Theory of parametrically amplified electron-phonon superconductivity

Ultrafast optical manipulation of ordered phases in strongly correlated materials is a topic of significant theoretical, experimental, and technological interest. Inspired by a recent experiment on light-induced superconductivity in fullerenes [M. Mitrano et al., Nature (London) 530, 461 (2016)], we develop a comprehensive theory of light-induced superconductivity in driven electron-phonon systems with lattice nonlinearities. In analogy with the operation of parametric amplifiers, we show how the interplay between the external drive and lattice nonlinearities lead to significantly enhanced effective electron-phonon couplings. We provide a detailed and unbiased study of the nonequilibrium dynamics of the driven system using the real-time Green's function technique. To this end, we develop a Floquet generalization of the Migdal-Eliashberg theory and derive a numerically tractable set of quantum Floquet-Boltzmann kinetic equations for the coupled electron-phonon system. We study the role of parametric phonon generation and electronic heating in destroying the transient superconducting state. Finally, we predict the transient formation of electronic Floquet bands in time- and angle-resolved photoemission spectroscopy experiments as a consequence of the proposed mechanism.

Figure 1

Parametric amplification of the phonon response. (Left) Phonon-mediated electron attraction in the absence of external drive. (Right) The external drive and lattice nonlinearities parametrically amplify lattice distortions which in turn mediate stronger attraction between the electrons.

Figure 2

Nonequilibrium evolution of the driven electron-phonon system obtain using the Floquet-Migdal-Eliashberg formalism. (a) Intensity of the external drive, (b) phonon spectral function $\rho (\nu ,t)$, showing the red-shift of the phonon peak and along with emergent oscillatory features, (c) electron distribution $n(\omega ,t)$ showing the smearing of the Fermi surface as the electrons heat up, (d) electron effective mass (black, left axis) and damping (red, right axis), (e) lowest eigenvalue of the Floquet-Migdal-Eliashberg gap functional, where N and SC correspond to normal conducting and superconducting (instability) intervals, (f) predicted time-resolved ARPES signal in the log scale as a function of electron frequency $\omega$ and kinetic energy $\xi$ at $t=15{\tau }_{\mathrm{ph}}$, showing the formation of electronic Floquet bands. The initial temperature is $T_i=0.04{\mathrm{\Omega }}_0$, the lattice nonlinearity is cubic type with ${\kappa }_3=0.1{\mathrm{\Omega }}_0$, and the drive frequency and amplitudes are ${\mathrm{\Omega }}_{\mathrm{drv}}=0.4{\mathrm{\Omega }}_0$ and $\mathcal{A}=0.75$ (refer to Sec. 6 for additional details).

Figure 3

Illustration of parametric amplification from classical phase-space trajectories. The classical phase-space trajectories correspond to a parametrically driven oscillator in response to a momentum jump with magnitude $P_0$ at $t=0$ for ${\mathrm{\Omega }}_{\mathrm{drv}}$ below resonance (left), on resonance (middle), and above resonance (right). Here, $X_0\equiv P_0/\left(2M{\omega }_{\mathbf{q}}\right)$ is a normalization constant and the Mathieu parameter is set to $\alpha =0.2$. The red circle is the periodic trajectory in the absence of the drive ($\alpha =0$). Note the significantly amplified response below resonance ${\mathrm{\Omega }}_{\mathrm{drv}}<{\omega }_{\mathbf{q}}/2$, the diverging response on resonance ${\mathrm{\Omega }}_{\mathrm{drv}}={\omega }_{\mathbf{q}}/2$, and suppressed response above resonance ${\mathrm{\Omega }}_{\mathrm{drv}}>{\omega }_{\mathbf{q}}/2$.

Figure 4

The effective electron-electron interaction $U\left(t\right)$ as a function of drive frequency ${\mathrm{\Omega }}_{\mathrm{drv}}$ and time. Here, ${\tau }_{\mathrm{drv}}=\pi /{\mathrm{\Omega }}_{\mathrm{drv}}$ and $U_{\mathrm{eq}}=-2{\left|g_{\mathbf{q}}\right|}^2/\left(\hslash {\omega }_{\mathbf{q}}\right)$ is attraction strength in the absence of the drive. The Mathieu parameter is $\alpha =0.2$ and we have set the damping rate to $\epsilon =0.1{\omega }_{\mathbf{q}}$. The red thick lines show $U\left(t\right)$ on the first two resonances ${\mathrm{\Omega }}_{\mathrm{drv}}/{\omega }_{\mathbf{q}}=\frac{1}{2},\frac{1}{4}$.

Figure 5

Time average and variance of the effective electron-electron interaction. Solid lines are numerical results obtained from the solutions of the Mathieu equation. Dashed lines correspond to the perturbative results given in Eq. (20). The Mathieu parameter is set to $\alpha =0.2$.

Figure 6

Heat map plot of $T_c^{\mathrm{drv}}/T_c^{\mathrm{eq}}-1$ based on the analytic Floquet-BCS analysis of Sec. 3a. The plot on the left is the result obtained from $U\left(t\right)=U_0+U_{1}cos\left(2{\mathrm{\Omega }}_{\mathrm{drv}}t\right)$. The plot on the right is obtained by neglecting the ac component and setting $U\left(t\right)=U_0=\mathrm{const}.$ The significant role of oscillations of $U\left(t\right)$ in enhancing $T_c$ is clearly noticeable. Note that $U_0$ and $U_1$ are both functions of $\alpha$ and ${\mathrm{\Omega }}_{\mathrm{drv}}$ as given in Eq. (22), and $T_c$ is calculated using Eqs. (27a) and (27b). In both cases, we have set $\nu \left(0\right)U_{\mathrm{eq}}=0.5$ and ${\omega }_c={\omega }_0$.

Figure 7

A flowchart for the Floquet-Migdal-Eliashberg quantum kinetic formalism. The external drive along the width of the initial electron propagators determines the evolution of the coherent (“classical”) lattice displacement and phonon propagators (“quantum”). Subsequently, the evolution of electronic energy distribution is calculated on the backdrop of the driven lattice. This procedure can be iterated until convergence if required. Finally, the Floquet-Migdal-Eliashberg pairing condition is assessed to determine whether the normal state exhibits a pairing instability during the evolution. The thin lines show the “procedural flow” of the calculations. The thick red lines show the “heat flow”, from the external drive to phonons, then to electrons, and finally back to phonons through self-consistency.

Figure 8

An illustration of Floquet-Boltzmann kinetic formalism. An arbitrary observable in a driven system is expected to have fast temporal variations on the scale of the driving frequency and a slowly varying envelope. By decomposing the observable into the harmonics of the driving frequency using short-time Fourier transforms.

Figure 9

Evolution of phonon propagators in response to a ramped-up external drive for drive frequency ${\mathrm{\Omega }}_{\mathrm{drv}}=0.4{\mathrm{\Omega }}_0$ and maximum drive amplitude $\mathcal{A}=0.75$. The physical parameters are set to ${\kappa }_3=0.1,{\kappa }_4=0$, and ${\gamma }_{\ell }=0.2{\mathrm{\Omega }}_0$. The leftmost panel shows $n=0$ (period-averaged) Keldysh phonon propagator. The red-shift of the phonon peak is clearly noticeable. The next two columns show the real and imaginary parts of $n=1,2$ propagators. Notice the absence of a single quasiparticle peak. Panels (d) and (e) show the period-averaged squeezing correlations and the density of phonon excitations, respectively. Both quantities increase as the external field is ramped up. Finally, panel (f) shows the time-dependent mass enhancement factor as defined in the text, along with its time average (green solid line) and the lower and upper envelopes (blue and red lines, respectively). Notice the significant increase in the mass enhancement factor, as well as its high-amplitude oscillations.

Figure 10

The relative change of Floquet superconducting transition temperature with respect to equilibrium, $T_c^{\mathrm{Floq}}/T_c^{\mathrm{eq}}-1$, in the equilibrium electron approximation. The left and right columns show the results for cubic and quartic nonlinearities ${\kappa }_3=0.1{\mathrm{\Omega }}_0,{\kappa }_4=0$ and ${\kappa }_3=0,{\kappa }_4=0.1{\mathrm{\Omega }}_0$, respectively. The top row is obtained using full Floquet phonon propagators whereas $n>0$ Floquet components (dynamical effects) are neglected in the bottom row. The electrons are kept in a thermal state at temperature $T=0.04{\mathrm{\Omega }}_0$.

Figure 11

Density of phonon excitations (left), period average (middle), and time variance (right) of the mass enhancement factor. The electrons are kept in a thermal state at temperature $T=0.04{\mathrm{\Omega }}_0$. These quantities are calculated in the stationary driven-dissipative state of phonons.

Figure 12

Time evolution of the energy statistics of electrons for ${\mathrm{\Omega }}_{\mathrm{drv}}=0.4{\mathrm{\Omega }}_0,\mathcal{A}=0.75$, and ${\tau }_{\mathrm{drv}}=5{\tau }_{\mathrm{ph}}$. From left to right, the plots show the period-averaged $(n=0)$ energy statistics, its first and second Floquet components, and the period-averaged scattering rate $\langle \mathrm{\Gamma }(\omega ,t)\rangle \equiv -2i\mathrm{Im}\left[{\mathrm{\Sigma }}_{n=0}^R(\omega ;t)\right]$ of electrons. The heating of electrons is noticeable in (a) as the drive is ramped up, as well as the increase in the scattering rate in (d).

Figure 13

Assessment of the Floquet-Migdal-Eliashberg (FME) pairing condition for a ramped-up external drive with different frequencies and amplitudes. The red segments indicate regions where the lowest eigenvalue of the FME gap functional is negative, signaling the pairing instability. The dashed lines show the hypothetical case if the electrons were to remain in their initial thermal state (no heating). The nonlinearity is cubic, the initial temperature is set to $T_i=0.04{\mathrm{\Omega }}_0\simeq 1.2T_c^{\mathrm{eq}}$, and the physical parameters are chosen as described in Sec. 6.

Figure 14

Evolution of the density of phonon excitations $\langle \mathrm{\Delta }{n}_{\mathrm{ph}}\left(t\right)\rangle$ and instantaneous mass enhancement factor $\lambda \left(t\right)$ for different drive frequencies and amplitudes. In the bottom row, the green, blue, and red lines correspond to the period-averaged, lower, and upper envelopes of $\lambda \left(t\right)$. The nonlinearity is cubic, the initial temperature is $T_i=0.04{\mathrm{\Omega }}_0\simeq 1.2T_c$. The choice physical parameters are as given in Sec. 6.

Figure 15

Evolution of the electronic effective mass $\langle m^{*}\left(t\right)\rangle$ and scattering rate $\langle \mathrm{\Gamma }\left(t\right)\rangle$ for different drive frequencies and amplitudes. The nonlinearity is cubic, the initial temperature is $T_i=0.04{\mathrm{\Omega }}_0\simeq 1.2T_c$. The choice physical parameters are as given in Sec. 6.

Figure 16

The energy distribution of electrons after the drive is ramped up. The physical parameters are the same as Fig. 13. The dashed lines show the Fermi-Dirac fits. The thick gray line is the Fermi-Dirac distribution at the initial temperature $T_i=0.04{\mathrm{\Omega }}_0$. The thin solid lines correspond to ${\mathrm{\Omega }}_{\mathrm{drv}}/{\mathrm{\Omega }}_0=0.3,0.4,0.5$ with decreasing slope, respectively. The measurement time is $t=30{\tau }_{\mathrm{ph}}$. The final effective temperatures are $T_f^{\mathrm{eff}}/{\mathrm{\Omega }}_0\approx 0.04,0.17,0.22$ for ${\mathrm{\Omega }}_{\mathrm{drv}}/{\mathrm{\Omega }}_0=0.3,0.4,0.5$, respectively.

Figure 17

Probing the formation of electronic Floquet bands via tr-ARPES experiments. The heat map plots show the intensity of the signal at different times in the logarithmic scale. The inset plots show the instantaneous amplitude of the drive during ramp-up. The white dashed lines indicate the dispersion of the main quasiparticle peak. The initial temperature is $T_i=0.04{\mathrm{\Omega }}_0$, the drive parameters are chosen as ${\mathrm{\Omega }}_{\mathrm{drv}}=0.4{\mathrm{\Omega }}_0$ and $\mathcal{A}=0.75$. The nonlinearity is cubic, and the physical parameters are chosen as described in Sec. 6. Note the progressive formation of Floquet quasiparticle bands and the softening of the polaronic kink in the main quasiparticle dispersion as the system heats up. The time-averaged effective mass at the Fermi surface is inversely proportional to the slope of the main quasiparticle dispersion at $\xi =0$ and is shown separately in Fig. 15 for better visibility.