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 magnetization 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, Hall effect, 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 superconductivity [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 magnetic field 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 inductance [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 electric field 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) electromagnetic field 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 Meissner effect and can be written as
L r p where s ¼ s × s is the magnetic moment 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