Environments
Bosonic Environments
- class BosonicEnvironment(T: float = None, tag: Any = None)[source]
The bosonic environment of an open quantum system. It is characterized by its spectral density and temperature or, equivalently, its power spectrum or its two-time auto-correlation function.
Use one of the classmethods
from_spectral_density,from_power_spectrumorfrom_correlation_functionto construct an environment manually from one of these characteristic functions, or use a predefined sub-class such as theDrudeLorentzEnvironment, theUnderDampedEnvironmentor theOhmicEnvironment.Bosonic environments offer various ways to approximate the environment with a multi-exponential correlation function, which can be used for example in the HEOM solver. The approximated environment is represented as a
ExponentialBosonicEnvironment.All bosonic environments can be approximated by various fitting methods, see the description of
approximate. Subclasses may offer additional approximation methods such as the analytical Matsubara or Pade expansions.- Parameters:
- Toptional, float
The temperature of this environment.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- approximate(
- method: Literal['cf'],
- tlist: ArrayLike,
- target_rmse: float = 2e-05,
- Nr_max: int = 10,
- Ni_max: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- full_ansatz: bool = False,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['ps'],
- wlist: ArrayLike,
- target_rmse: float = 5e-06,
- Nmax: int = 5,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['sd'],
- wlist: ArrayLike,
- Nk: int = 1,
- target_rmse: float = 5e-06,
- Nmax: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['aaa'],
- wlist: ArrayLike,
- tol: float = 1e-13,
- Nmax: int = 10,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
- tlist: ArrayLike,
- separate: bool = False,
- Nr: int = 3,
- Ni: int = 3,
- combine: bool = True,
- tag: Any = None,
Generates a multi-exponential approximation of this environment. The available methods are
"cf","ps","sd","aaa","prony","esprit","espira-I"and"espira-II". The methods and the parameters required per method are documented in the Users Guide.
- abstract correlation_function(
- t: float | ArrayLike,
- *,
- eps: float = 1e-10,
The two-time auto-correlation function of this environment. See the Users Guide on bosonic environments for specifics on the definitions used by QuTiP.
If no analytical expression for the correlation function is known, it will be derived from the power spectrum via a fast fourier transform.
If no analytical expression for the power spectrum is known either, it will be derived from the spectral density. In this case, the temperature of this environment must be specified.
- Parameters:
- tarray_like or float
The times at which to evaluate the correlation function.
- epsoptional, float
Used in case the power spectrum is derived from the spectral density; see the documentation of
BosonicEnvironment.power_spectrum.
- classmethod from_correlation_function(
- C: Callable[[float], complex] | ArrayLike,
- tlist: ArrayLike = None,
- tMax: float = None,
- *,
- T: float = None,
- tag: Any = None,
- args: dict[str, Any] = None,
Constructs a bosonic environment from the provided correlation function. The provided function will only be used for times \(t \geq 0\). At times \(t < 0\), the symmetry relation \(C(-t) = C(t)^\ast\) is enforced.
- Parameters:
- Ccallable or array_like
The correlation function. Can be provided as a Python function or as an array. When using a function, the signature should be
C(t: array_like, **args) -> array_likewhere
tis time andargsis a dict containing the other parameters of the function.- tlistoptional, array_like
The times where the correlation function is sampled (if it is provided as an array).
- tMaxoptional, float
Specifies that the correlation function is essentially zero outside the interval [-tMax, tMax]. Used for numerical integration purposes.
- Toptional, float
Environment temperature. (The spectral density of this environment can only be calculated from the correlation function if a temperature is provided.)
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- argsoptional, dict
Extra arguments for the correlation function
C.
- classmethod from_power_spectrum(
- S: Callable[[float], float] | ArrayLike,
- wlist: ArrayLike = None,
- wMax: float = None,
- *,
- T: float = None,
- tag: Any = None,
- args: dict[str, Any] = None,
Constructs a bosonic environment with the provided power spectrum.
- Parameters:
- Scallable or array_like
The power spectrum. Can be provided as a Python function or as an array. When using a function, the signature should be
S(w: array_like, **args) -> array_likewhere
wis the frequency andargsis a dict containing the other parameters of the function.- wlistoptional, array_like
The frequencies where the power spectrum is sampled (if it is provided as an array).
- wMaxoptional, float
Specifies that the power spectrum is essentially zero outside the interval [-wMax, wMax]. Used for numerical integration purposes.
- Toptional, float
Environment temperature. (The spectral density of this environment can only be calculated from the powr spectrum if a temperature is provided.)
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- argsoptional, dict
Extra arguments for the power spectrum
S.
- classmethod from_spectral_density(
- J: Callable[[float], float] | ArrayLike,
- wlist: ArrayLike = None,
- wMax: float = None,
- *,
- T: float = None,
- tag: Any = None,
- args: dict[str, Any] = None,
Constructs a bosonic environment with the provided spectral density. The provided function will only be used for frequencies \(\omega > 0\). At frequencies \(\omega \leq 0\), the spectral density is zero according to the definition used by QuTiP. See the Users Guide on bosonic environments for a note on spectral densities with support at negative frequencies.
- Parameters:
- Jcallable or array_like
The spectral density. Can be provided as a Python function or as an array. When using a function, the signature should be
J(w: array_like, **args) -> array_likewhere
wis the frequency andargsis a tuple containing the other parameters of the function.- wlistoptional, array_like
The frequencies where the spectral density is sampled (if it is provided as an array).
- wMaxoptional, float
Specifies that the spectral density is essentially zero outside the interval [-wMax, wMax]. Used for numerical integration purposes.
- Toptional, float
Environment temperature. (The correlation function and the power spectrum of this environment can only be calculated from the spectral density if a temperature is provided.)
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- argsoptional, dict
Extra arguments for the spectral density
J.
- abstract power_spectrum(
- w: float | ArrayLike,
- *,
- eps: float = 1e-10,
The power spectrum of this environment. See the Users Guide on bosonic environments for specifics on the definitions used by QuTiP.
If no analytical expression for the power spectrum is known, it will be derived from the spectral density. In this case, the temperature of this environment must be specified.
If no analytical expression for the spectral density is known either, the power spectrum will instead be derived from the correlation function via a fast fourier transform.
- Parameters:
- warray_like or float
The frequencies at which to evaluate the power spectrum.
- epsoptional, float
To derive the zero-frequency power spectrum from the spectral density, the spectral density must be differentiated numerically. In that case, this parameter is used as the finite difference in the numerical differentiation.
- abstract spectral_density(w: float | ArrayLike) float | ArrayLike[source]
The spectral density of this environment. For negative frequencies, a value of zero will be returned. See the Users Guide on bosonic environments for specifics on the definitions used by QuTiP.
If no analytical expression for the spectral density is known, it will be derived from the power spectrum. In this case, the temperature of this environment must be specified.
If no analytical expression for the power spectrum is known either, it will be derived from the correlation function via a fast fourier transform.
- Parameters:
- warray_like or float
The frequencies at which to evaluate the spectral density.
- class DrudeLorentzEnvironment(
- T: float,
- lam: float,
- gamma: float,
- *,
- Nk: int = 10,
- tag: Any = None,
Bases:
BosonicEnvironmentDescribes a Drude-Lorentz bosonic environment with the spectral density
\[J(\omega) = \frac{2 \lambda \gamma \omega}{\gamma^{2}+\omega^{2}}\](see Eq. 15 in [BoFiN23]).
- Parameters:
- Tfloat
Environment temperature.
- lamfloat
Coupling strength.
- gammafloat
Spectral density cutoff frequency.
- Nkoptional, int, defaults to 10
The number of Pade exponents to be used for the calculation of the correlation function.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- approximate(
- method: Literal['matsubara', 'pade'],
- Nk: int,
- combine: bool = True,
- compute_delta: Literal[False] = False,
- tag: Any = None,
- approximate(
- method: Literal['matsubara', 'pade'],
- Nk: int,
- combine: bool = True,
- compute_delta: Literal[True] = True,
- tag: Any = None,
- approximate(
- method: Literal['cf'],
- tlist: ArrayLike,
- target_rmse: float = 2e-05,
- Nr_max: int = 10,
- Ni_max: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- full_ansatz: bool = False,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['ps'],
- wlist: ArrayLike,
- target_rmse: float = 5e-06,
- Nmax: int = 5,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['sd'],
- wlist: ArrayLike,
- Nk: int = 1,
- target_rmse: float = 5e-06,
- Nmax: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['aaa'],
- wlist: ArrayLike,
- tol: float = 1e-13,
- Nmax: int = 10,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
- tlist: ArrayLike,
- separate: bool = False,
- Nr: int = 3,
- Ni: int = 3,
- combine: bool = True,
- tag: Any = None,
Generates a multi-exponential approximation of this environment. The available methods are
"matsubara","pade","cf","ps","sd","aaa","prony","esprit","espira-I"and"espira-II". The methods and the parameters required per method are documented in the Users Guide.
- correlation_function(
- t: float | ArrayLike,
- Nk: int = None,
- **kwargs,
Calculates the two-time auto-correlation function of the Drude-Lorentz environment. The calculation is performed by summing a large number of exponents of the Pade expansion.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- Nkint, optional
The number of exponents to use. If not provided, then the value that was provided when the class was instantiated is used.
- class UnderDampedEnvironment(
- T: float,
- lam: float,
- gamma: float,
- w0: float,
- *,
- tag: Any = None,
Bases:
BosonicEnvironmentDescribes an underdamped environment with the spectral density
\[J(\omega) = \frac{\lambda^{2} \Gamma \omega}{(\omega_0^{2}- \omega^{2})^{2}+ \Gamma^{2} \omega^{2}}\](see Eq. 16 in [BoFiN23]).
- Parameters:
- Tfloat
Environment temperature.
- lamfloat
Coupling strength.
- gammafloat
Spectral density cutoff frequency.
- w0float
Spectral density resonance frequency.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- approximate(
- method: Literal['matsubara'],
- Nk: int,
- combine: bool = True,
- compute_delta: Literal[False] = False,
- tag: Any = None,
- approximate(
- method: Literal['matsubara'],
- Nk: int,
- combine: bool = True,
- compute_delta: Literal[True] = True,
- tag: Any = None,
- approximate(
- method: Literal['cf'],
- tlist: ArrayLike,
- target_rmse: float = 2e-05,
- Nr_max: int = 10,
- Ni_max: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- full_ansatz: bool = False,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['ps'],
- wlist: ArrayLike,
- target_rmse: float = 5e-06,
- Nmax: int = 5,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['sd'],
- wlist: ArrayLike,
- Nk: int = 1,
- target_rmse: float = 5e-06,
- Nmax: int = 10,
- guess: list[float] = None,
- lower: list[float] = None,
- upper: list[float] = None,
- sigma: float | ArrayLike = None,
- maxfev: int = None,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['aaa'],
- wlist: ArrayLike,
- tol: float = 1e-13,
- Nmax: int = 10,
- combine: bool = True,
- tag: Any = None,
- approximate(
- method: Literal['prony', 'esprit', 'espira-I', 'espira-II'],
- tlist: ArrayLike,
- separate: bool = False,
- Nr: int = 3,
- Ni: int = 3,
- combine: bool = True,
- tag: Any = None,
Generates a multi-exponential approximation of this environment. The available methods are
"matsubara","cf","ps","sd","aaa","prony","esprit","espira-I"and"espira-II". The methods and the parameters required per method are documented in the Users Guide.
- correlation_function(
- t: float | ArrayLike,
- **kwargs,
Calculates the two-time auto-correlation function of the underdamped environment.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- class OhmicEnvironment(
- T: float,
- alpha: float,
- wc: float,
- s: float,
- *,
- tag: Any = None,
Bases:
BosonicEnvironmentDescribes Ohmic environments as well as sub- or super-Ohmic environments (depending on the choice of the parameter s). The spectral density is
\[J(\omega) = \alpha \frac{\omega^s}{\omega_c^{s-1}} e^{-\omega / \omega_c} .\]This class requires the mpmath module to be installed.
- Parameters:
- Tfloat
Temperature of the environment.
- alphafloat
Coupling strength.
- wcfloat
Cutoff parameter.
- sfloat
Power of omega in the spectral density.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- approximate(method: str, *args, **kwargs)
Generates a multi-exponential approximation of this environment. The available methods are
"cf","ps","sd","aaa","prony","esprit","espira-I"and"espira-II". The methods and the parameters required per method are documented in the Users Guide.
- correlation_function(
- t: float | ArrayLike,
- **kwargs,
Calculates the correlation function of an Ohmic environment using the formula
\[C(t)= \frac{1}{\pi} \alpha w_{c}^{1-s} \beta^{-(s+1)} \Gamma(s+1) \left[ \zeta\left(s+1,\frac{1+\beta w_{c} -i w_{c} t}{\beta w_{c}} \right) +\zeta\left(s+1,\frac{1+ i w_{c} t}{\beta w_{c}}\right) \right] ,\]where \(\Gamma\) is the gamma function, and \(\zeta\) the Riemann zeta function.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- class CFExponent(
- type: str | CFExponent.ExponentType,
- ck: complex,
- vk: complex,
- ck2: complex = None,
- tag: Any = None,
Represents a single exponent (naively, an excitation mode) within an exponential decomposition of the correlation function of a environment.
- Parameters:
- type{“R”, “I”, “RI”, “+”, “-”} or one of CFExponent.types
The type of exponent.
“R” and “I” are bosonic exponents that appear in the real and imaginary parts of the correlation expansion, respectively.
“RI” is a combined bosonic exponent that appears in both the real and imaginary parts of the correlation expansion. The combined exponent has a single
vk. Theckis the coefficient in the real expansion andck2is the coefficient in the imaginary expansion.“+” and “-” are fermionic exponents.
- ckcomplex
The coefficient of the excitation term.
- vkcomplex
The frequency of the exponent of the excitation term.
- ck2optional, complex
For exponents of type “RI” this is the coefficient of the term in the imaginary expansion (and
ckis the coefficient in the real expansion).- tagoptional, str, tuple or any other object
A label for the exponent (often the name of the environment). It defaults to None.
- Attributes:
- fermionicbool
True if the type of the exponent is a Fermionic type (i.e. either “+” or “-”) and False otherwise.
- coefficientcomplex
The coefficient of this excitation term in the total correlation function (including real and imaginary part).
- exponentcomplex
The frequency of the exponent of the excitation term. (Alias for vk.)
- All of the parameters are also available as attributes.
- rescale(alpha: float) CFExponent[source]
Rescale the coefficient of the exponent by a factor of alpha.
- types
alias of
ExponentType
- class ExponentialBosonicEnvironment(
- ck_real: ArrayLike = None,
- vk_real: ArrayLike = None,
- ck_imag: ArrayLike = None,
- vk_imag: ArrayLike = None,
- *,
- exponents: Sequence[CFExponent] = None,
- combine: bool = True,
- T: float = None,
- tag: Any = None,
Bases:
BosonicEnvironmentBosonic environment that is specified through an exponential decomposition of its correlation function. The list of coefficients and exponents in the decomposition may either be passed through the four lists ck_real, vk_real, ck_imag, vk_imag, or as a list of bosonic
CFExponentobjects.- Parameters:
- ck_reallist of complex
The coefficients of the expansion terms for the real part of the correlation function. The corresponding frequencies are passed as vk_real.
- vk_reallist of complex
The frequencies (exponents) of the expansion terms for the real part of the correlation function. The corresponding coefficients are passed as ck_real.
- ck_imaglist of complex
The coefficients of the expansion terms in the imaginary part of the correlation function. The corresponding frequencies are passed as vk_imag.
- vk_imaglist of complex
The frequencies (exponents) of the expansion terms for the imaginary part of the correlation function. The corresponding coefficients are passed as ck_imag.
- exponentslist of
CFExponent The expansion coefficients and exponents of both the real and the imaginary parts of the correlation function as
CFExponentobjects.- combinebool, default True
Whether to combine exponents with the same frequency. See
combinefor details.- T: optional, float
The temperature of the environment.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- classmethod combine(
- exponents: Sequence[CFExponent],
- rtol: float = 1e-05,
- atol: float = 1e-07,
Group bosonic exponents with the same frequency and return a single exponent for each frequency present.
- Parameters:
- exponentslist of
CFExponent The list of exponents to combine.
- rtolfloat, default 1e-5
The relative tolerance to use to when comparing frequencies.
- atolfloat, default 1e-7
The absolute tolerance to use to when comparing frequencies.
- exponentslist of
- Returns:
- list of
CFExponent The new reduced list of exponents.
- list of
- correlation_function(
- t: float | ArrayLike,
- **kwargs,
Computes the correlation function represented by this exponential decomposition.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- power_spectrum(
- w: float | ArrayLike,
- **kwargs,
Calculates the power spectrum corresponding to the multi-exponential correlation function.
- Parameters:
- warray_like or float
The frequency at which to evaluate the power spectrum.
- rescale(
- alpha: float,
- tag: Any = None,
Returns a new environment where all exponents are scaled by the factor alpha. This corresponds to changing the coupling constant. The spectral density, the correlation function and the power spectrum are all scaled by the same factor.
- Parameters:
- alphafloat
Rescaling factor.
- tagoptional, str, tuple or any other object
An identifier (name) for the rescaled environment.
- Returns:
- ExponentialBosonicEnvironment
The new environment with the rescaled exponents.
- system_terminator(
- Q: Qobj,
- delta: float,
Constructs the terminator for a given approximation discrepancy.
- Parameters:
- Q
Qobj The system coupling operator.
- deltafloat
The approximation discrepancy of approximating an environment with a finite number of exponentials, see for example the description of the Matsubara approximation.
- Q
- Returns:
- terminator
Qobj A superoperator acting on the system Hilbert space. Liouvillian term representing the contribution to the system-environment dynamics of all neglected expansion terms. It should be used by adding it to the system Liouvillian (i.e.
liouvillian(H_sys)).
- terminator
Fermionic Environments
- class FermionicEnvironment(T: float = None, mu: float = None, tag: Any = None)[source]
The fermionic environment of an open quantum system. It is characterized by its spectral density, temperature and chemical potential or, equivalently, by its power spectra or its two-time auto-correlation functions.
This class is included as a counterpart to
BosonicEnvironment, but it currently does not support all features that the bosonic environment does. In particular, fermionic environments cannot be constructed from manually specified spectral densities, power spectra or correlation functions. The only types of fermionic environment implemented at this time are Lorentzian environments (LorentzianEnvironment) and environments with multi-exponential correlation functions (ExponentialFermionicEnvironment).- Parameters:
- Toptional, float
The temperature of this environment.
- muoptional, float
The chemical potential of this environment.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- abstract correlation_function_minus(
- t: float | ArrayLike,
The “-“-branch of the auto-correlation function of this environment. See the Users Guide on fermionic environments for specifics on the definitions used by QuTiP.
- Parameters:
- tarray_like or float
The times at which to evaluate the correlation function.
- abstract correlation_function_plus(
- t: float | ArrayLike,
The “+”-branch of the auto-correlation function of this environment. See the Users Guide on fermionic environments for specifics on the definitions used by QuTiP.
- Parameters:
- tarray_like or float
The times at which to evaluate the correlation function.
- classmethod from_correlation_functions(
- **kwargs,
User-defined fermionic environments are currently not implemented.
- classmethod from_power_spectra(
- **kwargs,
User-defined fermionic environments are currently not implemented.
- abstract power_spectrum_minus(
- w: float | ArrayLike,
The “-“-branch of the power spectrum of this environment. See the Users Guide on fermionic environments for specifics on the definitions used by QuTiP.
- Parameters:
- warray_like or float
The frequencies at which to evaluate the power spectrum.
- abstract power_spectrum_plus(
- w: float | ArrayLike,
The “+”-branch of the power spectrum of this environment. See the Users Guide on fermionic environments for specifics on the definitions used by QuTiP.
- Parameters:
- warray_like or float
The frequencies at which to evaluate the power spectrum.
- abstract spectral_density(w: float | ArrayLike) float | ArrayLike[source]
The spectral density of this environment. See the Users Guide on fermionic environments for specifics on the definitions used by QuTiP.
- Parameters:
- warray_like or float
The frequencies at which to evaluate the spectral density.
- class LorentzianEnvironment(
- T: float,
- mu: float,
- gamma: float,
- W: float,
- omega0: float = None,
- *,
- Nk: int = 10,
- tag: Any = None,
Bases:
FermionicEnvironmentDescribes a Lorentzian fermionic environment with the spectral density
\[J(\omega) = \frac{\gamma W^2}{(\omega - \omega_0)^2 + W^2}.\](see Eq. 46 in [BoFiN23]).
- Parameters:
- Tfloat
Environment temperature.
- mufloat
Environment chemical potential.
- gammafloat
Coupling strength.
- Wfloat
The spectral width of the environment.
- omega0optional, float (default equal to
mu) The resonance frequency of the environment.
- Nkoptional, int, defaults to 10
The number of Pade exponents to be used for the calculation of the correlation functions.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- approx_by_matsubara(
- Nk: int,
- tag: Any = None,
Generates an approximation to this environment by truncating its Matsubara expansion.
- Parameters:
- Nkint
Number of Matsubara terms to include. In total, the “+” and “-” correlation function branches will include Nk+1 terms each.
- tagoptional, str, tuple or any other object
An identifier (name) for the approximated environment. If not provided, a tag will be generated from the tag of this environment.
- Returns:
- The approximated environment with multi-exponential correlation
- function.
- approx_by_pade(
- Nk: int,
- tag: Any = None,
Generates an approximation to this environment by truncating its Pade expansion.
- Parameters:
- Nkint
Number of Pade terms to include. In total, the “+” and “-” correlation function branches will include Nk+1 terms each.
- tagoptional, str, tuple or any other object
An identifier (name) for the approximated environment. If not provided, a tag will be generated from the tag of this environment.
- Returns:
- The approximated environment with multi-exponential correlation
- function.
- correlation_function_minus(
- t: float | ArrayLike,
- Nk: int = None,
Calculates the “-“-branch of the two-time auto-correlation function of the Lorentzian environment. The calculation is performed by summing a large number of exponents of the Pade expansion.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- Nkint, optional
The number of exponents to use. If not provided, then the value that was provided when the class was instantiated is used.
- correlation_function_plus(
- t: float | ArrayLike,
- Nk: int = None,
Calculates the “+”-branch of the two-time auto-correlation function of the Lorentzian environment. The calculation is performed by summing a large number of exponents of the Pade expansion.
- Parameters:
- tarray_like or float
The time at which to evaluate the correlation function.
- Nkint, optional
The number of exponents to use. If not provided, then the value that was provided when the class was instantiated is used.
- power_spectrum_minus(
- w: float | ArrayLike,
Calculates the “-“-branch of the power spectrum of the Lorentzian environment.
- Parameters:
- warray_like or float
The frequency at which to evaluate the power spectrum.
- class ExponentialFermionicEnvironment(
- ck_plus: ArrayLike = None,
- vk_plus: ArrayLike = None,
- ck_minus: ArrayLike = None,
- vk_minus: ArrayLike = None,
- *,
- exponents: Sequence[CFExponent] = None,
- T: float = None,
- mu: float = None,
- tag: Any = None,
Bases:
FermionicEnvironmentFermionic environment that is specified through an exponential decomposition of its correlation function. The list of coefficients and exponents in the decomposition may either be passed through the four lists ck_plus, vk_plus, ck_minus, vk_minus, or as a list of fermionic
CFExponentobjects.Alternative constructors
from_plus_exponentsandfrom_minus_exponentsare available to compute the “-” exponents automatically from the “+” ones, or vice versa.- Parameters:
- ck_pluslist of complex
The coefficients of the expansion terms for the
+part of the correlation function. The corresponding frequencies are passed as vk_plus.- vk_pluslist of complex
The frequencies (exponents) of the expansion terms for the
+part of the correlation function. The corresponding coefficients are passed as ck_plus.- ck_minuslist of complex
The coefficients of the expansion terms for the
-part of the correlation function. The corresponding frequencies are passed as vk_minus.- vk_minuslist of complex
The frequencies (exponents) of the expansion terms for the
-part of the correlation function. The corresponding coefficients are passed as ck_minus.- exponentslist of
CFExponent The expansion coefficients and exponents of both parts of the correlation function as
CFExponentobjects.- T: optional, float
The temperature of the environment.
- mu: optional, float
The chemical potential of the environment.
- tagoptional, str, tuple or any other object
An identifier (name) for this environment.
- correlation_function_minus(
- t: float | ArrayLike,
Computes the “-“-branch of the correlation function represented by this exponential decomposition.
- Parameters:
- tarray_like or float
The times at which to evaluate the correlation function.
- correlation_function_plus(
- t: float | ArrayLike,
Computes the “+”-branch of the correlation function represented by this exponential decomposition.
- Parameters:
- tarray_like or float
The times at which to evaluate the correlation function.
- power_spectrum_minus(
- w: float | ArrayLike,
Calculates the “-“-branch of the power spectrum corresponding to the multi-exponential correlation function.
- Parameters:
- warray_like or float
The frequency at which to evaluate the power spectrum.
- power_spectrum_plus(
- w: float | ArrayLike,
Calculates the “+”-branch of the power spectrum corresponding to the multi-exponential correlation function.
- Parameters:
- warray_like or float
The frequency at which to evaluate the power spectrum.
- rescale(
- alpha: float,
- tag: Any = None,
Returns a new environment where all exponents are scaled by the factor alpha. This corresponds to changing the coupling constant. The spectral density, the correlation function and the power spectrum are all scaled by the same factor.
- Parameters:
- alphafloat
Rescaling factor.
- tagoptional, str, tuple or any other object
An identifier (name) for the rescaled environment.
- Returns:
- ExponentialBosonicEnvironment
The new environment with the rescaled exponents.