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Three-Level Atom Micromaser Analysis Using Quantum Trajectory Method Thesis Propsal by Tareq Ahmed Mokhiemer Electronics Research Institute • An overview of Micromaser ( Single atom maser) • Quantization of electromagnetic field • Interaction of an atom with a quantized field • Quantum Trajectory analysis • Three level atom • Research proposal An overview of Micromaser •a collimated beam of Rubidium atoms is injected inside a high quality microwave cavity. •The cavity dimensions are set to select only one radiation mode that is resonant with the maser transition. •The system can be used to study the interaction between a single two level atom and a photon of the electromagnetic field inside the cavity. The atom field interaction is probed by observing the population in the upper and lower maser levels after the atoms have left the cavity. Typical parameters: 10 Quality of cavity ~ 10 ِAverage photon lifetime ~ 0.2 s Cavity Temperature:~0.3 K ِAverage number of thermal photons: 0.1 Rydeberg atoms : 63 p 1/2 to 61d 3/2. Corresponds to a transition frequency of 21.5GHz Interaction time 35s The importance of the micromaser • A basic prototype to study the interaction between light and matter (in the level of single atoms and photons). • A tool to generate non-classical states of electromagnetic fields, e.g. Fock states and single photon on demand. • Has applications in quantum information processing, e.g. a source to generate entangled photons. ( The photon gun ) • A platform for testing the postulates of quantum mechanics (e.g. testing the nonlocal features of quantum mechanics) ِQuantization of the electromagnetic field: Beginning with Maxwell’s field equations in free space we find that the vector potential A(r,t) satisfies the homogenous wave equation : 2 1 ( 2 . 2 ) A(r , t ) 0 c t Expanding A(r,t) into a complete set of wave functions uk, (r ) A( r , t ) Al ql (t )u (r ) l l such that 1 VlVl ' 3 ( r )u ( r )dr u l ,l ' l' l From Maxwell’s equations time evoultion of the basis functions is given by : 2 q l (t ) c | kl | ql (t ) 0 .. 2 (A harmonic oscillator equation) The expansion of the electric and magnetic fields is given by: . E A(r , t ) Al q l (t )ul (r ) t B A Al ql (t ) u l (r ) The total energy inside the resonator is given by 1 1 2 3 2 H ( 0E B )dr 2 20 1 Taking Al it follows directly from the columb gauge 0Vl 1. 2 1 2 2 H H l q l l ql ; where l c | kl | 2 2 From the Harmonic Oscillator theory: . (ql , q l ) (qˆl , pˆ l ) [qˆl , pˆ l ' ] i l ,l ' ql qˆl . ˆl ql p Field operators are given by : ˆ E (r , t ) Al pˆ l (t ) ul (r ) ˆ B(r , t ) Al qˆl (t ) u l (r ) 2 1 1 2 ˆ ˆ H H l pˆ l l qˆl 2 2 2 Creation and Annihilati on operators a l l / 2 (ql i pl / l ) a l l / 2 (ql i pl / l ) ˆ ˆ Let nl a l aˆ l Hˆ Hˆ l l (nˆl 1 / 2) l ˆ l E (r , t ) i Al (aˆl aˆl )ul (r ) 2 ˆ 2 B(r , t ) Al (aˆl aˆl ) ul (r ) l Notes : The electromagnetic energy is quantized in terms of photons whose energy equals Field operators don' t commute, so E and B can' t be measured simultaneously upto arbitrary accuracies . The electromagnetic field in vacuum is not identicall y null [ When n 0 H 0 (zero point energy ) ] The physical meaning of the aˆ, aˆ operators ˆa, aˆ are called annihilati on and creation operators respectively where nks is the eigenstate of the number operator nˆ correspond ing to eigenvalue n So what is created or annihilated is actually photons Interaction of an atom with a quantized field: • Consider 2-level atom e g Define the atomic transition operators : ˆ e g ( excitation operator) ˆ g e ( decay operator) ̂ 3 e e g g (atomic transition operator) e g H total H atom H field H atom field interaction If we define the zero energy level to be halfway between e , g 1 1 ˆ H Atom ( Ee E g )ˆ 3 0ˆ 3 2 2 Hˆ Field aˆ aˆ ( neglecting the zero - point energy term) Using the electric dipole approximation : Hˆ dˆ.Eˆ Interaction 1/ 2 ˆ (aˆ aˆ ) sin( kz) E e 0V where e is an arbitrary oriented polarizati on vector dˆ d (ˆ ˆ ) Hˆ Interaction (ˆ ˆ )( aˆ aˆ ) d 0V 1/ 2 sin( kz) / 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ H total 0 3 a a ( )( a a ) 2 The approximate time dependence of the operator products is as follows : i (0 ) t ˆ ˆ a ~ e ˆ aˆ ~ e i (0 ) t i ( 0 ) t ˆ ˆ a ~ e ˆ aˆ ~ e i ( 0 ) t When the time - dependent Schroding er' s equation is integrated we can neglect the terms containing (0 ) in the denominator w.r.t terms containing (0 ) , assuming near resonance condition This is called Rotating Wave Approximation 1 So H total 0ˆ 3 aˆ aˆ (ˆ aˆ ˆ aˆ ) 2 This is called Jayness - Cumming model So what is a micromaser? • The micromaser is a system in which single atoms and single modes of a cavity field interact via the Jaynes-Cummings Hamiltonian Consequences of atom-field interactions: The sponatneous emission rate of the atom inside the cavity can be modified ( Purcell effect ) Spontaneous emission in free space produces monotonic and irreversible decay of upper-level amplitude, whereas here we find the so-called vacuum Rabi oscillation,i.e, reversible spontaneous emission Vacuum Rabi Oscillations: if we call the ground state b , the excited state a then the combined eigen - states of the atom field unperturbe system will be either b n 1 or a n These are called bare - atom states. The total hamiltonia n of the system (J - C Hamiltonia n) is given by : 1 H 0ˆ 3 aˆ aˆ g (ˆ aˆ ˆ aˆ ) 2 writing the total hamiltonia n in matrix notation based on the bare atom states for a given n we obtain 1 1 0 2 g n 1 H n ( n ) 2 0 1 2 2 g n 1 where is the detuning - The eigen states of this Hamiltonia n are given by : 2n cos n a n sin n b n 1 1n sin n a n cos n b n 1 where cos n Rn ( Rn ) 4 g (n 1) 2 2 ; Rn (the quantized general Rabi flopping frequency) 2 4 g 2 (n 1) These are called dressed states. The relation between the bare state coeffecientcan , cbn1 and dresssed state coeffecients c1n , c2 n is given by the transformation matrix c2n (t ) cos n sin n can (t ) c (t ) sin c (t ) cos n n bn 1 2n for the resonant case ( 0) the time evolution of the bare state coeffecirnts is given by : can (t ) cos 2 ( g n 1t ) 2 cbn1 (t ) sin ( g n 1t ) 2 2 which shows that the atom flops between the lower and upper states with the Rabi frequency : g n 1 Rabi Oscillations: Collapse and revival of rabi oscillatio In the case of a coherent field, where the photon distribution is characterized by a poissonian distribution, Rabi Oscillations show a collapse and revival in the amplitude of the oscillations: The semiconductor analog of the micromaser [ A quantum dot embedded in a microcavity ] • One or more InAs quantum dots, surrounded by a GaAs matrix, are embedded in a micropillar optical cavity. • The optical microcavity serves to modify the spontaneous emission properties of the quantum dot through the Purcell effect • The DBR mirrors consist of alternating layers of GaAs and AlAs General notes • Much of the interest in the micromaser lies in determining the properties of the field inside the cavity, but experimentally, this field is not amenable to observation, at the very least because there do not exist good photon detectors at microwave frequencies. • By performing measurements on the atoms as they leave the cavity, information about the cavity field is obtained, since the states of the atoms and cavity field are entangled • For a given sequence of atomic measurements, the evolution of the state of the cavity field conditioned on the outcome of these measurements has been termed a ‘quantum trajectory’ • Dissipative processes are coupled with the micromaser, such as the loss of photons through the low reflecting mirror and the coupling to free space modes ( considered as a large reservoir ) Density operator approach Usually a state of a quantum system is expressed as a weighted sum of the basis functions : But some times a state vector is not enough when what is known about the system is only the probabilit ies of each state,then we use the density operator to describe the state of the system : where Pm is the probabilit y of being in state m Each operator A in quantum mechanics is represented by a matrix whose elements are given by: aij i A j So the density operator is represented by the density matrix Interpretation of matrix elements Diagonal elements = probabilities Off-diagonal elements = "coherences" (provide info. about relative phase) All the information about the system is contained in the density matrix : • The time evolution of the system is obtained from the evolution of the elements of the density matrix ( The Master Equation ) • The expectaion value of any observable (a) is obtained from the density matrix A Tr ( . A) • In many cases the system consists of two parts A, B (i.e, the atom and the field) whose basis functions : , A jB • The combined system has eigenstates denoted by: A jB • And a general wavefunction is given by: AB a , j A jB , j • If the total density operator is pAB then the density operator of A only is called the reduced density operator pA and is given by A a * 'j a j A ' A , ', j • Example: If p is the operator of the system+reservoir then ps , ,the operator of the system, is given by: And the master equation of its development is given by: Notes: • It is assumed here to be local in time, which means that ps(t) depends only on ps at the same time (Markov approximation). • All the system dynamics can be deduced from the master equation. • The number of variables involved in solving the Master equation for the density matrices is ~ N2 One of the main results obtained from the master equation is the photon statistics of the one atom micromaser: •P(n) is the probability of finding n photons (stored) in the micromaser cavity. • Nex is the average number of atoms that enter the cavity during , is the vacuum Rabi flopping frequency, and tint is the atom ( cavity interaction time. Non-Poissonian photon statistics Quantum Trajectory Method • Rather than solving the master equation for the density operator itself, an ensemble of random state functions is constructed. • Each member of the ensemble evolves stochastically, typically by intervals of smooth non-unitary evolution interrupted at random by discontinuous changes of state—‘quantum jumps • Examples of quantum jumps are atoms undergoing photon emissions, photons being lost from the cavities,…etc. • Parameters of the system are obtained by averaging over the whole ensemble. Example on a 2-level atom The Hamiltonian and the Master equation are given by The effective non-Hermitian Hamiltonian is then given by Assume a wavefunction in the form : First we calculate the probability of spontaneous emission between t and t + dt which is given by A random number is chosen to determine whether or not the spontaneous emission occurs Three-level atoms: New Phenemoena and applications • Fano interference: Fano Interference : Two excitation paths to the same ionising state within a continuum leads to a cancellati on in absorption as the two paths destructively interfere. Coherent population trapping Levels 1 , 2 are hyperfine states within the same ground state and both are populated The excitation to level 3 can occur through 2 routes 1 3, 2 3 If both interfere destructively , population would be trapped in the ground state and no excitation will occur ( hence it' s called " Dark state" ) CPT state serves as the basis for many other applications such as stimulated rapid adiabatic passage (STIRAP), lasing without inversion (LWI), electromagnetically induced transparancy (EIT), etc… . Electromagnetically Induced Transparancy (EIT) • It says propagate one laser beam through a medium and it will get absorbed; propagate two laser beams through the same medium and neither will be bsorbed. • Qualitatively,the interactions of the two beams (conventionally called the pump and the probe) with the atoms pumps the latter into the dark state or CPT state. Once the atoms are in the CPT state, no light absorption can take place. (i.e the the medium becomes transparent) Refractive index changes in EIT • Modification of the absorptive properties of a medium will result in a change in the refractive index properties as well, hence the group velocity of the light waves through it will change as well. Lasing Without Inversion • In conventional lasers the medium suffers from stimulated emission as well as stimulated absorption then the medium can never experience laser action without a population inversion. • However if the stimulated absorption is turned off, or significantly decreased then it should be possible to have inversionless laser. stimulated rapid adiabatic passage STIRAP • The technique of STIRAP is used to coherently transfer populations from one state to another (lower) state. • Spontaneous transitions are not allowed to occur, hence coherence of emitted photons is ensured. • Serves as a single photon source for quantum information systems Vacuum-Stimulated Raman Scattering Based on Adiabatic Passage in a High-Finesse Optical Cavity [Similar to Raman Laser ] Research Propsal • Studying the micromaser theory for a three-level atom mainly : 1- Photon statistics 2- Coherent effects 3- Collective effects using the Quantum Trajectory Analysis as a numerical tool. Thank you