Cavity Quantum Electrodynamics Lecture 1

Transcription

1 DÉPARTEMENT DE PHYSIQUE DE L ÉCOLE NORMALE SUPÉRIEURE LABORATOIRE KASTLER BROSSEL Cavity Quantum Electrodynamics Lecture 1 Michel BRUNE Les Houches

2 Quantum information and Cavity QED Principle of cavity QED experiments: Theoretical background: the Jaynes Cummings model Simple but contains already a lot of physics Fields of interest: Fundamental: better understanding of quantum theory. "Applications": using quantum physics to manipulate quantum information. Central role of "entanglement" Two level atoms interacting with a single mode of a high Q cavity In practice: Rydberg atoms superconducting microwave cavity The "strong coupling" regime: coupling >> dissipation Les Houches

3 Probing the EPR atom-field entanglement A B C Illustrating Bohr-Einstein dialog central concept: complementarity Massive slits: insensitive to collisions with single particles Interferences: mater behave as waves Experiment performed with photons, electrons, atoms, molecules. Light slits: recoil of the slit monitors which path information No interferences: mater behave as particles Les Houches

4 The "Schrödinger cat" Elementary formulation of the problem: Superposition principle in quantum mechanics: - Any superposition state is a possible state - Schrödinger: this is obviously absurd when applied to macroscopic objects such as a cat!! Up to which scale does the superposition principle applies? Les Houches

5 "Schrödinger cat" and quantum theory of measurement Hamiltonian evolution of a microscopic system coupled to a measurement apparatus: 1 Ψ at chat = + 2 Entangled atom-meter state Problem: real meters provide one of the possible results not a superposition of the two too much entanglement in QM? Need to add something? NO! "decoherence" does the work Les Houches

6 Can one do something "useful" from the strangeness of quantum logic? Yes: How? Quantum cryptography: does work Quantum communication: teleportation Quantum algorithms: Search problems (Grover) Factorization (Shor) by manipulating qubits with a quantum computer Practically: extreeeeemely difficult to realize: a quantum computer manipulates huge Schrödinger cat states. Entanglement and quantum information Qubit: two level system Classical bit: 0 and 1 0 ou 1 1 Anyway interesting for understanding the essence and the limits of quantum logic a + b Les Houches

7 CQED, brief history : Effect of boundary conditions on dipole radiation (Maxwell, Hertz, Sommerfeld). 1930: atom-metal surface interaction (London, Lennard Jones). 1946: Spins coupled with a tuned resonator (Purcel). Boundary effects on Synchrotron radiation (Schwinger). 1947: Vacuum fluctuations between two mirors, Casimir effect : Masers and Lasers: collective radiation of atoms in a cavity (Townes, Schalow ) 1974: Modification of molecular fluorescence near surfaces (Drexhage) : Rydberg atoms in microwave cavities, Masers (ENS, Munich) : Modification of spontaneous emission, experiments: ENS, MIT, Seattle, Yale, Rome CQED in the PERTUBATIVE regime: Low Q cavity or effect of a single mirror: coupling strength << dissipation rates Perturbation of atomic radiative properties which remains qualitatively the same as in free space. Les Houches

8 Cavity Quantum electrodynamics: strong coupling The strong coupling: Γ at,γ cav << Ω 0 Ω 0 "vacuum Rabi frequency" CQED in optics: direct detection of the field Caltech: J. Kimble Munich: G. Rempe - cold atoms "trapped" by vacuum forces -single photon gun difficulties: - control of atomic motion within λ opt - atomic lifetime: 10 ns also: Excitons and microcavities Microwave: detection of atoms Munich: H.Walther - closed cavities - Low l Rydberg atoms micromaser, trapping states, number states Paris: - open cavities - circular Rydberg atoms Main topic of this course Les Houches

9 Outline of the course Aim of this course: CQED with Rydberg atoms in the strong coupling regime. Lecture 1: the strong coupling regime Lecture 2: quantum gates and quantum logic Lecture 3: Quantum measurement and decoherence Lecture 4: perspectives Les Houches

10 Outline of Course 1 1. One atom, one mode, the Jaynes-Cummings model 2. Rydberg atoms in a cavity: the tools achieving the strong coupling regime The experimental setup Vacuum Rabi oscillations 3. Rabi oscillation in a small coherent field Direct observation of field graininess 4. Rabi oscillation Ramsey interferometry and complementarity Les Houches

11 1. One atom, one mode, the Jaynes-Cummings model Les Houches

12 Same procedure as in free space: 1- Find the classical eigenmodes of the resonator satisfying the boundary conditions. iωt Classical electric field: E ( r, t) = E. f ( r). e + cc α ω α 2- Each mode is quantized as an harmonic oscillator. ( ) * + Electric field operator: Eˆ α ( r, t) = Eω. f α ( r).ˆ aα + f α ( r). aˆ α aa, + = 1 Where: ω Eω = "vacuum electric field". 2ε 0 V cav 2 3 V = f r d r cav ( ) α volume of the mode. V cav is really a physical volume. Cavity Field quantization in a cavity f (r ) complex function of Normalization: α Max fα ( r ) = 1 (real functions will be enough for us) - Here the quantized object is a collective excitation of the field and all the electric charges at the surface of the mirror. - We now consider a single mode and drop the index α. Les Houches

13 The Jaynes Cummings model: e g ω at + a single two level atom, frequency ω at +a single field mode, frequency ω c + dipole coupling + negligible damping Atom-field Hamiltonian: H = H at +H cav +V at-cav H H at cav V d = at cav ω at = 2 = ω c = d. E ˆ ( r ) d eg [ e e g g ] + [ a a + 1 2] [ e g + g e ] Condition of validity: - ω c close to a single atomic transition: - small cavity: FSR >> δ δ = ω ω << ω, ω c at c at Les Houches

14 The Jaynes Cummings hamiltonian Rotating wave approximation (RWA): V at cav + + = Ω ( r) 2 ae g+ ag e+ a g e + a e g Non-resonant terms are neglected + Vat cav Ω ( r) 2 a e g + a g e Vacuum Rabi frequency: Ω ( r) = 2 d. f ( r). E =Ω. f ( r ) eg ω 0 Ω = 0 2 d. eg ω 2ε V 0 cav Validity of RWA: Ω << ω, ω at c Les Houches

15 Dressed energy levels at resonance (ω at =ω c ) Eigenvalues: E ± n = ωc n+ + ωat ± Ω0 n + ( 1/2) 2 1 Eigenstates: ± n = 1 2 e, n ± g, n+ 1 g,n+1> e,n> +,n> -,n> Ω 0 n + 1 Levels just couple by pairs (except the ground state) g,1> e,0> Ω 0 level splitting scales as the Field amplitude ω c g,0> Les Houches

16 Resonant atom-field coupling: dynamic point of view 2 Electric dipole coupling Ω 0 2 e g e,0 1 0 e g g,1 1 0 Ω 0 /2π=50 khz T rabi =20 µs Ω t Ω t e e i g ,0 cos,0 sin,1 Coherent Rabi oscillation Les Houches

17 Condition of observation of vacuum Rabi splitting Cavity damping rate: Atomic lifetime: Γ 1 cav cav cav The width of dressed levels must be smaller than the vacuum Rabi frequency:, T = Γ Γ, T =Γ 1 at at at Ω Γ Γ 0 cav, at +,n> (Γ at +Γ cav )/2 -,n> Ω 0 Les Houches

18 2. Rydberg atoms in a cavity: achieving the strong coupling regime One photon and one atom in a box: Photon box: superconducting microwave cavity circular" Rydberg atoms Les Houches

19 The cavity 5 cm a "photon box": - superconducting Niobium mirrors - microwave photons: λ=6 mm, ν cav =51GHz - photon lifetime: T cav =1ms ( Q = ) Microwave Power(dB) Measurement of cavity damping time τ cav = 1 ms Q = ν= GHz time (ms) - one mode - a few trapped photons Les Houches

20 The "circular" Rydberg atoms "Circular Rydberg atoms": l= m =n-1 g e ν cav n=51 ν at =51.1 GHz n=50 - radiative lifetime: 30 ms - dipôle: d= 1500 u.a. - ideal closed two level system Les Houches

21 "Circular" atoms as two level atoms Stark diagram of Rydberg levels: Good quantum number: m n=50 n=51 m= / c 50c ω c ω at The electric field removes the degeneracy between transitions: good isolation of the 51c to 50c transition ω Stark Linear Stark effect: ω Stark /2π = 100 MHz/(V/cm) Quadratic Stark shift of the 51c-50c transition: 255 khz/(v/cm) 2 used for fast tuning of the atom in or out of resonance Les Houches

22 Preparation of circular atoms 1 52f,m=2 2 σ - 52c E stat 5d 5/2 1.26µm m=2 3 σ c B stat 776nm E RF 5p 3/2 50c E stat 780nm 5s 1/2 F=3 F=2 85 Rb 1 - laser excitation of 52f, m= "Stark switching": E stat =0 2.5V/cm photons adiabatic transfer to 52c in duced by E RF (ν=250mhz). B stat removes degeneracy betwen σ + and σ "Purification": selective transfer to 51c or 50c. 300 circular atom/laser pulse. Purity > 99% L R F Purif 20µs Les Houches

23 Detection of Rydberg atoms (1) V = V e, V g ou V i 1 atom e - atomic beam Electron multiplier n=52 1 click Counter n=51 n=50 atoms detected one by one by selective ionization in an electric field measurement of internal energy state of the atom after interaction with C CW detection in a field gradient: efficiency 40% 125 V/ cm 136 V/ cm 148 V/ cm Les Houches Field

24 Detection of Rydberg atoms (2) New detector: atoms are ionized in a pulsed homogeneous electric field: V(t) e - atomic beam electrostatic lens Electron multiplier Improved detection efficiency: 70% +/- 10% Les Houches

25 Velocity selection by optical pumping: L 1 L 2 L 3 Rb oven detector e - e, g or i? 85 Rb 5p 3/2 ν Doppler F=4 F=3 F=2 ν F=1 Ghost hl1: depumping beam 5s 1/2 L 1 F=3 L 2 hl2: velocity selective repumping beam, tuned on ν F=2-F=3 + ν Doppler hl3: velocity selective depumping beam for removing "ghost"velocity pumped by L2 on the F=2-F=2 transition F=2 Les Houches

26 Velocity "superselection" optical pumpilg only: v=19m/s optical pumping+ time of flight "suprselection" v=4m/s Counts (A.U.) v=503.3m/s v=4m/s Velocity (m/s) Les Houches

27 Experimental set-up 85 Rb Laser velocity selection Circular atom preparation: - 53 photons process - pulsed preparation 0.1 to 10 atoms/pulse Cryogenic environment T=0.6 to 1.3 K weak blackbody radiation e - State selective detector One atom = one click Les Houches

28 Experimental setup Atom preparation detection lasers atomic beam Les Houches

29 Single photon induced Rabi oscillation 1,0 0,8 g,1 e P g (50 circ) 0,6 0,4 0,2 e,0 Ω 0 =47 khz T Rabi =20µs e ω ge = ω cav g e - 0, interaction time (µs) Coherent Rabi oscillation replaces irreversible damping by spontaneous emission Les Houches

30 3. Rabi oscillation in a small coherent field Direct observation of discrete Rabi frequencies Les Houches

31 Coherent field states Number state: N Quasi-classical state: N 2 α 2 α α = e N ; α = α e N! N iφ Photon number distribution: P ( N ) N N N = e ; N = α N! 2 P(N) 0,4 Poisson distribution 0,3 0,2 N=2 0,1 0, Photon number N 1 Pg () t = P 1 cos N N ( ) ( N ) ( Ω ) 0t Les Houches

32 Rabi oscillation in a small coherent field P g (t) 1,0 0,5 exp fit e S α e - 0,0 2 α = 0.85 photon interaction time (µs) Les Houches

33 Rabi oscillation in a small coherent field: observing discrete Rabi frequencies Fourier transform of the Rabi oscillation signal Ω 0 Ω 0 2 Ω 0 3 Discrete peaks corresponding to discrete photon numbers FFT (arb. u.) Direct observation of field quantization in a "box" Frequency (khz) Les Houches

34 Rabi oscillation in a small coherent field: Measuring the photon number distribution 1 P () ( ) 1 cos ( ) g t = P N Ω 0t N + 1. e 2 N ( ) t / τ Fit of P(n) on the Rabi oscillation signal: 0,5 0,4 Measured P(n) Poisson law P(n) 0,3 0,2 N = ,1 0, Photon number accurate field statistics measurement Les Houches

35 Rabi oscillation in small coherent fields (α) (a) (A) <n>=0.06 Probability P(n) 1,0 0,5 0,0 0,5 0,0 0,5 0,0 0,0 0,5 1,0 (β) (b) (B) <n>=0.4 (γ) (c) Fourier Transform Amplitude (C) <n>=0.85 e to g transfer rate 0,0 0,5 0,0 0,5 (δ) (d) (D) <n>=1.77 0,3 0,0 0,5 0, n Frequency (khz) Time (µs) Les Houches Phys. Rev. Lett. 76, 1800 (1996)

36 4. Rabi oscillation Ramsey interferometry and complementarity Entanglement and complementarity quantum eraser Les Houches

37 Probing the EPR atom-field entanglement A B C Illustrating Bohr-Einstein dialog central concept: complementarity Massive slits: insensitive to collisions with single particles Interferences: mater behave as waves Experiment performed with photons, electrons, atoms, molecules. Light slits: recoil of the slit monitors which path information No interferences: mater behave as particles Les Houches

38 More practical interferometers Ramsey interferometer e: 51c Mach Zender Interferometer g: 50c a M e S e B 1 R 1 φ R 2 b - R1 and R2: resonant π/2 pulses induced by CLASSICAL microwave fields. g M' φ B 2 D acts as "beam splitter" for the internal atomic state - The phase φ of the interferometer is scanned using a Stark pulse Detectd signal 1,0 0,5 0, φ/2π Les Houches

39 More practical interferometers Ramsey interferometer e: 51c Mach Zender Interferometer g: 50c P M a e S e B 1 R 1 φ R 2 b O φ - R1 and R2: resonant π/2 pulses induced by CLASSICAL microwave fields. acts as "beam splitter" for the internal atomic state - The phase φ of the interferometer is scanned using a Stark pulse g M' φ Small mass beam splitter: the recoil momentum P at reflexion of the particle on B1 keeps track of which path information No fringes for a microscopic beam splitter B 2 D Les Houches

40 Ramsey interferometry with a "quantum beam splitter" π/2 pulse R 1 : Rabi oscillation in a small coherent field injected in C. S injects in C a small coherent field α. S e R 1 R 2 C ( e e g g ) e α 1 α + α 2 φ g D φ - αe and α g are not coherent states - Generaly: α α < 1 e g - partial atom-field entanglement - partial which path information is stored in the final field state Les Houches

41 From classical to quantum beam splitters: π/2 pulse in a large coherent state: α α α e g n 1 e α ( e + g ) α n 2 α e e When α is large enough, R one more photon in the field does not 1 g make any difference on the field state NO which path information stored in the field: "classical beam splitter" n P(n) Poisson law n n π/2 pulse in vacuum: α=0 1 2 (,0,1 ) e 0 e + g Atom-field EPR pair: Hagley et al. PRL 79,1 (1997) The photon number is a perfect label of the atomic state FULL which path information stored in the field: "quantum beam splitter" Les Houches e,0 R 1 e,0 g,1

42 Practical realization V cav S ω eg ω cav e S R 1 R 2 C φ g D φ Time The atomic frequency is adjusted by Stark effect with V cav Timing: t int Inject a controlled coherent field. (average photon number <n>measured by measuring light shifts in an auxiliary experiment) Put atom on resonance at time t int. For each value of <n>, t int is adjusted for performing a π/2 Rabi oscillation pulse Vary φ by varying V cav at time t φ. t φ Les Houches

43 Fringes signal for various values of n: 1,0 0,8 0,6 0,4 0,2 α=0 n=0 Perfect distingishability: No fringes 0,8 P g 0,6 0,4 0,2 0,8 0,6 0,4 0,2 α=0.56 α=1.41 n=0.31 n=2 0,8 0,6 0,4 0,2 0,0 Saturated contrast: η=0.73 α=3.6 n=12.8 Bertet et al. Nature 411, 166 (2001) φ/π Les Houches

44 Quantitative interpretation in term of atom-cavity entanglement Variation of fringe e α Visibility V: R 1 R 2 V 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 experiment theory 0, Average photon number <n> η= ( e α ) e + g αg Reduced atom density matrix: ρ at * 1 1 αe α g = 2 αe αg 1 V = α α. η e g η: saturated contrast at large n Les Houches

45 Checking the atom-"beam splitter"entanglement: a quantum eraser C is initially empty: P 1 a M S B 1 e R 1 φ C R 2 g D M' b O φ P 2 B 2 D φ P g 1,0 0,8 0,6 0,4 0,2 0, φ/π The which path information is erased by the second interaction with C variations of fringe contrast can not be interpreted as simply resulting from noise added by the interaction with the beam-splitter Les Houches

46 A genuine quantum eraser: Which path information is erased by appropriate measurement of the "which path meter": e,0 R (,0,1 ) 2 e + g 1 1 C Measurement of the cavity field in the basis: 1 2 ( 0 ± 1 ) Projection of atomic state on: 1 2 e g ( ± ) R 1 is then equivalent to a classical π/2 pulse whose phase depends on the result of the measurement of the field. When R 2 is activated, one expects conditional Ramsey fringes. Les Houches

47 Quantum eraser: measuring the cavity field with a second atom S Atom 2 R 1 R 2 e D g g C 1. π pulse in C: swaps a 0 or 1 photon state into an atomic state: Maître et al. PRL 79,769 (1997) g,0 g,0 g,1 e,0 2. Detection of the atom after a classical π/2 pulse: equivalent to the measurement of the phase of the atomic dipole equivalent to the measurement of the phase on the initial field state. This performs the appropriate measurement of the cavity field state in C Les Houches

48 Quantum eraser: experimental results 0,7 G1ifG2 G1ifE2 conditional probability 0,6 0,5 0,4 0,3 0,2-1,0-0,5 0,0 0,5 1,0 φ/π High visibility is restored Les Houches

49 Conclusion of lecture 1: Cavity QED with microwave photons and circular Rydberg atoms:. a powerfull tool for: Achieving strong coupling between single atoms and single photons manipulating entanglement and complementarity. next lecture: Rabi oscillation in vacuum and quantum gates Les Houches

50 References Strong coupling regime in CQED experiments: F. Bernardot, P. Nussenzveig, M. Brune, J.M. Raimond and S. Haroche. "Vacuum Rabi Splitting Observed on a Microscopic atomic sample in a Microwave cavity". Europhys. lett. 17, (1992). P. Nussenzveig, F. Bernardot, M. Brune, J. Hare, J.M. Raimond, S. Haroche and W. Gawlik. "Preparation of high principal quantum number "circular" states of rubidium". Phys. Rev. A48, 3991 (1993). M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond and S. Haroche: "Quantum Rabi oscillation: a direct test of field quantization in a cavity". Phys. Rev. Lett. 76, 1800 (1996). J.M. Raimond, M. Brune and S. Haroche : "Manipulating quantum entanglement with atoms and photons in a cavity", Rev. Mod. Phys. vol.73, p (2001). P. Bertet, S. Osnaghi, A. Rauschenbeutel, G. Nogues, A. Auffeves, M. Brune, J.M. Raimond and S. Haroche : "Interference with beam splitters evolving from quantum to classical : a complementarity experiment". Nature 411, 166 (2001). E. Hagley, X. Maître, G. Nogues, C. Wunderlich, M. Brune, J.M. Raimond and S. Haroche: "Generation of Einstein-Podolsky-Rosen pairs of atoms", PRL 79,1 (1997). Les Houches