### Microwave setup

The microwave setup is described intimately in ref. ^{11}. We use a dual-feed waveguide antenna able to synthesizing arbitrary polarization utilizing two unbiased controllable feeds. The finite ellipticity and interference between these feeds lead to an noticed change in Rabi frequency of roughly ±4% when adjusting the relative section, thereby contributing to the systematic uncertainty.

By way of the management electronics, we now have upgraded the amplifiers to 100 W (Qualwave QPA-5600-5800-18-47, the gate voltage of which is supplied by a custom-made linear energy provide) to attain larger Rabi frequencies. Furthermore, we now have applied filter cavities to suppress section noise. Moreover, we now have included a voltage-controlled section shifter, enabling dynamic management of the relative section between the 2 feeds for fine-tuning the microwave ellipticity. To take care of a continuing output energy whereas adjusting the ellipticity, we monitor the facility in every feed utilizing an influence detector and use a suggestions management utilizing a voltage-controlled attenuator.

### Dimer loss close to the field-linked resonance

We experimentally map out the field-linked resonance by measuring the dimer loss. Prolonged Knowledge Fig. 1 exhibits the remained dimer quantity after a 100-ms maintain time at *Ω* = 2π × 29(1) MHz, *Δ* = 2π × 9.5 MHz, as a perform of ellipticity *ξ*. The loss dip place matches the theoretical resonance place *ξ* = 4.8°.

### Situations for environment friendly electroassociation

We experimentally determine the optimum situation for electroassociation. We acquire the tetramer quantity from the distinction between photos with and with out the tetramer removing course of outlined beforehand. First, we probe the timescale of the tetramer formation. We ramp the ellipticity from *ξ* = 0(1)° to eight(1)° and differ the ramp pace. As proven in Prolonged Knowledge Fig. 2a, we observe the formation of tetramers inside 0.3(1) ms and subsequently decay due to the finite lifetime. We estimate that the tetramers scatter on common greater than as soon as throughout the affiliation, bringing them near thermal equilibrium with the remaining dimers.

Subsequent, we examine the function of quantum degeneracy in environment friendly electroassociation. For magnetoassociation of Feshbach molecules, it has been proven {that a} low entropy pattern is essential to attain excessive conversion effectivity, due to the improved section–house overlap between the atoms^{49}. Right here we differ the degeneracy of our preliminary dimer samples by altering the ultimate entice depth of the evaporation^{42}. We observe a rise within the conversion effectivity *η*, that’s, the fraction of dimers transformed into tetramers, with quantum degeneracy of the dimer gasoline. We obtain a most *η* = 25(2)% conversion effectivity at *T* = 0.44(1)*T*_{F}. Much like that for magnetoassociation^{49}, a most unity conversion effectivity is anticipated at zero temperature.

### Affiliation spectra evaluation

We decide the binding power of the tetramers for various goal ellipticities (Fig. 2b) and discover wonderful settlement between the experimental information and coupled-channel calculations with out free parameters.

We assume the dimer loss within the modulation spectra to be proportional to the variety of fashioned tetramers. The road form may be modelled utilizing Fermi’s golden rule^{50}

$${N}_{{rm{T}}}(nu )propto {int }_{0}^{infty }{rm{d}}{{epsilon }}_{{rm{r}}}F({{epsilon }}_{{rm{r}}})g({{epsilon }}_{{rm{r}}}){{rm{e}}}^{-{(hnu -{E}_{{rm{b}}}-{{epsilon }}_{{rm{r}}})}^{2}/{sigma }^{2}}$$

(1)

the place *ν* is the modulation frequency and *E*_{b} is the binding power of the tetramer. The perform (g({{epsilon }}_{{rm{r}}})propto {{rm{e}}}^{-{{epsilon }}_{{rm{r}}}/{okay}_{{rm{B}}}T}) denotes the variety of colliding pairs per relative kinetic power interval d*ϵ*_{r}. Right here, the temperatures *T* are obtained from the information situated away from the affiliation transitions. The perform (F({{epsilon }}_{{rm{r}}})propto sqrt{{{epsilon }}_{{rm{r}}}}{(1+{{epsilon }}_{{rm{r}}}/{E}_{{rm{b}}})}^{-2}) denotes the Franck–Condon issue *F*(*ϵ*_{r}) between the unbound dimer state and the tetramer state, which we assume to take the identical type as for Feshbach molecules^{50}. The product *F*(*ϵ*_{r})*g*(*ϵ*_{r}) is convoluted with a Gaussian distribution with the width *σ* to account for the linewidth of the tetramer state and the finite power decision. The extracted linewidth exhibits the same pattern with ellipticity because the theoretical linewidth however barely bigger.

### Estimation of the elastic scattering charges

We estimate the elastic dipolar scattering charges of dimer–tetramer and tetramer–tetramer collisions. The scattering fee coefficient is given by *β* = *σ**v*, the place (v=sqrt{8{okay}_{{rm{B}}}T/{{uppi }}mu }) denotes the typical relative velocity and *σ* denotes the cross-section. Within the regime of huge dipole second (E > {hbar }^{6}/{mu }^{3}{d}_{1}^{2}{d}_{2}^{2}), the cross-section *σ* may be estimated utilizing the semiclassical system given by^{51}

$$sigma =frac{2}{3}frac{{d}_{1}{d}_{2}}{{{epsilon }}_{0}hbar }sqrt{frac{mu }{2E}}.$$

(2)

Right here *d*_{1} and *d*_{2} are the dipole moments of the 2 colliding particles, *μ* is the decreased mass and *E* is the kinetic power. We neglect the impact of a small ellipticity *ξ* and estimate the efficient dipole second of the dimers to be ({d}_{0}/sqrt{12(1+{(varDelta /varOmega )}^{2})}). The dipole second of tetramers is roughly twice as giant as that of dimers. With that, the above system gives an estimation for the elastic scattering charges to be 9.7 × 10^{−9} cm^{3} s^{−1} for dimer–tetramer and 1.9 × 10^{−8} cm^{3} s^{−1} for tetramer–tetramer. This suggests that tens of elastic collisions can happen throughout the lifetime of tetramers.

### Lifetime evaluation

For the measurements in time of flight, we confirm within the absence of tetramers that the two-body loss between dimers is negligible throughout the maintain time. Thus we match an exponential decay with a continuing offset given by the unpaired dimer quantity *N*(*t*) = 2*N*_{T}e^{−Γt} + *N*_{D}. The offset *N*_{D} is extracted from the information with ellipticity over 8°, during which the quantity undergoes a quick preliminary decay and stays fixed afterwards.

To analyze the collisional stability of tetramers, we additionally assess their lifetimes whereas the dipole entice stays energetic. Our observations point out a mixed one-body and two-body lack of the detected dimer quantity, and we affirm that the two-body loss arises from dimer–dimer collisions. Other than the information close to the collisional threshold *ξ* = 5(1)°, during which in-trap measurements are influenced by thermal dissociation, we don’t detect notable further lack of tetramers in in-trap measurements in contrast with these in time-of-flight experiments. The deduced inelastic collision charges are in line with zero throughout the error bar. We estimate that greater than ten elastic collisions can happen all through the lifetime of tetramers, which means that collisions with tetramers are predominantly elastic.

For measurements in a entice, we ramp up the entice depth by 50% concurrently with the affiliation, to compensate for the drive from the inhomogeneous microwave subject. The spatially various microwave adjustments the dressed state power, and thus exerts a drive on the molecules that lowers the entice depth and results in further loss within the trapped lifetime measurements.

We first measure the whole variety of tetramers and dimers, after which do a comparability measurement during which we take away the tetramers as described in the principle textual content. As proven in Prolonged Knowledge Fig. 3a, we observe a two-body decay within the dimer quantity, in distinction to the time-of-flight measurements. To account for this background loss, we first decide the two-body loss fee *Γ*_{2} and the preliminary dimer quantity *N*_{D,0} from the comparability measurement after which carry out a match of one-body plus two-body decay during which we repair *Γ*_{2} and *N*_{D,0}. The match perform is given by *N*_{D}(*t*) = 2*N*_{T,0}e^{−Γt} + *N*_{D,0}/(1 + *Γ*_{2}*t*). Prolonged Knowledge Fig. 3b,c exhibits that the tetramer decay in entice and in free house are comparable. The extracted decay charges differ by 9(9) × 10^{1} Hz, which we use to acquire an higher sure for the inelastic scattering fee coefficients. By assuming that the extra loss is both purely dimer–tetramer or tetramer–tetramer, we estimate the higher bounds of their inelastic collision fee coefficients to be 2(2) × 10^{−10} cm^{3} s^{−1} and 9(9) × 10^{−10} cm^{3} s^{−1}, respectively. Each values are in line with zero throughout the error bar. Even for the worst-case estimation, the inelastic collision fee coefficients stay orders of magnitude decrease than the estimated elastic dipolar scattering fee coefficients.

The lifetime of the long-range field-linked tetramers is for much longer than that noticed in polyatomic Feshbach molecules, that are both quick lived (<1 μs) (ref. ^{22}) or unstable within the presence of an optical entice^{37}. These options make them a promising candidate for realizing a BEC of polyatomic molecules. Utilizing the resonance at round polarization, the improved shielding will increase the tetramer lifetime to a whole lot of milliseconds. As our experiments recommend that they’re steady towards dimer–tetramer collisions, it appears promising to evaporatively cool tetramers to decrease temperatures^{52}.

### Affiliation timescale evaluation

We apply the next double-exponential match to the tetramer quantity as a perform of ramp time *t* in Prolonged Knowledge Fig. 2a

$${N}_{{rm{T}}}(t)={N}_{0}(1-{{rm{e}}}^{-t/tau }){{rm{e}}}^{-{t}_{{rm{T}}}/{tau }_{{rm{T}}}},$$

(3)

the place *τ* provides the timescale for affiliation and *τ*_{T} provides the timescale for tetramer decay. The time *t*_{T} ≈ 0.4(*t* + *t*_{disso}) is the time at which the ramp is above the field-linked resonance, which is a few issue of 0.4 of the affiliation time *t* and the dissociation time *t*_{disso} = 0.5 ms. We extract *τ* = 0.3(1) ms and *τ*_{T} = 2(1) ms.

### Hyperfine transitions within the modulation spectra

We observe the consequences of the hyperfine construction of NaK molecules within the modulation spectra. After we modulate the ellipticity of the microwave by section modulation, we generate two sidebands which can be offset from the service by the modulation frequency *ν*. When *ν* matches the ground- or excited-state hyperfine splitting of the dimer, a two-photon hyperfine transition happens. In Prolonged Knowledge Fig. 4b, we map out the transition spectrum by Landau–Zener sweeps, during which the modulation frequency is ramped from one information level to the subsequent. If a sweep is carried out over a hyperfine transition, molecules are transferred to a different hyperfine state inflicting a depletion of the detected variety of dimers. We observe three fundamental hyperfine transitions from 2 kHz to 200 kHz and some weaker ones. We confirm that these transitions aren’t affected by adjustments within the ellipticity, which confirms that they aren’t associated to the tetramer states. To acquire a transparent spectrum, when measuring the dissociation spectrum, we use a small modulation amplitude to attenuate energy broadening and be sure that we keep away from measuring close to these transitions.

### Tetramer dissociation spectrum evaluation

For modulation dissociation, we first create tetramers at *ξ* = 8(1)° utilizing electroassociation, then modulate the ellipticity for two ms to dissociate them. In the meantime, we flip off the entice to suppress additional affiliation of dimers. Afterwards, we take away the remaining tetramers and let the dissociated dimers develop for one more 6 ms earlier than absorption imaging.

Along with the hyperfine transitions talked about above, the affiliation of background dimers into tetramers additionally impacts the measurement of the dissociation spectrum. Nonetheless, it’s price noting that the affiliation spectra are significantly narrower than the dissociation spectrum, and their affect may be mitigated by utilizing a small modulation amplitude. To supply proof for this, we current a comparative measurement in Prolonged Knowledge Fig. 4a, performed below an identical experimental circumstances, besides that the ellipticity ramp is as quick as 0.5 μs in order that no tetramers are fashioned. Be aware that the modulation time is far shorter than for the affiliation spectra in Fig. 2a. The noticed fixed background on this measurement demonstrates that the frequencies at which we measure the dissociation spectrum stay unaffected by hyperfine transitions or affiliation.

We match the dissociation spectrum with a dissociation line form that’s just like the one used to explain the dissociation of Feshbach molecules^{39}

$${N}_{{rm{T}}}(nu )propto varTheta (nu -{E}_{{rm{b}}}/h)frac{sqrt{nu -{E}_{{rm{b}}}/h}}{{nu }^{2}+{gamma }^{2}/4},$$

(4)

the place *Θ*(*ν* − *E*_{b}/*h*) is the step perform and *γ* = 20(7) kHz accounts for the broadening of the sign.

### Imaging methodology for the dissociated tetramers

Right here we describe the measurement in Fig. 4b–d. We flip off the entice after the electroassociation and picture the cloud after 4.5 ms of growth time. To picture the molecules, we ramp the ellipticity again to round to quickly dissociate the tetramers in 0.3 ms, then flip off the microwave and reverse the stimulated Raman adiabatic passage to switch the dimers to the Feshbach molecule state. Lastly, we separate the sure atoms utilizing magnetodissociation, straight adopted by absorption imaging of the atoms to attenuate further cloud growth from residual launch power of the tetramer and Feshbach molecule dissociation.

### Angular distribution of the dissociation patterns

We common alongside the radial path of the dissociation patterns to acquire their angular distribution, as proven in Prolonged Knowledge Fig. 5. The distribution of the typical optical density exhibits a sinusoidal oscillation, which matches the *p*-wave symmetry. We extract the orientation angle *ϕ*_{0} by becoming a perform proportional to (1+ccos (2(widetilde{phi }-{phi }_{0}))), the place (widetilde{phi }) is the angle relative to the horizontal axis of the picture and *c* accounts for the finite distinction.

### Tetramer lifetime at round polarization

The lifetime of the tetramers may be improved by shifting the field-linked resonance in the direction of round polarization, during which the microwave shielding is extra environment friendly. With round polarization, two practically degenerate tetramer states emerge above the field-linked resonance at Rabi frequency *Ω* = 2π × 83 MHz and *Ω* = 2π × 85 MHz, which corresponds to the 2 *p*-wave channels with angular momentum projection *m* = 1 and *m* = −1, respectively, as proven in Prolonged Knowledge Fig. 6. For the *m* = 1 state, the lifetime at binding power *E*_{b} < *h* × 4 kHz exceeds 100 ms. As compared, we present the decay fee for *ξ* = 5° for which the resonance happens at *Ω* = 2π × 28 MHz. For a similar binding power, the lifetime is 10 instances shorter than that for the *m* = 1 state due to the smaller Rabi frequency.

### Rovibrational excitations of field-linked tetramers

We examine solely the primary field-linked sure state within the present experiment. At larger ellipticities and Rabi frequencies, the potential is deep sufficient to carry a couple of sure state, which corresponds to the rovibrational excitation of the tetramers. For vibrational (rotational) excitations, the radial (axial) wavefunction of the constituent dimers has a number of nodes^{53}. These excited field-linked states have extra complicated constructions, which may be probed equally with microwave-field modulation.

### Area-linked states of polyatomic molecules

Right here we talk about the applicability of field-linked resonances to complicated polyatomic molecules. For molecules during which the dipole second is orthogonal to one of many axes of inertia, the identical calculation may be carried out throughout the corresponding rotational subspace, as proven in ref. ^{10} for CaOH and SrOH. For extra complicated molecules during which the body-frame dipole second shouldn’t be orthogonal to any of the three axes of inertia, the microwave can induce the π transition between the bottom state and the *m*_{J} = 0 rotational excited state. Nonetheless, this detrimental π coupling may be suppressed by making use of a d.c. electrical subject to shift the *m*_{J} = 0 state away from the *m*_{J} = ±1 states, in order that the microwave may be off-resonant to the π transition, as proven in ref. ^{54}. With that, the same evaluation of field-linked resonances may be utilized.

### Concept

We apply coupled-channel calculations to check the scattering of molecules ruled by the Hamiltonian (widehat{H}=-{{nabla }}^{2}/M+{sum }_{j=1,2}{widehat{h}}_{{rm{in}}}(,j)+V({bf{r}})), the place the decreased Planck fixed *ħ* = 1.

The dynamics of a single molecule is described by the Hamiltonian ({widehat{h}}_{{rm{in}}}={B}_{{rm{rot}}}{{bf{J}}}^{2}+varOmega {{rm{e}}}^{-{rm{i}}{omega }_{0}t}left|{xi }_{+}rightrangle leftlangle 0,0right|/2+{rm{h.c.}}) with the rotational fixed *B*_{rot} = 2π × 2.822 GHz. Right here, we focus solely on the bottom rotational manifolds (*J* = 0 and 1) with the 4 states |*J*, *M*_{J}⟩ = |0, 0⟩, |1, 0⟩ and |1, ±1⟩, the place *M*_{J} denotes the projection of angular momentum with respect to the microwave wavevector. The microwave subject of frequency *ω*_{0} and the ellipticity angle *ξ* {couples} |0, 0⟩ and (| {xi }_{+}rangle equiv cos xi ,| 1,1rangle +sin xi ,| 1,-1rangle ) with the Rabi frequency *Ω*. Within the interplay image, the eigenstates of ({widehat{h}}_{{rm{in}}}) are (| 0rangle equiv | 1,0rangle ,| {xi }_{-}rangle equiv cos xi ,| 1,-1rangle -sin xi ,| 1,1rangle ,| ,+,rangle equiv u| 0,0rangle +v| {xi }_{+}rangle ) and (| ,-,rangle equiv u| {xi }_{+}rangle -v| 0,0rangle ), and the corresponding eigenenergies are ({E}_{0}={E}_{{xi }_{-}}=-varDelta ) and *E*_{±} = (−*Δ* ± *Ω*_{eff})/2, the place (u=sqrt{(1+varDelta /{varOmega }_{{rm{eff}}})/2}) and (v=sqrt{(1-varDelta /{varOmega }_{{rm{eff}}})/2}) with *Δ* > 0 being the blue detuning and ({varOmega }_{{rm{eff}}}=)(sqrt{{varDelta }^{2}+{varOmega }^{2}}) the efficient Rabi frequency.

The interplay of two molecules *V*(**r**) = *V*_{dd}(**r**) + *V*_{vdW}(**r**) comprises the dipolar interplay

$${V}_{{rm{dd}}}({bf{r}})=frac{{d}^{2}}{4{{uppi }}{{epsilon }}_{0}{r}^{3}}left[{widehat{{bf{d}}}}_{1}cdot {widehat{{bf{d}}}}_{2}-3({widehat{{bf{d}}}}_{1}cdot widehat{{bf{r}}})({widehat{{bf{d}}}}_{2}cdot widehat{{bf{r}}})right],$$

(5)

and the van der Waals interplay −*C*_{vdW}/*r*^{6} (*C*_{vdW} = 5 × 10^{5} arbitrary models; ref. ^{55}). We will mission the Schrödinger equation within the two-molecule symmetric subspace ({{mathcal{S}}}_{7}equiv { alpha rangle }_{alpha =1}^{7}={| +,+rangle , +,0rangle _{s},{| +,{xi }_{-}rangle }_{s},)( +,-rangle _{s}, -,0rangle _{s},{| -,{xi }_{-}rangle }_{s},| -,-rangle }) as ({sum }_{{alpha }^{{prime} }}{widehat{H}}_{alpha {alpha }^{{prime} }}{psi }_{{alpha }^{{prime} }}({bf{r}})=E{psi }_{alpha }({bf{r}})), the place ( i,jrangle _{s}=(| i,jrangle +)(| ,j,irangle )/sqrt{2}) is the symmetrization of (left|i,jrightrangle ). Beneath the rotating wave approximation, the Hamiltonian reads

$${widehat{H}}_{alpha {alpha }^{{prime} }}=left(-frac{{{nabla }}^{2}}{M}+{{mathcal{E}}}_{alpha }proper){delta }_{alpha {alpha }^{{prime} }}+{V}_{alpha {alpha }^{{prime} }}({bf{r}}),$$

(6)

the place ({{mathcal{E}}}_{alpha }={0,-frac{1}{2}(varDelta +{varOmega }_{{rm{eff}}}),-frac{1}{2}(varDelta +{varOmega }_{{rm{eff}}}),-{varOmega }_{{rm{eff}}},-frac{1}{2}(varDelta +3{varOmega }_{{rm{eff}}}),-frac{1}{2}(varDelta +3{varOmega }_{{rm{eff}}}),)(-2{varOmega }_{{rm{eff}}}}) are asymptotic energies of seven dressed states with respect to the very best dressed state channel (left|1rightrangle ) and ({V}_{alpha {alpha }^{{prime} }}({bf{r}})=leftlangle alpha proper|V({bf{r}})left|{alpha }^{{prime} }rightrangle ).

To acquire the binding power and the decay fee of the tetramer within the dressed state (left|1rightrangle ), we think about a pair of molecules with incident power ({{mathcal{E}}}_{2} < E < {{mathcal{E}}}_{1}), the angular momentum *l* and its projection *m* alongside the *z*-direction. We use the log-derivative methodology^{56} to numerically clear up the Schrödinger equation within the angular momentum foundation, that’s, ({psi }_{alpha }({bf{r}})={sum }_{lm}{psi }_{alpha lm}(r){Y}_{lm}(widehat{r})/r,) the place the loss induced by the formation of a four-body complicated is characterised utilizing the absorption boundary situation at *r*_{a} = 48.5*a*_{0}. By matching the numerical answer *ψ*_{αlm}(*r*) with the precise wavefunction within the asymptotic area *r* > *R*_{c}, we acquire the scattering amplitudes ({f}_{alpha lm}^{{alpha }^{{prime} }{l}^{{prime} }{m}^{{prime} }}) and the scattering cross sections ({sigma }_{alpha lm}^{{alpha }^{{prime} }{l}^{{prime} }{m}^{{prime} }}) from the channel (*α**l**m*) to the channel (({alpha }^{{prime} }{l}^{{prime} }{m}^{{prime} })). All outcomes are convergent for (*l*, ∣*m*∣) > 7 and *R*_{c} > 5 × 10^{4}*a*_{0}. We be aware {that a} completely different place of the absorption boundary (for instance, *r*_{a} = 32*a*_{0} and *r*_{a} = 64*a*_{0}) doesn’t have an effect on the end result as a result of the wavefunction has a negligible part contained in the shielding core.

With out lack of generality, we consider the cross-section ({sigma }_{210}^{210}) of the incident and outgoing molecules within the channel (210). When the incident power is resonant with the tetramer state, a peak seems within the cross-section ({sigma }_{210}^{210}), the place the width of the height is the decay fee of the tetramer. The cross-section ({sigma }_{210}^{210}) quantitatively agrees with the lineshape

$$sigma (E)=frac{2{{uppi }}}{{okay}_{2}^{2}}{left|{rm{i}}{g}^{2}G(E)+{S}_{{rm{bg}}}-1right|}^{2},$$

(7)

the place *G*(*E*) = 1/(*E* − *E*_{b} + i*Γ*/2) is the tetramer propagator, ({okay}_{2}=sqrt{M(E-{{mathcal{E}}}_{2})}) and *S*_{bg} are the incident momentum and the background scattering amplitude of molecules within the dressed state channel (left|2rightrangle ), respectively. By becoming ({sigma }_{210}^{210}) and *σ*(*E*), we acquire the binding power *E*_{b} and the decay fee *Γ* of the tetramer. We comment that for the incident and outgoing molecules in different channels *α* ≈ 3–7, the propagator *G*(*E*) in equation (7) doesn’t change. Due to this fact, becoming ({sigma }_{alpha lm}^{{alpha }^{{prime} }{l}^{{prime} }{m}^{{prime} }}) in a special scattering channel results in the identical binding power *E*_{b} and decay fee *Γ*.

For a tetramer with a small decay fee, its wavefunction *ψ*_{b}(**r**) may be obtained by fixing the Schrödinger equation ({H}_{{rm{eff}}}{psi }_{{rm{b}}}({bf{r}})={bar{E}}_{{rm{b}}}{psi }_{{rm{b}}}({bf{r}})). The one-channel mannequin *H*_{eff} = −*Δ*^{2}/*M* + *V*_{eff}(**r**) is decided by the efficient potential^{15}

$$start{array}{l}{V}_{{rm{eff}}}({bf{r}})=frac{{C}_{6}}{{r}^{6}}{sin }^{2}theta {1-{{mathcal{F}}}_{xi }^{2}(phi )+{[1-{{mathcal{F}}}_{xi }(phi )]}^{2}{cos }^{2}theta } ,+,frac{{C}_{3}}{{r}^{3}}[3{cos }^{2}theta -1+3{{mathcal{F}}}_{xi }(phi ){sin }^{2}theta ]finish{array}$$

(8)

for 2 molecules within the dressed state channel (left|1rightrangle ), the place ({{mathcal{F}}}_{xi }(phi ),=)(-sin 2xi cos 2phi ,theta ) and *ϕ* are the polar and azimuthal angles of **r**. The power ({C}_{3}={d}^{2}/left[48{{uppi }}{{epsilon }}_{0}(1+{delta }_{r}^{2})right]) of the dipole–dipole interplay relies upon solely on the relative detuning *δ*_{r} = ∣*Δ*∣/*Ω*, whereas the *C*_{6} time period describes an anisotropic shielding potential that stops harmful short-range collisions. Utilizing the B-spline algorithm, we acquire the binding power ({bar{E}}_{{rm{b}}}) and the wavefunction *ψ*_{b}(**r**) ≈ *Y*_{1−}(*r*)*φ*_{1}(*r*)/*r* of the primary tetramer sure state, the place ({Y}_{1-}(r)=({Y}_{11}(r)-{Y}_{1-1}(r))/sqrt{2}). The binding energies ({bar{E}}_{{rm{b}}}) and *E*_{b} obtained from the single-channel mannequin and the seven-channel scattering calculation agree with one another quantitatively for small *ξ* and *Ω*. For the biggest *ξ* and *Ω* in Fig. 1, the relative error of ({bar{E}}_{{rm{b}}}) is lower than 30%. The tetramer wavefunction within the momentum house is the Fourier remodel *ψ*_{b}(**okay**) = ∫*d***r**e^{−iokay·r}*ψ*_{b}(**r**)/(2π)^{3/2} of *ψ*_{b}(**r**).

For the modulation dissociation, the transition likelihood *p*_{okay} to the momentum state **okay** is decided by the coupling power ({g}_{{bf{okay}}}=int ,{rm{d}}{bf{r}}{psi }_{{bf{okay}}}^{* }({bf{r}}){partial }_{xi }{V}_{{rm{eff}}}({bf{r}}){psi }_{{rm{b}}}({bf{r}})). Right here, *ψ*_{okay}(**r**) represents the wavefunction of the scattering state. The coupling power *g*_{okay} is primarily influenced by ({Y}_{1-}(widehat{okay})), which characterizes the angular distribution of *ψ*_{b}(**okay**). This dominance arises as a result of ∂_{ξ}*V*_{eff} maintains mirror symmetry with respect to the *x*–*y* airplane. Due to this fact, by measuring ({p}_{{bf{okay}}}approx | {Y}_{1-}(widehat{okay}) ^{2}), we will successfully probe the angular dependence of the tetramer state within the momentum house.