1. Introduction

In recent years, a number of schemes for the production of ultracold molecules have been proposed [1–12]. These schemes include buffer-gas loading into magnetic traps [1], sequential cooling of rotation, translation and vibration [2], and far off-resonance trapping (FORT) [4] of molecules. One of the most promising technique is the photoassociation (PA) of cold atoms [5–12], which was recently demonstrated experimentally [10].

In a typical PA scheme, pairs of colliding atoms are radiatively excited to form bound molecules in an excited electronic state. These molecules are either allowed to decay spontaneously to the the ground molecular state [5–10], resulting in a vibrational population distribution, or subjected to a second appropriately tuned laser, which stimulates a bound-bound transition to a single target vibrational level in the ground electronic state [11,12].

In previous work we have developed an exact time-dependent formalism for treating photodissociation (PD) [13–15] and PA [11] processes using strong pulses. This formalism aims at extending the stimulated Raman adiabatic passage (STIRAP) technique of Bergmann et al. [16–22s] to the case of an initial or final continua. It was shown that enhanced two-photon association of ultracold sodium atoms can be attained in an efficient way, thereby producing υ = 0, J = 0 translationally cold Na₂ molecules.

In this article we show, by applying the same formalism, that adiabatic passage via two-photon dissociation of sodium dimers and the reverse two-photon association process is possible in a molecular beam environment. It is shown that complete dissociation of specific quantum states, such as the υ = 28, J = 0 to 10 Na₂ molecules is achievable using nanosecond laser pulses no stronger than a few MW/cm², and that stimulated Raman photoassociation of sodium atoms in a beam (translational temperature of 5–10 K) can be utilized for the production of translationally cold molecules in specific vib-rotational levels. These findings prove that coherent population transfer is possible, even for a final (or an initial) continuum state.

The organization of this article is as follows: In section 2 we review the theory of two-photon dissociation and association by strong laser pulses and in section 3 we apply this formalism to the simulation of resonantly enhanced two-photon dissociation of sodium dimers and stimulated two-photon association of Na atoms.

2. Theory of Two Photon Dissociation and Association

2.1 The Slowly Varying Continuum Approximation

We consider a system with two bound molecular states ∣ 1⟩ and ∣ 2⟩, and a continuum of scattering states ∣ E, n ^±) (it is convenient to use “incoming” scattering states ∣ E, n ^-) for the dissociation problem and “outgoing” scattering states ∣ E, n ⁺) for the association process), subjected to the combined action of two laser pulses of central frequencies ω ₁ and ω ₂. We assume that ω ₁ is in near resonance with the bound-bound transition, ∣ 1⟩ ↔ ∣ 2⟩, and that ω ₂ is in near resonance with the bound-free transition, ∣ 2⟩ ↔ ∣E, n ^±). Depending on the initial state of the system and on the pulse configuration, molecules in the bound manifold can dissociate to the continuum, or colliding atoms initialy in the continuum, may associate to form a bound molecular states. The situation is depicted in Fig. 1 for a Λ-type configuration. Other configurations such as the ladder system, may be equally treated.

The total Hamiltonian of the system is written as

where H is the radiation-free Hamiltonian, ϵ ₁(t) and ϵ ₂(t) are “slowly varying” electric field amplitudes and ${\overrightarrow{\mu }}_1$ and ${\overrightarrow{\mu }}_2$ are (electronic) transition dipole operators. The material wave function of the system may be expanded as,

where

 

Figure 1. Energy levels of the resonantly-enhanced 2-photon dissociation and association schemes.

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Substitution of the expansion of Eq. (2) into the time-dependent Schrodinger equation, iħ∂Ψ/∂t = H_tot Ψ, and use of the orthogonality of the ∣1⟩, ∣2⟩ and ∣ E, n ^±) basis states, results in an (indenumerable) set of first-order differential equations for the expansion coefficients. In the rotating wave approximation, and neglecting the low amplitude inter-continuum transitions, this set of equations is of the form,

where N is the number of (asymptotically) open channels,

and

and we have assumed for simplicity that ⟨2∣μ ₁∣1⟩ ϵ ₁(t) and ϵ ₂(t) are real. In the above we ignored spontaneous emission from ∣ 2⟩, assuming that the pulse intensities are such that the stimulated emission rates are much faster than the spontaneous emission rates.

In order to obtain a unique solution for Eqs. (4)–(6), we need to specify initial conditions, denoted here as,

Substituting the formal solution of Eq. (6)

into Eq. (5), we obtain,

If the molecular continuum is unstructured, as in the present Na-Na system at threshold energies, where the bound-free dipole matrix-elements vary with energy by less than 1% over the (nsec) pulse bandwidth, we can invoke the slowly varying continuum approximation (SVCA) [13–15] and replace the energy-dependent bound-free dipole-matrix elements at energies spanning the laser profile by their value at the pulse center, given (in the Λ configuration of Fig. 1) as E_L = E ₂ - ħ ω ₂,

The use of the SVCA (whose range of validity has been thoroughly researched [14]) greatly simplifies the equations because upon substitution of Eq. (7) and Eq. (12) into Eq. (11) we can perform the integration over E and t′ analytically. We obtain that,

where

The source term F ₂(t) is given as,

where

2.2 The Adiabatic Approximation

Equation (13) and Eq. (4), can be expressed in matrix notation as,

where

and

and the initial conditions are obtained from Eq. (9) and Eq. (17) as,

As a first step towards solving Eq. (16) we can diagonalize the 𝖧 matrix,

thereby defining an adiabatic basis set. In the above, the eigenvalue matrix, ε̂, is given as,

The complex-orthogonal eigenvector matrix 𝖴, satisfying the equation,

can be parameterized in the 2 × 2 case in terms of a complex “mixing angle” θ [15],

where

Operating with 𝖴(t) on Eq. (16), and defining,

we obtain that,

where

is the “non-adiabatic” coupling matrix. The source-vector 𝗀 is given as,

The adiabatic approximation amounts to ignoring 𝖠. This can be done whenever the rate of change of 𝖴 with time is slow. Equation (27) then becomes,

with the initial condition that

The adiabatic solution for Eq. (30) with the initial conditions given in Eq. (31) is of the form

where

and

2.3 Adiabatic Two-Photon Dissociation

We first consider the case of PD from the bound manifold into the continuum. In PD the entire population is taken to belong initialy to the bound manifold, i.e. $b_{E,\mathbf{m}}^0$ = 0 for all E and m. As a result ${\bar{\mu }}_2$(t) of Eq. (15) is zero and g(t) vanishes for all times. The adiabatic solution of Eq. (32) now becomes

Using Eq. (26) and Eq. (17), we obtain for the b ₁(t) and b ₂(t) coefficients:

If only state ∣1⟩ is initialy populated we have that 𝖻₀ = (1,0) and

2.4 Adiabatic Two-Photon Association

In the PA process, the initial conditions are such that 𝖺₀ = 0 (the entire population is initialy in the continuum). Hence the adiabatic solutions of Eq. (32) are of the form,

or, using the definitions of Eq. (26) and Eq. (17),

Given b ₂(t), the (channel specific) continuum coefficients b _E,n(t) are obtained directly via Eq. (10).

3. Numerical Results

3.1 Photodissociation of Na₂ Molecules

The formalism of section 2 enables an easy computation of PD and PA processes. In this section, we study the pulsed two-photon dissociation of Na₂ molecules in characteristic molecular-beam conditions. In order to perform the calculation, the transition-dipole matrix elements of Eq. (7), obtained by solving the radial Schrödinger equation with known [23] Na₂ potential curves, need be computed only once for all pulse configurations.

We consider the pulsed PD of molecules in the X ¹${\sum }_g^{+}$ (υ = 28,J) state with J in the range of 0 to 10, to the (E, 3s +3s) continuum, with the bound A ¹${\sum }_u^{+}$ (υ′ = 37,J+1) state acting as an intermediate resonance. Given the ab-initio [23] electronic dipole-moments and potential curves of Fig. 2, the bound eigenfunctions and eigenenergies are obtained using the renormalized Numerov method [24]. The continuum wavefunctions are expressed, to excellent accuracy, in terms of the Uniform Airy functions [25–29]. The overlap integrals between the bound states are calculated using Simpson quadrature. The calculation of the bound-continuum matrix elements is performed with a high-order Gauss-Legendre quadrature.

 

Figure 2. Potentials and vibrational wave functions used in the simulation of the Na₂ 2-photon dissociation and the Na+Na 2-photon recombination.

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Bound-bound and bound-continuum transition-dipole matrix elements for various choices of υ′ are plotted in Fig. 3. Choosing υ′ = 37 for the intermediate state clearly maximizes the bound-free transition probability without compromising the bound-bound transitions.

Since the pulses used in our simulations typically last 5–10 nsec, their small bandwidth allows for the resolution of individual rotational levels. Transition-dipole matrix elements for J in the range of 0 to 10 are plotted in Fig. 4. Both vibrational states, which lie well below their respective dissociation thresholds, are hardly affected by the centrifugal barrier. As a result, the variation in the bound-bound transition matrix elements with J is less than 1%.

 

Figure 3. Bound-bound (below) and bound-continuum (above) transition-dipole matrix elements for various intermediate vibrational levels. The initial bound state is taken at υ = 28, J = 10 and the kinetic energy of the dissociated atoms is E = 5K.

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Figure 4. Bound-bound (below) and bound-continuum (above) transition-dipole matrix elements for various rotational transitions. The initial vibrational level is υ = 28, the intermediate level is υ′ = 37 and the kinetic energy of the dissociated atoms is E = 5K.

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In contrast, the radial continuum wavefunctions are more sensitive to the rotational quantum number, resulting in the variation of the bound-free dipole matrix elements with J by as much as 5%. These small variations are found, however, to have only a marginal effect on the overall population transfer probabilities, which are very insensitive to changes in the Rabi frequencies Ω₁ and Ω₂.

Having computed all the input matrix elements, the dynamical equations are solved using either the Runge Kutta Merson (RKM) algorithm for direct integration of the full non-adiabatic equation (Eq. (16)) or the adiabatic solutions of Eqs. (38) and (39). For pulse parameters of relevance to this work, the adiabatic solutions are found to be practically indistinguishable from the numerically-exact RKM solutions.

The results of a PD process conducted in a “counter-intuitive” fashion, in which the “dump” ϵ ₁(t) pulse is applied before the “pump” ϵ ₂(t) pulse, are shown in Fig. 5. As shown by Bergmann et al. [16–22] for bound-bound Λ-type systems, this configuration enables complete population transfer from the initial state to the final state without ever populating the intermediate-resonance. In Fig. 5 we show, in agreement with our previous work on two-photon dissociation [15], that, with judicious choice of pulse parameters, a “counter-intuitive” pulse sequence is capable of dissociating every molecule in our ensemble, while keeping at all times the intermediate state population low. In this way, losses due to spontaneous emission from the intermediate state are avoided. Thus, although not following a perfect adiabatic passage scenario, population transfer to the continuum is nevertheless adiabatic. [As pointed out above, the adiabatic solutions of Eq. (16) (Eq. (38) and Eq. (39)) are in perfect agreement with the RKM solutions].

 

Figure 5. Results of a “counter-intuitive” dissociative pulse sequence. Shown are the population of the υ = 28, J = 10 initial state, the population of the υ = 37, J = 11 intermediate state, and the final integrated population of continuum states, vs. time. Dotted lines are the intensity profiles of the two Gaussian pulses whose maximum intensities are 5 × 10⁴ W/cm² and 5 × 10⁶ W/cm² for ϵ ₁ and ϵ ₂ respectively. Both pulses last 8.5 nsec. The central frequency of the pump pulse is chosen in resonance with the bound-bound transition (Δ₁ = 0).

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In Fig. 6 the calculated dissociation probability is plotted as a function of the Δ₁ detuning, at three pulse intensities. The resulting symmetric lineshapes have widths that increase with pulse intensity. This power-broadening is related to the saturation of the bound-continuum transition at high intensities. The lineshapes of Fig. 6 are smooth functions of the detuning and the pulse intensity. We see that at large enough intensities, the two-photon dissociation yield of the present scheme is stable with respect to fluctuations in laser frequency and power.

 

Figure 6. Calculated dissociative lineshapes at I ₁ = 5 × 10⁴ W/cm², I ₂ = 5 × 10⁶ W/cm² (—), I ₁ = 2 × 10⁵ W/cm², I ₂ = 2 × 10⁷ W/cm² (┄), and I ₁ = 5 × 10⁵ W/cm², I ₂ = 5 × 10⁷ W/cm² (╴∙╴∙╴), where I ₁ and I ₂ stand for the peak intensity of ϵ ₁ and ϵ ₂ respectively. Pulse durations are as in Fig. 5.

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3.2 Photoassociation of a Coherent Na+Na Wavepacket

We now turn to the reverse process: the pulsed photoassociation (PA) of a coherent wavepacket of cold Na atoms. In this process the initial wavefunction is described by a moving Gaussian wave packet composed of of radial waves:

where

and t ₀ is the peak time of the Na+Na wave packet (i.e. the time of maximum overlap with the ∣ 2⟩ state). In our simulations we have chosen the mean initial collision energies to be E ₀ = 1 - 10K and wave packet widths δ_E = 10^-4 - 10^-3cm^-1. Radial waves with J in the range of 0 to 10 were considered, keeping in mind that individual rotational transitions could be resolved due to the energetic-narrowness of the wavepacket and laser profiles.

As depicted in Fig. 1, the combined effect of the two laser pulses of central frequencies ω ₂ and ω ₁ (taken to be in resonance with the X ¹${\sum }_g^{+}$ (υ = 28,J) to A ¹${\sum }_u^{+}$ (υ′ = 37, J+1) transition), is the transfer of population from the continuum to a single vib-rotational state X ¹${\sum }_g^{+}$ (υ = 28, J), with the bound A ¹${\mathrm{\Sigma }}_u^{+}$ (υ′ = 37, J+1) state acting as an intermediate resonance.

It was shown in subsection 3.2 (Fig. 4) that the material transition-matrix elements of Eq. (7) are only slightly affected by the choice of initial radial wavefunctions. Therefore, the final PD population transfers are also insensitive to the initial state. We now examine the stability of the PA probabilities with respect to the initial state, i.e., the translational temperature of the atomic ensemble. Free-bound transition-dipole matrix elements for translational temperatures typical to the relative lateral motion within a Na beam of 1–10K, are plotted in Fig. 7. We find, as in the bound-bound case, that the variation of the free-bound dipole-matrix elements with collision energy, which over the 1–10K range of temperatures is of the order of 5%, has almost no effect on the population transfer efficiencies. In addition, as demonstrated in Fig. 7, the μ _2,E bound-continuum matrix elements are practically constant over the pulse spectral bandwidth (typically in the range of 5–10 mK), thus justifying the use of the SVCA of Eq. (12).

 

Figure 7. Bound-continuum PA transition-dipole matrix elements as a function of the initial collision temperature, for the J = 10 radial wave.

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Making use of the SVCA, we rewrite Eq. (15) as

and obtain from Eq. (44) and Eq. (45) that,

Choosing a pair of Gaussian pulses of the form

we can write Eq. (46) as,

The solutions of Eq. (16) are obtained using either the RKM algorithm or the PA adiabatic expressions of Eqs. (41) and (42). As in the PD calculations, adiabaticity is found to be maintained for the ${ϵ}_{1,2}^0$ and Δt _1,2 pulse parameters used in the calculations. In all the results presented here the adiabatic solutions are found to be virtually identical to the RKM numerical-solutions.

The photoassociative production of Na₂ molecules in the υ = 28, J = 10 level of the ground electronic state is shown in Fig. 8. Contrary to the PD process, in the PA process “counter-intuitive” pulse ordering means that the ϵ ₁(t) pulse, coupling the two bound states, is made to precede ϵ ₂(t) pulse, which couples the bound to the continuum states.

 

Figure 8. Results of the “counter-intuitive” associative pulse sequence. Symbols are the same as in Fig. 5. The maximum intensity of the dump pulse is 5 × 10⁴ W/cm² and that of the pump pulse is 6 × 10⁶ W/cm². Both pulses last 8.5 nsec. The pump pulse peaks at the peak of the Na+Na wave packet (t ₀ = 20 nsec) and the dump pulse peaks at 5 nsec before that time. Central frequencies are chosen so that Δ₁ = Δ_E0, = 0. The initial kinetic energy of the coherent wave packet is E ₀ = 5K and its bandwidth is δE = 10^-3 cm^-1.

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We find that population is transferred monotonically from the continuum into the “target” bound-state, almost without ever populating the intermediate (A ¹${\sum }_u^{+}$, υ′ = 37, J′ = 11) level. As in the adiabatic PD process, in this way the spontaneous emission from the intermediate state is eliminated, thus preventing the formation of molecules in vib-rotational states other than the the X ¹${\sum }_g^{+}$ (υ = 28, J = 10) level of choice. We observe that 98% of all J = 10 atom-pairs that collide during the pulse form υ = 28, J = 10 Na₂ molecules.

 

Figure 9. Cylinder of collisions with a given J, occurring when the PA pump pulse is on.

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Based on the above near unity association probability per pulse, we can estimate the PA yield in a thermal ensemble. In order to estimate the probability for a collision with a given J, we use the semiclassical relation between J and the impact parameter $b_J,b_J=\frac{ħ\left(J+\frac{1}{2}\right)}{\mathit{m\upsilon }}$, where m is the reduced mass of the collision pair and υ is their relative velocity. Due to the rotational selection rules for optical transitions, by tuning the laser central frequencies to a specific J → J ± 1 → J sequence, only those colliding pairs whose impact parameter lies between b _J-1 and b_J are affected by the laser. Hence all atoms contained in a cylinder (see Fig. 9) whose height is υΔt ₂ (where Δt ₂ is the duration of the pump pulse), and whose area is π($b_J^2$ - $b_{J-1}^2$) will be associated with a given atom. η_J (T) - the fraction of recombining atoms per pulse at temperature T is therefore given as,

where n is the number-density of Na atoms in the beam and δ_i,j is the Kronecker delta function. Taking the atom density and average lateral velocity in a typical Na atomic beam to be n = 10¹⁶ cm^-3 and υ = 1 × 10⁴ cm/sec (corresponding to a translational temperature of ~ 7K), and using a 20 nsec pump pulse, we find that the association yield to form J = 10 molecules is η ₁₀(7K) = 4 × 10^-3 per pulse. This means that with a 10Hz pulsed laser source we can recombine half of all the ensemble of Na atoms in about 15 seconds.

The above PA yields can be further increased using longer pulses. This is possible due to the existence of an exact scaling relations in Eq. (16). The initial wave packet energetic-width is inversely proportional to the size of the wavepacket, i.e., the effective Na-Na distance for which the association is complete. By scaling down the wavepacket’s energetic-width together with the pulse intensities as,

it is possible to scale up the duration of both pulses as,

since it follows from Eq. (48) and Eq. (14) that under these transformations,

and Eq. (16) becomes,

where 𝖻̄ denotes the vector of solutions of the scaled equations. We see that the scaled coefficients at time t are identical to the unscaled coefficients at times t/s. Thus, pulses’ durations can be made longer and their intensities concomitantly scaled down, without changing the final population-transfer yields. This behavior is demonstrated in Fig. 10 where pulse widths and intensities are scaled as above, with s = 10. It is evident that the resulting time evolution of the system is scaled up by a factor of 10, with the same final populations. As mentioned above, longer pulses increase the Na-Na distances for which collisions are effective in bringing about radiative recombination. Since the intermediate state population is low at all times, one needn’t worry about spontaneous emission losses when pulse durations are taken beyond the radiative lifetime of that state.

 

Figure 10. Results of a “counter-intuitive” associative pulse sequence. Symbols are the same as in Fig. 8. The maximum intensity of the dump pulse is 500 W/cm² and that of the pump pulse is 6 × 10⁵ W/cm². Both pulses last 85 nsec. The pump pulse peaks at the peak of the Na+Na wave packet (t ₀ = 200 nsec) and the dump pulse peaks at 50 nsec before that time. Central frequencies are chosen so that Δ₁ = Δ_E0 = 0. The initial kinetic energy of the coherent wave packet is E ₀ = 5K and its bandwidth is δE = 10^-4 cm^-1.

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4. Conclusions

We have established that complete population transfer to and from a continuum is achievable in cold thermal ensembles, using pulses of realistic durations and intensities. We have shown that, contrary to results based on a discretized quasi-continuum approach [30], the 3-level STIRAP technique may be extended to consider adiabatic passage into a final continuum, and that stimulated two photon association is an efficient mechanism for producing translationally cold molecules in specific vib-rotational states.

While an alternative “intuitive” approach may be taken in both processes [11], the “counter-intuitive” pulse scheme has the advantage of maintaining a low intermediate state population, thus minimizing spontaneous emission losses. Since the final population distribution is unaltered when longer pulses (of lower intensities) are used, this means that higher ensemble yields may be obtained from the “counter-intuitive” scheme, by increasing the pulse duration [11].

We have performed detailed calculations for the PA of cold Na atoms to produce X ¹${\sum }_g^{+}$(υ = 28, J = 10) molecules. Due to more favorable transition dipole matrix elements, the required intensities for a given pulse duration are almost two orders of magnitude lower than those required to produce X ¹${\sum }_g^{+}$ (υ = 0, J = 0) ultracold molecules, calculated in our previous work [11]. In addition, the molecular beam number-densities used in this work are four to five orders of magnitude higher than the those available in a magneto-optical trap (MOT). The resulting 4 × 10^-3 PA yield per 20 nsec pulse is three orders of magnitude higher than the efficiency we obtained for PA in a MOT.

Given the low intensities required in this work, a two-step scheme may be devised for the production of cold Na₂ molecules in the ground X ¹${\sum }_g^{+}$ (υ = 0, J = 0) state. Starting from a translationally cold ensemble of Na atoms, the first stage is the two-photon association process outlined in this article. Once enough X ¹${\sum }_g^{+}$ (υ = 28, J = 0) molecules are formed, a second 3-level STIRAP stage may be employed to transfer molecules from the X ¹${\sum }_g^{+}$ (υ = 28, J = 0) level into the groundX ¹${\sum }_g^{+}$(v = 0,J = 0) state. This four-photon, two-step approach is admittedly more complex from an experimental point of view than the two-photon one-step scheme [11] of our previous work. Nevertheless, the gained efficiency may compensate for the increase in experimental complexity.