Doppler-free coherent-control spectroscopy with a colliding pair of shaped pulses

Minhyuk Kim, Kyungtae Kim, Dewen Cao, 3 Fang Gao, Feng Shuang, 3, 4 and Jaewook Ahn Department of Physics, KAIST, Daejeon 305-701, Korea Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, Anhui 230031, China Department of Automation, University of Science and Technology of China, Hefei, Anhui 230026, China Department of Mechanical Engineering, Anhui Polytechnic University, Wuhu Anhui 241000, China (Dated: September 11, 2018)


I. INTRODUCTION
Light structured in the spectro-temporal domain is used in coherent control to enhance or suppress nonlinear material responses through engineered passages of optical transitions [1][2][3]. It has been demonstrated that laser pulses with programmed phase and amplitude can boost, for example, the two-photon transitions of atoms in two, three, and four energy-level structures [4][5][6][7]. Various functional light-matter interactions have also been designed with shaped lights; examples include dark pulses [8], laser-catalytic chemical reactions [9], quantum gates [10], and high-harmonic generations [11], to list a few. In particular, state-to-state controllability in coherent control allows for the selective excitation of otherwise unresolvable energy states of a complex quantum system [12], promising the use of coherent control methods in precision and/or functional spectroscopy.
The use of shaped-pulses in laser spectroscopy became more interesting following the recent demonstration of Doppler-free coherent-control spectroscopy [13]. Termed as ultrafast spatial coherent-control (USCC), this method uses a pair of colliding laser pulses, of which the spectrum is phase modulated in such a way that counter-propagating photons that satisfy each twophoton excitation meet at a specific location along the beam propagation path. With this method, laser pulses were successfully programmed to provide a spatiallymapped spectroscopic assessment of each two-photon transition of rubidium and cesium all at once [13]. In particular, in conjunction with the optical frequency comb [14,15], USCC holds a promise for ultra-high precision spectroscopy of atomic species for which the laser cooling method is unavailable [16].
Our previous work extended this method [13] to a twophoton transition system having an intermediate resonance using a special spectral-phase modulation [17] pathways may exist in quantum systems [18][19][20], we extend this approach [17] even further to a system of multiple transition pathways, where the crowding resonances require special spectral-phase modulations. We propose three such modulations, particularly designed to resolve the fine-structure energy levels of alkali atoms and experimentally demonstrate their performance. The experimental concept is shown in Fig. 1, where the two finestructure two-photon transition pathways, 5S 1/2 -5P 1/2 -5D and 5S 3/2 -5P 1/2 -5D, of atomic rubidium are separated in the excitation space.
In the rest of the paper, we first theoretically describe an imaging method using the USCC to revolve distinct nonlinear transition pathways in Sec. II, and briefly sum-arXiv:1710.03387v1 [physics.atom-ph] 10 Oct 2017 marize our experimental procedure in Sec. III. The experimental results are presented in Sec. IV, with a comparison of the performance of the three proposed modulations. The conclusion follows in Sec. V.

II. THEORETICAL CONSIDERATION
As the simplest model, we consider a four-level system in the diamond-type configuration. There are two two-photon transition pathways, |g → |a → |f and |g → |b → |f , where |g is the ground state, |a and |b are the intermediate states, and |f is the final state. The energies are 0, ω a , ω b , and ω f (in unit ), respectively. From the second-order perturbation theory, the two-photon transition probability amplitude c f g to the final state |f is given by Here, the resonant and non-resonant transition contributions are respectively defined as where µ ij is the transition dipole moments and ω ij = ω i −ω j is the resonant transition frequencies [6]. The c r f g,i term is the resonant transition contribution that only depends on the resonance frequency components of laser spectrum E(ω). Due to the narrow spectral response in c r f g,i , it is not possible to isolate the resonant excitation part in time (likewise in space). However, the c nr f g,i term, the non-resonant transition contribution, depends on all possible spectral pairs that satisfy the energy conservation ω i + ω j = ω f g . Note that only c nr f g,i is involved in USCC [17].
We first consider a single laser-pulse to be programmed in the frequency domain as where A(ω) and Φ(ω) are the programmed spectral amplitude and phase of the electric-field, respectively. In order to eliminate the resonant contribution c r f g,i , we program spectral holes near the resonant frequencies ω ag and ω bg , by (5) Here, A 0 (ω) is the spectral amplitude before programming and δ i=a,b is the width of each spectral hole, which is significantly smaller than the pulse bandwidth. With this spectral amplitude programming, the resonant transition background signal is removed (i.e., c r f g,i = 0), so only the non-resonant transition part will be considered, as When two pulses of the same electric-field spectrum E s (ω) counter-propagate along ±z, the combined electric field reads and Eq. (6) can be replaced by ,ω = ω f g − ω and the global phase factor exp(iω f g z/c) is omitted. Then, the spatial excitation probability is given by where the cosine term in the square bracket corresponds to the counter-propagating pulse contribution, and the remaining two terms correspond to the single-sided pulse contribution.
The spectral phase programming in Φ (ω)+Φ (ω) needs two strategies: a sign-flipping of the function f i (ω) at the intermediate resonances, and a maximizing of the ratio between the counter-propagating pulse and single-sided pulse contributions. After describing the experimental procedure in Sec. III, we introduce three such phaseprogramming solutions in Sec. IV, along with the corresponding experiments.

III. EXPERIMENTAL DESCRIPTION
Experiments were performed for the fine-structure transitions of atomic rubidium ( 85 Rb), where the four lowest energy levels are |g = 5S 1/2 , |a = 5P 1/2 , |b = 5P 3/2 , and |f = 5D. A schematic of the experimental setup is shown in Fig. 1, which is similar to the one from our earlier work [17]. We used a Ti:sapphire mode-locked laser oscillator, producing sub-picosecond laser pulses at a repetition rate of 82 MHz. The laser pulses were frequency-centered at ω 0 /2π = 384.3 THz (ω 0 = ω f g /2 and equal to 778 nm in wavelength) to be two-photon resonant to the 5S 1/2 -5D transition. The laser bandwidth was ∆ω/2π = 22.3 THz (with a full width at half maximum of 45 nm in wavelength), which was sufficient to cover all four transitions.
The laser pulse was programmed with a transmissive spatial light modulator (SLM) with an array of liquidcrystal pixels (128 pixels, 100 µm pitch) placed in the Fourier domain with a 4f geometry pulse shaper [21]. The focal length of the 4f geometry was f = 150 mm, and the spectral resolution of each pixel was 0.46 nm in wavelength. The groove density of the gratings was 1200 mm −1 . The spectral position of each SLM pixel was calibrated by scanning a π-step phase [6]. For the spectral amplitude modulation, two copper wires were placed on the Fourier plane: the first one was 140 µm in width to block the intermediate resonance ω ag , and the second one was 700 µm in width to block ω bg and also reduce the signal strength difference between the transition paths. The as-programmed pulses were then focused in the rubidium vapor cell and the fluorescence at 420 nm from the 5D state through the 6P state was imaged with a charge-coupled device (CCD) camera. The vapor cell was heated to around 50∼60 • C to enhance the fluorescence signal.
The numerical calculation and the experimental result for the double V -phase modulation are plotted in Fig. 2(c) and (d), where α 2 was increased from 0.4 to 1.6 ps with a step size of 0.1 ps, while α 1 = 0.4 ps was kept constant. For α 1 = α 2 = 0.4 ps, there are two sub-pulses split in the time domain, so two excitation peaks exist with overlapped D1 and D2 transitions, as shown in the upper part of Fig. 2(e). As α 2 increases, the sub-pulses having the spectral blocks corresponding to D2 move further from z = 0, so the excitation peaks are better resolved as in the lower part of Fig. 2(e).

B. Three phase-slopes
The second phase-programming solution utilizes three phase-slopes, which are given by: where B12 combines the spectral blocks B1 and B2. This phase modulation, plotted as the solid red line in Fig. 3(a), results in three sub-pulses in the timedomain, as shown in the spectrogram in Fig. 3(b). The first and third pulses (e.g., from the forward and backward propagating pulses, respectively) cause the twophoton transition through D1, and the second and the third through D2. As a result, the integrand of the counter-propagating pulse contribution term in Eq. (9) becomes e 2i(α1,2+α3)(ω−ω0) cos ((ω − ω) z/c), and it constructively interferes when α 1,2 + α 3 = 2z/c. The excitation positions are determined as z 1 = ±(α 1 + α 3 )c/2 and z 2 = ±(α 2 + α 3 )c/2, respectively. The numerical calculation and experimental result are plotted in Figs. 3(c) and (d), respectively, where α 1 and α 2 were fixed with α 1 = 0.1 ps and α 2 = 1.5 ps, while α 3 was increased from 0.1 to 1.6 ps with a step size of 0.1 ps. In this case, the spacing between the excitation peaks for the respective D1 and D2 transitions is fixed because α 1 and α 2 are constants, while the peak positions are gradually separated as α 3 increases.

C. Periodic square phase
The last phase modulation solution utilizes the mathematical relation e i π 2 [sgn(x)−1] cos(x) = | cos(x)| to include the counter-propagating pulse contribution in Eq. (9) (the first term in the square bracket), with all spectral pairs in-phase. The as-obtained solution reads: The last phase modulation solution utilizes the mathematical relation e i π 2 [sgn(cos x)−1] cos x = | cos x| to add the counterpropagating pulse contribution in Eq. (9) (the first term in the square bracket) all in phase. The as-obtained solution reads: where A 1,2 are the modulation amplitudes, sgn(x) is the signum function defined as sgn(x) = −1 for x ≤ 0 and +1 for x > 0, and β 1,2 are the modulation frequencies, or the period, of the square function for each spectral region. When the modulation amplitudes and frequencies satisfy A 1 = A 2 = π/4 and β 1,2 = 2z/c, as illustrated in Fig. 4(a), the integrand in Eq. (9) becomes e i π 2 sgn[cos(2(ω−ω0)z/c)] cos[2(ω 0 −ω)z/c] = e iπ/2 | cos(2(ω 0 − ω)z/c)|, which induces a complete constructive interference from the counter-propagating pulse contributions. As a result, excitation occurs at z 1 = ±β 1 c/2 and z 2 = ±β 2 c/2 (the spatially resolved two-photon excitation pattern). On the other hand, when A 1 = A 2 = π/2, the integral of the counter- propagating pulse contribution term is eliminated because e iπsgn[cos(β1,2(ω−ω0))] ≡ −1, and only the singlesided excitation term (the spatially independent signal) remains in Eq. (9). Therefore, there is no specific spatial excitation pattern. The numerical calculation of Eq. (9) with the solution of Eq. (13) is shown in Fig. 4(b), where the modulation depth is changed from A 1 = A 2 = 0 to A 1 = A 2 = π with 0.025π steps for δ = 0, with α 1 = 1.16 ps and α 2 = 2.32 ps. The experimental result for the periodic square phase solution is shown in Fig. 4(c), where A 1,2 were scanned together from A 1 = A 2 = 0 to π with a step of 0.05π. The optimal modulation amplitude with which the single-side contribution is minimized was found to be A 1 = A 2 = 0.35π, slightly shifted from the theoretically predicted A 1 = A 2 = π/4. This discrepancy mainly comes from the finite size of the phase modulation pixels of our SLM, as the modulation functions of two transition pathways (Φ 1 and Φ 2 in Eq. (13)) could not both be centered on the two-photon center ω 0 simultaneously. Another discrepancy originates from the spatial background intensity distribution by beam focusing. This is more apparent for the modulation depths where the single-side contribution is dominant. The combined numerical calculation with these constructive interference leakage effects in the experimental system is illustrated in Fig. 4(d), which better matches the result in Fig. 4(c). In spite of these system-atic errors, the overall behavior where the single-sided contribution is recovered at A 1 = A 2 = π/2 and the minimized point exists between A 1 = A 2 = 0 and π/2 remains.
The effects of atomic motion in vapor and the imaging blurring due to Abbe diffraction are taken into account to reconstruct Fig. 4(e). To investigate the extent of each excitation peak broadening, the convolution integral of the numerical calculation results of excitation patterns with these effects were performed. For the atomic motion effects, the position distribution along the pulse propagation axis after a 5D state lifetime of 240 ns from the velocity distribution of Rubidium (for the pulse propagation axis) at a temperature 55 • C was considered. For the diffraction effect from the imaging aperture, an impulse response function with a lens aperture size of 25.4 mm was taken into account. The width of the broadened excitation peak after the convolution integral with atomic motion effects was approximately 120 µm, which is comparable to the average width of each excitation pattern in the experimental result in Fig. 4(c) with Gaussian fitting (R-square above 0.965). The width of the broadened excitation peak by diffraction was on the order of a few µm, which is smaller than the pixel width of our CCD camera, and therefore negligible. Then, comparing the width of the Gaussian fittings of excitation patterns (R-square above 0.99) before and after convolution, we found that this broadened excitation pattern could be effectively reconstructed by multiplying a Gaussian with a FWHM of approximately 30 µm to each broadened excitation peak.
By Gaussian fitting and multiplying the Gaussian function found above, the broadening effect of the experimental results in Fig. 4(c) is effectively reconstructed. This result is shown in Fig. 4(e). Each excitation peak narrowed, agreeing well with the measurement in Fig. 4(d).

V. CONCLUSION
In summary, we performed ultrafast spatial coherentcontrol experiments to resolve the fine-structure twophoton transitions (5S 1/2 -5P 1/2 -5D and 5S 1/2 -5P 3/2 -5D pathways) of atomic rubidium with various phase programming solutions. This work not only extended our earlier work [17] of combining spectral amplitude and phase programmings to deal with atomic transitions with multiple intermediate resonances, but also newly proposed and demonstrated two additional phase programming solutions. Compared to the previously introduced double-V spectral phase, the three phase slopes and the periodic square phases provided for simpler programming and the possibility for finer spectral resolution spectroscopy, respectively. In experiment, counterpropagating ultrafast optical pulses as-shaped with the spectral phase and amplitude programming solutions successfully induced the given two-photon transitions, simultaneously and at distinct spatial locations, agreeing well with the theoretical analysis.