System–Environment Coupling

No system is truly isolated

Closed-system quantum mechanics — a state vector $|\psi\rangle$ evolving by a unitary $U(t) = e^{-iHt/\hbar}$ — is an idealization. Every real quantum system, a trapped ion, a superconducting qubit, an electron spin in a solid, is embedded in a larger world: stray photons, phonons, fluctuating fields, neighbouring atoms. We call the system of interest $S$ and everything it touches the environment or bath $B$ . The combined object $S+B$ is closed and evolves unitarily, but $S$ alone does not.

The universal decomposition

The total Hilbert space factorizes as a tensor product,

\mathcal{H} = \mathcal{H}_S \otimes \mathcal{H}_B,

and the most general total Hamiltonian is written as a sum of three terms:

H = H_S \otimes \mathbb{1}_B + \mathbb{1}_S \otimes H_B + H_{SB}.

Here $H_S$ generates the free evolution we would like the system to have, $H_B$ governs the bath's internal dynamics, and the interaction $H_{SB}$ is what couples them. If $H_{SB} = 0$ the two never talk and $S$ stays closed; all of open-system physics lives in $H_{SB}$ .

The structure of the coupling

A coupling term that acts non-trivially on both factors can always be expanded in a sum of product operators,

H_{SB} = \sum_\alpha A_\alpha \otimes B_\alpha,

where each $A_\alpha$ acts on the system and each $B_\alpha$ on the bath. A canonical example is a single qubit dephased by a bosonic bath of harmonic oscillators (the spin–boson model),

H_{SB} = \sigma_z \otimes \sum_k g_k\,(b_k + b_k^\dagger),

in which the system operator is $A = \sigma_z$ , the bath operators are the field quadratures $B_k = g_k(b_k + b_k^\dagger)$ , and the $g_k$ set the coupling strengths to each mode $k$ .

Why the system stops being unitary

Suppose at $t=0$ the system and bath are uncorrelated, $|\Psi(0)\rangle = |\psi_S\rangle \otimes |\phi_B\rangle$ . Under the joint unitary $U(t)$ , the interaction $H_{SB}$ generically entangles them, so for $t>0$

|\Psi(t)\rangle = U(t)\,\bigl(|\psi_S\rangle \otimes |\phi_B\rangle\bigr) \neq |\psi_S(t)\rangle \otimes |\phi_B(t)\rangle .

Once $S$ and $B$ are entangled, no pure state of $S$ alone reproduces all its measurement statistics: the system must be described by a mixed state. The mathematical object that captures this — the reduced density matrix obtained by tracing out the bath — is the subject of the next lesson, and the dynamical law it obeys, once we make suitable approximations, is the Lindblad master equation that names this module.

The two regimes

Two limits organize everything that follows. When the bath forgets its interaction with the system far faster than the system evolves — short bath correlation time, weak coupling — the system's future depends only on its present, and the dynamics are Markovian, described by a time-local master equation. When the bath retains memory and feeds information back, the dynamics are non-Markovian. We build the Markovian theory first (Born–Markov, Lindblad) and contrast it with the non-Markovian case at the end of the module.