1. Quantum information: From foundations to experiments Second Lecture Luiz Davidovich Instituto de Física Universidade Federal do Rio de Janeiro BRAZIL

2. Possible level schemes

3. Measurement of the parity of the field

4. Atom in superposition of two states  superposition of two refraction indices (two media)  superposition of two fields with different phases M. Brune, J.M. Raimond, S. Haroche, L.D. et N. Zagury, PRA 45, 5193 (1992) Measuring decoherence in cavity QED

5. HOW TO DETECT THE COHERENCE ? Send a second atom! [L.D., A. Maali, M. Brune, J.M. Raimond, and S. Haroche, PRL 71, 2360 (1993); L.D., M. Brune, J.M. Raimond, and S. Haroche, PRA 53, 1295 (1996)]. Results for phase difference equal to  : Coherent superposition: preparation and probing atoms detected in the same state  P ee Coherent superposition: preparation and probing atoms detected in the same state  P ee  Statistical mixture: second atom detected in  e  or  g  with 50 % chance  P ee  /2Statistical mixture: second atom detected in  e  or  g  with 50 % chance  P ee  /2

6. PHYSICAL INTERPRETATION FOR  : DETECTION OF FIELD PARITY  /2 rotation Even number of photons: 2k  rotation (dispersive interaction)  /2 rotation Bloch sphere

7. A VARIANT Displace field in the cavity by  (by turning on the microwave field): What about dissipation? Exit of just one photon is enough to destroy superposition! If damping time of field is t cav, then it takes t cav /  n  t cav /4  2 for one photon to leave the cavity if state is  2 . Since there is only a 50% chance that the field is in this state, the time should be twice as large: t cav /2  2 Superposition of dark and lighted cavity

8. EFFECT OF DISSIPATION    t cav  n  Decoherence time: t cav  D  n   average number of photons in cavity L. D., M. Brune, J.M. Raimond, and S. Haroche, Phys. Rev. A 53, 1295 (1996).

9. EXPERIMENTAL RESULTS [Brune et al., PRL 77, 4887 (1996)] Plot of P ee  P eg

10. What about state of the field in the cavity? For classical particle with position q and momentum p, state is defined by distribution of points in phase space (just one point if one has precise information on q and p). Could this be done for the quantized electromagnetic field? Can one measure this phase-space representation?

11. PHASE-SPACE REPRESENTATION Look for representation with following properties: Pure state: Property must remain true if axes are rotated:

12. RADON TRANSFORM (1917) P(q  ) uniquely determines W(q,p)!  Radon inverse transform  tomography P(q  ) uniquely determines W(q,p)!  Radon inverse transform  tomography Cormack and Hounsfield: Nobel prize in Medicine (1979) Quantum mechanics: P(q  )  Wigner function (Bertrand and Bertrand, 1987)

13. THE WIGNER DISTRIBUTION Wigner, 1932: Quantum corrections to classical statistical mechanics Moyal, 1949: Average of operators in symmetric form: Density matrix in terms of W:

14. PAULI’S QUESTION Handbuch der Physik, 1933 – “The mathematical problem, as to whether for given functions W(x) and W’(p) [position and momentum probability densities], the wave function , if such a function exists, is always uniquely determined has not been investigated in all its generality.”

15. EXAMPLES OF WIGNER DISTRIBUTIONS Ground state Fock state n=3 Mixture  Superposition   Experimentally produced (ions, cavities)

16. MEASUREMENT OF THE MOTIONAL QUANTUM STATE OF A TRAPPED ION Wineland’s group – PRL 77, 4281 (1996)

17. Field quadratures

18. Phase-shift operator and generalized quadratures Generalized quadratures: Special cases:

19. MEASUREMENT OF QUADRATURES Risken and Vogel, 1989: homodyne measurements  P(q  )  Wigner function for EM field

20. EXPERIMENTAL RESULTS Smithey et al., PRL 70, 1244 (1993) SqueezedVacuum Breitenbach et al, Nature 387, 471 (1997)

21. MEASUREMENT OF THE WIGNER FUNCTION FOR ONE PHOTON Lvovsky et al, PRL 87, 050402 (2001)

22. WIGNER FUNCTION AND THE CLASSICAL LIMIT OF QUANTUM MECHANICS Dissipation leads to disappearance of interference fringes plus evolution towards ground state Decay time for fringes =dissipation time/2|  | 2 Evolution of coherent superposition of coherent states of harmonic oscillator, with dissipation    Fast decoherence: one needs a snapshot!

23. Another expression for the Wigner function

24. Displacement operator Translates position and momentum (or quadratures) in phase space Corresponds to action of external force for harmonic oscillator, or external current for the field

25. DIRECT MEASUREMENT OF THE WIGNER DISTRIBUTION L.G. Lutterbach and L.D., PRL 78, 2547 (1997) Based on following expression for Wigner function (Cahill and Glauber, 1969): Displacement operator Parity operator (phase shift of the field) Banaszek and Wódkiewicz (1996), Wallentowitz and Vogel (1996), Messina, Manko and Tombesi (1998), Banaszek et al (1999). Also used by Wineland’s group to measure Wigner function for vibrational state of trapped ion.

26. EXPERIMENTAL PROPOSAL 1.Displace field to be measured (turn on microwave) 2. Send atom, displace phase of the field by  iff atomic state is  e  3. Detect atomic state 4. Produce field anew, repeat procedure Problem: must produce shift equal to 

27. Quantum circuit for measuring the Wigner function

28. MEASUREMENT OF THE QUANTUM STATE OF A PHOTON IN A CAVITY – ENS Measurement of sub-Planck phase-space structure

29. A sensitive instrument… W. Zurek, Nature 412, 712 (2001) X p PXPX

30. Using this instrument for measuring small displacements and rotations

31.  System is prepared in a known input state  which experiences a small displacement transforming X into X+  We want to infer with minimum error from measurement performed on the displaced state  ’  exp(  i P) , where P is the momentum operator  If one measures X, then precision in the determination of the displacement is limited by  (width of wavepacket)  For coherent state, Measuring small displacements and rotations - standard quantum limit

32. Measurement of weak classical forces Classical force acting for a fixed time on a simple harmonic oscillator displaces the complex amplitude of the oscillator in phase space Action of the force in the interaction picture: Must resolve displacement in order to measure the force

33. Interference regions in Wigner function (“sub-Planck” structures) In order to have Minimum translation: Effect of small displacement Weak-force detection Much better than standard limit!

34. Wigner function of unperturbed states Product of Wigner functions: integration over blue areas cancels out integration over red areas Wigner function of perturbed states Small rotations

35. General strategy for measurement: couple oscillator to two-level system LOSCHMIDT ECHO!

36. Revisiting collapses and revivals J.H. Eberly, N.B. Narozhny, and J.J. Sanchez Mondragon, Phys. Rev. Lett. 44, 1323 (1980) J. Gea-Banacloche, Phys. Rev. A 44, 5913 (1991) Initial state  e , resonant interaction, described by Jaynes-Cummings model: Atom gets disentangled from field at time Field is left in a superposition of two coherent states  Displace it!

37. Echoes Echoes arise when through suitable manipulations in a system the dynamics is reversed and a more or less complete recovery of the initial state is achieved (ex: acoustical echoes arising from reflections of sound at walls) J. Loschmidt, Sitzungsber. Kais. Akad. Wiss. Wien Math. Naturwiss. Klasse 73 128–142 (1876) A. Peres, Phys. Rev. A 30, 1610–15 (1984) - Application to chaos

38. How to invert motion? Apply percussive 2  pulse to state  e  Effect on state:  e   e  Effect on operators: ee gg ii pp G. Morigi, E. Solano, B.-G. Englert, and H. Walther, Phys. Rev. A 65, 040102(R) (2002)

39. Atoms and photons as qubits Two-level atoms ee gg Cavity with zero or one photon Measurement of atom: Measurement of field ionization

40. Information transfer and entanglement How to calibrate interaction time: apply potential between mirrors, taking through Stark effect atoms in and out of resonance with field mode Cavity mode: quantum databus

41. CNOT with cavity (not quite…)

42. TELEPORTATION Alice wants to transmit to Bob quantum state of system in her possession (example: photon polarization state). Alice and Bob share an entangled state: Alice has serious problems! Bennett et al, PRL (1993) Alice implements two binary measurements on her pair of spins and informs Bob, who applies appropriate transformations to his spin so as to reproduce original state Just two bits!

43. TELEPORTATION II Three qubits state: Alice measures her Bell states and informs Bob, who applies appropriate transformations to his qubit Just two bits!

44. DETECTION OF BELL STATES Hadamard gate: H Reversible: Production and analysis of Bell states

45. Teleportation with cavities L.D., N. Zagury, et al, PRA 50, R895 (1994) “Teleportation machine”

46. Recent implementation Zeilinger et al, Nature 430, 849 (2004)

47. Conclusions Cavity QED offers the possibility of exploring fundamental phenomena in quantum mechanics Realization of quantum gates, proposals for experiments on teleportation of quantum states Study of the dynamics of the decoherence process Direct measurement of the quantum state of the electromagnetic field, Heisenberg-limited measurements