APS march meeting 2020

1. Theory of time-resolved optical conductivity of
superconductors: comparing two methods for its
evaluation
John Revelle, Ankit Kumar, Alexander F. Kemper,
North Carolina State University

2. The Optical Conductivity
● The optical conductivity 𝛔 is the linear proportionality constant between applied electric field E(t)
and resulting current J(t).
○ Commonly used to study quantum materials driven out of equilibrium.
● For a system in equilibrium we express the conductivity as a ratio in the
frequency domain as

3. The Pump-Probe Setup
● Pump-probe spectroscopy is used to study systems driven out of
equilibrium. We send in two short pulses:
○ Pump: stronger pulse, used to excite the system, breaking equilibrium
○ Probe: weaker pulse used to measure resulting changes in optical properties

4. Optical Conductivity out of Equilibrium
● For a system out of equilibrium, calculating
𝛔 is more complicated. We have 3 time
points to consider:
○ Pump time: time at which the pump
arrives, tpump
○ Probe time: time at which the probe
arrives, tprobe
○ Gate time: the time at which the current is
measured, tgate
● For convenience we define tpp = tprobe-
tpump to be the pump-probe separation.

5. The Dilemma
● Along which time axis should we take
Fourier transform? There are really 2
options: do we consider the relative motion
of the probe time or the gate time?
Traditionally we consider the motion of
gate time, but we consider both cases.

6. Two Definitions of Conductivity
Method I. Take FT along the tgate axis (we hold
tpp constant):
Method II. Now, we fix tgate and sweep the
probe backwards to obtain

7. Interpolation Procedure
● We obtained the current as a
function of tgate for several
values of tpp .
○ Note that at this point we
may readily do Method I by
taking FT and holding tpp
constant.
.

8. Interpolation Procedure
● We then arranged our currents in
(tgate,tpp) plane and interpolated
to obtain better resolution for
taking FT while holding tgate
constant:
○ Less computationally
intensive than generating
more data sets directly.
○ We used Akima spline
interpolation

9. Comparison of Conductivity Data from Two Methods

10. Video: Evolution of 𝝈 over Time

11. Comparison of the Methods
● Experimentally there appears to be a preference towards method 1 (taking FT
with pump-probe separation held fixed) due to its simplicity.
● Method 2 (taking FT with gate time held fixed) is more complex since many
pump-probe data sets must be generated. It is possible to use data generated
for method 1 and do interpolation (we used spline interpolation).
● Method 2 offers an advantage: the system dynamics, which are driven by the
pump, are always in the same state when the current is measured.
● Our results seem to suggest that method 2 offers improved time resolution
over method 1, and there is less smearing overall in calculated quantities in
frequency domain.