Abstract
Threewave mixing in secondorder nonlinear optical processes cannot occur in atomic systems due to the electricdipole selection rules. In contrast, we demonstrate that secondorder nonlinear processes can occur in a superconducting quantum circuit (i.e., a superconducting artificial atom) when the inversion symmetry of the potential energy is broken by simply changing the applied magnetic flux. In particular, we show that difference and sumfrequencies (and second harmonics) can be generated in the microwave regime in a controllable manner by using a single threelevel superconducting flux quantum circuit (SFQC). For our proposed parameters, the frequency tunability of this circuit can be achieved in the range of about 17 GHz for the sumfrequency generation, and around 42 GHz (or 26 GHz) for the differencefrequency generation. Our proposal provides a simple method to generate secondorder nonlinear processes within current experimental parameters of SFQCs.
Introduction
NOnlinear optical effects have many fundamental applications in quantum electronics, atom optics, spectroscopy, signal processing, communication, chemistry, medicine, and even criminology. These phenomena include optical Raman scattering, frequency conversion, parametric amplification, the Pockels and Kerr effects (i.e., linear and nonlinear electrooptical effects), optical bistability, phase conjugation, and optical solitons^{1,2}. Threewave mixing (including the generations of the sumfrequency, differencefrequency, and second harmonics) and fourwave mixing are important methods to study nonlinear optics. It is wellknown that materials without inversion symmetry can exhibit both second and thirdorder nonlinearities. However, materials with inversion symmetry usually exhibit only thirdorder nonlinearities. Thus, threewave mixing (which requires the secondorder nonlinearity) cannot occur in atomic systems with welldefined inversion symmetry, because the electricdipole transition selection rules produce a zero signal^{1} with mixed frequencies. Although chiral molecular threelevel systems without inversion symmetry can be used to generate threewave mixing in the microwave domain^{3,4,5,6}, such wave mixing cannot be tuned because the energy structure of the systems is fixed by nature.
Recently, superconducting charge, flux, and phase quantum circuits based on Josephson junctions have been extensively explored as basic building blocks for solidstate quantum information processing^{7,8,9,10}. These circuits can also be considered as artificial atoms^{9,11}. In contrast to natural atoms, the quantum energy structure and the potential energy of these artificial atoms can usually be tuned by external parameters. Thus, they can possess new features and can be used to demonstrate fundamentally new phenomena which cannot be found in natural threelevel atoms. For example, with the tunable potential energy of superconducting flux quantum circuits (SFQCs) by varying the bias magnetic flux, threelevel (qutrit) SFQCs can have a Δtype (cyclic) transition^{12}. Twolevel SFQCs are also known as superconducting flux qubits^{13}. Threelevel SFQCs (i.e., superconducting flux qutrits) can be used to demonstrate the coexistence of single and twophotons^{12,14}, which does not occur in natural threelevel atomic systems with electricdipole interaction. Such Δtype atoms can also be used to cool quantum systems^{15}, or generate microwave singlephotons^{16}.
In solidstate quantum information processing, microwave signals are usually employed for measuring and controlling the qubits. Moreover, these signals can also be used to detect the motion of nanomechanical resonators^{17} and to read out the spin information in nitrogenvacancy centers in diamonds^{18}. Therefore, the controllable generation, conversion and amplification of microwave signals play a very important role in solidstate quantum information processing. The generation of microwave Fock's states^{19,20,21}, superpositions of different Fock's states^{22}, squeezed states^{23}, nonclassical microwave^{24} and giant Kerr nonlinearities^{25,26} have been studied in the microwave domain via circuit quantum electrodynamics (QED)^{7,8,9}. Microwave parametric amplification^{27} has also been studied by using threewave mixing^{28} in superconducting circuits with four Josephson junctions. Different from Ref. 28, here we propose another method to generate microwave threewave mixing, including the generation of the sum and differencefrequencies in a controllable way via a tunable single SFQC. This method also applies for phase^{29,30,31} and transmon^{32} qutrits. In our proposal, such threewave mixing can be switched off at the optimal point by the bias magnetic flux. We also discuss the possibility for the generations of second harmonics and zerofrequency using SFQCs.
Model
To be specific, our study below will focus on threelevel SFQCs, also called a qutrit or threelevel qudit. However, our results can also be applied to phase and transmon qutrits. As shown in Fig. 1(a), a SFQC consists of a superconducting loop interrupted by three Josephson junctions and controlled by a bias magnetic flux Φ_{e}. The Josephson energies (capacitances) of the two identical junctions and the smaller one are E_{J} (C_{J}) and αE_{J} (αC_{J}) with 0.5 < α < 1, respectively. If we assume that the SFQC is driven by the external timedependent magnetic flux with frequencies ω_{l}, then we can describe the system by this Hamiltonian with M_{p} = 2C_{J}[Φ_{0}/(2π)]^{2} and M_{m} = M_{p}(1 + 2α). The potential energy is with phases φ_{p} = (φ_{1} + φ_{2})/2 and where φ_{1} and φ_{2} are the gaugeinvariant phases of the two identical junctions (see Fig. 1). Here f = Φ_{e}/Φ_{0} is the reduced magnetic flux, and Φ_{0} = h/(2e) is the flux quantum. The interaction between the SFQC and the timedependent magnetic flux is described by V(t) = I(φ_{p}, φ_{m}, f)Φ(t), with the supercurrent inside the superconducting loop^{33,34} and I_{0} = 2πE_{J}/Φ_{0}. The supercurrent I ≡ I(φ_{p}, φ_{m}, f) and the external magnetic flux Φ(t) are equivalent to the electric dipole moment operator and timedependent electric field of the electric dipole interaction in atomic systems. It is obvious that U(φ_{p}, φ_{m}, f) in Eq. (1) can be tuned by the bias magnetic flux Φ_{e}. We have shown that one of two flux quits cannot work at the optimal point when both qubits are directly coupled through their mutual inductance^{34}, because of its selection rules^{12,33}. Such problem can be solved by introducing a coupler (e.g., see, Refs. 35,36,37).
We have shown^{12} that threelevel SFQCs have Δtype (cyclic) transitions among the three lowest energy levels i〉 when the inversion symmetry of the potential energy is broken, otherwise it has a cascade transition. Under the threelevel approximation of SFQCs, Eq. (1) becomes where E_{i} (i = 1, 2, 3) are three eigenvalues corresponding to the three lowest eigenstates i〉 of Eq. (1) with V(t) = 0. With this threelevel approximation of SFQCs, the interaction Hamiltonian V_{T}(t) in Eq. (5) can be generally written as with operators σ_{ij} = i〉 〈j and matrix elements I_{ij}(f) ≡ 〈iI(φ_{p}, φ_{m}, f)j〉 dipolelike moment operator. Here, the longitudinal coupling between the threelevel SFQC and the timedependent magnetic flux is neglected even though the reduced magnetic flux is not at the optimal point, i.e., f ≠ 0.5. We note that f = 0.5 is called as the optimal point or the symmetry point^{13}, where the influence of flux noise is minimal. When the relaxation and dephasing of the threelevel SFQC are included, the dynamics can be described by the master equation with ρ(t) ≡ ρ. Here, different energy levels are assumed to have different dissipation channels. The operator ρ(t) is the reduced density matrix of the threelevel SFQC. We will study the steadystate response; thus, the thermal equilibrium state for V(t) = 0 with matrix elements is added to the master equation. Also, γ_{ii} is the pure dephasing rate of the energy level i〉, while γ_{ij} = γ_{ji} (with i ≠ j) are the offdiagonal decay rates.
Sum and differencefrequency generations
We assume that the SFQC is in the thermal equilibrium state when V(t) = 0. To study the steadystate response of the threelevel SFQC to weak external fields, we have to obtain the solution of the reduced density matrix ρ for the threelevel SFQC in Eq. (7) by solving the following equations: with the parameters Γ_{12} = γ_{12}, Γ_{13} = γ_{13} + γ_{23} + γ_{33} and Γ_{23} = γ_{12} + γ_{13} + γ_{23} + γ_{22} + γ_{33}, derived from Eq. (7). Note that Γ_{ij} = Γ_{ji}. Here we define . Because the external fields are weak, the solution of ρ(t) can be obtained by expressing ρ(t) in the form of a perturbation series in V_{T}(t), i.e., with the density matrix operator in the zerothorder approximation. We define the magnetic polarization P due to the external field as P = Tr[ρ(t)I], in analogy to the electric polarization^{1}, then the secondorder magnetic polarization can be given as P^{(2)} = Tr[ρ_{2}(t)I], and then the secondorder magnetic susceptibility can be given by In our study, since the condition (with i ≠ j) is satisfied, then the system is in its ground state 1〉 in the thermal equilibrium state, i.e., .
Sumfrequency generation
To study the microwave generation of the sumfrequency, we now assume that the two external magnetic fluxes are applied to the threelevel SFQC. As schematically shown in Fig. 1(b), one magnetic flux with frequency ω_{1} (ω_{2}) induces the transition between the energy levels 1〉 and 2〉 (2〉 and 3〉). In this case, the interaction Hamiltonian V_{T}(t) between the threelevel SFQC and the two external fields is given by under the rotatingwave approximation. On replacing V_{T}(t) in Eq. (7) by V_{1}(t), and using the perturbation theory discussed above, we can obtain the reduced density matrix of the threelevel SFQC, up to second order in V_{1}(t), and find the secondorder magnetic susceptibility as for the sumfrequency generation with ω_{+} = ω_{1} + ω_{2}, and , with i > j. Equation (12) obviously shows that the secondorder magnetic susceptibility is proportional to the product of the three different electric dipolelike matrix elements (or transition matrix elements) I_{ij}(f), with i ≠ j. Therefore, for a given reduced magnetic flux f, the maximum value of the susceptibility in Eq. (12) is when ω_{+} = ω_{31} and ω_{1} = ω_{21}.
Differencefrequency generation
Similarly, the differencefrequency can also be generated by using a threelevel SFQC. We assume that a magnetic flux with frequency ω_{1} (ω_{2}) is applied between the energy levels 1〉 and 3〉 (2〉 and 3〉) as shown in Fig. 1(c). In this case, the interaction between the threelevel SFQC and the external magnetic fields can be described by under the rotatingwave approximation.
Using the same calculation as for Eq. (12), we can also obtain the secondorder magnetic susceptibility of the differencefrequency ω_{−} = ω_{2} − ω_{1} as For a given reduced magnetic flux f, the maximum amplitude of the susceptibility in Eq. (14) for the differencefrequency can be obtained under the resonant driving conditions: ω_{−} = ω_{21} and ω_{1} = ω_{31}.
Numerical simulation
Both Eqs. (12) and (14) show that the susceptibilities of the sum and differencefrequencies can be controlled by the bias magnetic flux Φ_{e}. According to the analysis of the inversion symmetry for flux quantum circuits^{12}, we know that the threelevel SFQC has a welldefined symmetry at the optimal point f = 0.5 and it behaves as natural threelevel atoms with the Ξtype (or laddertype) transition. In this case, the transition matrix elements between the energy levels 1〉 and 3〉 is zero, i.e., I_{13}(f = 0.5) = I_{31}(f = 0.5) = 0, and both susceptibilities, χ^{(2)}(ω_{+}) in Eq. (12) and χ^{(2)}(ω_{−}) in Eq. (14), are zero. Thus, the microwave sum or differencefrequencies cannot be generated at the optimal point as for natural threelevel atoms with the electricdipole selection rule. Equations (12) and (14) also tell us that the amplitudes of the susceptibilities for both the sum and differencefrequencies are proportional to the modulus R(f) of the product of the three different transition matrix elements, i.e., Thus, the maximum value R^{(max)}(f) of R(f) corresponds to the maximal susceptibilities under the resonant driving condition. To show clearly how the bias magnetic flux Φ_{e} can be used to control the sum and differencefrequency generations, the three transition elements I_{12}, I_{23} and I_{13} versus the reduced magnetic flux f are plotted in Fig. 2(a). Also, the fdependent product I_{12}I_{23}I_{31} is plotted in Fig. 2(b). Here, we take experimentally accessible parameters, for example, α = 0.8, E_{J}/h = 192 GHz, and E_{J}/E_{c} = 48, where E_{c} is the charging energy and h is the Planck constant. These data are taken from the RIKENNEC group for their most recent, unpublished, experimental setup. Figures 2(a) and 2(b) clearly show that the bias magnetic flux Φ. i.e., f = Φ/Φ_{0}, can be used to tune the transition elements, and then R(f) is also tunable. We find that R(f) is zero, at the optimal point corresponding to the zero signal for the sum and differencefrequency generations, because the transition selection rule at this point makes the transition element I_{13} = 0, as shown in Fig. 2(a). That is, the transition between the energy levels 1〉 and 3〉 is forbidden. However, the sum and differencefrequencies can be generated when f ≠ 0.5, and the maximum R^{(max)}(f) corresponds to two symmetric points with f = 0.4992 and f = 0.5008. To show the tunability of the frequency generation, we now define a maximum variation of the sum and differencefrequency generation as for a given range of the reduced magnetic flux f. Here, denotes the transition frequency between the energy levels i〉 and j〉 at the optimal point.
Figure 2(c) shows that the maximum variation of the sumfrequency is for 0.5 < f < 0.53. However, the maximum variation or of the differencefrequency is or for 0.5 < f < 0.53. Thus, the tunability for the sum and differencefrequency generations can be, in principle, over a very wide GHz range, by using the bias magnetic flux Φ_{e}.
Secondharmonic generation
From Eqs. (12) and (14), we find that the secondharmonic and zerofrequency signals can also be generated in threelevel SFQCs when two applied external fields have the same frequency and satisfy the condition Let us now discuss secondharmonic generation. As shown in Fig. 3(a), we can find two values of the reduced magnetic flux, f = 0.4878 or f = 0.5122, such that ω_{31} = 2ω_{21} = 2ω_{32}. In this case, the susceptibility of the second harmonic reaches its maximum, when an external field with the same frequency as ω_{21} = ω_{32} is applied to the threelevel SFQC. However, the secondorder susceptibility becomes small when the magnetic field deviates from the points f = 0.4878 or f = 0.5122 because of the anharmonicity of the energylevel structure for the SFQC. If we assume that the anharmonicity is characterized by then the secondorder susceptibility for the secondharmonic generation can be approximately written as We note that this equation for the secondorder susceptibility is a rough approximation when , i.e., δ = 0. Because the independentenvironment assumption for the decays of different energy levels might not always hold and the dissipation rates Γ_{12} and Γ_{13} should be modified. However, the main physics is not changed. In Fig. 3(b), as an example, the amplitude of , which is given by is plotted as a function of f for given parameters, e.g., Γ_{12}/2π = 50 MHz and Γ_{13}/2π = 30 MHz. It clearly shows that the maximum amplitude of the susceptibility corresponds to the reduced magnetic flux f = 0.4878 or f = 0.5122, in which the three energy levels have a harmonic structure. It should be noted that we take Γ_{21} and Γ_{31} as the findependent parameters for convenience when Fig. 3(b) is plotted. In practice, they should also depend on f.
Measurements
We now take the sumfrequency generation as an example to show how to measure the frequency generation by coupling the threelevel SFQC to the continuum of electromagnetic modes confined in a 1D transmission line as for measuring the resonance fluorescence of single artificial atoms^{38,39}. As discussed in Ref. 40, if the three transition frequencies of the threelevel SFQC are much larger than the decay rates, then we can consider that the decays of different energy levels occur via different dissipation channels. In this case, the interaction Hamiltonian between the threelevel SFQC and the continuum modes in the transmission line can be modeled as under the Markovian approximation with the bosonic commutation relation [α(ω), β^{†}(ω′)] = δ_{α,β}δ(ω − ω′) with α, β = a, b, c for the three kinds of different continuum mode operators. According to the inputoutput theory^{41}, the output field centered at the sumfrequency ω_{1} + ω_{2} = ω_{+} can be given as since Tr[ρσ_{13}(t)] = Tr[ρ(t)σ_{13}] = ρ_{31}(t). Therefore, up to second order in V_{1}(t) for the sumfrequency generation, we can approximately obtain the output of the sumfrequency generation as where the input field for the continuum mode c(ω) is in the vacuum. Equation (23) shows that the amplitude of the output field is proportional to the intensities Φ(ω_{1}) and Φ(ω_{2}) of the two external magnetic fields, the modulus of the product of two transition matrix elements I_{21}(f) and I_{32}(f), and the square root of the decay rate γ_{13}. It is obvious that the intensity of the output field can be tuned by the bias magnetic flux Φ_{e}. Similarly, the amplitude of the output field for the differencefrequency generation described in Eq. (14) is proportional to the modulus of the product of two transition matrix elements I_{13}(f) and I_{32}(f). The moduli R_{1}(f) ≡ I_{21}(f)I_{32}(f) and R_{2}(f) ≡ I_{13}(f)I_{32}(f) versus f are plotted in Figs. 4(a) and (b), which show that the amplitude of the output fields for the sum and differencefrequency generations can also be tuned by f. However, the maximum value, corresponding to maximum secondorder susceptibility under resonant condition, of R^{(max)}(f) does not correspond to the maximum value of R_{1}(f) for the sumfrequency, or R_{2}(f) for the differencefrequency.
Conclusions
We have proposed and studied a controllable method for generating sum and difference frequencies by using threewave mixing in a single threelevel SFQC driven by two weak external fields. Thus, in perturbation theory, the noise and frequency shifts introduced by the driving fields can be neglected and we can obtain all the response functions of different frequencies. We point out that the threewavemixing signal can only be generated when the inversion symmetry of the potential energy for the SFQC is broken, that is, the SFQC cannot work at the optimal point. Otherwise, the transition between the ground state and the secondexcited state is forbidden, so threewave mixing cannot be generated as in naturalatom systems. We have shown that the generated microwave signal can be tuned in a very large GHz range. We have also discussed how to generate secondharmonics in the single SFQC. We note that threewave mixing can also occur in superconducting phase^{29,30,31} and transmon^{32} qutrits, when the inversion symmetry of their potential energies is broken. In particular, the phase qutrits might be better for secondharmonic generation because of their small anharmonicity. It should be pointed out that the microwave signal with the sumfrequency might exceed the highfrequency cutoff of the cryogenic amplifier^{38}. Thus, the differencefrequency generation should be easier to be experimentally accessed.
In contrast to Ref. 28, with a frequency tunability of about 500 MHz, we show that the tunability of the output frequency using single flux qubit circuits can be a few GHz. Our proposal is valid not only for nondegenerate threewave mixing, but it can also be applied for secondharmonic generation by changing the bias magnetic flux. Also, contrary to Ref. 28, where the circuit itself is in the classical regime, in our study, the threewave mixing is generated using excitations of real quantized energy levels of the artificial atoms. Such excitation will result in a strong nonlinearity. Thus, the threewave mixing in single artificial atoms can be used to generate entangled microwave photons and act as entanglement amplifier or correlated lasing. These could be important toward future quantum networks.
In summary, our study could help generating three or multiwave mixing using single artificial atoms. The proposed method is simple and could be used for manipulating secondorder and other nonlinear processes in the microwave regime by using single superconducting artificial atoms. Our proposal is realizable using current experimental parameters of superconducting flux qubit circuits.
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Acknowledgements
Y.X.L. is supported by the National Basic Research Program of China Grant No. 2014CB921401, the NSFC Grants No. 61025022, and No. 91321208. A.M. is supported by Grant No. DEC2011/03/B/ST2/01903 of the Polish National Science Centre. F.N. is partially supported by the RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics, and a GrantinAid for Scientific Research (S). Z.H.P. and J.S.T. were supported by Funding Program for WorldLeading Innovative R & D on Science and Technology (FIRST), MEXT KAKENHI “Quantum Cybernetics”.
Author information
Affiliations
Institute of Microelectronics, Tsinghua University, Beijing 100084, China
 Yuxi Liu
 & HuiChen Sun
Tsinghua National Laboratory for Information Science and Technology (TNList), Beijing 100084, China
 Yuxi Liu
CEMS, RIKEN, Saitama 3510198, Japan
 Yuxi Liu
 , HuiChen Sun
 , Z. H. Peng
 , Adam Miranowicz
 , J. S. Tsai
 & Franco Nori
Faculty of Physics, Adam Mickiewicz University, 61614 Poznań, Poland
 Adam Miranowicz
NEC Green Innovation Research Laboratories, Tsukuba, Ibaraki 3058501, Japan
 J. S. Tsai
Physics Department, The University of Michigan, Ann Arbor, Michigan 481091040, USA
 Franco Nori
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Contributions
Y.X.L. proposed the main idea. Y.X.L., H.C.S., Z.H.P., A.M. and F.N. contributed to the findings of this work and wrote the manuscript. J.S.T. participated in the discussions.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Yuxi Liu.
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