Preparation of a GHZ state

Transcription

1 Preparation of a GHZ state Cavity QED Gilles Nogues Experimental realization A first atom in e 1 performs a π/2 pulse ψ 1 = 1 2 ( e 1, 0 + g 1, 1 ) A second atom in 1/ 2( i 2 + g 2 ) performs a QPG gate without affecting the field state (QND) ψ 2 = 1 e 1, 0 1 ( i 2 + g 2 ) + g 1, 1 1 «( i 2 g 2 ) cavity QED with Rydberg atoms Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state An atom-field-atom GHZ state A third atom in g 3 can perform a π pulse in order to read the field state ψ 3 = 1 e 1, g 3 1 ( i 2 + g 2 ) + g 1, e 3 1 «( i 2 g 2 ) 2 2 2

2 Detection of the GHZ state Cavity QED Gilles Nogues cavity QED with Rydberg atoms 2nd Ramsey pulse for atom 2 at frequency ν 0 It corresponds to the QND detection conditions: 1 2 ( i 2 + g 2 ) i 2 Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state 1 2 ( i 2 g 2 ) g 2

3 Detection of the GHZ state 2nd Ramsey pulse for atom 2 at frequency ν 0 It corresponds to the QND detection conditions: 1 2 ( i 2 + g 2 ) i ( i 2 g 2 ) g 2 Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state Results Probability G1G G1E3 E1G3 E1E3 no atom #2 0 Coincidences between atom #1 and #3

4 Detection of the GHZ state 2nd Ramsey pulse for atom 2 at frequency ν 0 It corresponds to the QND detection conditions: 1 2 ( i 2 + g 2 ) i ( i 2 g 2 ) g 2 Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state Results Probability G1G G1E3 E1G3 E1E3 atom #2 detected in g atom #2 detected in i no atom #2 0 Coincidences between atom #1 and #3

5 Correlation in another basis Cavity QED Gilles Nogues EPR correlation for atoms 1 and 3 Measurement of σ x,1 σ φ,3 as in previous experiments. Pe 1 e 3 - Pe 1 g 3 - Pg 1 e 3 + Pg 1 g 3 0,3 0,2 0,1 Two atoms signal signal if i 2 signal if g 2 signal if no at 2 cavity QED with Rydberg atoms Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state 0, ,1 Frequency (khz) of The e-i two photon transition -0,2 QPG action of atom 2 changes the sign of the correlation of atoms 1 and 3.

6 General conclusion Cavity QED Gilles Nogues cavity QED with Rydberg atoms Simple quatum gates demonstrated most complicated algorithm uses up to 4 qubits and entangles 3 of them complete measurement of the density matrix not realised (see ion trap experiment) early experiment data acquisition time too long due to the very low probability for detecting coincidences (20h for the 3 atom experiment) Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state

7 Decoherence in Quantum Mechanics A cavity QED approach Gilles NOGUES Laboratoire Kastler Brossel Ecole normale supérieure, Paris

8 A century of quantum mechanics And yet an intriguing theory which defies our «classical intuition» Many paradoxes as soon as one tries to interpret its results Entanglement: after interaction 2 subsystems, although separated in space, cannot be considered as independant Superposition principle: Any linear combination of physical state is a physical state

9 The Schrödinger cat A gedanken experiment A cat in a box with an excited atom If the atom desexcite, a setup kills the cat Three related questions Why do we need the box? Why do we found the cat dead or alive? Why can t we predict the outcome of a single realization? What is the state of the system after a half-lifetime of the atom? Environment Preferred basis?

10 Outline A cavity QED experiment How to create and destroy and restore a coherent superposition The decoherence in quantum mechanics Effect of the size of the system and of the environment Monitoring the decoherence What can be an optical Schrödinger cat? Observing the decoherence Beyond decoherence Pushing the limits?

11 A cavity QED experiment on complementarity How to wash out fringes in an interferometer by gaining a «which-path» information? How to restore the interference by manipulating the information

12 The strangeness of the quantum Superposition principle and quantum interferences The sum of quantum states is yet another possible state A system suspended between two different classical realities D a Ψ = Ψ 1 + Ψ 2 d I = I + 2 Re Ψ Ψ a = λ D d Feynman: Young s slits experiment contains all the mysteries of the quantum

13 Interferometers for photons and atoms Mach-Zender Ramsey: a M B 1 R 1 R 2 b M' B 2 D 2 energy path separated by interaction with classical field 2 optical path separated by beamsplitters P g Fréquence relative (khz)

14 Destroying the fringes By gaining a which-path information on the system One slit is mounted on springs Microscopic slit: set in motion when deflecting particle. Which path information and no fringes Macroscopic slit: impervious to interfering particle. No which path information and fringes Wave and particle are complementary aspects of the quantum object. (From Einstein-Bohr at the 1927 Solvay congress)

15 Bohr s experiment with a Ramsey interferometer Illustrating complementarity: Store one Ramsey field in a cavity Atom-cavity interaction time Tuned for π/2 pulse Possible even if C empty Initial cavity state α Intermediate atom-cavity state e Ramsey fringes contrast Large field α α α e g α α e ( e, αe g, α g ) 1 Ψ = + 2 g S R 1 R 2 C φ FRINGES g D φ Small field α e = 0, α = 1 g NO FRINGES

16 Interferences versus field size N=0 N=0.31 N=2 A microscopic field is strongly affected by the interaction Stores an information about the atomic state For larger fields the «which path» information is lost Fringes restored N=12.8 Nature, 411, 166 (2001)

17 Interference versus field size (2) Maximum contrast 72% Mean photon number final contrast due to external imperfections (velocity dispersion, 2 atom events ) Photon number acts as the mass of the slit in the Young s experiment Classical behaviour observed for about 10 photons

18 Important remarks No need to measure the field state The mere fact that the information exists destroy the interferences Close link with entanglement Atom and field no longer separable Local operation on the atom cannot recover the whole information Fringes are recoverable If one operates on the atom and the field in order to erase the information

19 e Ramsey quantum eraser A second interaction with the mode erases the atom-cavity entanglement R 1 φ C S R 2 g e,0 g,1 1 (,0,1 ) 2 e + g D Ramsey fringes without fields! Quantum interference fringes without external field A good tool for quantum manipulations Pe

20 Intermediate conclusions Any coherent superposition can be created and probed in a interferometer experiment The fringe contrast is a direct measurement of how much «which-path» information one has about the system It is in principle possible to manipulate this amount of information to restore the coherence Cavity QED is an appropriate tool to manipulate those concepts in the case of an atom coupled to a few tens of photons

21 A general frame for decoherence The environment as an unavoidable «whichpath» meter Effect of the size of the system Link with the measurement theory

22 Why the box? Hide the cat Physically: prevent diffusion of light on the system Isolate from its environment Coupling with the outside world Metallic box? Blackbody radiation Surrounding gaz Collision: brownian motion Vibration: noise All those interactions provide a «which-path» information

23 Why the cat? What is the difference between a photon in an interferometer and a cat in a box Strength of the coupling with the environment Distance (in Hilbert space) between states T dec =T rel /Distance How to evaluate the distance? Depends on the nature of the coupling Diffusion of photons: distance = (distance in space) / wavelength Not a trivial relaxation mechanism But is explained by relaxation theory for simple models Final state Trace over the environment: statistical mixture

24 Link with measurement theory What is a good meter? Coupled to the observed system Gives an answer that you can see with your eye Quantum system Open, macroscopic system - 0 x + A two step process System-meter coupling May be unitary leads to an entangled state «which path» information gathered by environment Coherence is lost And unrecoverable if the environment is huge Environment

25 Pointer states Only a small fraction of the Hilbert space is observed Example: or, but no superpositions This «preferred basis» correspond to meter state which do not get entangled with the environment Pointer states

26 What is the final state of the cat The coherence leaks into the environment One has to account for the lack of knowledge about the environment Density matrix description with a trace over the environment ρfin =1/2( >< + >< ) Statistical mixture of two pointer state The meter is either pointing one way or the other

27 A short summary: what is important in decoherence An open system whose interaction with the environment selects a preferred basis Pointer states which can be discriminated by a classical experience Large distance in Hilbert space The larger the distance, the faster the process This is the case of every large system Demonstration: One never observes a cat dead and alive at the same time

28 Monitoring the decoherence Requirements to observe the phenomenon A system as isolated as possible Long relaxation time A mesoscopic system Whose «size» can be varied between macroscopic and microscopic dimensions An easy way to prepare and analyse coherent superpositions of the system A small coherent field in a cavity is a perfect example

29 Preparing and observing a Schrödinger kitten Decoherence for a superposition of coherent states of light How to prepare such a state? An interferometric experiment to test the coherence.

30 Cavity relaxation Due to diffraction on the surface Not perfectly spherical over a few mm State at time t: At T=0K: αe -γt/2 Π β i (t) Mode i A coherent state remains disentangled Pointer state Energy conservation Interaction Hamiltonian: H=Σ g i (ab i + + a + b i ) Σ β i (t) 2 = α 2 (1-e -γt ) = n(1-e -γt )

31 Decoherence of a coherent state superposition Initial state 1/N( 0 + α ) Π 0 i N= 2+ 0 α = 2+e - α 2 /2 2 if α >>1

32 Decoherence of a coherent state superposition At time t: 0 Π 0 i + αe -γt/2 Π β i (t) Entangled system Which-path information has leaked to the environment Contrast: C=Π i 0 β i (t) =Πe - βi(t) 2 /2 = e - Σ βi(t) 2 /2 =e - α 2 (1-e -γt )/2 e - α 2 γt/2 Decoherence time T dec = 2 T r / α 2

33 A single atom in a coherent field Non resonant interaction e ω 2 No exchange of energy ω 0 ω g 1 0 Uncoupled Coupled If δ>>ω: e,n +,n g,n and E +,n =-Ω 2 n/4δ g,n+1 e,n-1 g,n ω ω+δω -Ω 2 (n+1)/4δ -Ω 2 n/4δ Cavity frequency ω ω e ω g e,0 g,1 g,0 δ ω ω+δω -Ω 2 /4δ Atom position in cavity mode Atom acts as a dielectric that phase shifts the field in the cavity depending on its state

34 Another experiment on complementarity e Cavity as an external detector in the Ramsey interferometer C S φ "Which path" information: Small phase shift (large δ) (smaller than quantum phase noise) R 1 R 2 Cavity contains initially a coherent field α Non-resonant atom-field interaction: Atom modifies the cavity field phase (index of refraction effect) e g e g 1 g D field phase almost unchanged No which path information Standard Ramsey fringes Large phase shift (small δ) (larger than quantum phase noise) Cavity fields associated to the two paths distinguishable Unambiguous which path information No Ramsey fringes Phase shift α 1/ δ ( δ:atom-cavity detuning)

35 Ramsey Fringe Signal Fringes and field state Complementarity 712 khz Vacuum 712 khz, 9.5 photons 347 khz 104 khz ν (khz) PRL 77, 4887 (96) State transformations R1 1 R2 C Before R1 Before C After C After R2 ( ) e e + g 2 1 iϕ e ( e + e g ) 2 1 g e e + g 2 Detection probabilities 1 ( ), 1 Re i ϕ i i Pg e = e +Φ αe Φ αe Φ 2 ± Ramsey fringes signal multiplied by ( iϕ ) e, α e e, αe g, α g, αe 1 2 iφ iφ iφ e, α 1 ( ) 2 e + g α 1 e e, αe + g, αe 2 i ( Φ i Φ i Φ ) ( ϕ ) { α α } e e e e e iφ iφ i +Φ iφ ϕ { α α } 1 + g e e + e e 2 αe i( ϕ +Φ) iφ i( +Φ) iφ iφ αe iφ

36 Signal analysis Fringe signal multiplied by αe Modulus iφ αe iφ Fringes contrast and phase 60 n=9.5 (0.1) 6 e Contrast reduction Phase 2 2 = e 2n sin Φ D / 2 2n sin Φ Phase shift corresponding to cavity light shifts Phase leads to a precise (and QND) measurement of the average photon number D Fringe Contrast (%) φ (radians) Excellent agreement with theoretical predictions. Not a trivial fringes washing out effect Calibration of the cavity field 9.5 (0.1) photons φ (radians) 4 2 Fringe Shift (rd)

37 Probing the coherence Field state after atomic detection 1 2 ( ) A coherent superposition of two 'classical' states. Decoherence will transform this superposition into a statistical mixture Is it possible to study the dynamics of this phenomenon Requires to perform an interferometry experiment with the cavity state A second atom to probe the field First atom + Φ Second atom Two indistinguishable quantum paths to the same final state e 1 g 2 and e 2 g 1 2Φ 2Φ Φ Quantum interferences D

38 Atomic correlations A correlation signal η = Π Π e, e g, e e, e = Independent of Ramsey interferometer frequency when Φis neither 0 nor π/2 0.5 for a quantum superposition η P = 1 Re 2 α α g, e e, e e, g g, e g, g 0 for a statistical mixture P P + P P + P correlation signal Principle of the experiment Send a first atom to prepare the cat Wait for a delay τ Send a second probe atom Measure η versus τ Raw correlation signals τ=40 µs for an empty cavity -0.1 ν (khz) coincidences

39 Seeing the decoherence over time Two-Atom Correlation Signal Effect of atomic phase shift n=3.3 δ/2π =70 and 170 khz τ/t r correlation signal correlation signal Effect of field amplitude δ/2π =70 khz n= n=3.3 t/t PRL 77, 4887 (1996) t/t r r

40 What can we say from those results It is possible to prepare mesocopic superpositions of states Schrödinger kitten However the coherence is lost very very very very fast The larger the cat, the faster the phenomenon Experimental data in good agreement with theory T dec =T rel /D In our case D goes up to 9 photons

41 How to obtain larger cat states? Increase cavity lifetime (i.e. T rel ) Increase coupling Dispersive interaction Ω2 / δ with δ>>ω Resonant interaction Ω Increase interaction time (lower atomic velocity) What is the effect of a resonant atom on a mesoscopic coherent field?

42 Rabi oscillation in a mesoscopic field Intermediate regime of a few tens of photons. A first insight A simple theoretical problem

43 Collapse and revival Collapse: dispersion of field amplitudes due to dispersion of photon number Revival: rephasing of amplitudes at a finite time such that oscillations corresponding to n and n+1 come back in phase Revival is a genuinely quantum effect

44 Oscillations in a small coherent field 1,0 0,8 0,6 0,4 <n>=0,4 Brune et al., PRL 76, 1800 (1996) Initial state e,α α 2 =<n> 0,2 0,0 Pe <n>=0,85 1,0 0,8 0,6 0,4 0,2 Observation of the collapse and revival A fourier transform reveals the different frequencies Ω (n)=ω 0 n 1,0 0,8 0,6 0,4 0, <n>=1,77 0,0 Open questions How is the field affected? What is happening if <n> increases (classical limit) 0, Interaction time (µs)

45 All results N= P e (t) FFT Amplitude P(n) n th = 0.06 n = 0.4 coh n coh = 0.85 n coh = Time ( µs) Frequency (khz) n

46 Remark at short time time necessary for doing a π/2 pulse Complementarity experiment shows that the field is not affected Maximum contrast 72% Mean photon number

47 A more detailed analysis of Rabi oscillation Valid in the case of mesoscopic fields n<<n Expansion of Ψ around n=<n> (Gea Banacloche PRL 65, 3385, Buzek et al PRA 45, 8190) First non-trivial order: ± iω0 nt / 2 i Ψ ± a = e e ± Φ e i g Atomic states slowly rotating in the equatorial plane of the Bloch sphere (<n> times slower than Rabi oscillation) A slowly rotating field state in the Fresnel plane Graphical representation of the joint atom-field evolution in a plane t=0: 1 2 Ψ c e e ± iω0 nt / 4 ± iφ = α Ψ ( t) = Ψ a ( t) Ψ c ( t) + Ψ a ( t) Ψc ( t) 2 both field states coincide with original coherent state Atomic states are the classical eigenstates Φ = Ω 4 0 t n

48 Atom-field states evolution + Ψ c Ψc + Ψ a Ψ a Initial state: a coherent superposition of two states

49 Atom-field states evolution + Ψ c Ψc + Ψ a Ψ a At most times: Ψ c- Ψ c+ =0 an atom-field entangled state In spite of large photon number: considerable reaction of the atom on the field

50 Link with collapses and revival Rabi oscillation: interference between Ψ a + and Ψ a - Contrast vanishes when Ψ c- Ψ c+ A direct link between Rabi collapse and complementarity 1 P e (t) t/

51 Link with collapses and revival Rabi oscillation: interference between Ψ a + and Ψ a - Contrast vanishes when Ψ c- Ψ c+ A direct link between Rabi collapse and complementarity 1 P e (t) t/ Loss of contrast as the field gets which-path information

52 Link with collapses and revival Rabi oscillation: interference between Ψ a + and Ψ a - Contrast vanishes when Ψ c- Ψ c+ A direct link between Rabi collapse and complementarity 1 P e (t) t/ t rev /2: atom disentangled from a field in a Schrödinger cat state

53 Link with collapses and revival Rabi oscillation: interference between Ψ a + and Ψ a - Contrast vanishes when Ψ c- Ψ c+ A direct link between Rabi collapse and complementarity 1 P e (t) t/ revival: field phase information is erased

54 Phase distribution measurement Heterodyning a coherent field S Inject a coherent field α> Atom in g> Add a coherent amplitude αe iφ Resulting field (within global phase) α(1-e iφ )> Zero final amplitude for φ=0 Probe final field amplitude with atom in g P g =1 for a zero amplitude P g =1/2 for a large amplitude More generally: P g (φ) reveals field phase distribution In technical terms, P g (φ)=q distribution

55 Phase distribution measurement Heterodyning a coherent field S Inject a coherent field α> Atom in g> Add a coherent amplitude αe iφ Resulting field (within global phase) α(1-e iφ )> Zero final amplitude for φ=0 Probe final field amplitude with atom in g P g =1 for a zero amplitude P g =1/2 for a large amplitude More generally: P g (φ) reveals field phase distribution In technical terms, P g (φ)=q distribution

56 Phase distribution measurement Heterodyning a coherent field S Inject a coherent field α> Atom in g> Add a coherent amplitude αe iφ Resulting field (within global phase) α(1-e iφ )> Zero final amplitude for φ=0 Probe final field amplitude with atom in g P g =1 for a zero amplitude P g =1/2 for a large amplitude More generally: P g (φ) reveals field phase distribution In technical terms, P g (φ)=q distribution

57 Phase distribution measurement Heterodyning a coherent field S Inject a coherent field α> Add a coherent amplitude αe iφ Resulting field (within global phase) α(1-e iφ )> Zero final amplitude for φ=0 Probe final field amplitude with atom in g P g =1 for a zero amplitude P g =1/2 for a large amplitude More generally: P g (φ) reveals field phase distribution In technical terms, 2xP g (φ)-1=q(α) distribution

58 Pg Coherent field phase distribution 0,9 0,8 0,7 0,6 0,5 0,4 Homodyning signal Microwave Attenuation injection 76dB 73dB 70dB 67dB width (in degreas) Width versus <n> 64 db - 23 ph 76 db ph Phase ( ) 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 1 n One can apply the same phase measurement after interaction with an atom Detection of states Ψ c+ and Ψ c-

59 Phase splitting in quantum Rabi oscillation Timing S 2nd atom 1st atom Inject a coherent field Send a first atom: Rabi oscillation and phase shift Inject a phase tunable coherent amplitude Send a second atom in g: final amplitude read out

60 Phase splitting of a mesoscopic field versus <n> and interaction time 335 m/s 200 m/s ,7 0,6 0,5 S g (φ) n (degrees) (degrees) Φ = Ω 0 t 4 n A. Auffèves et al., Phys. Rev. Lett. 91, (2003)

61 Phase splitting in quantum Rabi oscillation Observed phase versus theoretical phase Phase (degrees) y x Φ = Ω 4 0 t n Numerical simulation of field master equation φ + (degrees) Large Shrödinger cat states (up to 40 photons separation)

62 How to test that one has a coherent superposition for the field state Interference between the two states Waiting for the revival will do the trick If atomic populations oscillates again then it proves the coherence But interaction time is limited Maximum interaction time 100µs for atoms at 100 m/s Revival time for n=25: 300µs

63 Test of coherence: induced quantum revivals Initial Rabi rotation, Collapse And slow phase rotation Stark pulse (duration short compared to phase rotation). Equivalent to a Z rotation by π Reverse phase rotation Recombine field components and resume Rabi oscillation

64 Test of coherence: induced quantum revivals Initial Rabi rotation, Collapse And slow phase rotation Stark pulse (duration short compared to phase rotation). Equivalent to a Z rotation by π Reverse phase rotation Recombine field components and resume Rabi oscillation

65 Test of coherence: induced quantum revivals Initial Rabi rotation, Collapse And slow phase rotation Stark pulse (duration short compared to phase rotation). Equivalent to a Z rotation by π Reverse phase rotation Recombine field components and resume Rabi oscillation

66 Test of coherence: induced quantum revivals Initial Rabi rotation, Collapse And slow phase rotation Stark pulse (duration short compared to phase rotation). Equivalent to a Z rotation by π Reverse phase rotation Recombine field components and resume Rabi oscillation

67 Induced quantum revivals (I) Effect on the field observed by homodyning 0,80 t=0 15 photons injected t=t π =22 µs t=2t π signal 0,70 0,60 0, phase ( ) Two components clearly resolved Separated in phase space by a distance of 9 photons Associated decoherence time 100 µs

68 Induced quantum revivals (II) Atomic population as a function of π pulse delay Pg 1,0 0,8 0,6 0,4 0,2 1,0 T π =18.5 µs T π =23.5 µs 0,8 0,6 0,4 0,2 0, Interaction time Contrast decreases when T π increases simulation of the experimental data Interaction time Main limitation due to inhomogeneous effect (velocity dispersion) Additional decoherence term 0,0

69 Conclusion Study of decoherence in the frame of cavity QED A Schödinger cat for a «trapped field» In good agreement with theory Effect of the size of the cat Close relation with the theory of measurement in quantum physics Possible application and testground for basic Quantum information processing experiments

70 Lecture 4: future directions Gilles NOGUES Laboratoire Kastler Brossel Ecole normale supérieure, Paris

71 The team Frédérick.Bernardot Paulo Nussenzweig Abdelhamid Maali Jochen Dreyer Xavier Maître Gilles Nogues Arno Rauschenbeutel Patrice Bertet Stefano Osnaghi Alexia Auffeves Paolo Maioli Tristan Meunier Philippe Hyafil * Sébastien Gleyzes Jack Mozley * Christine Guerlin Thomas Nirrengarten* PhD Post doc Ferdinand Schmidt-Kaler Edward Hagley Christof Wunderlich Perola Milman Stefan Kuhr Angie Quarry* Luiz Davidovich Nicim Zagury Wojtek Gawlik Daniel Estève Gilles Nogues * Michel Brune Jean-Michel Raimond Serge Haroche Collaboration Permanent *: atom chip team

72 Outline More about the field QND detection of more than one atom Measurement of the Wigner distribution More about the atom QND detection of the atomic state A Schrödinger cat state for a mesoscopic ensemble of atoms New experimental tools A two-cavity setup

73 Outline More about the field QND detection of more than one atom Measurement of the Wigner distribution More about the atom QND detection of the atomic state A Schrödinger cat state for a mesoscopic ensemble of atoms New experimental tools A two-cavity setup

74 QND detection experiment Only works for 0> and 1> If one has 2 photons in the cavity Rabi frequency Ω xΩ 0 A 3π pulse is performed g,2> e,1> One photon is absorbed and atom is neither in g or i 2 main problems A measured qubit can only provide one bit of information Dispersive interaction is required to prevent energy exchange for all photon number

75 Dispersive interaction Dispersive regime : δ= ω at ω cav > No energy exchange Ω/2 ω cav e> ω at g> δ But : light shift E E 2 e, n + = Ω (n 1) 4δ 2 = n 4δ Ω g, n 1 ( ) 2 e g Phase shift + i Φ( n) of Ramsey fringes on the e-g transition 1 P(e) 1 ( e e g ) 2 + Φ(n)=Φ 0 n φ(n) Φ 0= Ω 2 / δt int Empty cavity 0> Fock state n> 0 φ

76 Parity Measurement Φ 0 =π For φ=φ* : If N even, detection in e If N odd, detection in g 1 P(e) 0 φ N even Vacuum 0> N odd State 1> State ρ C φ The measurement of the final atomic state gives the parity operator value P=e ia+ a P=+1 for state 2n> P=-1 for state 2n+1> ^ = n φ n P ( 1) P(n) = Pφ *(e)-p *(g) = C

77 Photon number decimation Ramsey interferometer set on a bright fringe for 0> P(n) Initial distribution Φ(n)=Φ 0 n=π.n n P g (n) e> g> P(n) P(n) Final conditional probabilities n n

78 A complete QND measurement Adapt the phase shift per photon and the interferometer phase at each step Φ 0 =π P(n) P g (n) n Number of steps: P(n) D étection dans e> P(n) D étection dans g> N s =Log 2 (<n>) Φ 0 =π/ 2 P g (n) n n P(n) D étection dans e> P(n) D étection dans g> Φ 0 =π/ 4 P g (n) n n P (n) Détection dans e> P(n) D étection dans g> n n

79 The first step of the QND measurement Measurement of the Wigner distribution

80 Wigner function: an insight into a quantum state A quasi-probability distribution in phase space. Characterizes completely the quantum state Negative for non-classical states. p Im(α) Describes the motion of a particle or a quantum single mode field q Re(α) Motion of a particle Electromagnetic field

81 Properties Definition By inverse Fourier transform In particular The probability distribution of x is obtained by integrating W over p This property should obviously be invariant by rotation in phase space W q cosθ p sin θ, q sinθ + p cosθ dp ( ) θ = P q = q Uˆ ˆUˆ q ( ) ( θ ) ρ ( θ ) θ All elements of density matrix derived from W: contains all possible information on quantum state.

82 Examples of Wigner functions Coherent state β> Vacuum 0> Thermal field nth=1 (β= i) Fock state 1> photons Fock state

83 How to measure W for the electromagnetic field? Propagating fields : «Tomographic» methods Principle : - Homodyning measures marginal distributions P(q θ ) for different θ - inverse Radon transform allows reconstruction of W(q,p) ( medical tomography) Refs : - Coherent and squeezed states : - Smithey et al., PRL 70, 1244 (1993) - Breitenbach et al., Nature 387, 471 (1997) - One-photon Fock state : Lvovsky et al., PRL 87, (2001) - α 0>+β 1> : Lvovsky et al., PRL 88, (2002)

84 RESULTATS EXPERIMENTAUX Smithey et al., PRL 70, 1244 (1993) Breitenbach et al, Nature 387, 471 (1997) Comprimé Vide

85 MESURE COMPLETE DE LA DISTRIBUTION DE WIGNER POUR UN PHOTON Lvovsky et al, PRL 87, (2001)

86 Other methods Use the link between W and parity operator ^ W(α) = 2Tr(D( α)ρ D(α) ( 1) N ) ^ ˆ Displace the field and measure parity by determination of photon number probability Direct counting (Banaszek et al for coherent states) Quantum Rabi oscillations for an ion in a trap (Wineland) A demanding method. Much more information than the mere average parity needed

87 Wigner distribution for a trapped ion Etat nombre 1 n = ( ) 1 2 Matrice densité D. Liebfried et al, PRL 77, 4281 (1996), NIST, Boulder Same outcome for trapped neutral atom: - G.Drobny and V. Buzek, PRA (2002) From the data of C. Salomon et I. Bouchoule

88 Our approach - Proposed by Lutterbach and Davidovich (Lutterbach et al. PRL 78 (1997) 2547) - Based on : ρ ^ W(α) = 2Tr(D( α)ρd(α)( 1) D(-α) ρ ρ(-α) (-α) W is the expectation value of the Parity operator the displaced state ρ( α) ^ ˆ ( 1) N n = Nˆ ) ( 1) A) Apply D(-α) Inject α in cavity mode OK B) Parity measurement directly gives ( 1) «parity» operator + n if n=2k n Nˆ if n=2k+1 Nˆ in

89 Testing the method: vacuum state Wigner function e-g detection position Atomic frequency ν cav D(-α) π/2 R1 g,0> δ Dispersive interaction φ π/2 R2 Cavity mode time Use Stark effect to tune interferometer phase No phase information in cavity field: injected field phase irrelevant Finite intrinsic contrast of the Ramsey interferometer

90 Wigner function of the "vacuum" P(e) α= P(e) α=0.6 πw(α) 2 (norm.) N phot α= P(e) φ/π α 2

91 Single photon Wigner function measurement e-g detection φ π/2 R2 position Atomic frequency ν cav e,0> π D(-α) π/2 R1 g,1> δ Dispersive interaction Cavity mode time Preparation of cavity state Wigner function measurement scheme

92 Wigner function of a "one-photon" Fock state 0,7 0,6 α=0 1,0 P(e) P(e) 0,5 0,4 0,3 0,7 0,6 0,5 0, α=0.3 Φ/ π (norm.) πw(α) 0,5 0,0-0,5-1,0 0,0 0,5 1,0 1,5 2,0 α P(e) 0,3 0,7 0,6 0,5 0,4 0, α= Φ/ π Φ/ π Nphot

93 Towards other states - Cavity QED setup : direct measurement of the field Next improvements : - better isolation - better detectors 1 0 More complex states : ex ( )/ In the future : «movie» of the decoherence of a Schrödinger cat

94 Outline More about the field QND detection of more than one atom Measurement of the Wigner distribution More about the atom QND detection of the atomic state New experimental tools A two-cavity setup

95 Phase shift with dispersive atom-field interaction Atom in g Non resonant atom: no energy exchange but cavity mode frequency shift (atomic index of refraction effect). Phase shift of the cavity field (slower than in the resonant case) 0,70 0,65 0,60 0,55 0,50 0,45 Atom in g 0,75 0,70 0,65 No atom 1 atom in g 2 atoms in g No atom 0,75 0,70 0,65 0,60 0,55 0, No atom 0,60 0,55 0, ,70 0,65 0,50 Atom in e 0,60 0,55 Atom in e 0,45 0,50 0, Opposite values for e and g Phase ( ) Proportional to atom number

96 Absolute measurement of atomic detection efficiency Histogram of field phase reveals exact atom count Comparison with detected atom counts provides field ionization detectors efficiency in a precise and absolute way 0.4 atoms samples: % detection efficiency

97 Towards a 100% efficiency atomic detection Inject a very large coherent field in the cavity Im(α) Send an atomic sample Different phase shifts for e, g or no atom φ φ g e Re(α) Inject homodyning amplitude Zero amplitude for e. Larger for no atom. Still larger for g g e Read final field amplitude by sending a large number of atoms in g Final number of atoms in e proportional to photon number

98 Preliminary experimental results Probability 0,18 0,16 0,14 0,12 0,10 0,08 Atom 1 prepared in g no atom 1 atom in g Atom1 prepared in e no atom 1 atom in e Experimental conditions: 75 photons initially v=200 m/s δ=50 khz 70 absorber atoms 0,06 0,04 0,02 0, Total number of excited atoms detection efficiency: 87% error probability: 0 atom detected as 1: 10% (main present limitation) e in g: 1.6% g in e: 3% 100% detection efficiency within reach with slower atoms: v=150 m/s.experiment in progress.

99 Outline More about the field QND detection of more than one atom Measurement of the Wigner distribution More about the atom QND detection of the atomic state New experimental tools A two-cavity setup

100 A two-cavity experiment Rydberg atoms and superconducting cavities: Towards a two-cavity experiment Creation of non-local mesoscopic Schrödinger cat states Non-locality and decoherence (real time monitoring of W function) Complex quantum information manipulations Quantum feedback Simple algorithms Three-qubit quantum error correction code

101 Teleportation of an atomic state C R 2 D a beam 2 P b beam 3 P a R R 3 A 1' Davidovich et al, PHYS REV A 50 R895 (1994) B C 1 3' EPR pair 3 C 2 0 D c D b beam 1 This scheme works for massive particles Detection of the 4 Bell states and application of the "correction" to the target is possible using a C-Not gate (beam 2 and 3) The scheme can be compacted to 1 cavity and 1 atomic beam

102 Entangling two modes of the radiation field Principle: First atom Initial state e,0,0 Single photon beats 1.0 P e (T) 0.5 (a) π/2 pulse in M a π pulse in M b 1 2 e g 1 0,1 1, 0 2 g + (,0,0 +,1,0 ) ( ) (b) 0 δ Second atom: A s probes field states Final transfer rate modulated versus the delay at the beat note between modes π/2 π D A p π (b) π/2 0 π/2ω 3π/2Ω T Τ+π/Ω M a M b D t (c) (d) T(µs)

103 Implementation of 3 qubit error correction R encoding R R R R R R Error? R decoding σ z2 σ z3 Correction α S S Détection Correction Détection 0 + β 1 error detection Ramsey π/2 pulses Error encoding and decoding: preparation of a GHz triplet all the tools exist!

104 New non-locality explorations Use a single atom to entangle two mesoscopic fields in the cavity A non-local Schrödinger cat or a mesoscopic EPR pair Easily prepared via dispersive atom-cavity interaction

105 Mesoscopic Bell inequalities A Bell inequality form adapted to this situation Here, Πis the parity operator average. Dichotomic variable for which the Bell inequalities argument can be used (transforms the continuous variable problem in a spin-like problem) Maximum violation for parity entangled states:

106 Bell inequalities violation Optimum Bell signal versus γ

107 Probing the Wigner function A second atom to read out both cavities (same scheme as for single mode Wigner function)

108 A difficult but feasible experiment Bell signal versus time T c =30 and 300 ms

109 Outline More about the field QND detection of more than one atom Measurement of the Wigner distribution More about the atom QND detection of the atomic state New experimental tools A two-cavity setup