Edge states of periodically kicked quantum rotors

We present a quantum localization phenomenon that exists in periodically kicked three-dimensional rotors, but is absent in the commonly studied two-dimensional ones: edge localization. We show that under the condition of a fractional quantum resonance there are states of the kicked rotor that are strongly localized near the edge of the angular momentum space at $J=0$. These states are analogs of surface states in crystalline solids, and they significantly affect resonant excitation of molecular rotation by laser pulse trains.

Figure 1

Projection of the quasienergy states on the angular momentum states, for the case of the third order resonance $\tau =t_{\mathrm{rev}}/3$ and an interaction strength of $P=3$. (a) The calculated quasienergy states; almost all states are extended over the full angular momentum space. (b) One of the extended states separately, as an example. In (a) one can see two states that are localized at the lower edge of the momentum space; these states are shown separately in (c). Only states of even parity are shown.

Figure 2

Spectrum of the quasienergy states for a rigid linear rotor kicked periodically at the fractional resonance $\tau =t_{\mathrm{rev}}/3$, as a function of the kick strength $P$. The markers correspond to the states shown in Fig. 1. Only states of even parity are included. The color axis depicts the numerical density, in particular the number of states per pixel; note its logarithmic scale.

Figure 3

Overlap $O\left(P\right)$ [see Eq. (7)] of different initial states $|\mathrm{\Psi }\left(0\right)\rangle =|J_0,0\rangle$ with edge states as a function of the effective interaction strength $P$. Shown is the case for the third order resonance $\tau =t_{\mathrm{rev}}/3$.

Figure 4

Time-dependent population of the angular momentum states $|J,0\rangle$ for a rotor kicked periodically at the fractional resonance $\tau =t_{\mathrm{rev}}/3$ with kick strength $P=3$. The initial state is (a) $\left|\mathrm{\Psi }\right(0)\rangle =|0,0\rangle$ and (b) $\left|\mathrm{\Psi }\right(0)\rangle =|40,0\rangle$. Shown are only the states of even parity. Note the logarithmic scale of the color axis.

Figure 5

Rotational energy as a function of the number of kicks, for a rotor initially in its ground state $|0,0\rangle$ (solid line) and an excited state $|40,0\rangle$ (dashed line). The kicking period is $\tau =t_{\mathrm{rev}}/3$, the kick strength $P=3$.

Figure 6

Density of quasienergy states for a rigid linear rotor kicked periodically at the fractional resonance $\tau =t_{\mathrm{rev}}/2$, as a function of the kick strength $P$. Only states of even parity are included. The color axis depicts the numerical density, in particular the number of states per pixel; note its logarithmic scale.

Figure 7

Time-dependent population of the angular momentum states $|J,0\rangle$ for a rotor kicked periodically at the fractional resonance $\tau =t_{\mathrm{rev}}/2$ with kick strength $P=3$. The initial state is (a) $\left|\mathrm{\Psi }\right(0)\rangle =|0,0\rangle$ and (b) $\left|\mathrm{\Psi }\right(0)\rangle =|40,0\rangle$. Shown are only the states of even parity. Note the logarithmic scale of the color axis.

Figure 8

Density of quasienergy states for a rigid linear rotor kicked periodically at the high order fractional resonance $\tau =t_{\mathrm{rev}}/17$, as a function of the kick strength $P$. Only states of even parity are included. The color axis depicts the numerical density, in particular the number of states per pixel; note its logarithmic scale.

Figure 9

Density of quasienergy states for a rigid linear rotor kicked periodically at the fractional resonance $\tau =t_{\mathrm{rev}}/3$, as a function of the kick strength $P$. The projection quantum number is $M_J=10$. Only states of even parity are included. The color axis depicts the numerical density, in particular the number of states per pixel; note its logarithmic scale.

Figure 10

Simulated population of the angular momentum levels $J$ for ${}^{129}\mathrm{I}{}^{35}\mathrm{Cl}$ molecules kicked periodically at (a) the full resonance ($\tau =t_{\mathrm{rev}}=146.1$ ps), and (b) the third order resonance ($\tau =t_{\mathrm{rev}}/3$). The pulse duration is 500 fs (full width at half maximum), the peak intensity is 1.5 ${\mathrm{TW}/\mathrm{cm}}^2$. This corresponds to a kick strength of $P=10$. The initial rotational temperature is set to 5 K.

Figure 11

Absolute value of the Fourier transform of the molecular alignment signal $\langle {cos}^2\theta \rangle \left(t\right)$ for ICl molecules kicked at $\tau =t_{\mathrm{rev}}/3$ [same conditions as for Fig. 10]. The results are shown after $N$ pulses, for $N=2,4,6,8,10$. The zero-frequency component (time-averaged alignment) is removed from the signal. We added a broadening of 0.33 ${\mathrm{cm}}^{-1}$, which corresponds to a measurement window of about 100 ps.