Detecting superconductivity out-of-equilibrium

TL;DR

This paper explores how to reliably detect transient superconductivity out-of-equilibrium, emphasizing the limitations of optical conductivity measurements and proposing time-resolved ARPES as a better probe.

Contribution

It demonstrates that optical conductivity alone is insufficient for identifying superconductivity out-of-equilibrium and advocates for using time-resolved ARPES to detect condensate formation.

Findings

Optical conductivity evolution is not enough to confirm superconductivity out-of-equilibrium.

Time-resolved ARPES can effectively probe for condensate formation.

Current experimental methods may misidentify transient superconducting states.

Abstract

Recent pump-probe experiments on underdoped cuprates and similar systems suggest the existence of a transient superconducting state above $\mathrm{T}_c$. This poses the question how to reliably identify the emergence of long-range order, in particular superconductivity, out-of-equilibrium. We investigate this point by studying a quantum quench in an extended Hubbard model and by computing various observables, which are used to identify (quasi-)long-range order in equilibrium. Our findings imply that, in contrast to current experimental studies, it does not suffice to study the time evolution of the optical conductivity to identify superconductivity. In turn, we suggest to utilize time-resolved ARPES experiments to probe for the formation of a condensate in the two-particle channel.

Figures

a](#S0.F1.sf1)) and imaginary ([Fig. 1b](#S0.F1.sf2)) parts of $\sigma(\omega,\Delta t)$ for delays up to $\Delta t=10$ and a system with $L=64$ sites. For the real part, we find a sudden transfer of spectral weight from the CDW signal at around $\omega\approx 3$ towards $\omega\approx 1.7$, which is due to the sudden change of the Hamiltonian. At the same time, in the imaginary part a peak forms near $\omega=0$. We compare this to the SC ground state and realize that, at the first glance, similar behavior is induced. However, in particular in $\sigma_{1}$, clear differences appear, and $\sigm

b](#S0.F2.sf2), which shows dispersive, incoherent doublons. We observe similar behavior in the corresponding pairing spectral function of the $t$-$J_{\perp}$ model [supmat ](#bib.bib53), so that we expect this to be a generic feature, at least for quenches to SC phases.

a](#S0.F3.sf1)) there is a shift of spectral weight towards $q=\nicefrac{{\pi}}{{2}}$. This is to be contrasted with the distribution of weights in the CDW ground state, where the maximum value is at $q=0$. Comparing to the spectral function after the quench ([Fig. 3c](#S0.F3.sf3)), a weak transfer of spectral weight from $q=0$ towards $q=\nicefrac{{\pi}}{{2}}$ can be identified. However, from the numerical data it is not possible to clearly identify this as a signal for a (transient) SC phase. Note that the observed behavior is not reproduced for a quench within the CDW phase, as the two-elec

Equations

$$\hat{H}$$

$$\sigma (\omega ,\Delta t)$$

$$S_{\hat{O}}(q,\omega ,\Delta t)$$

$${\chi }_{\hat{O}}$$

Full text

Introduction

Superconductivity (SC) is one of the hallmarks of condensed-matter systems and has inspired researchers since its discovery in 1911, and later by the advent of high-temperature SC in cuprate materials highTc_original ; Wu1987 ; dagotto ; Lee2006 . While, in particular for the latter class of materials, a lot of questions are subject of ongoing research, many characteristics of the SC phase are by now well established as long as the system is in equilibrium. However, recent experiments (e.g., Foerst2014 ; Mankowsky2014 ; Hu2014 ; Hunt2016 ; Mitrano2016a on copper oxides, or on $\mathrm{K}_{3}\mathrm{C}_{60}$) report the observation of possible photo-induced transient SC phases, which can exist at elevated temperatures, even above the equilibrium-critical temperature $\mathrm{T}_{c}$ Foerst2011 ; Fausti189 ; Kaiser2014 ; buzzi2020photomolecular . In these investigations, ultrashort THz pulses excite single phonon modes, which decay very slowly compared to the typical time scale of the electron dynamics and thereby offer the possibility to control the interaction parameters of the electronic system Singla2015 . Subsequently, the $\omega$-dependent optical conductivity is determined as a function of time via reflectivity measurements using a probe pulse, and SC correlations are identified by the emergence or enhancement of a signal at $\omega\rightarrow 0$. This is by now a standard experimental procedure, which, however, leaves many questions open, in particular concerning the characterization of the state induced by the pump excitation (see, e.g., Refs. Eckstein2010, ; PhysRevB.96.054506, ; PhysRevB.96.235142, ; Bittner2019, ; Fauseweh2017, ; schwarz2020momentumresolved, ).

In this Letter, we address this issue regarding further experimental measures to probe SC in such non-equilibrium setups. For the sake of simplicity, we focus on one-dimensional (1D) systems, for which powerful numerical techniques are available in terms of matrix-product states (MPS) White1992 ; Schollwoeck2010 ; Paeckel2019 . We argue that it does not suffice to study only the optical conductivity, since the pump as well as the probe pulse can induce currents, which can modify the low-frequency behavior, without being a direct proof for SC. Nevertheless, we are able to provide evidence for the emergence of SC in the course of time by studying the time evolution of spectral functions, which are accessible to variants of time- and angle-resolved photoemission spectroscopy (tr-ARPES) experiments Damascelli2003 ; Damascelli2004 ; Lynch2005 ; Eckstein2008 ; Freericks2009 ; Ligges2018 ; wang2018theoretical . We propose to study the usual single-particle and a pairing spectral function, which we introduce below. We find particularly in the latter quantity clear signatures for the accumulation of weight at $k=0$. This raises the question, whether a condensation or quasicondensation of pairs is achieved, which would realize a transient non-equilibrium SC state. It is interesting to ask, whether non-equilibrium situations can be beyond the realm of validity of the Mermin-Wagner-Hohenberg theorem merminwagner_orig ; merminwagner_erratum ; hohenberg_orig , which inhibits the formation of true long-range order (LRO) in 1D systems. The scope of this Letter is, therefore, three-fold: To demonstrate that the time evolution of the optical conductivity does not suffice to unambiguously establish transient SC order, to present spectral functions as a more reliable probe, and to further characterize the observed transient state by investigating correlation matrices. The general validity of our findings is supported by comparing the extended Hubbard model voit ; PRBVoit2 ; Jeckelmann2002 ; Tsuchiizu2002 ; Sandvik2004 ; Ejima2007 ; PhysRevB.85.205127 and a variant of the 1D $t$-$J$ model tJoriginal0 ; tJoriginal1 ; tJoriginal2 ; dagotto ; PhysRevB.83.205113NO ; PhysRevLett.107.115301 ; PhysRevA.96.043618 .

Model and Methods

We study the time evolution of Hubbard chains Hubbard1963 ; Gutzwiller1963 ; Kanamori1963 ; HubbBook following a quantum quench Eisert2015a . Recent experiments Fausti189 ; Hu2014 ; Hunt2016 ; Mitrano2016a on high-$\mathrm{T}_{c}$ superconductors suggest that if there exists preformed pairing, e.g., in the normal state slightly above $\mathrm{T}_{c}$, pumping particular phonon modes induces charge coherences, which drive the system into a transient superconducting state. Therefore, our starting point is to assume that lattice distortions modify the strength of the couplings Foerst2011 ; Singla2015 and thereby alter the nearest-neighbor interaction between the electrons. Thus, we consider a quench in the 1D extended Hubbard model,

[TABLE]

with $\hat{T}=-t_{\rm hop}\sum_{j,\sigma}\left(\hat{c}^{\dagger}_{j,\sigma}\hat{c}^{\phantom{\dagger}}_{j+1,\sigma}+\mathrm{h.c.}\right)$ being the kinetic energy. Therein, $\hat{c}^{(\dagger)}_{j,\sigma}$ are $S-\nicefrac{{1}}{{2}}$ fermionic ladder operators, which obey the canonical anticommutation relations $\left\{\hat{c}^{\phantom{\dagger}}_{i,\sigma},\hat{c}^{\dagger}_{j,\sigma^{\prime}}\right\}=\delta_{i,j}\delta_{\sigma,\sigma^{\prime}}$, $\left\{\hat{c}^{\phantom{\dagger}}_{i,\sigma},\hat{c}^{\phantom{\dagger}}_{j,\sigma^{\prime}}\right\}=\left\{\hat{c}^{\phantom{\dagger}}_{i,\sigma},\hat{c}^{\phantom{\dagger}}_{j,\sigma^{\prime}}\right\}=0$, and we denote by $\hat{n}_{j}=\hat{n}_{j,\uparrow}+\hat{n}_{j,\downarrow}$ the total electron occupation at site $j$. For later convenience, we also define doublon ladder operators $\hat{d}^{\phantom{\dagger}}_{j}\equiv\hat{c}^{\phantom{\dagger}}_{j,\uparrow}\hat{c}^{\phantom{\dagger}}_{j,\downarrow}$. As motivated above, we start in a charge-density wave (CDW), which favors double occupancies ($U/t_{\rm hop}=-4$, $V/t_{\rm hop}=\nicefrac{{1}}{{4}}$) Bittner2019 ; BittnerPhD . We perform a sudden quench in the nearest-neighbor interaction $V/t_{\rm hop}=\nicefrac{{1}}{{4}}\rightarrow-\nicefrac{{1}}{{4}}$ into the s-wave superconducting phase at zero temperature supmat , keeping the local Hubbard interaction fixed at $U/t_{\rm hop}=-4$. Note that this value of the attractive interaction is comparable to the electronic bandwidth. Although this is out of the scope for cuprate high-$T_{c}$ superconductors, it is, however, similar to the electronic structure in alkali-doped fullerides, where the effective phonon-mediated intraorbtial interaction is comparable to the electronic bandwidth coming from $t_{1u}$ bands Nomurae1500568 .

We then calculate the real-time evolution using a combined single- and two-site time-dependent variational principle (TDVP) scheme in the MPS formulation of the density-matrix renormalization group (DMRG) for lattices White1992 ; Schollwoeck2010 ; PhysRevB.94.165116 ; Paeckel2019 with up to $L=80$ sites, open boundary conditions, and a maximal bond dimension of $m_{\mathrm{max}}=1000$ states. By comparing calculations with different values of $m_{\mathrm{max}}$ we estimate the accuracy of the presented results to be of a few percent supmat . To investigate the formation and stability of transient SC, we studied the differential optical conductivity after a probe pulse Lenar2014 ; Shao2016 , spectral functions, and the correlation matrices supersolid_penrose ; rigol:031603 of single- and two-particle excitations Damascelli2003 ; Kouzakov2003 ; Damascelli2004 ; Lynch2005 ; Eckstein2008 ; Freericks2009 ; Truetzschler2017 ; Ligges2018 ; Stahl2019 . We complement our studies by considering a similar quench in the 1D $t$-$J_{\perp}$ model PhysRevLett.107.115301 ; PhysRevA.96.043618 , \hat{H}_{\text{t$-$J_{\perp}}}=\hat{T}+\nicefrac{{J_{\perp}}}{{2}}\sum_{j}\left(\hat{S}_{j}^{+}\hat{S}_{j+1}^{-}+\hat{S}_{j}^{-}\hat{S}_{j+1}^{+}\right), at filling $n=0.2$ by quenching $J_{\perp}/t_{\rm hop}=2\rightarrow 6$, i.e., from a Luttinger liquid giamarchi to a singlet SC phase 111The operators $S_{j}^{\pm}$ are the usual $S-\nicefrac{{1}}{{2}}$ ladder operators on site $j$. Note that double occupancies are forbidden in the $t$-$J$ model.. We choose our energy and time units by setting $t_{\rm hop}\equiv 1$ and $\hbar\equiv 1$.

Time-dependent Optical Conductivity

The experimental setups Fausti189 ; Hu2014 ; Hunt2016 ; Mitrano2016a we refer to typically measure the reflectivity after pump-probe excitations, from which the optical conductivity is extracted. We follow Refs. Lenar2014, ; Shao2016, and compute the differential optical conductivity

[TABLE]

after the quench at $t=0$. $A(\omega,\Delta t)$ is the vector potential of the Fourier-transformed Gaussian probe pulse222For the width of the probe pulse we chose $\tau=0.05$ throughout this letter $A(t,\Delta t)=A_{0}e^{-\frac{(t-\Delta t)^{2}}{2\tau^{2}}}\cos(\omega_{0}t)$ in Peierls substitution PhysRevB.14.2239 applied after the time delay $\Delta t$. The overall response current $j(t,\Delta t)=\mathrm{i}\left\langle\sum_{j,\sigma}\left(e^{-\mathrm{i}A(t,\Delta t)}\hat{c}^{\dagger}_{j,\sigma}\hat{c}^{\phantom{\dagger}}_{j+1,\sigma}-\mathrm{h.c.}\right)\right\rangle$ was calculated from a real-time evolution of the perturbed system, and the $\omega$-dependence $j(\omega,\Delta t)$ is obtained by a Fourier transform 333In Shao2016 the non-equilibrium response function is obtained by considering the Fourier transform of the difference $j(t,\Delta t)-j_{0}(t)$ with $j_{0}(t)$ the current without probe pulse. As in our quench protocol without probe pulse there is always $j_{0}(t)\equiv 0$ we drop this term for brevity. including values up to time $t=25/t_{\mathrm{hop}}$ after the probe pulse. We show the real (Fig. 1a) and imaginary (Fig. 1b) parts of $\sigma(\omega,\Delta t)$ for delays up to $\Delta t=10$ and a system with $L=64$ sites. For the real part, we find a sudden transfer of spectral weight from the CDW signal at around $\omega\approx 3$ towards $\omega\approx 1.7$, which is due to the sudden change of the Hamiltonian. At the same time, in the imaginary part a peak forms near $\omega=0$. We compare this to the SC ground state and realize that, at the first glance, similar behavior is induced. However, in particular in $\sigma_{1}$, clear differences appear, and $\sigma_{2}$ shows additional features. Also, it is hard to decide whether $\sigma_{2}(\omega,\Delta t)$ is diverging as $\nicefrac{{1}}{{\omega}}$ for $\omega\to 0$, since we are limited in the frequency resolution. We conclude that, as in the experiments, the question whether the accumulation of spectral weight near $\omega=0$ is due to induced SC or an enhanced metallicity after a pump pulse is hard to decide supmat . However, in contrast to the experimental situation we have direct access to the time dependence of the current induced by the probe pulse. The properties of this current are further illustrated by the inset of Fig. 1c, where we display the response electron density $\Delta n_{i}(t)=\braket{\hat{n}_{i}(t)}_{\mathrm{probe}}-\braket{\hat{n}_{i}}_{0}$, which compares the time evolution of the local density in the SC phase with and without probe pulse. As can be seen, the effect of the probe pulse is to accumulate charge at the edges of the system. After passage of the probe pulse, this causes the measured current. In the SC phase the probe pulse induces a long-living current, while in non-equilibrium we find no clear evidence for a comparable response (see Fig. 1c). In turn, in our simulations the induced charge flow decays on time scales of at least $t\sim 25/t_{\mathrm{hop}}$ which sets the scale of a low-frequency response $2\pi/t\approx 0.25$ in the imaginary part $\sigma_{2}$. Thus, a strengthening of the response at $\omega\to 0$ alone, as observed here, does not suffice to demonstrate SC. This is supported by studying a quench within the CDW phase, in which a similar behavior of the optical conductivity is observed, but without further indication for SC in other observables supmat .

Spectral Functions

From now on we consider postquench states at $\Delta t=15/t_{\mathrm{hop}}$. This is justified, since for times $t>4/t_{\mathrm{hop}}$ a transient state is reached, as seen in the time evolution of the eigenvalues of the correlation matrix, which we find to be non-thermal supmat . Motivated by tr-ARPES, we consider the in- and out-of-equilibrium time-dependent lesser Greens functions for $t>\Delta t$: $C_{\hat{O}}(j,t,\Delta t)=\braket{\psi(\Delta t)}{\hat{O}^{\dagger}_{j}(t)\hat{O}^{\phantom{\dagger}}_{\nicefrac{{L}}{{2}}}(0)}{\psi(\Delta t)}$ in the single- and two-particle channel, i.e., $\hat{O}^{\phantom{\dagger}}_{j}=\hat{c}^{\phantom{\dagger}}_{j,\uparrow}$ and $\hat{O}^{\phantom{\dagger}}_{j}=\hat{d}^{\phantom{\dagger}}_{j}$, respectively, and we indicate the equilibrium case by setting $\Delta t\equiv 0$. We refer to the Fourier transform to momentum and frequency space supmat

[TABLE]

as the differential spectral function, where we have introduced a spectral broadening $\eta=0.1$ Paeckel2019 . Note that here we explicitly do not restrict ourselves to single-electron excitations, but also study processes that may excite double occupations, i.e., doublons. This can be related to ongoing experiments Truetzschler2017 , in which photoemission of pairs of electrons is studied, and is theoretically corroborated in a BCS-type picture Kouzakov2003 .

Fig. 2 displays the spectral functions for double occupations $S_{\hat{d}}(q,\omega,\Delta t)$ for both equilibrium phases (SC and CDW) and after the quench. We obtain a clear accumulation of weight at $q=0$ in the postquench state, which renders the result similar to the one of the SC ground state displayed. This is the central statement of this Letter: A (quasi-)condensate of s-wave (Cooper-)pairs seems to form after the quench, which is clearly detectable in $S_{\hat{d}}(q,\omega,\Delta t)$, and which has a striking similarity to the spectral function in the SC ground state. We emphasize that this coherence between the charges is dynamically created after the quench, as seen in the comparison with Fig. 2b, which shows dispersive, incoherent doublons. We observe similar behavior in the corresponding pairing spectral function of the $t$-$J_{\perp}$ model supmat , so that we expect this to be a generic feature, at least for quenches to SC phases.

Current tr-ARPES experiments usually investigate the time evolution of the spectral functions for single-electron excitations $S_{\hat{c}}(q,\omega,\Delta t)$, which we show in Fig. 3. The signatures to discriminate the SC phase from the CDW phase are not as prominent as for the double occupations. Nevertheless, we find that in the SC ground state (Fig. 3a) there is a shift of spectral weight towards $q=\nicefrac{{\pi}}{{2}}$. This is to be contrasted with the distribution of weights in the CDW ground state, where the maximum value is at $q=0$. Comparing to the spectral function after the quench (Fig. 3c), a weak transfer of spectral weight from $q=0$ towards $q=\nicefrac{{\pi}}{{2}}$ can be identified. However, from the numerical data it is not possible to clearly identify this as a signal for a (transient) SC phase. Note that the observed behavior is not reproduced for a quench within the CDW phase, as the two-electron excitations do not show a sharp signal at small momenta, indicating the absence of a transient SC phase there supmat .

Correlation Matrices

Now we turn to the question on how to further characterize the observed transient state. To do so, we attempt a standard analysis for the investigation of LRO in equilibrium systems and apply it to the present non-equilibrium situation. In Ref. supersolid_penrose, , Onsager and Penrose suggest to detect off-diagonal LRO by determining the eigenvalues $\lambda_{\nu}$ of single-particle correlation matrices,

[TABLE]

i.e., $\chi_{\hat{O}}\mathbf{v}_{\nu}=\lambda_{\nu}\mathbf{v}_{\nu}$444We use the bold symbol $\mathbf{e}_{j}$ to denote canonical unit vectors. This concept was generalized by Yang RevModPhys.34.694 to include also two-particle correlation matrices to describe superconductivity. The correlation matrix determines the order parameter if the dominating eigenvalue scales extensively in the system size $\lim_{L\rightarrow\infty}\frac{\lambda_{L}}{L}\rightarrow\mathcal{O}(1)$, which also implies an extensive separation of the dominating eigenvalue $\lambda_{L}$ from the bulk supmat .

Due to the Mermin-Wagner-Hohenberg theorem merminwagner_orig ; merminwagner_erratum ; hohenberg_orig , in 1D and in equilibrium, for pair formation only quasi-long-range order (qLRO) can be realized, which translates to a SC order parameter vanishing in the thermodynamic limit. Here, we studied the time evolution of the correlation matrix $\chi_{\hat{d}}(i,j)=\braket{\psi(t)}{\hat{d}^{\dagger}_{i}\hat{d}^{\phantom{\dagger}}_{j}}{\psi(t)}$, which provides the SC order parameter.

Figure 4 shows the difference between the two largest eigenvalues $\Delta\lambda/L=\frac{\lambda_{L}-\lambda_{L-1}}{L}$ as a function of time. At later times we find this separation to be about an order of magnitude larger than in the initial state. We have estimated the saturation values of the dominating eigenvalue by averaging over the accessible time scales, in which we assumed quasi-stationarity, $\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu_{L}=\frac{1}{t_{1}-t_{0}}\sum_{t=t_{0}}^{t_{1}}\lambda_{L}(t)$ with $t_{0}=10,t_{1}=32$. In the inset of Fig. 4, we compare the scaling behavior to the one of the SC ground state, where $\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu_{L}/L\sim L^{-0.50}=\nicefrac{{1}}{{\sqrt{L}}}$ is realized, with saturation value $\beta_{\mathrm{SC}}=-1\times 10^{-3}\approx 0$, as expected. After the quench, the scaling is best described by fitting the asymptotic behavior with $\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu_{L}/L\sim L^{-0.92}\sim\nicefrac{{1}}{{L}}$, and in this case we extracted a value $\beta=9\times 10^{-4}$, whose magnitude is even smaller than the one obtained in equilibrium, and hence zero within the error bars of our scaling analysis. This scaling to zero indicates that no LRO is obtained on the time scales investigated but cannot be excluded for later times Lemonik2017 ; Lemonik2018 .

Conclusion and Outlook

Our results for the differential optical conductivity in the extended Hubbard model after a quench indicate that the enhancement of $\sigma_{2}(\omega,\Delta t)$ in the low-frequency regime does not suffice to uniquely identify SC: The currents induced by a probe pulse cannot unambiguously be identified as supercurrents. In contrast, the pairing spectral function shows a clear accumulation of weights at $q\to 0$, which is absent in our initial state, but present in the SC ground state. This provides stronger evidence for the formation of a non-equilibrium SC state than the reflectivity measurements.

Our study indicates that reflectivity measurements need to be complemented by tr-ARPES -type experiments in future investigations. Readily available tr-ARPES setups measure the single-particle spectral function, which shows a shift of weights towards the SC state. We propose to also investigate the time evolution of pairing spectral functions, which provide the clearest evidence for the formation of a transient SC state. Our work therefore supplies an important theoretical result for the growing field of double photoemission spectroscopy.

Acknowledgements

We thank B. Lenz, L. Cevolani, K. Harms, N. Abeling, F. Heidrich-Meisner, S. Kaiser, G. Uhrig, and K. Schönhammer for insightful discussions. B.F. and D.M. thank the Max Planck-UBC-UTokyo Center for Quantum Materials for financial support. T.K.’s contribution was partially funded through an ERC Starting Grant from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 758935. We acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through SFB/CRC1073 (projects B03 and B07) and Research Unit FOR 1807 (project P7), and TU Clausthal for providing access to the Nuku computational cluster as well as the GWDG for providing HPC resources. This work was supported in part by the Los Alamos National Laboratory LDRD Program. S.P., T.K., and S.R.M. thank MPI-FKF for their hospitality.