# Higgs spectroscopy of superconductors in nonequilibrium

B. Fauseweh<EMAIL_ADDRESS>Max-Planck-Institut für
Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany  L.
Schwarz  Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1,
D-70569 Stuttgart, Germany  N. Tsuji  RIKEN Center for Emergent Matter Science
(CEMS), Wako 351-0198, Japan  N. Cheng  Department of Physics and Astronomy,
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
N. Bittner  Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1,
D-70569 Stuttgart, Germany  Department of Physics, University of Fribourg,
1700 Fribourg, Switzerland  H. Krull  Lehrstuhl für Theoretische Physik I,
Technische Univerität Dortmund, Otto-Hahn Straße 4, 44221 Dortmund, Germany
M. Berciu  Department of Physics and Astronomy, University of British
Columbia, Vancouver, British Columbia, Canada V6T 1Z1  Quantum Matter
Institute, University of British Columbia, Vancouver, British Columbia, Canada
V6T 1Z4  G. S. Uhrig  Lehrstuhl für Theoretische Physik I, Technische
Univerität Dortmund, Otto-Hahn Straße 4, 44221 Dortmund, Germany  A. P.
Schnyder  Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1,
D-70569 Stuttgart, Germany  S. Kaiser  Max-Planck-Institut für
Festkörperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany  D. Manske
Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, D-70569
Stuttgart, Germany

###### Abstract

The collective non-equilibrium dynamics of superconducting condensates could
not be studied systematically so far. The response in conventional optical
pump-probe experiments contains almost no information on the actual ground
state, since it is limited to high-energy processes. Recent development of
pulsed THz laser technology has made it possible to excite superconducting
Higgs oscillations, without melting the superconducting order. Here we develop
a theory for tailored quantum quenches which can excite all possible
symmetries of Higgs oscillations. In analogy to phonon spectroscopy, these
oscillations allow for a complete characterization of the superconducting gap
function. Simulations for realistic laser pulses reveal spectroscopic
information of these Higgs oscillations in the optical conductivity. By
exemplary calculating the nonequilibrium response of s- and d-wave
superconductors we show, that such nonequilibrium Higgs spectroscopy opens a
unique approach to distinguish between different symmetries of the condensate,
even for new and unknown superconductors.

##  I Introduction

Spectroscopy is one of the most used tools in physics in order to examine the
dynamics and structure of matter, from the level of atomic spectroscopy, which
allows one to determine the energy structure of single atoms and molecules,
through nuclear magnetic resonance (NMR) spectroscopy to phonon spectroscopy
of crystals using Raman scattering. In all these techniques, symmetry plays a
crucial role.
In crystals one of the most important symmetries is the point group of the
underlying lattice, because the allowed phonon modes correspond to the
irreducible representations of the point group. Thus the maximum number of
expected modes is predicted by group theory. Using the polarization selection
rules for Raman scattering RevModPhys.79.175  allows us to directly address
different symmetry channels, because the excitations in the material have the
symmetry imposed on them by the light orientation. Thus we can directly map
and classify the symmetries of ground and excited states based on light-matter
scattering.
While this approach is well established in phonon spectroscopy, it is not
straightforward to generalize it to other symmetry broken states, such as
superconducting condensates. Here the Higgs mode is the collective excitation
similar to the phonon of lattice systems. Unfortunately the Higgs mode is not
easily examinable, because it cannot be probed by linear-response-type
measurements, as it is charge neutral and therefore does not couple directly
to the electromagnetic field. There is, however, a much weaker nonlinear
coupling PhysRevB.92.064508 ; PhysRevB.93.180507 ; annurev.varma2015  of the
Higgs mode to the electromagnetic field, see Fig. 1a.

Figure 1:  Illustration of Higgs oscillations in a superconductor. a) Feynman
diagram describing the excitation of a Higgs mode $H$ by a light field
$\vec{A}$. b) Free energy landscape $F$ of a superconductor as function of the
real and imaginary part of the superconducting gap. After a quench the free
energy is suddenly changed, exciting the superconducting condensate and
leading to collective Higgs oscillations, indicated by a black arrow. The red
arrow indicates a quench by a THz pump pulse. c) Higgs excitation mechanism
using a THz pump pulse. The pump pulse only slightly overlaps with the quasi-
particle continuum indicated in blue. The Mexican hat shrinks due to the
breaking of Cooper pairs. To excite the Higgs oscillation the pulse must
fulfil the nonadiabacity condition in time domain.

The Higgs oscillation can be understood as a massive excitation along the
radial direction in the Mexican hat potential of the free energy
PhysRevLett.13.508 ; PhysRevB.84.174522 ; PhysRevB.87.054503 ;
Pashkinscience14 . Several mechanisms have been investigated in order to start
Higgs oscillations in superconducting systems PhysRevLett.45.660 ; Sherman2015
; PhysRevB.89.060503 ; PhysRevB.91.214505 .
Phenomenologically the following instructive picture can be drawn: Higgs
oscillations can be excited by taking the superconductor out of equilibrium,
for example by an ultrafast intense THz laser pulse kopninBook ; Orenstein12 ;
Papen07 ; Papen08 ; Schny11 ; Krull14 ; PhysRevB.95.104507 ; volkov1974 ;
YuzJofPhys05 ; zachmannNJP13 ; PhysRevB.92.224517 ; PhysRevB.96.184518 ;
2017arXiv171104923K . In this case, Cooper pairs are partially broken and the
landscape of the free energy changes suddenly, i.e. the Mexican hat shrinks.
Thus the THz laser pulse acts like a quantum quench, reflecting the impulsive
character of the light pulse. As long as this process is faster than the time
scale of the condensate, given by $\tau_{\Delta}=\hbar/(2\Delta)$, where
$2\Delta$ is the energy gap of the superconductor, the condensate is unable to
follow the minimum of free energy adiabatically krull_nat_commun_2016 ; Akba13
. Consequently, the collective Higgs oscillation of the condensate is excited,
as sketched in Fig. 1b.
In order to excite Higgs oscillations, the laser pulse must fulfil two
conditions. On the one hand, the pulse should only excite a small fraction of
the Cooper pairs, enough to have a significant quench of the Mexican hat, but
not too many that the superconducting signatures would be screened by hot
electrons. More specifically, a short optical pulse far above gap frequencies
induces a strong Drude-Peak in the optical conductivity, which would overlap
with the weak signal of the Higgs oscillations. Instead a suitable pulse
corresponds to a peak located in or close to the gap, which only slightly
overlaps with the continuum of quasi-particles, see left side of Fig. 1c. On
the other hand the pulse must fulfil the non-adiabacity condition, which
implies a short laser pulse (right side of Fig. 1c) and hence requires the
broad spectrum in energy domain (rather than a narrow band multicycle pulse
tuned close to the gap). For typical gaps in the meV regime, a single cycle
THz laser pulse is exactly on the brink of these two regimes Fulop12 ,
allowing for an excitation of the Higgs oscillations without heating the
system too much. For thin films of the s-wave superconductor Nb1-xTixN, this
approach has already been implemented successfully in experiments
matsunagaPRL12 ; matsunagaPRL13 ; Matsunagascience14 ; Pashkinscience14 .
Note that the breaking of Cooper pairs is accompanied by transfer of momentum
$2\bf q$. This reflects the finite quadrupol moment of the excitations. For a
THz pump pulse the corresponding wavelength is in the same order of magnitude
as the coherence length of the Cooper pairs $\xi^{-1}\propto|{2\bf q}|$.
In this article we show that nonequilibrium Higgs spectroscopy offers a unique
way to investigate the symmetry and collective excitation spectrum of
superconductors. Depending on the symmetry of the excitation mechanism and the
ground state of the system, Higgs oscillations with different frequencies can
be induced and identified. In strong correspondence to phonon spectroscopy,
this allows us to completely characterize the nature of a superconducting
condensate with a single class of experiments, repeated at various momentum
points.

##  II Higgs spectroscopy

Figure 2: Illustration of d-wave Higgs oscillation symmetries. Possible Higgs oscillation symmetries for a $d_{x^{2}-y^{2}}$-wave superconductor with point group symmetry $D_{4h}$ of the underlying lattice. The arrows indicates the motion of the lobes as a function of time. The notation of the symmetry stresses the initial state, from which the modes are excited. Note that during such a quantum quench, quasi-particles are excited as well, which are in general gapless for a d-wave superconductor. Thus the collective Higgs mode is located within a continuum of quasi-particle pairs making it a resonance with some width.  | Symmetry |  |
---|---|---|---
Gap Function | Oscillation | Quench | Modes | Excitation geometry
$s$ | $A^{1g}_{s}$ | $\left[2\left(x^{2}-y^{2}\right)^{2}-1\right]$ |  |  $\phi=$any
| $A^{2g}_{s}$ | $\left[xy\left(x^{2}-y^{2}\right)\right]$ |  |  $\phi=$any
| $B^{1g}_{s}$ | $\left[x^{2}-y^{2}\right]$ |  |  $\phi=$any
| $B^{2g}_{s}$ | $\left[xy\right]$ |  |  $\phi=$any
$d_{x^{2}-y^{2}}$ | $A^{1g}_{x^{2}-y^{2}}$ | $\left[x^{2}-y^{2}\right]$ |  | $\phi=\pi/4$
| $A^{2g}_{x^{2}-y^{2}}$ | $\left[xy\right]$ |  | $\phi=0$
| $B^{1g}_{x^{2}-y^{2}}$ | $\left[2\left(x^{2}-y^{2}\right)^{2}-1\right]$ |  | $\phi=0$
| $B^{2g}_{x^{2}-y^{2}}$ | $\left[xy\left(x^{2}-y^{2}\right)\right]$ |  | $\phi=\pi/4$
$d_{xy}$ | $A^{1g}_{xy}$ | $\left[xy\right]$ |  | $\phi=0$
| $A^{2g}_{xy}$ | $\left[x^{2}-y^{2}\right]$ |  | $\phi=\pi/4$
| $B^{1g}_{xy}$ | $\left[2\left(x^{2}-y^{2}\right)^{2}-1\right]$ |  | $\phi=\pi/4$
| $B^{2g}_{xy}$ | $\left[xy\left(x^{2}-y^{2}\right)\right]$ |  | $\phi=0$
Table 1: Higgs spectroscopy Possible Higgs oscillations for a $D_{4h}$ point
group symmetry of the lattice. Depending on quench symmetries, the possible
modes are shown for $s$, $d_{x^{2}-y^{2}}$ and $d_{xy}$ gap functions. The
number and position of the modes are indicated by symbols. For d-wave we
expect one or two modes, depending on the quench symmetry. To excite the modes
with a maximal response on the quench, the incident angle $\phi$ between the
pump pulse momentum $\mathbf{q}$ and the axis $\mathbf{k_{x}}$ is given in the
last column, see also Section III.

While the nonequilibrium probe of collective excitations in conventional
s-wave superconductors has been studied intensively barankov2004 ;
matsunagaPRL12 ; matsunagaPRL13 ; krull_nat_commun_2016 , the response for
unconventional superconductors is still at its infancy. In view of the
promising results so far, the question arises: How can we transfer the phonon
spectroscopy technique to collective modes in superconductors? These systems
often exhibit very complicated correlations DalScience2013 ;
PhysRevLett.108.227003 ; PhysRevX.4.041046  and a variety of different
mechanisms which can lead to superconductivity. If we want to examine the
nonequilibrium response of the coherent condensate of such superconductors in
general, we have to go back to the fundamental properties of these systems:
The symmetry of the lattice.
According to group theory, every configuration of the condensate at a given
time can always be decomposed with respect to the different irreducible
representations of the point group symmetry of the lattice. As an example, we
take a superconductor on a lattice with $D_{4h}$ space group symmetry, which
is the lattice symmetry of cuprate high-$T_{c}$ superconductors. Based on this
argument, there are four different irreducible representations for the
possible gap symmetries: $A_{1g}$, $A_{2g}$, $B_{1g}$ and $B_{2g}$. The Higgs
oscillations can always be decomposed into the contributions from these
sectors.
The strong excitation is achieved experimentally by short intense laser pulses
Aku-Leh:04 ; PerfettiPRL07 ; Demsar2003 ; mihailovic_PRL_09 ;
PhysRevB.96.020505 ; mansartPNAS13 . Theoretically, we start by considering a
quench of the initial state. Every quantum quench deforms the gap function
from its equilibrium value, which then can be decomposed into contributions
from different irreducible representations. Taking for example a momentum
independent quench would only probe the $A_{1g}$ channel of the condensate
peronaci_capone_PRL_15 , but does not couple to other allowed modes. The
solution to address also the other possible modes is to modify the quench and
make it momentum depended, so that we can probe other symmetry sectors as
well. For example in case of $d_{x^{2}-y^{2}}$-wave superconductivity, the
possible modes are shown in Fig. 2.
In order to illustrate this concept, we performed numerical exact simulations
for s- and d-wave BCS superconductors to study the nonequilibrium response to
k-dependent quantum quenches. The Hamiltonian we are investigating is given by

$\displaystyle H_{\text{BCS}}=H_{0}-W\sum_{{\bf k},{\bf k}^{\prime}}f({\bf
k})f({\bf
k}^{\prime})c^{\dagger}_{\textbf{k},\uparrow}c^{\dagger}_{-\textbf{k},\downarrow}c^{\
}_{-\textbf{k}^{\prime},\downarrow}c^{\ }_{\textbf{k}^{\prime},\uparrow},$ (1)

with $f(\bf k)$ describing the symmetry of the interaction, $W$ the
interaction strength, and the normal state Hamiltonian
$H_{0}=\sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}}c^{\dagger}_{\textbf{k},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma}$,
where $c^{\dagger}_{\textbf{k},\sigma}$ creates electrons with momentum ${\bf
k}$ and spin $\sigma$. Within the BCS solution, the gap is determined by

$\displaystyle\Delta_{0,\textbf{k}}=Wf({\bf
k})\sum_{\textbf{k}^{\prime}\in\mathcal{W}}f({\bf k}^{\prime})\langle
c^{\phantom{\dagger}}_{-\textbf{k}^{\prime},\downarrow}c^{\phantom{\dagger}}_{\textbf{k}^{\prime},\uparrow}\rangle.$
(2)

Now we perform a quantum quench, by changing the symmetry of the quasi-
particle distribution $\langle
c^{\phantom{\dagger}}_{-\textbf{k}^{\prime},\downarrow}c^{\phantom{\dagger}}_{\textbf{k}^{\prime},\uparrow}\rangle$
slightly away from its equilibrium value. Note that we do not change the
symmetry of the gap, it always remains in a single symmetry sector. For
details, please see supplemental material S1. After the quench we calculate
the Higgs oscilllation of the order parameter as function of time. For the
temporal evolution we use the density matrix formalism Papen07 , which is
exact for the Hamiltonian in (1). Our main results are summarized in Tab. 1.
For a given gap function, there are different oscillations possible for the
Higgs mode, which can be excited depending on the symmetry of the quench. This
approach closely follows the idea of Raman and phonon spectroscopy, but due to
the nature of the Higgs mode, it can only be observed in a nonequilibrium
setup. Thus a modified notation is required, which takes the gap symmetry into
account. We add as an additional information the gap symmetry as as subscript
and the oscillation symmetry as a superscript to the group theoretic notation.
The resulting classification for the $D_{4h}$ point group can be found in the
second column of Tab. 1. We observe, that not only the symmetric $A^{1g}$ mode
is excited, but a second low energy mode can appear in the spectrum of the gap
oscillations. Two examples for this observation are shown in Fig. 3. On the
left hand side, we see the Higgs oscillations of the $d_{x^{2}-y^{2}}$ gap
after a quench in the $A_{x^{2}-y^{2}}^{1g}$ channel and in the
$B_{x^{2}-y^{2}}^{1g}$ channel. A symmetric $A_{s}^{1g}$ quench for a s-wave
superconductor is shown in black for comparison. We highlight that the d-wave
oscillations decay much faster than the s-wave oscillations. This can be
traced back to the stronger dephasing of the mode due to coupling to the
gapless quasi-particles in the d-wave case. The final value of the gap, i.e.
$\Delta_{\infty}$, depends on the strength of the quench, i.e. how strongly
the initial states deviate from the equilibrium state Yuz05 ; Yuz06 . The
right hand side of Fig. 3 shows the Fourier transform of the Higgs
oscillations. The large peak at $2\Delta_{\infty}$ corresponds to the
symmetric $A_{x^{2}-y^{2}}^{1g}$ mode. Most importantly a second low energy
peak is visible in the $B_{x^{2}-y^{2}}^{1g}$ channel for d-wave
superconductors. This mode does not exist for pure s-wave superconductors and
it is also not excitable by a highly symmetric $A_{x^{2}-y^{2}}^{1g}$ quench.
Similarly for other combinations of gap and quench symmetries additional modes
can be identified. Thus there exists a direct connection between the symmetry
of the gap and the existence of low energy Higgs modes. Note that although the
$B_{x^{2}-y^{2}}^{2g}$ mode is not a highly symmetric quench, it shows no low
energy peak. Hence, the important conclusion is possible, that for a full
analysis of the gap structure, information from all possible quench symmetries
is required.

Figure 3:  d-wave Higgs oscillations. a) Numerically exact simulation of the
Higgs oscillations induced by various quench symmetries for a
$d_{x^{2}-y^{2}}$-wave superconductor. The blue lines show the gap
oscillations as function of time. The black solid line shows an s-wave quench
for an s-wave superconductor for comparison . b) Fourier spectrum of the Higgs
oscillations. The $A^{1g}_{x^{2}-y^{2}}$ oscillation shows a single peak,
similar to the s-wave case (shown in solid black). The peak position
corresponds to $2\Delta_{\infty}$ which is two times the maximum of the gap
for $t\rightarrow\infty$ after the quench. In the channel
$B^{1g}_{x^{2}-y^{2}}$ a second peak at low energy appears, which is absent in
$A^{1g}_{x^{2}-y^{2}}$.

Figure 4:  d-wave optical conductivity after excitation with a pump pulse.
Real part of the optical conductivity Re $\sigma(\Delta t,\omega)$ after a
realistic pump pulse for incident angles a) $\phi=0$ and b) $\phi=\pi/4$ for a
$d_{x^{2}-y^{2}}$-wave superconductor. The pump pulse parameters are
$\tau_{\textrm{p}}=0.4$ ps, $|{\bf A}_{\textrm{p}}|$ = $7\cdot 10^{-8}$ Js/Cm
and $\hbar\omega=3$ meV for the pump pulse and $\hbar\omega=2.5$ meV. The
vertial axis denotes the time delay between the excitation of the system with
the pump pulse and the probe pulse. The horizontal axis denotes frequency
$\omega$. The oscillation frequencies in $\Delta t$ correspond to the Higgs
modes given in Tab. 1, see also supplemental material S5.

To understand the nature of the second mode in more detail, we performed a
linear analysis of the gap dynamics after a quantum quench. Specifically, we
analytically compute the dynamics of the gap according to the expansion

$\displaystyle\Delta(t)=\Delta(0)+\delta\Delta(t),$ (3)

in first order of $\delta\Delta(t)$ for different initial states. Here
$\Delta(0)$ is the gap for time $t=0+\epsilon$ for very small $\epsilon$
directly after the quantum quench. Transforming into Laplace space allows us
to identify the leading contributions to the gap dynamics and leads to the
expression

$\displaystyle\frac{\delta\Delta(s)}{\Delta_{0}}=\frac{F_{2}(s)}{1-F_{1}(s)}$
(4)

where $\Delta_{0}$ is the equilibrium gap and

$\displaystyle F_{2}(s)\propto$ (5)
$\displaystyle\int_{0}^{2\pi}\mathrm{d}\varphi\frac{f(\varphi)(\Delta_{0}f^{\prime}(\varphi)-\Delta(0)f(\varphi))}{\sqrt{s^{2}+4\Delta_{0}^{2}(f(\varphi)^{2}-f^{\prime}(\varphi)^{2})}\sqrt{s^{2}+4\Delta_{0}^{2}f(\varphi)^{2}}}\dots,$

where the dots imply a weight function. In the denominator of $F_{2}$ we see a
low energy scale appearing, $f(\varphi)^{2}-f^{\prime}(\varphi)^{2}$, which
corresponds to the second mode in the numerical calculations. Thus we can
clearly trace back the second mode to the difference in the symmetry between
the quench and the condensate.
Note that $f^{\prime}(\varphi)$ describes the symmetry of the quantum quench.
For example in a $A^{1g}_{x^{2}-y^{2}}$ quench the symmetry of the quantum
quench is proportional to $\cos(2\varphi)$. In these expressions we assumed,
that the symmetry functions primarily depend on the angle $\varphi$ between
$\mathbf{k}$ and $\mathbf{k_{x}}$. For details, please see supplemental
material S1.
In this section we have shown, that nonequilibrium Higgs oscillations can be
used as an efficient tool, to completely classify the ground state symmetry of
a superconducting condensate. We used a lattice with $D_{4h}$ space group
symmetry to compute the possible $s$\- and $d$-wave Higgs oscillations and
introduced a new notation to account for the symmetry of the quench. Our
analysis given in Tab. 1, clearly shows that the presence of Higgs modes
serves as an identifier for the symmetries of the ground state. This scheme
can be easily generalized to all possible superconducting gaps. It is also
applicable in multi-band scenarios, where even more effects determine the
response of the condensate krull_nat_commun_2016 .

##  III Realistic pulse.

So far we concentrated our analysis on quantum quenches, which we classified
according to the symmetry of their initial deformation. To show that these
results also carry over to more realistic scenarios, we calculate the response
of a $d_{x^{2}-y^{2}}$-wave BCS superconductor coupled to a laser field. So
called pump-probe experiments have been used to study the excitation and
relaxation processes of superconductors Ave01 ; PerfettiPRL07 ; Demsar2003 ;
mihailovic_PRL_09 ; leitensdorferPRL10 . In a pump-probe experiment the pump
pulse excites the system and after a delay time the probe pulse measures
various properties of the transient dynamics of the system. Varying the delay
time, the temporal evolution of the system after a perturbation can be
studied. The Hamiltonian describing the pump and probe pulses, is given in the
supplemental material S1. We assume a short and intense THz laser pump pulse
which excites the condensate in an anisotropic fashion and the superconducting
gap starts to oscillate. For all pulses we fixed the pulse duration to
$\tau=0.4$ ps. With this choice we are in the nonadiabatic regime, where a
generation of collective modes is possible. Further, we varied the direction
of the light wave vector ${\bf q}$ to study the dependence of the optical
conductivity on the pump pulse. Thus we define the angle $\phi$ between ${\bf
q}$ and the $\bf k_{x}$ axis of the superconductor. By choosing the angle of
the pump pulse, the $A^{1g}_{x^{2}-y^{2}}$ as well as the
$B^{1g}_{x^{2}-y^{2}}$ Higgs oscillations can be addressed selectively (see
last column of Tab. 1 and Fig. S2 in the supplemental material). We compared
both methods, quantum quench and pump pulse, in the supplemental material S3.
Besides the time evolution of the gap, the pump-probe optical conductivity
provides an experimental fingerprint to observe Higgs oscillations as well.
This was explicitly shown in case of s-wave symmetry, i.e. as the oscillation
of the conductivity depending on the delay time Krull14 . Thus we use a pump
pulse to induce the Higgs oscillations and a much weaker probe pulse to
measure the optical conductivity. The specific form of the single cycle pump
pulse was measured experimentally and is implemented in the equation of
motion, see also supplemental material S4.

In Fig. 4 the real part of the optical conductivity Re $\sigma(\Delta
t,\omega)$ is shown for the angles $\phi=0$, along the anti-nodal direction,
and $\phi=\pi/4$, along the nodal direction, as a function of frequency
$\omega$ and time delay $\Delta t$. These angles correspond to the maximal
response angles in Tab. 1. Note that the probe pulse is applied in the same
direction as the pump pulse.
For $\phi=\pi/4$ the symmetric $A^{1g}_{x^{2}-y^{2}}$ mode is excited, and
most of the weight is located at or above the gap. For $\phi=0$ the low energy
Higgs mode and the symmetric mode are excited and the spectrum is dominated by
an in-gap response. Most importantly the signal oscillates with respect to the
time delay between pump and probe pulse, reflecting the excitation of Higgs
oscillations, see supplemental material S5. Thus the angle-resolved pump-probe
experiments should be able to see Higgs oscillations in the optical
conductivity, which can address different oscillation symmetries depending on
the incident angle.

##  IV Conclusion

Summarizing we have introduced the concept of nonequilibrium Higgs
spectroscopy that allows to characterize the ground state of superconducting
condensates. Our analytical calculations show, that depending on the symmetry
of the quench and of the gap function, low-lying modes exist, which can be
directly identified with the different oscillations of the condensate. We
introduced a new notation to combine the information of the ground state with
the quench symmetry in order to distinguish the different Higgs oscillations.
This analysis is applicable for all superconductors and requires only the
knowledge of the symmetry of the crystal. It is a natural extension of the
group theoretical notation to the case of nonequilibrium excitation of the
system. To exemplary demonstrate the approach, we fully characterized all
possible Higgs oscillations for the important $D_{4h}$ point group, relevant
for e.g. for high-temperature cuprate superconductors. It is important to note
that we did not introduce additional energy scales nor other degrees of
freedom, such as sub-leading channels: the observed oscillations and the
corresponding frequencies are intrinsic to the pure d-wave superconductor.
Beyond our analytical results, we showed that a realistic pulse can excite
these low-lying modes and that the Higgs oscillations should be observable in
the optical conductivity of a pump-probe experiment. These findings pave the
way for investigations and classifications of the dynamics of known and
unknown superconductors directly within the framework of the presented Higgs
spectroscopy.
In particular it opens the possibility to determine the symmetry of the order
parameter in systems that are difficult to access with momentum dependent
probes otherwise. This allows to trace the dynamics of the condensates
directly rather than probing the pair breaking dynamics via the absorption
spectrum. In principle our theory for Higgs spectroscopy is not limited to
characterize equilibrium condensates but may also be used to investigate
possible light induced superconducting states in transient states of matter
Mitrano2016 ; Kaiser_Scripta_2017 . Beyond that, Higgs spectroscopy could also
be extended to investigate collective excitations of non-superconducting,
symmetry broken phases, such as order parameter oscillations in excitonic
insulators 2016arXiv161101053W  or Higgs modes in antiferromagnets
PhysRevLett.119.067201 .

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Acknowledgements
We thank the Max Planck Centre for Quantum Materials for fruitful
collaborations and financial support. N. T. is supported by JSPS KAKENHI
(Grant No. 16K17729).

Methods
In order to compute the nonequilibrium response of the superconductor we use
the Anderson pseudo spin description and evaluate the Bloch equation
numerically. The linear analysis is based on an expansion of the Bloch
equation for small perturbations. The coupling to the light field was
implemented based on a semi-numerical method which computes the time dependent
Bogoliubov quasi-particle densities as function of time. The equations are
solved in a time independent Bogoliubov-de-Gennes basis and derived from the
Heisenberg equation of motion. The pump-probe spectra of the optical
conductivity are obtained by calculating the response of the pumped
superconductor to a weak probe pulse.

Supplemental Materials: Higgs spectroscopy of superconductors in
nonequilibrium

##  S1 Extended Methods

Model

The Hamiltonian we are investigating is given by,

$\displaystyle H_{\text{BCS}}$ $\displaystyle=H_{0}-W\sum_{{\bf k},{\bf
k}^{\prime}\in\mathcal{W}}f({\bf k})f({\bf
k}^{\prime})c^{\dagger}_{\textbf{k},\uparrow}c^{\dagger}_{-\textbf{k},\downarrow}c^{\
}_{-\textbf{k}^{\prime},\downarrow}c^{\ }_{\textbf{k}^{\prime},\uparrow},$
(S1) $\displaystyle H_{0}$
$\displaystyle=\sum_{\textbf{k},\sigma}\epsilon_{\textbf{k}}c^{\dagger}_{\textbf{k},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma}.$
(S2)

The normal state Hamiltonian was taken to be a free electron gas with an
effective mass $m$. The interaction strength $W$ is normalized to the system
size. The energy dispersion $\epsilon_{\textbf{k}}$ is measured relative to
the Fermi level. We apply the BCS solution in order to describe the
superconducting phase annett2004superconductivity . The superconducting gap
equation reads

$\displaystyle\Delta_{0,\textbf{k}}=Wf({\bf
k})\sum_{\textbf{k}^{\prime}\in\mathcal{W}}f({\bf k}^{\prime})\langle
c^{\phantom{\dagger}}_{-\textbf{k}^{\prime},\downarrow}c^{\phantom{\dagger}}_{\textbf{k}^{\prime},\uparrow}\rangle.$
(S3)

The sums in Eqs. (S1) and (S3) are taken over the set $\mathcal{W}$ of all
${\bf k}$ vectors with $|\epsilon_{\bf k}|\leq\hbar\omega_{c}$, $\omega_{c}$
being the frequency cutoff. For a phononic glue, this corresponds to the
Debeye frequency. We can define a momentum independent gap by

$\displaystyle\Delta_{0}=W\sum_{\textbf{k}^{\prime}\in\mathcal{W}}f({\bf
k}^{\prime})\langle
c^{\phantom{\dagger}}_{-\textbf{k}^{\prime},\downarrow}c^{\phantom{\dagger}}_{\textbf{k}^{\prime},\uparrow}\rangle.$
(S4)

For s-wave superconductors the function $f(\bf k)$ is a constant. In case of
the $D_{4h}$ group the irreducible representations for $f(\bf k)$ contain the
basis functions in Table 1 dresselhaus2007group .

Table 1: Even parity basis functions for $D_{4h}$ point group. $A_{1g}$ |  $1$, $\left(\cos(k_{x})-\cos(k_{y})\right)^{2}-1$
---|---
$A_{2g}$ | $\sin(k_{x})\sin(k_{y})\left(\cos(k_{x})-\cos(k_{y})\right)$
$B_{1g}$ | $\cos(k_{x})-\cos(k_{y})$
$B_{2g}$ | $\sin(k_{x})\sin(k_{y})$

In our calculations we assume, that the order parameter has full d-wave
symmetry in the $B_{1g}$ channel. In the ground state the expectation values
for the electron and quasi-particle distribution read

$\displaystyle\left\langle
c^{\dagger}_{\textbf{k},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma}\right\rangle=-\frac{\epsilon_{\textbf{k}}}{2E_{\bf
k}},$ (S5) $\displaystyle\left\langle
c^{\phantom{\dagger}}_{-\textbf{k},\downarrow}c^{\phantom{\dagger}}_{\textbf{k},\uparrow}\right\rangle=\frac{\Delta_{0}f(\textbf{k})}{2E_{\bf
k}}.$

Where $E_{\bf k}=\sqrt{\epsilon^{2}_{\bf k}+|\Delta_{\bf k}|^{2}}$ is the
quasi-particle energy. In our simulations we used the parameters
$\Delta_{0}=1.35$ meV, $E_{F}=9470$ meV and $m=1.9$ $m_{e}$.

Anderson pseudo spin description

We define the Nambu-Gorkov spinors

$\displaystyle\Psi_{\vec{k}}=\begin{pmatrix}c_{\mathbf{k},\uparrow}\\\
c^{\dagger}_{-\mathbf{k},\downarrow}\end{pmatrix}$ (S6)

and the Anderson pseudospin PhysRev.112.1900

$\displaystyle\vec{\sigma}_{\mathbf{k}}=\frac{1}{2}\Psi_{\mathbf{k}}^{\dagger}\vec{\tau}\Psi_{\mathbf{k}}$
(S7)

where $\vec{\tau}$ are the Pauli matrices. We rewrite the BCS Hamiltonian to

$\displaystyle
H_{\mathrm{BCS}}=2\sum\limits_{\mathbf{k}}\vec{b}_{\mathbf{k}}\cdot\vec{\sigma}_{\mathbf{k}}$
(S8)

with

$\displaystyle\vec{b}_{\mathbf{k}}=\left(-\Delta(t)f(\mathbf{k}),0,\epsilon_{\mathbf{k}}\right)$
(S9)

We assume a fixed phase of the gap $\Delta\in\mathbb{R}$. For
$d_{x^{2}-y^{2}}$-wave we can approximate $f(\mathbf{k})$ by

$\displaystyle f(\mathbf{k})\approx\cos(2\varphi)$ (S10)

where $\varphi$ is the angle of the $\bf k$ wave vector in polar coordinates.
The Bloch equation for the pseudo spins read PhysRevB.92.064508_2

$\displaystyle\partial_{t}\vec{\sigma}_{\mathbf{k}}=i\left[H_{\mathrm{BCS}},\sigma_{\mathbf{k}}\right]=2\vec{b}_{\mathbf{k}}\times\vec{\sigma}_{\mathbf{k}}.$
(S11)

At $t=0$, we apply a state quench, i.e. we change the state of the system
suddenly, and choose

$\displaystyle\sigma_{\mathbf{k}}^{x}(0)$
$\displaystyle=\frac{\Delta_{0}f^{\prime}(\mathbf{k})}{\omega^{\prime}_{\mathbf{k}}},$
(S12) $\displaystyle\sigma_{\mathbf{k}}^{y}(0)$ $\displaystyle=0,$ (S13)
$\displaystyle\sigma_{\mathbf{k}}^{z}(0)$
$\displaystyle=\frac{-\epsilon_{\mathbf{k}}}{\omega^{\prime}_{\mathbf{k}}},$
(S14) $\displaystyle\Delta(0)$
$\displaystyle=W\sum\limits_{\mathbf{k}}f(\mathbf{k})\sigma_{\mathbf{k}}^{x}(0)=W\sum\limits_{\mathbf{k}}f(\mathbf{k})\frac{\Delta_{0}f^{\prime}(\mathbf{k})}{\omega^{\prime}_{\mathbf{k}}},$
(S15)

where
$\omega^{\prime}_{\mathbf{k}}=2\sqrt{\epsilon_{\mathbf{k}}^{2}+\Delta_{0}^{2}f^{\prime}(\mathbf{k})^{2}}$
and $f^{\prime}(\mathbf{k})$ is the change of the symmetry of the state. For
example, for an $A^{1g}_{x^{2}-y^{2}}$ quench the symmetry of the quantum
quench is proportional to $\cos(2\varphi)$. Note the the gap $\Delta(0)$ at
$t=0$ is different from the equilibrium gap $\Delta_{0}$. For numerical
purposes we choose a gap value of $2\Delta_{0}=2.7$ meV. However all our
computations can be rescaled to any energy scale for the gap. The equations
are then computed on a finite size grid in momentum space.

Linear analysis

To probe the different symmetry sectors, we add small deviations to the basis
function $f(\textbf{k})$ of the d-wave order parameter in the distributions of
Eq. (S5). Then we calculate the time-dependence of the density expectation
values using the Heisenberg equation of motion. For example for the
$B^{1g}_{x^{2}-y^{2}}$ channel we evolve the system with an initial
$f^{\prime}(\textbf{k})$ given by

$\displaystyle f^{\prime}(\textbf{k})_{t=0}$
$\displaystyle=\frac{1}{2}(\cos(k_{x})-\cos(k_{y}))$
$\displaystyle+\delta\left(\left[\cos(k_{x})-\cos(k_{y})\right]^{2}-1\right)$
(S16)

with $\delta\ll 1$. In order to gain an analytical insight into the evolution
of the nonequilibrium equation of motion, we perform a linear analysis. We
linearize the equation of motion (eom) around this initial condition.

$\displaystyle\sigma_{\mathbf{k}}^{\alpha}(t)=\sigma_{\mathbf{k}}^{\alpha}(0)+\delta\sigma_{\mathbf{k}}^{\alpha}(t)$
(S17) $\displaystyle\Delta(t)=\Delta(0)+\delta\Delta(t)$ (S18)

The eom reads

$\displaystyle\partial_{t}\delta\sigma_{\mathbf{k}}^{x}(t)$
$\displaystyle=-2\epsilon_{\mathbf{k}}\delta\sigma_{\mathbf{k}}^{y}(t)$ (S19)
$\displaystyle\partial_{t}\delta\sigma_{\mathbf{k}}^{y}(t)$
$\displaystyle=2\epsilon_{\mathbf{k}}\left(\frac{\Delta_{0}f^{\prime}(\mathbf{k})}{\omega^{\prime}_{\mathbf{k}}}+\delta\sigma_{\mathbf{k}}^{x}(t)\right)$
$\displaystyle+2\Delta(0)f(\mathbf{k})\delta\sigma_{\mathbf{k}}^{z}(t)$
$\displaystyle+2\delta\Delta(t)f(\mathbf{k})\frac{-\epsilon_{\mathbf{k}}}{\omega^{\prime}_{\mathbf{k}}}$
(S20) $\displaystyle\partial_{t}\delta\sigma_{\mathbf{k}}^{z}(t)$
$\displaystyle=-2f(\mathbf{k})\Delta(0)\delta\sigma_{\mathbf{k}}^{y}(t)$ (S21)

Going into Laplace space, we can solve the ensuing equations for the gap and
end up with

$\displaystyle\delta\Delta(s)$ $\displaystyle=\frac{F_{2}(s)}{1-F_{1}(s)}$
(S22)

where

$\displaystyle F_{1}(s)$
$\displaystyle=\mathrm{const}-W\sum\limits_{\mathbf{k}}f(\mathbf{k})^{2}\frac{4\Delta_{0}^{2}f(\mathbf{k})^{2}+s^{2}}{\omega^{\prime}_{\mathbf{k}}\left(s^{2}+4\epsilon_{\mathbf{k}}^{2}+4\Delta_{0}^{2}f(\mathbf{k})^{2}\right)}$
(S23) $\displaystyle F_{2}(s)$
$\displaystyle=\frac{\Delta_{\mathrm{const}}}{s}+\frac{1}{s}\left[W\sum\limits_{\mathbf{k}}\frac{f(\mathbf{k})(\Delta_{0}f^{\prime}(\mathbf{k})-\Delta(0)f(\mathbf{k}))}{\omega^{\prime}_{\mathbf{k}}}\frac{s^{2}+4\Delta_{0}^{2}f(\mathbf{k})^{2}}{s^{2}+4\epsilon_{\mathbf{k}}^{2}+4\Delta_{0}^{2}f(\mathbf{k})^{2}}\right].$
(S24)

Note that $\mathrm{const}\in\mathcal{O}(1)$ and
$\Delta_{\mathrm{const}}\in\mathcal{O}(f^{\prime}(\mathbf{k})-f(\mathbf{k}))$.
We focus on the nominator and further evaluate

$\displaystyle F_{2}(s)$
$\displaystyle=\frac{\Delta_{\mathrm{const}}}{s}+\frac{U}{s}\int_{0}^{\infty}\mathrm{d}\epsilon\int_{0}^{2\pi}\mathrm{d}\varphi\frac{f(\varphi)(\Delta_{0}f^{\prime}(\varphi)-\Delta(0)f(\varphi))}{2\sqrt{\epsilon^{2}+\Delta_{0}^{2}f^{\prime}(\varphi)^{2}}}\frac{s^{2}+4\Delta_{0}^{2}f(\varphi)^{2}}{s^{2}+4\epsilon^{2}+4\Delta_{0}^{2}f(\varphi)^{2}}$
$\displaystyle=\frac{\Delta_{\mathrm{const}}}{s}+\frac{U}{s}\int_{0}^{2\pi}\mathrm{d}\varphi\frac{f(\varphi)(\Delta_{0}f^{\prime}(\varphi)-\Delta(0)f(\varphi))}{\sqrt{s^{2}+4\Delta_{0}^{2}(f(\varphi)^{2}-f^{\prime}(\varphi)^{2})}\sqrt{s^{2}+4\Delta_{0}^{2}f(\varphi)^{2}}}$
$\displaystyle\cdot\sinh^{-1}\left(\frac{\sqrt{s^{2}+4\Delta_{0}^{2}(f(\varphi)^{2}-f^{\prime}(\varphi)^{2})}}{2\Delta_{0}\left|f(\varphi)\right|}\right)$
(S25)

For simplicity, we make the usual assumption, that $f$ only depends on the
angle $\varphi$ in polar coordinates of the Brillouin zone. Without explicitly
solving the integral over $\varphi$, we already see a low-energy scale
emerging in the denominator. This low energy scale corresponds to the low
energy peak, as observed in the main text.

Quasi-particle density time evolution for pump pulses

The Hamiltonian describing the coupling between superconductor and pump pulse,
which brings the system out of equilibrium, is modelled by

$\displaystyle H_{\text{Laser}}$
$\displaystyle=\frac{e\hbar}{2m}\sum_{\textbf{k},\textbf{q},\sigma}(2\textbf{k}+\textbf{q})\textbf{A}_{\textbf{q}}(t)c^{\dagger}_{\textbf{k}+\textbf{q},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma}$
(S26)
$\displaystyle+\frac{e^{2}}{2m}\sum_{\textbf{k},\textbf{q},\sigma}\left(\sum_{\textbf{q}^{\prime}}\textbf{A}_{\textbf{q}-\textbf{q}^{\prime}}(t)\textbf{A}_{\textbf{q}^{\prime}}(t)\right)c^{\dagger}_{\textbf{k}+\textbf{q},\sigma}c^{\phantom{\dagger}}_{\textbf{k},\sigma},$

where ${\bf A}_{\bf q}(t)$ is the transverse vector potential Papen07_2 ;
Krull14_2 . Working within the Coulomb gauge, the pump pulse is expressed in
terms of the transverse vector potential

$\displaystyle{\bf A}_{\bf q}(t)={\bf
A}_{\textrm{p}}e^{-\left(\frac{2\sqrt{\ln
2}t}{\tau_{\textrm{p}}}\right)^{2}}\left(\delta_{{\bf q},{\bf
q}_{p}}e^{-i\omega_{\textrm{p}}t}+\delta_{{\bf q},-{\bf
q}_{p}}e^{+i\omega_{\textrm{p}}t}\right).$ (S27)

The pump pulse is of Gaussian shape with photon frequency
$\omega_{\textrm{p}}$ , photon wave vector ${\bf q}_{\textrm{p}}$ , full width
at half maximum (FWHM) $\tau_{\textrm{p}}$, and amplitude ${\bf
A}_{\textrm{p}}$. For our simulations we consider various directions of the
photon wave vector ${\bf q}_{\textrm{p}}$ and with this concomitantly various
directions of the quench induced by the pulse. In order to simulate the
evolution of the system, we use methods based on an expansion of Heisenberg’s
equation of motion. For the temporal evolution of the order-parameter we use
the density matrix formalism Ross02 . The main task of this technique is to
derive equations of motion for quasi-particle densities. Within this formalism
it is advantageous to perform a Bogoliubov transformation of the electron
operators which diagonalizes the Hamiltonian $H_{\text{BCS}}$ in the initial
state. We introduce new fermionic operators $\alpha_{\bf k}$ and $\beta_{\bf
k}$, with

$\displaystyle\alpha_{\bf k}=u_{\bf k}^{*}c_{{\bf k},\uparrow}+v_{\bf
k}c^{{\dagger}}_{-{\bf k},\downarrow},\quad\beta_{\bf k}^{{\dagger}}=-v_{\bf
k}^{*}c_{{\bf k},\uparrow}+u_{\bf k}c^{{\dagger}}_{-{\bf k},\downarrow},$
(S28)

where $v_{\bf k}=\Delta_{0}/|\Delta_{0}|\sqrt{(1-\epsilon_{\bf k}/E_{\bf
k})/2}$ and $u_{\bf k}=\sqrt{(1+\epsilon_{\bf k}/E_{\bf k})/2}$. We emphasize
that the coefficients $u_{\bf k}$ and $v_{\bf k}$ do not depend explicitly on
time, i.e., the temporal evolution of the quasi-particle densities is computed
with respect to a fixed time-independent Bogoliubov-de Gennes basis in which
the initial state is diagonal.

All physical observables, such as the order parameter amplitude $|\Delta_{\bf
k}(t)|$ can now be expressed in terms of the new Bogoliubov quasi-particle
densities $\langle\alpha^{{\dagger}}_{\bf k}\alpha^{\ }_{{\bf
k}^{\prime}}\rangle$, $\langle\beta^{{\dagger}}_{\bf k}\beta^{\ }_{{\bf
k}^{\prime}}\rangle$, $\langle\alpha^{{\dagger}}_{\bf
k}\beta^{{\dagger}}_{{\bf k}^{\prime}}\rangle$, and $\langle\alpha^{\ }_{\bf
k}\beta^{\ }_{{\bf k}^{\prime}}\rangle$. Applying the density matrix formalism
for these quasi-particle densities, we get a closed set of differential
equations. However, in order to reduce computational effort, all densities
with $|\bf k-{\bf k}^{\prime}|>4|\bf q_{p}|$ are set to zero, because they
only contribute in order $\mathcal{O}\left({\bf A}_{\textrm{p}}^{5}\right)$.
The ensuing differential equations are then solved on a finite size grid in
momentum space.

Optical conductivity

To calculate the optical conductivity, we calculate the temporal evolution of
the current density as function of time delay

$\displaystyle\mathbf{j}_{\bf q_{\mathrm{pr}}}(\Delta t,t)$
$\displaystyle=\frac{-e\hbar}{2mV}\sum\limits_{\bf k,\sigma}\left(2\bf k+\bf
q_{\mathrm{pr}}\right)\left\langle c^{\dagger}_{\bf
k,\sigma}c^{\phantom{\dagger}}_{\bf k+\bf
q_{\mathrm{pr}},\sigma}\right\rangle(\Delta t,t)$
$\displaystyle-\frac{e^{2}}{mV}\sum\limits_{\bf k,\bf q,\sigma}\mathbf{A}_{\bf
q_{\mathrm{pr}}-\bf q}\left\langle c^{\dagger}_{\bf
k,\sigma}c^{\phantom{\dagger}}_{\bf k+\bf q,\sigma}\right\rangle(\Delta t,t),$
(S29)

where $V$ is the normalization volume and $\bf q_{\mathrm{pr}}$ is the wave-
vector of the probe pulse Papen07_2 . We neglect the second term, because it
only leads to an offset of the imaginary part of the optical conductivity.
Then the optical conductivity can be calculated PhysRevB.93.195144 ; Krull14_2
by computing

$\displaystyle\sigma(\Delta t,\omega)=\frac{j_{\bf q_{\mathrm{pr}}}(\Delta
t,\omega)}{i\omega A_{\bf q_{\mathrm{pr}}}(\omega)}.$ (S30)

##  S2 Microscopic description of Higgs excitations

The microscopic mechanism to excite Higgs oscillations with an intense THz
laser pulse is shown in Fig. S1. Note that the external field couples in a
nonlinear fashion to the broken Cooper pairs of the superconducting condensate
annurev.varma2015_2 . The time dependence of the propagating quasi-particles
corresponds to the time dependence of the superconducting condensate. The
transferred momentum ${\bf q}$ is much smaller than the Fermi wave vector,
$|{\bf q}|\ll|\bf k_{\mathrm{F}}|$. Therefore the pulse is only breaking a
small fraction of the Cooper pairs but it does not lead to Fulde-Ferrel-
Larkin-Ovchinnikov (FFLO) oscillations. Note that this diagram only describes,
how the light breaks the Cooper pairs. The actual calculation of the gap
dynamics is computed using an expansion of Heisenberg’s equation of motion for
the quasi-particle densities.
Note that we make no assumptions on the actual glue function. Within Higgs
spectroscopy it is not possible to distinguish different pairing mechanism,
which can affect the broadening of the Higgs oscillations. But the presence of
the Higgs modes is independent of the pairing and results only from the
symmetry of the condensate.

Figure S1:  Feynmann diagram describing the elementary excitation mechanism of
the Higgs oscillations via a laser field The external laser field
$A_{\mathbf{q}}(t)$ couples quadratically to the broken Cooper pairs, which
stem from the superconducting condensate. Within the time-dependent BCS
theory, the condensate is described by the self-consistent Hartree-Fock
diagram.

##  S3 Comparison between analytical calculations and numerical results

Since we can not simplify the result (S25) further, we calculated the gap
dynamics of the linearized equation numerically and compare it to the solution
of the full equation and the simulation with a light pulse in Fig. S2. All
solutions show a low energy peak, however the full nonlinear equations further
amplify the intensity of the low energy mode. The pulse solution shows a
stronger broadening then the analytical calculations, but is in qualitative
agreement. Note that we do not expect quantitative agreement, since a quantum
quench is different from a laser pulse.

Figure S2:  Comparison between full solution, the linear analysis and the
pulse solution of the d-wave Higgs oscillations. Fourier transform of the gap
oscillations for a $B^{1g}_{x^{2}-y^{2}}$ quench. Comparison between the full
exact solution and the linearized equations. Although the linear analysis
underestimates the low energy peak intensity, it is clearly visible.

##  S4 Analysis of realistic pump pulse

In order to verify, that our description of the experimental pulse is valid,
we performed measurements of the electric field of a single cycle THz laser
pulse. From the theoretical parametrization in Eq. (S27) we deduced the field
$E(t)$ and fitted it to the experimental results in Fig. S3. The fit provides
an excellent approximation to the pulse used in experiment. In our simulations
we rescaled the pulse width to $\tau=0.4$ ps and the pump frequency to
$\hbar\omega_{\textrm{p}}=3.0$ meV to adjust it to the gap parameter used.

Figure S3:  Comparison between experimental pulse and theoretical
parametrization. The electrical field of a single cycle THz pump pulse as used
in experiment. The parameters of a fit with a Gaussian shape, as given in Eq.
(S27), are given by $\hbar\omega_{\textrm{p}}=1.33$meV and
$\tau_{\textrm{p}}=1.63$ps.

##  S5 Fourier Transform of the optical conductivity

The optical conductivity shown in Fig. 4 of the main text, displays Higgs
oscillations as function of time delay $\Delta t$ between pump and probe
pulse. In Fig. S4 we analyze the Fourier spectrum of the oscillations, to
verify that they correspond to the Higgs oscillations of the gap. To achieve
this, we fixed the frequency $\omega$ of the optical conductivity and Fourier
transformed the optical conductivity as function of time delay $\Delta t$. We
denote the corresponding frequency axis with $\omega^{\prime}$ to distinguish
it from the frequency $\omega$. Clearly the optical conductivity shows the
same peak structure as the Higgs oscillations, depending strongly on the
incident angle $\phi$. Thus the optical conductivity serves as an efficient
tool to perform Higgs spectroscopy.

Figure S4:  Fourier transform in the delay time $\Delta t$ of the optical
conductivity for a d-wave superconductor after a pump pulse. Fourier transform
of the real part of the optical conductivity for fixed frequency slices in
Fig. 4 of the main text. The red and the blue line display Higgs oscillations
for a pump pulse with $\phi=0$. The red curve corresponds to $\phi=\pi/4$.

##  S6 Fluence dependence of the gap for realistic pump pulses

Figure S5:  Fluence dependence of the gap dynamics for $\phi=0$. Fourier
spectrum of the Higgs oscillations as a function of energy and laser fluence.
The curves indicate the position of the peak at higher energies and of the
lower-lying peak.

Besides the angular dependence of the Higgs mode in simulations with realistic
pump pulses, it also depends on the intensity of the pump pulse. In Fig. S5
the Fourier transform of $\Delta(t)$ is shown for various pump pulse
intensities. The $A_{x^{2}-y^{2}}^{1g}$ Higgs mode depends on the asymptotic
value of the order parameter. With increasing intensity more and more quasi-
particles are excited and thereby the order parameter is reduced. Due to this
mechanism the frequency of the symmetric Higgs mode decreases with increasing
intensity. This behaviour is similar to the $s$-wave superconductor Yuz06_2 .
In contrast, the frequency of the second mode at low energy is increased at
higher fluence. But we can only observe this mode as long as its energy is
lower than the energy of the first mode, because otherwise it can no longer be
resolved. The direction of the frequency shift of the lower mode can be
understood by investigating the main result in Eq. 5 of the main text. Here
the low energy scale represents the second mode also visible in the pump-probe
calculation. For increasing strength of the quench, i.e. intensity of the
laser, the difference $f(\phi)^{2}-f^{\prime}(\phi)^{2}$ increases. This
factor also determines the position of the second mode in the energy domain.
Thus with increasing intensity the energy of the low energy mode is increased
as well.

## References

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  * (2) Dresselhaus, M., Dresselhaus, G. & Jorio, A.  _Group Theory: Application to the Physics of Condensed Matter_.  SpringerLink: Springer e-Books (Springer Berlin Heidelberg, 2007).  URL https://books.google.com/books?id=sKaH8vrfmnQC.
  * (3) Anderson, P. W.  Random-phase approximation in the theory of superconductivity.  _Phys. Rev._ 112, 1900–1916 (1958).  URL https://link.aps.org/doi/10.1103/PhysRev.112.1900.
  * (4) Tsuji, N. & Aoki, H.  Theory of anderson pseudospin resonance with higgs mode in superconductors.  _Phys. Rev. B_ 92, 064508 (2015).  URL https://link.aps.org/doi/10.1103/PhysRevB.92.064508.
  * (5) Papenkort, T., Axt, V. M. & Kuhn, T.  Coherent dynamics and pump-probe spectra of bcs superconductors.  _Phys. Rev. B_ 76, 224522 (2007).
  * (6) Krull, H., Manske, D., Uhrig, G. S. & Schnyder, A. P.  Signatures of nonadiabatic BCS state dynamics in pump-probe conductivity.  _Phys. Rev. B_ 90, 014515 (2014).
  * (7) Rossi, F. & Kuhn, T.  Theory of ultrafast phenomena in photoexcited semiconductors.  _Rev. Mod. Phys._ 74, 895–950 (2002).
  * (8) Shao, C., Tohyama, T., Luo, H.-G. & Lu, H.  Numerical method to compute optical conductivity based on pump-probe simulations.  _Phys. Rev. B_ 93, 195144 (2016).  URL https://link.aps.org/doi/10.1103/PhysRevB.93.195144.
  * (9) Pekker, D. & Varma, C.  Amplitude/higgs modes in condensed matter physics.  _Annu. Rev. Condens. Matter Phys_ 6, 269–297 (2015).
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\caption{The irreducibles that have no vertex of degree 2.}
\label{fig_no_deg_two}
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\end{tabular}
\caption{CCA representations of the 12 irreducible torus graphs
with no vertices of degree two: \( C^i_j \) is a CCA representation of
\( G^i_j \) from Figure \ref{fig_no_deg_two}.}
\label{fig_CCArepsnodegree2}
\end{figure}

Finally we observe that allowing divalent vertex additions we have the following inductive
construction for \( (2,2) \)-tight torus graphs.

\begin{thm}
If \( G \) is a \( (2,2) \)-tight torus graph then \( G \) can be constructed
from one of the torus graphs in Figure \ref{fig_no_deg_two} by a sequence
of moves each of which is either a digon split, triangle split,
quadrilateral split or a divalent vertex addition. \qed
\end{thm}

\clearpage

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