⇦Chinese Physics Letters, 2019, Vol. 36, No. 5, Article code 054203 Effect of Carrier Envelope Phase on High-Order Harmonic Generation from Solid * Jie Shao (邵洁)1,2, Cai-Ping Zhang (张彩萍)1,2**, Jing-Chao Jia (贾景超)1,2, Jun-Lin Ma (马俊琳)1,2, Xiang-Yang Miao (苗向阳)1,2** Affiliations 1College of Physics and Information Engineering, Shanxi Normal University, Linfen 041004 2Key Laboratory of Spectral Measurement and Analysis of Shanxi Province, Shanxi Normal University, Linfen 041004 Received 7 January 2019, online 17 April 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11404204 and 11504221, the Program for the Top Young Academic Leaders of Higher Learning Institutions of Shanxi Province, the Natural Science Foundation for Young Scientists of Shanxi Normal University under Grant No ZR1805, and the Project for Graduate Research Innovation of Shanxi Normal University.
**Corresponding author. Email: zhangcp@sxnu.edu.cn; xymiao@sxnu.edu.cn Citation Text: Shao J, Zhang C P, Jia J C, Ma J L and Miao X Y et al 2019 Chin. Phys. Lett. 36 054203    Abstract The dependence of harmonic emission from a solid on the carrier envelope phase (CEP) is discussed by numerically solving the time-dependent Schrödinger equation. The harmonic spectra periodically exhibit three distinct oscillating structures, which indicate the different dependences of the cutoff energies on the CEP. Furthermore, with time-dependent population imaging and the populations of different energy bands, the underlying physical mechanism is explored. DOI:10.1088/0256-307X/36/5/054203 PACS:42.65.Ky, 42.65.Re, 78.47.-p © 2019 Chinese Physics Society Article Text High-order harmonic generation (HHG) from gases has been extensively studied for a few decades[1–6] and has become one of the most important research fields in ultrafast atomic and molecular physics, realized by a semi-classical three-step model.[7] First, the electron tunnels through a barrier formed by the Coulomb potential and the laser field. Secondly, the ionized electron is accelerated in the oscillating laser field. Thirdly, when the laser field reverses its direction, the electron can return and recombine with the parent ions and emit harmonic photons. The cutoff energy is near $I_{\rm p}+3.17U_{\rm p}$, where $I_{\rm p}$ is the ionization energy, and $U_{\rm p}$ is the ponderomotive energy. HHG provides a new type of coherence light source of extremely short wavelength, which can be widely used in attosecond pulse generation,[8,9] plasma spectroscopy,[10] x-ray fluorescence analysis,[11] and so on. However, the low harmonic yields from the atomic and molecular systems may confine the development of nonlinear attosecond science. Compared with gas systems, the special characteristics of solid systems such as high density and diverse spatial structure may make it possible to achieve a high-intensity ultraviolet light source. Moreover, solid HHG can also provide an efficient way to achieve full optical reconstruction of the band structure[12–14] and control ultrafast electron dynamics in solids.[15–17] In experimental and theoretical studies, ultrafast laser-driven crystal materials have produced many important advances in solid HHG. Burnett et al.[18] found harmonic generation in plasmas formed by the interaction of a strong laser and the solid. Ghimire et al.[19] indicated that the higher harmonics can also be emitted under the interaction between a mid-infrared laser and the semiconductor ZnO. In addition, solid HHG was also experimentally generated by bulk crystal.[19–21] With the rapid development of solid harmonics, more novel features were discovered. Only one plateau can be seen from the harmonic spectra generated in solid Ar and Kr at low laser intensities, while it exhibits a multi-plateau structure with the increase of the laser intensity.[22] Moreover, many methods have been proposed to enhance the efficiency of solid harmonics. In addition to the spatially inhomogeneous laser field[23] and two-color field,[24] Guan et al.[25] proved that the carrier envelope phase (CEP) can significantly improve the efficiency of the second plateau. Du et al.[26] introduced a model in the coordinate space combined with the motion of the Bloch electron wave packets moving at group and phase velocities, which also proves that the second HHG plateau has a magnitude enhancement of six to seven orders with the variation of the CEP. Liu et al.[27] revealed that not only the efficiency of the second plateau but also the cutoff energy is sensitive to the CEP. To obtain a deeper insight into the CEP effect on solid harmonic emission, we further explore the electronic dynamics under different CEPs. Our results show that the cutoff energy of the harmonic spectra periodically exhibits three different structures in different regions of the CEP. Based on time-dependent population imaging, the underlying physical mechanism is provided.
cpl-36-5-054203-fig1.png
Fig. 1. Energy band structure of five energy bands calculated with diagonalization in coordinate space.
In our simulation, we describe the interaction of a laser with crystal based on single-electron approximation. The linearly polarized laser is in the direction of the crystal axis. In the length-gauge treatment, the one-dimensional time-dependent Hamiltonian is written as $\hat{H}(t)=\hat{H}_{0}+x\hat{E}(t)$, with $\hat{H}_{0}=\hat{P}^{2}/2+V(x)$ being the field-free Hamiltonian, where $\hat{P}$ is the momentum operator, and $V(x)$ is the periodic lattice potential. Herein, we use the Mathieu-type potential[28] $V(x)=-V_{0}[1+\cos(2\pi x/a_{0})]$, with $V_0=0.37$ a.u. and lattice constant $a_{0}=8$ a.u., which has been used extensively in the optical lattice research area.[29,30] The linearly polarized laser pulse with cosine-squared envelope is $E(t)=E_{0}f(t)\cos({\omega}t+\varphi)$, where $E_{0}$, $\omega$ and $\varphi$ are the peak amplitude, laser frequency and CEP, respectively. The free-field Schrödinger equation is written as $\hat{H}_{0}\phi_{n}(x)=E_{n}\phi_{n}(x)$, where $n$ is the eigenstate number, and $\phi_{n}(x)$ is the corresponding eigenstate wave function. We numerically solve this equation by diagonalizing $\hat{H}_{0}$ on a coordinate grid. The energy band structure is shown in Fig. 1, which contains two valence bands (VBs) and three conduction bands (CBs). Specifically, the five bands are denoted as VB0, VB, CB1, CB2 and CB3, respectively. The state numbers corresponding to the five bands are 1–59, 60–120, 121–179, 180–240, and 241–300, respectively. Moreover, the VB is chosen as the initial state to make most electrons tunnel into the CB1 with the minimum energy gap.
cpl-36-5-054203-fig2.png
Fig. 2. (a) The dependence of high-order harmonic spectra (with a logarithmic color scale) on CEP from $-1.5\pi$–2.5$\pi$ in increments of 0.05$\pi$. The wavelength, intensity and duration are 3.2 µm, $8.087\times10^{11}$ W/cm$^{2}$ and 2 o.c., respectively. (b) The enlargement of the oblique structure, and the symbols A, B and C correspond to the photon energies of 9.5 eV, 11.0 eV and 12.5 eV, respectively.
The time-dependent wave function $\psi(t)$ is obtained by solving the time-dependent Schrödinger equation (TDSE) using the second-order splitting operator method.[31–33] Additionally, we perform all calculations in the coordination space with the region [$-$240 a.u., 240 a.u.], and then the harmonic spectrum is obtained by calculating the Fourier transform of the laser-induced current: $H(\omega)=\frac{2}{3\pi c^3}\mid\int j(t)e^{i\omega t}dt\mid ^{2}$, where $c$ is the speed of light in a vacuum. The laser-induced current is written as $j(t)=-\langle\psi(t)\mid\hat{p} \mid\psi(t)\rangle$.[25] Unless otherwise stated, atomic units are used throughout this work.
cpl-36-5-054203-fig3.png
Fig. 3. The sketch of the laser electric field (solid lines) and the related vector potential (short dashed lines) for different CEPs: (a) $-1.2\pi$, (b) $-1.1\pi$ and (c) $-1.0\pi$, respectively. (d)–(f) The corresponding time-dependent population imaging pictures. The vertical solid line arrow represents the energy difference of the oscillating electron in CB1 and VB.
Figure 2(a) shows the harmonic spectra as a function of the CEP in a linearly polarized pulse with two optical cycles (o.c.) and the intensity of $8.087\times10^{11}$ W/cm$^{2}$. Additionally, the harmonic spectra with three different oscillating structures change periodically in the range of [$-1.5\pi$, 2.5$\pi$] in increments of 0.05$\pi$, and $n$ is an integer. To obtain the underlying physical mechanism, the results in the region of [$-1.5\pi$, $-0.5\pi$] are taken as an example. For photon energies below 16.5 eV and a CEP in the region of [$-1.5\pi$, $-0.9\pi$], the cutoff energy depends linearly on the CEP value, named as the oblique structure, as indicated by the solid line in Fig. 2(b). For photon energy in the region of [5 eV, 16.5 eV] and a CEP in the region of [$-0.7\pi$, $-0.5\pi$], the cutoff energy of the harmonic is almost independent of the CEP, called the similar horizontal structure. Moreover, a vertical structure appears from 16.5 eV to 28.5 eV, and the related CEP in the region of [$-0.9\pi$, $-0.7\pi$]. Next, the related physical mechanisms of the three structures will be explored. As shown in Fig. 2(b), three CEP values (i.e., $-1.2\pi$, $-1.1\pi$, $-1.0\pi$) are selected to research the mechanism of the oblique structure. The symbols A, B and C correspond to 9.5 eV, 11.0 eV and 12.5 eV, respectively, i.e., the cutoff energy of the first plateau. It indicates that the cutoff energy of the first plateau is sensitive to the variation of CEP in the region of [$-1.5\pi$, $-0.9\pi$]. To explain the related physical mechanism, sketches of the laser field (solid lines) and the related vector potential (short-dashed lines) are shown in Figs. 3(a)–3(c), and the time-dependent population imaging in different energy bands is shown in Figs. 3(d)–3(f). The peaks of population oscillations correspond to the zero point of the laser field (i.e., the peak of vector potential), as indicated by the dash-dotted line. Conversely, the minima of oscillations correspond to the peak of the laser field (i.e., the zero point of vector potential), as indicated by the thick dashed line. Furthermore, we calculate the energy differences between the peak of the oscillation in CB1 and the lowest one in VB, as depicted by the vertical solid arrow. Obviously, the peak of vector potential gradually increases with the CEP (marked by the thin dashed line in Figs. 3(a)–3(c)). The electrons in CB1 are distributed in the higher eigenstate, resulting in greater energy differences. For the three cases, the electrons are distributed in eigenstates $n=156$, 164 and 171 in CB1, and then return back to the eigenstates $n=83$, 74 and 70 in VB, respectively. The corresponding energy differences are 9.5 eV, 11.0 eV and 12.5 eV, respectively, which are in good agreement with the photon energies marked with symbols A, B and C in Fig. 2(b). In other words, the emitted photon energy is equal to the energy difference between different bands,[27] i.e., the inter-band process. As in the above analysis, with the variation of the CEP, electrons can be pumped into higher eigenstates in CB1, and the energy difference between CB1 and VB gives rise to the cutoff energy of the first plateau. Specifically, the cutoff energy is proportional to the CEP in the region of [$-1.5\pi$, $-0.9\pi$], which can be fitted by the function $y=13.39x+25.87(\pm0.24)$ as shown in Fig. 2(b). To further investigate the underlying electronic dynamics, the populations of different conduction bands with CEP = $-1.2\pi$ are described in Fig. 4. The first plateau is formed by three steps: tunneling, Bloch electron oscillation in the CB1 and recombination. Initially, the electron is mainly concentrated in the VB, and the energy gap (4.2 eV) is relatively small. After the solid interacts with the laser field, the electron can be excited more easily from VB to CB1 around $t_1=1.1$ o.c. (i.e., around the negative peak of the laser field) as shown in Fig. 4(d), corresponding to the electronic transition moment from VB to CB1 in Fig. 3(d) as depicted by the thick dashed line. However, the electron oscillates back and forth in the CB1, and then it can return to the VB to contribute to the first plateau. The related cutoff energy corresponds to the energy difference of 9.5 eV in Fig. 3(d). With a similar mechanism, the oblique structure can be obtained in the region of [$-1.5\pi$, $-0.9\pi$]. Moreover, it can be seen that the population of the higher conduction band is gradually pumped from the lower energy band step by step.[34] The electron in CB1 is driven to the Brillouin zone boundary at around $t_2=1.35$ o.c., corresponding to the electronic transition moment from CB1 in Fig. 3(d) as shown by the dash-dotted line. Since the amplitude of the electric field is close to zero, only a small portion of the electron can be excited to CB2 as illustrated in Fig. 4(c). After a quarter optical period, the electron in CB2 is driven to the center of the Brillouin zone at around $t_3=1.6$ o.c. (i.e., around the positive peak of the laser field), and it can be easily pumped into CB3. Finally, the electrons in CB2 and CB3 may contribute to the second plateau. Due to the lack of electron populations in CB2 and CB3, the first plateau is mainly focused on an oblique structure. Additionally, the oblique structure still exists in the region of [$-1.5\pi+k\pi$, $-0.9\pi+k\pi$] for different wavelengths and laser intensities, with $k$ being an integer. However, the related slope tends to be larger with the increase of the wavelength and intensity, respectively. Due to the similar physical mechanism, the numerical simulation is not provided here.
cpl-36-5-054203-fig4.png
Fig. 4. (a) The sketch of the laser electric field for CEP = $-1.2\pi$. (b)–(d) Populations of conduction bands CB1, CB2 and CB3 as a function of time with CEP = $-1.2\pi$. Laser parameters are the same as those in Fig. 3(a).
cpl-36-5-054203-fig5.png
Fig. 5. (a) The enlargement of the horizontal structure, and the symbol D corresponds to the photon energy of 16.5 eV. (b) and (c) The time-dependent population imaging pictures with CEP = $-0.6\pi$ and $-0.5\pi$, respectively. The vertical dashed arrow represents the energy difference of the oscillating electron in CB2 and VB.
cpl-36-5-054203-fig6.png
Fig. 6. (a) The enlargement of the vertical structure, with the symbol E corresponding to the photon energy of 28.5 eV. (b) The harmonic spectra with CEP equaling $-0.9\pi$, $-0.8\pi$ and $-0.7\pi$, respectively. (c) The time-dependent population imaging picture for the case of CEP = $-0.8\pi$.
Next, we will discuss the physical mechanism of a similar horizontal structure. Figure 5(a) is an enlarged view of the spectra in the range of [$-1.5\pi$, $-0.5\pi$]. Figures 5(b) and (c) show the time-dependent population imaging for the cases of CEP = $-0.6\pi$ and $-0.5\pi$, respectively. We choose CEP = $-0.7\pi$, $-0.6\pi$, $-0.5\pi$ to discuss the mechanism of this structure, as shown by the vertical dashed line. The obvious structure indicated by the solid line is selected as an example. As shown in Fig. 5(a), the photon energy is always near the position of 16.5 eV as indicated by the symbol D. It indicates that the position of the harmonic energy is insensitive to the change of CEP in the range of [$-0.7\pi$, $-0.5\pi$]. As can be seen from Fig. 5(b), when the oscillating electron reaches the top of CB1, the electron can easily tunnel through into CB2 due to the quite narrow band gap between CB1 and CB2, and then the tunneling electron in CB2 will continue to oscillate back and forth in the laser field. Moreover, the cutoff energy of the first plateau cannot be limited by the energy range between CB1 and VB. As depicted in Figs. 5(b) and 5(c), the maximum energy difference is from the oscillation peak of CB2 (i.e., $n=187$) to that of VB (i.e., $n=70$), corresponding to the cutoff energy of the first plateau as the vertical dashed arrow indicates. Therefore, the cutoff energy of the first plateau is independent of the CEP in the region of [$-0.7\pi+k\pi$, $-0.5\pi+k\pi$], and presents different CEP dependences to that in the region of [$-1.5\pi+k\pi$, $-0.9\pi+k\pi$]. How about the electronic dynamics for the vertical structure? A partial enlarged view of the vertical structure is shown in Fig. 6(a), and the harmonic spectra for the cases of CEP = $-0.9\pi$, $-0.8\pi$ and $-0.7\pi$ are shown in Fig. 6(b). Clearly, the cutoff energy is always close to 28.5 eV as indicated by the vertical dashed line. Figure 6(c) shows the time-dependent population imaging for the case of CEP = $-0.8\pi$. It shows that the excited electron gradually tunnels to CB3. The highest peak of CB3 ($n=247$) to the lowest one of VB ($n=115$) forming the maximum energy difference contributes to the cutoff energy of the second plateau (as marked by the symbol E in Fig. 6(b)). In other words, the vertical structure is derived from the cutoff energy of the second plateau. This is similar to the mechanism of the similar horizontal structure that was contributed to by the cutoff energy of the first plateau as discussed above. Thus the cutoff energy of the second plateau is also insensitive to the variation of the CEP in the region of [$-0.9\pi+k\pi$, $-0.7\pi+k\pi$]. In summary, the dependence of the harmonic spectra from solids on the CEP is investigated by solving the TDSE, and the simulation shows that cutoff energy is prominently differently dependent on the CEP. For the similar horizontal structure and the vertical structure, the cutoff energy of HHG is independent of the CEP. It can be attributed to the similar electronic dynamics for the cutoff energies of the first plateau and the second one, which are not limited by the energy range of the involved bands. The former originates from the transitions from CB2 to VB in the region of [$-0.7\pi+k\pi$, $-0.5\pi+k\pi$], and the latter from CB3 to VB in the region of [$-0.9\pi+k\pi$, $-0.7\pi+k\pi$]. However, for oblique structures, the cutoff energy of the first plateau originating from the transitions from CB1 to VB shows a linear dependence on the CEP in the region of [$-1.5\pi+k\pi$, $-0.9\pi+k\pi$]. Furthermore, based on the discussions of these three structures, we can find that the CEP can change the population of high conduction bands to adjust the photon energy via the inter-band process. We sincerely thank Professor Keli Han and Dr. Ruifeng Lu for providing us the LZH-DICP code. References Theoretical study of high-order-harmonic generation from asymmetric diatomic moleculesAsymmetric Molecular Imaging through Decoding Odd-Even High-Order HarmonicsMultichannel recombination in high-order-harmonic generation from asymmetric molecular ionsMultiple rescattering processes in high-order harmonic generation from molecular systemTheoretical research on multiple rescatterings in the process of high-order harmonic generation from a helium atom with a long wavelengthGeneration of Linear Isolated Sub-60 Attosecond Pulses by Combining a Circularly Polarized Pulse with an Elliptically Polarized PulsePlasma perspective on strong field multiphoton ionizationObservation of a Train of Attosecond Pulses from High Harmonic GenerationAmplitude and Phase Control of Attosecond Light PulsesPhase-matching analysis of high-order harmonics generated by truncated Bessel beams in the sub-10-fs regimeSoft X-ray Microscope Constructed with 130-nm Spatial Resolution Using a High Harmonic X-ray SourceAll-Optical Reconstruction of Crystal Band StructureMapping the electron band structure by intraband high-harmonic generation in solidsDetermination of band structure from the intra-band power spectrum of high harmonic generation in crystalAttosecond band-gap dynamics in siliconReal-time observation of interfering crystal electrons in high-harmonic generationQuantum-trajectory analysis for charge transfer in solid materials induced by strong laser fieldsHarmonic generation in CO 2 laser target interactionObservation of high-order harmonic generation in a bulk crystalSub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillationsExtreme ultraviolet high-harmonic spectroscopy of solidsSolid-state harmonics beyond the atomic limitEnhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fieldsEnhancement of the second plateau in solid high-order harmonic spectra by the two-color fieldsHigh-order-harmonic generation from periodic potentials driven by few-cycle laser pulsesHigh-order-harmonic generation from solids: The contributions of the Bloch wave packets moving at the group and phase velocitiesTime-dependent population imaging for high-order-harmonic generation in solidsA Soluble Problem in Energy BandsBloch–Zener oscillationsObserving the Onset of Effective MassAttosecond-resolution quantum dynamics calculations for atoms and molecules in strong laser fieldsCorrelation Quantum Dynamics between an Electron and D 2 + Molecule with Attosecond ResolutionTheoretical exploration of the interference minimum in molecular high-order harmonic generationQuasi-classical analysis of the dynamics of the high-order harmonic generation from solids
[1] Miao X Y and Du H N 2013 Phys. Rev. A 87 053403
[2] Chen Y J, Fu L B and Liu J 2013 Phys. Rev. Lett. 111 073902
[3] Miao X Y and Zhang C P 2014 Phys. Rev. A 89 033410
[4] Zhang C P, Xia C L et al 2016 Opt. Express 24 20297
[5] Zhang C P, Pei Y N et al 2017 Laser Phys. Lett. 14 015301
[6] Xia C L and Miao X Y 2015 Chin. Phys. Lett. 32 043202
[7] Corkum P B 1993 Phys. Rev. Lett. 71 1994
[8] Paul P M, Toma E S, Breger P et al 2001 Science 292 1689
[9] López-Martens R, Varjú K, Johnsson P et al 2005 Phys. Rev. Lett. 94 033001
[10] Altucci C, Bruzzese R, de Lisio C et al 2003 Phys. Rev. A 68 033806
[11] Kim D S, Park J J, Lee K H et al 2009 Jpn. J. Appl. Phys. 48 026506
[12] Vampa G, Hammond T J, Thiré N et al 2015 Phys. Rev. Lett. 115 193603
[13] Lanin A A, Stepanov E A, Fedotov A B et al 2017 Optica 4 516
[14] Wang C M, Ho T S and Chu S I 2016 J. Phys. B: At. Mol. Opt. Phys. 49 225401
[15] Schultze M, Ramasesha K, Pemmaraju C D et al 2014 Science 346 1348
[16] Hohenleutner M, Langer F, Schubert O et al 2015 Nature 523 572
[17] Jiang S, Yu C, Yuan G et al 2017 J. Phys.: Condens. Matter 29 275702
[18] Burnett N H, Baldis H A, Richardson M C et al 1977 Appl. Phys. Lett. 31 172
[19] Ghimire S, DiChiara A D, Sistrunk E et al 2011 Nat. Phys. 7 138
[20] Schubert O, Hohenleutner M, Langer F et al 2014 Nat. Photon. 8 119
[21] Luu T T, Garg M, Kruchinin S Y et al 2015 Nature 521 498
[22] Ndabashimiye G, Ghimire S, Wu M X et al 2016 Nature 534 520
[23] Du T Y, Guan Z, Zhou X X et al 2016 Phys. Rev. A 94 023419
[24] Li J B, Zhang X, Yue S J et al 2017 Opt. Express 25 18603
[25] Guan Z, Zhou X X and Bian X B 2016 Phys. Rev. A 93 033852
[26] Du T Y, Huang X H, Bian X B 2018 Phys. Rev. A 97 013403
[27] Liu X, Zhu X S, Lan P F et al 2017 Phys. Rev. A 95 063419
[28] Slater J C 1952 Phys. Rev. 87 807
[29] Breid B M, Witthaut D and Korsch H J 2006 New J. Phys. 8 110
[30] Chang R, Potnis S, Ramos R et al 2014 Phys. Rev. Lett. 112 170404
[31] Lu R F, Zhang P Y and Han K L 2008 Phys. Rev. E 77 066701
[32] Hu J, Han K L and He G Z 2005 Phys. Rev. Lett. 95 123001
[33] Miao X Y 2016 J. At. Mol. Sci. 7 1
[34] Du T Y and Bian X B 2017 Opt. Express 25 151