Signatures of nonadiabatic BCS state dynamics in pump-probe conductivity

We theoretically study the pump-probe response of nonequilibrium BCS superconductors coupled to optical phonons. For ultrashort pump pulses a nonadiabatic regime emerges, which is characterized by oscillations of the superconducting order parameter as well as by the generation of coherent phonons. Using the density-matrix formalism, we compute the pump-probe response in the nonadiabatic regime of the coupled Bogoliubov quasiparticle-phonon system and determine the signatures of the order parameter and the phonon oscillations in the pump-probe conductivity. We find that the nonadiabatic dynamics of the BCS superconductor reflects itself in oscillations of the pump-probe response as functions of delay time $\delta t$ between pump and probe pulses. We argue that from the analysis of this oscillatory behavior both frequency and decay time of the algebraically decaying order-parameter oscillations can be inferred. Similarly, the coherent phonons are evidenced in the pump-probe conductivity by oscillations with the frequency of the phonons. Remarkably, we find that the oscillatory response in the pump-probe conductivity is resonantly enhanced when the frequency of the order-parameter oscillations is tuned to the phonon energy.

Figure 1

(a) Real part of the pump-probe response, $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=8\times {10}^{-8}$ Js/(Cm)] in the absence of phonons. (b),(c) Pump-probe signal $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$ as a function of delay time $\delta t$ for (b) $\omega ={\omega }_{{\Delta }_{\infty }}$ and (c) $\omega =0.81{\omega }_{{\Delta }_{\infty }}$. The gray lines in panels (b) and (c) represent the best fits of Eq. (11) to the numerical data; see discussion in the text. (d),(e) Temporal evolution of $\left|\Delta \right(t\left)\right|$ and spectral distribution of the gap oscillation, respectively, for the same parameters as in panel (a).

Figure 2

(a) Real part of the pump-probe conductivity, $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the adiabatic regime [${\tau }_{\mathrm{p}}=20$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=0.5\times {10}^{-8}$ Js/(Cm)] in the absence of phonons. Panel (b) shows a detail of the pump-probe response for frequencies just below the pump frequency ${\omega }_{\mathrm{p}}=1.21{\omega }_{{\Delta }_{\infty }}$. The small dip near $\omega \approx 1.15{\omega }_{{\Delta }_{\infty }}$ is due to Pauli blocking. (c) Time dependence of $\left|\Delta \right(t\left)\right|$ for the same parameters as in panel (a).

Figure 3

(a) Absorption spectrum $\text{Re}\left[\sigma \right(\delta t,\omega \left)\right]$ versus $\omega$ and $\delta t<0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=13\times {10}^{-8}$ Js/(Cm)] in the absence of phonons. (b) Fourier transform of the data in panel (a), i.e., $\text{Re}\left(\sigma \right)$ as a function of Fourier frequency ${\omega }_{\delta t}$ and absorption frequency $\omega$. (c),(d) Time dependence of $\left|\Delta \right(t\left)\right|$ and spectral distribution of the gap oscillation, respectively, for the same parameters as in panel (a).

Figure 4

Real part of the pump-probe signal, $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=10\times {10}^{-8}$ Js/(Cm)] in the presence of an optical-phonon mode with energy $\hslash {\omega }_{\mathrm{ph}}=2$ meV and coupling strength $g_{\text{ph}}=0.1$ meV. (b),(c) Pump-probe response $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$ as a function of delay time $\delta t$ for (b) $\omega ={\omega }_{{\Delta }_{\infty }}$ and (c) $\omega ={\omega }_{\text{ph}}$. The gray lines in panels (b) and (c) represent the best fits of Eqs. (11) and (12), respectively, to the numerical data. (d),(e) Time evolution of the order-parameter amplitude $\left|\Delta \right(t\left)\right|$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (f) Fourier spectra of the order-parameter and coherent-phonon oscillations (dashed blue and solid red, respectively).

Figure 5

(a) Real part of the the pump-probe response, $\mathrm{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=11.64\times {10}^{-8}$ Js/(Cm)] in the presence of an optical-phonon mode with energy at resonance with the order-parameter oscillations, i.e., ${\omega }_{\text{ph}}={\omega }_{{\Delta }_{\infty }}=2$ meV. Here the electron-phonon coupling strength is $g_{\text{ph}}=0.1$ meV. (b),(c) Pump-probe signal $\mathrm{Re}\left[\sigma \right(\delta t\left)\right]$ as a function of delay time $\delta t$ for (b) $\omega ={\omega }_{{\Delta }_{\infty }}={\omega }_{\text{ph}}$ and (c) $\omega =0.875{\omega }_{{\Delta }_{\infty }}$. The gray lines in panels (b) and (c) represent the best fits of Eqs. (12) and (11), respectively, to the numerical data. (d),(e) Time dependence of the order-parameter amplitude $\left|\Delta \right(t\left)\right|$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (f) Spectral distribution of the order-parameter and coherent-phonon oscillations (dashed blue and solid red, respectively).

Figure 6

(a) Absorption spectrum $\text{Re}\left[\sigma \right(\delta t,\omega \left)\right]$ versus $\omega$ and $\delta t>0$ for the adiabatic regime [${\tau }_{\mathrm{p}}=20$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=0.5\times {10}^{-8}$ Js/(Cm)] in the presence of an optical-phonon mode with energy $\hslash \omega =2$ meV and coupling strength $g_{\text{ph}}=0.1$ meV. (b),(c) Time dependence of the order-parameter amplitude $\left|\Delta \right(t\left)\right|$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (d) Spectral distribution of the coherent-phonon oscillations.

Figure 7

(a) Real part of the pump-probe signal, $\text{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the adiabatic regime [${\tau }_{\mathrm{p}}=20$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=0.5\times {10}^{-8}$ Js/(Cm)] in the presence of an optical-phonon mode that is in resonance with pump pulse energy, i.e., $\hslash {\omega }_{\mathrm{ph}}=\hslash {\omega }_{\mathrm{p}}=3$ meV. The electron-phonon coupling strength is taken to be $g_{\text{ph}}=0.1$ meV. (b),(c) Time dependence of the order-parameter amplitude $\left|\Delta \right(t\left)\right|$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (d) Spectral distribution of the coherent-phonon oscillations.

Figure 8

(a) Real part of the pump-probe signal, $\text{Re}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t<0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=13\times {10}^{-8}$ Js/(Cm)] in the presence of a phonon mode with energy $\hslash {\omega }_{\mathrm{ph}}=2$ meV $\approx 1.19\hslash {\omega }_{{\Delta }_{\infty }}$ and coupling strength $g_{\text{ph}}=0.1$ meV. (b) Fourier transform of the data in panel (a), i.e., $\text{Re}\left(\sigma \right)$ as a function of Fourier frequency ${\omega }_{\delta t}$ and absorption frequency $\omega$. (c),(d) Time dependence of the order-parameter amplitude $\left|\Delta \right(t\left)\right|$ and the lattice displacement $U(0,t)$, respectively, for the same parameters as in panel (a). (e) Spectral distribution of the order-parameter and coherent-phonon oscillations (dashed blue and solid red lines, respectively).

Figure 9

Sketch of the two-dimensional momentum space. The red line represents the Fermi surface $k=k_{\text{F}}$ and the solid black lines are the boundary given by the radial component $k=k_{\text{F}}\pm k_{\text{c}}$ ($k_{\text{c}}$, radial component of the cutoff vector). The chosen $k_y$ values are indicated by the dotted line. In the inset the discretization of the $k_x$ values is shown. Only $k_x$ values with a distance of $q_0$ couple to each other.

Figure 10

Temporal evolution of the pump field $A_{{\mathbf{q}}_{\mathrm{p}}}\left(t\right)$ (solid lines) and and probe field $A_{{\mathbf{q}}_{\mathrm{pr}}}$(t) (dashed lines) with pulse widths ${\tau }_{\mathrm{p}}=0.5\text{ps}$ and ${\tau }_{\mathrm{pr}}=0.25\text{ps}$, respectively. The black solid (dashed) line represents the Gaussian envelope of the pump (probe) pulse. The blue and red traces show the real and imaginary parts of the pulse fields, respectively. The pump pulse is centered at $t=0\text{ps}$ and the probe pulse at $t=\delta t=2$ ps.

Figure 11

(a) Imaginary part of the pump-probe response, $\mathrm{Im}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=8\times {10}^{-8}$ Js/(Cm)] in the absence of phonons. (b),(c) Pump-probe signal $\mathrm{Im}\left[\sigma \right(\delta t,\omega \left)\right]$ as a function of delay time $\delta t$ for (b) $\omega ={\omega }_{{\Delta }_{\infty }}$ and (c) $\omega =0.81{\omega }_{{\Delta }_{\infty }}$. The gray lines in panels (b) and (c) represent the best fits of the numerical data with Eqs. (B1) and (11); see discussion in the text.

Figure 12

Imaginary part of the pump-probe signal, $\mathrm{Im}\left[\sigma \right(\delta t,\omega \left)\right]$, versus $\omega$ and $\delta t>0$ for the nonadiabatic regime [${\tau }_{\mathrm{p}}=0.5$ ps, $|{\mathbf{A}}_{\mathrm{p}}|=10\times {10}^{-8}$ Js/(Cm)] in the presence of an optical-phonon mode with energy $\hslash {\omega }_{\mathrm{ph}}=2$ meV and coupling strength $g_{\text{ph}}=0.1$ meV.