An example of non-positivity in the Redfield equation
Huo Chen, Jenia Mozgunov
In this notebook, we will construct an example where the Redfield equation becomes non-positive. We will also show how to use the positivity-check routine to stop the solver when this happens.
Ohmic bath
We first create an Ohmic bath with the following parameters:
using OpenQuantumTools, OrdinaryDiffEq, Plots, Printf, LaTeXStrings β = 4 T = β_2_temperature(β) η = 0.1 fc= 10/(2π) bath = Ohmic(η, fc, T)
Ohmic bath instance: η (unitless): 0.1 ωc (GHz): 1.5915494309189535 T (mK): 1.9095587777458247
The spectral density $\gamma$ is plotted below:
plot(bath, :γ, range(0,10,length=100), linewidth=2, label="")
The time scales (defined in [1] Completely positive master equation for arbitrary driving and small level spacing) of the bath are:
τsb, err_τsb = τ_SB((x)->correlation(x, bath)) @printf("τ_sb of the Ohmic bath is %.6f with error estimation %.2e \n", τsb, err_τsb) τb, err_τb = τ_B((x)->correlation(x, bath), 100, τsb) @printf("τ_b of the Ohmic bath is %.6f with error estimation %.2e \n", τb, err_τb)
τ_sb of the Ohmic bath is 0.666454 with error estimation 6.48e-09 τ_b of the Ohmic bath is 0.201395 with error estimation 1.91e-10
Evolution
We construct the Hamiltonian by:
Hp = 0.5*σz⊗σi - 0.7*σi⊗σz + 0.3*σz⊗σz Hd = standard_driver(2) H = DenseHamiltonian([(s)->1-s, (s)->s], [-Hd, Hp], unit=:ħ)
DenseHamiltonian with Complex{Float64}
with size: (4, 4)
The spectrum of the Hamiltonian during the evolution is
plot(H, range(0,1,length=100), 4, linewidth=2) xlabel!("s") ylabel!(L"P(s)")
Closed system
We now run the closed-system simulation:
tf = 20 ρ0 = (σi+σx)⊗(σi+σx)/4 coupling = ConstantCouplings([σz⊗σi, σi⊗σz], unit=:ħ) annealing = Annealing(H, ρ0, bath=bath, coupling=coupling) close_sol = solve_von_neumann(annealing, tf, alg = Tsit5(), abstol=1e-6, reltol=1e-6);
The population of the instantaneous ground state is:
plot(close_sol, H, 1, range(0,tf,length=100), linewidth=2) xlabel!("t") ylabel!(L"P_G(s)")
The populations of the computational states are:
t_axis = range(0,tf,length=100) p_computational_basis = [real(diag(close_sol(s))) for s in t_axis] p_computational_basis = hcat(p_computational_basis...) plot(t_axis, p_computational_basis', linewidth=2, label=[L"\rho_{00}" L"\rho_{11}" L"\rho_{22}" L"\rho_{33}"]) xlabel!("t") ylabel!(L"\rho")
Redfield equation
We solve the Redfield equation:
tf = 20 U = solve_unitary(annealing, tf, alg = Tsit5(), abstol=1e-7, reltol=1e-7); redfield_sol = solve_redfield(annealing, tf, U, alg = Tsit5(), abstol=1e-7, reltol=1e-7);
We plot the populations of the computational basis states:
t_axis = range(0,tf,length=100) p_computational_basis = [real(diag(redfield_sol(s))) for s in t_axis] p_computational_basis = hcat(p_computational_basis...) plot(t_axis, p_computational_basis', linewidth=2, label=[L"\rho_{00}" L"\rho_{11}" L"\rho_{22}" L"\rho_{33}"]) xlabel!("t") ylabel!(L"\rho")
We can see that the density matrix becomes negative during evolution.
Positivity check
We can add a callback to stop the ODE solver when the density matrix becomes negative.
redfield_sol = solve_redfield(annealing, tf, U, alg = Tsit5(), abstol=1e-7, reltol=1e-7, callback=PositivityCheckCallback())
retcode: Terminated
Interpolation: specialized 4th order "free" interpolation
t: 38-element Array{Float64,1}:
0.0
9.999999999999999e-5
0.0010999999999999998
0.00786551568397621
0.016932240388251633
0.026303763546061412
0.03727043304206125
0.049257562452163145
0.06292827203759598
0.07875976726667351
⋮
1.6713448602411505
1.8638813298254504
2.0645751797094887
2.274212968610653
2.4922159175945793
2.7193240163467145
2.9544081226732986
3.19854539102924
3.4527816734933183
u: 38-element Array{Array{Complex{Float64},2},1}:
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⋮
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