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Cavity QED with Rydberg Atoms Serge Haroche, Collège De France & Ecole Normale Supérieure, Paris

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Cavity QED with Rydberg Atoms Serge Haroche, Collège De France & Ecole Normale Supérieure, Paris Cavity QED with Rydberg Atoms Serge Haroche, Collège de France & Ecole Normale Supérieure, Paris A three lecture course Goal of lectures Manipulating states of simple quantum systems has become an important field in quantum optics and in mesoscopic physics, in the context of quantum information science. Various methods for state preparation, reconstruction and control have been recently demonstrated or proposed. Two-level systems (qubits) and quantum harmonic oscillators play an important role in this physics. The qubits are information carriers and the oscillators act as memories or quantum bus linking the qubits together. Coupling qubits to oscillators is the domain of Cavity Quantum Electrodynamics (CQED) and Circuit Quantum Electrodynamics (Circuit- QED). In microwave CQED, the qubits are Rydberg atoms and the oscillator is a mode of a high Q cavity while in Circuit QED, Josephson junctions act as artificial atoms playing the role of qubits and the oscillator is a mode of an LC radiofrequency resonator. The goal of this course is to present the field of CQED, whose concepts are readily extended to circuit QED, and to analyze various ways to measure, manipulate and reconstruct quantum fields in cavities. These lectures will give us an opportunity to review basic concepts of measurement theory in quantum physics. Outline of lectures (tentative) •Lecture 1 (July 17th): Introduction to Cavity QED with Rydberg atoms interacting with microwave fields in a high Q superconducting resonator. • Lecture 2 (July 19th): Review of measurement theory illustrated by the description of quantum non-demolition photon counting in Cavity QED. • Lecture 3 (July 20th): State reconstruction and quantum feedback experiments in Cavity QED. Link with Circuit QED. I-A The basic ingredients of Cavity QED: qubits and oscillators Two-level system Field mode (qubit) (harmonic oscillator) Description of a qubit (or spin 1/2) |0> Any pure state of a qubit (0/1) is parametrized by two polar angles !," and is represented by a point on the Bloch sphere : ! ! ! !," = cos e#i" /2 0 + sin ei" /2 1 2 2 " A statistical mixture is represented by a density operator: ! = " " (pure state) P qubit qubit (i) (i) ! = # pi "qubit "qubit (# pi = 1) (mixture) |1> i i which can be expanded on Pauli matrices with coeffcients defining a Bloch vector: "0 1% "0 (i% "1 0 % 1 ! ! ! ! = ; ! = ; ! = ! = I + P." ; P # 1 x $ ' y $ ' z $ ' 2 ( ) #1 0& # i 0 & #0 (1& 2 ! i = I (i = x, y,z) ! hermitian, with unit trace and ! 0 eigenvalues ! ! ! P =1: pure state ; P <1: mixture ; P >1:non physical (negative eigenvalue) A qubit quantum state (pure or mixture) is fully determined by its Bloch vector Description of a qubit (cont’d) The Bloch vector components are the expectation values of the Pauli operators: 1 $ ' Tr! i! j = 2"ij Pi = Tr!" i = " i (i = x, y,z) ; ! = &I + # " i " i ) ( ) 2 % i ( The qubit state is determined by performing averages on an ensemble of realizations: the concept of quantum state is statistical. Calling pi+ et pi- the probabilities to find the qubit in the eigenstates of "i with eigenvalues ±1, we have also: 1 % ( Pi = pi+ ! pi! ; " = 'I + $( pi+ ! pi! )# i * 2 & i ) Manipulation and measurement of qubits: Qubit rotations are realized by applying resonant pulses whose frequency, phase and durations are controlled. In general, it is easy to measure the qubit in its energy basis ("z component). To measure an arbitrary component, one starts by performing a rotation which maps its Bloch vector along 0z, and then one measures the energy. Qubit rotation induced by microwave pulse Coupling of atomic qubit with a classical field: !eg "# "# H = H + H (t) ; H = ! " ; H (t) = #D.E mw (t) at int at 2 z int Microwave electric field linearly polarized with controlled phase #0: E0 Emw (t) = E0 cos(!mwt " #0 ) = [expi(!mwt " #O )+ exp" i(!mwt " #O )] 2 Qubit electric-dipole operator component along field direction is off-diagonal and real in the qubit basis (without loss of generality): Dalong field = d! x (d real) Hence the Hamiltonian: !eg $mw H = ! " z # ! " x [expi(!mwt # %O )+ exp# i(!mwt # %O )] ; $mw = d.E0 / ! 2 2 and in frame rotating at frequency $mw around Oz: d ! $ " # ' d !" i! = H ! ; !" = exp&i z mw t) ! * i! = H" !" ; dt % 2 ( dt " z#mwt " z#mwt #eg +#mw , i +i " ( ) mw 2 2 . i(#mwt +-O ) +i(#mwt +-O ) 0 H = ! " z + ! e " xe /e + e 1 2 2 Bloch vector rotation (ctn’d) # z!mwt # z!mwt !eg "!mw $ i "i ! ( ) mw 2 2 & i(!mwt "%O ) "i(!mwt "%O ) ( H = " # z " " e # xe 'e + e ) ! " t ! " t 2 2 i z mw #i z mw $! + i! ' $! # i! ' 2 2 x y i"mwt x y #i"mwt e ! xe = ! x cos"mwt # ! y sin"mwt = & )e + & )e % 2 ( % 2 ( The rotating wave approximation (rwa) neglects terms evolving at frequency ±2$mw: !eg "!mw $ ! ( ) mw i%0 "i%0 H rwa = " # z " " (# +e +# "e ) ; # ± = (# x ± i# y ) / 2 2 2 The rwa hamiltonian is t-independant. At resonance ($eg=$mw), it simplifies as: " " ! mw i$0 !i$0 mw H rwa = !" (# +e +# !e ) = !" (# x cos$0 !# y sin$0 ) 2 2 A resonant mw pulse of length % and phase #0 rotates Bloch vector by angle &mw% around direction Ou in Bloch sphere equatorial plane making angle -#0 with Ox: # ! Rotation angle U! (! ) = exp("iH! ! / ") = exp("i mw $ ) rwa 2 u control by pulse length $ u = $ x cos%0 "$ y sin%0 Rotation A method to prepare arbitrary pure qubit state from axis state |e> or |g>. By applying a convenient pulse prior to control by pulse detection in qubit basis, one can also detect qubit phase state along arbitrary direction on Bloch sphere. Description of harmonic oscillator a + a† a ! a† X = ; P = (phonons or photons) 2 2i p (E ) i 2 [X,P] = I Mechanical or 2 electromagnetic n oscillation. x (E1) Coupling qubits to Gaussian x Hermite oscillators is an !"#$%&# Pol. important ingredient in quantum Gaussian 0 information. x Phase space (with conjugate Particle in a parabolic potential or coordinates x,p or E ,E ) field mode in a cavity 1 2 Basic formulae with photon annihilation and creation operators †n P(n) † † a "!a,a $# = I ; a n = n n %1 ; a n = n +1 n +1 ; n = 0 n! † iN&t %iN&t %i&t N = a a ; H field = !&N ; e ae = e a † * Displacement operator : D(') = e'a %' a n n % ' 2 ' Coherent state : ' = D(') 0 = e /2 n ( Photon (or phonon) number n n! distribution in a coherent state: Poisson law Coherent state Coupling of cavity mode with a small resonant classical antenna located at r=0: ! ! V J(t).A(0) ; J(t) cos t ; A(0) a a† = ! " " " + Hence, the hamiltonian for the quantum field mode fed by the classical source: † † i!t #i!t † HQ = !!a a +V = !!a a + "(e + e )(a + a ) (" : constant proportional to current amplitude in antenna) Interaction representation: d !! † † ! † field ! ! ! i"t %i"t i"a at † %i"a at ! field = exp(i"a at) ! field # i" = HQ ! field with HQ = $(e + e )e (a + a )e dt Rotating wave approximation (keep only time independent terms): H! (rwa) = ! a + a† Q ( ) Field evolution in cavity starting from vacuum at t=0: ( + * † * ! # $ †& † * ".. /2 .a ". a ! field (t) = exp*"i %a + a 't- 0 = exp(.a ". a) 0 = e e e 0 (. = "i#t / ") ) " , We have used Glauber formula to split the exponential of the sum in last expression. Expanding exp('a†) in power series, we get the field in Fock state basis: 2 n ! " # /2 # ! field (t) = e $ n n n! Coupling field mode to classical source generates coherent state whose amplitude increases linearly with time. Coherent state (ctn’d) Produced by coupling the A Poissonian superposition of cavity to a photon number states: source (classical n 2 ! oscillating ! = C n , C = e" ! /2 , current) # n n n n! n 2 2 "n n n n = ! , P(n) = C = e n n! = " n = n Complex plane representation ! Im(# ) Uncertainty circle («!width!» of | The larger n, |# Wigner function- the more see next page) classical the " = " 0$ % c t field is Amplitude (relative fluctuations Re(#) n=0 become small) Representations of a field state Density operator for a field mode: (i) (i) ! = " field " field (pure state) ; ! = # pi " field " field (# pi = 1) (mixture) i i Matrix elements of ! can be discrete (!nn’ in Fock state basis) or continuous † (!xx’ in quadrature basis where |x> are the eigenstates of a+a ). Going from one representation to the other is easy knowing the amplitudes <x|n> expressing the oscillator energy eigenstates in the x basis (Hermite polynomial multiplied by gaussian functions). Representation in phase space: the Wigner function: 1 W (x, p) = du exp("2ipu) x + u / 2 $ x " u / 2 ! # W is a real distribution in (x,p) space (equivalently, in complex plane), whose knowledge is equivalent to that of !. The «shadow» of W on p plane yields the x- distribution in the state. Gaussian Gaussian Positive and function ' function negative translated rings W<0 : Ground state Coherent state Fock state non-classical I-B Coupling a qubit to a quantized field mode: the Jaynes-Cummings Hamiltonian Matter Radiation Translation in Classical current Quantum field mode phase space and Semi- Coherent states classical Rotation in Bloch Quantum (qubit) Classical field sphere and arbitrary qubit states Fully Quantum (qubit) Quantum field mode Cavity quantum QED Qubit-quantum field mode coupling Coupling qubit dipole to quantum electric field operator: Absorption and emission ! ! † of photons while qubit V = !D. EQ (0) ; EQ (0) = iE0 (a ! a ) jumps between its energy states.
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