System–Bath Hamiltonians

A system–bath Hamiltonian is a microscopic model of an open quantum system before any tracing, coarse graining, or master-equation approximation has been made. It specifies the degrees of freedom kept as the system, the degrees of freedom treated as the environment, and the operators through which they interact.

Helpful background. Composite-system operators explain tensor factors, and reduced dynamics explains how the joint unitary produces subsystem predictions.

The standard structure is

$$H = H_S + H_E + H_I,$$

with interaction

$$H_I = \sum_\alpha S_\alpha\otimes B_\alpha.$$

Here $S_\alpha$ acts on the system Hilbert space and $B_\alpha$ acts on the environment Hilbert space. Tensor-product identity operators are usually omitted:

$$H_S \equiv H_S\otimes I_E, \qquad H_E \equiv I_S\otimes H_E.$$

For finite-dimensional systems these are Hermitian matrices. Oscillator and field models require a self-adjoint total Hamiltonian on a suitable domain, finite relevant correlation functions, and any stated high-frequency cutoff; a formal sum of unbounded terms does not establish these properties by itself.

System, environment, and interaction energies

The system Hamiltonian $H_S$ defines the closed dynamics of the degrees of freedom of interest. It determines energy eigenstates, Bohr frequencies, coherent oscillations, and which transitions can be resonant with a bath.

The environment Hamiltonian $H_E$ defines the free dynamics of the ignored degrees of freedom. It determines bath modes, correlation functions, spectra, temperature dependence, and memory times.

The interaction Hamiltonian $H_I$ defines the boundary between them. It answers:

• which system observables the environment monitors;
• which transitions exchange energy with the environment;
• which bath operators carry noise or dissipation;
• which symmetries are preserved or broken by the coupling.

Different $H_I$ choices can produce pure dephasing, relaxation, thermalization, measurement backaction, leakage, or non-Markovian memory.

Interaction Decomposition

An interaction can often be written as

$$H_I = \lambda \sum_\alpha S_\alpha\otimes B_\alpha,$$

where $\lambda$ is a dimensionless parameter used to track perturbative order. Unless displayed explicitly below, it is absorbed into the coupling coefficients. With dimensionless $S_\alpha$, each $B_\alpha$ has energy units.

If $H_I$ is Hermitian, the decomposition may be chosen with Hermitian operators $S_\alpha$ and $B_\alpha$, but this is not required. A non-Hermitian exchange form such as

$$H_I = \sigma_+\otimes B + \sigma_-\otimes B^\dagger$$

is Hermitian because the two terms are adjoints.

The decomposition is not unique. Operators can be linearly recombined:

$$\sum_\alpha S_\alpha\otimes B_\alpha = \sum_\beta \widetilde S_\beta\otimes \widetilde B_\beta.$$

An exact change of operator basis leaves predictions unchanged. Truncating terms or applying frequency averaging inconsistently after such a change can alter an approximate model.

Mean Bath Forces

If the bath reference state is $\rho_E$ and

$$\langle B_\alpha\rangle_E = \operatorname{Tr}_E(B_\alpha\rho_E) \ne0,$$

then

$$S_\alpha\otimes B_\alpha = S_\alpha\otimes \left( B_\alpha-\langle B_\alpha\rangle_E I_E \right) + \langle B_\alpha\rangle_E S_\alpha\otimes I_E.$$

The second term is a system Hamiltonian correction. It is usually cleaner to define centered bath operators

$$\delta B_\alpha = B_\alpha-\langle B_\alpha\rangle_E I_E$$

and absorb the mean force into

$$H_S \mapsto H_S + \sum_\alpha \langle B_\alpha\rangle_E S_\alpha.$$

This prevents a first-order coherent shift from being mistaken for a dissipative effect. If an explicit $\lambda$ is retained in $H_I$, the shift is $\lambda\sum_\alpha\langle B_\alpha\rangle_E S_\alpha$. Recompute the interaction picture using whichever system Hamiltonian is being treated as the unperturbed one.

Interaction Picture

Master-equation derivations usually move to the interaction picture with respect to

$$H_0=H_S+H_E.$$

Then

$$H_I(t) = e^{iH_0t/\hbar} H_I e^{-iH_0t/\hbar} = \sum_\alpha S_\alpha(t)\otimes B_\alpha(t),$$

where

$$S_\alpha(t) = e^{iH_St/\hbar} S_\alpha e^{-iH_St/\hbar}, \qquad B_\alpha(t) = e^{iH_Et/\hbar} B_\alpha e^{-iH_Et/\hbar}.$$

For the following correlation convention, choose Hermitian, centered bath operators. The bath enters second-order reduced dynamics through

$$C_{\alpha\beta}(t-s) = \operatorname{Tr}_E \left[ B_\alpha(t)B_\beta(s)\rho_E \right]$$

when $\rho_E$ is stationary under $H_E$.

These correlations determine rates, Lamb shifts, memory kernels, and the validity of Markov approximations.

Bohr-Frequency Decomposition

Let the system Hamiltonian have spectral projectors

$$H_S = \sum_\epsilon \epsilon\,\Pi(\epsilon).$$

For weak-coupling master equations, a system coupling operator is decomposed into Bohr-frequency components:

$$S_\alpha(\omega) = \sum_{\epsilon'-\epsilon=\hbar\omega} \Pi(\epsilon)S_\alpha\Pi(\epsilon').$$

These satisfy

$$[H_S,S_\alpha(\omega)] = -\hbar\omega\,S_\alpha(\omega).$$

With the Fourier convention $\int d\tau\,e^{i\omega\tau}C_{\alpha\beta}(\tau)$, positive $\omega$ labels system energy loss in $S_\alpha(\omega)$. The corresponding bath spectrum controls that downward transition. This is the bridge from a microscopic Hamiltonian to relaxation rates and dephasing rates. For a coordinate-coupled oscillator bath where the spectral density is usually interpreted as damping and force noise, see Caldeira–Leggett Model.

If $S_\alpha$ commutes with $H_S$, only the $\omega=0$ component is present. Then $[H_S,H_I]=0$, so the bare system energy is exactly conserved for a time-independent total Hamiltonian. Such a coupling can dephase different eigenvalues of the monitored operator; degeneracies can leave entire coherent subspaces unaffected.

Example: Pure Dephasing Spin-Boson Coupling

For a qubit with

$$H_S = \frac{\hbar\omega_0}{2}Z,$$

a pure-dephasing spin-boson interaction has the schematic form

$$H_I = Z\otimes \sum_k \left( g_k b_k^\dagger + g_k^* b_k \right),$$

with environment Hamiltonian

$$H_E = \sum_k \hbar\omega_k b_k^\dagger b_k.$$

Since

$$[H_S,Z]=0,$$

the coupling distinguishes the $Z$ alternatives without directly driving transitions between them. In suitable regimes this produces dephasing in the energy basis.

This model is useful for frequency noise, elastic scattering, and which-path information. It is not a model of energy relaxation unless additional noncommuting couplings are included. The broader two-state dissipative model is Spin-Boson Model.

Example: Relaxation Coupling

Let

$$H_S = \hbar\omega_0 \lvert e\rangle\langle e\rvert, \qquad \sigma_-=\lvert g\rangle\langle e\rvert, \qquad \sigma_+=\lvert e\rangle\langle g\rvert.$$

An energy-exchange interaction may be written as

$$H_I = \sigma_+\otimes B + \sigma_-\otimes B^\dagger.$$

Because

$$[H_S,\sigma_-] = -\hbar\omega_0\sigma_-, \qquad [H_S,\sigma_+] = \hbar\omega_0\sigma_+,$$

the bath spectrum near $\omega_0$ controls downward and upward transition rates. With a broadband vacuum environment and weak coupling, this structure leads to spontaneous-emission and amplitude-damping models. A single mode or a structured band can instead give coherent exchange, revivals, or incomplete decay.

A coupling such as $\sigma_x\otimes B$ contains both $\sigma_+$ and $\sigma_-$ components, so it can cause transitions even though it is written with a Hermitian system operator.

Example: Oscillator Bath and Brownian Coupling

A common oscillator-bath model uses

$$H_S = \frac{p^2}{2m} + V(x),$$

and

$$H_E = \sum_k \left( \frac{P_k^2}{2m_k} + \frac12m_k\omega_k^2Q_k^2 \right).$$

The interaction is often taken to be linear in position:

$$H_I = x\otimes \sum_k c_k Q_k.$$

Completing the square shows the potential shift explicitly:

$$\frac12m_k\omega_k^2Q_k^2+c_kxQ_k =\frac12m_k\omega_k^2\left(Q_k+\frac{c_kx}{m_k\omega_k^2}\right)^2 -\frac{c_k^2x^2}{2m_k\omega_k^2}.$$

To retain the specified bare potential $V(x)$, include the positive counterterm $H_{\mathrm{ct}}=x^2\sum_k c_k^2/(2m_k\omega_k^2)$. For one mode and $V(x)=m\Omega^2x^2/2$, omitting it changes the static curvature to $m\Omega^2-c^2/(m_B\omega_B^2)$, which can even become negative. The counterterm is a physical convention about the chosen bare potential, not an arbitrary noise correction.

The bath is summarized by a spectral density such as

$$J(\omega) = \sum_k \frac{c_k^2}{2m_k\omega_k} \delta(\omega-\omega_k),$$

in a convention without a factor of $\pi$. It gives $H_{\mathrm{ct}}=x^2\int_0^\infty J(\omega)\,d\omega/\omega$, when this integral exists. A high-frequency cutoff or a specified renormalization prescription is essential when it diverges. Ohmic, sub-Ohmic, super-Ohmic, and structured spectra describe different low-frequency and resonant behavior.

For conventions and classifications of $J(\omega)$, see Spectral Densities. For the distinction between oscillator mode spectral density and ordered noise spectra used in rate formulas, see Noise Spectra.

Example: Atom-Field Coupling

For a two-level atom coupled to electromagnetic field modes, a rotating-wave interaction has the form

$$H_I = \sum_k \left( g_k\sigma_+a_k + g_k^*\sigma_-a_k^\dagger \right).$$

The term $\sigma_-a_k^\dagger$ describes atomic de-excitation with photon creation. The term $\sigma_+a_k$ describes absorption. In vacuum, absorption is absent at the level of ordinary transition rates, while emission remains possible.

The rotating-wave approximation has already removed counter-rotating terms. Whether that step is justified depends on coupling strength, detuning, bandwidth, and timescale.

Example: Central Spin Model

In a central spin model, a distinguished spin couples to many environmental spins:

$$H_I = \sum_k A_k\,\mathbf S\cdot\mathbf I_k.$$

The bath is not a collection of independent harmonic modes. It may have long memory, strong finite-size effects, and non-Gaussian fluctuations. Such models appear in spin qubits, color centers, nuclear-spin environments, and magnetic resonance.

They are useful reminders that “bath” does not automatically mean thermal, Gaussian, Markovian, or infinite.

From Hamiltonian to Reduced Dynamics

The Hamiltonian is the starting point, not the final open-system model. The exact reduced state is

$$\rho_S(t) = \operatorname{Tr}_E \left[ U(t)\rho_{SE}(0)U^\dagger(t) \right].$$

To obtain a closed master equation one usually adds assumptions:

• a specified initial state, often $\rho_S(0)\otimes\rho_E$;
• a stationary or controlled environment state;
• weak coupling or another expansion parameter;
• bath correlation functions with known decay;
• Markov, secular, or coarse-graining approximations when justified.

The Reduced Dynamics page explains the exact partial-trace step. The Born Approximation page explains the weak-coupling expansion used before many Markovian derivations, Markov Approximation covers the short-memory step, and Secular Approximation covers Bohr-frequency averaging. The Approximation Checklist lists the assumptions that must be checked before trusting a master equation.

Boundary Choices and Renormalization

The system–bath split is a modeling choice. Moving a strongly coupled mode from the bath into the system can turn a non-Markovian problem into a larger Markovian one. This is the idea behind reaction-coordinate and pseudomode methods.

Similarly, Hamiltonian terms can be redistributed:

• mean bath forces can be absorbed into $H_S$;
• Lamb shifts can be reported as Hamiltonian corrections;
• counterterms can preserve a chosen bare potential;
• driving fields may be included in $H_S$ or treated as part of the environment.

These choices are not merely cosmetic. They affect perturbative order, resonance conditions, and which approximation is controlled.

Exercises

Centering a bath operator

Suppose

$$H_I=S\otimes B$$

and $\langle B\rangle_E=b_0$. Rewrite the interaction using $\delta B=B-b_0I_E$ and identify the correction to $H_S$.

Solution

Write

$$B=\delta B+b_0I_E.$$

Then

$$H_I = S\otimes\delta B + b_0 S\otimes I_E.$$

The second term acts only on the system, so it can be absorbed into

$$H_S \mapsto H_S+b_0S.$$

The centered interaction is $S\otimes\delta B$, with $\langle\delta B\rangle_E=0$.

Dephasing or relaxation

Let $H_S=(\hbar\omega_0/2)Z$. Which coupling is pure-dephasing-like in the energy basis, $Z\otimes B$ or $X\otimes B$? Explain.

Solution

Since

$$[Z,H_S]=0,$$

the coupling $Z\otimes B$ does not connect different energy eigenstates of $H_S$. It monitors the energy-basis alternatives and is pure-dephasing-like.

By contrast,

$$X=\sigma_+ + \sigma_-,$$

where the raising and lowering components connect the two energy eigenstates. The coupling $X\otimes B$ can drive transitions if the bath has spectral weight near $\omega_0$.

Bohr components of a qubit operator

For

$$H_S = \hbar\omega_0 \lvert e\rangle\langle e\rvert,$$

decompose $X=\lvert e\rangle\langle g\rvert+\lvert g\rangle\langle e\rvert$ into Bohr-frequency components.

Solution

The projectors are

$$\Pi_g=\lvert g\rangle\langle g\rvert, \qquad \Pi_e=\lvert e\rangle\langle e\rvert.$$

The lowering component is

$$S(\omega_0) = \Pi_gX\Pi_e = \lvert g\rangle\langle e\rvert = \sigma_-,$$

because it maps an energy $\hbar\omega_0$ state to energy $0$ and satisfies

$$[H_S,\sigma_-]=-\hbar\omega_0\sigma_-.$$

The raising component is

$$S(-\omega_0) = \Pi_eX\Pi_g = \lvert e\rangle\langle g\rvert = \sigma_+,$$

with

$$[H_S,\sigma_+]=+\hbar\omega_0\sigma_+.$$

The sign convention for labeling $\omega$ varies; the commutator equation is the safest way to check.

Hermitian exchange coupling

Show that

$$H_I = \sigma_+\otimes B + \sigma_-\otimes B^\dagger$$

is Hermitian.

Solution

Using $\sigma_+^\dagger=\sigma_-$,

$$\left( \sigma_+\otimes B \right)^\dagger = \sigma_-\otimes B^\dagger,$$

and

$$\left( \sigma_-\otimes B^\dagger \right)^\dagger = \sigma_+\otimes B.$$

Therefore

$$H_I^\dagger = \sigma_-\otimes B^\dagger + \sigma_+\otimes B = H_I.$$

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983). DOI.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer (2004). Publisher.
• A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Reviews of Modern Physics 59, 1–85 (1987). DOI.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).