Cold molecules: Theory. Viatcheslav Kokoouline Olivier Dulieu

Transcription

1 Cold molecules: Theory Viatcheslav Kokoouline Olivier Dulieu

2 Summary (1) Born-Oppenheimer approximation; diatomic electronic states, rovibrational wave functions of the diatomic cold molecule molecule () Some formation, theory optical of transitions in diatomic molecules (PA example) (3) three-body states near dissociation

3 Adiabatic or Born-Oppenheimer approximation Electrons move much faster than nuclei. We separate nuclear and electronic motions and solve the Schrödinger equation first for electronic degrees of freedom at fixed nuclei and then treat vibrational motion as a perturbation =T R T r V r, R H ℏ T r = i r i R and r are vectors here, µ, r ~ electrons ℏ T R = j Mj Rj Solving the equation tot r, R =E tot r, R H Mj, R ~ nuclei Our approximation: TR is small. So, we split H in two parts = H 0 T R H H 0 = T r V r, R and (1) obtain adiabatic molecular states ϕn(rf,r) and energies for a fixed Rf H 0 n R f, r = n R n R f, r n=1,, step: Ψtot(r,R) is represented as an expansion in the basis of ϕn(r,r) depends on R tot r, R = c n n R, r = n R n R, r n n In more complete form (with continuum states) we would obtain exact functions Ψtot(r,R), but it is difficult to do tot r, R = R R, r R R, r d n n n

4 discrete continuum states tot r, R = n R n R, r R R, r d n If Λmn are small, we have T R =E [ R m ] m m we write [ T R m R ] 0mv =E 0mv 0mv we'll write simply r, R = n R n n Plugging the expansion in the complete Molecular Schrödinger equation H0+TR, multiplying terms it with <m and integrating over r: tot m H 0 T R n = m E n m R [ T R m R E ] m R = mn n R n where non-adiabatic couplings Λmn are Energies E0nv and states tot r, R 0nv R n R, r are adiabatic or BornOppenheimer solutions of the = ℏ m n m T n mn R Rj Rj the molecular Hamiltonian. j Mj The BO approximation is valid The two above equations are exact if only if the whole (discrete and continuum) 0 dr E 0 E 0 spectrum in included: If the complete 0 n'v' mn nv nv n' v ' basis ϕn(r,r) is used. Usually, one truncates the basis.

5 Near equilibrium In normal conditions, molecules are in their ground electronic BO state ϕ0(r,r) and vibrational state can be approximated by a harmonic oscillator state. Close to the equilibrium, we can use normal coordinates and write 0 R = 0 R 0 1 s s s We can also neglect rotation of the molecule. Then T = ℏ and the Hamiltonian becomes R s s with energies E 0 n and wave functions 1 ℏ H s 0 R0 s s 1, n = 0 ℏ s n s s=1 1 0 ; n 1, n = 0 R0, r n s 0 s s If the molecule is in an exited state a similar procedure can be applied.

6 Diatomic molecule Born-Oppenheimer (adiabatic) approx.

7 Diatomic molecule Molecular Quantum numbers for electronic states of diatomic molecule: parity g/u, sign +/, spin S (spin multiplicity S+1), projection Λ of electronic angular momentum on the axis. The quantum numbers are combined in the form: term S 1 g /u/

8 Example of potential surfaces H3 Symmetric stretch coordinate asymmetric stretch coordinate (Cv is kept) Na

9 Notations O Na

10 Rovibrational wave function of diatomic molecule (in the BO approximation) U(R) does not depend on -> radial (R) and angular ( ) motions are ℏ separable: H R = U R = m ℏ 1 l = R U R m R R dr m R reduced mass m=m1m/(m1+m) So the total wave function is also R, = R Y lm, where l Y =ℏ l l 1 Y separable lm lm Schrödinger equation becomes [ Spherical harmonic ] ℏ 1 l l 1 R U R R =E R m R R dr R changing (R) > (R)= (R)R we obtain: [ ] ℏ l l 1 ℏ ' ' U R =E m mr

11 Vibrational wave functions (in BO approximation) With effective potential ℏ l l 1 becomes m R U l R =U R the equation ℏ ' ' U l R =E m Dissociation Solutions R for a given energy U( ) l=0,1,,3,4... (especially continuum states) are often called as s,p,d,f waves. Reminder: total wave function in the BO approximation has the form tot = n r, R Y lm R = n r, R Y lm R / R Normalization of bound states: 0 =c 0 = R R R= 0 =0 nv' t, t ' = t t ' = t t ' d r R dr d = n n ' d r Y lm Y l ' m ' d v v' R dr R R 0 for bound states nn' ll' mm' v,v ' = v v' R dr= v v ' dr

12 Vibration and rotation structure of diatomic molecules Close to the minimum of the molecular potential, we can use the harmonic approximation for U(n)(R): m ℏ l l 1 U l R =U R0 R R0 m R n Particular molecular term (curve) n E v =U R 0 ℏ v Contribution from 1 centrifugal potential: ℏ l l 1 ℏ v v =B v l l 1, where Bv mr m R0 Then, total energy near the minimum: E n,v,l =U n R0 ℏ 1 Bv l l 1 E el E vib E rot because mel mnuclei v

13 Continuum states We need to solve the Schrödinger equation for E>0 (we choose the energy origin: U( )=0). Let E=E kin = ℏ k m m l l 1 ' ' U R k =0 ℏ R [ ] For l=0 and U(R)=0, we have ' ' k =0 = n r Y lm n,l, v R / R n,l, v 0 =0 with two independent solutions Asin kr and B cos kr which is unphysical because cos(0)=1. Normalization (momentum) of continuum states 0 sin kr dr= how to choose A? Consider first normalization of eikr and the inverse Fourier transform of a function f f p =F 1 [ f x ]= 1 f x e ix p dx = 1 f p ' eix p ' p dx dp ' 1 f p ' eix p ' dp' f x = f p = f p ' p ' p dp ' definition of the δ-function, therefore ikr ik ' R 1 ix p ' p e = k ' k so, for sin(kr) with k>0: e p ' p = e dx 1 sin kr sin k ' R = eikr e ikr eik ' R e ik' R = k ' k R = sin kr, ir R = cos kr 8

14 Continuum states So the ular and irular functions are sin kr, ir R = sin kr R =, ir R = R R = Now l is arbitrary ' ' we have A [ cos kr cos kr R ] l l 1 k =0 R R = kr l Spherical Bessel and Neumann functions n,l, v 0 =0 l sin kr R = j l kr = 1 l k R kr = n r Y lm n,l, v R / R consider now at large R ' ' [ sin kr l / R ir R cos kr l / R ] m l l 1 U R k =0 ℏ R if U(R) is not zero for small R but =0 at R>R0, we can write Due to Phase shift R =R R = R A j kr B l kr normalization l due to U(R) A B =1 A sin kr l / B cos kr l / = sin kr l / l cos l =A,sin l =B R l

15 Different normalizations 1. So the ular and irular momentum-normalized functions are A B =1 R A sin kr l / B cos kr l / = sin kr l / l irr R A cos kr l / A sin kr l / = cos kr l / l k ir ir ir k ' k = k ' k, k ' k = k ' k, k ' k =0 k cos l = A, sin l = B There are two other normalizations of continuum states that are used in applications. Energy normalized states = A E k property of δ(x) function:. Consider the such that E ' E =. E' E de de ℏ k f E ' = E E ' f E de = E E ' f E k dk = f E k ' k = E = E dk dk m m m 1 k then we have E ' E = E ' E = A k ' k = k ' k E = k = ℏv ℏ k ℏ k 3. Current density normalized states : E = A E.sin(kR) function can be written iℏ d d as sin kr = i e ikr eikr. We can calculate the flux j= in the m dr dr incoming wave e ikr and then normalize the function such that flux=1 (one particle per unit time) j[sin kr ]= v j[ ]= v 1 = 1 E 4 4 ℏv ℏ = ℏ E E

16 Resonances (WKB way of looking at) Semiclassical calculation gives a non-zero probability c P=exp b dr m [U R E ] 1 ℏ to overcome the barrier once per oscillation. The (semiclassical) period Shape resonance System will decay with the probability Γ=P/T per unit time. b b b T = a dt = a dr /v R = dr a dr m[ E U R ] Now consider an ensemble of such molecules. At time t0 we have N0 molecules. We can write dn = N dt, solving for N, we obtain the exponential law of decay of the ensemble t t 0 N=N0e Another type of resonance Γ cannot be obtained semiclassically, but the same exponential law applies Feshbach resonance

17 Theory of Feshbach resonances We go beyond the BO approximation. l=0 and we keep two terms in tot = n r, R n R / R Then the equations for χn(r) are n= 1 [ [ ] ] ℏ U 1 R 1 R 1 R =E 1 R m dr ℏ U R R 1 1 R =E R m dr We start with BO solutions [ [ and assume that χn(r) =Aχ0(R) [ ] ] ℏ U R 0 R =E 0 0 R m dr R ] ℏ E U 1 R 1 R = A 1 0 R m dr A E E 0 0 = 1 1 R We use Green's function which is ℏ U R R =E R 1 m dr [ ] m ℏ k ℏ E U 1 R G R, R ' = R R ' m dr G R, R ' = R irr R ' for R R ' G R, R ' = R' irr R for R ' R sin kr 0

18 1 R = A G 1 0 = A G R, R ' 1 0 R ' dr ' 0 We insert it into and obtain A A E E 0 0 = 1 1 R 0 1 A= E E G 1 0 multiply with <χ0(r) where 1 0 = dr dr ' 0 R 1 R G R, R ' 1 R ' 0 R ' 0 1 G So R = G 1 0 = 1 0 E E G 0 1 = dr ' G R, R ' 1 R ' 0 R ' E E G We are interested in χ1(r) at large values of R in χ1(r). The integral over R' is not zero only for R>R' -> we use G R, R ' = R' irr R and evaluate it dr ' G R, R ' 1 R ' 0 R ' = dr ' R ' irr R 1 R ' 0 R ' = 0 1 irr R 0 0 = R tan E irr R 1 R = R irr R = E E G m sin kr 0 E cos E ℏ k

19 We introduce notations and = 0 1 E = arctan = 0 1 G 1 0 / / = arctan E E 0 E E R The energy normalized version of 1 R =cos E R sin E irr R = m ℏ k R tan E irr R sin kr 0 E At the energy near a resonance the phase-shift δ(ε) changes by π. Position of the resonance is found from the maximum of dδ/de d E = de / E E R The maximum (so the position of the resonance) is at ER=E0+. The resonance width is Γ. So Γ can be found from dδ/de (in calculation) of from widths of observables (in experiment) is

20 two-body states near dissociation Ultra-cold atomic gas: atomic collisions with energy close to dissociation. Molecules are formed from two colliding atoms. On the theoretical language it means that a continuum state is coupled to a bound state by a field. So, it is a Feshbach resonance. The process of molecule formation can be considered as the process inverse to the decay of a resonance. Physics of Feshbach resonances can be used tp explains two mechanisms of cold molecule formation: photoassociation and magnetic Feshbach resonances.

21 Methods to form of cold molecules: 1. using magnetic field At low energies δ-> 0 (usually) and the expansion of δ(e) in series of k is the following k / tan 0 k = 1/ a r 0 k / o k Scattering Due to change of the magnetic field B For s-wave scattering: k = k l 1 sin l k l =0 Effective potential If you have a Feshbach resonance at energies of the experiment and you can change Un, or/and Λ1, you can tune δ(e) 4 el k = 4 length 1 1 r 0 k k a and, therefore, change collisional properties of colliding particles

22 Methods to form of cold molecules: 1. using magnetic field Rb molecular potentials Magnetic field B is changing B=0 Simplified two potential (channel) picture If you have a Feshbach resonance at energies of the experiment and you can change Un, or/and Λ1, you can tune δ(e) and, therefore, change collisional properties of colliding particles

23 Methods to form of cold molecules:. Using lasers - Photoassociation

24 Photoassociation spectroscopy of cold molecules

25 Optical transitions in diatomic molecules 1. Spontaneous emission. Molecule is in an excited state i> can emit a photon (resonant state) and go to state f>. Probability to emit a photon per unit time 4 e 3 P fi = r fi 3 3 ℏc if several final states are available then the total emission probability is Pi = f P fi and the lifetime of the initial state is =1 / Pi. Induced emission. The emission in a presence of an external EM field. In this 4 e case P fi = 3 ℏ r fi I =B fi I 3. Absorption of a photon is characterized by the cross-section: Vector of polarization of external field Intensity of the field Einstein coefficient B 4 e = f r i E f E i ℏ c Vector of electronic coordinate Energy conservation law

26 How to deal with optical transitions? Let us derive theory of photoassociation. PA is the the process of absorption of a photon when the diatomic systems changes its continuum vibrational state to a bound vibrational state. Rotational and/or electronic states are changed too. The main thing to calculate is f r i Let assume that the PA laser is polarized along z' (lab) axis r =r ' cos '= r ' Y 10 ' ' = r D1 0,,0 Y = 1 Coord. of electron in lab system We rewrite the same quantity in the molecular system using Wigner functions Coord. of electron in mol. System, r'=r Electr. Initial and final states are i = n r, R RJ, m, v, n R / R= n, v, J, m, f = n ', v ', J ', m ', ' with RJ, m, = J 1 8 D J m, 0 momentum J = L l Total momentum Nucl. momen.

27 Combining all factors together into the matrix element 1 4 f r i = n ' r, R R J ', m ', ' v', n ' R / R r D 1 r, R RJ, m, v, n R / R 0,,0 Y 1 n 3 = 1 We use the property D J ' m ', ' 1 0 D,,0 D J m, 8 = C J10,' mjm' C 1J ', J' J ' 1 and introducing the matrix element of dipole moment transition r n ' ' ;n R = n ' r, R 4 r Y 1 n r, R 3 It gives selection rules for electronic transitions: L'=L,L±1, Λ'=Λ+µ So, we obtain 1 f r i = = 1 u->g, g->u J 1 J ' m ' J ' ' C 10,Jm C 1, J v', n ' R / R r R v, n R / R J ' 1 It gives selection rules for rotational transitions: J'=J,J±1, Λ'=Λ+µ m'=m 1 f r i = = 1 For given Λ' and Λ only one term µ=λ'-λ is not zero J 1 J ' m' J ' ' C 10,Jm C 1, J 0 v', n ' R r n ' ' ; n R v, n R dr= J ' 1 J 1 J ' m ' J ' ' C 10, Jm C 1, J 0 v', n ' R r n ' ' ;n R v, n R dr, J ' 1 = '

28 Therefore, for photoabsorption cross-section we obtain J 1 J ' m ' J ' ' f C 10,Jm C 1, J 0 v ', n ' R r n ' ' ; n R v, n R dr r i = J ' 1 4 e fi = f r i E f E i ℏ = c 4 e J 1 J ' m ' J ' ' = C 10, Jm C 1,J 0 v ', n ' R r n ' ' ;n R v, n R dr E f E i ℏ c J ' 1 Consider an atom that collides with others atoms in presence of EM field. The number of absorbed photons (number of events) per unit time in the interval of frequencies dω= fi j p= fi I d ℏ We have to multiply with the number of incident atoms = Current in the vibrational wf 1 f E n de j vib Density of Maxwell- atomic cloud Boltzmann distr. over collision velocities to obtain probability of absorption per unit time dp fi = fi I d 1 f E n de ℏ j vib Integrating over w and E and multiplying with n, we obtain the total number of PA events in a unit volume per sec 4 e n J 1 J ' m' J ' ' C C de c J ' 1 10,Jm 1, J [ 0 v ',n ' dr R r n' ' ; n I E f E 1 R E, n R f E E f E j vib ]

29 Franck-Condon overlap Considering transitions between low vibrational states the dipole transition matrix element rµ(r) could be considered as constant. Then we can write 0 v', n ' R r R v, n R dr=r R0 0 v', n ' R v, n R dr This integral is Franck-Condon overlap or factor Many observables in molecular physics are determined by FC factor

30 Methods to form of cold molecules:. Using lasers - Photoassociation First and second transitions should have a good Franck-Condon overlap

31 On 3 body states near dissociation Hyperspherical coordinates Three inter particle distances are represented by two hyperangles Changing hyperangles and the hyper radius. Changing hyper radius it represents the size of the molecule represents the shape of the molecule

32 Adiabatic separation in hyperspherical coordinates To treat the vibrational dynamics in hyperspherical coordinates we represent the vibrational wave function ( ) as the expansion,, = k y k,, c k in the basis of non orthogonal basis functions y k,, = j a, j, k {a, j} where j( ) are some convenient basis functions, and a,j( ) are hyperspherical adiabatic states calculated at fixed hyper radii j: ℏ 15 4 V i,, a, j, =U a i a, j,. 0 with the corresponding eigenvalue Ua( j); V( ) is the molecular (three body) potential.

33 Efimov resonances (quasi Feshbach resonances) Adiabatic potentials Ua( ) Comparison of our results to those of Nielsen et al. [PRA 66, 01705]. Eigenenergy is given as Ei = E i /, where is the linewidth of the resonance and E its energy. E1 x 10 (a.u.) E x 10 4 (a.u.) E3 x 10 7 (a.u.) E4 x 10 9 (a.u.) Nielsen et al i i i i adiabatic states 96.0 i i i i adiabatic states 96.0 i i i i adiabatic states 96.0 i i i i 1.77

34 Shape resonances Fedorov et al. Few-Body Sys. 33, 153 (003). Only one three body resonance with E= 5.31 i0.1 MeV. Fedorov et al. found E= 5.96 i0.4

35 On 3 body bound states and resonances near dissociation (Efimov states) Universality, r0 effective range of body potential, a body scattering length. If r0«a, the wave function in the ion r0 «r«a does not depend on r0 or a. Effective 3b potential in the ion is ~1/. Thus, 3 body bound states may exist even if there is no body bound states. When a +, the number of 3 body bound states Effective range of a potential: if k > 0 k / tan k = 1 /a r 0 k / o k

36 On Efimov states (1970) When a= the hyper radial ℏ 15 4 V R i,, a R j,, =Ua R a R j,,. R 0 Equation is si is a transcendental constant. The lowest si is s0=1.0064i. Spectrum for s0 is When a the spectrum: g is the interaction parameter, such that at g=1, a=

37 Observation of Efimov states: No direct observation. Kramer et al. see the increase of the 3 body recombination rate very close to 3 body dissociation limit as predicted by theory (Esry, Greene). This is an indirect evidence for Efimov states.

38 Observation of Efimov states: Theory Experiment

39 Concepts that are important to understand cold molecule formation (minimum theorist's checklist) Born-Oppenheimer approximation and electronic states of molecules. Principles of ab initio calculation Group theory, irreducible representations How to treat rotation and vibration of molecules (different levels of approximation) Scattering theory (partial wave scattering, Born approximation) Feshbach resonances, interaction of electronic (and nuclear) spins with the magnetic field Optical transitions, selection rules, Franck-Condon overlap Long-range interactions between atoms and molecules