Nguyen Khoi Nguyen (Alan), Boston University
Advised by Prof. Luca Dal Negro, EC 585 / EC 777

Computational study of quantum light-matter interaction in the Jaynes-Cummings (JC) model. All dynamics are computed from first principles using QuTiP, with no phenomenological approximations.

Five principal results:

1. Time-resolved Wigner function snapshots showing Schrödinger cat-state formation at $t = t_r/2$ with Wigner negativity $\delta = 0.43$
2. Systematic comparison of atom-field entanglement entropy across coherent, thermal, squeezed, and Fock initial field states
3. Quantitative decoherence study: cavity decay $\kappa/g = 0.02$ reduces cat-state Wigner negativity by over 80%
4. Cat-state survival phase diagram: 2D parameter sweep of $\delta(\bar{n}, \kappa/g)$ mapping the boundary of observable quantum coherence, plus even-cat-state fidelity
5. Photon blockade and second-order coherence: $g^{(2)}(0)$ as a function of coupling strength, drive power, and detuning, demonstrating the quantum-to-classical crossover

• Animations
  • Wigner Function Evolution
  • Entanglement, Inversion, and Purity
  • Decoherence Destroys the Cat State
  • Dressed-State Avoided Crossing
  • Photon Number Distribution Dynamics
  • Bloch Sphere Trajectory
  • Wigner vs Husimi Q-Function
  • Cat-State Decoherence Sweep
  • Photon Blockade Transition
• Static Figures
  • Dressed-State Avoided Crossing and Scaling
  • Photon Number Distribution at Six Key Times
  • Bloch Sphere Dynamics
  • Wigner vs Husimi Q-Function Comparison
  • Cat-State Survival Phase Diagram
  • Cat-State Survival: Parameter Slices
  • Photon Blockade: g(2) vs Coupling Strength
  • Photon Statistics vs Drive Strength
  • Photon Blockade Spectrum
  • g(2) Combined Summary
  • Cat-State Detail
  • Entanglement Across Field States
  • Entropy Scaling with Photon Number
  • Dissipative Entanglement
  • Decoherence Table
  • Coherent vs Thermal Inversion
  • Mollow Triplet
  • Static Wigner Functions for Fock States
• Extended Theory
  • 1. Husimi Q-Function and the Phase-Space Smoothing Theorem
  • 2. Schrodinger Cat-State Wigner Function and Fidelity
  • 3. Microscopic Derivation of Enhanced Decoherence
  • 4. Second-Order Coherence and the Photon Blockade Effect
  • 5. Bloch Vector Dynamics and the Geometry of Entanglement
  • 6. Photon Number Distribution Dynamics and the Number-Space Cat Signature
  • 7. Cat-State Survival Phase Diagram: Interpretation
• Simulation Details
  • Parameters
  • Figure-to-Script Map
• Repository Structure
• Quick Start
• Citation
• License
• Acknowledgments

A coherent state $|\alpha = \sqrt{10}\rangle$ evolves under the resonant JC Hamiltonian. During the collapse of Rabi oscillations, the intracavity field splits into a superposition of two phase-space components — a Schrödinger cat state. Left: Wigner function $W(x,p)$ of the reduced cavity field state $\rho_\text{field} = \mathrm{Tr}_\text{atom}[\rho]$, computed on a 200×200 phase-space grid. Interference fringes between the two coherent components produce negative regions ($W < 0$, blue), the hallmark of non-classicality. Right: Atomic inversion $\langle\sigma_z\rangle(t)$ with a moving time marker. The animation pauses at the cat-state time ($t = t_r/2$) where the field is maximally entangled with the atom, and at the first revival ($t = t_r = 2\pi\sqrt{\bar{n}}/g$) where the system approximately refactorizes.

Simultaneous evolution of three complementary observables. Left: Wigner function $W(x,p)$. Center: Atomic inversion $\langle\sigma_z\rangle(t)$. Right: Von Neumann entanglement entropy $S(\rho_\text{atom})$ (red) and field-state purity $\mathrm{Tr}[\rho_\text{field}^2]$ (green). Entropy saturates at 1 bit during the collapse window (atom and field become maximally entangled, cat state forms), while purity drops to ~0.5 — consistent with a statistical mixture of two near-orthogonal coherent components. Both quantities recover partially at the first revival as the composite state approximately refactorizes.

Cavity photon loss via the Lindblad dissipator $\kappa \mathcal{D}[a]$ erases quantum coherence on a timescale $\sim 1/(\kappa \bar{n})$, far shorter than the bare cavity lifetime $1/\kappa$. Left: Wigner function at the cat-state time $t = t_r/2$ as the decay rate $\kappa/g$ increases from 0 to 0.15. The interference fringes vanish first (they involve high-order coherences), while the two Gaussian lobes persist — the state decoheres into a classical mixture. Right: Wigner negativity volume $\delta$ tracking the continuous loss of non-classicality.

The JC dressed states $|n, \pm\rangle$ are the exact eigenstates of the coupled atom-cavity system. As the atom-cavity detuning $\Delta = \omega_a - \omega_c$ is swept, the bare-state energies (dashed grey) would cross, but the JC interaction opens an avoided crossing with a gap of $2g\sqrt{n+1}$. This animation sweeps $\Delta/g$ from $-10$ to $+10$ for manifolds $n = 0, 1, 5, 10$ simultaneously, making the $\sqrt{n+1}$ scaling of the vacuum Rabi splitting directly visible. At large detuning the dressed states approach the bare (uncoupled) states; on resonance the hybridization is maximal.

Time evolution of the intracavity photon number distribution $P(n, t) = \langle n|\rho_\text{field}(t)|n\rangle$ for an initial coherent state with $\bar{n} = 10$. At $t = 0$ the distribution is Poissonian (red dashed envelope). During Rabi oscillations it develops a bimodal structure — photon numbers near $\bar{n}$ split into two peaks separated by $\sim 2\sqrt{\bar{n}}$, the photon-number signature of the cat state at $t = t_r/2$. The distribution partially recovers toward Poissonian at the first revival $t = t_r$, though it never fully returns due to the anharmonic $\sqrt{n+1}$ Rabi spectrum.

The reduced atomic state $\rho_{\text{atom}} = \mathrm{Tr}_{\text{field}}[\rho]$ traces a trajectory inside the Bloch sphere. A pure atomic state sits on the surface ($|\mathbf{r}| = 1$); entanglement with the field pulls the Bloch vector toward the center ($|\mathbf{r}| \to 0$, maximally mixed). Left: 3D Bloch sphere trajectory color-coded by time. The atom starts at the excited state (red dot, north pole), spirals inward during collapse, reaching near the origin at $t = t_r/2$ (gold star), then spirals partially outward at the first revival (green triangle). Center: Bloch vector length $|\mathbf{r}|(t)$. Right: Von Neumann entropy $S(\rho_{\text{atom}})$.

Side-by-side comparison of the Wigner function $W(x,p)$ (top row) and Husimi Q-function $Q(\alpha) = \langle\alpha|\rho|\alpha\rangle/\pi$ (bottom row) during JC evolution with $\bar{n} = 10$. The Wigner function takes negative values — the negativity volume $\delta$ (shown in each panel) quantifies non-classicality. The Husimi Q is a Gaussian-smoothed Wigner function ($Q = W * G_\text{vacuum}$) and is non-negative by construction ($Q \geq 0$ always). At $t = t_r/2$ the Wigner function resolves the interference fringes sharply, while the Q-function shows only two smooth lobes. This demonstrates that Wigner negativity, not Q-function structure, is the proper witness of quantum coherence in phase space.

The cat state at $t = t_r/2$ is progressively destroyed as cavity decay $\kappa/g$ increases from 0 to 0.12, with $\bar{n} = 10$ held fixed. Left: Wigner function $W(x,p)$ of the reduced cavity field. At $\kappa = 0$, the full interference pattern is visible between the two coherent lobes — deep negative fringes certifying a macroscopic quantum superposition. As $\kappa$ increases, the fringes wash out first (they are encoded in high-order off-diagonal elements $\langle n|\rho|n + 2k\rangle$ with $k \gg 1$), while the two classical lobes persist longer. By $\kappa/g \approx 0.04$ the Wigner function is everywhere positive — the cat has decohered into a classical mixture. Right top: Wigner negativity volume $\delta(\kappa/g)$ tracing the quantitative loss of non-classicality; the horizontal grey line marks $\delta = 0.05$, our operational threshold for observability. Right bottom: Field-state purity $\mathrm{Tr}[\rho_\text{field}^2]$, which drops from ~0.96 (near-pure cat state) to ~0.38 (highly mixed), confirming that decoherence (fringe erasure) proceeds much faster than energy dissipation (photon loss).

The transition from weak to strong coupling as $g/\kappa$ is swept from 0.1 to 12. Left: Energy-level diagram of the first three JC manifolds ($n = 0, 1, 2$) in units of $\hbar g$, with linewidth bands (shaded) that visibly shrink as $\kappa/g$ decreases. A red arrow marks the coherent drive; the red ✗ marks the blocked second-photon transition once the anharmonic splitting exceeds the linewidth. Right: Cavity transmission spectrum $\langle n \rangle(\Delta)$ computed from the steady-state Lindblad equation at each $g/\kappa$. At weak coupling ($g \ll \kappa$), the spectrum is a single Lorentzian centered at $\Delta = 0$ — the atom is too weakly coupled to modify the cavity response. As $g$ crosses $\kappa$, the peak broadens and flattens. At $g/\kappa \gtrsim 2$, the spectrum splits into two resolved peaks at $\Delta = \pm g$ — the vacuum Rabi doublet, the spectroscopic signature of strong coupling. The splitting grows as $2g$ (red annotation), directly mirroring the dressed-state gap in the energy ladder. Numerical readouts track the on-resonance $g^{(2)}(0)$ (dropping from ~1 to deep antibunching) and $\langle n \rangle_\text{res}$ (suppressed by the blockade).

Left: Dressed-state energy eigenvalues $E_{n,\pm}$ as a function of detuning $\Delta/g$ for photon manifolds $n = 0$ (blue), $1$ (orange), $5$ (green), $10$ (red). At resonance ($\Delta = 0$), each manifold exhibits an avoided crossing with splitting $\Omega_n(0) = 2g\sqrt{n+1}$, indicated by colored arrows. Right: On-resonance splitting $\Omega_n(0)/g$ vs photon number $n$ (black dots), overlaid with the analytic curve $2\sqrt{n+1}$ (dashed pink). The $\sqrt{n+1}$ dependence is the quantum-mechanical fingerprint of the quantized field: a classical drive would produce a splitting independent of intensity.

Snapshots of the intracavity photon number distribution $P(n,t)$ at six characteristic times during JC evolution ($\bar{n} = 10$, $\Delta = 0$). (a) $t = 0$: initial Poissonian distribution. (b) $t = 0.5,t_c$: early Rabi oscillations, still approximately unimodal. (c) $t = 2,t_c$: collapse onset, broadening as different Fock components oscillate at incommensurate $\sqrt{n+1}$ frequencies. (d) $t = t_r/2$ (cat state): bimodal structure with peaks separated by $\sim 2\sqrt{\bar{n}} \approx 6$ photons. (e) $t = 0.75,t_r$: partial recombination. (f) $t = t_r$ (first revival): partially recovered unimodal shape, broader than the initial Poissonian.

Left: 3D trajectory of the reduced atomic Bloch vector $\mathbf{r} = (\mathrm{Tr}[\rho_\text{atom}\sigma_x],, \mathrm{Tr}[\rho_\text{atom}\sigma_y],, \mathrm{Tr}[\rho_\text{atom}\sigma_z])$ during one full collapse-revival cycle. The atom starts at the excited state (red dot, $|\mathbf{r}| = 1$), spirals inward to the origin at $t = t_r/2$ (gold star, maximally mixed / maximally entangled), and partially re-emerges at $t = t_r$ (green triangle). Center: Bloch vector length $|\mathbf{r}|(t)$. Right: Von Neumann entropy $S(\rho_\text{atom})$. Vertical dashed lines mark $t_c$ (blue), $t_r/2$ (pink), and $t_r$ (purple).

Side-by-side snapshots at seven characteristic times. Each Wigner panel is annotated with the negativity volume $\delta$. At $t = t_r/2$, the Wigner function exhibits oscillatory fringes with $\delta = 0.85$, while the Q-function shows only two smooth, positive peaks. At $t = t_r$, residual negativity ($\delta = 0.31$) reflects imperfect refactorization.

Systematic 2D parameter sweep (16 × 16 = 256 independent Lindblad simulations) mapping cat-state survival in the $(\bar{n},, \kappa/g)$ plane at $t = t_r/2$. (a) Wigner negativity volume $\delta(\bar{n}, \kappa/g)$. The cyan contour marks $\delta = 0.05$ — the practical boundary below which cat-state interference fringes are unobservable. At $\kappa = 0$, negativity grows with $\bar{n}$ (more photons → sharper fringes → more negative Wigner values). Any nonzero $\kappa$ destroys the cat state, with the critical decay rate scaling as $\kappa_\text{crit} \sim g / \bar{n}$ — the decoherence rate is $\kappa\bar{n}$, not $\kappa$. (b) Fidelity $F = \langle\text{cat}^+|\rho_\text{field}|\text{cat}^+\rangle$ against the ideal even cat state $|\text{cat}^+\rangle = \mathcal{N}(|\alpha\rangle + |-\alpha\rangle)$. High fidelity is concentrated at small $\bar{n}$ and small $\kappa/g$, confirming that the JC interaction produces near-ideal cat states only in the few-photon regime where the rotating-wave approximation is excellent.

1D slices through the phase diagram. Left: $\delta$ vs $\bar{n}$ at fixed $\kappa/g$. Without dissipation ($\kappa = 0$, blue), negativity grows monotonically with $\bar{n}$ and saturates near $\delta \approx 0.9$ for $\bar{n} \gtrsim 15$. Even modest decay ($\kappa/g = 0.03$, orange) limits the useful range to $\bar{n} \lesssim 5$. The horizontal grey line marks $\delta = 0.05$. Right: $\delta$ vs $\kappa/g$ at fixed $\bar{n}$. The decay is approximately exponential, with the $1/e$ decay point scaling as $\kappa_{1/e} \propto g / \bar{n}$ — directly confirming the enhanced decoherence rate of macroscopic superpositions.

Equal-time second-order coherence $g^{(2)}(0)$ of the intracavity field as a function of the vacuum Rabi coupling $g/\kappa$, for several drive amplitudes $\varepsilon/\kappa$. At weak coupling ($g \ll \kappa$), the cavity acts as a passive filter and $g^{(2)}(0) \to 2$ (thermal statistics). As $g$ increases past $g \approx \kappa$ (strong-coupling threshold, red dashed line), the photon blockade mechanism activates: the anharmonic JC ladder prevents simultaneous absorption of two photons, driving $g^{(2)}(0) \to 0$. Weaker drives produce deeper antibunching because the blockade condition requires $\varepsilon \ll g$. Bottom: Mean intracavity photon number $\langle n \rangle$ — the blockade is accompanied by a dramatic suppression of transmission.

$g^{(2)}(0)$ as a function of drive strength $\varepsilon/g$ for several values of $g/\kappa$. At weak drive ($\varepsilon \ll g$), the photon blockade suppresses multi-photon occupation and $g^{(2)}(0) \ll 1$ (antibunched / sub-Poissonian). As the drive increases and overcomes the blockade, $g^{(2)}(0)$ rises through 1 (Poissonian) and can exceed 1 (bunched / super-Poissonian) before settling back toward the classical limit. Stronger coupling ($g/\kappa = 10$, green) maintains antibunching to higher drive powers.

Top: $g^{(2)}(0)$ as a function of laser-cavity detuning $\Delta/g$ for $g/\kappa = 5$. At the dressed-state resonances $\Delta = \pm g$ (red dashed lines), the drive is resonant with the $|0\rangle \to |1,\pm\rangle$ transitions, producing peaks in $\langle n \rangle$ but also sharp structure in $g^{(2)}(0)$. Between the two polariton peaks, the photon blockade produces a deep antibunching dip ($g^{(2)}(0) \ll 1$). The 2-photon resonance condition $\Delta \approx \pm g(\sqrt{2} - 1)$ (orange dashed lines) marks where two-photon absorption becomes possible, creating localized bunching features. Bottom: Vacuum Rabi doublet in the cavity transmission spectrum $\langle n \rangle(\Delta)$ — the splitting of $2g$ is the spectroscopic signature of strong coupling.

Four-panel summary of photon blockade physics: (a) blockade transition vs $g/\kappa$, (b) quantum-to-classical crossover vs drive strength, (c) blockade spectrum, (d) vacuum Rabi splitting in transmission.

Cross-section $W(x, p{=}0)$ through the Wigner function at the cat-state time $t = t_r/2$. Deep negative fringes reach $W \sim -0.22$, with fringe spacing $\sim \pi / \sqrt{2\bar{n}} \approx 0.7$.

Von Neumann entanglement entropy $S(\rho_\text{atom})$ for four initial field states with $\bar{n} = 10$. Coherent (blue): clean collapse to ~1 bit followed by a revival dip at $t_r$. Fock (orange): periodic oscillations at $2g\sqrt{n+1}$. Thermal (green): permanent saturation — no revival. Squeezed (red): intermediate. Only the coherent state supports rephasing for revivals.

The collapse time $t_c \sim 1/g$ is independent of $\bar{n}$, while $t_r = 2\pi\sqrt{\bar{n}}/g$ scales as $\sqrt{\bar{n}}$. Clean collapse-revival structure emerges for $\bar{n} \geq 9$.

Effect of cavity dissipation on atom-field entanglement. By $\kappa/g = 0.05$ the revival is absent entirely.

Atomic inversion $\langle\sigma_z\rangle(t) = \sum_n P(n)\cos(2g\sqrt{n{+}1},t)$ for coherent (Poisson) and thermal (Bose-Einstein) initial fields at $\bar{n} = 10$.

Resonance fluorescence spectrum via the quantum regression theorem. Sidebands at $\pm\Omega$ with HWHM $= 3\gamma/4$, peak ratio 3:1.

$W_n(x,p) = \frac{(-1)^n}{\pi} L_n(2r^2) e^{-r^2}$ for Fock states $|n\rangle$. Ring structure and negativity grow with $n$.

Note: The full theoretical framework — JC Hamiltonian derivation, rotating-wave approximation, dressed states and eigenvalues, collapse and revival timescales, Wigner function formalism, Lindblad master equation, and von Neumann entropy — is presented in the companion paper (Sections II–III). The sections below cover new theoretical material developed for the extended computational results in this repository.

The Husimi Q-function is defined as the diagonal matrix element of the density operator in the coherent-state basis:

$$Q(\alpha) = \frac{1}{\pi}\langle\alpha|\rho|\alpha\rangle$$

Since $\rho$ is a positive operator and $|\alpha\rangle$ is a normalized state, it follows immediately that $Q(\alpha) \geq 0$ for all $\alpha$. This is in sharp contrast to the Wigner function, which can take negative values.

The precise relationship between $Q$ and $W$ is given by the Gaussian convolution theorem: writing $\alpha = x_\alpha + ip_\alpha$ in phase-space coordinates,

$$Q(x_\alpha, p_\alpha) = \frac{2}{\pi}\int!!\int W(x', p'),\exp!\left[-2\left((x' - x_\alpha)^2 + (p' - p_\alpha)^2\right)\right]dx',dp'$$

The convolution kernel is a Gaussian of variance $\sigma^2 = 1/4$ in each quadrature — precisely the vacuum-state Wigner function $W_{|0\rangle}(x,p) = \frac{1}{\pi}e^{-(x^2 + p^2)}$. In other words, $Q$ is obtained by smoothing $W$ with a minimum-uncertainty Gaussian of width equal to the vacuum fluctuation.

This smoothing has an irreversible information-theoretic consequence. Any phase-space feature of $W$ with a spatial frequency above $k_\text{max} \sim 1/\sigma = 2$ is exponentially attenuated. For a Schrödinger cat state with coherent-component separation $2|\alpha|$, the interference fringes in the Wigner function have spatial frequency $k_\text{fringe} = 2|\alpha|$. The ratio $k_\text{fringe}/k_\text{max} = |\alpha|$ determines the suppression factor: the fringes are damped by $\exp(-|\alpha|^2/2)$ in the Q-function. For $\bar{n} = |\alpha|^2 = 10$, this is a suppression of $e^{-5} \approx 0.007$, explaining why the Q-function shows two smooth blobs while the Wigner function resolves detailed oscillatory structure (as demonstrated in our Wigner vs Q comparison figures).

This can also be understood information-theoretically through the Wehrl entropy $h_W = -\int Q(\alpha)\ln Q(\alpha),d^2\alpha$, which satisfies the Lieb bound $h_W \geq 1$, with equality only for coherent states. The Wehrl entropy is always larger than the von Neumann entropy, $h_W \geq S(\rho) + 1$, with the gap quantifying the information lost by the Gaussian smoothing.

The practical implication is that Wigner negativity, not Q-function structure, is the correct witness of quantum coherence in phase space. The Q-function is useful for visualization but fundamentally cannot distinguish a quantum cat state $|\alpha\rangle + |-\alpha\rangle$ from a classical mixture $|\alpha\rangle\langle\alpha| + |-\alpha\rangle\langle-\alpha|$.

At time $t = t_r/2$, the JC interaction produces a cavity field state that closely approximates an even cat state. The ideal even cat state and its normalization are

$$|\text{cat}^+\rangle = \mathcal{N}_+\bigl(|\alpha\rangle + |-\alpha\rangle\bigr), \qquad \mathcal{N}_+ = \frac{1}{\sqrt{2(1 + e^{-2|\alpha|^2})}}$$

The Wigner function of this state decomposes exactly into three terms:

$$W_\text{cat}(x,p) = \frac{\mathcal{N}_+^2}{2}\Bigl[W_\alpha(x,p) + W_{-\alpha}(x,p)\Bigr] + \mathcal{N}_+^2,W_\text{int}(x,p)$$

where $W_\alpha$ and $W_{-\alpha}$ are the Gaussian Wigner functions of the two coherent components, and the interference term is

$$W_\text{int}(x,p) = \frac{1}{\pi}\cos!\bigl(4,p,\mathrm{Re},\alpha\bigr),e^{-(x^2 + p^2)}$$

for $\alpha$ real. This interference term is responsible for the oscillatory fringes between the two lobes and is the sole source of Wigner negativity. The fringe spacing in the $p$-direction is

$$\Delta p_\text{fringe} = \frac{\pi}{2,\mathrm{Re},\alpha} = \frac{\pi}{2\sqrt{\bar{n}}}$$

which for $\bar{n} = 10$ gives $\Delta p \approx 0.50$. In the $x$-direction (along the cross-section $W(x, p{=}0)$), the fringes are determined by the overlap of the two Gaussian tails, with spacing $\Delta x \approx \pi/\sqrt{2\bar{n}} \approx 0.70$, consistent with our simulation result.

The maximum negative value of $W_\text{int}$ at the origin is $W_\text{int}(0,0) = -1/\pi$ when $4\cdot 0 \cdot \mathrm{Re},\alpha$ produces $\cos = -1$ (which occurs at $p = \pi/(4\mathrm{Re},\alpha)$). In practice, the Gaussian envelope $e^{-p^2}$ modulates this, giving the observed minimum $W \approx -0.22$ at $\bar{n} = 10$.

A key result is that larger cat states have finer fringes. Since $\Delta p \propto 1/\sqrt{\bar{n}}$, the fringe frequency grows with $\bar{n}$, making the interference pattern increasingly vulnerable to any smoothing process — whether instrumental (finite detector resolution) or physical (decoherence). This is the phase-space manifestation of the fragility of macroscopic superpositions.

We quantify the quality of the dynamically generated cat state using the fidelity against the ideal even cat:

$$F(\bar{n}, \kappa/g) = \langle\text{cat}^+|,\rho_\text{field}(t_r/2),|\text{cat}^+\rangle$$

Our phase diagram shows that $F > 0.5$ is achievable only for $\bar{n} \lesssim 4$ at $\kappa = 0$. The fidelity is low even without dissipation at large $\bar{n}$ because the JC interaction does not produce a perfect cat state — the anharmonic $\sqrt{n+1}$ spectrum introduces phase errors that grow with the width of the photon-number distribution.

The paper cites the decoherence rate $\Gamma_\text{dec} \sim \kappa\bar{n}$ following Zurek [22]. Here we derive it explicitly from the Lindblad dynamics.

Consider the master equation $\dot{\rho} = \kappa\mathcal{D}[a]\rho$ acting on the cavity field alone (ignoring the atom for this argument). We wish to compute the decay rate of the off-diagonal coherence $\langle\alpha|\rho|-\alpha\rangle$ between two coherent states separated by $2\alpha$ in phase space.

Using the coherent-state matrix elements of the Lindblad superoperator, and the eigenvalue relation $a|\alpha\rangle = \alpha|\alpha\rangle$:

$$\frac{d}{dt}\langle\alpha|\rho|-\alpha\rangle = \kappa\left[\alpha(-\alpha)^* - \frac{|\alpha|^2 + |-\alpha|^2}{2}\right]\langle\alpha|\rho|-\alpha\rangle$$

$$= \kappa\left[-|\alpha|^2 - |\alpha|^2\right]\langle\alpha|\rho|-\alpha\rangle = -2\kappa|\alpha|^2,\langle\alpha|\rho|-\alpha\rangle$$

This gives the exact result:

$$\langle\alpha|\rho(t)|-\alpha\rangle = \langle\alpha|\rho(0)|-\alpha\rangle,\exp!\left(-2\kappa|\alpha|^2,t\right)$$

The decoherence rate is therefore

$$\boxed{\Gamma_\text{dec} = 2\kappa|\alpha|^2 = 2\kappa\bar{n}}$$

This is a factor of $2\bar{n}$ faster than the energy decay rate $\kappa$. The physical interpretation is that each photon lost from the cavity carries "which-component" information about the cat state: a measurement of the photon's phase would distinguish $|\alpha\rangle$ from $|-\alpha\rangle$. The rate of such information leakage is proportional to the photon number.

The decoherence time is

$$t_\text{dec} = \frac{1}{2\kappa\bar{n}}$$

For the cat state to be observable, we need $t_\text{dec} \gtrsim t_r/2$, i.e., the cat must survive long enough to form. This requires

$$\frac{1}{2\kappa\bar{n}} \gtrsim \frac{\pi\sqrt{\bar{n}}}{g} \qquad \Longrightarrow \qquad \frac{\kappa}{g} \lesssim \frac{1}{2\pi\bar{n}^{3/2}}$$

For $\bar{n} = 10$, this gives $\kappa/g \lesssim 0.005$, consistent with the $\delta = 0.05$ contour in our phase diagram lying near $\kappa/g \approx 0.01$ at $\bar{n} = 10$. The approximate $\kappa_\text{crit} \propto \bar{n}^{-3/2}$ scaling (combining the $\bar{n}$-dependent decoherence rate with the $\bar{n}^{1/2}$-dependent formation time) is steeper than the naive $\propto 1/\bar{n}$ estimate, reflecting the double penalty of requiring both a longer formation time and surviving faster decoherence.

The equal-time second-order correlation function is defined as

$$g^{(2)}(0) = \frac{\langle a^\dagger a^\dagger a,a\rangle}{\langle a^\dagger a\rangle^2} = \frac{\langle \hat{n}(\hat{n}-1)\rangle}{\langle \hat{n}\rangle^2} = 1 + \frac{\mathrm{Var}(\hat{n}) - \langle\hat{n}\rangle}{\langle\hat{n}\rangle^2}$$

This classifies photon statistics:

The connection to the paper's Sec. V is that sub-Poissonian statistics ($g^{(2)}(0) < 1$) and photon antibunching are among the definitive criteria for non-classical light. Here we develop the full theory of how the JC nonlinearity produces antibunching via the photon blockade.

Adding a coherent drive of amplitude $\varepsilon$ at frequency $\omega_L$ and moving to the rotating frame:

$$H_\text{driven} = \Delta_c,a^\dagger a + \frac{\Delta_a}{2}\sigma_z + g\bigl(a^\dagger\sigma^- + a,\sigma^+\bigr) + \varepsilon\bigl(a^\dagger + a\bigr)$$

where $\Delta_c = \omega_c - \omega_L$ and $\Delta_a = \omega_a - \omega_L$. The steady state $\rho_\text{ss}$ is found from the Lindblad equation $0 = -i[H, \rho_\text{ss}] + \kappa\mathcal{D}[a]\rho_\text{ss} + \gamma\mathcal{D}[\sigma^-]\rho_\text{ss}$, and $g^{(2)}(0)$ is computed directly from $\rho_\text{ss}$.

The photon blockade arises from the anharmonicity of the JC energy ladder. On resonance ($\Delta_c = \Delta_a = 0$), the dressed-state energies (from the paper's Eq. 14) give transition frequencies:

$$\omega_{0 \to 1,\pm} = \omega_c \pm g \qquad \text{(first photon)}$$

$$\omega_{1,\pm \to 2,\pm} = \omega_c \pm g(\sqrt{2} - 1) \quad \text{or} \quad \omega_c \pm g(2 - \sqrt{2}) \qquad \text{(second photon)}$$

The energy cost of the first photon differs from the second by

$$\delta E_\text{block} = \hbar g\bigl(2 - \sqrt{2}\bigr) \approx 0.59,\hbar g$$

A drive laser tuned to the first transition (say $\omega_L = \omega_c - g$) is therefore detuned from the second transition by $\delta E_\text{block}/\hbar$. If this detuning exceeds the dressed-state linewidth ($\sim \kappa$), the second photon cannot be absorbed while the first is still in the cavity. This is the photon blockade.

The blockade condition is:

$$\frac{g(2 - \sqrt{2})}{\kappa} \gg 1 \qquad \Longleftrightarrow \qquad \frac{g}{\kappa} \gg \frac{1}{2 - \sqrt{2}} \approx 1.71$$

In practice, our simulations show that $g^{(2)}(0) < 0.1$ requires $g/\kappa \gtrsim 5$ at weak drive ($\varepsilon/\kappa = 0.05$), accounting for the finite linewidth and multi-level structure.

At fixed $g/\kappa$, increasing the drive $\varepsilon$ eventually overwhelms the blockade. In the limit $\varepsilon \gg g$, the cavity is driven classically and $g^{(2)}(0) \to 1$. Our simulations show an intermediate regime where $g^{(2)}(0)$ can exceed 1 (bunching) before settling to the coherent-state value — this occurs because the multi-photon dressed states become populated non-thermally when the drive competes with the nonlinearity.

Sweeping the laser detuning $\Delta$ at fixed $g, \kappa, \varepsilon$ maps out the spectral structure of the blockade. The transmission spectrum $\langle n\rangle(\Delta)$ shows the vacuum Rabi doublet (peaks at $\Delta = \pm g$), while $g^{(2)}(0)(\Delta)$ reveals:

• Antibunching dips near $\Delta = 0$, between the two polariton peaks, where the anharmonic detuning $\delta E_\text{block}$ is maximal
• Bunching spikes at specific detunings where multi-photon resonances align, particularly near $\Delta = \pm g(\sqrt{2}-1)$ where the two-photon transition $|0\rangle \to |2,\pm\rangle$ becomes resonant

This spectral structure is the fingerprint of the quantized JC ladder and has been directly observed in circuit QED experiments.

The reduced atomic state is a $2 \times 2$ density matrix, completely characterized by the Bloch vector:

$$\mathbf{r} = \bigl(\langle\sigma_x\rangle,, \langle\sigma_y\rangle,, \langle\sigma_z\rangle\bigr) = \bigl(\mathrm{Tr}[\rho_\text{atom}\sigma_x],, \mathrm{Tr}[\rho_\text{atom}\sigma_y],, \mathrm{Tr}[\rho_\text{atom}\sigma_z]\bigr)$$

A pure atomic state lies on the surface of the Bloch sphere ($|\mathbf{r}| = 1$); a maximally mixed state sits at the center ($|\mathbf{r}| = 0$). The Bloch vector length is related to the atomic purity by

$$\mathrm{Tr}[\rho_\text{atom}^2] = \frac{1 + |\mathbf{r}|^2}{2}$$

and to the von Neumann entropy by the binary entropy function:

$$S(\rho_\text{atom}) = h!\left(\frac{1 + |\mathbf{r}|}{2}\right), \qquad h(p) = -p\log_2 p - (1{-}p)\log_2(1{-}p)$$

This provides a geometric interpretation of entanglement dynamics: the atom's entanglement with the field is encoded in how far the Bloch vector has retreated from the surface toward the center. At the collapse time, $|\mathbf{r}|$ drops rapidly from 1 to near 0 as the atom becomes maximally entangled with the field. At the cat-state time $t = t_r/2$, the Bloch vector is at the origin — the atom is in a maximally mixed state, and the field is in a cat state. These are two descriptions of the same physical event: maximal bipartite entanglement.

The trajectory itself is not a simple radial contraction. In 3D, the Bloch vector traces a spiral because the coherent Rabi dynamics (rotation about an axis in the $xz$-plane) compete with the dephasing caused by the spread of Fock-state Rabi frequencies. The spiral structure is visible in our Bloch sphere animations: at early times the atom precesses rapidly (Rabi oscillations), while the envelope of the spiral contracts (collapse). The spiral partially re-expands at the revival, but does not return to the surface due to residual entanglement.

The photon number distribution $P(n,t) = \langle n|\rho_\text{field}(t)|n\rangle$ provides complementary information to the Wigner function. For the initial state $|e\rangle \otimes |\alpha\rangle$ evolving under the resonant JC Hamiltonian, the reduced field state at time $t$ has diagonal elements

$$P(n,t) = \sum_{m=0}^{\infty} p(m)\left|\langle n|\bigl[\cos(g\sqrt{m{+}1},t)|m\rangle\langle m| + \ldots\bigr]|\alpha\rangle\right|^2$$

At $t = 0$, $P(n,0) = e^{-\bar{n}}\bar{n}^n/n!$ (Poisson). At $t = t_r/2$, the conditional dynamics split the distribution into two peaks. This can be understood as follows: at the cat-state time, the field is approximately $|\alpha_+\rangle + |\alpha_-\rangle$ with $\alpha_\pm$ separated by $\sim 2\sqrt{\bar{n}}$ in amplitude. The photon-number distribution of such a superposition is

$$P_\text{cat}(n) \approx \frac{1}{2}\left[P_{\alpha_+}(n) + P_{\alpha_-}(n)\right] + \text{interference}$$

The two Poisson distributions centered at $|\alpha_+|^2$ and $|\alpha_-|^2$ produce the bimodal structure visible in our simulations, with peak separation $\sim 2\sqrt{\bar{n}} \approx 6$ photons at $\bar{n} = 10$. The interference terms produce an even-odd oscillation: for an ideal even cat state, only even Fock numbers are populated ($P(n) = 0$ for odd $n$). In the JC-generated cat state, this even-odd asymmetry is approximate but measurable — it provides an experimentally accessible signature of cat-state parity that does not require full Wigner tomography.

At the revival time $t_r$, the distribution partially recombines toward a unimodal shape, but the anharmonicity of the JC spectrum ($\sqrt{n+1}$ rather than linear) prevents perfect recurrence. The distribution at $t = t_r$ is broader and more irregular than the initial Poissonian, with a variance that exceeds $\bar{n}$ — the field has acquired super-Poissonian statistics through its interaction with the atom.

The 2D parameter sweep over $(\bar{n}, \kappa/g)$ is the central new computational result. The phase diagram encodes two competing effects:

Negativity grows with $\bar{n}$ at $\kappa = 0$. In the ideal (lossless) case, larger $\bar{n}$ produces cat states with sharper interference fringes and larger $\delta$. The negativity saturates near $\delta \approx 0.9$ for $\bar{n} \gtrsim 15$, approaching the ideal even-cat-state value.

Decoherence accelerates with $\bar{n}$ at $\kappa > 0$. From Sec. 3, the decoherence rate $\Gamma_\text{dec} = 2\kappa\bar{n}$ grows linearly with $\bar{n}$, while the formation time $t_r/2 = \pi\sqrt{\bar{n}}/g$ grows as $\sqrt{\bar{n}}$. The product $\Gamma_\text{dec} \cdot t_r/2 = 2\pi\kappa\bar{n}^{3/2}/g$ grows as $\bar{n}^{3/2}$, meaning that larger cat states are exponentially harder to observe at any fixed nonzero $\kappa$.

The critical decay rate for cat-state survival scales as

$$\frac{\kappa_\text{crit}}{g} \sim \frac{C}{\bar{n}^{3/2}}$$

where $C$ is a threshold constant that depends on the observability criterion for $\delta$. Our phase diagram confirms this: the $\delta = 0.05$ contour follows an approximate $\kappa \propto \bar{n}^{-3/2}$ power law, steeper than the naive $1/\bar{n}$.

The parameter slices provide practical design guidance. To observe a cat state with $\delta > 0.05$ at $\bar{n} = 10$, one requires $\kappa/g < 0.02$, i.e., $g/\kappa > 50$. This condition is achievable in microwave cavity QED ($g/\kappa \sim 10^4$) and circuit QED ($g/\kappa \sim 100\text{–}300$), but remains challenging in optical cavities where typical $g/\kappa \sim 1\text{–}10$.

git clone https://github.com/alanknguyen/QSOL_CQED.git
cd QSOL_CQED
pip install -r requirements.txt

Static figures:

cd simulations
python sim_wigner_evolution.py            # ~2 min
python sim_entanglement_dynamics.py       # ~5 min
python sim_avoided_crossing.py            # ~30 sec
python sim_photon_number_distribution.py  # ~1 min
python sim_bloch_sphere.py                # ~2 min
python sim_q_vs_wigner.py                 # ~3 min
python sim_phase_diagram.py              # ~10 min (256-point sweep)
python sim_g2_coherence.py               # ~5 min
python jaynes_cummings_comparison.py      # ~10 sec
python mollow_triplet.py                  # ~1 min
python wigner_fock_states.py              # ~5 sec

Animations:

cd animations
python anim_wigner_evolution.py           # ~5 min
python anim_entanglement.py              # ~5 min
python anim_decoherence.py              # ~3 min
python anim_avoided_crossing.py          # ~2 min
python anim_photon_number.py             # ~3 min
python anim_bloch_sphere.py              # ~3 min
python anim_q_vs_wigner.py              # ~8 min
python anim_phase_diagram.py            # ~5 min
python anim_g2_blockade.py              # ~3 min
python thumbnail_banner.py              # ~30 sec

@article{nguyen2025quantum,
  author  = {Nguyen, Nguyen Khoi},
  title   = {Quantum States of Light in Cavity {QED}: A Computational Study
             of {Wigner} Function Dynamics and Atom-Field Entanglement
             in the {Jaynes-Cummings} Model},
  journal = {arXiv preprint arXiv:XXXX.XXXXX},
  year    = {2025},
}

MIT. See LICENSE.

Prepared under the guidance of Prof. Luca Dal Negro at Boston University (EC 585, EC 777). Simulations use QuTiP by J. R. Johansson, P. D. Nation, and F. Nori.