Efficient prediction of time- and angle-resolved photoemission spectroscopy measurements on a nonequilibrium BCS superconductor

We study how time- and angle-resolved photoemission (tr-ARPES) reveals the dynamics of BCS-type, $s$-wave superconducting systems with time-varying order parameters. Approximate methods are discussed, based on previous approaches to either optical conductivity or quantum dot transport, to enable computationally efficient prediction of photoemission spectra. One use of such predictions is to enable extraction of the underlying order parameter dynamics from experimental data, which is topical given the rapidly growing use of tr-ARPES in studying unconventional superconductivity. The methods considered model the two-time lesser Green's functions with an approximated lesser self-energy that describes relaxation by coupling of the system to two types of baths. The approach primarily used here also takes into consideration the relaxation of the excited states into equilibrium by explicitly including the level-broadening of the retarded and advanced Green's functions. We present equilibrium and nonequilibrium calculations of tr-ARPES spectrum from our model and discuss the signatures of different types of superconducting dynamics.

Figure 1

Nonequilibrium tr-ARPES calculation of constant superconducting gap with a quench, described by Eq. (11) with a BCS superconducting gap $\mathrm{\Delta }=0.01$. The lesser Green's function is calculated using Eq. (10) with ${\epsilon }_k=k,t_p=500,t_0=0,t=1000,\sigma =400$. The black curve shows $\omega \left(k\right)=\pm \sqrt{k^2+{\mathrm{\Delta }}^2}$.

Figure 2

Equilibrium tr-ARPES calculations of metal, with ${\epsilon }_k=k,\gamma =0.0001,t_p=500,t=1000,\sigma =200$ (left panel), $\sigma =400$ (right panel). The black curve shows $\omega \left(k\right)={\epsilon }_k$.

Figure 3

Equilibrium tr-ARPES calculations of a BCS superconducting system at equilibrium, with $\mathrm{\Delta }=0.005$ (left panel) and $\mathrm{\Delta }=0.01$ (right panel), ${\epsilon }_k=k,\gamma =0.0001,t_p=500,t=1000,\sigma =400$. The black curve shows $\omega \left(k\right)=-\sqrt{{\epsilon }_k^2+{\mathrm{\Delta }}^2}$.

Figure 4

Schematic plot of various gap profiles considered in this work.

Figure 5

Nonequilibrium tr-ARPES calculations with BCS order parameter featuring constant with a quench, as mentioned in Eq. (20), with $\mathrm{\Delta }=0.01$. The calculation is with ${\epsilon }_k=k$ and $\gamma =0.0001$. The width of the probe pulse is $\sigma =0.02\times 2/\gamma$. The two different $t_p$'s, short and long time after quench, are indicated in the figures.

Figure 6

Occupation number, $n_k\left(t\right)$, of the quench system Eq. (20) with $\mathrm{\Delta }=0.01$. The times when $n_k$ peaks, $t_m$'s, are shown in the caption along with their corresponding $k$ values.

Figure 7

Nonequilibrium tr-ARPES calculations with BCS order parameters featuring quenched order parameters that decay, as mentioned in Eq. (21), with $\mathrm{\Delta }=0.01,T=0.05\times 2/\gamma$ (left column) and $1\times 2/\gamma$ (right column). The calculations are with ${\epsilon }_k=k$ and $\gamma =0.0001$. The width of the probe pulse is $\sigma =0.02\times 2/\gamma$, and various $t_p$'s are indicated in the figures. The insets of the panels show gap profiles, $\mathrm{\Delta }\left(t\right)$, as functions of time in unit of $2/\gamma$ in blue curves, and the instantaneous gap at $t_p$'s in red dots.

Figure 8

Nonequilibrium tr-ARPES calculations with BCS order parameters featuring Gaussian order parameters, as mentioned in Eq. (22), with $\mathrm{\Delta }{\left(t\right)}_{\text{max}}=0.01,T=0.05\times 2/\gamma$ (left column) and $3\times 2/\gamma$ (right column), $t_d=0.035\times 2/\gamma$ (left column) and $0.5\times 2/\gamma$ (right column). The calculations are with ${\epsilon }_k=k$ and $\gamma =0.0001$. The width of the probe pulse is $\sigma =0.02\times 2/\gamma$, and various $t_p$'s are indicated in the figures. The insets of the panels show gap profiles, $\mathrm{\Delta }\left(t\right)$, as functions of time in unit of $2/\gamma$ in blue curves, and the instantaneous gap at $t_p$'s in red dots.

Figure 9

Equilibrium calculations of metal at nonzero temperature: $k_B{T}_e=50\gamma$. Upper panel: tr-ARPES signal with ${\epsilon }_k=k,\gamma =0.0001,t_p=500,t=1000,\sigma =400$. The black curve shows $\omega \left(k\right)={\epsilon }_k$. Lower panels: $I_{\omega }(k,t)$ (left) and $I_k(\omega ,t)$ (right) at $k_B{T}_e=0$ (solid line with dotted data points) and $k_B{T}_e=50\gamma$ (dashed line with dotted data points). The vertical dash-dotted lines show where $k$ (left) or $\omega$ (right) is zero.

Figure 10

Equilibrium calculations of a BCS superconducting system with $\mathrm{\Delta }=0.01$ at nonzero temperature: $k_B{T}_e=50\gamma$. Upper panel: tr-ARPES signal with ${\epsilon }_k=k,\gamma =0.0001,t_p=500,t=1000,\sigma =400$. The black curve shows $\omega \left(k\right)=-\sqrt{{\epsilon }_k^2+{\mathrm{\Delta }}^2}$. Lower panels: $I_{\omega }(k,t)$ (left) and $I_k(\omega ,t)$ (right) at $k_B{T}_e=0$ (solid line) and $k_B{T}_e=50\gamma$ (dashed line). The vertical dash-dotted lines show where $k$ (left) or $\omega$ (right) is zero.

Figure 11

Nonequilibrium calculations with BCS order parameter featuring constant with a quench, as mentioned in Eq. (20), with $\mathrm{\Delta }=0.01$. Upper panel: tr-ARPES signals with ${\epsilon }_k=k,\gamma =0.0001$, and $\sigma =400=0.02\times 2/\gamma$. The two different $t_p$'s, short and long time after quench, are indicated in the figures. Lower panels: $I_{\omega }(k,t)$ (left) and $I_k(\omega ,t)$ (right) at $k_B{T}_e=0$ (solid line) and $k_B{T}_e=50\gamma$ (dash line), both at short (black line) and long (blue line) time after quench. The vertical dash-dotted lines show where $k$ (left) or $\omega$ (right) is zero.