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.

Choosing a fixed projection does not itself approximate the dynamics. The identity remains exact with its full kernel and initial-state source. Approximations enter when those terms are truncated or discarded, or when an assumed initial state differs from the physical preparation.

Helpful background. Reduced dynamics defines the subsystem state; memory kernels explain time-nonlocal equations. The time-convolutionless method gives a complementary local representation.

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 $\rho(t)$ includes system and environment. The derivation below uses a time-independent linear projection and a finite-dimensional model, or an infinite-dimensional model with sufficiently well-defined propagators and operator domains. A time-dependent projection introduces additional derivative terms.

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 fixed positive normalized 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 choice of variables, not a claim that the total physical state equals $\mathcal P\rho$. In particular, $\mathcal Q\rho$ need not be positive and is not a second density matrix. More elaborate correlated projections can retain additional 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

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

Formally,

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

where $\mathcal T$ denotes time ordering. On the range of $\mathcal Q$, $\mathcal Q$ is the identity. This convention makes explicit that $\mathcal G_Q$ propagates only the eliminated operator components; it is not generally a positive quantum channel.

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:

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

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

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

in the interaction picture with respect to the free Hamiltonian; there is no remaining free $\mathcal L_0$ in this picture. Since the two outer interaction factors already contribute $\lambda^2$, replace $\mathcal G_Q(t,s)$ by its zeroth-order value $\mathcal Q$ to obtain

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

For the standard projection, centered stationary bath operators give $\mathcal P\mathcal L_I(t)\mathcal P=0$, so the internal $\mathcal Q$ may be replaced by the identity between the two interaction factors. If also $\mathcal Q\rho(t_0)=0$, 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.

Worked example: eliminating one correlation

Let two qubits evolve with $H=\hbar gZ_S\otimes Z_E$, and define

$$x(t)=\langle X_S\otimes I_E\rangle_t, \qquad y(t)=\langle Y_S\otimes Z_E\rangle_t.$$

The Heisenberg commutators give $\dot x=-2gy$ and $\dot y=2gx$. The first is a local system expectation; the second involves a system–environment correlation. Integrating the second equation and substituting into the first eliminates that correlation:

$$\dot x(t)=-2g\,y(0)-4g^2\int_0^t x(s)\,ds.$$

This is an exact scalar memory equation with a constant kernel and an initial-correlation source. For an initially factorized state with $\langle Z_E\rangle=0$, $y(0)=0$. If $x(0)=1$, its solution is $x(t)=\cos(2gt)$: coherence disappears and returns because the environment has not forgotten.

More generally $x(t)=x(0)\cos(2gt)-y(0)\sin(2gt)$. Dropping a nonzero initial $y(0)$ changes the answer even though the kernel is exact. This small example displays the same elimination mechanism as the full operator equation, without evaluating a large projected propagator.

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.

Exact projection versus an approximate kernel

A fixed projection can place slow variables in the eliminated sector and thereby make the exact kernel difficult to compute, but it does not invalidate the identity. Weak-coupling truncation, omitted initial sources, short-memory replacement, and secular averaging are separate approximations.

The exact unitary model gives a physical reduced trajectory for each physical joint initial state. It gives a universal CPTP map on all system inputs when the environment preparation is fixed and uncorrelated. Initial correlated families need their own preparation-domain analysis. Neither an approximate kernel nor the projection notation alone proves complete positivity.

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

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• H. Grabert, Projection Operator Techniques in Nonequilibrium Statistical Mechanics, Springer (1982).
• S. Nakajima, “On quantum theory of transport phenomena,” Progress of Theoretical Physics 20, 948–959 (1958). DOI.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• R. Zwanzig, “Ensemble method in the theory of irreversibility,” Journal of Chemical Physics 33, 1338–1341 (1960). DOI.