Impact of damping on superconducting gap oscillations induced by intense Terahertz pulses
Tianbai Cui Xu Yang Chirag Vaswani Jigang Wang Rafael M. Fernandes Peter P. Orth






School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA



Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011,
USA



Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011,
USA



Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011,
USA



School of Physics and Astronomy, University of Minnesota, Minneapolis,
Minnesota 55455, USA



Department of Physics and Astronomy, Iowa State University, Ames,
Iowa 50011, USA

Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iowa 50011,
USA

We investigate the interplay between gap oscillations and damping
in the dynamics of superconductors taken out of equilibrium by strong
optical pulses with sub-gap Terahertz frequencies. A semi-phenomenological
formalism is developed to include the damping within the electronic
subsystem that arises from effects beyond BCS, such as interactions
between Bogoliubov quasiparticles and decay of the Higgs mode. Such
processes are conveniently expressed as T_1 and T_2 times
in the standard pseudospin language for superconductors. Comparing
with data on NbN that we report here, we argue that the superconducting
dynamics in the picosecond time scale, after the pump is turned off,
is governed by the T_2 process. 

Impact of damping on superconducting gap oscillations induced by
intense Terahertz pulses
    Peter P. Orth
    December 30, 2023
=========================================================================================



Introduction.– The coherent control of non-equilibrium
states of interacting quantum matter promises far-reaching capabilities
by turning on (or off) desired electronic material properties. A particular
focus in this field has been the manipulation of superconductivity
by non-equilibrium probes. While earlier works showed that microwave
pulses could be used to enhance the superconducting transition temperature
T_c of thin superconducting films <cit.>,
recent advances in ultrafast pump-and-probe techniques opened the
possibility of investigating superconductivity in the pico- and femto-second
timescales by coherent light pulses <cit.>.
Such coherent pulses have been employed to manipulate the electronic
and lattice properties of quantum materials, resulting in transient
behaviors that are consistent with the onset of non-equilibrium superconductivity
above T_c <cit.>.
Alternatively, coherent pulses have also been employed to assess the
coherent dynamics of the superconducting state <cit.>.

To maintain coherence and avoid excess heating, it is advantageous
to apply pulses at energies below twice the superconducting gap 2Δ,
where quasi-particle (Bogoliubov) excitations are absent. As the superconducting
gap energy scale lies in the Terahertz (THz) regime, this requires
the application of intense and coherent sub-gap THz light pulses <cit.>.
In Ref. <cit.>, a monocycle intense THz pulse
was applied to a thin film of the conventional s-wave superconductor
NbN, reporting coherent oscillations of the superconducting gap with
frequency 2Δ.

Such oscillations arise naturally from the solution of the time-dependent
BCS (Bardeen-Cooper-Schrieffer) equation <cit.>, which can be conveniently recast in terms of Anderson pseudospins <cit.>
precessing around a pseudo magnetic field that is changed by the optical
pulse. While this coherent evolution describes qualitatively well
the behavior of the system in a restricted time window, there is also
damping present in the system, which is absent in this BCS approach.

Here, we develop a semi-phenomenological model that captures not only
the coherent evolution of the gap function, in the picosecond time
scale, but also damping effects in the time scale of tens to hundreds
of picoseconds. Since this time scale precedes the thermalization
with the lattice, the relevant relaxation processes arise within the
electronic subsystem from effects not captured by BCS. These include
interactions between Bogoliubov quasiparticles and the coupling between
the Higgs (amplitude) mode and the continuum. In the pseudospin notation,
we identify two types of relaxation process: the longitudinal relaxation
T_1, corresponding to relaxation of quasiparticles, and the transverse
relaxation T_2, corresponding to relaxation of the gap.

We apply this formalism to elucidate the dynamics of superconducting
NbN, which was measured at very low temperatures using intense THz
fields with sub-gap spectra. Our data reveals the gap oscillating
at a frequency corresponding to twice the pump frequency. When the
pump is turned off, however, the gap oscillations quickly disappear,
and the amplitude of the gap continues to be suppressed. Such a behavior
is at odds with the nonequilibrium dynamics given by the time-dependent
BCS equation, where the gap displays coherent oscillation with very
slow collisionless relaxation <cit.>. We show instead
that this behavior is well captured by our semi-phenomenological model,
and arises from a dominant T_2 relaxation process whose time
scale is of the same order as the duration of the pump.



Experimental results.– The data was acquired using
an intense THz pump, weak THz probe ultrafast spectroscopy setup.
A Ti-Sapphire amplifier was used to generate pulses of energy 3
mJ, duration 40 fs, 1 KHz repetition rate, and 800 nm center
wavelength. The pulses were split into three paths: pump, probe and
sampling. The intense THz pump pulses were generated by the tilted-pulse-front
phase matching through a 1.3% MgO doped LiNbO_3 crystal. The
weak THz probe pulses, generated by optical rectification, were detected
by free space electro-optic sampling through a 1mm thick (110) ZnTe
crystal. Pump and probe THz pulses with orthogonal polarizations were
combined by a wire grid polarizer in a collinear geometry and focused
on the sample at normal incidence. The pump was blocked by another
wire grid polarizer and the probe electric field E was sampled
by a 800 nm pulse. The peak E field of the narrow-band 1
THz pump was observed to be as large as 109 kV/cm.

The sample studied here was a 120 nm NbN film grown on (100)-oriented
MgO single crystalline substrates via pulsed laser deposition, as
previously reported in Ref. <cit.>. The equilibrium
and non-equilibrium optical conductivity were extracted from the complex
transmission using a scanning gate pulse delay t_gate.
The ultrafast dynamics was extracted by scanning the optical delay
t_pp between the pump and the probe.

Figs. <ref> (A)-(B) show the behavior of the real and imaginary
parts of the optical conductivity, σ_1(ω)
and σ_2(ω), respectively. In equilibrium
(gray cruves), the onset of superconductivity below T_c≈13.4
K is signaled by the opening of a gap 2Δ≈4.2 meV in
σ_1(ω), and by a 1/ω dependence
of σ_2(ω) at low frequencies. The post-pump
state (red curve) exhibits larger values of σ_1(ω)
within the 2Δ range, and slightly reduced values of σ_2(ω),
presumably due to the THz-induced quench of the SC condensate <cit.>.

To extract the ultrafast dynamics of the gap function, we measure
the change in the transmitted field Δ E/E, which was shown
in Ref. <cit.> to faithfully
reflect the transient behavior of Δ(t). Fig. <ref>(C)
shows the ultrafast time evolution of Δ E/E∝1-|Δ(t)|/Δ_0,
with Δ_0≡Δ(t=0), well inside the superconducting
state (blue curve, at T=4 K), superimposed with the applied pump
pulse (red curve). Interestingly, we find oscillations on Δ(t)
only while the pump pulse is on. After it is turned off, the oscillations
disappear quickly, but Δ(t) continues to decrease on the time
scale of tens of picoseconds. The Fourier decomposition of Δ E/E
(not shown) indeed demonstrates that the gap oscillations do not scale
with the gap function, unlike reported for shorter monocycle pulses <cit.>,
but instead correspond to twice the pump frequency <cit.>.

Theoretical modeling and analysis.– To model and
elucidate these experimental results, we need to consider relaxation
processes beyond the standard coherent time evolution predicted in
BCS theory. Within BCS, the quench dynamics of Δ(t) can display
three different behaviors <cit.>:
(i) overdamped decay of Δ(t)→0 (phase I); (ii) underdamped
oscillations with frequency 2Δ_∞ that decay algebraically
(∝ t^-1/2) towards a finite asymptotic value Δ(t)→Δ_∞
(phase II); and (iii) persistent undamped oscillations (phase III).

In contrast to these predictions, our experimental observation is
that although the gap oscillations are rapidly damped out, the gap
remains finite after the pump pulse is off (see Fig. <ref>(C)).
Moreover, it continues to show a slow decay between 10 ps and
20 ps, a behavior that presumably persists into the time scale
of hundreds of picoseconds. The gap eventually returns to its initial
equilibrium value on even longer nanosecond time scales via
equilibration with phonons. This regime is not discussed in this paper.

To explain this discrepancy, one must include damping within the electronic
subsystem. Before discussing possible microscopic mechanisms for damping,
we employ a phenomenological approach that is best expressed within
the pseudospin description of the BCS model. The standard BCS Hamiltonian
is given by:

    H_BCS=∑_,σξ_+e_0c_,σ^†c_,σ-∑_(Δ c_,↑^†c_-,↓^†+h.c.)+|Δ|^2/V_0


Here we consider the square-lattice dispersion ε_=-2J(cos k_x+cos k_y),
and ξ_=ε_-μ, with chemical potential
μ=-1.18J corresponding to quarter-filling, and electron charge
e_0. The superconducting order parameter obeys the self-consistent
equation Δ=-V_0∑_c_-,↓c_,↑,
where V_0<0 denotes an attractive interaction. For the calculations
in this paper, we set V_0=-3J and the Debye frequency ω_D=J/2,
yielding Δ_0=0.08J and T_c=0.048J.

The vector potential (t) is related to the electric field
of the pump via _pump=-∂/∂ t.
In our experiment, it takes the form (t)=A_0θ(-t)θ(τ-t)_pumpe^-(t-τ/2)^2/2σ^2cos(ω_pumpt)
with center frequency ω_pump, temporal width σ,
linear polarization vector _pump, and duration
τ. For the calculations in Fig. <ref>(D) and  <ref>,
which refer to our experiments on NbN, we consider a long pulse with
τ=10π/Δ_0, σ=τ/5, A_0=√(0.75Δ_0)
and ω_pump=1.41Δ_0, corresponding to a subgap
frequency. To compare with previous experiments involving short pulses,
such as Ref. <cit.>, in Fig. <ref>
we consider a Gaussian-shaped short pulse (t)=A_0θ(-t)θ(τ-t)_pumpe^-(t-τ/2)^2/2σ^2
with τ=5/Δ_0, A_0=√(1.5Δ_0).

To describe the gap dynamics, it is convenient to use Anderson pseudospins
_=ψ_^†σ/2ψ_,
with Nambu spinor ψ_=(c_,↑,c_-,↓^†)^T
and Pauli matrices σ. The Hamiltonian then takes the
simple form H_BCS=-∑__·_+|Δ|^2/V_0,
with a pseudo magnetic field _=2(Δ',-Δ”,-ξ̅_+e_0),
where Δ=Δ'+iΔ”, and ξ̅_+e_0=1/2(ε_+e_0+ε_-e_0)-μ.
Importantly, the magnetic field depends itself on the state of the
pseudospins via Δ=-V_0∑_S_^-.

In equilibrium, all spins are aligned with the field direction and
their expectation value is given by _,eq=1/2_,eqtanh(E_/2T_i).
Here, T_i denotes the initial temperature, E_=√(|Δ|^2+ξ_^2)
is the Bogoliubov quasiparticle dispersion, and _,eq=(cosϕsinθ_,-sinϕsinθ_,-cosθ_)
is a unit vector denoting the direction of the pseudospins. The polar
angle is determined by the ratios sinθ_=|Δ|/(2E_)
and cosθ_=ξ_/(2E_), whereas ϕ is
the phase Δ=|Δ|e^iϕ.

The pump pulse (t) changes the band dispersion, which in turn
changes the z-component of the pseudo magnetic field B_^z.
Within BCS, the spins precess around the new _ according
to d_/dt=_×_.
Importantly, the pseudospin dynamics is immediately fed back into
the magnetic field via the gap equation. Due to parity symmetry, only
even-order terms of (t) appear <cit.>,
and the oscillation frequency of the gap during the pump is a multiple
of 2ω_pump.



By setting 2ω_pump≈2Δ_0, the pump pulse
resonantly drives the coherent 2Δ gap oscillations after the
pump pulse is turned off (i.e. t>τ), similarly to interaction
quenches <cit.>. However, as none of the quench dynamics predicted
by time-dependent solutions of the BCS Hamiltonian (phases I-III described
above) is observed experimentally in NbN (see Fig. <ref>),
we go beyond this description and include phenomenologically damping
in the pseudospin equations of motion. The microscopic origin of these
terms will be discussed below. In analogy with the general problem
of spin precession, we introduce longitudinal (T_1) and transverse
(T_2) relaxation rates:


    d_/dt=    _×_
    -_·_∥,^*-|_^*|/T_1_∥,^*
        -∑_i=1^2_·_⊥,^*,i/T_2_⊥,^*,i


Here, _^*[T_*(t)]=1/2_∥,^*[T_*(t)]tanh(√(ξ_^2+Δ_*^2)/2T_*(t))
is the thermalized pseudospin configuration at time t at an effective
temperature T_*. The two vectors _⊥,^*,i
span the plane perpendicular to the equilibrium pseudospin direction
_∥,^*. Physically, the time scale T_1
is related to a redistribution of the quasiparticles, whereas the
time scale T_2 is related to the relaxation of the gap to the
thermalized value Δ_*.

To compute _∥,^* and the effective temperature
T_*, we consider that all the energy deposited in the electronic
subsystem by the pump is converted into a change in the internal energy
ℰ(t)=[H_BCS(t)_A=0-H_BCS_i]
(see also Ref. <cit.>). Here, the expectation
value is calculated in the time-evolved BCS state according to Eq. (<ref>)
and H_BCS_i is the initial ground state energy.
From ℰ(t), we extract both T^* and Δ^*,
which are themselves function of time while the pump is turned on.
Once the pump is turned off, energy is no longer deposited in the
electronic subsystem, and thus T^*(t>τ)=T^*(τ)≡ T_f.
Fig. <ref>(A) shows ℰ(t) for different values
of T_1 and T_2. The parameters used are the same as in Fig.
<ref>(D). Clearly, the effects of T_1 and T_2 kick
in when the pump is weak, as the relaxation processes redistribute
the energy within the electronic subsystem. In Fig. <ref>(B),
we show how the “final” temperature T^*(τ)≡ T_f
depends on the pump fluence. As expected, for sufficiently strong
pumps, the superconducting state can be completely melted by heating.



Using this semi-phenomenological approach, we can capture, as shown
in Fig. <ref>(D), the experimentally-observed gap dynamics
of NbN shown in Fig. <ref>(C). In this calculation, we set
T_1=2T_2=1.5τ. In contrast to the case with no damping,
T_1=T_2=∞ (Fig. <ref>(D)), we find that the oscillations
of |Δ(t)| are quickly suppressed after the pulse is turned
off, and that a continuous and slow increase of 1-|Δ(t)|/Δ_0
takes place over the time scale of tens of picoseconds. This characteristic
behavior has also been recently observed in ultraclean samples of
Nb_3Sn, with a larger post-pump suppression of the gap <cit.>.

To correctly capture the experimental observations, it is crucial
to restrict T_2 to the time scale of the order of the pump duration. To further
elucidate how T_1 and T_2 affect the time-evolution of |Δ(t)|,
in Fig. <ref> we explore different parameter regimes.
In order to highlight the effects of T_1 and T_2, and also
to make connection with experiments using short pulses <cit.>,
we consider a short Gaussian-shaped pulse of duration τ=5/Δ_0.
As expected, when T_1,2≫τ, the behavior of |Δ(t)|
is essentially the same as of the system without damping (Fig. <ref>A).
As T_1,2 decrease, the damping increases and the gap oscillations become noticeably damped for T_1,2 of the same order as the pump duration τ. To disentangle the
contributions of T_1,2, we show |Δ(t)|/Δ_0 for
fixed T_2 (T_1) and changing T_1 (T_2) in panel
B (C). It is evident that the oscillatory behavior of |Δ(t)|
is much more sensitive on the transverse relaxation T_2 than
on the longitudinal relaxation T_1, which only affects weakly
the asymptotic value of the gap. 

We therefore conclude that our experimental observations using long
pulses suggest a dominant T_2 process in NbN. It is interesting
to note that signatures of damping were also present in previous experiments
on the same material but using short pulses <cit.>.
Although oscillations were observed in that case after the pump was
off, their decay was reported to be much stronger than the polynomial
1/√(t) decay predicted by the coherent BCS dynamics. Comparison
with our results in Fig. <ref>A reveals that this effect
may be explained by the same damping processes revealed in our experiment.

Although T_1 and T_2 are phenomenological quantities, it
is important to discuss their possible microscopic origins. As we
explained above, T_1,2 processes arise within the electronic
subsystem, before equilibration with the lattice. Because the BCS
Hamiltonian is integrable <cit.>, any damping
must arise from non-BCS effects. Residual interactions between the
Bogoliubov quasiparticles, which are neglected in the mean-field BCS
approach, could provide a mechanism for quasiparticle relaxation,
which affects T_1. Moreover, the Higgs (amplitude) mode excited
resonantly by the laser pump disperses into the quasiparticle continuum
<cit.>. As a result, one expects
damping of the amplitude mode, which should affect the T_2 process.

Conclusions.– In this paper, we established a semi-phenomenological
framework that allows us to incorporate damping in the picosecond
time-evolution of the gap function of an s-wave superconductor
subject to an intense THz pulse. In the pseudospin language, damping
arises from a longitudinal process T_1 (related to quasiparticle
relaxation) and from a transverse process T_2 (related to relaxation
of the gap). Our experimental results reveal that, in NbN, for large-amplitude
long pump pulses, the picosecond evolution of the gap function is
different than that expected for coherent BCS-like dynamics. Instead,
we showed that the experimental behavior is consistent with a dominant
T_2 process that arises within the electronic subsystem, and
that has the same time scale as the duration of the pump. Future application
of this approach to different superconductors will allow one to distinguish
the type of relaxation processes dominant in each system. 

We thank M. Schtt for fruitful discussions and N. P. Armitage for
providing the sample. T.C. and R.M.F. are supported by the Office
of Basic Energy Sciences, U.S. Department of Energy, under award DE-SC0012336.
J.W. and X.Y. acknowledge supported by the Army Research office under
award W911NF-15-1-0135 (THz spectroscopy). P.P.O. acknowledges support from Iowa State University Startup Funds.





