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Quantum Interference Effects In Atom-Atom And Ion-Atom Cold Collisions In The Presence Of External Fields THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (SCIENCE) IN CHEMISTRY (PHYSICAL CHEMISTRY) BY ARPITA RAKSHIT Department of Chemistry University of Calcutta 2014 To my parents . . . . . . To Be An Achiever, You Got To Be A Believer First . . . A. Einstein Acknowledgements Before writing the thesis I want to take the opportunity to thank all the people who provided enormous support, continuous encouragement and inspiration throughout my Ph.D. research work. First of all, I would like to express my earnest gratitude to my supervisor Professor Bimalendu Deb for his continuous support, encouragement and advices during my study and research work at I.A.C.S. He has also taught me to aim high, to be persevering, and above all, to embrace scientific challenges with passion and enthusiasm. I thank him for his insight, guidance and caring support, and for so much I have learned from him. I will never forget his strong support during my difficult times. I express my gratitude to Professor Deb Sankar Ray, Professor S. P. Bhattacharyya, Professor Debashis Mukherjee and Professor Kamal Bhattacharyya for their kind help. I thank Sumantada ( Dr. Sumanta Kumar Das, post-Doctoral fellow, The Niels Bohr Institute) and Saikatda (Dr. Saikat Ghosh, I.I.T., Kanpur) as collaborators for their valuable discussions, suggestions and help when I needed. I thank Indian Association for the Cultivation of Science (IACS) for providing all types facilities needed to carry out my work. I thank all the faculty and staff members of Material Science Department for help and support. The fellowship provided by the Council of Scientific and Industrial Research (CSIR) is gratefully acknowledged. My thanks also go to the current and former members of our lab, with whom I have fond memories and from whom I have learned much and received a lot of help. Jishadi, Debashree, Biswajit, Partha, Somnathda, Arpita, Somnath, Farzana and Soham, thank you! I would like to acknowledge my heartiest gratitude to my excellent teachers of my undergraduate and master degree classes. I thank specially Dr. Rana Sen, Dr. Anupa Saha, Dr. Pradipta Ghosh, Dr. Samrajnee Dutta and Dr. Durba Barik for their care, love and continuous encouragement. I thank Antaradi, Moumitadi, Shrabanidi, Namrata, Partha, Tuluma, Ananya, Avijit and my other numerous friends who make my stay at IACS enjoyable. I specially thank Shreetama and Shubhro for their valuable friendship, constant support and help in critical situations. In this joyful moment, I want to mention specially Sohini, Kamalika and Anindita from B. Sc. batch of Scottish Church College who have provided me constant motivation. I would like to thank my college, Sidhu Kanhu Birsa Polytechnic, Keshiary and Department of Technical Education and Training, Govt. of West Bengal for giving me the permission for continuing my Ph.D. work after my joining as lecturer in May, 2012. I would like to acknowledge Dr. Parijat De, Director of Technical Education and Training, Govt. of West Bengal, Dr. Amit Ranjan Ghatak, exPrincipal and Mr. Asok Kumar Deb, Officer-In-Charge of my college for allowing me to devote my off-time to research. I thank all my colleagues at I.C.V. Polytechnic, Jhargram and S. K. B. Polytechnic, Keshiary. I want to mention specially Mr. Ushnish Sarkar, Mr. Arunanshu Das, Mr. Satyam Paul and Dr. Subrata Kamilya for their constant encouragement. Lastly I thank my parents. They have contributed much more than I can say. I am overwhelmed having the scope to acknowledge them. I am thankful to my husband, Dr. Uttam Kumar Das whose continuous moral support and love motivated me to remain focused towards achieving this milestone of my journey. IACS Kolkata June, 2014 ARPITA RAKSHIT Contents Acknowledgements 1 Introduction 1.1 Preliminary Discussions . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 2 Quantum Interference in Atoms and Molecules: A Review 7 2.1 Fano Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Spontaneous Emission and Vacuum-Induced Coherence . . . . . . . 12 2.3 Quantum Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Fano-Feshbach Resonances 3.1 Ultracold Scattering and Resonances . . . . . . . . 3.1.1 Elastic Scattering . . . . . . . . . . . . . . . 3.1.2 Scattering Length in Ultracold Gases . . . . 3.1.3 Scattering Resonances at Low Energy . . . . 3.2 Single Channel Resonance . . . . . . . . . . . . . . 3.2.1 Shape Resonance . . . . . . . . . . . . . . . 3.2.2 Potential Resonance . . . . . . . . . . . . . 3.3 Feshbach Resonance : A Multichannel Resonance . 3.3.1 Magnetic Feshbach Resonance . . . . . . . . 3.3.2 Scattering Length and Feshbach Resonance . 3.4 Photoassociation and Optical Feshbach Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Quantum Interference in Photoassociation in the Presence of Feshbach Resonance 4.1 Perspective of The Work . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Formulation of the Problem and Solution . . . . . . . . . . . . . . . 4.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Numerical Results and Discussion . . . . . . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 20 21 22 23 24 24 25 26 26 27 28 35 35 36 43 43 47 48 Contents 5 Vacuum- and Light-Induced and Molecules 5.1 Perspective of The work . 5.2 Scheme 1 . . . . . . . . . . 5.3 Solution . . . . . . . . . . 5.4 Results and Discussions . 5.5 Scheme 2 . . . . . . . . . . 5.6 Master equation . . . . . . 5.7 Solution . . . . . . . . . . 5.8 Results and discussions . . 5.9 Conclusion . . . . . . . . . Coherences in Cold Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Atom-Ion Cold Collisions: Formation of Cold 6.1 Theory of Atom-Ion Collision . . . . . . . . . 6.1.1 Atom-Ion Interaction Potential . . . . 6.1.2 Elastic Collisions . . . . . . . . . . . . 6.1.3 Radiative Transfer . . . . . . . . . . . 6.1.4 Spin Exchange Collision . . . . . . . . 6.2 Fomation of Cold Molecular Ion . . . . . . . . 6.2.1 Formulation of Problem . . . . . . . . 6.3 Results and Discussion . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . 7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 52 53 56 59 62 69 72 74 83 . . . . . . . . . 88 89 91 92 94 97 98 99 103 109 115 Chapter 1 Introduction 1.1 Preliminary Discussions The development of laser cooling and trapping of atoms (Nobel Prize 1997) [1] in the 80’s and early 90’s, culminating in the realization of Bose-Einstein-Condensation in dilute atomic gases (Nobel Prize 2001) [2] in 1995, has opened a new doorway for atomic, molecular and optical sciences. Using laser, Doppler and evaporative cooling cold atoms in quantum degenerate regime are now routinely obtained. Coherent control of external and internal degrees-of-freedom of cold atoms are made possible by optical precision spectroscopy (Nobel Prize 2005) [3]. At sub-Kelvin temperatures, atoms move extremely slowly, offering larger observation time required for high resolution spectroscopy. Following the great success in cooling atoms and exploring physics and chemistry with cold atoms, now the focus of interest has also encompased cold molecules. Cold molecules with ro-vibrational structures can reveal many novel physics which can not be observed in atomic systems. Now the problem is that producing cold molecules in a straightforward way is difficult. Though most successful for atoms, the method of laser cooling is not in general applicable to molecules due to their complicated level structures. Laser cooling requires a cyclic transition which repeats absorption of cooling lasers and spontaneous emission. Laser cooling of a molecule is extremely difficult since a large number of extra repumping lasers would be required to avoid optical pumping to the ro-vibrational levels irrelevant to the cooling transition. Different technologies like supersonic expansion, cryogenic cooling technology etc. have been used to cool molecules with limited success. 1 Chapter 1. General Introduction 2 Alternatively, two indirect methods are now used for producing cold molecules by associating cold atoms, namely photoassociation (PA) [4] and Feshbach Resonance (FR) [5]. In both processes, translationally cold atoms are used as an initial source and they are transferred to translationally cold molecules by association process using external electromagnetic fields. Since collisions between cold atoms occur for lower partial waves, the rotational temperature of formed molecules is also low. Though this approach is limited to molecules whose constituent atoms can be cooled by laser cooling, it may be regarded as one of the more promising ways to achieve quantum degenerate molecular gases. Cold atoms and molecules are important for exploring new physics and chemistry [6]. Research areas such as molecular chemistry [7], condensed matter physics [8] and astrochemistry [9] are revitalized with advancement in cold atom physics. Cold atoms and molecules can serve as good candidates for observing quantum interference and coherence. We are quite familiar with interference phenomena in our everyday life. For example, if we throw two pebbles in a quiet pond, waves produced by them interact and produce an intricate pattern of crests and troughs as a result of constructive and destructive interferences between them. This can be well described with help of classical physics. Idea of quantum interference can be described similarly. When the atoms are driven by electromagnetic fields, they undergo transitions from an initial to a final state. In some instances, these transitions can occur through multiple indistinguishable but independent pathways which may interfere leading to the destructive or constructive interference effects. Interference effects are mostly studied in atomic systems and seldom in molecular systems. But ultracold atom-molecule coupled systems obtained via free-bound transitions have not yet been addressed adequately so far. Main objective of this thesis is to explore different quantum interference phenomena in these systems. 1.2 Scope of the Thesis Main objective of this thesis is to investigate quantum interference (QI) phenomena in an ultracold atom-molecule coupled system in the presence of external fields [10] within the framework of celebrated Fano theory [11]. Fano effect arises due to interaction between configurations of discrete levels and continuum states. Two excitation pathways leading to continuum states result in coherent interference which leads to asymmetric absorption spectra. Chapter 1. General Introduction 3 To show the effects of QI in the atom-molecule coupled system, we investigate photoassociation (PA) in the presence of magnetic Feshbach resonance (MFR) [12]. We work in dressed basis using Fano method. Solution of the problem gives the analytical expression for linewidth and shift for any arbitrary laser intensity. We show that power broadening of line width can be suppressed even at high laser intensities by tuning the magnetic field close to Fano minimum. We also demonstrate that Fano effect can lead to large light shift near Fano minimum. These features arising from quantum interference are useful for efficient tuning of scattering length by optical means. We propose a novel PA scheme for realization of vacuum induced coherence (VIC) [13] in atom-molecule coupled systems [20]. To the best of our knowledge, the possibility of VIC in a PA system has not been discussed earlier. We consider PA transitions from the collisional continuum of two atoms to two excited rovibrational levels belonging to same excited molecular electronic state and the excited levels decay spontaneously due to interaction with the electromagnetic vacuum. The two spontaneous emission pathways interfere resulting in VIC. We take bosonic Ytterbium (174 Yb) atoms for numerical illustration of our theoretical proposal. We solve our model using both Wigner-Weisskopf and master equation approaches and demonstrate theoretically an interesting interplay between VIC and PA. We show that VIC between two ro-vibrational levels arises due to the quantum interference between spontaneous emission pathways from the rovibrational levels to the electronic ground molecular state. We also investigate this system driven by two lasers using master equation approach. We study the temporal dynamics of the excited bound states and demonstrate quantum beats in emission spectra as a signature of QI. We also demonstrate laser induced coherence (LIC) between two excited molecular states [15]. Thus it is possible to control decoherences and spontaneous decay. Finally, we investigate elastic and inelastic (charge transfer) collisions between atoms and ions at low temperatures and discuss formation of cold molecular ions by atom-ion photoassociation. Cold molecular ions may be obtained by various processes such as photoassociative ionization, buffer gas and rotational cooling, sympathetic cooling etc. Possibly, this is the first theoretical demonstration of creating translationally and rotationally cold molecular ions by photoassociation [16]. Chapter 1. General Introduction 1.3 4 Outline of the Thesis Before studying in detail, we briefly provide a survey of the literature on quantum interference in atoms and molecules in chapter 2. QI phenomena like electromagnetically induced transperancy, coherent population trapping, stimulated Raman adiabatic Passage (STIRAP), slow light that appear in three level systems are well known. Fano interference in ultracold atom-molecule coupled systems will lead to analogous QI effects in new parameter regimes. We discuss in short Fano interference, vacuum-induced coherence and quantum beating. The continuum-bound spectra in atom-molecule coupled systems are discussed in chapter 3. Description of different single- and multi-channel resonances are also given. We mainly focus on Fano-Feshach resonance and discuss its tunability in the presence of magnetic and optical field. Photoassociation in the presence of magnetic Feshbach resonance is addressed in chapter 4. We present a theoretical model and solve it analytically. Then we apply it to a realistic system to verify the prediction of our theory and finally analyze our numerical results. In chapter 5 vacuum-induced and light-induced coherences in atom-molecule coupled systems are studied. We consider that two excited ro-vibrational levels in the same electronic molecular potentials are coupled to the continuum of scattering states of two ground-state atoms. We solve the problem in two ways. In the first way we include the spontaneous decay terms using Wigner-Weisskopf approach. In other case we solve it using master equation approach. Chapter 6 is devoted to atom-ion scattering at cold and ultracold temperature regimes. We also discuss different radiative and non-radiative processes. Here we propose a new formalism for the formation of translationally and rotationally cold molecular ions photoassociation. Lastly, we conclude and give an outlook of our studies in chapter 7. References [1] C. N. Cohen-Tannoudji, Rev. Mod. Phys. 70, 707 (1998); W. D. Phillips, Rev. Mod. Phys. 70, 721 (1998); S. Chu, Rev. Mod. Phys. 70, 685 (1998) [2] W. Ketterle, Rev. Mod. Phys. 74, 1131 (2002); E. A. Cornell and C. E. Wieman, Rev. Mod. Phys. 74, 875 (2002). [3] J. L. Hall, Rev. Mod. Phys. 78, 1279 (2006); R. J. Glauber, Rev. Mod. Phys. 78, 1267 (2006); T. W. Hansch, Rev. Mod. Phys. 78, 1297 (2006). [4] P. Fedichev, Y. Kagan, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. lett. 77, 2913 (1996); K. M. Jones, E. Tiesinga, P. D. Lett and P. S. Julienne, Rev. Mod. Phys. 78, 483 (2006). [5] H. Feshbach, Ann. Phys. 5, 357 (1958); T. Köhler, K. Góral and P. S. Julienne, Rev. Mod. Phys. 78, 1311 (2006). [6] O. Dulieu, R. Krems, M. Weidemüller and S. Willitsch, Phys. Chem. Chem Phys. 13, 18703 (2011); L. Carr, D. DeMille, R. V. Krems, J. Ye, New J. Phys. 11, 055049 (2009) and references therein. [7] R. Krems, Physics 3, 10 (2010) and references therein. [8] G. Pupillo, A. Micheli, H. P. Büchler and P. Zoller, arXiv:0805.1896 (2008); W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet and M. Greiner, Science 329, 547 (2010); M. Aidelsburger, M. Atala, M. Lohse, J. T. Barrelro, B. Paredes and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013). [9] I. W. M. Smith, Low Temperature and Cold Molecules (Imperial College Press, 2008). [10] B. Deb, A Rakshit, J Hazra and D. Chakraborty, Pramana J. Phys. 80, 3 (2013). 5 Chapter 1. General Introduction 6 [11] U. Fano, Phys. Rev. 124, 1866 (1961). [12] B. Deb and A. Rakshit, J. Phys. B: At. Mol. Opt. Phys. 42, 195202 (2009). [13] G. S. Agarwal, Springer Tracts in Mosern Physics: Quantum Optics (Springer-Verlag, Berlin, 1974). [20] S. Das, A. Rakshit, and B. Deb, Phys. Rev. A 85, 011401(R) (2012) [15] A. Rakshit, S. Ghosh and B. Deb, J. Phys. B: At. Mol. Opt. Phys. 47, 115303 (2014). [16] A. Rakshit and B. Deb, Phys. Rev. A 83, 022703 (2011) Chapter 2 Quantum Interference in Atoms and Molecules: A Review Quantum interference is one of the most profound effects of quantum mechanics. Feynman referred it as ‘the only mystery’ of quantum mechanics. The field of optical interference has a very old history back in early nineteenth century when Thomas Young performed his famous double slit experiment[1–3]. In general, interference means the superposition of two or more coherent waves resulting in reinforcing or neutralizing effects. Atoms or molecules interacting with electromagnetic fields undergo transitions from initial to final states. Sometimes these may occur through multiple indistinguishable pathways which may interfere enhancing a desired process or suppressing the undesired one under suitable conditions. The coherence signifies the correlation or matching between phases or amplitudes of interfering pathways. Quantum interference has applications in devices such as different types of interferometers [2], super computing quantum device(SQUID), quantum cryptography, quantum computing [4] etc. Light interacting with a two-level system is a primitive case and has been widely studied [1]. When a multilevel system interacts with electromagnetic fields it can display non-linear optical behaviour which is of great interest for quantum optics researchers. The most simple system is a three-level atomic system. A three-level system can be of three different types: 1) Vee , 2) Lambda and 3) Cascade which are shown in Fig.2.1. Now let us consider a three-level system interacting with two nearly-resonant coherent fields. Each field connects a separate transition, but both transitions share a common energy state. These two pathways can interact 7 Chapter 2. Quantum Interference (a) Vee-type 8 (b) Lambda-type (c) Cascade-type Figure 2.1: Different types of three-level systems. with each other resulting in new and counter-intuitive observations. Interference occurring within a single atom or molecule in the presence of electromagnetic fields can lead to many interesting physical effects such as electromagnetically induced transparency (EIT)[5], coherent population trapping (CPT)[6], lasing without inversion(LWI) [7], Rabi oscillation (RO)[8], Autler Townes splitting (AT) [9], slow light [10], STIRAP [11] , vacuum-induced coherence (VIC)[12] , quantum beating (QB) [13] etc. Most of the studies on quantum interference have been done in case of atoms. With the development of the fields of cold and ultra atoms and ions, continuum-bound transitions in atom-molecule coupled system have started to draw attention. In this thesis, our aim is to discuss the quantum interference effects in case atommolecule coupled systems. Our primary focus is on Fano interference. In 1961, Ugo Fano studied the interference among the configurations of discrete level(s) to a continuum [14]. Two ionization pathways interfere leading to an asymmetric peak which is known as Fano profile. This method is useful to discuss the interference effects in atom-molecule coupled system which involves at least one transition from continuum to a bound state. Therefore, we can obtain dressed continuum as discussed in Fano’s theory. In next section, Fano interference will be discussed in Chapter 2. Quantum Interference 9 10 q=0 q=1 q=3 8 Sq(E) 6 4 2 0 -10 -5 0 ε 5 10 Figure 2.2: natural line shapes for different values of q [14]. short. Apart from Fano interference, another two interesting quantum interference effects will play key role in chapter 5 and hence they need short introduction in the perspective of atom-molecule coupled systems. They are vacuum induced coherence (VIC) and quantum beating. In the following subsections, we shall discuss these three effects in short. 2.1 Fano Interference Fano interference [14] is named after famous scientist Ugo Fano. In 1961, Fano studied the coupling of discrete configurations with a continuum of states in case of autoionization. Let us consider a simple case of an autoionizing discrete level interacting with a continuum. Two ionization pathways, one direct and another through autoionizing state, interfere resulting in asymmetric spectral profile. First, Let us consider an atomic system with discrete level φ of energy Eφ and a continuum of states ψE ′ . Each of these states is assumed to be non degenerate. Next, the energy sub-matrix belonging to the subset of states φ, ψE ′ will be Chapter 2. Quantum Interference 10 diagonalized. Its elements form a square sub-matrix: hφ|H|φi = Eφ , (2.1) hψE ′ |H|φi = VE ′ , (2.2) hψE |H|ψE ′ i = E ′ δ(E − E ′ ). (2.3) The discrete energy level Eφ lies within the continuous range of values. Fano treated the problem in dressed picture. The dressed eigenvector of the energy matrix is assumed to have the form: χE = aφ + Z dE ′ bE ′ ψE ′ , (2.4) where the dressed coefficients a and bE ′ are the function of E. These coefficients are obtained as solutions of the system of eqs. (2.1) to (2.3): Eφ a + Z dE ′ VE∗′ bE ′ = Ea′ , VE ′ a + E ′ bE ′ = EbE ′ . The formal solution can be written as 1 ′ bE ′ = + z(E)δ(E − E ) VE ′ a. E − E′ (2.5) (2.6) (2.7) The asymptotic behaviour of χE is now compared to the continuum. If ψE ∝ sin(kr), where E = Z k 2 ~2 2m and m is the mass of the system, then at asymptotic limit dE ′ bE ′ ψE ′ ∝ −π cos(kr) + z(E) sin(kr) = sin(kr + ∆). (2.8) Here, ∆ = − arctan[π/z(E)] represents the phase shift due to the interaction between the continuum of states ψE and discrete level φ. z(E) can be expressed as z(E) = E − Eφ − F (E) |VE |2 (2.9) where, F (E) = P Z dE ′ |VE ′ |2 . E − E′ (2.10) Chapter 2. Quantum Interference 11 P stands for the principal value of the integral. The phase shift ∆ varies by ∼ π as E covers the intervals ∼ |VE |2 about the resonance at E = Eφ + F (E). Therefore, F (E) represents shift from resonance position of discrete level φ. With proper normalization, the dressed amplitudes are given by sin ∆ , πVE VE ′ sin ∆ = − δ(E − E ′ ) cos ∆. ′ πVE E − E a = bE ′ (2.11) (2.12) Now if a transition takes place between a state |ii and the dressed state χE and T be the transition operator, the transition probability amplitude is given by hχE |T |ii = sin ∆ hΦ|T |ii − hψE |T |ii cos ∆ πVE (2.13) where, Φ=φ+P Z dE ′ |VE ′ ψE ′ E − E′ (2.14) is an admixture of the discrete level and the states of the continuum. The sharp variation of ∆ as E passes through resonance induces a sharp variation of hχE |T |ii. Since sin ∆ is an even function and cos ∆ is an odd function of (E − Eφ − F (E)), their contributions to hχE |T |ii by hΦ|T |ii and hψE |T |ii, respectively, ‘interfere with opposite phase on the two sides of the resonance’ [14], which is a characteristic feature of Fano resonances. The general line shape of of Fano resonance can be written as (ǫ + q)2 Sq (E) = 2 , (ǫ + 1) (2.15) Here, the reduced energy ǫ is given by ǫ = − cot ∆ = E − Eφ − F (E) Γ/2 (2.16) and q is the ratio of indirect resonant scattering and background scattering. The expression (2.15) shows that it gives rise to asymmetric line shapes as described in Fig.2.2. This line shape profile is known as ‘Fano profile’. q describes the degree of asymmetry in resonance. Chapter 2. Quantum Interference 12 We will apply this well known Fano theory to treat the atom-molecule coupled systems. 2.2 Spontaneous Emission and Vacuum-Induced Coherence Vacuum-induced coherence (VIC) takes place due to the interference between two pathways of transitions in system-vacuum interaction [12]. Vacuum does not mean some absolutely empty space. It is actually quantized three-dimensional multimode electromagnetic field. Normally, a vacuum state is denoted as |0ks i, where k and s denote vacuum field wave vector and polarization. Let us consider a two level atom interacting with the vacuum field. Just as light field drives an excited atom to emit stimulated emission, interaction of an atom with the electromagnetic vacuum results in spontaneous emission. As a result, state of the field changes from |0ks i → |1ks i. 1 signifies the emission of photon and the system moves from an excited to a ground state as shown in Fig.2.3. The overall state vector can be written as [1] |ψ(t)i = ca (t)|a, 0k i + X k cb,k (t)|b, 1ks i, (2.17) where ca and cb,ks are the coefficients of corresponding states. The interaction Hamiltonian is given by Hint = X [gks Ŝ + âks ei(ω−νks )t + H.C.]. (2.18) k Here ω and νks is the atomic transition frequency and frequency of the field, âks is the lowering operator of field and Ŝ + is the raising operator of the atom. ~E ~ vac |bi is the dipole coupling with d~ being the atomic dipole moment gks = −ha|d· p ~ vac = ~νks /2ǫ0 V ǫ~s . ~νks /2ǫ0 V is the amplitude of vacuum field, ǫ~s is the and E electric field polarization vector and V is the quantization volume. |gks |2 may be written as |gks|2 = ~νks |d1 |2 cos2 θ 2ǫ0 V (2.19) Chapter 2. Quantum Interference 13 Figure 2.3: two-level system interacting with electromagnetic vacuum. where, θ is the angle between d~ and ǫ~s . From the Schrödinger equation |ψ̇(t)i = −(i/~)Hint |ψ(t)i, one can obtain ċa (t) = − X ks |gks | 2 Z t ′ dt′ e(ω−νks )(t−t ) ca (t′ ). (2.20) 0 The summation over the modes of vacuum fields can be converted to an integral X k V →2 (2π)3 ∞ Z dkk 0 2 Z π dθ sin θ 0 Z 2π dφ. (2.21) 0 Integrating over θ and φ and putting k = νks /c, the Eq. (2.20) becomes 4d2 ċa (t) = − (2π)2 6~ǫ0 c3 Z ∞ 3 dνks νks 0 Z t ′ dt′ e(ω−νks )(t−t ) ca (t′ ). (2.22) 0 Considering that the frequency of the vacuum field is centered around atomic transition frequency, νk3 is replaced by ω 3 and the lower limit in the νks integration by −∞. The dνks integral can be solved as Z ∞ −∞ ′ dνks ei(ω−νks )(t−t ) = 2πδ(t − t′ ). (2.23) Chapter 2. Quantum Interference 14 So, under Wigner-Weisskopf approximation one can write γ ċa (t) = − ca (t) 2 (2.24) 1 4ω 3 d2 γ= 4πǫ0 3~c3 (2.25) where, is the spontaneous decay constant. So this is the origin of spontaneous emission. Now for demonstration of VIC, let us consider a Vee-type system consisting of two excited states |2i and |3i and a ground state |1i (as depicted in Fig.2.1(a)). The coupling of the system with the background vacuum fields causes spontaneous decay from the excited states to the same ground state. Now these two spontaneous emission pathways can interfere resulting in VIC. This effect can modify and even quench the spontaneous emission. Several studies have suggested to control the spontaneous emission by using external fields in atomic systems [12, 15–22]. Using VIC, Hegerfeldt and Plenio showed periodic dark states and quantum beats in a near-degenerate Vee-system [15]. VIC may lead to the modification and sometimes suppression of resonance fluorescence [15–17]. Elimination of spectral line and even the cancellation of spontaneous emission have been demonstrated as an application of VIC [18]. It may lead to phase sensitive absorption and emission profile [23–25]. It also has been found effective for controlling decoherence in quantum information processing [26]. All these schemes relies on two stringent conditions. First, The frequency splitting between two excited states must not exceed the natural line width of transitions, which means that the excited states must be near-degenerate. Second, the dipole moments d~1 and d~2 associated with transitions |2i → |1i and |3i → |1i, respec- tively must be non-orthogonal. If these two conditions are fulfilled, the resultant interference term can be written as γij = √ γi γj d~i .d~j , |d~i||d~j | (2.26) here i is not equal to j. This term changes the system dynamics. To meet up both the requirements is very hard for atomic systems. A possible realization of VIC for an excited atom interacting with an anisotropic vacuum [27–29] and utilizing the j = 1/2 → j = 1/2 transition in 198 Hg+ and 139 Ba+ [30, 31] has been Chapter 2. Quantum Interference 15 suggested. On the other hand molecules are the natural candidates for observing VIC [32] . The required orthogonality criteria is easily satisfied if the two excited states belong to same electronic configuration, but differs only in rotational or vibrational quantum numbers. But only a few ventures have been made in case of molecules [33]. Atom-molecule coupled system may provide itself as a better candidate. Using photoassociation (PA) spectroscopy, low-lying rotational levels in an excited electronic level can be selectively populated. VIC will be significant if (i) the ground state has no hyperfine interaction, (ii) these is no bound state close to ground state dissociation threshold and (iii) excited molecular levels have a long lifetime. To the best of our knowledge VIC in such PA systems has not been addressed. In this thesis, VIC in the context of atom-molecule coupled systems will be discussed for the first time by us [32]. In chapter 5, this will be demonstrated in detail. 2.3 Quantum Beats The phenomena of quantum beats are important for studying the quantum interference in the multilevel atomic or molecular systems. Quantum beats in radiation intensity arise from coherent superposition of two long-lived excited states. Such state superpositions and their manipulations are of considerable recent interest to create long-lived molecular-state qubits. The possibility of using quantum beats as a spectroscopic measure for quantum superposition was discussed as early as in 1933 [13]. Experimentally, spectroscopic study of quantum beats started since 1960s [34]. The use of lasers to create quantum superposition and detect resulting quantum beats in fluorescence started in early 1970s [35]. Forty years ago, Haroche, Paisner and Schawlow [36] demonstrated quantum beats in florescence light emitted from the excited hyperfine levels of a Cs atom as a signature of quantum superposition between the excited atomic states. Since then quantum beats in fluorescence spectroscopy have been studied in a variety of physical situations [1, 15]. These techniques open up new possibilities for studying excited state properties, state preparation and manipulation as well as collisional and spectroscopic aspects of ultra-cold atoms and molecules. Let us consider a Vee-type system, as shown in Fig.2.1(a). When the excited states |2i and |3i being coherently excited by an external source decay to the Chapter 2. Quantum Interference 16 same final state |1i with slightly different radiation frequencies, the interference between these two transition pathways gives rise to a periodic modulation of the intensity of emitted radiation with a modified frequency given by the difference of two frequencies. This phenomena is known as quantum beating. Quantum beats are manifested as oscillations in the emitted radiation intensity Iqb from two correlated excited states as a function of time which is given by [2, 15, 37] Iqb (t) = γ(ρ33 (t) + ρ22 (t) + 2Re[ρ23 (t)]). (2.27) Here ρnn is the population of the excited state | ni and ρ23 is the coherence between | 2i and | 3i. We consider, γ2 = γ3 = γ, where γn is the spontaneous line width of | nith excited state. References [1] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997). [2] Z. Ficek and S. Swain, Quantum Interference and Coherence (Springer, New York, 2007) [3] L. Mandel and E. Wolf, Optical coherence and quantum optics (Cambridge University Press, London, 1995) [4] M. Keyl, Phys. Rep. 369, 431 (2002). [5] O. Kocharovskaya, Y. I. Khanin, Sov. Phys. JETP 63, 945 (1986); S. Harris, Phys. Today 50, 36 (1997). [6] E. Arimondo and G. Orriols, Lettere al Nuovo Cimento 17, 333 (1976); E. Arimondo, Prog. in Opt. 35, 257 (1937); H. R. Gray, R. M. Whitley and C. R. Stroud, Jr. Opt. Lett. 3, 218 (1978). [7] S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989), M. O. Scully, S. Y. Zhu and A. Gavrielides, Phys. Rev. lett. 62, 2813 (1989). [8] I. I. Rabi, Phys. Rev. 51, 652 (1937) [9] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955). [10] W. J. Cromie, Physicists Slow Speed of Light (The Harvard University Gazette Retrieved) (1999). [11] N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann, Adv. At. Mol. Opt. Phys. 46, 55 (2001) 3 [12] G. S. Agarwal, Quantum Optics, Springer Tracts in Modern Physics, (Springer Verlag, Berlin, 1974). 17 Chapter 2. Quantum Interference 18 [13] G. Breit, Rev. Mod. Phys. 5, 91 (1933). [14] U. Fano, Phys. Rev. 124, 1866 (1961) [15] G. C. Hegerfeldt and M. B. Plenio, Phys. Rev. A 46, 373; ibid 47, 2186 (1993). [16] D. A. Cardimona, M. G. Raymer and J. C. R. Stroud, J. Phys. B. 15, 55 (1982). [17] P. Zhou and S. Swain, Phys. Rev. Lett. 77, 3995 (1996); Z. Ficek and S. Swain, Phys. Rev. A 69, 023401 (2004) [18] S. Y. Zhu, R. C. F. Chan and C. P. Lee, Phys. Rev. A 52, 710 (1995); S. Y. Zhu and M. O. Scully, Phys. Rev. lett. 76, 388 (1996); H. Huang, S. Y. Zhu and M. S. Zubairy, Phys. Rev. A 55, 744 (1997); H. Lee, P. Polykin, M. O. Scully and S. Y. Zhu, Phys. Rev. A 55, 4454 (1997). [19] G. S. Agarwal, Phys. Rev. A 55, 2457 (1997). [20] M. V. G. Dutt, J. Cheng, B. li, X. Xu, X. Li, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, S. E. Economou, R. Liu and L. J. Sham, Phys. Rev. Lett. 94, 227403 (2005). [21] D. J. gauthier, Y. Zhu, T. W. Mossberg, Phys. Rev. Lett. 66, 2460 (1991). [22] B. M. Garraway, M. S. Kim and P. L. Knight Opt. Commun. 117, 560 (1995). [23] S. Menon and G. S. Agarwal, Phys. Rev. A 57, 4014 (1998). [24] M. A. G. Martinez et al., Phys. Rev. A 55, 4483 (1997). [25] E. Paspalakis and P. L. Knigh, Phys. Rev. Lett. 81, 293 (1998); E. Paspalakis, C. H. Keitel and P. L. Knight, phys. Rev. A 58, 4868 (1998). [26] S. Das and G. S. Agarwal, Phys. Rev. A 81, 052341 (2010). [27] G. S. Agarwal, Phys. Rev. Lett. 84, 5500 (2000). [28] Y. Yang, J. Xu, H. Chen and S. Zhu, Phys. Rev. Lett. 100, 043601 (2008); J. P. Xu and Y. P. Yang, Phys. Rev. A. 81, 013816 (2010). [29] S. Evangelou, V. Yanopapas and E. Paspalakis, Phys. Rev. A 83, 023819 (2011). [30] M. Kiffner, J. Evers and C. H. Keitel, Phys. Rev. Lett. 96, 100403 (2006). Chapter 2. Quantum Interference 19 [31] S. Das and G. S. Agarwal, Phys. Rev. A 77, 033850 (2008). [32] S. Das, A. Rakshit and B. Deb, Phys. Rev. A 85, 011401(R) (2012). [33] H. R. Xia, C. Y. Ye and S. Y. Zhu, Phys. Rev. Lett. 77, 1032 (1996), L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi and A. M. Lyyra, et al., ibid 84, 4016 (2000). [34] A Corney and G. W. Series, Proc. Phys. Soc., 83, 213 (1964); J. N. Dodd, R. D. Kaul, and D. M. Warrington, Proc. Phys. Soc., London 84, 176 (1964). [35] T. W. Hansch, Appl. Opt. 11, 895 (1972); W. Gornik et al., Opt. Commun. 6, 327 (1972). [36] S. Haroche, J. A. Paisner and A. L. Schawlow, Phys. Rev. Lett. 30 948 (1973). [37] P. Zhou and S. Swain, J. Opt. Soc. Am. B 15, 2593 (1998). Chapter 3 Fano-Feshbach Resonances This chapter provides an overview of atom-atom scattering at low energy along with atom-molecule transitions in the presence of external fields. At the outset, a brief discussion on the ultracold scattering [5] is presented in section 3.1. The next section provides discussion on different scattering resonances [6–8]. We focus on Fano-Feshbach resonance. Both magnetic and optical Feshbach resonance are discussed. Feshbach resonances are used for controlling interactomic interactions [9], production of cold molecules [10, 11] and BEC-BCS crossover [12]. 3.1 Ultracold Scattering and Resonances Scattering theory provides the theoretical frame-work to describe collisions between particles. In our case we restrict our discussion to atom-atom scattering at low energy only. In elastic scattering kinetic energy of the system remains constant before and after the collision and thus the system remains in its initial state. This case may be treated as single channel scattering. In inelastic scattering the kinetic energy after a collision is not equal to that of the initial state and the system changes its state. This can be treated by the methods of multichannel scattering. 3.1.1 Elastic Scattering Let us consider two atoms colliding in the presence of an interaction potential V (r). If V (r) is spherically symmetric, the Hamiltonian is decoupled into radial 20 Chapter 3. Resonances 21 part and angular part. Using partial-wave decomposition, the effective potential can be written as ~2 ℓ(ℓ + 1) Vef f (r) = V (r) + . 2µr 2 (3.1) where, r is separation between two atoms and µ is the reduced mass of the system. ℓ denotes the quantum number corresponding to relative motion of two atoms or partial wave. The centrifugal barrier VCB = ~2 ℓ(ℓ+1) 2µr 2 is in general significantly higher than kinetic energy of colliding cold atoms. So, it suppresses the collisions with ℓ > 0 at low energy. The time independent partial-wave Schrödinger equation describing the system is given by 2 2 ~ d − + Vef f (r) ψk (r) = Eψk (r) 2µ dr 2 (3.2) where, E = ~2 k 2 /(2µ) and k denotes the wavenumber of atoms. For r → ∞, the total wave function Ψ(r) take the form ~ Ψ(r) ∼ eik·~r + f (k, θ) eikr . r (3.3) The first term represents the incoming plane wave and second term represents the scattered spherical wave. Here the scattering amplitude, f (k, θ) depends on potential V . The angle θ is between the direction of incidence and the direction of observation. For central potentials, the scattering amplitude can be expanded in terms of partial waves as f (k, θ) = X (2ℓ + 1)fℓ (k)Pℓ (cosθ) (3.4) ℓ where, Pℓ (cos θ) is Legendre polynomial and represents the angular part and the partial wave scattering amplitude is fℓ (k) = 2iδℓ (k) element Sℓ (k) = e 1 (Sℓ (k) − 1) 2ik with scattering matrix . δℓ (k) is the phase shift in outgoing wave generated due to scattering. The total cross section σ can be written as, σ(k) = Z |f (k, θ)|2dΩ = X ℓ 4π(2ℓ + 1)|fℓ (k)|2 = X 4π ℓ k (2l + 1) sin2 (δℓ ) (3.5) Chapter 3. Resonances 22 or, σ(k) = X σℓ (k) ℓ So we can say that the total cross section is the sum of partial cross sections, σℓ . 3.1.2 Scattering Length in Ultracold Gases As discussed before, at low energy only a few partial waves contribute to collision. For ultracold atom-atom collisions only s-wave scattering is important. At near zero energy the s-wave radial wave function at long range goes as u0 ∼ sin(kr + δ0 ) ≈ sin k(r − a0 ) (3.6) with δ0 being s-wave phase shift and 1 tan(δ0 ) k→0 k a0 = − lim (3.7) is the s-wave scattering length. Scattering length contains all the physical information about the scattering process at low energy. A negative (positive) scattering length implies attractive (repulsive) interaction. The scattering length depends on the nature of potential and the position of highest bound molecular state in it. The energy of highest bound state Eb may be related to s-wave scattering length a0 by Eb = −~2 /2µa20 [7, 23]. If this bound state is just below the continuum, the scattering length is large and positive. On the other hand, if the highest bound state is deeply bound in such a way that a new bound state may be stemmed in if the depth of the potential is slightly increased, then the value of scattering length is large negative. When a bound state occurs just on or near the threshold (zero energy bound state) of the continuum, the scattering length diverges. The divergence of scattering length signifies the resonance condition. 3.1.3 Scattering Resonances at Low Energy Occurrence of resonances is one of the most interesting phenomena of quantum scattering. Scattering resonances are continuations of bound states in the continuum. In other word, resonance states may be defined as the quasi-bound state Chapter 3. Resonances 23 with small finite lifetime. At resonance the two colliding particles at a given energy spend some time in a virtual bound-like state, and then they get separated. The imaginary part of resonance energy i.e the width of the resonance is inversely related to the life time of quasibound resonant state. Hence the resonance states are basically the projections of real states in complex plane. In general the phase shifts and hence scattering lengths are slowly varying functions of the collision energy and the strength of the applied field. However, under resonance condition phase shift goes through a rapid variation, from zero through the value of π/2, over a small range of collision energy. As a result, the cross section changes dramatically in that energy range. If a simple resonance occurs, ℓ-th partial wave cross section can also be written as σℓ = 4π(2ℓ + 1) 2 4π(2ℓ + 1) (Γ/2)2 sin (δ ) = . ℓ k2 k2 (E − ER )2 + Γ2 /4 (3.8) This is known as Breit Wigner profile. Here, ER is the resonance energy and Γ is the full width at half maximum, provided the resonance is reasonably narrow. From Eq. (3.8), it is clear that the resonance cross section is Lorentzian in shape. When the system reaches resonance condition, i.e E → ER , then the cross section attains its maximum value. It is to be noted here that the enhancement of cross section requires the phase shift δℓ = π/2. Scattering resonances can be broadly divided into two categories. Two atoms may collide with each other elastically remaining in the same channel as the incoming or incident one, or they may undergo inelastic collision and go to other channels. So the scattering in one channel may be modified by the effect of other channels and thereby may result in resonances. First one is single channel resonance occurring during elastic single channel scattering. Second one is multichannel resonance [13, 14]. Feshbach resonance which involves at least two coupled channel is the best known example of multichannel resonance. Again the Feshbach resonance (FR) in cold atoms can be classified into two categories –magnetic and optical Feshbach resonance. As discussed in previous chapter, Fano’s Theory [15] has the same essence as Feshbach resonance. Both methods deal with a continuum interacting with one (more than one) bound state(s). Therefore both the methods are related, though the formalism and physical contexts in which they are discussed are different. That’s why Feshbach resonance can be referred to as Fano-Feshbach resonance. In the following sections, different types of resonances Chapter 3. Resonances 24 will be discussed in detail with special emphasis on magnetic Feshbach resonance (MFR) and photoassociation as optical Feshbach resonance (OFR). 3.2 Single Channel Resonance Single channel resonances occurs when the particles scatter back into the incident channel. In following sub sections shape resonance and potential resonance are discussed in short as examples of single channel resonances. 3.2.1 Shape Resonance Shape resonance [7, 8] is one of the most prominent example of single channel resonance. Shape resonance is a metastable bound state trapped due to the shape of a potential barrier of interparticle potential. If the potential barrier is infinitely high, then a bound state can easily be accommodated behind it. As the barrier is finite, the particles may tunnel through it though the presence of barrier helps to form a quasi-bound state, in correspondence to the energy where the real bound state were exist. This increases the scattering cross section implying a resonance. Shape resonance ubiquitously occurs in case of nonzero partial wave scattering. The long range potential in the expression (3.1) modifies due to the presence of repulsive centrifugal barrier for nonzero angular momentum. Thus effective potential of single channel supports the resonance state. The phase shift passes through the value of π/2 as the incident collision energy gradually changes and the partial cross section σℓ pass through maximum value 4π(2ℓ + 1)/k 2 at resonance. For s-wave, the effective potential becomes same as the normal long range potential, hence there is no question of shape resonance to occur. 3.2.2 Potential Resonance Potential Resonance [6, 7] occurs in the absence of any potential energy barrier and is therefore a purely s-wave phenomenon. This resonance occurs due to the presence of a bound state or a virtual bound state close to the collision threshold of single channel. For an attractive potential the scattering length is negative. As 25 Potential Energy Chapter 3. Resonances Internuclear Separation (r) Figure 3.1: Schematic diagram for Feshbach resonance. the depth of the potential increases the phase shift increases and the scattering length becomes more and more negative. Then the phase shift passes through π/2 and a new real bound state appears in the continuum and the scattering length diverges to negative infinity. If the depth is again increased then the scattering length changes sign and eventually decreases to finite positive value, then goes through zero and again through negative infinity as another new bound state is added to the potential. 3.3 Feshbach Resonance : A Multichannel Resonance In late fifties Herman Feshbach (1917-2000) first introduced the concept of Feshbach resonance (FR) in the field of nuclear physics. He developed a new theory of nuclear reactions [16] to treat resonant nucleon scattering that occurs when the energy of initial scattering is equal to that of a bound state between nucleon and nucleus [17]. In 1990s, physicists began to use this concept to realize resonant scattering between ultracold alkali atoms. Such resonances in collisions of alkali Chapter 3. Resonances 26 atoms were predicted for the first time by Tiesinga et al. [18, 19] and first experimentally realized in case of 23 Na and 85 Rb in the year 1998 [20–22]. Now FR has been observed in most of alkali atoms and many heteronuclear mixtures. Feshbach resonance is intrinsically multichannel scattering resonance. It occurs when a pair of cold colliding atoms is coupled to a quasi bound state in higher lying molecular potential and the scattering state is shifted to resonance with a bound molecular level. A Fano resonance is essentially equivalent to a Feshbach resonance. The term Fano resonance is usually associated with the asymmetric line shape as a function of energy where as Feshbach resonance is normally associated with magnetic field tuning of the scattering length. But the origin of both the resonance is same: the interference between a background and a resonant scattering process. 3.3.1 Magnetic Feshbach Resonance Feshbach Resonance, tuned by magnetic field, is called magnetic Feshbach resonance (MFR) [10, 11]. For demonstration, let us consider a simple two channel model as depicted in Fig.3.1. Let us consider two molecular potentials Vbg (r) and Vc (r) in different hyperfine states. At large separations Vbg (r) connects two free colliding atoms. It is known as incident or open channel. In this channel motion is unbound and the threshold energy of this channel is lower than the given energy of the system. The wave function associated with this channel is continuum or scattering wave function. On the other hand Vc (r) supports the molecular bound states. The scattering energy of the two incident atoms lies below the dissociation threshold of this channel.That is why it is called closed channel. For large internuclear separations, Vc (r) connects to a continuum that corresponds to atom pair with higher internal states than that of Vbg (r) i.e atoms in higher hyperfine states than that of Vbg (r). The origin of Feshbach resonance comes from the hyperfine interaction Ehf , which mixes the singlet or triplet states . The hyperfine energy Ehf is obtained by summing the hyperfine energy of individual atom. Hyperfine enrgy of a single atom in the absence of magnetic field is given by atom Ehf = ahf [fa (fa + 1) − sa (sa + 1) − ia (ia + 1)] . ~2 (3.9) Chapter 3. Resonances 27 Now the highest lying vibrational level supported by the closed channel may lie below or above the continuum of open channel A bound (or virtual )state of closed channel just below (or above) the continuum of the open channel gives rise to a large positive (or negative) scattering length. The molecular state has different magnetic moment from that of the two free atoms. So both the potentials can be tuned by applying an external magnetic field, provided the atoms must be paramagnetic. Hence by varying applied magnetic field, a situation may appear when the continuum state of the open channel coincides energetically with the bound state of closed channel resulting in Feshbach resonance and the scattering length then diverges. Due to the coupling between the two channels, the effective collisional potential gets modified. Zeeman effect allows mixing of the potential Vbg (R) and Vc (R) changing the relative energies of the internal states. If the potential gets modified then the scattering wave function is also modified as scattering wave function also depends on the shape of the scattering potential. As a consequence the scattering phase shift δ changes which in tern results in a varying scattering length a. The varying scattering length can also be correlated to the varying position of last bound state, as the binding energy EB of the last bound state is given by EB = ~2 /(2µa2 ), where µ is the reduced mass of the system [7, 23]. 3.3.2 Scattering Length and Feshbach Resonance Feshbach resonance is the most appreciated tool for the tuning of scattering length via external magnetic field in ultracold atomic collision. Near a Feshbach resonance the scattering length a varies as [9] a(B) = abg 1 − ∆B B − B0 (3.10) where, abg is the background scattering length at far-off resonant condition, ∆B is the resonance width and B0 is the resonant magnetic field. ∆B depends on the coupling between the two channels and the shift of two potentials as a function of varying mangnetic field B. The entrance channel’s last bound vibrational level determines abg . Feshbach molecules can be formed from two free cold atoms by sweeping the applied magnetic field adiabatically across a Feshbach resonance. In 1999, Abeelen Chapter 3. Resonances 28 et al. [24] pointed out that it is possible to create ultracold molecules near FR. In 2002, Donley et al. were first to observe a signature of molecules created near FR [25]. The molecules formed by FR tend to be in highly excited vibrational states, though they are rotationally and translationally cold. Ultracold molecules [26, 27] can be created from both bosonic [28–30] and fermionic [2, 3] atoms. Boson-boson and boson-fermion molecules are very short lived. But for the fermionic case, the molecules are quite long lived, because the inelastic atom-molecule or molecule-molecule scattering are suppressed due to Pauli blocking [31, 32]. Such long lived molecules may allow us to study of ro-vibrational states, relaxation processes, low temperature chemical reactivity, BEC-BCS crossover and s-wave superfluidity etc. 3.4 Photoassociation and Optical Feshbach Resonance Photoassociation (PA) [37] (depicted in Fig.3.2) is one of the most admired continuumbound process for the formation of ultracold molecule. The idea of using ultracold and trapped atoms to make cold molecule through photoassociation was first suggested by Thorsheim et al. in 1987 [34]. PA is a process of resonant excitation in which translationally cold colliding atom pair in the presence of a laser of appropriate frequency is transferred to a molecular bound level via free-bound electric dipole transition with the aid of a single photon. The molecule is formed in a ro-vibrational level near the threshold of excited electronic state and hence is long-ranged as compared to normal diatomic molecule. Binding energy of such loosely bound molecule lies in the range of sub mK to hundreds of mK. Therefore the initial temperature of the atomic gas should be cold enough. As the molecule is being formed from initial translationally cold atom pair, hence the molecule is translationally cold. Now at low energy, only collisions of the lowest few partial waves are allowed. Hence through PA the excited molecules is formed in low rotational levels. Hence the photoassociated molecule is translationally and rotationally cold. 29 Potential Energy Chapter 3. Resonances Internuclear Separation (r) Figure 3.2: Schematic diagram showing photoassociation. The stimulated linewidth due to PA between the scattering state | ψ(E, ℓ)i and the excited bound state | φ(v, J)i is given by the expression Γ= πI |hφ(v, J) | D(r)) | ψ(E, ℓ)i|2 ǫ0 c (3.11) where, D(r) is the transition dipole moment, I is the intensity of applied laser and c is the velocity of light. The rate of photoassociation can be written as [35] KP A Z ∞ 1 X (2ℓ + 1) |SP A |2 e−E/kB T dE = hQT ℓ 0 (3.12) where, kB is the Boltzmann constant, QT = (2πµkB T /h2 )3/2 is the translational partition function with T being temperature of atomic gas. SP A , the scattering S-matrix element, is given by |SP A |2 = γΓ/4 . (E + ~ωℓ − Eb )2 + (Γ + γ)2 (3.13) Here, ωℓ is frequency of applied laser, Eb is the bound state energy and γ is the spontaneous line width of the excited state. Chapter 3. Resonances 30 Though the molecule formed by the single photon PA is translationally and rotationally cold, but vibrationally it is very unstable. The excited molecule may spontaneously decay to some ground bound state or to the ground continuum. Now this photoassociated molecule may be driven by using a second laser of appropriate frequency to a low vibrational bound state of the ground electronic configuration in stimulated manner. Thus in comparison with one-photon PA and the problems with high vibrational quantum numbers, two-color photoassociation is a big improvement. PA can also be used to tune atom-atom interaction of cold atoms in the same way as MFR. As PA uses optical field to couple the colliding atoms to excited bound state, it can be referred as optical Feshbach resonance (OFR). In MFR, the energy difference between the open and closed channels is controlled by external magnetic field. The necessary criteria for this is that both channels must have different magnetic moments and ground state should be degenerate in the absence of any field. On the other hand, OFR tunes the atom-atom interaction by coupling two-atom scattering state to excited bound state via photoassociation. Hence OFR can be observed for all sorts of atoms where as MFR is observed for atoms having permanent magnetic moment. Optical fields can be switched and controlled much faster and can be focused much better than magnetic fields. OFR provides a new way of research for alkaline earth metal atoms and similar systems with nondegenerate ground state. It may be used to control the scattering wave function to modify the PA rate and modulate the thermalization and loss rate. The use of light fields to modify the atom-atom interaction and the scattering length in atomic collisions has been first proposed by Fedichev et al. [36]. and has been further explored by Bohn and Julienne [37, 38] using quantum defect approach. Optical s-wave scattering resonances have been first observed in sodium vapour by Fatemi et al. [39] using one-color PA spectroscopy. Later several experiments have been designed to observe optical Feshbach resonance (OFR) in one-color and two-color scheme [40–44]. To modify the scattering length by OFR, in one photon PA scheme, laser is tuned close to PA resonance which couples ground scattering state and excited molecular state. The optical field dresses the excited bound state and by modulating the PA laser frequency the dressed state can be tuned below, at or above the collisional threshold which results in dramatic change in s-wave scattering length. But it also leads to atomic loss due to spontaneous decay via bound state. Hence the s-wave scattering length in the presence Chapter 3. Resonances 31 of light field can be expressed as α − iβ. Here α denotes the modified scattering length in the presence of OFR and the imaginary part β describes the inelastic loss rate due to two body collision. For the weak coupling limit i.e for Γ ≤ γ, α and β are given by [37, 38] 1 Γ Γ∆ ∆ α = abg − = abg 1 − 2k ∆2 + (γ/2)2 2kabg (∆2 + (γ/2)2 ) 1 Γγ β = 2 k ∆ + (γ/2)2 (3.14) (3.15) where, ∆ is the laser detuning and abg is the non-resonant background scattering length when there is no optical coupling. Till now, it is found that OFR is not as efficient as MFR, as the excited photoassociated molecule may eventually decay leading to drastic loss of atoms from trap. If an efficient all optical method could be devised, it would prove itself advantageous over MFR, in particular to manipulate p or d partial wave interaction [45–47]. References [1] I. Bloch, J, Dalibard and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008) [2] M. Greiner, C. A. Regal and D. S. Jin, Nature 426, 357 (2003) [3] S. Jochim Science 302, 2101 (2003) [4] A. J. Daley, M. 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Deb, Phys. Rev. A 86, 063407(2012) [47] S. Saha, A. Rakshit, D. Chakraborty, A. Pal and B. Deb, arXiv:1405.1674 (2014) Chapter 4 Quantum Interference in Photoassociation in the Presence of Feshbach Resonance In previous two chapters, we have discussed about quantum interference effects with an emphasis on Fano effect, magnetic Feshbach resonance and photoassociation as optical Feshbach resonance. This chapter is devoted for the discussion of quantum interference effects in the context of photoassociation (PA) in the presence of magnetic Feshbach resonance (MFR) in the light of celebrated Fano effect. Here we examine the effects of interference on PA spectrum and our aim is to obtain large light shift along with an extremely narrow line width which will be effective for the manipulation of atom-atom interactions and cold collisions. The chapter is designed as follows: in section 4.1, we present the motivation and perspective of present work. Formulation of the problem and its solution are discussed in section 4.2. In the next section results and their interpretations are discussed taking 7 Li as an example. 4.1 Perspective of The Work In the previous chapter, we have seen the manipulation of atom-atom interaction at low energy can be achieved by either magnetic Feshbach resonance (MFR) or optical Feshbach resonance (OFR). Now it would be interesting to investigate 35 Chapter 4. PA in the presence of FR 36 what happens if PA occurs in the presence of MFR. Several experimental studies on PA in the presence of MFR [1–4]) have been carried out over last 16 years. In recent years, it has attracted a lot of interests both experimentally [5–10] as well as theoretically [11–16] revealing significant effects of MFR on PA and cold collision properties. According to of Franck Condon principle a continuum-bound or boundbound transition is most probable when the prominent anti-nodes of initial and final states are located at a comparable internuclear separation. Probability for such transition would be least when the anti-node of either of the states lies at a separation close to the node of the other. Hence, in the presence of Feshbach resonace, enhancement [7] and suppression in PA spectral intensity profile can take place due to quantum interference between PA and Feshbach resonances resulting in Fano-type [23] asymmetric spectral profile. Such quantum interferences can be used for coherent control of cold atom-molecule conversion and ultracold collisions. In this chapter we explore the possibility of suppression of power broadening in strong-coupling PA by manipulation of continuum-bound coherences with a Feshbach resonance. We consider that two optically coupled bound states interacts with a common continuum as shown in Fig.4.1. We solve the problem following Fano’s Theory where the system is exactly diagonalised leading to a ‘dressed’ continuum state. We demonstrate that by tuning the magnetic field close to Fano minimum where excitation probability vanishes, it is possible to obtain line narrowing in the PA spectrum with large shifts at high laser intensities which may be useful in efficient tuning of elastic scattering length by optical means. 4.2 Formulation of the Problem and Solution To model PA occurring in the presence of MFR, we first consider a simple three channel model as depicted in Fig.4.1 [9]. It consists of two asymptotic hyperfine channels of which one is closed channel |1i having higher threshold energy than the asymptotic collision energy and other one is open channel |2i having lower threshold energy than that. In the presence of appropriate magnetic field strength, scattering state associated with open channel can couple to a quasibound state supported by closed channel resulting in Feshbach molecular (FM) state. So the presence of MFR modifies the continuum states of open channel and vice versa. As the applied magnetic field is varied, this quasibound state can move across the collision energy. Our model also consists a third channel |3i which Chapter 4. PA in the presence of FR 37 Vex(r) Excited bound state |3 〉 (S+P) 0 |3〉 LPA |2〉 Bound−bound |E〉 Continuum−bound LPA Vg(r) Bound state |2〉 Closed channel 2 (S+S) 0 Continuum |E〉 Open channel 1 Inter−channel coupling V r Figure 4.1: A schematic diagram showing the coupling between the bound state (magenta) | 2i and and continuum (green) | Ei with the excited bound state (blue) | 3i via same laser LP A ( magenta and green double-arrow vertical lines). Red dashed line indicates the hyperfine coupling between the open and closed channel. corresponds to the excited photoassociated molecular (PM) state in the excited electronic state. Now when Photoassociation laser of appropriate frequency is applied, bound-bound dipole transition between FM state and PA state may occur along with normal PA transition between open channel continuum and PM state. We now assume that the energy spacing in FM states is much larger than the line width of the PA laser, so PA laser effectively couples only one FM bound state to PM state. We further assume that rotational spacing of PM states is much larger than PA laser line width so that only one rotational level J of a particular vibrational state v is coupled by the PA laser. So, now there are three competing pathways. Two of these are continuum-bound types and one is bound-bound dipole transition. So quantum interference may naturally arise between any two of these three competing transition pathways. Chapter 4. PA in the presence of FR 38 The dressed state of a system where PA of two colliding atoms is taking place in the presence of MFR can be written as: " # X 1 | ΨE i = χ | 1i + Φi (r) | ii r i=2,3 (4.1) where, r is the relative coordinate of the two atoms. The continuum state has the R form χ = dE ′ bE ′ ψE ′ (r), where ψE ′ is an energy normalised scattering state of collision energy E ′ and bE ′ is density of states of the unperturbed continuum. E is the eigen energy of the dressed state, Φi s (i = 2, 3) are the wave functions of perturbed bound states. The Hamiltonian for the system can be written as H = Hkin + Helec + Hhf s + HB + HL (4.2) where,Hkin corresponds to total kinetic energy and Helec depends on electronic coordinate of two atoms, Hhf s is the hyperfine interaction term, HB and HL represent respectively the magnetic and laser field interactions between atomic and molecular states. From the time-independent Schrödinger equation HΨE = EΨE under Born-Oppenheimer approximation we obtain the coupled differential equations: ~2 d 2 − + BJ (r) Φ3 + [Ve (r) − ~δ − E − i~γ/2] Φ3 2µ dr 2 = −Ω1 χ − Ω2 Φ2 , (4.3) 2 2 ~ d − + V2 (r) − E Φ2 = −Ω∗2 Φ3 − V χ, 2µ dr 2 (4.4) ~2 d 2 + V1 (r) − E χ = −Ω∗1 Φ3 − V ∗ Φ2 . − 2 2µ dr (4.5) Here BJ = ~2 J(J + 1)/2µr 2 is the rotational term of the excited state and δ = ωL − ωA is the laser-atom detuning. Vi (i = 1, 2) are the potentials including hyperfine and Zeeman terms and Ve is the excited state molecular potential. The hyperfine spin coupling between the cha5nnels 1 and 2 is denoted by V (r). Ω1 (r) and Ω2 (r) represent the molecular Rabi couplings of the excited state | 3i with the ground states | 1i and | 2i, respectively. In the absence of couplings, let the unperturbed bound states be denoted by φ3 (r) Chapter 4. PA in the presence of FR 39 and φ2 (r) with bound state energies E3 and E2 , respectively and the unperturbed continuum states by ψE ′ with asymptotic collision energy E ′ . With the use of these unperturbed solutions, we construct three Green’s functions GE (r, r ′ ), G2 (r, r ′ ) and G3 (r, r ′ ) which correspond to channels 1, 2 and 3, repectively. The continuum Green’s function GE (r, r ′) can be written as GE (r, r ′ ) = −πψEreg (r< )ψE+ (r> ), where r<(>) implies either r or r ′ whichever is smaller (greater) than the other. Here ψE+ (r) = ψEirr + iψEreg where ψEreg and ψEirr represent regular and irregular scattering functions, respectively. Asymptotically, ψE0,reg (r) ∼ j0 cos η0 − n0 sin η0 and ψE0,irr (r) ∼ −(n0 cos η0 +j0 sin η0 ), where j0 and n0 are the spherical Bessel and Neu- mann functions for partial wave ℓ = 0 (s-wave) and η0 is the s-wave phase shift in the absence of laser and magnetic field couplings. The other two Green’s functions correspond to bound states are of the form G3 (r, r ′ ) = − ~δ+E−E13 +i~γ/2 φ3 (r)φ3 (r ′ ) 1 and G2 (r, r ′ ) = − E−E φ2 (r)φ2 (r ′ ). Now using these above Green’s functions, we 2 can write down eqs (2) and (3) in the form Φ3 = − Φ2 = − Z Z dr ′ [Ω1 (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ2 (r ′ )] G3 (r, r ′ ) dr ′ [V ∗ (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ3 (r ′)] G2 (r, r ′ ). (4.6) These equations may be also expressed as Φ3 = AP A φ3 and Φ2 = ACC φ2 , where AP A and ACC is given by following expressions dr ′ [Ω1 (r ′ )χ(r ′ ) + Ω2 (r ′ )Φ2 (r ′ )] φ3 ~δ + E − E3 + i~γ/2 R ∗ ′ [V (r )χ(r ′ ) + Ω∗2 (r ′ )Φ3 (r ′ )] φ2 (r ′) = − E − E2 R ′ ′ ∗ ′ dr φ2 (r )V (r )χ(r ′ ) + Ω∗32 (r ′ )AP A ) = − E − E2 AP A = − ACC R (4.7) (4.8) dr ′ φ3 (r ′ )Ω2 (r ′ )φ2 (r ′ ) is bound-bound Rabi coupling. The continuum R state can be written as χ = dr ′ bE ′ ψE ′ (r). Thus we obtain Here, Ω32 = − R ~2 d 2 ψE ′ (r) + [V1 (r) − E]ψE ′ (r) = −Ω∗1 (r)ÃP A φ3 (r) − ÃCC V ∗ (r) 2µ dr 2 (V ∗ + ÃP A Ω∗32 ) V (r)φ2(r) = −Ω∗1 (r)ÃP A φ3 (r) − 2E E − E2 (4.9) where. ÃP A = Ω3E (E − E2 ) + Ω32 V2E . D(E − E2 ) − |Ω32 |2 (4.10) Chapter 4. PA in the presence of FR 40 6 6 2 q = 1.0 q = 0.5 0 −5 0 4 q = −1.0 q = − 0.5 2 0 −5 5 0 −1 3 q = 0.5 ∆E/ΓPA ∆E/ΓPA (kares) 0 −1 q = 1.0 −2 (c) q = 2.0 −3 −5 5 −1 (kares) 1 q = − 2.0 (b) PA 4 (a) Γ/Γ Γ/Γ PA q = 2.0 0 2 1 (d) q = − 0.5 0 q = −1.0 −1 −5 5 q = − 2.0 0 −1 5 −1 (kares) (kares) Figure 4.2: Subplots (a) and (b) show Γ/ΓP A Vs (kares )−1 and subplots (c) and (d) exihibit ∆E/ΓP A Vs (kares )−1 . Plots are for different values q as 2 k2 indicated in the figure. k is defined as ~2m = KB T Here, D = (~δ + E − E3 + i~γ/2), V2E = R drφ3 (r)Ω1 (r)ψE ′ (r). R drφ2 (r)V (r)ψE ′ (r) and Ω3E = Now Eq. (4.9) can be solved by constructing the Green’s function GE (r, r ′ ) with the scattering solutions of homogeneous part. This Green’s function as discussed earlier can be written as: GE (r, r ′ ) = −π ψE0,reg (r)ψE0,irr (r ′ ) + iψE0,reg (r)ψE0,reg (r ′ ) (r ′ ≥ r) GE (r, r ′ ) = −π ψE0,reg (r ′ )ψE0,irr (r) + iψE0,reg (r)ψE0,reg (r ′ ) (r ′ ≤ r). (4.11) Now, we can express the solution of Eq. (4.9) in the following form ψE ′ = + exp(iη0 )ψE0.reg ′ + Z h dr ′ GE (r, r ′ ) Ω∗1 (r ′ )ÃP A φ3 (r ′ ) # ∗ V2E + ÃP A Ω∗32 V (r ′ )φ2 (r ′ ) . (E − E2 ) (4.12) Chapter 4. PA in the presence of FR 41 Substituing ψE ′ in the expression of V2E , we obtain V2E = Ω32 0 2 0 0 P A V32 ∆E2 − i πAE−E |V2E | exp(iη0 )V2E + V32 AP A − iπV2E Ω3E AP A + AP A E−E 2 2 0 2 (E − E2 − ∆E2 + iπ|V2E | )/(E − E2 ) . Here, V32 = RR (4.13) dr ′ drφ2(r)V ∗ (r)Re[GE (r ′ , r)]V (r ′ )φ2 (r ′ ) represents continuum- mediated an effective magneto-optical coupling between photoassociated and FeshRR ′ bach molecular bound states. ∆E2 = dr drφ2(r)V ∗ (r)Re[GE (r ′ , r)]V (r ′ )φ2 (r ′ ) is the energy shift of the Feshbach molecular state due to its coupling with the con- tinuum. The Stark energy shift due to laser coupling of Photoassociated molecular RR ′ ′ (PM) state with the continuum is given by ∆E3 = dr drφ3 (r)Ω1 (r)Re[GE (r , r)] q R 0 0 2 Ω∗1 (r ′ )φ3 (r ′ ). V2E and Ω03E are defined as V2E = drφ3(r)V (r)ψE0,reg (r) = hΓ 2π q R hΓ3 and Ω03E = drφ3 (r)Ω1 (r)ψE0,reg (r) = . Here Γ2 and Γ3 are the Feshbach 2π resonance linewidth and stimulated PA linewidth, respectively. Substituting Eqs. (4.13) and (4.12) in Eq. (4.10), we finally obtain ÃP A = 0 exp(iη0 ) [Ω03E + GV2E ] . (4.14) [Ω32 + V32 − iπV2E Ω3E ] , [E − E2 − ∆E2 + iπ|V2E |2 ] (4.15) D− |Ω32 |2 E−E2 − (Bp + GBf ) Here, G= [−V32 + iπV2E Ω3E ] (E − E2 ) (4.16) [−∆E2 + iπ|V2E |2 ] . (E − E2 ) (4.17) 0 2 Bp = Eshif t − iπ|Ω3E | − Ω32 and Bf = V32 − iπV2E Ω3E − Ω32 Now, let us introduce a low-energy dimensionless interaction parameter β(k) which is defined as β = (E − E2 − ∆E2 )/(Γ2 /2). It may be described as shifted energy of the Feshbach molecule and related to the Feshbach resonance phase shift ηres (k) as β(k) = − cot ηres (k) ∼ (kares )−1 + 21 re k. re is effective range and is related to Feshbach resonance linewidth. ares related to the applied magnetic field B by ares = −(abg ∆)/(B − B0 ) [19], where abg is the background scattering length, Chapter 4. PA in the presence of FR 42 Γ/Γ PA 30 weak Γ Γ 20 10 0 −30 −20 −10 0 10 20 30 −1 (kares) 3 weak Γ/ΓPA Γ 2 Γ 1 0 −4 −3 −2 −1 0 (ka 1 2 3 4 −1 ) res Figure 4.3: Solid lines represent the linewidth Γ (in unit of ΓP A ) given by the expression (4.20) as a function of (kares )−1 for q = −5.0 (upper panel) and q = −0.1 (lower panel). Dotted lines represent the low coupling expression of (4.27) for C1 = 6 and C2 = 1 (upper panel); and C1 = 0.1 and C2 = 0 (lower panel). The values of C1 and C2 are so chosen such that in the limit ares → 0, Γ → Γweak . B0 is the resonance magnetic field, ∆ is resonance width. So, we can say that β(k) ∼ −(B − B0 )/(kabg ∆). Let us also introduce another parameter analogous to Fano asymmetry parameter q which is defined as q= Ω32 + V32 . πΩ3E Ω2E (4.18) It is to be noted that q is independent of the laser intensity. In the limit k → 0, both Ω32 and V32 become energy-independent while Ω3E V2E ∼ k. Thus at low energy, q ∼ 1/k. Chapter 4. PA in the presence of FR 43 Now substituting the expressions of G, Bf and Bp in Eq.(4.14) and using the expressions β(k) and q, we finally obtain the expression of ÃP A as given below, exp (iη0 )πΩ3E (β(k) + q)/(β(k) + i) i ÃP A = h (q−i)2 0 ~δ + E − E3 + Eshif + i~(γ + Γ )/2 − ~Γ 3 3 (β(k)+i) t = exp (iη0 )πΩ3E (β(k) + q) . (β(k) + i) [∆p + i~(γ + Γ3 )/2] − ~Γ3 (q − i)2 (4.19) 0 0 Here, ∆p = ~δ + E − (E3 + Eshif t ), Eshif t is the energy shift in the absence of FeshRR ′ 0 bach resonance and given by the expression Eshif dr drφ3 (r)Ω∗1 (r)Re[GE (r ′ , r)] t = Ω1 (r ′ )φ3 (r ′ ). 4.3 4.3.1 Results and Discussions Analytical Results Since the rate of Photoassociation (PA) is proportional to |AP A |2 , from Eq. (4.19), we obtain PA linewidth in the presence of MFR in the form of Γ = f (q, β)Γ3 = (β(k) + q)2 Γ3 . β(k)2 + 1 (4.20) Here Γ is the stimulated linewidth without MFR. From above equation, it is clear that Γ depends on a nonlinear function f (q, β) of q and β. Note that, when the magnetic field is far off resonant, i.e at β → ∞, Γ → Γ3 . On the other hand, when β = −q, then Γ = 0 and AP A = 0. So by tuning β(k) close to −q, Γ may be made arbtrarily small. Γ3 is proportional to laser intensity I. So it is possible to suppres power broadening at increased laser intensities by tuning the magnetic field close to Bmin where the PA rate is minimum. This point is given by β = −q. From the Eq. (4.19), we also obtain the extra shift due to MFR in the form of 1 (q 2 − 1)β(k) − 2q Γ3 . ∆E = 2 β(k)2 + 1 (4.21) which again goes to zero as β → ∞. Again, for β ≃ −q, ∆E ≃ −qΓ3 /2, which is proportional to I. Hence we can conclude that when line broadening is suppressed 0 by the tunability of MFR, total shift Etot = Eshif t + ∆E still remains proportional 0 to I. Normally the shift Eshif t is negative in the low energy regime. However, the Chapter 4. PA in the presence of FR 44 total shift Etot in the presence of MFR can be positive or negative depending on the values of q. So it may become possible to obtain arbitrarily small linewidth with large shift at high intensities by appropriate tuning the magnetic field. Next, we discuss the weak-coupling limit of (4.20) and (4.21) when laser intensity is low. For this we first find the dressed continuum state in the limit Γ3 → 0. From Eq. (4.8), we can write the expression of ÃCC as ÃCC = ÃP A Ω32 + R drψE (r)V ∗ (r)φ2 (r) E − E2 (4.22) When, Γ3 → 0, ÃP A → 0 and ∆E = 0, then ÃCC may be expressed as ÃCC = R drψE (r)V ∗ (r)φ2 (r) E − E2 (4.23) Now substituting ψE (r) by Eq. (4.12) and considering ∆E = 0 as Γ3 → 0, ÃCC may be written as r ÃCC = − 2 exp i(η0 + ηres ) sin ηres πΓ2 (4.24) and Eq. (4.12), at weak coupling limit, reduces to ψE (r) = exp i(η0 + ηres )[ψE0,reg cos ηres + ψE0,irr sin ηres ] (4.25) The dressed continuum state in Eq. (4.1) reduces in the limit of Γ3 → 0 to 1 | ΨE i0 = [ÃCC φ2 (r) | 2i + r Z bE ′ ψE ′ (r)dE ′ | 1i] (4.26) Taking bE ′ = δ(E − E ′ ), the stimulated linewidth Γweak in the weak coupling limit is given by the Fermi golden rule expression Γ weak 2π | = ~ Z rφvJ h3 | Ω1 (r) | ΨE i0 dr|2 r 2 2π 0 2 1 Ω32 Ωirr 3E = |Ω3E | 1 + − sin ηres + 0 tan ηres 0 ~ πΓ2 Ω3E Ω3E = Γ3 |1 + C1 tan ηres + C2 sin ηres |2 (4.27) q 0,irr 2 0 )Ω32 /Ω03E . Here Ωirr i. The So, C1 = Ωirr /Ω and C = (− 2 3E = h3 | Ω1 | ψE 3E 3E πΓr expression (4.27) is in agreement with (6) of [6]. When ηres → π/2, Γweak diverges Chapter 4. PA in the presence of FR 45 −1 (cm sec ) 0.2 0.1 0.05 K PA × 10 10 3 0.15 0 −100 0 δ p 100 200 300 (MHz) Figure 4.4: KP A in cm3 sec−1 Vs. detuning δp in MHz. Each pair of dashed and solid curves are obtained for ΓP A = 1.0 MHz (left pair), ΓP A = 10.0 MHz (middle pair) and ΓP A = 25.0 MHz (right pair) for the fixed q = −6.36. For solid curves, magnetic fields are B = 705.00 G (left), B = 708.56 G (middle) and B = 709.12 G (right). For dash-dotted curves, these are B = 713.19 G (left), B = 711.09 G (middle) and B = 710.68 G (right). The magnetic fields are so chosen such that the linewidth Γ remains fixed at 0.04 MHz. and hence Eq. (4.27) is not valid near ηres = π/2. In other words, Eq. (4.27) is not applicable close to Feshbach resonance. Finally we prove that line narrowing in one-photon PA is not possible in the absence of coupling between the open and closed channel. we analyze PA rate. It describes the loss of atoms due to the decay of the excited state into decay channels. The expression is given by KP A 1 = hQT Z dEk ~2 γΓ exp(−Ek /KB T ) (Ek − ∆Es + ~δp )2 + ~(γ + Γ)2 /4 (4.28) 0 as a function of B and the detuning parameter δp = δ − (E3 + Eshif t − Eth )/~, where Eth is the threshold of the open channel. Here QT = (2πµKB T /h2 )(3/2) , Ek = E − Eth and KB is the Boltzmann constant. Now it can be noted that the PA laser can be tuned either near continuum-bound frequency or near bound- bound transition frequency. In first case, ~δp ≃ Ek and in second case −~δp = Chapter 4. PA in the presence of FR 46 (E2 + ∆E2 − Eth ). In the limit Γ2 → 0, ~2 γΓP A KP A ≃ n 0 (E − E3 − Eshif t ) + ~δ − ~2 Ω232 E−E2 o2 (4.29) + i~2 (γ + ΓP A )2 /4 which is in agreement with the expression of |S1g |2 of [20], if we identify ∆1 and ∆2 of [20] with −~δ and E2 , respectively. In our case there is only one laser coupling between the continuum and the excited bound state and also between the two bound states. It is clear from the above expression that in the absence of coupling between open and closed channels the narrowing of PA linewidth is not be possible. before going to numerical demonstration, finally, we discuss one important consequence of line narrowing. The asymptotic form of the perturbed scattering state is given by ΨE (r) = exp(iη0 )ψEreg′ (r) + T exp(−iη0 )ψE+′ (r). (4.30) From it, we can deduce the scattering T -matrix element in the following form T = T0 + exp(2iη0 )Tf + exp[2i(η0 + ηres )]Tpf . (4.31) Here, T0 , back ground T -matrix element, and Tf , T -matrix element in the presence of magnetic field only, are expressed as follows: T0 = exp(iη0 ) sin η0 and Tf = −1/(β + i). Tpf , T -matrix element due to photoassociation in the Fes- hbach resonance is given by Tpf = −(Γ/2)/[(Ek + ~δp − ∆Es ) + i(γ + Γ)/2]. The terms T0 and Tf are independent of laser field while Tpf explicitly depends on β, q and laser intensity. The amplitude of closed-channel bound state is R ÃCC = 2[ÃP A Ω∗32 + drψE (r)Ω1 (r)φ2(r)]/[Γ2 β + 2∆E2 ], where ∆Er is the shift of the closed-channel bound state due to its coupling with the open channel. Note that in the limit β → ±∞, ÃCC → 0, Γ → Γp . In the limit Γp → 0, ÃP A ≃ 0 and the problem reduces to that of two coupled-channel Feshbach resonance [7]. In the absence of spontaneous emission (γ = 0), we obtain T = [exp(2iηtot ) − 1]/2i, where ηtot = η0 + ηres + ηpf with ηpf = − tan−1 Γ/2 . Ek +~δp −∆E It then follows that the scattering matrix S = 1 + 2iT is unitary. The s-wave scattering amplitude (0) (f ) is f0 = T /k = f0 + exp(2iη0 )f0 (f ) f0 = Tf /k and (pf ) f0 (q) (0) + exp[2i(η0 + ηres )]f0 , where f0 = T0 /k, = Tpf /k. In the presence of spontaneous emission, the real Chapter 4. PA in the presence of FR 47 (nm) 720 20 10 700 710 720 B (Gauss) 710 ∆ E (MHz) B (Gauss) a res 715 30 705 150 100 50 0 700 0 10 0 10 20 ΓPA (MHz) 20 30 30 40 40 Figure 4.5: The locus of B and ΓP A for which the linewidth Γ remains fixed at 0.04 MHz. The lower inset shows the variation of ∆E (in MHz) against ΓP A (in MHz) and the upper inset exhibits the variation ares (in nm) against B (in Gauss) at the fixed Γ = 0.04 MHz. (pf ) and imaginary part of f0 are −Γ × (Ek + ~δp − ∆Es ) and Γ(Γ + γ), respec- tively. In the limit k → 0 and δp → 0; and if Γ is suppressed and the shift ∆E is enhanced such that Γ << γ and |∆E| >> γ, the elastic scattering will dominate over the inelastic part. This is what exactly required for efficient manipulation of scattering length by optical Feshbach resonance [21]. 4.3.2 Numerical Results and Discussion For numerical illustration, we consider a model system of two ground-state (S1/2 ) 7 Li atoms undergoing PA from the ground molecular configuration 3 Σ+ u to the vi- brational state v = 83 of the excited molecular configuration 1 3 Σ+ g which correlates asymptotically to 2S1/2 + 2P1/2 free atoms [22, 23]. All the relevant parameters 0 γ, Eshif t , ∆, abg and Γr are estimated from [22], [24] and [7]. In Fig.4.2, we have plotted Γ/ΓP A and ∆E/ΓP A against 1/kares for positive and negative q values. The maximum and minimum values of linewidth would be observed for β = 1/q and β = −q, respectively. The magnitude of the change in shift due to PA in the Chapter 4. PA in the presence of FR 48 presence of MFR is significant near β = −q. Fig.4.3 clearly shows that the stim- ulated linewidth in the weak coupling limit, represented by dashed lines deviate appreciably from nonperturbative results as shown by solid lines. The deviations are the most prominent in the region (kares )−1 ≃ 0 (ηres ≃ π/2). Furthermore, for lower q values these two results deviate most significantly. Fig.4.4 illustrates how to suppress power-broadening by the appropriate tuning of magnetic field near Bmin and thereby to keep the total linewidth close to the natural linewidth. There are two values of β(B) and correspondingly two values of ares where the linewidth Γ can be kept fixed at a small value at an increased laser intensity. In Fig.4.5, we show how to vary ΓP A (or laser intensity) and the magnetic field in order to keep Γ fixed at 0.04 MHz which is much smaller than the natural linewidth γ although ΓP A can be many orders of magnitude higher than γ. The lower inset in Fig.4.5 shows that the extra shift ∆E can exceed γ by many orders of magnitude while power-broadening is suppressed. 4.4 Conclusions In this chapter, we have demonstrated that linewidth of photoassociation spectrum can be narrowed down close to the natural linewidth by tuning the magnetic field near Feshbach resonance along with a large shift. This enhencement of the life time of excited molecular state may be beneficial for population transfer from ground state collisional continuum to ground molecular state by two-photon Raman-type PA. Furthermore, narrow linewidth with large shift will be useful for efficient manipulation of scattering length by optical Feshbach resonance [21]. This will be particularly important for altering scattering amplitude of higher partial waves [26]. References [1] Ph. Courtellle, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998) [2] van Abeelen, D. J. Heinzen and B. J. Verhaar, Phys. Rev. A 57, R4102 (1998) [3] V. Vuletic, C. Chin, A. J. Kerman and B. J. Verhaar, Phys. Rev. Lett. 83, 943 (1999) [4] C. Chin, A. J. Kerman, V. Vuletic and S. Chu, Phys. Rev. Lett. 90, 033201 (2003) [5] M. Mackie, F. Matthew, D. Savage and J. Kesselman, Phys. Rev. Lett. 101, 040401 (2008) [6] P. Pellegrini and R. Côté, New J. Phys. 11, 055047 (2009) [7] P. Pellegrini, M. Gacesa and R. Côté, Phys. Rev. Lett. 101, 053201 (2008) [8] E. Kuznetsova, M. Gacesa, P. Pellegrini, F. Y. Susanne and R. Cote, New J. Phys. 11, 055028 (2009) [9] B. Deb and A. Rakshit, J. Phys. B: At. Mol. Opt. Phys. 42, 195202 (2009) [10] B. Deb and G. S. Agarwal, J. Phys. B: At. Mol. Opt. Phys. 42, 215203 (2009) [11] C. Chin, V. Vuletic, A. J. Kerman, S. Chu, E. Tiesinga, P. J. Leo and C. J. Williams, Phys. Rev. A 70, 032701 (2004) [12] M. Junker, D. Dries, C. Welford, J. Hitchcock, Y. P. Chen and R. G. Hulet, Phys. Rev. Lett. 101, 060406 (2008). [13] K. Winkler, F. Lang, G. Thalhammer, R. Straten and H. Denschlag, Phys. Rev. Lett. 98,043201 (2007). 49 Chapter 4. PA in the presence of FR 50 [14] K K Ni, S. Ospelkaus, M. H. G. de Miranda, A. Peer, B. Neyenheis, J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin and J. Ye, Science 322, 231 (2008). [15] D. M. Bauer, M. Lettner, C. Vo, G. Rempeand S. Durr, Nature Phys. 5, 339 (2009) [16] D. M. Bauer, M. Lettner, C. Vo, G. Rempeand S. Dürr, Phys. Rev. A 79 062713 (2009) [17] M. Gacesa, S. Ghosal, J. N. Boyd and R. Côté, Phys. Rev. A 88,063418 (2013) [23] U. Fano, Phys. Rev. 124, 1866 (1961). [19] A. J. Moerdjik, B. J. Verhaar and A. Axelsson, Phys. Rev. A 51, 4852 (1995). [20] J. L. Bohn and P. S. Julienne, Phys. Rev. A 54, R4637 (1996). [21] P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996); F. K. Fatemi, K. M. Jones and P. D. Lett, Phys. Rev. Lett. 85, 4462 (2000). [22] I. D. Prodan, M. Pichler, M. Junker, R. G. Hulet and J. L. Bohn, Phys. Rev. Lett. 91 080402 (2003). [23] E. R. I. Abraham, W. I. McAlexander, C. A. Sackett and R. G. Hulet, Phys. Rev. Lett. 74, 1315 (1995). [24] S. E. Pollack, D. Dries, M. Junker, Y. P. Chen, T. A. Corcovilos and R. G. Hulet, Phys.Rev. Lett. 102, 090402 (2009). [7] C. Chin, R. Grimm, P. Julienne and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). [26] B. Deb and J. Hazra, Phys. Rev. Lett. 103, 023201 (2009). Chapter 5 Vacuum- and Light-Induced Coherences in Cold Atoms and Molecules Most of the studies on quantum interferences and coherences have been done in case of atomic systems and sometimes in case of molecular systems [1, 2]. But the developments in the fields of free-bound spectra open up a new research area to study quantum interferences in atom-molecule coupled systems. In this chapter, we study vacuum-induced and light-induced coherences in case of ultracold photoassociation and propose novel photoassociation (PA) schemes for the realization of vacuum-induced coherence in atom-molecule coupled system. In the section 5.1, the perspective of our work is discussed. Then, we describe two schemes that are proposed to discuss vacuum- and light-induced coherences in Veetype atom-molecule coupled system. The first scheme and its results have been discussed in sections 5.2, 5.3 and 5.4. In first scheme, we employ Wigner-Weisskopf approach to solve the problem where as in second scheme, master equation method demonstrated by Agarwal et al. in [3] to treat spontaneous emission of continuumbound coupled autonizing Fano state, is adopted. It is discussed in sections 5.5, 5.6, 5.7 and 5.8. We demonstrate that it is possible to generate and manipulate coherence between the excited states by PA lasers. We conclude in section 5.9. 51 Chapter 5. VIC and LIC 5.1 52 Perspective of The work Vacuum-induced coherence (VIC) [3] results from the quantum interference between two spontaneous emission pathways. On the other hand, the light induced coherence (LIC) occurs due to interplay of applied light fields. As we change the relative intensity, relative phase shift and the detunings of both lasers, the coherence between the states changes. VIC can lead to the population trapping in the excited state [2, 3]. This can be utilized in manipulating environment-induced relaxation processes in a wide variety of systems, such as atoms, molecules, quantum dots [3–10]. It has also been found effective against decoherence in quantum information processing [11]. In this chapter, our aim is to show how VIC can be modulated by using LIC. The necessary criteria for VIC to occur is the non-orthogonality of dipole transitions of two participating pathways. For atomic systems, nonorthogonality is a stringent condition to achieve for atoms. For possible realization of VIC for an excited atom interacting with an anistopic vacuum [12–14], utilization of j = 1/2 → j = 1/2 transition in 198 Hg + and 139 Ba+ ions has been suggested [15, 16]. Recently, a proof-of-principle experiment verifying its presence has been performed in quantum dots [17]. In spite of these attempts, a clear signature of VIC in atomic systems has yet to be obtained. On the other hand, molecules are the natural candidates for VIC as the non-orthogonality criteria is satisfied naturally. Spontaneous transitions from two excited states, which belong to same electronic state differing only in rotational or vibrational quantum numbers, to same ground state have transition dipole moments which are nonorthogonal to each other. The interference between these two spontaneous emission pathways results in VIC. Now, with the tremendous progress in the fields of free-bound PA spectroscopy [18, 19], the low lying ro-vibrational levels in excited electronic states can be selectively populated. This occurs due to PA transitions from the collisional continuum of two ground-state cold atoms. VIC would be significant in such an atom-molecule interface system provided (i) there is no hyperfine interaction in the atoms that are photoassociated, (ii) there is no bound state close to the dissociation continuum of the ground molecular state and (iii) excited molecular levels have a long lifetime. To the best of our knowledge, the possibility of observing vacuum induced coherence in PA system has not been addressed so far. In this chapter, our Chapter 5. VIC and LIC 53 J2 J1 Ve(r) |φ2 ! ωL1 ωL2 |φ1 ! γ2 γ1 Vg(r) |E!b Continuum Figure 5.1: A schematic diagram for creation of LIC and VIC. The twoatom continuum |Eb i is coupled to two rovibrational levels |v , J1 i(|φ1 i) and |v , J2 i(|φ2 i) of an excited molecular state via the lasers of frequencies ωL1 and ωL2 , respectively. γ1 and γ2 are the spontaneous decay rates of |φ1 i and |φ2 i, respectively due to their vacuum coupling with continuum. aim is to show VIC in atom-molecule interface system and to investigate whether quenching of spontaneous emission is possible for such system or not. 5.2 Scheme 1 The basic idea of the first scheme is depicted in Fig.5.1 [20]. We consider as our model a system which consists two excited molecular ro-vibrational levels |v , J1i(|φ1 i) and |v , J2 i(|φ2i) , belonging to same electronic state, coupled to the two-atom ground unperturbed continuum |Eib via lasers. Initially either |φ1 i or |φ2 i is populated or partially both are populated via photoassociation of cold atoms using two lasers L1 and L2 of frequencies ωL1 and ωL2 , tuned near | Eib →| φ1 i and | Eib →| φ2 i transitions, respectively. Both the excited levels |φ1 i and |φ2 i decay spontaneously to the same ground continuum with decay rates γ1 and γ2 , Chapter 5. VIC and LIC 54 respectively. We show that coherence between the excited ro-vibrational states builds up due to their interaction with the background electromagnetic vacuum. Moreover, we demonstrate that VIC would be affected by PA lasers. The total Hamiltonian governing the dynamics of the system is given by H = Hcoh + Hincoh where Hcoh is the coherent part involving PA couplings and is given by, Hcoh = 2 X n=1 ~ωbn | φn ihφn | + Z E ′ | E ′ ib b hE ′ | dE ′ Z X 2 n o + ΛnE ′ e−iωLn t ŜE† ′ + H.C. dE ′ . (5.1) n=1 Here ~ωbn are the binding energies of the bound states |φn i(n = 1, 2); |E ′ ib is the ~n · E ~ Ln | E ′ ib is the laser coupling for the bare continuum state and ΛnE ′ = hφn | D ~n transition from the n-th bound state to the bare continuum |E ′ ib . The vectors D ~ Ln are the dipole moment and electric field of the laser associated with the and E n-th transition, respectively. The operator ŜE† ′ = |φn ib hE ′ | is a raising operator. Hcoh is exactly diagonalizable [20–22] in the spirit of Fano’s theory [23] and we obtain Hcoh = Z E | Eid d hE | dE (5.2) where | Eid is the dressed continuum state with dressed energy E with the nor- malization condition hE ′′ |Ei = δ(E − E ′′ ). It is given by | Eid = A2E |φ2 i + A1E |φ1 i + Z CE ′ (E) | E ′ ib dE ′ . (5.3) AnE and CE ′ (E) are the dressed amplitudes. Solving time independent Schrödinger equation HΨE = EΨE under Born-Oppenheimer approximation, we obtain the coupled differential equations: 2 2 ~ d + BJ (r) A1E φ1 + [Ve − (E − ~δ1 )] A1E φ1 − 2µ dr 2 Z = −Λ1E CE ′ |E ′ ib dE, (5.4) Chapter 5. VIC and LIC 55 ~2 d 2 + BJ ′ (r) A2E φ2 + [Ve − (E − ~δ2 )] A2E φ2 − 2µ dr 2 Z = −Λ2E CE ′ |E ′ ib dE, (5.5) Z 2 2 ~ d + Vg (r) − E CE ′ |E ′ ib dE = −Λ∗1E A1E φ1 − Λ∗2E A2E φ2 . − 2 2µ dr (5.6) Here BJ = ~2 J(J + 1)/2µr 2 is the rotational term of the excited state and δ = ωbn − ωLn is the laser-atom detuning. Ve and Vg are excited and ground molecular potentials, respectively. These equations can be solved in a similar manner as depicted in previous chapter using Fano’s method [23]. The dressed amplitudes are given by AnE = (q + ǫn′ ) , n = 1, 2 πΛnE [(ǫ1 + i)(ǫ2 + i) − (q − i)2 ] (5.7) and CE ′ (E) = δ(E − E ′ ) + A1E Λ1E ′ + A2E Λ2E ′ . (E − E ′ ) (5.8) Here, n 6= n′ . ǫn = (ωE + ωLn − ωbn )/[Γn (E)/2]. The term Γn (E) = 2π|ΛnE |2 /~, is the stimulated linewidth of the n-th bound state due to continuum-bound laser coupling. Here q = V12 /(πΛ1E Λ2E ) is analogous to the well-known Fano’s q paR rameter [23] with V12 = P dE ′ Λ1E ′ Λ2E ′ /(E − E ′ ) where ‘P’ stands for principal value. The incoherent part of the Hamiltonian Hincoh describes the interaction of vacuum field with the system and is given by, Hincoh = Z dE ′ " 2 XX n=1 κ,σ gn,σ (E ′ , κ)ŜE† ′ âκ,σ e−iωκ t + H.c. # (5.9) ~ vac and gn,σ (E ′ , κ) = where âκ,σ is the annihilation operator of the vacuum field E p ~i · E ~ vac (κ) | E ′ ib is the dipole coupling with E ~ vac (κ) = ~ωκ /2ǫ0 V ~εσ , −hφn | D p κ being the wave number, σ the polarization of the field and ~ωκ /2ǫ0 V the amplitude of the vacuum field. ρ22 ρ11 ρnn Chapter 5. VIC and LIC 1 0.8 0.6 0.4 0.2 0 0 1 0.8 0.6 0.4 0.2 0 0 1 0.8 0.6 0.4 0.2 0 0 56 ρ11 ρ22 2 γt 4 6 8 10 δ2 = 0 MHz δ2 = 1 MHz δ2 = 3 Mhz 2 γt 4 6 8 10 δ2 = 0 MHz δ2 = 1 Mhz δ2 =3 MHz 2 4 γt 6 8 10 Figure 5.2: Uppermost Panel shows the plots of ρ11 and ρ22 in the absence of laser which is the case of normal VIC. Both ρ11 and ρ22 go to zero in the long time limit. In the middle and lower panel, the excited state population ρ11 and ρ22 are plotted as a function of dimensionless time γt for different values of the detuning of the second laser, delta2 , keeping δ1 = 0. At δ2 = 1 MHz, the population is trapped between the two excited states. The intensities of L1 and L2 are 50 mW cm−2 and 0.1 mW cm−2 , respectively. 5.3 Solution Let the joint state of the system-reservoir at a time t be expressed as, | Ψ(t)i = + X n Z an (t) | φn , {0}i dE ′ X κ,σ bE ′ ,κ,σ (t) | E ′ ib | {1κ,σ }i (5.10) where an and bE ′ ,κ,σ are the amplitudes of n-th excited state and ground continuum, respectively. The state |φn , {0}i corresponds to molecular excited state with field in vacuum and |E ′ ib |{1κσ }i refers to ground bare continuum state with energy E ′ Chapter 5. VIC and LIC 57 1 I1 = 0.1 W/cm ρ11 0.8 I1 = 2.0 W/cm 0.6 I1 = 5.0 W/cm 2 2 2 0.4 0.2 0 0 3 6 9 γt 12 15 12 15 1 ρ22 0.8 I1 = 0.1 W/cm 0.6 I1 = 2.0 W/cm 0.4 I1 = 5.0 W/cm 2 2 2 0.2 0 0 3 6 γt 9 Figure 5.3: Same as in Fig.5.2, but for different values of intensity of first laser, I1 , keeping the intensity of second laser fixed at 0.1 mW/cm−2 , keeping δ1 = δ2 = 0. The population is trapped at excited states for intensity of 2 W/cm−2 . and one photon in mode κ of polarization σ. Now solving the time dependent of Schrodinger equation i | Ψ̇(t)i = − H | Ψ(t)i ~ (5.11) we obtain ȧn (t) = −iω̃n an − i Z dE ′ gn,σ bE ′ ,κ,σ e−i(ωκ −ωLn )t (5.12) and ḃE ′ ,κ,σ = −iω̃b (E ′ )bE ′ ,κ,σ − i X n ∗ gn,σ (κ, E ′ )an ei(ωκ −ωLn )t (5.13) Chapter 5. VIC and LIC 58 where dressed frequencies are given by ω̃n = 1 ~ R |AnE |2 EdE and ω̃b = 1 ~ R |CE ′ (E)|2 EdE. Putting the value of bE ′ ,κ,σ , obtained after integrating Eq.(5.13) in Eq. (5.12) and using the standard Wigner-Weisskopf approach [2], we finally obtain G2 G12 iδ12 L )a1 − e t a2 2 2 G1 G12 −iδ12 L t ȧ2 = (−iω̃2 − )a2 − e a1 . 2 2 ȧ1 = (−iω̃1 − (5.14) (5.15) Detuning δ12 in the above equation is given by δ12 = (ωL1 − ωL2 ) + (ω̃1 − ω̃2 ). Let us consider, ãn is the modified amplitude related to an by transformations, ã1 = a1 exp [i(ω̃1 − δ12 )t] and ã2 = a2 exp [iω̃2 t]. Hence the modified equations become ã˙ 1 = (−G1 − iδ12 ) ã1 − G12 ã2 ã˙ 2 = −G2 ã2 − G21 ã1 (5.16) (5.17) and Gn , the decay constant of the nth bound state, is given by Gn 1 = 2 ~ Z dE ′ Z dt′ X κ,σ |gnσ (E ′ , κ)|2 E′ ′ exp i ωLn + ω̃n − − ωκ (t − t ) ~ (5.18) and Gnn′ , cross damping term, reponsible for coupling between amplitudes can be written as Gnn′ 1 ≃ 2 ~ Z dE ′ Z dt′ X gnσ (E ′ , κ)gn∗ ′σ (E ′ , κ) κ,σ E′ ′ exp i ωLn + ω̃n − − ωκ (t − t ) ~ (5.19) where, n is not equal to n′ . For simplicity in writing the above equation we have assumed (ωL1 + ω̃1 ) ≃ (ωL2 + ω̃2 ). It is important to understand that G12 arises due to quantum interference of the spontaneous emission pathways resulting in VIC between the excited states amplitudes. Summing over the vacuum modes and then carrying out the time integral under Born Markov approximation, we finally obtain 3 −1 Gn = (3πǫ0 ~c ) Z ~ n |E ′ ib |2 dE ′ (ωLn + ω̃n − E ′ /~)3 |hφn |D (5.20) Chapter 5. VIC and LIC 59 and 3 G12 = G21 = (3ǫ0 ~c π) −1 Z ~ 1 |E ′ ihE ′ |D ~ 2 |φ2 i dE ′ hφ1 |D (ωL1 + ω̃1 − E ′ /~)3/2 (ωL2 + ω̃2 − E ′ /~)3/2 . 5.4 (5.21) Results and Discussions Solving the coupled Eqs. (5.16) and (5.17) analytically, we obtain ã1 (t) = c1− ez− t + c1+ ez+ t (5.22) ã2 (t) = c2− ez− t + c2+ ez+ t (5.23) where, z± = 21 [−iδ12 −G+ ±Ω] and c1± = [±(−iδ12 −G− ±Ω)ã1 (0)∓2G12 ã2 (0)]/(2Ω) p 2 and c2± = [±(iδ12 +G− ±Ω)ã2 (0)∓2G12 ã2 (0)]/(2Ω) with Ω = (G− + iδ12 )2 + 4G12 , G− = G1 − G2 and G+ = G1 + G2 . In the limit when laser intensities going to zero (weak coupling), we find ω˜n → (ωbn − ωLn ), thus Gn reduces to the usual damping constant γn . Moreover, for low energy we have G12 ≃ γ12 p (ωb1 ωb2 )3 ~1 · D ~ 2 | φ2 i. = hφ1 | D (3πǫ0 ~c3 ) (5.24) Thus in the absence of lasers, the model reduces to normal VIC case in V-type system [2]. The above equation shows that γ12 vanishes if the molecular transition ~1 and D ~2 are orthogonal. In our model D ~1 and D ~2 are the dipole moments D transition dipole moments between the same ground and excited electronic states, therefore they are essentially nonorthogonal. ′ The excited state populations ρ11 = |ã1 |2 , ρ22 = |ã2 |2 and the coherence ρ12 = ã1 ã2∗ can be obtained from Eqs. (5.22) and (5.23). Explicitly ρnn (t) = e−2G1 t 2 2 2 AΩ − Bδ12 ρnn (0) 2 4Ω 2 −8BG12 ρn′ n′ (0) + 8BG12 δ12 Im[ρ12 (0)] −8G12 ΩRe[ρ12 (0)] sinh(Ωt)] , (5.25) Chapter 5. VIC and LIC 60 where n′ 6= n, A = [1 + cosh(Ωt)] and B = [1 − cosh(Ωt)] and we have consid2 2 ered G1 = G2 . At 4G12 = [4G12 − δ12 ], it follows from above equation that ρ11 and ρ22 become time-independent in long time limit meaning coherent population trapping in the excited states. When δ12 = 0, ρ11 (t → ∞) = ρ22 (t → ∞) = [ρ11 (0) + ρ22 (0) − 2Re[ρ12 (0)]] /4 become exactly same as normal VIC case [2]. It is worthwhile to emphasize that the results given in Eqs. (5.22), (5.23) and (5.25) are general because they are applicable to any PA coupling regime. For experimental realization of VIC, our model can be applied to the spin forbidden intercombination transition 1 S0 −3 P1 of bosonic 174 Yb [24–26] which has no hyperfine interaction. The only molecular ground electronic state of 174 Yb is 1 Σg which corresponds to 1 S0 +1 S0 at long separation and represents the only bare continuum |Eib of our model. The excited states |φn i can be chosen as ro-vibrational levels in long range Ou+ state that can be populated by PA. For illustration, we specifically consider excited ro-vibrational levels |φ1 i = |v = 118, J = 1i and |φ2 i = |v = 118, J = 3i [25]. According to the selection rules of continuum-bound transitions, the minimum partial wave (l) that be coupled to |φ2 i by PA is d wave (l = 2). Usually at ultracold temperatures, d wave scattering amplitude becomes insignificant due to large centrifugal barrier. But ground state scattering properties of 174 Yb are exceptional in the sense that it exhibits a prominent d-wave shape resonance at temperatures as low as 25 µK [24, 27]. We now discuss our numerical results. In Fig.5.2, we show the dynamical behavior of populations ρ11 and ρ22 as a function of scaled time γt. The upper most panel shows populations in the absence of the lasers assuming γ1 = γ2 = γ12 = γ = 2.29 MHz [28]. The short time dynamics clearly shows exchange of population between |φ1 i and |φ2 i due to VIC. In the lower two panels, we plot ρ11 and ρ22 of Eq. (5.25) for different values of δ2 , keeping δ1 fixed. We find that, as δ2 increases upto an optimum frequency, the lifetime of both the excited levels also increases. Then at an optimum frequency , the population gets trapped in the excited state. For the parameters of Fig.5.2, this optimum frequency is found to be 1 MHZ. When δ2 increases beyond the optimum value, population falls off. Since the value of dressed frequency ω̃n depends upon the PA laser intensities, we expect the dynamics to be intensity dependent. Hence by varying the laser intensity of one of the PA lasers while keeping all other parameters fixed, we can achieve excited state population trapping for an optimum intensity of that laser. In Fig.5.3. we show this explicitly for an optimized intensity I1 = 2 W cm−2 . Note that at this 61 0 0.06 -0.05 0.03 -0.1 Im[ρ12] Re[ρ12] Chapter 5. VIC and LIC -0.15 0 -0.03 -0.2 -0.06 -0.25 0 2 4 γt 6 8 10 0 -0.09 0 2 4 γt 6 8 10 0 2 4 γt 6 8 10 0.06 Im[ρ12] Re[ρ12] -0.05 0.03 -0.1 -0.15 -0.2 0 -0.25 0 2 4 γt 6 8 10 -0.03 Figure 5.4: Plot of real and imaginary part of ρ12 against γt for different detunings of second laser (two upper subplots ) and different intensities of the first laser (two lower subplots). All other parameters of the upper subplots is same as in Fig.5.2 while the other parameters of lower subplots are the same as in Fig.5.3. In the two upper subplots, the detuning δ2 = 0 (black solid line), δ2 = 1 MHz (red dotted line) and δ2 = 3 MHz (blue dashed lines) while in the lower two subplots the intensity I1 = 0.1 W cm−2 (black solid line), I1 = 2.0 W cm−2 (red dotted line) and I1 = 5.0 W cm−2 (blue dashed lines) laser intensity, PA stimulated linewidth Γ1 is much larger than γ1 meaning that the system is in the strong-coupling regime. In Fig.5.4, we plot the dynamical behavior of the coherence ρ12 as a function of γt. It is clearly visible that the imaginary part is much more smaller than the real part. The upper panel of Fig.5.4 shows that Re[ρ12 ] becomes steady in the long time limit for an optimum frequency. Lower panel of Fig.5.4 shows that Re[ρ12 ] becomes time-independent in the long time limit for the optimum parameters for which population in Fig.5.3 becomes trapped. Using scheme 1, we have shown that it is possible to generate and manipulate coherence between two excited ro-vibrational states of a molecule by using the Chapter 5. VIC and LIC 62 Figure 5.5: A schematic diagram showing the two-atom continuum |Eib is coupled to two rovibrational levels |φ1 i and |φ2 i of an excited molecular state via the lasers of frequencies ωL1 and ωL2 , respectively. γ1 and γ2 are the spontaneous decay rates of |φ1 i and |φ2 i, respectively due to their vacuum coupling with continuum to bound state |b0 i in ground molecular state. technique of PA spectroscopy. 5.5 Scheme 2 Scheme 2 is depicted in Fig.5.5. Our model consists of two excited diatomic molecular ro-vibrational states |φ1 i and |φ2 i (belonging to the same molecular electronic state) coupled to the ground-state bare continuum |Eib of scattering states, by the lasers 1 and 2, respectively. Initially either |φ1 i or |φ2 i or partially both are populated due to two photoassociation lasers L1 and L2 of frequencies ωL1 and ωL2 , tuned near | Eib →| φ1 i and | Eib →| φ2 i transitions, respectively. The ground continuum is assumed to have only one internal molecular state with only Chapter 5. VIC and LIC 63 ρ11 ρ22 Re[ρ12] Im[ρ12] 0.25 0.2 ρnn(0), Re[ρ12 (0)], Im[ρ12 (0)] 0.15 0.1 0.05 0 0 20 40 I (W/cm2) 60 1 80 100 20 40 I (W/cm2) 60 2 80 100 0.2 0.15 0.1 0.05 0 0 Figure 5.6: ρnn (0) (n = 1, 2) and the real and imaginary parts of ρ12 (0) are plotted against I1 (upper panel) and I2 (lower panel) in unit of W cm−2 , keeping the intensity of the other laser fixed at 1 W cm−2 . The other parameters are φ = 0 and δ1 = δ2 = 0. one threshold and no hyperfine interaction. We assume that the two free-bound PA transitions between the ground-state continuum and the two excited ro-vibrational states are strongly driven so that the spontaneous emissions from these two bound states to the continuum are negligible as compared to the corresponding stimulated ones. However, these two driven bound states can spontaneously decay to other bound state |b0 i of binding energy ~ωb0 in the ground electronic configuration. Our aim is to discuss the creation of laser-induced coherence and its implications in decay dynamics within the framework of master equation approach. The work shows exciting possibilities of manipulating excited state coherences using the relative phase between two lasers. The Hamiltonian governing the dynamics of this system can be written as H = HS + HSR , where HS = Hcoh + ~ωb0 | b0 ihb0 | is the system Hamiltonian with two parts: the first part Hcoh describes coherent dynamics with the two strong Chapter 5. VIC and LIC 64 5 0 shift /hγ -5 En -10 n=1 n=2 -15 -20 -25 10 1 1 10 10 2 10 3 4 10 10 5 6 10 7 10 8 10 9 10 10 10 11 10 0 Γn(E)/γ 10 10 -1 -2 10 -3 10 -4 10 1 10 10 2 10 3 4 10 10 5 6 10 E (Hz) 7 10 8 10 9 10 10 10 11 10 Figure 5.7: Light shifts (scaled by ~γ) and free-bound stimulated line widths (scaled by γ) of the two excited bound states n = 1 (J = 1) (solid) and n = 2 (J = 3) (dashed) - both having the same vibrational quatum number v = 106 of 174 Yb2 (see text) are plotted as a function of collision energy E (in Hz) in upper and lower panels, respectively; for I1 = I2 = 1 W cm−2 and the detunings δ1 = δ2 = 0. PA couplings. On the other hand, the second part HSR is the interaction part of the system with a reservoir of vacuum electromagnetic modes. Explicitly, one can write Hcoh = 2 X n=1 ~(ωbn − ωLn ) | φn ihφn | + Z E ′ | E ′ ib b hE ′ | dE ′ Z X 2 n o † + ΛnE ′ ŜnE + H.C. dE ′ , ′ (5.26) n=1 HSR = XX n=1,2 κ,σ âκ,σ e−i(ωκ +ωLn )t Vn0 (κσ) | φn ihb0 | +H.c. (5.27) Chapter 5. VIC and LIC 65 Here ~ωbn are the binding energies of the bound states |φn i(n= 1, 2); |E ′ ib is the bare continuum state. In deriving the above Hamiltonian, we have used rotating wave approximation (RWA) [29]. In RWA, one works in a frame rotating with the frequency of the sinusoidally oscillating field interacting with a two-level system (TLS) and neglects the counter-rotating terms that oscillate with the sum of the field and the system frequencies. It primarily relies on two conditions: (i) the system relaxation time is much larger than the time period of oscillation of the field and (ii) Rabi frequency or the system-field coupling is much smaller than the transition frequency of TLS. These conditions are in general fulfilled in most cases of a TLS interacting with a monochromatic optical field and therefore RWA can be regarded as a cornerstone for studying quantum dynamics of TLS. Nevertheless, RWA may break down in case of intense laser fields or short pulses when the Rabi frequency or the coupling becomes comparable with the system frequency. Generally, this may happen when the laser intensity is of the order of 1012 W cm−2 or higher. In PA experiments the laser intensity is much lower, typically in the W cm−2 or kW cm−2 . Strong-coupling regime in ultracold PA can be reached with laser intensities higher than 1 kW cm−2 but much lower than 1 MW cm−2 . For driven TLS, corrections beyond RWA and in terms of Bloch-Siegert shift [30] have been discussed by Grifoni and Hanggi [31]. The corrections to RWA can be formulated as a systematic expansion in terms of the ratio of Rabi frequency to the field frequency [32]. In case of two coupled TLS, there exists a parameter regime where leading order term in the expansion vanishes rendering the next higher order term to be significant [32]. However, such situation does not arise in our case and so RWA remains valid. With electric dipole approximation, the laser coupling ΛnE ′ for the absorptive transition from the bare continuum |E ′ ib to the nth excited bound state | φn i is given by ~ n · ELn | E ′ ib ΛnE ′ = ei(kLn ·R+φLn ) hφn | D (5.28) where, kLn , ELn and φLn are the wave vector, electric field and phase of the nth laser, respectively; R is the center-of-mass position vector of the two atoms and ~ n is the free-bound molecular dipole moment associated with the nth bound D state. The electric dipole approximation here dictates that kLn r << 1, where r is the separation between the two atoms. We have thus used exp(ikLn · r) ≃ 1 in † ′ writing the above equation. The operator ŜnE ′ = |φn ib hE | is a raising operator, Chapter 5. VIC and LIC 66 1.5 shift E12 Γ12 0.5 E12 shift /hγ, Γ12/γ 1 0 -0.5 -1 1 10 2 10 3 10 10 4 5 10 6 7 10 10 E (Hz) 10 8 9 10 10 10 10 11 12 10 shift in unit of hγ (black solid) and Γ Figure 5.8: Plotted are E12 12 in unit of hγ (red dashed) are plotted as a function of collision energy E (in Hz). Other parameters are as same as in Fig.5.7. ~ vac and Vn0 (κσ) = âκ,σ denotes the annihilation operator of the vacuum field E p ~ n0 · E ~ vac (κ) | b0 i is the dipole coupling with E ~ vac (κ) = −hφn | D ~ωκ /2ǫ0 V ~εσ , ~ n0 the transition dipole moment between nth excited ωκ being the wave number, D bound state and the ground bound state | b0 i, σ the polarization of the field and p ~ωκ /2ǫ0 V the amplitude of the vacuum field and ~ωb0 is the binding energy of the bound state | b0 i. The Hamiltonian Hcoh is exactly diagonalizable [20, 33] in the spirit of Fano’s theory [23]. The eigenstate of HS is a dressed continuum expressed as | Eidr = 2 X n=1 AnE |φn i + with the normalization condition dr hE and CE ′ (E) are derived in Ref [33]. ′′ Z CE ′ (E) | E ′ ib dE ′ (5.29) |Eidr = δ(E − E ′′ ). The coefficients AnE Chapter 5. VIC and LIC 67 0.2 Re[ρ12] 0.1 0 -0.1 -0.2 0 10 20 30 40 50 0.15 φ=0 φ=π φ = π/2 Im[ρ12] 0.1 0.05 0 -0.05 0 10 20 30 40 50 2 I1 (W/cm ) Figure 5.9: Re[ρ12 (0)] and Im[ρ12 (0)] are plotted against I1 (in unit of W cm−2 ) for different values φ of the difference between the phases of the two lasers in upper and lower panels, respectively. The other parameters are I2 = 1 W cm−2 and δ1 = δ2 = 0 P By using partial-wave decomposition of the bare continuum | E ′ ib = ℓmℓ | P ℓmℓ ′ ′ E ℓmℓ′ ibr , we have ΛnE ′ = exp[i(kLn · R + φLn )] ℓmℓ ΛJn Mn (E ) where Jn and Mn are the rotational and the magnetic quantum number, respectively, of the nth excited bound state in the space-fixed (laboratory) coordinate system. Note ℓ that Λℓm Jn Mn (E) represents amplitude for free-bound transition from (ℓmℓ ) incident partial-wave state to the nth bound state. To denote the amplitude for reverse J n Mn (E). Accordingly, we can write (bound-free) transition, we use the symbol Λℓm ℓ ′ P P P ℓ mℓ′ ℓ′ mℓ′ AnE = ℓ′ mℓ′ AnE Yℓ,mℓ′ (k̂) and CE ′ (E) = ℓmℓ ℓ′ mℓ′ CE ′ ,ℓm (E)Yℓ′ mℓ′ (k̂) where ℓ k̂ represents a unit vector along the incident relative momentum between the two atoms. Explicitly, ℓ′ m ℓ′ m AnE ℓ′ = ℓ′ m LL iθn′ ℓ′ (E) eiθn ΛJn Mℓ′n (E) + ξn−1 ΛJn′ M ′ Knn′ e n′ ξn − LL LL ξn−1 ′ Knn′ Kn′ n , n′ 6= n (5.30) Chapter 5. VIC and LIC 68 and ′ ℓ mℓ′ Jn Mn X AnE Λℓmℓ (E ′ ) ′ = δℓℓ′ δmℓ mℓ′ δ(E − E ) + E − E′ n=1,2 ℓ′ mℓ′ (E) CE ′,ℓm ℓ (5.31) where θn = kLn · R + φLn , and ξn (E) = ~(δnE + iΓn (E)/2), (5.32) ~δnE = E + ~δLn − (En + Enshift ) (5.33) with En being the binding energy of nth excited bound state measured from the threshold of the excited state potential, Enshift is the light shift of the nth bound state and δLn = ωLn − ωA with ωLn is the laser frequency of n-th laser and ωA the atomic transition frequency. The two lasers interacting with the system results in an effective coupling LL Knn ′ 1 = Vnn′ − i ~Gnn′ 2 (5.34) between the two bound states where Vnn′ = exp[i(θn − θn′ )] Gnn′ X ℓmℓ P Z dE ′ J n Mn ′ ℓ (E ′ )Λℓm Λℓm Jn′ Mn′ (E ) ℓ E − E′ 2π X JnMn ℓ (E)Λℓm Λ = exp[i(θn − θn′ )] Jn′ Mn′ (E). ~ ℓm ℓmℓ , (5.35) (5.36) ℓ The term Γn (E) = 2π|ΛnE |2 /~ = 2π P ℓ,mℓ J n Mn (E ′ )|2 /~, is the stimulated |Λℓm ℓ linewidth of the n-th bound state due to continuum-bound laser coupling. Note P shift that the light shift Enshift = ℓ Enℓ is the sum over all the partial light shifts shift Enℓ = X mℓ P Z dE ′ J n Mn ′ ℓ (E ′ )Λℓm Λℓm Jn Mn (E ) ℓ . E − E′ Here, P stands for Principal value integral. (5.37) Chapter 5. VIC and LIC 69 0.1 Re[ρ12] 0.05 0 -0.05 -0.1 0 0.1 10 20 30 40 50 40 50 φ=0 φ=π φ = π/2 Im[ρ12] 0.05 0 -0.05 0 10 20 30 2 I2 (W/cm ) Figure 5.10: Same as in Fig.5.9, but as a function of I2 keeping I1 = 1 W cm−2 5.6 Master equation The system Hamiltonian can be written in dressed basis as H0 = Z EdE | Eidr dr hE | +~ωb0 | b0 ihb0 | . (5.38) To derive master equation we work in the dressed continuum basis of the system Hamiltonian. We express bare basis in terms of dressed basis as follows | bn i = Z dE | Eidr dr hE | bn i = Z dEA∗nE | Eidr (5.39) Z dECE∗ ′ (E) | Eidr (5.40) and ′ | E ibr = Z ′ dE | Eidr dr hE | E i = Chapter 5. VIC and LIC 70 0.17 2 I1 = 1.30 W/cm 2 I1 = 39.70 W/cm 2 ρ11 (t) 0.16 I1 = 43.63 W/cm 0.15 ρ11 (t) 0.13 0.14 I1 = 1.3 W/cm 0.12 0.11 0.13 2 0 0 2 4 γt 8 6 2 4 h 10 6 γt 8 10 Figure 5.11: ρ11 as a function of dimensionless unit γt for different values I1 for I2 = 1 W cm−2 , δ1 = δ2 = 0 and φ = 0. The inset shows the plot of ρ11 when laser-2 is switched off and I1 = 1.3 W cm−2 , δ1 = 0 and θ1 = 0. By substituting all bare basis states with there expansions in terms of dressed basis, we can write system-reservoir interaction Hamiltonian in terms of dressed basis. In the interaction picture, the effective system-reservoir interaction Hamiltonian I HSR = eiH0 t/~HSR e−iH0 t/~ of the driven system interacting with a reservoir of vacuum modes can be written as I HSR = X κ,σ + H.c −iωκ t e 2 X n=1 i(ωb0 −ωLn )t e âκ,σ Z † dEA∗nE Vn0 (κσ)eiωE t Ŝ0E (5.41) where the superscript ‘I’ refers to interaction picture, Ŝ0E =| b0 i dr hE | and ωn0 = ωbn − ωb0 Let ρS+R (t) denote the system-reservoir joint density matrix. Following Agarwal [3], the projection operator P is defined by PρS+R (t) = ρR (0)ρS (t) (5.42) Chapter 5. VIC and LIC 71 where ρR and ρS are the density matrices of vacuum and the dressed system (S) system, respectively. With the use of this projection operator, Liouville equation under Born approximation can be expressed [3] as ∂ PρIS+R (t) = − ∂t t Z dτ PLIS (t)LIS (t − τ )PρIS+R (t − τ ) 0 (5.43) where ρI = eiH0 t/~ρe−iH0 t/~ (5.44) is the density matrix in the interaction picture. Here LIS (t) · · · = X −iωκ t e κ,σ h i + âκσ Σ̂κσ (t), · · · + H.c. (5.45) where Σ̂+ κσ (t) = 2 X −iωLn t e n=1 Z † dE Ŝ0E ei(ωE −ωb0 )t A∗nE Vn0 (κσ). (5.46) Tracing over the vacuum states, we obtain h i XZ t n ∂ I + − I −iωκ τ ρ (t) = − Σ̂κσ (t), Σ̂κσ (t − τ )ρS (t − τ ) dτ e ∂t S κ,σ 0 h io + I + Σ̂− (t), Σ̂ (t − τ )ρ (t − τ ) + H.c. (5.47) κσ κσ S From Eq. (5.47), making use of Markoff approximation, we derive the equations of motion of reduced density matrix elements in dressed basis. These are ρ̇EE ′ Z dE ′ [AEE ′ ρEE ′ + C.c] Z Z ′ = −iωE0 ρE0 − dE AE ′ E ρE ′ 0 − dE ′ AE ′ E ′ ρE0 Z = −iδEE ′ ρEE ′ − dE ′′ AE ′′ E ρE ′′ E ′ dE ′ ρEE ′ Z − dE ′′ AE ′ E ′′ ρEE ′′ ρ̇00 = ρ̇E0 Z dE (5.48) (5.49) (5.50) where δEE ′ = (E − E ′ )/~ and AEE ′ ≃ 1X γnn′ (ωn − ωb0 ) exp[iδnn′ t]AnE A∗n′ E ′ 2 nn′ (5.51) Chapter 5. VIC and LIC 72 with δnn′ = ωLn − ωLn′ being the difference between n-th and n′ -th lasers and γnn′ (ωn − ωb0 ) ≃ ~ n0 D ~ 0n′ (ωn − ωb0 )3 D . 3πǫ0 c3 ~ (5.52) γnn‘ (x) is a function of x. γnn is the spontaneous linewidth of nth excited state and γ12 = γ21 is the vacuum-induced coupling between the two excited states [2, 20]. Note that in Eq. (5.52) we have neglected the light shift of the excited levels in comparison to the transition frequency ωn0 = ωn − ωb0 which is in the optical frequency domain while the typical light shifts as shown in Fig.5.7 are of the order of MHz. The expression (5.52) is obtained in the following way: We first substitute Eq. (5.46) into Eq. (5.47) and express the vacuum coupling Vn0 ~ n0 . The sum in terms of corresponding bound-bound transition dipole moment D over κ and σ is replaced by an integral over the infinite vacuum modes. Using standard Markoffian approximation, one can carry out first the integration over τ and then over the vacuum modes to arrive at the expression for γnn′ as given in Eq. (5.52). The normalization condition is ρ00 + Z ρEE dE = 1 (5.53) Eqs. (5.48)-(5.50) form a set of three integro-differential equations for the density matrix elements expressed in the dressed continuum basis. 5.7 Solution The density matrix elements can be expressed in bare basis by the transformation ρnn′ = Z dE Z dE ′ AnE A∗n′ E ′ ρEE ′ . (5.54) In interaction picture, ρIEE ′ = exp(iδEE ′ t)ρEE ′ and the Eq. (5.50) can be rewritten as ρ̇IEE ′ = − Z dE ′′ AE ′′ E ρIE ′′ E ′ eiδEE ′′ t − Z dE ′′ AE ′ E ′′ ρIEE ′′ eiδE ′′ E ′ t . (5.55) Chapter 5. VIC and LIC 73 The solution of the above equation can be formally expressed as ρIEE ′ (t) ′ Z t Z ′ = δ(E − E ) − dE ′′ AE ′′ E (t′ )eiδEE ′′ t ρIE ′′ E ′ (t′ ) dt 0 Z t Z ′ − dt′ dE ′′ AE ′ E ′′ (t′ )ρIEE ′′ (t′ )eiδE ′′ E ′ t . ′ (5.56) 0 The delta function on the right hand side is the initial value ρIEE ′ (0). The quantity AEE ′ (t) given in Eq. (5.51) is expressed in terms of the product AnE A∗n′ E of the amplitudes of the nth and n′ th bound states in energy-normalized dressed continuum of Eq. (5.29). If vacuum couplings are neglected, the bound-state R probability densities are given by ρnn = dE|AnE |dE and the coherence terms R ρnn′ = dEAnE A∗n′ E with n′ 6= n. It is important to note that, apart from causing spontaneous decay of the nth bound-state probability with decay constant γnn , vacuum couplings of the two excited bound states | b1 i and | b2 i with the ground bound-state | b0 i effectively give rise to vacuum-induced coherence (VIC) [3] between the two excited bound states with coupling constant γ12 . Recently, atom-molecule coupled photoassociative systems are shown to be better suited for realizing VIC [20]. Though the quantities γnn′ are calculable from Eq. (5.52) when the molecular transition dipole moments Dn0 are given, for simplicity of our model calculations, we have set γ11 = γ22 = γ12 = γ21 = γ. In fact, since we consider that both the excited bound states belong to the same vibrational level but differing only in rotational quantum number, the spontaneous linewidths γ11 √ and γ22 would not differ much. Furthermore, since γ12 = γ21 ≃ γ11 γ22 , we have γ12 = γ21 = γ for the case considered here. The stimulated line width Γn (E) is a function of the collision energy E for the ground state scattering between the two ground state atoms. Both in the limits E → 0 and E → ∞, Γn vanishes. Let us fix an energy Ē near which both Γ1 (Ē) and Γ2 (Ē) attain their maximum values. It is then possible to write Eq. (5.51) in the form AEE ′ (t) = 1X γ̄nn′ exp[iδnn′ t]ĀnE Ā∗n′ E ′ ~ nn′ (5.57) p p where γ̄nn′ = γ/ Γn (Ē)Γn′ (Ē) and ĀnE = AnE ~Γn (Ē)/2 are the dimensionless quantities. The absolute value of ĀnE is less than unity. Supposes, the intensities of the two lasers are high enough so that Γn (Ē) >> γ for both the excited bound states. In that case, using γ̄nn′ or the product γ̄nn′ ĀnE Ā∗n′ E ′ as a small parameter, Chapter 5. VIC and LIC 74 we can expand Eq. (5.55) in a time-ordered series ρIEE ′ (t) Z ′ t ′ ′ iδEE ′ t′ Z t ′ = δ(E − E ) − − dt AE ′E (t )e dt′ AE ′ E (t′ )eiδEE ′ t 0 0 Z t′ Z t Z ′ ′′ + dt′ dE ′′ AE ′′ E (t′ )eiδEE ′′ t × dt′′ AE ′ E ′′ (t′′ )eiδE ′′ E ′ t 0 0 Z t′ Z t Z ′ ′′ + dt′ dE ′′ AE ′′ E (t′ )eiδEE ′′ t dt′′ AE ′ E ′′ (t′′ )eiδE ′′ E ′ t 0 0 Z t Z Z t′ ′′ ′ dt′′ AE ′′ E (t′′ )eiδEE ′′ t + dt′ dE ′′ AE ′ E ′′ (t′ )eiδE ′′ E ′ t 0 0 Z t′ Z t Z ′ ′′ + dt′ dE ′′ AE ′ E ′′ (t′ )eiδE ′′ E ′ t dt′′ AE ′′ E (t′′ )eiδEE ′′ t 0 0 + ··· (5.58) It is worthwhile to point out that this method of solution is similar in spirit to that of time-dependent perturbation, however it differs in essence because we have used dressed state amplitude as a small parameter and not the atom-field coupling. If a large number of terms are taken, then the expansion essentially provides solution for any time. However, numerically calculating higher order terms becomes increasingly involved because of larger number of multiple integrals in energy variable appearing in higher order terms. We therefore restrict our numerical studies to a few leading order terms as described in the next section. 5.8 Results and discussions Driven by the two strong lasers, the system is prepared in a dressed continuum given by Eq. (5.29). Since this state is an admixture of the two excited bound states, it is subjected to spontaneous emission. We include spontaneous emission by considering the dressed levels to decay to a third bound level, thereby neglecting the decay of the excited states to the ground-state continuum inside the dressedstate manifold. To discuss the effects of the phase-difference φ = θ1 − θ2 between the two lasers, the laser intensities I1 and I2 , and the detunings δ1 and δ2 on decay dynamics, we ℓ first rewrite the dressed-state amplitude Aℓm nE of Eq. (5.30) in the form ℓ Aℓm nE ℓmℓ −i(θn −θn′ ) + Ann ′ e En + iGn /2 ℓmℓ iθn ΛJn Mn (E) =e (5.59) Chapter 5. VIC and LIC 75 0.155 0.13 2 I2 = 1.3 W/cm I2 = 3.5 W/cm2 2 I2 = 6.05 W/cm ρ22 0.15 2 I2 = 1.3 W/cm 0.12 0.11 0 2 4 ρ22 (t) 0.145 γt 6 8 10 0.14 0.135 0.13 0 1 2 3 4 5 γt 6 7 8 9 10 Figure 5.12: ρ22 (t) is plotted as a function of γt for different values of I2 for I1 = 1 W cm−2 , δ1 = δ2 = 0 and φ = 0. The inset shows the plot of ρ22 when only laser-2 is switched on at intensity I2 = 1.3 W cm−2 , δ2 = 0 and θ2 = 0. ℓmℓ −1 LL where Ann ′ = ξn′ Knn′ and ′ shift En = E + ~δn − (En + Enshift + Enn ′ ), n 6= n. (5.60) The additional shift for the nth excited bound state due to laser-induced cross coupling with the other (n′ ) excited bound state is −1 LL LL shift Enn ′ = Re[ξn′ Knn′ Kn′ n ]. (5.61) LL LL Here Gn = Γn + Γnn′ with Γnn′ = −2Im[Re[ξn−1 ′ Knn′ Kn′ n ] being the contribution to the total stimulated line width due to the cross coupling. In expression (5.59), the first term in the numerator corresponds to single-photon transition amplitude due to nth laser while the second term describe a net 3-photon transition amplitude with 2 photons coming from the n′ th laser and the other one from nth laser. Chapter 5. VIC and LIC 76 0.14 φ=0 φ=π φ = π/2 ρ22 0.14 ρ11 0.13 0.13 0.12 0.11 0 0.1 1 2 3 4 0 1 2 -0.1 0 1 2 3 4 5 3 4 5 0.05 Im[ρ12] 0.05 Re[ρ12] 0.12 5 0 0 -0.05 -0.05 -0.1 0 1 2 Figure 5.13: γt 3 4 5 γt ρnn′ (t) are plotted against γt for different values of φ but for fixed I1 = I2 = 1 W cm−2 and δ1 = δ2 = 0. The foregoing discussion has so far remained quite general. Now, we apply our method to ultracold 174 Yb atoms. For numerical illustration, we use realistic pa- rameters following the recent experimental [24, 26–28] and theoretical [25] works on PA with 174 Yb. We have chosen 174 Yb system because this offers some advan- tages compared to other systems. For instance, it has no hyperfine structure and the ground-state molecular potential of 174 Yb2 is spin-singlet only. Furthermore, it has spin-forbidden inter-combination transitions. The total rotational quantum number is given by J~ = J~e + ~ℓ where Je is the total electronic angular momentum. For numerical work, we specifically consider a pair of 174 Yb atoms being acted upon by two co-propagating linearly polarized cw PA lasers. The polarizations of both lasers are assumed to be same. This geometry is the same as used in Ref [33] for manipulation of d-wave atom-atom interactions. For our numerical work, we consider that the two lasers drive transitions to the molecular bound states 1 and 2 characterized by the rotational quantum numbers J1 = 1 and J2 = 3, respectively; of the same vibrational level v = 106. The two bound states belong to 0+ u (Hund’s Chapter 5. VIC and LIC 77 0.5 0.4 0.3 Iqb φ=0 φ=π φ = π/2 0.2 0.1 0 0 1 2 3 4 5 0.5 I2 = 1.30 W/cm 0.45 I2 = 3.50 W/cm Iqb 0.4 I2 = 6.05 W/cm 0.35 2 2 2 0.3 0.25 0.2 0 2 4 γt 6 8 10 Figure 5.14: In upper panel, we plot Iqb against γt for different values of φ when other parameters are fixed as I1 = I2 = 1 W/cm2 and δ1 = δ2 = 0 MHz. In lower panel Iqb is plotted as a function of γt for different values of I2 for I1 = 1 W/cm2 , φ = 0 and δ1 = δ2 = 0 MHz. case c) molecular symmetry meaning that the projection of Je on the internuclear axis being zero. Since 174 Yb atoms are bosons, only even partial waves are al- lowed for the scattering between the two ground state atoms. Free-bound dipole transition selection rules then dictate that the bound state 1 can be accessed from s- and d-wave scattering states, while the bound state 2 is accessible from d- and g-wave only. Thus d-wave ground scattering state is coupled to both the excited LL states by the two PA lasers resulting in the laser-induced coupling term Knn . In general, d-wave scattering amplitude is small at low energy. But, fortunately for 174 Yb atoms, there is a d-wave shape resonance [24, 27] in the µK temperature regime leading to significant enhancement in d-wave scattering amplitude at relatively short separations where PA transitions are possible. In our calculations we neglect g-wave contributions. As we prepare the system in a desired dressed continuum, the populations of Chapter 5. VIC and LIC 78 the two excited bound states and the coherence between them depend on the relative intensity and phase between the two lasers. In the absence of spontaneous emission (idealized situation), the dressed state properties correspond to the initial conditions for our model. Fig.5.6 shows variation of the initial populations and the coherence as a function of the intensity of either laser for the intensity of the other laser being fixed at 1 W cm−2 . For all our numerical work, we set the spontaneous line width γ = 2.29 MHz [28]. The variation of stimulated line widths and light shifts of the two bound states of 174 Yb2 as a function of collision energy E for the laser intensities I1 = I2 = 1 W cm−2 and zero detunings are displayed in Fig.5.7. The shift E1shift (stimulated line width Γ1 ) is a sum of s- and d-wave partial shifts ( stimulated line widths) while the shift E2shift and the stimulated line width Γ2 are made of mainly d-wave partial shift and width, respectively; with no contribution from s-wave. From Fig.5.7 we notice that the shifts of both bound states as a function of energy change rapidly from negative to positive value near E = Ē = 194 µK and the stimulated line widths of both bound states exhibit prominent peaks at that energy. This can be attributed to a d-wave shape resonance [24, 25]. We have found that the d-wave partial stimulated line widths of both the bound states near shape resonance are comparable. For the first bound state, the value of the d-wave partial stimulated line width near the resonance is found to exceed the s-wave partial line width by shift about 2 orders of magnitude. In Fig.5.8, mutual light shift E12 and stimulated line width Γ12 due to the coupling between the bound states are shown as a function of collision energy E in Hz. The mutual shifts and widths arise from the coupling of the d-wave scattering state with the two bound states by the two lasers. Owing to the existence of the d-wave shape resonance, the laser couplings of the d-wave scattering state to both bound states become significant, and so are the mutual shifts and stimulated line widths. Figs.5.9 and 5.10 exhibit intensity-dependence of the coherence ρ12 (0) for different relative phases. The purpose of plotting these two figures is to assert that it is possible to prepare the dressed system with a desirable coherence between the two excited bound states by judiciously selecting relative intensities and phases between the two lasers. It is interesting to note that d-wave shape resonance has a drastic effect on the properties of dressed continuum. Because of this resonance, the d-wave contributions to the amplitudes of transition to both the bound states are large even at a low temperature allowing an appreciable cross coupling to Chapter 5. VIC and LIC 79 develop between the two bound states. For very large laser intensities at δ1 = δ2 = 0, light shifts would be so large that the system will be effectively far off resonant and therefore ρnn′ (0) ≃ 0. For calculating time-dependence of the density matrix elements for all the times, we need to calculate a large number of terms appearing on the right hand side of Eq. (5.58) order by order in γ̄nn′ . This is a laborious and time-consuming exercise. Instead, to demonstrate the essential dynamical features arising from quantum superposition of the two rotational states, we restrict our study of decay dynamics to relatively short times. Inserting Eq. (5.51) in Eq. (5.58), retaining the terms up to first order in γ̄nn′ , we have h i t ρnn (t) = Ãnn (0) − γ dt′ |Ã12 (t′ )|2 + |B̃12 (t′ )|2 + |Ãnn (t′ )|2 + |B̃nn (t′ )|2 0 Z t n o ′ ′ ′ ′ ′ − 2γ dt Re Ã12 (t )Ãnn (t ) + B̃nn (t )B̃12 (t ) cos(δ12 t′ ) + · · · (5.62) Z 0 R dEA∗nE An′ E cos(ωE t) and B̃nn′ (t) = dEA∗nE An′ E sin(ωE t) with R ωE = E/~. Here Ãnn (0) = dE|AnE |2 = ρnn (0). Similarly, the coherence term where Ãnn′ (t) = R ρ12 can be calculated up to the first order in γ̄nn′ . These solutions hold good for γt < |Ãnn (0)|−2 or equivalently, γt < |ρnn (0)|−2 for both n = 1, 2. The decay dynamics of the populations ρ11 (t) and ρ22 (t) as a function of the scaled time γt for different laser intensities are shown in Figs. 5.11 and 5.12, respectively. These results clearly exhibit that, when the system is strongly driven by two lasers, the decay is non-exponential and has small oscillations. The oscillation are particularly prominent for short times. In the long time limit the oscillations slowly die down. However, the oscillations can persist for long times if couplings are stronger. We have chosen the values of the laser intensities I1 and I2 such that the initial values of dressed population ρ11 (t = 0) or ρ22 (t = 0) are the same for those intensities. We notice that, though the values ρ11 (0) (or ρ22 (0)) for a set of I1 values for a fixed I2 value (or a set of I2 values for a fixed I1 ) are the same, their time evolution is quite different and strongly influenced with the relative intensity of the two lasers. That the population oscillations result from the laser-induced coherence between the two bound states can be inferred by observing the decay of the populations when either of the lasers is switched off. Plots of ρ11 and ρ22 against γt for only laser-1 and laser-2 switched on, respectively are illustrated in the insets of Figs. 5.11 and 5.12 which show exponential decay of the populations Chapter 5. VIC and LIC 80 0.52 δ1 = 0 MHz, δ2 = 0 MHz δ1 = 1 MHz, δ2 = 1MHz δ1 = 1 MHz, δ2 = -1 MHz δ1 = -1 MHz, δ2 = 1 MHz δ2 = -1 MHz, δ2 = -1 MHz 0.5 0.48 0.46 Iqb 0.44 0.42 0.4 0.38 0.36 0.34 0 2 4 γt 6 8 10 Figure 5.15: Iqb is plotted against γt for different values of detuning parameters. Other parameters are fixed as I1 = I2 = 1 W/cm2 and φ = 0. ρ11 (t) and ρ22 (2) with no oscillations. When only one laser is tuned near the resonance of either bound state, we do not have any coherence between the two bound states. As the two excited bound states are about 57 MHz apart, one of the bound states remains far off-resonant in case of single-laser driving. As a result, the decay of the driven bound state occurs independent of the other bound state. The laser-induced coherence between the two bound states is developed only when we apply both the lasers. The dynamical characteristics of population decay can be interpreted by analyzing the time-dependence and relative contributions of the two expressions within the third and second brackets on the RHS of Eq. (5.62). Since δ12 ≃ −57 MHz and the free-bound couplings are most significant near E ≃ Ē ∼ 4 MHz as can be noticed from Fig.5.7, we may perform the time integration on the terms associated with Chapter 5. VIC and LIC 81 cos(δ12 t) in Eq. (5.62) in the slowly varying envelope approximation to obtain n o − 2γRe Ã12 (t)Ãnn (t) + B̃nn (t)B̃12 (t) sin(δ12 t)/δ12 . (5.63) Further, since in energy integrations the major contributions will come from energies near E ≃ Ē, we may approximate Ãnn′ (t) = Z dEA∗nE An′ E cos(ωE t) ≃ cos(ωĒ t)ρnn′ (0). (5.64) Similarly, B̃nn′ ≃ sin(ωĒ t)ρnn′ (0). Using these approximations, we get ρnn (t) ∼ ρnn (0) − γt |ρ12 (0)|2 + ρnn (0)2 − 2γRe {ρ12 (0)ρnn (0)} sin(δ12 t)/δ12 + · · · . (5.65) This expression clearly shows that when the quantities (|ρ12 (0)|2 + ρnn (0)2 ) and 2Re(ρ12 (0)ρnn (0)) are of comparable magnitude, we expect oscillations in population decay with time period τosc =∼ 2π/|δ12 | ≃ 0.11 in unit of γ −1 . When the laser intensities are not too high to induce large shifts, we would expect the qualitative features of the oscillations will be largely governed by one time scale which is τosc . In fact, the solid black curves in Figs. 5.11 and 5.12 clearly demonstrate oscillatory modulations with time scale τosc . However, when the laser intensities are high enough so that the energy-dependent shifts and stimulated line widths are appreciable for a range of energies around E = Ē, then expression (5.65) would not be useful to indicate correct qualitative features. In that case, we need to retain full time dependence which will introduce another time scale 2π/ωĒ which is, in the present context, roughly equal to 2π in unit of γ −1 . In such situations the net result would be a competition between oscillations with the two time scales. The plots in Figs. 5.11 and 5.12 at larger laser intensity or intensities clearly demonstrate such oscillatory modulations of the population decay with two time scales. It is particularly important to note that for larger intensity and appropriate detunings, the early population decay can be made much slower for an appreciable time duration. It is worthwhile to point out that, this analysis is done only to gain insight into the physics of the decay dynamics of the system, all the results presented here are obtained by numerically integrating over time t′ and the entire range of energy. Fig.5.13 shows the effects of φ on the temporal evolution of the populations ρnn Chapter 5. VIC and LIC 82 and the coherence terms ρnn′ with n 6= n′ . Though φ does affect the behavior of the oscillations in population decay, the magnitude of the populations at a time t is not altered much with the change of φ. In contrast, the magnitudes of the real and imaginary parts of the coherence term ρ12 are largely influenced by φ. When φ is altered by π, the sign of both real and imaginary parts of ρ12 changes. Finally, we discuss quantum beats by studying the temporal evolution of the intensity of light emitted from the two correlated excited bound states. Quantum beats are manifested as oscillations in the emitted radiation intensity Iqb as a function of time [2, 34], which is given by Iqb (t) = γ(ρ11 (t) + ρ22 (t) + 2Re[ρ12 (t)]). (5.66) In lower panel of Fig.5.14 we show the effects of laser phase φ on quantum beats in time-dependent fluorescent intensity. The effects of different intensities of laser-2 on quantum beats are illustrated in the lower panel of same figure. We demonstrate the effects of different detunings on quantum beats in Fig.5.15. Because of the shift LL mutual light shift E12 between the two bound states due to coupling term K12 , the resonance conditions in case of two PA lasers are altered in comparison to those in single PA laser case. This leads to the non-monotonous effects of detunings on quantum beats as shown in Fig.5.15. Before ending this section, we wish to make a few remarks on the possibility of experimental demonstration of the physical effects discussed here. Our model can be easily realisable with currently available technology of high precision PA spectroscopy. Ultracold bosonic Yb or Sr atoms appear to be most suitable for this purpose. Because, they offer several advantages. First, their electronic ground state is purely singlet and has no hyperfine structure. This means that the bare continuum has no multiplet structure and so there is only one ground-state channel. Second, they have narrow line singlet-triplet inter-combination transitions. Third, they have long-range excited bound states which are accessible via PA [25]. These bound states have relatively long life time (∼ microsecond). It is possible to selectively drive two rotational levels as required for the model. Furthermore, since both the excited bound states have the same vibrational quantum number, their outer turning points will lie almost at the same separation. Because of long-range nature of these excited bound states that are strongly driven by the two lasers from the bare continuum, these two excited bound states are expected to have the largest Franck-Condon overlap with the near-zero energy or the last bound Chapter 5. VIC and LIC 83 state in the ground electronic potential. It is therefore quite natural that these two driven bound states will predominantly spontaneously decay to the last bound state. In fact, the scattering length of Yb atoms have been experimentally determined by detecting the last bound state via two-colour PA spectroscopy [26], since the energy of the last bound state and the scattering length are closely related. All these facts indicate that our model is a realistic one and the predicted excited state coherence and the resulting quantum beats can be experimentally realisable. 5.9 Conclusion We have demonstrated that it is possible to generate and manipulate coherence between two excited ro-vibrational states of a molecule by using the technique of PA spectroscopy. Though both the schemes seem to be same, but they have some basic differences. In first scheme, the excited states populated by PA from ground continuum state are assumed to decay to the same continuum only whereas in the second scheme, the dressed continuum decays spontaneously outside the system, to the last bound state of the ground configuration. In first case, we use WignerWeisskopf method and the spontaneous emission is considered phenomenologically, while in second one, master equation approach is applied and the spontaneous decay is taken into account from first principle calculation. In first scheme, only s-wave scattering is considered while both s- and d-wave are taken in the second scheme. The first scheme is quite simplified whereas second scheme is more realistic as it considers decay of the system outside the dressed continuum. In both cases we use 174 Yb for numerical demonstration though the excited states chosen are different. Moreover the second scheme shows exciting possibilities of manipulating excited state coherences using the relative phase between two lasers. But from both the models, we have demonstrated that the spontaneous emission from two correlated excited bound states are strongly influenced by the coherence between them. We have also discussed the possibility of experimental realization of our models. A promising candidate for exploring such excited state coherence is the bosonic Yb atoms. Finally, our work may be useful in stimulating further studies on laser manipulation of the continuum and the bound states between ultracold atoms. 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B 1, 102 (1984); A. Lami and N. K. Rahman Phys. Rev. A 33, 782 (1986); E. Kyrola J. Phys. B. 19, 1437 (1986). [37] S. L. Haan and G. S. Agarwal, Phys. Rev. A 35, 4592 (1987). [38] J. L. Liu, P. McNicholl, D. A. Harmin, J. Ivri, T. Berefman and H. J. Metcalf, Phys. Rev. Lett. 55, 189 (1985); S. Feneuille, S. Liberman, E. Luc-Koenig, J. Pincard and A. Taleb, J. Phys. B 15, 1205 (1985). [39] B. Deb, A. Rakshit, J. Hazra and D. Chakraborty, Pramana Journal of Physics 80, 3 (2013). [40] S. Saha, A. Rakshit, D. Chakraborty, A. Pal and B. Deb, arXiv:1405.1674 (2014) Chapter 6 Atom-Ion Cold Collisions: Formation of Cold Molecular Ion In recent years, both the fields of cold and ultracold neutral atomic gases and cold trapped atomic ions have developed greatly. Total control have been acquired over both the fields down to the quantum level. The progresses in both the fields provide us a new opportunity for exploring atom-ion quantum dynamics, charge transfer and spin exchange reactions. Understanding ion-atom cold collision [1–12] is important for realizing a charged quantum gas, studying charge transport [13] at low energy, exploring polaron physics [14–16] and producing ion-atom bound states [17] and cold molecular ions [18–20]. Molecular ions have lots of prospects for exploring the new fields of physics and chemistry. Cold molecular ions may be used as natural candidates for the measurement of electron dipole moment (EDM) [21, 22]. The study of cold molecular ions has relevance in diverse areas such as metrology [23, 24] and astrochemistry [25]. Recently, molecular ions have been cooled into rovibrational ground states by all optical [20], laser and symphathetic cooling methods [18, 19]. A large variety of diatomic and triatomic molecular ions are also cooled by symphathetic methods [26–28]. Other methods such as photoassociative ionization [29–33], buffer gas [34] and rotational cooling [35] have been widely used for producing low-energy molecular ions. In this chapter, a new method for preparing cold molecular ion is proposed. A new formalism to prepare cold molecular ion from colliding cold atom-ion pair in a hybrid trap [1– 3, 36] by photoassociation (PA) [37] process is discussed. Both homonuclear and heteronuclear collisions are studied with an emphasis on heteronuclear rediative processes. It is shown that it is possible to form translationally and rotationally 88 Chapter 6. Atom-Ion Cold Collisions 89 Figure 6.1: Schematic diagram of possible physical processes which can take place during atom-ion collision at low energy. cold molecular ions by PA, starting from a cold alkaline earth metal ion and an ultracold alkali-metal atoms. In section 6.1, we outline the theoretical background for understanding of atom-ion cold collision dynamics. Then we focus on elastic and inelastic processes in section 6.2. We propose new scheme for the formation of cold molecular ion. In section 6.3, we present results and discussion. In section 6.4, conclusions are drawn. 6.1 Theory of Atom-Ion Collision In this section, a brief discussion is presented on theoretical framework for understanding atom-ion interaction in the cold and ultracold regimes where a full Chapter 6. Atom-Ion Cold Collisions 90 E (a.u.) -21.65 -21.7 1 + 2Σ 1 + 1Σ -21.75 -21.8 -21.85 + 2 + Σu -14.55 E (a.u.) 2 Σg -14.5 2 Πu -14.6 -14.65 -14.7 -14.75 0 5 10 15 20 25 30 r (a.u.) Figure 6.2: Upper panel shows 11 Σ+ (solid) and 21 Σ+ (dashed) model poten2 + tials of (LiBe)+ system. Lower panel shows 2 Σ+ g (solid)), Σu (dashed-dotted) + and 2 Πu (dashed) potentials of Li2 . quantum mechanical treatment becomes necessary. A point to be noted that we are focusing on two body collisions and ions are singly charged. Atom-ion two body collisions can generally be divided into two categories. First one is the normal elastic collision where there is no change in internal structures of colliding atom-ion. If the collision is taking place between atom A and ion B + , then elastic collision may be denoted as A + B+ → A + B+. (6.1) But elastic collision is not the sole possible outcome of the collision. Due to the interaction between two colliding particles, an electron may hop from atom A to ion B + provided that colliding atom and ion are close enough. This type of collision is known as inelastic collision in which the internal structure is changed. Chapter 6. Atom-Ion Cold Collisions 91 This is given by A + B + → A+ + B. (6.2) Now it is important to know that the inelastic collision can be divided into various categories such as radiative and nonradiative charge transfer collision, photoassociative transfer or molecule formation [38–42] and spin-exchange collision. In following subsections we discuss in detail about elastic and inelastic collisions. But before that atom-ion interaction potential is discussed. 6.1.1 Atom-Ion Interaction Potential A neutral atom placed in the electric field produced by an ion have an induced dipole moment or higher induced multiple moments and the polarised neutral atom is attracted towards the ion. To derive the corresponding interaction potential we consider an atom with an induced dipole moment p. We can write p = αε, where ε is the electric field of the ion and α is the atomic polarizability. The potential energy of the atom in the presence of electric field of the ion ε is given by 1 1 U = − pε = − αε2. 2 2 (6.3) Now using the well-known expression for the electric field of an ion, we get V (r) = − C4 2r 4 (6.4) where C4 = q 2 α/4πǫ0 . This provides an attractive potential between the atom and ion. Here r is the distance between the atom and ion and q is the charge of ion. So the long range potential [10, 11] is given by the expression 1 V (r) = − 2 C4 C6 C8 + 6 + 8 ··· r4 r r (6.5) where, C6 , C8 correspond to quadrupole, octupole polarisabilities of atom concerned and C4 /2r 4 is dominating term. The polarisation interaction falls off much more slowly than van der Waals interaction which represents the long range part of interaction between neutral atoms. Hence collision between atom and ion is dominated by the long range polarization interaction. The qualitative feature of this long range interaction of atom-ion is governed by effective length β4 . It is Chapter 6. Atom-Ion Cold Collisions 92 3.0 2.5 D(r) (a.u.) 2.0 1.5 1.0 0.5 0.0 0 5 10 15 20 25 30 r (a.u.) Figure 6.3: Radial transition dipole matrix element as a function of separation r for (LiBe)+ system. calculated by equating centrifugal energy with the potential in (6.4). It is found p that β4 = µC4 /~2 6.1.2 Elastic Collisions In case of purely elastic scattering the internal states of atom and ion donot change as stated above. Continuum wave function for atom-ion cold collision can be obtained by solving the partial wave Schrödinger equation given by d2 2µ ℓ(ℓ + 1) 2 ψℓ (kr) = 0 + k − 2 V (r) − dr 2 ~ r2 (6.6) where, r is the ion-atom separation. The wave function ψℓ (kr) has the asymptotic form ψℓ (kr) ∼ sin [kr − ℓπ/2 + ηl ] with ηℓ (k) being the phase shift for ℓ-th partial Chapter 6. Atom-Ion Cold Collisions 93 8 linear fit log10σel (a.u.) 7 6 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 log10E (K) Figure 6.4: Total elastic scattering cross-section σel for Li + Be+ (21 Σ+ ) collision is plotted against collision energy E in K. The dashed curve is a linear fit for energies greater than 10−6 K. wave. The total elastic scattering cross section is expressed as ∞ 4π X (2ℓ + 1) sin2 (ηℓ ) σel = 2 k ℓ=0 where k = (6.7) p (2mE/~2 ). As the energy gradually increases more and more partial waves start to contribute to total elastic scattering cross sections and the scattering cross section at large energy is [10] σel ∼ π µC42 ~2 13 1 π2 E− 3 . 1+ 16 (6.8) As k → 0, according to Wigner threshold laws ηℓ (k) ∼ k 2ℓ+1 if ℓ ≤ (n − 3)/2 with n being the exponent of long-range potential behaving as ∼ 1/r n as r → ∞. If ℓ > (n − 3)/2 then the threshold law is ηℓ (k) ∼ k n−2 . Since the long-range part of Chapter 6. Atom-Ion Cold Collisions 94 6.5 linear fit log10σel (a.u.) 6 5.5 5 4.5 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 log10E(K) Figure 6.5: Same as in Fig.6.4 but for Li+ + Be (11 Σ+ ). ground as well as excited ion-atom potentials goes as ∼ 1/r 4 as r → ∞, Wigner threshold laws tell us that s-wave (ℓ = 0) ion-atom scattering cross section should be independent of k while all the higher partial wave scattering cross sections should go as ∼ k 2 in the limit k → 0. 6.1.3 Radiative Transfer Charge Transfer process between atom A and ion B + may take place in following ways: (1) radiative charge transfer process A + B + → A+ + B + ~ω, (6.9) Chapter 6. Atom-Ion Cold Collisions 95 4 10 σl (a.u) 10 3 10 l=0 l=2 1 0 10 -1 10 -3 10 -9 10 10 -8 10 -7 -6 10 10 -5 -4 10 -3 10 -2 10 -1 10 10 0 E (K) Figure 6.6: Partial wave cross sections for Li+ + Be (11 Σ+ ) collision are plotted as a function of E (in K) for ℓ = 0 (solid) and ℓ = 2 (dashed). (2) radiative association process or photoassociative charge trasfer A + B + → (AB)+ + ~ω, (6.10) (3) nonradiative charge transfer A + B + → A+ + B. (6.11) Since at low energy nonradiative charge transfer is dominated, we are mainly interested in the first two processes. The radiative processes occurs due to the interaction of the system with the radiation field. The direct charge transfer arises through the transition between different molecular states due to perturbation caused by the nuclear motion. The radiative charge transfer cross section [38–40] Chapter 6. Atom-Ion Cold Collisions 96 is given by σct = Z ωmax ωmin dσct dω dω (6.12) where, ω is the angular frequency of emitted photon and dσct 8ω 3 π 2 X 2 ℓMℓ,ℓ−1 (km , kn ) = 2 dω 3c3 km l 2 + (ℓ + 1)Mℓ,ℓ+1 (km , kn ) where, Mℓ,ℓ′ (km , kn ) = Z 0 (6.13) ∞ drψℓm (km r)D(r)ψℓn′ (kn r). (6.14) D(r) is the magnitude of the molecular transition dipole moment. Here km = p p 2µ [E − Vm (∞)] and kn = 2µ [E − Vn (∞) − ~ω] are the momentum of en- trance and exit channels, respectively; and E is collision energy of entrance (m) channel. Vm and Vn are the potential energies of the entrance (m) and exit (n) channels, respectively. ψℓi (kir) is the wave function of ℓ-th partial wave for i-th channel of momenum ki . The total radiative decay [39] from the upper state (m) to the lower state (n) is given by σrt = ∞ π X (2ℓ + 1) [1 − exp(−4ζℓ )] 2 km (6.15) ℓ where, π ζℓ = 2 Z ∞ 0 |ψℓm (km r)|2Anm (r)dr (6.16) is a phase shift and |Vn (r) − Vm (r)|3 4 Anm (r) = D 2 (r) 3 c3 (6.17) is the transition probability. The difference between the total radiative transfer cross section and the radiative chrage transfer cross section is the cross section is the radiative association cross section. Chapter 6. Atom-Ion Cold Collisions 10 97 5 4 10 σl (a.u.) 10 3 10 1 0 l=0 l=2 10 -1 10 -3 10 10 -4 -8 10 10 -7 -6 10 10 -5 -4 10 -3 10 -2 10 -1 10 10 0 E (K) Figure 6.7: Same as in Fig.6.6 but for Li + Li+ ground state collision in 2 Σg state. 6.1.4 Spin Exchange Collision Another outcome of collisions between an alkali atom and an alkali type ion, both in 2 S1/2 state is spin exchange collision. The interatomic potential is differnt for the both spin parallel case (triplet state) than both spin anti parallel case (singlet state). Hence this spin exchange interaction originates from the overall symmetry requirement of the electron wave function. The quantum mechanical cross section for spin exchange, in the degenarate internal state approximation [11, 43], is given by σ= π l Σl=0 (2l + 1) sin2 (δℓtriplet − δℓsinglet ). 2 k (6.18) where, δℓtriplet and δℓsinglet are the phase shifts due to scattering from triplet and singlet state, respectively. A point to be noted that, though the validity of the approximation is for the collision energies larger than the hyperfine splitting, it Chapter 6. Atom-Ion Cold Collisions 98 8 7.5 2 + 2 + log10 σel (a.u.) Σu Σg 7 6.5 6 5.5 5 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 log10 E (a.u.) Figure 6.8: Same as in Fig.6.4 but for Li + Li+ collision in 2 Σ+ g (dashed) and 2 Σ+ (solid) potentials. u usually gives right order of magnitude at smaller energies [44]. It is also observed that spin exchange collision rate increases as the temperature gradually decreases . All possible processes except the spin exchange collosion between an alkali atom and an alkali like ion are shown in Fig.6.1 [45] 6.2 Fomation of Cold Molecular Ion In our proposed mechanism for formation of cold molecular ion, cold alkali atom and alkali-type ion are used as starting materials. Formation of ground molecular ion requires a three-step radiative reaction process. in the first step, the ionatom pair in the excited elecronic state undergoes radiative charge transfer process to form ground state ion-atom pair. In the second step, these ion-atom pair is Chapter 6. Atom-Ion Cold Collisions 99 10 σct (a.u.) 8 6 4 2 -1 0 10 10 10 1 E (mK) Figure 6.9: Charge transfer scattering crosssection σct (in a.u.) of (LiBe)+ system is plotted against collisional energy E (in mK). photoassociated to the excited electronic bound state in the presence of laser of suitable frequency. In the last step, the excited molecular ion is stimulated by another lasser to deexcite to form ground molecular ion. since the molecular ion is formed from translationally cold atom and ion, the molecular ion is translationally and rotationally cold. 6.2.1 Formulation of Problem To illustrate atom-ion radiative cold collisions, we consider a model system of 7 Li + Be+ undergoing elastic and radiative charge transfer collisions. The possible experimental situation can be imagined as a single Be+ ion immersed in a BoseEinstein condensate of 7 Li atoms in a hybrid trap. The molecular potentials 11 Σ+ (ground) and 21 Σ+ (excited) of (LiBe)+ system as shown in Fig.6.2 (upper inset) asymptotically go to 1 S +1 S (Li+ + Be) and 2 S Chapter 6. Atom-Ion Cold Collisions 100 2 7 |ηJl| x 10 (a.u.) 250 200 150 100 50 400 2 |ηJl| (a.u.) 600 200 0 -8 10 10 -7 -6 10 10 -5 10 -4 -3 10 -2 10 -1 10 0 10 1 10 E (K) Figure 6.10: Square of Franck Condon overlap integral |ηJℓ |2 (in a.u.) for Li-Li+ (upper) and (LiBe)+ (lower) is plotted against E (in K). In the upper panel, |ηJℓ |2 is mutiplied by a factor of 107 . +2 S (Be+ + Li), respectively. We construct model potentials 11 Σ+ (ground) and 21 Σ+ (excited) of (LiBe)+ system using spectroscopic constants given in Ref.[46]. Short range potential is approximated using Morse potential and the long range potential [10, 11] is given by the expression (6.5).The short range and long range parts of the potentials are smoothly joined by spline. Since Li+ may be formed due to charge transfer collision between Be+ and Li, we need to consider the interaction between this Li+ and other Li atoms present + 2 + 2 in the condensate. The data for 2 Σ+ g , Σu and Πu potentials of Li2 are taken from Ref.[47]. Dissociation energy De , equilibrium position re and effective range β4 of the ground and excited state potentials of (LiBe)+ and LiLi+ systems are given in Table 6.1. A comparison of potentials of these two systems reveals that ground state potential 11 Σ+ of (LiBe)+ is much shallower than 2 Σ+ g potential of Li+ 2 . The equilibrium positions of both ground and excited state potentials of Chapter 6. Atom-Ion Cold Collisions 101 Table 6.1: Dissociation energies De in a.u., equilibrium positions re and the effective lengths β4 in Bohr radius for excited and ground state potentials (V (r)) of (LiBe)+ and (LiLi)+ systems. system (LiBe)+ (LiLi)+ (LiBe)+ (LiLi)+ V(r) 21 Σ+ 2 Πu 1 + 1Σ 2 + Σg De 0.06 0.01 0.02 0.05 re 5.46 7.50 5.03 6.00 β4 1083.4 1019.8 515.9 1019.8 (LiBe)+ system lie almost at the same separation. Unlike the asymptotic behavior of the excited 21 Σ+ potential of (LiBe)+ system, the excited state potential 2 Πu + of homonuclear Li+ 2 molecular ion asymptotically corresponds to one Li ion in the electronic ground S state and one neutral Li atom in the excited P state. The equilibrium positions re of ground and excited state potentials of Li+ 2 system are shifted by 1.5 Bohr radius. For of (LiBe)+ system, we notice that β4 of excited (21 Σ+ ) potential is almost twice that of the ground (11 Σ+ ) potential. Let us first consider cold collision between Li and Be+ with both of them being in 2 S electronic state. So, our initial system corresponds to the continuum of 21 Σ+ potential. Due to charge transfer collision neutal Be atom and Li+ ion are generated. In the separated two-particle limit of this system, dipole transition to ground state at the single particle level is forbidden. Furthermore, since at low energy non-radiative charge transfer is suppressed, the dominant inelastic channel is the radiative charge transfer transition that occurs at intermediate or short separations. Electronic transition dipole moment between two ionic molecular electronic states vanishes at large separation. Therefore, transitions occur at short range where hyperfine interaction is negligible in comparison to central(Coulombic) interaction. The ground continuum atom-ion pair, formed by radiative charge transfer process, can be photoassociated to form excited molecular ion. This process is basically one photon PA process. The photoassociation rate coefficient is given by KP A = * ∞ πvr X (2ℓ + 1)|SP A (E, ℓ, wL )|2 2 k ℓ=0 + (6.19) where, vr = ~k/µ is the relative velocity of the two particles and h· · · i implies averaging over thermal velcity distribution. Here SP A is S matrix element given Chapter 6. Atom-Ion Cold Collisions 102 by |SP A |2 = δE2 γΓℓ + (Γℓ + γ)2 /4 (6.20) where δE = E/~+δvJ , δvJ = ωL −ωvJ with EvJ = ~ωvJ being binding energy of the excited ro-vibrational state, ωL being the laser frequency and γ the spontaneous line width. Thus PA rate is primarily determined by partial wave stimulated line width Γℓ given by ~Γℓ = πI h(J, ℓ)|DvJ,l |2 ǫ0 c (6.21) where, DvJ,l = hφvJ | D(r) | ψℓ (kr)i (6.22) is the radial transition dipole matrix element between the continuum and bound state wave functions ψℓ (kr) and φvJ (r), respectively. I is the intensity of laser, c is the speed of light and ǫ0 is the vacuum permittivity. Here h(J, ℓ) is Hönl London factor [48] which in the present context is given by h(J, ℓ) = (1 + δΛ′ 0 + δΛ′′ 0 − 2δΛ′ 0 δΛ′′ 0 ) !2 J 1 ℓ (2J + 1)(2ℓ + 1) −Λ′ Λ′ − Λ′′ Λ′′ (6.23) where Λ′ and Λ′′ are the projections of the total electronic orbital angular momentum of the excited and ground states, respectively, on molecular axis and (· · · ) is the Wigner 3j symbol. The spontaneous line width γ of the excited state (v, J) is given by Z 1 ~γ = (∆E)3 |hφvJ | D(r) | ψE i|2 dE 3πǫ0 c3 # X ∆3v′ J ′ |hφvJ | D(r) | φv′ J ′ i|2 + (6.24) v′ ,J ′ where ∆E = (EvJ − E)/~, ∆v′ J ′ = (EvJ − Ev′ ,J ′ )/~, ψE is the scattering wave function and | φv′ J ′ i stands for all the final bound states to which the excited state can decay spontaneously. ~ +L ~ + ~ℓ where S and The total molecular angular momentum is given by J~ = S Chapter 6. Atom-Ion Cold Collisions 103 L are the total electronic spin and orbital quantum number, respectively; and ℓ stands for the angular quantum number of the relative motion of the two atoms. For the particular model for (LiBe)+ system chosen here, we have L = 0 and S = 0 for both the ground and the excited electronic states. Thus here the total angular momenta for both the ground and excited states are given by J = ℓ. However, it is more appropriate to denote total angular quantum number of a molecular bound state by J and that of the continuum or collisional state of this atom-ion system by simply ℓ. The parity selection rule for the electric dipole transition between the ground and excited states dictates ∆J = ±1. In next step, a ground-state molecular ion is formed from excited molecular ion by a stimulated Raman-type process by applying a second laser tuned near a boundbound transition between the excited and the ground potentials. Necessary criteria is that the turning points of the two selected bound states should lie in almost same distances, which results in large Franck Condon overlap between them. Rabi frequency Ω for such bound-bound transition is given by ~Ω = I 4πcǫ0 12 ′ ′ ~ |hv, J | D(r).ǫ̂ L | v , J i| (6.25) where ǫ̂L is the unit vecot of laser polarization and | v, Ji and | v ′ , J ′ i are the two bound states with hr | v, Ji = φvJ (r). 6.3 Results and Discussion Standard renormalized Numerov-Cooley method [49] is used to calculate the bound and scattering state wave functions. The molecular transition dipole matrix element of (LiBe)+ system is calculated using GAMESS. This matrix element strongly depends upon separation and goes to zero at a large r as shown in Fig.6.3. In Figs.6.4 and 6.5, we have plotted the excited and ground state elastic scattering cross section σel as a function of energy E for Li + Be+ and Li+ + Be collisions, respectively. We find that at least 35 partial waves are required to get converging results on elastic scattering at 1 µK. As the energy gradually increases larger numbers of partial waves would be required to get converging results. At high 1 energies, for both the cases, σel decreases as E − 3 . The proportionality constant c 1 in the expression σel (E → ∞) = cE − 3 calculated using Eq. (6.8) for excited 21 Σ+ and ground 11 Σ+ potentials are 2936 and 1091 a.u., respectively, whereas linear Chapter 6. Atom-Ion Cold Collisions 104 ψl (kr) (a.u.) 40 20 0 -20 -40 0 5 10 15 20 25 0 5 10 15 20 25 φvJ (r) (a.u.) 0.8 0.4 0 -0.4 r (a.u.) Figure 6.11: Energy-normalized s-wave ground scattering (upper) and unit normalized excited bound (lower) wave functions of (LiBe)+ system are plotted as a function of separation r. fit to σel vs. E curves provides c = 3548 and 1335 a.u., respectively. Figs.6.6 and 6.7 exhibit s- and d-wave partial scattering cross section as a function of energy for Li+ +Be and Li+Li+ collisions, respectively. In Fig.6.8, we have plotted total 2 + elastic scattering cross section for Li+Li+ collisions in 2 Σ+ g and Σu potentials. Starting from the low energy continuum state of Li + Be+ collision in the 21 Σ+ potential, there arise two possible radiative transitions by which the system can go to the ground electronic state 11 Σ+ . One is continuum-continuum and the other is continuum-bound dipole transition. The transition dipole moment as a function of separation as shown in Fig.6.3 shows that the dipole transition probability will vanish as the separation increases above 20a0 . So, a dipole trasition has to take place at short separations. Let us consider radiative transfer processes from the upper (21 Σ+ ) to the lower (11 Σ+ ) state of (LiBe)+ . We then need to apply the formulae (6.12) and (6.15) where m ≡ 21 Σ+ and n ≡ 11 Σ+ in Chapter 6. Atom-Ion Cold Collisions 105 ψl (kr) (a.u.) 40 20 0 -20 -40 0 5 10 15 20 25 0 5 10 15 20 25 φvJ (r) (a.u.) 0.4 0.2 0 -0.2 -0.4 r (a.u) Figure 6.12: Same as in Fig.6.11 but for Li-Li+ system. our case. Continuum-continuum charge transfer cross section σct between 21 Σ+ and 11 Σ+ states of (LiBe)+ system is plotted against E in Fig.6.9. We evaluate the photoassociative (continuum-bound) transfer cross section by subtracting σct from the total radiative transfer cross section σrt calculated using the formula (6.15). At energy E = 0.1 mK, σct and the photoassociative transfer cross section are found to be 10.39 a.u. and 0.03 a.u., respectively. Thus we infer that the continuum-continuum radiative charge transfer process dominates over the radiative association process. Also, we notice that σct is smaller than both the excited and ground state elastic scattering cross sections σel (as given in Figs.6.4 and 6.5, respectively) by several orders of magnitude. Molecular dipole transitions between two ro-vibrational states or between continuum and bound states are governed by Franck-Condon principle. According to this principle, for excited vibrational (bound) states, bound-bound or continuumbound transitions primarily occur near the turning points of bound states. In general, highly excited vibrational state wave functions of diatomic molecules or Chapter 6. Atom-Ion Cold Collisions 106 Table 6.2: Ro-vibrational energy (Ev J ), inner (ri ) and outer turning points (ro ) of two selected bound states of (LiBe)+ molecular ion - one bound state in excited (21 Σ+ ) and the other in ground (11 Σ+ ) potential. The energy Ev J is measured from the threshold of the respective potential. Potential 21 Σ+ 11 Σ+ v 68 29 J 1 0 EvJ (a.u.) -3.30×10−3 -0.25×10−3 ri (a.u.) 3.4 3.8 ro (a.u.) 16.3 16.6 molecular ions have their maximum amplitude near the outer turning points. Spectral intensity is proportional to the overlap integral.This means that the spectral intensity for a continuum-bound transition would be significant when the continuum state has a prominent node near the outer turning point of the bound state. For transitions between two highly excited bound states, Franck-Condon principle implies that the probability of such transitions would be significant when the outer turning points of these two bound states lie nearly at the same separation. The upper panel of Fig.6.10 shows the variation of the square of franck Condon overlap integral |ηJℓ |2 between the ground s-wave (ℓ = 0) scattering and the ex- cited ro-vibrational (v = 26, J = 1/2) states of Li-Li+ system as a function the collision energy E. The lower panel of Fig.6.10 displays the same as in the upper panel but for (LiBe)+ system with v = 68 and J = 1. The excited ro-vibrational state v = 26, J = 1/2 of Li-Li+ is very close to dissociation threshold while the excited ro-vibrational state v = 68, J = 1 of (LiBe)+ system is a deeper bound state. Thses two excited states are so chosen such that free-bound Franck-Condon overlap integral for both the systems become significant. Comparing these two plots, we find that |ηJℓ |2 of Li-Li+ system is smaller than that of (LiBe)+ sys- tem by seven orders of magnitude. To understand why the values |ηJℓ |2 for the two systems are so different, we plot the the energy-normalized s-wave ground scattering and the bound state wave functions of (LiBe)+ system in Fig.6.11 and those of Li-Li+ system in Fig.6.12. A comparison of Figs.6.11 and 6.12 reveals that, while in the case of (LiBe)+ the maximum of the excited bound state wave function near the outer turning point coincides nearly with a prominent antinode of the scattering wave fucntion, in the case of Li-Li+ the maximum of the bound state wave function near the outer turning point almost coincides with a minimum (node) of the scattering wave function. These results indicate that the possibility of the formation of excited LiLi+ molecular ion via PA is much smaller than that of (LiBe)+ ion. We henceforth concentrate on PA of (LiBe)+ system only. We next explore the possibility of PA in Li+ -Be cold collision in the presence Chapter 6. Atom-Ion Cold Collisions 107 7 l=0 l=2 6 4 2 | DvJ, l | (a.u.) 5 3 2 1 0 -8 10 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 -1 10 10 0 10 1 10 2 E (K) Figure 6.13: Square of free-bound radial transition dipole moment (|DvJ,l |2 ) (in a.u.) for ground continuum states with ℓ = 0 (solid) and ℓ = 2 (dashed) and excited bound ro-vibrational level with v = 68 and J = 1 of laser light. As discussed before, continuum-bound molecular dipole transition matrix element depends on the degree of overlap between continuum and bound states. PA rate (6.19) is proportional to the square of free-bound radial transition dipole moment element |DvJ,l |2 . In Fig.6.13 we plot |DvJ,l |2 against E for s- (ℓ = 0) and d-wave (ℓ = 2) ground scattering states and v = 68 , J = 1 excited molecular state. It is clear from this figure that the contributions of both ℓ = 0 and ℓ = 2 partial waves are comparable above enegy corresponding to 0.1 mK. At lower energy (E < 0.1 mK), only s-wave makes finite contribution to the the dipole transition. Fig.6.14 exhibits |DvJ,l |2 as a fucntion of E for the transition from s- wave (ℓ = 0) scattering state of the excited (21 Σ+ ) continuum to the ground (11 Σ+ ) ro-vibrational state with v = 36 , J = 1. A comparison between the Figs.6.13 and 6.14 reveals that the probability for the transition from the upper continuum to the ground bound state is smaller by several orders of magnitude Chapter 6. Atom-Ion Cold Collisions 108 -4 10 2 | DvJ, l | (a.u.) -5 10 -6 10 10 -7 10 -8 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 E (K) Figure 6.14: Same as in Fig.6.13 but for excited continuum state with ℓ = 0 and ground ro-vibrational state with v = 36 and J = 1. than that from ground continuum to an excited bound state. In Fig.6.15, we have plotted the rate of photoassociation KP A as a function of temperature T for laser frequency tuned at PA resonance. The ion-atom PA rate as depicted in Fig.6.15 is comparable to the typical values of rate of neutral atom-atom PA at low laser intesities. In Fig.6.16 we have plotted the rate of photoassociation as a function of laser intensity at a fixed temperature T = 0.1 mK to show the saturation effect that occurs around intensity I = 50 kW/cm2 . Thus the formation of excited (LiBe)+ molecular ion by photoassociating colliding Li+ with Be with a laser of moderate intensity appears to be a feasible process. Now we discuss the possibility of formation of ground state molecular ion by stimulated Raman-type process. Let us consider two selected bound states whose salient features are given in Table 6.2. The outer turning points of these two bound state almost coincide implying large Franck Condon overlap integral. Rabi frequency corresponding to this bound-bound transition is found to be 285 MHz Chapter 6. Atom-Ion Cold Collisions 109 -13 10 -14 10 3 -1 KPA (cm s ) -15 10 -16 10 10 10 -17 -18 -19 10 10 -5 -4 -3 10 10 10 -2 T (K) Figure 6.15: Rate of photoassociation KP A (in cm3 s−1 ) of (LiBe)+ is plotted against temperature (in K) at I = 1 W/cm2 and δvJ = ωL − ωvJ = 0 for laser intensity I = 1 kW/cm−2 . Comparing this value with the spontaneous line width γ = 57 kHz of the excited bound state calculated using the formula (6.24), we infer that even at a low laser intensity which is far below the saturation limit, bound-bound Rabi frequency Ω exceeds γ by several orders of magnitude. This indicates that it may be possible to form ground molecular ion by stimulated Raman-type process with two lasers. 6.4 Conclusions In this chapter, we have shown that alkaline earth metal ions immersed in an ultracold gas of alkali atoms can give rise to a variety of cold chemical reactions considering system of a Beryllium ion interacting with cold Lithium atoms. We have predicted the formation of translationally and rotationally cold (LiBe)+ molecular Chapter 6. Atom-Ion Cold Collisions 3 -1 KPA (cm s ) 10 10 110 -11 -12 10 -13 -14 10 0 10 20 30 40 50 60 2 I (kW/cm ) Figure 6.16: KP A (in cm3 s−1 ) of (LiBe)+ is plotted as a function of laser intensity I (in kW/cm2 ) at temperature T = 0.1 mK with laser tuned at PA resonance. ion by photoassociation. Theoretical understanding of low energy atom-ion scattering and reactions may be important for probing dynamics of quantum gases. Since both Bose-Einstein condensation and fermionc superfluidity have been realized in atomic gases of Lithium, understanding cold collisions between Lithium and Beryllium ion may be helpful in probing both bosonic and fermionic superfluidity. In particular, this may serve as an important precursor for generating and probing vortex ring in Lithium quantum gases. 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We have showed that it is possible to generate and manipulate coherence between excited ro-vibrational states of a molecule by PA lasers. We have also analyzed atom-ion cold collisions and predicted a new pathway for the formation of translationally and rotationally cold molecular ion by cold collisions between atoms and ions in the presence of laser light. Such studies may be beneficial for modification of decoherence and dissipation in cold molecules, controlling the dynamics of quantum gas, creation of artificial media with negative refractive index [1] (i.e with simultaneous negative electric and magnetic permittivity) etc. Now, we would like to mention some of our future plans. First of all, we wish to study cold atom-ion collisions for different heteronuclear and homonuclear alkali atom and alkali ion pairs and alkali atom and alkaline earth metal ion pairs and to check the possibility of formation of cold molecular ions in these systems. We also have plans to study quantum interference phenomena like field induced Stüeckelberg oscillations [2] and Landau-Zener [3] transitions in atom-ion cold collisions. A transition between two potential curves at an avoided crossing due to motional effects is known as Landau-Zener transition. Under certain simple 115 Chapter 7. Conclusion and Outlook 116 assumptions, the transition probability can be written as P = exp − 2πV122 v(V11 − V22 ) (7.1) Here V11 and V22 denote the slopes of the diabatic potential curves near pseudocrossing and V12 is the diabetic coupling potential. V12 is estimated from the adiabatic energies Ei (R) as V12 = 21 ∆E = 12 |E1 (rx ) − E2 (rx )|. rx is the distance at pseudo-crossing takes place and v is the velocity at the of the colliding particles. Let us now consider a reaction process, in which system enters along the adiabatic curve E2 (r) (diabetic curve V22 (r)) and leaves the reaction on curve E1 (r) (diabetic curve V11 (r)). Transition between these two curves may happen on either the incoming or the outgoing side of the reaction. If both paths have the same scattering angle, they are indistinguishable and therefore interfere. The resulting interference pattern has been first recognized by Stückelberg in 1932. We have plans to study these effects in ultracold atom-ion collisional systems like in (SrX)+ and (MgX)+ , where X = Li, Na, K, Rb, Cs. Stückelberg oscillation may be measured as a function of scattering angle. These types of studies may be helpful for solving problems in fast and slow collisions between atoms and ions without going into numerical detail of multichannel close-coupling method [4]. Another project which we wish to do in near future is to study photoassociation (PA) in the presence of Feshbach resonance (FR) in bosonic 7 Li and fermionic 6 Li atoms as multichannel scattering. In chapter 4, we have already studied the same problem in the framework of Fano’s theory. Now our aim is to check it using multichannel close-coupled scattering calculation [5]. The coupled equations introduce a n × n interaction matrix. The diagonal elements of the interaction matrix define the diabtic channel. The off-diagonal terms of the interaction matrix describe the interaction between different channels. We plan to apply quantum defect theoretic approach [6] to solve multichannel scattering problem. On the other hand, a multichannel problem may be simplified to a model of two-channel configuration interaction in some cases [5]. The model consists of a single open-channel continuum state and a single isolated resonance closed-channel which incorporates the exact closed-coupled interactions between multichannel closed states. For illustration, two ground state Li atoms undergoing collision in the presence of magnetic field may be considered. For atomic 6 Li, electronic spin and nuclear spin quantum number ~s and ~i are 1/2 and 1, respectively. Hence the possible hyperfine quantum numbers f~ are 3/2 and 1/2 [7]. In the absence of magnetic field their Chapter 7. Conclusion and Outlook 117 projections mf s are degenerate. But the applied magnetic field splits different mf states at asymptotic limit. So for two-atom collision, one threshold would split into a number of channels in the long-range. Our objective is to calculate the modified scattering wave functions at the magnetic field at FR with the help of multichannel quantum defect theory and then to demonstrate PA in comparison to the results obtained in chapter 4. This work may help to develop insight into the research areas of atom-molecule conversion and optical and magneto-optical Feshbach resonance at ultralow energy. Our work may stimulate further studies on laser manipulation of continuum and bound states. It may become useful for the studies of controlling decoherence and dissipation in cold molecules. 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