PHYSICAL REVIEW LETTERS 127, 087001 (2021)

Microwave-Induced Inverse Faraday Effect in Superconductors

A. Hamed Majedi * Department of Electrical and Computer Engineering, Department of Physics and Astronomy, Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received 28 November 2020; accepted 19 July 2021; published 20 August 2021)

The inverse Faraday effect (IFE) in superconductors is proposed, where a static is generated under the influence of a circularly polarized microwave field. Classical modeling of the IFE explicitly provides superconducting gyration coefficient in terms of its complex conductivity. The IFE is then considered as a source of nonlinearity and gyrotropy even at a low-power microwave regime giving rise to a spectrum of phenomena and applications. Microwave-induced gyroelectric conductivity, , microwave birefringence, flux quantization, and a vortex state are predicted and quantitatively analyzed. A peculiar microwave birefringence in gyrotropic superconductors due to radical response of superelectrons has been highlighted.

DOI: 10.1103/PhysRevLett.127.087001

Introduction.—Nonlinear microwave response of super- such as Hall effect and microwave birefringence but also conducting structures is a core subject not only in probing to derive type II superconductors to vortex state. the physics of [1] but also implemented In this Letter I first develop a classical formalism to find in numerous applications ranging from quantum metrology the microwave induced static through a to superconducting qubits and microwave quantum optics gyration coefficient that is proportional to the complex [2]. The microwave nonlinearities in superconducting conductivity of the superconductor. The critical microwave structure possess diverse origins but mostly are attributed field to suppress superconductivity is derived in terms of to nonlinear kinetic [3,4] and Josephson [5,6] the critical magnetic field. Turning to IFE consequences in including Kerr-type [7,8], Duffing and anharmonicity superconductors, the gyroelectric conductivity and micro- [9,10], weak links [11–13], phase slip formation [14], wave-induced Hall effect, as experimental tools to measure and vortex dynamics [15,16]. superconducting gyration coefficient are analytically dis- Borrowing from optomagnetics, a less-explored field in cussed. Gyroelectric conductivity is also used for linear and nonlinear optics [17], a microwave-induced nonlinearity in circular birefringence for a microwave pump-probe sce- superconductors based on the inverse Faraday effect (IFE) nario. Finally, I propose an embodiment of the dynamic is proposed in this Letter. The IFE refers to the generation and controllable flux quantization and vortex state in of a static magnetic field by not linearly polarized, e.g., type II superconductors using a circularly polarized micro- circularly polarized, light [18]. The purely nonlinear effect wave field. arising from the IFE is solely based on the gyration of the Modeling of inverse Faraday effect in superconductors.— time-varying and it does not directly link to The electrodynamic response of a superconductor is any linear electromagnetic properties of the materials such considered in the two-fluid model. The equation of motion as Kerr-type that is related to the linear refractive index. The for superelectrons and normal electrons under the influence IFE in a superconductor is based on angular momentum of electric field Eðr;tÞ and its associated magnetic field transfer between the circularly polarized microwave field Bðr;tÞ can be phenomenologically described by the and a superconductor. Electric field gyration creates an London equations encircling supercurrent and a normal current associated d with a local static magnetic field opposing the microwave m v eE r;t ev B r;t dt s ¼ ð Þþ s × ð Þð1Þ field. The IFE can be employed not only to make the tunable gyrotropic superconductor prevailing upon phenomena d m v mΓ v eE r;t e v B r;t dt h niþ h ni¼ ð Þþ h ni × ð Þð2Þ

where m is the mass of an electron, e is charge of an Published by the American Physical Society under the terms of electron, Γ ¼ð1=τÞ is the inverse of momentum relaxation the Creative Commons Attribution 4.0 International license. v v Further distribution of this work must maintain attribution to time for normal electrons, and s and h ni are the the author(s) and the published article’s title, journal citation, superelectron and average normal electron velocities, and DOI. respectively. We consider a circularly polarized plane

0031-9007=21=127(8)=087001(5) 087001-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 127, 087001 (2021) with pumping frequency, ωp, travel- Equation (7) is in the form of Pitaevskii’s relationship [18] ing normal to the surface of a semi-infinite superconductor, indicating that the dc magnetization solely depends on the i.e., at z ¼ 0, in the following form: gyration of the electric field, through the relationship of ˜ ˜ 2 −2αpz iE × E ¼ 2zjEoj e . This fact is signified by the ˜ iωpt −αpz iðωpt−βpzÞ E ¼ RefEe g¼RefEoðx þ iyÞe e gð3Þ gyration coefficient, γðωp;TÞ, reminiscent of magneto- gyration coefficients in magnetooptic materials [22]. The B B˜ eiωpt B y − ix e−αpzeiðωpt−βpzÞ ; ¼ Ref g¼Ref oð Þ g ð4Þ microwave-induced dc magnetization in a superconductor is inherently a nonlinear electrodynamic process rooted in where x, y, z are Cartesian unit vectors, 0 < ℏωp < 2Δ, Δ the IFE but the gyration coefficient can be approximated in being a superconducting energy gap, and αp and βp are the the low frequency regime, i.e., ω ≪ Γ, in terms of the linear propagation loss and the propagation constant, respectively. complex conductivity, σ1 − iσ2 based on London equations The low-frequency propagation characteristics can be [23]. Note that the magnetic field associated with the derived based on the two-fluid model [19]. When the microwave field has no contribution to dc magnetization, circularly polarized wave interacts with the ensemble of ruling out the direct magnetization of the superconductor. free superelectrons and normal electrons, its angular In fact, the linear response of the superconductor due to momentum generates a local circulating supercurrent and the incident electromagnetic field, i.e., Eqs. (3) and (4), normal current in the superconductor that produces a creates a time-dependent supercurrent and normal current magnetic field parallel to the microwave field in the according to Maxwell’s and London equations, i.e., opposite direction. This is a manifestation of the IFE where ˜ ˜ ˜ ˜ J Js Jn σ1 − iσ2 E. For a circularly polarized high-intensity circularly polarized light generates dc mag- ¼ þ ¼ð Þ electric field there is a time-dependent circular supercurrent netization in matter, a theoretical prediction by L. Pitaevskii having x and y components. In addition, the microwave- in 1961 [18] and its subsequent experimental demonstra- induced dc magnetization forms a supercurrent density due tion in 1965 [20]. The magnetization in a superconductor, M to the nonlinear response based on the IFE and is shown by s can be found as [17] ind Js . The induced supercurrent due to dc magnetization n e n e opposes to the incident electric field to maintain the M s L n L s ¼ 2m s þ 2m n ð5Þ and can be written as

L r p where s ¼ s × s is the of super- ∂fðx; yÞ ∂fðx; yÞ L r p Jind ∇ M M x − y ; electrons and n ¼ n × h ni is the magnetic moment of s ¼ × dc ¼ dc ∂y ∂x ð9Þ normal electrons in terms of their momenta ps and hpni, respectively. For the intermediate temperature range, 0

087001-2 PHYSICAL REVIEW LETTERS 127, 087001 (2021) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 2 1=2 ð 1 þ 16ηoγ ðω;TÞHc − 1Þ Ec ¼ pffiffiffi ð11Þ 2 2γðω;TÞηo where ηo is the free space characteristic impedance. Gyroelectric conductivity and Hall effect.—The micro- wave-induced dc magnetization breaks the directional symmetry making a superconductor a gyrotropic material represented by a conductivity tensor, the so-called gyro- electric conductivity. Referring to the London model, one can find the gyroelectric conductivity tensor, σ¯, relating the total J to the applied weak electric field E, ¯ ˜ ˜ ˜ ˜ i.e., J ¼ σ¯E. Considering E ¼ xEx þ yEy þ zEzz, with angular frequency ω, the gyroelectric conductivity tensor FIG. 1. A superconducting wire is illuminated by the circularly can be written as polarized microwave pump signal in the presence of a dc electric Hall 0 1 field, and Ex develops Hall V . ωc σs þ σn −i ω σs þ τωcσn 0 B C σ¯ ω ωc ð sÞ¼@ i ω σs − τωcσn σs þ σn 0 A Hall field along the y direction, where the current cannot y J 0 00σ1 − iσ2 flow out of the wire along the direction, i.e., y ¼ . The Hall resistance is then given by ð12Þ

iωc 1 σs − τωcσn where R ω 16 Hall ¼ 2 iωc 2 ð Þ lz ðσs þ σnÞ þð σs − τωcσnÞ ω 0 iωe2n ω ¼ σ s s ¼ 2 2 ð13Þ mðωc − ω Þ where the microwave field decay has not been considered, 2 therefore, the length should be chosen less than the inverse nne τð1 þ iωτÞ −1 σ of the field attenuation constant, i.e., lz < αp . The Hall n ¼ m 1 iωτ 2 ω2τ2 ð14Þ ½ð þ Þ þ c resistance shows the contribution from both superelectrons and the microwave-induced cyclotron angular frequency is and normal electrons in the superconducting wires and well into the superconducting state the Hall resistance will be 2eμ ω ≜ o γ ω ;T E 2: c m ð p Þj oj ð15Þ μ λ2 R o L ω Hall ¼ l c ð17Þ Note that the cyclotron frequency should be smaller than z the gap frequency, otherwise the energy of the microwave- λ induced magnetic moments is larger than Cooper pair where L is the London penetration depth. Equation (17) binding energy leading to suppressed superconductivity. offers a way to measure the gyration coefficient of Therefore, the IFE is pronounced in the regime where the the superconductor through the Hall resistance measure- microwave pump and signal frequencies are smaller than ment. Close to the critical temperature the Hall resistance R μ M =n el the cyclotron and gap frequencies and the field amplitudes tends to its normal value of Hall ¼ð o dc n zÞ¼ 2μ γ ω ;T =n el E 2 are smaller than the critical electric field. The gyroelectric ½ o ð p Þ n zj oj . This result conforms the linear conductivity can be tracked down due to the contribution of dependency of the Hall resistance to the microwave- microwave-induced magnetization to the longitudinal con- induced dc magnetization while the longitudinal resistance ductivities, σxx ¼ σyy ¼ σs þ σn through the cyclotron is unaffected by the microwave signal. — frequency ωc. Gyroelectric conductivity also leads to the Gyroelectric birefringence. Another feature of the Hall effect where the microwave-induced magnetization gyroelectric conductivity in a superconductor is the micro- ’ causes a dc electric field to develop across the super- wave birefringence. According to Maxwell s equations, the conducting wire perpendicular to the direction of micro- conductivity tensor, Eq. (12), yields the anisotropic per- wave propagation, i.e., the z direction [24]. Consider a long mittivity tensor as superconducting wire along the x direction with the 0 1 0 00 0 00 l l l ϵr − iϵr ϵxy − iϵxy 0 corresponding length scales at x, y, and z, and in the i B C ¯ ¯ 0 00 0 00 presence of a longitudinal dc electric field along the x ϵ¯r ¼ 1 − σ¯ ¼ @ −ϵxy þ iϵxy ϵr − iϵr 0 A: ωϵo direction and a circularly polarized microwave signal 0 00 00ϵzz − iϵzz propagating along the z direction as shown in Fig. 1. The microwave-induced dc magnetization produces the ð18Þ

087001-3 PHYSICAL REVIEW LETTERS 127, 087001 (2021)

Note that the superconductor’s relative consists supercurrent equation about a closed contour within the of a large negative real part where the electromagnetic field superconductor where the path is either in the bulk super- is weak with a frequency significantly smaller than its gap conducting region or the multiply connected region in the frequency and below its critical temperature. Now, if a presence of a circularly polarized field. Assuming that weak linearly polarized wave with a frequency of ω < ωc is jΨðr;tÞj is a well-defined function then the line integration launched to the superconductor along the z axis copropa- of supercurrent equation along the contour C encircling the gating with the microwave pump field with frequency ωp,it surface S yields the following fluxoid quantization expres- experiences birefringence leading to polarization rotation. sion: If the permittivity tensor is diagonalized in the coordinate I pffiffiffi e 1= 2 x iy ˜ ˜ system with orthogonal unit vectors ¼ð Þð Þ, iΛγðω;TÞ∇ × ðfðx; yÞE × EÞ:dl then the linearly polarized wave has two normal propaga- C Z ϵ0 ϵ00 − tion modes with relative ð xx xyÞ μ iγ ω;T f x; y E˜ E˜ :dS nΦ 00 0 þ o ð Þ ð Þ × ¼ o ð22Þ iðϵxx ∓ ϵxyÞ while the z axis acts as the uniaxial optical S symmetry. The weak signal entering the gyrotropic super- conductor decomposes into the slow component with the where Φo is the flux quantum, n represents a winding complex propagation constants of number of the macroscopic wave function. The left-hand side of Eq. (22) represents the electric field-induced fluxoid 00 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ϵxx − ϵxy in the superconductor. Thus, if the superconductor is o 0 00 α1 iβ1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iko ϵxx ϵxy 19 þ ¼ 0 00 þ þ ð Þ magnetized by the circularly polarized microwave field, 2 ϵxx þ ϵxy once the field is removed the trapped flux inside the and the fast component having the following complex superconductor is quantized. This might offer a new way propagation constant of to magnetize and demagnetize superconductors with inci- dent microwave field. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 0 — k ϵxx ϵxy Vortex state. Consider a type II superconductorffiffiffi where 00 0 o þ p α2 þ iβ2 ¼ ko ϵxy − ϵxx þ i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð20Þ the Ginzburg-Landau parameter κ ¼ðλ=ξÞ > ð1= 2Þ, that 2 ϵ00 − ϵ0 xy xx is defined as the ratio of its penetration depth λ to its ξ k 2π=λ ω=c coherence length . In the mixed state, the where o ¼ð oÞ¼ð Þ is the free space wave penetrates the type II superconductor starting at lower number in terms of wavelength λo and c as the speed of 2 critical field Hc1 ¼ðΦo=4πμoλ Þ lnðλ=ξÞ in the form of light. The dependence of the real and imaginary parts of the triangular array of vortices until reaches its upper critical relative permittivity on the microwave induced magneti- 2 field Hc2 ¼ðΦo=2πμoξ Þ. In order to derive the type II zation leads to linear birefringence (Cotton-Mouton effect) superconductor to its mixed states by the microwave- and a circular birefringence (Faraday effect), respectively induced IFE, we need to satisfy the inequality relation [25]. In the case of ω < ωc < ωs one can find the rotation for the electric field amplitude as Ec1

087001-4 PHYSICAL REVIEW LETTERS 127, 087001 (2021) gyrotropy. They offer novel applications to control super- Black-Box Superconducting Circuit Quantization, Phys. Rev. conductivity in a dynamic and fully controllable fashion Lett. 108, 240502 (2012). that is solely enabled by a microwave polarization degree of [11] R. Kümmel, U. Gunsenheimer, and R. Nicolsky, Andreev freedom. New readout electronics for superconducting scattering of quasiparticle wave packets and current-voltage qubits and cavity QED circuits can be envisioned where characteristics of superconducting metallic weak links, Phys. Rev. B 42, 3992 (1990). a train of linearly and circularly polarized microwave [12] B. Abdo, E. Arbel-Segev, O. Shtempluck, and E. Buks, pulses can potentially control the timing of qubit initial- Observation of bifurcations and hysteresis in nonlinear NbN ization, computation, and measurement. superconducting microwave , IEEE Trans. Appl. 16 The author acknowledges the financial support of Supercond. , 1976 (2006). [13] B. G. Ghamsari and A. H. Majedi, Current-voltage charac- Natural Sciences and Engineering Research Council of teristics of superconductive heterostructure arrays, IEEE Canada (NSERC) through its Discovery Grant. The author Trans. Appl. Supercond. 19, 737 (2009). thanks Dr. Sheva Naahidi for her continuous support. [14] A. Jafari-Salim, A. Eftekharian, A. H. Majedi, and M. H. Ansari, Stimulated quantum phase slips from weak electro- magnetic radiations in superconducting nanowires, AIP * 6 Corresponding author. Adv. , 035209 (2016). [email protected] [15] A. Gurevich and V. M. Vinokur, Nonlinear Electrodynamics Also at Perimeter Institute for Theoretical Physics, of Randomly Inhomogeneous Superconductors, Phys. Rev. 83 Waterloo, Ontario, Canada N2L 2Y5. Lett. , 3037 (1999). [1] A. Lara, F. G. Aliev, A. V. Silhanek, and V. V. Moshchalkov, [16] S. K. Yip and J. A. Sauls, Nonlinear Meissner Effect in CuO 69 Microwave-stimulated superconductivity due to presence of Superconductors, Phys. Rev. Lett. , 2264 (1992). vortices, Sci. Rep. 5, 9187 (2015). [17] A. H. Majedi and B. Lounis, Nonlinear optics of optomag- 102 [2] J. You and F. Nori, Atomic physics and quantum optics netics: Quantum and classical treatments, Phys. Rev. B, , using superconducting circuits, Nature (London) 474, 589 214401 (2020). (2011). [18] L. Pitaevskii, Electric forces in a transparent dispersive 12 [3] S. Anlage, H. Snortland, and M. Beasley, A current medium, Sov. Phys. JETP , 1008 (1961). controlled variable delay superconducting transmission line, [19] M. J. Lancaster, Passive Microwave Device Applications of IEEE Trans. Magn. 25, 1388 (1989). High-Temperature Superconductors (Cambridge University [4] H. R. Mohebbi and A. H. Majedi, Periodic superconducting Press, Cambridge, England, 2006), chap. 1. microstrip line with nonlinear kinetic inductance, IEEE [20] J. Van der Ziel, P. S. Pershan, and L. Malmstrom, Optically- Trans. Appl. Supercond. 19, 930 (2009). Induced Magnetization Resulting from the Inverse Faraday 15 [5] B. Rossignol, T. Kloss, and X. Waintal, Role of Quasipar- Effect, Phys. Rev. Lett. , 190 (1965). ticles in an Electric Circuit with Josephson Junctions, Phys. [21] O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, Empirical Rev. Lett. 122, 207702 (2019). model of the microwave properties of high-temperature 46 [6] H. R. Mohebbi and A. H. Majedi, Analysis of series- superconductors, IEEE Trans. Microwave Theory Tech. , connected discrete josephson transmission line, IEEE Trans. 469 (1998). Microwave Theory Tech. 57, 1865 (2009). [22] B. E. Saleh and M. C. Teich, Fundamentals of Photonics [7] C. W. Robson, K. A. Fraser, and F. Biancalana, Giant (John Wiley & Sons, New York, 2019), chap. 6. ultrafast Kerr effect in superconductors, Phys. Rev. B 95, [23] M. Tinkham, Introduction to Superconductivity (Dover 214504 (2017). Publications Inc., 2004). [8] Y. Krupko, V. D. Nguyen, T. Weißl, É. Dumur, J. Puertas, R. [24] Y. B. Band, Light and Matter: , Optics, Dassonneville, C. Naud, F. W. Hekking, D. M. Basko, O. Spectroscopy and Lasers (John Wiley & Sons, New York, Buisson, N. Roch, and W. Hasch-Guichard, Kerr non- 2006), Vol. 1, chap. 2. linearity in a superconducting Josephson , [25] Y.-R. Shen, The Principles of Nonlinear ooptics (John Phys. Rev. B 98, 094516 (2018). Wiley and Sons, New York, 1984), chap. 4,5. [9] V. E. Manucharyan, E. Boaknin, M. Metcalfe, R. Vijay, I. [26] M. Battiato, G. Barbalinardo, and P. M. Oppeneer, Quantum 89 Siddiqi, and M. Devoret, Microwave bifurcation of a theory of the inverse Faraday effect, Phys. Rev. B , Josephson junction: Embedding-circuit requirements, Phys. 014413 (2014). Rev. B 76, 014524 (2007). [27] T. P. Orlando and K. A. Delin, Foundation of Applied [10] S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Superconductivity (Addison-Wesley, Reading, MA, 1991), Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, chap. 4,5.

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