Excitation-Conserved Eigenbasis

• Excitation-Conserved Eigenbasis is a set of quantum states with a fixed total excitation number, partitioning the system into invariant subspaces.
• It enables block-diagonalization of both Hamiltonian and Liouvillian operators, allowing exact analytic treatment of system dynamics.
• This framework, exemplified in the Jaynes–Cummings model, facilitates precise decoherence analysis and efficient numerical benchmarks in open quantum systems.

An excitation-conserved eigenbasis is a set of eigenstates for a quantum or mathematical system in which each basis vector possesses a well-defined, conserved "excitation" quantum number, with the system's dynamical or structural evolution restricted to subspaces labeled by this quantum number. This concept is of particular significance when both the system's Hamiltonian and any dissipative or stochastic processes commute with an excitation number operator, guaranteeing that transitions between different excitation sectors are forbidden. Such a structure allows for block-diagonalization of the evolution operators, enabling exact solvability within smaller, well-characterized invariant subspaces. The rigorous construction and implications of excitation-conserved eigenbases are exemplified in open quantum systems such as the dissipative Jaynes–Cummings model (Torres et al., 2011), where engineered dissipation coexists with quantum number conservation, yielding tractable dynamics and facilitating analytical insight into decoherence and dissipation at fixed excitation number.

1. Conservation of Excitation Number and Model Hamiltonian Structure

The Jaynes–Cummings (JC) Hamiltonian under the rotating-wave approximation admits an additional conserved quantity, the total excitation number:

$$I = a^\dagger a + \frac{1}{2}(\sigma_z + \mathbb{I})$$

where $a, a^\dagger$ are bosonic annihilation and creation operators for the field mode, and $\sigma_z$, $\sigma_+$, $\sigma_-$ are Pauli operators associated with a two-level atom. The JC Hamiltonian,

$$H = \delta \sigma_z + g(a \sigma_+ + a^\dagger \sigma_-)$$

commutes with $I$, which partitions the full Hilbert space into subspaces with fixed values of the excitation number. In these subspaces, each labeled by $n$, the restricted Hamiltonian is a $2 \times 2$ block:

$$H_n = \begin{pmatrix} \delta & g \sqrt{n} \ g \sqrt{n} & -\delta \end{pmatrix}$$

The eigenstates within each block form the excitation-conserved eigenbasis, often referred to as the "dressed states".

2. Formulation of Dissipative Dynamics with Excitation Conservation

In inclusion of open system effects, the master equation for the system density matrix $\rho$ is often written in Lindblad form:

$$\dot{\rho} = -i[H, \rho] - \sum_j \frac{\gamma_j}{2} \left( O_j^\dagger O_j \rho + \rho O_j^\dagger O_j - 2 O_j \rho O_j^\dagger \right)$$

Exact solvability with excitation conservation at the level of dissipative dynamics requires that all dissipation operators $O_j$ commute with the excitation number operator $I$:

$[O_j, I] = 0, \ \forall j$

In (Torres et al., 2011), the specific choice $O_1 = a \sigma_+,\ O_2 = a^\dagger \sigma_-$ ensures this commutation, since both operators only transfer excitations between atom and field without changing their sum. As a result, both coherent and incoherent evolution conserve total excitation.

3. Block Structure and Spectral Representation of the Liouville Operator

The conservation of excitation number implies that the Liouvillian superoperator $\mathcal{L}$ governing the evolution of $\rho$ becomes block-diagonal. The density matrix itself is decomposed according to left and right excitation quantum numbers:

$\rho = \sum_{n,m} \rho_{n,m}$

with

$[I, \rho_{n,m}] = (n-m) \rho_{n,m}$

The Liouvillian acts independently within each $(n,m)$ sector, and for a two-level atom these blocks can be represented (after vectorization) as $4 \times 4$ matrices:

$$\mathcal{L}_{n,m} = \begin{pmatrix} -\sqrt{mn}\gamma_0 & ig\sqrt{m} & -ig\sqrt{n} & \sqrt{mn}\gamma_1 \ ig\sqrt{m} & -2i\delta-\tfrac{1}{2}\sqrt{mn}(\gamma_1-\gamma_0) & 0 & -ig\sqrt{n} \ -ig\sqrt{n} & 0 & 2i\delta-\tfrac{1}{2}\sqrt{mn}(\gamma_1-\gamma_0) & ig\sqrt{m} \ \sqrt{mn} \gamma_0 & -ig\sqrt{n} & ig\sqrt{m} & -\sqrt{mn}\gamma_1 \end{pmatrix}$$

Diagonalization of these blocks yields the spectrum of the evolution, and explicit steady-state characterization is possible in the resonant case ($\delta=0$). This reduction is only possible because of the excitation-conserved eigenbasis structure.

4. Implications of the Excitation-Conserved Eigenbasis

The excitation-conserved eigenbasis dramatically simplifies the analysis of both closed and open system dynamics:

• Dimensional reduction: Once excitation number is fixed, all computations are performed in small invariant subspaces (e.g., size 2 for the JC model), scaling only linearly with the maximal excitation considered.
• Spectral decomposition: The time evolution operator (or Liouvillian) can be explicitly diagonalized within each sector, yielding analytic solutions for observables and coherence decay, which are otherwise intractable for large Hilbert spaces.
• Decoupling of dynamics: If a system is initialized in a definite excitation sector, it remains confined to that sector throughout the evolution, both under coherent and engineered dissipative processes.

5. Practical Applications and Numerical Benchmarks

Excitation-conserved eigenbases provide powerful tools for both analytical and computational studies across various open quantum system models:

Application	Context	Utility
Cavity QED	Atom-photon interactions in a cavity	Analytic/numeric decoherence and dissipation benchmarking
Quantum information	Error-protected subspace engineering	Design of symmetry-respecting loss channels
Numerical validation	Large Hilbert space open systems	Exact subspace diagonalization for benchmarking approximate solvers

Their utility is constrained by the requirement that both Hamiltonian and dissipative operators commute with the excitation number. For more general forms of dissipation (e.g., spontaneous emission alone), the block-diagonal structure is destroyed, and the excitation number is no longer a good quantum number.

6. Limitations and Generalizations

Limitations include:

• Dissipation engineering constraint: The method requires dissipators that preserve excitation number (which may not always be experimentally achievable).
• Applicability to higher-dimensional systems: The model and eigenbasis construction are readily applicable for the JC model or two-level atom + single mode; extension to multi-level, multi-mode, or multi-atom cases may require significantly heavier computational machinery and may lack analytic closed forms.
• Physical constraints: Realistic systems may be susceptible to processes not captured by excitation-preserving dissipators (e.g., environment-induced photon loss).

Nevertheless, excitation-conserved eigenbases serve as essential analytical benchmarks and inspire symmetry-preserving numerical approaches for more complex systems.

7. Summary and Conceptual Significance

The excitation-conserved eigenbasis in open quantum systems, as exemplified by a dissipation-augmented Jaynes–Cummings model (Torres et al., 2011), arises whenever both the Hamiltonian and dissipation are constructed to strictly conserve a quantum number (here: total excitation). This leads to block-diagonality of the evolution generator, tractable spectral analysis, and explicit insight into dynamics and steady-state formation. While the exact conservation requirement may be idealized, such models are invaluable conceptual and computational frameworks for exploring the interplay of coherence, dissipation, and quantum number conservation in quantum mechanics.