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Optical control of non-equilibrium superconducting phase transition below Tc in a cuprate

Photoinduced phase transitions from insulating to metallic states, accompanied by structural re-arrangements, have been recently reported in complex transition-metal oxides. However, the optical control of a purely electronic phase transition, where the thermodynamic phase is determined by the distribution of excitations, has remained elusive. Here we report optical control of the electronic phase in an underdoped Bi2212 crystal through impulsive photoinjection of quasiparticles (QP) via ultrashort laser pulses, avoiding significant laser heating. An abrupt transition of the transient optical electronic response is observed at a critical fluence of Ipump=70 uJ/cm^2. We show that the measured dynamics is consistent with an inhomogenous first-order superconducting-to-normal state phase transition, triggered by a sudden shift of the chemical potential.

The basic idea of non-equilibrium superconductivity is to photo-inject an excess of excitations through an ultrashort pump pulse, modifying the equilibrium electron distribution on a timescale faster than the impulsivelyinduced thermal heating. Under these conditions the free energy of the superconducting system FSC(T,n) can be varied along non-equilibrium pathways, by suddenly varying the number of excitations n, whereas for thermal phase transitions n(T,μ=EF) is univocally determined by the temperature and the chemical potential, through the Fermi-Dirac statistics (8). Non-equilibrium superconductivity has been studied in BCS systems, evidencing the possibility of a non-thermal quenching of the superconducting phase, inducing a first-order phase transition to a metastable, inhomogenous state (9,10). However, these experiments have been performed without sufficient time resolution to follow the onset of the SNPT.
The unconventional SNPT in high-TC superconductors (HTSC) is a physical phenomenon that has been the subject of intense studies (11) yet remains poorly understood. Broadly, the fascinating physics of HTSC is characterized by four unexplained features: (i) the ground state of the undoped compounds (12); (ii) the pseudogap phase (13) where Cooper pairs are probably formed locally in the absence of a macroscopic condensate (14); (iii) the intrinsic inhomogeneity of the SC phase at the nanometer scale (15); (iv) the basic interaction responsible for the binding potential necessary to form Cooper pairs (16). So far, the SC phase has been studied either at equilibrium in the frequency domain (17) or out of equilibrium in the time-domain by externally modifying the temperature of the system (18,19).
The control of the dynamics of the SNPT in general and, in particular, in high-TC cuprates, has never been achieved by direct interaction with photons, mainly due to problems related to sample heating (19). In the cuprates, the condensation energy of the SC state is small as compared to the energy gain of the insulating phase in complex metal oxides and one does not expect significant changes of the optical properties in the visible spectral range, where femtosecond resolutions are generally achieved. The delicate balance between the small variations of the optical properties and the low threshold expected (18, 20) to photoinduce a merely electronic SNPT has prevented scholars from investigating this physics.
In this work we perform time-resolved reflectivity measurements in the near-infrared with a tunable repetition rate femtosecond laser source. By setting the proper repetition rate (<100 kHz), we are able to impulsively excite an underdoped Bi2Sr2CaCu2O8+δ crystal (TC=81 K) with pump fluences ranging from 10 μJ/cm 2 to 700 μJ/cm 2 while avoiding a significant heating of the material. Under these conditions, above a critical fluence of Ipump≅70 μJ/cm 2 , we observe an abrupt transition of the electronic response. This behavior is an unambiguous signature of a photoinduced phase transition from the superconducting state to a new metastable state. Fig. 1A shows the time-resolved reflectivity variation measured at room temperature. The impulsive variation, at zero delay, reflects the non-equilibrium excitation of the electronic system, due to the pump pulses. In agreement with the literature (21), the decay of the reflectivity in the normal state is well described by the two temperature model, in which the fast dynamics of ∼350 fs is related to the electron-phonon relaxation. The inset of Fig. 1A displays the maximum reflectivity variation ΔR/R(0) as a function of the pump fluence. A linear dependence is observed over the entire experimentally available range. When the sample is cooled below TC, the relaxation time increases up to few picoseconds, as shown in Fig. 1B. This well-established (18, 22) decrease of the decay rate of photo-injected excitations below TC is generally attributed to the suppression of the QP recombination process related to the opening of the superconducting gap (23,24) (Fig. 2) and, on the μs timescale, to the energy and momentum conservation rules of nodal QP (25). These observations confirm that time-resolved reflectivity, in the visible spectral range, is sensitive to the density of the photoinjected excitations (19,20). We also note that at T<TC, the rise time of the ΔR/R signal increases up to ∼300 fs. The inset of Fig. 1B reports the dependance of ΔR/R(0) on the pump intensity, at a repetition rate of 108 kHz. The fluence range so far explored (19) with higher repetition rate sources is highlighted by the red square. Outside this region, at repetition rates higher than 100 kHz, the average heating of the sample can not be prevented. In this region we measured a linear scaling of the signal. At larger fluences we observed a progressive saturation of the signal up to the point at which a different linear behaviour sets in. Exactly the same phenomenon is observed repeating the measurements at half the repetition rate (54 kHz), i.e. with the same fluence but with half the average power impinging on the sample. This observation excludes disturbing effects connected to the average heating of the excited area.  23,24). In this intensity regime, the results of the fitting procedure indicate that the pump energy is mainly absorbed through excitation of the phonon population. This finding satisfactorily explains both that the decay dynamics is dominated by the inelastic scattering rate of high-energy phonons (γesc≈2 ps -1 (21)) without any intensity dependence (19) and that a constant delay of ∼300 fs of the maximum of the signal is measured (23). At absorbed energy densities larger than ∼70 μJ/cm 2 , a fast peak emerges (Fig. 3B)  and causes a decrease of the signal rise-time (Fig. 3C), while the following slow dynamics results progressively delayed (Fig. 3D). Above this intensity threshold the data can no longer be fitted with the RTE.

T=20 K
The numerical solution of the heat equation and the calculated temperature increase profile, show that, at the threshold fluence, the maximum temperature increase is <10 K. In Fig. 4 we report the time-resolved reflectivity data on the same sample, at T>70 K. When the laser repetition rate is increased from 54 kHz to 540 kHz and the estimated average temperature exceeds Tc=81 K, the slow dynamics disappears and a flat signal after the fast peak, is measured. This is regarded as the evidence that the slow increase of ΔR/R, at Ipump>70 μJ/cm 2 and shown in Fig. 3A, originates from a local average temperature smaller than TC. Therefore, the observed transition of ΔR/R at Ipump>70 μJ/cm 2 is a genuine signature of a non-thermal phase transition and the slow increase of the signal, on the ps timescale, is related to the recovery of the superconducting phase. Since ΔR/R is related to the excitations number (18), we conclude that, after the photo-induced phase transition, the number of excitations during the superconductivity recovery increases, reaching a maximum and decreasing again on the picosecond timescale (Fig. 3A). This is in contrast with the kinetics of the order parameter η in a second-order phase transition (27). When the average number of Cooper pairs, related to the order parameter, is impulsively decreased, at T<Tc the system monotonically recovers the equilibrium value η.
This reflects the fact that the reconstruction of the SC gap implies a monotonic decrease of the excitations in the system. The measured increase of the signal after the fast decay suggests that an inhomogeneous metastable state is formed with the SC fraction increasing in time. The perturbation of the electron distribution is generally mimicked either by introducing an effective temperature Teff (28) or an effective chemical potential μeff (29). Both the models predict a decrease of the superconducting gap Δ(n,T) with the excitation number, but with important differences. In the Teff model the gap dependence can be approximated by Δ(n,0)/Δ(0,0)≈1-32(3n) 3/2 /π 3 , n being measured in units of 4N(0) Δ(0,0) (8). A complete closing of the gap is obtained at ncr≈0.33, causing a second-order phase transition to the normal state (8). In the μeff model the gap closing is slower and, before the complete collapse at n≈0.65, the free energy of the superconducting state equals the normal state one, causing a first-order phase transition from the superconducting to the normal state (8). We underline that the differences between the predictions of the two models are not related to the number of excitations injected into the system, but to their energy distribution. We  (24,25). The combination of these two mechanisms leads to the accumulation of antinodal quasiparticles on the top of the gap. This non-thermal energy distribution causes a first-order phase transition to the normal state before the complete closing of the gap predicted by the Teff model can take place (8,29).
Within this framework, the slow increase of the transient reflectivity, after the fast dynamics, contains the information on the final state after the impulsive photo-excitation and on the superconducting phase recovery.
The observed dynamics can be interpreted as the signature of a coalescence process triggered by the photoinduced first-order phase transition. Within the first hundreds of femtoseconds the free energy of the superconducting phase equals the free energy of the normal phase, leaving the system in an inhomogenous mixture of the two phases. In the superconducting volume (VSC) the relaxation dynamics is slow and described by the RTE, whereas in the normal fraction (1-VSC) the decay of the excitations is extremely fast and is completed in a few hundreds of femtoseconds. On the picosecond timescale the metastable system can thus be schematized as superconducting regions surrounded by a normal volume where the free-energy of the superconducting phase is smaller than the free energy of the normal state and a coalescence process can start, in analogy with the case of a supersaturated solution (27). In this picture, the transient reflectivity is ΔR(t)/ R∝VSC(t)⋅n(t)+(1-VSC(t))⋅fN(t), fN(t) being the transient reflectivity signal related to the normal phase (Fig. 1A). The dynamics of VSC(t) is modelled as the precipitation of the solute in a supersaturated solution (Supporting Online Material). At the beginning of the coalescence process the ΔR(t)/R signal is dominated by the growth of the superconducting volume, resulting in the increase of the signal, whereas on longer timescales the supersaturation tends to zero and the dynamics is regulated by the RTE. The fit of this model to the data is reported in Fig. 3A. A very good agreement with the measurements is obtained, confirming our interpretation.
The possibility to photoinduce a first-order electronic phase transition in a superconducting high-Tc system opens intriguing possibilities. First, the present measurements underline the interesting question of the real nature of the metastable final state after photo-excitation of the SNPT and its relationship with the underlying pseudogap state where the ΔR/R dynamics drastically changes (22). The detailed investigation of the electronic dynamics of the final state should help in discriminating between a pseudogap phase competing and co-existing with the superconducting phase. Second, a weak first-order SNTP is expected at |T-TC|/TC≈10 -6 K, because of the effects of the order parameter fluctuations (30). In the present experiment the first order phase transition is photoinduced by impulsively varying the order parameter, the thermodynamic temperature T<<TC being fixed.

Experimental set-up
Time-resolved reflectivity measurements have been performed through a standard pump-probe set-up.
The laser source is a cavity-dumped Ti:sapphire oscillator producing 120 fs-1.5 eV light pulses. The output energy is ∼50 nJ/pulse at repetition rates tunable from 543 kHz to single shot. The sample is an underdoped Bi2Sr2CaCu2O8+δ crystal with Tc=81 K, characterized through AC magnetic susceptibility. The sample is mounted on a cold finger and covered by a coverslip to protect the surface during the cooling.

Fitting procedures
The low fluence data have been fitted with the Rothwarf-Taylor equations (RTE), dn dt = I n (t) + 2pγ − βn 2 dp dt = I p (t) − pγ + β 2 n 2 − (p − p eq )γ esc (1) describing the number of excitations n coupled to phonons, p being the gap-energy phonon number. The coupling of the electronic and phonon populations is obtained through a) the annihilation of a Cooper pair via gap phonon absorption (pγ term) and b) the emission of gap phonons during the two-body direct recombination of excitations to form a Cooper pair (βn 2 term). The most relevant prediction of the model is that the decay dynamics of impulsive photo-injected electronic and phonon excitations (In(t) and Ip(t) source terms) is ultimately regulated by the escape rate of the non-equilibrium gap-phonons ((p-peq)γesc term). We fixed β≈0.1 cm 2 /s (1) and γ=3 ps -1 , as obtained from room temperature measurements.

Average thermal heating
The average thermal heating has been simulated by integrating the heat equation on a 1 mm-thick slab, whose bottom temperature is kept equal to the cold finger one. Thermal insulation conditions are used on the other sides of the slab. The stationary solution has been calculated in cylindrical coordinates with a heat source modelled by: I(x, y, z) = I 0 αe −αz 4 ln 2 πf whm 2 e −4 ln 2 r 2 f whm 2 (2) α=6⋅10 6 m -1 being the absorption coefficient, fwhm the full-width-half-maximum of the beam profile, and I0 the average power impinging on the sample. Both the pump and probe contributions and the temperature dependence of the anisotropic thermal conductivity (2) were taken into account.

Coalescence
The dynamics of the superconducting fraction VSC(t) is modelled by: where c and t0 are coefficients related to the physical properties of the system and to the distribution of the grain sizes when the coalescence process starts at t=0 (3).