A tutorial on the polaron transformed Redfield equation
Huo Chen
Correlation function in the polaron frame
This tutorial demonstrates how to use the polaron transformed Redfield equation (PTRE) in HOQST. For more details on the PTRE, see [1] Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach.
We solve both the Redfield equation and the PTRE for a single qubit Hamiltonian
\[ H_\mathrm{S}=\epsilon \sigma_z + \Delta \sigma_x \]
coupled to an Ohmic bath via $\sigma_z$ interaction:
\[ H = H_\mathrm{S} + \sigma_z \otimes B + H_\mathrm{B}\ . \]
Loosely, the main difference between the Redfield equation and PTRE is that they have different bath correlation functions. For the Redfield equation, the bath correlation function is
\[ C(t_1, t_2) = \langle B(t_1)B(t_2) \rangle \ . \]
In the polaron frame, however, the bath correlation function becomes
\[ K(t_1, t_2) = \exp\Big\{ -4 \int_0^t \int_{-\infty}^{0}C(t_1, t_2) \mathrm{d}t_1 \mathrm{d}t_2 \Big\} \ . \]
Interested readers can refer to [2] Macroscopic Resonant Tunneling in the Presence of Low Frequency Noise and [3] Dynamics of the dissipative two-state system for more details.
Error bound on the second-order master equation
The most straightforward analysis is to compare the error bounds given in [4] Completely positive master equation for arbitrary driving and small level spacing between the Redfield equation and PTRE. We define the error scaling parameter as
\[ error = \frac{\tau_\mathrm{B}}{\tau_\mathrm{SB}} \ . \]
Then we plot the error ratio between the Redfield equation and the PTRE
\[ R = \frac{error_{\mathrm{Redfield}}}{error_{\mathrm{PTRE}}} \ , \]
vs. the system bath coupling strength $\eta g^2$ while fixing other parameters in the Ohmic bath.
using OrdinaryDiffEq, OpenQuantumTools, Plots using LaTeXStrings function err_bound(tf, cfun) tsb, esb = τ_SB(cfun) tb, eb = τ_B(cfun, tf, tsb) tb / tsb end fc = 4; T =12; tf = 1000; ηlist = log_uniform(1e-3, 5, 1000) err_ratio = [] err_clist = [] err_klist = [] for η in ηlist bath = Ohmic(η, fc, T) cfun = (x)->correlation(x, bath) pfun = (x)->polaron_correlation(x, bath) err_c = err_bound(tf, cfun) err_k = err_bound(tf, pfun) push!(err_clist, err_c) push!(err_klist, err_k) push!(err_ratio, err_c/err_k) end idx = findfirst((x)->x>=1, err_ratio) plot(ηlist, err_ratio, xscale=:log10, yscale=:log10, label="", linewidth=2) vline!([ηlist[idx]], label="", linestyle=:dash, linewidth=2) annotate!([(0.5, 1.0, Plots.text("polaron")), (0.01, 1.0, Plots.text("Redfield"))]) xlabel!(L"\eta g^2") ylabel!("R")
From the above figure we observe that when the system-bath coupling strength is larger than $10^{-1}$, the PTRE should have better error scaling than the standard form of the Redfield equation. We also plot the corresponding error values for both the Redfield equation and the PTRE:
plot(ηlist, err_clist, xscale=:log10, yscale=:log10, label="Redfield", linewidth=2) plot!(ηlist, err_klist, xscale=:log10, yscale=:log10, label="PTRE", linewidth=2) xlabel!(L"\eta g^2") ylabel!("error")
The above figure confirms that the Redfield equation applies to the weak-coupling regime while the PTRE applies to the strong coupling regime.
Solving PTRE
Since the Redfield equation and the PTRE have identical forms, solve_redfield can also be used for the PTRE. To see this, let's first write down the PTRE for our example.
\[ \dot{\rho}_\mathrm{S} = \epsilon σ_z + [ \sigma_i, \Lambda_i(t) \rho_\mathrm{S}(t) ] + h.c. \]
where $i,j \in [+, -]$, $i \neq j$ and
\[ \Lambda_i(t)=\Delta^2 \int_0^{t} K(t-\tau)U(t,\tau) \sigma_j U^\dagger(t,\tau) \mathrm{d}\tau \ . \]
From the above equations, it is clear that the following steps are needed to define an evolution in the polaron frame:
define a new Hamiltonian $H = \epsilon σ_z$;
define new coupling operators $\sigma_-$ and $\sigma_+$;
define a new correlated bath with two-point correlation $K_{i,j}(t_1, t_2)$;
The following code block illustrates how this can be done in HOQST:
# assume ϵ = 1 const Δ = 0.1 # define the Ohmic bath in the polaron transformed frame η = 0.5; bath = Ohmic(η, fc, T) K(t1, t2) = Δ^2 * polaron_correlation(t1-t2, bath) cfun = [nothing K; K nothing] pbath = CorrelatedBath(((1,2),(2,1)), correlation=cfun) # define coupling as σ+ and σ- operators σp = [0 1;0 0.0im]; σm = [0 0;1 0.0im] coupling = ConstantCouplings([σp, σm]) # manually define the unitary operator U(t) = exp(-2.0im * π * σz * t) H = DenseHamiltonian([(s)->1.0], [σz]) u0 = PauliVec[3][1] annealing = Annealing(H, u0, coupling = coupling, bath = pbath) tf = 100 sol_ptre = solve_redfield(annealing, tf, U, alg=Tsit5(), Ta=2, reltol=1e-5) pop_e = [real(s[1,1]) for s in sol_ptre.u] plot(sol_ptre.t, pop_e, xlabel=L"t\ (\mathrm{ns})", ylabel=L"P_0(t)", label="", linewidth = 2)
For historical reasons, this is known as an example of "incoherent tunneling". The off-diagonal elements of the density matrix in the computational basis (the Z-basis) vanish during the entire evolution (shown in the next section).
Redfield equation
What happens to the Redfield equation in this regime? We can also try:
H = DenseHamiltonian([(s)->1.0], [σz+0.1*σx]) coupling = ConstantCouplings(["Z"]) annealing = Annealing(H, u0, coupling = coupling, bath = bath) tf = 100 # manually define the unitary operator U(t) = exp(-2.0im * π * (σz+0.1*σx) * t) sol_redfield = solve_redfield(annealing, tf, U, alg=Tsit5(), Ta=40, reltol=1e-5, callback=PositivityCheckCallback())
retcode: Terminated
Interpolation: specialized 4th order "free" interpolation
t: 22-element Vector{Float64}:
0.0
0.003928759496976375
0.005712702105779054
0.00868538284690315
0.010748436797295107
0.013343972909612483
0.015630574875531853
0.018040070229891184
0.020327710050893358
0.022614512268462634
⋮
0.03126733644196358
0.03336262989102297
0.03544603050483967
0.03752137951197164
0.03959565980312375
0.04167340886800694
0.04375982866539185
0.04585917830352944
0.04797576487851534
u: 22-element Vector{Matrix{ComplexF64}}:
[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im]
[0.9999942655705363 + 0.0im 2.3405168745021195e-5 + 0.00219374300574346im;
2.3405168745021195e-5 - 0.00219374300574346im 5.734429463642047e-6 + 0.0im
]
[0.9999885526527795 + 0.0im -1.6098894434093695e-5 + 0.0028681886770785987
im; -1.6098894434093695e-5 - 0.0028681886770785987im 1.1447347220274872e-5
+ 0.0im]
[0.9999762824915798 + 0.0im -0.00026798716109694765 + 0.003654533478940986
im; -0.00026798716109694765 - 0.003654533478940986im 2.3717508419724734e-5
+ 0.0im]
[0.9999662031906074 + 0.0im -0.0005842937299489406 + 0.004120304221198509i
m; -0.0005842937299489406 - 0.004120304221198509im 3.379680939219504e-5 + 0
.0im]
[0.9999517825461793 + 0.0im -0.001115605899581189 + 0.004729454751473413im
; -0.001115605899581189 - 0.004729454751473413im 4.82174538193727e-5 + 0.0i
m]
[0.9999373816728743 + 0.0im -0.0016835737348890997 + 0.005297796662131235i
m; -0.0016835737348890997 - 0.005297796662131235im 6.261832712481899e-5 + 0
.0im]
[0.9999204146273409 + 0.0im -0.0023717853091745696 + 0.005910253580108264i
m; -0.0023717853091745696 - 0.005910253580108264im 7.958537265799662e-5 + 0
.0im]
[0.9999025905694068 + 0.0im -0.003105230981241366 + 0.006488770467415236im
; -0.003105230981241366 - 0.006488770467415236im 9.740943059240977e-5 + 0.0
im]
[0.999883126735672 + 0.0im -0.003913250858523132 + 0.007054790877644963im;
-0.003913250858523132 - 0.007054790877644963im 0.00011687326432621205 + 0.
0im]
⋮
[0.9997954625649402 + 0.0im -0.007593216989156528 + 0.009016925074664294im
; -0.007593216989156528 - 0.009016925074664294im 0.000204537435058375 + 0.0
im]
[0.9997711556620502 + 0.0im -0.008619684963591303 + 0.009442538470109785im
; -0.008619684963591303 - 0.009442538470109785im 0.00022884433794879795 + 0
.0im]
[0.9997459018236546 + 0.0im -0.009687884470117376 + 0.009845730915410916im
; -0.009687884470117376 - 0.009845730915410916im 0.00025409817634431124 + 0
.0im]
[0.9997197219479761 + 0.0im -0.010796892837049153 + 0.010227599257367563im
; -0.010796892837049153 - 0.010227599257367563im 0.0002802780520233289 + 0.
0im]
[0.9996925858837132 + 0.0im -0.011948024515996868 + 0.010589794153091936im
; -0.011948024515996868 - 0.010589794153091936im 0.0003074141162859749 + 0.
0im]
[0.9996644830460419 + 0.0im -0.01314181747724239 + 0.010933448746627866im;
-0.01314181747724239 - 0.010933448746627866im 0.0003355169539566007 + 0.0i
m]
[0.9996353847581846 + 0.0im -0.014379622631485891 + 0.011259742442199539im
; -0.014379622631485891 - 0.011259742442199539im 0.0003646152418134799 + 0.
0im]
[0.999605267069751 + 0.0im -0.015662623261621465 + 0.011569632687661376im;
-0.015662623261621465 - 0.011569632687661376im 0.0003947329302471738 + 0.0
im]
[0.9995740985607838 + 0.0im -0.016992346716545714 + 0.011864024725645031im
; -0.016992346716545714 - 0.011864024725645031im 0.0004259014392143201 + 0.
0im]
The Redfield equation becomes non-positive in a very short evolution time. To fix this issue, we use the coarse-grained master equation (CGME) instead.
tf = 50 solc = solve_cgme(annealing, tf, U, alg=Tsit5(), reltol=1e-3, int_atol=1e-3, int_rtol=1e-3) t_axis = range(0,50,length=200) pop_e = [real(sol_ptre(t)[1,1]) for t in t_axis] pop_e_cg = [real(solc(t)[1,1]) for t in t_axis] plot(t_axis, pop_e, xlabel=L"t\ (\mathrm{ns})", ylabel=L"P_0(t)", label="PTRE", linewidth = 2) plot!(t_axis, pop_e_cg, xlabel=L"t\ (\mathrm{ns})", ylabel=L"P_0(t)", label="CGME", linewidth = 2)
The PTRE gives a much stronger decay than the Redfield equation for the parameters chosen in this example. One can also verify the amplitude of the off-diagonal elements during the evolution. Unlike the PTRE, the solution of the CGME has non-vanishing off-diagonal elements of the density matrix.
t_axis = range(0, 0.2, length=100) off_diag_ptre = [abs(sol_ptre(t)[1,2]) for t in t_axis] off_diag_cg = [abs(solc(t)[1,2]) for t in t_axis] plot(t_axis, off_diag_ptre, xlabel=L"t\ (\mathrm{ns})", ylabel=L"\lvert \rho_{12} \rvert|(t)", label="PTRE", linewidth = 2, legend=:right) plot!(t_axis, off_diag_cg, xlabel=L"t\ (\mathrm{ns})", ylabel=L"|\rho_{12}(t)|", label="CGME", linewidth = 2)
Appendix
This tutorial is part of the HOQSTTutorials.jl repository, found at: https://github.com/USCqserver/HOQSTTutorials.jl.
To locally run this tutorial, do the following commands:
using HOQSTTutorials
HOQSTTutorials.weave_file("introduction","04-polaron_transformed_redfield.jmd")Computer Information:
Julia Version 1.6.0
Commit f9720dc2eb (2021-03-24 12:55 UTC)
Platform Info:
OS: Windows (x86_64-w64-mingw32)
CPU: Intel(R) Core(TM) i7-6700K CPU @ 4.00GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-11.0.1 (ORCJIT, skylake)
Package Information:
Status `tutorials\introduction\Project.toml`
[2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91] StatsBase 0.33.4
[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.52.2
[e429f160-8886-11e9-20cb-0dbe84e78965] OpenQuantumTools 0.6.2
[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 1.11.2
[b964fa9f-0449-5b57-a5c2-d3ea65f4040f] LaTeXStrings 1.2.1
[1fd47b50-473d-5c70-9696-f719f8f3bcdc] QuadGK 2.4.1