Magneto-optical Feshbach resonance: Controlling cold collision with quantum interference
aa r X i v : . [ phy s i c s . a t o m - ph ] A p r Magneto-optical Feshbach resonance: Controllingcold collision with quantum interference
Bimalendu Deb
Department of Materials Science, Indian Association for the Cultivation of Science,Jadavpur, Kolkata 700032. INDIA
Abstract.
We propose a method of controlling two-atom interaction using bothmagnetic and laser fields. We analyse the role of quantum interference betweenmagnetic and optical Feshbach resonances in controlling cold collision. In particular,we demonstrate that this method allows us to suppress inelastic and enhance elasticscattering cross sections. Quantum interference is shown to modify significantlythe threshold behaviour and resonant interaction of ultracold atoms. Furthermore,we show that it is possible to manipulate not only the spherically symmetric s-wave interaction but also the anisotropic higher partial-wave interactions which areparticularly important for high temperature superfluid or superconducting phases ofmatter.PACS numbers: 34.50.Cx, 34.80.Dp, 32.70.Jz, 34.80.Pa agneto-optical Feshbach resonance
1. Introduction
Two-particle interaction is a key to describing interacting many-particle systems at amicroscopic level. Means of manipulating this interaction enable us to explore physicsof such systems with controllable interaction. In solid state systems, the scope ofexternally controlling inter-particle interactions is limited due to crystalline structures.By contrast, ultracold atomic gases offer a unique opportunity since their interatomics-wave interaction is widely tunable by a magnetic Feshbach resonance (MFR) [1]. Newinsight into the exotic phases of interacting electrons in solids can be gained from theexperiments involving ultracold atoms with tunable interactions. Atom-atom interactioncan also be manipulated by an optical Feshbach resonance (OFR) [2], albeit withlimited efficiency. Over the last decade, MFR [3, 4] has been extensively used to studyinteracting Bose[5, 6, 7, 8] and Fermi gases[9, 10, 11] of atoms. Electric fields[12, 13]can also be used to alter interatomic interaction.MFR relies on the interplay of Zeeman effects and hyperfine interactions while OFRis based on photoassociation (PA)[14, 15, 16] of two colliding ground state atoms into anexcited molecular state. OFR has been demonstrated in recent experiments[17, 18, 19].Recently, PA spectroscopy in the presence of an MFR has attracted a lot of attentionboth experimentally[20, 21, 22, 23] and theoretically[24, 25, 26, 27, 28]. Junker etal. [20] have observed asymmetric profile in PA spectrum under the influence of anMFR. This spectral asymmetry results from Fano-type quantum interference[29] incontinuum-bound transitions[26]. The use of quantum interference to control Feshbachresonance had been suggested earlier by Harris[30]. Of late, quantum interference hasbeen observed in two-photon PA[31, 32, 33] and coherent atom-molecule conversion[34].It has also been shown that Fano’s theory[29] can account for PA spectrum[35, 36] evenin the absence of any MFR.Here we demonstrate theoretically a new method of altering two-atom interaction.Let us consider that a laser field is tuned near a PA transition of two atoms which aresimultaneously influenced by a magnetic field-induced Feshbach resonance. There aretwo competing resonance processes occurring in this system. One is the MFR attemptingto associate the two ground state atoms into a quasi-bound state embedded in theground continuum. The other one is the PA resonance tending to bind the two atomsinto an excited molecular state. PA transitions can occur in two competing pathwayswhich originate from the perturbed and unperturbed continuum states. The Fano-typequantum interference between these two pathways can be used to control atom-atominteraction. This quantum control of two-body interaction due to applied magnetic andoptical fields is what we call “magneto-optical Feshbach resonance” (MOFR). In strong-coupling regime of PA transitions, s-wave scattering state gets coupled to higher partial-wave states[37, 38] via two-photon continuum-bound dipole coupling. Since s-wavescattering amplitude is largely enhanced due to the applied magnetic field, amplitudesof the higher partial-waves coupled to s-wave will also be largely modified. By resortingto a model calculation, we present explicit analytical expressions for phase shifts, elastic agneto-optical Feshbach resonance
2. The model
As a simple model, we consider three-channel time-independent scattering of twohomonuclear Alkali atoms in the presence of a magnetic and a PA laser field. Herechannel implies asymptotic hyperfine or electronic states of the two atoms. There aretwo ground hyperfine channels of which one is energetically open (labeled as channel ‘1’)and the other one is closed (channel ‘2’) in the separated atom limit. Channel 3 belongsto an excited molecular state which asymptotically corresponds to two separated atomswith one ground and the other excited atom. We assume that the collision energy is closeto the binding energy of a quasi-bound state supported by the ground closed channel. Itis further assumed that the rotational energy spacing of the excited molecular levels ismuch larger than PA laser linewidth so that PA laser can effectively drives transitions toa single ro-vibrational level ( v, J ) of the excited molecule, where v stands for vibrationaland J for rotational quantum numbers. The angular state of the two atoms in themolecular frame of reference can be written as | J Ω M i = i J q J +18 π D ( J ) M Ω (ˆ r ) where Ω isthe projection of the electronic angular momentum along the internuclear axis and M is the z-component of J in the space-fixed coordinate (laboratory) frame. D ( J ) M Ω (ˆ r ) is therotational matrix element with ˆ r representing the Euler angles for transformation frombody-fixed to space-fixed frame. In our model, we assume that the PA laser is tunednear resonance of J = 1 level of the excited molecule.The energy-normalized dressed state of these three interacting states with energyeigenvalue E can be written asΨ E = X M φ vJM ( r ) r | e i | J Ω M i + χ ( r ) r | g i | i + Z dE ′ β E ′ X ℓm ℓ ψ E ′ ℓm ℓ ( r ) r | g i | ℓ m ℓ i (1)where φ vJM ( r ) is the radial part of the excited molecular state, χ ( r ) is the bound statein the closed channel and ψ E ′ ℓm ℓ ( r ) represents energy-normalized scattering state of thepartial wave ℓ with m ℓ being the projection of ℓ along the space-fixed z-axis. | g i i and | e i denote the internal electronic states of i -th ground and excited molecular channels,respectively. Here E ′ = ¯ h k / (2 µ ) is the collision energy, where k and µ are the relativemomentum and reduced mass of the two atoms, respectively. β E ′ denotes density ofstates of the unperturbed continuum. Note that φ vJM ( r ) and χ ( r ) are the perturbedbound states. In the limit r → ∞ , we have r Ψ E → R dE ′ β E ′ P ℓm ℓ ψ E ′ ℓm ℓ | g i | ℓ m ℓ i and thus the scattering properties in MOFR are determined by the asymptotic behaviorof ψ E ′ ℓm ℓ . agneto-optical Feshbach resonance h ˆ h J + V e ( r ) − ¯ hδ − E − i ¯ hγ J / i φ vJM = − X ℓ,m ℓ Λ (1) ℓm ℓ ,JM ˜ ψ Eℓm ℓ + Λ (2)00 ,JM χ, (2) h ˆ h + V ( r ) − E i χ = − X M Λ (2) JM, φ vJM − V ˜ ψ E (3) h ˆ h ℓ + V ( r ) − E i ˜ ψ Eℓm ℓ = − X M Λ (1) ℓm ℓ ,JM φ vJM − δ ℓ V χ, (4)where ˜ ψ Eℓm ℓ = R β E ′ dE ′ ψ E ′ ℓm ℓ , ˆ h J ( ℓ ) = − ¯ h µ d dr + B J ( ℓ ) ( r ) with B J ( ℓ ) ( r ) = ¯ h / (2 µr ) X J ( ℓ ) being the rotational term corresponding to J ( ℓ ). If the excited molecular potential V e belongs to Hund’s case (a) and (c), then X J = [ J ( J + 1) − Ω ], otherwise X J = J ( J + 1)and X ℓ = ℓ ( ℓ + 1). The laser couplings between different angular states are denoted byΛ ( i ) ℓm ℓ ,JM = −h J M Ω | ~D i . ~ E P A | ℓm ℓ i , where ~D i is the transition dipole moment betweenthe excited and the ground i -th channel molecular electronic states. For homonuclearatoms, V e ( r ) goes as − /r and the ground potentials V and V behave as − /r in thelimit r → ∞ . Here δ = ω − ω A is the detuning between the laser frequency ω andthe atomic resonance frequency ω A , V is the interatomic potential in channel 1(2), δ ℓ stands for Kronecker- δ and V denotes spin-spin coupling between the two groundchannels. We have here phenomenologically introduced the term − i ¯ hγ J / v, J ). The zero of the energy scale istaken to be the threshold of channel 1 and the atomic frequency ω A corresponds to thethreshold of the channel 3 (threshold of excited molecular potential). For simplicity, weassume that the excited state belongs to the Σ symmetry. Then the dipole couplingbetween angular states provides m ℓ = M and thus we can solve the above coupledequations for a given value of M . For notational convenience, we henceforth suppressthe subscripts M and m ℓ .
3. The solution
The coupled equations (2-4) can be conveniently solved by the method of Green’sfunction. Let φ vJ be the excited bound state solution of the homogeneous part of (2)with binding energy E vJ . Using the Green’s function G vJ ( r, r ′ ) = − φ J ( r ) φ J ( r ′ )∆ E vJ + i ¯ hγ J / where∆ E vJ = ¯ hδ + E − E vJ , we can write φ vJ ( r ) = R E ′ dE ′ β E ′ P ℓ Λ E ′ ℓ,J + Λ bb ∆ E vJ + i ¯ hγ J / φ vJ ( r ) (5)where Λ E ′ ℓ,J = R dr ′ Λ (1) J,ℓ ( r ′ ) φ J ( r ′ ) ψ E ′ ℓ ( r ′ ) is the free-bound dipole coupling betweenthe unperturbed bound state φ vJ and the perturbed scattering state ψ E ′ ℓ and Λ bb = R dr ′ Λ (2) J, ( r ′ )( r ′ ) φ vJ ( r ′ ) χ ( r ′ ) is the bound-bound dipole coupling between φ vJ and the agneto-optical Feshbach resonance χ . Let χ ( r ) be the solution of the homogeneous part of (3) withbinding energy E χ . Writing φ vJ in the form φ vJ = R dE ′ β E ′ A E ′ φ vJ , we can express χ = 1 E − E χ Z dE ′ β E ′ (cid:16) A E ′ | Λ bb | + V E ′ (cid:17) χ ( r ) (6)where Λ bb is the Rabi frequency between the two bound states φ vJ and χ and V E ′ = R dr ′ ψ E ′ ( r ′ ) V ( r ′ ) χ ( r ′ ). Using this one can express Λ bb in terms of Λ bb and V E ′ . After having done some minor algebra, we obtain A E ′ = ( E − E χ ) P ℓ Λ (1) E ′ ℓ,J + V E ′ Λ bb ( E − E χ )(∆ E v + i ¯ hγ J / − | Λ bb | . (7)Note that the right hand side of (7) involves the laser coupling Λ (1) E ′ ℓ,J with the perturbedcontinuum states. Here A E ′ is related to the coefficient of φ vJ in the energy-normaliseddressed state (1) of three interacting states of which two are bound states and one isground continuum state. Since φ vJ is unit-normalised, A E ′ has the dimension of inverseof square root of energy. Physically, PA excitation probability for collision energiesranging from E ′ to E ′ + dE ′ is given by | A E ′ | dE ′ . Now, substituting (7) into (5) and(6) and then using the resultant form of φ vJ and χ into (4), it is easy to see that theequation of motion for particular ℓ -wave function gets coupled to other ℓ -wave functions.The Green’s function for the homogeneous part of (4) can be written as K ℓ ( r, r ′ ) = − πψ ,regEℓ ( r < ) ψ + Eℓ ( r > ) where r < ( > ) implies either r or r ′ whichever is smaller (greater)than the other. Here ψ + Eℓ ( r ) = ψ ,irrEℓ + iψ ,regEℓ where ψ ,regEℓ and ψ ,irrEℓ representregular and irregular scattering wave functions, respectively, in the absence of opticaland magnetic fields. Asymptotically, ψ ,regEℓ ( r ) ∼ j ℓ cos η ℓ − n ℓ sin η ℓ and ψ ,irrEℓ ( r ) ∼− ( n ℓ cos η ℓ + j ℓ sin η ℓ ), where j ℓ and n ℓ are the spherical Bessel and Neumann functionsfor partial wave ℓ and η ℓ is the phase shift in the absence of laser and magnetic fieldcouplings. According to Wigner threshold laws, as k → η ℓ ∼ k ℓ +1 for ℓ ≤ ( n − / η ℓ ∼ k n − with n being the exponent of the inverse power-law potential atlarge separation. Using K ℓ ( r, r ′ ), the perturbed wave function ψ E ′ ℓℓ ′ can be formallyexpressed in terms of V E ′ , A E ′ and Λ bb and the partial-wave free-bound dipole transitionmatrix elements Λ E ′ ℓ,vJ = R drφ J ( r )Λ (1) ℓ,J ( r ) ψ E ′ ( r ). Next, substituting this into (7) andthe expression for V E ′ , we can express A E ′ exclusively in terms of couplings betweenunperturbed states. Explicitly, we have A E ′ = e iη ( q f + ǫ ) / ( ǫ + i )Λ + P ℓ ≥ e iη ℓ Λ E ′ ℓ,vJ D − E shiftq + i ¯ h ( γ J + Γ q + P ℓ ≥ Γ Jℓ ) / = Λ (1) E ′ ,vJ , ǫ = [ E − E χ − E shiftχ ] / (Γ mf /
2) with E shiftχ =Re R drV (r) χ (r ′ ) R dr ′ K (r , r ′ )V ∗ (r ′ ) χ (r ′ ) and Γ mf = 2 π | R drψ reg, E ′ , ( r ) V ( r ) χ ( r ) | =2 π | V E ′ | being the MFR shift and line width, respectively. Here q f = V eff + Λ bb π Λ V E ′ (9) agneto-optical Feshbach resonance
650 700 750 80010 −12 −10 −8 σ , σ i ne l ( c m ) B (Gauss)
650 700 750 80010 −12 −10 −8 B (Gauss) σ , σ i ne l ( c m )
700 720 74010 −12 −10 −8 −6 B (Gauss) σ , σ i ne l ( c m )
650 700 750 800−5000500 R e [ a m o f ], a m f ( n m ) B (Gauss) σ σ inel σ σ inel σ inel σ Re[a mof ] a mf (a) (b)(c) (d) Figure 1.
Subplots (a) and (b) show elastic and inelastic scattering cross sections σ (solid line) and σ inel (solid-dotted line), respectively, in unit of cm as a function ofmagnetic field B in Gauss (G) for Γ J /γ = 0 . J /γ = 10 . E = 10 µ K and q f = − .
89. Subplot (c) displays σ Vs. B (solid and dashedlines) and σ inel Vs. B (dotted and solid-dotted lines) plots for Γ J /γ = 10 . J /γ = 0 . E = 100 nKand q f = − .
88. Subplot (d) exhibits the variation of Re[ a mof ] (solid line) and a mf (dashed lines) as a function of B for Γ J /γ = 10 . E = 10 µ K and q f = − .
89. Theother fixed parameters for all the subplots are Γ mf = 16 .
67 MHz and γ = 11 . is Fano’s q -parameter which is, in the present context, called ‘Feshbach asymmetryparameter’[26] with V eff = Re Z dr φ (r)Λ (1)J ℓ =0 (r) Z dr ′ K (r , r ′ )V (r ′ ) χ (r ′ )being an effective potential acting between the two bound states as a result of theirinteractions with the s-wave part of the continuum states. In (8), D = ∆ E vJ − P ℓ E shiftJℓ ,Γ Jℓ = 2 π | Λ E ′ ℓ,vJ | , E shiftJℓ = Re R drΛ (1) ℓ, J (r) φ ′ ℓ (r ′ ) R dr ′ K ℓ (r , r ′ )Λ (1)J ,ℓ (r ′ ) φ ′ ℓ (r ′ ),Γ q = " ( q f + ǫ ) ǫ + 1 Γ J (10)and E shiftq = " ǫ ( q f − − q f ǫ + 1 ¯ h Γ J . (11)Finally, we have ψ E ′ ℓ = e iη ℓ ψ E ′ ℓ + e iη V E ′ + A E ′ ( q f − i ) π Λ V E ′ ( ǫ + i )Γ f / δ ℓ × Z dr ′ K ( r, r ′ ) V ( r ′ ) χ ( r ′ ) + A E ′ Z dr ′ K ℓ ( r, r ′ )Λ (1) ℓ,J ( r ′ ) φ vJ ( r ′ ) (12)where ψ Eℓ = ψ ,regEℓ . The equations (8) and (12) constitute the solutions of our model.The elastic scattering amplitude is given by f ℓℓ ′ = (1 / ik )( δ ℓℓ ′ − S ℓℓ ′ ) = T ℓℓ ′ /k wherethe S -matrix element S ℓℓ ′ is related to the T − matrix element T ℓℓ ′ by S ℓℓ ′ = δ ℓℓ ′ − iT ℓℓ ′ . agneto-optical Feshbach resonance
700 710 720 730 740 75010 −14 −13 −12 −11 −10 −9 −8 pa r t i a l sc a tt. c r o ss s e c t i on ( c m ) B (Gauss)
700 720 7400204060 E q s h i ft Bp−waved−wave
Figure 2.
Partial-wave scattering cross section σ ℓ is plotted as a function of B for ℓ = 1 (solid line) and ℓ = 2 (dashed lines) for Γ J /γ = 10 .
0, Γ J = 0 . J ,Γ J = 10 − Γ J , E = 10 µ K and q f = − .
89. The inset shows E shiftq (in unit of ¯ h Γ mf )as a function of B for the same parameters as in the main figure. The other parametersare same as in figure1 We can now derive T ℓℓ ′ from the asymptotic behaviour the wave function of (12) whichis given by ψ E ′ ℓ ( r → ∞ ) ∼ sin( kr − ℓ ′ π/ δ ℓℓ ′ − T ℓℓ ′ exp( ikr − ℓπ/ σ el = P ℓ ′ ,m ℓ ′ P ℓ,m ℓ σ ℓℓ ′ where σ ℓℓ ′ = 4 πg s | T ℓℓ ′ | /k , with g s = 1 for two distinguishable atoms and g s = 2 if the atoms are indistinguishable.
4. Results and discussions
We first consider the s-wave ( ℓ = 0) scattered wave function. From the asymptoticform ψ E ′ , ∼ e iη ψ ,regE ′ , − e i ( kr + η ) [ e iη + A E ′ ( q f + ǫ ) π Λ ] / ( ǫ + i ), we find T = T + exp(2 iη ) T mf + exp[2 i ( η + η mf )] T q = (1 − S ) / i where T = − exp( iη ) sin η , T mf = 1 / ( ǫ + i ) = − exp( iη mf ) sin η mf where the MFR phase shift η mf is given bycot η mf = − ǫ , T q = Γ q / [ D − E shiftq + i ¯ h ( γ J + Γ J )]. Here Γ J = P ℓ Γ Jℓ . In the limit k →
0, Γ Jℓ ∼ k ℓ +1 and hence Γ J >> Γ Jℓ =0 for all ℓ ≥
1. The S -matrix elementis S = exp(2 iη tot ), where η tot = η + η mf + η q with η q being a complex phase shift.Since in the limit k → q f ∼ /k , near MFR ( ǫ ≃
0) the stimulated linewidthΓ J ≃ Γ q ≃ q Γ J ∼ /k , E shiftq ≃ q f ¯ h Γ J and cot η q = − [ D − E shiftq + iγ J ] / Γ q . Thus inthe limit γ → k → T fulfills unitarity.The s-wave elastic scattering cross section is σ = g s π | − S | /k andthe inelastic cross section is σ inel = g s π (1 − | S | ) /k . The corresponding ratecoefficients are given by K el = h v rel σ el i and K inel = h v rel σ inel i where h· · ·i stands forthermal averaging over the relative velocity v rel = ¯ hk/µ . Far from MFR ( ǫ → ±∞ )we have T mf → E shiftq → q → Γ J . In this limit T q reduces to the form agneto-optical Feshbach resonance T of = − Γ J / [ D + i ¯ h ( γ J + P ℓ Γ Jℓ ] which is the T -matrix element of standard OFR forwhich both elastic and inelastic scattering rates increase as laser intensity increases [42].We can define an energy-dependent complex MOFR scattering length by a mof = − tan η tot /k . In the limit k → a mof ≃ a mf + q f ¯ h Γ J / [ k ( D − E shiftq + i ¯ hγ J )]1 + ka mf q f ¯ h Γ J / ( D − E shiftq + i ¯ hγ J ) (13)where a mf = − lim k → tan η mf /k is the MFR scattering length. Since ( kq f Γ J )tends to be independent of k at ultralow energy, it is possible to have the conditionRe[ ka mf q f ¯ h Γ J / ( D − E shiftq + i ¯ hγ J )] >> a mf → ±∞ ) andPA resonance ( D ≃
0) in the strong-coupling regime (Γ J >> γ J ). Note that D = ∆ E vJ − P ℓ E shiftJℓ = 0 is the PA resonance condition in the absence of MFR.Furthermore, it is to be noted that E shiftq as given by (11) is independent of k in thelimit k → ǫ → γ J in thestrong-coupling regime[27]. Under such conditions, we can write a mof ≃ D − E shiftq kq f ¯ h Γ J + 1 k a mf ! + i γ J kq f Γ J ! . (14)Let us recall that a mf = − / ( kǫ ) = − ¯ h Γ mf / [2 k ( E ′ − ˜ E χ )], where ˜ E χ = E χ + E shiftχ and E ′ = ¯ h k / (2 µ ). Therefore, in the case of finite ˜ E χ > E ′ , the real part of a mof (Re[ a mof ]) becomes inversely proportional to energy and hence σ el ∼ /k as k →
0. Inthe case of ˜ E χ = 0, Re[ a mof ] goes to a constant in the limit k →
0. In both the cases, theimaginary part of a mof (Im[ a mof ]) becomes independent of k but inversely proportionalto laser intensity suggesting that K inel can be made very small by increasing the laserintensity. On the other hand, for D = 0, the (14) indicates that Re[ a mof ] becomesindependent of laser intensity. Thus we can infer that the inelastic scattering rate canbe suppressed while elastic rate can be enhanced by using quantum interference in thestrong-coupling regime at ultralow temperatures. Very recently, Bauer et al. [23, 43]have experimentally demonstrated the effect of suppression of inelastic rate in PA dueto the influence of a magnetic Feshbach resonance.The amplitudes of higher partial-wave scattered wavefunctions can also be enhancedby MOFR. The higher partial waves that can be manipulated are given by the condition ~J = ~L + ~S + ~ℓ . In the case of singlet to singlet PA transition for J = 1, the maximumpartial-wave that can be significantly affected is ℓ = 2 (d-wave), while in the case oftriplet to triplet transition it is ℓ = 3. For ℓ = 0, we have T ℓ = πA E ′ exp( iη ℓ )Λ Eℓ,vJ .Using (8), in the leading order in dipole coupling at ultralow energy we have T ℓ ≃ e i ( η + η ℓ ) ( q f + ǫ ) / ( ǫ + i ) π Λ Λ Eℓ,vJ
D − E shiftq + i ¯ h ( γ J + Γ q ) / ǫ → ∞ , T ℓ, reduces to that of OFR[37] for ℓ ≥ agneto-optical Feshbach resonance −13 −12 −11 −10 −9 −8 −7 −6 −5 σ , σ i ne l ( c m ) E (nK) −5051015 E q s h i ft −10 −5 σ , σ i ne l σ σ σ σ inel σ inel σ inel Figure 3. σ and σ inel are plotted as a function of collision energy E (in nK) for B = 730 G (solid and dashed curves), B = 700 G (plus-solid and plus curves) and B = 800 G (solid-dotted and dotted curves) with Γ J / Γ mf = 1 .
4. The upper insetshows the same but for Γ J / Γ mf = 0 .
07. In the lower inset, E shiftq (in unit of ¯ hγ ) isplotted against E for B = 730 G (solid line), B = 700 G (dotted line) and B = 800 G(dashed lines) with Γ J / Γ mf = 1 .
4. The other parameters remain same as in figure1
To illustrate further the analytical results discussed above, we present selective numericalresults. As a model system, we consider Li atoms with PA transition Σ + u → Σ + g .The parameter ǫ is related[44] to the magnetic field B , the resonance width ∆ andthe background scattering length a bg by ǫ ≃ − ( B − B ) / ( ka bg ∆), where B is theresonance magnetic field. We use the realistic parameters taken or estimated fromearlier experimental results[45, 46]. These parameters are the spontaneous line width γ J = 11 . − . a bg = − . a ( a is Bohr radius).We take B = 730 . q f = − .
89 at E = 10 µ K. Using low energy behaviour q f ∼ /k , we extrapolate q f atother collision energies. The Feshbach resonance line width Γ mf is taken to be 16.66MHz for E = 10 µ K. In all our numerical plots we set D = 0.In figure 1 (a-c), σ as a function of B is compared with σ inel . We notice that,compared to weak-coupling results of figure 1(a), the strong-coupling result σ in figure1(b) largely exceeds σ inel in almost entire range of B . Because of interference betweenthe two resonances, two closely spaced maxima appears near B in figure 1(b). Evenin figure 1(a), there is a prominent maximum at and near which σ exceeds σ inel . Thereason for such feature is that, as can be inferred from (14), for a given collision energyand D = 0, Re[ a mof ] becomes independent of laser intensity as ǫ → a mof ]goes to zero in the strong-coupling regime. Figure 1(c) shows that at much lower energy( E = 100 nK) inelastic scattering rates are further suppressed while elastic ones areenhanced both in weak- and strong-coupling regimes. figure 1(d) illustrates how MFR agneto-optical Feshbach resonance B . The minimum at B = 710 G arises due to Fano minimum atwhich PA transition amplitude vanishes.We show the partial p- and d-wave scattering amplitudes in figure 2 in the strongcoupling regime. Typically, the higher partial-wave stimulated line width Γ Jℓ =1 andΓ Jℓ =2 are smaller than Γ Jℓ =0 by one and four order of magnitudes, respectively[37].Comparing figure 2 with figure 1(b), we notice that p- and d-wave scattering crosssections show a maximum near B at which σ ℓ =1 , is of the same order of σ whileΓ Jℓ =2 is 3 order of magnitude smaller than that σ . The minimum near B ≃
730 Gcan be attributed to the quantum interference induced anomalously large positive shiftas shown in the inset of figure 2.Figure 3 shows energy dependence of elastic and inelastic scattering cross sectionsat three different values of B in both the strong- (main figure) and weak-coupling (upperinset) regimes. The main figure and the upper inset clearly show that when B = 730G which is close to B , the elastic part of scattering cross section largely exceeds theinelastic part in the low energy regime. We notice that elastic scattering cross section σ (solid curve) at E = 10 nK and B = 730 G exceeds the inelastic scattering crosssection σ inel (dashed curve) by two orders of magnitudes. In contrast, this does nothappen if B is tuned far away from B . For instance, when B = 700 G and E = 10 nK, σ (plus solid curve) is smaller than σ inel (plus curve) by two orders of magnitude. Theeffect of laser intensity on the scattering cross sections at low energy can be understoodby comparing the main figure with the upper inset of figure 3. The stimulated line width(Γ J ) in the strong-coupling regime (main figure) is taken to be twenty times larger thanthat in the weak-coupling regime (upper inset). In other words, PA laser intensity forstrong-coupling case is taken to be twenty times larger compared to the weak-couplingcase. Let us now compare the plots of the main figure with the corresponding plots ofthe upper inset: When B is tuned close to B or MFR, the elastic scattering cross section σ (solid curve) for strong- (main figure) as well as weak-coupling (upper inset) regimetends to be equal as the energy E decreases. At E = 10 nK, we find σ ≃ . × − cm in both the regimes. In contrast, when B = 700 G which is away from MFR, σ (plus solid curves) at E = 10 nK for weak- and strong-coupling regimes are 1 . × − cm and 9 . × − cm , respectively. Thus in conformity with our previous analysis,by comparing the plots in the main and in the upper inset of figure 3, we can infer thatwhen B is tuned near B , the elastic cross section at low energy becomes independent oflaser intensity. The minimum at B ≃ B in σ Vs. E plots of figure 3 can be attributedto the large positive shift E shiftq as depicted in the lower inset of this figure.
5. Conclusions and outlook
Quantum interference is shown to change threshold and resonance behviour significantly.This may in turn change the character of near-zero energy dimer states. Therefore, thecrossover physics between Bardeen-Cooper-Schrieffer (BCS) state of atoms and Bose- agneto-optical Feshbach resonance
Appendix-A
We discuss how to derive (8). Using K ℓ we first convert (4) (with the index M = m ℓ being suppressed) into an integral equation of the form˜ ψ Eℓ ( r ) = exp( iη ℓ ) ˜ ψ Eℓ + Z dr ′ K ℓ ( r, r ′ ) × h Λ (1) ℓ,J ( r ′ ) φ vJ ( r ′ ) + V ( r ′ ) χ ( r ′ ) δ ℓ i (A.1)Substituting φ vJ = R dE ′ β E ′ A E ′ φ vJ and (6) into (A.1), we get ψ E ′ ℓ = e iη ℓ ψ E ′ ℓ + Λ bb A E ′ + V E ′ E − E χ δ ℓ × Z dr ′ K ( r, r ′ ) V ( r ′ ) χ ( r ′ )+ A E ′ Z dr ′ K ℓ ( r, r ′ )Λ (1) Jℓ ( r ′ ) φ vJ ( r ′ ) (A.2)Putting the above equation for ℓ = 0 ( ψ E ′ ) into the equation V E ′ = R drψ E ′ ( r ) V ( r ) χ ( r ) and after a minor algebra we obtain V E ′ = ( E − E χ ) [ e iη V E ′ + A E ′ ( V eff − iπ Λ V E ′ )] E − ( E χ + E shiftχ ) + i Γ f / A E ′ Λ bb (cid:16) E shiftχ − i Γ f / (cid:17) E − ( E χ + E shiftχ ) + i Γ mf / A E ′ . Now, substituting (A.2) and (A.3) into (7) and using ǫ = [ E − ( E χ + E shiftχ )] / (Γ mf /
2) and the parameter q f defined by (9), we obtain (8). Thus (12) isfinally expressed in terms of all the known or unperturbed parameters. References [1] Tiesinga E., Verhaar B. J. and Stoof H. T. C. 1993
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