Parametric control of Meissner screening in light-driven superconductors

We investigate the Meissner effect in a parametrically driven superconductor using a semiclassical $U(1)$ lattice gauge theory. Specifically, we periodically drive the $z$-axis tunneling, which leads to an enhancement of the imaginary part of the $z$-axis conductivity at low frequencies if the driving frequency is blue-detuned from the plasma frequency. This has been proposed as a possible mechanism for light-enhanced interlayer transport in YBa$_2$C$_3$O$_{7-\delta}$ (YBCO). In contrast to this enhancement of the conductivity, we find that the screening of magnetic fields is less effective than in equilibrium for blue-detuned driving, while it displays a tendency to be enhanced for red-detuned driving.


I. INTRODUCTION
Optical driving of solids opens up the possibility to induce superconducting-like features in their response to electric fields. This was first achieved in several cuprates by the excitation of specific phonon modes [1,2] or near-infrared excitation [3,4]. Later, signatures of a superconducting state were induced in fullerides and organic salts by exciting molecular vibrations [5][6][7]. In all these experiments, the imaginary part σ 2 (ω) of the optical conductivity exhibited a 1/ω divergence at low frequencies following optical excitation at temperatures above the equilibrium critical temperature T c . In the case of YBCO, an enhancement of the low-frequency conductivity σ 2 (ω) along the c axis was also observed below T c [2,8]. Several mechanisms have been proposed to explain the enhancement of interlayer transport, including nonlinear lattice dynamics [9], parametric driving [10][11][12][13], and suppression of competing orders [14,15]. While the transient optical response of the light-driven cuprates and organic materials is consistent with enhanced or induced superconducting states, their response to magnetic fields has remained largely unexplored. That is due to the limited lifetimes of the excited states, which make experimental measurements of the magnetic response challenging [16]. Therefore, it is an open question whether the experimental observations of the light-induced transport properties indeed correspond to light-enhanced or light-induced superconductivity in the sense of an enhanced Meissner effect [17][18][19][20][21].
In this paper, we theoretically study the Meissner effect in a parametrically driven superconductor. We consider a specific mechanism of parametric driving, where the Cooper pair tunneling along the z axis is periodically modulated in time [12,22]. This type of driving enhances the imaginary part of the optical conductivity along the z axis at low frequencies. Based on analytical and numerical calculations, we find that the screening of DC magnetic fields is less effective than in equilibrium for slightly blue-detuned driving. This is due to the generation of electromagnetic waves by the parametric driving. For red-detuned parametric driving, there is no transmission of electromagnetic waves into the bulk and the Meissner screening is enhanced on a length scale that depends on the driving strength and the driving frequency at the order of our analytical investigation. The enhancement of the Meissner screening is particularly effective when the driving frequency is close to the plasma frequency. Notably, the imaginary part of the optical conductivity is reduced in this regime of driving frequencies.
This paper is organized as follows. After introducing our semiclassical method in Section II, we discuss the optical conductivity of a parametrically driven superconductor in Section III. In Section IV, we first investigate the Meissner effect in a parametrically driven superconductor from an analytical perspective. Furthermore, we present numerical results for parametrically driven superconductors with isotropic and anisotropic lattice parameters.
We conclude this work in Section V.

II. METHOD
Here, we give an overview of the semiclassical U (1) lattice gauge theory that we utilize to simulate the dynamics of a parametrically driven superconductor [23][24][25]. The static part of the Lagrangian is the Ginzburg-Landau free energy [26] on a three-dimensional lattice. As depicted in Fig. 1(a), the superconducting order parameter ψ r (t) is located on the sites of a cubic lattice with lattice constant d, where r = (x, y, z) is the lattice site. The components of the electromagnetic vector potential A j,r (t) are defined on the lattice bonds, which connect each site r with its nearest neighbor in the j ∈ {x, y, z} direction. We employ the temporal gauge such that the electric field components are calculated according to E j,r = −∂ t A j,r . The magnetic field components B j,r = jkl (A l,r (k) − A l,r )/d are found on the lattice plaquettes, with r (k) as the neighboring site of r in the k direction. The Lagrangian of the lattice gauge model is

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where µ and g are the Ginzburg-Landau coefficients and the coefficient K describes the magnitude of the dynamical term [27,28]. The dynamical term is of the form |∂ t ψ r | 2 , which supports the particle-hole symmetry of the Lagrangian, i.e., L is invariant under ψ r → ψ * r and e → −e. The coupling of the unitless vector potential a j,r = −2edA j,r / to the phase of the order parameter ensures the local gauge-invariance of the Lagrangian. Note that the charge of a Cooper pair is −2e. The tunneling coefficients t x = t y = t xy and t z determine the plasma frequencies of the superconductor, with the equilibrium Cooper pair density n 0 = µ/g. The superconductor is isotropic for t xy = t z and anisotropic for t xy = t z .
We derive the Euler-Lagrange equations from Eq. (1) and include damping terms, where γ sc and γ el,j are phenomenological damping coefficients of the order parameter and the electric field, respectively. We note that these equations are the zero-temperature limit of the Langevin equations used in Ref. [25].
We numerically solve the equations of motion employing periodic boundary conditions along the y axis. With this boundary condition, we take the superconducting sample to be spatially homogeneous along the y axis, rather than having open boundary conditions. We assume open boundary conditions in the x and z direction and impose a spatially uniform magnetic field B = B extŷ at the surfaces in x and z direction. For this purpose, we add one numerical layer outside the sample; see Fig. 1(a). On the external plaquettes, we fix the magnetic field to zero, ramp it up to a non-zero constant or specify a temporally oscillating value in the following. Thus, we model different physical scenarios. To simulate the vacuum, we set the order parameter and the tunneling coefficients to zero outside the sample. Inside the sample, we initialize the order parameter and the vector potential in the ground state, where ψ r ≡ µ/g and A r ≡ 0, and integrate the differential equations using Heun's method with a step size of ∆t = 2.5 ns.

III. CONDUCTIVITY OF PARAMETRICALLY DRIVEN SUPERCONDUCTORS
We measure the optical conductivity by adding a weak probe current J ext (t) = J 0 cos(ω pr t) to the equations of motion for the z component of the electric field, as depicted in Fig. 1(b). For this measurement, we fix the surface magnetic field to B ext = 0, neglecting radiation from the sample due to the probe current. Thus, the dynamics is spatially homogeneous along the x axis and independent of the sample width. The optical conductivity is σ(ω pr ) = J ext (ω pr )/E(ω pr ), where E(ω pr ) is the Fourier transform of the spatial average of the electric field in the steady state. Additionally, we drive the sample by periodically modulating the tunneling coefficients of all z-axis junctions, Experimentally, this could be achieved by resonantly exciting an infrared-active phonon mode; see Refs. [2,12,22,29].
The effect of this parametric driving on the imaginary part σ 2 (ω pr ) of the z-axis conductivity is displayed in Fig. 2(a). While σ 2 is reduced for ω dr < ω pl at probe frequencies ω pr |ω pl − ω dr |, it is enhanced for ω dr > ω pl . Figure 2(b) reveals that σ 2 approaches a 1/ω pr behavior at low probe frequencies, regardless of whether the superconductor is driven or not. These results are consistent with the findings of Refs. [12,22]. To quantify the superconducting character of the optical response, we use the superconducting weight following the definition given in Ref. [30]. For an infinitely large sample, the analytical expression for the superconducting weight is D 0 = π 0 ω 2 pl . An analytical prediction for the parametrically driven case was derived in Ref. [12], respectively. According to our semiclassical U (1) gauge theory, the supercurrent density along the z axis is given by Neglecting fluctuations of the superconducting order parameter, this can be simplified to In the previous step, we fixed the gauge such that arg(ψ r ) ≡ 0, which complies with the temporal gauge in a charge neutral system. For weak fields, we linearize the above expression and rewrite it using the expression for the plasma frequency ω pl from Eq. (2), In the following, we treat A and J as continuous fields and drop the subscript r. As Neglecting the current contribution from the damping term, we obtain for the curl of the free current density. On the other hand, Maxwell's equations imply Combining Eqs. (12) and (13) yields the minimal model as This leads to where λ = c/ω pl is the London penetration depth of the undriven superconductor. To determine B 0 (x) and B 1 (x), we use the ansatz The solutions for B 0 and B 1 are superpositions of four exponentials with While 1 is generally real-valued and smaller than λ, 2 is real-valued only for ω dr ω pl and imaginary for ω dr > ω pl . The absolute value of 2 is larger than λ for ω dr ∼ ω pl . The solution for B 2 (x) is of the form where is real-valued for ω dr < ω pl and imaginary for ω dr > ω pl .
In the red-detuned case of ω dr < ω pl , we exclude exponentially growing solutions and write the magnetic field inside the superconductor as The corresponding electric field is We note that red-detuned parametric driving induces an AC contribution to the magnetic field, which is less effectively screened than the DC magnetic field. For blue-detuned parametric driving, the induced AC part of the magnetic field leads to the formation of two standing waves. As these standing waves are induced at the surface, we use the ansatz for ω dr > ω pl . The electric field has the form In general, parametric driving of a superconductor in the presence of a magnetic field causes emission of electromagnetic waves. Here, we consider the emission of electromagnetic waves from the left edge of the sample, Using the continuity of B y (x, t) and E z (x, t) at the surface of the sample, we determine the coefficients β 1 , β 2 and β 3 ; see supplementary material for details of the calculation. In the red-detuned case, we obtain where In the blue-detuned case, we find In Figs. 3(a) and 3(b), we show how red-and blue-detuned driving modifies the spatial dependence of the DC magnetic field inside the superconductor. In the red-detuned case, the DC part of the magnetic field is the sum of two exponentially decaying contributions.
As the driving frequency approaches the plasma frequency, the length scale of the first decay converges to a value below the equilibrium penetration depth, This feature of the response by itself indicates a parametric enhancement of the Meissner screening. However, the length scale 2 of the second exponential decay is generally larger than the equilibrium penetration depth. Thus, the enhancement of the Meissner screening is lessened or reverted. While 2 diverges for ω dr → ω pl , the prefactor of the second exponential decay vanishes in this limit. Taking both into account, the Meissner screening is enhanced as the driving frequency is slightly below the plasma frequency. For larger detuning, the enhanced screening is effective only on a short length scale. Further away from the surface, the slower decaying contribution dominates such that the DC magnetic field is larger than in the absence of driving. This is visible in Fig. 3(a).
For blue-detuned driving, the DC part of the magnetic field is the sum of a contribution that decays exponentially on the length scale 1 < λ and a spatially oscillating contribution. As evidenced by Fig. 3(b), the spatially oscillating contribution reduces the Meissner screening such that the DC magnetic field is larger than in the absence of driving. In the supplementary material, we present the spatial dependence of the DC magnetic field explicitly, using a higher driving amplitude of M = 0.6. In our simulations, we apply a static magnetic field at the surface of the superconductor. The analytical solution for this boundary condition is provided in the supplementary material. We find that the solution for the DC magnetic field inside the superconductor is not affected in the case of blue-detuned driving. However, the modified boundary condition suppresses the enhancement regime for red-detuned driving.

B. Numerical results for an isotropic superconductor
To simulate the Meissner effect, we apply a small surface magnetic field along the y axis, i.e., B = B extŷ . Throughout this paper, we use B ext = 1 mT. Note that we obtain consistent results for B ext = 0.1 mT, which confirms that the linear response is measured. In Fig. 4, we present equilibrium results for an isotropic superconductor with the same parameters as in Section III, except for the sample size. The plasma frequency is ω pl /2π = 100 THz and the edge length is L = 6 µm along both axes. We see in Fig. 4(a) that the magnetic field B y (x, z) is screened away from the surface, which is the characteristic response of a superconductor to a magnetic field. As shown in Fig. 4(b), the decay of the magnetic field from the sample surfaces is well captured by the exponential fit functions B y (x, 0) = B ext exp (−x/λ eq ) and In the remainder of this section, we investigate the response of an isotropic superconductor to an external magnetic field in the presence of parametric driving as defined in Eq. (6).
We characterize the Meissner screening in the driven state by the attenuation lengths R x and R z . The attenuation length R x quantifies the Meissner screening at the center of the left sample surface, i.e., for x L x and z = L z /2. To resolve changes of R x below the discretization length d, we interpolate the magnetic field linearly between the plaquettes left and right of x = R x . The attenuation length along the z axis, R z , is determined analogously. In equilibrium, the attenuation lengths equal the London penetration depth, i.e., R x = R z = λ eq .
We consider a superconducting sample with the same parameters as before but with a sample size of 12 × 12 µm 2 to ensure the convergence of our results. First, we choose the driving frequency ω dr /2π = 110 THz and the driving amplitude M = 0.3, consistent with the conductivity measurements shown in Fig. 2. Once a steady state is reached, the mag-netic field inside the superconductor oscillates with the driving frequency. Snapshots of the time evolution of the magnetic field during one driving cycle are displayed in Fig. 5(b). The parametric driving with ω dr > ω pl has two main effects. Firstly, electromagnetic waves generated at the left and right surfaces are transmitted into the bulk of the sample. However, the magnitude of the magnetic field inside the superconductor is strongly suppressed compared to the surface field B ext . Secondly, the attenuation lengths are no longer isotropic and exhibit an oscillatory behavior in time. As evidenced by Fig. 5(a), R x exhibits a pronounced oscillation, while R z has a small oscillation amplitude. Remarkably, we find that there is  frequencies ∼1 THz; see supplementary material. We note that the relative phase between the oscillation of R x and the oscillation of R z depends on the lateral sample size. This suggests that the modulation of R z is due to the transmission of electromagnetic waves from the left and right surfaces.
We proceed by varying the driving strength and the driving frequency. Figure 6(a) demonstrates that the attenuation length R x grows monotonically with increasing driving amplitude. The data points in Fig. 6 of blue-detuned driving.

C. Numerical results for an anisotropic superconductor
In this section, we study the effect of parametric driving on the magnetic response of an anisotropic superconductor. Since our analytical arguments in Section IV A are not limited to an isotropic superconductor, we expect a similar reduction of the Meissner screening for an anisotropic superconductor. In cuprate superconductors, the ratio between the inplane plasma frequency and the (lower) c-axis plasma frequency is of the order of 100.
Due to numerical constraints, we choose plasma frequencies with a smaller ratio. In the following, we consider a superconductor with the plasma frequencies ω x /2π = 300 THz and ω z /2π = 50 THz along the x axis and the z axis, respectively. Consistent with the relations λ x = c/ω x and λ z = c/ω z , we find the attenuation lengths R x = λ z = 954 nm and R z = λ x = 159 nm in equilibrium. The sample size is 24 × 6 µm 2 .
We then add parametric driving with frequency ω dr /2π = 55 THz and amplitude M = 0.3.
We show in Fig. 7(b) that the time evolution of the magnetic field during one driving cycle is comparable to the isotropic case. However, the spatial patterns are sharper and more pronounced, especially towards the top and bottom of the sample. While the oscillation amplitude of R x compared to its temporal average is similar to the isotropic case, the oscillation amplitude of R z is significantly larger as shown in Fig. 7(a). This further indicates that the modulation of R z is a consequence of electromagnetic waves transmitted into the bulk. Here, the attenuation lengths of the time-averaged magnetic field are R x = 977 nm and R z = 161 nm in the driven state. The increase of R x by approximately 2% is in good agreement with our observation for an isotropic superconductor, where we also used ω dr = 1.1ω z and M = 0.3. The increase of R z by more than 1% is considerably larger than in the isotropic case.

V. CONCLUSION
In conclusion, we have presented the response of light-driven superconductors to magnetic fields for the scenario of parametrically driven z-axis tunneling. For driving with a frequency blue-detuned from the plasma frequency, we find an enhancement of z-axis transport and a reduction of the Meissner screening along the x axis, the direction perpendicular to the parametric drive and the applied magnetic field. This key result is in contrast to the equilibrium behavior of superconductors. In the absence of driving, both London theory [31] and our model in Eq. (1) predict that an enhancement of the low-frequency σ 2 (ω) along the z axis implies an enhancement of the Meissner screening along the x axis. Our simulations demonstrate the breakdown of this general relation in a driven superconductor. In fact, the screening of DC magnetic fields is reduced for slightly blue-detuned driving, which can be understood analytically based on a minimal model that we derived in this work. According to our analytical calculations, the screening of DC magnetic fields shows a tendency to be enhanced for slightly red-detuned driving. This enhanced screening is enabled by emission of electromagnetic waves from the superconductor. If the emission of electromagnetic waves is suppressed, as in our simulations, the Meissner screening for red-detuned driving is generally less effective than in the absence of driving. We emphasize that we observe similar behavior for isotropic and anisotropic superconductors.
Our findings suggest that the enhancement of the low-frequency conductivity is nat- As shown in Fig. 1, the real part σ 1 of the optical conductivity exhibits a maximum around ω * = |ω pl − ω dr | for red-detuned driving, while it exhibits a minimum around ω * for bluedetuned driving. These extrema of σ 1 correspond to parametric attenuation/amplification as discussed in Ref. [1]. Following the notation in Ref. [2], we write the optical conductivity as The total spectral weight is then given by Here, we evaluate the spectral weight difference with ω a /2π = 2 THz and ω b /2π = 100 THz. For red-detuned driving, we find ∆W red ≈ −0.84 · (D red − D 0 )/2, while we obtain ∆W blue ≈ −0.94 · (D blue − D 0 )/2 for blue-detuned driving. Note that the absolute value of ∆W is underestimated in both cases due to the lower cutoff at ω a . These results indicate that the parametric driving redistributes spectral weight, while the total spectral weight is conserved.
The continuity of E z at x = 0 implies We find Figure 2 displays the analytical solution for the spatial dependence of the DC magnetic field explicitly. Here, the driving amplitude is chosen relatively high such that the effect of the parametric driving is visible. For comparison with the numerical simulations, we apply the boundary condition B y (x = 0, t) = B ext . With the general ansatz for the magnetic field inside the superconductor from Section 4.1 of the main text, we then obtain

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for both red-and blue-detuned driving. In the case of blue-detuned driving, the DC part of the magnetic field is the same as for the emitting boundary condition considered in the previous section. For red-detuned driving, however, the modified boundary condition qualitatively changes the solution. In contrast to the emitting case, β 2 does not vanish in the limit of ω dr → ω pl such that the divergence of 2 implies a less effective Meissner screening than in the absence of driving; see

III. SIMULATION PARAMETERS
The parameters of the simulated superconductors are summarized in Table I. Our choice of µ and g implies a Cooper pair density of n 0 = µ/g = 2 × 10 21 cm −3 in equilibrium. The discretization length d is of the order of the Ginzburg-Landau estimate for the coherence length, for the isotropic sample. The Higgs frequency is

IV. FINITE SIZE ANALYSIS
In the main text, we present results for a parametrically driven superconductor with a sample size of 12 × 12 µm 2 . Here, we investigate the magnetic response of isotropic

VI. AC MEISSNER EFFECT IN A PARAMETRICALLY DRIVEN SUPERCON-DUCTOR
Finally, we briefly discuss the response of a parametrically driven isotropic superconductor to AC magnetic fields, i.e., B ext → B ext cos(ω pr t). Once a steady state is reached, we record the time evolution of the magnetic field for 10 ps with a detection rate of 5 PHz. We then compute the Fourier transform of the magnetic field to evaluate the component of B y (x, z) that oscillates with the probe frequency ω pr . Eventually, we determine the attenuation lengths R x and R z as in the case of a static magnetic field. Figure 6 displays the attenuation lengths as a function of the probe frequency. Importantly, the attenuation lengths of an AC magnetic field with ω pr /2π = 1 THz approach the attenuation lengths of a static magnetic field. Therefore, it is sufficient to probe the sample with a static magnetic field in order to obtain the low-frequency limit of its magnetic response. This is particularly relevant when comparing the magnetic response to the optical response in Section 3 of the main text. For increasing probe frequency, the equilibrium In equilibrium, the attenuation length equals the London penetration depth λ eq . The driving frequency is ω dr = 1.1ω pl and the driving amplitude is M = 0.3. The sample size is 16 × 16 µm 2 . penetration depth also increases, which can be understood from an analytical perspective.
As mentioned in Section 4.1 of the main text, Maxwell's equations imply We insert the London equation ∇ × J = −B/(µ 0 λ 2 ) into Eq. (28), where λ is the London penetration depth of a static magnetic field. This leads to implying λ(ω pr ) = λ For example, the numerical value of λ eq = 487 nm for ω pr /2π = 20 THz agrees with the analytical prediction based on Eq. (30). In the presence of parametric driving, the attenuation lengths follow a qualitatively similar dependence on the probe frequency as in equilibrium.