Effective mass in cavity QED

We consider propagation of a two-level atom coupled to one electromagnetic mode of a high-$Q$ cavity. The atomic center-of-mass motion is treated quantum mechanically and we use a standing wave shape for the mode. The periodicity of the Hamiltonian leads to a spectrum consisting of bands and gaps, which is studied from a Floquet point of view. Based on the band theory, we introduce a set of effective mass parameters that approximately describe the effect of the cavity on the atomic motion, with the emphasis on one associated with the group velocity and on another one that coincides with the conventional effective mass. Propagation of initially Gaussian wave packets is also studied using numerical simulations and the mass parameters extracted thereof are compared with those predicted by the Floquet theory. Scattering and transmission of the wave packet against the cavity are further analyzed, and the constraints for the effective mass approach to be valid are discussed in detail.

Figure 1

The lowest lying bands (solid lines) of the Hamiltonian given by Eq. (3) for the first Bril- louin zone. The dressed energy bands are marked on the $y$ axis with the dominant bare state index $\mu$. Crosses shows the bare energy bands for excited states $\mid +⟩$ and diamonds bands for ground bare states $\mid -⟩$. In the last figure (d), the coupling is so strong that the bare and dressed energies starts differ considerable. The parameters are given on top of each figure.

Figure 2

The expansion coefficients $c_{\mu }^{\nu }$ for the four lowest dressed states $\nu =1$, 2, 3, 4, Eq. (10), of bare states $\mu$ for (a) $k=0$ and (b) $k=1∕4$. Black bars corresponds to even $\mu$’s with ground state atom, and white bars to odd $\mu$’s with excited atom. The coefficients are symmetric around $\mu =0$ only for $k_0=0$. The parameters are in both plots the same as for Fig. 1: $\Delta =0$ and $g_0=0.05$.

Figure 3

Error estimate $\delta (g_0,\Delta ;n)$ for $n=5$, as defined in Eq. (15). The quasi-momentum $k=0$. The errors for the $k=1∕4$ case is almost identical.

Figure 4

The evolution of (a) an initial Gaussian dressed state and (b) an initial Gaussian bare state. The dressed state stays localized around an average position, while the bare state clearly splits up. The three subpackets in (b) correspond to the bare states $\mid -,k_0⟩,\mid +,k_0-q⟩$ and $\mid +,k_0+q⟩$, which is also seen in the inset showing the final momentum distribution ${\mid ⟨k\mid \Psi ⟩\mid }^2$. The parameters are the same in both plots, $g_0=0.001,\Delta =0,x_0=0,k_0=1∕4$, and ${\Delta }_k^2=1∕10000$, and in (b) the atom is initially in its lower state. The contour bar shows relative values.

Figure 5

The scale mass parameters $1∕\left(m_2\right)$ and $v_g=k_0∕m_1$ as functions of $g_0$ and $\Delta$. The first two plots (a) and (b) show $1∕\left(m_2\right)$ for $k=0$ and $k=q∕4$, respectively, and (c) the group velocity $v_g$ for $k=q∕4$, where the unperturbed velocity $(g_0=0)$ should be $v_0=1∕4$. The result derives from numerical diagonalization of the matrix (11) and numerical calculation of the coefficients of the dis- persion curve around the point $k$ of interest. The bare energies of states $\mu =0,-1$ cross for $\Delta =kq-q^2∕2$ and the dominant bare states of the lowest band changes. This leads to a drastic change in the effective parameters, as seen in the figures.

Figure 6

The variation of the overlap (a) $F^{1,0}\left(0\right)$ and (b) $F^{1,0}(q∕4)$, as defined in Eq. (22), with respect to $g_0$ and $\Delta =0$. For positive $k$, a huge change in the overlap occurs when $\Delta =kq-q^2∕2$, which was also seen in the previous figure of the mass parameters. For such $\Delta$, the corresponding quasi- momentum $k$ is located at a gap, and as $\Delta$ changes more, the lowest dressed states $\nu =1$ will have a larger overlap with bare states with $\mu =\pm 1$ rather than $\mu =0$. This can be clearly under- stood from Figs. 1b, 1c. The photon momentum $q$, which was used to define the characteristic length scale, is, as earlier 1.

Figure 7

The evolution of an initial Gaussian bare wave packet prepared inside the cavity with ${\Delta }_x^2=500$ and $k_0=1∕2$, corresponding to a quasi-momentum of the first forbidden energy gap. The upper plot shows the wave packets of the ground, excited and total (ground plus excited) atomic state in the $x$- representation. The lower figure displays the wave packet of the ground and excited atomic state in the momentum $k$-representation. It is clear that we have a spreading in $x$ and also a small drift of the lower atomic wave packet. The population “Rabi” oscillates between the two states with a period $T_R\approx \pi ∕g_0$, corresponding to the $2\pi$ divided by the energy gap size $2g_0$ in first order. The remaining parameters are $\Delta =0$ and $g_0=0.05$. The contour bars give the relative values.

Figure 8

The group velocity $v_g$ as function of the coupling, obtained, both from the wave-packet simulation, circles shows the result from bare state propagation and crosses from dressed state propagation, and the Floquet theory (solid line), when $k_0=q∕4$. A clear agreement is seen, but unexpectedly the bare state result coincide better with the Floquet one, which might be due to a not fully adiabatic preparation of the dressed state. The parameters are $t_f=1000,\Delta =0$, and ${\Delta }_x^2=2500$.

Figure 9

The effective mass $m_2$ as function of the coupling, obtained, both from the wave-packet simulation of bare (circles) and dressed states (crosses) and the Floquet theory (solid line), when $k=0$. The main reason for the deviation of the result of the bare state propagation is that the wave packet of the lower atomic state, which is used in the calculation, does not stay Gaussian and the tails of the distribution are cut off for $\mid x\mid >15$ in the calculation of ${\Delta }_x^2$. The parameters are $t_f=1000,\Delta =0$, and ${\Delta }_x^2=1500$.

Figure 10

Scattering of an atom against a cavity. The cavity is located within $\mid x\mid \lesssim x_{\mathrm{l}}$ for $x_{\mathrm{l}}=1500$ (marked with dashed line) and $x_{\mathrm{e}}=50$. In (a), we show the ground state atomic wave function ${\mid ⟨-\mid ⟨x\mid \Psi ⟩\mid }^2$ and in (b) the upper state ${\mid ⟨+\mid ⟨x\mid \Psi ⟩\mid }^2$ as function of position $x$ and time $t$. Note that the atom starts to propagate in its lower state at $x_0=-2500$ at the time $t=0$ with momentum $k_0=q∕2$ until it reaches the cavity boundary at $x=-1500$. It is reflected and its internal state is flipped to the upper state $\mid \uparrow ⟩$. The reflection is due to the fact that its energy coincides with an energy gap; hence the atom absorbs the photon from the cavity field and the impact changes the direction of propagation. Note the scale of the contours, which indicate that almost everything is reflected, as confirmed in Fig. 13 for the atomic inversion. The parameters are here are $g_0=0.01,\Delta =0$, and ${\Delta }_x^2=2500$. The contour bar shows relative values.

Figure 11

This figure shows the same as Figs. 10a, 10b, except that now the momentum distribution is much wider, ${\Delta }_x^2$ is chosen 100 instead of 2500 as in the previous plot. The consequence is that energy spread exceeds the band gap and the tails of the momentum wave packet are transmitted rather than reflected. The contour scale is the same as in Fig. 10, and a closer look shows that a much larger part of the wave packet is propagating inside the cavity. Note how the wave packets spreads much faster in these plots.

Figure 12

The momentum distribution of the lower atomic state, ${\mid ⟨-\mid ⟨k\mid \Psi ⟩\mid }^2$, corresponding to Fig. 11, as a function of time $t$. The forbidden energies are burned out when the packet hits the cavity. The contour bar gives relative values.

Figure 13

The atomic inversion $⟨{\sigma }_3⟩$ as function of time $t$ for the example in the previous Fig. 10 (solid line), and Fig. 11 (dashed line). It is clear that at the cavity boundary, the atom flips from $\mid \downarrow ⟩$ to $\mid \uparrow ⟩$ in the first example, but when the momentum width becomes larger than the gap, parts of the wave packet will not be reflected and therefore will not be flipped.

Figure 14

Transmission of the atom through the cavity. The plots are the same as in Figs. 10a, 10b, but now the initial momentum is $k_0=q∕4$, and the atom energy falls on the first allowed energy band, contrary to Figs. 10, 11. The overlap between the initial bare state $\mid {\Psi }_0^{k_0=1∕4}⟩$ and the dressed state $\mid {\varphi }_1^{k_0=1∕4}⟩$ is 0.998. Thus, the propagation can be described using the effective parameters and the wave packet remains Gaussian; only a small amount of population is transferred into the $\mid +⟩$-state and the Gaussian wave packet spreads according to the effective mass $m_2$ and traverse the cavity with a velocity $v_g$. The contour scale is the same as in Fig. 10 and the cavity boundaries are marked by the dashed lines.

Figure 15

The atomic inversion $⟨{\sigma }_z⟩$, corresponding to Fig. 14. Note that when the atom has tra- versed the cavity, it regains its original state. The asymmetry is due to broadening of the wave packet while traversing the cavity, leading to a longer transit time as it leaves the cavity compared to when it enters.