Laguerre-Gaussian light induction of orbital currents and Kapitza stabilization in superconducting circuits
Hennadii Yerzhakov Tien-Tien Yeh Alexander Balatsky




Nordita, Stockholm University, and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

Nordita, Stockholm University, and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

Nordita, Stockholm University, and KTH Royal Institute of Technology, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA




 
  
 We investigate the effects of a Laguerre-Gaussian (LG) beam on the superconducting state. We show that the vortex angular momentum of a LG beam  affects the superconducting state and induces currents.  The induction of the current by light is illustrated on a Josephson loop and SQUID devices. 
 In particular, we establish that coupling a dc SQUID to the AC magnetic flux of a LG beam can stabilize π phase in the SQUID. This can happen via developing a global or local minimum in the effective potential at π. In the latter case, this happens via the Kapitza mechanism.


Laguerre-Gaussian light induction of orbital currents and Kapitza stabilization in superconducting circuits 
    Alexander Balatsky
    April 29, 2024
============================================================================================================





§ INTRODUCTION




The process of interaction of quantum fields where one transfers quantum numbers from one to another is a well known phenomenon in quantum mechanics. We focus on the specific case of transfer of quantum numbers of light to coherent electron quantum state.  Examples of these transfer processes include energy, momentum, angular momentum transfers as illustrated, e.g., by light-induced momentum change of a particle and induction of magnetization due to circular polarization of the photons <cit.>
. Recently, it has been suggested that interaction of light carrying non-zero angular momentum can induce vortices in superconductors <cit.>

In this paper, we look at a newer example of the interaction of quantum matter with structured light and microwave radiation: the induction of the circular orbital  motion of the superconducting electrons illuminated by the so-called Laguerre-Gaussian (LG) beam, see <ref>. A LG beam has the vortex in light field phase <cit.>. The assumption  hence is that even in linear response the phase coherent superconducting (SC) electrons will develop a coherent response while interacting  with the beam that carries vorticity. We indeed  find that LG beams do induce SC currents.  One can say that vorticity of the beam is imprinted on the coherent quantum fluid, in our case chosen to be SC fluid.   Qualitative explanation of the effect is direct: to carry orbital momentum, an electromagnetic (EM) wave propagating in z-direction must have its amplitude of the electric and magnetic fields to be dependent on (x,y) coordinates, or in polar angle θ in case of LG beam, on which we concentrate. This means that z-component of the magnetic field of the beam is non-zero. This results in non-zero flux through an area normal to the beam propagation, <ref>, which oscillates with the frequency of the LG beam.



The effect  is expected to be present  in broader contexts;  here we choose to illustrate it on the SQUID - the SC device that contain loop currents that are interacting with the light. 

To test these ideas  one  can use sensing of the induced flux by SQUID devices via Aharonov-Bohm like setup. We show that this can lead to the dynamically stabilized effective π Josephson junction. While the effect we discuss has similarities with the Aharonov-Bohm effect, there is a key difference – in the light-induced flux, the light field needs to enter into a SC and hence plays the role of the physical field that is sampled by SC current. In the traditional Aharonov-Bohm effect, the magnetic field Ḇ is nonzero inside the solenoid and does not enter into the superconductor. Hence, for the setups we consider, it is important that the light field directly “touches" the superconducting electron liquid. 


The structure of the paper is as follows:  we give a brief introduction into a LG beam in <ref>.  In <ref>, we start  with the simpler example of a magnetic field and flux induced by a LG beam in the SC loop. In <ref>, we
expand the calculation to the dc SQUID geometry. In <ref>, we consider the full nonlinear dynamics of the coupled light-SQUID setup and find that, with the proper tuning, one can induce π Josephson phase shift as a stable point of the junction potential. We also  elaborate on the light-induced Kapitza engineered π junction. We conclude in <ref>.



§ MAGNETIC FLUX GENERATED BY LG BEAMS



To the zeroth order in the paraxial parameter, the electric field of a linearly polarized LG higher mode with a wavevector ḵ and wavelength λ propagating in z-direction can be considered purely transverse and written as <cit.>


    E̱(ṟ) = E_0(r,z) e^i α(r,z) e^i(k z - ω t) e^i ℓθx̱̂̂,

where ṟ=(r,θ,z) is the radius-vector written in cylindrical coordinates, x̱̂̂ is a unit vector in x-direction, and

    E_0(r, z) =    E_0 w_0/w(z)(r √(2)/w(z))^|l| e^-r^2/w^2(z) L_p^|l|(2 r^2/w^2(z)), 
    α(r,z) =     - k r^2/2 R(z) + ψ(z),

where w(z)=w_0 √(1+(z/z_R)^2) is the spot size at z, where z_R=π w_0^2 n/λ is the Rayleigh range with n being the refraction index of the medium, R(z)=z[1+(z_R/z)^2] is the radius of curvature of the wavefront of the Gaussian beam at z, ψ(z)=arctan(z/z_R) is the Gouy phase, and ℓ is an integer corresponding to the order of the LG mode. In fact, z-component of the electric field is not zero: From ∇·E̱ = 0, it can be estimated that E_z(r,z) ∼λ/w(z) E_x, which is small compared to E_x in the paraxial approximation.

where ṟ=(r,θ,z) is the radius-vector written in cylindrical coordinates, E̱_̱0̱(r,z) and α(r,z) are real vector and scalar functions of r and z, respectively, ℓ is an integer corresponding to the order of LG mode, and we use the convention of not writing the operation of taking real part explicitly.


Correspondingly, the vector potential is the real part of


    A̱(ṟ) = E_0(r,z)/i ω e^i α(r,z) e^i(k z - ω t) e^i ℓθx̱̂̂,

and the flux through a disk-area of radius r_0 normal to the beam propagation is 


    Φ_e = ∮A̱· dḻ = E_0(r_0,z)/ω e^i α(r_0,z) e^i(k z - ω t)π  r_0 δ_ℓ,±1.

This simple observation implies the possibility of generating high-frequency oscillating electromotive force with a LG beam via Faraday effect. For modern lasers, such flux is rather small -  on the order 10^-13-10^-12 Wb, which is only 10-10^2 times larger than the magnetic flux quantum, Φ_0. However, the Josephson phase is sensitive to the magnetic flux which is on the order of Φ_0, and in the next sections we study the interaction of such a flux from LG beams with superconducting rings with Josephson junctions (JJ).

We note that the flux is contour dependent, and specifically chosen contours for other type of beams also can contain non-zero flux. However, a circular-shaped contours look particularly convenient to us.








§ A LG BEAM SHONE ON A RF SQUID



We start with shining LG light into the orifice of an rf SQUID structure, see <ref>. In the following, we neglect the physics that can lead to Shapiro effect <cit.> due to the electric component of the EM wave and photon-assisted tunneling <cit.>. This is a reasonable approximation if the JJs are positioned in such a way that the polarization of the electric field is perpendicular to the Josephson current through the junctions (i.e., it is parallel to the surfaces of the electrodes that constitute the junctions) and the power of the beam is sufficiently small.



The RCSJ model for the dynamics of the rf SQUID then reads <cit.>

    I    =C d^2 Φ/d t^2+1/Rd Φ/d t+I_0 sin(2 πΦ/Φ_0), 
    Φ   = Φ_e - LI,

where Φ is the full flux through the orifice of the SQUID, C, R, and L are the capacitance, resistance, and self-inductance of the rf SQUID, respectively, I_0 is the Josephson critical current, I is the full current in the loop, and Φ_0 is the magnetic flux quantum. Combined, eqs. in <ref> give

    ϕ̈+ Γϕ̇+ ω_0^2/2πsin (2 πϕ)  = ω_L^2 ϕ_e - ω_L^2 ϕ,

where ϕ = Φ/Φ_0, ϕ_e = Φ_e/Φ_0, ω_0^2= 2 π I_0/Φ_0 C, ω_L^2 = 1/LC, and Γ = 1/RC. This model with oscillating in time ϕ_e studied in <cit.> exhibits complex dynamic behavior including chaos.






§ A LG BEAM IS SHONE INSIDE A DC SQUID SHUNTED BY AN EXTERNAL CAPACITOR



We can couple ϕ_e(t) to the dynamic variable ϕ in the dc SQUID setup.
Let us consider a symmetric dc SQUID shunted by an external capacitor as depicted in <ref>. We start with the simplest model neglecting the self-inductances of the electrodes comprising the dc SQUID and the capacitances and resistances of the JJs. In this case, the dc SQUID behaves effectively like a JJ with the amplitude of the Josephson critical current modulated by the external flux through the dc SQUID <cit.>.



Indeed, the flux through the SQUID is

    ϕ_1-ϕ_2=2 πϕ_e - 2 π/Φ_0 2L I_c,

where ϕ_1,2 are phase differences across the JJs, ϕ_e=Φ_e/Φ_0, and I_c is the current circulating in the loop of the dc SQUID. Neglecting 2LI_c term in <ref>, the interference of the currents I_1 and I_2 through JJs gives

    I=I_1+I_2=2 I_0 cos (πϕ_e) sinϕ_+,

where ϕ_+=ϕ_1+ϕ_2/2.

Ultimately, we can think about the dc SQUID as an effective JJ with flux modulated critical current I_0,eff(ϕ_e)=2 I_0 cos (πϕ_e). Under the assumptions, the voltage across the SQUID is

    V=Φ_0/2πϕ̇_+.

Combinination of <ref>, and I+Q̇ =I_b, where Q is the charge on the shunting capacitor and I_b is the bias current (which we do not show in <ref>) assumed to be constant, leads to the equation for ϕ_+ analogous to the RCSJ model

    ϕ̈_+ + Γϕ̇_+ + ω^2(ϕ_e) sinϕ_+ = ω_0^2 I_b/2 I_0,

where we have also added a dissipative term Γϕ̇ and ω^2(ϕ_e) = 2 π I_0,eff(ϕ_e)/C Φ_0 = 2 ω_0^2 cos (πϕ_e), where ω_0^2=2 π I_0/C Φ_0. 

We assume now, according to <ref>, the time-dependence for the external flux is cosinusoidal, such that ϕ_e(t) = ϕ_0 cos(ω t) with ϕ_0 = maxΦ_e/Φ_0, and I_b=0. 
Using the Jacobi–Anger expansion <cit.> 

    cos (x cosα) = J_0(x) + 2 ∑_p=1^∞ (-1)^p J_2p(x) cos(2 p α),

we rewrite <ref> as

    ϕ̈+ Γϕ̇+ 2 ω_0^2 J_0(ϕ_0) sinϕ   = 
    
        -4 ω_0^2 ∑_p=1^∞ (-1)^p J_2p(ϕ_0) cos(2 p ω t) sinϕ,

which we recognize as an equation of an oscillator in a fast oscillating field. Then, for sufficiently large ω, we can expect the Kapitza-pendulum scenario <cit.>, i.e., the stabilization upon drive  of the formerly unstable state. We investigate such a possibility in the next section, where, based on the simple idea layed out in this section, we systematically study Kapitza-pendulum scenario in the dc SQUID geometry without shunting the SQUID by an external capacitor. The case with an external capacitor without neglecting the JJ's capacitances and inductances is briefly discussed in <ref>.

To investigate this we employ the idea of time-scale separation <cit.>: given ω^2/ω_0^2 J(ϕ_0)≫ 1, we expect the ϕ(t)-vs-t curve can be described as a slowly varying with time (on the scale of (ω_0 J(ϕ_0)^1/2)^-1) curve with small-amplitude fast oscillations (on the scale of ω^-1) on top of it, i.e., we represent

    ϕ=θ + ψ,

where θ and ψ are the slow and fast parts, respectively. Substituting this in <ref>, expanding in ψ, and collecting fast terms together, we obtain an equation for ψ



    ψ̈+ Γψ̇+ ω_0^2 J_0(ϕ_0) cosθ·ψ =
     -2 ω_0^2 ∑_p=1^∞ (-1)^p J_2p(ϕ_0) cos(2 p ω t) sinθ.


For simplicity, let us neglect the dissipative term, which we may do not only because Γ≪ 1 but also because for large ω the second derivative dominates. Then


    ψ(t) = 2 ω_0^2/ω^2∑_p=1^∞ (-1)^p J_2p(ϕ_0) cos(2 p ω t) sinθ +At +B,

in which we set A=B=0 to comply with the physical picture of the solution.

For slow terms, after averaging over time, we have


    θ̈+ Γθ̇+ ω_0^2 J_0(ϕ_0) sinθ = 
    
        -2 ω_0^2 ∑_p=1^∞ (-1)^p J_2p(ϕ_0) cos(2 p ω t) cosθ·ψ(t)
     
        = -2 ω_0^4/ω^2∑_p=1^∞ (-1)^p J_2p(ϕ_0) cos(2 p ω t) sinθcosθ,

where overbar stands for time-averaging.
When J_0(ϕ_0)>0, <ref> is an oscillation equation for a mathematical pendulum in an effective potential


    U_eff=ω_0 J_0(ϕ_0) ( -cosθ + ω_0^2/ω^2∑_p=1^∞J_2p^2(ϕ_0)/J_0(ϕ_0)sin^2 θ).


This potential develops a local minimum at θ = π when the coefficient in front of sin^2 θ in <ref> is greater than 1/2:


    ω_0^2/ω^2∑_p=1^∞J_2p^2(ϕ_0)/J_0(ϕ_0) > 1/2.


It is easy to see that such a condition can be satisfied for ϕ_0 sufficiently close to the roots of zero-order Bessel function of the first kind. Simultaneously, the natural frequency of the oscillator in <ref> becomes small making assumption ω/ω_0≫ 1 less strict, i.e., we rather have to require ω/J_0(ϕ_0)ω_0≫ 1. We plot regions in ϕ-ω plane for which condition <ref> is satisfied in <ref>.

oscillation equation for a mathematical pendulum

Obtained analytically under certain assumptions, criterion <ref> serves only as a guide. The results of numerical investigation (see <ref>) shows that the actual regions for which the condition is satisfied are somewhat larger. *** Give a figure and explanation of types of pendulum ???






§ DC SQUID



Based on the ideas in the previous section, we investigate the possibility of the Kapitza scenario in a dc SQUID subjected to the flux from a LG beam without it being shunted by an external capacitor. A vast literature is devoted to the study of dynamics in dc SQUIDs <cit.>. Kapitza engineering <cit.> of a SQUID has not been widely discussed to date. 






Consider an equivalent scheme of a symmetric dc SQUID  depicted in <ref>. The relation for the flux and phases reads


    Φ_0/2 π( ϕ_1-ϕ_2 )=Φ_e - 2 L I_c,

where I_c=(I_1-I_2)/2 is the current circulating in the loop of the dc SQUID. For the time-dependent ϕ_e, this equation assumes instantaneous adjustment of the phases ϕ_1,2. This is approximately true if the time-scale on which ϕ_e changes is significantly slower than the relevant time-scale for the superconductor dynamics, τ_r ∼ħ/Δ, where ħ and Δ are the Plank constant and the superconducting order parameter, respectively.

The set of Kirhoff's laws for currents reads


    I_0 sinϕ_1 + Φ_0/2 π Rϕ̇_1 + C Φ_0/2 πϕ̈_1 = I_1, 
    
        I_0 sinϕ_2 + Φ_0/2 π Rϕ̇_2 + C Φ_0/2 πϕ̈_2 = I_2, 
    
        I_1+I_2 = I_b,

where I_b is the bias current assumed to be constant.
And the voltage across the dc SQUID is given by (see <ref> for an example of the derivation)


    V=Φ_0/2 πϕ̇_1+ ϕ̇_2/2 - L/2İ_b.


Taking sum and difference of the first two equations in <ref> and introducing notation ϕ_± = ϕ_1 ±ϕ_2/2, ω_0^2=2 π I_0/Φ_0 C, Γ = 1/CR, and ω_L^2 = 1/LC, we obtain


    ϕ̈_+ + Γϕ̇_+ + ω_0^2 cos(πϕ) sinϕ_+ = ω_0^2 I_b/2 I_0, 
       ϕ̈+ Γϕ̇+ ω_0^2/πcosϕ_+ sin (πϕ)  = ω_L^2 ϕ_e - ω_L^2 ϕ,

where ϕ = Φ/Φ_0 is the full flux in units of Φ_0 and relates to ϕ_- via <ref>.

Note that the second equation in <ref> is an equation for the flux inside the SQUID and is essentially the same as <ref> for the dynamics of the flux inside an rf SQUID.

We can deem the considered system as a black-box device and ask a question about its I-V characteristic. Under typical assumption that I_b is created by an external source of constant current, the output voltage, according to <ref>, is V=Φ_0/2 πϕ̇_+. Thus, in the Kapitza scenario of inverted pendulum, where ϕ_+ oscillate around π, such a black-box device may be considered as an effective π-phase Josephson junction.

For L=0, two equations in <ref> decouple. We start with this case in the following section.


    2 I_0 cos(Φ_e + LI_c) sinϕ_+ +  2 C Φ_0/2 πϕ̈_+ = I_b,


    2 I_0 cosϕ_+ sin (Φ_e + LI_c) +  2 C Φ_0/2 π( Φ̈_e + 2 L Ï_c ) = 2 I_c,





 §.§ Neglecting the self-inductance: L=0


When L=0, ϕ is simply given by the external flux: ϕ = ϕ_e. For generality, we will include a dc component of the external flux


    ϕ_e=ϕ_1 + ϕ_0 cos(ω t).

Substituting this into the first equation in <ref> and using the Jacobi-Anger expansion, we obtain


    ϕ̈_+    + Γϕ̇_+ + ω_0^2 cosπϕ_1 J_0(πϕ_0) sinϕ_+ = 
       -2 ω_0^2 ∑_p=1^∞ J_p(πϕ_0) cos(πϕ_1+π/2 p) cos(p ω t) sinϕ_+,

where we also set I_b=0. Non-zero I_b cases then can be taken into account in the context of the washboard potential.

To investigate this, we employ the idea of separation of time scales <cit.>: given ω/ω_0 √(J_0(ϕ_0))≫ 1, we expect the  solution ϕ_+(t) can be described as a slowly varying component (on the scale of (ω_0 √(J(ϕ_0)))^-1) with small-amplitude fast oscillations (on the scale of ω^-1) on top of it [An analytical method of wider applicability can be used <cit.>], i.e., we represent

    ϕ_+=θ + ψ,

where θ and ψ are the slow and fast components, respectively. Substituting this in <ref>, expanding in ψ, and collecting the dominant fast terms together, we obtain an equation for ψ



    ψ̈+ Γψ̇=
     -2 ω_0^2 ∑_p=1^∞ J_p(πϕ_0) cos(πϕ_1+π/2 p) cos(p ω t) sinθ,

where we kept the dissipative term in <ref> to incorporate the overdamped regime, Γ≫ω_0 √(J(ϕ_0)). The solution for the fast component is 


    ψ(t) =    ∑_p=1^∞2 ω_0^2 J_p(πϕ_0) cos(πϕ_1 + π/2p)/√(p^4 ω^4 +Γ^2 p^2 ω^2)cos(p ω t +α_p) sinθ
       + A e^-Γ t +B,

where the phase shifts α_p satisfy tanα_p = Γ/(p ω). We set A=B=0 to comply with the physical picture of the solution (small-amplitude fast oscillations about the slow-mode component).

For the slow term, after averaging over time, we have


    θ̈+ Γθ̇+ ω_0^2 cosπϕ_1 J_0(πϕ_0) sinθ = 
     
        -2∑_p=1^∞ω_0^4 J_p^2(πϕ_0) cos^2(πϕ_1 + π/2p)/√(p^4 ω^4 +Γ^2 p^2 ω^2)sinθcosθ,


When cosϕ_1 J_0(πϕ_0)>0, <ref> is an equation of motion for a mathematical pendulum in an effective potential


    U_eff=ω_0^2 cosπϕ_1 J_0(πϕ_0) ( -cosθ + K sin^2 θ),

where

    K=ω_0^2/ω^2∑_p=1^∞J_p^2(πϕ_0) cos^2(πϕ_1 + π/2p)/p^2√(1 +Γ^2/ (p^2 ω^2)) J_0(πϕ_0) cosπϕ_1.



This potential develops a local minimum at θ = π when K>1/2. It is easy to see that such a condition can be satisfied for πϕ_0 sufficiently close to the roots of the zero-order Bessel function of the first kind or/and for πϕ_1 such that cosπϕ_1 is sufficiently small. Simultaneously, the natural frequency of the oscillator in <ref> becomes small, which allows for the Kapitza effect to occur at frequencies ω comparable to ω_0, the plasma frequency of the Josephson junction.


When cosπϕ_1 J_0(πϕ_0)<0, the coefficient in the third term in <ref> becomes negative. This sign can be absorbed by shifting θ by π in <ref>: θ→θ + π. Correspondingly, the effective potential shifts by π as well, moving the global minimum to θ=π. We call this regime the inverted pendulum regime. The Kapitza effect, i.e., the stabilization of the oscillation around the emergent local minimum, then is realized for K<-1/2. We call this regime the inverted Kapitza pendulum regime. Setting πϕ_1=0, we illustrate these regimes in <ref>, where we plot K-vs-πϕ_0 dependence at ω=ω_0.





Found analytically under the assumptions explained above, <ref> is not exact and does not allow to determine the smallest values of ω at which the Kapitza effect still occur. In <ref>, we plot a curve in πϕ_0-ω plane at πϕ_1=0 at which K=1/2 and numerically determined region, where the Kapitza effect takes place
. We see that the theoretical curve fit the numerically determined boundaries of Kapitza region quite well for sufficiently large ω. In <ref>, we plot K=1/2 curves in πϕ_0-πϕ_1 plane for four values of ω: ω/ω_0=1,2,3,10.










 §.§ Relevant parameters of the LG beam


The model we have considered above have certain bounds of applicability. 
As follows from the discussion after <ref>, the frequency of ϕ_e should satisfy ω≪Δ/ħ.

Considering typical T_c in high-temperature superconductors being about 100 K, the frequency of the LG beam then has to satisfy ω≪ 10^12 Hz, i.e., ω has to lie in the microwave range. Correspondingly, the wavelength of the LG beam is at least on the order of 10^-3 m. Consequently, the waist of the beam is also at least 10^-3 m. This would likely imply that in a realistic experiment the paraxial parameter f=λ/2 π w_0 is not small, and the formulas in <ref> are not quantitatively correct. Nevertheless, the value of the transverse component remains of the same order as the paraxial approximation predicts <cit.>. Thus, since the total flux through the ring is determined by the transverse component of the vector potential, we proceed with the estimation of the flux through the ring at z=0 (i.e., at the beam's waist) using the electric field given by <ref>


    πϕ_0= E_0 w_0/ωΦ_0 f(r),

where f(r)=√(2)π^2 ( r/w_0)^2 e^-r^2/w_0^2, which has a maximum at r/w_0 = 1.

According to <ref>, the smallest values of flux at which the  Kapitza effect is observed is around πϕ_0 ≈ 2.405, the first root of J_0(πϕ_0)=0. And thus, from <ref>, we find that for r/w_0=1 this value is achieved at E_0 w_0 ≈ 10^-15 [V·s] ·ω, which for ω = 10^10 Hz becomes E_0 w_0 ≈ 10^-5 V.


In the paraxial approximation, the total power of the ℓ=1 LG mode is given by


    P=π (E_0 w_0)^2/4 μ_0 c,

which is the same formula as for the power of the fundamental Gaussian mode.

The power output of contemporary continuous-wave masers is much lower than that of lasers, typically on the order of picoWatts. For P ∼ 10^-12 W, from <ref>, we obtain E_0 w_0 ∼ 2 · 10^-5 V, which is enough to achieve the desirable amount of flux. In case of a maser based on nitrogen vacancies in diamond, which produces one of the highest output power of P ≈ 2.2 · 10^-9 W <cit.>, we estimate
E_0 w_0 ≈ 2 · 10^-3 V.





 §.§ The effect of self-inductance: L ≠ 0


When L ≠ 0, two equations in <ref> are coupled. Here we discuss the results of numerical investigation of this set of equations at non-zero L.

If L is non-negligible, then the screening effect from the current inside the closed-loop SQUID circuit affects the dynamics of the effective phase ϕ_+. We utilize the Crank–Nicolson method to solve for the time trace of the phase ϕ_+ and flux ϕ in <ref>. Adopting the screening parameter β_L = 2L I_0/Φ_0, we plot the phase diagram in terms of the normal, Kapitza (encompassing both the Kapitza and the inverted Kapitza), and inverted pendulum regimes in <ref> for ω/ω_0=1 and ω/ω_0=2, respectively, without distinguishing between the Kapitza and the inverted Kapitza regimes.

For both ratios of ω/ω_0 after small finite value of πϕ_0, the phase diagrams display ray-like structures of normal and inverted regimes with origins on the horizontal line at finite β_L. These “rays" are separated by the “rays" with a mix of the Kapitza and “Other" regimes (see <ref>). Here, it is possible that some of the inclusions (especially in the lower part of the plots) of regions of “Other" regimes is due to instabilities in our numerical solutions. For πϕ_0 < πϕ_0^* (≈ 0.36 for ω/ω_0=1 and ≈ 0.31 for ω/ω_0=2), there is only normal pendulum regime of the dynamics of ϕ_+.













§ CONCLUSION AND DISCUSSION



In this study we present the results for stirring of the circular motion of the superconducting fluid with structured light, a LG beam. In particular, we find Laguerre-Gaussian beams with ℓ=1 provide non-zero AC magnetic flux through a disk-area of finite radius. This flux can be used to induce currents in superconducting rings and superconducting rings with Josephson junctions.

In the dc SQUID geometry, a SC ring with two JJ, we have found that coupling of periodic flux might lead to the Kapitza effect – the stabilization of oscillations about emergent local minimum at phase equal to π. In the limit of negligible self-inductance, the ranges of the flux amplitude for which Kapitza effect 
is found are relatively narrow with the widest being the first one from zero flux. Numerical investigations suggest that these intervals for the Kapitza effect shift to smaller values of fluxes, and the width of such intervals can even enlarge (see <ref>). Unlike the case of the well-known mechanical oscillator, within the time-scales separation approach, the effect can be observed for driving frequencies that are on the order of the natural frequency, ω∼ω_0.

It is known that inverted regime of the oscillations of a dc SQUID, in which the global minimum shifts by π, is achieved by biasing the SQUID with proper amount of DC flux. Here we have shown that analogical regime can be achieved with purely AC magnetic flux. 

Our estimations suggest that the frequency of a LG beam should be in the microwave regime for the employed model to be correct. However, the effect might survive beyond the applicability of the model used. While the power output of current masers is much lower compared to that of lasers, we estimate that it is enough to produce the Kapitza effect. On the other hand, the model of the dynamics does not rely on the particular source of the AC magnetic flux, and the predicted dynamics can be tested in the lab with conventional electronics. The possibility of utilizing the Kapitza and inverted regime of a dc SQUID in applications, e.g., sensing, is not fully clear to us at this point.

The appearance of the local minimum in the effective potential allows to speculate about the possibility of utilizing it for constructing a novel Floquet-type qubit <cit.> based on Josephson junctions.

Based on a simple idea similar to the one explained in <ref>, we expect the possibility of stabilizing the effective phase at ϕ_+ ≠ 0,π in an asymmetric dc SQUID. We will investigate such a possibility in a follow-up paper.




§ ACKNOWLEDGMENTS

We are grateful to A. Dalal, G. Jotzu, I. Khaymovich, D. Kuzmanovski, I. Sochnikov, and P. Volkov for useful discussions. 
This work was supported by the European Research Council ERC
HERO-810451 synergy grant, KAW foundation 2019-0068 and the Swedish Research Council
(VR). Work at University of Connecticut was supported by the OVPR  Quantum CT. 






§ A DC SQUID SHUNTED BY AN EXTERNAL CAPACITOR



In this section, we formulate the equations that govern the dynamics of a symmetric dc SQUID shunted by an external capacitor. The equivalent scheme is depicted in <ref>. The flux equation is 

    ϕ_1-ϕ_2 = 2 π/Φ_0( Φ_e(t) - 2 L_J I_c ),

where I_c=( I_1 - I_2 )/2 is the current circulating in the dc SQUID loop. The Kirchhoff's laws take the form

    I_0 sinϕ_1 + C_J Φ_0/2 πϕ̈_1 + Φ_0/2 π R_Jϕ̇_1 =    I_1, 
        
        I_0 sinϕ_2 + C_J Φ_0/2 πϕ̈_2 + Φ_0/2 π R_Jϕ̇_2 =    I_2, 
        
        I_1 + I_2 =    I_3, 
    
        I_3 + Q̇ =    I_b,

where ϕ_1,2 are the Josephson phases of the JJs J_1,2, I_1,2 are currents through the JJs J_1,2, I_3 is the total current through the dc SQUID, I_b is the bias current assumed to be constant, I_0, C_J, and R_J are the critical current, capacitance, and the resistance of JJs, L_J is the self-inductance of the parts with JJs of the circuit between points A and B (see <ref>), and Q = C ( V_2 -V_1 ) is the charge of the external capacitor C, where V_1,2 are the voltages at points 1,2 (see <ref>).



To find the expression for V_1 - V_2, let us consider two paths connecting points 1 and 2: 1-A-J_1-B-2 and 1-A-J_2-B-2. Then for V_2 we can write


    V_2 = V_1 + Φ_0/2 πϕ̇_1 - L_J İ_1 - ϕ̇_e/2, 
    
        V_2 = V_1 + Φ_0/2 πϕ̇_2 - L_J İ_2 + ϕ̇_e/2.

From which we find

    V_2-V_1 = Φ_0/2 πϕ̇_1 + ϕ̇_2/2 -L_J/2İ_3.

Using ϕ_± = ϕ_1 ±ϕ_2/2 as before, summing and taking the difference of <ref>, and adding to them <ref>, we get the following system of equations:


    I_3 + C Φ_0/2 πϕ̈_+ - C L_J/2Ï_3 = I_b, 
    
        2I_0 cosπϕsinϕ_+ + C_J Φ_0/πϕ̈_+ + Φ_0/π R_Jϕ̇_+ = I_3, 
    
        2I_0 cosϕ_+ sinπϕ + C_J Φ_0 ϕ̈+ Φ_0/R_Jϕ̇= Φ_0/L_J( ϕ_e - ϕ).


Substituting I_3 from <ref> in <ref>, we obtain a fourth order differential equation for ϕ_+. This prohibits direct analysis of this system in terms of an effective potential. However, the idea of separation of time scales remains valid.





