First- and Second-Order Digital Quantum Simulation of Three-Level Jaynes-Cummings Dynamics on Superconducting Quantum Processors

TL;DR

The results demonstrate that calibrated gate operations and noise-aware circuit design enable reliable digital simulation of multi-level light-matter interactions on noisy intermediate-scale quantum platforms.

Abstract

This work presents a digital quantum simulation of a three-level atomic system interacting with a single-mode electromagnetic field based on the Jaynes-Cummings model, implemented on IBM Quantum superconducting processors. A qutrit is encoded using two physical qubits to represent the atomic states, while an additional qubit encodes the truncated field mode, enabling the realization of effective $Λ$-type atomic dynamics.The continuous-time light-matter interaction is implemented in a digital form by discretizing the evolution using Suzuki-Trotter decomposition. In contrast to an analog realization, the digital simulation replaces the continuous evolution with a sequence of quantum gates whose parameters are explicitly controlled. Phase evolution arising from the interaction Hamiltonian is digitally encoded using calibrated $R_Z$ gates, whose rotation angles are fixed by the physically relevant coupling scale and the chosen Trotter time step.State preparation is achieved using Hadamard and parametrized rotation gates, while the interaction dynamics are implemented through controlled operations. A comparative analysis between first- and second-order Trotter implementations reveals a trade-off between digital accuracy and hardware-induced noise. Overall, the results demonstrate that calibrated gate operations and noise-aware circuit design enable reliable digital simulation of multi-level light-matter interactions on noisy intermediate-scale quantum platforms.

Figures (7)

• Figure 1: (a) The initial coherent field state, where the atom is prepared in the excited state and the cavity field is in a coherent state with mean photon number $\bar{n}$. Since a coherent state is a superposition of many Fock states, each photon-number component evolves with a slightly different Rabi frequency. This results in dephasing among the components, causing the atomic von Neumann entropy to increase and exhibit oscillatory behavior. The entropy does not return exactly to zero within the plotted time range because the revival time is longer than the displayed interval. No decoherence or dissipation is included in the numerical simulation; the observed behavior arises purely from coherent, unitary Jaynes--Cummings dynamics. (b) Time evolution of the populations of the three atomic levels under anti–Jaynes--Cummings couplings. The populations $P_0$, $P_1$, and $P_2$ exhibit coherent oscillations and population redistribution, indicating reversible excitation exchange and $\Lambda$-type three-level atomic dynamics.
• Figure 2: First-order Suzuki--Trotter simulation of the three-level Jaynes--Cummings model. (a) Digital quantum circuit implementing the first-order decomposition of the interaction Hamiltonian. (b) Measurement probability distribution obtained from 1024 shots on real hardware, showing dominant populations within allowed excitation subspaces. In comparison with the corresponding ideal (noise-free) simulator, small but nonzero populations appear in classically forbidden states, arising from gate imperfections and decoherence. (c) Q-sphere plots are generated from simulator-derived statevectors (or reconstructed quantum states), whereas hardware executions directly yield measurement counts in selected bases. Consequently, Q-sphere visualizations should be interpreted as simulation-based diagnostics of phase and amplitude structure, unless full or partial quantum state tomography or an entanglement witness is additionally performed.
• Figure 3: Second-order Suzuki--Trotter simulation of the three-level Jaynes--Cummings model. (a) Symmetric second-order digital circuit with half-step free evolutions and a full interaction block. (b) Measurement probability distribution obtained from 1024 shots on real hardware. In comparison with the corresponding ideal (noise-free) simulator, the experimental distribution preserves the dominant population structure and suppresses dynamically forbidden states, while showing residual population leakage due to gate imperfections, decoherence, and readout errors inherent to NISQ devices.
• Figure 4: Measurement outcomes from the first-order digital simulation of the three-level Jaynes--Cummings system executed on the ibm_torino, ibm_marrakesh, and ibm_fez processors. The probability distributions exhibit population leakage and asymmetry across backends, reflecting the sensitivity of first-order Trotterization to digital errors and hardware noise. Bitstrings are ordered as $|q_2 q_1 q_0\rangle$, where $q_2$ encodes the truncated cavity photon number.
• Figure 5: Measurement outcomes from the second-order digital simulation of the three-level Jaynes--Cummings system executed on the ibm_torino, ibm_marrakesh, and ibm_fez processors. Compared to the first-order implementation, improved confinement within the excitation-conserving subspace and reduced population leakage are observed, particularly on higher-fidelity hardware. Bitstrings are ordered as $|q_2 q_1 q_0\rangle$, where $q_2$ encodes the truncated cavity photon number.