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OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Preparation of two-qubit steady entanglement through driving a single qubit Li-Tuo Shen, Rong-Xin Chen, Zhen-Biao Yang, Huai-Zhi Wu, and Shi-Biao Zheng* Laboratory of Quantum Optics, Department of Physics, Fuzhou University, Fuzhou 350002, China *Corresponding author: [email protected] Received June 30, 2014; revised August 9, 2014; accepted September 19, 2014; posted September 22, 2014 (Doc. ID 214980); published October 15, 2014 Inspired by a recent paper [J. Phys. B 47, 055502 (2014)], we propose a simplified scheme to generate and stabilize a Bell state of two qubits coupled to a resonator. In the scheme only one qubit is needed to be driven by external classical fields, and the entanglement dynamics is independent of the phases of these fields and insensitive to their amplitude fluctuations. This is a distinct advantage as compared with the previous ones that require each qubit to be addressed by well-controlled classical fields. Numerical simulation shows that the steady singlet state with high fidelity can be obtained with currently available techniques in circuit quantum electrodynamics. © 2014 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.2500) Fluctuations, relaxations, and noise; (270.5585) Quantum information and processing; (270.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.006046
In recent years, the preparation of entanglement through dissipation has attracted much attention [1–5]. This approach does not require preparation of specific initial states and precise timing and allows automatic feedback control, as compared to conventional methods based on the unitary dynamics [6–13]. The dissipative preparations of steady entanglement have been reported in different quantum systems, for example, in an ion trap [14,15] and a system consisting of two atomic ensembles [16]. Recently, schemes for dissipative preparation and stabilization of a Bell state between two superconducting qubits without any measurement have been theoretically proposed [17–19] by coupling these qubits with a quantized field, whose energy decaying rate is far larger than that of the qubits. In [17] the qubits are dispersively coupled to the field mode, and six continuous-wave microwaves are used, with four driving the two qubits and two driving the resonator. Following this scheme, twoqubit entanglement with the fidelity 67% was experimentally achieved [20]. Reiter et al. [18] showed that two transmon qubits can be driven into the steady Bell state by combining the effective two-photon process induced by microwave driving with the photon loss. We proposed an alternative scheme for generating and stabilizing a Bell state for two qubits coupled to a decaying resonator, requiring neither specifically designed three-level systems nor additional drives applied to the resonator [19]. All of the previous schemes need to use four or more classical fields to drive two qubits simultaneously, and the relative amplitudes and phases of the driving fields applied to different qubits should be accurately set. Based on [19], here we propose a simplified scheme for producing and stabilizing a Bell state of two qubits coupled to a field mode by dissipation, where only one qubit is needed to be driven by classical fields. The strong resonant coupling between two qubits and the field mode produces the dressed states, and two classical fields with well-chosen frequencies applied to one of the two qubits drive the system to the two-excitation dressed state, which is the product of the qubit singlet state and onephoton state. Due to the photon loss, this dressed state decays to the steady state with the two qubits in the singlet 0146-9592/14/206046-04$15.00/0
state and the resonator in the vacuum, which is not affected by both the qubit-reconator coupling and the microwave driving. With currently achievable experiment parameters, numerical simulations demonstrate that a Bell state of two qubits with high fidelity can be obtained in an environment close to zero temperature. Compared with previous schemes, the present one does not require driving both qubits with two or more pairs of classical fields of specific relative amplitudes and phases, which largely reduces the complexity of experimental implementation. Another important advantage of the present scheme over the previous ones is that the parameters of the driving fields do not need to be accurately set. In particular, the two-qubit entanglement evolution is completely independent of the phases of the driving fields. Our present stabilization mechanism is applicable to different qubit-resonator systems, including cavity quantum electrodynamics (QED) [21], trapped ions [14], and superconducting qubits [20,22–25].
Fig. 1. (a) Schematic of the setup. Two qubits resonantly interact with a common quantized field, with the coupling strength, and the first qubit is driven by two classical fields. (b) Resonant transitions (denoted by the solid arrow lines) driven by two classical lasers in the dressed-state picture. (c) Competition between the coherent and incoherent processes for producing and stabilizing the singlet state. The interaction between system and environment is characterized by the photon loss and qubit energy decay with the rates κ and γ, respectively. © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
The system, drawn schematically in Fig. 1(a), contains two identical qubits resonantly coupled to a common quantized field. We assume that one of the qubits, i.e., the first qubit, is driven by two classical fields. Under the rotating-wave approximation, the coherent dynamics of the total system is described by the Hamiltonian H H 0 H g H l (ℏ 1), where H 0 ω0 je1 ihe1 j je2 ihe2 j ω0 a† a;
(1)
H g gS 1 S 2 a H:c:;
(2)
Hl
X j1;2
Ωj e−iwj t S 1 H:c:.
(3)
S 1 je1 ihg1 j and S 2 je2 ihg2 j, jej i and jgj i are the excited and ground states of the jth qubit, respectively, ω0 denotes the qubit’s level spacing, a is the annihilation operator of the field mode, g is the coupling between the qubits and the field mode, and Ωj and ωj represent the amplitude and frequency of the jth driving field. Note that during the preparation process there is no requirement on the phase difference between these two classical fields. For simplicity we here assume that their phases are zero. This model is a simplified version of that proposed in [19]. The only difference is that the second qubit is not driven, thus there is no contribution of the S 2 terms in Eq. (3). Without classical drivings, the system Hamiltonian H nl H 0 H g , expanded in the qubit-field basis Γ ≡ fjg1 g2 ij0i;jg1 g2 ij1i; je1 g2 ij0i; jg1 e2 ij0i; jg1 g2 ijm 2i; je1 g2 i jm 1i; jg1 e2 ijm 1i;je1 e2 ijmi m 0; 1;2;…g with jmi being the m-photon Fock state, can be alternatively expressed in the dressed-state picture:
H nl λ0 jΦ0 ihΦ0 j ∞ X X
X p0;
λp1 jΦp1 ihΦp1 j
λpn jΦpn ihΦpn j;
(4)
λ0 0; jΦ0 i jg1 g2 ij0i;
(5)
λ01 w0 ; jΦ01 i jψ − ij0i;
(6)
n2 p1;2;
where
λ 1 w0
p 1 2g; jΦ 1 i p jψ ij0i jg1 g2 ij1i; 2
λ1n λ2n nw0 ; jΦ1n i jψ − ijn − 1i;
jΦ2n i
rr n−1 n je e ijn − 2i − jg1 g2 ijni ; 2n − 1 n−1 1 2
(7)
(8)
(9)
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p λ 4n − 2g; n nw0 " r 1 n−1 ijn − 1i p i jΦ jψ je e ijn − 2i n 2n −1 1 2 2 r n jg g ijni ; (10) 2n − 1 1 2 p with jψ i je1 g2 i jg1 e2 i∕ 2. Note that Eqs. (5)–(8) and Eqs. (9) and (10) for n 2 have been given in [19]. Moreover, the cases for n > 2 are not important to understand the generation of entangled states, but they are included for the completeness of our presentation. The resonant coupling between qubits and the field leads to unequal spacing of the energy levels of the qubit-field dressed states. As plotted in Fig. 1(b), there are two degenerate dressed states in each conserved Hilbert subspace with excitation number n > 1, one of which is the product of a two-qubit Bell state with the n − 1photon Fock state, and the spacing of energy splitting has a quadratic dependence on the excitation number n, which provides the prerequisite for choosing appropriate drivings. Applying two drivings with carefully selected detunings on the first qubit, we can drive the two-qubit system to experience the specific resonant and offresonant transitions. We focus the discussion on the dynamics induced by the coherent driving H l and the dissipative dynamics in the dressed-state picture. The detunings Δ1 ω0 − ω1 p and Δ2 ω2 − ω0 are chosen to be 2g. With this choice of detunings, the classical field Ω1 resonantly drives the ground state jΦ0 i to the one-excitation dressed state jΦ−1 i, and Ω2 resonantly drives jΦ−1 i to the degenerate two-excitation dressed states jΦ12 i and jΦ22 i, as shown in Fig. 1(b). Due to unequal spacings of energy levels of the dressed states, transitions to the Hilbert subspaces with three or more excitations are off-resonance with the two classical fields. When the Rabi frequencies of the classical fields are much smaller than the qubit-field coupling strength, populations of the dressed states with more than two excitations are negligible [26]. Therefore, we can restrict our analysis to the physics mechanism of dissipative stabilization in the Hilbert subspace up to two excitations. The competition between the coherent driving and dissipation is shown in Fig. 1(c). Without the classical drivings, all the excited states would decay to jΦ0 i after a long time due to energy relaxation. To generate the required Bell state, we here require the qubit dissipation to be much slower than other dynamical processes, so that it can be ignored. The driving Ω1 resonantly pumps the population in jΦ0 i to the state jΦ−1 i. There are three transition channels associated with this state: the coherent processes jΦ−1 i↔jΦ12 i and jΦ−1 i↔jΦ22 i induced by the driving field Ω2 and the incoherent process jΦ−1 i → jΦ0 i due to the photon loss, which is followed by repumping by Ω1 . The photon loss results in the decaying channel jΦ12 i → jΦ01 i jψ − ij0i. The state jΦ01 i is unaffected by the qubit-resonator interaction and photon decay and the transitions associated with it are off-resonant with the classical drivings, so that it is a steady state. On the other hand, the photon loss, together with coherent
OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
κ ρ_ t −iH; ρt Daρt 2 2 X γϕ γ j − DS j ρt DS z ρt ; 4 2 j1
(11)
where DOρ 2OρO† − ρO† O − O† Oρ, κ is the photon loss rate, γ is the qubit-energy-decay rate, γ ϕ is the qubit dephasing rate, and S jz jej ihej j − jgj ihgj j. This master equation, compared with that of [19], includes additionally a pure dephasing term. To produce and stabilize the Bell state jψ − i, the qubit-field system should fulfill two basic requirements. First, the decay rate of field mode needs to be far larger than the spontaneous emission rate of qubits, i.e., γ ≪ κ;
(12)
second, the decay rate of field mode should be comparable with the weak driving strengths, i.e., κ ≃ Ω1 ;
Ω2 ≪ g:
(13)
When the above requirements (12) and (13) are satisfied, we can efficiently generate and protect the singlet state jψ − i by dissipation and coherent drivings. We note that the values of the Ω1 and Ω2 do not need to be accurately set. More importantly, the entanglement dynamics for the two qubits is completely independent of the phases of the driving fields. This can be explained as follows. Since only one qubit is driven, each coherent coupling between the relevant dressed states is induced by a single driving without competition associated with driving different qubits, the corresponding coupling strengths, and hence the transition rates do not depend on the driving phases. These features are in distinct contrast with previous schemes, in which the relative amplitudes and phases of the fields addressing different qubits should be accurately set. As has been analyzed above, the system is hardly driven to the subspaces with three or more excitations, so that it is reasonable to restrict the simulation in the subspaces with no more than three excitations. We take the parameters reported in a recent circuit QED experiment [20]: χ∕2π ≃ 6 MHz, κ 1.7 MHz, T 1 ≃ 9 μs, and T ϕ ≃ 36 μs, where T 1 is the qubit energy relaxation time, T ϕ is the pure dephasing time, and χ g2 ∕Δ, with Δ being the qubit-resonator detuning. As this experiment
uses dispersive qubit-resonator interaction to conditionally shift the resonator frequency, which is valid under the large detuning condition g ≪ Δ. Therefore, it is reasonable to set Δ 10g, yielding g∕2π ≃ 60 MHz, κ ≃ 2.8 × 10−2 g, γ ≃ 2.72 × 10−4 g, and γ ϕ ≃ 6.8 × 10−5 g. With these parameters, in Fig. 2(a) we plot the fidelity Ft Trjψ − ihψ − j ⊗ I c ρt
(14)
for the steady state with respect to the target Bell state jψ − i as a function of the drivings Ω1 and Ω2 . The optimized Rabi frequencies are Ω1 0.084g and Ω2 0.035g, with the corresponding fidelity F 88.25%. Without considering the effect of dephasing, the fidelity is about 90%, which is about 7% lower than that of [19]. This is due to the fact that for the single-qubit driving case the optimized Rabi frequencies of the driving fields should be increased to pump the qubit system to the steady state in a reasonable time as compared with the two-qubit driving case, so that the weak driving condition is not satisfied so well and the system has a higher probability of being driven to undesired states. Figure 2(a) shows that the optical fidelity is insensitive to deviations of the parameters Ω1 and Ω2 from the optimal values (when the deviations are 5%, the fidelity is only decreased by a value of less than 0.36%). Numerical simulation also verifies that the qubit entanglement dynamics is independent of the phases of the driving fields (not shown here). In Fig. 2(b), we set κ 2.8 × 10−2 g and plot the evolutions of the optimized fidelity of the two-qubit state with respect to the singlet state jψ − i for different qubit decay rates. We see that, for the currently available decay rates κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g 6.8 × 10−5 with optimized Rabi frequencies of the drivings, the two qubits reach the steady state after a time t ≃ 1000∕g. The results show that both the time needed to arrive at the steady state and the corresponding fidelity decrease as γ increases. This can be explained as follows. Since the steady state is determined by the combined effect of the coherent driving and dissipation, the optimal driving amplitudes Ω1 and Ω2 decrease with γ. When Ω1 and Ω2 are decreased, the dynamics becomes slower and hence the time needed to reach stabilized state increases. On the other hand, as γ decreases, the transition rate from the singlet state to the ground state (b) 1
2 1
(a) 0.15 0.8 0.1
0.7
0.05
0.5
1
qubit-resonator coupling, results in the process jΦ22 i → jΦ 1 i → jΦ0 i, again followed by repumping. Therefore, the coherent driving and dissipation processes continue until all of the qubit population is driven to the singlet state jψ − i, with the resonator left in the vacuum state j0i. For simplicity, in the above analysis we do not consider dephasing, which would result in the population transfer between singlet state jψ i and the triplet state jψ − i, decreasing the fidelity of the steady state. This effect will be included in the numerical simulation. To demonstrate the feasibility of our Bell-state stabilization mechanism, we assess the performance of our scheme by numerically simulating the Lindblad master equation,
Ω /g
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F
0.5
0 0
0.6
0.01 0.01
2.8
0.4 (c) 0.05
0.1 Ω2/g
1000 2000 3000 4000 5000 gt 2
0.15
1
S
−4
1
→γ/g=2.72×10 ,Ω /g=0.084,Ω /g=0.035
2
→γ/g=1.75×10−5,Ω1/g=0.055,Ω2/g=0.024
1
2
0
2 1000 2000 3000 4000 5000 gt
Fig. 2. (a) Fidelity F of the stabilized two-qubit state with respect to the target Bell state jψ − i as a function of the parameters Ω1 ∕g and Ω2 ∕g, with the choice κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g ≃ 6.8 × 10−5 . Evolution of the fidelity (b) and the Bell signal (c) for κ∕g 2.8 × 10−2 and γ ϕ ∕g 6.8 × 10−5 . The initial state is jg1 g2 ij0i.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
jg1 g2 i becomes smaller so that the fidelity increases. The Bell signal for the steady state can be defined as [27] St TrOCHSH ρt; −σ 2y − σ 2x −σ 2y − σ 2x p σ 1x p 2 2 2 2 2 σy − σx σ y − σ 2x σ 1x p − σ 1y p : 2 2
OCHSH σ 1y
(15)
Here σ 1k and σ 2k (k x; y) are the Pauli operators σ k for the first and second qubits, respectively. The evolution of the Bell signal in Fig. 2(c) is similar to that of the fidelity. For κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g 6.8 × 10−5 , the Bell signal for the steady state is about 2.45, clearly exceeding the maximum value of 2 allowed by the local hidden variable theories. In conclusion, based on the work of [19] we have presented a simplified scheme to produce and stabilize a Bell state for two qubits by dissipation engineering. The distinct feature is that only one of these qubits needs to be driven by classical control fields. This not only greatly simplifies experimental implementation, but also makes the scheme robust against the amplitudes fluctuations and immune from the phase fluctuations of the control fields. Using presently available experiment parameters, the steady Bell state with a high fidelity can be obtained by optimizing the driving amplitudes in an environment close to zero temperature. The present approach is generally suitable for different qubit-resonator systems. Our work is supported by the Major State Basic Research Development Program of China under Grant No. 2012CB921601, the National Natural Science Foundation of China under Grant No. 11374054, No. 11305037, and No. 11347114, the Natural Science Foundation of Fujian Province under Grant No. 2013J01012, and the funds from Fuzhou University under Grant No. 022513, No. 022408, and No. 600891. References 1. M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011). 2. L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, Phys. Rev. A 84, 064302 (2011). 3. J. Ma, Z. Sun, X. G. Wang, and F. Nori, Phys. Rev. A 85, 062323 (2012).
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4. J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. B 68, 024510 (2003). 5. J. Zhang, Y. X. Liu, C. W. Li, T. J. Tarn, and F. Nori, Phys. Rev. A 79, 052308 (2009). 6. S. B. Zheng and G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000). 7. S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. 87, 037902 (2001). 8. K. Maruyama and F. Nori, Phys. Rev. A 78, 022312 (2008). 9. Y. Ota, S. Ashhab, and F. Nori, J. Phys. A Math. Theor. 45, 415303 (2012). 10. L. F. Wei, Y. X. Liu, and F. Nori, Phys. Rev. Lett. 96, 246803 (2006). 11. D. Burgarth, K. Maruyama, and F. Nori, Phys. Rev. A 79, 020305(R) (2009). 12. D. Burgarth, K. Maruyama, M. Murphy, S. Montangero, T. Calarco, F. Nori, and M. B. Plenio, Phys. Rev. A 81, 040303(R) (2010). 13. D. Burgarth, K. Maruyama, and F. Nori, New J. Phys. 13, 013019 (2011). 14. Y. Lin, J. P. Gaebler, F. Reiter, T. R. Tan, R. Bowler, A. S. Sørensen, D. Leibfried, and D. J. Wineland, Nature 504, 415 (2013). 15. J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, Nature 470, 486 (2011). 16. H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, Phys. Rev. Lett. 107, 080503 (2011). 17. Z. Leghtas, U. Vool, S. Shankar, M. Hatridge, S. M. Girvin, M. H. Devoret, and M. Mirrahimi, Phys. Rev. A 88, 023849 (2013). 18. F. Reiter, L. Tornberg, G. Johansson, and A. S. Sørensen, Phys. Rev. A 88, 032317 (2013). 19. S. B. Zheng and L. T. Shen, J. Phys. B 47, 055502 (2014). 20. S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Nature 504, 419 (2013). 21. J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001). 22. J. Q. You and F. Nori, Nature 474, 589 (2011). 23. J. Q. You and F. Nori, Phys. Today 58(11), 42 (2005). 24. I. Buluta, S. Ashhab, and F. Nori, Rep. Prog. Phys. 74, 104401 (2011). 25. Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013). 26. A. Miranowicz, M. Paprzycka, Y. X. Liu, J. Bajer, and F. Nori, Phys. Rev. A 87, 023809 (2013). 27. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
Preparation of two-qubit steady entanglement through driving a single qubit Li-Tuo Shen, Rong-Xin Chen, Zhen-Biao Yang, Huai-Zhi Wu, and Shi-Biao Zheng* Laboratory of Quantum Optics, Department of Physics, Fuzhou University, Fuzhou 350002, China *Corresponding author: [email protected] Received June 30, 2014; revised August 9, 2014; accepted September 19, 2014; posted September 22, 2014 (Doc. ID 214980); published October 15, 2014 Inspired by a recent paper [J. Phys. B 47, 055502 (2014)], we propose a simplified scheme to generate and stabilize a Bell state of two qubits coupled to a resonator. In the scheme only one qubit is needed to be driven by external classical fields, and the entanglement dynamics is independent of the phases of these fields and insensitive to their amplitude fluctuations. This is a distinct advantage as compared with the previous ones that require each qubit to be addressed by well-controlled classical fields. Numerical simulation shows that the steady singlet state with high fidelity can be obtained with currently available techniques in circuit quantum electrodynamics. © 2014 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.2500) Fluctuations, relaxations, and noise; (270.5585) Quantum information and processing; (270.1670) Coherent optical effects. http://dx.doi.org/10.1364/OL.39.006046
In recent years, the preparation of entanglement through dissipation has attracted much attention [1–5]. This approach does not require preparation of specific initial states and precise timing and allows automatic feedback control, as compared to conventional methods based on the unitary dynamics [6–13]. The dissipative preparations of steady entanglement have been reported in different quantum systems, for example, in an ion trap [14,15] and a system consisting of two atomic ensembles [16]. Recently, schemes for dissipative preparation and stabilization of a Bell state between two superconducting qubits without any measurement have been theoretically proposed [17–19] by coupling these qubits with a quantized field, whose energy decaying rate is far larger than that of the qubits. In [17] the qubits are dispersively coupled to the field mode, and six continuous-wave microwaves are used, with four driving the two qubits and two driving the resonator. Following this scheme, twoqubit entanglement with the fidelity 67% was experimentally achieved [20]. Reiter et al. [18] showed that two transmon qubits can be driven into the steady Bell state by combining the effective two-photon process induced by microwave driving with the photon loss. We proposed an alternative scheme for generating and stabilizing a Bell state for two qubits coupled to a decaying resonator, requiring neither specifically designed three-level systems nor additional drives applied to the resonator [19]. All of the previous schemes need to use four or more classical fields to drive two qubits simultaneously, and the relative amplitudes and phases of the driving fields applied to different qubits should be accurately set. Based on [19], here we propose a simplified scheme for producing and stabilizing a Bell state of two qubits coupled to a field mode by dissipation, where only one qubit is needed to be driven by classical fields. The strong resonant coupling between two qubits and the field mode produces the dressed states, and two classical fields with well-chosen frequencies applied to one of the two qubits drive the system to the two-excitation dressed state, which is the product of the qubit singlet state and onephoton state. Due to the photon loss, this dressed state decays to the steady state with the two qubits in the singlet 0146-9592/14/206046-04$15.00/0
state and the resonator in the vacuum, which is not affected by both the qubit-reconator coupling and the microwave driving. With currently achievable experiment parameters, numerical simulations demonstrate that a Bell state of two qubits with high fidelity can be obtained in an environment close to zero temperature. Compared with previous schemes, the present one does not require driving both qubits with two or more pairs of classical fields of specific relative amplitudes and phases, which largely reduces the complexity of experimental implementation. Another important advantage of the present scheme over the previous ones is that the parameters of the driving fields do not need to be accurately set. In particular, the two-qubit entanglement evolution is completely independent of the phases of the driving fields. Our present stabilization mechanism is applicable to different qubit-resonator systems, including cavity quantum electrodynamics (QED) [21], trapped ions [14], and superconducting qubits [20,22–25].
Fig. 1. (a) Schematic of the setup. Two qubits resonantly interact with a common quantized field, with the coupling strength, and the first qubit is driven by two classical fields. (b) Resonant transitions (denoted by the solid arrow lines) driven by two classical lasers in the dressed-state picture. (c) Competition between the coherent and incoherent processes for producing and stabilizing the singlet state. The interaction between system and environment is characterized by the photon loss and qubit energy decay with the rates κ and γ, respectively. © 2014 Optical Society of America
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
The system, drawn schematically in Fig. 1(a), contains two identical qubits resonantly coupled to a common quantized field. We assume that one of the qubits, i.e., the first qubit, is driven by two classical fields. Under the rotating-wave approximation, the coherent dynamics of the total system is described by the Hamiltonian H H 0 H g H l (ℏ 1), where H 0 ω0 je1 ihe1 j je2 ihe2 j ω0 a† a;
(1)
H g gS 1 S 2 a H:c:;
(2)
Hl
X j1;2
Ωj e−iwj t S 1 H:c:.
(3)
S 1 je1 ihg1 j and S 2 je2 ihg2 j, jej i and jgj i are the excited and ground states of the jth qubit, respectively, ω0 denotes the qubit’s level spacing, a is the annihilation operator of the field mode, g is the coupling between the qubits and the field mode, and Ωj and ωj represent the amplitude and frequency of the jth driving field. Note that during the preparation process there is no requirement on the phase difference between these two classical fields. For simplicity we here assume that their phases are zero. This model is a simplified version of that proposed in [19]. The only difference is that the second qubit is not driven, thus there is no contribution of the S 2 terms in Eq. (3). Without classical drivings, the system Hamiltonian H nl H 0 H g , expanded in the qubit-field basis Γ ≡ fjg1 g2 ij0i;jg1 g2 ij1i; je1 g2 ij0i; jg1 e2 ij0i; jg1 g2 ijm 2i; je1 g2 i jm 1i; jg1 e2 ijm 1i;je1 e2 ijmi m 0; 1;2;…g with jmi being the m-photon Fock state, can be alternatively expressed in the dressed-state picture:
H nl λ0 jΦ0 ihΦ0 j ∞ X X
X p0;
λp1 jΦp1 ihΦp1 j
λpn jΦpn ihΦpn j;
(4)
λ0 0; jΦ0 i jg1 g2 ij0i;
(5)
λ01 w0 ; jΦ01 i jψ − ij0i;
(6)
n2 p1;2;
where
λ 1 w0
p 1 2g; jΦ 1 i p jψ ij0i jg1 g2 ij1i; 2
λ1n λ2n nw0 ; jΦ1n i jψ − ijn − 1i;
jΦ2n i
rr n−1 n je e ijn − 2i − jg1 g2 ijni ; 2n − 1 n−1 1 2
(7)
(8)
(9)
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p λ 4n − 2g; n nw0 " r 1 n−1 ijn − 1i p i jΦ jψ je e ijn − 2i n 2n −1 1 2 2 r n jg g ijni ; (10) 2n − 1 1 2 p with jψ i je1 g2 i jg1 e2 i∕ 2. Note that Eqs. (5)–(8) and Eqs. (9) and (10) for n 2 have been given in [19]. Moreover, the cases for n > 2 are not important to understand the generation of entangled states, but they are included for the completeness of our presentation. The resonant coupling between qubits and the field leads to unequal spacing of the energy levels of the qubit-field dressed states. As plotted in Fig. 1(b), there are two degenerate dressed states in each conserved Hilbert subspace with excitation number n > 1, one of which is the product of a two-qubit Bell state with the n − 1photon Fock state, and the spacing of energy splitting has a quadratic dependence on the excitation number n, which provides the prerequisite for choosing appropriate drivings. Applying two drivings with carefully selected detunings on the first qubit, we can drive the two-qubit system to experience the specific resonant and offresonant transitions. We focus the discussion on the dynamics induced by the coherent driving H l and the dissipative dynamics in the dressed-state picture. The detunings Δ1 ω0 − ω1 p and Δ2 ω2 − ω0 are chosen to be 2g. With this choice of detunings, the classical field Ω1 resonantly drives the ground state jΦ0 i to the one-excitation dressed state jΦ−1 i, and Ω2 resonantly drives jΦ−1 i to the degenerate two-excitation dressed states jΦ12 i and jΦ22 i, as shown in Fig. 1(b). Due to unequal spacings of energy levels of the dressed states, transitions to the Hilbert subspaces with three or more excitations are off-resonance with the two classical fields. When the Rabi frequencies of the classical fields are much smaller than the qubit-field coupling strength, populations of the dressed states with more than two excitations are negligible [26]. Therefore, we can restrict our analysis to the physics mechanism of dissipative stabilization in the Hilbert subspace up to two excitations. The competition between the coherent driving and dissipation is shown in Fig. 1(c). Without the classical drivings, all the excited states would decay to jΦ0 i after a long time due to energy relaxation. To generate the required Bell state, we here require the qubit dissipation to be much slower than other dynamical processes, so that it can be ignored. The driving Ω1 resonantly pumps the population in jΦ0 i to the state jΦ−1 i. There are three transition channels associated with this state: the coherent processes jΦ−1 i↔jΦ12 i and jΦ−1 i↔jΦ22 i induced by the driving field Ω2 and the incoherent process jΦ−1 i → jΦ0 i due to the photon loss, which is followed by repumping by Ω1 . The photon loss results in the decaying channel jΦ12 i → jΦ01 i jψ − ij0i. The state jΦ01 i is unaffected by the qubit-resonator interaction and photon decay and the transitions associated with it are off-resonant with the classical drivings, so that it is a steady state. On the other hand, the photon loss, together with coherent
OPTICS LETTERS / Vol. 39, No. 20 / October 15, 2014
κ ρ_ t −iH; ρt Daρt 2 2 X γϕ γ j − DS j ρt DS z ρt ; 4 2 j1
(11)
where DOρ 2OρO† − ρO† O − O† Oρ, κ is the photon loss rate, γ is the qubit-energy-decay rate, γ ϕ is the qubit dephasing rate, and S jz jej ihej j − jgj ihgj j. This master equation, compared with that of [19], includes additionally a pure dephasing term. To produce and stabilize the Bell state jψ − i, the qubit-field system should fulfill two basic requirements. First, the decay rate of field mode needs to be far larger than the spontaneous emission rate of qubits, i.e., γ ≪ κ;
(12)
second, the decay rate of field mode should be comparable with the weak driving strengths, i.e., κ ≃ Ω1 ;
Ω2 ≪ g:
(13)
When the above requirements (12) and (13) are satisfied, we can efficiently generate and protect the singlet state jψ − i by dissipation and coherent drivings. We note that the values of the Ω1 and Ω2 do not need to be accurately set. More importantly, the entanglement dynamics for the two qubits is completely independent of the phases of the driving fields. This can be explained as follows. Since only one qubit is driven, each coherent coupling between the relevant dressed states is induced by a single driving without competition associated with driving different qubits, the corresponding coupling strengths, and hence the transition rates do not depend on the driving phases. These features are in distinct contrast with previous schemes, in which the relative amplitudes and phases of the fields addressing different qubits should be accurately set. As has been analyzed above, the system is hardly driven to the subspaces with three or more excitations, so that it is reasonable to restrict the simulation in the subspaces with no more than three excitations. We take the parameters reported in a recent circuit QED experiment [20]: χ∕2π ≃ 6 MHz, κ 1.7 MHz, T 1 ≃ 9 μs, and T ϕ ≃ 36 μs, where T 1 is the qubit energy relaxation time, T ϕ is the pure dephasing time, and χ g2 ∕Δ, with Δ being the qubit-resonator detuning. As this experiment
uses dispersive qubit-resonator interaction to conditionally shift the resonator frequency, which is valid under the large detuning condition g ≪ Δ. Therefore, it is reasonable to set Δ 10g, yielding g∕2π ≃ 60 MHz, κ ≃ 2.8 × 10−2 g, γ ≃ 2.72 × 10−4 g, and γ ϕ ≃ 6.8 × 10−5 g. With these parameters, in Fig. 2(a) we plot the fidelity Ft Trjψ − ihψ − j ⊗ I c ρt
(14)
for the steady state with respect to the target Bell state jψ − i as a function of the drivings Ω1 and Ω2 . The optimized Rabi frequencies are Ω1 0.084g and Ω2 0.035g, with the corresponding fidelity F 88.25%. Without considering the effect of dephasing, the fidelity is about 90%, which is about 7% lower than that of [19]. This is due to the fact that for the single-qubit driving case the optimized Rabi frequencies of the driving fields should be increased to pump the qubit system to the steady state in a reasonable time as compared with the two-qubit driving case, so that the weak driving condition is not satisfied so well and the system has a higher probability of being driven to undesired states. Figure 2(a) shows that the optical fidelity is insensitive to deviations of the parameters Ω1 and Ω2 from the optimal values (when the deviations are 5%, the fidelity is only decreased by a value of less than 0.36%). Numerical simulation also verifies that the qubit entanglement dynamics is independent of the phases of the driving fields (not shown here). In Fig. 2(b), we set κ 2.8 × 10−2 g and plot the evolutions of the optimized fidelity of the two-qubit state with respect to the singlet state jψ − i for different qubit decay rates. We see that, for the currently available decay rates κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g 6.8 × 10−5 with optimized Rabi frequencies of the drivings, the two qubits reach the steady state after a time t ≃ 1000∕g. The results show that both the time needed to arrive at the steady state and the corresponding fidelity decrease as γ increases. This can be explained as follows. Since the steady state is determined by the combined effect of the coherent driving and dissipation, the optimal driving amplitudes Ω1 and Ω2 decrease with γ. When Ω1 and Ω2 are decreased, the dynamics becomes slower and hence the time needed to reach stabilized state increases. On the other hand, as γ decreases, the transition rate from the singlet state to the ground state (b) 1
2 1
(a) 0.15 0.8 0.1
0.7
0.05
0.5
1
qubit-resonator coupling, results in the process jΦ22 i → jΦ 1 i → jΦ0 i, again followed by repumping. Therefore, the coherent driving and dissipation processes continue until all of the qubit population is driven to the singlet state jψ − i, with the resonator left in the vacuum state j0i. For simplicity, in the above analysis we do not consider dephasing, which would result in the population transfer between singlet state jψ i and the triplet state jψ − i, decreasing the fidelity of the steady state. This effect will be included in the numerical simulation. To demonstrate the feasibility of our Bell-state stabilization mechanism, we assess the performance of our scheme by numerically simulating the Lindblad master equation,
Ω /g
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F
0.5
0 0
0.6
0.01 0.01
2.8
0.4 (c) 0.05
0.1 Ω2/g
1000 2000 3000 4000 5000 gt 2
0.15
1
S
−4
1
→γ/g=2.72×10 ,Ω /g=0.084,Ω /g=0.035
2
→γ/g=1.75×10−5,Ω1/g=0.055,Ω2/g=0.024
1
2
0
2 1000 2000 3000 4000 5000 gt
Fig. 2. (a) Fidelity F of the stabilized two-qubit state with respect to the target Bell state jψ − i as a function of the parameters Ω1 ∕g and Ω2 ∕g, with the choice κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g ≃ 6.8 × 10−5 . Evolution of the fidelity (b) and the Bell signal (c) for κ∕g 2.8 × 10−2 and γ ϕ ∕g 6.8 × 10−5 . The initial state is jg1 g2 ij0i.
October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
jg1 g2 i becomes smaller so that the fidelity increases. The Bell signal for the steady state can be defined as [27] St TrOCHSH ρt; −σ 2y − σ 2x −σ 2y − σ 2x p σ 1x p 2 2 2 2 2 σy − σx σ y − σ 2x σ 1x p − σ 1y p : 2 2
OCHSH σ 1y
(15)
Here σ 1k and σ 2k (k x; y) are the Pauli operators σ k for the first and second qubits, respectively. The evolution of the Bell signal in Fig. 2(c) is similar to that of the fidelity. For κ∕g 2.8 × 10−2 , γ∕g 2.72 × 10−4 , and γ ϕ ∕g 6.8 × 10−5 , the Bell signal for the steady state is about 2.45, clearly exceeding the maximum value of 2 allowed by the local hidden variable theories. In conclusion, based on the work of [19] we have presented a simplified scheme to produce and stabilize a Bell state for two qubits by dissipation engineering. The distinct feature is that only one of these qubits needs to be driven by classical control fields. This not only greatly simplifies experimental implementation, but also makes the scheme robust against the amplitudes fluctuations and immune from the phase fluctuations of the control fields. Using presently available experiment parameters, the steady Bell state with a high fidelity can be obtained by optimizing the driving amplitudes in an environment close to zero temperature. The present approach is generally suitable for different qubit-resonator systems. Our work is supported by the Major State Basic Research Development Program of China under Grant No. 2012CB921601, the National Natural Science Foundation of China under Grant No. 11374054, No. 11305037, and No. 11347114, the Natural Science Foundation of Fujian Province under Grant No. 2013J01012, and the funds from Fuzhou University under Grant No. 022513, No. 022408, and No. 600891. References 1. M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. 106, 090502 (2011). 2. L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, Phys. Rev. A 84, 064302 (2011). 3. J. Ma, Z. Sun, X. G. Wang, and F. Nori, Phys. Rev. A 85, 062323 (2012).
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