Nakajima–Zwanzig Projection

The Nakajima–Zwanzig projection method derives an exact time-nonlocal equation for selected variables of a larger dynamical system. In open quantum systems, those selected variables are usually the reduced system state, while the eliminated variables contain environmental degrees of freedom and system-environment correlations.

The output has the memory-kernel form

$$\frac{d}{dt}\mathcal P\rho(t) = \mathcal P\mathcal L(t)\mathcal P\rho(t) + \int_{t_0}^t ds\, \mathcal K(t,s)\mathcal P\rho(s) + \mathcal I(t).$$

Here $\mathcal P$ is a projection superoperator, $\mathcal L$ is the total Liouvillian, $\mathcal K$ is the memory kernel, and $\mathcal I(t)$ is an inhomogeneous term determined by the initially irrelevant part of the state.

The method is exact until one approximates the kernel, chooses a simplified projection, assumes a factorized initial state, or takes a Markov limit.

For the time-local counterpart, see Time-Convolutionless Master Equations.

Liouville Equation

Let the total state obey

$$\frac{d}{dt}\rho(t) = \mathcal L(t)\rho(t),$$

where, for Hamiltonian evolution,

$$\mathcal L(t)X = -\frac{i}{\hbar}[H(t),X].$$

The total state $\rho(t)$ may include both system and environment. The goal is to derive a closed equation for a relevant part of $\rho(t)$.

Projection Superoperators

A projection superoperator satisfies

$$\mathcal P^2=\mathcal P.$$

Define the complementary projection

$$\mathcal Q = \mathcal I-\mathcal P,$$

so that

$$\mathcal Q^2=\mathcal Q, \qquad \mathcal P\mathcal Q = \mathcal Q\mathcal P = 0.$$

For a system $S$ and environment $E$, the standard factorizing projection is

$$\mathcal P X = \operatorname{Tr}_E(X)\otimes\rho_E,$$

where $\rho_E$ is a chosen reference environment state. Then $\mathcal P\rho(t)$ stores the reduced system state together with the fixed reference bath:

$$\mathcal P\rho(t) = \rho_S(t)\otimes\rho_E.$$

The complementary part

$$\mathcal Q\rho(t) = \rho(t)-\mathcal P\rho(t)$$

contains whatever is not represented by $\rho_S(t)\otimes\rho_E$: system-environment correlations, bath deviations, and other discarded variables.

The projection is a modeling choice. More elaborate correlated projections can keep additional relevant variables.

Splitting the Dynamics

Apply $\mathcal P$ and $\mathcal Q$ to the Liouville equation:

$$\frac{d}{dt}\mathcal P\rho(t) = \mathcal P\mathcal L(t)\mathcal P\rho(t) + \mathcal P\mathcal L(t)\mathcal Q\rho(t),$$

and

$$\frac{d}{dt}\mathcal Q\rho(t) = \mathcal Q\mathcal L(t)\mathcal P\rho(t) + \mathcal Q\mathcal L(t)\mathcal Q\rho(t).$$

The first equation is not closed because it contains $\mathcal Q\rho(t)$. The second equation tells how the irrelevant part is driven by the relevant part and by its own dynamics.

Solving the Irrelevant Part

Define the propagator inside the $\mathcal Q$ subspace by

$$\frac{\partial}{\partial t} \mathcal G_Q(t,s) = \mathcal Q\mathcal L(t)\mathcal G_Q(t,s), \qquad \mathcal G_Q(s,s)=\mathcal I.$$

Formally,

$$\mathcal G_Q(t,s) = \mathcal T \exp \left[ \int_s^t du\, \mathcal Q\mathcal L(u) \right],$$

where $\mathcal T$ denotes time ordering.

The solution of the $\mathcal Q$ equation is

$$\mathcal Q\rho(t) = \mathcal G_Q(t,t_0)\mathcal Q\rho(t_0) + \int_{t_0}^t ds\, \mathcal G_Q(t,s) \mathcal Q\mathcal L(s)\mathcal P\rho(s).$$

The first term propagates initially irrelevant information. The second term is the irrelevant part generated after $t_0$ by coupling to the relevant variables.

Exact Nakajima–Zwanzig Equation

Substitute the solution for $\mathcal Q\rho(t)$ into the $\mathcal P$ equation:

$$\begin{aligned} \frac{d}{dt}\mathcal P\rho(t) =& \mathcal P\mathcal L(t)\mathcal P\rho(t) \\ & + \int_{t_0}^t ds\, \mathcal P\mathcal L(t) \mathcal G_Q(t,s) \mathcal Q\mathcal L(s) \mathcal P\rho(s) \\ & + \mathcal P\mathcal L(t) \mathcal G_Q(t,t_0) \mathcal Q\rho(t_0). \end{aligned}$$

This is the Nakajima–Zwanzig equation.

The memory kernel is

$$\mathcal K(t,s) = \mathcal P\mathcal L(t) \mathcal G_Q(t,s) \mathcal Q\mathcal L(s) \mathcal P,$$

and the inhomogeneous term is

$$\mathcal I(t) = \mathcal P\mathcal L(t) \mathcal G_Q(t,t_0) \mathcal Q\rho(t_0).$$

If $\mathcal Q\rho(t_0)=0$, the inhomogeneous term vanishes. This is the projection-operator version of a factorized or projection-compatible initial condition. For the preparation-domain issue behind this term, see Initial Correlations.

Reduced Equation for the System

With the standard projection

$$\mathcal P\rho(t)=\rho_S(t)\otimes\rho_E,$$

the $\mathcal P$ equation can be traced over $E$ to obtain an equation for $\rho_S(t)$. Its general form is the memory-kernel equation described in Memory Kernels:

$$\frac{d}{dt}\rho_S(t) = \int_{t_0}^t ds\, \mathcal K_S(t,s)\rho_S(s) + \mathcal I_S(t),$$

possibly with an additional time-local term if $\mathcal P\mathcal L\mathcal P$ does not vanish.

In an interaction picture with centered bath operators, the $\mathcal P\mathcal L_I\mathcal P$ term often vanishes. Then the leading reduced dynamics is carried by the memory kernel.

Second-Order Weak-Coupling Kernel

Let

$$\mathcal L(t) = \mathcal L_0+\lambda\mathcal L_I(t)$$

in the interaction picture with respect to the free dynamics. To second order in $\lambda$, the projected kernel is approximated by

$$\mathcal K^{(2)}(t,s) = \lambda^2 \mathcal P\mathcal L_I(t) \mathcal Q\mathcal L_I(s) \mathcal P.$$

For the standard factorizing projection and centered bath operators, this gives the familiar Born memory term:

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

The Markov Approximation then replaces $\rho_S^{I}(s)$ by $\rho_S^{I}(t)$ under short-memory assumptions. The Redfield Equation is a common time-local weak-coupling result before full secularization.

What Is Exact and What Is Approximate

The exact projection identity is not itself a weak-coupling approximation. The approximations enter through later choices:

• choosing a projection that discards important variables;
• assuming $\mathcal Q\rho(t_0)=0$;
• truncating the kernel in powers of the interaction;
• replacing the $\mathcal Q$ propagator by a simpler propagator;
• taking a Markov limit;
• applying secular or coarse-graining approximations;
• ignoring positivity constraints of the resulting reduced map.

This distinction matters. A memory-kernel equation may be exact in principle and still unusable in practice unless the kernel can be computed or approximated consistently.

Physical Interpretation

The relevant part $\mathcal P\rho$ is the information retained in the reduced model. The irrelevant part $\mathcal Q\rho$ stores everything else.

The memory kernel says:

the system at time s creates irrelevant correlations;
those correlations evolve in the Q subspace;
later they feed back into the relevant state at time t.

This is why the kernel contains

$$\mathcal P\mathcal L(t) \mathcal G_Q(t,s) \mathcal Q\mathcal L(s) \mathcal P.$$

Read from right to left:

• $\mathcal P$ selects the relevant state at time $s$;
• $\mathcal Q\mathcal L(s)$ sends part of it into irrelevant variables;
• $\mathcal G_Q(t,s)$ propagates those variables;
• $\mathcal P\mathcal L(t)$ brings their influence back into the relevant sector.

Projection Choices

The standard projection

$$\mathcal P X = \operatorname{Tr}_E(X)\otimes\rho_E$$

is natural for weak coupling to a large bath near a fixed reference state. It is not mandatory.

Other projections may keep:

• several bath sectors;
• classical populations of environmental states;
• slow collective coordinates;
• correlated system-environment reference states;
• relevant observables rather than the full reduced density operator.

A better projection can shorten the memory kernel by keeping slow variables explicitly. A poor projection can create long memory because it throws away variables that should have remained part of the effective system.

Common Mistakes

Calling Nakajima–Zwanzig an approximation

The projection identity is exact once $\mathcal P$ is chosen. Approximating or truncating the kernel is a separate step.

Dropping the inhomogeneous term without checking the initial state

The term $\mathcal I(t)$ vanishes only when $\mathcal Q\rho(t_0)=0$. Initial correlations or bath deviations can make it important.

Confusing projection with measurement

$\mathcal P$ is a mathematical projection in operator space. It is not a physical projective measurement unless an additional measurement model is specified.

Assuming the kernel guarantees complete positivity

The exact kernel from a microscopic unitary model gives physical reduced dynamics. An approximate kernel may not. Complete positivity must be checked after approximation.

Using a projection that discards slow variables

If an environmental mode keeps returning information to the system, putting it in $\mathcal Q$ creates long memory. Including it in the system may produce a simpler time-local model.

Exercises

Standard projection is idempotent

Show that

$$\mathcal P X = \operatorname{Tr}_E(X)\otimes\rho_E$$

satisfies $\mathcal P^2=\mathcal P$ when $\operatorname{Tr}_E\rho_E=1$.

Solution

Apply $\mathcal P$ twice:

$$\mathcal P^2 X = \mathcal P \left[ \operatorname{Tr}_E(X)\otimes\rho_E \right].$$

Taking the environment trace gives

$$\operatorname{Tr}_E \left[ \operatorname{Tr}_E(X)\otimes\rho_E \right] = \operatorname{Tr}_E(X)\operatorname{Tr}_E\rho_E = \operatorname{Tr}_E(X).$$

Thus

$$\mathcal P^2X = \operatorname{Tr}_E(X)\otimes\rho_E = \mathcal P X.$$

Derive the split equations

Starting from $\dot\rho=\mathcal L\rho$ and $\mathcal I=\mathcal P+\mathcal Q$, derive the coupled equations for $\mathcal P\rho$ and $\mathcal Q\rho$.

Solution

Write

$$\rho = \mathcal P\rho+\mathcal Q\rho.$$

Apply $\mathcal P$:

$$\frac{d}{dt}\mathcal P\rho = \mathcal P\mathcal L(\mathcal P\rho+\mathcal Q\rho) = \mathcal P\mathcal L\mathcal P\rho + \mathcal P\mathcal L\mathcal Q\rho.$$

Apply $\mathcal Q$:

$$\frac{d}{dt}\mathcal Q\rho = \mathcal Q\mathcal L\mathcal P\rho + \mathcal Q\mathcal L\mathcal Q\rho.$$

Inhomogeneous term

Under what condition does the inhomogeneous term

$$\mathcal P\mathcal L(t)\mathcal G_Q(t,t_0)\mathcal Q\rho(t_0)$$

vanish?

Solution

It vanishes if

$$\mathcal Q\rho(t_0)=0.$$

For the standard factorizing projection, this means the initial state lies in the range of $\mathcal P$, such as $\rho(t_0)=\rho_S(t_0)\otimes\rho_E$ for the chosen reference environment state.

Read the kernel

Interpret the factors in

$$\mathcal K(t,s) = \mathcal P\mathcal L(t) \mathcal G_Q(t,s) \mathcal Q\mathcal L(s) \mathcal P.$$

Solution

Read the expression from right to left. $\mathcal P$ selects the relevant state at time $s$. $\mathcal Q\mathcal L(s)$ sends part of the dynamics into irrelevant variables. $\mathcal G_Q(t,s)$ propagates those variables inside the irrelevant subspace. $\mathcal P\mathcal L(t)$ maps their later influence back into the relevant sector.

References

• S. Nakajima, “On quantum theory of transport phenomena,” Progress of Theoretical Physics 20, 948–959 (1958).
• R. Zwanzig, “Ensemble method in the theory of irreversibility,” Journal of Chemical Physics 33, 1338–1341 (1960).
• H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer, 1982.
• 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.