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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Landau-Zener tunneling in the process of sum frequency generation

Baihong Li, Yongfang Li

Published Aug 21, 2014• 1 citation
Publisher

Abstract

Based on an analogy between the macroscopic phenomena and quantum physics, we study the problem of energy conversion for the sum frequency generation (SFG) process in a linear chirped quasi-phase-matched (QPM) crystal. We show that the process of energy conversion from the signal to the sum frequency field is very similar to the process of Landau-Zener tunneling (LZT) in a two-level system. Therefore, the energy conversion mechanism of SFG process can be explained by the theory of LZT. In addition, we obtain the asymptotic and analytical solution of the sum frequency field by solving the coupled equations with the model of Landau-Zener.

DOI
10.1117/12.2065384

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Based on an analogy between the macroscopic phenomena and quantum physics, we study the problem of energy conversion for the sum frequency generation (SFG) process in a linear chirped quasi-phase-matched (QPM) crystal. We show that the process of energy conversion from the signal to the sum frequency field is very similar to the process of Landau-Zener tunneling (LZT) in a two-level system. Therefore, the energy conversion mechanism of SFG process can be explained by the theory of LZT. In addition, we obtain the asymptotic and analytical solution of the sum frequency field by solving the coupled equations with the model of Landau-Zener.
According to an analogy of the macroscopic phenomena to quantum physics, the physical mechanism that the sum frequency field intensity oscillate with the transmission distance, which is similar Rabi oscillation of the atom transition process in the two-level system, is explained with the interference theory. The energy transfer mechanism of the optics parametric conversion process is revealed from the points of Landau-Zener tunneling (LZT), and the producing process of sum frequency signal is showed detailedly with asymptotic, analytical solution.
An efficient broadband sum-frequency generation (SFG) technique using the two cascaded optical parametric processes ${\ensuremath{\omega}}_{3}={\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{2}$ and ${\ensuremath{\omega}}_{4}={\ensuremath{\omega}}_{1}+{\ensuremath{\omega}}_{3}$ is proposed. The technique uses chirped quasi-phase-matched gratings, which, in the undepleted pump approximation, make SFG analogous to adiabatic population transfer in three-state systems with crossing energies in quantum physics. If the local modulation period first makes the phase match occur for ${\ensuremath{\omega}}_{3}$ and then for ${\ensuremath{\omega}}_{4}$ SFG processes then the energy is converted adiabatically to the ${\ensuremath{\omega}}_{4}$ field. Efficient SFG of the ${\ensuremath{\omega}}_{4}$ field is also possible by the opposite direction of the local modulation sweep; then transient SFG of the ${\ensuremath{\omega}}_{3}$ field is strongly reduced. Most of these features remain valid in the nonlinear regime of depleted pump.
Sum-frequency generation in nonlinear optics is a key technology enabling optical frequency conversion. However, how to most effectively enhance the frequency conversion efficiency in nonlinear crystals remains a fundamental problem; shorter crystal length and lower pump intensity while maintaining high-efficiency frequency conversion are critical for enhancing device integration potential and improving device performance in practical sum-frequency generation (SFG) device design. This study employed the Pontryagin maximum principle to investigate the SFG process. A cost functional was formulated that complies with the minimum constraint during SFG, enabling the derivation of optimal control parameters. A central finding revealed an inverse proportionality between the coupling coefficient and the crystal length: for a given crystal length, the pump intensity in the SFG is minimized, and vice versa. The optimal evolution trajectory follows a geodesic path on the Bloch sphere surface, representing the minimal-length evolutionary path from the signal field to the sum-frequency field. This geodesic behavior also indicates that the loss and absorption of the optical field in the crystal medium vanish during SFG. Furthermore, the crystal modulation period for the SFG is determined using the optimal phase-matching function. These results offer valuable insights for the design of compact optical chips and nonlinear optical devices.
Based on the QPM technology and the counter diabatic compensation approach in quantum physics, we propose an optimal scheme that the signal field completely converts into the sum frequency field under the condition of the superadiabaticity. With the geometric rotation plane designed, we subtly reveals the physical mechanism of the systems before and after adding a counter diabatic term. The property of the energy conversion which is analogous to the elastic collision is explained for the addition counter diabatic compensation. With the help of Bloch vectors in atoms physics, we can clearly show the physics process of the complete energy conversion and the physical mechanism for the shortcuts of the superadiabaticity from a new perspective. This new scheme can optimize adiabatic process, shorten the coupling region and nonlinear crystal length, reduce the influence of the dispersion effect, and also implement the robust efficient energy conversion under the condition of relatively weak pump intensity. Not only does this scheme provide a good theoretical basis for the design of nonlinear optical devices, but also its physical thoughts can be widely used in many other fields.
We present an asymptotic approach towards the standard Landau-Zener problem based on two linearly independent elementary waves of constant amplitude but time-dependent phase. The two contributions to this phase are quadratic and logarithmic in time and result from the linear chirp of the energies and the lowest order correction in the coupling between the two levels in the long-time limit. Indeed, our solutions subjected to initial conditions at a large but finite time in the past, are valid for large negative and large positive times. Due to their asymptotic nature they are not valid in the neighborhood of the moment when the levels cross. However, as the starting point of the dynamics moves further into the past, the time interval of the break-down of our asymptotic solutions shrinks and vanishes in the limit of the infinite past which corresponds to the standard Landau-Zener situation. Our approach explains not only every feature of the exact solution but yields deeper insights into the origin of the effects. In particular, it (i) brings to light the subtleties involved in the asymptotic limit leading to the standard expressions for the Landau-Zener transition amplitudes, (ii) identifies the logarithmic phase as the origin of the exponential transition probability amplitude, and (iii) reveals the structure of the lowest order corrections to the Landau-Zener result when the starting point is not in the infinite past.