Energy-level quantization in YBa2Cu3O7-x phase-slip nanowires

Significant progress has been made in the development of superconducting quantum circuits, however new quantum devices that have longer decoherence times at higher temperatures are urgently required for quantum technologies. Superconducting nanowires with quantum phase slips are promising candidates for use in novel devices that operate on quantum principles. Here, we demonstrate ultra-thin YBa2Cu3O7-x nanowires with phase-slip dynamics and study their switching-current statistics at temperatures below 20 K. We apply theoretical models that were developed for Josephson junctions and show that our results provide strong evidence for energy-level quantization in the nanowires. The crossover temperature to the quantum regime is 12-13 K, while the lifetime in the excited state exceeds 20 ms at 5.4 K. Both values are at least one order of magnitude higher than those in conventional Josephson junctions based on low-temperature superconductors. We also show how the absorption of a single photon changes the phase-slip and quantum state of a nanowire, which is important for the development of single-photon detectors with high operating temperature and superior temporal resolution. Our findings pave the way for a new class of superconducting nanowire devices for quantum sensing and computing.

(PSN) to refer to a superconducting nanowire that has a finite critical current, with the resistive state occurring due to phase slippage.
However, there is a significant difference between a JJ and a low-Tc PSN, because the physical mechanisms that determine the frequency of plasma oscillations have a very different nature. The zero-bias plasma frequency of a JJ ωp0 = (2eIc/Cħ) 1/2 (referred to as a Josephson plasma frequency) is given by the resonant frequency of an LJC circuit consisting of a Josephson inductance LJ = ħ/2eIccosφ and a junction capacitance C 15 , where Ic is the junction critical current, e is the electron charge, ħ is the reduced Planck constant and φ is the phase difference across the junction. When compared with tunnel JJs, the nanowires typically have very small intrinsic capacitances. In a low-Tc nanowire, the energy of Josephson plasma oscillations is much higher than the superconducting energy gap 2Δ, which makes such oscillations impossible, as shown in Figure 1a. In a pioneering study 14 ,Giordano proposed that the plasma frequency of low-Tc PSNs is limited by another physical mechanism and scales with Δ. This was experimentally confirmed by measurements of the crossover temperature between MQT and thermal activation (TA) escape mechanisms for different low-Tc PSNs 12,13 . As a result of the very high plasma frequency, it is therefore unlikely that more than one energy level exists in a low-Tc PSN. In contrast, in high-temperature (high-Tc) superconductors the superconducting energy gap is much larger and Josephson plasma oscillations are allowed, as shown in Figure 1a. In Figure 1| a, Energy diagrams for low-Tc and high-Tc nanowires. Energies below and above 2Δ are shown in blue and red, respectively. b, Scanning electron micrograph of an YBCO nanowire shaped by two FIB cuts across a microbridge.
The YBCO film, STO substrate and FIB cuts are shown in gray, blue and black, respectively. c, IV curve of a 55-nmwide YBCO nanowire at 4.2 K. Inset: temperature dependences of the average switching Is (orange circles) and retrapping Ir (blue squares) currents. addition to having a large superconducting energy gap of Δ = 25-30 meV 16 , YBa2Cu3O7-x (YBCO) nanowires meet the requirements for the "ideal" Josephson effect 2 : they exhibit Josephson behavior 17 and show a single-valued sine-like current-phase relationship, even at temperatures close to zero 18 .
Here, we show that YBCO nanowires are promising candidates for realizing superconducting quantum circuits. Our measurements of switching-current statistics for ultra-thin YBCO nanowires with phase-slip dynamics 19 provide clear evidence of ELQ in the nanowires.

Prospect of energy-level quantization in YBCO phase-slip nanowires
We performed electrical transport measurements on 2-μm-long 8.2-nm-thick YBCO nanowires with widths below 160 nm on a (100) SrTiO3 (STO) substrate. Figure 1b shows a scanning electron micrograph of a representative nanowire patterned using focused ion beam (FIB) milling across a 10μm-wide microbridge. All of the nanowires showed current-voltage (IV) curves that were characteristic of phase slippage. Based on the linear dependence of switching current on nominal nanowire width, we determined the effective nanowire width and thickness. Significantly, the switching-current statistics of nanowires with effective widths of below 100 nm cannot be explained using models developed for low-Tc nanowires. For a nanowire with an effective thickness and width of 4.3 and 55 nm, respectively, we observe current-voltage characteristics that show direct voltage switching from the superconducting to the resistive state and large current hysteresis over an extended temperature range of up to 18-20 K, as shown in Figure 1c.
The zero-bias plasma frequency of the 55-nm-wide nanowire is calculated as ωp0/2π ≈ 1.6 THz for a nanowire capacitance of C = 4.8 fF (see the Methods section) and on the assumption that the switching current is close to the critical current. The energy of the plasma oscillations ωp0ħ = 6.6 meV is significantly lower than 2Δ0 = 50-60 meV. Therefore, several quantized energy levels can exist in the nanowire. The detection of a macroscopic quantum tunneling event, which can be considered as particle escape from a well of a tilted washboard potential, depends on the quality factor Qn of the JJ/PSN in the resistive state. For Qn > 0.8382 the particle continues to move along a tilted washboard potential after escape, even when the washboard potential has local minima (referred to as a running state) 20 . This running state is observed as a jump from the superconducting to the resistive state and is straightforward to detect. Here, we use the motion of a particle in a tilted washboard potential as a mechanical analog of the RCSJ model. We calculate the quality factor of the nanowire in the resistive state as Qn = ωpsRnCps = 2ωpsRnCSΛQ ≈ 1, where Rn = 120 Ω is the phase-slip line resistance, ωps = 2eVs/ħ is the phase-slip oscillation frequency, Vs = 2.7 mV is the voltage switching amplitude, and Cps = 1 fF is the capacitance that shunts the phase-slip oscillations.
In order to calculate the capacitance Cps, we take into account that the phase-slip oscillations take place in a 2ΛQ-long section of the nanowire 9 , where ΛQ = 200 nm is the charge imbalance distance 19 .
Furthermore, we expect a crossover between TA and MQT escape mechanisms at a temperature of Tcr ≈ ωp0ħ/2πk = 12.4 K, where k is the Boltzmann constant 15 . Based on these calculations, we conclude that the energy levels in our YBCO phase-slip nanowires must be quantized and can be revealed using switching-current measurements.

Experimental results and discussion
We measured switching-current statistics for a 55-nm-wide 4.3-nm-thick YBCO nanowire, both under equilibrium conditions and under illumination with 77 K black body radiation (BBR) as well as visible light, inducing non-equilibrium state of the wire. In order to reach thermal equilibrium of the nanowire with external radiation, we kept the radiation shield surrounding the nanowire and the nanowire itself at the same temperature. We recorded 1500 IV curves with a current sweep rate of dI/dt = 0.55 mA/s for each of eight distinct temperatures in the 4.2-18 K temperature range, determined the switching current values and extracted the switching-current distributions (SCDs), which are shown in Figure   2a. At the highest temperature of 18 K, the SCD shows a single peak at a switching current of 156.12 μA. As the temperature is decreased to 16-14 K, the SCD peak shifts to higher switching currents, broadens and develops an asymmetry with a fine structure of closely-spaced peaks with spacings of 74±20 nA and 116±31 nA, respectively (enlarged SCDs are available in the Supplementary Information). Below 14 K, the SCDs show new peaks on either side of the main peak.
This three-peaked structure is most pronounced close to 10 K and gradually disappears at 6-4.2 K.
Within the framework of the RCSJ model, the switching and retrapping processes are affected by external noise in a similar way. We measure retrapping current distributions (RCDs) simultaneously Figure 2| a, Switching-current distributions for a 55-nm-wide YBCO nanowire. b-g, Washboard potential with quantum states (b-d) and corresponding switching current distributions (e-f) measured with a "non-adiabatic" sweep rate at temperatures below, close to and above the crossover temperature between the MQT and TA regimes.
with the SCD and find only a single peak with standard deviation σr = 62.8±3.6 nA for all temperatures, indicating that the SCD transformation is caused by intrinsic nanowire dynamics rather than by external noise (Supplementary Information).
The broadening of the SCD at temperatures slightly above the calculated crossover temperature Tcr = 12.4 K is consistent with the temperature dependence of SCDs for the tunnel JJ measured using a "non-adiabatic" current sweep rate 21  the lifetime in the excited state τ ≈ Qs/ωp0 > 1.5 msec at T ≈ Tcr. We attribute the presence of smaller peaks on both sides of the main peak in the SCD at T < Tcr to interaction of the nanowire plasma oscillations with an external resonant system, i.e., with geometric resonances of the 10-μm-long microbridge or the bow-tie antenna of our device (see Methods), with resonant frequencies of 1.2 THz and 40 GHz, respectively.
In order to probe the energy level structure in the PSN under equilibrium conditions, we applied external noise to the nanowire via the cables connecting the Dewar insert to the measurement equipment. We found that external noise has a significant effect on the SCDs, as shown in Figure 3a for three different temperatures at a current sweep rate dI/dt = 2.75 mA/s. At 4.2 K, the SCD consists of four nearly-equally-spaced peaks, which are marked Is1-Is4 in Figure 3a. At 5.9 K, the spacing between the peaks decreases slightly. At 8 K, only two peaks remain in the SCD. The noise-affected RCD shows only a single peak with σr = 51±3 nA (Supplementary Information), indicating that the applied low frequency noise does not cause the multiple peak structure in the SCD, but acts as a trigger for another physical mechanism, which is discussed below.
The nearly-evenly-spaced peaks Is1-Is4 in Figure 3a are signatures of tunneling from different energy levels, as shown in Figure 3b. If the damping is moderate, i.e., if the quality factor is close to unity, then a particle that escaped from the potential well due to external noise activation (magenta arrow) can be trapped in the upper energy level of the lower potential well. This retrapping process in a PSN is similar to that of thermal or quantum phase diffusion in underdamped tunnel-JJs 22,23 . When a particle is trapped in the upper energy level, it can decay to lower energy levels in the same well (blue arrows) or escape from the potential well by tunneling (red arrows), resulting in the presence of multiple peaks in the SCD. Within the framework of this model, the Is1-Is4 values are given by a system of equations ΔU(Isi) = ΔUtun + En(Isi) (1), where i is an integer and ΔU(Isi) and En(Isi) are the energy barrier height and the energy of the populated energy level corresponding to escape at current Isi, respectively. Here, we assume that the particle tunnels through the barrier when the barrier height for the populated energy level is decreased to ΔUtun. We approximate the nanowire by a harmonic quantum oscillator with energy levels En(Isi) = ωp(Isi)ħ(n+1/2), where ωp is the current-dependent plasma frequency. We also use the approximate expressions for barrier height ΔU(I) = (hIc/2πe)[(1-(I/Ic) 2 ) 1/2 -(I/Ic)arccos(I/Ic)] 24 and plasma frequency ωp(I) = ωp0[1-(I/Ic) 2 ] 1/4 15 . By solving this system of equations, we obtain ωp0/2π = 744 GHz. We analyze the stability of the solution under small perturbations of the initial parameters and find that the real-valued solution disappears when the Isi values are varied by more than 100 nA from their initial values. Hence, we conclude that the observed peaks Is1-Is4 can indeed be assigned to energy levels in a tilted washboard potential. We note that the calculated zero-bias plasma frequency is only approximately half of our previous estimate based on the nanowire capacitance and the crossover temperature. Using the numerically simulated eigenvalues En/ωpħ of the Josephson junction in a cubic approximation of the washboard potential, which take values of 0.5, 1.45, and 2.37 for n = 0, 1 and 2 24 , we obtain the ωp0/2π value as 1.5 THz, which is close to our theoretical estimate. The reduced number of peaks in the SCD at 8 K can be attributed to a decrease in energy level lifetime with increasing temperature when transitions to the lower energy level become more probable than tunneling through the barrier.
In order to probe the energy level structure of the nanowire using external radiation, we illuminated the device with 77 K BBR, which has a broad continuous spectrum peaked at a frequency of 8 THz and can populate energy levels up to 2Δ, resulting in a non-equilibrium state of the nanowire. We When the temperature is close to the crossover temperature (Figure 4b), the SCD has a single peak at the lowest current sweep rate dI/dt = 0.055 mA/s, which broadens towards higher currents with increasing current sweep rate, eventually transforming into a distribution with two peaks at dI/dt = 2.75 mA/s. This transformation of the SCD with current sweep rate reflects a transition from "adiabatic" to "non-adiabatic" measurements, as lower energy levels become accessible. Below the crossover temperature, at 10.9 K, we observe a strong dependence of the spectral weight of the peak Is2 on the current sweep rate, as shown in Figure 4c. At even lower temperature (5.4 K in Figure 4d), the spectral weight dependence of the Is2 peak with current sweep rate is less pronounced and the Is1 peak is hardly observable.
We interpret the difference in spectral weight between the peaks at Is1 and Is2 in terms of a population inversion resulting from the decay of higher-lying energy levels, similar to observations in superconducting quantum circuits based on tunnel JJs 25,26 . We identify the Is1 and Is2 peaks in Figure   4c-d with the ground and first exited energy levels, respectively. The effects of temperature and current sweep rate on the spectral weights of the Is1 and Is2 peaks can then be assessed using the tilted washboard potential model, as shown in Figure 4e. Tunneling from the ground, first and second excited energy levels occurs at currents I1, I2 and I3, respectively. Population inversion in the steady state is possible for currents I < I3 when three or more energy levels exist in the potential well. At Additional illumination of the nanowire using optical radiation with an LED led to seemingly counterintuitive results: the number of switching events with higher switching current was observed to increase with increasing LED intensity. Representative SCDs measured at different intensities of a blue LED with 460 nm wavelength for a current sweep rate of dI/dt = 0.55 mA/s are shown in Figure   5a. When the LED is turned off, the nanowire is still subject to 77 K BBR and switches into the resistive state in current range II (see Figure 5a), which corresponds to tunneling from the first exited state of the nanowire. At an LED intensity of ILED = 0.7 W/m 2 , switching events occur not only in region II but also in region I of the current range in Figure 5a, which corresponds to tunneling from the ground energy state. As the LED intensity rises, the number of switching events in current region I increases and some switching events start to appear in the gap between current regions I and II, forming a peak at I = 186.5 μA when ILED = 8.3 W/m 2 .
The interaction of optical photons with a superconducting nanowire has been studied widely because of its practical importance for the development of superconducting nanowire single-photon detectors.
Here, we use a refined hotspot model to analyze the effects of optical radiation on the YBCO nanowire 27 . An optical photon whose energy is much higher than the superconducting energy gap disrupts tens of Cooper pairs, resulting in the appearance of non-equilibrium quasiparticles. The absorbed photon induces a normal-state domain (hotspot) across the nanowire when the number of non-equilibrium quasiparticles reaches Nq = nsWd(πDτth) 1/2 (1-I/Ic), where ns is the local density of paired electrons, d is the nanowire thickness, D is the quasiparticle diffusion coefficient, τth is the quasiparticle thermalization time and Ic is the nanowire critical current 27

. If the entire photon energy
Eph is transferred to the quasiparticles, their actual number is given by Nq = Eph/Δ. The boundary for hotspot appearance can then be calculated as IHS = Ic -[jcEph/nsΔ(πDτth) 1/2 ], where jc is the critical current density. Photon absorption below and above IHS has qualitatively different consequences. For I < IHS, the photon creates non-equilibrium quasiparticles, but the normal-state domain across the nanowire does not appear. For I > IHS, photon absorption results in a hotspot across the nanowire, which leads to local collapse of the order parameter. The PSN evolves from this transient state towards a state with a phase-slip process, corresponding to switching of the nanowire from the superconducting to the resistive state, as described in Ref. [28].  Figure 5a, we highlight this current region III in red, in which the nanowire can switch to the resistive state only by the 460 nm wavelength photons. We assume that the critical current in the refined hotspot model corresponds to the switching current of the nanowire in the ground energy state. Figure 5b illustrates schematically the interaction of optical photons with a PSN that has quantized energy levels and is prepared in the excited state. When the LED is turned off, the nanowire switches to a resistive state at current I2 due to tunneling from the first excited energy state. At a low LED intensity (ILED = 0.7 W/m 2 in Figure 5a), the nanowire can absorb the photon before reaching current I2. Since photon absorption takes place at I < IHS, it creates a number of non-equilibrium quasiparticles, resulting in an increase in nanowire losses, which is shown in Figure 5b by a broadening of the energy levels and, hence, faster decay from the excited to the ground energy state. Switching into the resistive state then occurs at the higher current I1 by tunneling from the ground energy level. Therefore, the nanowire quantum state can be changed by single-photon absorption. When the radiation intensity is increased (corresponding to high LED intensity in Figure 5b and ILED = 1.4-5.5 W/m 2 in Figure 5a), the nanowire can absorb a second photon at bias currents of I > IHS, where hotspot conditions are met. As a result of the local collapse of the order parameter, the oscillating phase-slip process appears and the nanowire switches into the resistive state. Since the absorbed photon changes the number of phase-slip processes in the nanowire, i.e., the phase-slip state of the nanowire, we refer to this process as the phase-slip mechanism for photon detection.
At very high radiation intensities, switching of the nanowire occurs predominantly at the boundary of the hotspot region, forming a peak in the SCD at I = IHS. We observe this very high intensity regime at ILED = 8.3 W/m 2 in Figure 5a, where a peak at the edge of region III is visible. We treat the observed interaction of the optical photons with the YBCO nanowire as a single-photon process because the position of the photon-induced SCD peak corresponds to the single-photon energy and the radiation power is much lower than that required for the two-photon process, taking into account the short quasiparticle recombination time in YBCO 30,32 .
In this work, we have implicitly considered a nanowire made from YBCO, which has dx2-y2-wave symmetry of the order parameter as a fully gapped superconductor. We find that a small (Nq ≤ 100) number of non-equilibrium quasiparticles generated by an optical photon is significantly larger than the number of equilibrium quasiparticles in the nanowire, resulting in fast decay from the excited to the ground state. This behavior is expected for a fully gapped superconductor with an exponentially small number of equilibrium quasiparticles at a temperature well below the critical temperature.
Deviations from dx2-y2-wave symmetry in our YBCO nanowires can arise from size or doping effects, which have recently been observed in different cuprate superconductors 33,34 .

Conclusions
We have fabricated sub-100-nm-wide YBCO nanowires with phase-slip dynamics and measured their switching-current statistics under equilibrium and non-equilibrium conditions. Our experimental data show energy-level quantization in YBCO phase-slip nanowires. The YBCO nanowires have a high crossover temperature between thermal activation and quantum regimes of 12-13 K and their lifetime in the excited state exceeds 20 ms at 5.4 K, which is at least one order of magnitude longer than in low-Tc tunnel Josephson junctions 1 . We also show that the absorption of a single-photon changes the quantum and phase-slip states of YBCO nanowires. Our findings demonstrate that phase-slip YBCO nanowires are promising systems for quantum technology applications, including quantum sensing and computing.

Methods
Nanowire fabrication. YBCO nanowires were fabricated from an 8.2 nm (7 unit cell) thick YBCO film deposited on a TiO2-terminated (100) STO substrate by dc sputtering at high (3.4 mbar) oxygen pressure. YBCO deposition followed a procedure that is described elsewhere 35 . 100-nm-thick Au contact pads were deposited ex situ using room temperature dc magnetron sputtering with a shadow mask. Following contact pad deposition, nanowires were fabricated in a two-stage process. In the first stage, 10-μm-wide 10-μm-long microbridges integrated with a bow-tie antenna and leads were patterned using optical UV contact lithography with a PMMA resist and Ar ion beam etching. In the second stage, 2-m-long nanowires aligned along STO crystallographic axes were fabricated across the microbridges with two cuts made with focused ion beam (FIB) milling using a Au/PMMA protective layer. A sketch of the device is shown in Extended Data Fig. 1a. More details about the patterning process can be found in Ref. 19 .
Experimental setups. We performed the measurements using two experimental setups. The first experimental setup was based on a liquid helium storage Dewar insert filled with He exchange gas, in which the nanowire and surrounding radiation shield had the same temperature. The second setup was based on an HLD-5 liquid helium cryostat (Infrared Laboratories, Inc.). The sample was placed on a sample holder mounted on the 4 K stage of the cryostat and shielded by a radiation shield with a quartz window. The sample was illuminated through the window using continuous optical radiation emitted by LEDs and 77 K blackbody radiation from a 77 K radiation shield. The LEDs were placed in front of the window at a distance of 2 cm from the substrate and cooled to 77 K. The sample holder temperature was maintained to an accuracy of ±5 mK at 4.2 K and ±20 mK at 20 K for the Dewar-insert-based setup and ±10 mK for the cryostat-based setup over the whole temperature range. The wiring inside the Dewar insert was made with twisted-pair cables and had a bandwidth of 3.9 MHz. The nanowires inside the cryostat were connected to room temperature measuring equipment using high-frequency SMA coax cables and a 10 GHz probe. We used battery-operated low-noise analog electronics with a 100 kHz frequency bandwidth to sweep the bias current and amplify the voltage across the nanowire. The root-mean-square noise of the current source was 1 nA. The output signals of the analog electronics, proportional to the current and voltage across the nanowire, were digitized using a simultaneous 16-bit data acquisition board DT9832 (Data Translation). All electrical connections, apart from the HF coax cables, were filtered using low-frequency feedthrough filters. Cables between the analog electronics and cryogenic units were as short as possible to eliminate electromagnetic interference. The frequency spectrum of the output voltage signal was controlled before the measurement to ensure that no low-frequency external noise was present in the measurement system.
Switching-current measurement. In order to measure the switching current-and retrapping-current statistics, the bias current through the nanowire was ramped linearly up and down over the range (0.85-1.05)Is and IV curves were recorded with 2·10 4 points per curve. The spacing between independent current points of 15.3 nA was determined by the 16-bit resolution of the data acquisition board. The switching and retrapping currents were determined by post-processing of the recorded data. Standard deviations of the retrapping current were computed in the standard way.
Nanowire capacitance calculation. The nanowire layout shown schematically in Extended Data Fig. 1b is similar to that of a coplanar waveguide with ground. As a result of this similarity, we used a coplanar waveguide calculator 10 to calculate the nanowire capacitance. The following parameters were used for the capacitance calculations: spacing S = 200 nm, substrate thickness H = 1 mm, and dielectric index of the STO substrate  = 300. The spacing S consisted of the 60-nm-wide FIB cut and two 70-nm-wide damaged insulating regions of YBCO on the sides of the FIB cut 19 . By using the coplanar waveguide calculator 36 , we obtained a nanowire specific capacitance per unit length of CS = 2.4 fF/μm and a total capacitance of the 2-μm-long nanowire of C = 4.8 fF. Supplementary Fig. S1| Switching-current distributions for a 55-nm-wide YBCO nanowire measured under equilibrium conditions at temperatures of 14 and 16 K. Red arrows indicate the bias current value at which the height of the energy level coincides with the barrier height. Supplementary Fig. S2| Retrapping-current distributions for a 55-nm-wide YBCO nanowire measured under equilibrium conditions. The numbers above the peaks are the nanowire temperature and the shift along the current axis.