Stimulated Raman adiabatic passage (STIRAP) is a coherent manipulation technique that allows robust population transfer between discrete quantum states.^[1,2] The principle of STIRAP can be understood using a three-level system involving two ground states and one excited state. By coupling the two ground states with the excited state using a pump light and a Stokes light, respectively, and adiabatically controlling the Rabi frequencies in a counterintuitive way, the system can be transferred from one ground state to the other one without any losses. Recently, the STIRAP technique has been employed to prepare ultracold alkali-metal diatomic molecules in the rovibrational ground state.^[3–10] To prepare the ultracold ground-state molecules, weakly bound Feshbach molecules are first created in an ultracold atomic gas in the vicinity of an atomic Feshbach resonance, and then are transferred to the ground state via a molecule excited state by the STIRAP. An illustration of the STIRAP in the ²³Na⁴⁰K system is shown in Fig. 1.

Fig. 1.

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	Fig. 1. Illustration of the stimulated Raman adiabatic passage in the 23Na40K system. By choosing the different Feshbach states, the proper hyperfine levels of the excited states, and the laser polarizations, the various hyperfine level of the ground-state molecule can be populated.

The Feshbach molecule is a kind of long-range molecule, and the long-range wave function is simply determined by the large magnetically-tunable scattering length.^[11] This kind of universal property has been employed to study the BEC–BCS crossover in degenerate fermion gases,^[12] the dynamics of solitons and votices in Bose-Einstein condensates (BEC),^[13,14] and the universal few-body physics.^[15,16] However, to prepare the ground-state molecule, we need to understand the short-range wave function of the Feshbach molecule, since the excited state is a conventional rovibrational state of the excited electronic state, and thus the coupling between the Feshbach state and the excited state occurs at the short range.

In the ²³Na⁴⁰K system, there are various atomic Feshbach resonances that can be used to create Feshbach molecules.^[17,18] The projection of the total angular momentum along the magnetic field and the short-range wave function of these Feshbach molecules are different. The electronic excited state is a combination of the singlet and triplet electronic states, and it has many hyperfine levels due to the coupling between the electron spin and the nuclear spins.^[19,20] These rich energy level structures offer the opportunity to create different hyperfine levels of the ground state by choosing the proper hyperfine excited states, the polarization of the pump and Stokes lasers, and the different Feshbach states. In our recent work, we have prepared the molecules in several different hyperfine levels of the rovibrational ground state to study the atom–molecule Feshbach resonances.^[10] Besides, it has been demonstrated that in the ²³Na⁴⁰K system, the interference between a desired resonant STIRAP via resonantly coupled hyperfine level of the excited state and an unexpected detuned STIRAP via other off-resonantly coupled hyperfine levels can induce the oscillation of the round-trip STIRAP efficiency and affect the purity of the hyperfine ground states.^[21] Therefore, it is important to understand the coupling between the Feshbach states and different hyperfine levels of the excited states.

In this paper, we present a detailed theoretical study of the coupling between the Feshbach states and different hyperfine levels of the excited states in the ²³Na⁴⁰K system. We will consider the broad Feshbach resonances between |f_Na,m_fNa⟩ = |1,1⟩ and |f_K,m_fK⟩ = |9/2,m_fK⟩ states with m_fK = −9/2, −7/2, −5/2. For each spin combination, there are two broad atomic Feshbach resonances. We will calculate the short-range wave function of the Feshbach state, and calculate the coupling between these Feshbach states and the hyperfine levels of the excited state, which is a mixture of the electronic excited states B¹ Π |v = 12, J = 1⟩ and c³Σ |v = 35, J = 1⟩. Our results can be used to prepare the molecules in various hyperfine levels of the rovibrational ground states.

We first consider the s-wave Feshbach molecule in the vicinity of the Feshbach resonance between the |f_Na,m_fNa⟩ = |1,1⟩ and |f_K,m_fK⟩ = |9/2,m_fK⟩ states. The Feshbach state is labelled by |1,1;9/2,m_fK⟩. The binding energy and the wave function of the Feshbach molecule can be calculated using the Hamiltonian

The internal states can be expressed in terms of the uncoupled basis |m_sNa, m_sK, m_iNa, m_iK⟩. The Hamiltonian couples all the internal states with the same projection along the magnetic field M_F = m_fNa + m_fK = m_sNa + m_sK + m_iNa + m_iK. For a given M_F and B, we first diagonalize the hyperfine and Zeeman Hamiltonian H_hf + H_z to obtain the internal eigenstate |χ_i⟩ and the threshold energy of these channels. Expanding the wave function in terms of the new bases |ψ_F(r) ⟩ = ∑_iψ_i(r)|χ_i⟩, we obtain the coupled-channel Schrödinger equation

The calculated wave function can be expanded in terms of the spin-coupled basis |σ⟩ = |S, M_S, m_iNa, m_iK⟩ by a unitary transformation

Fig. 2.

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	Fig. 2. (a) The spatial wave function of the Feshbach state |1,1;9/2, −7/2⟩ at B = 104 Gs (1 Gs = 10−4 T). Each color curve represents the wave function in a certain basis |σj⟩. (b) The triplet component of the spatial wave function at the short range. The inset is the normalized short-range wave function , with the cut off length a0.

It can be easily seen that only several channels have the long-range components. For these channels, the long-range wave function can be expressed as

For the STIRAP transfer, the short-range wave function is important. The short-range wave function includes the contribution from the the singlet and triplet potentials. In most cases, the contribution from the singlet potential is negligibly small, and thus the wave function of the Feshbach molecule is dominated by the triplet component. That is the reason why we usually need to find an excited state which is a mixture of triplet and singlet electronic excited state to perform the STIRAP.

The short-range wave function of the triplet component can be written as

For the Feshbach state |1,1;9/2,−7/2⟩ at B = 104 Gs, we have plotted the short-range wave function and the normalized wave function in Fig. 2(b). The normalized wave function can be approximated by the highest vibrational state of the triplet potential. We have calculated the short-range coefficients for the Feshbach molecules in the vicinity of different Feshbach resonances. The results are shown in 1. The coefficients will be used to calculate the coupling between the Feshbach states and the hyperfine levels of the excited states. Since the triplet s-wave bound state has good quantum numbers N = 0, S = 1, and J = 1, in the following discussions, we use the molecule quantum numbers to represent the basis , and the short-range wave function of the Feshbach state can be described by

Table 1.

Table 1.

Table 1. The coefficients of the short-range wave function for different Feshbach molecules |1,1;9/2,mf⟩ with mf = −9/2, −7/2, −5/2. . |1,1;9/2,−9/2⟩ at B = 85.7 Gs |1,1;9/2,−9/2⟩ at B = 78 Gs J MJ miNa miK J MJ miNa miK 1 −1 −3/2 −1 −0.092 1 −1 −3/2 −1 0.113 1 −1 −1/2 −2 0.198 1 −1 −1/2 −2 −0.378 1 −1 1/2 −3 0.154 1 −1 1/2 −3 0.345 1 −1 3/2 −4 0.128 1 −1 3/2 −4 0.035 1 0 −3/2 −2 −0.098 1 0 −3/2 −2 0.126 1 0 −1/2 −3 −0.214 1 0 −1/2 −3 0.287 1 0 1/2 −4 −0.712 1 0 1/2 −4 −0.344 1 1 −3/2 −3 0.345 1 1 −3/2 −3 −0.604 1 1 −1/2 −4 0.481 1 1 −1/2 −4 0.378 |1,1;9/2, −7/2⟩ at B = 104 Gs |1,1;9/2, −7/2⟩ at B = 89.9 Gs J MJ miNa miK J MJ miNa miK 1 −1 −3/2 0 0.103 1 −1 −3/2 0 −0.150 1 −1 −1/2 −1 −0.168 1 −1 −1/2 −1 0.449 1 −1 1/2 −2 −0.254 1 −1 1/2 −2 −0.288 1 −1 3/2 −3 −0.289 1 −1 3/2 −3 −0.156 1 0 −3/2 −1 0.132 1 0 −3/2 −1 −0.195 1 0 −1/2 −2 0.180 1 0 −1/2 −2 −0.238 1 0 1/2 −3 0.625 1 0 1/2 −3 0.149 1 0 3/2 −4 0.298 1 0 3/2 −4 0.266 1 1 −3/2 −2 −0.268 1 1 −3/2 −2 0.634 1 1 −1/2 −3 −0.459 1 1 −1/2 −3 −0.261 1 1 1/2 −4 −0.048 1 1 1/2 −4 −0.102 |1,1;9/2, −5/2⟩ at B = 128 Gs |1,1;9/2, −5/2⟩ at B = 104 Gs J MJ miNa miK J MJ miNa miK 1 −1 −3/2 1 −0.108 1 −1 −3/2 1 −0.172 1 −1 −1/2 0 0.147 1 −1 −1/2 0 0.483 1 −1 1/2 −1 0.301 1 −1 1/2 −1 −0.204 1 −1 3/2 −2 0.402 1 −1 3/2 −2 −0.238 1 0 −3/2 0 −0.159 1 0 −3/2 0 −0.263 1 0 −1/2 −1 −0.173 1 0 −1/2 −1 −0.196 1 0 1/2 −2 −0.553 1 0 1/2 −2 0.033 1 0 3/2 −3 −0.346 1 0 3/2 −3 0.284 1 1 −3/2 −1 0.215 1 1 −3/2 −1 0.632 1 1 −1/2 −2 0.423 1 1 −1/2 −2 −0.158 1 1 1/2 −3 0.074 1 1 1/2 −3 −0.152 1 1 3/2 −4 −0.056 1 1 3/2 −4 0.048

Table 1. The coefficients of the short-range wave function for different Feshbach molecules |1,1;9/2,mf⟩ with mf = −9/2, −7/2, −5/2. .

The excited state used in the STIRAP transfer is a mixture of the singlet electronic excited state B¹Π |v = 12, J = 1⟩ and the triplet electronic excited state c³Σ |v = 35, J = 1⟩, and thus the excited state can be written as a superposition state |ψ_e⟩ = |ψ_B¹Π⟩ + |ψ_c³Σ ⟩, where |ψ_B¹Π⟩ and |ψ_c³Σ⟩ represent the singlet and triplet excited states, respectively. The hyperfine level structures of the excited states have been discussed in Refs. [19,20].

The triplet component |ψ_c³Σ⟩ accounts for the coupling between the Feshbach state and the electronic excited state, since the short-range wave function of the Feshbach molecule is dominated by the triplet component. The state |ψ_c³Σ⟩ can be expanded in the basis |NSJ M_J m_iNam_iK⟩ with N = 1, S = 1, and J = 1. The hyperfine and Zeeman interactions for the c³Σ state are described by

The singlet component |ψ_B¹Π⟩ is responsible for the coupling between the excited state and the ground state, since the ground molecule state is a singlet state. The |ψ_B¹Π⟩ state can be expanded in the basis |JΩM_J m_iNa m_iK⟩ with J = 1 and Ω = 1. The B¹Π state has negligible hyperfine interaction and the Zeeman interaction is diagonal. The diagonal matrix element is given by ⟨JΩM_J m_iNam_iK| H^B|JΩM_J m_iNam_iK⟩ = μ_B M_J B/(J(J + 1)). The |NSJ M_J m_iNa m_iK⟩ state of c³Σ and the |JΩM_J m_iNa m_iK⟩ state of B¹Π are mixed by the spin–orbit coupling with a coupling coefficient ξ_Bc = 0.54899 cm⁻¹. Therefore, the energy and the wave function of the hyperfine levels of the excited states are obtained by diagonalizing the coupled Hamiltonian

Fig. 3.

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	Fig. 3. The energy level structures of the excited states. The quantum numbers F1 = J + iNa and mF1 are approximately good quantum numbers. Each solid line represents 9 nearly degenerate hyperfine states with the same F1 = J + iNa and mF1 but different miK.

The triplet component of the wave function of the hyperfine excited state can be expressed as

The energies of the hyperfine levels of the excited state, relative to a reference frequency of 17700.64 cm⁻¹, and the relative coupling strength between the hyperfine excited states and the Feshbach states in the vicinity of different Feshbach resonances are listed in Table 2. For each Feshbach state with M_F, three hyperfine levels of the excited states with the total projection can be coupled by choosing the polarization of the lasers. To perform the STIRAP, we can choose a resolvable hyperfine level of the excited state, which should have a strong coupling with the Feshbach state and should be well separated from neighbouring hyperfine excited states. The good candidates are the states with F₁ ≈ 5/2, m_F1 ≈ 5/2,…,−5/2, the states with F₁ ≈ 3/2, m_F1 ≈ −3/2, and the states with F₁ ≈ 1/2, m_F1 ≈ ± 1/2.

Table 2.

Table 2.

Table 2. The energies of the hyperfine excited states Ee, relative to a reference frequency of 17700.64 cm−1, and the absolute value of the relative coupling strength |Ωef| between different Feshbach states and different hyperfine levels of the excited states. For each Feshbach state with the projection of the total angular momentum MF, the hyperfine levels of the excited states with can be coupled by the pump light. . |1,1;9/2, −9/2⟩ at B = 85.7 Gs |1,1;9/2, −9/2⟩ at B = 78 Gs Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz 107 0.43 100 0.32 58 0.18 63 0.24 55 0.33 58 0.31 8 0.06 9 0.20 11 0.07 7 0.06 9 0.33 10 0.20 −45 0.01 −48 0.07 −50 0.15 −44 0.08 −46 0.22 −106 0.03 −113 0.06 −90 0.01 −97 0.01 −104 0.05 −224 0.33 −226 0.15 −231 0.20 −229 0.24 −234 0.05 −232 0.21 −246 0.14 −243 0.28 −240 0.11 −249 0.12 −246 0.15 −244 0.12 −324 0.07 −328 0.18 −333 0.38 −318 0.07 −323 0.23 −327 0.43 −446 0.11 −450 0.02 −453 0.63 −442 0.21 −445 0.40 −448 0.43 −512 0.04 −517 0.03 −505 0.10 −510 0.06 |1,1;9/2,−7/2⟩ at B = 104 Gs 1,1;9/2,−7/2⟩ at B = 89.9 Gs Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz 166 0.19 150 0.2 117 0.38 123 0.01 105 0.19 110 0.04 65 0.11 69 0.23 74 0.31 57 0.33 61 0.27 65 0.02 9 0.06 11 0.16 13 0.08 7 0.08 8 0.36 10 0.21 −51 0.03 −53 0.08 −56 0.14 −45 0.01 −47 0.10 −49 0.27 −109 0.01 −118 0.04 −126 0.09 −95 0.01 −103 0.03 −110 0.08 −222 0.39 −220 0.03 −225 0.15 −223 0.01 −228 0.25 −226 0.26 −224 0.01 −232 0.08 −230 0.20 −228 0.01 −242 0.17 −238 0.28 −235 0.07 −247 0.17 −244 0.17 −241 0.07 −331 0.08 −336 0.19 −341 0.32 −322 0.10 −327 0.29 −331 0.42 −453 0.10 −457 0.13 −462 0.59 −445 0.26 −449 0.40 −452 0.28 −523 0.06 −528 0.14 −534 0.18 −510 0.09 −516 0.04 −521 0.2 |1,1;9/2, −5/2⟩ at B = 128 Gs |1,1;9/2, −5/2⟩ at B = 104 Gs Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz Ee/MHz 186 0.24 193 0.05 162 0.24 169 0.05 132 0.33 138 0.01 145 0.01 113 0.08 119 0.07 126 0.02 74 0.06 79 0.22 85 0.03 62 0.30 66 0.21 71 0.01 11 0.05 14 0.12 17 0.09 7 0.09 9 0.37 11 0.21 −58 0.05 −61 0.09 −63 0.13 −50 0.04 −52 0.12 −54 0.32 −126 0.02 −135 0.06 −144 0.12 −103 0.01 −112 0.04 −121 0.13 −218 0.44 −217 0.09 −215 0.10 −223 0.16 −221 0.01 −220 0.02 −223 0.28 −222 0.33 −219 0.13 −230 0.10 −228 0.30 −225 0.00 −235 0.17 −232 0.26 −229 0.03 −244 0.23 −241 0.20 −237 0.04 −342 0.07 −348 0.20 −354 0.28 −328 0.14 −333 0.32 −338 0.40 −463 0.06 −469 0.20 −474 0.55 −450 0.28 −455 0.34 −459 0.17 −538 0.06 −545 0.25 −551 0.22 −519 0.07 −525 0.15 −531 0.24

Table 2. The energies of the hyperfine excited states Ee, relative to a reference frequency of 17700.64 cm−1, and the absolute value of the relative coupling strength |Ωef| between different Feshbach states and different hyperfine levels of the excited states. For each Feshbach state with the projection of the total angular momentum MF, the hyperfine levels of the excited states with can be coupled by the pump light. .

The coupling between the excited state and the ground state can be easily calculated once the singlet component of the excited state is obtained. The ground state can be expressed as |NSJM_J m_iNam_iK⟩ with N = S = J = M_J = 0. The singlet component accounts for the coupling between the excited state and the ground state, and it can be expressed as,

In summary, we have presented an intensive investigation of the coupling between the Feshbach states and the hyperfine levels of the excited states in the adiabatic creation of the ²³Na⁴⁰K molecule. The ²³Na⁴⁰K ground state molecules have 36 hyperfine levels. By employing the different Feshbach molecules discussed in the present work, in principle we can address 21 hyperfine levels of the ground state by choosing the proper laser polarization and hyperfine levels of the excited states. This is more efficient and convenient than using the microwave pulses to prepare different hyperfine levels of the molecular ground states. This is because a microwave Raman π pulse or two resonant microwave π pulses can only change the projection of the nuclear spin m_iNa or m_iK by 1. This means many π pulses may be required if we prepare one hyperfine ground state by the STIRAP and use microwave pulses to transfer it to other hyperfine states. However, by choosing proper hyperfine excited states and laser polarizations, various hyperfine states can be populated by simply preforming the STIRAP. By preparing the different hyperfine levels of the molecule states, we can study the atom-molecule scattering resonances in different spin state combination, precisely change the number of open channels, and investigate the role of nuclear spins in the ultracold molecule collisions.