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We report the polarization dependence of the spectrum in modulation transfer spectroscopy for the transitions from the lower ground state (Fg = 1) of 87Rb atoms. We measured the spectra for the two polarization configurations where the carrier and probe beams were linearly polarized in parallel or perpendicular directions. The measured spectra were in excellent agreement with calculated results. The spectra were strongly dependent on the polarization configurations. In particular, the signal for parallel polarization configuration was generated via an incoherent process mediated by spontaneous emission.
© 2013 Optical Society of America
1. Introduction
Active stabilization of laser frequency to an atomic resonance line is one of the most important ingredients in high-resolution laser spectroscopy and laser manipulation of atoms. Many spectroscopic methods have been developed, such as saturated absorption spectroscopy (SAS) [1], polarization spectroscopy [2], dichroic atomic vapor laser lock (DAVLL) [3], sub-Doppler DAVLL [4], frequency modulation spectroscopy [5], and modulation transfer spectroscopy (MTS) [6]. SAS is commonly used in frequency locking due to its simplicity. Recently, MTS has been used for this purpose due to its ability to provide a robust lock. Of note, the MTS signal is not sensitive to experimental conditions owing to background-free characteristic. In addition, in MTS only the dispersive signal for the cycling transition survives. This is an important feature for the atoms with small frequency spacing of the excited states such as potassium.
Since MTS was first proposed in 1980 [6], many theoretical and experimental reports have been published [7–12]. Recently, McCarron et al. reported MTS spectra for rubidium atoms [13]. Mudarikwa et al. reported an experimental work on frequency modulation spectroscopy and MTS for K atoms [14]. They observed an enhanced MTS signal when the polarizations of all laser beams were circular. Very recently, Negnevitsky and Turner reported the results for MTS with an acousto-optic modulator [15]. The authors reported MTS for two-level [16] and real 87Rb atoms, in which a huge dispersive signal for the cycling transition line (Fg = 2 → Fe = 3 of 87Rb atoms) was evident, and compared the results with calculated results [17]. In contrast, a huge signal for the other cycling transition line (Fg = 1 → Fe = 0) was not observed. In the previous paper [17], we mentioned that this was due to the same linear polarization of the laser fields.
In this paper, we show that the MTS signals for the transitions from the lower ground state (Fg = 1) strongly depend on the polarization configurations. We also show that the MTS signal for the parallel polarization configuration is generated by an incoherent process mediated by spontaneous emission from the excited state in contrast to usual MTS signal due to coherent four-wave mixing process. In section 2, we describe the theory for calculating the MTS spectra for both polarization configurations. Experimental results and discussion are presented in section 3. The final section summarizes the results.
2. Theory
The energy level diagram for the D2 transition line of 87Rb atoms is shown in Fig. 1(a). In this paper we consider the transitions Fg = 1 → Fe = 0, 1, 2. As shown in Fig. 1(b), the schematics of the MTS setup are similar to a typical SAS setup. A difference is that a linearly polarized pump beam is frequency modulated at the frequency of Ω. Therefore, a carrier and two sidebands exist, whose angular frequencies are ω and ω ± Ω, respectively, where ω is the angular frequency of the unmodulated pump beam. The sidebands of higher harmonics are not included in the calculation. The linearly polarized probe beam of frequency ω propagates in an opposite direction with the carrier or the sidebands waves. The polarization of the pump beams (a carrier and two sidebands) was fixed to z direction, whereas that of the probe beam was either along the z or x direction. Therefore, if the probe beam was polarized in the z (x) direction, the setup was in the parallel (perpendicular) polarized configuration. From a nonlinear interaction via an atomic medium, new probe beams modulate at frequencies of ω ± Ω are generated, which then beat with the original probe beam to create the MTS signals. The experimental and calculating methods were previously detailed [17]. Here we explain the method of calculation regarding on the polarization dependence. Figure 1(c) shows the basic energy level diagrams for the parallel and perpendicular polarization configurations. In the figure, c, s, and p denote the carrier, sidebands, and probe beams, respectively. The frequencies of the carrier, sidebands, and probe beams in the rest frame of an atom moving at velocity of v are ω + kv, ω + kv ± Ω, and ω − kv, respectively, where k (= ω/c) is the wave vector. The detuning is given by δ = ω − ω0 where ω0 is the resonant frequency corresponding to the transition Fg = 1 → Fe = 2. The effective detunings of the carrier and the probe beams are given by δ1 = δ + kv and δ2 = δ − kv, respectively. For later purpose, we define δp ≡ δ2 − δ1 = −2kv, which is the frequency difference in the atomic rest frame between the carrier and the probe beams. In the calculation, density matrix elements are calculated as a function of δ1 and δp, and then averaged over the velocity distribution and the various transit times.
Fig. 1 (a) Energy level diagram of 87Rb atoms. (b) Schematic diagram of the MTS. (c) Simple energy level diagrams for the parallel and perpendicular polarization configuration.
Since the electric field direction (z direction) of the pump beams was chosen as a quantization axis, the polarization vector of the probe beam (ε̂p) can be decomposed in the spherical bases as follows
where ε̂± = ∓2−1/2 (x̂± iŷ) and ε̂0 = ẑ. In Eq. (1), x̂, ŷ, and ẑ are the unit vectors for the cartesian coordinates x, y, and z, respectively. In the parallel polarization configuration, b0 = 1 and b± = 0 because ε̂p = ẑ. In contrast, since ε̂p = x̂ = 2−1/2 (ε̂− − ε̂+) in the perpendicular polarization configuration, b±1 = ∓2−1/2 and b0 = 0. In the parallel polarization configuration all the beams are linear and parallel to one another. So, the dipole interactions between the states with Δm = 0 are allowed where Δm is the difference in the magnetic quantum numbers between the magnetic sublevels in the excited and the ground states, when the direction of the electric field is chosen as a quantization axis. In contrast, in the perpendicular polarization configuration, the probe beam excites the transitions with |Δm| = 1, while the carrier and sideband beams excite the transitions with Δm = 0.As reported in the previous paper [17], because many oscillation frequencies exist, we have to consider the oscillation frequencies for relevant density matrix elements. In both configurations, the populations have three oscillation frequencies, 0 and ±Ω. Thus, the populations can be written as;
where σij = 〈i|σ|j〉 are the matrix elements of the slowly-varying density operator (σ), and i and j run over all magnetic sublevels in the excited and ground states, respectively. , , and are the time-independent real values for the diagonal density matrix elements σjj. The oscillation frequencies of the optical coherences and Zeeman coherences are different for each polarization configuration. In the case of parallel polarization configuration, there are no Zee-man coherences and the optical coherences between the magnetic sublevels in the excited and ground states with Δm = 0 have oscillation frequencies of −δp ± Ω, −δp, 0, ±Ω, and ±2Ω up to the three-photon interactions. In contrast, the coherences in the perpendicular polarization configuration have complicated oscillation frequencies. As mentioned above, the optical coherences and Zeeman coherences become non-vanishing. The oscillation frequencies for the optical coherences with |Δm| = 1 are given by −δp ± Ω, ±δp, δp ± Ω, and δp ± 2Ω; those for the optical coherences with |Δm| = 0 are given by 0, ±Ω, ±2Ω, −2δp, and −2δp ± Ω; those for the optical coherences with |Δm| = 2 are given by 0, ±Ω, −2δp, and −2δp ± Ω; the oscillation frequency for the optical coherences with |Δm| = 3 is given by −δp; those for Zeeman coherences with |Δm| = 1 are given by ±δp, −δp ± Ω, and δp ± Ω; and the oscillation frequency for Zeeman coherences with |Δm| = 2 is given by 0.The in-phase (I0) and the quadrature (Q0) components of the detected signal are given by
where Cij is the relative transition strength between the states i and j[17], and b0,±1 are defined in Eq. (1). In Eq. (3), and ( and ) are the real (imaginary) parts of the optical coherence between the states |i〉 and |j〉 oscillating at the frequencies of −δp − Ω and −δp + Ω, respectively. I0 and Q0 in Eq. (3) are functions of velocity v and time t. The time dependence originates from the fact that the time-dependent differential equations are numerically solved. The velocity dependence results from the definitions of δ1 = δ + kv and δp = −2kv. The in-phase (I) and the quadrature (Q) signals, averaged over a Maxwell-Boltzmann velocity distribution and various transit times, are then given by respectively, where u(= (2kBT/M)1/2) is the most probable velocity of the atom (T is the temperature of the cell; M is the mass of an atom) and is the average transit time to cross a laser beam of diameter d[18]. Finally, the detected signal is given by where ϕ is the phase of the detector relative to the phase of the modulated field.3. Experimental results and discussion
The experimental and calculated results for the parallel polarization configuration are shown in Figs. 2(a) and 2(b), respectively. In the experiment, unwanted strong magnetic field produced by Earth’s magnetic field was reduced down to ∼10 mG by using a three-layer μ-metal sheet. Therefore, the population transfer between Zeeman sublevels performed by the precession of the atomic magnetic moment in an external magnetic field can be negligible compared to the population transfer due to optical pumping. The Rabi frequencies of the carrier, sidebands, and the probe beams are given by Ωc = 0.74Γ, Ωs = 0.39Γ, and Ωp = 0.70Γ, respectively, where Γ (= 2π ×6.065 MHz) is the decay rate of the excited state [19]. The diameter of the laser beams is approximately 2.8 mm and the modulation frequency is Ω = 2π × 3 MHz. From the bottom of each figure, the MTS signals for ϕ = 0, 30, 60, 90, 120, 150, and 180 degrees are shown. The traces for ϕ = 0 and 90 degrees correspond to the in-phase and quadrature components of the signal, respectively. In Fig. 2, the SAS spectra are shown to identify the location of the resonance and crossover signals. In the SAS spectra, Sμ denotes the resonance signal for the transition Fg = 1 → Fe = μ, and COμν represents the crossover signal with the transitions Fg = 1 → Fe = μ and Fe = ν. Comparison of the experimental and calculated results revealed an excellent agreement between the two. As discussed in the previous paper, a large signal for the crossover CO10 is evident [17]. This is discussed in the next section. By measuring the slope of the dispersive signals, we found that the optimum angle for providing the largest slope was ϕ = 150 degree. This information will be useful when the laser lock is performed for the transition from the lower ground state of 87Rb atoms. It should be noted that the optimum angle depends on experimental parameters such as laser intensities and modulation frequency.
The results for the perpendicular polarization configuration are shown in Fig. 3. The schemes are the same as in Fig. 2. We observed good agreement between the experimental and calculated results. In contrast to the results in Fig. 2, the signals for Fg = 1 → Fe = 0 and Fg = 1 → Fe = 1 are large, and the crossover signal CO10 that appeared in Fig. 2 disappeared.
Fig. 3 (a) Experimental and (b) calculated results for the perpendicular polarization configuration.
To identify the origin of the signals we decomposed the in-phase MTS signals into three parts corresponding to Fg = 1 → Fe = 2, 1, and 0. The results for parallel and perpendicular polarization configurations are depicted in Figs. 4 and 5, respectively. In Fig. 4, the crossover signal results from both probe components Fg = 1 → Fe = 1 and Fg = 1 → Fe = 0, where the carrier beam (with sidebands) is tuned to Fg = 1 → Fe = 0 and Fg = 1 → Fe = 1, respectively. In Fig. 4, all the signals except for CO10 are very weak. This is because none of the excited transitions are cycling. However, the populations (including the oscillating components) of the ground state (Fg = 1) are not efficiently depleted as depicted in Fig. 4. Figure 4(b) shows the energy level diagrams for the signal CO10 where the thick solid (thin dotted) line denotes the carrier and sidebands (probe) beams. In the case of (ii), since the transition from |Fg = 1, mg = 0〉 to |Fe = 1, me = 0〉 is not allowed, the population in the state |Fg = 1, mg = 0〉 accumulates to a certain amount despite the optical pumping to the other ground state (Fg = 2). Then, the population (accurately speaking, two components oscillating at the frequencies of ±Ω) is detected by the probe beam tuned to Fg = 1 → Fe = 0. In the other case (i), the populations at |Fg = 1, mg = ±1〉 accumulate due to the carrier and sidebands tuned to Fg = 1 → Fe = 0, which is the cycling transition, and then detected by the probe beam tuned to Fg = 1 → Fe = 1. It should be noted that the MTS signal for this crossover signal is not a four-wave mixing process but the wholly incoherent process mediated by spontaneous emission. This is discussed below.
Fig. 4 (a) Decomposition of the calculated MTS spectrum for the parallel polarization configuration. (b) Energy level diagrams responsible for the crossover signal CO10.
Fig. 5 (a) Decomposition of the calculated MTS spectrum for the perpendicular polarization configuration. (b) Energy level diagrams responsible for the resonance signals S1 and S0. (c) Energy level diagrams for the crossover signal CO10 for both polarization configurations.
The resonance signal for Fg = 1 → Fe = 2 was weak (Fig. 5). This is because this transition is not a cycling transition line. However, the other two resonance signals for Fg = 1 → Fe = 1 and Fg = 1 → Fe = 0 are strong. As shown in Fig. 5(b) [(i)], in the case of the transition Fg = 1 → Fe = 1, the accumulated population at |Fg = 1, mg = 0〉 is detected by the probe beam. In the case of the transition Fg = 1 → Fe = 0, since this is a cycling transition line, the optically pumped populations at |Fg = 1, mg = ±1〉 are detected by the probe beam as shown in Fig. 5(b) [(ii)]. Figure 5(c) represents why the crossover signal CO10 disappears in contrast to the parallel polarization configuration. In the parallel polarization configuration, the populations at |Fe = 1, me = ±1〉 decay spontaneously to |Fg = 1, mg = 0〉 and the populations at |Fe = 0, me = 0〉 decay spontaneously to |Fg = 1, mg = ±1〉. As shown in Fig. 5(c), the branching ratios for these transitions are and , respectively. Thus, the populations of the excited states can transfer to the other ground states. This is an essential condition required to construct a MTS signal via spontaneous emission. However, this is not the case for the perpendicular polarization configuration; the decay from the state |Fe = 1, me = 0〉 into the state |Fg = 1, mg = 0〉 is not allowed. Since the optical pumping from the excited to the ground state is prohibited, the MTS signal for the crossover CO10 is not seen for the perpendicular polarization configuration.
To account for the role of spontaneous emission in the construction of the MTS signal, we calculate the MTS signal for a simple four-level model as shown in Fig. 6(a). The carrier and sidebands are tuned to the transition between |1〉 and |3〉, while the probe is tuned to the transition between |2〉 and |4〉. As shown in Fig. 6(b), we define the branching ratios, b1 and b2. In Fig. 6(b), we show the MTS spectra for several combinations of b1 and b2. In Fig. 6(b), we see that when either b1 = 1 or b2 = 1, the MTS signal vanishes. We observe also that the signal is largest when b1 = b2 = 0, i.e., when the transferring of the populations of the excited state to the other ground states is maximized. Therefore, we are assured that the transferring of the population of the excited state for one transition line to the ground state for the other transition is an essential condition to construct the MTS signal which is mediated by spontaneous emission. When the phase differs from zero, the lineshapes in Fig. 6(b) will be modified although the overall magnitude does not vary significantly. The detailed study on the effect of phase on the MTS signals for a simple four-level atomic system is currently underway.
Fig. 6 (a) An energy level diagram for simple four-level model. (b) Calculated in-phase MTS signals for various values of the branching ratios, b1 and b2.
4. Conclusions
We present the MTS spectra for the transitions Fg = 1 → Fe = 0, 1, 2 for 87Rb atoms in two polarization configurations where the linear polarization axis of the carrier beam is either parallel or perpendicular to that of the probe beam. In parallel polarization configuration, the MTS signal for CO10 is prominent, whereas the signals for S1 and S0 are strong for the perpendicular polarization configuration. We investigated the reasons for the presence or absence of the signal for specific transition lines. It is interesting that the strong MTS signal for CO10 in parallel polarization configuration does not originate from a coherent process but from an incoherent process mediated by spontaneous emission. We experimentally observed that the MTS signal for the transitions Fg = 1 → Fe = 0, 1, 2 depended strongly on the intensities of the laser beams. The details of this study will be reported elsewhere.
Acknowledgments
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology( 2011-0009886). This work was also supported by the Korea Research Institute of Standards and Science under the project ‘Establishment of National Physical Measurement Standards and Improvements of Calibration/Measurement Capability,’ grant 13011001.
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