Revisiting the Jaynes-Cummings model with time-dependent coupling

TL;DR

This work addresses the resonant Jaynes–Cummings model with time-dependent atom–field coupling, deriving an exact evolution operator via the coupling-area A(t) and analyzing two explicit modulations—linear and hyperbolic secant—as well as atomic motion in a partially cooled cavity. It computes population inversion and atom–field entanglement using von Neumann entropy, and employs the Bloch vector to study state purity and dipole alignment under motion-induced periodic coupling. The linear modulation shifts revival times and alters entanglement dynamics, while the sech modulation enables transient turning-off with controllable asymptotic entanglement. Atomic motion induces periodic Bloch-vector trajectories even in a thermal field, with p and ζ_3 tuning frequency and purity, and increasing ⟨n⟩ producing population trapping without destroying periodicity. Overall, the paper demonstrates practical time-dependent control of entanglement and population in TDJC, with implications for cavity QED experiments and quantum technologies.

Abstract

The Jaynes-Cummings (JC) model stands as a fully quantized, fundamental framework for exploring light-matter interactions, a timely reflection on a century of quantum theory. The time-dependent Jaynes-Cummings (TDJC) model introduces temporal variations in certain parameters, which often require numerical methods. However, under the resonance condition, exact solutions can be obtained, offering insight into a variety of physical scenarios. In this work, we study the resonant TDJC model considering different modulations of the atom-field coupling. The model is presented and an analytical solution derived in a didactic way, allowing us to examine how time-dependent couplings affect atomic population inversion and atom-field entanglement. We also consider an atom traversing a partially cooled cavity, which induces periodicity and reveals the combined effects of atomic motion and thermal fluctuations. The Bloch vector is used to analyze the dynamics of the system, including the atomic state purity, and reveals phenomena such as atomic dipole alignment with the field due to the oscillating coupling, as well as atomic population trapping, which arises by increasing the initial mean thermal photon number.

Figures (10)

• Figure 1: A schematic description of the JC model. The atom is represented as a TLS, namely $|e\rangle$ and $|g\rangle$, with a transition frequency $\omega$, while the cavity mode is modeled as a simple harmonic oscillator with frequency $\nu$ and characterized by the Fock states $|n\rangle$, $n=0,1,2,\hdots$.
• Figure 2: Scheme for the relation between the von Neumann entropy and entanglement. We begin with an initial pure state $\ket{\Psi(t)}\in \mathcal{H}_{BC}=\mathcal{H}_B \otimes \mathcal{H}_C$, and describe it using the density operator formalism to obtain $\hat{\rho}(t) = |\Psi(t)\rangle\langle\Psi(t)|$. Sequentially, we arbitrarily trace out the degree of freedom $C$, resulting in the reduced density matrix $\hat{\rho}_B(t) = \operatorname{Tr}_C\left[\hat{\rho}(t)\right]$. We then compute the von Neumann entropy $S_B(t) = -\operatorname{Tr}\left[\hat{\rho}_B(t) \log \hat{\rho}_B(t)\right]$ to quantify the amount of mixedness. A zero entropy indicates that the reduced density matrix $\hat{\rho}_B(t)$ is pure, and hence the state $|\psi_{BC}\rangle$ is separable. Conversely, a nonzero entropy implies that $\hat{\rho}_B(t)$ is a mixed state, signaling that the state $|\Psi(t)\rangle$ is entangled. In this scenario, the mixedness is proportional to the entanglement, as it reflects the uncertainty in the individual subsystem.
• Figure 3: The population inversion (solid blue line) and von Neumann entropy (dotted red line) as a function of the dimensionless time, when considering the initial state $|\Psi (0) \rangle = |e,\alpha \rangle$, an average photon number of $\langle n \rangle = 25$, and a constant coupling parameter $\lambda_0 = 1$.
• Figure 4: The population inversion (solid blue line) and von Neumann entropy (dotted red line) as a function of dimensionless time, when considering the initial state $|\Psi (0) \rangle = |e,\alpha \rangle$, an average photon number of $\langle n \rangle = 25$, $\lambda_0=1$ and the linear modulation. In (a), we depict the sudden change in the coupling, with $\zeta_1=0.16$, while in (b) we represent the adiabatic change, $\zeta_1=0.01$. As an inset in both plots, we present the behavior of the time-dependent coupling. We observe a change in the entanglement dynamics, with its minimum and maximum occurring at different times compared to the constant coupling scenario.
• Figure 5: The population inversion (solid blue line) and von Neumann entropy (dotted red line) as a function of dimensionless time, when considering the initial state $|\Psi (0) \rangle = |e,\alpha \rangle$, an average photon number of $\langle n \rangle = 25$, $\lambda_0=1$ and the hyperbolic secant modulation. In (a), we present a faster decay in the coupling, with $\zeta_2=0.3$, while in (b) we represent a slower stabilization, $\zeta_2=0.1$. As an inset in both plots, we present the behavior of the time-dependent coupling. We observe an asymptotic value for both the population inversion and the entanglement.