Ultrafast Coherent Control of Condensed Matter with Attosecond

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Cite This: Acc. Chem. Res. XXXX, XXX, XXX−XXX

Ultrafast Coherent Control of Condensed Matter with Attosecond Precision Hiroyuki Katsuki,*,† Nobuyuki Takei,‡,§ Christian Sommer,∥,‡ and Kenji Ohmori*,‡,§ †

Graduate School of Science and Technology, Nara Institute of Science and Technology, Ikoma 630-0192 Japan Department of Photo-Molecular Science, Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585 Japan § The Graduate University for Advanced Studies (SOKENDAI), Myodaiji, Okazaki 444-8585, Japan ∥ Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany ‡

CONSPECTUS: Coherent control is a technique to manipulate wave functions of matter with light. Coherent control of isolated atoms and molecules in the gas phase is well-understood and developed since the 1990s, whereas its application to condensed matter is more difficult because its coherence lifetime is shorter. We have recently applied this technique to condensed matter samples, one of which is solid para-hydrogen (p-H2). Intramolecular vibrational excitation of solid p-H2 gives an excited vibrational wave function called a “vibron”, which is delocalized over many hydrogen molecules in a manner similar to a Frenkel exciton. It has a long coherence lifetime, so we have chosen solid p-H2 as our first target in the condensed phase. We shine a time-delayed pair of femtosecond laser pulses on p-H2 to generate vibrons. Their interference results in modulation of the amplitude of their superposition. Scanning the interpulse delay on the attosecond time scale gives a high interferometric contrast, which demonstrates the possibility of using solid p-H2 as a carrier of information encoded in the vibrons. In the second example, we have controlled the terahertz collective phonon motion, called a “coherent phonon”, of a single crystal of bismuth. We employ an intensity-modulated laser pulse, whose temporal envelope is modulated with terahertz frequency by overlap of two positively chirped laser pulses with their adjustable time delay. This modulated laser pulse is shined on the bismuth crystal to excite its two orthogonal phonon modes. Their relative amplitudes are controlled by tuning the delay between the two chirped pulses on the attosecond time scale. Two-dimensional atomic motion in the crystal is thus controlled arbitrarily. The method is based on the simple, robust, and universal concept that in any physical system, two-dimensional particle motion is decomposed into two orthogonal one-dimensional motions, and thus, it is applicable to a variety of condensed matter systems. In the third example, the double-pulse interferometry used for solid p-H2 has been applied to many-body electronic wave functions of an ensemble of ultracold rubidium Rydberg atoms, hereafter called a “strongly correlated ultracold Rydberg gas”. This has allowed the observation and control of many-body electron dynamics of more than 40 Rydberg atoms interacting with each other. This new combination of ultrafast coherent control and ultracold atoms offers a versatile platform to precisely observe and manipulate nonequilibrium dynamics of quantum many-body systems on the ultrashort time scale. These three examples are digested in this Account.

1. INTRODUCTION Coherent control is a technique that manipulates wave functions by utilizing coherence of light. Typical examples are as follows. When an atom or molecule is excited with multiple laser beams through multiple different pathways to a common final state, those excited wave functions interfere with each other. Their constructive or destructive interference leads to the enhancement or depletion of the final-state population, respectively, depending on the relative phase of the laser beams. This scenario was proposed theoretically by Brumer and Shapiro1 in the 1980s and is now called the “Brumer−Shapiro scheme”. They showed that the branching ratios of molecular photodissociation can be controlled by irradiating a molecule © XXXX American Chemical Society

simultaneously with two continuous-wave (CW) laser beams with different wavelengths. In the same decade, Rice, Tannor, and Kosloff2 theoretically proposed another scheme, known as the “Tannor−Kosloff−Rice scheme”. They showed that the motion of a nuclear wave packet in a molecule can be steered with two time-delayed femtosecond laser pulses, so that branching ratios of different reaction pathways can be controlled. The rapid development of laser technology, including picoand femtosecond pulsed lasers in the 1990s, has enabled the Received: January 9, 2018

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DOI: 10.1021/acs.accounts.7b00641 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 1. Experimental setup for the interference of vibrons in solid p-H2. (a) Preparation scheme for the p-H2 crystal. The dotted circle indicates the p-H2 crystal inside the cryostat. (b) Pulse sequence and measurement scheme for vibron interference. (c) Feynman diagrams representing the two CARS signals generated by the first (pump-1, stokes-1) and second (pump-2, stokes-2) IRE pulses. (d) Fifth-order Raman signal that can be observed under a different phase-matching condition. Adapted with permission from ref 25. Copyright 2013 American Physical Society.

implementation of these schemes experimentally and moreover added another promising scheme of coherent control. Scherer and co-workers conducted an experiment in which a pair of vibronic wave packets (WPs) was produced in the iodine molecule with a pair of femtosecond laser pulses.3 They actively controlled the interference of those two WPs by tuning the relative phase of the laser pulses. Similar experiments have been performed for other molecules as well as for Rydberg WPs in alkali atoms.4−7 Another important technique for coherent control is laserpulse shaping with a liquid-crystal spatial light modulator or acousto-optic device. The optimal pulse shape can be estimated by quantum optimal control theory8 or a learning algorithm.9 This scheme has been applied to the control of a variety of unimolecular processes.9−12

In a series of papers, we have reported high-precision coherent control with attosecond precision that has been applied to vibronic WPs in isolated molecules in the gas phase.13−18 The interferometric superposition of two WPs is created by a pair of femtosecond laser pulses whose timing is manipulated with attosecond precision by utilizing a highly stabilized Michelson-type interferometer. The spatiotemporal images of the WP interference are visualized and actively designed on the picometer and femtosecond scales.13,19 This high-precision coherent control has been applied to ultrafast computing with vibrational WPs of the iodine molecules, where a discrete Fourier transform is securely executed on the femtosecond time scale.14,15 In addition to such isolated atoms and molecules in the gas phase, condensed matter has also been the target of coherent control over the past 20 years.20−24 Coherent control in the B


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common probe pulse. Other nonlinear contributions such as fifth-order Raman signals are spatially and spectrally filtered out with an iris and band-pass filters, respectively. Figure 1c,d shows Feynman diagrams representing those two CARS and fifth-order Raman signals. The intensity of this anti-Stokes pulse is proportional to the square of the coherence between the ground and vibron states. It can be seen from eq 1 that the coherence oscillates as a function of τIRE as a result of the interference of vibrons created by the time-delayed pair of IRE pulses. The details of the experimental conditions are described in ref 25. The merit of the current CARS scheme compared with the fluorescence detection scheme demonstrated before13−16 is that it is straightforward to utilize the retrieved anti-Stokes pulse for secondary optical processing such as an amplification or cascade process in another p-H2 crystal.

condensed phase is generally more difficult than in the gas phase, as its coherence lifetime is shorter. However, the decay of this coherence could provide important information on interactions among atoms and molecules. In this Account, we digest three recent applications of our ultrahigh-precision coherent control with attosecond precision to condensed matter systems. They are vibrational wave functions, called “vibrons”, delocalized in solid para-hydrogen (p-H2),25−27 coherent phonons in a single crystal of bismuth,28 and manybody electron dynamics in a strongly correlated ultracold Rydberg gas.29,30

2. CONTROL OF VIBRONS IN SOLID P-H2 Solid p-H2 has long vibrational dephasing time.31 This is due to the weak electrostatic interaction between p-H2 molecules. This long coherence lifetime motivated us to choose solid p-H2 as our first target in the condensed phase.

2.2. Results and Discussion

Figure 2 shows the anti-Stokes intensity measured as a function of the delay τIRE scanned with a piezo stage and normalized by

2.1. Experimental Scheme

Figure 1 shows a schematic illustration of our experiment. The preparation of an almost-pure p-H2 crystal was described in our previous paper.25 Briefly, normal hydrogen was preconverted to almost pure p-H2 by contact with powder of the catalyst (FeO(OH)). The p-H2 gas was flowed into a Cu cell fixed in a liquid-helium-cooled optical cryostat. The temperature of the crystal was kept at ∼5.4 K during the measurement. We utilized coherent anti-Stokes Raman scattering (CARS) to prepare and detect the vibrational coherence. A femtosecond Ti/sapphire regenerative amplifier was used as the light source. The output was split into three pulses, namely, the pump, Stokes, and probe pulses, with their central wavelengths tuned at ∼600 nm, ∼800 nm, and ∼560 nm, respectively. The pump and Stokes pulses were shined simultaneously on the p-H2 crystal and induced an impulsive Raman transition to the v = 1 state, where v is the vibrational quantum number of a p-H2 molecule. The angular frequencies of the pump and Stokes pulses, denoted as ωpump and ωst, respectively, were tuned so that ωpump − ωst was resonant with the v = 1 ← 0 transition of the p-H2 molecule. These pump and Stokes pulses are hereafter called an “impulsive Raman excitation (IRE) pulse”. Because of the selection rule of a Raman transition, only the k = 0 state at the bottom of the v = 1 band can be excited by the IRE pulse. The wave function of this state is delocalized over many hydrogen molecules in a manner similar to a Frenkel exciton and is called a “vibron”. A pair of IRE pulses were prepared with a stable Michelsontype interferometer25 and shined on the p-H2 crystal with their delay τIRE. After the second IRE pulse, the vibrational wave function is described as

Figure 2. Interference of vibrons in solid p-H2. The delay τprobe of the probe pulse is fixed at ∼1 ns. The delay τIRE between the two IRE pulses is scanned around (a) 10 ps and (b) 500 ps. Adapted with permission from ref 25. Copyright 2013 American Physical Society.

the anti-Stokes intensity measured with a single IRE pulse. The delay τprobe is fixed at ∼1 ns. Perfect interference of vibrons should give intensity oscillating between the values 0 and 4. The measured fringe contrast, defined by the parameter A/Y0, where A and Y0 are given by the sinusoidal fit y = Y0 + A sin(ωt + ϕ), is 0.994 in Figure 2a, demonstrating almost perfect interference at τIRE around 10 ps. The period of the oscillation is estimated to be 8.05 fs, which agrees well with the value of 8.04 fs derived from the v = 1 ← v = 0 Raman transition frequency.32 The fringe contrast is derived to be 0.926 at τIRE ∼ 500 ps in Figure 2b, which is slightly worse than in Figure 2a. This degradation is probably due to pure dephasing between the ground and vibron states caused mainly by the impurity ortho-hydrogen (o-H2). It has been reported that the vibrational coherence lifetime of p-H2 is strongly influenced by the concentration of o-H2. The main mechanism of the dephasing could be elastic scattering of the v = 1 vibron by the o-H2 impurity.33 The result shown in Figure 2 suggests that the coherence between the ground and vibron states generated by the first IRE pulse survives at least on the nanosecond time

|Ψ(t , τIRE)⟩ = |v = 0⟩ + c|v = 1, k = 0⟩e−iω1t (1 + e iω1τIRE)

(1)

where |v = 0⟩ is the vibrational ground-state wave function, |v = 1, k = 0⟩ is the vibron wave function, c (≪1) is its amplitude, and ω1 is the angular frequency between the ground and vibron states. The probe pulse, having the angular frequency ωprobe, was shined on the crystal with its delay τprobe from the first IRE pulse, and the scattered anti-Stokes pulse with kas = kpump − kst + kprobe at ωas = ωpump − ωst + ωprobe was detected by a CCD camera coupled to a spectrometer. This anti-Stokes pulse is the result of the interference of two third-order CARS signals generated by the first and second IRE pulses followed by their C


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Accounts of Chemical Research scale. Such a long coherence lifetime of p-H2 is also expected from the narrow bandwidth of the relevant Raman transition.32 We also measured the real-time evolution of the vibron interference. Figure 3 shows the anti-Stokes intensity measured

Figure 3. Real-time observation of the interference of vibrons in solid p-H2. The delay τIRE between the two IRE pulses is tuned to ∼500 ps and is different by only ∼4 fs between the constructive (red) and destructive (blue) traces. The black trace is a reference obtained without the second IRE pulse. Adapted with permission from ref 25. Copyright 2013 American Physical Society.

by scanning τprobe with the delay τIRE set at ∼500 ps. The delay τIRE is stabilized to the timing around the maximum or minimum of the vibron interferogram shown in Figure 2b. It can be seen from the red and blue traces in Figure 3 that the evolution changes drastically from constructive to destructive when τIRE is changed by only 4 fs.

3. STEERING AND VISUALIZING ATOMIC MOTION IN A SINGLE CRYSTAL OF BISMUTH Coherent phonons are collective oscillations of atoms in the solid state. They can be triggered with an optical pulse whose temporal width is shorter than the periods of lattice vibrations. Coherent phonons have been observed in a variety of materials,34 and their amplitudes have been controlled onedimensionally.35−37 Two-dimensional (2D) control of atomic motion in the solid state has been performed by superposing energetically degenerate orthogonal phonon modes with two identical laser pulses polarized perpendicular to each other.38 More universal control of 2D atomic motion should be useful to study the optical response of solid-state materials and thereby develop their novel photoinduced functionalities. That universal 2D control requires another new methodology to superpose nondegenerate orthogonal phonon modes, which is more common in solid-state materials. We employed a single crystal of bismuth to demonstrate such a new methodology. The crystal structure of bismuth is rhombohedral, and its unit cell consists of two atoms. There exist three optical phonon modes of A1g and Eg symmetry, the latter of which is doubly degenerate.39 They are schematically depicted in Figure 4a, which shows that the atoms move longitudinally and laterally in the A1g and Eg modes, respectively, according to our definition of crystal axes as shown in Figure 4a, so that they are orthogonal to each other. At the center of the reciprocal space (Γ point), the frequencies of the A1g and Eg modes are ∼3.0 and ∼2.1 THz, respectively.

Figure 4. Scheme of the bismuth experiment. (a) Unit cell structure of bismuth and orientation of the atomic distortion by phonon excitation. (b) Experimental setup for the pump−probe reflectivity measurement. Abbreviations: CM, chirp mirror; PD, photodiode; ND filter, neutral density filter; SQ plates, window plates of synthesized quartz substrate; Bi, bismuth. (c) Coherent phonon signal observed with τmod ∼ 0 fs. The origin of the probe delay τprobe = 0 ps is defined as the top of the first positive peak of the oscillatory phonon signal. Adapted with permission from ref 28. Copyright 2013 Nature Publishing Group.

precompensated with a pair of chirped mirrors and then split into two pulses with a 9:1 ratio, which were used as pump and probe pulses, respectively. The pump pulse was introduced into our homemade highly stabilized Michelson-type interferometer to generate a phased pair of laser pulses (pulse width ∼ 30 fs in their Fourier transform limits). These pulses were actively chirped with fused silica (SQ) plates and hereafter called “chirped subpulses” (CSPs). The CSPs were temporally superposed with a delay τmod to produce a pump pulse whose temporal intensity envelope was periodically modulated. The modulated pump pulse was shined vertically on the (0001) surface of the bismuth crystal, which was fixed inside a liquidhelium-cooled optical cryostat. The probe pulse was split into two pulses, one of which was detected with a Si photodiode and used as a reference. The other one was shined on the bismuth crystal, and its reflection was detected with another Si photodiode. The differential signal from the two detectors was amplified by a current amplifier and collected by an oscilloscope. The details of the experiments are described in ref 28.


A schematic diagram of the experimental setup is shown in Figure 4b. The output of a Ti:sapphire oscillator was D


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Figure 5. Control of 2D atomic motions in a bismuth crystal. (a) Simulated temporal envelopes of the CSP pair: green, τmod = 49.4 fs; blue, τmod = 92.2 fs; red, τmod = 93.6 fs. The dotted curve is the temporal intensity of the CSP pair with τmod = 0 fs. (b) Fourier-transformed terahertz spectra calculated from the temporal envelopes in (a). Black arrows show the frequencies of the A1g and Eg coherent phonons. (c) Coherent phonon signals observed at three different τmod similar to those used for the simulations in (a) and (b). (d) Fourier-transformed terahertz spectra of the coherent phonon signals shown in (c). (e−g) Two-dimensional atomic motion within a unit cell of the bismuth crystal reconstructed from the coherent phonon signals shown in (c) within the time window from τprobe = 0.82 to 10.48 ps. Adapted with permission from ref 28. Copyright 2013 Nature Publishing Group.


its Supplementary Information. Details of the coherent phonon generation and decay mechanism are given in the previous paper.40 The colored traces in Figure 5a show the temporal intensity envelopes of the modulated pump pulse simulated with the CSP delays τmod = 49.4, 92.2, and 93.6 fs, whose Fourier transform spectra are shown in Figure 5b. It can be seen from these simulations that one can control the relative intensities of the frequency components of the pump pulse at ∼3.0 and ∼2.1 THz, which correspond respectively to the A1g and Eg phonons, by tuning τmod on the 1 fs time scale. Figure 5c shows the coherent phonon signals observed at three different τmod similar to those for the simulations shown in Figure 5a,b. Their Fourier transforms are shown in Figure 5d, demonstrating that the relative intensities of the A1g and Eg phonons at ∼3.0 and ∼2.1 THz were clearly controlled by tuning τmod. Figure 5e−g shows the real-space atomic motions reconstructed from the fitted coefficients Aa and Ae in eq 2 derived from the measured phonon signals shown in Figure 5c on the basis of the correspondence between the nuclear displacement and the reflectivity change ΔR. This correspondence was obtained by linear response density functional theory (DFT) calculations,41

The typical coherent phonon signal observed with τmod ∼ 0 fs at 5.2 K is plotted in Figure 4c, which shows the reflectivity change ΔR divided by the total reflectivity R as a function of the delay τprobe between the pump and probe pulses. The strong nonresonant signal that appears around τprobe = 0 is filtered out by the bandpass filter function of the current amplifier. It can be seen from the oscillatory coherent phonon signal shown in Figure 4c that its oscillation amplitude is modulated periodically, indicating two contributions from the A1g and Eg modes. The oscillation is thus reasonably fitted with a linear combination of two damped oscillators that correspond to A1g (longitudinal) and Eg (lateral) phonons:28 ΔR(t ) = A 0 + A a exp( −Γat ) sin(2πνat + δa) R + Ae exp( −Γet ) sin(2πνet + δe)

(2)

where A0 is the baseline and Ai, Γi, νi, and δi with {i} = {a, e} are the phonon amplitudes, damping factors, oscillation frequencies, and initial phases of the A1g and Eg phonons, respectively. The value of each parameter is given in ref 28 and E


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Figure 6. Schematic diagram of the experimental setup and Rydberg excitation. (a) Sketch of the experimental setup. (b) Two-photon pump (probe) excitation of the Rb atom to its Rydberg states. (c) Rydberg states resolved by field ionization with a slowly ramped electric field. (d) Sketch of the two-body interaction compared with the excitation bandwidth accompanied by calculated nearest-neighbor distributions of Rydberg atoms for the peak atom density of 1.3 × 1012 cm−3 (pink and red traces) and the averaged density over the whole atoms (light- and dark-blue traces). Adapted with permission from ref 29. Copyright 2016 Nature Publishing Group.

whose details are given elsewhere.28 It can clearly be seen from Figure 5e−g that ultrafast 2D atomic motions in the bismuth crystal were actively controlled and visualized on the femtosecond time scale by all-optical methods.

number of Rydberg excitations per unit volume can be larger than the ones in the previous experiments with narrow-band lasers by 2 orders of magnitude.29 Accordingly, the Rydberg gas should become a strongly correlated system in which many Rydberg atoms interact simultaneously with each other. Although broad-band pulsed lasers were previously used to observe Rydberg WPs in isolated atoms,4,5 we utilized them to investigate how coherent dynamics is affected by many-body interactions.

4. MANY-BODY ELECTRON DYNAMICS IN A STRONGLY CORRELATED ULTRACOLD RYDBERG GAS Many-body interactions in the condensed phase govern a variety of important physical and chemical phenomena ranging from the emergence of superconductivity and magnetism in solid materials to chemical reactions in liquids. One of the latest developments of the studies on many-body interactions is experimental efforts on long-range interactions among ultracold particles, including dipolar quantum gases,42 ion crystals,43 polar molecules,44 and Rydberg atoms.45 Rydberg atoms distinguish themselves by their large dipole moments, which induce strong interactions, as well as high tunability of the nature and strength of those interactions. One of the prominent effects due to the large dipole moments of Rydberg atoms is “Rydberg blockade”,45 which suppresses simultaneous Rydberg excitations of a pair of atoms whose Rydberg interaction is larger than the line width of the excitation laser. This blockade condition determines the smallest distance between neighboring Rydberg atoms, which was typically on the order of several micrometers in previous experiments on ultracold Rydberg atoms usually performed with narrow-band CW lasers. Alternatively, an off-resonant narrow-band excitation gave Rydberg excitations at a smaller but specific interatomic distance.46 Here we combined an ultracold Rydberg gas with a broadband picosecond laser pulse that can induce Rydberg excitations over a wide range of interatomic distances from less than 1 μm to the isolated-atom limit. Therefore, the


Figure 6a shows a schematic illustration of the experiment. We prepared a cold ensemble of 87Rb atoms in an optical dipole trap. Its temperature and highest peak atom density were estimated to be ∼70 μK and ∼1.3 × 1012 cm−3, respectively. The atoms in the 5S1/2 ground state were excited to Rydberg states νD5/2, where ν is the principal quantum number, via a two-photon transition using broad-band picosecond laser pulses with their center wavelengths tuned to ∼779 nm and ∼481 nm (Figure 6b). The population of the Rydberg state was measured by field ionization. Figure 6c shows a field ionization spectrum in which almost a single Rydberg state, 42D5/2, which was the main target state of the current experiment, was mainly populated by the present picosecond-pulse excitation. From this field ionization spectrum, the bandwidth of our Rydberg excitation was determined to be about 150 GHz (fwhm), which is wide enough to excite a pair of Rb atoms simultaneously to their 42D5/2 Rydberg states even at interatomic distances shorter than 1 μm, as schematically shown in Figure 6d. We measured interferograms similar to those shown in Figure 2 with a pair of two-photon excitations, hereafter called the “pump” and “probe” excitations, whose delay τ was scanned on the attosecond time scale with our highly stabilized interferometer.25 Those interferograms are hereafter called “time-domain Ramsey interferograms”. Figure 7 shows the F


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Figure 7. Ramsey oscillation of the 42D5/2 state. The field ionization signals for the higher-density (red traces) and lower-density (blue traces) ensembles are plotted as functions of the pump−probe delay τ scanned over a range of ∼3 fs around (a) 75, (b) 245, and (c) 500 ps, showing clear oscillations. The signal intensities are normalized by the mean value of the sinusoidal function fitted to each oscillation. The origin of the pump− probe delay τ = 0 is arbitrary and is taken to be the left edge of each plot. Adapted with permission from ref 29. Copyright 2016 Nature Publishing Group.

Figure 8. Measured Ramsey contrasts and phase shifts compared with the numerical ones based on the analytical continuum approximation. The black and blue points show the measured (a, b) Ramsey contrasts and (c, d) phase shifts for 42D5/2 state populations of (a, c) ∼1.2% and (b, d) ∼3.3%. In (a) and (b), the measured Ramsey contrasts are compared with the numerical ones based on the analytical continuum approximation for average numbers of Rydberg atoms N = 20 and N = 40 (purple and blue semitransparent lines, respectively) as well as for the limit N → ∞ (black and blue solid lines). Similarly, the measured and numerical phase shifts are compared in (c) and (d). Adapted with permission from ref 29. Copyright 2016 Nature Publishing Group.

fitted with a sinusoidal function to obtain their contrasts and

Rydberg population measured after the pair of excitations tuned to the 42D5/2 state and plotted as a function of the delay τ. As can be seen in Figure 7, the population oscillates with a period of ∼1 fs, which corresponds to the transition frequency between the 5S1/2 and 42D5/2 states. This oscillation corresponds to the temporal oscillation of the 42D5/2 wave function and to the motion of an electronic WP composed of the 5S1/2 and 42D5/2 states superposed coherently by the excitation pulses.17 The measured Ramsey interferograms were

phases. The measurements were made for two different peak atom densities, which were estimated to be ∼1.3 × 1012 and ∼4 × 1010 cm−3. In the lower-density ensemble, the interaction effects are negligible, so the contrasts and phases measured in this ensemble were used as references to be compared with those measured in the higher-density ensemble. G


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Figure 9. Principal quantum number and atom density dependences of the Ramsey contrast. (a) Measured Ramsey contrasts plotted as functions of τ for three different Rydberg levels (ν = 38, 42, and 50). (b) Ramsey contrast measured as a function of the peak atom density at two different pump−probe delays (τ = 300 and 510 ps) for ν = 42 with its population being ∼3.5%. Adapted with permission from ref 29. Copyright 2016 Nature Publishing Group.

around a particular position r.30 This approximation leads to the following expression for the Ramsey signal averaged over the whole atomic ensemble:


Figure 8 shows the contrast ratio and phase shift between Ramsey interferograms measured for the excitation tuned to the Rydberg state 42D5/2 at the two different atom densities given above, hereafter called the “Ramsey contrast” and “phase shift”, respectively. These results are plotted as functions of the pump−probe delay τ for two different Rydberg populations, ∼1.2% and ∼3.3%, showing that the Ramsey contrast decays and the phase shift accumulates as a function of τ. The offsets of the phase shift at τ = 0 in Figure 8c,d are essentially due to the difference between AC Stark shifts of the atomic levels in the higher- and lower-density ensembles, and its details are given elsewhere.29 Figure 8 shows that the Ramsey contrast decay and phase shift are accelerated by increasing the Rydberg population. Figure 9 shows the principal quantum number and atom density dependences of the Ramsey contrast decay, demonstrating that the decay is accelerated also by increasing the Rydberg orbital size and by decreasing the interatomic distances. A set of these dependences on the Rydberg population, Rydberg orbital size, and interatomic distance indicates that the measured Ramsey contrast decay and phase shift are induced by interactions of Rydberg atoms. The Ramsey contrast decay and phase shift shown in Figure 8 are not reproduced by assuming only nearest-neighbor interactions or by a mean-field approximation.29 It is therefore concluded that our experimental observations demonstrate many-body interactions beyond mean-field approximations and that the effect of those interactions on the electron dynamics was actively controlled. We then performed a theoretical analysis beyond mean-field approximations by using an exactly solvable model with an Ising-type interaction Hamiltonian.30,47 With an exact solution for the time evolution with this Hamiltonian, we can calculate the time evolution of the Ramsey signal P(τ) for any strength of interactions.30,47 Further analytical progress is possible by using an approximation, hereafter called the “continuum approximation”, in which a continuum function n(r) is considered for the density distribution of Rydberg atoms. We assume the density distribution to be homogeneous in a small volume

P(τ ) = 2pg pe [1 + |g (τ )| cos(ωτ + α(τ ) + ϕ)]

(3)

where pg and pe represent the populations of the 5S1/2 and 42D5/2 states, respectively, and the contrast decay |g(τ)| and phase shift α(τ) are obtained from the function g(τ) ≡ |g(τ)| eiα(τ), which depends on the character of the Rydberg interaction, the average number of interacting Rydberg atoms N, and the density distribution of the atomic ensemble.30 Figure 8 shows comparisons between the measured Ramsey contrast decay and phase shift for the 42D5/2 state and the numerical results based on the analytical continuum approximation assuming an isotropic van der Waals interaction U(r) = −C6/r6 and a Gaussian atom density distribution. The figure demonstrates that the analytical results obtained for C6 = 34 GHz μm6 in the limit N → ∞ (keeping the atom density distribution fixed so that the volume goes to infinity simultaneously) are in good agreement with the measured Ramsey contrasts and phase shifts for both of the Rydberg populations, pe ∼ 1.2% and ∼ 3.3%. Figure 8 also shows numerical results for finite average numbers of interacting Rydberg atoms (N = 20 and 40). The agreement between the numerical and experimental results improves monotonically, indicating that interactions among more than 40 Rydberg atoms are necessary to reproduce our experimental observations.

5. SUMMARY AND OUTLOOK In this Account, we have digested three examples of our recent experiments on ultrafast coherent control with attosecond precision in the condensed phase. In the first example with solid p-H2, we have shown that long-lived phase information can be encoded into the wave functions of solid p-H2 with laser light and can be processed and retrieved on the femtosecond time scale. We have recently implemented such ultrafast information processing in a p-H2 crystal,26,27 demonstrating that solid p-H2 could be a promising quantum-information medium in the solid state. H


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Accounts of Chemical Research Notes

In the second example, we have controlled and visualized ultrafast 2D atomic motions in a single crystal of bismuth on the femtosecond time scale by all-optical methods. Our control−visualization scheme is based on the simple, robust, and universal concept that in any physical system, 2D motion of a particle is determined by the projections of its pathway onto two orthogonal axes. The scheme is thus applicable to a variety of condensed matter systems and therefore will be useful in investigating the correlation between atomic motions and functionalities of solid-state materials. One of the most striking features of the condensed phase is the effect of many-body interactions. This effect has been investigated in the third example. We have successfully combined our ultrafast coherent control with an ultracold Rydberg gas to observe and manipulate many-body electron dynamics on the attosecond time scale. The observed electron dynamics has been analyzed beyond mean-field approximations, demonstrating many-body interactions among at least 40 Rydberg atoms. Our fruitful combination of ultrafast and ultracold approaches thus offers a novel platform to study nonequilibrium dynamics of quantum many-body systems on the ultrashort time scale. One of the intriguing targets for our ultrafast−ultracold approach will be a more strongly correlated system where Rydberg orbitals are spatially overlapped between neighboring atoms, so that electrons are shared by many atoms,48 as schematically shown in Figure 10. This “metal-like”

The authors declare no competing financial interest. Biographies Hiroyuki Katsuki received his Ph.D. (Science) in 2002 from Kyoto University for work on high-resolution spectroscopy of molecules in solid hydrogen. In 2004 he moved to the Institute for Molecular Science as an assistant professor to work with Prof. K. Ohmori. In 2012 he moved to the Nara Institute of Science and Technology as an associate professor. Currently he is interested in applying the techniques of ultrafast coherent control to exciton−polariton systems prepared in a microcavity. Nobuyuki Takei obtained his Ph.D. from the University of Tokyo in 2005 for studies of continuous-variable quantum information processing. After a one-year postdoctoral stay at The University of Tokyo, he joined the JST-ERATO Ueda Macroscopic Quantum Control Project as a postdoctoral fellow. In 2008 he became an assistant professor in the group of Prof. K. Ohmori. His research interests include experimental atomic, molecular, and optical physics and ultrafast coherent control of many-body quantum dynamics. Christian Sommer received his Ph.D. from the Technical University of Munich in association with the Max Planck Institute of Quantum Optics in 2011 for the construction and operation of a cryogenic source for cold polar molecules. After his Ph.D. work, he joined the group of Prof. K. Ohmori as a postdoctoral fellow and as an assistant professor. He is currently working at the Max Planck Institute for the Science of Light in Germany. His current research interests are collective effects in cavity quantum electrodynamics with a focus on many-body Rydberg systems. Kenji Ohmori is a professor at the Institute for Molecular Science (IMS) of the National Institutes of Natural Sciences in Okazaki, Japan, where he is also the chair of the Department of Photo-Molecular Science. After receiving his Ph.D. from The University of Tokyo in 1992, he was a research associate and an associate professor at Tohoku University. In 2003 he was appointed as a full professor at IMS. His research interests focus on exploring the quantum−classical boundary and developing quantum technology based on the wave nature of matter. He is currently leading a JSPS Specially Promoted Research Project that involves combining an ensemble of ultracold Rydberg atoms and ultrafast coherent control to look into the dynamics in the transition from a delocalized wave function to a localized particle. He has been honored with the Japan Academy Medal (2007) and the JSPS Prize (2007), is a Fellow of the American Physical Society (2009), and has received the Humboldt Research Award (2012), the Hiroshi Takuma Memorial Prize (2017), and the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology of Japan (2018).

Figure 10. Schematic of ultracold Rydberg atoms in an optical lattice where Rydberg orbitals are spatially overlapped between neighboring atoms. Adapted with permission from ref 48. Copyright 2014 Springer Science+Business Media New York.

state is expected to decay quickly through Penning ionization, in which one of the neighboring atoms is ionized while the other atom is de-excited. The time scale of such a decay of two 85 Rb atoms has recently been estimated numerically to be about 10 ns for n = 53.49 This is shorter than the measurement time scale of most of the ultracold Rydberg experiments performed with narrow-band CW lasers by 2 orders of magnitude, whereas it is longer than our measurement time scale by 2 orders of magnitude. Observation of that exotic metal-like phase should thus be possible using our ultrafast−ultracold approach.

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ACKNOWLEDGMENTS

The authors thank Y. Kayanuma, M. E. Garcia, E. S. Zijlstra, K. G. Nakamura, M. Kitajima, M. Weidemüller, G. Pupillo, C. Genes, D. Jaksch, and M. Kiffner for fruitful discussions. This work was partially supported by CREST-JST, a JSPS Grant-inAid for Specially Promoted Research (Grant 16H06289), JSPS KAKENHI Grant 19684014, and the Photon Frontier Network Program by MEXT. K.O. thanks the Alexander von Humboldt Foundation, the University of Heidelberg, and the University of Strasbourg for supporting this international collaboration.

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. I


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Ultrafast Coherent Control of Condensed Matter with Attosecond

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