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### Abstract

A general theoretical framework of two-dimensional time-domain second-order and third-order terahertz spectroscopy has been presented. The theoretical treatment is based on a classical and phenomenological model with weak nonlinearities. Three types of nonlinearity sources, anharmonicity, nonlinear coupling, and nonlinear damping, were considered. The second-order THz spectroscopy has an exact correspondence to fifth-order off-resonance Raman spectroscopy, and it has been shown that the present treatment gives exactly the same results as of the quantum mechanical theory under the weak nonlinearity condition. General expressions for the nonlinear signal have been obtained for a single-mode system, and numerical calculations for delta-function incident terahertz pulses were shown. For the third-order signal, two-level systems were also considered for comparison. Contributions of two types of incident pulse sequences have been studied separately in the third-order signals. Profiles of the two-dimensional signals were found to depend on the origin and order of the nonlinearity and also on the pulse sequence. The results of the present study show that the two-dimensional signal features of second- and third-order nonlinear terahertz spectroscopy can clarify the nature of the system which is not accessible using linear spectroscopy.

I. INTRODUCTION

II. MODEL

A. Model experiment

B. Theoretical model

C. Fifth-order and seventh-order Raman scattering

III. SECOND-ORDER PROCESSES

A. Anharmonicity

B. Nonlinear coupling

C. Nonlinear damping

IV. THIRD-ORDER PROCESSES

A. Anharmonicity

B. Nonlinear coupling

C. Two-level systems

V. DISCUSSION

VI. CONCLUSION

### Key Topics

- Terahertz spectroscopy
- 34.0
- Polarization
- 26.0
- Raman spectroscopy
- 26.0
- Nonlinear spectroscopy
- 17.0
- Terahertz radiation
- 11.0

## Figures

Timing of the incident THz pulses, *E* _{1} and *E* _{2}, and the probe in the two-dimensional time-domain THz spectroscopy.

Timing of the incident THz pulses, *E* _{1} and *E* _{2}, and the probe in the two-dimensional time-domain THz spectroscopy.

Two-dimensional profiles of second-order signals for a system with an anharmonic potential and nonlinear damping. (a) Second-order polarization for an anharmonic potential, which corresponds to the response function for the fifth-order Raman scattering with an anharmonic potential. (b) Second-order THz field for an anharmonic potential. (c) Second-order THz field for nonlinear damping. Each image is normalized to its maximum magnitude.

Two-dimensional profiles of second-order signals for a system with an anharmonic potential and nonlinear damping. (a) Second-order polarization for an anharmonic potential, which corresponds to the response function for the fifth-order Raman scattering with an anharmonic potential. (b) Second-order THz field for an anharmonic potential. (c) Second-order THz field for nonlinear damping. Each image is normalized to its maximum magnitude.

Second-order two-dimensional signals for systems with nonlinear coupling. (a) Second-order polarization for nonlinear coupling, which corresponds to the response function for the fifth-order Raman scattering with nonlinear coupling. (b) Second-order THz field for the nonlinear coupling case. Each image is normalized to its maximum magnitude.

Second-order two-dimensional signals for systems with nonlinear coupling. (a) Second-order polarization for nonlinear coupling, which corresponds to the response function for the fifth-order Raman scattering with nonlinear coupling. (b) Second-order THz field for the nonlinear coupling case. Each image is normalized to its maximum magnitude.

Two contributions for third-order two-dimensional signals in collinear geometry. (a) (*T* _{1}, 0, *t*) sequence. (b) (0, *T* _{1}, *t*) sequence.

Two contributions for third-order two-dimensional signals in collinear geometry. (a) (*T* _{1}, 0, *t*) sequence. (b) (0, *T* _{1}, *t*) sequence.

Third-order THz field for a system with anharmonicity. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

Third-order THz field for a system with anharmonicity. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

Third-order THz field for a system with nonlinear coupling. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

Third-order THz field for a system with nonlinear coupling. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

Third-order THz field for a two-level system. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

Third-order THz field for a two-level system. (a) Contribution of (*T* _{1}, 0, *t*) sequence. (b) Contribution of (0, *T* _{1}, *t*) sequence. (c) Sum of (a) and (b). (d) Difference of (a) and (b).

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