qit.markov.MarkovianBath

class qit.markov.MarkovianBath(bath_type: str, stat: str, TU: float, T: float)

Markovian heat bath.

Supports bosonic and fermionic canonical ensembles at absolute temperature T, with an ohmic spectral density. The bath couples to the system via a single-term coupling $H_{\text{int}}=A\otimes \sum _k{\lambda }_k(a_k+a_k^{\prime })$.

Parameters

• bath_type (str) – bath type, determines the spectral density, in {‘ohmic’, ‘photon’}.

• stat (str) – bath statistics, in (‘boson’, ‘fermion’)

• TU (float) – bath time unit (in s)

• T (float) – bath temperature (in K)

The bath spectral density is either ohmic or photonic, with a cutoff:

$$\begin{array}{r}{J}_{\text{ohmic}}(\omega )=\omega \mathrm{cut}(\omega )\mathrm{\Theta }(\omega ),\\ J_{\text{photon}}(\omega )={\omega }^3\mathrm{cut}(\omega )\mathrm{\Theta }(\omega ).\end{array}$$

Three types of cutoffs are supported:

$$\begin{array}{rl}{\mathrm{cut}}_{\text{exp}}(x{\omega }_c)& =\exp (-x),\\ {\mathrm{cut}}_{\text{smooth}}(x{\omega }_c)& =\frac{1}{1+x^2},\\ {\mathrm{cut}}_{\text{sharp}}(x{\omega }_c)& =\mathrm{\Theta }(1-x).\end{array}$$

The effects of the bath on the system are contained in the complex spectral correlation tensor $\mathrm{\Gamma }$. Computing values of this tensor is the main purpose of this class. It depends on three parameters: the inverse temperature of the bath $s=\beta \hslash$, the spectral cutoff frequency of the bath ${\omega }_c$, and the system frequency $\omega$. It has the following scaling property:

$${\mathrm{\Gamma }}_{s,{\omega }_c}(\omega )={\mathrm{\Gamma }}_{s/a,{\omega }_{c}a}(\omega a)/a.$$

Hence we may eliminate the dimensions by choosing a = TU:

$${\mathrm{\Gamma }}_{s,{\omega }_c}(\omega )=\frac{1}{\text{TU}}{\mathrm{\Gamma }}_{\hat{s},\widehat{{\omega }_c}}(\hat{\omega }),$$

where the hat denotes a dimensionless quantity. Since we only have one coupling term, $\mathrm{\Gamma }$ is a scalar. We split it into its hermitian and antihermitian parts:

$${\mathrm{\Gamma }}_{s,{\omega }_c}(\omega )=\frac{1}{2}\gamma (\omega )+iS(\omega ),$$

where $\gamma$ and $S$ are real, and

$$\begin{array}{rl}\gamma (\omega )& =2\pi (1\pm n(\omega ))\mathrm{cut}(|\omega |)\{\begin{array}{l}\omega \text{(bosons)}\\ |\omega |\text{(fermions)}\end{array}\\ S(\omega )& =P{\int }_0^{\mathrm{\infty }}\mathrm{d}\nu \frac{J(\nu )}{(\omega -\nu )(\omega +\nu )}\{\begin{array}{l}\omega \coth (\beta \hslash \nu /2)+\nu \text{(bosons)}\\ \omega +\nu \tanh (\beta \hslash \nu /2)\text{(fermions)}\end{array}\end{array}$$

where $n(\omega ):=1/(e^{\beta \hslash \omega }\mp 1)$ is the Planc/Fermi function and $\beta =1/(k_{B}T)$. Since $\mathrm{\Gamma }$ is pretty expensive to compute, we store the computed results into a lookup table which is used to interpolate nearby values.

Private attributes (set automatically):

Data member	Description
cut_func	Spectral density cutoff function.
pf	Planck or Fermi function depending on statistics.
corr_int	Integral transform for the dimensionless bath correlation function.
g_func	Spectral correlation tensor, real part. $\gamma (\omega /\text{TU})\text{TU}=\mathrm{g}\mathrm{\_}\mathrm{func}(\omega )$.
s_func	Spectral correlation tensor, imaginary part.
g0	$\underset{\omega \to 0}{lim}\gamma (\omega )$.
s0	$\underset{\omega \to 0}{lim}S(\omega )$.
omega	Lookup table.
gs_table	Lookup table. $\gamma (\text{omega[k]}/\text{TU})\text{TU}$ = gs_table[k, 0].

Attributes

type	Bath type.
stat	Bath statistics.
TU	Time unit (in s).
T	Absolute temperature of the bath (in K).
scale	Dimensionless temperature scaling parameter $\hslash /(k_{B}T\text{TU})$.
cut_type	Spectral density cutoff type, in {'sharp', 'smooth', 'exp'}.
cut_omega	Spectral density cutoff angular frequency ${\omega }_c$ (in $1/\text{TU}$).

Methods

build_LUT([om])	Build a lookup table for the spectral correlation tensor Gamma.
compute_gs(x)	Computes the spectral correlation tensor.
corr(x)	Bath spectral correlation tensor, computed or interpolated.
desc([long])	Bath description string for plots.
fit(delta, T1, T2)	Qubit-bath coupling that reproduces given decoherence times.
plot_bath_correlation()	Plot the bath correlation function $C_{s,{\omega }_c}(t)=\frac{1}{{\hslash }^2}⟨B(t)B(0)⟩$.
plot_spectral_correlation()	Plot the spectral correlation tensor components $\gamma$ and $S$ as a function of omega.
plot_spectral_correlation_vs_cutoff([boltz])	Plot spectral correlation tensor components as a function of cutoff frequency.
set_cutoff(cutoff_type, cut_omega)	Set the spectral density cutoff.
setup()	Initializes the g and s functions, and the LUT.

type

Bath type. ‘ohmic’ or ‘photon’.

stat

Bath statistics. Either ‘boson’ or ‘fermion’.

TU

Time unit (in s). All Hamiltonians have been made dimensionless by multiplying with $\text{TU}/\hslash$.

T

Absolute temperature of the bath (in K).

scale

Dimensionless temperature scaling parameter $\hslash /(k_{B}T\text{TU})$.

cut_type

Spectral density cutoff type, in {'sharp', 'smooth', 'exp'}.

cut_omega

Spectral density cutoff angular frequency ${\omega }_c$ (in $1/\text{TU}$).

desc(long: bool = True) → str

Bath description string for plots.

Parameters

long (bool) –

Return type

str

set_cutoff(cutoff_type: Optional[str], cut_omega: Optional[float]) → None

Set the spectral density cutoff.

Passing None leaves the corresponding property unchanged.

Parameters

• cutoff_type (Optional[str]) – cutoff type, in {'sharp', 'smooth', 'exp'}

• cut_omega (Optional[float]) – omega cutoff value

Return type

None

setup() → None

Initializes the g and s functions, and the LUT.

Must be called after parameters change.

Return type

None

_pad_LUT() → None

Add limits at infinity to the lookup tables.

Return type

None

build_LUT(om: Optional[np.ndarray[float]] = None) → None

Build a lookup table for the spectral correlation tensor Gamma.

Parameters

om (Optional[np.ndarray[float]]) – vector of omegas denoting the points to compute

Return type

None

_plot(x, om, q, gs, odd_s, even_s, f=<function plot>)

Plotting utility.

Parameters

• x (array[float]) – x coordinates of the points

• om (array[float]) –

• q (array[float]) –

• gs (array[float]) – gamma and S, the real and imaginary components of the spectral correlation tensor

• odd_s (array[float]) – odd part of S

• even_s (array[float]) – even part of S

• f (callable) – plotting function

Returns

the plot

Return type

Axes

plot_spectral_correlation() → None

Plot the spectral correlation tensor components $\gamma$ and $S$ as a function of omega.

Additionally plots the even and odd parts of $S$, and compares them to analytical expressions. Uses the spectral correlation tensor LUT.

Return type

None

plot_spectral_correlation_vs_cutoff(boltz: float = 0.5) → None

Plot spectral correlation tensor components as a function of cutoff frequency.

The angular frequency $\omega$ at which the spectral correlation tensor is evaluated is fixed by giving the Boltzmann factor $e^{-\beta \hslash \omega }$.

Parameters

boltz (float) – Boltzmann factor

Return type

None

plot_bath_correlation() → None

Plot the bath correlation function $C_{s,{\omega }_c}(t)=\frac{1}{{\hslash }^2}⟨B(t)B(0)⟩$.

It scales as

$$C_{s,{\omega }_c}(t)=\frac{1}{a^2}{C}_{s/a,{\omega }_{c}a}(t/a).$$

Choosing a = TU, we obtain

$$C_{s,{\omega }_c}(t)=\frac{1}{{\text{TU}}^2}{C}_{\hat{s},\widehat{{\omega }_c}}(\hat{t}).$$

Return type

None

compute_gs(x: float) → Tuple[float, float]

Computes the spectral correlation tensor. See corr for usage.

Parameters

x (float) – angular frequency [1/TU]

Returns

Real and imaginary parts of the spectral correlation tensor at x [1/TU]

Return type

Tuple[float, float]

corr(x: float) → Tuple[float, float]

Bath spectral correlation tensor, computed or interpolated.

$$\mathrm{\Gamma }(x/\text{TU})\text{TU}=\frac{1}{2}g+is$$

Parameters

x (float) – angular frequency [1/TU]

Returns

Real and imaginary parts of the spectral correlation tensor at x [1/TU]

Return type

Tuple[float, float]

fit(delta: float, T1: float, T2: float)

Qubit-bath coupling that reproduces given decoherence times.

Parameters

• delta (float) – qubit energy splitting (in units of $\hslash /TU$)

• T1 (float) – qubit decoherence times T1 and T2 (in units of $TU$)

• T2 (float) – qubit decoherence times T1 and T2 (in units of $TU$)

Returns

qubit Hamiltonian, qubit-bath coupling operator

Return type

array[complex], array[complex]

Returns the qubit Hamiltonian H and the qubit-bath coupling operator D that reproduce the decoherence times T1 and T2 for a single-qubit system coupled to the bath.

The bath object is not modified.