Laser Cooling of Molecules: A Theory of Purity Increasing Transformations
|
|
- Belinda Marsh
- 5 years ago
- Views:
Transcription
1 Laser Cooling of Molecules: A Theory of Purity Increasing Transformations COHERENT CONTROL Shlomo Sklarz Navin Khaneja Alon Bartana Ronnie Kosloff LASER COOLING QUANTUM INFORMATION/ DECOHERENCE
2 The Challenge: Direct Laser Cooling of Molecules ATOMS MOLECUL ES Why traditional laser cooling fails for molecules
3 3 Laser Cooling Schemes I) DOPPLER COOLING T D =hγ/k B 40µK II) SISYPHUS COOLING T R =h k /MK B.5µK Force III) VELOCITY SELECTIVE COHERENT POPULATION TRAPPING (VSCPT) T=0? nk Normalized velocity ν,σ+ a,p> ν,σ b -,p-hk> b +,p+hk> Energy Atomic Position
4 I) Atom Cooling Schemes Questions: Each new scheme seems to come out of the blue. Is there a systematic approach? Can the efficiency be improved? Where is the thermodynamics? II) Optimal Control Theory. Tannor and Rice 985 (Calculus of variations) Peirce, Dahleh and Rabitz 988 Kosloff, Rice, Gaspard, Tersigni and Tannor 989
5 Introduction to Optimal Control ih ψ t = H[ε(t)]ψ(t) J = lim t <ψ(t) P ψ(t) > equations of motion with control (penalty) objective ψ(0) χ(0) χ(t) ψ(t) ε(t) = i λ [<ψ b µ χ a > < χ b µ ψ a >] optimal field Iteration! Tannor, Kosloff, Rice (985-89) Rabitz et al. (988)
6 Optimal Control of Cooling ρ t = ih [ H[ ε(t)], ρ] + Γρ Bartana, Kosloff and Tannor, 993, 997, 00 Γρ = FρF + [F + Fρ + ρf + F] dissipation J = lim ρ(t)â T  0 0 ρ(0) Â(0) t  + = L (Â) ρ(t) Â(T) ε(t) = i λ Tr[ ˆ ρ c ˆ µ A ˆ e + ( ˆ ρ g ˆ µ ˆ µ ˆ ρ e ) A ˆ c ˆ µ ˆ ρ ˆ c B g ] optimal field
7 Laser Cooling of Molecules: Vibrations + Rotations Optimal Control meets Laser Cooling VIBRATIONS ROTATIONS Absorption Stimulated Emission Spontaneous Emission
8 Rotational Selective Coherent Population Trapping Ĥ Ĥ = e/g = B j e/g Ĥ ε j e (t) * µ ˆ l(l + ) j j ε Ĥ j (t) µ ˆ g j --Projection onto 0><0 --Largest eigenvalue of ρ --Purity Tr(ρ )
9 What is Cooling? = n n P n < P n Tr(ρ ) = n n P n = P n Tr(ρ) is a measure of coherence. The essence of cooling is increasing coherence!
10 PHASE SPACE PICTURE ( ) A ˆ = Tr ρa ˆ ˆ ρ = Tr( ρ ) = Tr ( W ρ ) W = πh dpdqρ W (p,q)
11 Bombshell: Hamiltonian Manipulations Cannot Increase Tr(r)! ih [ H[ ε(t ρ] ρ & = )], Control Tr( ρ& ) = Tr( ρρ& ) Need Dissipation: H ρ & = [, ρ] + Γρ i h = ih Tr( ρ[h, ρ]) (Ketterle + Pritchard 99) Tr( ρ& ) = Γρ = Tr( ργρ) i = 0 {[Vρ,V i 0 + i Tr(ρ ) = ] + [V, ρv BUT DISSIPATION (Γ) CANNOT BE CONTROLLED! i + i ]}
12 Questions: How can cooling be affected by external fields? What are the general rules for when spontaneous emission leads to heating and when to cooling? Tr( ρ& γ + ) γ Tr( ρ ) 0,0 d 0,0 d..99 a ρ = c b d 0 a + d = bc ad bc = γcd 0 γ
13 Interplay of control fields and spontaneous emission γ Tr( ρ& ) Tr( ρ γ ) 0,0 Optimal cooling strategy d Strange but interesting form! 0,0 max Tr( ρ& ) d, γ d + λtr( ρ ) Physical significance of optimal strategy = T T Algorithm: optimal trajectory Tr( ρ& ) = Tr( ρ keep coherence off the off-diagonal. ) + d, γ Tr( ρ Tr( ρ ) ) Diff. eq. for Tr(ρ) vs t: 3rd law of thermodynamics! ~ d, ~ γ Tr( ρ& )
14 Purity Increasing Transformations: Bloch Sphere Representation Purity decreasing Tr(r.) Dissipative Tr(r ) Unitary ρ = ρ ρ ρ ρ. Purity increasin g Tr(r ) does depend on the control E(t) indirectly
15 Constant T (uncontrollable) Constant S (controllable) Carnot cycle Spontaneous emission (uncontrollable) Coherent Fields (controllable) Laser Cooling Thermalization, Collisions (uncontrollable) Trap Lowering (controllable) Evaporative Cooling Universality of the interplay of controllable + uncontrollable in cooling
16 Beyond two-level systems: Two simplifying assumptions Instantaneous unitary control U= e ih[e]t is infinitely fast compared with G Criterion: w ij ³g Complete unitary control Any U in SU(N) can be produced by e ih[e]t Lie algebra criterion: dim {H, H } LA =N - è Complete and Instantaneous Unitary Control
17 Representation of the problem in terms of spectral transformations r r L r L L=U+rU L L=U+rU
18 Modified Control problem I II Eqn. of Motion ρ = i [H,ρ] + Γρ h Control E(t) U(t) Objective Tr(ρ ), ρ
19 Hamilton-Jacobi-Bellman Theorem (Dynamical Programming) V(l,t) l t
20 Hamilton-Jacobi-Bellman Theorem Guaranteed to give GLOBAL maximum. Capable of giving analytic optimal solutions. Very Computationally expensive. A possible method of solution: guess optimal strategy and prove that HJB equations are satisfied.
21 Greedy strategy for 3 level L system is optimal The Greedy strategy: Maximize dp/dt at each instant Maintain maximal population of the excited state Keep r Diagonal (Q={P} ) (No coherences) and Ordered (P=I) (Ordered Eigenvalues) Theorem: The greedy trajectorydiag(r)=l is optimal
22 time Greedy strategy for N+ level g ñ ñ g g 3 4 3ñ 4 ñ g n n ñ Greedy=. No coherences Q={P i }. Ordered Eigenvalues P i =I l (populations) system; Spectral evolution 4ñ g g ñ ñ g 3 3ñ
23 4 levels G=[0.05, 0.045, 0.000] l (populations) Investment Return 4ñ g g ñ ñ g 3 3ñ
24 Summary The Greedy cooling strategy is optimal for the three-level L system Investment & Return strategies rather than Greedy are optimal for N>3 level systems Coherences are required for optimality
25 THERMODYNAMICS Definition of Cooling Tr(ρ ) 0 th law of thermo Tr(ρ )=0 for Hamiltonian manipulations Optimal Control Theory nd law of thermo 3 rd law of thermo
26 Conclusions New frontier for optimal control Increasing Tr(ρ )= increasing coherence is relevant to more than laser cooling! It may be profitable to reexamine existing laser cooling schemes in light of purity increase. There is the potential for great improvement in rate/ efficiency by exploiting all spontaneous emission. New strategy for cooling molecules. Experiments, anyone? Thermodynamic analysis of laser cooling 0 th, nd + 3 rd law Cooling and Lasing as complementary Processes Lasing as cooling light! ρ33 LASING IWL LWI COOLING Re ρ Kocharovskaya + Khanin 988
Quantum control of dissipative systems. 1 Density operators and mixed quantum states
Quantum control of dissipative systems S. G. Schirmer and A. I. Solomon Quantum Processes Group, The Open University Milton Keynes, MK7 6AA, United Kingdom S.G.Schirmer@open.ac.uk, A.I.Solomon@open.ac.uk
More informationIntroduction to optimal control theory
Introduction to optimal control theory Christiane P. Koch Laboratoire Aimé Cotton CNRS, France The Hebrew University Jerusalem, Israel Outline 0. Terminology 1. Intuitive control schemes and their experimental
More informationSpin-Boson Model. A simple Open Quantum System. M. Miller F. Tschirsich. Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012
Spin-Boson Model A simple Open Quantum System M. Miller F. Tschirsich Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012 Outline 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson
More informationQuantum optics of many-body systems
Quantum optics of many-body systems Igor Mekhov Université Paris-Saclay (SPEC CEA) University of Oxford, St. Petersburg State University Lecture 2 Previous lecture 1 Classical optics light waves material
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More informationS.K. Saikin May 22, Lecture 13
S.K. Saikin May, 007 13 Decoherence I Lecture 13 A physical qubit is never isolated from its environment completely. As a trivial example, as in the case of a solid state qubit implementation, the physical
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationProtecting coherence in Optimal Control Theory: state dependent constraint approach
Protecting coherence in Optimal Control Theory: state dependent constraint approach José P. Palao, 1 Ronnie Kosloff, 2 and Christiane P. Koch 3 1 Departamento de Física Fundamental II, Universidad de La
More information7 Three-level systems
7 Three-level systems In this section, we will extend our treatment of atom-light interactions to situations with more than one atomic energy level, and more than one independent coherent driving field.
More informationCONTROLLABILITY OF QUANTUM SYSTEMS. Sonia G. Schirmer
CONTROLLABILITY OF QUANTUM SYSTEMS Sonia G. Schirmer Dept of Applied Mathematics + Theoretical Physics and Dept of Engineering, University of Cambridge, Cambridge, CB2 1PZ, United Kingdom Ivan C. H. Pullen
More informationOptimal Controlled Phasegates for Trapped Neutral Atoms at the Quantum Speed Limit
with Ultracold Trapped Atoms at the Quantum Speed Limit Michael Goerz May 31, 2011 with Ultracold Trapped Atoms Prologue: with Ultracold Trapped Atoms Classical Computing: 4-Bit Full Adder Inside the CPU:
More informationThe Two Level Atom. E e. E g. { } + r. H A { e e # g g. cos"t{ e g + g e } " = q e r g
E e = h" 0 The Two Level Atom h" e h" h" 0 E g = " h# 0 g H A = h" 0 { e e # g g } r " = q e r g { } + r $ E r cos"t{ e g + g e } The Two Level Atom E e = µ bb 0 h" h" " r B = B 0ˆ z r B = B " cos#t x
More informationEnsembles and incomplete information
p. 1/32 Ensembles and incomplete information So far in this course, we have described quantum systems by states that are normalized vectors in a complex Hilbert space. This works so long as (a) the system
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationEinselection without pointer states -
Einselection without pointer states Einselection without pointer states - Decoherence under weak interaction Christian Gogolin Universität Würzburg 2009-12-16 C. Gogolin Universität Würzburg 2009-12-16
More informationLyapunov-based control of quantum systems
Lyapunov-based control of quantum systems Symeon Grivopoulos Bassam Bamieh Department of Mechanical and Environmental Engineering University of California, Santa Barbara, CA 936-57 symeon,bamieh@engineering.ucsb.edu
More informationA geometric analysis of the Markovian evolution of open quantum systems
A geometric analysis of the Markovian evolution of open quantum systems University of Zaragoza, BIFI, Spain Joint work with J. F. Cariñena, J. Clemente-Gallardo, G. Marmo Martes Cuantico, June 13th, 2017
More informationPhase control in the vibrational qubit
Phase control in the vibrational qubit THE JOURNAL OF CHEMICAL PHYSICS 5, 0405 006 Meiyu Zhao and Dmitri Babikov a Chemistry Department, Wehr Chemistry Building, Marquette University, Milwaukee, Wisconsin
More informationQuantum control: Introduction and some mathematical results. Rui Vilela Mendes UTL and GFM 12/16/2004
Quantum control: Introduction and some mathematical results Rui Vilela Mendes UTL and GFM 12/16/2004 Introduction to quantum control Quantum control : A chemists dream Bond-selective chemistry with high-intensity
More informationWhen Worlds Collide: Quantum Probability From Observer Selection?
When Worlds Collide: Quantum Probability From Observer Selection? Robin Hanson Department of Economics George Mason University August 9, 2001 Abstract Deviations from exact decoherence make little difference
More informationQuantum Reservoir Engineering
Departments of Physics and Applied Physics, Yale University Quantum Reservoir Engineering Towards Quantum Simulators with Superconducting Qubits SMG Claudia De Grandi (Yale University) Siddiqi Group (Berkeley)
More informationGeometric control for atomic systems
Geometric control for atomic systems S. G. Schirmer Quantum Processes Group, The Open University Milton Keynes, MK7 6AA, United Kingdom S.G.Schirmer@open.ac.uk Abstract The problem of explicit generation
More informationControl of Dissipative Quantum systems By a Non-Markovian Master equation approach. Christoph Meier. LCAR-IRSAMC Université Paul Sabatier, Toulouse
Control of Dissipative Quantum systems By a Non-Markovian Master equation approach Christoph Meier LCAR-IRSAMC Université Paul Sabatier, Toulouse September 2012 Context: Femtosecond laser interaction with
More informationOptimal Control. Macroeconomics II SMU. Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112
Optimal Control Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 112 Review of the Theory of Optimal Control Section 1 Review of the Theory of Optimal Control Ömer
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationOptimal control of time-dependent targets
Optimal control of time-dependent targets Diploma Thesis Submitted to the Free University Berlin, Physics Department Ioana Şerban Advisor: Professor Dr. Eberhard Groß Berlin, August 3, 24 Title page: time-dependent
More informationRotation and vibration of Molecules
Rotation and vibration of Molecules Overview of the two lectures... 2 General remarks on spectroscopy... 2 Beer-Lambert law for photoabsorption... 3 Einstein s coefficients... 4 Limits of resolution...
More informationPhysics 581, Quantum Optics II Problem Set #4 Due: Tuesday November 1, 2016
Physics 581, Quantum Optics II Problem Set #4 Due: Tuesday November 1, 2016 Problem 3: The EPR state (30 points) The Einstein-Podolsky-Rosen (EPR) paradox is based around a thought experiment of measurements
More informationLaser Induced Control of Condensed Phase Electron Transfer
Laser Induced Control of Condensed Phase Electron Transfer Rob D. Coalson, Dept. of Chemistry, Univ. of Pittsburgh Yuri Dakhnovskii, Dept. of Physics, Univ. of Wyoming Deborah G. Evans, Dept. of Chemistry,
More informationUniversity of New Mexico
Quantum State Reconstruction via Continuous Measurement Ivan H. Deutsch, Andrew Silberfarb University of New Mexico Poul Jessen, Greg Smith University of Arizona Information Physics Group http://info.phys.unm.edu
More information(Dynamical) quantum typicality: What is it and what are its physical and computational implications?
(Dynamical) : What is it and what are its physical and computational implications? Jochen Gemmer University of Osnabrück, Kassel, May 13th, 214 Outline Thermal relaxation in closed quantum systems? Typicality
More information1 Measurement and expectation values
C/CS/Phys 191 Measurement and expectation values, Intro to Spin 2/15/05 Spring 2005 Lecture 9 1 Measurement and expectation values Last time we discussed how useful it is to work in the basis of energy
More informationFrom laser cooling to BEC First experiments of superfluid hydrodynamics
From laser cooling to BEC First experiments of superfluid hydrodynamics Alice Sinatra Quantum Fluids course - Complement 1 2013-2014 Plan 1 COOLING AND TRAPPING 2 CONDENSATION 3 NON-LINEAR PHYSICS AND
More informationThermodynamical cost of accuracy and stability of information processing
Thermodynamical cost of accuracy and stability of information processing Robert Alicki Instytut Fizyki Teoretycznej i Astrofizyki Uniwersytet Gdański, Poland e-mail: fizra@univ.gda.pl Fields Institute,
More informationLie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains
.. Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains Vladimir M. Stojanović Condensed Matter Theory Group HARVARD UNIVERSITY September 16, 2014 V. M. Stojanović (Harvard)
More informationAtomic Coherent Trapping and Properties of Trapped Atom
Commun. Theor. Phys. (Beijing, China 46 (006 pp. 556 560 c International Academic Publishers Vol. 46, No. 3, September 15, 006 Atomic Coherent Trapping and Properties of Trapped Atom YANG Guo-Jian, XIA
More informationBose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation
Bose Gases, Bose Einstein Condensation, and the Bogoliubov Approximation Robert Seiringer IST Austria Mathematical Horizons for Quantum Physics IMS Singapore, September 18, 2013 R. Seiringer Bose Gases,
More informationControl of bilinear systems: multiple systems and perturbations
Control of bilinear systems: multiple systems and perturbations Gabriel Turinici CEREMADE, Université Paris Dauphine ESF OPTPDE Workshop InterDyn2013, Modeling and Control of Large Interacting Dynamical
More informationMolecular spectroscopy
Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More informationMixing Quantum and Classical Mechanics: A Partially Miscible Solution
Mixing Quantum and Classical Mechanics: A Partially Miscible Solution R. Kapral S. Nielsen A. Sergi D. Mac Kernan G. Ciccotti quantum dynamics in a classical condensed phase environment how to simulate
More information6. Cosmology. (same at all points) probably true on a sufficiently large scale. The present. ~ c. ~ h Mpc (6.1)
6. 6. Cosmology 6. Cosmological Principle Assume Universe is isotropic (same in all directions) and homogeneous (same at all points) probably true on a sufficiently large scale. The present Universe has
More information1 More on the Bloch Sphere (10 points)
Ph15c Spring 017 Prof. Sean Carroll seancarroll@gmail.com Homework - 1 Solutions Assigned TA: Ashmeet Singh ashmeet@caltech.edu 1 More on the Bloch Sphere 10 points a. The state Ψ is parametrized on the
More informationBOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 4, P AGES 655 664 BOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION Guang-Da Hu and Qiao Zhu This paper is concerned with bounds of eigenvalues of a complex
More informationFelix Kleißler 1,*, Andrii Lazariev 1, and Silvia Arroyo-Camejo 1,** 1 Accelerated driving field frames
Supplementary Information: Universal, high-fidelity quantum gates based on superadiabatic, geometric phases on a solid-state spin-qubit at room temperature Felix Kleißler 1,*, Andrii Lazariev 1, and Silvia
More informationQuantum Entanglement and Error Correction
Quantum Entanglement and Error Correction Fall 2016 Bei Zeng University of Guelph Course Information Instructor: Bei Zeng, email: beizeng@icloud.com TA: Dr. Cheng Guo, email: cheng323232@163.com Wechat
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 3: Basic Quantum Mechanics 26th September 2016 School of Informatics, University of Edinburgh Resources 1. Quantum Computation and Quantum Information
More information2 The Density Operator
In this chapter we introduce the density operator, which provides an alternative way to describe the state of a quantum mechanical system. So far we have only dealt with situations where the state of a
More informationAlgebra C Numerical Linear Algebra Sample Exam Problems
Algebra C Numerical Linear Algebra Sample Exam Problems Notation. Denote by V a finite-dimensional Hilbert space with inner product (, ) and corresponding norm. The abbreviation SPD is used for symmetric
More informationLaser Cooling of Gallium. Lauren Rutherford
Laser Cooling of Gallium Lauren Rutherford Laser Cooling Cooling mechanism depends on conservation of momentum during absorption and emission of radiation Incoming photons Net momentum transfer to atom
More informationIntroduction to quantum information processing
Introduction to quantum information processing Measurements and quantum probability Brad Lackey 25 October 2016 MEASUREMENTS AND QUANTUM PROBABILITY 1 of 22 OUTLINE 1 Probability 2 Density Operators 3
More informationContrôle direct et indirect des systèmes quantiques ouverts
Contrôle direct et indirect des systèmes quantiques ouverts Andreea Grigoriu LJLL - Université Paris Diderot - Paris 7 Groupe de Travail Contrôle - LJLL 22 Mars 2013 1 / 49 Outline 1 Introduction Closed
More informationOptimal Control of Quantum Systems
University of California, Santa Barbara Optimal Control of Quantum Systems PhD Thesis Symeon Grivopoulos 1 Average populations.8.6.4.2 1 3 4 6 9 1.1.2.3.4.5.6.7.8.9 1 Scaled time Profile intensities.1.2.3.4.5.6.7.8.9
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationWhen Worlds Collide: Quantum Probability From Observer Selection?
When Worlds Collide: Quantum Probability From Observer Selection? arxiv:quant-ph/0108070v1 14 Aug 2001 Robin Hanson Department of Economics George Mason University August 9, 2001 Abstract Deviations from
More informationFast Adiabatic Control of a Harmonic Oscillator s Frequency
Fast Adiabatic Control of a Harmonic Oscillator s Frequency Peter Salamon Dept. of Math & Stats San Diego State University KITP; June, 2009 Ensemble of Independent Harmonic Oscillators Sharing a Controlled
More informationNumerical Integration of the Wavefunction in FEM
Numerical Integration of the Wavefunction in FEM Ian Korey Eaves ike26@drexel.edu December 14, 2013 Abstract Although numerous techniques for calculating stationary states of the schrödinger equation are
More informationA prime factorization based on quantum dynamics on a spin ensemble (I)
A prime factorization based on quantum dynamics on a spin ensemble (I) arxiv:quant-ph/0302153v1 20 Feb 2003 Xijia Miao* Abstract In this paper it has been described how to use the unitary dynamics of quantum
More informationconventions and notation
Ph95a lecture notes, //0 The Bloch Equations A quick review of spin- conventions and notation The quantum state of a spin- particle is represented by a vector in a two-dimensional complex Hilbert space
More informationPhysics 550. Problem Set 6: Kinematics and Dynamics
Physics 550 Problem Set 6: Kinematics and Dynamics Name: Instructions / Notes / Suggestions: Each problem is worth five points. In order to receive credit, you must show your work. Circle your final answer.
More informationQuantum decoherence: From the self-induced approach to Schrödinger-cat experiments
Quantum decoherence: From the self-induced approach to Schrödinger-cat experiments Maximilian Schlosshauer Department of Physics University of Washington Seattle, Washington Very short biography Born in
More informationDark pulses for resonant two-photon transitions
PHYSICAL REVIEW A 74, 023408 2006 Dark pulses for resonant two-photon transitions P. Panek and A. Becker Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden,
More informationSymmetries and Supersymmetries in Trapped Ion Hamiltonian Models
Proceedings of Institute of Mathematics of NAS of Ukraine 004, Vol. 50, Part, 569 57 Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models Benedetto MILITELLO, Anatoly NIKITIN and Antonino MESSINA
More informationQuantum Chaos and Nonunitary Dynamics
Quantum Chaos and Nonunitary Dynamics Karol Życzkowski in collaboration with W. Bruzda, V. Cappellini, H.-J. Sommers, M. Smaczyński Phys. Lett. A 373, 320 (2009) Institute of Physics, Jagiellonian University,
More informationOPTI 511R, Spring 2018 Problem Set 10 Prof. R.J. Jones Due Thursday, April 19
OPTI 511R, Spring 2018 Problem Set 10 Prof. R.J. Jones Due Thursday, April 19 1. (a) Suppose you want to use a lens focus a Gaussian laser beam of wavelength λ in order to obtain a beam waist radius w
More informationIntroduction to Quantum Spin Systems
1 Introduction to Quantum Spin Systems Lecture 2 Sven Bachmann (standing in for Bruno Nachtergaele) Mathematics, UC Davis MAT290-25, CRN 30216, Winter 2011, 01/10/11 2 Basic Setup For concreteness, consider
More informationA Simple Model of Quantum Trajectories. Todd A. Brun University of Southern California
A Simple Model of Quantum Trajectories Todd A. Brun University of Southern California Outline 1. Review projective and generalized measurements. 2. A simple model of indirect measurement. 3. Weak measurements--jump-like
More informationQuantum mechanics in one hour
Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might
More informationUnder evolution for a small time δt the area A(t) = q p evolves into an area
Physics 106a, Caltech 6 November, 2018 Lecture 11: Hamiltonian Mechanics II Towards statistical mechanics Phase space volumes are conserved by Hamiltonian dynamics We can use many nearby initial conditions
More informationProject: Vibration of Diatomic Molecules
Project: Vibration of Diatomic Molecules Objective: Find the vibration energies and the corresponding vibrational wavefunctions of diatomic molecules H 2 and I 2 using the Morse potential. equired Numerical
More informationPhysics 239/139 Spring 2018 Assignment 6
University of California at San Diego Department of Physics Prof. John McGreevy Physics 239/139 Spring 2018 Assignment 6 Due 12:30pm Monday, May 14, 2018 1. Brainwarmers on Kraus operators. (a) Check that
More informationIntroduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871
Introduction to Quantum Information Processing QIC 710 / CS 768 / PH 767 / CO 681 / AM 871 Lecture 9 (2017) Jon Yard QNC 3126 jyard@uwaterloo.ca http://math.uwaterloo.ca/~jyard/qic710 1 More state distinguishing
More information11 Perturbation Theory
S.K. Saikin Oct. 8, 009 11 Perturbation Theory Content: Variational Principle. Time-Dependent Perturbation Theory. 11.1 Variational Principle Lecture 11 If we need to compute the ground state energy of
More informationRepresentation theory of SU(2), density operators, purification Michael Walter, University of Amsterdam
Symmetry and Quantum Information Feruary 6, 018 Representation theory of S(), density operators, purification Lecture 7 Michael Walter, niversity of Amsterdam Last week, we learned the asic concepts of
More informationIntroduction to Cold Atoms and Bose-Einstein Condensation. Randy Hulet
Introduction to Cold Atoms and Bose-Einstein Condensation Randy Hulet Outline Introduction to methods and concepts of cold atom physics Interactions Feshbach resonances Quantum Gases Quantum regime nλ
More informationNon-stationary States and Electric Dipole Transitions
Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i
More informationRobotics. Control Theory. Marc Toussaint U Stuttgart
Robotics Control Theory Topics in control theory, optimal control, HJB equation, infinite horizon case, Linear-Quadratic optimal control, Riccati equations (differential, algebraic, discrete-time), controllability,
More informationarxiv: v1 [cond-mat.other] 12 Jul 2007
Optimal control of charge transfer Jan Werschnik a and E.K.U. Gross a a Freie Universität, Arnimallee 14, Berlin, Germany; arxiv:77.1874v1 [cond-mat.other] 12 Jul 27 ABSTRACT In this work, we investigate
More informationOPTI 511, Spring 2016 Problem Set 9 Prof. R. J. Jones
OPTI 5, Spring 206 Problem Set 9 Prof. R. J. Jones Due Friday, April 29. Absorption and thermal distributions in a 2-level system Consider a collection of identical two-level atoms in thermal equilibrium.
More informationQuantum Dynamics. March 10, 2017
Quantum Dynamics March 0, 07 As in classical mechanics, time is a parameter in quantum mechanics. It is distinct from space in the sense that, while we have Hermitian operators, X, for position and therefore
More information2.1 Definition and general properties
Chapter 2 Gaussian states Gaussian states are at the heart of quantum information processing with continuous variables. The basic reason is that the vacuum state of quantum electrodynamics is itself a
More informationIntroduction to cold atoms and Bose-Einstein condensation (II)
Introduction to cold atoms and Bose-Einstein condensation (II) Wolfgang Ketterle Massachusetts Institute of Technology MIT-Harvard Center for Ultracold Atoms 7/7/04 Boulder Summer School * 1925 History
More informationWhat are we going to talk about: BEC and Nonlinear Atom Optics
What are we going to talk about: BEC and Nonlinear Atom Optics Nobel Prize Winners E. A. Cornell 1961JILA and NIST Boulder, Co, USA W. Ketterle C. E. Wieman 19571951MIT, JILA and UC, Cambridge.M Boulder,
More informationEnergy Level Sets for the Morse Potential
Energy Level Sets for the Morse Potential Fariel Shafee Department of Physics Princeton University Princeton, NJ 08540 Abstract: In continuation of our previous work investigating the possibility of the
More informationA microscopic approach to nuclear dynamics. Cédric Simenel CEA/Saclay, France
A microscopic approach to nuclear dynamics Cédric Simenel CEA/Saclay, France Introduction Quantum dynamics of complex systems (nuclei, molecules, BEC, atomic clusters...) Collectivity: from vibrations
More informationThe Quantum Theory of Atoms and Molecules
The Quantum Theory of Atoms and Molecules The postulates of quantum mechanics Dr Grant Ritchie The postulates.. 1. Associated with any particle moving in a conservative field of force is a wave function,
More informationWeek 13. PHY 402 Atomic and Molecular Physics Instructor: Sebastian Wüster, IISERBhopal, Frontiers of Modern AMO physics. 5.
Week 13 PHY 402 Atomic and Molecular Physics Instructor: Sebastian Wüster, IISERBhopal,2018 These notes are provided for the students of the class above only. There is no warranty for correctness, please
More information1 Time-Dependent Two-State Systems: Rabi Oscillations
Advanced kinetics Solution 7 April, 16 1 Time-Dependent Two-State Systems: Rabi Oscillations a In order to show how Ĥintt affects a bound state system in first-order time-dependent perturbation theory
More informationTheoretical Biophysics. Quantum Theory and Molecular Dynamics. Pawel Romanczuk WS 2017/18
Theoretical Biophysics Quantum Theory and Molecular Dynamics Pawel Romanczuk WS 2017/18 http://lab.romanczuk.de/teaching/ 1 Introduction Two pillars of classical theoretical physics at the begin of 20th
More informationThe interaction of light and matter
Outline The interaction of light and matter Denise Krol (Atom Optics) Photon physics 014 Lecture February 14, 014 1 / 3 Elementary processes Elementary processes 1 Elementary processes Einstein relations
More informationWhat is thermal equilibrium and how do we get there?
arxiv:1507.06479 and more What is thermal equilibrium and how do we get there? Hal Tasaki QMath 13, Oct. 9, 2016, Atlanta 40 C 20 C 30 C 30 C about the talk Foundation of equilibrium statistical mechanics
More informationStorage of Quantum Information in Topological Systems with Majorana Fermions
Storage of Quantum Information in Topological Systems with Majorana Fermions Leonardo Mazza Scuola Normale Superiore, Pisa Mainz September 26th, 2013 Leonardo Mazza (SNS) Storage of Information & Majorana
More informationQuantum Systems Measurement through Product Hamiltonians
45th Symposium of Mathematical Physics, Toruń, June 1-2, 2013 Quantum Systems Measurement through Product Hamiltonians Joachim DOMSTA Faculty of Applied Physics and Mathematics Gdańsk University of Technology
More informationVibronic quantum dynamics of exciton relaxation/trapping in molecular aggregates
Symposium, Bordeaux Vibronic quantum dynamics of exciton relaxation/trapping in molecular aggregates Alexander Schubert Institute of Physical and Theoretical Chemistry, University of Würzburg November
More informationMechanics and Thermodynamics fundamentally united by density operators with an ontic status obeying a locally
G.P. Beretta, PIAF '09 "New Perspectives on the Quantum State", Perimeter Institute, Sept.27-Oct.2, 2009 Mechanics and Thermodynamics fundamentally united by density operators with an ontic status obeying
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Quantum Dynamics Lecture #2 Recap of Last Class Schrodinger and Heisenberg Picture Time Evolution operator/ Propagator : Retarded
More informationELECTROMAGNETICALLY INDUCED TRANSPARENCY
14 ELECTROMAGNETICALLY INDUCED TRANSPARENCY J.P. Marangos Quantum Optics and Laser Science Group Blackett Laboratory, Imperial College London, United Kingdom T. Halfmann Institute of Applied Physics Technical
More informationOn The Power Of Coherently Controlled Quantum Adiabatic Evolutions
On The Power Of Coherently Controlled Quantum Adiabatic Evolutions Maria Kieferova (RCQI, Slovak Academy of Sciences, Bratislava, Slovakia) Nathan Wiebe (QuArC, Microsoft Research, Redmond, WA, USA) [arxiv:1403.6545]
More informationNumerical Simulation of Spin Dynamics
Numerical Simulation of Spin Dynamics Marie Kubinova MATH 789R: Advanced Numerical Linear Algebra Methods with Applications November 18, 2014 Introduction Discretization in time Computing the subpropagators
More informationEigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator.
PHYS208 spring 2008 Eigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator. 07.02.2008 Adapted from the text Light - Atom Interaction PHYS261 autumn 2007 Go to list of topics
More information