Density Profile

Note:
This discrete numerical simulation is based on:
L. Wilen and R. Giannetta, Impedance Methods for Surface State Electrons, J. Low Temp. Phys. 72, 353–369 (1988). https://doi.org/10.1007/BF00682147
Intermediate steps follow CGS units; final results are converted to SI.

Step 0: Define the simulation domain.

Step 1: Generate Green's Functions

• Use Create_Green_Function_njit.ipynb.

⚠️ Note: The use of @njit (from Numba) is essential for speeding up core numerical loops. It compiles Python functions into fast machine code, enabling efficient computation.

The Green's function computed with a threshold error of $10^{-10}$ is stored as:

GreenFunction/Green_njit_R0pt75_H0pt2_N500_M200_posiE100_epsE-10.npy

This file was used in both the Rydberg FM paper and the Plasmon paper.

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Step 2-ver1: Calculate Density Profiles for Vmb Voltage Sweeps

• Use Calculate_DensityProfile_Vmb_sweep.ipynb to compute electron density profiles for your desired voltage configurations.
• Use Calculate_DensityProfile_Vmb_sweep_cupy.ipynb if you have a GPU.

⚠️ Note: Please note that there may be more effective ways to optimize the program using njit or cupy more appropriately than what we implemented above.
It is important not to use njit and cupy together.
njit compiles functions to run on the CPU, while cupy manages data on the GPU.
Using only cupy may result in longer execution times, depending on the situation.
(Here is a link where this issue is discussed: numba/numba#8784)

This notebook includes an example with the following settings:

• $V_\mathrm{ib} = 10$ V
• $V_\mathrm{ob} = -32$ V
• $V_\mathrm{mb}$ is swept from $-32$ V to $10$ V, while keeping the total number of electrons fixed
  (determined from the saturation condition at $V_\mathrm{ib} = 10$ V and $V_\mathrm{mb} = V_\mathrm{ob} = -32$ V).

The DC potential created by the applied electrode voltages, $\tilde{\phi}$, is stored as:

VmbSweep/Vib10V_Vobm32V/Phi_R0pt75_H0pt2_N500_M200_posiE100_epsE-10.0_Vmb_step=0pt2_Vmb_start=-32_Vmb_stop=10.npy

The density profile obtained for each value of $V_\mathrm{mb}$ is plotted as shown below
(red corresponds to $V_\mathrm{mb} = 9.8$ V and blue to $V_\mathrm{mb} = -32$ V):

The data is stored in:

VmbSweep/Vib10V_Vobm32V/fixed_electron_number_density/

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Step 2-ver2: Calculate Density Profiles for Vob Voltage Sweeps

• Use Calculate_DensityProfile_Vob_sweep.ipynb to compute electron density profiles for your desired voltage configurations.
• Use Calculate_DensityProfile_Vob_sweep_cupy.ipynb if you have a GPU.

This notebook includes an example with the following settings:

• $V_\mathrm{ib} = 17$ V
• $V_\mathrm{mb} = 17$ V
• $V_\mathrm{ob}$ is swept from $0$ V to $-160$ V, while keeping the total number of electrons fixed
  (determined from "Pencentage" % of the saturation condition at $V_\mathrm{ob} = 0$ V and $V_\mathrm{ib} = V_\mathrm{mb} = 17$ V).

The DC potential created by the applied electrode voltages, $\tilde{\phi}$, is stored as:

VobSweep/Vib17V_Vmb17V/Phi_R0pt75_H0pt2_N500_M200_posiE100_epsE-10.0_Vob_step=-0pt3_Vob_start=0_Vob_stop=-160.npy

The data for "Pencentage"=100 is stored in:

VobSweep/Vib17V_Vmb17V/fixed_electron_number_density_100_percent_SatENum/

and the data for "Pencentage"=80 is stored in:

VobSweep/Vib17V_Vmb17V/fixed_electron_number_density_80_percent_SatENum/

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Step 2-ver3: Calculate Density Profiles for Vib Voltage Sweeps

• Use Calculate_DensityProfile_Vib_sweep.ipynb to compute electron density profiles for your desired voltage configurations.
• Use Calculate_DensityProfile_Vib_sweep_cupy.ipynb to compute electron density profiles for your desired voltage configurations.

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Step 3: Use the Results

• Use the computed density profiles to calculate the electric field experienced by the electron in the Rydberg detection simulation,
  or to calculate the plasmon impedance for the plasmon simulation.

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⚠️ Due to the large file size, the generated folders GreenFunction/, VmbSweep/ and VobSweep/ could not be uploaded to GitHub.

Please refer to the following Google Drive link to access the data:

🔗 Google Drive – DensityProfile