Micromotion in trapped atom ion systems

MICROMOTION IN TRAPPED ATOM-ION SYSTEMS LE HUY NGUYEN B.Sc. (Hons.), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS Graduate School for Integrative Sciences and Engineering National University of Singapore 2012 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Acknowledgments I would like to express my utmost gratitude and appreciation to my supervisor, Professor Berthold-Georg Englert, for his support, encouragement and patience. Were it not for his guidance and advice I would not have been able to complete this research project on time. Also of great importance is the contribution of my collaborator Amir Kalev, who has worked with me on writing the numerical codes and many analytical calculations. His inputs in our discussions are vital to the completion of the project. I appreciate very much the help of Professor Murray D. Barrett, who introduced the micromotion problem to me and provided valuable information related to the experimental aspects of the problem. Murray also participated in many discussions and suggested many improvements in the writing of the manuscript. Special thank is due to Z. Idziaszek and T. Calarco for reading through the manuscript and giving helpful comments. I wish to thank Professor Gong Jiangbin for his interesting remark on the chaotic behaviour of the trapped atom-ion system. I would also like to acknowledge the useful discussions and kind hospitality of P. Zoller, P. Rabl, and S. Habraken at the Institute for Quantum Optics and Quantum Information (IQOQI). I am grateful to Ua for drawing some of the figures and Kean Loon for showing me how to prepare my thesis in Latex. This work is financially supported by the NUS Graduate School for Integrative Sciences and Engineering (NGS) and the Centre for Quantum Technologies (CQT). Finally, I want to say thank to my family and Ua for their understanding and i ii ACKNOWLEDGMENTS support throughout the period of my graduate study. The main results presented in this thesis were published in Phys. Rev. A 85, 052718 (2012). Contents Acknowledgments i Abstract v List of Figures vii List of Symbols ix List of Abbreviations xiii Introduction Motion of a trapped ion 2.1 Ion micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The harmonic approximation . . . . . . . . . . . . . . . . . . . . . 10 13 Trapped atom-ion systems 15 Floquet formalism 21 4.1 Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Floquet Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 iii iv CONTENTS Micromotion effect in one dimension 35 5.1 The unperturbed system . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Atom-ion quantum gate . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Micromotion-induced coupling . . . . . . . . . . . . . . . . . . . . . 52 5.4 Numerical calculation of the quasienergies . . . . . . . . . . . . . . 59 5.5 Adiabatic evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.6 A scheme to bypass the micromotion effect . . . . . . . . . . . . . . 77 5.7 Excess micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.8 A more realistic 1D model . . . . . . . . . . . . . . . . . . . . . . . 86 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Micromotion effect in three dimensions 93 6.1 Micromotion Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 93 6.2 Quasi-1D trap configuration . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Minimization of micromotion effect . . . . . . . . . . . . . . . . . . 102 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Conclusions 107 A Numerov method 111 B Floquet states of a trapped ion 117 C Floquet adiabatic process 125 D Derivation of Eq. (6.26) 131 Abstract We examine the validity of the harmonic approximation, where the radio-frequency ion trap is treated as a harmonic trap, in the controlled collision of a trapped atom and a single trapped ion. This is equivalent to studying the effect of the micromotion since this motion must be neglected for the trapped ion to be considered as a harmonic oscillator. By applying the transformation of Cook et al. we find that the micromotion can be represented by two periodically oscillating operators. In order to investigate the effect of the micromotion on the dynamics of a trapped atomion system, we calculate (i) the coupling strengths of the micromotion operators by numerical integration and (ii) the quasienergies of the system by applying the Floquet formalism, a useful framework for studying periodic systems. It turns out that the micromotion is not negligible when the distance between the atom trap and the ion trap is shorter than a characteristic distance. Within this range the energy diagram of the system changes dramatically when the micromotion is taken into account. The system exhibits chaotic behaviour through the appearance of numerous avoided crossings in the energy diagram when the micromotion coupling is strong. Excitation due to the micromotion leads to undesirable consequences for applications that are based on an adiabatic process of the trapped atom-ion system. We suggest a simple scheme for bypassing the micromotion effect in order to successfully implement a quantum-controlled phase gate proposed previously and create an atom-ion macromolecule. The methods presented in this thesis are v vi ABSTRACT not restricted to trapped atom-ion systems and can be readily applied to studying the micromotion effect in any system involving a single trapped ion. 133 which can be solved quite easily to obtain C1 = C0 D2 , D1 D2 − (D.10) C2 = C0 . D1 D2 − (D.11) and Similarly, by letting n = −2 and then n = −1 in the recurrence relation, one will be able to obtain C−1 = C0 D−2 , D−1 D−2 − (D.12) C−2 = C0 . D−1 D−2 − (D.13) and In view of the fact that β is of the first order in q, we have D2 q[(β + 4)2 − a] = D1 D2 − [(β + 4)2 − a][(β + 2)2 − a] − q q ¯ 3) + O(q = (β + 2) q ¯ ), = (1 − β) + O(q (D.14) and q2 = D1 D2 − [(β + 4)2 − a][(β + 2)2 − a] − q q2 ¯ 3) = 2 + O(q 42 q2 ¯ ). = + O(q 64 (D.15) Thus q ¯ ), C1 = (1 − β)C0 + O(q (D.16) 134 APPENDIX D. DERIVATION OF EQ. (6.26) and q2 ¯ ). C0 + O(q 64 C2 = (D.17) In essentially the same way we obtain q ¯ ), C−1 = (1 + β)C0 + O(q (D.18) and C−2 = q2 ¯ ). C0 + O(q 64 (D.19) The condition f (0) = leads to Cn = 0, (D.20) n which requires q q2 ¯ ) = 1. + + O(q 32 (D.21) ¯ ). + O(q + q/2 + q /32 (D.22) C0 + Hence we arrive at C0 = From the results just found for the coefficients Cn we have + (q/2) [cos(ωt) − iqβ sin(ωt)] + (q /32) cos(2ωt) ϕ(t) = + O(q ). + q/2 + q /32 (D.23) ¯ Notice that the terms indicated by the notation O, unlike those represented by O, can be time dependent. For later calculations we compute |ϕ(t)|2 = + (q /8) + q cos(ωt) + (3q /16) cos(2ωt) + O(q ). + q + 5q /16 (D.24) Having obtained the periodic function ϕ(t) of the special Floquet solution up to the second order in q, we may now calculate the matrix element u0x | x2i |u0x 135 of Eq. (6.26). By switching to the Heisenberg picture we may write this matrix element as 0| xi (t)2 |0 , where the state |0 is defined in Eq. (B.19). In the absence of the excess micromotion, we may set xp (t) = in Eq. (B.16). Using this equation and Eq. (B.18) we have X(t) = 2νmi f˙(t)∗ A + f˙(t)A† . (D.25) From the well-known relations for the ladder operators √ n |n − , √ A† |n = n + |n + , A |n = (D.26) one can show that 0| xi (t)2 |0 = 2mi ν |f (t)|2 = 2mi ν |ϕ(t)|2 , (D.27) and using the result of Eq. (D.24) one would have + q /8 + q cos(ωt) + (3q /16) cos(2ωt) + O(q ). 0| xi (t) |0 = 2mi ν + q + 5q /16 (D.28) It can be seen from Eq. (B.6) and Eq. (D.2) that ν= ω β+ 2nCn . (D.29) n Upon inserting the values found for the coefficients Cn we get ν= βω ¯ ) = µ − q + O(q ¯ 2) . − C0 q + O(q (D.30) 136 APPENDIX D. DERIVATION OF EQ. (6.26) The results of Eq. (D.28) and Eq. (D.30) yield + q /8 + q cos(ωt) + (3q /16) cos(2ωt) + O(q ) 2 ¯ 2mi µ [1 + q + 5q /16] − q + O(q ) + q /8 + q cos(ωt) + (3q /16) cos(2ωt) + O(q ) = ¯ 2) 2mi µ + O(q ¯ ) + q cos(ωt) + q cos(2ωt) + O(q ). = + O(q 2mi µ 16 0| xi (t)2 |0 = (D.31) Notice that in the above expression we keep the exact value for the Floquet critical exponent µ, which is itself a function of the trap parameters a and q. For the calculation of the similar matrix element for the motion along the y direction, it is necessary to express the dependence on the trap parameters explicitly; hence we will write µ as µ(a, q) from this point onward. The matrix element 0| xi (t)2 |0 should also be written as 0(a, q)| xi (t)2 |0(a, q) to reflect the dependence of the state |0 on the traps parameter. So, we rewrite Eq. (D.31) as 0(a, q)| xi (t)2 |0(a, q) = 2mi µ(a, q) ¯ ) + q cos(ωt) + × + O(q q cos(2ωt) + O(q ). 16 (D.32) ¯ ) are not determined in our calAlthough the time-independent constants in O(q culation, it will be clear later that these constants contribute to a small timeindependent perturbation that can be ignored. Only the oscillating terms need to be considered since they result in the micromotion effect. Recall that for a linear Paul trap we have qx = −qy , and ax = ay . The motion along the y direction of an ion in a linear Paul trap is also described by the Mathieu equation for the motion along the x direction, but with the parameter q replaced with −q. Therefore, by a simple change of the sign of q in Eq. (D.32), we can derive 137 the corresponding matrix element for the motion along the y direction, which is 0(a, −q)| yi (t)2 |0(a, −q) = 2mi µ(a, −q) ¯ ) − q cos(ωt) + × + O(q q cos(2ωt) + O(q ). 16 (D.33) As mentioned in Eq. (D.4), the constant β(a, q), and hence the Floquet critical exponent µ(a, q), does not depend on the sign of q, that is, µ(a, q) = µ(a, −q). (D.34) Upon replacing µ(a, −q) in Eq. (D.33) with µ(a, q) and adding the resulting equation to Eq. (D.32) we obtain 0(a, q)| xi (t)2 |0(a, q) + 0(a, −q)| yi (t)2 |0(a, −q) = ¯ ) + q cos(2ωt) + O(q ). + O(q mi µ(a, q) 16 (D.35) As the time-independent terms in the bracket are dominated by the factor 1, the ¯ ) are negligible. Furthermore, since µ(a, q) ≈ ω0 with ω0 the secular terms in O(q frequency of the motion along the radial direction, we finally arrive at 0(a, q)| xi (t)2 |0(a, q) + 0(a, −q)| yi (t)2 |0(a, −q) ≈ li2 + which is indeed Eq. (6.26) written in the Heisenberg picture. q cos(2ωt) . 16 (D.36) 138 APPENDIX D. DERIVATION OF EQ. (6.26) Bibliography [1] C.H. Bennett and D.P. DiVincenzo, Quantum information and computation, Nature 404, 247 (2000). doi:10.1038/35005001. [2] D.P. 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[...]... quantum motion of a trapped ion in the presence of the so-called excess micromotion is derived and the effect of this excess micromotion in trapped atom- ion systems is considered We also suggest a simple scheme to bypass the micromotion effect in order to realize some interesting applications based on the interaction of trapped atoms and ions After studying three-dimensional (3D) systems in chapter 6,... based on trapped atom- ion systems Most of these rely on the strong collisional interaction between trapped ions and atoms The first application considered has direct implication for quantum computing: an atom- ion entangling gate [13, 14] The entanglement of a trapped atom and a trapped ion is achieved through the adiabatic collision process shown in Fig 1.1 If the qubits are encoded in the hyperfine binaries... positive ions and neutral atoms is also predicted to exhibit interesting phenomena such as the transition from an almost insulating to a conducting phase at ultralow temperature [26] Figure 1.3: A trapped ion (red) is immersed in a Bose-Einstein Condensate The atoms (blue) in the surrounding are attracted to the ion and form a large molecular ion Many experiments have been done on trapped atom- ion systems. .. 3 we describe the ion s micromotion and the trapped atom- ion systems Chapter 4 is on the Floquet formalism and how we use it to study the effect of a periodically oscillating potential on the energy structure of a quantum system In chapter 5 we study the micromotion effect in one-dimensional (1D) trapped atom- ion systems by considering the micromotion- induced coupling and computing the exact quasienergies... immersed in a Bose-Einstein Condensate (BEC), the atoms in the surrounding are attracted to the ion to form a large molecular ion 4 CHAPTER 1 INTRODUCTION (see Fig 1.3) It is predicted that the number of captured atoms can be as large as a few hundreds for the example of a sodium ion in a BEC of sodium atoms Other intriguing proposals based on atom- ion systems include sympathetic cooling of ions [19,... conclusions and present some technical material in the appendices 8 CHAPTER 1 INTRODUCTION Chapter 2 Motion of a trapped ion The classical and quantum dynamics of a single trapped ion and its interaction with a radiation field have been studied extensively [40–43] Here we describe the key points in the motion of trapped ions which are relevant to our study and then explain the reasoning behind the... time-independent part of the secular motion and the time-dependent one of the micromotion are separated, makes it more convenient to work with when one wishes to study the effect of the micromotion In the next chapter we make use of this transformation to investigate the trapped atom- ion system Chapter 3 Trapped atom- ion systems The system is composed of an atom in a harmonic trap interacting with an ion. .. (t) + Vint , (3.1) Figure 3.1: A trapped atom- ion system O is the center of the atom trap which is chosen as the coordinate origin; ra and ri are the position vectors of the atom and ion, respectively; and d is the position vector of the center of the ion trap 15 16 CHAPTER 3 TRAPPED ATOM- ION SYSTEMS where Ha is the Hamiltonian of an atom in an atom trap, Hi (t) is the Hamiltonian of an ion in a rf... approximation However, there is a concern about the validity of this approximation since the kinetic energy of the micromotion can be comparable or even much larger than that of the ion s harmonic motion [36] More importantly, it is found in experiments that the ion s micromotion plays an important role in the dynamics of atom- ion collisions [27, 29, 31] Although the effect of the micromotion in the collision... cold atom- ion collisions in traps, particularly the scattering cross section for different types of collision The experiments on a single trapped ion immersed in a BEC also give evidence for the sympathetic cooling of the trapped ion On the theoretical aspects, a great effort has been made on studying the ultracold collisions of free atoms and ions [32–35], but the situation when the particles are trapped, . to trapped atom- ion systems and can be readily applied to studying the micromotion effect in any system involving a single trapped ion. List of Figures 1.1 Adiabatic collision of a trapped atom. ultracold atom- ion collisions is looking at some of the interesting proposals based on trapped atom- ion systems. Most of these rely on the strong collisional interaction between trapped ions and atoms trap, in the controlled collision of a trapped atom and a single trapped ion. This is equivalent to studying the effect of the micromo- tion since this motion must be neglected for the trapped ion