^{1}and Maxim Sukharev

^{2}

### Abstract

We introduce a non-Hermitian approximation of Bloch optical equations. This approximation provides a complete description of the excitation, relaxation, and decoherence dynamics of ensembles of coupled quantum systems in weak laser fields, taking into account collective effects and dephasing. In the proposed method, one propagates the wave function of the system instead of a complete density matrix. Relaxation and dephasing are taken into account via automatically adjusted time-dependent gain and decay rates. As an application, we compute the numerical wave packet solution of a time-dependent non-Hermitian Schrödinger equation describing the interaction of electromagnetic radiation with a quantum nano-structure, and compare the calculated transmission, reflection, and absorption spectra with those obtained from the numerical solution of the Liouville-von Neumann equation. It is shown that the proposed wave packet scheme is significantly faster than the propagation of the full density matrix while maintaining small error. We provide the key ingredients for easy-to-use implementation of the proposed scheme and identify the limits and error scaling of this approximation.

E.C. would like to acknowledge useful and stimulating discussions with O. Atabek and A. Keller from Université Paris-Sud (Orsay) and with E. Shapiro from the University of British Columbia (Canada). M.S. is grateful to the Université Paris-Sud (Orsay) for the financial support through an invited Professor position in 2011. E.C. acknowledges supports from Agence Nationale Research (ANR) (Contract Attowave ANR-09-BLAN-0031-01), and from the European Union (EU) (Project ITN-2010-264951, CORINF). We also acknowledge the use of the computing facility cluster GMPCS of the LUMAT federation (FR LUMAT 2764).

I. INTRODUCTION

II. THEORETICAL MODEL AND APPLICATIONS

A. Atomic two-level system

B. Non-Hermitian two-level wave packet approximation

C. Application to a uniform nano-layer of atoms

D. Generalization to multi-level systems

E. Application to a uniform nano-layer of molecules

III. SUMMARY AND CONCLUSIONS

### Key Topics

- Absorption spectra
- 24.0
- Maxwell equations
- 23.0
- Excited states
- 22.0
- Wave functions
- 13.0
- Dephasing
- 10.0

## Figures

Transmission *T*(*E*) (a), (d), and (g); reflection *R*(*E*) (b), (e), and (h); and absorption *A*(*E*) (c), (f), and (i) spectra of an atomic layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The atomic density is *n* = 2.5 × 10^{25} m^{−3} in the first column, *n* = 2.5 × 10^{26} m^{−3} in the second column, and *n* = 2.5 × 10^{27} m^{−3} in the last column. The decay rate and pure dephasing rate are Γ = 10^{12} s^{−1} and γ* = 10^{15} s^{−1}, respectively. The atomic transition energy is ℏω_{ B } = 2 eV and the transition dipole moment is 2 D. The solution of Maxwell-Liouville-von Neumann equations is shown as a blue solid line, while the open red squares are from the solution of our approximate non-Hermitian Schrödinger model.

Transmission *T*(*E*) (a), (d), and (g); reflection *R*(*E*) (b), (e), and (h); and absorption *A*(*E*) (c), (f), and (i) spectra of an atomic layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The atomic density is *n* = 2.5 × 10^{25} m^{−3} in the first column, *n* = 2.5 × 10^{26} m^{−3} in the second column, and *n* = 2.5 × 10^{27} m^{−3} in the last column. The decay rate and pure dephasing rate are Γ = 10^{12} s^{−1} and γ* = 10^{15} s^{−1}, respectively. The atomic transition energy is ℏω_{ B } = 2 eV and the transition dipole moment is 2 D. The solution of Maxwell-Liouville-von Neumann equations is shown as a blue solid line, while the open red squares are from the solution of our approximate non-Hermitian Schrödinger model.

Log-log plot of the maximum excited state population (red solid line with circles) and the relative error (blue lines with squares) in the calculation of the absorption spectrum *A*(*E*) at the transition energy *E* _{ B } = ℏω_{ B } using the Schrödinger approximation, when compared to the solution of the full Liouville-von Neumann equation as a function of the incident field intensity in atomic units. The blue solid line is for the lowest atomic density *n* = 2.5 × 10^{25} m^{−3} and the blue dashed line is for the highest atomic density *n* = 2.5 × 10^{27} m^{−3}. All other parameters are as in Fig. 1 .

Log-log plot of the maximum excited state population (red solid line with circles) and the relative error (blue lines with squares) in the calculation of the absorption spectrum *A*(*E*) at the transition energy *E* _{ B } = ℏω_{ B } using the Schrödinger approximation, when compared to the solution of the full Liouville-von Neumann equation as a function of the incident field intensity in atomic units. The blue solid line is for the lowest atomic density *n* = 2.5 × 10^{25} m^{−3} and the blue dashed line is for the highest atomic density *n* = 2.5 × 10^{27} m^{−3}. All other parameters are as in Fig. 1 .

Reflection probability *R*(*E*) of an atomic layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The atomic density is *n* = 2.5 × 10^{27} m^{−3}. The solution of Maxwell-Liouville-von Neumann equations is shown as a blue solid line, while the dotted line with red squares is from the solution of our approximate non-Hermitian Schrödinger model. The exciting field amplitude is chosen such that the maximum excited state population reaches 35%. All other parameters are as in Fig. 1 .

Reflection probability *R*(*E*) of an atomic layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The atomic density is *n* = 2.5 × 10^{27} m^{−3}. The solution of Maxwell-Liouville-von Neumann equations is shown as a blue solid line, while the dotted line with red squares is from the solution of our approximate non-Hermitian Schrödinger model. The exciting field amplitude is chosen such that the maximum excited state population reaches 35%. All other parameters are as in Fig. 1 .

Population dynamics: (a) excited state population as a function of time, (b) squared modulus of the system's coherence as a function of time, (c) effective ground state gain rate γ_{0}(*t*) as a function of time, and (d) real part of the system's coherence as a function of time. The results obtained from the solution of Liouville-von Neumann equations are shown with a blue solid line, while the results obtained from the non-Hermitian Schrödinger approach are shown using red dashed lines. The atomic density is *n* = 2.5 × 10^{27} m^{−3}. All other parameters are as in Fig. 1 .

Population dynamics: (a) excited state population as a function of time, (b) squared modulus of the system's coherence as a function of time, (c) effective ground state gain rate γ_{0}(*t*) as a function of time, and (d) real part of the system's coherence as a function of time. The results obtained from the solution of Liouville-von Neumann equations are shown with a blue solid line, while the results obtained from the non-Hermitian Schrödinger approach are shown using red dashed lines. The atomic density is *n* = 2.5 × 10^{27} m^{−3}. All other parameters are as in Fig. 1 .

Absorption spectra *A*(*E*) of a Li_{2} molecular layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The molecular density is *n* = 2.5 × 10^{25} m^{−3} in the left column (panels (a) and (b)) and *n* = 2.5 × 10^{27} m^{−3} in the right column (panels (c) and (d)). The solutions of Maxwell-Liouville-von Neumann equations are shown as blue solid lines in the first row (panels (a) and (c)) while the red solid lines (inverted spectra, panels (b) and (d)) are from the solutions of our approximate non-Hermitian Schrödinger model. All other parameters are as in Fig. 1 .

Absorption spectra *A*(*E*) of a Li_{2} molecular layer of thickness Δ*z* = 400 nm as a function of the incident photon energy *E*. The molecular density is *n* = 2.5 × 10^{25} m^{−3} in the left column (panels (a) and (b)) and *n* = 2.5 × 10^{27} m^{−3} in the right column (panels (c) and (d)). The solutions of Maxwell-Liouville-von Neumann equations are shown as blue solid lines in the first row (panels (a) and (c)) while the red solid lines (inverted spectra, panels (b) and (d)) are from the solutions of our approximate non-Hermitian Schrödinger model. All other parameters are as in Fig. 1 .

Computation process time necessary on a Intel Xeon E5-1650 processor for the calculation of the absorption spectrum of a Li_{2} molecular nano-layer of thickness Δ*z* = 400 nm as a function of the number of quantum levels included in the calculation. The blue line with circles is for the solution obtained from Maxwell-Liouville-von Neumann equations, while the red line with squares is for our proposed Schrödinger-type approximation. The spatial grid has a total size of 2560 nm with a spatial step of 1 nm. The time propagation is performed on a temporal grid of total size 1.7 ps with a time step of 1.7 as.

Computation process time necessary on a Intel Xeon E5-1650 processor for the calculation of the absorption spectrum of a Li_{2} molecular nano-layer of thickness Δ*z* = 400 nm as a function of the number of quantum levels included in the calculation. The blue line with circles is for the solution obtained from Maxwell-Liouville-von Neumann equations, while the red line with squares is for our proposed Schrödinger-type approximation. The spatial grid has a total size of 2560 nm with a spatial step of 1 nm. The time propagation is performed on a temporal grid of total size 1.7 ps with a time step of 1.7 as.

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