Floquet-Multiple Andreev Reflections

The gist: The quartet and higher-order multipair processes yield characteristic Floquet-multiple Andreev reflection (Floquet-MAR) finite-bias conductance and noise resonances that are parameterized by the bias voltage and electrochemical potential.

Theoretical Framework

The physics of time periodic phenomena can be described by the Floquet formalism [1]. In voltage-biased multiterminal Josephson junctions, the Josephson relation causes superconducting phase differences to evolve periodically in time, thereby providing an intrinsic Floquet drive. The authors develop a Floquet-based transport description for two-dimensional multiterminal Josephson junctions where the electrochemical potential µN of the normal region serves as an independent spectroscopic control parameter in addition to the bias voltage V and magnetic flux Φ. They use Keldysh Green’s functions to calculate the current susceptibility χI = ∂ I(eV,µN)/∂ µN and identify resonances interpreted as Floquet processes dressed by multiple Andreev reflections, which they call Floquet-MARs.

Floquet-MAR Resonances

The term Floquet-MAR resonances refers to the finite-bias resonant features in the differential current and quantum noise susceptibilities, arising from Floquet processes dressed by MAR-like energy shifts. These broadened spectral structures are generated by coherent multichannel propagation through the ballistic continuum, rather than discrete bound-state levels. The Floquet-MAR finite-bias resonances in the current susceptibility can be understood as arising from the underlying Andreev-tube structure of the multichannel ballistic configurations. These Andreev tubes correspond to families of quasione-dimensional semiclassical trajectories formed by counterpropagating electron and Andreev-reflected hole components.

Higher-Order Multipair Processes

The framework is extended to higher-order multipair processes that emerge from the higher-order snake diagrams. These Floquet-MARs are viewed as dressing the quartet diagrams at higher-order in tunneling. The resulting overall energy-conserving higher-order snake diagrams form the Ooctets and Q'-quartets. These structures yield the DC-current susceptibility of the corresponding Floquet-MARs.

Experimental Signatures and Noise

The study combines a microscopic picture based on Andreev-tube trajectories, a spectroscopic theory of Floquet-MAR resonances in nonlinear conductance, and a quantum noise diagnostic of correlated multipair transport. The dimensionless spectral current iQ(ω) is deduced from the Keldysh Green’s function calculations. The quantum noise cross-correlations are proportional to the dissipative current, and the Fano factor is calculated as FN ≡ ∂SL,R(eV,µN)/∂ µN / ∂ IL(eV,µN)/∂ µN = 2N in the main text.

Key Results for Different Multiplets

The paper presents maps of the dimensionless current susceptibility ∂ i′(eV,µN)/∂ µN as a function of (log(eV), logµN) for Q-quartets (a), O-octets (b), and Q'-quartets (c). The results show expected µN = 0, 2eV Floquet-MAR finite-bias resonances in the current susceptibility for the O-octets at µN = 0 and V = 0 and µN = eV, 3eV in Fig. 4c. The final expression for the cosΨO-component of the O-octet supercurrent demonstrates Eq. (4) in the main text and shows a complex dependence on bias voltage and electrochemical potential.

Conclusion

The authors developed the concept of Floquet-MARs via Keldysh diagrammatic expansions, obtaining compact expressions for the current susceptibility χI and quantum noise Sa,b within this theory. The value of µN can be experimentally monitored by adjusting the voltage on an attached noninvasive tunneling tip. This finding opens up the possibility to engineer Floquet states in Josephson junctions based on 2D conductors.

Supplemental Material Details

The Supplemental Material organizes technical details, including Hamiltonians for the BCS superconductors, the 2D conductor, and interfacial hopping amplitude in section I. The methods detail how current and noise are calculated using Keldysh Green’s functions in section II. The calculations for Q-quartets, O-octets, and Q'-quartets are presented in sections V, VI, and VII respectively. The Fano factor is calculated in section VIII. A toy-model for the emerging Floquet resonances is presented in section IX. The Hamiltonians include the BCS Hamiltonian (Eq. 1), the 2D conductor Hamiltonian (Eq. 2), and the interfacial hopping Hamiltonian (Eq. 3). The calculation of Andreev pair transmission across a ballistic 2D conductor involves nonlocal advanced electron and hole Nambu Green’s functions in section IV D. The derivation of the Fano factor is demonstrated by showing that BN/AN = 2N in section VIII A. The final expression for the Q'-quartet supercurrent demonstrates Eq. (67) which shows a complex dependence on bias voltage and electrochemical potential. The calculation of the energy-integral for IQ' leads to a sum of terms B1 through B8. The final expression for the cosΨQ-component of the Q'-quartet supercurrent demonstrates Eq. (67).

References

[1] G. Floquet, Sur les equations différentielles linéaires à coefficients périodiques, Ann. de l’ Ecole Norm. Sup. 12, 47–88

[2] T. Oka and S. Kitamura, Floquet Engineering of Quantum Materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019)

[3] M. Holthaus Floquet engineering with quasienergy bands of periodically driven optical lattices, J. Phys. B: At. Mol. Opt. Phys. 49, 013001 (2016)

[4] A. Gandon, C. Le Calonnec, R. Shillito, A Petrescu, and A Blais, Engineering Control and Longitudinal Readout of Floquet Qubits Phys. Rev. Appl. 17, 064006 (2022)

[5] L.B. Nguyen et al., Programmable Heisenberg interactions between Floquet qubits Nat. Phys. 20, 240 (2024)

[6] S. Park et al., Steady Floquet–Andreev states in graphene Josephson junctions Nature 603, 421 (2022)

[7] J.C. Cuevas and H. Pothier, Voltage-induced Shapiro steps in a superconducting multiterminal structure Phys. Rev. B 75, 174513 (2007)

[8] A Freyn et al., Production of non-local quartets and phase-sensitive entanglement in a superconducting beam splitter Phys. Rev. Lett. 106, 257005 (2011)

[9] R. Melin et al., Simple Floquet-Wannier-Stark-Andreev viewpoint and emergence of low-energy scales in a voltage-biased three-terminal Josephson junction Phys. Rev. B 95, 085415 (2017)

[10] R. Melin et al., Engineering the Floquet spectrum of superconducting multiterminal quantum dots Phys. Rev. B 100, 035450 (2019)

[18] S. Duhot et al., Cross correlation of incoherent multiple Andreev reflections Phys. Rev. Lett. 102, 086804 (27)

[26] A.S. Rashid et al., Nonequilibrium Andreev resonances in ballistic graphene Andreev interferometers Phys.

1. textbf Floquet-MAR resonance spectroscopy via electrochemical potential control: The system can be tuned to probe nonequilibrium Floquet-Andreev spectra by independently controlling both bias voltage V and electrochemical potential µN, which is experimentally accessible via adjusting the voltage on an attached noninvasive tunneling tip.

2. textbf Higher-order multipair process characterization: The AI system can resolve complex spectral features like Ooctets and Q'-quartets by calculating their corresponding dimensionless current susceptibilities, such as those derived from Eq. (4) and Eq. (5), which reveal more complex spectra in Fig. 2d, f.

3. textbf Quantum noise diagnostics via Fano factor measurement: The system can quantify the granularity of Floquet-MAR resonances by measuring the Fano factor, which is calculated as FN ≡ ∂SL,R(eV,µN)/∂ µN / ∂ IL(eV,µN)/∂ µN = 2N in Eq. (6), providing a measure of quantum coherence.

4. textbf Engineering Floquet states in ballistic conductors: The AI system can be used to engineer Floquet states in Josephson junctions based on 2D conductors by analyzing the derived expressions for the current susceptibility, which are "compatible with Fig. 4a calculated for eV > 0 and µN > 0."

5. textbf Multi-scale transport analysis: The system can distinguish between different physical regimes by analyzing the results of averaging over scales, such as averaging over disorder or multichannels, which is necessary to capture the resonances, as demonstrated by comparing ballistic calculations with those involving averaging over the short-scale λF-oscillations.

Floquet theory describes quantum systems governed by time-periodic Hamiltonians, much as Bloch theory describes spatially periodic solids. In voltage-biased multiterminal Josephson junctions, the Josephson relation causes superconducting phase differences to evolve periodically in time, thereby providing an intrinsic Floquet drive. In this Letter, we consider three-terminal Josephson junctions formed on a ballistic two-dimensional normal conductor with a continuum of electronic states. We show that the quartet and higher-order multipair processes yield characteristic Floquet-multiple Andreev reflection (Floquet-MAR) finite-bias conductance and noise resonances that are parameterized by the bias voltage and electrochemical potential. This microscopic picture opens a route toward implementing and probing Floquet-MAR physics in ballistic multiterminal Josephson junctions.