Multiple wave packets running in the photon number space

If a two-level system coupled to a single-mode cavity is strongly driven by an external laser, instead of a continuous accumulation of photons in the cavity, oscillations in the mean photon number occur. These oscillations correspond to peaks of finite width running up and down in the photon number distribution, reminiscent of wave packets in linear chain models. A single wave packet is found if the cavity is resonant to the external laser. Here, we show that for finite detuning, multiple packet structures can exist simultaneously, oscillating at different frequencies and amplitudes. We further study the influence of dissipative effects resulting in the formation of a stationary state, which, depending on the parameters, can be characterized by a bimodal photon number distribution. While we give analytical limits for the maximally achievable photon number in the absence of any dissipation, surprisingly, dephasing processes can push the photon occupations towards higher photon numbers.

Figure 1

(a) Time-dependent mean photon number and (b) the absolute value of its Fourier transform calculated from solutions of Eq. (2) using a driving strength of $f=5\text{g}$ and detunings $\delta =0\text{g}$ (blue line) and $\delta =0.1\text{g}$ (red line). (c)–(f) Photon number distribution $P_n\left(t\right)$ (left panel) at fixed time and the corresponding Wigner function $W\left(z\right)$ obtained from solutions of Eq. (2) with a driving strength of $f=5\text{g}$ and detunings $\delta =0\text{g}$ (c),(d) and $\delta =0.1\text{g}$ (e),(f). A Poisson distribution (blue boxes) is fit to each packet in the photon number distribution by calculating the mean and norm of the individual packets directly from the numerical solution (gray bars). (g),(h) Photon number distribution $P_n\left(t\right)$ resolved in both photon number $n$ as well as time $t$ obtained from solutions of Eq. (2) with a driving strength of $f=5\text{g}$ and detunings $\delta =0\text{g}$ (g) and $\delta =0.1\text{g}$ (h).

Figure 2

(a),(b) Energy level diagrams in the laser- and cavity-dressed state bases, respectively, in the case of a positive detuning $\delta >0$. The blue arrows mark energy differences between states, whereas the red ones indicate possible transitions. (c) Sketch of the eigenfrequencies ${\omega }_{n,\pm }$ of the cavity-dressed states as a function of the excitation number $n$ for multiple values of the detuning $\delta \ge 0$. The blue (red) background marks the area of wavelike propagation $0\ge {\omega }_{n,-}\ge -2f$ ($0\le {\omega }_{n,+}\le 2f$) of the wave packet corresponding to an initial state ${\mathrm{\Phi }}_0^{+}$ (${\mathrm{\Phi }}_0^{-}$). The accordingly colored markers indicate the respective upper turning points.

Figure 3

Photon number distribution $P_n\left(t\right)$ resolved in both photon number $n$ as well as time $t$ (right) and at fixed time $gt=18$ (left) with a driving strength of $f=5g$ and a detuning of $\delta =g^2/f=0.2g$. The gray solid line indicates $gt=18$, while the red dashed lines mark the intermediate turning point ${\tilde{N}}_{-}^{<}$ given by Eq. (27a) and the upper turning point ${\tilde{N}}_{-}^{>}$ given by Eq. (27b).

Figure 4

Maximum of the mean photon number (solid line) as a function of the detuning $\delta$ with a driving strength of $f=5\text{g}$ compared to the turning points in the CDS-decoupled regime given by Eqs. (25b) and (26) (dashed line). The gray dashed line indicates the amplitude of the approximate mean photon number in the LDS-decoupled regime shown in Eq. (12), whereas the red solid line shows the position $\delta =-g^2/f$ of the area around which the split of packets occurs. All calculations are performed with initial state $\mathrm{\Psi }\left(0\right)={\mathrm{\Phi }}_0^{+}$. The inset shows the same results plotted for a smaller range of $\delta$ and higher photon numbers.

Figure 5

Photon number distribution $P_n\left(t\right)$ calculated by numerically integrating the Liouville-von Neumann equation (4) with a driving strength of $f=5g$ and cavity loss rate $\kappa =0.01g$. Any dissipation of the TLS itself is neglected ${\gamma }_{\mathrm{RD}}={\gamma }_{\mathrm{PD}}=0$. Both the stationary distribution in dependence of the detuning $\delta$ (a) as well as its temporal evolution for values of the detuning of $\delta =0.02g$ (b) and $\delta =0.05g$ (c) are shown. In (a), the white dashed lines are graphs of the analytical expressions for the mean photon number of the packets given in Eqs. (31) and (32), whereas the blue line indicates the parameter region $\delta =g^2/4f$ [cf. Eq. (30)] around which two packets are present in the stationary distribution.

Figure 6

(a),(b) Dynamical evolution of the photon number distribution $P_n\left(t\right)$ with a driving strength of $f=5\text{g}$ and a detuning $\delta =0.1\text{g}$. Any dissipation is neglected with the exception of radiative decay with ${\gamma }_{\mathrm{RD}}=0.1\text{g}$ (a) or pure dephasing with ${\gamma }_{\mathrm{PD}}=0.05\text{g}$ (b). (c) Wigner function $W\left(z\right)$ at fixed time $gt=20$ starting from the initial state $\mathrm{\Psi }\left(0\right)={\mathrm{\Phi }}_0^{\mathrm{G}}$ in the LDS-decoupled regime ($f=10\text{g}, \delta =0.2\text{g}$) without any dissipation. (d) Difference between the Wigner functions with ${\gamma }_{\mathrm{PD}}=0.005\text{g}$ and without any dissipation at fixed time $gt=20$ starting from the initial state $\mathrm{\Psi }\left(0\right)={\mathrm{\Phi }}_0^{+}$ in the LDS-decoupled regime ($f=10, \delta =0.2\text{g}$).

Figure 7

Photon number distribution $P_n\left(t\right)$ with a driving strength of $f=5\text{g}$, a detuning $\delta =0.05\text{g}$, and decay rates $\kappa =0.01\text{g}, {\gamma }_{\mathrm{RD}}=0.005\text{g}$, and ${\gamma }_{\mathrm{PD}}=0.0025\text{g}$.