Abstract
We develop a theory for the quantum circuit consisting of a superconducting loop interrupted by four Josephson junctions and pierced by a magnetic flux (either static or timedependent). In addition to the similarity with the typical threejunction flux qubit in the doublewell regime, we demonstrate the difference of the fourjunction circuit from its threejunction analogue, including its advantages over the latter. Moreover, the fourjunction circuit in the singlewell regime is also investigated. Our theory provides a tool to explore the physical properties of this fourjunction superconducting circuit.
Similar content being viewed by others
Introduction
Superconducting quantum circuits based on Josephson junctions exhibit macroscopic quantum coherence and can be used as qubits for quantum information processing (see, e.g., refs 1, 2, 3, 4, 5, 6, 7, 8, 9). Behaving as artificial atoms, these circuits can also be utilized to demonstrate novel atomicphysics and quantumoptics phenomena, including those that are difficult to observe or even do not occur in natural atomic systems^{10}. As a rough distinction, there are three types of superconducting qubits, i.e., charge^{1,2}, flux^{4,11} and phase qubits^{5,6,12}. In the charge qubit, where the charge degree of freedom dominates, two discrete Cooperpair states are coupled via a Josephson coupling energy^{1,2}. In contrast, the phase degree of freedom dominates in both flux^{11} and phase qubits^{5,6}.
The typical flux qubit is composed of a superconducting loop interrupted by three Josephson junctions^{11}. Similar to other types of superconducting qubits, it exhibits good quantum coherence and can be tuned externally. Recent experimental measurements^{9} showed that the decoherence time of the threejunction flux qubit can be longer than 40 μs. Due to the convenience in sample fabrication (i.e., the doublelayer structure fabrication by the shadow evaporation technique^{13}), a superconducting loop interrupted by four Josephson junctions was also used as the flux qubit. The experiments^{14} showed that this fourjunction flux qubit behaves similar to the threejunction flux qubit. Also, two fourjunction flux qubits were interacting experimentally via a coupler^{15}, similar to the interqubit coupling mediated by a highexcitationenergy quantum object^{16}. The theory of the threejunction flux circuit with a static flux bias was well developed^{17}, but a theory for the fourjunction circuit lacks because adding one Josephson junction more to the superconducting loop makes the problem more complex.
In this paper, we develop a theory for the fourjunction circuit with either a static or timedependent flux bias. In addition to the similarity with the threejunction circuit, we demonstrate the difference from the threejunction circuit due to the different sizes of the two smaller Josephson junctions in the fourjunction circuit. We find that the fourjunction circuit with only one smaller junction has a broader parameter range to achieve a flux qubit in the doublewell regime than the threejunction circuit. Moreover, for the fourjunction circuit with two identical smaller junctions, the circuit can be used as a qubit better than the threejunction circuit, because it becomes more robust against the state leakage from the qubit subspace to the third level. This can be a useful advantage of the fourjunction circuit over the threejunction circuit when used as a qubit. Also, we study the fourjunction circuit in the singlewell regime, which was not exploited before. Our theory can provide a useful tool to explore the physical properties of this fourjunction superconducting circuit.
Results
The total Hamiltonian of the fourjunction superconducting circuit
Let us consider a superconducting loop interrupted by four Josephson junctions and pierced by a magnetic flux [see Fig. 1(a)], where the first and second junctions have identical Josephson coupling energy E_{J} and capacitance C (i.e., E_{Ji} = E_{J} and C_{i} = C, with i = 1, 2), while the third and fourth junctions are reduced as E_{J3} = αE_{J}, E_{J4} = βE_{J}, C_{3} = αC and C_{4} = βC, with 0 < α, β < 1. The phase drops φ_{i} (i = 1, 2, 3, 4) through these four Josephson junctions are constrained by the fluxoid quantization
where , with Φ_{tot}(t) being the total magnetic flux in the loop (which includes the externally applied flux, either static or timedependent and the inductanceinduced flux owing to the persistent current in the loop) and Φ_{0} = h/2e being the flux quantum.
The kinetic energy of the fourjunction circuit is the electrostatic energy^{18} stored in the junction capacitors, which can be written as
where is the voltage across the ith junction. Using the the fluxoid quantization condition in Eq. (1), we can rewrite the kinetic energy as
We introduce a phase transformation
where
with being the reduced static magnetic flux applied to the superconducting loop. The electrostatic energy can then be converted to a quadratic form
where
The total Josephson coupling energy of the fourjunction circuit is
Also, there is the inductive energy due to the inductance L of the superconducting loop^{19}:
where the reduced externallyapplied magnetic flux f_{ext} can generally be written as a sum of the static and timedependent fluxes, i.e., f_{ext} = f_{e} + f_{a}(t), with f_{a}(t) ≡ Φ_{a}(t)/Φ_{0} being the reduced timedependent magnetic field applied to the fourjunction loop. When including this inductive energy, the total potential energy of the fourjunction circuit is written as
The Lagrangian of the fourjunction circuit is
where we assign φ, φ_{±} and ξ as the canonical coordinates. The corresponding canonical momenta , and are
Therefore, the Hamiltonian of the fourjunction circuit is given by
where E_{C} = e^{2}/(2C) is the singleparticle charging energy of the Josephson junction. In comparison with the previous work in ref. 17 for the threejunctions flux qubit, a new degree of freedom ξ is included in the Hamiltonian, so that the Hamiltonian can also apply to the case when the superconducting loop contains a timedependent magnetic flux.
The reduced Hamiltonian of the fourjunction superconducting circuit
The total Hamiltonian of the fourjunction circuit can be rewritten as
where
Quantum mechanically, the canonical momenta can be written as , and in the canonicalcoordinate representation.
Note that the Hamiltonian H_{osc} in Eq. (15) can be rewritten as
i.e., a harmonic oscillator driven by a timedependent magnetic flux f_{a}(t). The angular frequency of this harmonic oscillator is
With the parameters achieved in experiments for the flux qubit^{14,20}, α ~ 0.7, C ~ 8 fF and L ~ 10 pH. Moreover, β ~ α, so ω_{osc}/2π ~ 1 × 10^{3} GHz. For the fourjunction flux qubit, the energy gap Δ between the lowest two levels is typically Δ ~ 1–10 GHz^{14,15}, which is much smaller than ω_{osc}/2π ~ 1 × 10^{3} GHz. Usually, the timedependent magnetic flux f_{a}(t) applied to the fourjunction loop is a microwave wave with ω_{a}/2π ~ 1–10 GHz, which is also much smaller than ω_{osc}/2π. Because Δ ≪ ω_{osc}/2π and the flux f_{a}(t) is also very off resonance from the harmonic oscillator (i.e., ω_{a} ≪ ω_{osc}), the oscillator is nearly kept in the ground state at a low temperature. Then, using the adiabatic approximation to eliminate the degree of freedom of the oscillator, the Hamiltonian of the fourjunction circuit can be reduced to
Also, both L and the persistent current I of the superconducting loop are small, so that^{17} IL/Φ_{0} ~ 10^{−3}. This inductanceinduced flux is much smaller than the externally applied magnetic flux f_{ext} = f_{e} + f_{a}(t). Therefore, the total flux f_{tot} can also be approximately written as f_{tot} ≃ f_{e} + f_{a}(t).
Below we first study the staticflux case, i.e., only a static magnetic flux is applied to the fourjunction loop. In this case, f_{tot} ≃ f_{e}, so ξ ≃ 0. The phase transformation in Eq. (4) becomes
and the Hamiltonian of the fourjunction circuit in Eq. (18) is further reduced to
with .
Figure 2 shows the contour plots of the potential in the twodimensional subspace spanned by φ_{1} and φ_{2} for f_{e} = 1/2, where φ_{i} (i = 1, 2, 3) are related to φ and φ_{±} by Eq. (19). For a threejunction flux qubit, α is usually in the range of 1/2 < α < 1. When 0 < α < 1/2, each double well in the potential is reduced to a single well^{17}, so the flux qubit in the doublewell regime is converted to a flux qubit in the singlewell regime. For the fourjunction circuit, there are wider ranges of parameters to achieve a flux qubit. For instance, in the case of three identical Josephson junctions (i.e., α = 1 and 0 < β < 1), when β > 1/3, the potential U (φ_{1}, φ_{2}, φ_{3}) has two energy minima in the unit cell of threedimensional periodic lattice at φ_{1} = φ_{2} = φ_{3} = ±φ* mod 2π, where
A flux qubit in the doublewell potential can then be achieved in the parameter range of 1/3 < β < 1, which is broader than the range of 1/2 < α < 1 for the threejunction flux qubit. Figure 2(a,b) show a section of U(φ_{1}, φ_{2}, φ_{3}) at φ_{3} = 0. Corresponding to the abovementioned two minima, a figureeightshaped double well exists in each unit cell of the periodic lattice in the twodimensional subspace. When β < 1/3, each figureeightshaped double well in the φ_{3} = 0 section of the potential is reduced to a single well [see Fig. 2(c)], with only one minimum in the unit cell at φ_{1} = φ_{2} = 0 mod 2π. This corresponds to a flux qubit in the singlewell regime achieved in the fourjunction superconducting circuit.
Energy spectrum
The energy spectrum and eigenstates of the fourjunction circuit are determined by
where φ ≡ (φ, φ_{+}, φ_{−}) = (φ_{1}, φ_{2}, φ_{3}) is a threedimensional vector in the phase space. Equation (22) is just like the quantum mechanical problem of a particle moving in a threedimensional periodic potential U(φ). Thus, the solution of it has the Blochwave form
where k is a wavevector and u(φ) is a periodic function in the phases of φ_{i} (i = 1, 2, 3). Also, Ψ(φ) should be periodic in the phases of φ_{i}. To ensure this, the wavefunction Ψ(φ) is constrained by k = 0. Then, Ψ(φ) can be written as
where K is a reciprocal lattice vector. Substituting Eq. (24) into Eq. (22), we then obtain an equation similar to the central equation in the theory of energy bands^{21}. Numerically solving this equation, we can obtain the energy spectrum and eigenstates of the Hamiltonian H_{0}.
For the threejunction flux qubit, an approximate tightbinding solution was obtained in ref. 17 by projecting the Schrödinger equation onto the qubit subspace, where the needed tunneling matrix elements were estimated using the WKB method. For the fourjunction case, such an approximate tightbinding solution can also be derived, but it is difficult to calculate the tunneling matrix elements via the WKB method, because a threedimensional potential is involved in the fourjunction circuit. Thus, we resort to the numerical approach to solve the Schrödinger equation in Eq. (22). With this numerical approach, we can obtain the results for both the flux qubit and the threelevel system.
Figure 3 shows the energy levels of the fourjunction circuit versus the reduced static flux f_{e}, in comparison with the threejunction circuit. In the case of fourjunction circuit, when the lowest two or three levels are considered, the energy spectrum with α = 1 and β = 0.6 is similar to the energy spectrum with α = 0.7 in the case of threejunction circuit [comparing Fig. 3(c) with Fig. 3(a)]. Because the lowest two levels are well separated from other levels, both three and fourjunction circuits can be utilized as quantum twolevel systems (i.e., flux qubits). In this case, the flux qubit can be modeled as
where the tunneling amplitude Δ corresponds to the energy difference between the two lowestenergy levels at f_{e} = 1/2 and ε = 2I_{p}Φ_{0}(f_{e} − 1/2) is the bias energy due to the external flux, with I_{p} being the maximal persistent current circulating in the loop. Here the maximal persistent current I_{p} can be approximately calculated as^{17} at a value of f_{e} considerably away from f_{e} = 1/2, where E_{0} is the energy level of the ground state of the system. The Pauli operators σ_{z} and σ_{x} are represented using the two (i.e., the clockwise and counterclockwise) persistentcurrent states. Moreover, similar to the threejunction circuit, the fourjunction circuit can also be used as a quantum threelevel system (qutrit) owing to the considerable separation of the third energy level from other higher levels as well. When reducing the smallest junction to, e.g., β = 0.3 in the fourjunction circuit [see Fig. 3(d)], only the lowest two levels are well separated from other levels, similar to the case of threejunction circuit in Fig. 3(b) where α = 0.4. Now the doublewell potential has been converted to a single well (see Fig. 2), so the circuit behaves as a flux qubit in the singlewell regime. Compared to the flux qubits in Fig. 3(a,c), the energy levels in Fig. 3(b,d) are less sensitive to the external flux f_{e}, so the obtained flux qubits in the singlewell regime are more robust against the flux noise. However, because the smallest Josephson junction in the loop is further reduced, the charge noise may become important^{22}. To suppress this charge noise, one can shunt a large capacitance to the smallest junction to improve the quantum coherence of the qubit^{9,22,23}.
Furthermore, let us consider the fourjunction circuit with two identical smaller Josephson junctions. In Fig. 3(e) where α = β = 0.6, the lowest two levels are also well separated from other levels, but the third level is not so separated from higher levels. Thus, from the energylevel point of view, this fourjunction circuit can be better used as a flux qubit than a threelevel system. In Fig. 3(f) where α = β = 0.3, the lowest three levels are well separated from other levels. It seems that the fourjunction circuit can be better used as a threelevel system. However, our calculations on transition matrix elements indicate that the circuit can still be better used as a qubit, because only the transition matrix element between the ground and first excited states is appreciably large (see the next section).
In addition, we further consider the case of two different smaller Josephson junctions (i.e., α ≠ β) in the fourjunction circuit. In the doublewell regime [see Fig. 3(g), where α = 0.5 and β = 0.6], the energy levels look similar to those in Fig. 3(e) and the lowest two levels can still be used as a qubit. Also, this qubit is less sensitive to the influence of the external magnetic field around the degeneracy point, because the energy levels are more flat than those in Fig. 3(e). In the singlewell regime [see Fig. 3(h), where α = 0.2 and β = 0.3], the lowest three levels are well separated from the higher levels. Moreover, in addition to the transition matrix element between the ground and first excited states, the transition matrix element between the first and second excited states is also larger (see the section below). Therefore, in the singlewell regime, the fourjunction circuit in the case of α ≠ β can be better used as a quantum threelevel system. This is different from the cases in Fig. 3(b,d,f).
Transition matrix elements
Now we consider the timedependent case with f_{tot}(t) ≃ f_{e} + f_{a}(t), i.e., in addition to a static flux f_{e}, a timedependent flux is also applied to the fourjunction loop. In this case, ξ ≃ f_{a}(t) when ignoring the very small inductanceinduced flux. For a small enough timedependent flux, only the firstorder perturbation due to ξ needs to be considered in Eq. (18). Then, the Hamiltonian of the fourjunction circuit in Eq. (18) can be expressed as
with H_{0} given in Eq. (20) and
The timedependent perturbation can be rewritten as
where
is the current in the superconducting loop. Because
we can express the current I as
where I_{i} = I_{c} sinφ_{i}, with i = 1, 2 and I_{c} = 2πE_{J}/Φ_{0}, I_{3} = αI_{c} sinφ_{3} and are Josephson supercurrents through the four junctions. The phase drops φ_{i} (i = 1, 2, 3) are related to φ and φ_{±} by Eq. (19) and φ_{4} is constraint by the fluxoid quantization condition in the staticflux case, i.e., .
Here we consider a microwave field with frequency ω_{a} applied to the superconducting loop. The timedependent magnetic flux in the loop can be written as . Then, with the current I available, the magneticdipole transition matrix elements are calculated by
where i〉 and j〉 are eigenstates of the Hamiltonian H_{0} in Eq. (20).
Figure 4 shows the transition matrix elements t_{01}, t_{02} and t_{12} of the three and fourjunction circuits as a function of the reduced static flux f_{e}, where the subscripts 0, 1 and 2 correspond to the ground state 0〉, the first excited state 1〉 and the second excited state 2〉 of the system, respectively. Similar to the threejunction circuit in Fig. 4(a) where α = 0.7, the fourjunction circuit with α = 1 and β = 0.6 (i.e., there is only one smaller Josephson junction in the circuit) behaves as a laddertype (namely, Ξtype^{24}) threelevel system at f_{e} = 1/2 and a cyclictype (Δtype^{25}) threelevel system at f_{e} ≠ 1/2 [see Fig. 4(c)]. For the Ξtype threelevel system achieved when f_{e} = 1/2, the transition between the ground state 0〉 and the second excited state 2〉 is not allowed, which is analogous to a natural atom. However, for the Δtype threelevel system at f_{e} ≠ 1/2, all transitions among 0〉, 1〉 and 2〉 are allowed. This is different from a natural atomic system^{25}. When the smallest Josephson junction is further reduced, t_{02} is greatly suppressed. Now both three and fourjunction circuits behave more like a Ξtype threelevel system in the whole region of f_{e} shown in Fig. 4(b,d).
As for the fourjunction circuit with two identical smaller Josephson junctions (α = β), while t_{01} remains appreciably large, the transition between 0〉 and 2〉 as well as the transition between 1〉 and 2〉 are greatly reduced (i.e., t_{02} ≈ 0 and t_{12} ≈ 0) in the whole region of f_{e} shown in Fig. 4(e,f). Now, in either double or singlewell regime, the fourjunction circuit can be well used as a qubit, because the state leakage from the qubit subspace to the third level is suppressed. This is an apparent advantage of the fourjunction circuit over the threejunction circuit when used as a qubit.
When the two smaller Josephson junctions in the fourjunction circuit become different (i.e., α ≠ β), in addition to t_{01}, both t_{02} and t_{12} become nonzero except for the degeneracy point [see Fig. 4(g,h)]. This circuit behaves very different from the circuit with two identical smaller junctions [comparing Fig. 4(g) with Fig. 4(e) and comparing Fig. 4(h) with Fig. 4(f)], but it is similar to the threejunction circuit and the fourjunction circuit with only one smaller junction [comparing Fig. 4(g) with Fig. 4(a,c) and comparing Fig. 4(h) with Fig. 4(b,d)]. However, when the distribution of the energy levels is also taken into account (see Fig. 3), the fourjunction circuit with α ≠ β can be better used as a quantum threelevel system (qutrit) in the singlewell regime. This is very different from the threejunction circuit and the fourjunction circuit with only one smaller junction, which can be better used as a qubit in the singlewell regime. Therefore, as compared to the threejunction circuit, the fourjunction circuit can provide more choices to achieve different quantum systems.
Summary
We have developed a theory for the fourjunction superconducting loop pierced by an externally applied magnetic flux. When the loop inductance is considered, the derived Hamiltonian of this fourjunction circuit can be written as the sum of two parts, one of which is the Hamiltonian of a harmonic oscillator with a very large frequency. This makes it feasible to employ the adiabatic approximation to eliminate the degree of freedom of the harmonic oscillator in the total Hamiltonian. Also, this theory can be used to study the case when the applied magneticflux bias becomes timedependent. In the case of static flux bias, the total Hamiltonian of the fourjunction circuit is reduced to the Hamiltonian of the superconducting qubit. When the flux bias is timedependent, the total Hamiltonian of the fourjunction circuit can be reduced to the Hamiltonian of the superconducting qubit plus a perturbation related to the applied timedependent flux. Then, we can calculate the energy spectrum and the transition matrix elements of the fourjunction superconducting circuit.
In conclusion, we have studied the fourjunction superconducting circuit in both double and singlewell regimes. In addition to the similarity with the threejunction circuit, we show the difference of the fourjunction circuit from its threejunction analogue. Also, we demonstrate its advantages over the threejunction circuit. Owing to the one additional Josephson junction in the circuit, the physical properties of the fourjunction circuit become richer than those of the threejunction circuit. For instance, in the case of fourjunction circuit with only one smaller Josephson junction, the circuit has a broader parameter range to achieve a flux qubit in the doublewell regime than the threejunction circuit does. Moreover, in the case of fourjunction circuit with two identical smaller junctions, the circuit can be used as a qubit better than the threejunction circuit in both double and singlewell regimes. This is because among the lowest three eigenstates of the fourjunction circuit, only the transition matrix element between the ground and first excited states is appreciably large, while other two elements become zero. These properties of the fourjunction circuit can suppress the state leakage from the qubit subspace to the second excited state and the circuit with these parameters is thus expected to have better quantum coherence when used as a qubit.
Methods
Threejunction circuit with a timedependent magnetic flux
To compare with our fourjunction results, we also consider a threejunction superconducting loop pierced by a timedependent total magnetic flux Φ_{tot}(t) [see Fig. 1(b)], because no explicit derivation exists in the literature for this timedependent case. The directions of the phase drops φ_{i} (i = 1, 2, 3) through the three Josephson junctions are chosen as in ref. 17, which are constrained by the following fluxoid quantization condition:
where . Here we assume that two larger junctions have identical capacitance C and coupling energy E_{J}, while the smaller junction has capacitance αC and coupling energy αE_{J}, with 0 < α < 1.
Similar to the fourjunction circuit, we introduce a phase transformation
where , with being the reduced static magnetic flux applied to the superconducting loop. The Hamiltonian of the threejunction circuit can be derived as
where E_{C} = e^{2}/(2C),
and
Quantum mechanically, the canonical momenta can be written as , and in the canonicalcoordinate representation.
The angular frequency of the harmonic oscillator given in Eq. (37) is
Using the parameters achieved in experiments^{14,20}, we have α ~ 0.7, C ~ 8 fF and L ~ 10pH, so one has ω_{osc}/2π ~ 10^{3} GHz, which is much larger than the energy gap Δ ~ 1–10 GHz of the threejunction flux qubit (see, e.g., ref. 4). If the timedependent magnetic flux is the usually applied microwave field, the oscillator can indeed be regarded as being in the ground state at a low temperature, as analyzed for the fourjunction flux qubit in the main text. Then, the Hamiltonian of the threejunction circuit can be reduced to
Because L is small in a threejunction flux qubit^{17}, we can ignore the flux generated by the loop inductance. Thus, when only a static flux is applied to the loop, f_{tot}(t) ≃ f_{e}, i.e., ξ ≃ 0. The phase transformation in Eq. (34) becomes
and the Hamiltonian of the circuit in Eq. (39) is reduced to
which is the Hamiltonian of the threejunction flux qubit derived in ref. 17.
For the timedependent case with , , where is the reduced timedependent magnetic flux applied to the threejunction loop. When the timedependent magnetic flux is small enough, only the firstorder perturbation due to ξ needs to be considered and the Hamiltonian of the circuit in Eq. (39) can be expressed as
with H_{0} given in Eq. (41) and , where
is the current in the threejunction loop^{26}. Using Eq. (40) and the fluxoid quantization condition in the staticflux case (i.e., ), the current I can also be rewritten as
where I_{i} is the Josephson supercurrent through each junction. Moreover, as in Eq. (32), the magneticdipole transition matrix elements are calculated by , where i〉 and j〉 are eigenstates of the Hamiltonian H_{0} in Eq. (41).
Additional Information
How to cite this article: Qiu, Y. et al. Fourjunction superconducting circuit. Sci. Rep. 6, 28622; doi: 10.1038/srep28622 (2016).
References
Nakamura, Y., Pashkin, Y. A. & Tsai, J. S. Coherent control of macroscopic quantum states in a singleCooperpair box. Nature 398, 786–788 (1999).
Bouchiat, V. et al. Quantum coherence with a single Cooper pair. Phys. Scr. T76, 165–170 (1998).
Vion, D. et al. Manipulating the quantum state of an electrical circuit. Science 296, 886–889 (2002).
Chiorescu, I., Nakamura, Y., Harmans, C. J. P. M. & Mooij, J. E. Coherent quantum dynamics of a superconducting flux qubit. Science 299, 1869–1871 (2003).
Yu, Y. et al. Coherent temporal oscillations of macroscopic quantum states in a Josephson junction. Science 296, 889–892 (2002).
Martinis, J. M., Nam, S., Aumentado, J. & Urbina, C. Rabi oscillations in a large Josephsonjunction qubit. Phys. Rev. Lett. 89, 117901 (2002).
Chiorescu, I. et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature 431, 159–162 (2004).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162–167 (2004).
Yan, F. et al. The flux qubit revisited. arXiv: 1508.06299 (2015).
You, J. Q. & Nori, F. Atomic physics and quantum optics using superconducting circuits. Nature 474, 589–597 (2011).
Mooij, J. E. et al. Josephson persistentcurrent qubit. Science 285, 1036–1039 (1999).
Martinis, J. Superconducting phase qubits. Quant. Inf. Proc. 8, 81 (2009).
Kuzmin, L. S. & Haviland, D. B. Observation of the Bloch oscillations in an ultrasmall Josephson junction. Phys, Rev. Lett. 67, 2890 (1991).
Bertet, P. et al. Dephasing of a superconducting qubit induced by photon noise. Phys, Rev. Lett. 95, 257002 (2005).
Niskanen, A. O. et al. Quantum coherent tunable coupling of superconducting qubits. Science 316, 723–726 (2007).
Ashhab, S. et al. Interqubit coupling mediated by a highexcitationenergy quantum object. Phys. Rev. B 77, 014510 (2008).
Orlando, T. P. et al. Superconducting persistentcurrent qubit. Phys, Rev. B 60 15398 (1999).
Makhlin, Y., Schön, G. & Shnirman, A. Quantumstate engineering with Josephson junction devices. Rev. Mod. Phys. 73, 357–400 (2001).
You, J. Q., Nakamura, Y. & Nori, F. Fast twobit operations in inductively coupled flux qubits. Phys. Rev. B 71, 024532 (2005).
van der Zant, H. S. J., Berman, D., Orlando, T. P. & Delin, K. A. Fiske modes in onedimensional parallel Josephsonjunction arrays. Phys. Rev. B 49, 12945 (1994).
Kittel, C. Introduction to Solid State Physics 8 ed. (Wiley, 2005).
You, J. Q., Hu, X. D., Ashhab, S. & Nori, F. Lowdecoherence flux qubit. Phys. Rev. B 75, 140515 (2007).
Steffen, M. et al. Highcoherence hybrid superconducting qubit. Phys. Rev. Lett. 105, 100502 (2010).
Scully, M. O. & Zubairy, M. S. Quantum Optics. (Cambridge University Press, Cambridge, 1997).
Liu, Y. X., You, J. Q., Wei, L. F., Sun, C. P. & Nori, F. Optical selection rules and phasedependent adiabatic state control in a superconducting quantum circuit. Phys. Rev. Lett. 95, 087001 (2005).
Liu, Y. X., Yang, C. X., Sun, H. C. & Wang, X. B. Coexistence of single and multiphoton processes due to longitudinal couplings between superconducting flux qubits and external fields. New. J. Phys. 16, 015031 (2014).
Acknowledgements
This work is supported by the NSAF Grant Nos U1330201 and U1530401, the National Natural Science Foundation of China Grant No. 91121015 and the National Basic Research Program of China Grant No. 2014CB921401.
Author information
Authors and Affiliations
Contributions
Y.Q. performed the calculations under the guidance of J.Q.Y. T.F.L., W.X. and X.L.H. also participated in the discussions. All authors contributed to the interpretation of the work and the writing of the manuscript.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Qiu, Y., Xiong, W., He, XL. et al. Fourjunction superconducting circuit. Sci Rep 6, 28622 (2016). https://doi.org/10.1038/srep28622
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/srep28622
 Springer Nature Limited
This article is cited by

Circulator function in a Josephson junction circuit and braiding of Majorana zero modes
Scientific Reports (2021)