Improving Efficiency of Sympathetic Cooling in Atom-Ion and Atom-Atom Confined Collisions
IImproving Efficiency of Sympathetic Cooling in Atom-Ion and Atom-Atom ConfinedCollisions
Vladimir S.Melezhik
1, 2, ∗ Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,Dubna, Moscow Region 141980, Russian Federation Dubna State University, 19 Universitetskaya Street,Dubna, Moscow Region 141982, Russian Federation (Dated: January 12, 2021)We propose a new way for sympathetic cooling of ions in an electromagnetic Paul trap: it impliesthe use for this purpose of cold buffer atoms in the region of atom-ion confinement-induced resonance(CIR). The problem is that the unavoidable micromotion of the ion and the long-range natureof its interaction with the environment of colder atoms in a hybrid atomic-ion trap prevent itssympathetic cooling. We show that the destructive effect of ion micromotion on its sympatheticcooling can however be suppressed in the vicinity of the atom-ion CIR. The origin of this is the”fermionization” of the atom-ion wave function near CIR, where the atom-ion pair behaves as a pairof noninteracting identical fermions. This prevents the complete approach of the atom with the ionnear resonance and does not enhance the ion micromotion, which interferes with its sympatheticcooling. We investigate the effect of sympathetic cooling around CIRs in atom-ion and atom-atomconfined collisions within the qusiclassical-quantum approach using the Li-Yb + and Li-Yb confinedsystems as an example. In this approach, the Schr¨odinger equation for a cold light atom is integratedsimultaneously with the classical Hamilton equations for a hotter heavy ion or atom during collision.We have found the region near the atom-ion CIR where the sympathetic cooling of the ion by coldatoms is possible in a hybrid atom-ion trap. We also show that it is possible to improve the efficiencyof sympathetic cooling in atomic traps by using atomic CIRs. PACS numbers: 32.60.+i,33.55.Be,32.10.Dk,33.80.Ps
I. INTRODUCTION
In the last decade there has been great interest in ul-tracold hybrid atom-ion systems, which is caused by newopportunities that arise here for control and simulationof various quantum processes and phenomena: formationof novel molecular states[1, 2], simulation of electron-phonon coupling in solid state physics [3], Feshbach reso-nances [4, 5], quantum information processing [6, 7] etc.[8]. However, a realization of the hot proposals with coldatoms and ions [8] is impeded by the unremovable ion mi-cromotion caused by the time-dependent radio frequency(RF) fields of the Paul traps used for confining ions inthe hybrid atom-ion systems [8–12]. Particularly, it isknown that the micromotion of the ion and the long-range nature of its interaction with the environment ofcolder atoms in a hybrid atomic-ion trap prevent the de-sired effect of sympathetic cooling of ions [8, 12]. De-spite the successes achieved in sympathetic cooling ofions in hybrid atomic-ion systems in the millikelvin rangeand above [8, 13–17], as well as the proposed promisingschemes for cooling to lower energies [7, 11, 12, 18–20],the problem of sympathetic cooling to lower energies inthese systems is still pending.In this paper, we propose a new way for sympatheticcooling of ions in an electromagnetic Paul trap: to ap- ∗ [email protected] ply for this purpose buffer cold atoms in the region ofthe atom-ion confinement-induced resonance (CIR). Weshow that the negative effect of micromotion on sympa-thetic ion cooling can be suppressed in the vicinity of theatom-ion CIR. Atom-ion CIRs were predicted in [21]and the influence of ion micromotion on the CIR posi-tion in Li-Yb + was investigated in the subsequent pa-per [22]. It was shown that the CIR occurs when the ra-tio of the transverse width of the atomic trap a ⊥ and thes-wave atom-ion scattering length in free space a s coin-cides with the value a ⊥ /a s = 1 .
46. Earlier, this conditionwas predicted [23] and subsequently confirmed in exper-iment [24] for atomic Cs waveguide-like traps. To de-scribe the dynamics of a quantum particle near CIR, the1D Fermi quasipotential with an effective coupling con-stant g D ( a ⊥ /a s ) proposed in [23] is successfully used.Atomic CIRs [23, 25–31] aroused great interest and stim-ulated research in this direction due to the possibility ofusing such resonances to tune effective interatomic in-teractions in a wide range - from super strong attraction g D → −∞ to super strong repulsion g D → + ∞ [24, 32–37]. It is also known that at the point a ⊥ /a s of CIR thedivergence of coupling constant g D ( a ⊥ /a s ) (and the to-tal reflection) leads to ”fermionization” of the relativewave-function of the colliding pair whose square modu-lus behaves the same as for two noninteracting identi-cal fermions [23, 34, 35, 37, 38]. This can lead to somecompensation of the long-range character of the atom-ion interaction and, as a consequence, to suppression ofthe micromotion-induced heating during collisions con- a r X i v : . [ phy s i c s . a t o m - ph ] J a n fined by the atom-ion trap. Here, we investigate howthe ”fermionization” can ”truncate” the effective atom-ion interaction and the possibility of using this effect forimproving the sympathetic cooling of the ions by buffercold atoms in hybrid atom-ion traps. We investigate theeffect of sympathetic cooling around CIRs in atom-ionconfined collisions within the qusiclassical-quantum ap-proach [22, 39–42] using the Li- Yb + pair in the hy-brid atom-ion trap as an example, which is currently un-der intense experimental investigations [18, 43, 44]. Itis assumed that this specific atom-ion pair is most per-spective for sympathetic cooling and reaching the s-waveregime of ions in Paul traps [12, 18]. The followingproblem is considered: a hot ion confined in a time-dependent RF Paul trap with linear geometry collideswith the cold atom constrained to move into a quasi-one-dimensional waveguide within the ion trap (see Fig. 1).In our approach [22, 45], the Schr¨odinger equation fora cold light atom is integrated simultaneously with theclassical Hamilton equations for hotter heavy ion duringcollision. We have found the regions near the atom-ionCIR where the sympathetic cooling of the ion by coldatoms is possible in a hybrid atom-ion trap. The possi-bility of sympathetic cooling of a heavy hot atom by lightcold atoms near atomic CIR is also investigated for thecase of a Li-Yb mixture confined by an atomic trap. Weshow that it is possible to improve the efficiency of sym-pathetic cooling in atomic traps by using atomic CIRs.In the next section, our theoretical approach and theprincipal elements of the computational scheme are dis-cussed. In Section III, the results and discussions arepresented. The concluding remarks are given in the lastsection. Some technical details of the computations arediscussed in the Appendix. II. PROBLEM FORMULATION ANDCOMPUTATIONAL SCHEME
A schematic view of the system under investigation isgiven in Fig. 1. A Li atom of cold atomic cloud confinedin the transverse direction by a harmonic potential of anoptical trap V ( r a ) = m a ω ⊥ (cid:0) x a + y a (cid:1) (1)collides with the Yb + ion in the potential created by thelinear RF Paul trap [18, 43, 44, 46] U ( r i , t ) = m i ω i (cid:18) z i − x i + y i (cid:19) + m i Ω rf q cos(Ω rf t ) (cid:18) y i − x i (cid:19) . (2)The interaction potentials (1),(2) of an atom and an ionwith a hybrid trap depend on the transverse frequency ofthe atomic trap ω ⊥ , which determines its transverse size a ⊥ = (cid:112) (cid:126) / ( m a ω ⊥ ), and the frequencies of the Paul trap +RF electrodes D.C. electrodes xy z FIG. 1: (color online) Schematic representation of the atom-ion system confined in a hybrid trap. Here, the ion is situatedin the cloud of cold atoms confined by an optical atomic trapinside the electromagnetic Paul trap. The time-dependent RFfield confines the ion transversally, whereas longitudinally astatic confinement is formed by the DC field. The dimensionsof the confinement region of the ion are determined by the fre-quencies of the Paul trap ω i and Ω rf . The atomic waveguidealong the longitudinal axis, z , of the linear Paul trap con-fines the atoms in the transverse x, y directions. The width ofthe atomic trap a ⊥ = (cid:112) (cid:126) / ( m a ω ⊥ ) is determined by the fre-quency ω ⊥ of the harmonic approximation for the trap shape.Inside the hybrid trap occur paired atom-ion collisions. ω i , Ω rf . Here, ω i = Ω rf (cid:112) a/ q and a are dimensionless geometric pa-rameters (i.e. a = 0 . (cid:28) q = 0 . < rf , causes RF oscilla-tions of the ion, i.e. sets its micromotion. The vectors r a and r i set the coordinates of the atom and the ion,and m a and m i are the masses of the atom and the ion,respectively. We assume that the axis of the waveguidein which, the colliding atom is travelling, is precisely the z -axis of the Paul trap (see Fig. 1). The origin is at thecenter of the Paul trap.In our work [22], the quantum-quaiclassical ap-proach [39–42] was extended and adapted for quantita-tive description of pair collisions of light slow Li atomswith heavy Yb + ions in the confined geometry of the hy-brid atom-ion trap defined by potentials (1),(2). In thisapproach, the problem is reduced to the simultaneousintegration of a system of coupled quantum and classi-cal equations: the time-dependent Schr¨odinger equation,that describes the collisional dynamics of an atom con-fined in an optical trap (1) with an ion, and the classicalHamilton equations, describing the vibrations of an ionin a Paul trap (2)and its perturbation during collisionwith an atom.The time-dependent Schr¨odinger equation for the wavefunction of the atom ψ ( r a , t ) is written in the form i (cid:126) ∂∂t ψ ( r a , t ) = [ − (cid:126) m a (cid:52) a + V ( r a )+ V ai ( | r a − r i ( t ) | )] ψ ( r a , t ) , (3)where the potential V ai defines the atom-ion interaction.The classical Hamiltonian describing an ion in a Paultrap is given by H trapi ( p i , r i , t ) = p i m i + U ( r i , t ) , (4)where p i ( t ) is the ion momentum. When the atom isconfined in the optical waveguide within the Paul trap,the ion experiences its presence via the atom-ion interac-tion V ai ( | r a − r i ( t ) | ) during the collision. Therefore, thefull classical ion Hamiltonian is given by H i ( p i , r i , t ; r a ) = H trapi ( p i , r i , t ) + (cid:104) V ai ( | r a − r i ( t ) | ) (cid:105) , (5)where (cid:104) V ai ( | r a − r i ( t ) | ) (cid:105) = (cid:104) ψ ( r a , t ) | V ai ( | r a − r i ( t ) | ) | ψ ( r a , t ) (cid:105) is the quantum mechanical average of the atom-ion inter-action over the atomic density instantaneous distribution | ψ ( r a , t ) | . Thus, the ion Hamiltonian (5) defined in sucha way has a parametric dependence on the atom position r a ( t ). It leads at the moment of the atom-ion collision tothe strong non-separability of the Hamilton equations ddt p i = − ∂∂ r i H i ( p i , r i , t ; r a ) ddt r i = ∂∂ p i H i ( p i , r i , t ; r a ) (6)describing the ion dynamics and its strong coupling withthe time-dependent Schr¨odinger equation (3). As a con-sequence, it requires sufficient stability of the computa-tional scheme (see Appendix).The set of classical equations (6)for the ion variables r i together with the Schr¨odinger equation (3) for theatomic wave function ψ ( r a , r i ( t )) form a complete setof dynamical equations for describing the atom-ion col-lision dynamics in a hybrid confining trap [22]. In thepresent study we consider collisions of a light atom witha much heavier ion in the range of very low atomic col-liding energies E coll (ultracold atoms), where the relation p a = √ m a E coll (cid:28) p i for their momentums is satisfied.In addition, we require that E i = p i / (2 m i ) (cid:29) (cid:126) ω i , whichfurther justifies the application of the classical descrip-tion for the ion.At the instant of collision ( r = | r a − r i ( t ) | → V ai ( r ) = − ( r − c )( r + c ) C ( r + b ) , (7) and decay into independent equations for an atom andan ion in the asymptotic region before and after the col-lision, where r → ∞ and V ai - interaction disappears: V ai ( r ) → − C /r →
0. The dispersion coefficient C isa known parameter for the Li-Yb + pair, and by vary-ing the free parameters b and c we change the inten-sity of the atomic-ion interaction, i.e. define the s-waveatomic-ion scattering length in free space a s , which canbe experimentally tuned. Thus, for tuning the inter-atomic interactions (the scattering length a s ) in atomictraps the magnetic Feshbach resonances are successfullyused [24, 37, 48]. The prospects for their use in hybridatomic-ion systems are also discussed [5, 18].To integrate simultaneously equations (3, 6), we needproper initial conditions with physical significance. Atthe beginning of the collisional process, the atom and theion are assumed to be far away from each other so thatthey do not interact ( V ai = 0). In particular, the atom isinitially in the ground state of the atomic trap with thelongitudinal colliding energy, that is, E coll (cid:28) (cid:126) ω ⊥ ψ ( r a , t = 0) = N φ ( ρ a ) e − ( za − z az e ikz a , (8)whereas the ion performs fast (with respect of atom mo-tion) oscillations in the Paul trap with mean transversal (cid:104) E ⊥ (cid:105) = (cid:104) E ix (cid:105) + (cid:104) E iy (cid:105) and longitudinal (cid:104) E (cid:107) (cid:105) = (cid:104) E iz (cid:105) en-ergies, which can be fixed by the proper choice of the ioninitial conditions r i ( t = 0) = r p i,x ( t = 0) = (cid:113) m i E (0) ix ,p i,y ( t = 0) = (cid:113) m i E (0) iy ,p i,z ( t = 0) = (cid:113) m i E (0) iz . (9)In Eq.(8) φ ( ρ a ) is the wave function of the ground state ofa two-dimensional harmonic oscillator approximating anatomic trap potential and N is the normalization coeffi-cient. The parameter a z specifies the width of the initialwave packet of the atom (8) in the longitudinal direction,which must be wide enough for the wave packet to be suf-ficiently monochromatic and its spreading in time couldbe neglected [22]. The initial position of the wave packet(8) is set by the value of z so that at the initial moment t = 0 the atomic-ion interaction V ai can be neglected.The initial conditions (8,9) set the initial state of anoninteracting atom-ion system: an ion performing a fi-nite motion in a Paul trap with given mean energies (cid:104) E (cid:107) (cid:105) and (cid:104) E ⊥ (cid:105) and a slow atom in the ground state φ ( ρ a ),which moves in the longitudinal direction of the opticaltrap with a velocity v a = (cid:126) k/m a = (cid:112) E coll /m a . Sincethe atom approaches the region of interaction with theion very slowly ( E coll / (cid:126) (cid:28) ω ⊥ (cid:28) ω i , Ω rf ), the initial po-sition of the ion does not influence the scattering processitself, which depends only on (cid:104) E ⊥ (cid:105) and (cid:104) E (cid:107) (cid:105) .As a result of the integration of the system of equations(3, 6), the wave packet ψ ( r a , t ) is calculated. Asymptot-ically it has the following behavior at t → + ∞ ψ ( r a , t ) −→ z a → + ∞ (1 + f + ) N φ ( ρ a ) χ ( z a − z ) e − ik f z a (10)in the asymptotic region r a → + ∞ , where f + ( a ⊥ /a s ) isthe forward scattering amplitude describing the atom-ioncollision confined by the hybrid trap. The longitudinalpart χ ( z a − z ) of the atomic wave packet describes theatom motion in z -direction, namely the spreading of theinitial Gaussian wave packet exp[ − ( z a − z )2 a z ] (8). Due tothe choice of the longitudinal width a z of the initial wavepacket rather large we achieve its sufficient monochro-maticity along z-direction k (cid:39) k f . This provides insignif-icant deformation of the envelope χ ( z a − z ) in (10) withrespect of its initial form in (8), what permits to calculatewith enough accuracy the scattering amplitude [22] (cid:104) ψ (0)+ ( t ) | ψ ( t ) (cid:105) −→ t → + ∞ f + ( k ) . (11)Here, the wave packet ψ (0)+ ( t ) is calculated by indepen-dent integration of the Schr¨odinger equation (3) with thesame initial conditions (8) and V ai = 0 [22], which hasduring the special choice of a z mentioned above the fol-lowing asymptotic behavior ψ (0)+ ( r a , t → + ∞ ) = N ϕ ( ρ a ) χ ( z a − z )) e ikz a . (12)The scattering amplitude determines the quasi-1Datomic-ion coupling constant by [23, 25] g D = lim k → (cid:126) km a Re [ f + ( k )] Im [ f + ( k )] . (13)The constant g D is the most relevant parameter foranalysing confined scattering close to a CIR, where g D → ±∞ [23, 25, 27, 29]. and the transmission T ( a ⊥ /a s ) = | f + ( a ⊥ /a s ) | → . (14)The trajectory of the ion r i ( t ), its momentum p i ( t )and kinetic energy E i ( t ) = p i ( t ) / (2 m i ) are also calcu-lated, which enter stable trajectories after colliding withan atom in the asymptotic region, if the ion does notleave the Paul trap after the collision. III. RESULTS AND DISCUSSIONA. Stability of ion motion near CIR in hybrid trap
To test the assumption about the possibility of sup-pressing micromotion-induced heating of an ion near the”fermionization” point a ⊥ /a s = 1 .
46, we first have con-sidered the collision of a cold Li atom with an
Yb ionat rest in the center of the hybrid trap (see Fig.1), i.e.with r = E (0) i = 0 in the initial conditions for ion (9),for different parameters of the effective atomic-ion inter-action by varying the ratio a ⊥ /a s . The ratio a ⊥ /a s was varied at a fixed width of atomic trap a ⊥ = (cid:112) (cid:126) / ( m a ω ⊥ )by changing the value of the atom-ion scattering length a s . This was achieved by varying the parameters b and c in the atom-ion interaction potential (7) at the in-terval 0 . R ∗ ≤ b ≤ . R ∗ and c ∼ . R ∗ . Here-after, we use the units of the problem: R ∗ = √ µC / (cid:126) , E ∗ = (cid:126) / (2 µR ∗ ), p ∗ = √ µE ∗ , ω ∗ = E ∗ / (cid:126) and t ⊥ =2 π/ω ⊥ , where µ is atom-ion reduced mass. The calcula-tions were performed for the atom initially in the groundstate with transversal E ⊥ = (cid:126) ω ⊥ = 0 . E ∗ and longitu-dinal colliding energy E coll /k B = 0 . E ∗ /k B = 11nKfor ω ⊥ = 0 . ω ∗ = 2 π × . rf = 2 π × ω i = 2 π × f + ( a ⊥ /a s , t ) (11), the transmission coeffi-cient T ( a ⊥ /a s , t ) (14), the effective coupling constant g D ( a ⊥ /a s , t ) (13) and the deviation of the Yb + ion fromthe center of the Paul trap r i ( t ) = (cid:113) x i ( t ) + y i ( t ) + z i ( t ) (15)are presented as a function of time for three differentvalues a ⊥ /a s .One can see that after the collision which occurs atthe time-interval t ⊥ (cid:46) t (cid:46) t ⊥ all the calculated scat-tering parameters and the value r i ( t ) reach stable regimeat the times t ∼ t ⊥ independently of the intensity ofthe atomic-ion interaction. The left graphs show the re-sults of calculations at a ⊥ /a s = 1 .
56 near the CIR, wherethe coupling constant diverges ( g D (cid:39) +65) and total re-flection is observed ( T → r i ( t ) as a result of thecollision and the absence of ion heating induced by mi-cromotion. The reason for this is the ”fermionization” ofthe relative atom-ion motion at the CIR point [37]: theatom-ion wave function is rearranged in such a way thatits modulus squared at small atom-ion distances repeatsthe modulus squared of the wave function of a pair of non-interacting fermions, i.e. the atom and the ion can notfully approach each other, what leads to T ( t → ∞ ) → V ai ( | r a − r i ( t ) | ) and, as a con-sequence, prevents the ion micromotion-induced heatingduring collision. Actually, here the deviation of the Yb + ion from the center of the Paul trap after the collisiondoes not exceed the value 5 × − R ∗ .The picture changes dramatically when we move fromthe resonance region around the CIR (region of ”fermion-ization”). It is illustrated by the central and right graphsin Fig.2. The central graphs illustrate the atom-ion col-lision with the repulsive atom-ion interaction V ai giv-ing the ratio a ⊥ /a s = 2 .
64 and the coupling constant g D = 1 .
82. We observe that outside of ”fermionization”the atom-ion interaction permits a closer approach be-tween the atom and ion during collision ( T ( t → ∞ ) (cid:39) .
1) and significant micromotion-induced heating of theion: the amplitude of deviation of the ion from the cen-ter of the Paul trap increases by more than an order
I m [ f + ( t ) ] a ^ / a s = 1 . 5 6 T ( t ) R e [ f + ( t ) ] I m [ f + ( t ) ]R e [ f + ( t ) ]T ( t ) a ^ / a s = 2 . 6 4 R e [ f + ( t ) ]I m [ f + ( t ) ]T ( t ) a ^ / a s = - 2 6 0
03 06 09 01 2 0 g1D(t) (units of 2E*R*) g = 6 5 g = 1 . 8 2 g1D(t) (units of 2E*R*) - 3- 2- 10 g1D(t) (units of 2E*R*) g = - 1 . 5 4 ri(t)/R* t / t ^ t / t ^ ri(t)/R* t / t ^ ri(t)/R* FIG. 2: (color online) The calculated atom scattering amplitude f + ( t ), transmission coefficient T ( t ), coupling constant g D ( t )and the ion deviation from the center of the trap r i ( t ) for the ion being initially at rest (i.e. with zero energy before the collisionwith the atom) for three different values of the ratio a ⊥ /a s . This parameter fixes the coupling constant g D (13) for threedifferent cases: resonant atom-ion repulsion near the atom-ion CIR (left panel: a ⊥ /a s = 1 .
56 at b = 0 . , c = 0 . a ⊥ /a s = 2 .
64 at b = c = 0 . a ⊥ /a s = 2 .
64 at b = 0 . , c = 0 . of magnitude in comparison with the case of CIR, upto the value ∼ × − R ∗ . The right graphs illus-trate the case of weak attraction between the atom andthe ion a ⊥ /a s = −
260 giving negative coupling constant g D = − .
54. Here, we also observe a considerable de-viation of the ion from the center of the Paul trap af-ter the collision, where r i ( t → ∞ ) approach the values ∼ × − R ∗ . This significant perturbation of the ionat close to zero atomic-ion scattering length a s → − a s → − V ai ( | r i − r a | ) → − C / | r i − r a | , which leadsto significant perturbation g D = − .
54 of the ion atthe moment of collision and to its noticeable heating as a result of the collision (see the time-dynamics of thequantity r i ( t ) in the right panel of Fig.2).We have to note the visible oscillations in the couplingconstant g D ( t ) and in the ion deviation from the cen-ter of the Paul trap r i ( t ) at a ⊥ /a s = 1 .
56 (i.e. nearCIR). It is noteworthy that their time period ∼ . t ⊥ =0 . π/ω ⊥ ) corresponds to the frequency 2 ω ⊥ of the vir-tual transitions between the input channel ( n = 0 , I ) tothe closed first excited channel ( n = 2 , I ), given by thetransverse oscillations of the atom in the optical trap (seeFig3). This is consistent with the physical interpretationof the CIR as a Feshbach-like resonance in the first closedtransverse channel [25]. Here, n and I define the quan-tum numbers of the atom in the atomic trap and the setof ion quantum numbers in the Paul trap correspond-ingly. When displaced from the resonance region (seethe central and right panels in Fig.2), these oscillationsof g D ( t ) and r i ( t ) disappear, since the resonance condi-tions between the transverse quantum states of the atomare violated (see Fig3). In this case, the higher-frequencyoscillations of the values g D ( t ) and r i ( t ) visible in thenonresonant cases are determined by the frequencies ω i and Ω rf of the Paul ion trap, which are much higher thanthe frequency of the atomic trap ω ⊥ .The performed analysis demonstrates the suppressionof micromotion-induced heating of an ion in a collisionwith a slow atom in the CIR region a ⊥ /a s (cid:39) . B. Improving efficiency of sympathetic cooling inatom-ion confined collisions near CIR
To analyze the possibility of improving sympatheticcooling of ions in hybrid atomic-ion traps (see Fig.1) withcold atoms, we have calculated the mean kinetic energyof the Yb + ion after collision with a cold Li atom insuch a trap (cid:104) E ( out ) i (cid:105) = 1 t max − t out (cid:90) t max t out E i ( t ) dt , (16)depending on the parameter a ⊥ /a s near CIR and out-side the resonant area. In the above formula, the ionenergy E i ( t ) = p i ( t ) / (2 m i ) was calculated by integrat-ing a coupled system of equations (3),(6). The limit ofintegration of the system of equations t max was chosenfrom the condition of the calculated parameters reachingstable values after the collision (see the previous subsec-tion), in the region t (cid:38) t ⊥ . The lower limit t out ofintegration in formula (16) was chosen in a similar way.In the calculation the following values t out = 9 t ⊥ and t max = 10 t ⊥ for these parameters were used.The initial conditions for a cold atom were chosen sim-ilarly to the previous subsection. The initial mean ionenergy (cid:104) E ( in ) i (cid:105) = 1 t in (cid:90) t in E i ( t ) dt = 0 . E ∗ , (17)was chosen significantly exceeding the longitudinal E coll = 0 . E ∗ as well as transverse E ⊥ = (cid:126) ω ⊥ =0 . E ∗ energy of the atom. The upper limit at calculat-ing the initial mean ion energy t in (cid:46) t ⊥ was chosen fromthe region before the collision, where an atom and an ionwere not interacting V ai ( | r a − r i ( t in ) | ) →
0. Here wehave considered two fundamentally different cases whichhowever correspond to the same mean initial energy ofthe ion (cid:104) E ( in ) i (cid:105) . FIG. 3: (color online) Schematic representation of the spec-trum of the atom-ion system confined in hybrid atom-ion trapas a function of the ratio a ⊥ /a s . The pair ( n, I ) indicates theatom quantum number n and the set of ion quantum numbers I . The point of the cross of the energy curve of the first ex-cited state with respect to the atomic motion (2 , I ) with thethreshold of the entrance channel of the system (0 , I ) definesthe position of the CIR on the a ⊥ /a s -axis. The first case: in the initial state, the ion has only onetransverse momentum component r i ( t = 0) = 0 p i,x ( t = 0) = (cid:113) m i E (0) ix ,p i,y ( t = 0) = 0 ,p i,z ( t = 0) = (cid:113) m i E (0) iz , (18)which leads to a “head-on collision” of an ion oscillatingin one xz plane with an incident atom moving along the zaxis. In our calculations E (0) ix = E (0) iz were chosen equal to0 . E ∗ , which gives the initial mean energy (cid:104) E ( in ) i (cid:105) =0 . E ∗ (cid:29) E ⊥ + E coll = 0 . E ∗ .The second case: not a head-on collision with the samemean initial ion energy, which was simulated by the fol-lowing initial conditions for the ion x i ( t = 0) = x i y i ( t = 0) = y i ,z i ( t = 0) = z i , p i ( t = 0) = 0 , (19)specifying its initial position as x i = z i = 0 . R ∗ , y i =0 . R ∗ . In this, case the initial condition generates 3Dmotion of the ion in the Paul trap before the collision.The results of calculations of the final mean ion en-ergy (cid:104) E ( out ) i (cid:105) (16) after the collision with the cold atomfor these two cases are given in Fig.4 as a functionof the ratio a ⊥ /a s . In this figure the calculated cou-pling constant g D (13), the transmission coefficient T (14) and the molecule formation probability P mol (de-fined below by Eq.(21)) corresponding to the same val-ues a ⊥ /a s are also presented. One can see that the cal-culated curves (cid:104) E ( out ) i ( a ⊥ /a s ) (cid:105) presented in Fig.4a have < E i( o u t ) ( e l ) >< E i( in ) > a ^ / a s
697 noticeably decreases its meanenergy as a result of the collision with a cold atom slightlyto the right of the CIR (bottom left graph in Fig.5). Aswe can see from the left column of the graphs, this occursdue to a decrease in its amplitude of oscillations in thetransverse directions after the collision. The amplitude ofoscillations in the longitudinal direction remains almostunchanged. There is also a decrease in the mean energy ofthe ion, oscillating in the plane before the collision, aftercollision with a cold atom at a ⊥ = 1 .
576 slightly to theright of CIR (see central column of the graphs in Fig.5).In this case, its oscillations remain two-dimensional, theorientation of the plane of oscillations does not change,and the energy decreases due to a decrease in the am-plitude of longitudinal vibrations. The graphs from theright column in Fig.5 demonstrate the heating of an ionupon collision with a cold atom at a ⊥ /a s = 1 .
796 outsidethe CIR region. As a result of the collision, the 2D os-cillations of the ion before the collision become 3D afterthe collision.Thus, the performed analysis demonstrates that onecan control the sympathetic cooling of ions in a hybridatom-ion trap by tuning the ratio a ⊥ /a s to the resonantregion around the CIR, where cooling can reach the mostoptimal conditions. In the cases considered, the maxi-mum loss of energy by an ion reaches the energy corre-sponding to the loss of energy by a heavy ball with a massof Yb + in its elastic head-on collision with a light slowball with a mass of Li.
C. Sympathetic cooling in atom-ion confinedcollisions in secular approximation
Here, we investigate the effect of a CIR on the efficiencyof sympathetic cooling in a hybrid atomic-ion trap in theframework of the time-independent secular approxima-tion [22, 46] with constant frequencies for the ion-trapinteraction (2). Since the secular approximation for thepotential of interaction of an ion with a trap does notcontain the RF part and, therefore, does not cause mi-cromotion of an ion, the calculation within its frameworkallows one to estimate the effect of micromotion-inducedheating at the collision with a cold atom by comparingwith the result of the previous subsection including themicromotion. The secular equation for the ion trap (2)gives the following expression [22]: U sec ( r i ) = m i ω xy ( x i + y i ) + ω z z i ] (22)with the frequencies ω z = 2 π × ω xy = 2 π × E ( out ) i ( a ⊥ /a s ), g D ( a ⊥ /a s ), T ( a ⊥ /a s ) and P mol ( a ⊥ /a s ) similarly to thecalculation performed above. In this case, the initial con-ditions (19) setting 3D motion of the ion before collisionswere used. However, in the case of a stationary secularpotential (2), these conditions rigidly fix the mean initialenergy of the ion as (cid:104) E ( in ) i (cid:105) = 12 E ( max ) i ( t → a ⊥ /a s (cid:39) . g D ( a ⊥ /a s )(Fig.6b) and T ( a ⊥ /a s ) (Fig.6c), but leads to a noticeableincrease in the probability P mol ( a ⊥ /a s ) of formation ofionic molecules (Fig.6d).In conclusion of this study, we can summarize thatreplacing the time-dependent Paul trap with the widelyused secular approximation [46] enhances the effect of in-creasing the efficiency of sympathetic cooling in the CIRregion: this region expands, and the ion cooling depthincreases compared to the values achieved in the Paultrap below the limit for the sympathetic cooling followingfrom the model of absolutely elastic head-on collision ofmechanical balls. We explain this effect by the absence inthe stationary potential (22) of the time-dependent RFpart, which is responsible for the micromotion of ionsand, as a consequence, for their possible heating uponcollisions with cold atoms, which prevents sympatheticcooling of ions. a ) t / t ^ pix(t)/p* b ) t / t ^ pix(t)/p* c ) pix(t)/p* t / t ^ - 6- 3036 piy(t)/p* - 8 x 1 0 - 1 1 - 4 x 1 0 - 1 1
04 x 1 0 - 1 1 - 1 1 piy(t)/p* - 8- 4048 piy(t)/p* - 202 piz(t)/p* - 202 piz(t)/p* - 6- 3036 piz(t)/p* a ^ / a s = 1 . 6 9 7 t / t ^ Ei(t)/E* a ^ / a s = 1 . 5 7 6 t / t ^ Ei(t)/E* a ^ / a s = 1 . 7 9 5 Ei(t)/E* t / t ^ FIG. 5: The calculated time-evolution of the ion momentum p i ( t ) and its kinetic energy E i ( t ) during the collision with theatom. The left column of graphs (graphs a)) presents the result of the calculation with the potential of atomic-ion interaction V ai fixing the ratio a ⊥ /a s = 1 .
697 slightly to the right of the CIR and with the initial condition (19) specifying the 3D-motionof the ion in the Paul trap before the collision. The results presented in the central and right columns of the graphs (graphs b)and c)) were obtained with the initial condition (18) specifying the 2D-motion of the ion in the Paul trap before the collision.The results shown in the central graphs (graphs b)) were obtained with the potential V ai giving the ratio a ⊥ /a s = 1 .
576 in theresonance region slightly to the right of the CIR. The right graphs (graphs c)) illustrate the dynamics of the ion outside theresonance region for a ⊥ /a s = 1 . a )
Finally, we evaluate the effect of the long-range charac-ter of the atom-ion interaction on the process of sympa-thetic cooling. For this purpose we replace the long-rangetail in the interparticle interaction (7) by more short-range Van-der-Waals potential V aa ( r ) = (cid:40) − ( r − c )( r + c )( r + b ) , r ≤ R ∗ − ( R ∗ − c )( R ∗ + c )( R ∗ + b ) R ∗ r , r > R ∗ (cid:41) . (23)With this potential we have performed the calculationof the mean kinetic energy of the heavy particle af-ter the collision (cid:104) E ( out ) i ( a ⊥ /a s ) (cid:105) , the coupling constant g D ( a ⊥ /a s ), the transmission coefficient T ( a ⊥ /a s ) andthe probability of formation of two-body bound state dur-ing the collision P mol ( a ⊥ /a s ). Results of the calculationsare presented in Fig.7. They were obtained with the po-tentials (1) and (22) describing interactions of the collid-ing particles with the trap. Herewith, a trap for a heavyparticle was described by the secular approximation fromthe previous subsection (22). The initial conditions forthe heavy particle were specified in the form (19), thesame as for the ion in the previous subsection.The curve (cid:104) E ( out ) i ( a ⊥ /a s ) (cid:105) given in Fig.7a qualitativelyrepeats the behavior of the curve calculated in the previ-ous subsection (Fig.6a) in the secular approximation forthe ion-trap interaction (22). However, the replacementof the long-range tail in the interparticle interaction (7)by the short-range Van-der-Waals tail (23) leads to a sig-nificant narrowing of the region of the minimum of thecalculated curve to the right of the CIR and its broaden-ing and deepening to the left of the CIR. We explain thiseffect by the narrowing of the splitting of CIR into com-ponents (2 , i ) and (0 , i + 1), which arises in a quasi-1Dcollision of distinguishable quantum particles, when thelong-range potential (22) is replaced by a shorter-rangeone (23). This splitting is developed in the calculatedcurves g D ( a ⊥ /a s ) and T ( a ⊥ /a s ) given in Figs. 7b and7c. Due to the narrowing of the distance between theCIRs (2 , i ) ( a ⊥ /a s =1.51)and (0 , i +1) ( a ⊥ /a s (cid:39) P mol of the formation of a boundtwo-particle system in a collision. Moreover, since thesplitting between the components (2 , i ) and (0 , i + 1) ofthe CIR is very narrow, the resonant amplification of P mol in the point of CIR (0 , i + 1) also covers the re-gion of the resonance (2 , i ). Note also that the replace-ment of the long-range tail in the interparticle interactionby the short-range Van-der-Waals potential (23) leadsto a significant shift of the CIR resonances to the left:the left resonance (2 , i ) shifts from point a ⊥ /a S = 1 . a ⊥ /a S = 1 .
51 (Fig.7b).In conclusion, the calculation performed shows thatthe replacement of the long-range tail in the atom-ion in-teraction (7) by the short-range Van-der-Waals potential(23), which simulates the atom-atom interaction, does not fundamentally change the picture of enhanced sym-pathetic cooling of a heavy hot particle near CIR. More-over, in this case we get a deeper minimum for the curve (cid:104) E ( out ) i ( a ⊥ /a s ) (cid:105) to the left of CIR, which is much deeperthan the limit following from the model of absolutelyelastic central collision of two balls.The above consideration can be regarded as a roughmodel of the quasi-1D Li-Yb scattering in an optical trapin the vicinity of the atomic CIR, which demonstratesthat the effect of enhancing the sympathetic cooling ofthe + Yb-ion in confined collisions with Li-atoms in theresonant region also remains in the confined collision ofatomic Yb with cold Li-atoms. An obvious improve-ment of the model is the use of more realistic parameters r ∗ (cid:39) . R ∗ for matching the Van-der-Waals tail withthe inner part of the potential (23) instead of r ∗ = R ∗ weuse here and C -coefficient, as well as more realistic pa-rameters ω x,y and ω z in the interaction of Yb-atom withthe optical trap (22). Such a modification, however, re-quires a significant improvement in the convergence andaccuracy of the computational scheme we use here. IV. CONCLUSION
We have investigated the effect of sympathetic coolingaround CIRs in atom-ion and atom-atom collisions withqusiclassical-quantum approach using the Li-Yb + and Li-Yb confined systems as an example. In this approach, theSchr¨odinger equation for a cold light atom is integratedsimultaneously with the classical Hamilton equations fora hotter heavy ion (atom) during collision. We haveshown in the framework of this model that the region nearthe atom-ion CIR is the most promising for the sympa-thetic cooling of an ion by cold atoms in a hybrid atom-ion trap due to suppression of the micromotion-inducedheating. The origin of this suppression is the “fermion-ization” effect of the relative distribution of the atom-ionprobability density near CIR, where the atom-ion pairbehaves as a pair of noninteracting identical fermions,which partially compensates the long-range character ofthe atom-ion interaction and, as a consequence, the pos-sibility to enhance the ion micromotion due to collisions.Moreover, it was also demonstrated that in the absenceof micromotion of a heated particle, as in the case ofthe secular approximation for an ion in a hybrid atom-ion trap or in the case of atoms in an optical trap, theefficiency of its sympathetic cooling in the CIR regionincreases.Thus, based on the performed analysis we propose anew way for sympathetic cooling of ions in an electromag-netic Paul trap: to use for this purpose cold buffer atomsin the region of atom-ion confinement-induced resonance(CIR). It was also shown that it is possible to improvethe efficiency of sympathetic cooling in atomic traps byusing atomic CIRs.In conclusion, atom-ion CIRs have not been experi-mentally discovered yet. Therefore, it seems to us that2it is easier to varify experimentally the predicted mech-anism of sympathetic cooling in atomic systems, wheremagnetic Feshbach resonances are successfully used totune atomic CIRs. Nevertheless, the implementationand tuning of the effective atomic-ion interaction onthe atom-ion CIRs may turn out to be a simpler ex-perimental problem than in pure atomic confined sys-tems that does not require the use of the Feshbach mag-netic resonance technique. Indeed, the nonresonant s-wave atomic scattering lengths a s in free space are muchsmaller than the transverse dimensions of the existingatomic optical traps a ⊥ , that is, in the nonresonant case a ⊥ (cid:29)| a s | . Therefore, the realization of the resonancecondition a ⊥ = 1 . a s for atomic systems required theuse of the technique of magnetic Feshbach resonances fora sharp increase in the value of a s . In atom-ion sys-tems, the scattering lengths, even in the nonresonantcase, significantly exceed the characteristic nonresonantatomic scattering lengths due to the long-range nature ofatomic-ion interactions compared to atomic interactions,which leads to the fulfillment of the relation a ⊥ ∼ | a s | for atom-ion systems already in the nonresonant case.Therefore, the necessary fine tuning of the atomic-ionscattering length a s to satisfy the resonance condition a ⊥ = 1 . a s can here be replaced by just a slight varia-tion of the atomic trap width a ⊥ without using the Fesh-bach resonance technique for enhancement of a s . In thiscontext, the task of experimental detection of the atom-ion CIR and its use for sympathetic cooling of ions ina hybrid atom-ion trap seems to us quite feasible andrelevant.Note also that a fully quantum consideration of theproblem would be very useful, especially in the case ofcomparable masses of an ion and an atom, when quan-tum effects become significant. However, its realizationis rather challenging technical problem but realistic one. V. ACKNOWLEDGEMENTS
The author thanks P.Schmelcher, A. Negretti, Z. Idzi-aszek, and O. Prudnikov for fruitful discussions. Thework was supported by the Russian Foundation for Ba-sic Research, Grants No. 18-02-00673.
VI. APPENDIX
The coupled system of quantum (3) and classical (6)equations describes collisional dynamics of a Li-Yb + pair confined in a hybrid trap with three absolutely differenttime-scales 2 π/ Ω rf (cid:28) π/ω i (cid:28) t ⊥ = 2 π/ω ⊥ definedby Paul trap frequencies of (Ω rf = 2 π × ω i = 2 π ×
63 kHz) and atomic waveguide ( ω ⊥ (cid:39) π × ∼ × t ⊥ = 10 × π/ω ⊥ and, on the otherhand, it must accurately handle fast oscillations definedby the frequency Ω rf of the RF-field, as well as the res-onant behavior of the the atom-ion interaction potential V ai ( | r a − r i ( t ) | ) (7) on collision.To integrate the coupled system of equations of motion(3) and (6), we applied the splitting-up method with a2D discrete-variable representation (DVR) [52–54]. Foran accurate inclusion of the atom-ion interaction poten-tial (7) in the numerical integration procedure of theSchr¨odinger equation (3) at the moment of the resonantatom-ion collision, a tailored splitting-up procedure inthe 2D-DVR representation was developed in [22].Simultaneously to the forward in time propaga-tion t n → t n +1 = t n + ∆ t of the atom wavepacket ψ ( r a , t n ) → ψ ( r a , t n +1 ) when integrating thetime-dependent Schr¨odinger equation (3), we integratethe Hamilton equations of motion (6), which in-volve three different scales of frequencies, namely Ω rf , ω i , as well as ω ⊥ in the quantum mechanical aver-age (cid:104) ψ ( r a , t ; r i ) | V ai ( | ˆ r a − r i ( t ) | ) | ψ ( r a , t ; r i ) (cid:105) . 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