Collge de France abroad Lectures Quantum information with real or - PowerPoint PPT Presentation

Collège de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities Serge Haroche, Collège de France & Ecole Normale Supérieure, Paris www.college-de-france.fr A series of six lectures at NUS in the Course « ! QT5201E Special Topics in quantum information: advanced quantum optics ! »

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 these lectures is to analyze various ways to synthesise non- classical states of qubits or quantum oscillators, to reconstruct these states and to protect them against decoherence. Experiments demonstrating these procedures will be described, with examples from both CQED and Circuit-QED physics. These lectures will give us an opportunity to review basic concepts of measurement theory in quantum physics and their links with classical estimation theory.

Outline of lectures • Lecture 1 (February 6 th ): Introduction to Cavity QED with Rydberg atoms interacting with microwave fields stored in a high Q superconducting resonator. • Lecture 2 (February 8 th ): Review of measurement theory illustrated by the description of quantum non-demolition (QND) photon counting in Cavity QED • Lecture 3 (February 10 th ): Estimation and reconstruction of quantum states in Cavity QED experiments; the cases of Fock and Schrödinger cat stats. • Lecture 4 (February 13 th ): Quantum feedback experiments in Cavity QED preparing and protecting against decoherence non-classical states of a radiation field. • Lecture 5 (February 15 th ): An introduction to Circuit-QED describing Josephson junctions as qubits and radiofrequency resonators as quantum oscillators. • Lecture 6 (February 17 th ): Description of Circuit-QED experiments synthesizing arbitrary states of a field oscillator.

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 ! e i " /2 1 ! 2 2 A statistical mixture is represented " by a density operator: P ! = " qubit " qubit ( pure state ) ! = p i = 1) # ( i ) ( i ) # p i " qubit " qubit ( ( mixture ) |1> i i which can be expanded on Pauli matrices: ! ! P . ! ( ) ! = 1 " % " % " % I + ! x = 0 1 ' ; ! y = 0 ( i ' ; ! z = 1 0 " ; P # 1 $ $ $ ' 2 ( 1 1 0 i 0 0 # & # & # & 2 = I ( i = x , y , z ) ! hermitian, with unit trace and " 0 eigenvalues ! i ! ! ! 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: ( ) $ ' Tr ! i ! j = 2 " ij ! = 1 i = Tr !" i = " i ( i = x , y , z ) I + # P ; " 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 p i+ et p i- the probabilities to find the qubit in the eigenstates of " i with eigenvalues ±1, we have also: % ( " = 1 ( ) # i i = p i + ! p i ! I + $ p i + ! p i ! P ; ' * 2 & ) i A useful formula: overlap of two qubit states described by their Bloch vectors: ! ! } = 1 { S 12 = Tr ! 1 ! 2 1 + " $ P 1 . P # % 2 2 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: "# " # H at = ! ! eg H = H at + H int ( t ) H int ( t ) = # D ; " z ; . E mw ( t ) 2 Microwave electric field linearly polarized with controlled phase # 0 : E mw ( t ) = E 0 cos( ! mw t " # 0 ) = E 0 [ ] exp i ( ! mw t " # O ) + exp " i ( ! mw t " # O ) 2 Qubit electric-dipole operator component along field direction is off-diagonal and real in the qubit basis (without loss of generality): D along field = d ! x ( d real ) Hence the Hamiltonian: H = ! ! eg " z # ! $ mw [ ] " x exp i ( ! mw t # % O ) + exp # i ( ! mw t # % O ) $ mw = d . E 0 / ! ; 2 2 and in frame rotating at frequency $ mw around Oz: d " ! i ! d ! $ ' ! = exp i " z # mw " = " H " = H ! i ! ; t ) ! * ! ; & dt % 2 ( dt ( ) i " z # mw t + i " z # mw t H = ! # eg + # mw " z + ! , mw " 2 " x e e i ( # mw t + - O ) + e + i ( # mw t + - O ) . 0 e 2 / 1 2 2

Bloch vector rotation (ctn’d) ( ) i # z ! mw t " i # z ! mw t H = " ! eg " ! mw # z " " $ mw ! 2 # x e e i ( ! mw t " % O ) + e " i ( ! mw t " % O ) & ( e 2 ' ) 2 2 ! y sin " mw t = ! x + i ! y i ! z " mw t # i ! z " mw t $ ' $ ' ) e i " mw t + ! x # i ! y 2 ! x e = ! x cos " mw t ) e # i " mw t e 2 # & & 2 2 % ( % ( The rotating wave approximation (rwa) neglects terms evolving at frequency ± 2 $ mw : ( ) H rwa = " ! eg " ! mw ( ) # z " " $ mw # + e i % 0 + # " e " i % 0 ! # ± = ( # x ± i # y ) / 2 ; 2 2 The rwa hamiltonian is t-independant. At resonance ( $ eg = $ mw ), it simplifies as: ( ) = ! " " mw ( ) H rwa = ! " " mw # + e i $ 0 + # ! e ! i $ 0 ! # 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 H rwa ! / " ) = exp( " i # mw ! U ( ! ) = exp( " i ! ! $ u ) control by pulse 2 length $ u = $ x cos % 0 " $ y sin % 0 Rotation A method to prepare arbitrary pure qubit state from axis control by state |e> or |g>. By applying a convenient pulse prior to pulse detection in qubit basis, one can also detect qubit phase state along arbitrary direction on Bloch sphere.

Description of harmonic oscillator (phonons or photons) X = a + a † P = a ! a † ; 2 2 i p (E 2 ) ] = i [ X , P I Mechanical or 2 electromagnetic oscillation. x (E 1 ) n !"#$%&# Coupling qubits to Gaussian x oscillators is an Hermite important ingredient Pol. in quantum Gaussian 0 information. x Phase space (with conjugate Particle in a parabolic potential or coordinates x,p or E 1 ,E 2 ) field mode in a cavity Basic formulae with photon annihilation and creation operators P(n) n = a † n n n % 1 ; a † n = $ = I a n = n + 1 n + 1 ! # a , a † ; ; 0 " n ! e iN & t ae % iN & t = e % i & t a N = a † a H field = ! & N ; ; n Displacement operator : D ( ' ) = e ' a † % ' * a ' n % ' 2 /2 Coherent state : ' = D ( ' ) 0 = e ( 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 = ! A (0) " a + a † J ( t ) " cos " t J ( t ). A (0) ; ; Hence, the hamiltonian for the quantum field mode fed by the classical source: H Q = ! ! a † a + V = ! ! a † a + " e i ! t + e # i ! t ( ) a + a † ( ) ( " : constant proportional to current amplitude in antenna ) Interaction representation: d ! ! field H Q = $ e i " t + e % i " t ( ) e i " a † at ( a + a † ) e % i " a † at = ! ! ! H Q ! ! field = exp( i " a † at ) ! field # i " ! field with dt Rotating wave approximation (keep only time independent terms): ( ) ! H Q ( rwa ) = ! a + a † Field evolution in cavity starting from vacuum at t=0: ( + ( ) 0 = e " .. * /2 e . a † e " . * a 0 ! field ( t ) = exp " i # - 0 = exp . a † " . * a ! a + a † ( . = " i # t / " ) $ & ' 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: # n " # 2 /2 ! ! field ( t ) = e $ n n ! n Coupling field mode to classical source generates coherent state whose amplitude increases linearly with time.