Validity of adiabaticity in cavity QED

This paper deals with the concept of adiabaticity for fully quantum mechanical cavity QED models. The physically interesting cases of Gaussian and standing wave shapes of the cavity mode are considered. An analytical approximate measure for adiabaticity is given and compared with numerical wave packet simulations. Good agreement is obtained where the approximations are expected to be valid. Usually for cavity QED systems, the large atom-field detuning case is considered as the adiabatic limit. We, however, show that adiabaticity is also valid, for the Gaussian mode shape, in the opposite limit. Effective semiclassical time-dependent models, which do not take into account the shape of the wave packet, are derived. Corrections to such an effective theory, which are purely quantum mechanical, are discussed. It is shown that many of the results presented can be applied to time-dependent two-level systems.

Figure 1

The adiabaticity parameter ${\mathcal{A}}^0\left(x\right)$ as function of $x$ and $\Delta$ in scaled dimensionless units (starting from $\Delta =0.0001$) for a Gaussian coupling (38) with amplitude $A=1$, pulse width $a=50$, and initial momentum $p_0=10$.

Figure 2

The location of the maximum $\max \left\{{\mathcal{A}}^0\left(x\right)\right\}$ as a function of the dimensionless parameter $\Delta$ for three different couplings. The dimensionless pulse width is again $a=50$ and the initial momentum $p_0=10$.

Figure 3

The adiabaticity parameter ${\mathcal{A}}^0\left(x\right)$ as a function of $x$ and $\Delta$ (scaled units) for the short-optical-wavelength situation (42). It is seen that ${\mathcal{A}}^0\left(x\right)$ becomes singular at $x=0$ when $\Delta \to 0$. The other dimensionless parameters are $A=1$, $q=1$, and $p_0=2$.

Figure 4

This and the next Fig. 5 show the fidelity (50) as a function of $⟨x⟩$ and $\Delta$ for the Gaussian mode shape (38) with $A=10$ (a) and 1 (b). It is seen that the adiabaticity and fidelity is increased for smaller couplings $A$ when the detuning $\Delta$ becomes large. The other parameters are ${\Delta }_x=10$, $x_0=-200$, $p_0=5$, and $a=50$. All parameters are dimensionless.

Figure 5

The same as Fig. 4, but for small detunings. The inset shows the location of the maximum of the adiabaticity parameter $\mathcal{A}\left(x\right)$ as a function of the relevant detunings and the two coupling amplitudes $A=10$ (a) and 1 (b). The wave packet is propagated to $x=300$ not $x=200$ as in the previous figure. The reason is to see the drop in the fidelity around $x\approx 200$ where $\mathcal{A}\left(x\right)$ has a maximum. The inset indicates that wave packets propagating with larger $\Delta$ will experience more of the maximum of $\mathcal{A}\left(x\right)$. The other dimensionless parameters are as in Fig. 4.

Figure 6

The fidelity (50) as a function of $⟨x⟩$ and $\Delta$ for the short-optical-wavelength mode shape (42) with $A=1$ (a) and 0.1 (b). It is seen that the adiabaticity and fidelity is increased for smaller couplings $A$ when the detuning $\Delta$ becomes large. The other dimensionless parameters are ${\Delta }_x=10$ (the width of the wave packet covers several wavelengths), $x_0=-25$, $p_0=5$, and $q=1$.

Figure 7

The same as Fig. 6 but for a long wavelength; $\lambda =20\pi$ and ${\Delta }_x=4$. $A=\left(\mathrm{a}\right)$ 1 and (b) 0.1. Note that around $x=-10\pi ,0,10\pi$, the fidelity changes the most. The initial position and momentum are $x_0=-50$ and $p_0=5$. All parameters are dimensionless.

Figure 8

These figures show the adiabaticity parameter (32) calculated from wave packet simulations (solid line), the corresponding approximated result (35) (dashed line), and the same with the ${\partial }^2\theta$ term included in (35) (circles). In (a) and (b) the coupling amplitude is $A=100$ which makes $p$ vary considerably, while in (c) and (d) $A=1$ giving that $p\approx p_0$. The left plots show the large-detuning situation $\Delta =10$, while $\Delta =0.05$ for the right ones. Note that the dashed and dotted lines are very similar, indicating that the main contribution comes from the first derivative of $\theta$. The remaining parameters are (a) $p_0=0.6$, $x_0=-200$, (b) $p_0=1.5$, $x_0=-300$, (c) $p_0=1.5$, $x_0=-200$, (d) $p_0=1.5$, $x_0=-300$, and $a=50$ and ${\Delta }_x=10$. All parameters are scaled to dimensionless units.

Figure 9

This shows the same as the previous Fig. 8, but for the standing wave mode of Eq. (42). Again, the dashed line displays the approximated result (35) and the circles the same but with the ${\partial }^2\theta$ term added. The broadening of the peaks coming from averaging over the atomic wave packet is clearly seen in (b) where the peaks are much narrower. The solid line gives the results from a simulation of a wave packet with ${\Delta }_x=4$, while the dotted line displays the results for ${\Delta }_x=3$. The faster broadening of the initial narrower packet can be seen. The dimensionless parameters are (a) $A=0.1$, $\Delta =0.5$, (b) $A=1$, $\Delta =0.05$, and for both plots $x_0=-50$, $p_0=5$, and $q=0.1$.

Figure 10

The approximated adiabaticity parameter of Eq. (54), which takes into account the broadening of the wave packet, as a function of the scaled dimensionless time (dotted lines). The solid lines $a$ and $b$ give the numerical results for the situation using Eq. (43) in the calculation when $\Delta =0.05$ and 0.2, respectively, which shows the effect of the width of the “spikes” in the adiabaticity parameter.