DÉPARTEMENT DE PHYSIQUE DE L ÉCOLE NORMALE SUPÉRIEURE LABORATOIRE KASTLER BROSSEL Cavity Quantum Electrodynamics Lecture : entanglement engineering with quantum gates Michel BRUNE Les Houches 003 1
CQED with Rydberg atoms and quantum information e> > 1> g> Atom qubit 0> Field qubit Vacuum Rabi oscillation: strongly couples two qubits achieves quantum gates: Quantum phase gate, CNOT gate allows for step by step preparation of complex entangled states: Les Houches 003
Vacuum Rabi oscillation and quantum gates 1,0 0,8 g,1 e P g (50 circ) 0,6 0,4 0, e,0 Ω 0 =47 khz T Rabi =0µs e ω ge = ω cav g e - 0,0 0 0 40 60 80 100 π interaction time (µs) π Phase gate, QND detection of a single photon Atom-field state exchange EPR pair preparation Les Houches 003 3
outline 1. Cavity qubit as a quantum memory. Quantum phase gate (QPG) and CNOT gate 3. CNOT as QND detection of one photon 4. Step by step preparation of a GHZ triplet Les Houches 003 4
1. Cavity qubit as a quantum memory Les Houches 003 5
Vacuum Rabi oscillation: the field as a quantum memory 1,0 0,8 g,1 e P g (50 circ) 0,6 0,4 0, 0,0 e,0 0 0 40 60 80 100 π interaction time (µs) Atom-field state exchange: Writing: Reading: Ω 0 =47 khz T Rabi =0µs e ω ge = ω cav g ( c e + c g ) 0 g ( ic 1 + c 0 ) e g e g ( c e + c g ) 0 g ( ic 1 + c 0 ) e g e g e - X. Maître et al. PRL 79, 769(1997) EPR pair preparation Les Houches 003 6
Quantum memory: experimental realization writing and reading the field state: e> ω R ω at g> S R g e pulse R 1 pulse R e g Probe atom: Reader Source atom: Writer R 1 : preparation of arbitrary e-g superpositions R : analysis of arbitrary e-g superpositions Exp 1: no pulse R1 : writing and reading field energy Exp : pulse R 1 on, storing a superposition state Les Houches 003 7
Quantum memory: storing energy 1,0 Quantum memory, X. Maître et al. PRL 79, 769(1997) conditional probability Π e/g1 0,8 0,6 0,4 0, 0,0 0 1 3 4 5 T/T r Measurement of the cavity damping time with 1 injected photon! Les Houches 003 8
. Quantum phase gate (QPG) and CNOT gate Les Houches 003 11
Single photon induced Rabi oscillation 1,0 0,8 g,1 e P g (50 circ) 0,6 0,4 0, e,0 Ω 0 =47 khz T Rabi =0µs e ω ge = ω cav g e - 0,0 0 0 40 60 80 100 interaction time (µs) π Phase gate, C-Not gate: e,0 e,0 π Atom-field state exchange EPR pair preparation A. Rauschenbeutel et al., PRL 83, 5166 (1999) Les Houches 003 1
Principle of the Quantum Phase Gate (QPG) uses three level atoms: e : 51c g : 50c i : 49c ω cav ω R e : ancillary level qubit 1: atom qubit 1 = g 0 = i qubit : field qubit 1 : one photon 0 : cavity vacuum Truth table of a π pulse in C: i,0 i,0 i,1 i,1 g,0 g,0 g,1 iπ e g,1 0,0 0,0 0,1 0,1 1, 0 1, 0 e π i 1,1 1,1 π phase shift if control and target =1 Les Houches 003 13
From QPG to CNOT gate Position (cm) D 10 8 6 π Atom π 4 π QPG in C Time 0 Classical pulse = Hadamar transform D Detection Two pulses: Ramsey interferometer on the g-i transition Les Houches 003 14
QPG as a CNOT The field sate controls the phase of Ramsey fringes: π e Ramsey interferometer g g g i i i 0 4 6 8 10 Les Houches 003 15 P i (ω R ) 1,0 0,8 0,6 0,4 0, 0,0 Ramsey fringes: C-Not operation: the photon number controls the final atomic state. N=1 N=0 φ R For a proper choice of the phase of the interferometer: 0, i 0, i 0, g 0, g 1, i 1, g 1, g 1, i
CNOT as QND detection of one photon The field sate controls the phase of Ramsey fringes: π e g g g i i i P i (ω R ) 1,0 0,8 0,6 0,4 0, 0,0 Ramsey fringes: 0 4 6 8 10 For a proper choice of the phase of the interferometer the atom state is perfectly correlated to the photon number 0 or 1 N=1 N=0 φ R Ramsey interferometer 0, g 0, g 1, g 1, i QND detection of one photon Les Houches 003 16
3. CNOT as QND detection of one photon Les Houches 003 17
Input-output characterization of the QND measurement 1 Signal: atom detected in g or i Input: 0 ou 1 photon QND measurement Output: 0 ou 1 photon 3 Les Houches 003 18
QND detection of one photon: experimental timing Position (cm) D D 10 8 6 π π 4 Atom # 1 Atom # 0 Initial field state: N=0, prepared with a bunch of atoms in g. (Cavity Cooling ) atom # 1 prepares one photon atom # performs a QND detection How good is the fidelity of the QND measurement? Les Houches 003 19
QND detection of one photon: Ramsey fringes signal 1,0 0,9 Probability P i 0,8 0,7 0,6 0,5 0,4 0,3 0, 0,1 0,0 0 photon 1 photon 0 10 0 30 40 50 60 frequency (ω R -ω gi )/π (khz) G. Nogues et al., Nature 400, 39 (1999) A. Rauschenbeutel et al., PRL 83, 5166 (1999) Les Houches 003 0
Input-output characterization of the QND measurement 1 Signal: atom detected in g or i Input: 0 ou 1 photon QND measurement Output: 0 ou 1 photon 3 Les Houches 003 1
QND detection: the photon is still there Position (cm) D D 10 8 6 Atom # 1: g π π 4 Atom # : g 0 Atom # 1 detects a small blackbody field in C: n th =0.5 photon Atom # prepared in g checks the result by absorbing the field lock at two atom correlation to check if the QND detected photon is still there Les Houches 003
QND detection: Checking the result i 1 1 photon 0,45 0,40 P(e if i 1 ) Probability 0,35 0,30 0,5 0,0 i 1 0 photon 0,15 0,10 0 10 0 30 40 50 60 Frequency ω R (khz) The absorption rate of atom is modulated depending whether Detection of atom 1 in i 1 corresponds to 0 or 1 photon in C The photon is still here! Les Houches 003 3
Input-output characterization of the QND measurement 1 Signal: atom detected in g or i Input: 0 ou 1 photon QND measurement Output: 0 ou 1 photon 3 Les Houches 003 4
QND measurement: field input-output correlation Position (cm) D D D 10 8 6 Atom # 1 Atom # π π 4 Atom # 3 0 Time Depending on the detected state of atom # 1, 0 or 1 photon is prepared in C atom # performs the QND measurement atom #3 checks the final state of the field Les Houches 003 5
Input-output correlation without QND meter Position (cm) D D D 10 8 6 Atom # 1 Atom # π π 4 Atom # 3 0 Time Depending on the detected state of atom # 1, 0 or 1 photon is prepared in C atom # is not prepared atom #3 checks the final state of the field Les Houches 003 6
Input-output correlation without QND meter Atom#1 - atom#3 correlation 1 photon 0 photon 0.5 e1g3 0.4 Probability 0.3 0. 0.1 g1g3 g1e3 e1e3 no atom # 0 atom #1 and #3 coincidence Les Houches 003 7
Input-output correlation with QND meter Atom#1 - atom#3 correlation 1 photon 0 photon 0.5 e1g3 Probability 0.4 0.3 0. 0.1 g1g3 g1e3 g g e1e3 i atom # detected in g atom # detected in i no atom # 0 i atom #1 and #3 coincidence Easy quantitative analysis of performances: absorption rate of atom : 10% fidelity of QND measurement: 80% Les Houches 003 8
Factor of merit of the QND detection detection fidelity η 1,0 0,8 0,6 0,4 0, Ideal QND QND classical 0,0 0,0 0, 0,4 0,6 0,8 1,0 absoption rate ε Les Houches 003 9
4. Step by step preparation of a GHZ triplet Les Houches 003 30
Three qubits entanglement : experimental sequence V e - Position (cm) field Atom # 1 Atom # π entanglement D D 10 8 6 4 Atome# 1 e, 0 ( e, 0 g, 1 ) Time 1 1 1 + 1 EPR Pair preparation 1 g i + g 1 C-Not e 1, 0 i + g + g 1, 1 i g gate Atome # ( ) π ( ) ( ) Les Houches 003 31 0
The "GHZ" state" prepared state: In term of qubits: 1, 0 1 e g 1 In termofspin 1/: 1 ( i + g ) +, ( i g ) 1 1 ( 0,0,0 + 1,1,1 ) ( +, +, + +, ) 1 c 1, " GHZ triplet " (Greenberger Horne Zeilinger) c Les Houches 003 3
Caracterization of the prepared state idealcase: Density matrix of the prepared state: ρ triplet 1 ψ triplet = + + ++... *...... *. *........ *........ *.... =.... *........ *........ *. *...... * ( +,, + +,, ) 1 c 1 performed measurements: * measurement of σ z1. σ z. σ z3 * measurement of σ x1. σ x. σ x3 c Les Houches 003 33
Measurement of σ z1. σ z. σ z3 : practical realization Step 1:transfer of the field state to a third atom performing a π absorption pulse in C: 1 e ( ) ( ) 1, 0 g + i + g1, 1 g i g3 1 e ( ) ( ) 1 g + i g3 + g1 g i e 3 1 ( + ) 1, +, + 3 + 1,, 3 0 step : detection of each atom for measuring σ z1. σ z. σ z3 - atoms 1 et 3 : direct measurement of energy - atome : measurement of energy after applucation of an external pulse: 1 ( g + i ) i 1 1,, 3 1,, 3 e i g + g g e ( ) 1 ( ) g i g Les Houches 003 34
Full set of operations for measurement of σ z1. σ z. σ z3 Position (cm) D D D 10 8 6 Atom # 1 Atom # π π 4 Atom # 3 0 θ Rabi oscillation in C Time State before detection: D Detection Classical p/ pulse 1 e i g + g g e (,,,, ) 1 3 1 3 Les Houches 003 35
Measurement results: measurement of σ z1. σ z. σ z3 P long =P eig + P gge = 0.58 (0.0) 0.4 0.3 0. - 1,-,- 3 + 1,+,+ 3 0.1 0 Pgig Pgie Pggg Pgge Peig Peie Pegg Pege Rauschenbeutel et al., Science 88, 04 (000) Les Houches 003 36
Full set of operations for measurement of σ x1. σ x. σ x3 Position (cm) D D D 10 8 6 Atom # 1 Atom # π π 4 θ D Time Rabi oscillation in C Detection Atom # 3 Classical p/ pulses, phase of detection pulses adjusted to measure σ x1 and s x3 0 Les Houches 003 37
Fidelity of preparation of the GHZ state measurement of σ z1. σ z. σ z3 P long =P eig + P gge = 0.58 (0.0) measurement ofσ x1. σ x. σ x3 A= σ x1. σ x. σ x3 = -0.8 (0.03) fidelity: F ψ ρ ψ = = triplet triplet 0.54 (0.03) F > 0.3 garanties non-separability see also: Sacket et al. Science 88, 04 (000) preparation of a 4 ions GHZ state in one step Les Houches 003 38
Engineered versus "spontaneous" entanglement first experiments on Bell inequalities: use of "spontaneous" entanglement relying on a symetry of the system atomic cascade Aspect, Grangier σ σ + σ + σ Parametric down conversion: Mandel, Zeilinger Gisin cavity QED: provide many tools for step by step entanglement engineering Nonlinear cristal the most complex sequence of gates applied on individually addressable and measurable qubits Position (cm) Atom # 1 Atom # Atom # 3 π π D D D 10 8 6 4 0 Time Where is the limit? answer not clear..but there are still interesting things to do! Les Houches 003 39
References (1) QND measurement in microwave CQED experiments: M. Brune, S. Haroche, V. Lefevre-Seguin, J.M. Raimond and N. Zagury: "Quantum nondemolition measurement of small photon numbers by Rydberg-atom phase sensitive detection", Phys. Rev. Lett. 65, 976 (1990). M.Brune, S. Haroche, J.M. Raimond,L. Davidovich and N. Zagury. "Manipulation of photons in a cavity by dispersive atom-field coupling: QND measurement and generation of "Schrödinger cat"states". Phys Rev A45, 5193, (199). S. Haroche, M. Brune and J.M. Raimond. "Manipulation of optical fields by atomic interferometry: quantum variations on a theme by Young".Appl. Phys. B, 54, 355, (199). S. Haroche, M. Brune and J.M. Raimond. "Measuring photon numbers in a cavity by atomic interferometry: optimizing the convergence procedure". Journal de Physique II Les Houches 003 40
References () Gates: QPG or C-Not, algorithm: M. Brune et al., Phys. Rev. Lett, 7, 3339(1994). Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995). C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995). A. Reuschenbeutel et al., PRL. G. Nogues et al. Nature 400, 39 (1999). S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond and S. Haroche, Phys. Rev. Lett. 87, 03790 (001) F. Yamaguchi, P. Milman, M. Brune, J-M. Raimond, S. Haroche: "Quantum search with two-atom collisions in cavity QED", PRA 66, 01030 (00). Q. memory: X. Maître et al., Phys. Rev. Lett. 79, 769 (1997). Atom EPR pairs: CQED: E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997). Ions: Q.A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998). Les Houches 003 41
4. Non-resonant gate entanglement directly generated by a two atom "collision" catalyzed by the cavity application to a non-resonant phase gate Les Houches 003 43
Two atoms and one mode Non-resonant coupling Zheng et al PRL 85 39 (000) One photon coupling eg,,0 ge,,0 eg,,0 Ω0 Ω R = δ Ω R ge,,0 gg,,1 ν at ν cav δ = ν ν cav at Atoms can exchange energy by virtually emitting a photon in C. Two atom EPR state preparation for a "Raman" pulse: gg,,0 1 ψ EPR = ( eg, + ge, ) 0 Les Houches 003 44
eg,,0 Advantage of non-resonant method of entanglement: Sensitivity to cavity damping Ω0 Ω R = δ Ω R ge,,0 gg,,1 effect of cavity damping: projection on g,g,0> Full loss of entanglement probability of error: P col err Ω δ Γ cav. T int Ω. = π T R int gg,,0 δ Γ cav Resonant case: P res err Γ cav. T res error rate reduced as: P P col err res err int Ω δ res Ω. T = π int efficient with slower atoms Les Houches 003 45
Advantage of non-resonant method of entanglement: Sensitivity to blackbody radiation coupling in the presence of N photons: egn,, Ω N + 1 een,, 1 Ω R Ω N gen,, Due to destructive interference between two probability amplitudes, the effective coupling is to first order independent of N: ggn+,, 1 ( ) Ω 0. N + 1 Ω0. N Ω0 ΩR = δ δ δ The method works even in the presence of blackbody radiation Similar to "hot" gate for ions: Moelmer et al PRL 8 1835 (000) Les Houches 003 46
Principle of the experiment Two atoms with different velocity "collide" in C. Position atom #1 atom # δ,θ entanglement cavity center Time Les Houches 003 47
Cavity assisted "collision": experimental signal pulse: preparation of: 1 ψ EPR = ( eg, + ge, ) 0 1,0 0,8 Pe 1 -g Pg 1 -e Probability 0,6 0,4 0, 0,0 0 1 3 4 η (x10-6 ) Osnaghi et al., PRL 87, 03790 (001) Solid line: second order coupling doted line: numerical integration Les Houches 003 48
Measuring the EPR entanglement: 1,0 Tranverse EPR correlation 0,5 σ 1,x σ,φ 0,0-0,5-1,0 φ/π -1 0 1 3 Fidelity of the EPR state: 0,79 Les Houches 003 49
Quantum phase gate using a cavity assisted collision Definition of logical qubits: e : 51c g : 50c ω cav 1 0 i : 49c ω gi 0 1 atom 1 atom result of a π two atom "Raman" pulse (taking into account light shift of levels: ig, ig, ii, ii, eg, i, g iπ ei, e ei, 0,0 0,0 0,1 0,1 1, 0 1, 0 e π i 1,1 1,1 Application to Grover algorithm: F. Yamaguchi, et al. "Quantum search with two-atom collisions in cavity QED", PRA 66, 01030 (00) The cavity field is not affected Les Houches 003 50