Time-Convolutionless Master Equations

A time-convolutionless master equation is a time-local equation for reduced dynamics:

$$\frac{d}{dt}\rho_S(t) = \mathcal K_{\mathrm{TCL}}(t)\rho_S(t).$$

It is called “convolutionless” because the equation does not contain an explicit integral over earlier states. Memory can still be present: it is encoded in the time dependence, singularities, and sign structure of the generator $\mathcal K_{\mathrm{TCL}}(t)$.

This makes TCL equations the natural counterpart of Memory Kernels and the Nakajima–Zwanzig Projection:

Nakajima–Zwanzig:  time-nonlocal kernel with rho(s)
TCL:               time-local generator with rho(t)

For the broader distinction between memory kernels, CP divisibility, and information backflow, see Non-Markovian Dynamics.

Exact Map Formula

Suppose the reduced dynamics from $t_0$ to $t$ is represented by a linear map

$$\rho_S(t) = \Phi(t,t_0)\rho_S(t_0).$$

If $\Phi(t,t_0)$ is invertible on the relevant operator space, then an exact time-local generator exists:

$$\mathcal K_{\mathrm{TCL}}(t) = \dot\Phi(t,t_0)\Phi^{-1}(t,t_0).$$

Indeed,

$$\frac{d}{dt}\rho_S(t) = \dot\Phi(t,t_0)\rho_S(t_0) = \dot\Phi(t,t_0)\Phi^{-1}(t,t_0)\rho_S(t).$$

This formula is exact. It does not say the dynamics is Markovian in the physical sense. It says the same reduced map can be written as a time-local differential equation while the map remains invertible.

If $\Phi(t,t_0)$ becomes noninvertible, the time-local generator can become singular even when the finite-time map remains well defined.

Relation to Memory Kernels

A memory-kernel equation has the form

$$\frac{d}{dt}\rho_S(t) = \int_{t_0}^t ds\, \mathcal K_{\mathrm{NZ}}(t,s)\rho_S(s).$$

A TCL equation has the form

$$\frac{d}{dt}\rho_S(t) = \mathcal K_{\mathrm{TCL}}(t)\rho_S(t).$$

Both can describe the same reduced dynamics when the needed objects exist. The difference is where the history is stored:

• in Nakajima–Zwanzig form, the history is explicit in $\rho_S(s)$;
• in TCL form, the history is compressed into $\mathcal K_{\mathrm{TCL}}(t)$.

This is why “time local” should not be read as “memoryless.” A time-local generator can describe recoherence, revivals, and information backflow through time-dependent coefficients.

Projection-Operator TCL Form

Projection methods can also derive TCL equations directly. With a projection $\mathcal P$, the desired structure is

$$\frac{d}{dt}\mathcal P\rho(t) = \mathcal K_{\mathrm{TCL}}(t)\mathcal P\rho(t) + \mathcal I_{\mathrm{TCL}}(t),$$

where $\mathcal I_{\mathrm{TCL}}(t)$ contains initially irrelevant information if $\mathcal Q\rho(t_0)\ne0$. This is the TCL version of the initial-correlation caveat.

For projection-compatible initial states, $\mathcal Q\rho(t_0)=0$, the homogeneous TCL equation is

$$\frac{d}{dt}\mathcal P\rho(t) = \mathcal K_{\mathrm{TCL}}(t)\mathcal P\rho(t).$$

The formal construction expands $\mathcal K_{\mathrm{TCL}}(t)$ in ordered cumulants or powers of the system-bath coupling. This is often more practical than computing a full memory kernel.

Weak-Coupling Expansion

Let the interaction-picture Liouvillian be

$$\mathcal L_I(t) = -\frac{i}{\hbar} [H_I(t),\,\cdot\,].$$

For a coupling-strength parameter $\lambda$, the TCL generator is expanded as

$$\mathcal K_{\mathrm{TCL}}(t) = \sum_{n=1}^{\infty} \lambda^n \mathcal K_n(t).$$

For centered bath operators and a standard factorizing projection, the first-order term often vanishes:

$$\mathcal K_1(t)=0.$$

The second-order TCL equation for the reduced state has the form

$$\frac{d}{dt}\rho_S^{I}(t) = -\frac{\lambda^2}{\hbar^2} \int_{t_0}^t ds\, \operatorname{Tr}_E \left[ H_I(t), \left[ H_I(s), \rho_S^{I}(t)\otimes\rho_E \right] \right].$$

Compare this with the second-order memory-kernel equation, where $\rho_S^{I}(s)$ appears inside the integral. TCL keeps the equation local in the current reduced state but keeps time-dependent coefficients generated by the past integration range.

Markov and Redfield Limits

If bath correlations decay quickly and the upper limit can be extended,

$$\int_{t_0}^t ds \longrightarrow \int_0^\infty d\tau,$$

the second-order TCL generator approaches the usual Born-Markov generator. Before full secularization, this is closely related to the Redfield Equation. After secularization and positivity-preserving assumptions, one obtains the standard weak-coupling Lindblad–GKSL form.

The conceptual chain is:

$$\text{exact reduced map} \to \text{exact TCL if invertible} \to \text{perturbative TCL} \to \text{Born-Markov} \to \text{secular Lindblad}.$$

Each arrow adds assumptions or approximations.

Time-Dependent Rates

Many TCL equations can be written in instantaneous Lindblad-like form:

$$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H(t),\rho] + \sum_\mu \gamma_\mu(t) \left( L_\mu(t)\rho L_\mu^\dagger(t) - \frac12 \{L_\mu^\dagger(t)L_\mu(t),\rho\} \right).$$

If all rates $\gamma_\mu(t)\ge0$ and the operators are well behaved, the dynamics is CP-divisible. If some rates become negative, the evolution is not a standard Markovian Lindblad semigroup. But a negative instantaneous rate does not automatically mean the finite-time map is unphysical. It means the integrated map must be checked.

This is the same distinction used in the Approximation Checklist: instantaneous generator structure and finite-time complete positivity are related but not identical.

Example: Pure Dephasing

A qubit pure-dephasing map may have

$$\rho_{01}(t) = \eta(t)\rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

When $\eta(t)\ne0$, the coherence obeys a time-local equation

$$\frac{d}{dt}\rho_{01}(t) = \frac{\dot\eta(t)}{\eta(t)}\rho_{01}(t).$$

Writing

$$\frac{\dot\eta(t)}{\eta(t)} = -\gamma_\phi(t)-i\Omega(t),$$

one obtains a TCL dephasing generator with time-dependent dephasing rate $\gamma_\phi(t)$ and frequency shift $\Omega(t)$.

If $|\eta(t)|$ temporarily increases, then $\gamma_\phi(t)$ becomes negative. The map may still be completely positive if $|\eta(t)|\le1$ for all times, but it is not CP-divisible during the revival interval.

Example: Amplitude Damping

For a zero-temperature amplitude-damping model, the excited-state amplitude may be written as $G(t)$. When $G(t)\ne0$, the exact time-local generator has coefficients of the form

$$\gamma(t) = -2\,\operatorname{Re} \frac{\dot G(t)}{G(t)}, \qquad S(t) = -2\,\operatorname{Im} \frac{\dot G(t)}{G(t)}.$$

The rate $\gamma(t)$ controls decay or temporary backflow, while $S(t)$ contributes a frequency shift. If $G(t)=0$, the generator becomes singular even though the amplitude-damping map can remain well defined as a finite-time channel.

Common Mistakes

Equating time local with Markovian

TCL equations are local in the current state. Their coefficients can still carry memory of earlier system-environment exchange.

Treating negative rates as automatically unphysical

Negative instantaneous rates indicate failure of CP-divisibility in that representation. The finite-time map must still be tested before declaring the dynamics unphysical.

Ignoring singular times

The formula $\dot\Phi\Phi^{-1}$ requires invertibility. Noninvertible maps can produce singular TCL generators even when $\Phi(t,t_0)$ itself is physical.

Truncating without positivity checks

A perturbative TCL generator can violate positivity when used beyond its regime. As with Redfield equations, one must check the intended state domain and time interval.

Confusing TCL with a Markov approximation

Second-order TCL and second-order memory-kernel equations differ by whether the current state or delayed state appears inside the integral. A Markov approximation is an additional short-memory step.

Exercises

Derive the exact TCL generator

Assume $\rho(t)=\Phi(t,t_0)\rho(t_0)$ and $\Phi(t,t_0)$ is invertible. Derive $\mathcal K_{\mathrm{TCL}}(t)=\dot\Phi(t,t_0)\Phi^{-1}(t,t_0)$.

Solution

Differentiate the map:

$$\dot\rho(t) = \dot\Phi(t,t_0)\rho(t_0).$$

Since $\rho(t)=\Phi(t,t_0)\rho(t_0)$, invertibility gives

$$\rho(t_0) = \Phi^{-1}(t,t_0)\rho(t).$$

Substitute:

$$\dot\rho(t) = \dot\Phi(t,t_0)\Phi^{-1}(t,t_0)\rho(t).$$

Thus the time-local generator is $\mathcal K_{\mathrm{TCL}}(t)=\dot\Phi(t,t_0)\Phi^{-1}(t,t_0)$.

Pure-dephasing rate

If $\eta(t)=e^{-\Gamma(t)}e^{-i\Theta(t)}$, find $\gamma_\phi(t)$ and $\Omega(t)$ in

$$\frac{\dot\eta(t)}{\eta(t)} = -\gamma_\phi(t)-i\Omega(t).$$

Solution

Take the logarithmic derivative:

$$\frac{\dot\eta}{\eta} = -\dot\Gamma(t)-i\dot\Theta(t).$$

Therefore

$$\gamma_\phi(t)=\dot\Gamma(t), \qquad \Omega(t)=\dot\Theta(t).$$

If $\dot\Gamma(t)\lt0$, the coherence magnitude is increasing and the instantaneous dephasing rate is negative.

Compare second-order NZ and TCL

In the second-order weak-coupling equations, what is the main difference between the memory-kernel and TCL forms?

Solution

The memory-kernel form contains the delayed state:

$$\rho_S^{I}(s)$$

inside the integral. The TCL form contains the current state:

$$\rho_S^{I}(t).$$

Both integrate over bath correlation history, but only the memory-kernel equation is explicitly time nonlocal in the reduced state.

Negative rate

A TCL equation has a temporarily negative rate but the finite-time map remains completely positive on the interval of interest. Is the model automatically invalid?

Solution

No. A negative instantaneous rate means the dynamics is not CP-divisible during that interval and should not be called a standard Markovian Lindblad semigroup. It does not by itself prove that the finite-time map is unphysical. The finite-time channel must be checked directly.

References

• F. Shibata, Y. Takahashi, and N. Hashitsume, “A generalized stochastic Liouville equation,” Journal of Statistical Physics 17, 171–187 (1977).
• S. Chaturvedi and F. Shibata, “Time-convolutionless projection operator formalism for elimination of fast variables,” Zeitschrift für Physik B 35, 297–308 (1979).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017).