DOCSLIB.ORG

1902-05814.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1902-05814.Pdf Periodic thermodynamics of the Rabi model with circular polarization for arbitrary spin quantum numbers Heinz-J¨urgen Schmidt1, J¨urgen Schnack2, and Martin Holthaus3 1Universit¨at Osnabr¨uck, Fachbereich Physik, D-49069 Osnabr¨uck, Germany 2Universit¨at Bielefeld, Fakult¨at f¨ur Physik, D-33501 Bielefeld, Germany and 3Carl von Ossietzky Universit¨at, Institut f¨ur Physik, D-26111 Oldenburg, Germany We consider a spin s subjected to both a static and an orthogonally applied oscillating, circularly polarized magnetic field while being coupled to a heat bath, and analytically determine the quasi- stationary distribution of its Floquet-state occupation probabilities for arbitrarily strong driving. This distribution is shown to be Boltzmannian with a quasitemperature which is different from the temperature of the bath, and independent of the spin quantum number. We discover a remarkable formal analogy between the quasithermal magnetism of the nonequilibrium steady state of a driven ideal paramagnetic material, and the usual thermal paramagnetism. Nonetheless, the response of such a material to the combined fields is predicted to show several unexpected features, even allowing one to turn a paramagnet into a diamagnet under strong driving. Thus, we argue that experimental measurements of this response may provide key paradigms for the emerging field of periodic thermodynamics. I. INTRODUCTION where the choice of the “canonical representative” distin- guished by setting ν = 0 is a matter of convention. A quantum system governed by an explicitly time- The significance of these Floquet states (2) rests in the dependent Hamiltonian H(t) which depends periodically fact that, as long as the Hamiltonian depends on time in on time t, such that a strictly T -periodic manner, every solution ψ(t) to the time-dependent Schr¨odinger equation can be| expandedi H(t)= H(t + T ) , (1) with respect to the Floquet basis, possesses a complete set of Floquet states, that is, of solu- ψ(t) = c u (t) exp( iε t) , (6) tions to the time-dependent Schr¨odinger equation having | i n | n i − n n the particular form X where the coefficients cn do not depend on time. Hence, ψ (t) = u (t) exp( iε t) . (2) | n i | n i − n the Floquet states propagate with constant occupation probabilities c 2, despite the presence of a time-periodic The Floquet functions un(t) share the T -periodic time | n| dependence of their Hamiltonian,| i drive. Under conditions of perfectly coherent time evo- lution these coefficients cn would be determined solely u (t) = u (t + T ) ; (3) | n i | n i by the system’s state at the moment the periodic drive is turned on. However, if the periodically driven sys- the quantities εn, which accompany their time evolu- tion in the same manner as energy eigenvalues accom- tem is interacting with an environment, as it happens in pany the evolution of unperturbed energy eigenstates, many cases of experimental interest [4–9], that environ- are known as quasienergies [1–3]. Here we assume that ment may continuously induce transitions among the sys- the quasienergies constitute a pure point spectrum, asso- tem’s Floquet states, to the effect that a quasistationary ciated with square-integrable Floquet states in the sys- distribution pn of Floquet-state occupation probabil- ities establishes{ } itself which contains no memory of the tem’s Hilbert space S; we also adopt a system of units H initial state, and the question emerges how to quantify arXiv:1902.05814v1 [quant-ph] 15 Feb 2019 such that both the Planck constant ~ and the Boltzmann this distribution. constant kB are set to one. Evidently the factorization of a Floquet state (2) into In a short programmatic note entitled “Periodic Ther- a Floquet function and an exponential of a phase which modynamics”, Kohn has drawn attention to such quasi- stationary Floquet-state distributions p , emphasizing grows linearly in time is not unique: Defining ω =2π/T , { n} and taking an arbitrary, positive or negative integer ν, that they should be less universal than usual distribu- one has tions characterizing thermal equilibrium, depending on the very form of the system’s interaction with its envi- iνωt un(t) exp( iεnt)= un(t)e exp i[εn+νω]t , (4) ronment [10]. In an earlier pioneering study, Breuer et | i − | i − al. had already calculated these distributions for time- where u (t)eiνωt again is a T -periodic Floquet function, n periodically forced oscillators coupled to a thermal os- representing| thei same Floquet state as u (t) . There- n cillator bath [11]. For the particular case of a linearly fore, a quasienergy is not to be regarded| as justi a sin- forced harmonic oscillator these authors have shown that gle number equipped with the dimension of energy, but the Floquet-state distribution remains a Boltzmann dis- rather as an infinite set of equivalent representatives, tribution with the temperature of the heat bath, whereas ε ε + νω ν Z , (5) it becomes rather more complicated in the case of forced n ≡{ n | ∈ } 2 anharmonic oscillators. These investigations have been ideal paramagnet to a circularly polarized driving field. extended later by Ketzmerick and Wustmann, who have In Sec. VI we consider the rate of energy dissipated by demonstrated that structures found in the phase space of the driven spins into the bath, thus generalizing results 1 classical forced anharmonic oscillators leave their distinct derived previously for s = 2 in Ref. [29]. In Sec. VII we traces in the quasistationary Floquet-state distributions summarize and discuss our main findings, emphasizing of their quantum counterparts [12]. To date, a great va- the possible knowledge gain to be derived from future riety of different individual aspects of the “periodic ther- measurements of paramagnetic response to strong time- modynamics” envisioned by Kohn has been discussed in periodic forcing, carried out along the lines drawn in the the literature [13–23], but a coherent overall picture is present work. still lacking. In this situation it seems advisable to resort to mod- els which are sufficiently simple to admit analytical so- II. TECHNICAL TOOLS lutions and thus to unravel salient features on the one hand, and which actually open up meaningful perspec- A. Golden-rule approach to open driven systems tives for new laboratory experiments on the other. To this end, in the present work we consider a spin s ex- Let us consider a quantum system evolving according posed to both a static magnetic field and an oscillating, to a T -periodic Hamiltonian H(t) on a Hilbert space S circularly polarized magnetic field applied perpendicu- which is perturbed by a time-independent operatorHV . lar to the static one, as in the classic Rabi set-up [24], Then the transition matrix element connecting an initial and coupled to a thermal bath of harmonic oscillators. Floquet function ui(t) to a final Floquet function uf (t) The experimental measurement of the thermal paramag- can be expanded| into ai Fourier series, | i netism resulting from magnetic moments subjected to a static field alone has a long and successful history [25, 26], (ℓ) uf (t) V ui(t) = Vfi exp(iℓωt) , (7) having become a standard topic in textbooks on Statis- h | | i ℓ∈Z tical Physics [27, 28]. We argue that a future generation X of such experiments, including both a static and a strong and consequently the “golden rule” for the rate of tran- oscillating field, may set further cornerstones towards the sitions Γfi from a Floquet state labeled i to a Floquet development of full-fledged periodic thermodynamics. state f is written as [29] We proceed as follows: In Sec. II we collect the nec- Γ =2π V (ℓ) 2 δ(ω(ℓ)) , (8) essary technical tools, starting with a brief summary of fi | fi | fi ∈Z the golden-rule approach to time-periodically driven open Xℓ quantum systems in the form developed by Breuer et al. [11], thereby establishing our notation. We also sketch where a technique which enables one to “lift” a solution to the (ℓ) 1 ωfi = εf εi + ℓω . (9) Schr¨odinger equation for a spin s = 2 in a time-varying − magnetic field to general s. In Sec. III we discuss the Thus, a transition among Floquet states is not associated Floquet states for spins in a circularly polarized driving with only one single frequency, but rather with a set of field, obtaining the states for general s from those for frequencies spaced by integer multiples of the driving fre- s = 1 with the help of the lifting procedure. In Sec. IV 2 quency ω, reflecting the ladder-like nature of the system’s we compute the quasistationary Floquet-state distribu- quasienergies (5); this is one of the sources of the pecu- tion for driven spins under the assumption that the spec- liarities which distinguish periodic thermodynamics from tral density of the heat bath be constant, and show that usual equilibrium thermodynamics [10, 11]. this distribution is Boltzmannian with a quasitempera- Let us now assume that, instead of merely being per- ture which is different from the actual bath temperature; turbed by V , the periodically driven system is coupled the dependence of this quasitemperature on the system to a heat bath, described by a Hamiltonian H acting parameters is discussed in some detail. In Sec. V we bath on a Hilbert space B, so that the total Hamiltonian on determine the magnetization of a spin system which is the composite HilbertH space takes the form subjected to both a static and an orthogonally applied, HS ⊗ HB circularly polarized magnetic field while being coupled ½ Htotal(t)= H(t) ½ + Hbath + Hint . (10) to a heat bath. To this end, we first
Load more

Recommended publications
  • Atom-Light Interaction and Basic Applications
    ICTP-SAIFR school 16.-27. of September 2019 on the Interaction of Light with Cold Atoms Lecture on Atom-Light Interaction and Basic Applications Ph.W. Courteille Universidade de S~aoPaulo Instituto de F´ısicade S~aoCarlos 07/01/2021 2 . 3 . 4 Preface The following notes have been prepared for the ICTP-SAIFR school on 'Interaction of Light with Cold Atoms' held 2019 in S~aoPaulo. They are conceived to support an introductory course on 'Atom-Light Interaction and Basic Applications'. The course is divided into 5 lectures. Cold atomic clouds represent an ideal platform for studies of basic phenomena of light-matter interaction. The invention of powerful cooling and trapping techniques for atoms led to an unprecedented experimental control over all relevant degrees of freedom to a point where the interaction is dominated by weak quantum effects. This course reviews the foundations of this area of physics, emphasizing the role of light forces on the atomic motion. Collective and self-organization phenomena arising from a cooperative reaction of many atoms to incident light will be discussed. The course is meant for graduate students and requires basic knowledge of quan- tum mechanics and electromagnetism at the undergraduate level. The lectures will be complemented by exercises proposed at the end of each lecture. The present notes are mostly extracted from some textbooks (see below) and more in-depth scripts which can be consulted for further reading on the website http://www.ifsc.usp.br/∼strontium/ under the menu item 'Teaching' −! 'Cursos 2019-2' −! 'ICTP-SAIFR pre-doctoral school'. The following literature is recommended for preparation and further reading: Ph.W.
    [Show full text]
  • Eindhoven University of Technology BACHELOR Creating Rydberg
    Eindhoven University of Technology BACHELOR Creating Rydberg crystals in ultra-cold gases using stimulated Raman adiabatic passage schemes Plantz, N.W.M.; van der Wurff, E.C.I. Award date: 2012 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Eindhoven University of Technology Department of Applied Physics Coherence and Quantum Technology group CQT 2012-08 Creating Rydberg crystals in ultra-cold gases using Stimulated Raman Adiabatic Passage Schemes N.W.M. Plantz & E.C.I. van der Wurff July 2012 Supervisors: ir. R.M.W. van Bijnen dr. ir. S.J.J.M.F. Kokkelmans dr. ir. E.J.D. Vredenbregt Abstract This report is the result of a bachelor internship of two applied physics students.
    [Show full text]
  • Ultracold Atomic Gases in Artificial Magnetic Fields
    Ultracold Atomic Gases in Artificial Magnetic Fields Von der Fakultat¨ fur¨ Mathematik und Physik der Gottfried Wilhelm Leibniz Universitat¨ Hannover zur Erlangung des Grades Doktor der Naturwissenschaften — Doctor rerum naturalium — (Dr. rer. nat.) genehmigte Dissertation von Dipl. Phys. Klaus Osterloh geboren am 15. April 1977 in Herford in Nordrhein-Westfalen 2006 Referent : Prof. Dr. Maciej Lewenstein Korreferent : Prof. Dr. Holger Frahm Tag der Disputation : 07.12.2006 ePrint URL : http://edok01.tib.uni-hannover.de/edoks/e01dh06/521137799.pdf To Anna-Katharina “A thing of beauty is a joy forever, its loveliness increases, it will never pass into nothingness.” John Keats Abstract A phenomenon can hardly be found that accompanied physical paradigms and theoretical con- cepts in a more reflecting way than magnetism. From the beginnings of metaphysics and the first classical approaches to magnetic poles and streamlines of the field, it has inspired modern physics on its way to the classical field description of electrodynamics, and further to the quan- tum mechanical description of internal degrees of freedom of elementary particles. Meanwhile, magnetic manifestations have posed and still do pose complex and often controversially debated questions. This regards so various and utterly distinct topics as quantum spin systems and the grand unification theory. This may be foremost caused by the fact that all of these effects are based on correlated structures, which are induced by the interplay of dynamics and elementary interactions. It is strongly correlated systems that certainly represent one of the most fascinat- ing and universal fields of research. In particular, low dimensional systems are in the focus of interest, as they reveal strongly pronounced correlations of counterintuitive nature.
    [Show full text]
  • Rabi Oscillations in Strongly- Driven Magnetic Resonance
    RABI OSCILLATIONS IN STRONGLY- DRIVEN MAGNETIC RESONANCE SYSTEMS by Edward F. Thenell III A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah May 2017 Copyright © Edward F. Thenell III 2017 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Edward F. Thenell III has been approved by the following supervisory committee members: Brian Saam , Chair 12/9/2016 Date Approved Christoph Boehme , Member 12/9/2016 Date Approved Mikhail E. Raikh , Member 12/19/2016 Date Approved Stephane Louis LeBohec , Member Date Approved Eun-Kee Jeong , Member 12/9/2016 Date Approved and by Benjamin C. Bromley , Chair of the Department of Physics and Astronomy and by David B. Kieda, Dean of The Graduate School. ABSTRACT This dissertation is focused on an exploration of the strong-drive regime in magnetic resonance, in which the amplitude of the linearly-oscillating driving field is on order the quantizing field. This regime is rarely accessed in traditional magnetic resonance experiments, due primarily to signal-to-noise concerns in thermally-polarized samples which require the quantizing field to take on values much larger than those practically attainable in the tuned LC circuits which typically produce the driving field. However, such limitations are circumvented in the two primary experiments discussed herein, allowing for novel and systematic exploration of this magnetic resonance regime. First, spectroscopic data was taken on 129Xe nuclear spins, hyperpolarized via spin- exchange optical pumping (SEOP).
    [Show full text]
  • A Generalization of the Quantum Rabi Model: Exact Solution and Spectral Structure
    A generalization of the quantum Rabi model: exact solution and spectral structure Hans-Peter Eckle1 and Henrik Johannesson2;3 1 Humboldt Study Centre, Ulm University, D{89069 Ulm, Germany 2 Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden 3 Beijing Computational Science Research Center, Beijing 100094, China Abstract. We consider a generalization of the quantum Rabi model where the two- level system and the single-mode cavity oscillator are coupled by an additional Stark- like term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. 107, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases. Keywords: quantum optics, quantum Rabi-Stark model, regular and exceptional spectrum 1. Introduction The integration of coherent nanoscale systems with quantum resonators is a focal point of current quantum engineering of states and devices. Examples range from trapped ions interacting with a cavity field [1] to superconducting charge qubits in circuit QED architectures [2]. The paradigmatic model for these systems is the Rabi model [3] arXiv:1706.02687v2 [quant-ph] 11 Sep 2017 which was first introduced 80 years ago to discuss the phenomenon of nuclear magnetic resonance in a semi{classical way. While Rabi treated the atom quantum mechanically, he still construed the rapidly varying weak magnetic field as a rotating classical field [4].
    [Show full text]
  • Charge Qubit Coupled to Quantum Telegraph Noise
    Charge Qubit Coupled to Quantum Telegraph Noise Florian Schröder München 2012 Charge Qubit Coupled to Quantum Telegraph Noise Florian Schröder Bachelorarbeit an der Fakultät für Physik der Ludwig–Maximilians–Universität München vorgelegt von Florian Schröder aus München München, den 20. Juli 2012 Gutachter: Prof. Dr. Jan von Delft Contents Abstract vii 1 Introduction 1 1.1 Time Evolution of Closed Quantum Systems . .... 2 1.1.1 SchrödingerPicture........................... 2 1.1.2 HeisenbergPicture ........................... 3 1.1.3 InteractionPicture ........................... 3 1.2 Description of the System of Qubit and Bath . ..... 4 2 Quantum Telegraph Noise Model 7 2.1 Hamiltonian................................... 7 2.2 TheMasterEquation.............................. 7 2.3 DMRGandModelParameters. 11 2.3.1 D, ∆, γ and ǫd ............................. 11 2.3.2 DiscretizationandBathLength . 11 3 The Periodically Driven Qubit 15 3.1 TheFullModel ................................. 15 3.2 RabiOscillations ................................ 15 3.2.1 Pulses .................................. 17 3.3 Bloch-SiegertShift .............................. 18 3.3.1 Finding the Bloch-Siegert Shift . .. 18 3.4 DrivenQubitCoupledtoQTN ........................ 21 3.5 Simulations of Spin Echo and Bang-Bang with Ideal π-Pulses. 23 4 Conclusion and Outlook 27 A Derivations 29 A.1 MasterEquation ................................ 29 A.1.1 ProjectionOperatorMethod. 29 A.1.2 Interaction Picture Hamiltonian . ... 31 A.1.3 BathCorrelationFunctions . 33 A.2 DiscreteFourierTransformation . ..... 36 vi Contents A.3 SolutionoftheRabiProblem . 39 A.3.1 Rotating-Wave-Approximation . .. 39 A.3.2 Bloch-SiegertShift ........................... 40 A.4 Jaynes-Cummings-Hamiltonian . ... 41 Bibliography 47 Acknowledgements 48 Abstract I simulate the time evolution of a qubit which is exposed to quantum telegraph noise (QTN) with the time-dependent density matrix renormalization group (t-DMRG).
    [Show full text]
  • Dynamics of Ultracold Quantum Gases and Interferometry with Coherent Matter Waves
    DYNAMICS OF ULTRACOLD QUANTUM GASES AND INTERFEROMETRY WITH COHERENT MATTER WAVES Dissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakultät für Naturwissenschaften der Universität Ulm vorgelegt von GERRIT PETER NANDI aus Bonn Institut für Quantenphysik Universität Ulm Leiter: Prof. Dr. Wolfgang P. Schleich Ulm 2007 Amtierender Dekan: Prof. Dr. Klaus-Dieter Spindler Erstgutachter: PD Dr. Reinhold Walser Zweitgutachter: Prof. Dr. Wolfgang Wonneberger Tag der Promotion: 25. Juni 2007 ÓÒØÒØ× ½º ÁÒØÖÓ ½ ¾º ÊÚÛ Ó Ò Ó×¹Ò×ØÒ 2.1. Quantumfieldtheoryinthemean-fieldlimit . ... 7 2.1.1. Quantizedatomicfields . 7 2.1.2. Mean-fieldapproximation . 10 2.1.3. TheGROSS-PITAEVSKII equationinscaledvariables . 13 2.1.4. Multi-component, vector-valued coupled BOSE-EINSTEIN con- densates ............................. 13 2.2. Healing length and THOMAS-FERMI limit................ 16 2.3. Low-dimensionalcondensates . 17 2.3.1. Quasione-dimensionalBEC . 18 2.3.2. Quasitwo-dimensionalBEC . 18 2.4. Hydrodynamicdescription . 18 2.5. Topological excitations – vortices in BOSE-EINSTEIN condensates . 19 2.6. Linear response analysis for multi-component BECs . ....... 20 2.6.1. One-componentBEC. 20 2.6.2. Two-componentBEC . 22 2.7. Beyond the mean-field limit – HAMILTONians with few quantized modes 23 2.7.1. TheJOSEPHSON HAMILTONian................. 23 2.7.2. Semiclassicalapproximation . 24 ¿º ÓÐ ÕÙÒØÙÑ ×× Ò Ý ¾ 3.1. Classical physics of a particle falling within the drop capsule . 28 3.1.1. Thedropcapsule ......................... 28 3.1.2. A single classical particle trapped within the drop capsule . 31 3.2. Quantum physics of identical atoms falling within the dropcapsule . 36 3.2.1. Static representation of states in EUCLIDean frames of reference 36 3.2.2.
    [Show full text]
  • A Generalization of the Quantum Rabi Model: Exact Solution and Spectral Structure
    A generalization of the quantum Rabi model: exact solution and spectral structure Hans-Peter Eckle1 and Henrik Johannesson2;3 1 Humboldt Study Centre, Ulm University, D{89069 Ulm, Germany 2 Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden 3 Beijing Computational Science Research Center, Beijing 100094, China Abstract. We consider a generalization of the quantum Rabi model where the two- level system and the single-mode cavity oscillator are coupled by an additional Stark- like term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. 107, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of two-fold near-degenerate levels as the Rabi coupling increases. Keywords: quantum optics, quantum Rabi-Stark model, regular and exceptional spectrum 1. Introduction The integration of coherent nanoscale systems with quantum resonators is a focal point of current quantum engineering of states and devices. Examples range from trapped ions interacting with a cavity field [1] to superconducting charge qubits in circuit QED architectures [2]. The paradigmatic model for these systems is the Rabi model [3] which was first introduced 80 years ago to discuss the phenomenon of nuclear magnetic resonance in a semi{classical way. While Rabi treated the atom quantum mechanically, he still construed the rapidly varying weak magnetic field as a rotating classical field [4].
    [Show full text]
  • The Floquet Theory of the Two-Level System Revisited
    Z. Naturforsch. 2018; 73(8)a: 705–731 Heinz-Jürgen Schmidt* The Floquet Theory of the Two-Level System Revisited https://doi.org/10.1515/zna-2018-0211 In the following decades it was realised [3, 4] that the Received April 24, 2018; accepted May 8, 2018; previously underlying mathematical problem is an instance of the published online June 14, 2018 Floquet theory that deals with linear differential matrix equations with periodic coefficients. Accordingly, analyti- Abstract: In this article, we reconsider the periodically cal approximations for its solutions were devised that were driven two-level system especially the Rabi problem with the basis for subsequent research. Especially the seminal linear polarisation. The Floquet theory of this problem can paper of Shirley [4] has been until now cited more than be reduced to its classical limit, i.e. to the investigation 11,000 times. Among the numerous applications of the the- of periodic solutions of the classical Hamiltonian equa- ory of periodically driven TLSs are nuclear magnetic res- tions of motion in the Bloch sphere. The quasienergy is onance [5], ac-driven quantum dots [6], Josephson qubit essentially the action integral over one period and the res- circuits [7] and coherent destruction of tunnelling [8]. On onance condition due to Shirley is shown to be equivalent the theoretical level the methods of solving the RPL and to the vanishing of the time average of a certain compo- related problems were gradually refined to include power nent of the classical solution. This geometrical approach series approximations for Bloch-Siegert shifts [9, 10], per- is applied to obtain analytical approximations to physical turbation theory and/or various limit cases [11–16] and the quantities of the Rabi problem with linear polarisation as hybridised rotating wave approximation [17].
    [Show full text]
  • Properties of Adiabatic Rapid Passage Sequences
    SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Properties of Adiabatic Rapid Passage Sequences A Thesis Presented by Benjamin Simon Gl¨aßle to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Master of Arts in Physics Stony Brook University August 2009 Stony Brook University The Graduate School Benjamin Simon Gl¨aßle We, the thesis committee for the above candidate for the Master of Arts degree, hereby recommend acceptance of this thesis. Harold J. Metcalf { Thesis Advisor Distinguished Teaching Professor, Department of Physics and Astronomy Todd Satogata Adjunct Professor, Department of Physics and Astronomy Thomas H. Bergeman Adjunct Professor, Department of Physics and Astronomy This thesis is accepted by the Graduate School. Lawrence Martin Dean of the Graduate School ii Abstract of the Thesis Properties of Adiabatic Rapid Passage Sequences by Benjamin Simon Gl¨aßle Master of Arts in Physics Stony Brook University 2009 Adiabatic rapid passage (ARP) sequences can be used to exert op- tical forces on neutral atoms, much larger than the radiative force. These sequences consist of counter-propagating and synchronized optical pulses whose frequency is swept through the resonance of the transition. The new results from the numerical studies discussed in chapter 3 shall be pointed out. The numerical integration of the optical Bloch equations show a superior stability of the force, when alternate sweep directions are used. This discovery was investigated in the parameter space of ARP and under various perturbations.
    [Show full text]
  • A Levitated Nanoparticle As a Classical Two-Level Atom
    A levitated nanoparticle as a classical two-level atom Martin Frimmer,1 Jan Gieseler,1 Thomas Ihn,2 and Lukas Novotny1 1Photonics Laboratory, ETH Zürich, 8093 Zürich, Switzerland∗ 2Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland (Dated: September 27, 2018) The center-of-mass motion of a single optically levitated nanoparticle resembles three uncoupled harmonic oscillators. We show how a suitable modulation of the optical trapping potential can give rise to a coupling between two of these oscillators, such that their dynamics are governed by a classical equation of motion that resembles the Schrödinger equation for a two-level system. Based on experimental data, we illustrate the dy- namics of this parametrically coupled system both in the frequency and in the time domain. We discuss the limitations and differences of the mechanical analogue in comparison to a true quantum mechanical system. I. INTRODUCTION 780 nm (a) EOM1 EOM2 One particularly interesting laboratory system for engi- 1064 nm neers and physicists alike are optically levitated nanoparti- cles in vacuum [1–3]. The fact that such a levitated particle at suitably low pressures is a harmonic oscillator with no FB mechanical interaction with its environment promises ex- ceptional control over the system’s dynamics [4, 5]. In fact, -18 S (b) at such low pressures, the only interaction expected to re- 10 z main between the particle and the environment is mediated by the electromagnetic field [6, 7]. Indeed, it has recently Sx Sy been shown that at sufficiently low pressures, the dominat- /Hz) 2 -19 ing interaction of the particle with its environment is deter- 10 (m u mined by the photon bath of the trapping laser [8].
    [Show full text]
  • Atom-Light Interaction and Basic Applications
    ICTP-SAIFR school on Interaction of Light with Cold Atoms Lecture on Atom-Light Interaction and Basic Applications Ph.W. Courteille Universidade de S~aoPaulo Instituto de F´ısicade S~aoCarlos 23/09/2019 2 . 3 . 4 Preface The following notes have been prepared for the ICTP-SAIFR school on 'Interaction of Light with Cold Atoms' held 2019 in S~aoPaulo. They are conceived to support an introductory course on 'Atom-Light Interaction and Basic Applications'. The course is divided into 5 lectures. Cold atomic clouds represent an ideal platform for studies of basic phenomena of light-matter interaction. The invention of powerful cooling and trapping techniques for atoms led to an unprecedented experimental control over all relevant degrees of freedom to a point where the interaction is dominated by weak quantum effects. This course reviews the foundations of this area of physics, emphasizing the role of light forces on the atomic motion. Collective and self-organization phenomena arising from a cooperative reaction of many atoms to incident light will be discussed. The course is meant for graduate students and requires basic knowledge of quan- tum mechanics and electromagnetism at the undergraduate level. The lectures will be complemented by exercises proposed at the end of each lecture. The present notes are mostly extracted from some textbooks (see below) and more in-depth scripts which can be consulted for further reading on the website http://www.ifsc.usp.br/∼strontium/ under the menu item 'Teaching' −! 'Cursos 2019-2' −! 'ICTP-SAIFR pre-doctoral school'. The following literature is recommended for preparation and further reading: Ph.W.
    [Show full text]