Steady Floquet-Andreev States Probed by Tunnelling Spectroscopy

Engineering quantum states through light-matter interaction has created a new paradigm in condensed matter physics. A representative example is the Floquet-Bloch state, which is generated by time-periodically driving the Bloch wavefunctions in crystals. Previous attempts to realise such states in condensed matter systems have been limited by the transient nature of the Floquet states produced by optical pulses, which masks the universal properties of non-equilibrium physics. Here, we report the generation of steady Floquet Andreev (F-A) states in graphene Josephson junctions by continuous microwave application and direct measurement of their spectra by superconducting tunnelling spectroscopy. We present quantitative analysis of the spectral characteristics of the F-A states while varying the phase difference of superconductors, temperature, microwave frequency and power. The oscillations of the F-A state spectrum with phase difference agreed with our theoretical calculations. Moreover, we confirmed the steady nature of the F-A states by establishing a sum rule of tunnelling conductance, and analysed the spectral density of Floquet states depending on Floquet interaction strength. This study provides a basis for understanding and engineering non-equilibrium quantum states in nano-devices.

physical systems by simply tuning the intensity, frequency and polarisation of the light.
Therefore, there has been a great deal of research effort expended on the realisation and manipulation of a wide variety of long-sought-after quantum states, such as chiral topological orders with no equilibrium counterpart 3 , including Floquet Majorana fermions 4 and a new braiding protocol in the energy dimension 5 . For example, the realisation and control of Floquet dynamics have been reported in both photonic and ultra-cold atomic systems 6,7 . Another important class of platforms are solid-state systems, in which transient Floquet states have been mainly investigated. As Floquet interaction strength is proportional to the ratio of the electric field to the square of the frequency of light 8

, previous condensed matter experiments realising
Floquet states at the optical frequency of 30-50 THz have applied pulsed lasers to achieve a large electric field of 2.4-4.0 × 10 7 Vm −1 (Refs. [9][10][11] ). However, Floquet states persist no longer than an order of picoseconds due to the transient nature of pulsed laser and their inherently short lifetime. This short timescale makes it difficult to fully investigate and understand these novel light-driven states and exploit them for practical applications. Another critical issue affecting most experimental realisations of Floquet states is heating by the energy absorption from time-periodic driving. The driving increases the entropy density of the system and eventually brings it to a featureless state with no local correlations 3,12 . Although a number of ways to reduce or avoid heating effects have been suggested 3,[13][14][15][16][17][18] , the heating problem still remains in photonic and ultra-cold gas systems because they are well isolated from the outside environment. On the other hand, condensed matter systems have well-defined electron cooling paths, such as electron-phonon coupling and Wiedemann-Franz cooling of conducting electrons. However, a large electric field in an optical domain, which has been used in previous condensed matter experiments, requires pulsed measurements to avoid heating problems. One 4 strategy to avoid thermal problems is to use lower frequency driving that requires a smaller electric field. There have been experimental studies in the microwave domain, but they relied on indirect methods for probing Floquet states either by tracing their time evolution 19,20 , qubitresonator resonance conditions 21 , magnetic resonance conditions 22 or AC Josephson effect [23][24][25] . process when bias voltage V < 0 is applied between the tunnelling probe and graphene, such that the occupied DOS peak of the tunnel probe matches the empty DOS peak ( + ) of ABS.
e, Colour-coded plot of differential conductance dI/dV as a function of V and B for device 1.
The top horizontal axis shows the superconducting phase difference corresponding to B.
Circles and solid lines represent the dI/dV peaks and corresponding theoretical fittings in a short junction limit, respectively. The line cut at = 0 plotted in the right panel shows dI/dV peaks coming from upper ( + ) and lower ( − ) bands of ABS.
In contrast to previous studies, we report the experimental realisation of truly steady Floquet-Andreev (F-A) states based on Andreev bound states formed in a graphene Josephson junction (GJJ) and their spectra based on direct tunnelling spectroscopy. Here, we generated steady F-A states by continuously applying a monochromatic microwave drive and probed them by superconducting tunnelling spectroscopy with high energy resolution. We investigated the behaviour of the F-A states at various microwave frequencies and powers, and showed that their spectral features agreed well with our theoretical calculations. For example, we corroborated the steady nature of the F-A states by establishing a sum rule of the measured tunnelling conductance. This study clearly demonstrated steady Floquet states and will have a substantial impact on different areas of physics, including topological condensed matter, cold atoms in optical traps and nonequilibrium quantum statistical physics. Our technique, when 6 combined with rapidly developing microwave technologies, can be extended to Floquet-based novel quantum device applications.
A GJJ consists of a non-superconducting graphene layer sandwiched between two superconductors, as shown in Fig. 1a. The GJJ allows the Josephson supercurrent by forming Andreev bound states (ABSs) of electron-and hole-like quasi-particles in the graphene, which are correlated by Andreev reflections at the graphene/superconductor interfaces. In the short junction limit where GJJ channel length L is much shorter than the superconducting coherence length = ℏ / ABS , a single pair of ABSs forms within the superconducting gap of ohmiccontacted Al electrodes ABS and oscillates with the macroscopic quantum phase difference between two superconductors as ± ( ) = ABS √(1 − sin 2 ( /2)) [26][27][28][29] . Here, ℏ is a reduced Planck's constant, =10 6 ms −1 is the Fermi velocity of graphene, and D is the contact transparency. The schematic in Fig. 1b shows the oscillation of ABS in a short junction limit and the corresponding density of state (DOS) at a fixed .
In this study, we performed tunnelling spectroscopy on ABS formed in graphene using Al superconducting tunnel contact with the graphene edge, as shown schematically in Fig. 1c (see Supplementary Fig. 1). Direct deposition of Al onto the graphene edge forms a sufficiently high potential barrier for the tunnelling probe due to the large inter-atomic distance between graphene and Al atoms 30 . In this structure, = 2π / 0 was controlled by the external magnetic flux threading the superconducting ring in which GJJ was embedded. Here, 0 = ℎ/2 is the superconducting flux quantum with Planck's constant h and electron charge e < 0.
We measured the voltage difference (V) between the Al tunnel probe and graphene while biasing the current (I). The output impedance of the current source (1 GΩ) was much larger than the largest tunnelling resistance (16 MΩ) measured in this experiment. We obtained the tunnelling differential conductance (dI/dV) as a function of V, which represents the convolution of DOS of the ABS in GJJ and that of a tunnel probe 28 . Figure

9
To generate steady F-A states of GJJ, the devices were continuously irradiated with microwaves and the tunnelling spectrum of graphene was measured using a superconducting tunnel probe.
As shown in Fig. 2a, additional peaks in dI/dV emerged above and below the original ABS peaks as the microwave power P increased. The average voltage spacing between the two adjacent dI/dV peaks ΔV = 51.7 ± 11.0 µV at P = -5. Nevertheless, the overall agreement with the sum rule supports the steadiness of F-A states.
We proceeded to calculate the differential conductance dI/dV theoretically, and compared the results with experimental data. Two major factors that determine tunnelling conductance are the spectral weight of F-A states and background differential conductance (dI/dV)BG from the The experimental data gradually deviates from the theoretical fitting at P > −7 dBm ( ≈ 1.7); this can be attributed to, for instance, the significant heating of electrons that leads to a non-Fermi-Dirac distribution of electrons. We now discuss the superconducting phase dependence of F-A states. Figure 3a shows synchronised oscillations of the original ABS and Floquet replica states with phase difference , which further confirms that the F-A states are replications of the original ABS. For better visualisation of the oscillations, we subtracted the background obtained by averaging dI/dV over several peaks in V (see Supplementary Fig. 8). With increasing P, the peak values of the background subtracted differential conductance (d /d ) BS of Floquet replica (n ≠ 0) states became larger, while that of the original ABS (n = 0) became smaller (Fig. 2e). In addition, the oscillation amplitude, which corresponds to ABS , becomes smaller with increasing P and vanished at P = 0 dBm. This can be explained by the significant electron heating due to strong microwave irradiation, such that electron temperature reaches the critical temperature of the ohmic-contacted Al superconductor (see Supplementary Fig. 6). Solutions of the timedependent Schrodinger equation of the GJJ under the unpolarised electromagnetic field allowed us to compute the quasi-energy spectrum of the F-A states. We found that our theoretical calculations well reproduced the observed oscillation of the F-A spectrum, as shown in Fig. 3a.
Here, dI/dV for V < 0 represents only the replicas of + ( ), as electrons in the superconducting tunnel probe can hop only to replicas of + ( ) that are empty (see Supplementary Fig. 9).
Finally, we discuss the microwave frequency dependence of F-A states. The power dependence of (dI/dV)BS at various microwave frequencies is shown in Fig. 3b. For all frequencies, the decreasing intensity of the peak from the original ABS (n = 0, red arrow) with increasing P is accompanied by increasing intensity of the Floquet replica states (n ≠ 0, white arrows), as expected from the sum rule and Bessel function behaviour. The overall peak position shift to lower voltages with increasing P is attributed to the heating of electrons. We found that the voltage spacing ΔV between adjacent (dI/dV)BS peaks was linearly proportional to f (Fig. 3c) with a slope of 4.34 ± 0.46 µV/GHz. This value is close to the slope expected for F-A states, h/e = 4.14 µV/GHz, confirming the nature of F-A states.
In sum, we have realised the steady F-A states in a Josephson junction device by continuous microwave irradiation without significant heating, and directly measured their energy spectra by superconducting tunnelling spectroscopy. This technique is readily applicable to other lowdimensional topological materials, such as topological insulators and topological semi-metals, for studying and engineering topological Floquet physics 5  Theoretical calculation of differential conductance. We theoretically calculated the timeaveraged differential conductance of the ABS. We first modelled the GJJ without any external driving and obtained the ABS. We then included microwave driving within the graphene region and obtained the Floquet states for a given polarisation of the light. Next, we computed the spectral function and differential conductance of the Floquet states using the Floquet-Kubo formula. 3 Finally, we averaged the differential conductance over the polarisation for comparison with our experiments where the microwaves were unpolarised. For further details, please see Supplementary Fig. 7 and Supplementary Section 9.
We first calculated the ABS spectra and corresponding wavefunctions using standard methods. 40,41 Essentially, the model is equivalent to "a (Dirac) particle in a box" under the superconducting boundary conditions. Formally, the Hamiltonian of graphene with superconducting electrodes is given as the Dirac-Bogoliubov-de Gennes (DBdG) equation. 42 To reflect the finite width of the GJJ, we allow several conduction modes to appear in the spectrum of ABS. Each mode is identified with a given y-directional momentum, i.e. the momentum along the width of the GJJ. We found that the 12 lowest modes are sufficient to explain the experiment, which gives small y-directional momentums compared to x-directional momentums. We numerically obtained the wavefunctions | , ⟩ and energy spectrum of the Thus, the time-averaged differential conductance is given as Derivation of the sum rule of differential conductance. Here, we describe our derivation of a sum rule for differential conductance, by following methods in the literature 3, 28, 31 that discuss closely related Floquet sum rules. The detailed derivation of the sum rule is given in the section 9 of the Supplementary Information.
The sum of the time-averaged differential conductance can be computed as This can be computed using the tunnelling current obtained above, as Note that ∫ ∞ −∞ ∑ ( ) is the sum of the density of states of the Floquet states; it is known to respect a sum rule 31 and should therefore satisfy a sum rule (see Supplementary Section 9-3 for details). Finally, invoking the particle-hole symmetry of the overall spectrum, we conclude that ∫ ∞ 0 ⟨ ⟩ = 1 2 also satisfies the sum rule, which was confirmed in our experimental data.

Data availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.