$N$-photon bundle statistics on different solid-state platforms

The term $N$-photon bundles has been coined for a specific type of photon emission, where light quanta are released from a cavity only in groups of $N$ particles. This emission leaves a characteristic number distribution of the cavity photons that may be taken as one of their fingerprints. We study this characteristic $N$-photon bundle statistics considering two solid-state cavity quantum electrodynamics (cQED) systems. As one example, we consider a semiconductor quantum-dot–microcavity system coupled to longitudinal acoustic phonons. There, we find the environmental influence to be detrimental to the bundle statistics. The other example is a superconducting qubit inside a microwave resonator. In these systems, pure dephasing is not important and an experimentally feasible parameter regime is found, where the bundle statistics prevails.

Figure 1

Sketch of a two-level system (2LS) embedded in a cavity resulting in the coupling to one cavity mode. The 2LS is driven by a continuous external excitation. It can decay radiatively, while the cavity is lossy. For particular sets of parameters, $N$-photon bundles leave the cavity. They are characterized by the specific temporal spacing between the constituent photons and their specific photon number statistics. Exemplary, four five-photon bundles are depicted.

Figure 2

Schematic sketch of the $N$-bundle mechanism. Black lines indicate the energetic position of systems states $|\pm ,n\rangle$ where the driven 2LS is in the upper (lower) laser-dressed state $|+\rangle \approx |X\rangle$ ($|-\rangle \approx |G\rangle$) and $n\in {\mathbb{N}}_{\text{0}}$ photons are inside the resonator. Note that in the rotating frame the contribution of a resonator photon $\hslash \mathrm{\Delta }{\omega }_{\text{CL}}$ to the total energy is negative in the regime where bundles are found (cf. Fig 3). The states $|-,0\rangle$ and $|+,N\rangle$ are in resonance and the resonator introduces an effective coupling ${\mathcal{V}}_{\mathcal{\text{eff}}}$ between them (blue arrows). Green (dashed black) arrows indicate the action of the resonator loss (radiative decay) with rate $n\kappa$ ($\gamma$) on a system state $|\pm ,n\rangle$.

Figure 3

Stationary probability $P_n$ of occupying the photon number states $|n\rangle$ in the QD–cavity system as a function of the laser–exciton detuning $\mathrm{\Delta }{\omega }_{\text{LX}}$ (a) without taking phonon effects into account, (b) including phonons initially at $T=4\mathrm{K}$ (the insets show the region marked by yellow boxes on a larger scale), (c) the corresponding energies of the laser-dressed states $|+\rangle$ and $|-\rangle$. The energy of a photon in the rotating frame is given by the cavity–laser detuning $\hslash \mathrm{\Delta }{\omega }_{\text{CL}}$, which is plotted as a shaded area to illustrate its modulus. Arrows indicate the number of photons involved in the processes leading to the various resonance peaks, while their length corresponds to their energy $\hslash \mathrm{\Delta }{\omega }_{\text{CL}}$. The circular arrow indicates a one-photon process with a photon energy (in the rotating frame) of $\hslash \mathrm{\Delta }{\omega }_{\text{CL}}=0$. The blue lines above panel (a) mark the energetic positions of the bundle resonances, starting for $N=1$ and quickly converging to $\hslash \mathrm{\Delta }{\omega }_{\text{CX}}$ for larger $N$. Since the bundle resonance is derived from the condition that $N$ cavity photons energetically fit between the two dressed states, an equation analogous to Eq. (7) can be found for the trivial case $N=1$.

Figure 4

The stationary probability $P_n$ of occupying the photon number states $|n\rangle$ normalized to its value at $n=1$ for the QD–cavity system. While the data labeled 'realistic losses' is obtained using the parameters listed in Sec. 2a1, weaker losses of $\gamma =0.01g$ and $\kappa =0.1g$ were chosen following Ref. [27] for the calculation shown in gray. Note that in the phonon-free case, the absolute values of the Fock state with $n=1$ are 0.016 for the weaker losses and 0.003 for the realistic parameter set.

Figure 5

The stationary ratio between the two- and the one-photon occupation probability in the QD–cavity system with the phonon influence approximated by a Lindblad operator with a phenomenological pure dephasing rate ${\gamma }_{\varphi }$ instead of the microscopic Hamiltonian $H_{\text{Ph}}$ in Eq. (6), cf. main text. (a) ${\gamma }_{\varphi }$ corresponding to the full driven Jaynes–Cummings model at $T=4\mathrm{K}$. (b) ${\gamma }_{\varphi }$ corresponding to a Jaynes–Cummings dynamics with $n=1$. (c) ${\gamma }_{\varphi }$ corresponding to a Jaynes–Cummings dynamics with $n=2$.

Figure 6

The stationary probability $P_n$ of occupying the photon number states $|n\rangle$ normalized to its value at $n=1$ for the superconducting qubit–microwave system. The data labeled $\kappa \ll \gamma$ is obtained using the parameters from Sec. 2a2. In dark blue, the result of a calculation with a cavity loss rate two orders of magnitude larger than in Sec. 2a2 is shown, namely $\kappa =0.1g=7.76\gamma$, cf. Fig. 4.

Figure 7

The stationary ratios of the two- to one-photon occupation probabilities and of the three- to one-photon occupation probabilities as functions of the resonator loss rate $\kappa$ (in units of $\gamma$) for the superconducting qubit–microwave system. The two vertical black lines mark those values of $\kappa$, which are used to obtain the corresponding data in Fig. 6. The dotted black line shows the target value of 0.5 for the ratio $r$.

Figure 8

Stationary probability $P_n$ of occupying the photon number states $|n\rangle$ in the QD–cavity system as a function of the laser-exciton detuning $\mathrm{\Delta }{\omega }_{\text{LX}}$ without taking phonon effects into account. This is a magnification of the resonance peak for $N\to \infty$ in Fig. 3. On this scale, the double-peak structure and the reversal of the photon order at its center are well visible.