Microscopic Model and Dynamical Simulation Microscopic ModelMicroscopic Model

LIST OF TABLES

5.3 Microscopic Model and Dynamical Simulation Microscopic ModelMicroscopic Model

due to a stronger electron-phonon coupling. First principle calculations suggested that all the 𝐴_𝑔 phonons in Ca₂RuO₄involve rotation or tilting of octahedra (M.-C.

Lee, C. H. Kim, Kwak, J. Kim, et al., 2018), which inevitably changes the Ru-O bond length and generates𝐸_𝑔-symmetry modulation of each octahedron. Therefore, the clear dichotomy shows that𝑄_𝜃 and𝑄_𝜖 eigenmodes of 3.7 THz phonon exhibit distinct behaviors as the QO is manipulated by light.

5.3 Microscopic Model and Dynamical Simulation

a

x z y

b

PES Qε

Qθ

Qε

Qθ

10 8

P Amp.6

-2 -1 0 1 2 F>Fc

F<Fc

2 1 0

2.0 1.5 1.0 0.5 0.0

-0.5 Time (ps) F>Fc

F<Fc

Q θQ ε

10 8 6 4 2

QOCP Amp. 0

1200 800 400

0 F

Q Q b

c

d ( )

( )

( ) ( )

Figure 5.7: Microscopic model simulation results. (a) Schematic of PES of the quadrupolar ordered phase calculated from the microscopic model (Methods). The corresponding real-space pseudospin distribution and lattice distortion for each valley are shown by side. (b)(c) Time evolution of 𝑄_𝜃 and 𝑄_𝑒𝑝 𝑠𝑖𝑙 𝑜𝑛 with pump fluence lower and higher than the critical fluence. (d) Pump fluence dependence of amplitude of QOCP along𝑄_𝜃 (red) and𝑄_𝜖 (blue) coordinates calculated based on the dynamical simulation. The dashed line denotes the first switching critical fluence.

and amplitude proportional to the fluence 𝐹 with a scaling factor 𝐴. By further including a damping term with constant𝛾which can be experimentally determined, we can write out the equation of motion of𝑄_𝜃and𝑄_𝜖:

𝑑2𝑄_𝜃/𝜖(𝑡) 𝑑 𝑡2 +2𝛾

𝑑𝑄_𝜃/𝜖(𝑡) 𝑑 𝑡

+ 𝑑𝑉(𝑄_𝜃(𝑡), 𝑄_𝜖(𝑡))

𝑑𝑄_𝜃/𝜖(𝑡) = 𝐴𝐹exp

4 ln(2)𝑡² 𝜎2

. (5.1) Simulation results

The time evolution of the structural order parameters𝑄_𝜃 and𝑄_𝜖 pumped with two characteristic fluences (𝐹 > 𝐹_𝑐 and 𝐹 < 𝐹_𝑐) are shown in Figures 5.7(b) and (c).

As 𝐹 > 𝐹_𝑐, a clear change of sign in offset of both 𝑄s can be unambiguously observed, which underlines a switch of QO from the initial valley to another hidden valley. By performing FFT to time traces after the system is stabilized as we vary the only free parameter 𝐹, we can track the fluence dependence of the amplitude of 𝑄_𝜃 and 𝑄_𝜖 modes. As expected, when 𝐹 < 𝐹_𝑐, the amplitude of both 𝑄_𝜃 and 𝑄_𝜖 coordinates exhibits the same linear dependence akin to a conventional phonon;

as the system transiently evolves into the two excited states accompanied by the QO switching (𝐹 > 𝐹_𝑐), phonons along both coordinates display an deviation from perfect linearity and the nonlinearity of𝑄_𝜖 is more drastic than𝑄_𝜃 [Figure 5.7(d)].

Not only does this result suggest that the nonlinear behavior of the coupled phonon signifies the ultrafast switch of QO, but it reproduces the probe-energy-dependent measurements in Figure 5.6, where𝑄_𝜖-sensitive probes indeed show a more dramatic deviation from linearity than the𝑄_𝜃-sensitive probes.

Note that there is an aperiodic oscillation of QOCP amplitude over fluence as 𝐹 > 𝐹_𝑐. The nonmonotonic behavior arises from the back-and-forth switching between different valleys as 𝐹 increases. This indicates 𝐹_𝑐 corresponds to the fluence where the first reversal occurs 𝐹_𝑐,𝑖=

1. When 𝐹 is slightly higher than 𝐹_𝑐

1, the system relaxes to and stays in the 𝑥/𝑦-compressed valley. When 𝐹 > 𝐹_𝑐

2, the system will relax back to the original valley. With even higher 𝐹, the system can switch between different valleys back and forth transiently before it settles into one valley. Now we provide an explanation on why the switch between valleys will generate the aperiodic oscillation. Let us consider two fluences: 𝐹

1 = 𝐹_𝑐

1−𝛿 and 𝐹2 = 𝐹_𝑐

1+𝛿, where 𝛿 is a small positive value. Since 𝐹

2 > 𝐹

1, the system will roll over the barrier and move to the new valley, which obviously travels a longer distance and thus the damping will decrease the amplitude for𝐹

2case more than𝐹

1

case. Thus we will find the phonon amplitude at𝐹

2is smaller than𝐹

1. As𝐹further increases, the larger energy dumped into the system will compensate the damping loss and the phonon amplitude will slowly increase until it reaches the boundary between the two valleys again at 𝐹 = 𝐹_𝑐

2. Then the new reversal occurs and the phonon amplitude will drop again. Therefore, the QOCP amplitude will show a sudden change at each𝐹_𝑐,𝑖 and thus depicts an oscillatory behavior as a function of 𝐹.

Also note that in the minimal model we neglect the intrinsic energetic difference between 𝑑_{𝑥 𝑦} and the nearly degenerate 𝑑_{𝑥 𝑧/𝑦 𝑧}-dominated states, which arises from noncubic environment induced by the octahedron distortion and the planar cornered- shared lattice structure. We can include the tetragonal splitting by phenomeno- logically adding a linear term 𝑘_𝐸𝑄_{𝐸 𝜃} to tilt the PES so that the ground state of compressed𝑧−axis is uniquely favored. Based on this assumption, the JT effect will become pseudo-JT effect due to the energetic nondegeneracy in the parent phase (Bersuker, 2006). After including the tetragonal splitting, we find that the original 𝑑_{𝑥 𝑦}-dominated valley will almost always be reached eventually after about 3 ps in-

1.0

0.8

0.6

0.4

0.2

0.0

FFT Amplitude (a.u.)

4.0 3.0 2.0

FFT Frequency (THz) QEθ sim.

α-peak probe exp. ^0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

FFT Amplitude (a.u.)

4.5 4.0 3.5 3.0 2.5 2.0

FFT Frequency (THz) 280K

220 K

130 K 80 K

β-peak probe exp. ^0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

4.5 4.0 3.5 3.0 2.5 2.0

FFT Frequency (THz) g = 0

g = 0.5g0

g = 0.85g0

g = g0

QEε sim.

(a) (b)

Figure 5.8: Illustration of the 2.5 THz hump. (a) Comparison of FFT spectra of QOCP between experiment and simulation with two𝑑_{𝑥 𝑦}→𝑑_{𝑥 𝑧/𝑦 𝑧}-peak probes (0.69, 0.92 eV) at 80 K. b. Comparison of FFT spectra of QOCP between experiment and simulation with 1.55 eV𝑑_{𝑥 𝑧/𝑦 𝑧}→𝑑_{𝑥 𝑧/𝑦 𝑧}-peak probe at various temperatures. All the experimental data were taken at a pump fluence of 15 mJ/cm². The simulated FFT spectra are calculated by changing𝑔in a form of√︁

1−𝑇/𝑇_𝑄𝑂. The relative value of 𝑔at each temperature is denoted. The data are scaled and offset vertically for better comparison.

dependent of 𝐹, but a transient switch between different QOs still occurs. We note that 𝐹_𝑐 in the presence of tetragonal splitting is larger than that without tetragonal splitting because𝑑_{𝑥 𝑧}and𝑑_{𝑦 𝑧}valleys are elevated and thus harder to reach. However, despite the discrepancy, the fluence dependence of QOCP amplitude and frequency is qualitatively identical to the case with tetragonal splitting ignored.

We can also simulate the temperature dependence of QOCP amplitude and 𝐹_𝑐 by assuming that microscopically the JT coupling constant 𝑔, which determines the spacing and depth of different valleys, decreases as temperature increases in an order-parameter form√︁

1−𝑇/𝑇_𝑄𝑂. By introducing a temperature dependence of𝑔, our model confirms a primary order-parameter-like behavior √︁

1−𝑇/𝑇_𝑄𝑂 of both QOCP amplitude and𝐹_𝑐, in perfect agreement with the experimental data in Figure 5.5(b). These results also underpin that𝐹_𝑐 can be used to characterize QO.

We note that in both the experimental and simulational results, a hump shows up at the low-frequency shoulder of the QOCP peak. According to previous Raman spectroscopy results (Rho et al., 2005), all the𝐴

1𝑔modes have been identified with energies all above 3.7 THz. Since our isotropic reflectivity measurement can only probe 𝐴

1𝑔 phonons according to symmetry, we can rule out the possibility that this damped mode is a phonon. Also, this mode exists even above𝑇_𝑁 but disappears at

𝑇_𝑄𝑂 [Figure 5.6(b)], indicating that it is not a magnon but should be related to QO.

We thus attribute this hump feature to the non-harmonicity of the PES. As SOC smoothens the barrier, the local curvature at the boundary between the two valleys decreases, rendering the PES a local anharmonicity. Therefore, when the system is transiently switched between the valleys, a low energy component emerges from the reduced curvature around the anharmonic part of PES. If this hypothesis is true, we would expect that this low frequency hump should emerge in both𝑄_{𝐸 𝜃} and𝑄_{𝐸 𝜖} only at𝑇 < 𝑇_𝑄𝑂 and 𝐹 > 𝐹_𝑐. This is indeed corroborated by both our simulation and experiments with good agreement [Figure 5.8]. We would also expect that the amplitude of the hump should increase with𝐹. Our current experimental setup can hardly resolve the dynamics of this mode unambiguously over the entire fluence and temperature range due to unideal SNR. This speculation thus calls for detailed measurements with better SNR in the future.