Quantum-state reconstruction of unidirectional molecular rotations

We present a quantum-state reconstruction procedure to retrieve a rotational wave packet created via nonadiabatic excitation by ultrashort pulses with controlled polarization. Here a unidirectionally rotating ensemble of CO molecules, adiabatically cooled to be populated only in $J$ = 0 as an initial state, has been prepared by skewed-polarization pulse-pair excitation. The two-dimensional sections of the molecular-axis probability distribution have been captured by time-delayed Coulomb explosion with the space-slice ion-imaging setup, recently developed in the present research group [K. Mizuse, R. Fujimoto, and Y. Ohshima, Rev. Sci. Instrum. 90, 103107 (2019)]. The time- and angle-dependent distribution has been subjected to the least-squares regression to provide a unique set of the amplitudes and phases for the eigenstates that constitute the wave packet. The reconstruction allows us to make a movie of the three-dimensional probability distribution and evaluate any physical properties, e.g., the magnitude of rotational angular momentum and its degree of orientation. The determined phases also serve as a sensitive probe for the excitation and the quantum interference to control the directionality of the rotational wave packet.

Figure 1

Schematic of the experimental setup. On the upper-left side, the present space-fixed axis system is shown. Experimentally recorded fragment-ion images at several delay times are also shown in black and white.

Figure 2

Time- and angle-dependent probability distribution for a rotational WP created by a single-pump pulse. The horizontal axis corresponds to delay time of the probe pulse relative to the pump pulse, and the vertical axis corresponds to the azimuthal angle $\varphi$. The pump intensity was estimated to be $\approx 15$ TW/${\mathrm{cm}}^2$. The polarization direction of the pump pulse is directed along $\varphi =0$ and ${180}^{\circ }$.

Figure 3

Time- and angle-dependent probability distribution for a UDR-WP created by a skewed-polarization pair of pump pulses. Time zero is set to the peak of the second pulse. The first pulse was introduced 3.87 ps before. The pulse intensities were estimated to be 6 TW/${\mathrm{cm}}^2$ or more. The polarization direction of the first pump pulse is directed along $\varphi =0$ and ${180}^{\circ }$ while that of the second pump is tilted by $+{45}^{\circ }$.

Figure 4

Time-dependent alignment parameters $\langle {cos}^2\varphi \rangle$ for (a) the rotational WP created by a single-pump pulse (corresponding to Fig. 2) and (b) the UDR-WP created by a skewed-polarization pair of pump pulses (corresponding to Fig. 3).

Figure 5

Power spectrum from the 2D Fourier transformation of the time- and angle-dependent probability distribution (shown in Fig. 2) from the single-pump experiment. The horizontal axis corresponds to the frequency in the $B$ unit and the vertical axis corresponds to the $M$ quantum number.

Figure 6

Power spectrum from the 2D Fourier transformation of the time- and angle-dependent probability distribution (shown in Fig. 3) from the double-pump experiment. Note the frequency resolution is four times better than in Fig. 5 due to the extended coverage in time.

Figure 7

A bar diagram representing the amplitudes $A_{J,M}$ of the rotational eigenstates $|J,M\rangle$ that constitute the rotational WP created by the single-pulse excitation. The horizontal axis is for the rotational states and the vertical axis is for the magnitudes of the amplitude. Colored bars on the left side for each $|J,M\rangle$ are those determined by the least-squares regression on the time- and angle-dependent probability distribution shown in Fig. 2. The error bars for the standard deviations of the fit are indicated. Gray bars on the right side are those derived from the TDSE calculation, in which the laser pulse with Gaussian envelope (120-fs duration, 11-TW/${\mathrm{cm}}^2$ peak power) is adopted.

Figure 8

A polar plots representing the phase angles ${\delta }_{J,M}$ of the rotational eigenstates $|J,M\rangle$ that constitute the rotational WP created by the single-pulse excitation. The value for the $|0,0\rangle$ state is fixed to zero. Left: Those determined by the least-squares regression on the time- and angle-dependent probability distribution shown in Fig. 2. The error bars for the standard deviations of the fit are buried within the thickness of the lines. Right: Those derived from the TDSE calculation.

Figure 9

A bar diagram representing the amplitudes $A_{J,M}$ of the rotational eigenstates $|J,M\rangle$ that constitute the rotational WP created by the skewed-polarization pulse-pair excitation. Colored bars on the left side for each $|J,M\rangle$ are those determined by the least-squares regression on the time- and angle-dependent probability distribution shown in Fig. 3. Gray bars on the right side are those derived from the TDSE calculation, in which the laser pulses with Gaussian envelope (120-fs duration, 5.3 and 5.5 TW/${\mathrm{cm}}^2$ for the first and second pulses) are adopted. The time interval between the two pump pulses is set to be 3.87 ps.

Figure 10

A polar plots representing the phase angles ${\delta }_{J,M}$ of the rotational eigenstates $|J,M\rangle$ that constitute the rotational WP created by the skewed-polarization pulse-pair excitation. Left: Those determined by the least-squares regression on the time- and angle-dependent probability distribution shown in Fig. 3. Right: Those derived from the TDSE calculation.

Figure 11

Snapshots of the observed (blue or dark gray line) and reconstructed (green or light gray line) 2D probability distribution for the UDR-WP at several delay times. The distribution is indicated in the polar plots for the $\varphi$ angle.

Figure 12

Schematic representation of complex coefficients in the skewed-polarization pulse-pair excitation. The coefficients of $J$ = 2 are indicated as arrows (blue, $M=0$; orange, $M=+2$; red, $M=-2$) in the complex plane (real and imaginary parts in the horizontal and vertical axes, respectively). The two plots in the upper left represent the creation of the WP by the first pulse and its time evolution. The plot in the upper right is for the WP creation by the second pulse. The vectorial addition of the two coefficients yields to the final states, represented in the lower plot.