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 the University of Buenos Aires
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 nonclassical 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 (tentative) Lecture 1 (April 9 th ): Introduction to Cavity QED with Rydberg atoms interacting with microwave fields stored in a high Q superconducting resonator. Lecture 2 (April 11 th ): Review of measurement theory illustrated by the description of quantum non-demolition (QND) photon counting in Cavity QED Lecture 3 (April 13 th ): Estimation and reconstruction of quantum states in Cavity QED experiments; the cases of Fock and Schrödinger cat states. Lecture 4 (April 16 th ):Quantum feedback experiments in Cavity QED preparing and protecting against decoherence non-classical states of a radiation field. Lecture 5 (April 18 th ): An introduction to Circuit-QED describing Josephson junctions as qubits and radiofrequency resonators as quantum oscillators. Lecture 6 (April 20 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 (qubit) Field mode (harmonic oscillator)
Description of a qubit (or spin 1/2) 0>! " P 1> 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! 2 e#i" /2 0 + sin! 2 ei" /2 1 A statistical mixture is represented by a density operator:! = " qubit " qubit (pure state) (i) (i)! = # p i " qubit " qubit i # ( p i = 1) (mixture) i which can be expanded on Pauli matrices: "! x = 0 1 % " $ ' ;! y = 0 (i % " $ ' ;! z = 1 0 % $ ' # 1 0& # i 0 & # 0 (1&! i 2 = I (i = x, y, z)! P =1: pure state! = 1 ( 2 I + P.! "! ) ;! P # 1! hermitian, with unit trace and! 0 eigenvalues ;! 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: P i = Tr!" i = " i (i = x, y,z) ;! = 1 $ 2 I + " ' & # i " 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: P i = p i+! p i! ; " = 1 % 2 I + p! p ( ' $ ( i+ i! )# i * & ) A useful formula: overlap of two qubit states described by their Bloch vectors: S 12 = Tr! 1! 2 { } = 1 2 1+! P 1.! P 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. i " # i $ % ( Tr! i! j = 2" ij )
Qubit rotation induced by microwave pulse Coupling of atomic qubit with a classical field: H = H at + H int (t) ; H at =!! eg 2 " z ; H int (t) = #D "#.E "# mw(t) Microwave electric field linearly polarized with controlled phase # 0 : [ ] E mw (t) = E 0 cos(! mw t " # 0 ) = E 0 2 expi(! t " # ) + exp" i(! t " # ) mw O mw O Qubit electric-dipole operator component along field direction is off-diagonal and real in the qubit basis (without loss of generality): Hence the Hamiltonian: D along field = d! x (d real) H =!! eg 2 " #! $ mw z 2 " x [ expi(! mw t # % O ) + exp# i(! mw t # % O )] ; $ mw = d.e 0 /! and in frame rotating at frequency $ mw around Oz: i! d! dt $ = H! ; "! = exp i " z# mw ' & t)! * i! d "! % 2 ( dt ( ) "H =! # eg +# mw 2 = " H "! ; " z +!, " z # mw t mw 2 ei 2 " x e +i " z# mw t 2. / e i(# mwt +- ) O + e +i(# mwt +- ) O 0 1
Bloch vector rotation (ctn d) ( )!H = "! "! eg mw # z " " $ # z! mw t mw e i! z" mwt 2! x e #i! z" mw 2 2 ei 2 # x e "i # z! mw t 2 & ' e i(! mwt "% ) O + e "i(! mwt "% ) O ( ) t $ 2 =! x cos" mw t #! y sin" mw t =! x + i! y ' $ & )e i" mwt +! x # i! y ' & )e #i" mwt % 2 ( % 2 ( The rotating wave approximation (rwa) neglects terms evolving at frequency ±2$ mw : (!H rwa = "! "! eg mw ) # z " " $ mw 2 2 A method to prepare arbitrary pure qubit state from state e> or g>. By applying a convenient pulse prior to detection in qubit basis, one can also detect qubit state along arbitrary direction on Bloch sphere. # + e i% 0 +# " e"i% 0 ( ) ; # ± = (# x ± i# y ) / 2 The rwa hamiltonian is t-independant. At resonance ($ eg =$ mw ), it simplifies as:!h rwa =!" " mw 2 # + e i$ 0 +#! e!i$ 0 ( ) =!" " mw 2 (# x cos$ 0!# y sin$ 0 ) 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:!U(! ) = exp("i!h rwa! / ") = exp("i # mw! 2 $ u ) $ u = $ x cos% 0 "$ y sin% 0 Rotation angle control by pulse length Rotation axis control by pulse phase
!"#$%&# Description of harmonic oscillator (phonons or photons) Gaussian 0 x Particle in a parabolic potential or field mode in a cavity Basic formulae with photon annihilation and creation operators n Gaussian x Hermite Pol. Mechanical or electromagnetic oscillation. Coupling qubits to oscillators is an important ingredient in quantum information.!a,a " # $ = I ; a n = n n %1 ; a n = n +1 n +1 ; n = a n n! 0 N = a a ; H field =!&N ; e in&t ae %in&t = e %i&t a Displacement operator : D(') = e 'a %' * a Coherent state : ' = D(') 0 = e % ' 2 /2 ( n ' n n! n p (E 2 ) x (E 1 ) Phase space (with conjugate coordinates x,p or E 1,E 2 ) P(n) X = a + a 2 ; P = a! a 2i [ X,P] = i 2 I n Photon (or phonon) number 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: ( )( a + a ) H Q =!!a a +V =!!a a + " e i!t + e #i!t (" : constant proportional to current amplitude in antenna) Interaction representation:! field = exp(i"a at)! field # i" d! field dt = H! Q! field with HQ! = $ ( e i"t + e %i"t )e i"a at (a + a )e %i"a at Rotating wave approximation (keep only time independent terms):!h Q (rwa) =! a + a ( ) Field evolution in cavity starting from vacuum at t=0: (! field (t) = exp *"i # $ % " a + & a ' ) t + - 0 = exp.a ". * a, ( ) 0 = e "..* /2 e.a e ".*a 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: $! field (t) = e " # 2 /2 # n n! n Coupling field mode to classical source generates coherent state whose amplitude increases linearly with time. n
n Coherent state (ctn d) Produced by coupling the cavity to a source (classical oscillating current) n=0 The larger n, the more classical the field is!n = " = n A Poissonian superposition of photon number states:! = C n n, C n = e "! 2 /2! n #, n n! n =! 2, P(n) = C n 2 = e "n n n n! Complex plane representation Im(# ) Amplitude # " = " 0 $ % c t Re(#) Uncertainty circle («!width!» of Wigner functionsee next page)
Representations of a field state Density operator for a field mode: (i) (i)! = " field " field (pure state) ;! = # p i " field " field i # ( p i = 1) (mixture) 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: W (x, p) = 1 # 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. i Gaussian function ' Gaussian function translated Positive and negative rings Ground state Coherent state Fock state W<0 : non-classical
I-B Coupling a qubit to a quantized field mode: the Jaynes-Cummings Hamiltonian Semiclassical Fully quantum Matter Classical current Quantum (qubit) Quantum (qubit) Radiation Quantum field mode Classical field Quantum field mode Translation in phase space and Coherent states Rotation in Bloch sphere and arbitrary qubit states Cavity QED
Qubit-quantum field mode coupling Coupling qubit dipole to quantum electric field operator: Rotating wave approximation: V =! D.! E! Q (0) ; E Q (0) = ie 0 ( a! a ) V =!ide 0 " x (a! a ) =!ide 0 (" + +"! )(a! a ) Absorption and emission of photons while qubit jumps between its energy states. with:! + =! x + i! y 2 V rwa =!i!" 0 2 (# + a!#! a ) ; " 0 = 2d.E 0 /! " = 0 1 % $ ' = e g ;! ( =! ( i! " x y = 0 0 % $ ' = g # 0 0& 2 # 1 0& e Vacuum Rabi frequency in cavity mode of volume V: 0 E Q 2 (0) 0 = E 0 2 0 aa +a a 0 = E 0 2! 0 V E 0 2 =!" 2 # E 0 =! 0 = 2d 2 " # 0!V!" 2! 0 V Requires large dipole d and small cavity volume V
Qubit-oscillateur coupling: the Jaynes-Cummings Hamiltonian " H =!! z qb 2 +!! oha a # i!$ % & 2 " +a #" # a ' ( g,n +1 e,n! n +1 ±,n g, 3 e,2! 3 ±,2 &! = " qb #" oh g,2 g,1 e,1 e,0! 2! ±,1 ±,0 Light shift (()0) e,n g,n +1 +,n!,n 0! n +1 e,n g,n +1 ( E ±,n = ( n +1/ 2)!! oh ±! 2 "2 + # 2 n +1 Anticrossing of dressed qubit ( ) g,0 uncoupled states ±,n = 1 2 e,n! "! cos # n +1t 2 g,n +1! "! $sin # n +1t 2 g,0 Coupled states (dressed qubit) at resonance ((=0) ( e,n ± i g,n +1 ) e,n + sin # n +1t 2 e,n + cos # n +1t 2 g,n +1 g,n +1
Non-Resonant coupling: light shifts in CQED g,n +1 e,n +,n!,n 0! n +1 e,n g,n +1 E ±,n = ( n +1 / 2)!! oh ±! 2 "2 + # 2 ( n +1) ( Second order perturbation theory: % E ±,n! ( n +1 / 2)!" C ±! # 2 + $2 (n +1) ( ' * & 4# ) Energy shift due to off-resonant coupling (distance between energy level and asymptot): E +,n! E e,n "! #2 (n +1) +... ; 4$ E!,n! E g,n+1 "!! #2 (n +1) +... 4$ Vacuum shift (Lamb-shift): Anticrossing of dressed qubit! g,0 = 0 ;! e,0 = E +,0 " E e,0 #! $2 4% Light shift induced on eg transition by n-photons: E +,n! E!,n!1 =!" eg +! #2 2$ n +... % &(" eg ) = #2 2$ n ; ' 0 = #2 t 2$ # 0 =Phase shift per photon accumulated during time t
Measuring qubit phase shift in CQED: the Ramsey interferometer R 1 C R 2 z z Qubit initially in e x R 1 pulse rotates Bloch vector by */2 around Ox Qubit phase shift # c during C crossing amounts to Bloch vector rotation around Oz $ exp!i " 4 # ' $ & x ) exp!i " C % ( 2 # ' & z ) % ( R 2 pulse realizes */2 rotation around Ou whose direction depends on R 2 -R 1 relative phase # r $ exp!i " 4 # ' & u ) = % ( $ exp!i " 4 # ' & +, x cos* r!# y sin* r -. ) % (
A useful formula: Ramsey interferometer (ctn d) exp(!i"# u ) = cos" I! isin" # u Hence the rotation induced by Ramsey interferometer: $ R = exp!i " 4 # ' $ & u )exp!i * C % ( 2 # ' & z % ( )exp $ &!i " % 4 # x ( for any Pauli operator ) ' ) ( Rotation induced by R 1 Rotation Cavity phase-shift induced by R 2 = 1 $ 1!ie i* r ' $ & ) e!i* C /2 0 ' $ & ) 1!i ' 2!ie!i* r % 1 ( 0 e i* C /2 & ) % (%!i 1 ( $ $ e i* r /2 sin * c + * r ' $ & ) e i* r /2 cos * c + * r '' & & )) % 2 ( % 2 ( =!i& ) & $ e!i* r /2 cos * c + * & r ' $ ) e!i* r /2 sin * c + * & r ') & )) % % 2 ( % 2 (( and the evolution of e> and g> states: # R e =!ie i" r /2 sin " + " & # c r % ( e! ie!i" r /2 cos " + " & c r % ( g $ 2 ' $ 2 ' # R g =!ie i" r /2 cos " + " & # c r % ( e! ie!i" r /2 sin " + " & c r % ( g $ 2 ' $ 2 ' #" P e!g = cos 2 c + " r & % ( $ 2 ' Ramsey fringes detected by sweeping # r. Fringe-phase used to measure cavity shift # C.
General scheme of cavity QED experiments Microwave pulses preparing (left) the atomic qubit and rotating it (right) in the detection direction Atom preparation Rydberg atoms High-Q-cavity storing the trapped field Detector of states e and g ENS experiments: Rydberg atoms in states e and g behave as qubits. They are prepared in B in state e and cross one at a time the high-q cavity C where they are coupled to a field mode. The atom-field system evolution is ruled by the Jaynes-Cummings hamiltonian. A microwave pulse applied in R 1 prepares each atom in a superposition of e and g. After C, a second pulse, applied in R 2, maps the measurement direction of the qubit along the Oz axis of the Bloch sphere, before detection of the qubit by selective field ionization in an electric field (in D). The R 1 -R 2 combination constitutes a Ramsey interferometer. This set-up has been used to entangle atoms, realize quantum gates, count photons non-destructively, reconstruct non classical states of the field and demonstrate quantum feedback procedure (lectures 1 to 4).
Analogy between Ramsey and Mach Zehnder interferometers In Ramsey interferometer, two resonant pulses split and recombine atomic state in Hilbert space. Qubit follows two pathes between R 1 and R 2 along which it undergoes different phase-shifts. By finally detecting qubit in e or g and sweeping phase, one get fringes which informs about differential phase shifts between the states. In Mach-Zehnder, the splitting and recombination occurs on particle trajectories. Beam splitters replace Ramsey pulses. Fringes inform about differential phase shifts induced on the pathes e # e D e B 1 a M R 1 R 2 b # g D g g B 2 M' P g, P e D 0 2! Detectors in the two outgoing channels detect fringes with opposite phases! r
I-C A special system: circular Rydberg atoms coupled to a superconducting Fabry-Perot cavity
A qubit extremely sensitive to microwaves:the circular Rydberg atom n! db = nh mv = 2"r R # r R = n! mv ; F = mv2 r R = # v = q 2 4"$ 0 r R 2 # r R = q2 4"$ 0 1 mv 2 Rydberg formulae q 2 c 4"$ 0!c n = % c fs n & 1 c 137 n # E = mv2 2 = 1 2n 2 mc2 2 % fs Atom in circular Rydberg state:! electron on giant orbit r R = n 2! fs mc = n2 a 0 (tenth of a micron diameter) Delocalized electron wave e (n=51) Atom in ground state: electron on 10-10 m diametre orbit Very long radiative lifetime T R! n,n"1! n "3 ; P rad = "! n,n"1 T R 2! (! n,n"1 r R ) 2! n "8 # T R! n 5 Electron is localised on orbit by a microwave pulse preparing superposition of two adjacent Rydberg states: e> & e> + g> g (n=50) The localized wave packet packet revolves around nucleus at the transition frequency (51 GHz) between the two states like a clock s hand on a dial. The electric dipole is proportional to the qubit Bloch vector in the equatorial plane of the Bloch sphere. Complex preparation with lasers and radiofrequency
The path towards circular states: an adiabatic process involving 53 photons
Controling the atom-cavity interaction time by selecting the atom s velocity via Doppler effect sensitive optical pumping Rubidium level scheme with transitions implied in the selective depumping and repumping of one velocity class in the F=3 hyperfine state In green, velocity distribution before pumping, in red velocity distribution of atoms pumped in F=3, before they are excited in circular Rydberg state
Detecting circular Rydberg states by selective field ionization n=51 n=50 Adjusting the ionizing field allows us to discriminate two adjacent circular states. The global detection efficiency can be > 80%.
The ENS photon box (latest version) In its latest version, the cavity has a damping time in the 100 millisecond range. Atoms cross it one at a time. Cavity half-mounted atom...and fully-mounted
w~+=6mm Orders of magnitude L=9+/2=2,7 cm V ~ *w 2 L/4 ~ 4+ 3 ~ 700 mm 3! "! 2d 2 # 0 "V"! q 2 $# 0 "c % r 2 a % & V! q 2 4$# 0 "c % r 2 a & = ' 1 * ), 2 ( 137 + 1/2 r a &! 10-6!! 10 "6 # = 2$ % 50kHz Parameter determined by geometric arguments Number of vacuum Rabi flops during cavity damping time (best cavity): N RF =!T C 2"! 5000 Atom-cavity interaction time (depends on atom velocity): t int! w! 10 to 50µs v at Number of vacuum Rabi flops during atom-cavity interaction time:!t int / 2"! 1 to 3 Strong coupling in CQED 1! < t int! T c 3.10 "6 s 3.10 "5 s 10 "1 s Many atoms cross one by one during T c
Cavity Quantum Electrodynamics: a stage to witness the interaction between light and matter at One atom interacts with one (or a few) photon(s) in a box the most fundamental level A sequence of atoms crosses the cavity,couples with its field and carries away information about the trapped light Photons bouncing on mirrors pass many many times on the atom: the cavity enhances tremendously the light-matter coupling 6 cm The best mirrors in the world: more than one billion bounces and a folded journey of 40.000km (the earth circumference) for the light! Photons are trapped for more than a tenth of a second!
I-D Entanglement and quantum gate experiments in Cavity QED
e g e Resonant Rabi flopping 0 photon R 1 R 2 0 photon R 1 R 2 e + 01 photon R 1 R 2 g cos (!t / 2) e,0 + sin (!t / 2) g,1 Spontaneous emission and absorption involving atom-field entanglement When n photons are present, the oscillation occurs faster (stimulated emission): cos (! n + 1t / 2 ) e,n + sin (! n + 1t / 2) g,n + 1 Simple dynamics of a two-level system ( e,n>, g,n+1>)
Rabi flopping in vacuum or in small coherent field: a direct test of field quantization p(n) "! n + 1t % n n P e (t) = ( p(n)cos 2 $ # 2 ' ; p(n) = e )n & n! n 0 1 2 3 n n = 0 (n th = 0.06) n = 0.40 (±0.02) n = 0.85 (±0.04) n = 1.77 (±0.15) P e (t) signal Fourier transform Inferred p(n) Brune et al, PRL,76,1800,1996.
Useful Rabi pulses (quantum «knitting») Initial state e,0> e,0>, e,0> + g,1> * / 2 pulse creates atom-cavity entanglement P e (t) 0.8 51.1 GHz 51 (level e) 0.6 50 (level g) 0.4 Brune et al, PRL 76, 1800 (96) 0.2 0.0 # e,0! cos "t & # $ % 2 ' ( e,0 + sin "t & $ % 2 ' ( g,1 µ time (? s) 0 30 60 90 Microscopic entanglement
Useful Rabi pulses (quantum knitting) e,0>, g,1> g,1>, e,0> g,0>, g,0> ( e> + g>) 0>, g> ( 1> + 0>) * pulse copying atomʼs state on field and back P e (t) 0.8 51.1 GHz 51 (level e) 0.6 50 (level g) 0.4 Brune et al, PRL 76, 1800 (96) 0.2 0.0 # e,0! cos "t & # $ % 2 ' ( e,0 + sin "t & $ % 2 ' ( g,1 µ time (? s) 0 30 60 90
Useful Rabi pulses (quantum knitting) e,0>, - e,0> g,1>, - g,1> g,0>, g,0> 2* pulse : atomic state changes sign conditioned to photon number: quantum gate P e (t) 0.8 51.1 GHz 51 (level e) 0.6 50 (level g) 0.4 Brune et al, PRL 76, 1800 (96) 0.2 0.0 # e,0! cos "t & # $ % 2 ' ( e,0 + sin "t & $ % 2 ' ( g,1 µ time (? s) 0 30 60 90
Atom pair entangled by photon exchange V(t) g 2 e 1 + Electric field F(t) used to tune atoms 1 and 2 in resonance with C for times t corresponding to ' / 2 or ' Rabi pulses + Hagley et al, P.R.L. 79,1 (1997)
Conclusion of first lecture We have learned how to describe a qubit and a field mode by their Bloch vector and Wigner function respectively, and how one can manipulate these systems by qubit rotations and oscillator state translations in phase space. The coupling of these two quantum systems is described by the Jaynes- Cummings Hamiltonian. At resonance, the combined qubit-oscillator undergoes a Rabi oscillation corresponding to the reversible emission and absorption of a photon by the qubit. Out of resonance, the qubit energy states undergo a light shift proportional to the photon number, resulting in a phase shift of the qubit coherence, which can be measured by Ramsey interferometry. Circular Rydberg atoms interacting with a microwave superconducting cavity realize ideally the Jaynes-Cummings model. The atomic and field relaxation rates are very slow, making it possible to observe coherent atom field coupling effects which occur faster than decoherence. We have analyzed how this system can be manipulated to produce atom-field and atom-atom entanglement, to realize elementary steps of quantum information processing.