Thermodynamical functions (quanguru.QuantumToolbox.thermodynamics)#

Contains methods to calculate certain quantities used in thermal states, open systems, and quantum thermodynamics. TODO update docstring examples and write some tests after writing some tutorials

Functions#

nBarThermal(angFreq, temp[, hbar, kb])

Calculates average excitation number \(\bar{n}(T) := 1/(e^{\hbar \omega / k_{b}T} - 1)\) of a bosonic field with frequeny \(\omega\) at a temperature T.

qubitPolarisation(freq, temp)

Returns the polarisation \(\langle\hat{\sigma}_{z}\rangle := P_{1} - P_{0}\) of a qubit with frequency \(\omega\) in a thermal state of temperature T.

HeatCurrent(Lindbladian, Hamiltonian, denMat)

Calculates the heat current \(\mathcal{J}:=Tr(\dot{\rho}\hat{H})\), where \(\dot{\rho} := \hat{\mathcal{L}}\rho\) is obtained using the given Lindbladian \(\hat{\mathcal{L}}\).

Function Name

Docstrings

Examples

Unit Tests

Tutorials

nBarThermal

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qubitPolarisation

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HeatCurrent

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nBarThermal(angFreq: float, temp: float, hbar: float = 1.0, kb: float = 1.0) → float[source]#

Calculates average excitation number \(\bar{n}(T) := 1/(e^{\hbar \omega / k_{b}T} - 1)\) of a bosonic field with frequeny \(\omega\) at a temperature T. Boltzmann and reduced Planck constants are by default \(\hbar = k_{B} = 1\). TODO Physical constants’ default values should be connected to simUnits.

Parameters:

  • angFreq (float) – (angular) frequency of the bosonic field

  • temp (float) – temperature

  • hbar (float) – reduced Planck’s constant

  • kb (float) – Boltzmann constant

Returns:

Average excitation number

Return type:

float

Raises:

ValueError – If average number is infinite.

Examples

# TODO

qubitPolarisation(freq: float, temp: float) → float[source]#

Returns the polarisation \(\langle\hat{\sigma}_{z}\rangle := P_{1} - P_{0}\) of a qubit with frequency \(\omega\) in a thermal state of temperature T. \(P_{1}\) and \(P_{0}\) are excited and ground state populations satisfying \(P_{1} + P_{0} = 1\), and thermal state populations also satisfy \(\frac{P_{1}}{P_{0}} := e^{\omega/T}\).

Parameters:

  • freq (float) – frequency of the qubit

  • temp (float) – temperature of the qubit

Returns:

qubit polarisation, i.e. difference betwennn ground and excited state populations.

Return type:

float

Examples

# TODO

HeatCurrent(Lindbladian: spmatrix | ndarray, Hamiltonian: spmatrix | ndarray, denMat: spmatrix | ndarray) → float[source]#

Calculates the heat current \(\mathcal{J}:=Tr(\dot{\rho}\hat{H})\), where \(\dot{\rho} := \hat{\mathcal{L}}\rho\) is obtained using the given Lindbladian \(\hat{\mathcal{L}}\). Here, \(\hat{H}\) is the system Hamiltonian, and the time derivative of density matrix is \(\dot{\rho}mathcal{L}\). It does not strictly speaking have to be a Lindbladian but any combination of terms from a Liouvillian. Disclaimer: physical meaning of those terms is not and cannot be interpreted by this function. TODO Write a bit of the theory here to better explain this function.

Parameters:

  • Lindbladian (Matrix) – a Lindbladian or any combination of terms from a Liouvillian

  • Hamiltonian (Matrix) – Hamiltonian of the system

  • denMat (Matrix) – Density matrix (state) of the system

Returns:

Heat current

Return type:

float

Examples

# TODO