Figure 1

Evolution of the imaginary parts Im$[{\psi }_{+0}(z,t)]$ (black solid line) and Im$[{\psi }_{-0}(z,t)]$ (red dotted line) of the wave packets ${\psi }_{\pm 0}(z,t)$ and the position probability density functions (PPDF) $\rho (z,t)$ for zero detuning. The first two rows refer to the initial velocity $v=3.9$ cm/s, the third and fourth rows to $v=27$ cm/s, and the fifth and sixth rows to $v=43$ cm/s. [(l) and (r)]A close-up of the reflected part of the PPDF is included. Time appears in units of $2\pi /g_0$ and the mode function $u(z)$ (blue dashedline) is drawn in all figures.Reuse & Permissions

Figure 2

The probability of finding the atom in the excited state at time $t$ for the initial velocities $v=3.9$ (black dash-dot line), $27$ (blue dashed line), and $43$ (red solid line) cm/s for zero detuning. Rabi-like oscillations are observable just for the highest velocity as can be observed in the close-up shown in the inside figure. Time appears in units of $2\pi /g_0$.Reuse & Permissions

Figure 3

(a) Entanglement of formation between the internal atom degrees of freedom and the cavity field; (b) linear entropy for the internal atom degrees of freedom and the cavity field; (c) linear entropy for the center of mass and internal atom degrees of freedom. Results are shown for zero detuning and initial velocities $v=3.9$ (black dash-dot line), $27$ (blue dashed line), and $43$ (red solid line) cm/s. Time is in units of $2\pi /g_0$.Reuse & Permissions

Figure 4

The first row illustrates the evolution of the imaginary parts Im$[{\psi }_{+0}(z,t)]$ (black solid line) and Im$[{\psi }_{-0}(z,t)]$ (red dotted line) of the wave packets ${\psi }_{\pm 0}(z,t)$ for blue detuning, ${\delta }_L=g_0/16$. The initial velocity is $v=3.9$ cm/s and the initial state is the same as in Fig. 1a. The real parts behave similarly. The second row illustrates the evolution of the PPDF for the same initial velocity and state used in Fig. 1d. Time appears in units of $2\pi /g_0$ and the mode function $u(z)$ (blue-dashed line) is drawn in all figures.Reuse & Permissions

Figure 5

The probability of finding the atom in the excited state for blue detuning, ${\delta }_L=g_0/16$. Each part of the figure illustrates the result obtained when the center-of-mass motion is quantized (red solid line), treated in the semiclassical adiabatic approximation (blue dash-dot line), or, finally, follows the classical trajectory $z=\mathit{vt}+\mu$, where $v$ is the initial velocity and $\mu$ the initial position (black dashed line). Results are shown for initial velocities (a) $v=3.9$, (b) $27$, and (c) $43$ cm/s. Time is in units of $2\pi /g_0$.Reuse & Permissions

Figure 6

For blue detuning, ${\delta }_L=g_0/16$, (a) illustrates the expected value of the position of the atom when the center-of-mass motion is quantized with the initial velocity $v=3.9$ cm/s (red solid line). Results for the adiabatic (blue dashed line) and $z=\mathit{vt}+\mu$ (black dash-dot line) treatments of the center-of-mass motion are also shown. (b) The expected values of the position of the atom with the initial velocity $v=27$ cm/s for the fully quantum treatment (blue dash-dot line) and for the SCAA (black dashed line). Also shown is $\langle Z\rangle$ for the initial velocity $43$ cm/s resulting from the full quantum treatment (red solid line) and for the SCAA (black dotted line). The horizontal dotted lines indicate the limits of the interaction region. (c) The standard deviation of the position of the atom when the center-of-mass motion is quantized with the initial velocities $v=3.9$ cm/s (black dash-dot line), $27$ cm/s (blue dashed line), and $43$ cm/s (red solid line). In all figures time is in units of $2\pi /g_0$.Reuse & Permissions

Figure 7

For blue detuning, ${\delta }_L=g_0/16$, (a) and (d) illustrate the evolution of the entanglement of formation and the linear entropy between the internal atom and field degrees of freedom, respectively. (b) The evolution of the linear entropy between the center-of-mass and internal atom degrees of freedom; it is exactly the same for the center-of-mass and field degrees of freedom. [(a), (b), and (d)] Results for initial velocities $v=3.9$ cm/s (black dash-dot line), $27$ cm/s (blue dashed line), and $43$ cm/s (red solid line). (c) The evolution of the square of the norm of the wave functions ${\psi }_{+0}(z,t)$ (red solid line) and ${\psi }_{-0}(z,t)$ (blue dash-dot line), the modulus of the overlap $|\langle {\psi }_{+0}(z,t)|{\psi }_{-0}(z,t)\rangle |$ (black dashed line), and the linear entropy between the internal atom and field degrees of freedom (blue dotted line) for the largest initial velocity $v=43$ cm/s. Time appears in units of $2\pi /g_0$.Reuse & Permissions

Figure 8

the probability of finding the atom in the excited state for red detuning, ${\delta }_L=-g_0/16$. Each part of the figure illustrates the result obtained when the center-of-mass motion is quantized (red solid line), treated in the semiclassical adiabatic approximation (blue dash-dot line), or, finally, follows the classical trajectory $z=\mathit{vt}+\mu$, where $v$ is the initial velocity and $\mu$ the initial position (black dashed line). $\mu$ is equal to the initial mean position of the atom in the quantum treatment. Results are shown for initial velocities (a) $v=3.9$ cm/s, (b) $27$ cm/s, and (c) $43$ cm/s. Time is in units of $2\pi /g_0$.Reuse & Permissions

Figure 9

For red detuning, ${\delta }_L=-g_0/16$, (a) the expected value of the position of the atom when the initial velocity is $v=3.9$ cm/s for a full quantum treatment (red solid line), for the SCAA (blue-dashed), and for the $z=\mathit{vt}+\mu$ treatment of the center-of-mass motion (black dash-dot line) ($\mu$ is equal to the initial mean position of the atom in the quantum treatment). (b) The expected values of the position of the atom with the initial velocity $v=27$ cm/s for the full quantum treatment (blue dash-dot line) and for the SCAA (black dashed line). It also shows $\langle Z\rangle$ for the initial velocity $43$ cm/s resulting from the full quantum treatment (red solid line) and for SCAA (black dotted line). The horizontal dotted lines indicate the limits of the interaction region. (c) The standard deviation of the position of the atom when the center-of-mass motion is quantized with the initial velocities $v=3.9$ cm/s (black dash-dot line), $27$ cm/s (blue dashed line), and $43$ cm/s (red solid line). In all figures time is in units of $2\pi /g_0$.Reuse & Permissions

Figure 10

For red detuning, ${\delta }_L=-g_0/16$, (a) and (d) illustrate the evolution of the entanglement of formation and the linear entropy between the internal atom and field degrees of freedom, respectively. (b) The evolution of the linear entropy between the center-of-mass and internal atom degrees of freedom, it is exactly the same for the center-of-mass and field degrees of freedom. [(a), (b), and (d)] Results for initial velocities $v=3.9$ cm/s (black dash-dot line), $27$ cm/s (blue dashed line), and $43$ cm/s (red solid line). (c) The evolution of the square of the norm of the wave functions ${\psi }_{+0}(z,t)$ (red solid line) and ${\psi }_{-0}(z,t)$ (blue dash-dot line), the modulus of the overlap $|\langle {\psi }_{+0}(z,t)|{\psi }_{-0}(z,t)\rangle |$ (black dashed line), and the linear entropy between the internal atom and field degrees of freedom (blue dotted line) for the largest initial velocity $v=43$ cm/s. Time appears in units of $2\pi /g_0$.Reuse & Permissions

Figure 11

(a) The probability $T(w_0)$ that the atom is transmitted through the cavity field, (b) the probability of finding the atom inside the interaction region $(-4w_0/\sqrt{2},4w_0/\sqrt{2})$, and (c) the probability of emission (equal to $1$ minus the probability of finding the atom in the excite state), all as a function of the Gaussian mode waist $w_0$ at time $t_f=21w_0\sqrt{M/K_0}$ for the initial mean kinetic energy $K_0=\hslash g_0/100$. Blue dashed lines correspond to a detuning ${\delta }_L=g_0/16>0$, black solid lines correspond to zero detuning, and red dashed-dotted lines correspond to a detuning ${\delta }_L=-g_0/16<0$. The waist $w_0$ is in nanometers.Reuse & Permissions

Figure 12

[(a), (b), and (c)] Respectively, the linear entropy of the center-of-mass density operator, the linear entropy of the field density operator, and the entanglement of formation of the density operator of the internal atom and field degrees of freedom as a function of the Gaussian mode waist $w_0$ at time $t_f=21w_0\sqrt{M/K_0}$ for the initial mean kinetic energy $K_0=\hslash g_0/100$. The linear entropy of the internal atom degrees of freedom is the same as in (b) [see Eq. (12) and the paragraph below it]. In the interval from $40$ to $70$ nm, the entanglement of formation for all values of the detuning decreases to zero (not shown for purposes of better illustration in the region presented). Blue dashed lines correspond to a detuning where ${\delta }_L=g_0/16$, black solid lines correspond to zero detuning, and red dashed-dotted lines correspond to a detuning where ${\delta }_L=-g_0/16$. The waist $w_0$ is in nanometers.Reuse & Permissions