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Quantum information processing and entanglement with Josephson charge qubits coupled through nanomechanical resonator
Physics Letters A 324 (2004) 484–488 www.elsevier.com/locate/pla
Quantum information processing and entanglement with Josephson charge qubits coupled through nanomechanical resonator XuBo Zou ∗ , W. Mathis Electromagnetic Theory Group at THT, Department of Electrical Engineering, University of Hanover, Germany Received 8 January 2004; received in revised form 3 February 2004; accepted 5 February 2004 Communicated by R. Wu
Abstract We present an experimental scheme to realize quantum phase gate between two Josephson charge qubits coupled through the nanomechanical resonator. In the scheme, two subsystems (qubits and resonator) in the time evolution operator are disentangled at the special time. Thus the scheme is insensitive to the state of resonator. We also demonstrate that the scheme can be directly used to create maximally entangled state of many charge qubits. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 85.25.Cp
The existence of quantum algorithms for specific problems shows that a quantum computer can in principle provide a tremendous speed up compared to classical computers [1,2]. This discovery motivated an intensive research into this mathematical concept which is based on quantum logic operations on multi-qubit systems [3]. In order to implement quantum computation into a real physical system, a quantum system is needed, which makes the storage and the read out of quantum information and the implementation of the required set of quantum gates possible. This system should be scalable and the isolation of the system from the environment should be very well in order to suppress decoherence processes. Several physical sys-
* Corresponding author.
E-mail address: [email protected] (X. Zou). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.079
tems have been suggested for implementing quantum computing: cavity QED systems [4], trapped ion systems [5] and nuclear magnetic resonance systems [6]. These systems have the advantage of high quantum coherence, but cannot be integrated easily to form largescale circuits. The quest for large scale integrability has stimulated an increasing interest in Josephson junctions as possible candidates for the implementation of the quantum computer [7]. Based on the charge and phase degrees of the freedom in Josephson junction devices, charge and phase qubits have been proposed [8], and the experimental implementation of charge and phase qubits have also been reported [9,10]. More recently, two charge qubits were capacitively coupled, and the entanglement [11] and quantum logic gate [12] have been observed in this coupled-qubit system. These experimental advances demonstrated the fea-
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
485
Fig. 1. Schematic diagram of N charge qubits coupled through the nanomechanical resonator. Each charge qubit consists of two symmetric Josephson junctions in a loop configuration, which can be tuned by an external classical magnetic flux φc .
sibility of realizing a large scale quantum computer with many charge qubits. Indeed, various theoretical schemes have been proposed for the implementation of quantum logic gate of two charge qubits by using electrical resonator [13] or superconducting resonator [14] as the data-bus. The electrical resonator can be either implemented by an L–C circuit or by a large Josephson junction [13]. Recently, the idea for coupling single charge qubit to the nanomechanical resonator has first been suggested by Armour et al. for creating the superposition of spatially separated resonator states [15]. The use of nanomechanical as opposed to electromagnetic resonators has the advantage that potentially much higher quality factors can be achieved, with significantly smaller dimensions, enabling a truly scalable approach. In Ref. [16], Irish et al. demonstrated that such system can be used to prepare the nanomechanical resonator in a Fock state. Cleland et al. described a quantum computational architecture based on integrating nanomechanical resonators with Josephson phase qubits [17]. The scheme is based on the transfer of quantum information between the phase qubits and the resonators. The scheme is sensitive to the state of the resonator. In this Letter, we propose a scheme for implementing quantum controlled-phase gate by using nanomechanical resonator to couple two charge qubits. Due to the feature that two subsystem (resonators and charge qubits) are disentangled at the special time, the scheme is insensitive to the state of the resonator. We also demonstrate that the scheme can be used to generate maximally entangled states of many charge qubits.
We consider the system of N charge qubits interacting with a resonator (see Fig. 1), where the interaction is given by the electrostatic force between the nanomechanical resonator and the charge qubits [15,16]. We assume that all charge qubits are same and each charge qubit consists of two symmetric Josephson junctions (each with capacitance CJ and Josephson coupling energy EJ ) in a loop configuration, which can be tuned by an external classical magnetic flux φc which is controlled by the current through the inductor loop. A controllable gate voltage Vb is coupled to the charge qubit via a gate capacitor Cb . We model the nanomechanical resonator as a single, simple harmonic mode with the resonant frequency ω0 . The Hamiltonian describing the system is [15,16] H = ω0 a † a
N 1 EJ (φc ) σj z − σj x 4Ec ng − n − + 2 2 j =1
+
N λ a + a † σj z
(1)
j =1
where a and a † are annihilation and creation operators of the nanomechanical resonator. ng = (Cb Vb + Cg Vg )/2e where Cg is the capacitance between the nanomechanical resonator and charge qubit, Vg is electrode voltage applied to the resonator. The control gate voltage Vb and electrode voltage Vg are appropriately chosen so that only charge states |↓ = |n and |↑ = |n + 1 play a role. Thus it is natural to use spin notation to describe the Josephson junc-
486
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
tions. σj z and σj x are Pauli spin matrices of the j th charge qubit. Ec = e2 /2(Cg + Cb + 2CJ ), EJ (φc ) = 2EJ cos(φc /φ0 ), where Φ0 = h¯ /2e is the flux quanNR tum, λ = −4Ec nNR g xzp /d where ng = Cg Vg /2e, xzp is the zero-point uncertainty of the ground state of the nanomechanical resonator, and d is the distance between the nanomechanical resonator and the charge qubits. If we tune the external classical magnetic flux φc to satisfy φc = πΦ0 /2, the Hamiltonian (1) reduces to H = ω0 a † a + 4∆c (ng )Jz + λ a + a † Jz , (2) where Jz = N we introduced the collective spin operator 1 σ , and ∆ (n ) = 4E (n − n − ). The time j,z c g c g j =1 2 evolution operator for the Hamiltonian (2) is
U (t) = exp −i∆c (ng )tJz
× exp −iω0 ta † a − iλt a + a † Jz . (3) With the relation λJz exp a † − a exp −iω0 ta † a ω0 = exp −iω0 ta † a † iω t −iω0 t λJz 0 × exp a e − ae ω0
(4)
and † iω t −iω0 t λJz † λJz 0 − ae exp a e exp a − a ω0 ω0 −iω t λ † iω0 t 0 −1 −a e −1 = exp − Jz a e ω0 λ2 sin ω0 t . × exp −iJz2 (5) ω02 We can rewrite Eq. (3) as follows
U (t) = exp −i∆c (ng )tJz 2 λ t λ2 sin ω0 t × exp iJz2 − ω0 ω02 × exp −iω0 a † at λ † iω0 t Jz a e −1 × exp ω0 − a e−iω0 t − 1 .
(6)
If we choose the interaction time τ to satisfy the condition τ = 2nπ/ω0 , where n is an integer, the time evolution operator (6) reduces to 2nπ∆c (ng ) 2inπλ2 2 Jz exp J U (τ ) = exp −i z . ω0 ω02 (7) It is easy to see, at the time τ , two subsystem are disentangled. The resonator mode is returned to its original state, be it the ground state or any excited state, and we are left with an internal state evolution, which is independent of the cavity state. Now we consider the implementation of a quantum phase gate of two charge qubits. In the case of two charge qubits, we choose the parameters ∆c (ng ) and λ to satisfy 1 1 ∆c (ng ) = ω0 . λ = √ ω0 , (8) 8n 4 n In this case, the time evolution operator (7) becomes iπ 2 U (τ ) = exp (9) J − 2Jz . 8 z It is easy to check that this time evolution operator represents a quantum phase gate 1 0 0 0 |↑1 |↑2 |↑1 |↑2 0 1 0 0 |↑1 |↓2 |↑1 |↓2 → . 0 0 1 0 |↓1 |↑2 |↓1 |↑2 0 0 0 −1 |↓1 |↓2 |↓1 |↓2 (10) A sequence of such gates supplemented by one-qubit rotations can serve as a universal element for quantum computing [3]. We now turn to the problem of generating an entangled state of N charge qubits
1 |Ψ = √ eiϕ↓ |↓ ↓ · · · ↓ + eiϕ↑ |↑ ↑ · · · ↑ (11) 2 irrespective of N even or odd. Quantum states of this kind were used to improve the frequency standard and test quantum mechanics against a local hidden variable theory [18]. Several schemes [19] were proposed to generate this kind of quantum states. We assume that each charge qubit √ is initially prepared in the state (|↓j + i|↑j )/ 2. Thus the initial state of the N charge qubits is N 1 N! j ψ(0) = √ i |Dickj , (12) j !(N − j )! 2N j =0
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
where |Dickj is the Dicke state with the j atoms in excited state, i.e., Jz |Dickj = (2j − N)|Dickj . The state of the system at the time τ = 2nπ/ω0 (n is the integer number) becomes ψ(τ ) = U (τ )ψ(0) N 1 j N! =√ i N j !(N − j )! 2 j =0 2inπλ2 × exp (2j − N)2 ω02 2nπ∆c (ng ) −i (2j − N) |Dickj . ω0 (13) If the parameters λ and ∆c (ng ) satisfy the Eq. (8), the state (13) can be rewritten in the form N
ψ(τ ) = √ 1 |↓j + (−i)N+1 |↑j eiπ/4 2N+1 j =1 + e−iπ/4 ×
N
|↓j − (−i)N+1 |↑j
j =1
(14) which is equal to the Eq. (11). We now give a brief discussion on the experimental realization of the proposed scheme. In order to implement a quantum phase gate, the coupling strength λ has to be adjusted to satisfy λ = 4√1 n ω0 , where n is the integer number. Based on the realistic parameters discussed in Ref. [20], this condition can be satisfied. We now consider a nanomechanical resonators with fundamental frequency 100 MHz (ω0 = 2π MHz = 0.5ν eV) and quality factor Q = 105 (corresponding to a photon storage time of 1.6 × 10−4 s). It is coupled to the charge qubit with the Coulomb charging energy Ec = 160ν eV. The gap between the resonator and charge qubit is d = 100 nm. The capacitance Cg between the nanomechanical resonator and the charge qubit is chosen to be Cg = 20 aF. Thus the coupling strength λ is determined by the gate voltage Vg . In order to obtain the condition λ = 14 ω0 , we require the voltage Vg to be Vg = 1.5 V. Such a voltage can be applied across a 100 nm gap [15,16,20]. In the scheme, the typical time for the entanglement and the two-qubit
487
operation is τ = 2π/ω0 . Thus, the interaction time between charge qubits is in the order 2π/ω0 ≈ 10−8 s, which is much shorter than the photon lifetime of the nanomechanical resonator. By choosing a suitable value of the integer n, we can achieve a quantum gate at a weak coupling, but a longer operation duration is required. On the other hand, charge qubits suffer decoherence due to the fluctuations of the voltage sources and fluxes. In Ref. [9], the experimental observation of the coherent oscillation in the charge qubit show that the phase coherence time is only about 2 ns. Recent experiment showed that the decoherence time of 5 × 10−7 s are available [21]. This experiments operates at the degeneracy point ∆g (ng ) = 0, but EJ (φc ) not equal zero so that a long decoherence time is achieved. However our scheme operates at the point EJ (φc ) = 0. At this condition, the low frequency noise affects the coherence of the charge qubit, and decoherence time is shorter than that at the point ∆g (ng ) = 0. Thus the main difficulty of the scheme in respect to an experimental demonstration comes from the decoherence of charge qubits, and considerable further improvements are to be expected. In summary, we proposed a scheme to implement a quantum phase gate and to entangle the N charge qubits. In the scheme, the charge qubits are coupled through the nanomechanical resonator. We calculate the exact time evolution operator and demonstrate, that the two subsystems are disentangled at the particular time τ = 2nπ/ω0 , where n is integer number. The photon-number dependent parts in the evolution operator are canceled and the scheme is insensitive to the resonator field. The practical constraint of the scheme arises from the decoherence of the charge qubit.
References [1] P. Shor, in: Proceeding of the 35th Annual Symposium on the Foundation of Computer Science, IEEE Comp. Soc. Press, Los Alomitos, CA, 1994, p. 124. [2] L.K. Grover, Phys. Rev. Lett. 80 (1998) 4329. [3] A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, H. Weinfurter, Phys. Rev. A 52 (1995) 3457. [4] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, H.J. Kimble, Phys. Rev. Lett. 75 (1995) 4710. [5] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 75 (1995) 4714.
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X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
[6] I. Chuang, N. Gershenfeld, M. Kubinec, Phys. Rev. Lett. 80 (1998) 3408. [7] Y. Makhlin, G. Schön, A. Shnirman, Rev. Mod. Phys. 73 (2001) 357. [8] Y. Makhlin, G. Schön, A. Shnirman, Nature (London) 398 (1999) 305; T.P. Orlando, J.E. Mooij, L. Tian, C.H. van der Wal, L.S. Levitov, S. Lloyd, J.J. Mazo, Phys. Rev. B 60 (1999) 15398. [9] Y. Nakamura, Y.A. Pashkin, J.S. Tsai, Nature (London) 398 (1999) 786. [10] J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, S. Lloyd, Science 285 (1999) 1036; C.H. van der Wal, A.C.J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd, J.E. Mooij, Science 290 (2000) 773. [11] Yu.A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D.V. Averin, J.S. Tsai, Nature (London) 421 (2003) 823. [12] T. Yamamoto, Yu.A. Pashkin, O. Astafiev, Y. Nakamura, J.S. Tsai, Nature 425 (2003) 941.
[13] J.Q. You, J.S. Tsai, F. Nori, Phys. Rev. Lett. 89 (2002) 197902; F. Plastina, G. Falci, Phys. Rev. B 67 (2003) 224514; D.V. Averin, et al., Phys. Rev. Lett. 91 (2003) 057003. [14] C.-P. Yang, et al., Phys. Rev. A 67 (2003) 042311; J.Q. You, F. Nori, Phys. Rev. B 68 (2003) 064509. [15] A.D. Armour, M.P. Blencowe, K.C. Schwab, Phys. Rev. Lett. 88 (2002) 148301. [16] E.K. Irish, et al., Phys. Rev. B 68 (2003) 155311. [17] A.N. Cleland, M.R. Geller, cond-mat/0311007. [18] D.M. Greenberger, M. Horne, A. Shimony, A. Zeilinger, Am. J. Phys. 58 (1990) 1131. [19] J.I. Bollinger, et al., Phys. Rev. A 54 (1996) R4649; K. Molmer, A. Sorensen, Phys. Rev. Lett. 82 (1999) 1835. [20] I. Martin, A. Shnirman, L. Tian, P. Zoller, cond-mat/0310229. [21] D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M.H. Devoret, Science 296 (2002) 886.
Quantum information processing and entanglement with Josephson charge qubits coupled through nanomechanical resonator XuBo Zou ∗ , W. Mathis Electromagnetic Theory Group at THT, Department of Electrical Engineering, University of Hanover, Germany Received 8 January 2004; received in revised form 3 February 2004; accepted 5 February 2004 Communicated by R. Wu
Abstract We present an experimental scheme to realize quantum phase gate between two Josephson charge qubits coupled through the nanomechanical resonator. In the scheme, two subsystems (qubits and resonator) in the time evolution operator are disentangled at the special time. Thus the scheme is insensitive to the state of resonator. We also demonstrate that the scheme can be directly used to create maximally entangled state of many charge qubits. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.Lx; 85.25.Cp
The existence of quantum algorithms for specific problems shows that a quantum computer can in principle provide a tremendous speed up compared to classical computers [1,2]. This discovery motivated an intensive research into this mathematical concept which is based on quantum logic operations on multi-qubit systems [3]. In order to implement quantum computation into a real physical system, a quantum system is needed, which makes the storage and the read out of quantum information and the implementation of the required set of quantum gates possible. This system should be scalable and the isolation of the system from the environment should be very well in order to suppress decoherence processes. Several physical sys-
* Corresponding author.
E-mail address: [email protected] (X. Zou). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.079
tems have been suggested for implementing quantum computing: cavity QED systems [4], trapped ion systems [5] and nuclear magnetic resonance systems [6]. These systems have the advantage of high quantum coherence, but cannot be integrated easily to form largescale circuits. The quest for large scale integrability has stimulated an increasing interest in Josephson junctions as possible candidates for the implementation of the quantum computer [7]. Based on the charge and phase degrees of the freedom in Josephson junction devices, charge and phase qubits have been proposed [8], and the experimental implementation of charge and phase qubits have also been reported [9,10]. More recently, two charge qubits were capacitively coupled, and the entanglement [11] and quantum logic gate [12] have been observed in this coupled-qubit system. These experimental advances demonstrated the fea-
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
485
Fig. 1. Schematic diagram of N charge qubits coupled through the nanomechanical resonator. Each charge qubit consists of two symmetric Josephson junctions in a loop configuration, which can be tuned by an external classical magnetic flux φc .
sibility of realizing a large scale quantum computer with many charge qubits. Indeed, various theoretical schemes have been proposed for the implementation of quantum logic gate of two charge qubits by using electrical resonator [13] or superconducting resonator [14] as the data-bus. The electrical resonator can be either implemented by an L–C circuit or by a large Josephson junction [13]. Recently, the idea for coupling single charge qubit to the nanomechanical resonator has first been suggested by Armour et al. for creating the superposition of spatially separated resonator states [15]. The use of nanomechanical as opposed to electromagnetic resonators has the advantage that potentially much higher quality factors can be achieved, with significantly smaller dimensions, enabling a truly scalable approach. In Ref. [16], Irish et al. demonstrated that such system can be used to prepare the nanomechanical resonator in a Fock state. Cleland et al. described a quantum computational architecture based on integrating nanomechanical resonators with Josephson phase qubits [17]. The scheme is based on the transfer of quantum information between the phase qubits and the resonators. The scheme is sensitive to the state of the resonator. In this Letter, we propose a scheme for implementing quantum controlled-phase gate by using nanomechanical resonator to couple two charge qubits. Due to the feature that two subsystem (resonators and charge qubits) are disentangled at the special time, the scheme is insensitive to the state of the resonator. We also demonstrate that the scheme can be used to generate maximally entangled states of many charge qubits.
We consider the system of N charge qubits interacting with a resonator (see Fig. 1), where the interaction is given by the electrostatic force between the nanomechanical resonator and the charge qubits [15,16]. We assume that all charge qubits are same and each charge qubit consists of two symmetric Josephson junctions (each with capacitance CJ and Josephson coupling energy EJ ) in a loop configuration, which can be tuned by an external classical magnetic flux φc which is controlled by the current through the inductor loop. A controllable gate voltage Vb is coupled to the charge qubit via a gate capacitor Cb . We model the nanomechanical resonator as a single, simple harmonic mode with the resonant frequency ω0 . The Hamiltonian describing the system is [15,16] H = ω0 a † a
N 1 EJ (φc ) σj z − σj x 4Ec ng − n − + 2 2 j =1
+
N λ a + a † σj z
(1)
j =1
where a and a † are annihilation and creation operators of the nanomechanical resonator. ng = (Cb Vb + Cg Vg )/2e where Cg is the capacitance between the nanomechanical resonator and charge qubit, Vg is electrode voltage applied to the resonator. The control gate voltage Vb and electrode voltage Vg are appropriately chosen so that only charge states |↓ = |n and |↑ = |n + 1 play a role. Thus it is natural to use spin notation to describe the Josephson junc-
486
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
tions. σj z and σj x are Pauli spin matrices of the j th charge qubit. Ec = e2 /2(Cg + Cb + 2CJ ), EJ (φc ) = 2EJ cos(φc /φ0 ), where Φ0 = h¯ /2e is the flux quanNR tum, λ = −4Ec nNR g xzp /d where ng = Cg Vg /2e, xzp is the zero-point uncertainty of the ground state of the nanomechanical resonator, and d is the distance between the nanomechanical resonator and the charge qubits. If we tune the external classical magnetic flux φc to satisfy φc = πΦ0 /2, the Hamiltonian (1) reduces to H = ω0 a † a + 4∆c (ng )Jz + λ a + a † Jz , (2) where Jz = N we introduced the collective spin operator 1 σ , and ∆ (n ) = 4E (n − n − ). The time j,z c g c g j =1 2 evolution operator for the Hamiltonian (2) is
U (t) = exp −i∆c (ng )tJz
× exp −iω0 ta † a − iλt a + a † Jz . (3) With the relation λJz exp a † − a exp −iω0 ta † a ω0 = exp −iω0 ta † a † iω t −iω0 t λJz 0 × exp a e − ae ω0
(4)
and † iω t −iω0 t λJz † λJz 0 − ae exp a e exp a − a ω0 ω0 −iω t λ † iω0 t 0 −1 −a e −1 = exp − Jz a e ω0 λ2 sin ω0 t . × exp −iJz2 (5) ω02 We can rewrite Eq. (3) as follows
U (t) = exp −i∆c (ng )tJz 2 λ t λ2 sin ω0 t × exp iJz2 − ω0 ω02 × exp −iω0 a † at λ † iω0 t Jz a e −1 × exp ω0 − a e−iω0 t − 1 .
(6)
If we choose the interaction time τ to satisfy the condition τ = 2nπ/ω0 , where n is an integer, the time evolution operator (6) reduces to 2nπ∆c (ng ) 2inπλ2 2 Jz exp J U (τ ) = exp −i z . ω0 ω02 (7) It is easy to see, at the time τ , two subsystem are disentangled. The resonator mode is returned to its original state, be it the ground state or any excited state, and we are left with an internal state evolution, which is independent of the cavity state. Now we consider the implementation of a quantum phase gate of two charge qubits. In the case of two charge qubits, we choose the parameters ∆c (ng ) and λ to satisfy 1 1 ∆c (ng ) = ω0 . λ = √ ω0 , (8) 8n 4 n In this case, the time evolution operator (7) becomes iπ 2 U (τ ) = exp (9) J − 2Jz . 8 z It is easy to check that this time evolution operator represents a quantum phase gate 1 0 0 0 |↑1 |↑2 |↑1 |↑2 0 1 0 0 |↑1 |↓2 |↑1 |↓2 → . 0 0 1 0 |↓1 |↑2 |↓1 |↑2 0 0 0 −1 |↓1 |↓2 |↓1 |↓2 (10) A sequence of such gates supplemented by one-qubit rotations can serve as a universal element for quantum computing [3]. We now turn to the problem of generating an entangled state of N charge qubits
1 |Ψ = √ eiϕ↓ |↓ ↓ · · · ↓ + eiϕ↑ |↑ ↑ · · · ↑ (11) 2 irrespective of N even or odd. Quantum states of this kind were used to improve the frequency standard and test quantum mechanics against a local hidden variable theory [18]. Several schemes [19] were proposed to generate this kind of quantum states. We assume that each charge qubit √ is initially prepared in the state (|↓j + i|↑j )/ 2. Thus the initial state of the N charge qubits is N 1 N! j ψ(0) = √ i |Dickj , (12) j !(N − j )! 2N j =0
X. Zou, W. Mathis / Physics Letters A 324 (2004) 484–488
where |Dickj is the Dicke state with the j atoms in excited state, i.e., Jz |Dickj = (2j − N)|Dickj . The state of the system at the time τ = 2nπ/ω0 (n is the integer number) becomes ψ(τ ) = U (τ )ψ(0) N 1 j N! =√ i N j !(N − j )! 2 j =0 2inπλ2 × exp (2j − N)2 ω02 2nπ∆c (ng ) −i (2j − N) |Dickj . ω0 (13) If the parameters λ and ∆c (ng ) satisfy the Eq. (8), the state (13) can be rewritten in the form N
ψ(τ ) = √ 1 |↓j + (−i)N+1 |↑j eiπ/4 2N+1 j =1 + e−iπ/4 ×
N
|↓j − (−i)N+1 |↑j
j =1
(14) which is equal to the Eq. (11). We now give a brief discussion on the experimental realization of the proposed scheme. In order to implement a quantum phase gate, the coupling strength λ has to be adjusted to satisfy λ = 4√1 n ω0 , where n is the integer number. Based on the realistic parameters discussed in Ref. [20], this condition can be satisfied. We now consider a nanomechanical resonators with fundamental frequency 100 MHz (ω0 = 2π MHz = 0.5ν eV) and quality factor Q = 105 (corresponding to a photon storage time of 1.6 × 10−4 s). It is coupled to the charge qubit with the Coulomb charging energy Ec = 160ν eV. The gap between the resonator and charge qubit is d = 100 nm. The capacitance Cg between the nanomechanical resonator and the charge qubit is chosen to be Cg = 20 aF. Thus the coupling strength λ is determined by the gate voltage Vg . In order to obtain the condition λ = 14 ω0 , we require the voltage Vg to be Vg = 1.5 V. Such a voltage can be applied across a 100 nm gap [15,16,20]. In the scheme, the typical time for the entanglement and the two-qubit
487
operation is τ = 2π/ω0 . Thus, the interaction time between charge qubits is in the order 2π/ω0 ≈ 10−8 s, which is much shorter than the photon lifetime of the nanomechanical resonator. By choosing a suitable value of the integer n, we can achieve a quantum gate at a weak coupling, but a longer operation duration is required. On the other hand, charge qubits suffer decoherence due to the fluctuations of the voltage sources and fluxes. In Ref. [9], the experimental observation of the coherent oscillation in the charge qubit show that the phase coherence time is only about 2 ns. Recent experiment showed that the decoherence time of 5 × 10−7 s are available [21]. This experiments operates at the degeneracy point ∆g (ng ) = 0, but EJ (φc ) not equal zero so that a long decoherence time is achieved. However our scheme operates at the point EJ (φc ) = 0. At this condition, the low frequency noise affects the coherence of the charge qubit, and decoherence time is shorter than that at the point ∆g (ng ) = 0. Thus the main difficulty of the scheme in respect to an experimental demonstration comes from the decoherence of charge qubits, and considerable further improvements are to be expected. In summary, we proposed a scheme to implement a quantum phase gate and to entangle the N charge qubits. In the scheme, the charge qubits are coupled through the nanomechanical resonator. We calculate the exact time evolution operator and demonstrate, that the two subsystems are disentangled at the particular time τ = 2nπ/ω0 , where n is integer number. The photon-number dependent parts in the evolution operator are canceled and the scheme is insensitive to the resonator field. The practical constraint of the scheme arises from the decoherence of the charge qubit.
References [1] P. Shor, in: Proceeding of the 35th Annual Symposium on the Foundation of Computer Science, IEEE Comp. Soc. Press, Los Alomitos, CA, 1994, p. 124. [2] L.K. Grover, Phys. Rev. Lett. 80 (1998) 4329. [3] A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, H. Weinfurter, Phys. Rev. A 52 (1995) 3457. [4] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, H.J. Kimble, Phys. Rev. Lett. 75 (1995) 4710. [5] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 75 (1995) 4714.
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