems (two nanomechanical cantilevers). One can also verify their entanglement with an entanglement witness or teleportation with higher than classically achievable fidelity. An interesting feature of the entangling of the cantilevers is that they are placed in a non-Gaussian continuous variable entangled state as a result of our scheme. Till date, only Gaussian entangled states have been used in continuous variable implementations of quantum information processing [20], and the scheme we suggest might enable one to realize a non-Gaussian entangled state. The scheme we suggest for detection of the non-Gaussian entanglement is equivalent to possibly the simplest realization of a quantum teleportation experiment with entangled nano-mechanical cantilevers. Positive features of the entangling scheme for the Cooper pair boxes are its applicability in entangling non-neighboring (not directly interacting) boxes in an array and its robustness to the thermal nature as well as decoherence of the states of the mediating cantilever. Most importantly, our schemes seek to extend the domain of quantum behavior by entangling two mesoscopic systems through a third mesoscopic system.
Entangling two nano-cantilevers: A Cooper pair box (CPB) is an example of a qubit with states |0 and |1 representing n or n + 1 Cooper pairs in the box [7,21]. It can be made to evolve under a Hamiltonian − EJ 2 σ x by the application of an appropriate voltage pulse [7,21], where σ x is the Pauli-X operator and the parameter E J is called the Josephson coupling. This gives rise to coherent oscillations between the |0 and |1 states as observed in Ref. [21]. A nano-mechanical cantilever (NC), on the other hand is a simple example of a quantum harmonic oscillator. We now proceed to the proposal for entangling two cantilevers based on their interaction with a single CPB. The setup is shown in Fig.1. The Hamiltonian required for the scheme is given by ergy of the Cooper pair box, σ z is the Pauli-Z operator for the CPB, operators a, a † and b, b † are the creation/annihilation operators for two oscillators and λ is a coupling strength. We assume that the NCs are prepared initially in their ground state (this is quite realistic for the GHz oscillators available now [19] by cooling, as suggested in Ref. [17]). Accordingly, we start with the cantilevers in the initial state |0 a |0 b , where subscripts a and b denote the two cantilevers, and the CPB in the state 1 √ 2 (|0 + |1 ) (This state can be prepared by using a voltage pulse to accomplish a π/2 rotation about x-axis through − EJ 2 σ x followed by local phase adjustments). The evolution that takes place in a time T = π/ω m is where β = λ/hω m is a dimensionless coupling and |±4β are coherent states. For simplicity, we will assume that 2EC T h is an integral multiple of 2π. We now measure the CPB in the basis |± = 1 √ 2 (|0 ± |1 ) to obtain the state where the upper and lower signs stand for the |+ and |− outcomes respectively. If β ∼ 1, as will happen, for example, if one takes the parameters of Ref. [7], then e −16β 2 ∼ O(10 −7 ) and both states |ψ(±) ab have nearly one ebit of entanglement and each outcome has a proba-bility of nearly 1/2 to occur. |ψ(±) ab are a class of non-Gaussian continuous variable entangled states known as entangled coherent states, proposed originally in the optical context [25]. It is trivial to check that the scheme also works if the cantilevers started in coherent states of non-zero amplitude.
Verifying the entanglement of the cantilevers by teleportation: An interesting question now is how to verify the entanglement of the states |ψ(±) ab . The non-local character can be ascertained in principle from Bell's inequality experiments [26]. However, these involve measurements in a highly non-classical (Schroedinger Catlike) basis [26], and could be rather difficult for a NC. For an NC, position/momentum measurements seem natural. Unfortunately, from joint uncertainties in position and momentum of the two NCs, the entangled nature of the state |ψ(±) ab cannot be inferred. We will thus use quantum teleportation through |ψ(±) ab to demonstrate its entangled nature. Note that the possibility of teleportation of Schrödinger Cat states of a third oscillator through the entangled coherent state of two oscillators has already been pointed out by van Enk and Hirota [27] in the quantum optical context. However, for NCs, preparing a third NC in a highly non-classical state such as a Schrödinger cat is challenging, making it directly interact with one of the entangled NCs is difficult and moreover, we do not want to increase the complexity of the system by adding an extra NC. We will thus concentrate on the teleportation of the state of a qubit through |ψ(±) ab with better than classically achievable (2/3) fidelity. This will prove the entangled nature of the state |ψ(±) ab .
For the teleportation protocol, first assume that the NCs were prepared in |ψ(+) ab as a result of the measurement of the CPB in the |± basis. The CPB is now, of course, disentangled from the state of the NCs. It is thus now prepared in the arbitrary state cos θ/2 |0 + e iδ sin θ/2 |1 which we want to teleport through |ψ(+) ab . The CPB interacts with cantilever a for a time T and the resulting evolution is: The position of the cantilever a and the state of the CPB in the |± basis are now measured. All the above corresponds to the Bell state measurement part of the teleportation procedure. As e −8β 2 << 1, there is a probability ∼ 1/2 that the cantilever is projected to the state |0 a . Let us, for the moment, concentrate on this outcome. Contingent on this outcome, the state of the CPB is projected to |+ and |− with 1/2 probability each, corresponding to which the state of cantilever b goes to cos θ/2 |−2β b + e iδ sin θ/2 |2β b and cos θ/2 |−2β b − e iδ sin θ/2 |2β b . Let us assume the state to be cos θ/2 |−2β b + e iδ sin θ/2 |2β b for the moment.
In some sense the above state of cantilever b already contains the teleported quantum information from the original state of the CPB. However, it is difficult to verify this information while it resides in the state of cantilever b. So we map it back from cantilever b to the CPB (which is now disentangled as a result of the previous measurement) by preparing the CPB in the state |+ , allowing for the evolution and then measuring the position of cantilever b. With a probability 1/2 it is |0 b , for which the CPB is projected to the state cos θ/2 |0 + e iδ sin θ/2 |1 , thereby concluding a chain of operations leading to teleportation with unit fidelity. In the case when the outcome |− |0 a is obtained during the Bell measurement procedure, a teleportation with unit fidelity can also be performed on obtaining |0 b in the mapping back stage followed by the correction of a known phase factor. For the outcomes |± |−4β a and |± |4β a in the Bell state measurement, the CPB is prepared in states |0 and |1 respectively, while for |−4β b and |4β b in the mapping back stage, it is prepared in states |0 and |1 respectively. This completes our teleportation protocol. The fidelity of the procedure is thus unity with probability 1/4, cos 2 θ/2 with probability (3/8) cos 2 θ/2 and sin 2 θ/2 with probability (3/8) sin 2 θ/2. Averaging over all possible initial states one then gets an average fidelity of 3/4, which is greater than the classical teleportation fidelity of 2/3. Let us clarify the sense in which the above is a bonafide teleportation procedure despite the systems being adjacent and the same CPB being reused. The CPB interacts with only cantilever a during the Bell state measurement procedure and hence this can be considered as a local action by a party holding cantilever a. The CPB is automatically reset in the process as a fresh qubit not bearing any memory of its initial state. In the mapping back stage it can thus be regarded as a local device used by the party holding cantilever b for extraction of the state.
Decoherence of the cantilever, if significant, will of course affect both the generation of the state |ψ(+) ab , as well as the teleportation. However, decoherence of a cantilever is in the coherent state basis and it will simply multiply the off diagonal term |−2β a |−2β b 2β| a 2β| b (and its conjugate) in |ψ(+) ab by a factor of the form e −Γ where e −Γ ∼ e −8β 2 π/Q in which Q is the quality factor of the cantilevers [7] (note that as physi-cally expected, higher the quality factor, lower the decoherence). Similarly, in evolutions given by Eq.(4) and Eq.(5), the off diagonal terms |0 |0 a |2β b 1| 0| a −2β| b and |0 |0 b 1| 0| b (and their conjugates) are multiplied by e −5Γ/2 and e −Γ/2 respectively. The net effect of decoherence at the end of the teleportation will then be a reduction of fidelity corresponding to the |± |0 a outcome of the Bell state measurement to (2+e −4Γ )/3, while the fidelity corresponding to other outcomes will remain unchanged. Thus unless all coherence is destroyed by decoherence i.e., e −4Γ ∼ 0, we have an average teleportation fidelity 2/3 + e −4Γ /12, which is better than 2/3. For example, for Q ∼ 1000 [7], we have e −Γ ∼ 0.975 (for β ∼ 1 [7]) and average teleportation fidelity is 0.74. In this paper we assume that the CPB hardly decoheres over the ns time-scale of experiments with a GHz NC [7].
Entangling two CPBs: The setting of our scheme of entangling two CPBs as depicted in Fig.2 is two CPBs coupled to a single NC. The Hamiltonian for this system, in the absence of the voltage pulse giving rise to − EJ 2 σ x , is well approximated (by straightforward extrapolation of Ref. [7]) as is a Pauli-Z operator of the ith Cooper pair box, a, a † are the annihilation-creation operators of the nanocantilever. We initially consider the NC to be starting in the coherent state |α (we shall generalize later to a thermal state) and the CPB's to be initialized in the state |0 1 |0 2 , where labels 1 and 2 stand for the two CPBs. At first, the Hamiltonian − EJ 2 σ x is applied to each CPB to rotate their states from |0 to 1 √ 2 (|0 + |1 ). Then evolution according to the Hamiltonian H kicks in and in a time T = π/ω m the evolution of the state can be calculated from Ref. [10] to be where φ(T, β, α) = 2βImα is a phase factor and |−α ,|−α − 4β and |−α + 4β are coherent states. The sign flip from α → −α in the above evolution occurs due to the oscillator evolution for half a time period. The production of states of the above type has been noted earlier in the context of cavity-QED [22] and very recently in the context of measurement based quantum computation [23]. In Ref. [23], it has been pointed out that for a large β, a measurement of the oscillator (NC in our case) will project the two qubits (CPBs in our case) probabilistically to the maximally entangled state |ψ + 12 = 1 √ 2 (|0 1 |1 2 + |1 1 |0 2 ). Such an entangled state can, of course, be verified through Bell's inequalities by measurements on the CPBs. However, here we want to go beyond this result and reduce the requirements necessary for observing entanglement between the CPBs. Suppose the cantilever is in a high temperature thermal state so that position measurements of the cantilever would be inefficient due to thermal noise. We thus ask the question as to whether we can observe any entanglement between the CPBs without the extra complexity of measurements on the NC. The reduced density matrix of the two CPBs, when the states of the NC are traced out will, for β large, be In deriving the above we have taken the overlap of coherent states −α| − α − 4β , −α| − α + 4β and −α + 4β| − α − 4β to be nearly zero.
Note that when decoherence of the states of the cantilever is taken into account, as it occurs in the coherent state basis [7], we can, without loss of generality, replace |−α ,|−α − 4β and |−α + 4β in Eq.(7) by |−α |ξ −α ,|−α − 4β |ξ −α−4β and |−α + 4β |ξ −α+4β where |ξ −α ,|ξ −α−4β and |ξ −α+4β are three distinct environmental states with pair-wise mutual overlap tending to zero in the limit of strong decoherence. Thereby, for β ∼ 1, the reduced density matrix of the two CPBs is unaffected by decoherence and still given by ρ 12 of Eq. (8). Also, note that ρ 12 does not, in any way, depend on the initial coherent state amplitude α. Thus even if we were to start in a thermal state of the cantilever given by d 2 α P (α)|α α|, the state of the two CPB qubits for large β will be ρ 12 for a time T = π/ω m .
Verification of the entanglement of the CPBs: ρ 12 is an entangled state, but not one that violates a Bell's inequality. So we have to check the entanglement of the CPBs through an entanglement witness [24]. Basically one has to measure the expectation value of the operator for the state of the CPB qubits. The expectation value of W is positive for all separable states, so if it is found to be negative, then we can conclude that the CPBs are entangled. In fact, for the predicted state ρ 12 at T = π/ω m , the expectation value of W is −0.25. Note that the operator W is a locally decomposable witness [24] which means that it is measurable by measuring only local operators in the same manner as Bell's inequalities. Its locally decomposable form is evident from Eq. (9). Thus no interactions between the CPBs are needed to verify their entanglement, and they can well be beyond the range of each other's interactions. We have thus proposed a way of entangling two CPBs through a cantilever in thermal state in the presence of decoherence without using any CANTILEVER CPB2 CPB1

SET2 SET1
FIG. 2: The figure shows a schematic diagram of the setup for entangling two Cooper pair boxes, denoted as CPB1 and CPB2 respectively, through a cantilever. For the entangling procedure, no measurements are required. For verification of the entanglement through a witness, measurements need to be performed on CPB1 and CPB2 through the single electron transistors SET1 and SET2 respectively.
measurements. This is an useful alternative to entangling the CPBs by direct interaction, as it will work even when the CPBs fall outside the range of each other's interaction. We have also proposed a method to verify their entanglement through local measurements on each of the CPBs. Of course, if the CPBs were allowed to resonantly exchange energy with a mode of the cantilever in analogy with Ref. [17], then not only entanglement, but any quantum computation would be possible in of low decoherence [17]. The presence of arbitrarily strong decoherence will, however, affect such a method. What we have shown is that even given the Hamiltonian of Ref. [7], arbitrarily strong decoherence and thermal states, entanglement between the CPBs is still possible.
Conclusions: In this paper we have proposed a scheme to entangle two mesoscopic systems of the same type through a third mesoscopic system. In this context we have also proposed a teleportation experiment in the mesoscopic setting using continuous variable entanglement for discrete variable teleportation.
SB gratefully acknowledges a visit to Oklahoma State University during which this work started.