Open Quantum Systems

An open quantum system is a chosen subsystem whose dynamics is influenced by degrees of freedom outside the retained description. The total system and environment may obey ordinary unitary quantum mechanics while the reduced state of the subsystem decoheres, dissipates energy, acquires noise, or retains memory of earlier interactions.

The exact starting point is simple:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U(t,t_0)\rho_{SE}(t_0)U^\dagger(t,t_0) \right].$$

The difficulty is not writing this formula. It is turning it into a useful closed equation for $\rho_S(t)$ while stating which correlations, timescales, spectral structures, and initial conditions have been neglected.

This chapter organizes that passage from exact composite dynamics to channels, Redfield equations, Markovian generators, and memory-sensitive descriptions.

System and Environment Are Modeling Choices

The boundary between system and environment is chosen according to the question. A cavity mode can be the system in one model and part of a structured reservoir in another. A vibrational mode can be traced out, promoted into the system, or absorbed into a reaction coordinate.

The words used outside the boundary carry different assumptions.

Term	Minimal meaning	Common additional assumption
environment	degrees of freedom excluded from the system	none by definition
bath	large or uncontrolled environment with a reference state	weak disturbance by the system
reservoir	bath that exchanges energy or particles	nearly fixed temperature or chemical potential
noise source	degrees of freedom represented through fluctuations	may be classical or quantum
record	degrees of freedom correlated with a measured alternative	may be read, ignored, or coarse-grained

Baths, Reservoirs, and Environments makes these distinctions precise. Size alone does not guarantee thermality, stationarity, weak coupling, or short memory.

Reading Path

Read this page	Use it for
Reduced Dynamics	Deriving subsystem evolution by tracing a larger unitary state.
System–Bath Hamiltonians	Specifying the microscopic split, coupling operators, and common models.
Baths, Reservoirs, and Environments	Stating what assumptions accompany each environment role.
Spectral Densities	Replacing mode sums by frequency-dependent coupling functions and identifying structure.
Born Approximation	Closing weak-coupling equations while keeping the bath near a reference state.
Markov Approximation	Replacing relevant history by present-time reduced dynamics under short-memory conditions.
Secular Approximation	Averaging rapidly rotating cross terms between separated Bohr frequencies.
Redfield Equation	Retaining nonsecular weak-coupling terms and assessing positivity risk.
Memory Kernels	Writing explicitly time-nonlocal reduced equations.
Nakajima–Zwanzig Projection	Deriving exact projected equations with memory and inhomogeneous terms.
Time-Convolutionless Master Equations	Encoding memory in a time-local, time-dependent generator.
Initial Correlations	Handling preparation dependence, assignment maps, and compatibility domains.

For the standard weak-coupling derivation, follow

$$\begin{gathered} \text{Hamiltonian model} \\ \longrightarrow\text{Born approximation}, \\ \text{Markov approximation} \\ \longrightarrow\text{Redfield equation}, \\ \text{secular approximation} \\ \longrightarrow \text{GKSL generator}. \end{gathered}$$

For memory-sensitive work, begin with reduced dynamics and then compare memory kernels, Nakajima–Zwanzig projection, time-convolutionless equations, and non-Markovian diagnostics.

Exact Reduced Dynamics

The total state satisfies the Liouville–von Neumann equation

$$\frac{d\rho_{SE}}{dt} = -\frac{i}{\hbar} [H_{SE},\rho_{SE}].$$

Tracing out $E$ gives an exact reduced state, but usually not a closed differential equation in $\rho_S$ alone. The derivative can depend on system-environment correlations that are absent from the reduced density operator.

If the initial state is factorized,

$$\rho_{SE}(t_0) = \rho_S(t_0)\otimes\eta_E,$$

with one fixed environment state $\eta_E$, then the finite-time reduced map

$$\begin{aligned} \Phi_{t,t_0}(\rho_S) &= \operatorname{Tr}_E \Bigl[ U(t,t_0) \\ &\quad{} (\rho_S\otimes\eta_E) U^\dagger(t,t_0) \Bigr] \end{aligned}$$

is completely positive and trace preserving. It need not be unitary, invertible, divisible, time homogeneous, or Markovian.

This distinction separates an exact finite-time channel from an approximate master equation. A family of valid channels can still contain memory and fail to have a regular time-local generator at some times.

Kraus Form from an Environment

Let the fixed environment state have spectral decomposition

$$\eta_E = \sum_\beta q_\beta|\beta\rangle\langle\beta|.$$

Choose an environment output basis $\{|\alpha\rangle\}$. Then

$$K_{\alpha\beta}(t) = \sqrt{q_\beta} \langle\alpha|U(t,t_0)|\beta\rangle$$

acts on the system, and

$$\Phi_{t,t_0}(\rho_S) = \sum_{\alpha,\beta} K_{\alpha\beta}(t) \rho_S K_{\alpha\beta}^\dagger(t).$$

This construction explains why factorized initial states naturally yield channels. It does not imply that the Kraus labels identify uniquely real environment events; the output basis and representation are not unique.

Microscopic Hamiltonian Structure

A standard open-system Hamiltonian is

$$\begin{aligned} H &=H_S\otimes I_E +I_S\otimes H_E +H_I, \\ H_I &= \sum_\alpha S_\alpha\otimes B_\alpha. \end{aligned}$$

The system operators $S_\alpha$ identify which observables the environment can monitor or change. The bath operators $B_\alpha$ determine fluctuations and response in the environment.

The split is not unique. Mean bath forces can be moved into a renormalized system Hamiltonian by writing

$$B_\alpha = \langle B_\alpha\rangle_E I_E + \widetilde B_\alpha, \qquad \langle\widetilde B_\alpha\rangle_E=0.$$

Counterterms, Lamb shifts, rotating frames, and reaction-coordinate mappings can all change which terms are called system, bath, and interaction. Every derivation should state the chosen partition.

Coupling Structure Predicts the Process

The commutator with $H_S$ is an immediate diagnostic.

• If a dominant $S_\alpha$ commutes with $H_S$, the coupling can randomize phase without changing system energy eigenstate populations. This is the pure-dephasing pattern.
• If $S_\alpha$ has off-diagonal components in the energy basis, the environment can induce transitions and energy relaxation.
• If several nearly degenerate transitions share bath correlations, their interference can make secularization delicate.
• If the environment contains a narrow resonant mode, it may exchange excitation coherently and should sometimes be promoted into the system.

The System–Bath Hamiltonians page works through spin-boson, oscillator-bath, atom-field, and central-spin examples before any master-equation approximation is imposed.

Correlation Functions and Spectral Densities

For a stationary reference bath, two-point functions have the form

$$C_{\alpha\beta}(\tau) = \operatorname{Tr}_E \left[ \widetilde B_\alpha(\tau) \widetilde B_\beta(0) \eta_E \right].$$

Their decay defines a bath correlation time $\tau_B$. Their Fourier transforms determine transition rates, noise asymmetry, and detailed-balance relations in weak-coupling limits.

For a bosonic mode continuum, a schematic spectral density is

$$J(\omega) = \sum_k |g_k|^2 \delta(\omega-\omega_k).$$

The spectral density summarizes mode frequencies and coupling strengths, not the complete bath state. Temperature occupation factors and convention-dependent prefactors enter correlation functions separately.

Ohmic, sub-Ohmic, and super-Ohmic labels describe low-frequency scaling. They do not by themselves establish Markovianity. Cutoffs, resonances, gaps, and low-frequency weight can dominate the actual memory structure. See Spectral Densities and Correlation Functions.

The Born Approximation

In the interaction picture, the exact state obeys

$$\frac{d\rho_{SE}^I(t)}{dt} = -\frac{i}{\hbar} [H_I^I(t),\rho_{SE}^I(t)].$$

Iterating once produces a second-order equation containing the full joint state at earlier times. The Born approximation closes that equation by replacing the joint state inside the second-order term with

$$\rho_{SE}^I(s) \approx \rho_S^I(s)\otimes\eta_E.$$

This does not assert that the system and bath never become correlated. It says that, to the perturbative order retained, those correlations need not be evolved as independent variables and the bath remains near the chosen reference state.

Typical requirements include weak dimensionless coupling, modest bath backaction, an appropriate reference state, and times short enough that neglected higher orders do not accumulate. Strong coupling, a small environment, a resonant mode, low temperature, or a correlated equilibrium preparation can invalidate the closure.

The Born approximation is distinct from the Born rule and distinct from the Born approximation in scattering theory.

The Markov Approximation

After Born closure, a typical reduced equation contains a history integral:

$$\frac{d\rho_S^I(t)}{dt} = \int_{t_0}^{t}ds\, \mathcal K_2(t,s)\rho_S^I(s).$$

If bath correlations decay on a time $\tau_B$ much shorter than the reduced evolution time $\tau_R$, then $\rho_S^I(s)$ changes little over the interval where the kernel matters. One replaces the delayed state by the present state in the appropriate interaction picture and often extends the upper integration limit.

Schematically,

$$\begin{aligned} \rho_S^I(t-\tau) &\approx \rho_S^I(t), \\ \int_0^t d\tau &\longrightarrow \int_0^\infty d\tau. \end{aligned}$$

The Markov approximation is therefore a controlled timescale claim, not a synonym for “the environment is large.” Structured spectra, slowly decaying correlations, feedback, recycled ancillas, and finite reservoirs can retain memory even when the environment has many degrees of freedom.

Markov approximation also does not automatically imply GKSL form. The resulting time-local Redfield equation may still contain nonsecular terms and may not preserve positivity outside its validity regime.

Redfield and Secular Dynamics

After Born and Markov steps, decomposing system operators into Bohr-frequency components gives terms oscillating as

$$e^{i(\omega'-\omega)t}.$$

Keeping these cross-frequency terms leads to a common time-local Redfield equation. It can capture coherence transfer and interference between nearly degenerate transitions that full secularization would remove. It is widely useful, but it is not generically in GKSL form and can produce nonpositive states if used beyond the weak-coupling, coarse-grained regime.

The secular approximation averages terms with well-separated frequencies:

$$|\omega-\omega'| \gg \Gamma,$$

where $\Gamma$ represents relevant dissipative rates. After resolving exact degeneracies and positive rate matrices correctly, secularization commonly yields a GKSL generator.

Choice	Advantage	Main risk
full secularization	transparent GKSL structure and complete positivity	removes physically relevant near-degenerate coherence coupling
nonsecular Redfield	retains more frequency interference	positivity can fail outside its regime
partial secularization	can retain selected near-degenerate blocks	requires an explicit coarse-graining criterion

Secularization cannot repair a bad Born or Markov approximation. It addresses fast oscillatory cross terms, not strong coupling or long bath memory.

Memory-Kernel Equations

An explicitly time-nonlocal equation has the form

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

The superoperator kernel $\mathcal K(t,s)$ carries history dependence. The inhomogeneous term $\mathcal I(t)$ can encode initially discarded correlations or components outside the chosen projected subspace.

A fitted memory kernel is not automatically physical. The resulting finite-time maps must still preserve trace, Hermiticity, and positivity on the relevant preparation domain. Approximate kernels can violate these properties even when the exact projected equation is physical.

Memory Kernels treats Laplace transforms, exponential kernels, auxiliary-mode embeddings, and positivity caveats.

Nakajima–Zwanzig Projection

Let $\mathcal P$ project joint operators onto the retained variables and let $\mathcal Q=1-\mathcal P$. For the standard open-system choice,

$$\mathcal PX = \operatorname{Tr}_E(X)\otimes\eta_E.$$

Splitting the Liouville equation into $\mathcal P$ and $\mathcal Q$ parts, formally solving the discarded part, and substituting back gives an exact projected equation with an instantaneous term, a memory kernel, and an initial-correlation term.

The method itself is not an approximation. Approximations enter when choosing a projection, truncating the kernel, expanding in coupling strength, or simplifying the memory integral. A poor projection can discard a slow collective coordinate and make the remaining kernel unnecessarily long-lived.

Nakajima–Zwanzig Projection is the canonical derivation page.

Time-Convolutionless Equations

When the reduced map is invertible, an exact time-local generator can be defined by

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

The state then satisfies

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

This equation is local in time but can describe memoryful dynamics. History is encoded in time-dependent coefficients, temporarily negative rates, or singularities when the map loses invertibility. Therefore

$$\text{time local} \quad\ne\quad \text{Markovian}.$$

Perturbative TCL generators can be convenient because they avoid storing the full history. Truncation still requires finite-time positivity checks. See Time-Convolutionless Master Equations.

Initial Correlations

If $\rho_{SE}(t_0)$ is correlated, the same reduced state $\rho_S(t_0)$ can be compatible with different joint states and therefore different future reduced states. A single map acting on arbitrary $\rho_S(t_0)$ is then not generally defined without an assignment rule that specifies the compatible joint preparation.

This does not make the exact evolution unphysical. It changes the domain of the reduced description. Important tools include:

• assignment maps from allowed reduced states to joint preparations;
• compatibility domains on which the reduced evolution is well defined;
• inhomogeneous terms in projection equations;
• slippage or preparation corrections in approximate weak-coupling models.

Correlated thermal equilibrium, prior measurements, strong coupling, and promoted reaction coordinates are common sources. Initial Correlations is the canonical home for the complete-positivity caveat.

Choosing a Description

Physical regime or question	Useful starting description
exact finite environment	direct unitary propagation and partial trace
weak coupling, short bath memory, separated transitions	Born–Markov–secular GKSL equation
weak coupling with important near degeneracies	Redfield or partial-secular treatment
explicit history dependence	memory-kernel or Nakajima–Zwanzig equation
time-local numerics with memory in coefficients	TCL equation
narrow environmental resonance	pseudomode or enlarged-system model
strong coupling to a collective bath coordinate	reaction-coordinate mapping or nonperturbative method
correlated initial preparation	assignment-domain or inhomogeneous projected dynamics

No method is universally most accurate. An enlarged Markovian model can outperform a complicated reduced memory kernel if it promotes the relevant slow mode into the system. Conversely, a simple weak-coupling generator can be more trustworthy than an overfit non-Markovian model when timescales are cleanly separated.

Approximation Checklist

Before trusting a reduced master equation, state and test:

1. the system-environment boundary and input preparation;
2. the coupling-strength parameter and perturbative order;
3. the bath reference state and whether it is stationary;
4. the correlation time $\tau_B$ and reduced evolution time $\tau_R$;
5. spectral cutoffs, gaps, resonances, and low-frequency weight;
6. relevant Bohr-frequency separations and dissipative rates;
7. whether initial correlations or initial transients matter;
8. whether the approximation preserves trace, Hermiticity, and positivity;
9. whether the result is compared against an exact limit or converged numerical benchmark;
10. the time interval on which the approximation is claimed to hold.

The Approximation Checklist provides a reusable version for calculations and simulations.

Canonical Boundaries

This chapter owns microscopic reduced-dynamics setup and the approximation methods that connect it to effective equations.

• Quantum Channels and Noise owns abstract finite-time maps and standard channel models.
• Decoherence and the Classical Transition owns interference suppression, preferred structures, and classical-record questions.
• Markovian Master Equations owns GKSL generators, semigroups, detailed balance, and standard rate equations.
• Non-Markovian Dynamics owns competing diagnostics, information backflow, divisibility, and constructive non-Markovian methods.
• Quantum Noise, Dissipation, and Baths owns noise spectra, fluctuation–dissipation relations, Langevin equations, input-output theory, and canonical bath models.
• Many-Body and QFT volumes own full nonequilibrium Green-function, Keldysh, and field-theoretic transport methods.

Cross-link rather than duplicating those canonical treatments.

Common Mistakes

• Treating the system-environment split as unique or physically automatic.
• Assuming every environment is thermal, stationary, or Markovian.
• Confusing the Born approximation with the Born rule.
• Saying Born closure means no system-bath correlations ever form.
• Equating a large reservoir with short memory.
• Assuming a Markov approximation automatically gives GKSL form.
• Applying full secularization across near-degenerate transitions.
• Calling the Nakajima–Zwanzig method itself an approximation.
• Equating a time-local equation with Markovian dynamics.
• Treating every negative time-local rate as an unphysical finite-time map.
• Ignoring initial correlations while using a correlated equilibrium state.
• Trusting a fitted memory kernel without positivity and convergence checks.
• Comparing spectral densities without matching conventions and cutoffs.

Exercises

Channel from a factorized environment

Using the Kraus construction above, prove that the reduced map from a fixed factorized environment state is trace preserving.

Solution

The Kraus operators are

$$K_{\alpha\beta} = \sqrt{q_\beta} \langle\alpha|U|\beta\rangle.$$

Sum their products:

$$\begin{aligned} \sum_{\alpha,\beta} K_{\alpha\beta}^\dagger K_{\alpha\beta} &= \sum_{\alpha,\beta} q_\beta \langle\beta|U^\dagger|\alpha\rangle \langle\alpha|U|\beta\rangle \\ &= \sum_\beta q_\beta \langle\beta|U^\dagger U|\beta\rangle \\ &= \left(\sum_\beta q_\beta\right)I_S =I_S. \end{aligned}$$

Completeness of the output basis, unitarity of $U$, and normalization of $\eta_E$ give trace preservation.

Centering a bath operator

For $H_I=S\otimes B$, write $B=\langle B\rangle_E I+\widetilde B$ with $\langle\widetilde B\rangle_E=0$. Show how the mean term changes the system Hamiltonian.

Solution

Substitution gives

$$H_I = \langle B\rangle_E S\otimes I_E + S\otimes\widetilde B.$$

The first term acts only on the system and can be absorbed into

$$H_S' = H_S+\langle B\rangle_E S.$$

The remaining interaction has zero bath mean. Removing the first-order mean force makes the fluctuation expansion and Lamb-shift bookkeeping clearer.

Testing the Markov step

A bath correlation decays as $C(\tau)=C_0e^{-\tau/\tau_B}$, while the reduced state changes appreciably over $\tau_R$. State the basic timescale requirement for replacing $\rho_S(t-\tau)$ by $\rho_S(t)$ in the memory integral.

Solution

The integral receives most of its weight from $\tau$ of order $\tau_B$. The reduced state is nearly constant over that interval when

$$\tau_B\ll\tau_R.$$

This is necessary for the simple short-memory replacement but may not be sufficient. Initial transients, strong coupling, structured resonances, low-frequency tails, and the chosen interaction picture must also be checked.

Secular or nonsecular

Two Bohr frequencies differ by $|\omega-omega'|=0.2\Gamma$, where $\Gamma$ is the dissipative scale. Should their cross terms be removed by full secularization?

Solution

No clear separation exists because

$$|\omega-\omega'| \not\gg \Gamma.$$

The cross term rotates no faster than the dissipative evolution. Full secularization can erase physically relevant coherence transfer between the nearly degenerate sectors. A Redfield, partial-secular, or block-secular treatment should be considered and checked against positivity and a more accurate model.

Time local does not mean Markovian

Suppose an invertible reduced map $\Phi_t$ has generator $\mathcal K(t)=\dot\Phi_t\Phi_t^{-1}$. Explain why the existence of this time-local equation does not prove Markovianity.

Solution

The exact map up to time $t$ already contains the full influence of the earlier system-environment history. Multiplying by $\Phi_t^{-1}$ repackages that history into the time-dependent generator. The generator can have temporarily negative decay rates, singularities, or intermediate maps that are not completely positive. Time locality is a representation property; Markovianity requires an additional criterion such as semigroup structure, CP divisibility, or a justified short-memory approximation.

Cross-Links

• Applications and Experimental Platforms
• Computational Notebooks
• Reference
• Reduced Dynamics
• System–Bath Hamiltonians
• Baths, Reservoirs, and Environments
• Spectral Densities
• Born Approximation
• Markov Approximation
• Secular Approximation
• Redfield Equation
• Memory Kernels
• Nakajima–Zwanzig Projection
• Time-Convolutionless Master Equations
• Initial Correlations
• Lindblad–GKSL Equation
• Non-Markovian Dynamics
• Approximation Checklist
• Quantum Noise, Dissipation, and Baths

References

• 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).
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004).
• E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).
• A. G. Redfield, “On the theory of relaxation processes,” IBM Journal of Research and Development 1, 19–31 (1957).
• 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).