REMOTE FIELD AND ATOMIC STATE PREPARATION

Transcription

1 International Journal of Quantum Information Vol. 6, No. (008) c World Scientific Publishing Company REMOTE FIELD AND ATOMIC STATE PREPARATION RAMEEZ-UL-ISLAM,, MANZOOR IKRAM, ASHFAQ H. KHOSA, and FARHAN SAIF Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan Photonics Division, National Institute of Lasers and Optronics, PINSTECH, Nilore, Islamabad, Pakistan Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan ashfaq khosa@yahoo.com Received 7 March 008 A scheme for remote preparation of field (atomic) states is proposed. Protocol execution requires cavity QED based atom-field interactions successively supplemented with Ramsey interferometry. The state to be remotely prepared at the receiver s end is acquired by deterministically manipulating the sender s component of the pre-shared entangled state. In the case of field entanglement, it is carried out with the help of an atom that passes through the sender s cavity and then traverses a classical external field for specified times prior to detection. However, for atomic entangled states, only interactions with the classical field suffice to complete the task. The scheme guarantees good success probability with high fidelity and requires one bit of classical communication. Keywords: Cavity QED; remote state preparation; quantum information. 1. Introduction Entangled states, a counter intuitive, nonlocal and characteristic trait of quantum theory, 1 6 is now being extensively exploited as a vital resource for various fields of quantum information. 7 Specifically important in our context are the phenomenon of quantum teleportation 8 and remote state preparation (RSP), 9 which can be employed to send (prepare) an arbitrary qubit faithfully over large distances without transporting any physical object, provided a prior entangled link exists between two spatially separated parties i.e. sender and receiver. In teleportation, an unknown quantum state is sent from Alice to Bob through a shared Einstein Podolosky Rosen (EPR) pair aided with two bit classical communication. According to protocol, Alice (sender) performs a joint Bell state measurement on the qubit along with her component of the EPR pair, and sends the result 393

2 394 R.-U.-Islam et al. thus obtained to Bob (receiver) through the classical channel. Quantum teleportation has been extensively investigated both theoretically as well as experimentally for various systems Execution of teleportation procedure generally does not require a prior knowledge of the state to be teleported. This implies that Alice can pick and teleport any arbitrary state with equal efficiency out of the ensemble of qubits presented to her. However in RSP, Alice precisely knows the state she wants to prepare at Bob s end. In this case, the remotely prepared state is engineered by the sender through sole manipulation of her part of the EPR pair using projective measurements in the qubit bases ( ψ (RSP ) A, ψ (RSP ) A ). 13 If the shared EPR state is ψ AB expressed in ( 0 A(B), 1 A(B) ) conventional basis, then the new qubit basis is related to the old one through the expressions; 0 A = α ψ (RSP ) A β ψ (RSP ) A and 1 A = β ψ (RSP ) A + α ψ (RSP ) A. Here α and β are probability amplitudes which satisfy the condition, α + β = 1. Therefore, the expression of shared EPR pair ψ AB under these transformations changes to ψ AB = 1 [ ψ (RSP ) A ψ (RSP ) B ψ (RSP ) A ψ (RSP ) B ]. (1) Thus, upon projective measurements accompanied by one bit classical communication, the state remotely prepared at Bob s end will be either, ψ (RSP ) B = α 1 B β 0 B, or ψ (RSP ) B = α 0 B + β 1 B, depending upon Alice s measurement result. Other Bell states can equivalently be used for RSP purpose. 13 The debate about known (unknown) state in the context of RSP (teleportation) seems trivial. 14 RSP can equally be used to prepare an unknown state if the sender uses chaotically modulated operational parameters during the state preparation procedure. For example, in our case as discussed below, chaotically modulated classical field amplitude may do this job. Therefore, in the comparison of RSP with teleportation, the real debatable issue is economy of resources, both classical and quantum. In this respect, Bennett et al. 15 have shown that given a large amount of entanglement, one classical bit is needed for RSP of a single qubit asymptotically, which is half that of the teleportation. Thus, it has been proven that for certain ensembles, RSP protocols are more economical compared to their sister teleportation protocols Resource economy of RSP becomes even more important in view of low optical detection efficiencies and the experimental problems related to Bell state measurement, an almost essential requirement for teleportation RSP is therefore being extensively explored on both theoretical 9,13,15,16,1 9 and experimental grounds Such investigations are expected to shed light on quantum-classical information linkage, and will help to find ways for efficient quantum information processing with optimum resource utilization. Our proposed RSP scheme relies on cavity QED techniques in conjunction with Ramsey interferometric method Although the protocol is described in a generalized cavity QED scenario, however, due to many experimental factors, its implementation is most feasible in the microwave regime. These, along with others,

3 Remote Field and Atomic State Preparation 395 include the availability of a similar setup comprised of two spatially separated high- Q cavities supplemented with required Ramsey zones in Haroche group at Paris. 36. Scheme and Mathematical Details In our proposed scheme, Alice and Bob are assumed to be in possession of any of the field Bell state that may be either single mode or two mode. Preparation techniques for such states have been proposed and experimentally implemented, again by Haroche s group. 40,41 For remote state preparation, Alice sends a two-level atom initially prepared in the ground state, resonant with her cavity mode, and controls the atom-field interaction time corresponding to the π Rabi pulse. This leaves her cavity into the vacuum state by effectively transferring the quantum information to the atom. This atom then passes through a Ramsey field for a specific, predetermined time to yield the desired ratio of probability amplitudes at Bob s end. Detection of the atom either in the ground or excited state after its interaction with the Ramsey field completes the operational scenario of RSP. Alice then sends one bit classical information (i.e. atom detected in ground or excited state) to Bob, who can recover the state simply, if necessary, through usual local transformations 4 of the state at his end (Fig. 1). Let us assume that Alice and Bob are sharing the Bell state ψ AB = 1 ( 0 A, 1 B 1 A, 0 B ). () State Selective Photo Ionization Detector Classical Field Cavity A Cavity B Two-level atom Fig. 1. Schematics of field RSP where Alice and Bob are sharing the entangled field state ψ AB. Alice sends a resonant two-level atom into the cavity in her possession where it interacts with the cavity field for a time corresponding to π-pulse. This atom, carrying the transferred entanglement, then passes through a classical Ramsey field for a predefined time and is finally detected through a state selective photoionization process.

4 396 R.-U.-Islam et al. Alice then sends a two-level atom, initially in its ground state g A, through her cavity, where it interacts resonantly with the cavity mode. The generalized state vector describing the atom-field interaction for such a system up to any arbitrary time t may be expressed as follows ψ AB (t) = C g A 0A,1 B (t) 0 A, g A, 1 B + C g A 1A,0 B (t) 1 A, g A, 0 B + C e A 0A,0 B (t) 0 A, e A, 0 B, with initial conditions C g A 0A,1 B (0) = C g A 1A,0 B (0) = 1/ and C e A 0A,0 B (0) = 0. The interaction picture Hamiltonian under dipole and rotating wave approximations describing resonant atom-field interaction is given as (3) Ĥ q I = ħµσ(a) + a A + H.C. (4) Here σ (A) ) = e A g A ( g A e A ) are the raising (lowering) operator for Alice s atom and a A (a A) denotes the field creation (annihilation) ladder operator for Alice s cavity mode. The atom-field coupling constant is denoted by µ. The solution of Schrodinger s equation for an interaction time t = π/µ yields the following state vector for the atom-field system + (σ (A) ψ AB = 1 ( g A, 1 B i e A, 0 B ) 0 A. (5) The atom, after completing its interaction with the cavity A, enters into a Ramsey field where it interacts, again resonantly, with the field for a predetermined time t R. The state of the system for such atom-field interaction may be expressed by ψ AB (t R ) = C g A 1B (t R ) g A, 1 B + C e A 1B (t R ) e A, 1 B + C g A 0B (t R ) g A, 0 B + C e A 0B (t R ) e A, 0 B. (6) This state vector is written, keeping the semi-classical theory in view. Thus, here the atom is quantized but the field is treated classically. Initial conditions applied to the system are; C g A 1B (t R = 0) = 1/, C e A 0B (t R = 0) = i/ and C g A 0B (t R = 0) = C e A 1B (t R = 0) = 0. The corresponding semi-classical interaction Hamiltonian under dipole and rotating wave approximations is 43 Ĥ (sc) I = ħω (e iφ σ (A) + + e iφ σ (A) ), (7) where Ω = ge ε/ is the Rabi frequency with ε and ge = e g A r e A representing the classical field amplitude and transition dipole matrix element having a relative phase φ respectively. Solution of the Schrodinger s equation results in four pair-wise coupled differential equations for the rate of change of probability amplitudes. This set of coupled differential equations, on simplification, lead to the final expression

5 Remote Field and Atomic State Preparation 397 of the state vector ψ AB (t R ) = 1 { ( Ω [ e iφ sin ) ( Ω 0 B cos t R { Ω i cos t R 0 B + e iφ sin t R ( Ω t R ) } 1 B g A ) 1 B } ] e A. (8) This expression shows that if Alice detects its atom in the ground state g A, then the field state prepared in Bob s cavity will be ψ (g A) Ω Ω B = sin t R 0 B e iφ cos t R 1 B. (9) Whereas if the atom at Alice s end gets detected in the excited state e A, then the corresponding remotely prepared state at the Bob s end will be ψ (e A) Ω Ω B = cos t R 0 B + e iφ sin t R 1 B. (10) Hence, the sender can remotely prepare any field superposition with adjustable probability amplitude ratios by controlling the interaction time t R of the atom with the Ramsey field. One bit communication through classical channel will help Bob to recover the original state. If the atom is detected in the ground state, then Bob need not to do anything further, as he has already received the prepared state in its full originality. However, if the atom is detected in the excited state, then the local operations of the phase gate followed by a NOT gate 4,44 will reincarnate the state at Bob s end. The states for which phase φ of the dipole matrix vanishes lie on the polar line, whereas manipulability of the phase in general allows us to span the complete surface of the Bloch sphere. Such manipulations can be carried out either by applying an additional external electric field or a magnetic field in an appropriate geometry. 45 The same task can be carried out by using a three-level atom in cascade configuration, such that the lower two-levels are resonant with the first Ramsey zone, whereas the upper two levels couple dispersively with an extra Ramsey zone. 46 The proposed scheme can be implemented with any field Bell state, whether it carries single or two-mode entanglement shared between the parties. Furthermore, as evident from the procedural discussion, the same setup can equally be implemented to remotely prepare an atomic superposition of the internal states, provided that sender and receiver are in possession of any one of the Bell atomic states, ψ ± AB = ( g A, e B ± e A, g B )/ or, φ ± AB = ( g A, g B ± e A, e B )/ with g A ( e A ) and g B ( e B ) representing the ground (excited) states of the atoms possessed by the sender and receiver respectively. Thus, in this case, Alice simply passes her part of the entangled atomic state through the Ramsey zone for a predetermined time, followed by detection of the atom either in the ground or excited

6 398 R.-U.-Islam et al. level. The same procedure can also be implemented to remotely prepare an entangled state. Consider the case where three spatially separated parties, say Alice, Bob and Charlie are initially assumed to share the following GHZ field state ψ ABC = 1 ( 0 A, 0 B, 0 C 1 A, 1 B, 1 C ), (11) then execution of the RSP protocol by Alice leaves Bob and Charlie entangled into the Bell state ψ (g A) Ω Ω BC = cos t R 0 B, 0 C e iφ sin 1 B, 1 C, (1) ψ (e A) Ω BC = e iφ sin t R 0 B, 0 C + cos t R ( Ω t R upon detection of the atom either into g A or e A respectively. ) 1 B, 1 C, (13).1. Success probability and fidelity The proposal presented above is based on the time-tested tools of cavity QED and Ramsey interferometry. 47 Good workability of a similar setup has already been experimentally demonstrated for some other task. 48 Success probabilities and fidelities of the schemes presented here mainly depend upon the degree of achievable precision in selection of the interaction times of the atom with both quantized cavity field as well as Ramsey classical fields. The success probability for atomic state RSP is unity if the supplied entangled state is truly a Bell state. However, for field RSP, the success probability mainly depends on the imprecision t incurred in the selection of resonant atom-field interaction times t, and is given by P s = 1 sin (µ t) (14) whereas fidelities of the field RSP comes to be Ϝ (f) g A = Ω Ω cos(µ t) sin t R sin (t R + t R ) Ω Ω + cos t R cos (t R + t R ), (15) and Here Ϝ (f) g A and Ϝ (f) e A Ϝ (f) e A = ( Ω cos(µ t) cos t R ( Ω + sin t R ) ( Ω cos ) ( Ω sin (t R + t R ) ) (t R + t R ) ). (16) stand for the fidelities of the remotely prepared field states when, after execution of the protocol, the atom is detected in the ground or excited state respectively. Interaction of the atom with the Ramsey field lasts for a predetermined

7 Remote Field and Atomic State Preparation 399 time t R with a controllable accuracy limited by t R. This imprecision in interaction time t R effectively incorporates almost all inaccuracies such as velocity spread of the atom, coupling dispersion and field on/off delays incurred during interactions. However, inaccuracies in atom-field interactional times are mainly contributed by the velocity spread ± v around the selected mean velocity v of the atom. Haroche et al. 41 have reported an atomic velocity spreading of about ± m/s if the mean velocity is taken to be 503 m/s. This minor width consequently yields an almost negligible imprecision in atom-field interaction times. More enhanced control over atomic motion can be achieved using pre-cooled atoms out of a magneto-optical trap. 49,50 Plots of fidelities versus Ω t and Ω t R, corresponding to time Ωt R / = π/4 for Ϝ (f) g A and Ωt R / = π/8 for Ϝ (f) e A are shown in Figs. and 3, respectively. 3. Conclusion The proposed scheme is expected to yield promising experimental results as compared to already published RSP data using either NMR 30 or measurement of field quadrature noise. 3 In the NMR case, the state prepared was just an angstrom distance away, whereas single-mode photonic qubit remotely prepared by measuring field quadrature noise has yielded only 55% cumulative efficiency due to optical losses, dark counts and other imperfections of the entire optical setup. Furthermore, the scheme was limited to single mode RSP. Our scheme, however, can incorporate Fig.. Fidelity of the remotely prepared state when atom is detected in ground state. Here we have taken Ωt R / = π/4.

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