Single-Defect-Induced Peculiarities in Inverse Faraday-Based Switching of Superconducting Current-Carrying States near a Critical Temperature

Abstract

The Inverse Faraday Effect (IFE) is a phenomenon that enables non-thermal magnetization in various types of materials through the interaction with circularly polarized light. This study investigates the impact of single defects on the ability of circularly polarized radiation to switch between distinct superconducting current states, when the magnetic flux through a superconducting ring equals half the quantum flux, ${\mathsf{\Phi }}_0/2$. Using both analytical methods within the standard Ginzburg–Landau theory and numerical simulations based on the stochastic time-dependent Ginzburg–Landau approach, we demonstrate that while circularly polarized light can effectively switch between current-carrying superconducting states, the presence of a single defect significantly affects this switching mechanism. We establish critical temperature conditions above which the switching effect completely disappears, offering insights into the limitations imposed by a single defect on the dynamics of light-induced IFE-based magnetization in superconductors.

1. Introduction

The Inverse Faraday Effect (IFE) is a phenomenon in which circularly polarized light induces non-thermal magnetization in a material [1,2]. This effect arises from the transfer of the light’s angular momentum to the magnetic/electronic subsystem of the material. The IFE has been experimentally observed in a variety of materials, particularly those with magnetic properties, and it has gained attention as a tool for ultrafast magnetization control using light [3,4,5,6]. It was experimentally demonstrated that illumination of the toroidal atomic Bose–Einstein condensate, an artificial superfluid system, by twisted light carrying a nonzero angular momentum produces DC persistent supercurrents [7,8]. The IFE has been documented in plasmonic systems, especially in gold (Au) nanoparticles [9,10,11] and heterostructures [12,13,14]. These results support a coherent mechanism for the transfer of angular momentum from the optical field to the electron gas [15,16,17]. Consequently, the IFE is a general effect that should be observed in metals [18,19] as well as in superconductors [20,21,22,23,24,25,26]. However, despite theoretical analyzes suggesting the potential for the IFE to function as an external magnetic field [15,16,17] and to generate current states in superconductors [20,21,22,23,24,25], experimental evidence of its manifestation in such materials remains scarce.

Recent studies [25,27] have explored the potential of circularly polarized light to switch between superconducting current states. This is particularly intriguing when the magnetic flux $\mathsf{\Phi }$ through a superconducting ring equals half the quantum flux, ${\mathsf{\Phi }}_0/2$. According to the Little–Parks effect [28], the critical temperature of the ring oscillates with the magnetic flux, exhibiting a period of ${\mathsf{\Phi }}_0$. At ${\mathsf{\Phi }}_0/2$, the ground state of the ring becomes degenerate, allowing it to exist in two superconducting states with different angular momenta, $n=0$ and $n=1$, with opposing current directions. Remarkably, numerical simulations based on the stochastic time-dependent Ginzburg–Landau (sTDGL) theory have demonstrated that circularly polarized radiation pulses can switch between the $n=0$ and $n=1$ states with nearly 100% probability [25,27]. However, the presence of a single defect [29,30,31,32] may exert a significant and detrimental influence on this switching mechanism [33,34]. As shown in Ref. [34], the extent of this influence depends on the strength of the defect and the temperature of the system.

This paper aims to investigate the effects of a single defect on the ability of circularly polarized radiation to distinguish between the two superconducting states in close proximity to the critical temperature. Additionally, we establish the critical temperature conditions above which the switching effect becomes negligible, thereby contributing to a deeper understanding of the limitations imposed by single defects on light-induced IFE-based magnetization dynamics in superconductors.

2. Model

We analyze the behavior of a thin superconducting ring with a radius R placed on a substrate at a fixed temperature $T_0<T_{c0}$, where $T_{c0}$ is the zero-field critical temperature $T_{c0}$ [35]. The ring’s thickness and width are much smaller than the temperature-dependent superconducting coherence length $\xi \left(T_0\right)$, effectively reducing the problem to one dimension, where the order parameter depends only on the curvilinear coordinate $s=R\theta$. We examine a scenario where local temperature fluctuations arise due to non-uniform thermal contact with the substrate. Specifically, a small segment of the ring, $-s_0/2<s<s_0/2$, experiences a different local temperature $T_{\mathrm{local}}$, which may be higher or lower than $T_0$. Thus, the temperature distribution along the ring can be written as follows:

$$T\left(s\right)=T_0+s_0\delta T\delta \left(s\right),$$

(1)

where $\delta T=\left(T_{\mathrm{local}}-T_0\right)$, and we assume $s_0\ll \xi \left(T\right)$. Here, is the temperature-dependent coherence length given by $\xi \left(T\right)={\xi }_0\sqrt{T_{c0}/(T_{c0}-T)}$ with ${\xi }_0$ being the coherence length at zero temperature. The local temperature variation, a single defect, is modeled by a Dirac delta function $\delta \left(s\right)$ function. In the Ginzburg-Landau framework, the reduced temperature parameter is defined as $ϵ=\left(T_{c0}-T\right)/T_{c0}$. For our system, this becomes:

$$ϵ={\displaystyle \frac{T_{c0}-T_0-s_0\delta T\delta \left(s\right)}{T_{c0}}}={\displaystyle \frac{\left(T_{c0}-s_0\delta T\delta \left(s\right)\right)-T_0}{T_{c0}}}$$

(2)

This is equivalent to the case of non-homogeneous critical temperature, such that $T_{c0}\to T_{c0}+$$\delta T_{c0}{s}_0\delta \left(s\right)$, where $\delta T_{c0}=-\delta T$. Therefore, the model also describes scenarios where the overall temperature is uniform, but local variations in the critical temperature occur due to fluctuations in the ring’s parameters. A localized increase in critical temperature can lead to superconducting states even at temperatures above the bulk critical temperature, leading, for instance, to twinning plane superconductivity in certain materials [36,37]. In our setup, the local defect can modify the ring’s critical temperature, potentially inducing a non-uniform superconducting order parameter profile, which influences the generation of orbital modes of the order parameter and affects transitions between current-carrying states.

The superconducting ring is also exposed to a Gaussian radiation pulse and an external magnetic field. A circularly polarized electromagnetic wave, characterized by its frequency $\omega$ and wavevector k, propagates perpendicular to the plane of the ring and the electric field in the plane of the ring, for circular polarizations (${\sigma }_{+}$ for right, ${\sigma }_{-}$ for left), is given by [38,39]:

$${\mathbf{E}}_{{\sigma }_{\pm }}\left(\mathbf{r},t\right)=E\left(t\right)\left\{cos\left({\omega }_{L}t\right),\pm sin\left({\omega }_{L}t\right),0\right\},$$

(3)

where $E\left(t\right)$ represents the time-dependent amplitude of the electric field, and ${\omega }_L=\omega {\tau }_{\Delta }$ is the dimensionless frequency, with ${\tau }_{\Delta }=\pi \hslash /\left(8T_{c0}{ϵ}_0\right)$ being the characteristic relaxation time of the system. In this expression, t is dimensionless ($t/{\tau }_{\Delta }\to t$) and ${ϵ}_0\equiv 1-T_0/T_{c0}=1-{\tilde{T}}_0$. The total vector potential A includes contributions from both the static magnetic field and the radiation, and in the curvilinear coordinate system, it has only an angular component:

$$A_{\theta }^{{\sigma }_{\pm }}\left(\theta ,t\right)={\displaystyle \frac{1}{\tilde{R}}}{\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}\pm {\displaystyle \frac{{\tilde{E}}_L\left(t\right)}{{\omega }_L}}cos\left(\theta \mp {\omega }_{L}t\right).$$

(4)

where $\tilde{R}=R/{\xi }_0$, ${\mathsf{\Phi }}_0$ is the magnetic flux quantum, $\mathsf{\Phi }$ is the magnetic flux through the ring, and ${\tilde{E}}_L\left(t\right)=E\left(t\right)2\pi c{\xi }_0/\left({\mathsf{\Phi }}_0{\tau }_{\Delta }^{-1}\right)$ is the dimensionless amplitude of the radiation field. The Gaussian radiation pulse [40,41] is centered at time $t=t_c$ with a width ${\tau }_E$, and the peak amplitude $E_L$ is large enough to destroy superconductivity:

$${\tilde{E}}_L\left(t\right)=E_{L}exp\left[-{\left({\displaystyle \frac{t-t_c}{{\tau }_E}}\right)}^2\right],$$

(5)

For our analysis, we assume $t_c=3{\tau }_E$, ensuring that the radiation is negligible at the initial time $t=0$.

We use the time-dependent Ginzburg–Landau (TDGL) equation to study the system’s response to the radiation, following the approach described in previous works [42,43,44]. We assume that the ring radius is smaller than the relaxation length of the electron–hole imbalance potential $l_E$, which permits us to neglect the scalar potential in the TDGL equation. This length $l_E\sim \xi \left(T\right){(1+4{\left|\Delta \right|}^2{\tau }_{ph}^2)}^{1/4}$ (where ${\tau }_{ph}$ is a characteristic time of inelastic scattering by phonons and $\Delta$ is the superconducting order parameter) [42,45,46] and it can be much larger than $\xi \left(T_0\right)$. Additionally, the magnetic field produced by the superconducting current may be ignored, as the ring’s cross-sectional area in ring geometry is much smaller than the square of the London penetration depth (${\lambda }^2>{\xi }^2$ for type II superconductor). In our model, we consider the ring radius $R\lesssim \xi \left(T_0\right)$, which permits us to limit the number of modes of order parameter taken into account in our calculations.

For the numerical analysis, we express the order parameter [47,48] in a Fourier series (the order parameter is dimensionless):

$$\psi \left(\theta ,t\right)=\sum _{n=-N_c}^{N_c}{\psi }_n\left(t\right)e^{-in\theta },$$

(6)

where $N_c$ is the maximum number of modes included. This transforms the TDGL equation into a set of coupled differential equations for the Fourier modes ${\psi }_n\left(t\right)$. The temperature variation in curvilinear coordinates is written as follows:

$$\tilde{T}\left(\theta \right)={\tilde{T}}_0+{\displaystyle \frac{{\tilde{s}}_0}{\tilde{R}}}{\displaystyle \frac{\delta T}{T_{c0}}}\delta \left(\theta \right)$$

(7)

with ${\tilde{T}}_0=$$T_0/T_{c0}$, $\tilde{s}=s/{\xi }_0$ and ${\tilde{s}}_0=s_0/{\xi }_0$.

In the presence of an external magnetic field and a radiation pulse, the TDGL equations for the n-th harmonic mode are:

$$\begin{array}{cc}\hfill {\mathsf{\Gamma }}_{\Delta }{\displaystyle \frac{\partial {\psi }_n\left(t\right)}{\partial t}}& =\left[1-2{\kappa }^2\left(t\right)-{\displaystyle \frac{1}{{ϵ}_0{\tilde{R}}^2}}{\left(n+{\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}\right)}^2\right]\hfill \\ \hfill & \times {\psi }_n\left(t\right)-{\displaystyle \sum _{l,m}}\left\{{\psi }_l\left(t\right){\psi }_m^{*}\left(t\right)\right\}{\psi }_{n+m-l}\left(t\right)\hfill \\ \hfill & -{\mathsf{\Lambda }}_n^{{\sigma }_{\pm }}\left(t\right)+{\zeta }_n\left(t\right)-{\displaystyle \frac{\gamma }{{ϵ}_0\tilde{R}}}{\displaystyle \sum _m}{\psi }_m\left(t\right),\hfill \end{array}$$

(8)

with

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_n^{{\sigma }_{\pm }}\left(t\right)& =\pm 2\left(n+{\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}+{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{1}{2\tilde{R}{ϵ}_0}}{\displaystyle \frac{E_L\left(t\right)}{{\omega }_L}}{e}^{\pm i{\omega }_{L}t}{\psi }_{n-1}\left(t\right)\hfill \\ \hfill & \pm 2\left(n+{\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}-{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{1}{2\tilde{R}{ϵ}_0}}{\displaystyle \frac{E_L\left(t\right)}{{\omega }_L}}{e}^{\mp i{\omega }_{L}t}{\psi }_{n+1}\left(t\right)\hfill \\ \hfill & +{\displaystyle \frac{E_L^2\left(t\right)}{4{ϵ}_0{\omega }_L^2}}{e}^{\pm 2i{\omega }_{L}t}{\psi }_{n-2}\left(t\right)+{\displaystyle \frac{E_L^2\left(t\right)}{4{ϵ}_0{\omega }_L^2}}{e}^{\mp 2i{\omega }_{L}t}{\psi }_{n+2}\left(t\right)\hfill \end{array}$$

(9)

and ${\kappa }_L\left(t\right)=E_L\left(t\right)/\left(2{\omega }_L\sqrt{{ϵ}_0}\right)$. Here, $\gamma ={\tilde{s}}_0{\displaystyle \frac{\delta T}{T_{c0}}}$ represents the effect of the local temperature variation, and ${\mathsf{\Gamma }}_{\Delta }=1+i\eta$ is a complex relaxation time, with $\eta$ being a small imaginary component from electron–hole asymmetry [42,45,49,50,51]. The stochastic force ${\zeta }_n\left(t\right)$ represents the Fourier transform of the Gaussian random fluctuations $\zeta \left(\theta ,t\right)$, which is assumed to be completely uncorrelated in space and time [52,53], i.e.,

$$\begin{array}{c}〈{\zeta }^{*}\left(\theta ,t\right)\zeta \left({\theta }^{\prime },t^{\prime }\right)〉={\displaystyle \frac{\sigma }{\tilde{R}}}\delta \left(\theta -{\theta }^{\prime }\right)\delta \left(t-t^{\prime }\right)\\ 〈\zeta \left(\theta ,t\right)〉=0,\end{array}$$

(10)

where the angular brackets $〈...〉$ denote the ensemble average, and $\sigma$ is the intensity of the noise determined by the fluctuation–dissipation theorem [54,55,56] as (in dimensionless units) [57,58,59,60,61]

$$\sigma \sim {\displaystyle \frac{T_0}{T_{c0}}}{\left({\displaystyle \frac{k{T}_{c0}}{E_F}}\right)}^2.$$

(11)

Thus, in the present model, the seeds for condensate growth come from the thermal fluctuations.

3. Analytical Consideration

We consider the case where an external magnetic field induces a half-quantum flux, $\mathsf{\Phi }=-{\displaystyle \frac{1}{2}}{\mathsf{\Phi }}_0$ through a superconducting ring at equilibrium. In the absence of circularly polarized radiation and defects, the ground states with $n=0$ and $n=+1$ become degenerate. Under these conditions, we approximate the order parameter, Equation (6), by retaining only the two dominant Fourier components:

$$\psi \left(\theta ,t\right)={\psi }_0\left(t\right)+{\psi }_1\left(t\right)e^{-i\theta }.$$

(12)

Substituting this into the stationary limit of the Equation (8), we derive the following system of coupled equations for the two Fourier modes:

$$\begin{array}{c}\hfill \begin{array}{c}0=\left[ϵ-{\displaystyle \frac{1}{{\tilde{R}}^2}}{\left(0-{\displaystyle \frac{1}{2}}\right)}^2\right]{\psi }_0\\ -{ϵ}_0\left({\left|{\psi }_0\right|}^2{\psi }_0+2{\left|{\psi }_1\right|}^2{\psi }_0\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right),\end{array}\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \begin{array}{c}0=\left[ϵ-{\displaystyle \frac{1}{{\tilde{R}}^2}}{\left(1-{\displaystyle \frac{1}{2}}\right)}^2\right]{\psi }_1\\ -{ϵ}_0\left({\left|{\psi }_1\right|}^2{\psi }_1+2{\left|{\psi }_0\right|}^2{\psi }_1\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right),\end{array}\end{array}$$

(14)

To simplify the system for the further analysis, we discard the stochastic terms and introduce the following parameter, $\tau =ϵ-{\displaystyle \frac{1}{4{\tilde{R}}^2}}$, which reduces the equations to the following:

$$\begin{array}{c}\hfill \begin{array}{c}\tau {\psi }_0-{ϵ}_0\left({\left|{\psi }_0\right|}^2{\psi }_0+2{\left|{\psi }_1\right|}^2{\psi }_0\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right)=0\\ \tau {\psi }_1-{ϵ}_0\left({\left|{\psi }_1\right|}^2{\psi }_1+2{\left|{\psi }_0\right|}^2{\psi }_1\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right)=0\end{array}\end{array}$$

(15)

Multiplying the first equation by ${\psi }_0^{*}$ and the second by ${\psi }_1^{*}$, and then subtracting the two, we obtain:

$$\begin{array}{c}\hfill \begin{array}{c}0=\tau \left({\left|{\psi }_0\right|}^2-{\left|{\psi }_1\right|}^2\right)-{ϵ}_0\left({\left|{\psi }_0\right|}^4-{\left|{\psi }_1\right|}^4\right)\\ -{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\left|{\psi }_0\right|}^2-{\left|{\psi }_1\right|}^2+\left({\psi }_1{\psi }_0^{*}-{\psi }_0{\psi }_1^{*}\right)\right)\end{array}\end{array}$$

(16)

In this expression, the term $\left({\psi }_1{\psi }_0^{*}-{\psi }_0{\psi }_1^{*}\right)$ is purely imaginary, while all others are real, so it should be 0. This implies that the phases of ${\psi }_0$ and ${\psi }_1$ are the same. We can always choose ${\psi }_0$ to be real, which means that ${\psi }_1$ is also real. In the case where ${\psi }_0^2\ne {\psi }_1^2$ we arrive at the following equation:

$$\tau -{ϵ}_0\left({\psi }_0^2+{\psi }_1^2\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}=0$$

(17)

Returning to Equation (15), which is now as follows

$$\begin{array}{cc}\hfill \tau {\psi }_0-{ϵ}_0\left({\psi }_0^3+2{\psi }_1^2{\psi }_0\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right)& =0\hfill \\ \hfill \tau {\psi }_1-{ϵ}_0\left({\psi }_1^3+2{\psi }_0^2{\psi }_1\right)-{\displaystyle \frac{\gamma }{\tilde{R}}}\left({\psi }_0+{\psi }_1\right)& =0\hfill \end{array}$$

(18)

we subtract the second equation from the first:

$$\tau \left({\psi }_0-{\psi }_1\right)-{ϵ}_0\left({\psi }_0^3-{\psi }_1^3-2{\psi }_0^2{\psi }_1+2{\psi }_1^2{\psi }_0\right)=0$$

(19)

or, considering ${\psi }_0-{\psi }_1\ne 0$

$$\tau -{ϵ}_0\left({\psi }_0^2+{\psi }_1^2-{\psi }_0{\psi }_1\right)=0$$

(20)

By summing the two equations instead, if ${\psi }_0+{\psi }_1\ne 0$, we obtain

$$\tau -{ϵ}_0\left({\psi }_0^2+{\psi }_1^2+{\psi }_0{\psi }_1\right)=2{\displaystyle \frac{\gamma }{\tilde{R}}}$$

(21)

Substracting Equation (20) from Equation (21), we find the following relation (valid when ${\psi }_0\pm {\psi }_1\ne 0$, or ${\psi }_0\ne \pm {\psi }_1$)

$${\psi }_1=-{\displaystyle \frac{\gamma }{{ϵ}_0\tilde{R}{\psi }_0}}$$

(22)

Substituting this into Equation (20) gives rise to:

$${\psi }_0^4-A\left(\tau -\tilde{\gamma }\right){\psi }_0^2+A^2{\tilde{\gamma }}^2=0$$

(23)

where we introduced $A={\displaystyle \frac{1}{{ϵ}_0}}$ and $\tilde{\gamma }={\displaystyle \frac{\gamma }{\tilde{R}}}$. This equation has the following solutions:

$${\psi }_0^2={\displaystyle \frac{A}{2}}\left[\left(\tau -\tilde{\gamma }\right)\pm \sqrt{{\tau }^2-2\tau \tilde{\gamma }-3{\tilde{\gamma }}^2}\right],$$

(24)

then

$${\psi }_1^2={\displaystyle \frac{A}{2}}\left[\left(\tau -\tilde{\gamma }\right)\mp \sqrt{{\tau }^2-2\tau \tilde{\gamma }-3{\tilde{\gamma }}^2}\right].$$

(25)

We must impose the condition ${\tau }^2-2\tau \tilde{\gamma }-3{\tilde{\gamma }}^2>0$. This and the ${\psi }_0^2>0$ condition imply $\tau >3\tilde{\gamma }$.

3.1. Case of the Current-Carrying State (CCS) with ${\psi }_0\ne \pm {\psi }_1$

We proceed to calculate the free energy associated with the expressions in Equation (18). The free energy for the current-carrying state is given by the following:

$$\begin{array}{cc}\hfill F_{\mathrm{CCS}}& =-\tau \left({\psi }_0^2+{\psi }_1^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2A}}\left({\psi }_0^4+{\psi }_1^4+4{\psi }_1^2{\psi }_0^2\right)+\tilde{\gamma }{\left({\psi }_0+{\psi }_1\right)}^2\hfill \end{array}$$

(26)

Substituting Equation (22),

$${\psi }_1=-{\displaystyle \frac{\gamma A}{{\psi }_0}}$$

(27)

with Equation (24), the free energy becomes:

$$F_{\mathrm{CCS}}=-{\displaystyle \frac{A}{2}}{\left(\tau -\tilde{\gamma }\right)}^2-A{\tilde{\gamma }}^2$$

(28)

with the condition $\tau >3\tilde{\gamma }$. As we will see in further discussions, under these conditions, a barrier arises between the $n=0$ and $n=+1$ dominated states, similar to the scenario without a defect, where the ring possesses two degenerate states with opposite current directions. When subjected to circularly polarized radiation, the Inverse Faraday Effect lifts this degeneracy between the $n=0$ and $n=+1$ dominated states, thereby enabling transitions between distinct current-carrying states.

3.2. Case of the Non-Current-Carrying State (NCS) Mixed State with ${\psi }_0=\pm {\psi }_1$

We now examine the solution to Equation (20) for the case where:

$${\psi }_0=-{\psi }_1$$

(29)

In this scenario, the equation does not include the parameter $\gamma$, indicating that for $\gamma >0$, the critical temperature $T_c$ remains unchanged compared with the case where $\gamma =0$. Using the original equation:

$$\tau {\psi }_0-{ϵ}_0\left({\psi }_0^3+2{\psi }_1^2{\psi }_0\right)=0$$

(30)

and applying ${\psi }_0=-{\psi }_1$, we find

$${\psi }_0^2={\displaystyle \frac{\tau A}{3}}$$

(31)

The free energy in this case is as follows:

$$F_{\mathrm{NCS}}^{\left(-1\right)}=-2{\displaystyle \frac{{\tau }^{2}A}{3}}+{\displaystyle \frac{1}{A}}{\displaystyle \frac{{\tau }^2{A}^2}{3}}=-{\displaystyle \frac{{\tau }^{2}A}{3}}$$

(32)

In the proximity to the case ${\psi }_0=-{\psi }_1$ when ${\psi }_0/{\psi }_1=-1+\epsilon$, the free energy is

$$F\approx -{\displaystyle \frac{A{\tau }^2}{3}}+{\displaystyle \frac{A{\tau }^2}{3}}{\epsilon }^2\left({\displaystyle \frac{\tilde{\gamma }}{\tau }}-{\displaystyle \frac{1}{3}}\right)$$

(33)

Thus, the critical temperature corresponds to ${\tau }_c=0$ and if $\tau >3\tilde{\gamma }$, the free energy $F<F_{\mathrm{NCS}}^{\left(-1\right)}$, thus solution ${\psi }_0=-{\psi }_1$ is stable for $\tau <3\tilde{\gamma }$. This is illustrated in the left panel of Figure 1.

Next, consider the solution to Equation (21)

$$\tau -{\displaystyle \frac{1}{A}}\left({\psi }_0^2+{\psi }_1^2+{\psi }_0{\psi }_1\right)=2\tilde{\gamma }$$

(34)

for the case where ${\psi }_0={\psi }_1$, corresponding to a scenario with $\gamma <0$. Equation (21) transforms to

$$\tau -{\displaystyle \frac{3}{A}}{\psi }_0^2=2\tilde{\gamma }$$

(35)

yielding the solution:

$${\psi }_0^2={\displaystyle \frac{\left(\tau -2\tilde{\gamma }\right)A}{3}}$$

(36)

The free energy in this scenario ($\tau <-\tilde{\gamma }$) is:

$$F_{\mathrm{NCS}}^{\left(+1\right)}=-{\displaystyle \frac{{\left(\tau -2\tilde{\gamma }\right)}^{2}A}{3}}$$

(37)

In proximity to the case ${\psi }_0=+{\psi }_1$ when ${\psi }_0/{\psi }_1=+1+\epsilon$, the free energy is

$$F\approx -{\displaystyle \frac{1}{3}}A{\left(2\tilde{\gamma }-\tau \right)}^2\left[1-{\epsilon }^2{\displaystyle \frac{\left(\tilde{\gamma }+\tau \right)}{3\left(2\tilde{\gamma }-\tau \right)}}\right]$$

(38)

The critical temperature is now determined from the condition $2\tilde{\gamma }-\tau =0$, thus ${\tau }_c=2\tilde{\gamma }$ and $T_c^{\left(\gamma <0\right)}>T_c$. If $\tau >-\tilde{\gamma }$, the free energy $F<F_{\mathrm{NCS}}^{\left(+1\right)}$, thus the solution ${\psi }_0={\psi }_1$ is stable for $\tau <-\tilde{\gamma }$. This is represented in the right panel of Figure 1. The presence of an inhomogeneity (a defect) strongly mixes the $n=0$ and $n=+1$ states near the critical temperature: for $\gamma >0$ the state with ${\psi }_0=-{\psi }_1$ is realized, while for $\gamma <0$, the state with ${\psi }_0={\psi }_1$ emerges. These mixed states do not carry any current and exist only within a narrow temperature range near $T_c$.

3.3. Potential Barrier

In the general case, ${\psi }_0\ne {\psi }_1$ the free energy is given by Equation (28). Therefore, in the case of $\gamma >0$, for $\tau >3\tilde{\gamma }$ the potential barrier is:

$${\displaystyle \frac{U_B^{\left(-1\right)}}{A/2}}={\displaystyle \frac{F_{\mathrm{CCS}}}{A/2}}-{\displaystyle \frac{F_{\mathrm{NCS}}^{\left(-1\right)}}{A/2}}$$

(39)

which explicitly becomes:

$${\displaystyle \frac{U_B^{\left(-1\right)}}{A}}={\displaystyle \frac{1}{6}}{\left(\tau -3\tilde{\gamma }\right)}^2$$

(40)

In the case of $\gamma <0$, for $\tau >-\tilde{\gamma }$, the potential barrier is

$${\displaystyle \frac{U_B^{\left(+1\right)}}{A/2}}={\displaystyle \frac{F_{\mathrm{CCS}}}{A/2}}-{\displaystyle \frac{F_{\mathrm{NCS}}^{\left(+1\right)}}{A/2}}$$

(41)

and explicitly:

$${\displaystyle \frac{U_B^{\left(+1\right)}}{A}}={\displaystyle \frac{1}{6}}{\left(\tau +\tilde{\gamma }\right)}^2$$

(42)

4. Results

We performed computations for rings with a normalized radius $\tilde{R}=4.5$, subjected to an external magnetic flux $\mathsf{\Phi }=-{\displaystyle \frac{1}{2}}{\mathsf{\Phi }}_0$, and a relaxation constant with an imaginary component $\eta =0.1$. The initial state considered was $n=0$. The temporal evolution of the order parameter ${\psi }_n\left(t\right)$ was simulated via numerical integration of the spectral representation of the stochastic Time-Dependent Ginzburg–Landau (sTDGL) equations.

In the absence of circularly polarized radiation, the states $n=0$ and $n=+1$ are degenerate, with a superconducting transition temperature given by $T_c=$$T_{c0}\left(1-1/\left(4{\tilde{R}}^2\right)\right)$, which, for our parameters, yields $T_c=0.988T_{c0}$. These states are separated by an energy barrier in the absence of defects, and both states are stable (This stability effectively enables the superconducting ring for use in various applications [62,63]). As the temperature decreases, the energy barrier increases [64]. The presence of a defect alters the barrier height, as depicted in Figure 2, which shows the free energy barrier as a function of defect strength, $\gamma$, for several temperatures. The calculations are based on Equations (40) and (42): This stability effectively enables the superconducting ring for use in various applications.

$$U_B\left(\tilde{\gamma }>0\right)=\left\{\begin{array}{cc}{U}_B={\displaystyle \frac{1}{6}}{\left(\tau -3\tilde{\gamma }\right)}^2,\hfill & \mathrm{for}\tau ⩾3\tilde{\gamma }.\\ 0,\hfill & \mathrm{for}\tau <3\tilde{\gamma }\hfill \end{array}\right.$$

(43)

and

$$U_B\left(\tilde{\gamma }<0\right)=\left\{\begin{array}{cc}{U}_B={\displaystyle \frac{1}{6}}{\left(\tau +\tilde{\gamma }\right)}^2,\hfill & \mathrm{for}\tau ⩾-\tilde{\gamma }.\\ 0,\hfill & \mathrm{for}\tau <-\tilde{\gamma }\hfill \end{array}\right.$$

(44)

For specific defect strengths, the barrier vanishes, leading to a strong mixing of the $n=0$ and $n=+1$ states near the critical temperature. Specifically, for $\gamma >0$ and $T_0>T_{\gamma >0}^{*}\simeq T_c-{\displaystyle \frac{3\gamma }{\tilde{R}}}{T}_{c0}$, the system realizes a state with ${\psi }_0=-{\psi }_1$. Conversely, for $\gamma <0$ and $T_0>T_{\gamma <0}^{*}\simeq T_c-{\displaystyle \frac{\left|\gamma \right|}{\tilde{R}}}{T}_{c0}$, the system realizes a state with ${\psi }_0={\psi }_1$. In both cases, the potential barrier is absent and these mixed states do not carry current. At lower temperatures, the potential barrier reappears, causing the mixed states to become unstable, leading to states with ${\psi }_0\ne \pm {\psi }_1$ as the ground states. In this scenario, the ring exhibits two degenerate states with opposite current directions. Under circularly polarized radiation, the Inverse Faraday Effect lifts the degeneracy between the $n=0$ and $n=+1$ dominated states, enabling transitions between current-carrying states if the temperature drops below $T^{*}$ [25,27].

To further explore the analytical results in the approximation of two harmonic modes, we performed numerical simulations with $N_c=8$, examining the system’s response to a Gaussian radiation pulse. The substrate temperature was set to ${\tilde{T}}_0=0.9<$${\tilde{T}}_{\gamma >0}^{*}=0.92$ and ${\tilde{T}}_0=0.95>$${\tilde{T}}_{\gamma >0}^{*}=0.92$ for $\gamma =+0.1$ and similarly for $\gamma =-0.1$ with substrate temperatures, ${\tilde{T}}_0=0.9<$${\tilde{T}}_{\gamma <0}^{*}=0.965$ and ${\tilde{T}}_0=0.95<$${\tilde{T}}_{\gamma <0}^{*}=0.965$. The substrate temperature was also set to ${\tilde{T}}_0=0.98>$${\tilde{T}}_{\gamma >0}^{*}=0.85$ for $\gamma =+0.2$ and similarly for $\gamma =-0.2$ with a substrate temperature, ${\tilde{T}}_0=0.98>$ ${\tilde{T}}_{\gamma <0}^{*}=0.94$. We tracked the evolution of the order parameter modes. Figure 3 shows typical examples of the time evolution of the order parameter modes ${\psi }_n$ for a single noise realization and defect strength $\gamma =\pm 0.1;\pm 0.2$. For $\gamma <0$, the condition $T_0<T^{*}$ is satisfied for temperatures ${\tilde{T}}_0=0.9;0.95$, resulting in no realization of the mixed state, and the system transitions into a current-carrying state with a dominant harmonic, where, as for ${\tilde{T}}_0=0.98$, the mixed state is realized. For $\gamma >0$, the mixed state is realized only at ${\tilde{T}}_0=0.95$ and ${\tilde{T}}_0=0.98$, whereas for ${\tilde{T}}_0=0.90$, the dominant harmonic state prevails. Our numerical results confirm this behavior.

Now, we investigate the probability of reaching a particular final state by analyzing multiple trajectories of the system under the influence of noise. The criterion used for state identification is based on the modes or harmonics of the system’s order parameter. Specifically, after the system reaches a relaxed state, we evaluate the magnitude of the order parameter, denoted as ${\left|{\psi }_n^{\left(i\right)}\right|}^2$, for the i-th noise realization. To ensure robustness, we verify that the system remains in this identified state over a subsequent time window, thus eliminating the possibility of transient fluctuations or oscillations being misidentified as stable states. This approach allows us to accurately assign each trajectory to a final state and subsequently calculate the probability as the fraction of trajectories that end up in each state out of the total number of simulated trajectories:

$$P\left[{\psi }_n\left({\sigma }_{\pm }\right)\right]={\displaystyle \frac{1}{N_c}}\sum _{i=1}^{N_c}{\left|{\psi }_n^{\left(i\right)}\left({\sigma }_{\pm }\right)\right|}^2$$

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Figure 4 illustrates the probability of generating the final state ${\psi }_{n=+1}$ as a function of defect strength for various radiation frequencies, with distinct panels showing the probability at temperatures ${\tilde{T}}_0=0.90$ and ${\tilde{T}}_0=0.95$. The radiation pulse duration is ${\tau }_E=10$. (Very short pulse durations ${\tau }_E\lesssim 1$ lead to a non-adiabatic regime of superconductivity generation, resulting in random realization of the final states with $P\simeq 0.5$ [65,66,67]). The radiation pulse has an amplitude $E_L/{\omega }_L=1.5$ and ${\sigma }_{+}$ polarization. The probability of generating the $n=0$ state is $P\left[{\psi }_0\left({\sigma }_{+}\right)\right]=1-P\left[{\psi }_{+1}\left({\sigma }_{+}\right)\right]$, with negligible probability of generating other states due to their higher energy.

These results show that the switching probability reaches $100\%$ at lower frequencies and ${\tilde{T}}_0=0.90$, consistent with our previous findings, and for low defect strengths. However, at ${\tilde{T}}_0=0.95$, the probability for large positive disorder strengths approaches $50\%$, confirming that the mixed state is realized at ${\tilde{T}}_0=0.95>$ ${\tilde{T}}_{\gamma >0}^{*}=0.92$, and the switching effect is suppressed.

5. Conclusions

In this work, we have investigated the scenario where an external magnetic field induces a half-quantum flux, $\mathsf{\Phi }=-{\displaystyle \frac{1}{2}}{\mathsf{\Phi }}_0$, through a superconducting ring at equilibrium. In the absence of circularly polarized radiation, the states with $n=0$ and $n=+1$ become degenerate. For $\gamma >0$ and temperatures $T_0>T_{\gamma >0}^{*}\simeq T_c-{\displaystyle \frac{3\gamma }{\tilde{R}}}{T}_{c0}$, the system realizes a state where ${\psi }_0=-{\psi }_1$, while all other harmonics vanish. The term involving $\gamma$ cancels out in the equations for ${\psi }_0$ and ${\psi }_1$, leading to a critical temperature $T_c$ that remains the same as in the absence of the defect, namely $\left(T_{c0}-T_c\right)/T_{c0}=1/4{\tilde{R}}^2$.

This result arises from the fact that, at ${\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}=-{\displaystyle \frac{1}{2}}$, the harmonics ${\psi }_0$ and ${\psi }_1$, along with their linear combinations, provide solutions with the same critical temperature $T_c$. The wave function for the state ${\psi }_0=-{\psi }_1$ is given by $\psi ={\psi }_0(1-exp\left(i\theta \right))$, with the modulus ${\left|\psi \right|}^2=4{\left|{\psi }_0\right|}^2\left(1-cos\theta \right)$, which vanishes at the location of the defect, $\theta =0$. As a result, the critical temperature is insensitive to the presence of the impurity. This conclusion holds exactly, independent of the two-harmonic approximation, although it is limited to the case of ${\displaystyle \frac{\mathsf{\Phi }}{{\mathsf{\Phi }}_0}}=-{\displaystyle \frac{1}{2}}$ and a single defect.

For $\gamma <0$ and temperatures $T_0>T_{\gamma <0}^{*}\simeq T_c-{\displaystyle \frac{\left|\gamma \right|}{\tilde{R}}}{T}_{c0}$, the system favors the state with ${\psi }_0={\psi }_1$. In both cases, the potential barrier vanishes, and these mixed states do not carry any current. At lower temperatures, a potential barrier re-emerges, as in the case without a defect, and the mixed states become unstable, transitioning into ground states where ${\psi }_0\ne \pm {\psi }_1$.

Our numerical calculations confirm that, for $T_0>T^{*}$, when the ring is in the mixed state and exposed to circularly polarized radiation, no switching occurs between current-carrying states, indicating the system remains in a non-current-carrying state. However, for $T_0<T^{*}$, the system develops two degenerate states. When subjected to circularly polarized radiation, the Inverse Faraday Effect lifts the degeneracy between the $n=0$ and $n=+1$ dominated states, enabling switching between different current-carrying states.