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Optically enhanced coherent transport in YBa2Cu3O6.5 by ultrafast redistribution of interlayer coupling

Nonlinear optical excitation of infrared active lattice vibrations has been shown to melt magnetic or orbital orders and to transform insulators into metals. In cuprates, this technique has been used to remove charge stripes and promote superconductivity, acting in a way opposite to static magnetic fields. Here, we show that excitation of large-amplitude apical oxygen distortions in the cuprate superconductor YBa2Cu3O6.5 promotes highly unconventional electronic properties. Below the superconducting transition temperature (Tc = 50 K), interbilayer coherence is transiently enhanced at the expense of intra-bilayer coupling. Strikingly, even above Tc a qualitatively similar effect is observed up to room temperature, with transient inter-bilayer coherence emerging from the incoherent ground state and similar transfer of spectral weight from high to low frequency. These observations are compatible with previous reports of an inhomogeneous normal state that retains important properties of a superconductor, in which light may be melting competing orders or dynamically synchronizing the interlayer phase. The transient redistribution of coherence discussed here could lead to new strategies to enhance superconductivity in steady state.

Superconductors at equilibrium display two characteristic physical properties: zero DC resistance and the expulsion of static magnetic fields. The first of these properties manifests itself as a zero-frequency delta function in the real part of the optical conductivity σ1(ω) and by a positive imaginary part σ2(ω) that diverges at low frequency as 1/ω.
In high-Tc cuprates, the layered structure gives rise to additional c-axis excitations of the superfluid, with the notable appearance of one or more longitudinal Josephson plasma modes due to tunneling of Cooper pairs between capacitively coupled superconducting planes.
In bi-layer cuprates, two longitudinal Josephson plasma modes are found 16,17 , reflected by two peaks in the energy loss function -Im[1/(ε1(ω)+iε2(ω))]. Within each family of cuprates, the longitudinal mode frequency quantifies the strength of the Josephson coupling between pairs of CuO2 layers (Fig.1a). In addition, a peak in the real part of the conductivity, the so-called transverse Josephson plasma mode 18 (Fig.   1a), is observed at ω = ωT 19,20,21,22,23,24,25 . This second dissipationless excitation of the superfluid 26 is characterized by simultaneous out of phase oscillations of the Josephson plasma within and between pairs of layers, and shares spectral weight with the zero-frequency conductivity peak.
In the specific case of YBa2Cu3O6.5, the two longitudinal Josephson plasma modes appear as reflectivity edges and as peaks in the loss function near 30 and 475 cm -1 ( Fig.1 b2 and b4). The transverse plasma mode ( Fig.1 b3) is observed near 400 cm -1 , and it is strongly coupled 27 to a 320-cm -1 phonon from which it gains oscillator strength with decreasing temperature.
Here, we measure the transient c-axis THz-frequency optical properties of YBa2Cu3O6.5 after excitation with mid-infrared optical pulses, both below and above Tc. Mid-infrared pump pulses of ~300 fs duration, polarized along the c direction and tuned to 670 cm -1 frequency (~15-µm, 83 meV, ± 15%), were made resonant with the infraredactive distortion shown in figure 1c. The 15-µm-wavelength pulses were generated by difference-frequency mixing in an optical parametric amplifier and focused onto the samples with a maximum fluence of 4 mJ/cm 2 , corresponding to peak electric fields up to ~3 MV/cm. At these strong fields, the apical oxygen positions are driven in an oscillatory fashion by several percent of the equilibrium unit-cell distance (see supplementary material S5).
For temperatures below and immediately above Tc, where the largest changes of the optical properties were observed, we interrogated the solid with broadband THz probe pulses generated by gas ionization, covering the 20 and 500 cm -1 frequency range. For temperatures far above Tc, smaller conductivity changes could be measured with sufficient signal to noise ratio only by using narrower-band pulses (20 -85 cm -1 ), generated by optical rectification in ZnTe.
The equilibrium low-frequency imaginary part of the optical conductivity σ2(ω) (Fig.   1b1) is positive and increasing at frequencies below 30 cm -1 . Note that because of the contribution from normal transport by non-condensed quasi-particles, the overall equilibrium σ2(ω) does not exhibit the 1/ω frequency dependence of a London superconductor. Such 1/ω frequency dependence can only be observed in this frequency range by measuring differential conductivity ∆ ( ) = ( , < ) − ( , > ).
The broadband photo-induced response of the superconducting state of YBa2Cu3O6.5 (T = 10 K) is reported in figures 2 and 3. The frequency-and time-delay-dependent optical properties were extracted from measurements of the amplitude and phase of the reflected electric field after photo-excitation, using the equilibrium optical properties of the material 21 (see Fig.1b) and taking into account the pump-probe penetration depth mismatch (see supplementary information S2). Immediately after excitation, a strong increase in the slope of σ2(ω) was observed. Since the superfluid density at equilibrium is quantified as ( )| → , the increase in the slope of σ2(ω) suggests a transient enhancement of the superfluid density of the superconductor (see upper panel of Fig. 2a). The frequency-dependent imaginary conductivity is shown for all pump-probe time delays in the color plot.
In Fig. 2b, we report the corresponding changes in the inter and intra-bilayer coupling by plotting time-and frequency-dependent energy-loss function -Im[1/(ε1(ω,τ)+iε2(ω,τ))]. The 30 cm -1 peak, which reflects the inter-bilayer longitudinal plasma mode at equilibrium, reduces in amplitude after photo excitation, as a second higher-frequency peak appears at 60 cm -1 . Simultaneously, the intrabilayer peak, which at equilibrium is observed at 475 cm -1 , shifts to the red. All transient shifts in the loss function relax back to the equilibrium spectrum with a 7 ps exponential time constant (black dashed line in Fig. 2b).
Finally, figure 3 displays the corresponding dynamics of the real part of the conductivity σ1(ω,τ). At equilibrium, σ1(ω) nears zero below 80 cm -1 , with several phonon peaks between 100-300 cm -1 . These phonon peaks remain virtually unaffected and only a small increase in σ1(ω) is detected below 80 cm -1 . The strongest light-induced changes in σ1(ω) are found at high frequency, where the transverse plasma mode at 400 cm -1 shifts to the red. Note that the red shift of this mode (σ1(ω) peak at 400 cm -1 ) is consistent with the red shift of the loss function peak at 475 cm -1 .
The transverse mode frequency follows and are the thickness of the inter-and intra-bilayer junctions, respectively, and and are the corresponding Josephson plasma frequencies. Because >> , the change in transverse plasma mode position is dominated by .
To analyze the photo-induced dynamics below Tc, we first note that the changes in optical properties are only partial. For example, the light induced 60-cm -1 loss function peak (figure 2b) takes only a fraction of the equilibrium spectral weight at 30 cm -1 . This is interpreted a signature of an inhomogeneous light induced phase, in which only a fraction of the equilibrium superconducting state is being transformed, a physical situation that can be well described by Bruggeman's effective medium The effective dielectric function ̃ ( ) is determined here by the dielectric function ̃ ( ) of the photo-transformed regions, which occupy a volume fraction f, and by the dielectric function ̃ ( ) of the remaining (1-f) volume, which we assumed to retain the properties of the equilibrium superconducting state.  We next turn to the response immediately above Tc, for which the signal was still large enough to allow for measurements with the same broadband source. Fig. 4a shows the equilibrium and light-induced σ2(ω) for T=60 K (> Tc = 50 K). We observe a shortlived enhancement of the low frequency σ2, which becomes positive and increases with decreasing frequency. Note that this transient σ2(ω) is very similar to that of the equilibrium superconductor at 10 K (grey line in Fig.2a). Figure 4b displays the corresponding loss function. The appearance of a broad loss function peak at 55 cm -1 , absent in the normal state, reflects the emergence of a longitudinal plasma mode 29 approximately at the same frequency where the blue shifted mode was observed below Tc. Furthermore, the 475 cm -1 loss function peak (Fig.4b) and the 400 cm -1 peak in σ1 (Fig.4c) both shift to the red, identifying a clear analogy between the below Tc and above Tc data. These broadband data at 60 K were also successfully fitted with the same effective medium model, where the unscreened inter-bilayer plasma frequency increases from = 0 cm -1 to & =250 cm -1 , and the unscreened intra-bilayer Josephson plasma frequency =1030 cm -1 reduces to & =960 cm -1 (see Supplementary Materials S3), again conserving the total coherent weight ! + ! .
The response over a broader range of temperatures below and above Tc was measured with narrowband THz pulses, which were more sensitive to smaller changes in optical properties. Figure 5 displays three representative sets of results at 10 K, 100 K and 300 K, as extracted from the experiment using the same procedure as for the broadband data. The 10 K results confirm the photo-induced enhancement in ( ) (Fig.5a1) and a weak increase in ( ) (Fig.5a2). For comparison, the differential measurement of the red-shift in ( ) near 400 cm -1 (extracted from the broadband measurements of Fig. 3) is displayed in Fig.5a3.
Above Tc, at 100 K ( Fig.5b1-5b3) we also observed an increase and change in ( ), which becomes positive below 30 cm -1 , as already shown for the 60 K data of Fig.4a.
The corresponding changes in the low frequency ( ) are negligible (Fig.5b2), and a transfer of spectral weight from the 400 cm -1 mode to lower frequencies is detected.
At 300 K ( Fig. 5c1- We next turn to a critical discussion of all the experimental results reported above. Below Tc, we observe strengthening of the low-frequency inter-bilayer coupling, occurring at the expense of that within the bilayers. A change in the Josephson coupling strength in cuprates may be ascribed to more than one physical origin. For example, the dynamical coupling between the layers may change because the charging energy of the planes is modified. However, if this were the case, both plasma modes below Tc should shift in the same direction, that is to lower/higher frequencies if the electronic compressibility 18 increased/decreased. Similarly, a reduction or increase of the interlayer coupling, either by ionization of the condensate across the gap or by an increase in the total number of Cooper pairs, would lead to a shift of the longitudinal modes in the same direction, either to the red or to the blue. As ! + ! is constant throughout the dynamics, we conclude that the light only rearranges the relative tunneling strengths, with coherence being transferred from the bilayers to the inter-bilayer region. This is further supported by a partial sum rule analysis for σ1(ω) over the measured spectral range. We compare the reduction in the spectral weight at finite frequencies (20-500 cm -1 ), which is dominated by the ωT peak, with the enhancement at low frequency. The zero frequency peak, proportional to the superfluid density, cannot be measured directly in σ1(ω) but can be quantified either by ωσ2(ω) for ω−>0 or, equivalently, as ! ! / 23,27 . Since ! ≈ , the c-axis superfluid density is approximately proportional to ! , which increases after photo- which is the highest frequency we could access experimentally) is comparable to the enhancement of the inter-bilayer coherence ∆ ! = 8.4 × 10 A ;< = of the same photo-perturbed region. Thus, the weakening of the intra-bilayer coupling alone seems to be responsible for the observed enhancement in superfluid density. Similar to the below Tc case, the finite-energy sum rule for the broadband data above Tc (at T = 60 K) shows the spectral weight loss in σ1 is 7 × 10 A cm = , and the emergence of inter-bilayer coherence is ∆ω EF = 6 × 10 A cm = for the photo-perturbed region (f = 19%).
Our effective medium analysis also shows that the phonon at 320 cm -1 , which is A more exotic effect that may give rise to extraordinarily high mobility and an anomalous dependence on the laser field may be conduction by a sliding onedimensional charge density wave (CDW) 31 . If a non-commensurate CDW in this density range 32 were to become de-pinned when the lattice is modulated, and if such a wave could slide along the c axis, it may pin again only a few picoseconds after excitation. Yet, it appears unlikely that the same carrier density as in the equilibrium superconducting state should contribute to such a CDW conductor. Transient superconducting coherence could not be caused by quasi-particle photoexcitation 33,34 , which was shown in the past to increase Tc at microwave 35 atomic positions more favorable to high temperature superconductivity, for example "melting" an ordered state that competes with superconductivity or cause a displacement of the apical oxygen away from the planes 43,44,45,46,47 . Similarly to what discussed for the below Tc data, the entire gain of coherent spectral weight above Tc can be accounted for by considering the red shift of the transverse plasma mode near 400 cm -1 . The appearance of this mode above equilibrium Tc has been discussed in the past as a signature of residual intra-bilayer superfluid density 48 in the normal state, and we speculate that redistribution of intra-bilayer coherence may explain our data.
We also mention the possibility that the observed effects result from dynamical         Josephson plasma resonance in the relative THz reflected field ratios (below T c divided by above T c ) for

YBa2Cu3O6.5.
The equilibrium optical properties were characterized by THz time-domain spectroscopy. Singlecycle THz pulses (20-85 cm -1 ) were focused onto the sample surface at 30° incidence, with polarization perpendicular to the Cu-O planes. The reflected electric field was measured by electro-optic sampling at different temperatures, below and above T c . Figure FS1.1B displays the absolute value of the frequency-dependent reflected electric field at a given temperature |_`( )|, normalized to the same quantity measured above T c . Below T c a strong edge appears ~ 30 cm -1 . This feature corresponds to the Josephson plasma resonance (JPR).

S2 Derivation of the complex conductivity from differential reflectivity
The transient optical properties were probed by broadband single cycle THz pulses generated in a laser-ionized plasma 2 , covering a bandwidth from 20 to 500 cm -1 . The source was driven by 800 nm pulses with 1 mJ energy and 35 fs duration from a 2 kHz Ti:sapphire laser. The broadband THz pulses were polarized along the c-axis of the YBa 2 Cu 3 O 6.5 single crystal. The out-of-plane optical properties were measured at normal incidence. Mid-infrared pump pulses with a wavelength of 15 µm, 300 fs duration, and pulse energies of 8 µJ were generated by optical parametric amplification and subsequent difference frequency generation in GaSe 3 .
The transient optical properties were also probed in reflection between 20 and 85 cm -1 by narrowband THz pulses, which were generated by optical rectification of 800 nm femtosecond pulses from a 1 kHz Ti:sapphire laser in a ZnTe crystal. The reflected pulses were electro-optically sampled by 800 nm pulses in a second ZnTe crystal. For the frequency range where the two different THz probe setups overlap, a good agreement of the transient optical constants was found, with a better signal-to-noise ratio for the data obtained from the narrowband THz probe setup.
The pump-induced change in the reflected field was measured at each time delay a during the dynamically evolving response of the material. For each a the relative delay between excitation and sampling pulse was kept fixed, and the THz transient was scanned with respect to these two, changing the internal delay t. Therefore each point in the THz profile probed the material at the same time delay a after excitation.
Note that with this method, the time resolution is not determined by the intensity envelope of the THz transient, which is generally frequency-chirped and can be > 1 ps long, but by the Fourier limit of the pulse, i.e. the bandwidth.   below and above Tc. In Figure FS2.2B we plot the time duration of the light-induced effect at 100 K (above Tc), measured by fixing the time delay t at the peak of the reflected field transient and scanning the pump-probe time-delay a. The pump-induced dynamics after excitation is well reproduced by a double exponential decay with time constants τ1 ~ 600 fs and τ2 ~ 7 ps (see Fig. FS2.2B). Note that the slow time constant measured here is approximately the same as the one measured below Tc (see Fig. 2 and Fig. 3 in the main text).
From these measurement we extract the changes in the normal-incidence reflectivity as Below Tc strong reflectivity changes appear around 30 cm -1 . That is the frequency where the JPR is located in the equilibrium reflectivity which is indicated with a black dashed line. We find positive changes at frequencies above the JPR and negative changes at lower frequencies.
The corresponding phase change of the transient reflected field evolves sharp around the shifted edge from 0 to approximately i/2.
Above Tc we observe the appearance of a reflectivity edge at 50 cm -1 indicative of the transient high mobility state upon photo-excitation. Also here the corresponding phase change evolves from 0 to approximately i/2. We note that for a change on a superconduting junction one would expect a i shift of the phase. However (see S3), the transient state is an inhomogeneous state of photoexcited regions and an unperturbed background. Therefore the THz transients probe an effective medium for which the phase shift is reduced. These "raw" reflectivity changes require reprocessing. Importantly, the measured changes above T c are only few percent in size, due to a mismatch between the several-µm penetration depth of the probe and that of the resonant 15-µm pump, which is tuned to the middle of the reststrahlen band for this particular phonon and is evanescent over a few hundred nm. At low frequencies, the probe interrogates a volume that is between 10 and 20 times larger than the transformed region beneath the surface, with this mismatch being a function of frequency (see figure FS2.3). ) and probe (20-500 cm -1 ).
To extract the optical constants, the penetration depth mismatch between the mid-infrared pump and the THz probe pulses was taken into account in the data analysis by assuming that the sample was photo-excited homogenously within a thickness corresponding to the midinfrared penetration depth d, and the total reflection coefficient R̃ is a combined contribution from the photo-excited top layer (reflection coefficient R j ) and the non-excited bottom layer where δ=2πd (n1+ik1)/λ, and λ is the wavelength of the probe pulse. The complex optical constants T e( , a) of the photo-exited layer were obtained from a numerical solution of the above complex Fresnel equation. For the experiment with the narrowband probe pulses a 30° angle of incidence in the experimental geometry was taken into account.
From the surface refractive index, we calculate the complex conductivity for a volume that is  The transient data in the normal state (at 60 K) displays strong resemblance to the below Tc data, and can be analyzed within a similar effective medium model with a filling factor of 19%.  To calculate the two components ̃ ( ) and ̃I 2 ( ) of the effective medium fit, we also use the conventional Drude-Lorentz model 6 . As shown in Fig. FS3 For the conventional Drude-Lorentz fit of the below Tc data, we use = 127 cm -1 to fit the low frequency Josephson plasma edge, and one Lorentz term centered at 376 cm -1 to fit the 400 cm -1 mode in . For the dielectric function ̃ ( ), & =392 cm -1 is used to fit the enhanced superconducting response at low frequencies, and the red-shifted 400 cm -1 mode and the corresponding spectral weight redistribution is fitted by a Lorentz oscillator at 330 cm -1 with a reduced width from 144 to 35 cm -1 .
For the above Tc data, we use a Drude term with b =14 Ω -1 cm -1 and a scattering rate of Γ=25.5 cm -1 to fit the small metallic background at low frequency. A Lorentz term at 368 cm -1 is used to fit the equilibrium transverse plasma mode in . For ̃ ( ), we replace the low frequency Drude contribution with an inter-bilayer Josephson plasma mode , =297 cm -1 , and shift the Lorentz peak from 368 to 335 cm -1 and reduce its width from 151 to 55 cm -1 to fit the red-shift of the transverse plasma mode.
The advantage of Munzar's multi-layer model is that no additional term is needed to fit the 400 cm -1 mode, which comes out directly from the bilayer structure. Its frequency is mainly determined by the intra-bilayer plasma frequency , and its spectral weight is strongly coupled to the 320 cm -1 planar-oxygen phonon. The light-induced red-shift and the reshaping of the 400 cm -1 mode can be directly explained by a reduction of the intra-bilayer plasma frequency & . The sharpening of the red-shifted peak in is results from a reduction of the 320-cm -1 phonon line width. Full fits to YBa2Cu3O6.5 narrowband data above Tc (Fig. 5 in the main text) are shown in Figure   FS3.4. Letting only the filling fraction of the high mobility state as free parameter allows describing the data at all temperatures. The photo-induced response can be fit equally well by assuming a perfect conductor with 7 ps lifetime or with infinitely long lifetime. If one assumes instead a carrier scattering time much shorter than 7 ps, for instance τs=1 ps, no agreement with the experimental results can be found. As shown in Figure FS3.5a, a conductor with τs = 1 ps and ~50 cm -1 would display, unlike the measured transient state, a significant increase in σ1(ω) over the whole measured range, no low-frequency divergence in σ2(ω), and no clear reflectivity edge. The lifetime of the transient state decreases with increasing temperature, down to ~2 ps at room temperature. We find that the transient optical properties remain consistent with a state in which its scattering rate is limited only by the lifetime of the state itself.
Also a small increase (< 10%) in the σ 1 (ω) of the normal state conductivity was necessary for the fit. Such a small Ohmic component suggests that incoherent carriers are also being excited.

S5 Evaluation of the peak lattice distortion
For these electric fields, the amplitude of the field-induced atomic displacement due to mid-