Numerics.LvNE.Lindblad

Lindblad formalism for open quantum systems

General theory

In reduced density representation of open quantum systems, the corresponding Liouville von Neumann equation has to include the coupling to the environment, thus accounting for time-directed irreversibility, i. e., dissipation and/or dephasing. The most general time-independent form that preserves complete positivity of the reduced density operator is the Lindblad Liouvillian superopator

where the operators C describe the effect of the environment on the system in Born-Markov approximation (weak coupling, no memory). In most applications, they are simply chosen to be projectors

with (phenomenological) rate constants given as inverse times Γ_i←j=1/T_i←j. With this choice (positive prefactors!) the above evolution is trace-preserving, i.e. the sum of populations remains constant, and completely positive, i. e., also the individual populations remain positive thus ensuring the probabilistic interpretation of densities in quantum mechanics. Typically, the upward rates are calculated from the downward ones using the principle of detailed balance

which ensures that the densities approach the Boltzmann distributions in the limit of infinitely long times. In marked contrast, Redfield formalism provides detailed balance automatically, but negative populations may occur.

The summation in the above equation for the Lindblad superoperator extends over all possible channels k=(i←j): While terms with i≠j involve both population relaxation and associated dephasing, terms with i=j result in pure dephasing. (Note that the terms decoherence and dephasing are inconsistently used in the literature.).

This formalism is implemented in the Matlab version of WavePacket inside the qm_abncd function.

Time evolution

The resulting time evolution for the matrix elements ρ_nm≡⟨n|ρ|m⟩of the reduced density operator can be written as

• The first term on the r.h.s. describes the (temporal) oscillations of the coherences (n≠m) governed by the Bohr frequencies ω_nm≡E_n-E_m. This term has no influence on the densities (n=m).

• The second term on the r. h. s. describes the influence of the external field F(t) through dipole matrix elements μ_nm≡⟨n|μ|m⟩ coupling various density matrix elements ρ_nm. In particular it also involves coupling between densities and coherences.

• The third term on the r. h. s. (diagonal in n=m) describes the gain in population of state n through relaxation of (other!) states j. For two states, no external fields (F(t)=0) and zero temperature (Γ_2←1=0), this is equivalent to the T₁ decay mechanism in the NMR literature with decay time T₁=1/Γ_1←2.

• The fourth term on the r. h. s. describes the loss of population (n=m) and/or the decay of coherences (n≠m) in terms of rates defined as

  

  The diagonal elements simplify to γ_nn=Σ_j≠nΓ_j←n which is the population loss from state n to states j, thus compensating for the gain in the third term of the r. h. s. of the above evolution equation. Together, these two expressions constitute a classical master equation which is dealing with populations only (which would be a valid model once all the coherences have decayed to zero).The off-diagonal elements γ_n≠m account for the dephasing associated with population relaxation, given by off-diagonal elements of Γ with j≠n,m. For two states, no external fields (F(t)=0) and zero temperature (Γ_2←1=0), the coherence ρ₁₂ decays exponentially with a time constant of 2/Γ_1←2 which is well known from the NMR literature.

• The fifth term on the r. h. s. describes the effect of pure dephasing which implies that it has no effect on populations, but leads to additional decay of the coherences. The corresponding rates are given by

  

  For two states and no external fields (F(t)=0), this is equivalent to the T₂^∗ decay mechanism in the NMR literature with decay time 1/T₂^∗=½(Γ_1←1+Γ_2←2). Hence, together with the above assumption of zero temperature (Γ_2←1=0), the coherence ρ₁₂ decays exponentially with a time constant T₂ which is given by

  

  which is the well known result from the NMR literature.

Population relaxation models

Specific models of the above Lindblad equations require - in principle - a microscopic knowledge of the system-bath coupling operator. In practice, however, this is most often not the case, and simplifying assumptions have to be made, e. g., a linear approach H_SB=Σ_iλ_iq_i where the summation extends over all bath modes q_i.

Using Fermi's golden rule for the weak coupling limit, and assuming equal masses and frequencies of the bath modes, it can be shown that the downward (population) relaxation rates fulfill the following relation

Hence, together with the principle of detailed balance given above, this model only requires knowledge of a single relaxation rate (often between ground and one excited state) which is then sufficient to determine all other rates, see e. g. work by the Saalfrank group on molecular vibration.

Other frequently used choices include models based on scaled Einstein coefficients for spontaneous emission (with A being a scaling factor)

see, e. g. work by the Saalfrank group on molecular electronic dynamics.

Pure dephasing models

A frequent choice for a model of the calculation of dephasing rates is based on the following Lindblad operator for pure dephasing (where κ is a scaling factor)

which leads to a quadratic energy gap dependence

see work by Lockwood, Ratner, Kosloff on vibrational dephasing rates for molecules interacting with a bath.

References

1. B. Schäfer-Bung, C. Hartmann, B. Schmidt, and Ch. Schütte: J. Chem. Phys. 135, 014112 (2011)