Light-induced Floquet spin-triplet Cooper pairs in unconventional magnets

This study demonstrates that time-periodic light drives, when applied to unconventional magnets with inherent momentum-dependent spin splitting, induce the formation of Floquet spin-triplet Cooper pairs and spin-triplet densities through photon-assisted processes, thereby enabling the dynamic creation and manipulation of nontrivial light-induced superconducting states.

The Gist

Imagine you have a very special, strange kind of magnet. Unlike the fridge magnets you know, which have a clear North and South pole, this "unconventional magnet" is a bit like a spinning top made of invisible dancers.

In a normal magnet, all the dancers spin the same way, creating a strong pull. In this new type, the dancers are arranged in a pattern where half spin one way and half spin the other, canceling each other out so there is no overall pull. However, if you look closely at their feet (their momentum), you see a secret: the dancers on the left spin differently than the dancers on the right. This is called spin-splitting.

Now, imagine you shine a strobe light (a laser) on these dancers. This isn't just a normal light; it's a rhythmic, flashing light that pulses up and down thousands of times a second.

Here is what happens when you combine this strange magnet with the rhythmic light and a special ingredient called superconductivity (a material where electricity flows with zero resistance, like a frictionless slide):

1. The "Time-Traveling" Dancers (Floquet States)

When the strobe light hits the dancers, something magical happens. The light doesn't just make them dance faster; it creates new versions of themselves.

Think of it like a video game with "levels."

• Level 0: The dancers in their normal state.
• Level +1: A version of the dancer who just absorbed a "photon" (a packet of light energy).
• Level -1: A version who just emitted a photon.

In physics, these are called Floquet sidebands. The light effectively gives the electrons a new "ID card" (the Floquet index) that tells us which "level" of light energy they are currently holding. This allows them to interact in ways that were impossible before.

2. The Magic Trick: Creating New Partners

In a normal superconductor, electrons usually pair up like a married couple: one "spin-up" and one "spin-down." This is a spin-singlet pair. They are very stable and stick together tightly.

But in this experiment, the rhythmic light acts like a matchmaker with a twist.

• The light hits the strange magnet.
• Because of the magnet's special "dance pattern" and the light's rhythm, the electrons are forced to pair up in a new, exotic way.
• Instead of the usual couple, they form spin-triplet pairs. Imagine three dancers holding hands instead of two. Or, more accurately, two electrons that are both spinning in the same direction, which is usually forbidden in normal materials.

The Analogy:
Imagine a dance floor where everyone is used to dancing in pairs (one left-foot, one right-foot). Suddenly, a DJ plays a beat that forces the dancers to switch partners and dance in a triplets formation, or dance with partners who are spinning the same way. The light provides the rhythm that makes this impossible dance possible.

3. The "Photon" Recipe

The paper discovers that the type of dance pair formed depends on how many photons (light packets) are involved in the process.

• Even Number of Photons (0, 2, 4...): These are like "background dancers." They can exist even if the light is turned off (though they are weaker). They are the "safe" pairs.
• Odd Number of Photons (1, 3, 5...): These are the stars of the show. They only exist because the light is on. If you turn off the laser, these pairs vanish instantly. They are purely "light-induced."

The researchers found that by changing the color of the light (circular vs. linear polarization) and the angle of the magnet, they could control exactly which type of pair forms. It's like having a remote control that switches the dance floor between "Waltz" and "Breakdance" just by turning a knob.

4. Why Does This Matter?

Why should we care about electrons dancing in the dark?

• New Electronics: These "spin-triplet" pairs are the holy grail for quantum computers. They are much more robust against noise and could help build computers that don't crash easily.
• Reading the Magnet: The way the light changes the dance tells us exactly what kind of magnet we are looking at. It's a new way to "see" the invisible structure of materials.
• Designing Reality: This proves we can use light to "engineer" new states of matter that don't exist in nature. We aren't just observing the universe; we are using light to write new rules for it.

Summary

In simple terms, the scientists took a weird magnet, shined a rhythmic laser on it, and found that the light forced the electrons to form new, exotic partnerships that only exist while the light is on. They discovered that the number of light particles involved determines the "personality" of these partnerships. This opens the door to creating superconductors and quantum devices that can be turned on and off with a light switch.

Technical Summary

1. Problem Statement

Unconventional magnets (such as altermagnets and p-wave magnets) are a newly discovered class of magnetic materials characterized by momentum-dependent spin splitting and zero net magnetization. While their static properties and interaction with conventional superconductors have been studied, the dynamic response of these systems to time-periodic external fields remains largely unexplored. Specifically, there is a lack of understanding regarding:

• How time-periodic light drives (Floquet engineering) modify the spin textures and superconducting correlations in unconventional magnets.
• Whether light can induce novel superconducting states, specifically spin-triplet Cooper pairs, in systems that are not naturally triplet superconductors.
• How the interplay between the momentum parity of the magnet (e.g., $d$-wave vs. $p$-wave) and the polarization of light dictates the symmetry and classification of emergent Floquet states.

2. Methodology

The authors employ Floquet theory to analyze unconventional magnets with $d$-wave (even parity) and $p$-wave (odd parity) symmetries, both in the normal state and when proximitized by a conventional spin-singlet $s$-wave superconductor.

• Hamiltonian Formulation:

  • The static Hamiltonian includes kinetic energy, a momentum-dependent magnetic field $J_q(\mathbf{k})$ (where $q$ denotes the angular momentum order), and a superconducting pairing term $\Delta$.
  • Light-Matter Interaction: Time-periodic drives (Circularly Polarized Light - CPL, and Linearly Polarized Light - LPL) are introduced via the Peierls substitution ($\mathbf{k} \to \mathbf{k} + e\mathbf{A}(t)/\hbar$).
  • The authors derive explicit time-dependent Hamiltonians for $d$- and $p$-wave systems, identifying non-trivial light-matter coupling terms that arise from the interplay between the light vector potential and the unconventional magnetic field.

• Floquet Formalism:

  • The time-dependent Schrödinger equation is mapped to an infinite-dimensional time-independent eigenvalue problem in an extended frequency space (Floquet space).
  • The Hamiltonian is expanded into Fourier components $H_n$, where $n$ represents the number of photons absorbed ($n>0$) or emitted ($n<0$).
  • The system is truncated to 11 Floquet sidebands ($n \in [-5, 5]$) for numerical calculations.

• Observables:

  • Floquet Spin Density: Calculated using the Floquet Green's function to analyze spin polarization in the normal state.
  • Floquet Cooper Pairs: Characterized via the anomalous Green's function in the Nambu space. The authors decompose pair amplitudes based on spin symmetry (singlet/triplet), frequency symmetry (even/odd), momentum parity, and Floquet index parity.

3. Key Contributions

A. Non-Trivial Light-Matter Interaction

The paper identifies that light drives induce specific, non-trivial interactions in unconventional magnets that are absent in static systems or conventional ferromagnets:

• $d$-wave magnets: Light induces both momentum-dependent terms (resembling Rashba spin-orbit coupling or $p$-wave fields) and momentum-independent Zeeman-like terms. The Zeeman-like term is sensitive to the angle between the magnetic order and light polarization.
• $p$-wave magnets: Light induces a momentum-independent Zeeman-like effect proportional to the light amplitude and magnetic strength.
• These interactions are governed by the specific momentum parity ($q$) of the magnet and the photon order ($n$).

B. Classification of Floquet Cooper Pairs

The most significant theoretical contribution is the systematic classification of eight distinct classes of Floquet Cooper pairs. The authors introduce the Floquet sideband index as an additional quantum number, broadening the symmetry classification beyond the static case.

• Even-Photon Processes ($|n| = 0, 2, \dots$):
  • Can exist in the static limit (zero-photon component).
  • Include spin-singlet pairs inherited from the parent superconductor and spin-triplet pairs induced by the interplay of magnetism and superconductivity.
• Odd-Photon Processes ($|n| = 1, 3, \dots$):
  • Strictly light-induced: These pairs vanish in the static regime.
  • They arise entirely from the interplay of light, unconventional magnetism, and superconductivity.
  • Symmetry Switching: The classification of spin-triplet pairs depends critically on the magnet's parity:
    • In $d$-wave magnets, odd-photon processes lead to specific triplet classes (e.g., Class 6 and 8).
    • In $p$-wave magnets, the same odd-photon processes lead to different triplet classes (e.g., Class 5 and 7) due to the inversion of momentum parity.

C. Mechanism of Induction

• Spin Density: Light drives create a Floquet spin density with distinct elliptical (for $d$-wave) or shifted circular (for $p$-wave) patterns in momentum space, directly linked to the Floquet sidebands.
• Triplet Pairing: The emergence of spin-triplet Cooper pairs is driven by photon-assisted transitions between Floquet sidebands. Odd-photon processes are shown to be the primary mechanism for generating new triplet correlations that do not exist in equilibrium.

4. Key Results

1. Floquet Spin Density:

   • In the normal state, light drives generate a rich structure of spin-degenerate nodes and spin-polarized Fermi surfaces.
   • For $d$-wave altermagnets, linearly polarized light (LPL) induces an effective Zeeman field that breaks spin degeneracy in a way dependent on the polarization angle $\phi_A$ relative to the magnetic order $\theta_J$.
   • The spin density can be used as a probe to identify the type and orientation of unconventional magnetism.

2. Floquet BdG Spectrum:

   • The superconducting spectrum exhibits multiple gaps opening at energies determined by photon energy differences ($\delta_n \hbar \Omega$).
   • In $d$-wave systems, gap centers are spin-dependent; in $p$-wave systems, they can be spin-degenerate depending on the magnetic orientation relative to the drive.

3. Symmetry of Cooper Pairs:

   • Table 1 in the paper provides a comprehensive map of the 8 symmetry classes.
   • Odd-photon spin-triplet pairs are the "smoking gun" of the light-matter-magnetism interplay. For example, in a driven $d$-wave magnet, the odd-photon triplet pairs exhibit even-frequency symmetry, whereas in a driven $p$-wave magnet, they exhibit odd-frequency symmetry (or vice versa depending on the specific class), a direct consequence of the underlying momentum parity.
   • The amplitude of these odd-photon pairs is strictly zero if either the light drive ($k_A$) or the magnetic field ($\alpha_q$) is zero.

4. Control via Polarization:

   • The amplitude of the induced Cooper pairs can be tuned by the angle between the linear polarization of the light and the magnetic order parameter.
   • For $p$-wave magnets, the pair amplitude follows a $\cos(\theta_J - \phi_A)$ dependence, allowing for the direct probing of the magnetic orientation angle $\theta_J$.

5. Significance and Implications

• New State of Matter: The work demonstrates that unconventional magnets can host light-induced superconducting states (specifically spin-triplet phases) that are entirely absent in equilibrium. This expands the landscape of Floquet engineering beyond topological insulators and Weyl semimetals.
• Symmetry Engineering: The ability to switch between different symmetry classes of Cooper pairs (singlet vs. triplet, even vs. odd frequency) simply by changing the light polarization or the magnetic order type offers a new degree of freedom for designing superconducting devices.
• Experimental Feasibility: The authors argue that these effects are observable using current technology:
  • Excitation: Mid-infrared (MIR) pump pulses ($\sim 10$ THz, intensities $10^{10} - 10^{14}$ W/m$^2$).
  • Detection: Time- and angle-resolved photoemission spectroscopy (TrARPES) for spin density in the normal state, and time-resolved transport (Andreev conductance) for triplet pairing in the superconducting state.
• Probing Tool: The distinct angular dependence of the induced pairs provides a novel spectroscopic method to determine the angular symmetry and orientation of unconventional magnetic orders, which is often difficult to measure directly.
• Robustness: The study addresses heating issues (a common limitation of Floquet systems) by suggesting that prethermal regimes in solid-state systems can sustain these states long enough for experimental detection.

In conclusion, this paper establishes a theoretical framework where Floquet engineering acts as a powerful tool to manipulate and induce exotic spin-triplet superconductivity in unconventional magnets, linking the fundamental symmetries of the magnet directly to the photon-assisted dynamics of the superconducting state.