A general approach to state-dependent optical tweezer traps for polar molecules

State-dependent optical tweezers can be used to trap a pair of molecules with a separation much smaller than the wavelength of the trapping light, greatly enhancing the dipole-dipole interaction between them. Here we describe a general approach to producing these state-dependent potentials using the tensor part of the ac Stark shift. The method is applicable to broad classes of molecules including bialkali molecules produced by atom association and those amenable to direct laser cooling.

An array of polar molecules formed using optical tweezer traps is an attractive platform for quantum simulation, quantum information processing, and the study and control of collisions and chemical reactions at ultracold temperatures. Long range electric dipole-dipole interactions between the molecules can be used to implement quantum gates [1][2][3][4][5][6][7] and a wide variety of Hamiltonians that model interesting many-body quantum systems [8][9][10][11][12][13][14]. Tweezer arrays are reconfigurable [15,16], so arbitrary geometries can be formed, defects can be removed or deliberately implanted, and interactions targeted to selected molecules by bringing them close to each other. There has been rapid recent progress in realizing such molecular arrays. Single NaCs molecules have been formed by association of Na and Cs atoms inside a tweezer trap [17][18][19], and single molecules of 85 Rb 87 Rb have been formed in a tweezer by microwave association assisted by coupling of the spin and motional degrees of freedom [20]. A tweezer array of laser-cooled CaF has been demonstrated and used to study collisions between pairs of molecules in selected states [21,22].
Recently, we introduced the idea of a state-dependent tweezer trap which confines two molecules in the same tweezer with controllable spacing on a sub-wavelength scale [7]. The method applies to molecules with electron spin, such as those produced by direct laser cooling, and relies on the interaction between the vector polarizability of the molecule and the polarization gradients around the focus of the tweezer. The tight focussing produces an elliptically polarized field with opposite handedness on either side of the focus. The resulting vector Stark shift depends on this handedness and on the orientation of the molecule's spin, so two molecules with opposite spin projections are trapped at different locations. Their separation can be controlled by superimposing another tweezer at a different wavelength. This method can enhance the dipole-dipole interaction by two orders of magnitude, enabling much faster and more robust gate operations. The scheme is ultimately limited by photon scattering which is higher than in a standard tweezer because one wavelength has to be tuned between the fine-structure components of an excited state, resulting in a comparatively small detuning, especially for light molecules. Furthermore, the scheme fails for spinless molecules, which is a major shortcoming since all molecules associated from ultracold alkali atoms are spinless in their ground states.
Here, we present a state-dependent tweezer based on the tensor Stark shift, which is large for most molecules. We produce two separated traps for molecules in two different states by using tweezers at two wavelengths, one where the scalar shift dominates and the other where the tensor shift is dominant. The sub-wavelength separation of the molecules is controlled through the separation and relative intensities of the tweezers. The scheme works for molecules with or without electron spin, greatly reduces the photon scattering rate, and enhances the dipoledipole interaction by a factor 100 or more. We show how this can be used to implement fast two-qubit gates using electric-field-induced dipoles or resonant dipoledipole interactions, and discuss the implementation of these ideas using NaCs and CaF molecules. We note that other methods for producing sub-wavelength potentials for molecules have recently been studied [23].
Method. To illustrate our scheme, we consider a simple 1 Σ molecule with dipole moment µ e , rotational constant B and no hyperfine structure. Figure 1(a) shows the Stark shift of the electronic ground state in an applied electric field along z, E z . The electric field mixes states with different rotational quantum number J but equal projection onto the z axis m J . We label the resulting states |J , m J where J is the rotational level the state connects to at zero field. In the absence of other fields, the states |J , m J and |J , −m J are degenerate and we refer to these pairs as an |m J | manifold. Consider the response of this molecule to a light field with polarization and intensity I. We assume that the dc Stark shift is much larger than the ac Stark shift. The molecule-light interaction has scalar, vector and tensor parts whose dependence on the frequency of the light can be factored out into three constants α (0) , α (1) and α (2) ; the scalar, vector and tensor polarizabilities. The scalar part of the interaction shifts every state by W 0 = − 1 2 0c α (0) I. In the absence of any electronic or nuclear spin, α (1) = 0 so there is no vector part. The tensor part causes the state-dependent shifts which we use in our scheme.
The values of α (0) and α (2) depend on the electronic structure of the molecule. Here, for simplicity, we assume they are dominated by a single rovibrational level from each of the first two electronic excited states -a Σ state and a Π state with energies ω Σ and ω Π above the ground state, as shown in Fig 1(b). If we define ∆, the detuning of the light from the midpoint of the two states, the polarizabilities can be written Here, d Σ and d Π are the transition dipole moments between the ground state and excited states, δ ΠΣ = ω Π −ω Σ and we have assumed |∆|, |δ ΠΣ | ω Π , ω Σ . Figure 1(c) shows α (0) and α (2) as a function of ∆ in this limit.
Our scheme uses optical tweezers at two different wavelengths. The first, which we call the scalar tweezer, has intensity I sc and wavelength λ sc such that ∆ = ∆ sc is negative with |∆ sc | > 2|δ ΠΣ |. Under these conditions, |α (0) | |α (2) |, so the scalar part of the interaction dominates and the shifts are nearly independent ofˆ . The second tweezer, which we call the tensor tweezer, has intensity I tens and wavelength λ tens such that Here whose magnitude can be large and whose sign is negative (positive) when the Π state lies above (below) the Σ state. The resulting state-dependent shifts depend on the polarization of the tensor tweezer. We choose this to be linearly polarized along either x or y. The diagonal matrix elements of the molecule-light interaction are equal for the two states of an |m J | manifold. The off-diagonal part can couple pairs of states with m J − m J = 2. Due to the electric field, the different |m J | manifolds are well separated, and the only states coupled by the light are pairs with |m J | = 1. These pairs are maximally mixed to give the two states |J , ± = 1 √ 2 (|J , 1 ± |J , −1 ) with energies E J ,+ and E J ,− . Forˆ along x (along y) and α (2) < 0, E J ,− lies above (below). The energy difference E J ,+ − E J ,− is proportional to I tens , as shown in the inset of Fig. 1(a), and is similar for all values of J . The off-diagonal matrix element is larger than the diagonal one so that, for each polarization, one state is trapped in the tensor tweezer while the other is repelled. A more complete treatment of the excited states changes the polarizability quantitatively but does not change the key principles essential to our scheme. Importantly, because the tensor tweezer is tuned between different electronic levels rather than between fine-structure components of the same state as in Ref. [7], the photon scattering rate R ph can be much lower for a given trap depth.
Two molecules can be brought to sub-wavelength separations using combinations of scalar and tensor traps. This reduced separation increases the dipole-dipole interaction between them, allowing faster two-qubit operations. We consider two schemes for controlling the separation r of a pair of molecules, one prepared in |J , + and the other in |J , − , and show how to implement two-qubit gates in both cases. In one scheme the two trap depths are different, while in the other they are the same. Throughout, we assume the molecules are cooled to the motional ground states of their traps [24].
Asymmetric traps. The first scheme is illustrated in the left hand column of Fig. 2. The |J , + molecule is loaded into a scalar trap and the |J , − molecule into a tensor trap withˆ along x. The two tweezers are brought together until the maximum intensity gradient from the tensor tweezer coincides with the center of the scalar tweezer. This can be done without tunneling between sites by ensuring the tensor tweezer is significantly deeper than the scalar tweezer. Once they reach this position, the separation between the potential minima for the two states is controlled by varying the relative intensity of the two tweezers. This scheme is suitable for implementing a gate that uses electric dipole moments induced by a static electric field [1]. As an example, we take |1, ± and |2, ± as our computational basis states. Their dipole moments, µ 1 and µ 2 , can be very different at electric fields that are quite easy to reach for most polar molecules. For example, when E z = 6B/µ e (dashed line in Fig. 1(a)), δµ = µ 1 − µ 2 = 0.43µ e . Single qubit operations are driven by microwaves polarized along z. Because the two trap depths are different, and the frequency of the transition |1, + ↔ |2, + has a different gradient with intensity from that of |1, − ↔ |2, − , the two qubits can be addressed separately. A CNOT gate can be implemented by driving a microwave transition resonant with |1, + |1, − ↔ |2, + |1, − with sufficient duration to resolve it from |1, + |2, − ↔ |2, + |2, − . Their frequencies differ by δµ 2 /4π 0 |r 3 | due to the dipole-dipole interaction.
Symmetric traps. The second scheme for reaching sub-wavelength separation is illustrated in the right hand column of Fig. 2. The |J , − molecule is loaded into a tensor tweezer with polarization along x and the |J , + molecule into another tensor tweezer with polarization along y. The two tweezers are brought together until they are separated by ∼ λ tens such that the maximum gradients of the two potentials coincide [25]. We then turn on a scalar tweezer located at the midpoint of the two tensor tweezers which squeezes the two molecules together. The separation can be tuned by varying the relative intensity of the scalar and tensor tweezers.
Two-qubit gates using permanent dipole moments can be carried out in the same way as described above. In addition, due to the symmetry of this scheme, the resonant dipole-dipole interaction can be utilized. As before, let us take |1, ± and |2, ± as our computational basis. If the scalar tweezer has no tensor contribution, the transitions |1, − ↔ |2, − and |1, + ↔ |2, + are exactly degenerate when the intensities of the two tensor tweezers are equal. For realistic parameters the scalar tweezer has a small tensor component, but the degeneracy can be recovered by tuning the relative intensity of the two tensor tweezers to cancel its contribution. The two-qubit states |1, − |2, + and |2, − |1, + are mixed by the dipole-dipole interaction and two-qubit gates can be carried out by driving transitions which resolve the energy splitting between the resultant pair of entangled states. In this case, the electric field need only be large enough to dominate over the interaction with the light fields. In practice, it can be useful to use a third manifold, say |3, ± . Then, |1, ± and |3, ± form the computational basis and for small E z have negligible transition dipole moment between them, so are (very nearly) stationary states, while |2, ± is used to implement the gate as described above. Suitable gate protocols are described in detail in Refs. [4,7]. To carry out single qubit operations, the intensities of the two tensor tweezers can be unbalanced so that the individual qubit frequencies can be resolved.
Minimum separation. The schemes illustrated in Fig. 2 can bring the molecules arbitrarily close together, but eventually their wavefunctions overlap and there are inelastic collisions. We consider tweezers with radial and axial trap frequencies of ω r and ω z and associated harmonic oscillator lengths of r 0 and z 0 . For unpolarized molecules the long range interaction, −C 6 /r 6 , has a negligible influence on the motional wavefunctions as long as r is a few times greater than a critical distance r c = (2C 6 / ω r ) 1/6 . For most cases of interest r c < 50 nm. In this case, the collision rate is where k 2 is the inelastic rate coefficient. Although k 2 can vary greatly from one system to another, several experiments (e.g. [22,[26][27][28]) have measured values close to that predicted by a simple single channel model with unity probability of loss at short range. For distinguishable particles near zero energy, this universal loss coefficient is [29] k 2 = 8π 2 Γ(1/4) 2 where m red is the reduced mass. For molecules with substantial dipole moments, C 6 is dominated by a rotational contribution giving C 6 ≈ 1/(4π 0 ) 2 × µ 4 e /(6B) [30,31]. Taking characteristic values of µ e = 3 D, B/h = 5 GHz, m red = 50 amu, ω r /2π = 100 kHz and ω z /2π = 23 kHz, Eqns. (4) and (5) give R col = 4 s −1 at r = 200 nm, rising to 270 s −1 at 150 nm. For separations in this range, the dipole-dipole interaction is enhanced by a factor of roughly 100 relative to normal tweezers. An electric field that induces aligned dipoles sets up a repulsive barrier that can greatly suppress k 2 [30,32,33], so that smaller values of r can be used to generate even larger enhancements of the dipole-dipole interaction.
Implementation with real molecules. The nuclear and electronic spins in real molecules couple to the rotational motion, complicating the structure. However, in a modest electric field the Stark interaction dominates over spin-rotation interactions so the rotational motion decouples from the spins. In this limit, we label the states |J , m J , G, m G where G and m G are the quantum numbers for the total spin and its projection onto the z axis. In the absence of light, |J , m J , G, m G and |J , −m J , G, −m G are degenerate. For |m J | = 1 manifolds, the tensor Stark shift mixes |J , m J , G, m G and |J , −m J , G, m G which are not degenerate in general. To obtain the linear shifts used in our scheme we either need the tensor light shifts to dominate over spin-rotation couplings, or need to take m G = 0. Below we discuss an example of each case. We note that although hyperfine structure adds complexity, it can also be useful. For example, the two qubits can use different spin manifolds so that the qubit frequencies differ due to the J -dependence of the hyperfine splittings. This does not change the dipole-dipole interaction used for the gates, but is helpful for single-qubit addressability.
Let us now consider how to implement the asymmetric trap scheme using 23 Na 133 Cs molecules, which have recently been prepared in tweezer traps [17][18][19]. The tensor Stark shifts of the |J = 1, m J = ±1 manifold are shown in Fig. 3(a). The manifold is split into 64 states due to the interaction of the nuclear spins (I Na = 3 2 , I Cs = 7 2 ) with each other and with the rotational motion. The splittings are small, typically 0.1−10 kHz, and I tens is large enough to separate the levels out into two groups of 32 states which behave identically to the |J , ± states described above. Each has at least one strong dipole-allowed transition to a state in the |J + 1, ± manifold. Many rovibrational levels of each excited electronic state contribute to the polarizability. To avoid high R ph , the light must be tuned to a place where the transition strengths to nearby excited states is negligible. In the Supplemental Material [34] we calculate that a suitable detuning exists at around 768 nm, between the A 1 Σ + and B 1 Π states. We estimateα (2) = α (2) /2 0 hc −8.4 × 10 −3 Hz W −1 m 2 , α (0) 0, and R ph 0.1 s −1 per µK of trap depth.
Next we consider the implementation of the symmetric trap scheme using 40 Ca 19 F, which has also recently been trapped in tweezers [21,22]. Figure 3(b) shows the ten- sor Stark shifts of the |J = 1, m J = ±1 manifold. The interaction of the electronic spin S = 1 2 and nuclear spin I F = 1 2 with each other and with the rotational motion splits each |m J | = 1 manifold into 8 states. The splittings are of order 10 MHz, large enough that, to a good approximation, the tweezer light does not mix different hyperfine states. In this limit, the only degenerate pairs that split linearly with I tens are those with m G = 0. There are two such pairs within each |m J | = 1 manifold, and they are suitable for our scheme. In CaF, the near diagonal Franck-Condon factors to the A 2 Π and B 2 Σ states almost realise the ideal case of a single vibrational contribution to the polarizability from each electronic state. A suitable wavelength for the tensor tweezer is λ tens 545 nm [34]. Here,α (2) 6.3 × 10 −3 Hz W −1 m 2 , α (0) 0, and R ph 0.1 s −1 per µK of trap depth.
Summary. We have presented a general approach to building state-dependent tweezer traps for molecules based on their large tensor Stark shifts, and shown how they can be used to control the separation of molecules on a sub-wavelength scale. These ideas should work for a wide variety of molecules, and we have described how they can be implemented in practice for the 1 Σ and 2 Σ molecules already prepared in tweezers. The methods will be useful for studying collisions and reactions between molecules with control over all degrees of freedom including the wavefunction overlap. The reduced separation increases the dipole-dipole interaction between the molecules by at least a factor 100, bringing typical interaction strengths above 100 kHz. The enhancement can be even greater if an electric field is used to suppress inelastic collisions. Photon scattering rates in our scheme are not much larger than for separated tweezers and are unlikely to be a limitation. The enhanced dipolar inter-actions will transform the utility of polar molecules for quantum information processing and the exploration of many-body quantum systems.
We are grateful to Jeremy Hutson and Simon Cornish for helpful discussions. This work was supported by EP-SRC grant EP/P01058X/1. that it is possible to find wavelengths with very low scattering rate between them.

B. CaF
For CaF we assume that the polarizability and scattering rate are dominated by the A 2 Π state and the B 2 Σ + state.
We use the spectroscopic constants from Ref. [36] and the transition dipole moments from Refs. [37,38]. The near-diagonal Franck-Condon factors from the ground state to each of these excited electronic states mean that it is only necessary to include the first three vibrational levels of each electronic state in the summation. At 545.08 nm we find α (0) = 4.3 × 10 −5 Hz W −1 cm 2 ,α (2) = 63 Hz W −1 cm 2 and R ph = 7.9 × 10 −5 s −1 W −1 cm 2 .