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∗ (Dated: December 1, 2020) 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. 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 vortex state are predicted and quantitatively analyzed. Peculiar microwave birefringence in gyrotropic superconductors due to radical response of superelectrons has been highlighted. Introduction- Nonlinear microwave response of supercon- microwave-induced Hall effect, as an experimental tool to ducting structures is a core subject not only in probing measure superconducting gyration coefficient are analyti- the physics of superconductivity [1] but also implemented cally discussed. Gyroelectric conductivity is also used for in numerous applications ranging from quantum metrol- linear and circular birefringence for a microwave pump- ogy to superconducting qubits and microwave quantum probe scenario. Finally, I propose an embodiment of the optics [2]. The microwave nonlinearities in superconduct- dynamic and controllable flux quantization and vortex ing structure possess diverse origins but mostly are at- state in type II superconductors using circularly polar- tributed to nonlinear kinetic [3, 4] and Josephson induc- ized microwave field. tance [5, 6] including Kerr-type [7, 8], Duffing and anhar- Modeling of Inverse Faraday Effect in Superconductors- monicity [9, 10], weak links [11{13], phase slip formation The electrodynamic response of a superconductor is con- [14] and vortex dynamics [15, 16]. sidered in the Two-Fluid model. The equation of motion Borrowing from optomagnetics, a less-explored field in for superelectrons and normal electrons under the influ- nonlinear optics [17], a microwave-induced nonlinearity ence of electric field E(r; t) and its associated magnetic in superconductors based on the inverse Faraday effect field B(r; t) can be phenomenologically described by the (IFE) is proposed in this letter. IFE refers to the gen- London equations eration of static magnetic field by not linearly polarized, e.g. circularly polarized, light [18]. Purely nonlinear ef- d m vs = eE(r; t) + evs × B(r; t) (1) fect arising from IFE is solely based on the gyration of dt d the time-varying electric field and it does not directly link m hv i + mΓhv i = eE(r; t) + ehv i × B(r; t) (2) to any linear electromagnetic properties of the materials dt n n n such as Kerr-type that is related to the linear refrac- where m is the mass of an electron, e is charge of an elec- tive index. IFE in superconductor is based on angular 1 momentum transfer between the circularly polarized mi- tron, Γ = is the inverse of momentum relaxation time τ crowave field and superconductor that is considered as for normal electrons and vs and hvni are the superelec- superelectron condensate and normal electrons. Electric tron and average normal electron velocities, respectively. field gyration creates encircling supercurrent and normal We consider a circularly polarized plane electromagnetic current associated with local static magnetic field op- field with pumping frequency, !p, traveling normal to the posing the microwave field. IFE can be employed not surface of a semi-infinite superconductor, i.e. at z = 0, only to make tunable gyrotropic superconductor prevail- in the following form ing upon phenomena such as Hall effect and microwave n o birefringence but also to derive type II superconductors i!pt −αpz i(!pt−βpz) E= RefE~e g = Re Eo(x + iy)e e (3) to vortex state. n o i!pt −αpz i(!pt−βpz) In this letter I first develop a classical formalism to find B= RefB~e g = Re Bo(y − ix)e e (4) the microwave induced static magnetic field through a arXiv:2011.13983v1 [cond-mat.supr-con] 27 Nov 2020 gyration coefficient that is proportional to complex con- where x; y; z are Cartesian unit vectors, 0 < ~!p < 2∆, ductivity of superconductor. The critical microwave field ∆ being a superconducting energy gap, and αp and βp to suppress superconductivity is derived in terms of the are the propagation loss and the propagation constant, critical magnetic field. Turning to IFE consequences respectively. The low-frequency propagation characteris- in superconductors, the gyroelectric conductivity and tics can be derived based on the Two-Fluid model [19]. When the circularly polarized wave interacts with the en- semble of free superelectrons and normal electrons, its an- ∗ Also at Perimeter Institute for Theoretical Physics, Waterloo, gular momentum generates local circulating supercurrent Ontario, Canada. and normal current in the superconductor that produces 2 ind a magnetic field parallel to the microwave field oppos- response based on IFE and is shown by Js . The in- ing in the direction. This is a manifestation of the IFE duced supercurrent due to DC magnetization opposes to where high-intensity circularly polarized light generates the incident electric field to maintain the Meissner effect DC magnetization in matter, a theoretical prediction by and can be written as L. Pitaevskii in 1961 [18] and its subsequent experimen- tal demonstration in 1965 [20]. The magnetization in ind @f(x; y) @f(x; y) Js = r × MDC = MDC x − y (9) superconductor, Ms can be found as [17] @y @x nse nne Ms = Ls + Ln (5) where we consider the magnetization profile dictated by 2m 2m microwave radiation in the xy plane to be considered by where Ls = rs × ps is the magnetic moment of super- an arbitrary function f(x; y). The induced DC magnetic ind electrons and Ln = rn × hpni is the magnetic moment vector potential, A , can also be defined as of normal electrons in terms of their momenta ps and hp i, respectively. For intermediate temperature range, ind n r × A = µoMDC (10) 0 < T < Tc, both superelectrons with density number n (T ) and normal electrons with density number n (T ) s n At temperatures much lower than the critical tempera- coexist in the form of ture, the microwave field propagates in low-loss regime T s T s penetrating the superconductor sample and the strength n = ns(T ) + nn(T ) = n 1 − ( ) + n (6) Tc Tc of DC magnetization decreases exponentially. Although the strength of DC magnetization is more pronounced where s is an empirical exponent, i.e. s = 4 for −3 at lower frequency, i.e. MDC / ! , but lower the fre- low-temperature superconductors and s = 2 for high- quency leads to a larger radiation area in the order of λ2, temperature superconductors and n is the total number where λ is the free-space wavelength of the microwave density [21]. Considering the microwave signal given in field. Eqs. (3) and (4), the solutions of Eqs. (1) and (2) can In type I superconductor, a magnetic field is screened yield the DC magnetization in the superconductor as until the critical field Hc is reached. In the case of ∗ microwave-induced DC magnetization, the critical field MDC = iγ(!p;T )E~ × E~ (7) Hc is reached where the superposition of DC magnetiza- where the gyration coefficient γ(! ;T ) is Bo p tion and the applied microwave field amplitude Ho = µo 3 −e ns nn adds up to the critical field. Then there is a critical elec- γ(!p;T ) = 2 2 + 2 2 4m !p !p !p + Γ tric field, Ec, where the superconducting phase is ther- modynamically unstable, and that can be written as −e σ σ ≈ 2 + 1 (8) 4m! ! Γ p p 1=2 p 4 2 2 Eq. (7) is in the form of Pitaevskii's relationship [18] 1 + 16ηoγ (!; T )Hc − 1 indicating that the DC magnetization is solely depends Ec = p (11) on the gyration of the electric field, through the relation- 2 2γ(!; T )ηo ∗ 2 −2αpz ship of iE~ × E~ = 2zjEoj e . This fact is signified where ηo is the free space characteristic impedance. by the gyration coefficient, γ(!p;T ), reminiscent to mag- netogyration coefficients in magneto-optic materials [22]. The microwave-induced DC magnetization in supercon- Gyroelectric Conductivity and Hall Effect- The ductor is inherently a nonlinear electrodynamic process microwave-induced DC magnetization breaks the rooted on the IFE but the gyration coefficient can be ap- directional symmetry making a superconductor a gy- proximated in the low frequency regime, i.e. ! Γ, in rotropic material represented by a conductivity tensor, terms of the linear complex conductivity, σ1 − iσ2 based the so-called gyroelectric conductivity. Referring to on London equations [23]. Note that the magnetic field the London model, one can can find the gyroelectric associated with the microwave field has no contribution conductivity tensor, σ¯, relating the total current density to DC magnetization ruling out the direct magnetization J to the applied weak electric field E, i.e. J = σ¯E. ~ ~ ~ ~ of the superconductor. In fact, the linear response of Considering E = xEx + yEy + zEzz, with angular the superconductor due to the incident electromagnetic frequency !, the gyroelectric conductivity tensor can be field, i.e. equations (3) and (4), creates a time-dependent written as supercurrent and normal current according to Maxwell's 0 !c 1 and London equations, i.e. J~ = J~s + J~n = (σ1 − iσ2)E~. σ + σ −i σ + τ! σ 0 s n ! s c n For a circularly polarized electric field there is a time- B !c C σ¯(!s) = B i σ − τ! σ σ + σ 0 C dependent circular supercurrent having x and y compo- @ ! s c n s n A nents.