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One-step schemes for multiqubit GHZ states and W-class states in circuit QED
Optics Communications 359 (2016) 359–363
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
One-step schemes for multiqubit GHZ states and W-class states in circuit QED Xin Liu a, Qinghong Liao b, Xuexin Xu a, Guangyu Fang a, Shutian Liu a,n a b
Department of Physics, Harbin Institute of Technology, Harbin 150001, China Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 August 2015 Received in revised form 1 October 2015 Accepted 3 October 2015
We propose two schemes to generate multiqubit GHZ states and W-class states in circuit QED. The schemes only need one step, and the operation times do not increase with the growth of the qubit number. Due to the virtual excitations of the resonator, the procedures are insensitive to the decay of the resonator. Numerical analysis shows that the schemes can be implemented with high fidelities. The schemes are suitable for large number of qubits under currently feasible circuit QED parameters. & 2015 Elsevier B.V. All rights reserved.
Keywords: GHZ states W-class states Circuit QED Multipartite entanglement
1. Introduction Entanglement has proven to be a key resource of quantum information processing (QIP) and a novel feature of quantum mechanics [1]. Many typical entanglement states attract great attention due to their wide application in QIP and demonstration of quantum nonlocality. After successful demonstrations of Bell-state preparation in many systems [2–4], more complicated multipartite entanglement intrigued the community such as GHZ-class states and W-class states [5,6]. Schemes for generating these states were proposed in cavity QED [7,5,8,9], photons [10], NV-center [11] and superconducting circuit [12–14]. However, owing to the fragile nature of multipartite entanglement states, the generation of those states remains a challenge for experiment. On the other hand, we notice that there are not many studies for generation of GHZ-class or W-class states in circuit QED which describes the interaction between microwave photons and superconducting qubits [15,16]. Bishopet al. proposed to generate multiqubit GHZ states employing probabilistic state preparation by measurement [17]. Wanget al. proposed a deterministic scheme for GHZ states which could work in the ultrastrong regime [14]. In Ref [18], a circle array of transmission line was designed to solve the problem of scalability in the GHZ states generation. However, in these schemes, the cavity is populated to mediate the energy exchange between superconducting qubits, which suffers from the n
Corresponding author. E-mail address: [email protected] (S. Liu).
http://dx.doi.org/10.1016/j.optcom.2015.10.012 0030-4018/& 2015 Elsevier B.V. All rights reserved.
adverse effect of cavity decay. In this work, we investigated schemes for generating GHZ states and W-class states via dispersive interaction, which will suppress the influence of cavity decay. The system is composed of gradiometer-type flux qubits [19] coupled to a transmission line resonator. Via a qubit–qubit interaction induced by the cavity, the GHZ states and W-class states are generated in one step. The operation time needed does not increase with the increase of the number of the qubits. Additionally, owning to the strong coupling nature of circuit QED, the generation time is quite short. We verify our schemes numerically under realistic parameters that currently reached in experiment, the results show that the aimed states can be generated with high fidelities. We also notice that to enhance the performance of our schemes, prolonging the decoherence time of qubits will be the main way. The rest of the paper is organized as follows: in Section 2, one-step schemes for generation of GHZ states and W-class states are presented. In Section 3, we analysis the performance of our schemes numerically under parameters currently reachable. Finally, a conclusion appears in Section 4.
2. Synthesis of multiqubit GHZ states and W-class states The schematic diagram of the investigated setup is shown in Fig. 1. N gradiometer-type four-junction flux qubits are coupled to a half-wavelength transmission line resonator (TLR) in the middle of the resonator where the magnetic field of the TLR have a maximal amplitude. Assuming that only the fundamental mode is
360
X. Liu et al. / Optics Communications 359 (2016) 359–363
⎡ ⎢ N He = λ ⎢ ∑ + ⎢ j=1 ⎢⎣
N j 〈+|
+
∑
⎤ ⎥
σ+j σ−k ⎥.
j, k = 1 j≠k
⎥ ⎥⎦
(4)
It can be found that the Hamiltonian in Eq. (4) is excitationnumber conserved. Therefore the population of different excitation-number states will not change. In another way, all the collective excited states are the eigenstates of Hamiltonian in Eq. (4) as
He ϕk = k (N + 1 − k ) λ ϕk .
(5)
where ϕk is the sum of all k (N + 1 − k ) terms with k qubits in the excited states. That is
−1, −2 , −3 , …, −N ( k = 0),
(6)
+1, −2 , −3 … , −N + −1, + 2 , −3 , … , −N + ⋯ + −1, −2 , −3 , … , +N
(k = 1 ) ,
+1, +2 , −3 , … , −N + +1, −2 , +3 , … , −N + ⋯ + −1, −2 , −3 , … , +N − 1 , +N
Fig. 1. The schematic diagram of the setup. (a) The system of flux qubits coupled to the transmission line resonator. (b) The circuit of the gradiometer-type flux qubit. The red represent the Josephson junctions. The external magnetic flux can be tuned by the offset lines.
considered, the TLR can be modeled as a one mode harmonic oscillator with the frequency ωr = π / LC , and L (C ) is the total inductance (capacitance) of the TLR. As shown in Fig. 1(b), we assume gradiometric-type flux qubits, whose symmetric design allows a tunable gap without crosstalk. On the basis of clockwise and counterclockwise persistent current states, the Hamiltonian of a qubit takes a form Hq = − (1/2)(ϵσ˜z + Δσ˜x ). The ϵ and Δ are tuned by magnetic flux Φϵ = Φ1 − Φ 2 and Φα , respectively. Tuning the qubits to the degeneracy point ( ϵ = 0) and applying the rotating-wave approximation, we take the eigenstates of the qubits as the new basis. Then, the Hamiltonian of the system is written as [18] N
H = ω r a†a +
∑ j=1
Δ j σz + 2
j=1
(2)
j=1
where δ = Δ − ωr is the detuning between the resonator and the qubits. When g ⪡δ , employing the perturbation approximation [20,21], an effective Hamiltonian is obtained as
⎡ ⎛ ⎢ N ⎜ He = λ ⎢ ∑ ⎜ + ⎢ j = 1 ⎜⎜ ⎢⎣ ⎝
j
However, the Hamiltonian in Eq. (4) generates different phase shift on those collective excited states. If every qubit is prepared in the state (1/ 2 )( g j + i e j ), the initial state of all qubits can be written as
⎞ ⎟ † † + aa − − − a a⎟ + ⎟⎟ ⎠
⎤ N ⎥ j k⎥ ∑ σ+ σ− , ⎥ j, k = 1 ⎥⎦ j≠k
2N
where λ = g 2/δ . Noticing that there is no energy exchange between the qubits and resonator, if the resonator is initially in the vacuum state, it will always remain in the vacuum state. Then the Hamiltonian changes to
∑ i k ϕk 〉 .
(10)
k=0
Due to different phase shift generated by Hamiltonian in Eq. (4), after a interaction time τ = (2n + 1) π /(2λ )(n = 0, 1, 2, …), the state of all qubits evolves to
φf 〉 =
1 2N
N
∑ e−ik (N + 1− k) λτ ik ϕk 〉 k=0
N ⎧ ⎪ e iπ /4 ⎨ ∏ [ −j 〉 + (−i) N +j 〉] ⎪ 2N + 1 ⎩ j=1
1
N ⎫ ⎪ − i ∏ [ −j 〉 − (−i) N +j 〉] ⎬ ⎪ ⎭ j=1
Since the state
↑ 〉j = −j 〉 +
(−i)N
(11) +j 〉 is orthogonal to the state
↓ 〉j = −j 〉 − (−i)N +j 〉, so we obtain a GHZ state. Now, we turn to the generation of W-class states. Despite the Hamiltonian in Eq. (4) does not change the population of different collective excitation-number states, it couples all states in the same excitation-number subspace. When the atoms start from a initial state −1, −2 , −3 , … , −N − 1 , +N 〉, the state of atoms under Hamiltonian in Eq. (4) evolves as
Ψ〉 =
(3)
N
1
φi 〉 =
(1)
N
HI = g ∑ (a†σ−j exp−iδt + aσ+j expiδt ),
(9)
+1, + 2 , +3 , … , +N (k = N )
=
where σ+ = + − and σ− = − + . We have assumed equal gap Δ and coupling strength g for all the qubits. The difference of the fabrication parameters can be compensated by tuning the magnetic flux or easily corrected by single-qubit operations. In the interaction picture, the Hamiltonian is written as
(8)
⋮
N
∑ g (a†σ−j + aσ+j ),
(k = 2),
(7)
e−iNλt + N − 1 −1 , −2 , −3 , …, −N − 1 , +N 〉 N e−iNλt − 1 + ( +1, −2 , −3 , …, −N 〉 N + ⋯ + −1, −2 , −3 , …, +N − 1 , −N 〉).
(12)
It is easy to find that, at any time t ≠ (2kπ ) /(Nλ )(k = 0, 1, 2…), all the coefficients of basis states in Eq. (12) are non-zero, thus the state Ψ 〉 is a W-class state. Particularly, at time t = (2m + 1) π /(Nλ )(m = 0, 1, 2…), the W-class state in Eq. (12) has
X. Liu et al. / Optics Communications 359 (2016) 359–363
361
a form
Φ〉 =
1⎡ ⎢ ( N − 2) −1, −2 , −3 …, −N − 1 , +N 〉 N⎣ − 2 ( +1, −2 , −3 , …, −N 〉
⎤ + ⋯ + −1, −2 , −3 , …, +N − 1 , −N 〉) ⎥. ⎦
(13)
3. Numerical simulation In this section, we analysis the performance of our schemes numerically under current feasible parameters. To be concrete, we consider the case of four qubits. In this situation, the aimed GHZ state and W-class state take the following forms:
GHZ 〉4 =
⎡ 4 ⎤ 4 1 iπ /4 ⎢ ( −j 〉 + +j 〉) − i ∏ ( −j 〉 − +j 〉) ⎥, e ∏ ⎢⎣ j = 1 ⎥⎦ 4 2 j=1
W − class〉4 =
1⎡ ⎢ −1, −2 , −3 , +4 〉 − −1, −2 , +3 , −4 〉 2⎣ ⎤ − −1, +2 , −3 , −4 〉 − +1, −2 , −3 , −4 〉⎥. ⎦
(14)
(15)
In the following, we will evaluate the property of the schemes using fidelity F (t ) = Tr [ρa ρ (t )], where ρa is the density matrix of the aimed state and ρ (t ) is the density matrix of the generated state. We choose a coupling strength g ¼100 MHz, which is typical in the circuit QED system [16]. The usual frequency of the TLR ranges from 1 2πto 10 2π GHz. Here, we choose ωr = 5 × 2π GHz . The time evolution of fidelity under Hamiltonian in Eqs. (2) and (4) is plotted in Fig. 2. Almost unity fidelities are both reached in Fig. 2(a) for GHZ state and Fig. 2(b) for W-class state. There are both slow-large and fast-tiny perturbations in Fig. 2(a) and (b). The slow perturbations of the dashed line and solid line indicate the exchange of energy between qubits. On the other hand, the fast perturbations of the solid lines are due to the hopping of excitation to the resonator which do not appear in the dashed lines because of approximation from Eq. (2) to Eq. (4). The fast perturbations in Fig. 2(b) are smaller, but the approximation induces a mismatch of the peaks of the dashed and solid lines which will be significant in Fig. 3(b). In practical implementation, the main factor that influences fidelity is the decoherence induced by coupling with environment. Benefiting from the fact that all the qubits are biased in the degeneracy point, the dephasing effect due to 1/f noise is largely suppressed. Therefore, the evolution under decoherence can be described by a master equation with only decay terms
Fig. 2. Fidelity as a function of the time, (a) GHZ state, (b) W-class state. The blue solid line is based on the Hamiltonian in Eq. (2), and the red dashed line is based on the effective Hamiltonian in Eq. (4). The used parameters are g ¼ 100 2π MHz, Δ = 10 g . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
(16)
Fig. 3. Deviation of the fidelity as a function of time, (a) GHZ state (b) W-class state. The GHZ state and W-class state can both be generated periodically. But the deviation of the fidelity will grow as the as the time passes. The used parameters are g ¼100 2π MHz, Δ = 10 g , κ ¼0.16 MHz, γ ¼0.67 MHz.
where κ and γ are decay rates of the resonator and qubits, respectively. A recent measured T1 time of gradiometer-type flux qubit is 1.5 μs [22], which corresponds to a decay rate γ ¼ 0.67 MHz. Current reachable quality factor of the TLR can be as high as 106 [23]. Here we choose a quality factor of Q = 2 × 105 for safety, i.e. a decay rate of κ ¼ 0.16 MHz. The deviation of fidelity ΔF = FI − Fd is plotted in Fig. 3, where FI is the fidelity of unitary evolution under Hamiltonian in Eq. (2) and Fd is the fidelity with decoherence. As expected, the deviation at preparation time grows larger and larger as the time increases. However, the deviation at the first preparation time is quite small as ΔF = 2.6% for the GHZ state and
ΔF = 0.95% for the W-class state. The shift of largest fidelity from the theoretical preparation time rises from the approximation of Eqs. (2) to Eq. (4), which will be unobvious if the detuning is increased to Δ = 20 g . To evaluate the influence of large-detuning approximation, we plot the variation of maximal fidelity under different detunings in Fig. 4. As expected, for both GHZ state and W-class state, a larger detuning results in a better approximation for unitary evolution. However, the maximal fidelity decreases for large detuning in the
κ γ ρ ̇ (t ) = − i [HI , ρ ( t ) ] + + [a] ρ + 2 2
4
∑ + [σj ] ρ, j=1
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X. Liu et al. / Optics Communications 359 (2016) 359–363
Fig. 4. Maximal fidelity as a function of detuning, (a) GHZ state (b) W-class state. The parameters are g ¼ 100 2π MHz, κ¼ 0.16 MHz, γ¼ 0.67 MHz.
case with decoherence. This is because a larger detuning induces a longer operation time, which will amplify the degradation effect caused by decoherence. For considered decay parameters, optimal detunings appear in Δ = 11.96 g for GHZ state and Δ = 5.35 g for W-class state. With optimal detunings that appeared in Fig. 4, we plot the fidelities as functions of decay rates κ and γ in Fig. 5. It is shown that for both GHZ state and W-class state the influence of resonator decay is greatly suppressed. Noticing that the qubit decay dominates the reduction of fidelity, to improve the fidelity of our schemes, increasing the relaxation time of qubit will be the prime method. Due to the fabrication and control signal imperfections, deviations of the Hamiltonian parameters usually appear in an experimental system of circuit QED. To evaluate the influences of these parameter disorders, we consider the random fluctuations of Δ or g and calculate their influences on the fidelities. The influences of parameter disorders are calculated separately by keeping one parameter in the expected value while another one fluctuates randomly. The fluctuation of a parameter x ( x = Δorg ) is assumed to be Gaussian distributed with mean x0 (x0 is the expected value of the parameter) and standard deviation δx . We simulate the evolution using the Hamiltonian in Eq. (2). Maximal fidelity in a single evolution is calculated for a sample array {xi } of one parameter. Mean value and standard deviation of the maximal fidelities in a set of single evolutions with the same δx are calculated which reflect the influence of the fluctuation of the parameter. For a fixed δx , the mean value of maximal fidelities is calculated as
Fmax =
1 nr
j , ∑ Fmax nr
(17)
where nr is the amount of the single evolution. The standard dej viation of {Fmax } is also calculated accordingly. Fig. 6 shows the variations of the mean maximal fidelity under the fluctuations Δ or g with the scaled δx between 0‰ and 5‰ . The
Fig. 5. The fidelity versus the decay rate of the resonator κ and the spontaneous emission rate of the qubit γ. (a) GHZ state Δ = 11.96 g , g = 100 × 2π MHz . (b) Wclass state Δ = 5.35 g , g = 100 × 2π MHz.
mean values of the maximal fidelities decrease with the increase of the standard deviation of parameters. Evidently, the influence of disordered g on the mean maximal fidelities is less than that of disordered Δ. On the other hand, the generation of GHZ state is more sensitive to the fluctuation of parameters. To obtain a mean maximal fidelity of GHZ state above 0.9, the standard deviation of Δ needs to be controlled below 2‰. We hope our results could be a reference in the parameter control of an experiment.
4. Conclusions In conclusion, we have proposed two schemes to generate the multiqubit GHZ states and W-class states in circuit QED. The GHZ states are generated by introducing phase shifts between different collective excited states, and W-class states are generated by changing the populations of states in the same subspace with one excited qubit. These procedures are insensitive to the decay of the TLR because of rare excitations of the TLR. The generations of GHZ states or W-class states are completed in one step, and the operation time does not increase with the growth of the qubit number. The performance of the schemes is analyzed in the four qubits case. Numerical results show that the evolutions are well described by the effective Hamiltonian. Taking account the influence of the decoherence, the deviations of fidelities are only 2.6% and 0.95% for GHZ state and W-class state, respectively. We find that there is an optimal detuning between the large detuning
X. Liu et al. / Optics Communications 359 (2016) 359–363
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Acknowledgements This work was supported by the National Basic Research Program of China under Grant no. 2013CBA01702, and the National Natural Science Foundation of China under Grant nos. 61575055, 61377016, 11104049 and 10974039, and Specialized Research Fund for the Doctoral Program of Higher education (Grant 20102302120009) and the Program for New Century Excellent Talents in University (NCET-12-0148).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Fig. 6. Influences of parameter imperfections with (a) disordered Δ for GHZ state, (b) disordered g for GHZ state, (c) disordered Δ for W-class state, (d) disordered g for W-class state. The mean maximal fidelities versus scaled standard deviations of j parameters are shown. The error bars show the standard deviation of {Fmax }. The parameters are g ¼100 2π MHz, Δ = 10 g , and nr ¼ 200.
approximation and the influence of decoherence. Fidelity variations under different decoherence rates and parameter disorders are also discussed.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
One-step schemes for multiqubit GHZ states and W-class states in circuit QED Xin Liu a, Qinghong Liao b, Xuexin Xu a, Guangyu Fang a, Shutian Liu a,n a b
Department of Physics, Harbin Institute of Technology, Harbin 150001, China Department of Electronic Information Engineering, Nanchang University, Nanchang 330031, China
art ic l e i nf o
a b s t r a c t
Article history: Received 7 August 2015 Received in revised form 1 October 2015 Accepted 3 October 2015
We propose two schemes to generate multiqubit GHZ states and W-class states in circuit QED. The schemes only need one step, and the operation times do not increase with the growth of the qubit number. Due to the virtual excitations of the resonator, the procedures are insensitive to the decay of the resonator. Numerical analysis shows that the schemes can be implemented with high fidelities. The schemes are suitable for large number of qubits under currently feasible circuit QED parameters. & 2015 Elsevier B.V. All rights reserved.
Keywords: GHZ states W-class states Circuit QED Multipartite entanglement
1. Introduction Entanglement has proven to be a key resource of quantum information processing (QIP) and a novel feature of quantum mechanics [1]. Many typical entanglement states attract great attention due to their wide application in QIP and demonstration of quantum nonlocality. After successful demonstrations of Bell-state preparation in many systems [2–4], more complicated multipartite entanglement intrigued the community such as GHZ-class states and W-class states [5,6]. Schemes for generating these states were proposed in cavity QED [7,5,8,9], photons [10], NV-center [11] and superconducting circuit [12–14]. However, owing to the fragile nature of multipartite entanglement states, the generation of those states remains a challenge for experiment. On the other hand, we notice that there are not many studies for generation of GHZ-class or W-class states in circuit QED which describes the interaction between microwave photons and superconducting qubits [15,16]. Bishopet al. proposed to generate multiqubit GHZ states employing probabilistic state preparation by measurement [17]. Wanget al. proposed a deterministic scheme for GHZ states which could work in the ultrastrong regime [14]. In Ref [18], a circle array of transmission line was designed to solve the problem of scalability in the GHZ states generation. However, in these schemes, the cavity is populated to mediate the energy exchange between superconducting qubits, which suffers from the n
Corresponding author. E-mail address: [email protected] (S. Liu).
http://dx.doi.org/10.1016/j.optcom.2015.10.012 0030-4018/& 2015 Elsevier B.V. All rights reserved.
adverse effect of cavity decay. In this work, we investigated schemes for generating GHZ states and W-class states via dispersive interaction, which will suppress the influence of cavity decay. The system is composed of gradiometer-type flux qubits [19] coupled to a transmission line resonator. Via a qubit–qubit interaction induced by the cavity, the GHZ states and W-class states are generated in one step. The operation time needed does not increase with the increase of the number of the qubits. Additionally, owning to the strong coupling nature of circuit QED, the generation time is quite short. We verify our schemes numerically under realistic parameters that currently reached in experiment, the results show that the aimed states can be generated with high fidelities. We also notice that to enhance the performance of our schemes, prolonging the decoherence time of qubits will be the main way. The rest of the paper is organized as follows: in Section 2, one-step schemes for generation of GHZ states and W-class states are presented. In Section 3, we analysis the performance of our schemes numerically under parameters currently reachable. Finally, a conclusion appears in Section 4.
2. Synthesis of multiqubit GHZ states and W-class states The schematic diagram of the investigated setup is shown in Fig. 1. N gradiometer-type four-junction flux qubits are coupled to a half-wavelength transmission line resonator (TLR) in the middle of the resonator where the magnetic field of the TLR have a maximal amplitude. Assuming that only the fundamental mode is
360
X. Liu et al. / Optics Communications 359 (2016) 359–363
⎡ ⎢ N He = λ ⎢ ∑ + ⎢ j=1 ⎢⎣
N j 〈+|
+
∑
⎤ ⎥
σ+j σ−k ⎥.
j, k = 1 j≠k
⎥ ⎥⎦
(4)
It can be found that the Hamiltonian in Eq. (4) is excitationnumber conserved. Therefore the population of different excitation-number states will not change. In another way, all the collective excited states are the eigenstates of Hamiltonian in Eq. (4) as
He ϕk = k (N + 1 − k ) λ ϕk .
(5)
where ϕk is the sum of all k (N + 1 − k ) terms with k qubits in the excited states. That is
−1, −2 , −3 , …, −N ( k = 0),
(6)
+1, −2 , −3 … , −N + −1, + 2 , −3 , … , −N + ⋯ + −1, −2 , −3 , … , +N
(k = 1 ) ,
+1, +2 , −3 , … , −N + +1, −2 , +3 , … , −N + ⋯ + −1, −2 , −3 , … , +N − 1 , +N
Fig. 1. The schematic diagram of the setup. (a) The system of flux qubits coupled to the transmission line resonator. (b) The circuit of the gradiometer-type flux qubit. The red represent the Josephson junctions. The external magnetic flux can be tuned by the offset lines.
considered, the TLR can be modeled as a one mode harmonic oscillator with the frequency ωr = π / LC , and L (C ) is the total inductance (capacitance) of the TLR. As shown in Fig. 1(b), we assume gradiometric-type flux qubits, whose symmetric design allows a tunable gap without crosstalk. On the basis of clockwise and counterclockwise persistent current states, the Hamiltonian of a qubit takes a form Hq = − (1/2)(ϵσ˜z + Δσ˜x ). The ϵ and Δ are tuned by magnetic flux Φϵ = Φ1 − Φ 2 and Φα , respectively. Tuning the qubits to the degeneracy point ( ϵ = 0) and applying the rotating-wave approximation, we take the eigenstates of the qubits as the new basis. Then, the Hamiltonian of the system is written as [18] N
H = ω r a†a +
∑ j=1
Δ j σz + 2
j=1
(2)
j=1
where δ = Δ − ωr is the detuning between the resonator and the qubits. When g ⪡δ , employing the perturbation approximation [20,21], an effective Hamiltonian is obtained as
⎡ ⎛ ⎢ N ⎜ He = λ ⎢ ∑ ⎜ + ⎢ j = 1 ⎜⎜ ⎢⎣ ⎝
j
However, the Hamiltonian in Eq. (4) generates different phase shift on those collective excited states. If every qubit is prepared in the state (1/ 2 )( g j + i e j ), the initial state of all qubits can be written as
⎞ ⎟ † † + aa − − − a a⎟ + ⎟⎟ ⎠
⎤ N ⎥ j k⎥ ∑ σ+ σ− , ⎥ j, k = 1 ⎥⎦ j≠k
2N
where λ = g 2/δ . Noticing that there is no energy exchange between the qubits and resonator, if the resonator is initially in the vacuum state, it will always remain in the vacuum state. Then the Hamiltonian changes to
∑ i k ϕk 〉 .
(10)
k=0
Due to different phase shift generated by Hamiltonian in Eq. (4), after a interaction time τ = (2n + 1) π /(2λ )(n = 0, 1, 2, …), the state of all qubits evolves to
φf 〉 =
1 2N
N
∑ e−ik (N + 1− k) λτ ik ϕk 〉 k=0
N ⎧ ⎪ e iπ /4 ⎨ ∏ [ −j 〉 + (−i) N +j 〉] ⎪ 2N + 1 ⎩ j=1
1
N ⎫ ⎪ − i ∏ [ −j 〉 − (−i) N +j 〉] ⎬ ⎪ ⎭ j=1
Since the state
↑ 〉j = −j 〉 +
(−i)N
(11) +j 〉 is orthogonal to the state
↓ 〉j = −j 〉 − (−i)N +j 〉, so we obtain a GHZ state. Now, we turn to the generation of W-class states. Despite the Hamiltonian in Eq. (4) does not change the population of different collective excitation-number states, it couples all states in the same excitation-number subspace. When the atoms start from a initial state −1, −2 , −3 , … , −N − 1 , +N 〉, the state of atoms under Hamiltonian in Eq. (4) evolves as
Ψ〉 =
(3)
N
1
φi 〉 =
(1)
N
HI = g ∑ (a†σ−j exp−iδt + aσ+j expiδt ),
(9)
+1, + 2 , +3 , … , +N (k = N )
=
where σ+ = + − and σ− = − + . We have assumed equal gap Δ and coupling strength g for all the qubits. The difference of the fabrication parameters can be compensated by tuning the magnetic flux or easily corrected by single-qubit operations. In the interaction picture, the Hamiltonian is written as
(8)
⋮
N
∑ g (a†σ−j + aσ+j ),
(k = 2),
(7)
e−iNλt + N − 1 −1 , −2 , −3 , …, −N − 1 , +N 〉 N e−iNλt − 1 + ( +1, −2 , −3 , …, −N 〉 N + ⋯ + −1, −2 , −3 , …, +N − 1 , −N 〉).
(12)
It is easy to find that, at any time t ≠ (2kπ ) /(Nλ )(k = 0, 1, 2…), all the coefficients of basis states in Eq. (12) are non-zero, thus the state Ψ 〉 is a W-class state. Particularly, at time t = (2m + 1) π /(Nλ )(m = 0, 1, 2…), the W-class state in Eq. (12) has
X. Liu et al. / Optics Communications 359 (2016) 359–363
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a form
Φ〉 =
1⎡ ⎢ ( N − 2) −1, −2 , −3 …, −N − 1 , +N 〉 N⎣ − 2 ( +1, −2 , −3 , …, −N 〉
⎤ + ⋯ + −1, −2 , −3 , …, +N − 1 , −N 〉) ⎥. ⎦
(13)
3. Numerical simulation In this section, we analysis the performance of our schemes numerically under current feasible parameters. To be concrete, we consider the case of four qubits. In this situation, the aimed GHZ state and W-class state take the following forms:
GHZ 〉4 =
⎡ 4 ⎤ 4 1 iπ /4 ⎢ ( −j 〉 + +j 〉) − i ∏ ( −j 〉 − +j 〉) ⎥, e ∏ ⎢⎣ j = 1 ⎥⎦ 4 2 j=1
W − class〉4 =
1⎡ ⎢ −1, −2 , −3 , +4 〉 − −1, −2 , +3 , −4 〉 2⎣ ⎤ − −1, +2 , −3 , −4 〉 − +1, −2 , −3 , −4 〉⎥. ⎦
(14)
(15)
In the following, we will evaluate the property of the schemes using fidelity F (t ) = Tr [ρa ρ (t )], where ρa is the density matrix of the aimed state and ρ (t ) is the density matrix of the generated state. We choose a coupling strength g ¼100 MHz, which is typical in the circuit QED system [16]. The usual frequency of the TLR ranges from 1 2πto 10 2π GHz. Here, we choose ωr = 5 × 2π GHz . The time evolution of fidelity under Hamiltonian in Eqs. (2) and (4) is plotted in Fig. 2. Almost unity fidelities are both reached in Fig. 2(a) for GHZ state and Fig. 2(b) for W-class state. There are both slow-large and fast-tiny perturbations in Fig. 2(a) and (b). The slow perturbations of the dashed line and solid line indicate the exchange of energy between qubits. On the other hand, the fast perturbations of the solid lines are due to the hopping of excitation to the resonator which do not appear in the dashed lines because of approximation from Eq. (2) to Eq. (4). The fast perturbations in Fig. 2(b) are smaller, but the approximation induces a mismatch of the peaks of the dashed and solid lines which will be significant in Fig. 3(b). In practical implementation, the main factor that influences fidelity is the decoherence induced by coupling with environment. Benefiting from the fact that all the qubits are biased in the degeneracy point, the dephasing effect due to 1/f noise is largely suppressed. Therefore, the evolution under decoherence can be described by a master equation with only decay terms
Fig. 2. Fidelity as a function of the time, (a) GHZ state, (b) W-class state. The blue solid line is based on the Hamiltonian in Eq. (2), and the red dashed line is based on the effective Hamiltonian in Eq. (4). The used parameters are g ¼ 100 2π MHz, Δ = 10 g . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
(16)
Fig. 3. Deviation of the fidelity as a function of time, (a) GHZ state (b) W-class state. The GHZ state and W-class state can both be generated periodically. But the deviation of the fidelity will grow as the as the time passes. The used parameters are g ¼100 2π MHz, Δ = 10 g , κ ¼0.16 MHz, γ ¼0.67 MHz.
where κ and γ are decay rates of the resonator and qubits, respectively. A recent measured T1 time of gradiometer-type flux qubit is 1.5 μs [22], which corresponds to a decay rate γ ¼ 0.67 MHz. Current reachable quality factor of the TLR can be as high as 106 [23]. Here we choose a quality factor of Q = 2 × 105 for safety, i.e. a decay rate of κ ¼ 0.16 MHz. The deviation of fidelity ΔF = FI − Fd is plotted in Fig. 3, where FI is the fidelity of unitary evolution under Hamiltonian in Eq. (2) and Fd is the fidelity with decoherence. As expected, the deviation at preparation time grows larger and larger as the time increases. However, the deviation at the first preparation time is quite small as ΔF = 2.6% for the GHZ state and
ΔF = 0.95% for the W-class state. The shift of largest fidelity from the theoretical preparation time rises from the approximation of Eqs. (2) to Eq. (4), which will be unobvious if the detuning is increased to Δ = 20 g . To evaluate the influence of large-detuning approximation, we plot the variation of maximal fidelity under different detunings in Fig. 4. As expected, for both GHZ state and W-class state, a larger detuning results in a better approximation for unitary evolution. However, the maximal fidelity decreases for large detuning in the
κ γ ρ ̇ (t ) = − i [HI , ρ ( t ) ] + + [a] ρ + 2 2
4
∑ + [σj ] ρ, j=1
362
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Fig. 4. Maximal fidelity as a function of detuning, (a) GHZ state (b) W-class state. The parameters are g ¼ 100 2π MHz, κ¼ 0.16 MHz, γ¼ 0.67 MHz.
case with decoherence. This is because a larger detuning induces a longer operation time, which will amplify the degradation effect caused by decoherence. For considered decay parameters, optimal detunings appear in Δ = 11.96 g for GHZ state and Δ = 5.35 g for W-class state. With optimal detunings that appeared in Fig. 4, we plot the fidelities as functions of decay rates κ and γ in Fig. 5. It is shown that for both GHZ state and W-class state the influence of resonator decay is greatly suppressed. Noticing that the qubit decay dominates the reduction of fidelity, to improve the fidelity of our schemes, increasing the relaxation time of qubit will be the prime method. Due to the fabrication and control signal imperfections, deviations of the Hamiltonian parameters usually appear in an experimental system of circuit QED. To evaluate the influences of these parameter disorders, we consider the random fluctuations of Δ or g and calculate their influences on the fidelities. The influences of parameter disorders are calculated separately by keeping one parameter in the expected value while another one fluctuates randomly. The fluctuation of a parameter x ( x = Δorg ) is assumed to be Gaussian distributed with mean x0 (x0 is the expected value of the parameter) and standard deviation δx . We simulate the evolution using the Hamiltonian in Eq. (2). Maximal fidelity in a single evolution is calculated for a sample array {xi } of one parameter. Mean value and standard deviation of the maximal fidelities in a set of single evolutions with the same δx are calculated which reflect the influence of the fluctuation of the parameter. For a fixed δx , the mean value of maximal fidelities is calculated as
Fmax =
1 nr
j , ∑ Fmax nr
(17)
where nr is the amount of the single evolution. The standard dej viation of {Fmax } is also calculated accordingly. Fig. 6 shows the variations of the mean maximal fidelity under the fluctuations Δ or g with the scaled δx between 0‰ and 5‰ . The
Fig. 5. The fidelity versus the decay rate of the resonator κ and the spontaneous emission rate of the qubit γ. (a) GHZ state Δ = 11.96 g , g = 100 × 2π MHz . (b) Wclass state Δ = 5.35 g , g = 100 × 2π MHz.
mean values of the maximal fidelities decrease with the increase of the standard deviation of parameters. Evidently, the influence of disordered g on the mean maximal fidelities is less than that of disordered Δ. On the other hand, the generation of GHZ state is more sensitive to the fluctuation of parameters. To obtain a mean maximal fidelity of GHZ state above 0.9, the standard deviation of Δ needs to be controlled below 2‰. We hope our results could be a reference in the parameter control of an experiment.
4. Conclusions In conclusion, we have proposed two schemes to generate the multiqubit GHZ states and W-class states in circuit QED. The GHZ states are generated by introducing phase shifts between different collective excited states, and W-class states are generated by changing the populations of states in the same subspace with one excited qubit. These procedures are insensitive to the decay of the TLR because of rare excitations of the TLR. The generations of GHZ states or W-class states are completed in one step, and the operation time does not increase with the growth of the qubit number. The performance of the schemes is analyzed in the four qubits case. Numerical results show that the evolutions are well described by the effective Hamiltonian. Taking account the influence of the decoherence, the deviations of fidelities are only 2.6% and 0.95% for GHZ state and W-class state, respectively. We find that there is an optimal detuning between the large detuning
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Acknowledgements This work was supported by the National Basic Research Program of China under Grant no. 2013CBA01702, and the National Natural Science Foundation of China under Grant nos. 61575055, 61377016, 11104049 and 10974039, and Specialized Research Fund for the Doctoral Program of Higher education (Grant 20102302120009) and the Program for New Century Excellent Talents in University (NCET-12-0148).
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Fig. 6. Influences of parameter imperfections with (a) disordered Δ for GHZ state, (b) disordered g for GHZ state, (c) disordered Δ for W-class state, (d) disordered g for W-class state. The mean maximal fidelities versus scaled standard deviations of j parameters are shown. The error bars show the standard deviation of {Fmax }. The parameters are g ¼100 2π MHz, Δ = 10 g , and nr ¼ 200.
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