Quantum jumps in amplitude bistability: Tracking a coherent and invertible state localization

TL;DR

This work analyzes quantum jumps between macroscopic metastable light states in the open driven Jaynes–Cummings model, focusing on the zero-system-size (strong-coupling) limit where spontaneous emission vanishes. It reveals a two-stage downward switching mechanism: a coherent localization to an unstable intermediate state, followed by decay to vacuum, with an invertible null-measurement record enabling time-reversed interpretation around the localization midpoint. The study contrasts JC bistability with Kerr dynamics, explores multiphoton resonances as precursors to amplitude bistability, and discusses how photon correlations, heterodyne readouts, and measurement context shape the observed jumps. Practical implications include using coherent-state localization as a basis for projective measurements via meter states and assessing coherence under realistic experimental nonidealities such as detector inefficiency and thermal baths. Overall, the results provide a detailed, trajectory-level account of bistable switching in strongly nonlinear light-matter systems and propose concrete experimental probes to observe coherent localization and its inversion.

Abstract

We investigate the nature of quantum jumps occurring between macroscopic metastable states of light in the open driven Jaynes-Cummings model. We find that, in the limit of zero spontaneous emission considered in [H. J. Carmichael, Phys. Rev. X 5, 031028 (2015)], the jumps from a high-photon state to the vacuum state entail two stages. The first part is coherent and modelled by the localization of a state superposition, in the example of a null-measurement record predicted by quantum trajectory theory. The underlying evolution is mediated by an unstable state (which often splits to a complex of states), identified by the conditioned density matrix and the corresponding quasiprobability distribution of the cavity field. The unstable state subsequently decays to the vacuum to complete the jump. Coherence in the localization allows for inverting the null-measurement photon average about its initial value, to account for the full switch which typically lasts a small fraction of the cavity lifetime; an asymptotic law for the jump time is established in high-amplitude bistability. This mechanism is contrasted to the jumps leading from the vacuum to the high-photon state in the bistable signal. Spontaneous emission degrades coherence in the localization, and prolongs the jumps.

Figures (12)

• Figure 1:
• Figure 2: Low-amplitude quantum bistable switching.(a) Sample realization of the conditioned photon number $\langle n_{\rm REC}(t)\rangle$ for $800$ (average) cavity lifetimes, showing bistable switching between the two metastable states, B and D. The upper (lower) row of vertical strokes indicates photon emissions (upward jumps). The inset depicts a schematic contour plot of $Q_{\rm ss}(x+iy)$(see Appendix \ref{['sec:MC']} and Ref. Tan1999). (b) Focus on the continuous segment of the B$\to$D jump plotted against the dimensionless time and modelled by a coherent localization which takes place between a bright state (B) with amplitude $\alpha_1=0.3-2.38i$ and an unstable state (U) with amplitude $\alpha_2=-0.175-0.53i$. The average conditioned photon number $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$ is plotted in blue, the part of $\langle n(t)\rangle_{\rm REC,\,NULL}$ [Eq. \ref{['eq:ntexact']}] from $t_{\rm mid}$ to $t^{\prime}\equiv(t_{\rm mid}+\Delta t_{\rm end})$ is plotted in grey, while the time-inverted part of $n_{\rm REC,\,NULL}(t)$ from $t_{\rm mid}$ to $t^{\prime \prime}\equiv(t_{\rm mid}-\Delta t_{\rm end})$ is plotted in orange. The pink square marks the inversion point. The inset plots the entire realization of $\langle n(t)\rangle_{\rm REC}$ for $800$ cavity lifetimes, where the jump B$\to$D is indicated by the left-pointing arrow in (a). The three insets (i--iii) depict contour plots of $Q_{\rm REC}(x+iy;t_k)$ corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=3.02, 2.92$ and $2.70$, for $k=1{\rm (i)}, 2{\rm (ii)}, 3{\rm (iii)}$, respectively. (c) Focus on the jump D$\to$B indicated by the right-pointing arrow in (a). Orange strokes mark upward jumps (the pair of emissions at $\kappa t \approx 590$ is not resolved on the scale of the figure). The two insets (i, ii) depict contour plots of the conditional distribution $Q_{\rm REC}(x+iy;t_k)$ corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=4.07$ and $7.08$, for $k=1 {\rm (i)}, 2{\rm (ii)}$, respectively. B, D, U, denote the bright, dim and unstable states, respectively. The operating parameters read: $g/\kappa=25$, $\varepsilon/\kappa=5.3$ and $\Delta\omega/\kappa=-8$.
• Figure 3: State localization in high-amplitude bistability.(a) Focus on the continuous segment of the B$\to$D jump, plotted against the dimensionless time and modelled by a coherent localization which takes place between a bright state with amplitude $\alpha_1=1.7-5.15i$ and an unstable state with amplitude $\alpha_2=-2.25-0.2i$. The average conditioned photon number $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$ is plotted in blue, the part of $\langle n(t)\rangle_{\rm REC,\,NULL}$ [Eq. \ref{['eq:ntexact']}] from $t_{\rm mid}$ to $t^{\prime}$ is plotted in grey, while the time-inverted part of $n_{\rm REC,\,NULL}(t)$ from $t_{\rm mid}$ to $t^{\prime \prime}$ is plotted in orange. The pink square frames the inversion point. The inset depicts the corresponding realization during $800$ cavity lifetimes in which the B$\to$ D jump in question is indicated by the arrow. (b) Broader perspective of the localization B$\to$U in question, continued beyond $t^{\prime}=(t_{\rm mid}+\Delta t_{\rm end})$ to show the decay to D. The inset zooms out even further and also indicates the up jumps (red strokes) as well as the scale of an average photon lifetime. Dominant states are marked along the jump. (c--e) Contour plots of $Q_{\rm REC}(x+iy;t_k)$, corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=19.39, 16.51$ and $11.19$, for $k=1, 2, 3$ in frames (c, d, e), respectively, with reference to frame (a). Schematic barplots of the real (Re) and imaginary (Im) parts of the corresponding conditioned cavity density matrix $(\rho_{\rm cav;\, REC}(t_k))_{mn}$ are drawn for the times $t_2, t_3$, to the right of frames (d, e), respectively. (f) Schematic contour of the steady-state distribution $Q_{\rm ss}(x+iy)$. B, D, U, denote the bright, dim and unstable states, respectively. The operating parameters read: $g/\kappa=60$, $\varepsilon/\kappa=13.5$ and $\Delta\omega/\kappa=-8$.
• Figure 4: An alternative localization between fluctuating states.(a) Focus on the quasicontinuous segment of the B$\to$D jump (interrupted only by one cavity emission) in $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$, plotted against the dimensionless time and modelled by a coherent localization which takes place between a bright state with amplitude $\alpha_1=1.8-5.45i$ and an unstable state with amplitude $\alpha_2=-1.8$. The average conditioned photon number $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$ is plotted in blue, the part of $\langle n(t) \rangle_{\rm REC,\,NULL}$ [Eq. \ref{['eq:ntexact']}] from $t_{\rm mid}$ to $(t_{\rm mid}+t_{\rm end})$ is plotted in grey, while the time-inverted part from $t_{\rm mid}$ to $(t_{\rm mid}-t_{\rm end})$ is plotted in orange. The pink square frames the inversion point. Inset (i) plots a schematic contour of the steady-state Q function $Q_{\rm ss}(x, y)$, and inset (ii) depicts the corresponding realization during $800$ cavity lifetimes in which the B$\to$ D jump in question is indicated by the arrow. The strokes underneath indicate the photon emissions corresponding to upward jumps. (b--e) Contour plots of the conditioned distributions $Q_{\rm REC}(x+iy;t_k)$ corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=23.06, 18.04, 14.60$ and $7.80$, for $k=1, 2, 3, 4$ in frames (b, c, d, e), respectively, with reference to frame (a). Schematic barplots of the real (Re) and imaginary (Im) parts of the corresponding conditioned cavity density matrix $(\rho_{\rm cav;\,REC}(t_k))_{mn}$ are drawn to the right of each frame. The operating parameters read: $g/\kappa=60$, $\varepsilon/\kappa=13.5$ and $\Delta\omega/\kappa=-8$. The Fock-state basis is truncated at $L_{\rm max}=60$.
• Figure 5: Upward jumps in the bistable switching.(a) Conditioned photon number $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$ focused on a fraction of a D$\to$B jump for $0.83$ cavity lifetimes, and indicated by the left arrow in (h). Orange strokes mark upward jumps. The framed inset plots a schematic contour of the steady-state Wigner function $W_{\rm ss}(x, y)$, and the arrow marks the excitation path. (b) Focus on the B$\to$D jump for $0.808$ cavity lifetimes, indicated by the right arrow in (h). The plot of Eq. \ref{['eq:ntexact']} is overlaid in gray, modelling a coherent localization between a bright state with amplitude $\alpha_1=2-4.2i$ and an unstable state with amplitude $\alpha_2=-1.95+1.55i$. The inverted part with respect to $\langle n(\kappa t_{\rm mid}=526.285)\rangle_{\rm REC,\, NULL}$ is plotted in orange, reaching $|\alpha_1|^2=21.64$. The framed inset depicts the schematic contour of the conditional $Q$-function corresponding to the maximum of $\langle n(t)\rangle^{\rm JCD}_{\rm REC}$. (c--e) Conditioned distributions $Q_{\rm REC}(x+iy;t_k)$ corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=8.37, 15.20$ and $29.31$, for $k=1, 2, 3$ in frames (c, d, e), respectively, with reference to the D$\to$B jump shown in frame (a). Schematic barplots of the real (Re) and imaginary (Im) parts of the corresponding conditioned cavity density matrix $(\rho_{\rm cav;\, REC}(t_k))_{mn}$ are drawn to the right of each quasiprobability distribution. (f--g) Contour plots of the conditioned distributions $Q_{\rm REC}(x+iy;t_k)$ corresponding to $\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=5.77$ and $11.08$, for $k=1, 2$ in frames (f, g), respectively, with reference to the B$\to$D jump shown in frame (b). These conditioned photon averages are attained immediately before [$\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=5.77$] and after [$\langle n(t_k)\rangle^{\rm JCD}_{\rm REC}=11.08$] the upward jump indicated by the arrow in (b). (h) The inset plots the entire trajectory of $800$ cavity lifetimes, where the D$\to$B jump of (a) is indicated by the left arrow and the the B$\to$D jump in (b) is indicated by the right arrow. Red strokes mark upward jumps. The operating parameters read: $g/\kappa=50$, $\varepsilon/\kappa=11.6$ and $\Delta\omega/\kappa=-8$. The Fock-state basis is truncated at $L_{\rm max}=50$.