Transition from electromagnetically induced transparency to Autler–Townes splitting in cold cesium atoms

Electromagnetically induced transparency (EIT) [1] is a quantum interference effect in which the absorption of a weak probe laser, interacting resonantly with an atomic transition, is reduced in the presence of a coupling laser, which (near-)resonantly couples the upper probe level to a third level. EIT is, for instance, crucial in optically controlled slowing of light [2] and optical storage [3]. Autler–Townes splitting (ATS) [4], proposed by Autler and Townes, is a linear (resonant) AC Stark effect in a two-level system that is strongly driven at resonance, and can be measured via a weak probe from a third level. ATS was observed originally in the microwave and later the light domain. EIT and ATS have been extensively investigated experimentally and theoretically in Λ−, V− and cascade-type three-level atoms [1, 5–10].

EIT and ATS may phenomenologically look similar, but they are different in nature, one being a quantum interference and the other a linear AC Stark effect. This leads to an interest in the establishment of criteria to discern them. In [9], threshold values for distinguishing EIT and ATS, Ω_t, are defined via the coherence decay rates in the Master equation of the various systems, according to which ATS is observed in four different three-level systems in a strong-coupling-field regime (Ω_c / Ω_t > 1), whereas EIT is observed only in Λ- and cascade EIT configurations in a weak-coupling-field regime (Ω_c/ Ω_t < 1). Anisimov et al [10] also propose an objective method based on Akaike's information criterion (AIC), to discern ATS from EIT in experimental data of a Λ-type three-level atom. Recently, a series of experiments [11–13] has been performed in which the AIC has been used to discriminate EIT from ATS in Λ-type three-level systems free of Doppler effects. Zhang et al [14] provided a theoretical model using a five-level system and investigated EIT and ATS based on the AIC. Another area of interest, in which EIT and ATS are important, is spectroscopy of highly exited Rydberg levels using all-optical methods. Rydberg-EIT, a cascade EIT case, was first observed in a vapor cell [15] and later in a rubidium MOT [16]. In Rydberg-EIT work, the phenomena under investigation are often due to strong Rydberg-atom interactions that are distinct from the optical couplings that give rise to EIT or ATS. For instance, Rydberg-EIT has been used to realize a single-photon transistor [17] and a single-photon source [18, 19] by employing a blockade effect [20–22], which results from the strong Rydberg-atom interactions. ATS has also been investigated in systems that have involved Rydberg states of rubidium [23, 24] and cesium [25, 26]. Exploiting the fact that Rydberg atoms strongly couple to radio-frequency (RF) fields, Rydberg EIT and ATS effects are used to realize precise measurements of such fields [27–31]. Holloway et al [32] have investigated the relationship between the Rabi frequency of resonant RF transitions between Rydberg states and the resultant ATS splitting in Rydberg-EIT spectra measured in room-temperature atomic vapor. In related work, the enhancement and suppression of multi-wave mixing (MWM) in five-level systems [33] has been studied, and the effect of Rydberg-atom interactions was explored [34]. Spectra and spatial images of probe and MWM signals were obtained for six-wave [35] and eight-wave implementations [36]. The role of Rydberg interactions as well as the co-occurrence of Rydberg-EIT and ATS were discussed in these works.

While numerous efforts have already dealt with discriminating EIT from ATS [9–14], as reported in the above overview, many of these earlier works are limited to Λ-type systems, deal with Doppler-effect-free cold-atom systems or all-optical equivalents of those, assume that the probe is in the weak-field limit, assume that the system is in steady-state, assume a vanishing laser linewidth, and/or assume that there is no interaction between the atoms (or their analogs) that exhibit EIT or ATS. In the present work we explore the EIT/ATS transition in a case where none of these ideal conditions are necessarily satisfied. We use cascade EIT and ATS, with a Rydberg upper-level state, in a cesium magneto-optical trap (MOT) as an experimental test system. We introduce two parameters, the Rydberg-EIT linewidth, γ_EIT, and the AT spilitting, γ_ATS, to characterize the EIT to ATS transition. In our model, we calculate spectra using methods provided in [32] to study the dependence of γ_EIT and γ_ATS on the Rabi frequencies of both the probe and coupling transitions. The transition between EIT and ATS effect is linked to a qualitative change in behavior of γ_EIT and γ_ATS as a function of coupler and probe Rabi frequencies. In our analysis, we expand the above cited previous studies to a cascade Rydberg EIT/ATS system. In the numerical methods used, we do not make any weak-field approximations. Also, the Doppler effect, residual Doppler mismatch, averaging over the Maxwell velocity distribution, and inhomogeneous broadening due to laser-frequency jitter are accounted for. The model does not include Rydberg-atom interactions, which are not relevant under the conditions (Rydberg quantum numbers, atom densities and interaction times) used in the experiment. We obtain readily accessible criteria to distinguish EIT from ATS.

We consider the cesium cascade three-level system shown in figure 1(a). The coupling laser drives the upper transition, $| 6{P}_{3/2}$, F' $=\,5\rangle$ ($| 2\rangle$) $\to | 35{S}_{1/2}\rangle$ ($| 3\rangle$). The weak probe laser couples the lower transition, $| 6{S}_{1/2}$, ${F}=4\rangle$ ($| 1\rangle$) $\to | 6{P}_{3/2}$, F' $=\,5\rangle$ ($| 2\rangle$). The respective wavelengths and Rabi frequencies are λ_c and Ω_c, and λ_p and Ω_p. In the rotating-wave approximation and the field picture, the Hamiltonian of the three-level atom represented in the space $\{| 1\rangle ,| 2\rangle ,| 3\rangle \}$ is

$$\begin{eqnarray}H=\displaystyle \frac{{\hslash }}{2}\left(\begin{array}{ccc}0 & {{\rm{\Omega }}}_{p} & 0\\ {{\rm{\Omega }}}_{p} & -2{{\rm{\Delta }}}_{p} & {{\rm{\Omega }}}_{c}\\ 0 & {{\rm{\Omega }}}_{c} & -2({{\rm{\Delta }}}_{p}+{{\rm{\Delta }}}_{c})\end{array}\right),\end{eqnarray} \tag{ 1 }$$

where Δ_c and Δ_p are the detunings of the coupling and probe beams, respectively. To account for decay and dephasing, the system is described using the Lindblad equation for the density matrix ρ

$$\begin{eqnarray}&&\dot{\rho }=-\displaystyle \frac{{\rm{i}}}{{\hslash }}[H,\rho ]+{\mathfrak{L}},\end{eqnarray} \tag{ 2 }$$

where ${\mathfrak{L}}$ is the Lindblad operator that accounts for the decay processes in the atom. In the space $\{| 1\rangle ,| 2\rangle ,| 3\rangle \}$, ${\mathfrak{L}}$ becomes [37]

$$\begin{eqnarray}{\mathfrak{L}}=\left(\begin{array}{ccc}{{\rm{\Gamma }}}_{{eg}}{\rho }_{22} & -\displaystyle \frac{1}{2}{\gamma }_{2}{\rho }_{12} & -\displaystyle \frac{1}{2}{\gamma }_{3}{\rho }_{13}\\ -\displaystyle \frac{1}{2}{\gamma }_{2}{\rho }_{21} & -{{\rm{\Gamma }}}_{{eg}}{\rho }_{22}+{{\rm{\Gamma }}}_{{re}}{\rho }_{33} & -\displaystyle \frac{1}{2}({\gamma }_{2}+{\gamma }_{3}){\rho }_{23}\\ -\displaystyle \frac{1}{2}{\gamma }_{3}{\rho }_{31} & -\displaystyle \frac{1}{2}({\gamma }_{2}+{\gamma }_{3}){\rho }_{32} & -{{\rm{\Gamma }}}_{{re}}{\rho }_{33}\end{array}\right),\end{eqnarray} \tag{ 3 }$$

where γ₂ and γ₃ are the dephasing rates of the intermediate and Rydberg states, respectively. It is γ₂ = γ_e + Γ_eg, where γ_e is a collision-induced dephasing rate of level $| 2\rangle$, and γ_e ≪ Γ_eg = 2π × 5.2 MHz. Further, γ₃ = γ_r + Γ_re. For the Rydberg level the population decay rate Γ_re is, typically, smaller than the dephasing γ_r, because Rydberg-atom lifetimes are long (lifetimes are ∼n³ and on the order of 100 μs) and interactions between Rydberg atoms in cold-atom clouds are often strong (van der Waals interactions scale as n¹¹).

Zoom In Zoom Out Reset image size

The spectrum is given by the probe-power transmission, $P={P}_{0}\exp (-\alpha L)$, with the probe-laser absorption coefficient, α = 2πIm(χ)/λ_p, the MOT size, L, and the susceptibility of the medium seen by the probe laser, χ. The susceptibility, χ, is

$$\begin{eqnarray}&&\chi =\displaystyle \frac{2N{\mu }_{12}}{{E}_{p}{\epsilon }_{0}}{\rho }_{12},\end{eqnarray} \tag{ 4 }$$

where N is the average atomic density, μ₁₂ is the dipole moment of transition $| 1\rangle \to | 2\rangle$, E_p is the amplitude of the probe, ₀ is the vacuum permittivity, and ρ₁₂ is the density matrix element between $| 1\rangle$ and $| 2\rangle$.

We numerically solve the equations (1)–(3) to obtain the absorption coefficient α for a range of values of Ω_c and Ω_p, including the case of a strong probe, Ω_p ≳ Γ_eg. The result is averaged over the thermal velocity distribution in the gas [32]; under our conditions (T = 100–200 μK) the thermal motion is marginally important. In the calculation we also assume γ_r = 0, which is admissible due to our very low experimental probe intensities and the low principal quantum number of the utilized Rydberg state. Further, in our initial discussion we assume zero laser linewidth; this assumption is dropped later-on.

In figure 2(a) we show α as a function of Ω_c and Δ_p, for the case of a fixed Ω_p = 2π × 1.05 MHz and Δ_c = 0. The shape of the spectrum clearly indicates two qualitatively different regimes, a regime of low Ω_c in which the overall linewidth is fixed at ≈Γ_eg and there is a very narrow transparency window whose width scales quadratically in Ω_c. This regime is the EIT regime. As Ω_c increases, the overall linewidth increases and the system enters the ATS regime. There, the line breaks up into two features that are narrower than the coupler-free absorption line and the peak-to-peak separation scales linearly in Ω_c. The wide transparency region between the AT lines is almost as wide as the peak-to-peak separation itself.

Zoom In Zoom Out Reset image size

For a quantitative analysis, we extract the full width at half maximum (FWHM) of the dip in the absorption, γ_EIT, as well as the peak-to-peak spacing of the line pair, γ_ATS, from the absorption coefficient data. Both parameters γ_EIT and γ_ATS can be calculated in both the EIT and the ATS regime. However, as detailed in the following, γ_ATS yields little useful information in the EIT regime and vice versa. In figure 2(b), both parameters are shown for a typical spectrum in the EIT regime, where Ω_p ≲ Ω_c = 2π × 2.03 MHz < Γ_eg. In EIT, the main interest lies in how narrow the EIT transparency window is, in comparison with the natural linewidth Γ_eg. Therefore, in the EIT regime the parameter γ_EIT is most important (hence the name); it measures the bandwidth over which the EIT quantum interference mechanism leading to reduced probe absorption is experimentally observable. In the EIT regime, the separation between the absorption maxima, γ_ATS, carries little useful information, as it just returns a value on the order of the overall coupler-free linewidth. Figure 2(c) shows the spectrum calculated for Ω_c = 2π × 13.01 MHz > Γ_eg, which is the ATS regime. Since ATS is a linear AC Stark effect, the interest mainly is in how far the AT-split line centers are separated from each other. Therefore, in the ATS regime a better parameter to quantify the behavior is the peak-to-peak spacing of the line pair, γ_ATS. As seen in figures 2(a) and (c), in the weak probe limit and for Δ_c = 0, in the ATS regime the peaks tend to have a width of Γ_eg/2 = 2π × 2.6 MHz, and a spacing γ_ATS ≈ Ω_c. (The Rydberg states are very long-lived (Γ_re ≪ Γ_eg) and do not contribute to the width of the AT peaks here).

Figure 3(a) shows the calculated γ_EIT as a function of both Ω_c and Ω_p. In the EIT domain, characterized by Ω_c, Ω_p ≲ Γ_eg and outlined by the dashed line, the separation between the contour lines decreases as a function of distance from the origin, reflecting that in the EIT regime ${\gamma }_{{\rm{EIT}}}\approx ({{\rm{\Omega }}}_{c}^{2}+{{\rm{\Omega }}}_{p}^{2})/{{\rm{\Gamma }}}_{{eg}}$. Further out, as the system enters the ATS regime, the separation between the contour lines tends to become fixed. Figure 3(b) shows that in the AT regime, characterized by Ω_c ≳ Γ_eg, the γ_ATS contour lines are equidistant circles, reflecting the fact that ${\gamma }_{{\rm{ATS}}}\approx \sqrt{{{\rm{\Omega }}}_{c}^{2}+{{\rm{\Omega }}}_{p}^{2}}$ in that regime. In figure 3(b), near the x-axis, and in figure 2(c) it is Ω_p ≪ Ω_c and therefore γ_ATS ≈ Ω_c. These trends for γ_ATS are also borne out by calculating the separation between the pair of non-dark eigenstates of the (decay-free) Hamiltonian in equation (1), when setting all detunings to zero.

Zoom In Zoom Out Reset image size

Calculations similar to those shown in figures 2 and 3 can be performed at any atom temperature (see [32] and references therein), and for both cascade and Λ-type three-level atoms. In the hot-atom cases (i.e., in vapor cells), the difference between coupler and probe wavelengths determines the size of residual Doppler shifts, which in turn determines the visibilities and detailed shapes of EIT and ATS spectra. In hot gases γ_EIT generally tends to be larger than in the cold-atom case, in particular in weak coupler and probe fields, and follows a different scaling (see figure 7 in [32]).

The EIT and ATS experiments are performed in a standard MOT with temperature ∼100 μK and atomic density up to ∼10¹⁰ cm⁻³. The coupling and probe lasers have linear and parallel polarizations and counter-propagate through the cold-atom cloud, as seen in figure 1(b). The details of the experiment have been described previously [38]. The probe beam is derived from a diode laser (DLpro, Toptica) that is locked to the ground-state transition, $| 1\rangle$ → $| 2\rangle$, using polarization spectroscopy [39]; the beam has a Gaussian waist ω_p0 = 10 μm at the MOT center. The strong coupling laser (Toptica TA-SHG110) has a Gaussian waist ω_c0 = 30 μm and drives the Rydberg transition $| 2\rangle \to | 3\rangle$. The frequency of the coupling laser is stabilized to the Rydberg transition using a Rydberg-EIT signal obtained from a cesium room-temperature vapor cell [40]. In each experimental cycle, after turning off the trap beams, we switch on the coupling and probe lasers for 25 μs. During the probe pulse the probe-laser frequency is swept across the $| 6{S}_{1/2},F=4\rangle$ → $| 6{P}_{3/2},{F}^{{\rm{{\prime} }}}=5 \rangle$ transition using a double-pass acousto-optic modulator (AOM) over a range of ±10 MHz relative to the transition center.

In order to avoid Rydberg excitation blockade and interaction effects, and to be able to reach high coupling Rabi frequencies, Ω_c, we have chosen a state with a low principal quantum number n (35S_1/2). Also, the MOT density is reduced to about ∼10⁸ cm⁻³ by reducing the repumping laser power. Further, to ensure that the effect of radiation pressure [38, 41] on the EIT and ATS spectral profiles is negligible, we use a single-photon counter module (SPCM) for probe-light detection, allowing us to use a very low probe power, P₈₅₂ = 200 pW. Under these conditions, the radiation-pressure-induced velocity change during the probe pulse has an upper limit of ≈7 cm s⁻¹, corresponding to Doppler shifts <100 kHz (which is negligible in the present work). The EIT and ATS spectra are recorded using a data acquisition card (NI-PCI-6542) and processed with a Labview program.

The absorption coefficient, α, is obtained from the measured SPCM count number, P(Δ_p), and the off-resonant (absorption-free) count number, P₀, using the relation $\alpha L=-\mathrm{ln}(P({{\rm{\Delta }}}_{p})/{P}_{0})$, where L is the effective MOT diameter along the probe beam path. In figure 4, we present experimental EIT and ATS sample spectra with indicated linewidth parameters γ_EIT and γ_ATS. We have determined γ_EIT and γ_ATS from experimental spectra covering a range of the coupling Rabi frequency Ω_c, for fixed Ω_p. In figure 5 we show the measured (hollow symbols) and calculated (lines) results for γ_EIT and γ_ATS versus Ω_c.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To compare experimental spectra with calculated ones, it is important to account for the laser linewidth in the model. The utilized diode laser systems exhibit a frequency jitter on the order of 1 MHz FWHM, whereas high-frequency laser phase noise is negligible. The frequency jitter causes inhomogeneous broadening of the spectra, because in every realization of the experiment the lasers have a slightly different frequency, that does not change significantly over the duration of a single realization. To account for the inhomogeneous broadening, subsequent to solving the Master equation and obtaining calculated spectra such as the ones in figure 2, the spectra are convoluted with a Gaussian, whose width is given by the laser-frequency jitter. It is noted that high-frequency laser noise would cause homogeneous broadening, which would have to be accounted for differently, namely via additional dephasing terms in the off-diagonal terms in equation (3). The γ_EIT and γ_ATS-values are obtained from the convoluted calculated spectra. In the present case, the calculations for 1.5 MHz FWHM of the laser-frequency jitter (bold solid lines in figure 5) reproduce the measurements best (also see discussion of figure 6 below).

Zoom In Zoom Out Reset image size

We first discuss the EIT linewidth γ_EIT. In the EIT domain, characterized by Ω_c, Ω_p ≲ Γ_eg, the value of γ_EIT in figure 5 exhibits a quadratic behavior, as expected from figure 3(a). For the experimental data in figure 5, we find that in the EIT domain the data are fit quite well by an equation ${\gamma }_{{\rm{EIT}}}\approx A+B({{\rm{\Omega }}}_{c}^{2}+{{\rm{\Omega }}}_{p}^{2})/{{\rm{\Gamma }}}_{{eg}}$, as shown by the black thin dashed line. The fitting parameters, which are introduced to account for the inhomogeneous line broadening caused by the laser-frequency jitter, are found to be A = 2π × (1.42 ± 0.10) MHz and B = 0.44 ± 0.03. Without the inhomogeneous broadening, it would be A ≈ 0 and B ≈ 1. It is seen that the laser jitter reduces the pre-factor B to a value significantly less than its ideal value of 1, and it adds an additive constant A on the order of the laser linewidth. In the AT regime, characterized by Ω_c ≳ Γ_eg, the γ_EIT-value follows a trend ${\gamma }_{{\rm{EIT}}}\approx {{\rm{\Omega }}}_{c}-2\pi \times 3.2\,\mathrm{MHz}$. (In the absence of laser-line broadening, in the AT regime it would be ${\gamma }_{{\rm{EIT}}}\approx {{\rm{\Omega }}}_{c}-{{\rm{\Gamma }}}_{{eg}}/2$ = Ω_c − 2π × 2.6 MHz). Over the entire Ω_c-range studied, the measured dependence γ_EIT(Ω_c) agrees quite well with a calculation in which a Gaussian laser-frequency jitter of 1.5 MHz FWHM is assumed.

Next we discuss γ_ATS. In the AT regime, Ω_c ≳ Γ_eg, and for Ω_p ≪ Ω_c, it is γ_ATS ≈ Ω_c. In the EIT regime, Ω_c, Ω_p ≲ Γ_eg, the value of γ_ATS tends to be larger than Ω_c; for Ω_c ≲ 2π × 2 MHz γ_ATS cannot be determined because the spectrum ceases to split into a separated line pair (also see figure 6). Inhomogeneous broadening due to the laser-frequency jitter generally increases the measured γ_ATS, because the outer wings of the split lines are wider than the inside wings, hence the spectral averaging will pull the line centers in figure 2(b) outward. The absorption minimum at Δ_p = 0 disappears entirely when Ω_c drops below a critical value, which is related to the laser linewidth and other broadening mechanisms. In our case this critical value is ≈2π × 2 MHz. Similar nonlinear behavior of the AT splitting is observed in [32], where an RF-induced ATS is used to measure a microwave electric field.

We note that Ω_c has a different significance in the EIT and AT domains. In the AT regime, Ω_c splits the lines but does not broaden them, whereas in the EIT regime Ω_c broadens the EIT line, as γ_EIT increases with (Ω_c² + Ω_p²). Conversely, laser jitter and other inhomogeneous broadening mechanisms also have a different significance in the two domains. In the EIT domain the spectra are very sensitive to inhomogeneous broadening, because even moderate laser jitter already strongly affects the parameters A and B in the fit function γ_EIT ≃ A + B(${{\rm{\Omega }}}_{c}^{2}+{{\rm{\Omega }}}_{p}^{2}$)/Γ_eg. (In the absence of broadening, it would be A = 0 and B = 1.) In the ATS domain, the broadening has a comparatively small effect on γ_EIT, and an even smaller one on γ_ATS. Generally, the agreement between theory and experiment in figure 5 is slightly better for γ_ATS than it is for γ_EIT. We believe this is because the details of the line broadening are harder to model than the AT splitting of the line centers.

In figure 5 we have included a similar measurement with 100 μs probe- and coupler-pulse sweeps (filled diamonds in figure 5). It is seen that γ_ATS for the longer pulses is less than in the 25 μs case. We attribute the reduction in γ_ATS in part to dephasing caused by Rydberg-atom interactions, which could be considered via the dephasing terms in equation (3). Longer pulses will generally lead to a higher Rydberg-atom number in the atom-field interaction volume, causing dephasing by Rydberg–Rydberg interactions. The dependence of the ATS splitting on Rydberg interactions is detailed in [26]; in our present work interactions are avoided by using a low principal quantum number, a low atom density and a relatively short interaction time. Longer times will also increase the likelihood of Penning and thermal ionization. Any ions in the sample would contribute to Rydberg-level dephasing via the ion electric fields. With increasing pulse duration, the spectra are also increasingly affected by radiation-pressure-induced Doppler shifts, causing changes in the γ_ATS-values derived from the spectra.

We finally consider the depth of the transparency window, which is an important measure for any practical application of EIT and ATS. The depth of the transparency window, H, is defined as the difference between the absorption coefficients on the peaks and at the center of the spectrum, Δ_p = 0 (see middle curve in figure 4). The depth H increases with Ω_c in the EIT regime and plateaus at a laser-jitter-dependent maximal value in the ATS regime, as shown by the calculated curves in figure 6. Assuming a laser jitter with 1.5 MHz FWHM, we achieve good agreement between experiment and calculation. This best-fit laser linewidth agrees well with manufacturer estimates for the utilized laser systems.

We have presented measured and calculated cold-atom EIT and AT absorption spectra of a cascade three-level atom involving the $35{S}_{1/2}$ Rydberg state. The measurements show good agreement with calculations. The spectra exhibit two regimes, an EIT regime for weak Ω_c and Ω_p, and an ATS regime for large Ω_c. While similar in appearance, the EIT and AT spectra are different in physical interpretation [9, 10]. We have defined widths γ_EIT and γ_ATS that facilitates making the distinction between EIT and ATS. In the cold-atom EIT regime, Ω_c, Ω_p ≲ Γ_eg, we find ${\gamma }_{{\rm{EIT}}}=A+B(({{\rm{\Omega }}}_{c}^{2}+{{\rm{\Omega }}}_{p}^{2})/{{\rm{\Gamma }}}_{{eg}})$, where the parameters A and B depend on inhomogeneous broadening due to laser jitter. The MOT magnetic field, which is always left on, may add to the inhomogeneous broadening. In the ATS regime, Ω_c ≳ Γ_eg, it is γ_ATS ≈ Ω_c and γ_EIT = γ_ATS − C, with a constant C ≳ Γ_eg/2. The lower limit of C is realized in the absence of inhomogeneous broadening. An emphasis of our analysis has been to include the effects of laser jitter, saturation of the probe transition, and Doppler mismatch caused by non-zero temperature and unequal probe and coupling laser wavelengths.

Practical criteria to distinguish between EIT and ATS, as developed in this work, are valuable in a wide variety of atomic-physics, quantum-optics and quantum information applications of these schemes. For instance, the details of ATS splitting and line broadening effects are important in applications that deal with quantitative, atom-based microwave field measurements using Rydberg-EIT and microwave-coupled ATS [27, 32]. Cold-atom, narrow-linewidth EIT and ATS are helpful in improving the accuracy and resolution of the atom-based field measurements.

The work was supported by the National Key R&D Program of China (Grant No. 2017YFA0304203), the National Natural Science Foundation of China (Grants No. 61475090, and No. 61675123, and No. 61775124), Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT13076), the State Key Program of National Natural Science of China (Grant No. 11434007) and 1331 KSC. GR acknowledges support by the NSF (PHY-1506093) and BAIREN plan of Shanxi province.