Landau-Zener transitions and adiabatic impulse approximation in an array of two Rydberg atoms with time-dependent detuning

We study the Landau-Zener (LZ) dynamics in a setup of two Rydberg atoms with time-dependent detuning, both linear and periodic, using both the exact numerical calculations as well as the method of adiabatic impulse approximation (AIA). By varying the Rydberg-Rydberg interaction strengths, the system can emulate different three-level LZ models, for instance, bow-tie and triangular LZ models. The LZ dynamics exhibits nontrivial dependence on the initial state, the quench rate, and the interaction strengths. For large interaction strengths, the dynamics is well captured by AIA. In the end, we analyze the periodically driven case, and AIA reveals a rich phase structure involved in the dynamics. The latter may find applications in quantum state preparation, quantum phase gates, and atom interferometry.

Figure 1

(a) The linear variation of detuning and the instantaneous energy eigenvalues. The dashed lines show the diabatic energy levels. Panel (b) shows the inverse of the energy gap between the adiabatic levels (top one), and we approximately identify the impulse, and the adiabatic regimes, separated at $\pm t^{\prime }$. The bottom plot shows the population in the excited state ($P_{+}={\left|a_{+}\right|}^2$) vs time $t$ for which the atom is initially prepared in the ground state and subject to a linear quench in the detuning. Solid line shows the exact result and dashed line is from the LZ model [Eq. (3)]. Panel (c) depicts the periodic time dependence of $\mathrm{\Delta }\left(t\right)$, and panel (d) shows the corresponding instantaneous energy eigenvalues $E_{\pm }$. At the avoided crossings, $t={\tau }_{2n}\left({\tau }_{2n+1}\right)$ LZT takes place, described by the operator ${\widehat{G}}_{LZ}\left({\widehat{G}}_{LZ}^T\right)$, and on either side of it the adiabatic evolution takes place, determined by ${\widehat{U}}_1$ and ${\widehat{U}}_2$. The shaded area indicates the accumulated phases (${\zeta }_{\pm }$) during the adiabatic evolution.

Figure 2

The transition probability to the excited state as a function of $\omega$ when the atom is initially prepared in the ground state for $\delta =20\mathrm{\Omega }, {\mathrm{\Delta }}_0=5\mathrm{\Omega }, t_i=\pi /2\omega$ after (a) one cycle and (b) 10 cycles. The solid line shows the exact results, and the dashed line is that from AIA. In panel (b), the peak at $\omega /\mathrm{\Omega }=2.5$ corresponds to the resonance $2\omega ={\mathrm{\Delta }}_0$. (c) Interferometric pattern using AIA: the long-time averaged population in the excited state (${\overline{P}}_{+}$) as a function of ${\mathrm{\Delta }}_0/\mathrm{\Omega }$ and $\delta /\mathrm{\Omega }$ for $\omega =0.32\mathrm{\Omega }$. The density peaks correspond to the resonances, and the solid lines mark the validity of AIA.

Figure 3

The excitation probability after 10 cycles as a function of $\alpha$ for $\delta /\mathrm{\Omega }=20, {\mathrm{\Delta }}_0=0$, and $\omega$ is varied. A given $\alpha$ is not associated with a unique value of $\omega$, leading to the scattered points, but bounded by a maximum value of ${sin}^{2}k\alpha$ shown by the solid line.

Figure 4

Energy eigenvalues $E_j$ as a function of the instantaneous detuning (the time dependence, $\mathrm{\Delta }\left(t\right)=vt$ is shown at the bottom) for (a) $V_0=0.1\mathrm{\Omega }$ and (b) $V_0=5\mathrm{\Omega }$. The dashed lines show the diabatic energy levels. The avoided crossings in panel (b) form a triangular LZ model. The inset in panel (b) shows the avoided crossing at $V_0/2$. The asymptotic states at $t\to \pm \infty$ are given in the left and right ends of the level diagrams. Panel (c) shows the energy gaps ${\overline{\mathrm{\Delta }E}}_{\alpha }=\mathrm{\Delta }{E}_{\alpha }/\mathrm{\Omega }$ at the avoided crossings as a function of $V_0$. The inset shows the schematic setup for the LZ interferometer in which the first ($S_1$) and the second ($S_2$) crossings act as beam splitters. At the last crossing, $O$ mixing takes place.

Figure 5

The dynamics of populations in the adiabatic [(a)–(d)] and the diabatic [(e)–(h)] states for the initial state $\left|\psi \right(t_i)\rangle =|1\rangle \sim |gg\rangle , v=(1{\mathrm{\Omega }}^2,10{\mathrm{\Omega }}^2$), and $V_0=(0.1\mathrm{\Omega },10\mathrm{\Omega })$. The first LZT takes place in the vicinity of $t=0$. The thin arrows show the times around which the second ($t_2$) and the third ($t_3$) LZTs occur. In panels (a), (b), (e), and (f), the LZTs are not resolvable, so a single arrow is shown. The dashed horizontal lines in the bottom row show the results from the noninteracting model.

Figure 6

The final population in (a) $|1\rangle \sim |rr\rangle$, (b) $|2\rangle \sim |s\rangle$, and (c) $|3\rangle \sim |gg\rangle$, after the linear quench, as a function of $v$ and $V_0$ for the initial state $\left|\psi \right(t_i)\rangle =|1\rangle , \mathrm{\Delta }\left(t_i\right)=-10\mathrm{\Omega }$, and $\mathrm{\Delta }\left(t_f\right)=30\mathrm{\Omega }+10v/\mathrm{\Omega }$.

Figure 7

(a) The final population in the adiabatic and diabatic states as a function of $v$ for $V_0=0.1\mathrm{\Omega }$ with the initial state $|1\rangle \sim |gg\rangle$. (b) The same as in panel (a) but as a function of $V_0$ for $v=2{\mathrm{\Omega }}^2$. The solid lines show exact results, and the filled squares, circles, and triangles are the theoretical predictions for small $V_0$ in the limit $t_f\to \infty$.

Figure 8

The dynamics of populations in the adiabatic [(a)–(d)] and the diabatic [(e)–(h)] states for the initial state $\left|\psi \right(t_i)\rangle =|2\rangle \sim |gg\rangle , v=(1{\mathrm{\Omega }}^2,5{\mathrm{\Omega }}^2$), and $V_0=(0.1\mathrm{\Omega },10\mathrm{\Omega })$. The first LZT takes place in the vicinity of $t=0$. The thin arrows show the times around which the second ($t_2$) and the third ($t_3$) LZTs occur. In panels (a), (b), (e), and (f), the LZTs are not resolvable, so a single arrow is shown. The dashed horizontal lines in panels (e) and (f) show the results from the noninteracting model.

Figure 9

The final population in (a) $|1\rangle \sim |rr\rangle$, (b) $|2\rangle \sim |s\rangle$, and (c) $|3\rangle \sim |gg\rangle$, after the linear quench, as a function of $v$ and $V_0$ for the initial state $\left|\psi \right(t_i)\rangle =|2\rangle , \mathrm{\Delta }\left(t_i\right)=-10\mathrm{\Omega }$, and $\mathrm{\Delta }\left(t_f\right)=30\mathrm{\Omega }+10v/\mathrm{\Omega }$.

Figure 10

The dynamics of population in the adiabatic [(a), (b)] and diabatic [(c), (d)] states for the initial state $\left|\psi \right(t_i)\rangle =|3\rangle \sim |rr\rangle$ with different values of $v$ and $V_0=10\mathrm{\Omega }$. The thin arrows show the times around which the second ($t_2$) and the third ($t_3$) LZTs occur.

Figure 11

The final population in (a) $|1\rangle \sim |rr\rangle$, (b) $|2\rangle \sim |s\rangle$, and (c) $|3\rangle \sim |gg\rangle$, after the linear quench, as a function of $v$ and $V_0$ for the initial state $\left|\psi \right(t_i)\rangle =|2\rangle , \mathrm{\Delta }\left(t_i\right)=-10\mathrm{\Omega }$, and $\mathrm{\Delta }\left(t_f\right)=30\mathrm{\Omega }+10v/\mathrm{\Omega }$.

Figure 12

(a) The beats in the dynamics of $P_s\left(t\right)$ for $V_0=2\mathrm{\Omega }, v=5{\mathrm{\Omega }}^2$, and different initial states. (b) The instantaneous energy eigenspectrum for $\mathrm{\Delta }\left(t\right)=vt$ for large $V_0$. The adiabatic and nonadiabatic regimes are marked by the operators ${\widehat{U}}_{1,2,3,4}$ and $\{\widehat{G}{1}_{LZ},\widehat{G}{2}_{LZ},\widehat{G}{3}_{LZ}\}$.

Figure 13

The final population in the adiabatic and diabatic states as a function of both $V_0$ and $v$ for the initial state $|1\rangle$ [(a), (b)], $|2\rangle$ [(c), (d)], and $|3\rangle$ [(e), (f)]. For the first column, $v=2{\mathrm{\Omega }}^2$ and for the second column, $V_0=2\mathrm{\Omega }$. The solid lines show exact results, and the filled squares, circles, and triangles are from AIA.

Figure 14

(a) The periodic time dependence of the detuning for $\delta =V_0/4-{\mathrm{\Delta }}_0$. The expected durations for adiabatic evolution are $T_a$ and $T_a^{\prime }$. Panel (b) shows the corresponding instantaneous energy eigenvalues. The instants (${\tau }_0^{\left(1\right)}, {\tau }_1^{\left(1\right)}, {\tau }_2^{\left(1\right)}$) at which the LZTs occur between states $|1\rangle$ and $|2\rangle$ are shown by shaded stripes. The operators ${\widehat{U}}_j$ and $\widehat{G}{1}_{LZ}$ indicate the adiabatic regimes and the impulse points, respectively. Between the origin and the dashed vertical line, we have one complete cycle.

Figure 15

The numerical results (solid lines) and that of AIA (dashed lines) for $P_1$ after (a) 10 and (b) 100 cycles, as a function of $\omega$ for the initial state $|1\rangle \sim |gg\rangle , {\mathrm{\Delta }}_0=-15\mathrm{\Omega }, V_0=40\mathrm{\Omega }$, and $\delta =25\mathrm{\Omega }$. Panel (b) shows that at longer times, AIA deviates from exact dynamics especially, at high $\omega$.

Figure 16

(a) The exact results (solid lines) for ${\overline{P}}_{1,2}$, and the same from the AIA (dashed lines) over a period of 100 cycles, as a function of $\omega$ for the initial state $|1\rangle \sim |gg\rangle , {\mathrm{\Delta }}_0=-15\mathrm{\Omega }, V_0=40\mathrm{\Omega }$, and $\delta =25\mathrm{\Omega }$. The dips (peaks) in ${\overline{P}}_1$ (${\overline{P}}_2$) indicate the resonances at $n\omega =|{\mathrm{\Delta }}_0|$. The six resonances seen at $\omega /\mathrm{\Omega }=15,7.5,5,3.75,3,2.5$ correspond to $n=1,2,3,4,5,6$, respectively. Panel (b) shows the coherent oscillation between $|gg\rangle$ and $|s\rangle$ at the resonance $\omega /\mathrm{\Omega }=7.5$ and panel (c) shows the same between $|1\rangle$ and $|2\rangle$ states.

Figure 17

(a) The exact results (solid lines) and that from AIA (dashed lines) of ${\overline{P}}_2$ over 100 cycles, as a function of $\omega$ for the initial state $|2\rangle \sim |s\rangle$. The dips indicate the resonances, and the five (broader) of them at $\omega /\mathrm{\Omega }=15,7.5,5,3.75,3$ correspond to $n\omega =|{\mathrm{\Delta }}_0|$ with $n=1,2,3,4,5$, respectively. The narrow ones at $\omega /\mathrm{\Omega }=18.33,13.75,11$ correspond to $n\omega =|{\mathrm{\Delta }}_0-V_0|$ with $n=3,4,5$, respectively, which are not captured by AIA.

Figure 18

(a) The periodic time dependence of the detuning for $\delta =3V_0/4-{\mathrm{\Delta }}_0$. The adiabatic durations are marked by $T_{a1}, T_{a2}$, and $T_{a1}^{\prime }$. Panel (b) shows the instantaneous energy eigenvalues. The instants (${\tau }_0^{\left(1\right)}, {\tau }_0^{\left(2\right)}, {\tau }_1^{\left(2\right)}, {\tau }_1^{\left(1\right)}, {\tau }_2^{\left(1\right)}, {\tau }_2^{\left(2\right)}$) at which the LZTs occur are shown by shaded stripes. The $\widehat{U}$ and $\widehat{G}$ operators indicate the adiabatic regimes and the impulse points, respectively. Between the origin and the dashed vertical line, we have one complete cycle.

Figure 19

The exact results (solid lines) for the time-averaged populations (${\overline{P}}_{1,2,3}$) and the same from the AIA (dashed lines) for a period of 100 cycles, as a function of $\omega$ for ${\mathrm{\Delta }}_0=-15\mathrm{\Omega }, V_0=40\mathrm{\Omega }$, and $\delta =45\mathrm{\Omega }$ with the initial state (a) $|1\rangle \sim |gg\rangle$, (b) $|2\rangle \sim |s\rangle$, and (c) $|3\rangle \sim |rr\rangle$. In panel (c), AIA completely failed to capture any resonances.

Figure 20

(a) The periodic time dependence of the detuning for $\delta =5V_0/4-{\mathrm{\Delta }}_0$ and (b) the instantaneous energy eigenvalues. The adiabatic durations are marked by $T_{a1}, T_{a2}, T_{a3}$, and $T_{a1}^{\prime }$. The instants (${\tau }_0^{\left(1\right)}, {\tau }_0^{\left(2\right)}, {\tau }_0^{\left(3\right)}, {\tau }_1^{\left(3\right)}, {\tau }_1^{\left(2\right)}, {\tau }_1^{\left(1\right)}, {\tau }_2^{\left(1\right)}, {\tau }_2^{\left(2\right)}, {\tau }_2^{\left(3\right)}$) at which the LZTs occur between different adiabatic states are shown by shaded stripes. The $\widehat{U}$ and $\widehat{G}$ operators represent the adiabatic regions and impulse points, respectively. Between the origin and the dashed vertical line, we have one complete cycle.

Figure 21

The numerical results (solid lines) for the time-averaged populations (${\overline{P}}_{1,2,3}$) in the adiabatic states, and the same from the AIA (dashed lines) over a period of 100 cycles, as a function of $\omega$ for the initial state (a) $|1\rangle$, (b) $|2\rangle$, and (c) $|3\rangle$. Other parameters are ${\mathrm{\Delta }}_0=-15\mathrm{\Omega }, V_0=40\mathrm{\Omega }$, and $\delta =65\mathrm{\Omega }$. In panel (a), the major peaks correspond to the resonances $n\omega =|{\mathrm{\Delta }}_0|$ (between $|gg\rangle$ and $|s\rangle$ states) and smaller ones indicate the resonances at $n\omega =|{\mathrm{\Delta }}_0-V_0|$ (between $|rr\rangle$ and $|s\rangle$ states). In panel (b), the peaks and dips indicate the resonances at $n\omega =|{\mathrm{\Delta }}_0|$ and $n\omega =|{\mathrm{\Delta }}_0-V_0|$, with no traces on the resonances at $n\omega =|2{\mathrm{\Delta }}_0-V_0|$. In panel (c), except the resonances at $n\omega =|{\mathrm{\Delta }}_0|$, other two types are seen. AIA results are in excellent agreement with the numerics in all three cases.