Beetroot is a project to compute the signal arising from a Single Electron Box.

It uses the Bessel function method to speedup the calculation and it already includes Lifetime Broadening.

The theory behind this module can be found in the related article in Quantum.

If you use Beetroot, please cite

Peri, L., Oakes, G. A., Cochrane, L., Ford, C. J. B., & Gonzalez-Zalba, M. F. (2024). Beyond-adiabatic Quantum Admittance of a Semiconductor Quantum Dot at High Frequencies: Rethinking Reflectometry as Polaron Dynamics. Quantum, 8, 1294. https://doi.org/10.22331/q-2024-03-21-1294

von Horstig, F.-E., Peri, L., Barraud, S., Shevchenko, S. N., Ford, C. J. B., Gonzalez-Zalba, M. F. Floquet Interferometry of a Dressed Semiconductor Quantum Dot. http://arxiv.org/abs/2407.14241.

Beetroot is available from pypi via

pip install Beetroot-SEB

To install this package from source, download this repo and simply

pip install .

To test Beetroot, install from source as

pip install .[test]

Please remember to escape the brackets if using zsh (i.e. pip install .\[test\])

Testing can now be conducted via

pytest

The simples calculation one might want to do is to compute the admittance of a single electron box at the N-th harmonic. This is done by the following code:

from Beetroot.signal import get_Signal

kt = 1
de = 0.5
omega = 1e-2
Gamma = 0.25

N = 1 # N-th harmonic

eps = np.linspace(-5* kt,  5 * kt, 1001)

Y = get_Signal(N, eps, de, Gamma, omega, kt)

The complete script can be found in examples/fundamental.py and generates this figure.

Note : Beetroot by default computes the non-normalized admittance. I.e. it does NOT include the phase. This can be fixed by including

from Beetroot import get_precoeff
Complex_Admittance = Y * get_precoeff(N, Gamma, omega)

It is sometimes useful to compute maps as a function of power. This is already implemented and parallelized.

from Beetroot.parallel import get_map

kt = 1
omega = 1e-2
Gamma = 0.25

N =1 # N-th harmonic

de = np.linspace(0, 10, 201)

eps = np.linspace(-12,  12, 501)

Y = get_map(N, eps, de, Gamma, omega, kt)

The complete script can be found in examples/map.py and generates this figure.

Functions in Beetroot.parallel are parallelized via multiprocessing.pool. To avoid recusive imports on non-linux platforms, please wrap the main file in

from multiprocessing import freeze_support

if __name__ == '__main__':
   freeze_support()

or consult the multiprocessing documentation.

It is desirable to obtain the small signal response. This can be done by specifying a small de in get_Signal. However, this can lead to slow conde and poor numerical precision. The fundamental submodule contains a analytical implementations of the small-signal response.

from Beetroot.fundamental import get_Small_Signal_Fundamental


kt = 1
omega = 1e-2
Gamma = 0.25

eps = np.linspace(-5* kt,  5 * kt, 1001)

Yss = get_Small_Signal_Fundamental(eps, Gamma, omega, kt)

The complete script can be found in examples/small_signal.py and generates this figure.

The small signal admittance is normalizetd to the input amplitude, so that Yss = Y/(2*de)

Beetroot also includes a module to compute the response of an SEB to a two-tone excitation.

The theory and experimental implementation behind this module can be found in this paper.

This is done by the following code:

from Beetroot.LZS import get_LZS_Signal

kt = 1
Gamma = 0.25

N = 1 # N-th harmonic

omega_rf = 2
de_rf = 0.4

omega_MW = 15
de_MW = 20

eps = np.linspace(-2.5 * de_MW,  2.5 * de_MW, 501)

Y = get_LZS_Signal(N, eps, de_rf, de_MW, Gamma, kt, omega_rf, omega_MW)

The complete script can be found in examples/two_tone.py and generates this figure.

One can compute the two-tone maps in parallel. To sweep the de_MW parameter, use

from Beetroot.LZS import get_LZS_map_MW

kt    = 1
Gamma = 4

N = 1 # N-th harmonic

omega_rf = 2
de_rf = 0.4

de_MW_array = np.linspace(0, 10*omega_MW, 101)
eps = np.linspace(-1.2 * de_MW_array.max(),  1.2 * de_MW_array.max(), 1001)

Y = get_LZS_map_MW(N,eps,de_MW_array, de_rf,Gamma, kt, omega_rf, omega_MW)

The complete script can be found in examples/two_tine_MW_map.py and generates this figure.

To sweep the de_rf parameter instead, use

from Beetroot.LZS import get_LZS_map_rf

kt    = 1
Gamma = 4

N = 1 # N-th harmonic

omega_rf = 2

omega_MW = 15
de_MW    = 2

de_rf_array = np.linspace(0, 10*omega_MW, 101)
eps = np.linspace(-1.2 * de_rf_array.max(),  1.2 * de_rf_array.max(), 501)

Y = get_LZS_map_rf(N, eps, de_MW, de_rf_array, Gamma, kt, omega_rf, omega_MW)

The complete script can be found in examples/two_tine_rf_map.py and generates this figure.