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Quantum information processing via superconducting quantum interference device coupled to imperfect cavity
Physica C 458 (2007) 58–63 www.elsevier.com/locate/physc
Quantum information processing via superconducting quantum interference device coupled to imperfect cavity Zheng-Yuan Xue *, Ping Dong, You-Min Yi, Zhuo-Liang Cao School of Physics and Material Science, Anhui University, Hefei 230039, China Received 3 December 2005; accepted 20 March 2007 Available online 14 April 2007
Abstract We investigate schemes for quantum information processing via radio frequency superconducting quantum interference device (rf-SQUID) coupled to imperfect cavity and liner optics devices. Our schemes combine two distinct advantages: rf-SQUID qubit with long coherent time sevres as memory bit and photonic qubit as flying bit, thus they are suitable for long distant quantum communication. Our schemes also requires less experimental demands, thus they can be demonstrated experimentally within current techniques. 2007 Elsevier B.V. All rights reserved. PACS: 03.67.Hk; 03.65.Ud; 42.50.Dv Keywords: Quantum information processing; rf-SQUID; Cavity QED
1. Introduction Recently, much attention has been paid to the realization of quantum computer [1], which can provide more powerful computational ability than a classical one. Later, various physical systems have been suggested for its Realization. Among them, cavity quantum electrodynamics (QED) system, where atoms interact with a quantized electromagnetic field, is proven to be a promising candidate [2]. However, the strong coupling limit is difficult to achieve with atoms in a microcavity. In contrast, it was realized with superconducting charge qubits and flux qubits [3,4] and semiconductor quantum dots [5] in a microcavity. The cavities usually act as memories, which store the information of an electronic system and transfer it back to that electronic system, thus the decoherence of the cavity field becomes one of the main obstacles for the implementation of quantum information in cavity QED system. Recently, Zheng and Guo *
Corresponding author. Address: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail address: [email protected] (Z.-Y. Xue). 0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.401
proposed a novel scheme [6], which greatly prolong the efficient decoherence time of the cavity with a virtually excited nonresonant cavity. Osnaghi et al. [7] had experimentally implemented the scheme using two Rydberg atoms crossing a nonresonant cavity. But, it is well known that atomic qubits are only ideal serving as stationary qubits, not suitable for long distance transmission. Thus, it is extremely hard experimental task to realize long distance quantum communication with atomic qubit serving as flying qubit. It is also well known that spontaneous and detected decay are unavoidable in practical quantum information processing in cavity QED system [8]. Therefore, one may not expect such a process to be helpful in quantum information processing. But, recent works [9] dispel this myth by showing that how the detection (or the non detection) of cavity decays can be used to entangle the states of distinct atoms. Furthermore, it can also be used for quantum teleportation [10], as photonic qubits are perfect candidates for flying qubits. SQUID qubits have recently attracted much attention in quantum information community as they are easy to scale up and have been demonstrated to have long decoherence time [4,11]. Here, we investigate
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schemes for implementing quantum information processing (QIP) and quantum communication via single-photon detected cavity decay using rf-SQUIDs trapped in different resonant imperfect cavities. Our protocol combines two distinct advantages: rf-SQUID qubit with long decoherent time sevres as memory bit and photonic qubit as flying bit, thus they are suitable for long distant quantum communication. This paper is organized as follows. In Section 2, we introduce the basic model of a three-state rf-SQUID coupled to a single-mode resonant leaky cavity or driven by a classical microwave pulse. In Section 3, we show how to entangle distant SQUIDs trapped in different cavities. In Section 4, we give a discussion on the realization of a Bell-state analyzer for QIP. In Section 5, we consider entanglement swapping without prior distribution of the entangled states. In Section 6, we present a quantum dense coding (QDC) scheme. A concluding summary and some discussions are given in Section 7. 2. Basic model In this section, we introduce the model of an rf-SQUID trapped in a single-mode microwave resonant leaky cavity, which was initially prepared in vacuum state, i.e., j0iC. The rf-SQUID consisting of a Josephson tunnel junction enclosed by a superconducting loop, the size of which is on the order of 10–100 lm. The Hamiltonian for an rfSQUID with junction capacitance C and loop inductance L is 2 Q2 ðU Ux Þ U Hs ¼ þ EJ cos 2p ; ð1Þ U0 2C 2L where U is the magnetic flux threading the ring, Q is the total charge on the capacitor, with the commutation relation [U, Q] = i h. Ux is the static (or quasistatic) external magnetic flux applied to the ring, EJ IcU0/2p is the Josephson coupling energy, with Ic the critical current of the junction and U0 = h/2e the flux quantum. The rf-SQUID considered in this paper has three-level K-type energy structure includes two lower flux states, i.e., j0i and j1i, which reside in two distinct potential valley and an upper state jei as shown in Fig. 1. Suppose the coupling of j0i, j1i and jei with other levels of the SQUID via the resonator is negligible, which can be readily satisfied by adjusting the level spacings of the SQUID. For a SQUID, the level spacings can be changed easily by varying the external flux Ux or the critical current Ic. Assuming the rf-SQUID is trapped in a specific position in the cavity, and the coupling coefficient g is constant during the interaction. The cavity is resonant with the j0i M jei transition while off-resonant with the j1i M jei transition and the j0i M j1i transition of the SQUID. The effective Hamiltonian (set h = 1) of the SQUID-cavity interacting system, in the interaction picture and after rotating-wave appropriation, is H I ¼ i½gðajeih0j aþ j0ihejÞ kaþ a:
ð2Þ
59
Fig. 1. Level structure of the rf-SQUID used in this paper. It has threelevel K-type energy structure includes two lower flux states j0i, j1i and an upper state jei.
where a and a+ are the annihilation and creation operators of the cavity mode; k is the photon decay rate from the cavity. If the rf-SQUID is initially in the state j0i and trapped in the cavity, a classical laser pulse, the frequency of which is j0i M jei and far-off resonant to the transitions j0i M j1i and j1i M jei, on the trapped rf-SQUID could switch on the effective Hamiltonian HI of the SQUID-cavity interacting system. Under the Hamiltonian in (2), the time evolution of the states of the system is described by j0ij0iC ! j0ij0iC ;
ð3aÞ
jeij0iC ! ðajeij0iC þ bj0ij1iC Þ:
ð3bÞ
The coefficients in (3b) are 1 k 1 1 a ¼ e2kt cos Xt þ sin Xt ; 2 X 2 2g 1kt 1 e 2 sin Xt; X 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with X ¼ 4g2 k 2 . Next, let us consider a SQUID driven by a classical laser pulse with the magnetic component B(r, t) = B(r)cos 2pmt. Here, B(r) and m are the magnetic field amplitude and frequency of the pulse. If it is resonant with the j1i M jei transition but far-off resonant with the transitions of j0i M jei and j0i M j1i of the SQUID, the interaction Hamiltonian in the interaction picture is b¼
i H C ¼ ðX12 j1ihej þ h:c:Þ; 2
ð4Þ
where X12 is the Rabi frequency of the pulse. So, a pulse with duration time t results in the following rotation 1 1 j1i ! cos X12 tj1i sin X12 tjei; 2 2 1 1 jei ! sin X12 tj1i þ cos X12 tjei: 2 2
ð5aÞ ð5bÞ
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3. Entanglement generation
q1;2 ¼
In this section, we consider the generation of a entangled state of two distant rf-SQUIDs trapped in diffident cavities. The two cavities are both initially prepared in vacuum state. The success or failure of the generation of the desired entangled state can be determined by detecting the photon leaking out of the cavity. The generating process is shown in Fig. 2. The two rf-SQUIDs are both initially prepared in the states jei. Thus the initial state of the two rf-SQUIDs and two cavities system is jwð0Þi ¼ jei1 j0iA jei2 j0iB :
ð6Þ
After an interaction time t1, the above state evolves to jwðt1 Þi ¼ ðajei1 j0iA þ bj0i1 j1iA Þ ðajei2 j0iB þ bj0i2 j1iB Þ: ð7Þ Classical laser pulse jei M j1i on the trapped rf-SQUIDs with time duration X12t = p lead the above state to jwðt1 Þi ¼ ðaj1i1 j0iA þ bj0i1 j1iA Þ ðaj1i2 j0iB þ bj0i2 j1iB Þ: ð8Þ Before one of the two detectors clicks, the joint state of atom-cavity system will evolve into jwðtÞi ¼ ðaj1i1 j0iA þ bekt j0i1 j1iA Þ ðaj1i2 j0iB þ bekt j0i2 j1iB Þ:
ð9Þ
While one of the detectors D± clicks after a time span of t2,pitffiffiffi corresponds to the action of the jump operators 1= 2ðaA aB Þ on the joint state jw(t)i, then the joint state of the entire system becomes b2 ekt2 j0i1 j0i2 jw iA;B þ abjW i1;2 j0iA j0iB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð10Þ b4 e2kt2 þ a2 b2 pffiffiffi pffiffiffi where jw iA;B ¼ 1= 2ðj0iA j1iB j1iA j0iB Þ; jW i1;2 ¼ 1= 2 ðj1i1 j0i2 j0i1 j1i2 Þ: Tracing over the state of cavities A and B, the reduced density matrix of rf-SQUIDs 1 and 2 is
jwðt2 Þi ¼
b4 e2kt j0i1 j0i2;1 h0j2 h0j þ a2 b2 jW i1;2;12 hW j b4 e2kt þ a2 b2
ð11Þ
The successful probability of obtaining the maximally bipartite entangled state in (11) is P = b4e2kt + a2b2. The fidelity of the desired states jW±i1,2 from the entangled states in (11) is F ¼1;2 hW jq1;2 jW i1;2 ¼
a2 b 2 : b4 e2kt þ a2 b2
ð12Þ
In the case of a/b 1, the fidelity will be F 1. In this way, we generate a bipartite entangled state of two distant rf-SQUIDs via single-photon detected cavity decay. The two rf-SQUIDs are trapped in diffident cavities and photonic qubit serves as flying bits in this generation scheme, thus this scheme can be used to entangle two distant rfSQUIDs. 4. Bell-state analyzer Bell-state measurement (BSM) plays an important role in QIP and is also a necessity in quantum communication protocols. But it is very difficult to realize BSM for atomic (ionic) states in experiment [12,13]. Usually, they will be converted into the product of separate measurements on single particle [6,15,14] in experimental demonstrations. Thus, we consider the implementation of BSM via cavity decay, which again can be succeeded probabilistically. But the two involved rf-SQUIDs can be in distant locations. The Bell states analyzer is similar to the generation process as shown in Fig. 2, but now the two rf-SQUIDs are initially entangled. The four Bell states are 1 jU i1;2 ¼ pffiffiffi ðj1i1 j1i2 j0i1 j0i2 Þ; 2 1 jW i1;2 ¼ pffiffiffi ðj1i1 j0i2 j0i1 j1i2 Þ: 2
ð13aÞ ð13bÞ
The two cavities are both initially prepared in the vacuum states j0iC. Classical laser pulse j1i M jei shining on rfSQUIDs 1 and 2 with time duration X12t = 3p, then both parties set the interaction time as tan Xt2 ¼ Xk , the states of the two rf-SQUIDs and two cavities system in (13) evolve into the following four corresponding unnormalized states 1 jU iA;B ¼ pffiffiffi ðb2 j11iA;B j00iA;B Þ; 2 b jW iA;B ¼ pffiffiffi ðj01iA;B j10iA;B Þ: 2
Fig. 2. Scheme for generating bipartite entangled state via cavity decay from two product state. A and B denote the two cavities, BS is a 50/50 beam splitter and D± are single-photon detectors.
:
ð14aÞ ð14bÞ
Here we have omitted the states of the two rf-SQUIDs, which are both in their ground states j0ii (i = 1,2) after the above controlled interactions. The joint state of the two cavities will evolve into
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1 jU iA;B ¼ pffiffiffi ðb2 e2kt j11iA;B j00iA;B Þ; 2 b jW iA;B ¼ pffiffiffi ekt ðj01iA;B j10iA;B Þ 2
ð15aÞ ð15bÞ
before one of the detectors D± clicks. If D+ clicks, one knows exactly that the initial bipartite entangled state is jW+i. If D clicks, then the initial bipartite entangled state is jWi. If none of the two detectors clicks then the process failed. So, our scheme is a probabilistic one, with the successful probability P = b2e2kt. In this way, we implement a probabilistic Bell state analyzer and it is well within current techniques. 5. Entanglement swapping In entanglement swapping [16], there are usually three spatially separate users, and two of them have shared one pair of entangled particles with the third user. Then the third user will operate a joint BSM on the two particles he possesses. Corresponding to the measurement result, the two particles possessed by the two spatially separate users will collapse into an entangled state without any entanglement before the joint measurement. Here, we will implement entanglement swapping via cavity decay without prior distribution of the entangled states, i.e., without distribution the two entangled states to the third party. Suppose two entangled states are both initially prepared in one of the four Bell states jW+i (see (13b)). The setup of entanglement swapping is shown in Fig. 3. Two rf-SQUIDs 1 and 3 are trapped in the optical cavities A and B, respectively. The initial state of the four rf-SQUIDs and two cavities system is 1 jwi ¼ ðj1i1 j0i2 þ j0i1 j1i2 Þj0iA ðj1i3 j0i4 þ j0i3 j1i4 Þj0iB : 2 ð16Þ Classical laser pulse j1i ! jei shining on rf-SQUIDs 1 and 3 with time duration X12t = 3p and an interaction time tan Xt2 ¼ Xk of the rf-SUQID-cavity systems, the state evolves into the following unnormalized state
Fig. 3. Setup to implement entanglement swapping without distribution the two entangled states to the third party Charlie. Two rf-SQUIDs 1 and 3 are trapped in the cavities A and B, respectively.
1 jwi ¼ ðbj0i2 j1iA þ jei2 j0iA Þ ðbj0i4 j1iB þ jei4 j0iB Þ: 2
61
ð17Þ
Again, we have omitted the state of the rf-SQUIDs 1 and 3, which are both in their ground states j0ii (i = 1, 3). Classical laser pulse jei ! j1i shining on rf-SQUIDs 2 and 4 with time duration X12t = p leads the above state to 1 jwi ¼ ðbj0i2 j1iA þ j1i2 j0iA Þ ðbj0i4 j1iB þ j1i4 j0iB Þ: 2 ð18Þ The above state will evolve into the following state before one of the two single-detectors clicks. 1 jwðtÞi ¼ ðbekt j0i2 j1iA þ j1i2 j0iA Þ ðbekt j0i4 j1iB þ j1i4 j0iB Þ: 2 ð19Þ
D+ or D clicks corresponds to the following two collapsed unnormalized states, respectively. b kt 1 jwðtÞi ¼ pffiffiffi e bekt j0 0i2;4 pffiffiffi ðj0 1i j1 0iÞA;B 2 2 1 þ pffiffiffi ðj1i2 j0i4 j0i2 j1i4 Þj0 0iA;B : ð20Þ 2 The successful probability of obtaining the entangled state in (20) is P = 1/2b2(e2kt + 1). The fidelity of the obtained state jW±i1,2 is F = b2/(b4e2kt + b2). In the case of b2 b, the fidelity will be F 1. In this way, we can entangle distant entangled state without prior distribution of the entangled state. 6. Quantum dense coding QDC [17] is a process to send two classical bits (cbits) of information from a sender (Alice) to a remote receiver (Bob) by sending only a single qubit. After the work of Bennett et al. [17], it attracts many public attentions. Schemes for implementing QDC are also been proposed [15,18] in cavity QED. But, it is well known that atomic qubit is only ideal stationary qubit, not suitable for long distance transmission. Thus the realization of long distance QDC with atomic qubit serving as flying qubit as suggested in [15,18] are extremely hard experimental tasks. Here we suggest a scheme for QDC via detected cavity decay. Similarly, with Bell state analyzer, we can also implementing quantum teleportation [19]. Assume the two parties, i.e., Alice and Bob, initially share a bipartite entangled state jW+i1,2 with rf-SQUIDs 1 and 2 belong to the two parties Alice and Bob, respectively. Information encoding. Alice performs one of the four local operations (I, rx, iry, rz) on her rf-SQUID. These operations denote 2 cbits of information, and will transfer the state jW+i1,2 into the four Bell states in (13). These single-qubit operations can be implemented readily by classical laser pulse on the rf-SQUID. Thus the information is encoded into the pure entangled state, which is shared
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between the two parties, the encoding of the two cbits information is completed. Information extracting. Alice and Bob send their rfSQUIDs into their cavities. According to the time evolution in Eq. (3), an interaction time of tan Xt2 ¼ Xk and some classical laser pulses lead the state (13) to the form of (14). Similarly to the above analysis dealing with Bell state analyzer, we can discriminate the four states in ((14)) with the successful probability P = b2e2kt. In other words, the QDC process succeed with probability P = b2e2kt. 7. Discussion and conclusion Finally we briefly address the experimental feasibility of the proposed scheme. The interaction time much shorter than the rf-SQUID decoherent time. After the very short time span of controlled interaction, both rf-SQUIDs are in their ground states, thus the spontaneous emission effect is greatly suppressed. Meanwhile, short time span of interaction is also very preferable in terms of decoherent. Relaxation rates of the system are small and well understood. The strong-coupling conditions are readily fulfilled [3,4]. The time constants involved are long enough to realize all the required manipulations. The quantum systems are separated by centimeterscale distances, thus can be individually addressed [7]. The state of the SQUIDs are initially in the excited state when spontaneous emission is inevitable. So in order to make the proposal more reasonable, we can first prepare the state in one of the ground state. Before we start the cavity-qubit interaction we use an classical laser field to excite it. The detection of photon in our schemes need further explanation. As the photons are in the radio frequency, there will be serious problem when detected with conventional optical detectors. But we can incorporate the photon to a cavity where we also have a SQUID prepared in the ground state, then measure the state of the SQUID [20]. If the SQUID still in its ground state, then there is no photon; when the SQUID is excited, we can conclude that there is one photon come into the cavity. It is worth to note that we reserved the word detector to denote the above-mentioned setup for detecting photon in this paper to simplify our presentation. For the sake of definitiveness, let us estimate the experimental feasibility of realizing the schemes with the parameters already available in present experiment [3,4,11]. Suppose the quality factor of the superconducting cavity is Q = 1 · 106 (cavity with Q = 3 · 108 has already been reported [7]) and the cavity mode frequency is xc = 50 GHz, the cavity decay time is k1 = Q/xc = 20 ls. The upper state energy relaxation constant is c1 e ¼ 2:5 ls. For a superconducting standing-wave cavity and a SQUID located at one of antinodes of the magnetism field, the coupling constant is g = 1.8 · 108 Hz, thus the resonant interaction time for our scheme is Tr = p/g ’ 17 ns. Meanwhile, the strong coupling condition can be perfectly achieved as g2/cek = 1.6 · 106 1. The Rabi frequency for the interaction of the rf pulse and the SQUID is X = 8.5 · 107 Hz, thus the interaction time for the single
photon detection is Tl = p/2X ’ 18 ns. Both interaction time (Tr and Tl) are much shorter than the cavity decay time and the relaxation time of the upper state. In summary, we have investigated schemes for quantum information processing and quantum communication via sinel-photon detected cavity decay and liner optics devices. The schemes are all probabilistic ones but well within current techniques. Our protocol combines two distinct advantages: atomic qubit sevres as stationary bit and photonic qubit as flying bit, thus it is suitable for long distant quantum communication. Compared with previously suggested schemes of quantum communication in cavity system, our scheme also requires less experimentally demands. Thus they are also easier in terms of experimental demonstration. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60678022, the Doctoral Fund of Ministry of Education of China under Grant No. 20060357008, the Key Program of the Education Department of Anhui Province under Grant Nos.: 2006KJ070A, 2006KJ057B and the Talent Foundation of Anhui University. References [1] D. Deutsch, Proc. R. Soc. London A 400 (1985) 97; D. Deutsch, Proc. R. Soc. London A 425 (1989) 73; P.W. Shor, in: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Santa Fe, NM, 1994; L.K. Grover, Phys. Rev. Lett. 79 (1997) 325. [2] J.M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565. [3] A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, Nature (London) 431 (2004) 162. [4] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans, J.E. Mooij, Nature (London) 431 (2004) 159. [5] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H.M. Gibbs, G. Rupper, C. Ell, O.B. Shchekin, D.G. Deppe, Nature (London) 432 (2004) 200; J.P. Reithmaier, G. Sk, A. Lo¨ffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, T.L. Reinecke, A. Forchel, Nature (London) 432 (2004) 197; A. Badolato, K. Hennessy, M. Atatu¨re, J. Dreiser, E. Hu, P.M. Petroff, A. Imamoglu, Science 308 (2005) 1158. [6] S.-B. Zheng, G.-C. Guo, Phys. Rev. Lett. 85 (2000) 2392. [7] S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. Lett. 87 (2001) 037902. [8] M.B. Plenio, P.L. Knight, Phys. Rev. A 53 (1996) 2986. [9] C. Cabrillo, J.I. Cirac, P. Garcła-Ferna´ndez, P. Zoller, Phys. Rev. A 59 (1999) 1025; M.B. Plenio, S.F. Huelga, A. Beige, P.L. Knight, Phys. Rev. A 59 (1999) 2468; G.J. Yang, O. Zobay, P. Meystre, Phys. Rev. A 59 (1999) 4012. [10] S. Bose, P.L. Knight, M.B. Plenio, V. Vedral, Phys. Rev. Lett. 83 (1999) 5158; B. Yu, Z.W. Zhou, Y. Zhang, G.Y. Xiang, G.C. Guo, Phys. Rev. A 70 (2004) 014302;
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Quantum information processing via superconducting quantum interference device coupled to imperfect cavity Zheng-Yuan Xue *, Ping Dong, You-Min Yi, Zhuo-Liang Cao School of Physics and Material Science, Anhui University, Hefei 230039, China Received 3 December 2005; accepted 20 March 2007 Available online 14 April 2007
Abstract We investigate schemes for quantum information processing via radio frequency superconducting quantum interference device (rf-SQUID) coupled to imperfect cavity and liner optics devices. Our schemes combine two distinct advantages: rf-SQUID qubit with long coherent time sevres as memory bit and photonic qubit as flying bit, thus they are suitable for long distant quantum communication. Our schemes also requires less experimental demands, thus they can be demonstrated experimentally within current techniques. 2007 Elsevier B.V. All rights reserved. PACS: 03.67.Hk; 03.65.Ud; 42.50.Dv Keywords: Quantum information processing; rf-SQUID; Cavity QED
1. Introduction Recently, much attention has been paid to the realization of quantum computer [1], which can provide more powerful computational ability than a classical one. Later, various physical systems have been suggested for its Realization. Among them, cavity quantum electrodynamics (QED) system, where atoms interact with a quantized electromagnetic field, is proven to be a promising candidate [2]. However, the strong coupling limit is difficult to achieve with atoms in a microcavity. In contrast, it was realized with superconducting charge qubits and flux qubits [3,4] and semiconductor quantum dots [5] in a microcavity. The cavities usually act as memories, which store the information of an electronic system and transfer it back to that electronic system, thus the decoherence of the cavity field becomes one of the main obstacles for the implementation of quantum information in cavity QED system. Recently, Zheng and Guo *
Corresponding author. Address: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail address: [email protected] (Z.-Y. Xue). 0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.401
proposed a novel scheme [6], which greatly prolong the efficient decoherence time of the cavity with a virtually excited nonresonant cavity. Osnaghi et al. [7] had experimentally implemented the scheme using two Rydberg atoms crossing a nonresonant cavity. But, it is well known that atomic qubits are only ideal serving as stationary qubits, not suitable for long distance transmission. Thus, it is extremely hard experimental task to realize long distance quantum communication with atomic qubit serving as flying qubit. It is also well known that spontaneous and detected decay are unavoidable in practical quantum information processing in cavity QED system [8]. Therefore, one may not expect such a process to be helpful in quantum information processing. But, recent works [9] dispel this myth by showing that how the detection (or the non detection) of cavity decays can be used to entangle the states of distinct atoms. Furthermore, it can also be used for quantum teleportation [10], as photonic qubits are perfect candidates for flying qubits. SQUID qubits have recently attracted much attention in quantum information community as they are easy to scale up and have been demonstrated to have long decoherence time [4,11]. Here, we investigate
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schemes for implementing quantum information processing (QIP) and quantum communication via single-photon detected cavity decay using rf-SQUIDs trapped in different resonant imperfect cavities. Our protocol combines two distinct advantages: rf-SQUID qubit with long decoherent time sevres as memory bit and photonic qubit as flying bit, thus they are suitable for long distant quantum communication. This paper is organized as follows. In Section 2, we introduce the basic model of a three-state rf-SQUID coupled to a single-mode resonant leaky cavity or driven by a classical microwave pulse. In Section 3, we show how to entangle distant SQUIDs trapped in different cavities. In Section 4, we give a discussion on the realization of a Bell-state analyzer for QIP. In Section 5, we consider entanglement swapping without prior distribution of the entangled states. In Section 6, we present a quantum dense coding (QDC) scheme. A concluding summary and some discussions are given in Section 7. 2. Basic model In this section, we introduce the model of an rf-SQUID trapped in a single-mode microwave resonant leaky cavity, which was initially prepared in vacuum state, i.e., j0iC. The rf-SQUID consisting of a Josephson tunnel junction enclosed by a superconducting loop, the size of which is on the order of 10–100 lm. The Hamiltonian for an rfSQUID with junction capacitance C and loop inductance L is 2 Q2 ðU Ux Þ U Hs ¼ þ EJ cos 2p ; ð1Þ U0 2C 2L where U is the magnetic flux threading the ring, Q is the total charge on the capacitor, with the commutation relation [U, Q] = i h. Ux is the static (or quasistatic) external magnetic flux applied to the ring, EJ IcU0/2p is the Josephson coupling energy, with Ic the critical current of the junction and U0 = h/2e the flux quantum. The rf-SQUID considered in this paper has three-level K-type energy structure includes two lower flux states, i.e., j0i and j1i, which reside in two distinct potential valley and an upper state jei as shown in Fig. 1. Suppose the coupling of j0i, j1i and jei with other levels of the SQUID via the resonator is negligible, which can be readily satisfied by adjusting the level spacings of the SQUID. For a SQUID, the level spacings can be changed easily by varying the external flux Ux or the critical current Ic. Assuming the rf-SQUID is trapped in a specific position in the cavity, and the coupling coefficient g is constant during the interaction. The cavity is resonant with the j0i M jei transition while off-resonant with the j1i M jei transition and the j0i M j1i transition of the SQUID. The effective Hamiltonian (set h = 1) of the SQUID-cavity interacting system, in the interaction picture and after rotating-wave appropriation, is H I ¼ i½gðajeih0j aþ j0ihejÞ kaþ a:
ð2Þ
59
Fig. 1. Level structure of the rf-SQUID used in this paper. It has threelevel K-type energy structure includes two lower flux states j0i, j1i and an upper state jei.
where a and a+ are the annihilation and creation operators of the cavity mode; k is the photon decay rate from the cavity. If the rf-SQUID is initially in the state j0i and trapped in the cavity, a classical laser pulse, the frequency of which is j0i M jei and far-off resonant to the transitions j0i M j1i and j1i M jei, on the trapped rf-SQUID could switch on the effective Hamiltonian HI of the SQUID-cavity interacting system. Under the Hamiltonian in (2), the time evolution of the states of the system is described by j0ij0iC ! j0ij0iC ;
ð3aÞ
jeij0iC ! ðajeij0iC þ bj0ij1iC Þ:
ð3bÞ
The coefficients in (3b) are 1 k 1 1 a ¼ e2kt cos Xt þ sin Xt ; 2 X 2 2g 1kt 1 e 2 sin Xt; X 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with X ¼ 4g2 k 2 . Next, let us consider a SQUID driven by a classical laser pulse with the magnetic component B(r, t) = B(r)cos 2pmt. Here, B(r) and m are the magnetic field amplitude and frequency of the pulse. If it is resonant with the j1i M jei transition but far-off resonant with the transitions of j0i M jei and j0i M j1i of the SQUID, the interaction Hamiltonian in the interaction picture is b¼
i H C ¼ ðX12 j1ihej þ h:c:Þ; 2
ð4Þ
where X12 is the Rabi frequency of the pulse. So, a pulse with duration time t results in the following rotation 1 1 j1i ! cos X12 tj1i sin X12 tjei; 2 2 1 1 jei ! sin X12 tj1i þ cos X12 tjei: 2 2
ð5aÞ ð5bÞ
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3. Entanglement generation
q1;2 ¼
In this section, we consider the generation of a entangled state of two distant rf-SQUIDs trapped in diffident cavities. The two cavities are both initially prepared in vacuum state. The success or failure of the generation of the desired entangled state can be determined by detecting the photon leaking out of the cavity. The generating process is shown in Fig. 2. The two rf-SQUIDs are both initially prepared in the states jei. Thus the initial state of the two rf-SQUIDs and two cavities system is jwð0Þi ¼ jei1 j0iA jei2 j0iB :
ð6Þ
After an interaction time t1, the above state evolves to jwðt1 Þi ¼ ðajei1 j0iA þ bj0i1 j1iA Þ ðajei2 j0iB þ bj0i2 j1iB Þ: ð7Þ Classical laser pulse jei M j1i on the trapped rf-SQUIDs with time duration X12t = p lead the above state to jwðt1 Þi ¼ ðaj1i1 j0iA þ bj0i1 j1iA Þ ðaj1i2 j0iB þ bj0i2 j1iB Þ: ð8Þ Before one of the two detectors clicks, the joint state of atom-cavity system will evolve into jwðtÞi ¼ ðaj1i1 j0iA þ bekt j0i1 j1iA Þ ðaj1i2 j0iB þ bekt j0i2 j1iB Þ:
ð9Þ
While one of the detectors D± clicks after a time span of t2,pitffiffiffi corresponds to the action of the jump operators 1= 2ðaA aB Þ on the joint state jw(t)i, then the joint state of the entire system becomes b2 ekt2 j0i1 j0i2 jw iA;B þ abjW i1;2 j0iA j0iB qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð10Þ b4 e2kt2 þ a2 b2 pffiffiffi pffiffiffi where jw iA;B ¼ 1= 2ðj0iA j1iB j1iA j0iB Þ; jW i1;2 ¼ 1= 2 ðj1i1 j0i2 j0i1 j1i2 Þ: Tracing over the state of cavities A and B, the reduced density matrix of rf-SQUIDs 1 and 2 is
jwðt2 Þi ¼
b4 e2kt j0i1 j0i2;1 h0j2 h0j þ a2 b2 jW i1;2;12 hW j b4 e2kt þ a2 b2
ð11Þ
The successful probability of obtaining the maximally bipartite entangled state in (11) is P = b4e2kt + a2b2. The fidelity of the desired states jW±i1,2 from the entangled states in (11) is F ¼1;2 hW jq1;2 jW i1;2 ¼
a2 b 2 : b4 e2kt þ a2 b2
ð12Þ
In the case of a/b 1, the fidelity will be F 1. In this way, we generate a bipartite entangled state of two distant rf-SQUIDs via single-photon detected cavity decay. The two rf-SQUIDs are trapped in diffident cavities and photonic qubit serves as flying bits in this generation scheme, thus this scheme can be used to entangle two distant rfSQUIDs. 4. Bell-state analyzer Bell-state measurement (BSM) plays an important role in QIP and is also a necessity in quantum communication protocols. But it is very difficult to realize BSM for atomic (ionic) states in experiment [12,13]. Usually, they will be converted into the product of separate measurements on single particle [6,15,14] in experimental demonstrations. Thus, we consider the implementation of BSM via cavity decay, which again can be succeeded probabilistically. But the two involved rf-SQUIDs can be in distant locations. The Bell states analyzer is similar to the generation process as shown in Fig. 2, but now the two rf-SQUIDs are initially entangled. The four Bell states are 1 jU i1;2 ¼ pffiffiffi ðj1i1 j1i2 j0i1 j0i2 Þ; 2 1 jW i1;2 ¼ pffiffiffi ðj1i1 j0i2 j0i1 j1i2 Þ: 2
ð13aÞ ð13bÞ
The two cavities are both initially prepared in the vacuum states j0iC. Classical laser pulse j1i M jei shining on rfSQUIDs 1 and 2 with time duration X12t = 3p, then both parties set the interaction time as tan Xt2 ¼ Xk , the states of the two rf-SQUIDs and two cavities system in (13) evolve into the following four corresponding unnormalized states 1 jU iA;B ¼ pffiffiffi ðb2 j11iA;B j00iA;B Þ; 2 b jW iA;B ¼ pffiffiffi ðj01iA;B j10iA;B Þ: 2
Fig. 2. Scheme for generating bipartite entangled state via cavity decay from two product state. A and B denote the two cavities, BS is a 50/50 beam splitter and D± are single-photon detectors.
:
ð14aÞ ð14bÞ
Here we have omitted the states of the two rf-SQUIDs, which are both in their ground states j0ii (i = 1,2) after the above controlled interactions. The joint state of the two cavities will evolve into
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1 jU iA;B ¼ pffiffiffi ðb2 e2kt j11iA;B j00iA;B Þ; 2 b jW iA;B ¼ pffiffiffi ekt ðj01iA;B j10iA;B Þ 2
ð15aÞ ð15bÞ
before one of the detectors D± clicks. If D+ clicks, one knows exactly that the initial bipartite entangled state is jW+i. If D clicks, then the initial bipartite entangled state is jWi. If none of the two detectors clicks then the process failed. So, our scheme is a probabilistic one, with the successful probability P = b2e2kt. In this way, we implement a probabilistic Bell state analyzer and it is well within current techniques. 5. Entanglement swapping In entanglement swapping [16], there are usually three spatially separate users, and two of them have shared one pair of entangled particles with the third user. Then the third user will operate a joint BSM on the two particles he possesses. Corresponding to the measurement result, the two particles possessed by the two spatially separate users will collapse into an entangled state without any entanglement before the joint measurement. Here, we will implement entanglement swapping via cavity decay without prior distribution of the entangled states, i.e., without distribution the two entangled states to the third party. Suppose two entangled states are both initially prepared in one of the four Bell states jW+i (see (13b)). The setup of entanglement swapping is shown in Fig. 3. Two rf-SQUIDs 1 and 3 are trapped in the optical cavities A and B, respectively. The initial state of the four rf-SQUIDs and two cavities system is 1 jwi ¼ ðj1i1 j0i2 þ j0i1 j1i2 Þj0iA ðj1i3 j0i4 þ j0i3 j1i4 Þj0iB : 2 ð16Þ Classical laser pulse j1i ! jei shining on rf-SQUIDs 1 and 3 with time duration X12t = 3p and an interaction time tan Xt2 ¼ Xk of the rf-SUQID-cavity systems, the state evolves into the following unnormalized state
Fig. 3. Setup to implement entanglement swapping without distribution the two entangled states to the third party Charlie. Two rf-SQUIDs 1 and 3 are trapped in the cavities A and B, respectively.
1 jwi ¼ ðbj0i2 j1iA þ jei2 j0iA Þ ðbj0i4 j1iB þ jei4 j0iB Þ: 2
61
ð17Þ
Again, we have omitted the state of the rf-SQUIDs 1 and 3, which are both in their ground states j0ii (i = 1, 3). Classical laser pulse jei ! j1i shining on rf-SQUIDs 2 and 4 with time duration X12t = p leads the above state to 1 jwi ¼ ðbj0i2 j1iA þ j1i2 j0iA Þ ðbj0i4 j1iB þ j1i4 j0iB Þ: 2 ð18Þ The above state will evolve into the following state before one of the two single-detectors clicks. 1 jwðtÞi ¼ ðbekt j0i2 j1iA þ j1i2 j0iA Þ ðbekt j0i4 j1iB þ j1i4 j0iB Þ: 2 ð19Þ
D+ or D clicks corresponds to the following two collapsed unnormalized states, respectively. b kt 1 jwðtÞi ¼ pffiffiffi e bekt j0 0i2;4 pffiffiffi ðj0 1i j1 0iÞA;B 2 2 1 þ pffiffiffi ðj1i2 j0i4 j0i2 j1i4 Þj0 0iA;B : ð20Þ 2 The successful probability of obtaining the entangled state in (20) is P = 1/2b2(e2kt + 1). The fidelity of the obtained state jW±i1,2 is F = b2/(b4e2kt + b2). In the case of b2 b, the fidelity will be F 1. In this way, we can entangle distant entangled state without prior distribution of the entangled state. 6. Quantum dense coding QDC [17] is a process to send two classical bits (cbits) of information from a sender (Alice) to a remote receiver (Bob) by sending only a single qubit. After the work of Bennett et al. [17], it attracts many public attentions. Schemes for implementing QDC are also been proposed [15,18] in cavity QED. But, it is well known that atomic qubit is only ideal stationary qubit, not suitable for long distance transmission. Thus the realization of long distance QDC with atomic qubit serving as flying qubit as suggested in [15,18] are extremely hard experimental tasks. Here we suggest a scheme for QDC via detected cavity decay. Similarly, with Bell state analyzer, we can also implementing quantum teleportation [19]. Assume the two parties, i.e., Alice and Bob, initially share a bipartite entangled state jW+i1,2 with rf-SQUIDs 1 and 2 belong to the two parties Alice and Bob, respectively. Information encoding. Alice performs one of the four local operations (I, rx, iry, rz) on her rf-SQUID. These operations denote 2 cbits of information, and will transfer the state jW+i1,2 into the four Bell states in (13). These single-qubit operations can be implemented readily by classical laser pulse on the rf-SQUID. Thus the information is encoded into the pure entangled state, which is shared
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between the two parties, the encoding of the two cbits information is completed. Information extracting. Alice and Bob send their rfSQUIDs into their cavities. According to the time evolution in Eq. (3), an interaction time of tan Xt2 ¼ Xk and some classical laser pulses lead the state (13) to the form of (14). Similarly to the above analysis dealing with Bell state analyzer, we can discriminate the four states in ((14)) with the successful probability P = b2e2kt. In other words, the QDC process succeed with probability P = b2e2kt. 7. Discussion and conclusion Finally we briefly address the experimental feasibility of the proposed scheme. The interaction time much shorter than the rf-SQUID decoherent time. After the very short time span of controlled interaction, both rf-SQUIDs are in their ground states, thus the spontaneous emission effect is greatly suppressed. Meanwhile, short time span of interaction is also very preferable in terms of decoherent. Relaxation rates of the system are small and well understood. The strong-coupling conditions are readily fulfilled [3,4]. The time constants involved are long enough to realize all the required manipulations. The quantum systems are separated by centimeterscale distances, thus can be individually addressed [7]. The state of the SQUIDs are initially in the excited state when spontaneous emission is inevitable. So in order to make the proposal more reasonable, we can first prepare the state in one of the ground state. Before we start the cavity-qubit interaction we use an classical laser field to excite it. The detection of photon in our schemes need further explanation. As the photons are in the radio frequency, there will be serious problem when detected with conventional optical detectors. But we can incorporate the photon to a cavity where we also have a SQUID prepared in the ground state, then measure the state of the SQUID [20]. If the SQUID still in its ground state, then there is no photon; when the SQUID is excited, we can conclude that there is one photon come into the cavity. It is worth to note that we reserved the word detector to denote the above-mentioned setup for detecting photon in this paper to simplify our presentation. For the sake of definitiveness, let us estimate the experimental feasibility of realizing the schemes with the parameters already available in present experiment [3,4,11]. Suppose the quality factor of the superconducting cavity is Q = 1 · 106 (cavity with Q = 3 · 108 has already been reported [7]) and the cavity mode frequency is xc = 50 GHz, the cavity decay time is k1 = Q/xc = 20 ls. The upper state energy relaxation constant is c1 e ¼ 2:5 ls. For a superconducting standing-wave cavity and a SQUID located at one of antinodes of the magnetism field, the coupling constant is g = 1.8 · 108 Hz, thus the resonant interaction time for our scheme is Tr = p/g ’ 17 ns. Meanwhile, the strong coupling condition can be perfectly achieved as g2/cek = 1.6 · 106 1. The Rabi frequency for the interaction of the rf pulse and the SQUID is X = 8.5 · 107 Hz, thus the interaction time for the single
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