Open Quantum Systems

Open-system theory starts with joint dynamics and asks which reduced equation accurately predicts the subsystem. The main work is identifying the information removed by each approximation: bath backaction, time history, rapidly oscillating terms, or initial correlations. This chapter separates those choices so that a convenient equation is not mistaken for an exact description.

Required background. Quantum Channels and Noise and Decoherence explain finite-time maps and reduced interference. Use Partial Trace and the Liouville–von Neumann Equation for the joint-to-reduced calculation.

Begin with the Hamiltonian and preparation

Begin with Closed vs Open Quantum Systems to choose the system boundary and distinguish reduced, conditional, and averaged states. Its two-qubit example shows why reduced dynamics can lose and regain purity.

Write the total Hamiltonian as

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

The system operators identify which energy sectors can couple; the bath operators and bath state determine the fluctuations and response. If all coupling operators commute with $H_S$, the interaction cannot transfer population between distinct system-energy eigenspaces. Couplings with off-diagonal energy components can induce transitions.

The exact state is $\rho_S(t)=\operatorname{Tr}_E[U_t\rho_{SE}(0)U_t^\dagger]$. It becomes a channel on arbitrary system inputs when the initial preparation is $\rho_S\otimes\eta_E$ with the same fixed $\eta_E$ for every input. No short-memory or weak-coupling assumption is needed for that channel statement.

Read Reduced Dynamics, System–Bath Hamiltonians, and Baths, Reservoirs, and Environments before selecting an approximation. A large environment need not be thermal, stationary, or rapidly forgetting.

Born, Markov, and secular are three distinct steps

Step	What is simplified	What must be small or well separated
Born Approximation	In a weak-coupling expansion, approximate the joint state inside the retained kernel by a system state times a reference bath state.	Corrections from bath disturbance and higher coupling orders on the stated time interval
Markov Approximation	Replace the delayed reduced state by its present value in the appropriate interaction picture and, when justified, extend the memory integral.	Bath correlation time $\tau_B$ relative to the reduced relaxation time $\tau_R$
Secular Approximation	Average cross terms rotating at different Bohr-frequency differences.	Inverse frequency differences relative to a coarse-graining time that remains short compared with relaxation

Born closure does not say that system–bath correlations never form. Their leading influence is precisely what produces reduced dissipation. Markov approximation does not automatically guarantee complete positivity, and secularization cannot repair strong coupling or a long-lived bath memory.

For a full secular treatment of two distinct transition frequencies, a useful window is

$$\tau_B\ll\Delta t\ll\tau_R, \qquad |\omega-\omega'|\Delta t\gg1.$$

If the transitions are too close, such a window may not exist. Exactly degenerate transitions must be handled within their common frequency block. Redfield Equation retains nonsecular interference, with corresponding positivity limitations; a justified partial-secular or coarse-grained treatment can sometimes retain the needed near-degenerate structure.

Bath spectra determine which timescales are plausible

Spectral Densities connects mode frequencies and coupling strengths to bath correlation functions. A spectral density alone is not the bath state: thermal occupations and other preparation information also enter the correlations. Fourier signs, frequency units, and normalization factors must be stated.

A narrow resonance, a gap edge, or slowly varying low-frequency noise can invalidate a simple short-memory picture. Calling a spectrum “Ohmic” specifies a scaling regime; it does not establish that every other relevant timescale is harmless. For ordered fluctuations and response, continue to Quantum Noise, Dissipation, and Baths.

Keep history or change the variables

Memory Kernels writes the reduced derivative as an integral over earlier states. Nakajima–Zwanzig Projection derives that structure exactly, including a term from initially discarded correlations. Choosing a projection is a choice of retained variables; approximation enters when the resulting equation is truncated or otherwise simplified.

Time-Convolutionless Master Equations instead uses a time-local generator. For a differentiable invertible reduced map it is formally $\dot\Phi_t\Phi_t^{-1}$. This can encode dynamics with memory; singular maps need additional care. A time-local form alone therefore does not establish Markovianity.

If a slow environmental mode dominates, promoting it into the system can simplify the remaining bath. The Non-Markovian Dynamics chapter develops pseudomodes, reaction coordinates, and numerical alternatives. Initial Correlations explains why different joint preparations with the same initial reduced state can have different futures.

Exercises

Short memory, but unresolved transitions. Let $\tau_B=10^{-7}\,\mathrm{s}$, a relaxation rate $\Gamma=10^4\,\mathrm{s}^{-1}$, and a transition-frequency difference $\Delta\omega=10^3\,\mathrm{s}^{-1}$. Assess the two timescale separations.

Solution

$\tau_R=1/\Gamma=10^{-4}\,\mathrm{s}$, so $\Gamma\tau_B=10^{-3}$ supports a short-memory approximation if the other weak-coupling assumptions hold. But $1/\Delta\omega=10^{-3}\,\mathrm{s}$ exceeds $\tau_R$. There is no coarse-graining time long enough to average this cross term while remaining short compared with relaxation. Full secularization of the pair is not justified by these scales.

A mean bath force. For $H_I=A\otimes B$ with bath mean $b=\operatorname{Tr}(\eta_E B)$, rewrite the interaction using a zero-mean bath operator.

Solution

Set $\widetilde B=B-bI$. Then $H_I=bA\otimes I+A\otimes\widetilde B$, so the first term can be included in $H_S'=H_S+bA$. The remaining bath operator has zero mean. Discarding the mean without shifting the Hamiltonian would omit a coherent force.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Microscopic derivations and projection methods.
• Davies, E. B. Quantum Theory of Open Systems. Academic Press, 1976. Weak-coupling limits and reduced generators.
• Rivas, Á., and S. F. Huelga. Open Quantum Systems: An Introduction. Springer, 2012. Dynamical maps, approximations, and memory.
• Weiss, U. Quantum Dissipative Systems. 4th edition, World Scientific, 2012. Structured environments and nonperturbative regimes.