Persistent Josephson Phase-Slip Memory with Topological Protection

Superconducting computing promises enhanced computational power in both classical and quantum approaches. Yet, efficient schemes for scalable and fast superconducting memories are still missing. On the one hand, the large inductance required in magnetic flux-controlled Josephson memories [1, 2] impedes device miniaturization. On the other hand, the use of ferromagnetic order to store information [3–8] often degrades superconductivity, and limits the operation speed to the magnetization switching rate of a few GHz. Here, we overcome the above limitations through a fully superconducting memory cell based on the topological transition driven by hysteretic phase slips [9, 10] existing in aluminum nanowire Josephson junctions [11, 12]. Our direct and nondestructive read-out scheme, based on local tunneling spectroscopy, ensures reduced dissipation (. 40 fW) and estimated response time below . 30 ps thereby yielding a maximum energy per bit consumption as low as ∼ 10−24 J. In addition, the memory topological index can be directly read by robust phase measurements thus further lowering dissipation whilst maximizing the stability against magnetic noise. The memory, measured over several days, showed no evidence of information degradation up to ∼ 1.1 K, i.e., ∼ 85% of the critical temperature of aluminum. The ease of operation combined with remarkable performance elects the Josephson phase-slip memory as an attractive storage cell to be exploited in advanced superconducting logic architectures. A Josephson junction (JJ) consists of a localized discontinuity (weak link) in the order parameter of two superconducting electrodes [13], where the Cooper pairs transport is ruled by the macroscopic quantum phase difference (φ) across the junction. Weak links are typically realized in the form of a thin insulator, a normal metal wire or a narrow superconducting constriction [11, 13]. The junction current-phase relation (CPR) strongly depends on the structural attributes of the constriction, i.e., on how its effective length (d, i.e., the distance between the superconducting leads), width (w), and thickness (t) compare to the superconducting coherence length (ξw) [11]. In a fully superconducting one-dimensional

Superconducting computing promises enhanced computational power in both classical and quantum approaches. Yet, efficient schemes for scalable and fast superconducting memories are still missing. On the one hand, the large inductance required in magnetic flux-controlled Josephson memories [1,2] impedes device miniaturization. On the other hand, the use of ferromagnetic order to store information [3][4][5][6][7][8] often degrades superconductivity, and limits the operation speed to the magnetization switching rate of a few GHz. Here, we overcome the above limitations through a fully superconducting memory cell based on the topological transition driven by hysteretic phase slips [9,10] existing in aluminum nanowire Josephson junctions [11,12]. Our direct and nondestructive read-out scheme, based on local tunneling spectroscopy, ensures reduced dissipation ( 40 fW) and estimated response time below 30 ps thereby yielding a maximum energy per bit consumption as low as ∼ 10 −24 J. In addition, the memory topological index can be directly read by robust phase measurements thus further lowering dissipation whilst maximizing the stability against magnetic noise. The memory, measured over several days, showed no evidence of information degradation up to ∼ 1.1 K, i.e., ∼ 85% of the critical temperature of aluminum. The ease of operation combined with remarkable performance elects the Josephson phase-slip memory as an attractive storage cell to be exploited in advanced superconducting logic architectures.
A Josephson junction (JJ) consists of a localized discontinuity (weak link) in the order parameter of two superconducting electrodes [13], where the Cooper pairs transport is ruled by the macroscopic quantum phase difference (ϕ) across the junction. Weak links are typically realized in the form of a thin insulator, a normal metal wire or a narrow superconducting constriction [11,13]. The junction current-phase relation (CPR) strongly depends on the structural attributes of the constriction, i.e., on how its effective length (d, i.e., the distance between the superconducting leads), width (w), and thickness (t) compare to the superconducting coherence length (ξ w ) [11]. In a fully superconducting one-dimensional * nadia.ligato@nano.cnr.it † e.strambini@sns.it ‡ federico.paolucci@nano.cnr.it § francesco.giazotto@sns.it JJ (w, t ξ w ) the CPR evolves from the single-valued distorted sinusoidal characteristic, typical of the shortjunction limit (d ξ w , Fig. 1a), to the multi-valued function obtained in the long regime (d ξ w , Fig. 1b) [11]. In the latter scenario, the Josephson current (I s ) flowing through the junction follows a hysteretic evolution in ϕ due to the presence of two energetically-stable states (corresponding to even and odd topological index) well separated by a strong phase-slip energy barrier [9], and accessible by going through the phase contour in the forward or backward direction (see Fig. 1b). These two states are topologically discriminated by the parity of the winding number of the superconducting phase along the wire [9,14] which reflects into opposite directions of the supercurrent flow [10], as shown in Fig. 1b. The transition between the even and odd state requires to pass through a gapless condition via a 2π phase-slip occurring in the superconducting nanowire [11,12]. We therefore take advantage of this topologically-protected hysteresis loop and well defined parity states to implement a robust and permanent superconducting memory: the Josephson phase-slip memory (PSM).
The design of a proof-of-concept PSM requires an architecture enabling the tuning of the phase ϕ, and the definition of an efficient readout scheme. The most effective way to finely control ϕ is by inserting the JJ in a superconducting loop, where an external magnetic field gives rise to a total flux Φ piercing the ring area. Stemming from fluxoid quantization [16], the superconducting phase difference across the weak link is given by ϕ = 2πΦ/Φ 0 (where Φ 0 2.067 × 10 -15 Wb is the flux-quantum). The phase together with the topological index determines the amplitude of the gap in the local density of states (DOS) of the wire [12] which can be probed with a metallic electrode tunnel-coupled to the middle of the junction (see the sketch on top of Fig. 1c), thereby implementing a superconducting quantum interference proximity transistor (SQUIPT) [17]. The DOS determines the tunneling current (I) flowing through the probing electrode which can vary from a continuous function of Φ for a short junction (Fig. 1c) to a discontinuous characteristic reflecting the hysteresis in the long regime (Fig. 1d). The parity of the topological index and Φ then determines the amplitude of the tunneling current that can read the logic [0] and [1] states of the PSM cell (see Fig. 1d). Sketch of the current-phase relation (Is vs ϕ) of a S-S'-S weak link in the short (a) and long (b) junction regime. The CPR evolves from a deformed sinusoid to a multi-valued function as the junction length increases. The transition between the two topologically-protected states (corresponding to even and odd topological index) [9] is due to phase-slips occurring in the wire [11,15], and corresponds to a vertical jump between the two current branches (indicated by coloured arrows). Dependence of the tunnel current (I) at fixed bias voltage (V ) on the normalized applied magnetic flux (Φ/Φ0, with Φ0 the flux quantum) for a SQUIPT in the short (c) and long (d) junction regime. In the latter case, the current evolution shows a hysteretical profile, which stems from the multi-valued CPR.
Top: scheme of a voltage-biased DC SQUIPT in a two-wire configuration. Φ is the magnetic flux piercing the ring. c Pseudo-colour scanning electron micrograph of a typical PSM. An Al nanowire (green) is inserted in a micron-size Al ring (yellow), whereas an Al0.98Mn0.02 probing electrode (red) is tunnel-coupled to the middle of the nanowire and to a second Al electrode (green) to allow the memory operation. Inset: blow-up of the weak-link region. The passive replicas due to the three-angle shadow-mask metal deposition are visible.
nm) with a length d ∼ 400 nm, embedded in a micronsized 70-nm-thick Al ring (yellow). In addition, a 20nm-thick normal metal electrode (red, Al 0.98 Mn 0.02 ) is tunnel-coupled to the center of the wire (with normalstate resistance R t1 65 kΩ). To measure the tunneling current, a second Al lead (green) is tunnel-coupled to the normal metal electrode (with normal-state resistance R t2 90 kΩ) [18]. Based on the actual device structural parameters, we estimate the ratio d/ξ w,0 5, where ξ w,0 80 nm the zero-temperature coherence length, thereby providing the frame of the long-junction regime [11] (see Methods for details).
To test the PSM transport properties and assess the operation parameters of the memory cell, we first performed a preliminary magneto-electric characterization at bath temperature T = 25 mK. Figure 2a shows the current vs voltage characteristics [I(V)] of a typical device measured at Φ = 0 (black curve) and Φ = 0.5Φ 0 (orange curve). At zero magnetic flux, the quasiparticle current is suppressed for |V | 400 µV due to the presence of the series connection of two S-I-N tunnel junctions. This behavior is consistent with the value of the bulk Al gap of both the read-out lead (∆ Al 200 µeV) and the weak link [∆ w (Φ = 0) 200 µeV]. The latter can be modulated by the external magnetic flux [17,18], showing a reduction of about 50% at Φ = 0.5Φ 0 which can be deduced from the I(V) characteristic (orange line), ∆ w (Φ = 0.5Φ 0 ) 100 µeV (see Fig. 6 for more details).
Differently from short-junction SQUIPTs [18,19], the I(Φ) characteristic is not only Φ 0 -periodic, but is also strongly hysteretic in Φ. This is highlighted in Fig. 2b, where the tunnel current measured at V = 300 µV as a function of increasing (forward, purple trace) and decreasing (backward, green trace) magnetic flux is shown. The forward trace exhibits periodic maxima followed by sudden jumps corresponding to the nucleation of phaseslip centers in the superconducting nanowire [11,12,20]. Accordingly, the backward trace evolves in a totally specular fashion. The stability and high reproducibility of these hysteretic curves is the key feature for the memory cell operation.
In order to find the optimal operating parameters for the PSM, we acquired the I(Φ) characteristics for different values of bias voltage, as shown in Fig. 2c. The hysteresis loop drawn by back and forth I(Φ) exhibits a reduction of its width (δΦ) by increasing V, as displayed in Fig. 2d. This trend is due to local overheating in the weak link induced by the quasiparticle current flowing through the probing junction, which increases ξ w [13] thereby deviating the CPR towards the single-valued non-hysteretical form [11,12].
The relative separation between the two I(Φ) branches can be quantified by a parameter (ζ) defined as the ratio between the current drop at the the phase-slip transition and the current at the hysteresis crossing point, ζ = δI/I cp . Similarly to δΦ, the increase of V induces a monotonic reduction of ζ, as shown in Fig. 2e.
The implementation of a PSM requires to bias the su-  perconducting loop with a flux (Φ B ) comprised within the hysteretic domain (Φ B min , Φ B max ). Writing (erasing) operations are performed by lowering (increasing) the total flux below (above) the hysteretic domain, as sketched in Fig. 3a. The two logic states [1] and [0] encoded in the topological index of the PSM (odd or even) can be simply read through the amplitude of the tunneling current measured with a low-power non-destructive scheme. Figure 3b shows the writing/erasing operations in the continuous read-mode, i.e., when a bias voltage V = 300 µV is permanently applied. The bias flux is set at 0.54Φ 0 , i.e., just above the crossing-point of the hysteresis (see Fig. 2c). The memory is initialized in the [0] state corresponding to a low current of I 43 pA. By applying a negative flux pulse down to Φ W = 0.33Φ 0 , the PSM logic state suddenly jumps to [1] as detected by the current jump to I 90 pA. Conversely, the logic state [0] is recovered via a positive erasing flux pulse up to Φ E = 0.75Φ 0 . The device unequivocally shows the typical behavior of a memory cell upon many erasing/writing cycles.
The ability of the PSM to retain the data even when the bias voltage is temporarily turned off (non-volatile memory) is displayed in Fig. 3c. Here, the reading voltage is set to V only during the readout operation. Non-volatility is a fundamental requirement for energy saving since power is dissipated only during the readout procedure without losing the stored information. The readout dissipation obtains values as low as P [0] 25 fW for logic state [0], and P [1] 40 fW for logic state [1]. Moreover, the operations have been performed several times always with comparable results, highlighting the relevant endurance of the device.
The intrinsic bandwidth limits of the PSM read and write operations were not achieved due to the strong cut-off (∼ 800 Hz) of our cryogenic filters and the large inductance of the superconducting coil used for the writing/erasing operations. However, a switching time of ∼ 1 ps for the writing/erasing process is expected, which is typical for small superconducting loops, while a readout time of τ R 30 ps can be estimated from the charging time of the tunnel junctions (see Methods for details). The PSM speed is therefore expected to be on par with current state-of-the-art superconducting memories [2,4,[6][7][8].
The combination of low power dissipation and fast access time yields tiny energy required per bit J [0] = P [0] τ R 0.8 yJ and J [1] = P [1] τ R 1.3 yJ for the readout operation of state [0] and [1], respectively. Notably, these values are several orders of magnitude smaller than any superconducting memory reported so far, and well below the requirements of rapid single flux quantum (RSFQ) logic [2,3,6]. The robustness of the PSM against flux fluctuations is tested by superimposing to the working biasing flux a sizable sinusoidal signal (Φ AC , see Fig. 4a). The PSM shows optimal stability with respect to flux oscillations, as shown in Fig. 4b for V = 300 µV and Φ B = 0.56Φ 0 . The memory preserves the stored state, and keeps the readout of the two logic states well separated for Φ AC 0.04Φ 0 , that is ∼ 80% of the separation between the working and the minimum erasing flux. Interestingly, thanks to the opposite magnetoconductance of the tunnel barrier in the two topological states, the AC flux modulation reflects in the AC response of the tunneling current that acquires a phase shift of π between the two logic states [0] and [1]. This phase shift provides an alternative and efficient method to probe the parity of the JJ winding number, which is not affected by the position of Φ B within the hysteretic domain, or by the low visibility of the DC readout signal (see Figs. 9 and 10 for more details). Figure 4c displays a persistency test of the PSM realized with the phase-based readout. The memory is initialized to logic state [1], and the readout is performed every 4 hours. No sign of signal degradation has been observed even after ∼ 3 days of measurement. This confirms the stability of the PSM as a permanent memory, which is guaranteed by the vanishing phase-slip rate (∼ e −10 3 s−1) expected at low temperature [12,20].
High temperature can degrade, in principle, the performance of PSM by increasing ξ w [13] thereby lowering the JJ effective length, and driving the nanowire junction towards the non-hysteretic single-valued CPR which is expected to occur for d 3.5ξ w [11,15]. In addition, thermal activation can substantially increase the phase-slip rate [12]. Figure 5a shows the evolution of the hysteresis loop at several bath temperatures (T). The hysteresis progressively fades out by increasing T and persist up to 1.1 K, which corresponds to ∼ 85% of the nanowire critical temperature, with δΦ reduced to ∼ 12% with respect to base temperature (see Fig. 5b). Figure 5c shows ζ vs T for different values of bias voltage, and highlights the drop of ζ by increasing T. The observed high-temperature operation of the PSM indicates a substantial protection of the topological state even in the presence of a sizable amount of hot quasiparticles [9].
In summary, we have envisioned and demonstrated an original persistent Josephson phase-slip memory cell which takes advantage of fluxoid quantization to codify two logic states in the topological index of the system, i.e., the superconducting winding number [14]. Differing from conventional superconducting loops, here the separation between the two topological states is provided by the large phase-slip barrier, which is unique to long superconducting JJs [9,12]. The memory exploits conventional superconductors thereby avoiding the use of complex ferromagnetic metals typical of present superconducting memories [3][4][5][6][7][8]. Notably, the PSM is characterized by reduced readout dissipation ( 40 fW), short estimated response time ( 30 ps), and ultralow energy per bit (∼ 10 −24 J). Moreover, the PSM state shows endurance, persistence, and high-temperature operation (up to ∼ 1.1 K), only limited by the Al critical temperature. The use of vanadium [19] or niobium [21], therefore, could push memory operation above liquid He temperature, and further improve miniaturization thanks to the low coherence length of these metals. In addition, our phase-based read-out scheme ensures stark protection against magnetic flux fluctuations, and provides ideal visibility in all the operation ranges. This makes the PSM a suitable candidate for the implementation of industrially-scalable classical memory cells in actual superconducting electronics technologies, such as rapid single flux quantum (RSFQ) [22], reciprocal quan- tum logic (RQL) [23], quantum flux parametrons (QFPs) [24], Josephson field-effect transistors (JoFETs) [25], and gate-controlled cryotrons (EF-trons) [26,27]. Yet, the strong topological protection and stability observed in the PSM make our approach promising in light of the implementation of phase-slip flux qubits [28] and quantum memories.

METHODS
Device fabrication details. The hybrid memory cells were realized by shadow-mask lithography technique.
The suspended resist-mask was defined by electron-beam lithography (EBL) onto a SiO 2 wafer. All metal-to-metal clean interfaces, and metal-to-oxide barriers were realized in an ultra-high vacuum (UHV) electron-beam evaporator (EBE) with a base pressure of 10 -11 Torr equipped with a tiltable sample holder suitable for multi-directional depositions. In order to obtain wire/ring transparent interfaces, which is crucial for the device operation, the use of the same material is strongly recommended [18]. Therefore, the nanowire and the ring of the PSM were realized with aluminum. Furthermore, the Al film evaporation is relatively simple, and its highquality native oxide allows the realization of good tunnel barriers through oxygen exposure at room temperature. At first, 15 nm of Al 0.98 Mn 0.02 were evaporated at an angle of -18 • to realize the normal metal electrode. Subsequently, the sample was exposed to 60 mTorr of O 2 for 5 min in order to form the thin insulating AlMnOx layer. Next, the sample holder was tilted to 10 • for the deposition of 20 nm of Al realizing the SQUIPT nanowire (length d = 400 nm, width w = 90 nm) and the superconducting electrodes. Finally, a thicker layer of Al (t R = 70 nm) was evaporated at 0 • to realize the superconducting loop of circumference ∼ 7.6 µm, and average width w R,ave 600 nm.
Magneto-electrical characterization. The magneto-electric characterization of the samples was performed at cryogenic temperatures in a 3 He-4 He dilution refrigerator (Triton 200, Oxford Instruments) equipped with RC-filters of resistance ∼ 2kΩ. The out-of-plane magnetic field was applied via a superconducting magnet driven by a low-noise current source (Series 2600, Keithley Instruments). The DC measurements were performed in a two-wire voltage-bias configuration through a low-noise voltage DC source (GS200, Yokogawa) coupled with a room-temperature current preamplifier (Model 1211, DL Instruments) (see Fig. 1-c). The AC characterization was performed via a combination of DC bias and low-frequency lock-in technique. A DC bias voltage (V) was applied to the device. A current given by the sum of a DC and AC sinusoidal modulation energized the superconducting magnet. The read-out current oscillations induced by variation of Φ, and the phase of the signal (with respect to the flux oscillations) were recorded by a lock-in amplifier. Further details can be found in the SI. Device parameters. Based on the device structure, we estimate the zero-temperature nanowire coherence length ξ w,0 = D/∆ w,0 80 nm, where is the reduced Planck constant, D = 22.5 cm 2 s -1 is the diffusion coefficient, and ∆ w,0 200 µeV is the zerotemperature gap in Al. The nanowire critical temperature is T C,w = ∆ w,0 /1.764k B 1.31 K, where k B is the Boltzmann constant. At low temperature, the ratio d/ξ w,0 5 confirming the frame of the long JJ regime for the PSM [11]. The single-valued CPR limit (achieved for ξ w,short d/3.5 ∼ 114 nm) [11] is reached at temperature T short = T C,w (1 − 0.852 2 ξ w,0 l/ξ w,short 2 ) ∼ 1.3 K [11,13], where l = 3D/v F 3.3 nm is the nanowire mean free path, and v F = 2.03 × 10 6 m/s is the Fermi velocity of Al.
The kinetic inductance (L K ) of a long JJ depends on the geometry and superconducting properties of the nanowire [12]. In our case, at 25 mK it takes the value [29]. The nanowire normalstate resistance is given by R N = d wtσ 14 Ω, where σ = DN f e 2 = 1.24 × 10 7 S is the Al film conductance (with N f = 2.15 × 10 47 J −1 m −3 the density of states at the Fermi energy of Al). Analogously, the ring total inductance takes the value L R 1.5 pH (with normal-state resistance R R 1.4 Ω). The contribution of the ring to the total inductance of the SQUIPT yields a screening parameter β = 2πL R I C /Φ 0 0.1, where I C 20 µA is the low-temperature nanowire critical current deduced from the weak link geometry [18]. The small β cannot account for the hysteretic behavior of the PSM, which stems, differently, from the long-junction regime of the Josephson nanowire [12,18].
The writing/erasing time (τ W,E ) is mainly due to the time required to polarize the SQUIPT with the external flux. It is given by τ W,E = L SQU IP T /R SQU IP T 1.1 ps, where L SQU IP T = L K + L R and R SQU IP T = R N + R R are the total inductance and resistance of the SQUIPT, respectively. The read out time (τ R ) is predominantly limited by the characteristic time of the two tunnel barriers, τ R = τ t1 + τ t2 31 ps, where τ t1 = R t1 C t1 19.5 ps is the characteristic time of the first tunnel junction, and τ t2 = R t2 C t2 11.5 ps is the time constant of the second junction. The junctions capacitances (C t1 0.3 fF and C t1 0.13 fF) are estimated from the area and the typical specific capacitance of AlOx tunnel barriers ∼ 50 fF/µm 2 [30].
AUTHOR CONTRIBUTIONS E.S. and F.G. conceived the experiment. N.L. fabricated the samples with inputs from F.P.. N.L. and E.S. performed the measurements. N.L. analysed the experimental data with inputs from E.S. and F.G.. N.L and F.P. wrote the manuscript with inputs from all the authors. All the authors discussed the results and their implications equally at all stages.

ADDITIONAL INFORMATION
The authors declare no competing financial interests.  ). I [1] and I [0] are negligible for V < 300 µV, therefore the PSM cannot be biased in that voltage range. Note that the memory works properly also applying multiple times the voltage bias, confirming the endurance of the PSM cell. All data were acquired at T = 25 mK.  [1] oscillates in phase with the magnetic flux (phase=0), while I [0] has the opposite dependence (phase=180 • ) allowing the acquisition of the two distinguishable signals of phase. All data were recorded at T = 25 mK.