1. Young's Double Slit Experiment

Fig. 1-1 Young's double slit experiment　

  Interference is a direct consequence of superposing waves. The most famous experiment demonstrating this characteristics is Young's double slit experiment. Fig. 1 shows the scheme of Young's double slit experiment. Due to the different optical path length between a-b-d path and a-c-d path in Fig. 1, the overlapped light waves interfere with each other at point d. If the phase shift is expressed as (2n+1)π, the two light waves are totally out of phase, so that destructive overlap occurs. On the other hand, if the phase shift is expressed as 2nπ, two waves are constructively overlapped to give bright area on the screen. The wavefunction which describes the quantum state of particles is also a kind of wave, so that we can observe the constructive and destructive overlaps. Such phenomena is called "quantum interference" compared to the normal interference of classical waves.

2. What is Coherent Control?

  Coherent control is a technique to manipulate the quantum wavefunction of atoms and molecules utilizing the coherent light sources. Through the light-matter interaction, the coherent characteristics of the light is imprinted on the wavefunctions of the target quantum system. The characteristics of the light, such as the temporal envelope and the spectrum, can be designed easily using the apparatus such as spatial light modulator and the Michelson interferometer. With such pulses, interference of wavefunctions can be manipulated so that the amplitude and phase distribution of the wavefunction can be controlled from outside.
  In our research, the wavefunction of the target quantum system is manipulated by femtosecond ultrashort laser pulses. Our final goal is to control the macroscopic characteristics of the target system by the coherent control techniques. Coherent control can be applied for the photo-reaction branching ratio control, and is also an important building block to manipulate the qubit states which is indispensable for quantum computing and quantum information technology.

  The most important characteristics necessary for the coherent control is the "long coherence life time." The photo-excited quantum state loses its quantum nature during the temporal evolution, due to the irreversible relaxation process called pure dephasing. Coherent control is effective only when the target system keeps the clear definition of the phase. If the phase is unclear, the interference doesn't bear any contrast.

　For coherent control, we need a highly coherent light sources, i.e. lasers. In our group, we use femtosecond ultrashort laser pulses to perform coherent control. Femtosecond laser pulses have broad spectral bandwidth, so that it is also convenient to produce a wave packet, which is a linear combination of quantum eigenfunctions. For example, the vibrational wave packet generated in the electronic excited state is described as follows,  

$$\phi(r,t)=\phi_{0}^{g}|g\rangle+\sum_{n} a_{n}\phi_{n}^{e}(r)e^{-i(\omega_{eg}+\omega_{n})t}|e\rangle$$

where Φ_0g is a vibrational eigenfunction with v=0 in the ground state. Φ_ne is the v=n state vibrational eigenfunction in the electronic excited state, ωeg is the transition angular frequency between the electronic ground and excited states, ωn is the vibrational angular frequency, an is the amplitude of v=n state, and |g> and |e> are the bra-ket notation of the electronic wave functions. Depend on the target quantum states, wave packet (WP) is called vibrational WP, electronic WP, rotational WP, etc.

In the following, I will mainly focus on the motion of vibrational WPs. What is shown below is the WP motion (probability distribution) in a 1D harmonic potential, where x-axis represents the internuclear distance. As you see, the motion of the packet is periodic, which is defined as classical vibrational period (Tvib). In realistic molecular potential, the potential is anharmonic so that the packet shows collapse and revival structure as temporal evolution.

Wave packet motion in a harmonic potential

  Next, we consider the situation where a pair of time-delayed femtosecond(fs) laser pulses is shined on the molecule with delay τ. Assuming that the laser pulse intensity is not so strong so that the depletion of the ground state population is negligible, the WP is described as,

\begin{align*} \phi(r,t) & =\phi_{0}^{g}(r)+\sum_{n} a_{n}\phi_{n}^{e}(r) e^{-i\omega_{n}t}+\sum_{n}a_{n}\phi_{n}^{e}(r)e^{-i\omega_{n}(t-\tau)} \\ & = \phi_{0}^{g}(r)+ \sum_{n} a_{n}\phi_{n}^{e}(r) e^{-i\omega_{n}t} (1+e^{i\omega_{n}\tau})\end{align*}

Squaring this equation gives the population distribution as

$$ |\phi(r,t)|^{2}=|\phi_{0}^{g}|^{2}+2\sum_{n} |a_{n}|^{2}|\phi_{n}^{e}|^{2} [1+\cos (\omega_{n}\tau)] $$

What is important in this equation is that the population of each vibrational level oscillate with angular frequency ω_n. In an actual experiments of iodine molecules where we use 530 nm optical pulse for pumping, the oscillation period is 1.8×10^-15s or 1.8 fs. Thus, the phase factor of the wavefunctions (especially for vibronic WP) oscillates vert rapidly. If we want to control the interference of such fast oscillating functions, we need very precise, sub femtosecond precision, control of the laser irradiation timings. Shown below is the result of WP simulation for constructive and destructive overlap. The arrow in each figure represents the timing of pulse injection.

Fig. 2-1 (a) Constructive case

Fig. 2-1 (b) Destructive case

In the following, the first excitation pulse is called "pump pulse", and the second one is called "control pulse." In Fig. 2-1(a), the delay between pump and control pulses is set as τ=499.7fs, and in Fig. 2-1(b), the delay is τ=498.8fs. Thus, only the shift of 0.9 fs results in a drastic difference of WP.  In the ideal case, the population generated by the double pulse oscillates between 0 to 4 times the single pulse population.

  My research topic is the coherent control of various quantum systems. Needless to say, simpler systems have longer coherence life time and it is much easier to demonstrate quantum interference. But sometimes the results are too simple, and not interesting scientifically. As shown in Fig. 2-2, we start from an isolated molecule, and now the target is extended for more complex systems such as ro-vibrational motion of solid para-hydrogen, coherent phonon motion, and polariton systems.

We will introduce some of the coherent control experiments below.

Fig. 2-2 Various target systems for coherent control

3. Coherent control of vibronic wave packets of isolated molecules in gas phase

    Iodine molecules in the gas phase have a small vibrational energy level spacing, so we can easily create vibrational wave packets with a laser having a temporal width of about 100 femtoseconds. If we create two wave packets in the electronically excited state of the iodine molecule with a femtosecond pump/control pulse pair as described in the previous section, they will interfere with each other to create the final state. To retrieve the detailed information of the actually created states, another laser pulse (probe pulse) must be injected to read out the state of the molecule. This type of experiment is called the pump-probe method, and is the most basic method of observing phenomena that occur in a target system in an ultrashort timescale.

    Figure 3-1 shows a schematic of such an experiment. After a delay time, the wave packet created by the pump/control pair is excited to a higher energy electronic state by probe light irradiation. The fluorescence emitted from this electronic state can then be observed by a photomultiplier tube to provide information about the state of the wave packet. In our study, we performed two different experiments depending on the time width of the probe pulses. The first is the readout of the population distribution using nanosecond probe pulses, and the second is the visualization of wave packet motion using femtosecond probe pulses. Each of them is described below.

Fig. 3-1 Scheme of wave packet interferometry experiment
in Iodine molecule

3-1. Read out population information by nanosecond laser pulse　

    When nanosecond lasers are used as probe pulses, the spectral width of the pulse is much narrower than the energy between different electronic vibrational transitions of the iodine molecule. In this case, transitions from the initial state will occur only when the wavelength of the nanosecond laser resonates with a particular transition, allowing transitions from the individual eigenlevels that make up the wave packet to be observed separately in the spectral domain. The data shown in Fig. 3-2 are plots of the change in the population of vibrational quantum number v=33 level in the B electronic state of the iodine molecule as a function of the delay time τ between pump and control pulses. In this experiment, the delay time between pump and control pulses was scanned ~ ±2 fs around 500 femtoseconds (~1.0 Tvib). The resulting signal intensity is found to oscillate between 0 and 1. The period of oscillation is approximately 1.8 fs, which corresponds to the transition energy from the v=0 level of the electronic ground state to the v=33 level of the B electronic state.

    By sweeping the wavelength of the probe pulse, one can visualize the population of all vibrational eigenstates in the wave packet and the relative phases of these states. As an application of this technique, we are also working on the implementation of the discrete Fourier transform using the vibrational eigenstates of molecules, and on shifting the relative phase between levels by distorting the potential with a strong electric field pulse.

Fig. 3-2 Ramsey fringe of Iodine molecule

3-2. Visualization of wave packet motion by fs probe pulse

    On the other hand, if a femtosecond laser is used as the probe pulse, the probability density of the wave packet density around a specific interatomic distance, determined by the wavelength, can be probed. Furthermore, by sweeping the wavelength of the probe, it is possible to plot the spatio-temporal wave packet density distribution in a visible manner. (Strictly speaking, we are not measuring the spatial distribution of the wave packet itself in the molecule, since transition probability and spectral intensity factors come into play.) 

    A plot of the spatio-temporal density distribution of the wave packet is shown below. The temporal interval between pump and control pulses is set around 1.5 Tvib with phase changes of 90° each. Although the spatial range observed in this measurement is very narrow (approximately 6 pm), a significant change in the spatio-temporal density distribution of the wave packet is observed as the relative phase between the two wave packets changes. For example, a comparison between 0° and 180° shows that the density peaks at 334 pm in 0° are reversed to troughs in 180°, indicating that the positions of the density peaks and troughs are reversed.

Fig. 3-3 Quantum carpets observed in iodine molecule

3-3. Strong laser induced quantum interference 

Under construction.

    The research on coherent control in the gas phase introduced here is not currently being conducted at NAIST. If you would like to know more details about the experiments, please refer to the following references.

【Related papers】

1. Visualizing picometric quantum ripples of ultrafast wave-packet interference
   H. Katsuki, H. Chiba, B. Girard, C. Meier, and K. Ohmori, Science 311, 1589-1592 (2006).
2. Real-time observation of phase-controlled molecular wave-packet interference
    K. Ohmori, H. Katsuki, H. Chiba, M. Honda, Y. Hagihara, K. Fujiwara, Y. Sato, and K. Ueda, Phys. Rev. Lett. 96, 093002 (2006).
3. READ and WRITE Amplitude and Phase Information by Using High-Precision Molecular Wave-Packet Interferometry
   H. Katsuki, K. Hosaka, H. Chiba, and K. Ohmori, Phys. Rev. A 76, 013403 (2007).
4. Actively tailored spatiotemporal images of quantum interference on the picometer and femtosecond scales
   H. Katsuki, H. Chiba, C. Meier, B. Girard, and K. Ohmori, Phys. Rev. Lett. 102, 103602 (2009).
5. Ultrafast Fourier transform with a femtosecond laser driven molecule
   K. Hosaka, H. Shimada, H. Chiba, H. Katsuki, Y. Teranishi, Y. Ohtsuki, and K. Ohmori, Phys. Rev. Lett. 104, 180501 (2010).
6. Strong-Laser-Induced Quantum Interference
   H. Goto, H. Katsuki, H. Ibrahim, H. Chiba, and K. Ohmori, Nature Phys. 7, 383-385 (2011).

4. Manipulation of rovibrational wavefunctions in a solid para-hydrogen

4-1. Amplitude control of coherent states by nonlinear Raman technique

　As a more complex system than isolated molecules in the gas phase, we will next introduce vibrational exciton control in solid para-hydrogen. Hydrogen molecule is classified into para- and ortho-hydrogen based on its nuclear spin symmetry. As a result of Pauli's exclusion rule, para (ortho) hydrogen can only take even (odd) rotational quantum numbers J. Because of the low temperature, almost all hydrogen molecules in a para-hydrogen crystal are considered to occupy the J=0 state, and since the eigenfunctions of the J=0 state have an isotropic distribution, each para-hydrogen molecule in a crystal can be treated like a sphere with an isotropic distribution, rather than each molecule having a specific axis direction.(see figure below). Furthermore, since there is no electrical multipole moment, the electrostatic interaction between molecules is also very weak. In particular, considering the pure vibrational transition of v=1←0, the coherence lifetime of the vibrationally excited state is very long because the spherical symmetry of the molecule is also maintained. Depending on the impurity concentration, a coherence lifetime of about 1 ns can be easily obtained.

Fig. 4-1 Characteristics of para-hydrogen crystal 

　Since para-hydrogen crystals are many-particle systems, their wavefunctions are very complex with many degrees of freedom. Starting from an individual molecular basis, a state in which molecule i is excited to v = 1 and all other molecules are in their ground state is given as

$$ |v_{I}=1,J=0 \rangle \equiv \prod_{j\neq I} |n_{j} =0, J_{j}=0\rangle |v_{i}=1, J_{i}=0 \rangle. $$

Using this state as a basis function, diagonalizing the intermolecular interactions yields delocalized vibrational eigenfunctions, which are labeled by the wavenumber vector k as follows

$$ |v=1, \mathbf{k}, J=0 \rangle = \sum_{I} \frac{1}{\sqrt{N}} e^{i\mathbf{k}\cdot \mathbf{R}_{i}}|v_{i}=1,J=0\rangle. $$

We performed wave packet interference control for such a system by superposition of wave packets in its vibrationally excited state. The scheme of the experiment is presented in the figure below. Since the electronic transitions of molecular hydrogen do not exist in the visible region and the vibrational transitions are not infrared active, direct absorption measurement cannot be used. Therefore, we applied the impulsive Raman transition to induce the v=1←0 transition by making the difference frequency of the two-color laser resonate with the vibrational transition energy of the hydrogen molecule. In the experiment, the vibrational transition was induced by superimposing a 600 nm pump pulse and an 800 nm Stokes pulse. As in the iodine case, the two wave packets can be superimposed by injecting the same pump-Stokes pulse pair after a delay time τ. The states produced are

$$ \Psi(t)=|0\rangle + ae^{-i\omega_{0}t} \left( 1+e^{i\omega_{0}\tau}\right) |v=1, \mathbf{k}=0, J=0 \rangle. $$

As in the case of the iodine molecule introduced in section 3, the amplitude of the vibrationally excited state oscillates as the delay time τ between excitation pulses is varied. The state of the generated wave packet is done by injecting a probe pulse after the delay time and observing the anti-Stokes pulses created by the scattering of the pulse. The method of observing the state of coherence created by the pump Stokes pulses with probe pulses is exactly the same as the CARS (coherent anti-Stokes Raman scattering) method. The scheme of the experiment is shown in Fig. 4-2.

Fig. 4-2 Experimental scheme for
para-H₂ experimentFig. 4-3 Quantum interference of
para-H₂ vibrational state 

    Since the excitation of the vibrational wave packet occurs impulsively, the pump pulse and the Stokes pulse must overlap in time. If the timing of the probe pulse (τ_probe) is fixed at 1 ns behind the first excitation pulse and the timing between the two Raman excitations (τ_IRE) is varied, the alternating strengthening and weakening seen in section 2 is observed. The period of the oscillation is estimated to be approximately 8 fs based on the difference in energy between the v=1 and v=0 states. 

The actual signal obtained from the measurement is shown in in Figure 4-3. By overlapping two excitations, the relative intensity fluctuates between 0 and 4, indicating that the superposition of wave packets can be controlled with very high accuracy. The coherence retention time in the solid is also very long, and the degree of degradation is suppressed to about 10% even after 500 ps.

4-2. Control of spatial phase distribution using a two-dimensional phase modulator

    Based on the results of 4-1, we established a method to control the spatial wavefront of pump Stokes pulses by a two-dimensional phase modulator to simultaneously control temporal and spatial interference, and to read out the result as a spatial distribution image of anti-Stokes pulses. As a result, arbitrary phase distribution can be realized by creating any wavefront distribution using a spatial phase modulator and writing it in a parahydrogen crystal. As a demonstration, a 2x2 dot pattern was written in solid para-hydrogen. By designing the mask, arbitrary phase distribution can be written on each of the four spots. Actually, three phase information, (0,0,0,0), (0,180,0,180), and (0,270,180,90), were written to the four spots, and the results of reading out their states are shown in Figure 4-4. It can be seen that the intensity of each spot oscillates in different phases, indicating that the spatial phase distribution has been successfully written and read out.

Fig. 4-4 Write and read the spatial phase distribution of wavefunctions

By applying the observation of the phase distribution, it is possible to read out phase shifts caused by external perturbations and differences in the vibration period that originate from the inhomogeneous environment in the solid.

 【Related papers】

1. Optically engineered quantum interference of delocalized wavefunctions in a bulk solid: The example of solid para-hydrogen
   H. Katsuki, Y. Kayanuma, and K. Ohmori, Phys. Rev. B 88, 014507 (2013).
2. Manipulation and visualization of two-dimensional phase distribution of vibrational wavefunctions in solid para-hydrogen crystal
   H. Katsuki, K. Ohmori, T. Horie, H. Yanagi, and K. Ohmori, Phys. Rev. B, 92, 094511 (2015). 
3. Simultaneous manipulation and observation of multiple ro-vibrational eigenstates in solid para-hydrogen
   H. Katsuki and K. Ohmori, J. Chem. Phys. 145, 124316 (2016).