⇦Chinese Physics Letters, 2017, Vol. 34, No. 4, Article code 043201 Controlling Three-Dimensional Electron–Electron Correlation via Elliptically Polarized Intense Laser Field * Jian-Xing Hao(郝建兴)1, Xiao-Lei Hao(郝小雷)1**, Wei-Dong Li(李卫东)1, Shi-Lin Hu(胡师林)2,3, Jing Chen(陈京)2,3 Affiliations 1Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006 2HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100871 3Institute of Applied Physics and Computational Mathematics, Beijing 100088 Received 19 January 2017 *Supported by the National Key Program for S&T Research and Development under Grant No 2016YFA0401100, the National Basic Research Program of China under Grant No 2013CB922201, and the National Natural Science Foundation of China under Grant Nos 11504215, 11374197, 11334009 and 11425414.
**Corresponding author. Email: xlhao@sxu.edu.cn Citation Text: Hao J X, Hao X L, Li W D, Hu S L and Chen J 2017 Chin. Phys. Lett. 34 043201 Abstract The three-dimensional electron–electron correlation in an elliptically polarized laser field is investigated based on a semiclassical model. Asymmetry parameter $\alpha$ of the correlated electron momentum distribution is used to quantitatively describe the electron–electron correlation. The dependence of $\alpha$ on ellipticity $\varepsilon$ is totally different in three directions. For the $z$ direction (major polarization direction), $\alpha$ first increases and reaches a maximum at $\varepsilon=0.275$, then it decreases quickly. For the $y$ direction in which the laser field is always absent, the ellipticity has a minor effect, and the asymmetry parameter fluctuates around $\alpha=-0.15$. However, for the $x$ direction (minor polarization direction), $\alpha$ increases monotonously with ellipticity though starts from the same value as in the $y$ direction when $\varepsilon=0$. The behavior of $\alpha$ in the $x$ direction actually indicates a transformation from the Coulomb interaction dominated correlation to the laser field dominated correlation. Therefore, our work provides an efficient way to control the three-dimensional electron–electron correlation via an elliptically polarized intense laser field. DOI:10.1088/0256-307X/34/4/043201 PACS:32.80.Rm, 32.80.Fb, 34.50.Rk, 42.50.Hz © 2017 Chinese Physics Society Article Text The correlated motion of few-body Coulomb systems provides a major theoretical challenge in quantum mechanics and has far-reaching practical importance in many fields of physics and chemistry. Strong field, laser-induced nonsequential double ionization (NSDI) is an ideal system to study field-affected electron–electron correlation (for reviews, see e.g., Ref. [1]). Thanks to the invention of the cold-target recoil-ion-momentum spectroscopy (COLTRIMS) technique,[2] the three-dimensional momenta of the electron and ion can be obtained and the electron–electron correlation can be revealed by plotting the correlated electron momentum distribution.[3-5] The correlated electron momentum distribution can provide detailed information about the mechanism of the ionization process. Now it has been accepted that rescattering[6,7] is the dominant mechanism for NSDI. Therein, the first electron is released via a tunneling process and then driven back by the laser field into a recollision with the ionic core after the field has reversed its direction. In the resulting inelastic collision, the second electron may be ionized directly (rescattering-impact ionization: RII) or be pumped to an excited state and then be freed by the field (rescattering excitation with subsequent ionization: RESI) at a later time. Experimental and theoretical studies on electron–electron correlation have mostly concentrated on the momentum components of the electrons and the ion in the polarization direction of the field (see references in Ref. [1]). These momenta are found to be dominated by the interaction of the particles with the strong external field after the electron impact ionization. The correlated electron momentum distribution in polarization direction of the field may show a correlation[3,4] or an anti-correlation,[8] and may also show some fine structures, such as the finger-like structure[9,10] and the cross-shape structure.[11] These results can therefore be used to obtain information on the time evolution of the field-induced ionization process with subfemtosecond time resolution. On the other hand, there are also a few investigations on the correlation in the perpendicular direction.[12-15] The virtue of the perpendicular momenta is evident since they are solely a product of the three-body interaction and not of the laser field. Thus the subtleties of momentum exchange in the recollision process should not be covered by the much larger momentum transfer taken from the field. The experimental measurements have shown that the electron–electron repulsion plays an important role in the ionization process. Precisely controlling the photon–photon correlation in atomic and atomic-like medium[16-18] as well as the Coulomb interaction like Rydberg–van der Waals interactions for molecules[19,20] can be achieved easily now in quantum optics. For the NSDI process in intense laser field, electron–electron correlation can also be controlled though only qualitatively by changing the laser parameters. For example, a transition from correlation to anti-correlation can be observed by decreasing the laser intensity to the below-threshold regime for Ar atoms,[8] and a cross-shape structure will arise if the pulse duration is shortened to a few cycles.[11] However, almost all these attempts have been carried out with linearly polarized laser pulse and electron–electron correlation can only be controlled in the direction of polarization. Actually, a three-dimensional controlling of the electron–electron correlation can be achieved if an elliptically polarized laser field is applied. Compared with the case of linear polarization, the kinematics caused by an elliptically polarized field are genuinely three-dimensional. According to the rescattering picture, the ellipticity of the polarized field will have significant effect on the NSDI process since its perpendicular component will drive the first tunneled electron away from the nucleus and the orbits will be distorted. As a result, the cross sections of RESI and RII pathways will change accordingly. On the other hand, the system in the case of linear polarization is cylindrical, while in the case of elliptical polarization the cylindrical symmetry is broken. All three directions are not equivalent and will show different correlations. More interestingly, the laser field in the minor polarization can be changed continuously from zero by varying the ellipticity. As a result, a transformation from a Coulomb interaction dominated correlation to a laser field dominated correlation can be achieved easily. In this work, we investigate the three-dimensional electron–electron correlation in the elliptically polarized laser field based on a semiclassical model. Our results show that the electron–electron correlations in both major and minor polarization directions can be controlled by varying ellipticity of the elliptically polarized laser field. Following the same procedure of the previous semiclassical model in the elliptically polarized laser field,[21,22] the ionization of the first outer electron from the bound state to the continuous state is described by the quantum tunneling ionization theory.[23] The subsequent propagation of this ionized electron and the bound electron is governed by the classical dynamics, in which the motions of the two electrons are described by the classical Newton equation (in atomic units $e=m=\hbar =1$) $$ \frac{d^{2}{\boldsymbol r}_{i}}{dt^{2}}={\boldsymbol E}(t)-{\boldsymbol \nabla}(V_{\rm ne}^{i}+V_{\rm ee}),~~ \tag {1} $$ where ${\boldsymbol E}(t)=(E_{x}(t),0,E_{z}(t))$ denotes the elliptically polarized laser field with $E_{z}(t)=f(t)E_{0z}\cos \omega t$ and $E_{x}(t)=f(t)E_{0x}\sin \omega t$ with a cosine square envelope. The ellipticity is defined as $\varepsilon \equiv E_{0x}/E_{0z} < 1$ ($\varepsilon =0$ for linearly polarized light, and $\varepsilon =1$ for circularly polarized light). The tunneled and bounded electrons, with ionization potentials $I_{\rm p1}$ and $I_{\rm p2}$, are denoted by $i=1$ and 2 respectively. The Coulomb potentials are $$ V_{\rm ne}^{i}=-\frac{Z_{\rm eff}}{r_{i}},~V_{\rm ee}=\frac{1}{|r_{1}-r_{2}|},~~ \tag {2} $$ where $Z_{\rm eff}=\sqrt{2I_{\rm p2}}$ is the effective charge of Ar$^{2+}$, and $r_{i}$ is the distance between the $i$th electron and nucleus. To solve Eq. (1), we have to determine the initial conditions for the two electrons. Assuming that the quasistatic approximation is valid for the tunneled electron here, we can obtain its initial conditions along with the method in Ref. [24]. After rotating the $z$ axis to the direction of the instantaneous external field, the tunneling process can be described by the following Schrodinger equation[23,25] in parabolic coordinates, $$ \frac{d^{2}\phi }{d\eta ^{2}}+\Big(\frac{I_{\rm p1}}{2}+\frac{1}{2\eta }+\frac{1}{4\eta ^{2}}+\frac{E\eta }{4}\Big)\phi =0,~~ \tag {3} $$ which describes the tunneling process for a single electron with energy $K=I_{\rm p1}/4$ within a one-dimensional effective potential $U(\eta)=-1/4\eta ^{2}-1/8\eta ^{2}-E\eta /8$ with $E$ being the uniform external field. At the moment $t_{0}$, the first electron tunnels the effective potential $U(\eta)$ through the turning point $\eta _{0}$, determined by $U(\eta)=K$. The initial position and velocity are expressed as $x_{0}=-\frac{1}{2}\eta _{0}\sin \{\arctan [\varepsilon \tan (\omega t_{0})]\}$, $y_{0}=0$, and $z_{0}=-\frac{1}{2}\eta _{0}\cos \{\arctan [\varepsilon \tan (\omega t_{0})]\}$, $v_{x0}=v_{\rm per}\cos \theta \cos \{\arctan [\varepsilon \tan (\omega t_{0})]\} $, $v_{y0}=v_{\rm per}\sin \theta $, and $v_{z0}=-v_{\rm per}\cos \theta \sin \{\arctan [\varepsilon \tan (\omega t_{0})]\}$, where $v_{\rm per}$ is the transverse velocity perpendicular to the instantaneous electric field, and $\theta$ is the angle between $v_{\rm per}$ and the $x$ axis after rotation. The weight of each trajectory is evaluated by $\omega(t_{0},v_{\rm per})=\omega (0)\overline{\omega (1)}$,[23] where $$\begin{align} \omega (0)=\,&\frac{4(2I_{\rm p1})^{2}}{E}\exp \Big[-\frac{2}{3E}(2I_{\rm p1})\Big],~~ \tag {4} \end{align} $$ $$\begin{align} \overline{\omega (1)}=\,&\frac{(2I_{\rm p1})^{1/2}}{E\pi }\exp \Big(-\frac{v_{\rm per}^{2}(2I_{\rm p1})^{1/2}}{E}\Big).~~ \tag {5} \end{align} $$ The initial condition of the second electron (bound electron) is determined by assuming that the electron is in the ground state of Ar$^{+}$ and its initial distribution is a microcanonical distribution.[26] All the results in the following are obtained under the laser parameters of $I=3.0\times 10^{14}$ W/cm$^{2}$ and $\lambda =800$ nm. In Fig. 1, we present the correlated momentum distributions in three directions at different ellipticities. Generally, the momentum distributions in the $z$ (major polarization direction) and $x$ direction (minor polarization direction) vary with ellipticity, while in the $y$ direction where the laser field is always absent, the momentum distribution always shows a maximum around the origin and almost does not change with ellipticity. For the $z$ direction (Figs. 1(a), 1(d), and 1(g)) the distribution shows a typical pattern of NSDI process, where symmetrical maxima in the 1st and 3rd quadrants at momenta are far from zero. Though the main shape of the distributions does not change, the fine structure and the relative amounts in the four quadrants vary with ellipticity. For the case of the $x$ direction (Figs. 1(b), 1(e), and 1(h)), the distribution is changed significantly by the elliptically polarized field. In the case of linear polarization ($\varepsilon=0$) in Fig. 2(b), the distribution shows a maximum around the origin similar to that in the $y$ direction, which is the product of the Coulomb singularity. When the ellipticity increases, the distribution extends to larger momentum and the distribution in the origin is dramatically reduced, which is no doubt due to the increasing laser field in the $x$ direction.
cpl-34-4-043201-fig1.png
Fig. 1. (Color online) Correlated electron momentum distributions at three directions: (a), (d) and (g) $z$ direction; (b), (e) and (h) $x$ direction; and (c), (f) and (i) $y$ direction. (a)–(c) $\varepsilon=0$, (d)–(f) $\varepsilon=0.2$, and (g)–(i) $\varepsilon=0.35$.
Electrons in different quadrants in the correlated momentum distribution may be liberated through different pathways. As discussed extensively in Refs. [3–5], electrons in RII pathway mainly concentrated in the 1st and 3rd quadrants. In contrast to RII, electrons liberated by the RESI mechanism are found in all four quadrants.[5,27-29] Here we extract the difference between the total yields in the 1st and 3rd quadrants and the 2nd and 4th quadrants and express it through the asymmetry parameter $\alpha=(Y_{1\&3}-Y_{2\&4})/(Y_{1\&3}+Y_{2\&4})$, where $Y_{1\&3}$ and $Y_{2\&4}$ denote the yields in the 1st and 3rd quadrants and in the 2nd and 4th quadrants, respectively. This asymmetry parameter allows us to retrieve quantitative information as shown in Ref. [29].
cpl-34-4-043201-fig2.png
Fig. 2. (Color online) (a)–(c) Asymmetry parameter versus ellipticity in the $z$ direction. Correlated electron momentum distributions for RESI pathway in the $z$ direction at different ellipticities indicated by red arrows in panel (c): (d) $\varepsilon=0$, (e) $\varepsilon=0.275$, and (f) $\varepsilon=0.35$. See text for details of the red dashed lines in (d)–(f).
The ellipticity dependence of the asymmetry parameter $\alpha$ in the $z$ direction is depicted in Fig. 2(a). With increasing the ellipticity, $\alpha$ first increases and reaches a maximum at $\varepsilon=0.275$, then it decreases quickly. Please note that the fast oscillations in Fig. 2 can be attributed to fluctuation in the Monte-Carlo simulation of the semiclassical model. In the semiclassical model, the contributions from RII and RESI can be distinguished by the time interval $\Delta t$ between the collision time and ionization time.[30-32] If $\Delta t < 0.1T$ ($T$ is the optical cycle), the DI event can be treated as an RII event, otherwise an RESI one. In Figs. 2(b) and 2(c) we present the asymmetry parameter corresponding to RII and RESI pathways, respectively. For the RII pathway, $\alpha$ increases monotonously with ellipticity slowly. However, for the RESI pathway, $\alpha$ shows a dependence on ellipticity similar to the total case in Fig. 2(a). To see this more clearly, in Figs. 2(d)–2(f) we present the correlated momentum distributions for the RESI pathway at three typical ellipticities indicated by red arrows in Fig. 2(c). It is evident that the distribution in the 2nd and 4th quadrants first decreases then increases with ellipticity especially in regions confined by red dashed lines in Figs. 2(d)–2(f). The above behavior can be interpreted by the following facts. With increasing the ellipticity, the collision between two electrons becomes weaker and the energy transferred from the first electron to the second electron upon collision is smaller. Therefore, the second electron will be pumped into excited states with lower energy and has to spend longer time before it is ionized by the field. Since the final momentum of the second electron depends on the laser phase when it is ionized, if the ionization time of the second electron is delayed for an optical cycle as the ellipticity increases, the direction of momentum will reverse twice. As a result, the correlated momentum distribution will shift from the 1st and 3rd quadrants to the 2nd and 4th quadrants and then shift back. In Fig. 3(a) we present the dependence of asymmetry parameter $\alpha$ on ellipticity in the $x$ and $y$ directions. For the $y$ direction, the ellipticity has a minor effect, the asymmetry parameter fluctuates around $\alpha=-0.15$. This is due to the fact that there is no laser field in the $y$ direction and the electron–electron correlation in this direction is mainly determined by the Coulomb attraction of the core and the Coulomb repulsion between the two electrons which do not depend on the ellipticity. The negative value of $\alpha$ results from the Coulomb repulsive interaction, which induces a back-to-back emission of the two electrons. For the $x$ direction, the situation is absolutely different. In the case of linear polarization ($\varepsilon=0$), the value of $\alpha$ is equal to that in the $y$ direction. When the ellipticity increases, the asymmetry parameter $\alpha$ begins to increase and more and more deviates from that in the $y$ direction. Obviously, it is the result of the increasing laser field in the $x$ direction competing with the Coulomb interaction.
cpl-34-4-043201-fig3.png
Fig. 3. (Color online) (a) Asymmetry parameter versus ellipticity in the $x$ and $y$ directions. (b) Asymmetry parameter versus ellipticity for RII and RESI pathways in the $x$ direction. Correlated electron momentum distributions for RII pathway in the $x$ direction at different ellipticities: (c) $\varepsilon=0$, and (d) $\varepsilon=0.35$.
To shed more light on the effect of ellipticity on the correlation in the $x$ direction, we also present the asymmetry parameter for RII and RESI pathways in Fig. 3(b). For the RESI pathway, since the two electrons depart from the ionic core at different times, which results in a much weaker Coulomb repulsion between the two electrons, the value of $\alpha$ is much closer to zero compared with the RII pathway as well as the total case in Fig. 3(a) at small ellipticity. The curve also shows a maximum at $\varepsilon=0.275$ similar to the RESI pathway in the $z$ direction. For the RII pathway, $\alpha$ increases quickly with ellipticity, which directly results in the increase of $\alpha$ in the total case in Fig. 3(a). Actually, the behavior of $\alpha$ for RII in the $x$ direction provides a clear picture of how the laser field competes with Coulomb repulsion between the two electrons in determining the electron–electron correlation. The Coulomb repulsion induces a concentration of electrons in the 2nd and 4th quadrants while the laser field-electron interaction favors distributions in the 1st and 3rd quadrants. In the RII pathway, the two electrons depart from the ionic core at the same time, thus the Coulomb repulsion is very strong. At small ellipticity, the laser field in the $x$ direction is very weak, the Coulomb repulsion dominates the correlation, which results in a small $\alpha$. As the ellipticity increases, the laser field increases and may overcome the Coulomb repulsion to become the dominant interaction, thus the asymmetry parameter $\alpha$ increases quickly. To see this more clearly, we present the correlated electron momentum distributions at $\varepsilon=0$ and $\varepsilon=0.35$ as seen in Figs. 3(c) and 3(d), respectively. For linear polarization ($\varepsilon=0$), the distributions are concentrated in the 2nd and 4th quadrants, while for $\varepsilon=0.35$, the bright spots in the 2nd and 4th quadrants disappear, and more electrons distribute in the 1st and 3rd quadrants. In conclusion, we have investigated the three-dimensional electron–electron correlation in elliptically polarized laser field based on a semiclassical model. The correlated electron momentum distributions in the $z$ (major polarization direction) and $x$ directions (minor polarization direction) vary with ellipticity, while in the $y$ direction where the laser field is always absent, the momentum distribution almost does not change. To give a quantitative description to the correlation, we also calculate the asymmetry parameter $\alpha$ of the correlated electron momentum distribution. For the $z$ direction, $\alpha$ first increases and reaches a maximum at $\varepsilon=0.275$, then it decreases quickly. For the $y$ direction, the ellipticity has a minor effect, the asymmetry parameter fluctuates around $\alpha=-0.15$. For the $x$ direction, the value of $\alpha$ is equal to that in the $y$ direction at small ellipticity. As the ellipticity increases, $\alpha$ begins to increase and more and more deviates from that in the $y$ direction. The behavior of $\alpha$ in the $x$ direction is the result of the increasing laser field in the $x$ direction competing with the Coulomb interaction. Therefore, it is actually a transformation from a Coulomb interaction dominated correlation to a laser field dominated correlation. Our work provides an efficient way to control the electron–electron correlation via elliptically polarized intense laser field. In addition to its unique advantage in controlling the electron–electron correlation in different directions, it is also easier to achieve since it is more convenient for experimentalists to vary the ellipticity in experiment without having to change the laser systems. References Theories of photoelectron correlation in laser-driven multiple atomic ionizationCold Target Recoil Ion Momentum Spectroscopy: a ‘momentum microscope’ to view atomic collision dynamicsCorrelated electron emission in multiphoton double ionizationMomentum Distributions of Ne n + Ions Created by an Intense Ultrashort Laser PulseSeparation of Recollision Mechanisms in Nonsequential Strong Field Double Ionization of Ar: The Role of Excitation TunnelingPlasma perspective on strong field multiphoton ionizationStrong-Field Double Ionization of Ar below the Recollision ThresholdCorrelated Two-Electron Momentum Spectra for Strong-Field Nonsequential Double Ionization of He at 800 nmBinary and Recoil Collisions in Strong Field Double Ionization of HeliumAttosecond tracing of correlated electron-emission in non-sequential double ionizationExperimental evidence for electron repulsion in multiphoton double ionizationCorrelated two-electron dynamics in strong-field double ionizationElectron-Electron Momentum Exchange in Strong Field Double IonizationFully Differential Rates for Femtosecond Multiphoton Double Ionization of NeonControlling four-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically induced transparencyParametric amplification of dressed multi-wave mixing in an atomic ensembleDressed four-wave mixing second-order Talbot effectNear-Threshold Low Rydberg Depletion Spectroscopy of H 2 , D 2 , and HDContinuum Structures at the Second Dissociation Limit of Hydrogen MoleculesNonsequential double ionization of Ne in an elliptically polarized intense laser fieldThe effect of electron initial longitudinal velocity on the non-sequential double ionization process in an elliptically polarized laser fieldEnergy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiationInterpretation of momentum distribution of recoil ions from laser-induced nonsequential double ionization by semiclassical rescattering modelComment on the classical-trajectory Monte Carlo method for ion-atom collisionsLaser-induced nonsequential double ionization at and above the recollision-excitation-tunneling thresholdStrong-field double ionization at the transition to below the recollision thresholdQuantum Effects in Double Ionization of Argon below the Threshold IntensityFrequency dependence of non-sequential double ionizationEffect of the electron initial longitudinal velocity on the nonsequential double-ionization process S -matrix and semiclassical study of electron-electron correlation in strong-field nonsequential double ionization of Ne
[1] Becker W, Liu X J, Ho P J and Eberly J H 2012 Rev. Mod. Phys. 84 1011
[2] Dörner R, Mergel V, Jagutzki O, Spielberger L, Ullrich J, Moshammer R and Schmidt-Böcking H 2000 Phys. Rep. 330 95
[3] Weber Th, Giessen H, Weckenbrock M, Urbasch G, Staudte A, Spielberger L, Jagutzki O, Mergel V, Vollmer M and Dörner R 2000 Nature 405 658
[4] Moshammer R, Feuerstein B, Schmitt W, Dorn A, Schröter C D and Ullrich J 2000 Phys. Rev. Lett. 84 447
[5] Feuerstein B, Moshammer R, Fischer D, Dorn A, Schroter C D, Deipenwisch J, Crespo Lopez-Urrutia J R, Hohr C, Neumayer P, Ullrich J, Rottke H, Trump C, Wittmann M, Korn G and Sandner W 2001 Phys. Rev. Lett. 87 043003
[6] Corkum P B 1993 Phys. Rev. Lett. 71 1994
[7]Perelomov A M, Popov V S and Terent'ev M V 1993 Sov. Phys. JETP 24 207
[8] Liu Y, Rudenko A, Tschuch S, Siegel M, Morgner U, Moshammer R and Ullrich J 2008 Phys. Rev. Lett. 101 053001
[9] Rudenko A, Jesus V L B D, Ergler T, Zrost K, Feuerstein B, Schröter C D, Moshammer R and Ullrich J 2007 Phys. Rev. Lett. 99 263003
[10] Staudte A, Ruiz C, Schoffler M, Schossler S, Zeidler D, Weber T, Meckel M, Villeneuve D M, Corkum P B, Becker A and Dörner R 2007 Phys. Rev. Lett. 99 263002
[11] Bergues B, Kübel M, Johnson N G, Fischer B, Camus N, Betsch K J, Herrwerth O, Senftleben A, Sayler A M, Rathje T, Pfeifer T, Ben-Itzhak I, Jones R R, Paulus G G, Krausz F, Moshammer R, Ullrich J and Kling M F 2012 Nat. Commun. 3 813
[12] Weckenbrock M, Hattass M, Czasch A, Jagutzki O, Schmidt L, Weber T, Roskos H, Löffler T, Thomson M and Dörner R 2001 J. Phys. B: At. Mol. Opt. Phys. 34 L449
[13] Moshammer R, Feuerstein B, López-Urrutia, Crespo J, Deipenwisch J, Dorn A, Fischer D, Hohr C, Neumayer P, Schroter C D, Ullrich J, Rottke H, Trump C, Wittmann M, Korn G and Sandner W 2002 Phys. Rev. A 65 035401
[14] Weckenbrock M, Becker A, Staudte A, Kammer S, Smolarski M, Bhardwaj V R, Rajner D M, Villeneuve D M, Corkum P B and Dörner R 2003 Phys. Rev. Lett. 91 123004
[15] Weckenbrock M, Zeidler D, Staudte A, Weber T, Schoffler M, Meckel M, Kammer S, Smolarski M, Jagutzki O, Bhardwaj V R, Rayner D M, Villeneuve D M, Corkum P B and Dörner R 2004 Phys. Rev. Lett. 92 213002
[16] Du Y G, Zhang Y P, Zuo C C, Li C B, Nie Z Q, Zheng H B, Shi M Z, Wang R M, Song J P, Lu K Q and Xiao M 2009 Phys. Rev. A 79 063839
[17] Chen H X, Qin M Z, Zhang Y Q, Zhang X, Wen F, Wen J M and Zhang Y P 2014 Laser Phys. Lett. 11 045201
[18] Chen H X, Zhang X, Zhu D Y, Yang C, Jiang T, Zheng H B and Zhang Y P 2014 Phys. Rev. A 90 043846
[19] Zhang Y P, Gan C L, Song J P, Yu X J, Ge H, Ma R Q, Li C S, Lu K Q and Eyler E E 2005 Chin. Phys. Lett. 22 1110
[20] Zhang Y P, Gan C L, Song J P, Yu X J, Ma R Q, Ge H, Jiang T, Lu K Q and Eyler E E 2005 Chin. Phys. Lett. 22 1114
[21] Hao X L, Wang G Q, Jia X Y, Li W D, Liu J and Chen J 2009 Phys. Rev. A 80 023408
[22] Hao X L, Li W D, Liu J and Chen J 2012 Chin. Phys. B 21 083304
[23] Delone N B and Krainov V P 1991 J. Opt. Soc. Am. B 8 1207
[24] Chen J, Liu J, Fu L B and Zheng W M 2000 Phys. Rev. A 63 011404(R)
[25]Landau L D and Lifshitz E M 1977 Quantum Mechanics (Oxford: Pergamon Press)
[26] Cohen J S 1982 Phys. Rev. A 26 3008
[27] Shaaran T, Nygren M T and Faria C F de M 2010 Phys. Rev. A 81 063413
[28] Ye F D and Liu J 2010 Phys. Rev. A 81 043402
[29] Hao X L, Chen J, Li W D, Wang B, Wang X and Becker W 2014 Phys. Rev. Lett. 112 073002
[30] Chen J, Kim J H and Nam C H 2003 J. Phys. B 36 691
[31] Hao X L, Li W D, Liu J and Chen J 2011 Phys. Rev. A 83 053422
[32] Jia X Y, Hao X L, Fan D, Li W D and Chen J 2013 Phys. Rev. A 88 033402