1. Logical gates and quantum processors with trapped ions and cavities MIGUEL ORSZAG FACULTAD DE FISICA PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE Course at the Institut Fourier, Universite Joseph Fourier, Grenoble, France May 2006

2. QUANTUM COMPUTATION AND INFORMATION 1982 Feyman: It is possible to improve the computation using Quantum Mechanics. 1985 Deutsch: Describes quantum computer model similar to the TURING MACHINE. 1994 Shor: Proposes a factorization algorithm.

3. Classical Information unit: Bit Quantum Information Unit: Quantum Bit.QUBIT Qubit: Microscopic system limited by two quantum states. Qudit: Microscopic system limited by N Quantum States. QUANTUM COMPUTING AND QUANTUM INFORMATION Superposition

5. 1 qubit gate

6. 01010101 Control Target C-NOT

7. 01010101 target 2 qubit gate control QUANTUM GATES C_NOT GATE

8. QUANTUM GATES Controlled Not (C-NOT) Qudit a a b b

9. Toffoli Control 1 Target 01010101 01010101 Control 2 3 qubit gate

10. QUANTUM GATES Toffoli a b c The target Only changes if both Controls J=k=1 3 qubit gate control1 control2 target

11. NO CLONING THEOREM Wooters-Zurek (1982) It is impossible to clone an arbitrary quantum state. UNIVERSAL QUANTUM COPYING MACHINE UQCM Buzek- Hillery (1996) Analize copies restricted by the no-cloning theorem. Quantum Copying Machines Fidelity: 2

12. Universal Quantum Copying Machine UQCM Ideal copying process: Analysis of the copies : Quantum Copying Machines 1. 2. 3.An ideal copy should be written as original blanc machine Hilbert-Schmidt NORM

13. QUANTUM COPYING MACHINE Result: WANTED STATE UNWANTED STATE Universal Quantum Copying MachineUQCM

14. IMPLEMENTED BY A CIRCUIT QUANTUM COPYING MACHINE

15. Preparation phase Quantum Copying Machine implemented by a circuit QUANTUM COPYING MACHINES

16. Implemented by a circuit QUANTUM COPYING MACHINES Copying phase

17. UQCM: Duplicator: Triplicator: QUANTUM COPYING MACHINES Analysis of the copies Universal, doesnot depend on the input state Universal,but restricted To real numbers(input coeff)

18. Apparatus capable of developing any function within the DATA REGISTER, such funtion being specified by a PROGRAM REGISTER. It is not possible to build a deterministic quantum processor with a finite number of resources, but it is possible to build a probabilistic one. CLASSICAL PROCESSOR QUANTUM PROCESSOR QUANTUM PROCESSORS

19. When implementing a set of inequivalent operations, the program state apace must contain a set of mutually orthogonal states. This means that the dimension of the Program Register must be as big as the number of unitary operators we want to implement.Since this number (of unitary operators) is infinite, it is not possible to build a processor with a finite number of resources. HOWEVER, WE CAN ALWAYS BUILD A PROBABILISTIC PROCESSOR (quantum measurements are implied) QUANTUM PROCESSORS Limitations due to Quantum Mechanics.

20. Fixed Unitary Operator (processor) State of the DATA Program State Residual State (independent of the data) This Type of Processor is necessarily stochastic U Unitary Operator ACTING ON THE DATA ONY

21. Stochastic Processor for a qubit We want: How to implement it???

22. A STOCHASTIC QUANTUM PROCESSOR 1.-G.Vidal,L.Masanes,J.I.Cirac,PRL,788,047905(2002) 2.-M.A.Nielsen,I.L.Chuang,PRL,79,321(1997) 3.-M.Hillary,V.Buzek,M.Ziman,PARA,65,022301(2002) Data qubit Program 1 qubit Program 2 qubit C_ NOT GATE Toffoli Gate

23. To understand the procedure, consider a single Program Register. First we define the program and the data PROGRAM STATE DATA STATE

24. + In this case, a measurement on the program register will cause a collapse On the data qubit with the outcome. If we measure 0 in the program, we get the good answer, If we measure 1, we get the wrong answer With a probability 1/2 + Bad result is for 1in P1

25. To improve upon this scheme, we introduce a Toffoli Gate, as in the Figure. The data line and the first program qubit are unchanged, however,If the output of the program register is Indicating a failure, the Toffoli gate effectively acts like a C-Not Gate Between the data line and a second program qubit There is again a probability 1/2 of getting this time THEREFORE, WE HAVE INCREASED THE SUCCESS PROBABILITY to 3/4 (and so on…) Bad result only in the 11 case

26. One can generalize the argument, including MORE GENERALIZED TOFFOLI GATES, and having a success probability of Where N is the number of Program qubits or generalized Toffoli Gates (also the number of measurements) The price we pay is the increase of gates and number of measurements

27. Efficiency of quantum processors

28. PROCESADORES CUANTICOS Processors performance

29. PROCESADORES CUANTICOS Processors performance I

30. State of Data Operation to implement: QUANTUM PROCESSORS Processor Performance I

31. Program State: Program Register base: PROCESADORES CUANTICOS Processors performance I

32. State that inputs the machine QUANTUM PROCESSORS Processors performance I

33. The output state is: QUANTUM PROCESSORS Processors performance I

34. QUANTUM PROCESSORS Processors performance

35. QUANTUM PROCESSORS Processors performance II

36. Program state: QUANTUM PROCESSORS Processors performance II Input state

37. Output State: QUANTUM PROCESSORS Processors performance II

38. QUANTUM PROCESSORS Processors performance II

39. Output State: QUANTUM PROCESSORS Processors performance II

40. QUANTUM PROCESSORS Processors performance II If we measure in: Success probability Error probability: 1. 2.

41. PROCESADORES CUANTICOS Processors performance

42. PROCESADORES CUANTICOS Processors performance III

43. Program State: Input State: QUANTUM PROCESSORS Processors performance III

44. Output State: QUANTUM PROCESSORS Processors performance III

45. Output State: QUANTUM PROCESSORS Processors performance III

46. QUANTUM PROCESSORS Hillery-Buzek (2002)Qubits Base de Bell P=1/4

47. QUANTUM PROCESSORS Hillery-Buzek (2002)Qudits

48. QUANTUM PROCESSORS Efficiency of processors

49. QUANTUM PROCESSORS Efficiency of the Vidal-Cirac 2002 processor

50. Every time one gets an error, it is possible to correct it, provided one has the l-th program state State of the Program: This state can be used to implement the Unitary U with a probability QUANTUM PROCESSORS Efficiency of the Vidal-Cirac 2002 processor

51. CONCLUSIONS Quantum Copying Machines Every preparation Program is realized with 2 C-Not and 3 rotations, for each preparation state one can find 8 possible sets of rotation angles.. In spite of the No-Cloning Theorem, it is always possible to build a copying machine with a fidelity different from one.(5/6 is the optimum) The machine described here can produce 2 or 3 copies according to the preparation. The copies obtained here have a fidelity factor of F =variable(max 5/6)

52. CONCLUSIONS Processors In spite of the fact that is impossible to build a deterministic processor (with finite resources), it is feasible to build a probabilistic one. The Buzek proposal for a processor, for qubits, can be also used to implement an unknown operation U, with a certain probability p. It is possible to generalize this processor and work with qudits of dimension N. The probability of implementation of the Cirac proposal can be incremented by a feedback procedure, approaching one. There is an effort to implement the probability with a single step, together with the factibility of implementing it with known gates….

53. CONCLUSIONS Copying machines, processors The cloning machine uses the same set of operators P31 P21 P13 P12 as compared to the processor. The same machine has two applications Quantum Copying Quantum Processor

55. QUANTUM PROCESSORS Effiviency of the processors Hillery-Buzek 2004 The processor is given by the unitary operator A jk is an operator in H d, {|j, |k are the basis for H p The required operation is The data and program states are It is useful to take c 1 = z c 0

56. QUANTUM PROCESSORS Efficiemcy of Processors Hillery-Buzek 2004 1.Case with only one program state, the operator A is Summ módulo 2 2.To improve the probability, one uses a vector program given by c k+1 = z c k In the two program case, the operator A Summ módulo 4

57. 3.To generalize this process, an N dimensional vector program is used QUANTUM PROCESSORS Rendimiento de Procesadores Hillery-Buzek 2004 In the case of N program states, A is given by Summ módulo N

58. QUANTUM PROCESSORS Rendimiento de Procesadores Hillery-Buzek 2004 Therefore, the processor is given by the operator Summ módulo N The success probability approaches to 1

59. PROPOSAL OF A UNIVERSAL QUANTUM COPYING MACHINE IN CAVITY QED Joanna Gonzalez Miguel Orszag Sergio Dagach Facultad de Física Pontificia Universidad Católica de Chile

60. The Jaynes Cummings model The Jaynes-Cummings Model describing the interaction Of a single two level atom with a single quantized cavity mode Of the radiation field plays a central role in quantum optics. In spite of the mathematical simplicity of this model, is physically Realistic. It describes purely quantum mechanichal phenomena like Rabi oscillations, collapses and revivals of the atomic inversion, subPoissonian statistics and squeezing of the cavity field. Also, experiments with highly excited Rydberg atoms in high Q Cavities have allowed to investigate experimentally the interaction of a single atom with a single cavity mode Thus proving experimentally the predictions of the JC Model.

61. The Hamiltonian in the dipole approximation can be written as Creation photon and destr atomic unit of en destr photon and creation atomic unit of en destr photon and destr atomic unit of en Creation photon and creation atomic unit of en JAYNES CUMMINGS MDEL TWO LEVEL ATOM AND DIPOLE APPROXIMATIONS Normally in QO this last two terma are neglected in the So-called RWA

62. a b Two level atom It is simple to prove that under the JC dynamics, and if The atom is initially in the a state, and the field with n photons (Fock state) Rabi oscillation and was originally studied in NMR

63. If initially we donot have a Fock state but a rather A combination of Fock states, a coherent state Then a curious phenomena takes place, Called COLLAPSE AND REVIVAL

64. Pa-Pb

65. Constructive and destructive interference between the various Rabi oscillations.Apparently the Poisson distribution has the correct Factors for total collapse. If we try a thermal state, will observe ONLY PARTIAL COLLAPSES

66. OSCILLATION VERSUS EXPONENTIAL DECAY For a long time the Jaynes-Cummings Model appeared to be A highly academic model. It described an atom and a field that periodically pass the energy From one to the other like two coupled pendulum or two coupled oscillators. On the other hand, people saw in the LAB a different reality. ATOMS DECAYED FROM EXCITED TO GROUND STATES. Of course there was the Wigner Weiskopf theory of spontaneous Emission (Also Einstein phenomenological theory) that explained Such a decay if one took AN INFINITE NUMBER OF OSCILLATORS COUPLED TO THE ATOM.

67. RYDBERG ATOMS However, the situation has changed over the last two decades. The introduction of highly tunable Dye Lasers, which can excite Large population of highly excited atomic states with a high Main quantum number n.These atoms are referred as Rydberg Atoms. Such excited atoms are very suitable for atom-radiation experiments Because they are very strongly coupled to the radiation field, since The transition rates between neighbouring levels scale as n^4. Also transitions are in the microwave region where photons live longer, thus allowing longer interaction times. Finally, Rydber atoms have long lifetimes with respect to spontaneous decay. THE STRONG COUPLING OF THE RYDBERG ATOMS TO THE FIELD CAN BE UNDERSTOOD SINCE THE DIPOLE MOMENT SCALES WITH n^2, (typical n~70 )

68. MICROMASER

69. A one atom maser is described in the previous figure. A collimated beam of Rubidium atoms is passed through A velocity selector. Before entering a high Q superconducting microwave cavity The atom is excited to a high n-level and converted In a Rydberg atom. The microwave cavity is made of niobium and cooled down To a low temperature. The Rydberg atoms are detected in the upper or lower level By two field ionization detectors with their fields Adjusted so that in the first detector, only atoms in the Upper state are ionized.

70. MICROMASER MASER OPERATION (Walther et al) WAS DEMONSTRATED BY TUNING THE CAVITY TO THE MASER TRANSITION AND RECORDING SIMULTANEOUSLY, THE FLUX OF ATOMS IN THE EXCITED STATE. AS SHOWN IN THE FIGURE, OM RESONANCE, A REDUCTION OF THE SIGNAL WAS OBSERVED FOR RELATIVELY SMALL ATOMIC FLUXES (1750 ATS^-1) HIGHER FLUXES PRODUCE POWER BROADENING AND A SMALL FRQUENCY SHIFT. ALSO THE TWO PHOTON MICROMASER WAS DEMONSTRATED(HAROCHE ET AL)

71. Meschede,Walther,Muller,PRL,54,551(1984), Rb85 63p3/2….. 61d3/2, Qcavity~10^8, Tcav~2K, nth~2

72. The No-Cloning Theorem (Wooters and Zurek,Nature299,802(1982)) showed that it is not possible to construct a device that will produce an exact copy of an arbitrary quantum state. This Theorem is an unexpected quantum effect due to the linearity of Quantum Mechanics, as opposed to Classical Physics, where the copying Process presents no difficulties, and this represents the most significant difference between Classical and Quantum Information. Thus, an operation like:

73. Is not possible, with: =INPUT QUBIT =initial state of cloner =Blank copy =final state of cloner Because of this Theorem, scientist ignored the subject up to 1996 when Buzek and Hillery (V.Buzek,M.Hillery,Phys.Rev.A,54,1844(1996) proposed the Universal Quantum Copying Machine(UQCM)-that produced two imperfect copies from an original qubit, the quality of which was independent of the input state.

74. UNIVERSAL QUANTUM COPYING MACHINE BASIS The quality of the copy is measured through the FIDELITY

75. In the present work, we propose a protocol that produces 2 copies from an input state, with Fidelity In the context of Cavity QED, in which the information is encoded in the electronic levels of Rb atoms, that interact with two Nb high Q cavities. SOME PREVIOUS BACKGROUND TO THE PROPOSAL Consider a two level atom that is prepared in a superposition state, using the Microwave pulses in a Ramsey Zone, with frequency Near the e(excited)-g(ground) transition. It generates superpositions Depends on the interaction time Is prop. detuning

76. On the other hand, the atom-field interaction is described by the Jaynes Cummings Hamiltonian Coupling constant

77. The atom-field state evolves like For example, for

79. Now, consider an external Classical pulse, interacting with the atom We use the dressed state basis that diagonalizes the J-C Hamiltonian:, The Energies of the dressed states are

80. In the limit Consider the external field in resonance with the (+,1)- (-,0) Transition, that is Where f(t) is some smooth function of time to represent the pulse shape, with(in the dispersive case)

81. The above Hamiltonian has been studied by several authors (Domokos et al;Giovannetti et al) and arrive to the conclusion that For a suitable pulse, a C-NOT gate can be achieved, where the photon Number (0 or 1) is the control and the atom the target The mechanism of the above C-NOT gate that forbids, for example the (g,0>-- (e,0> transition is the Stark Effect, caused by one photon in the cavity. In order to resolve these two transitions, we have to make sure that

83. Where Is the frequency difference between these two transitions.

84. The exchange IS POSSIBLE

85. C-NOT GATE N=0 ATOMIC STATE IS NOT CHANGED N=1 ATOMIC STATE IS EXCHANGED CONTROLTARGET

86. UQCM PROPOSED PROTOCOL

87. ATOM 1 A1, initially at is prepared in a superposition, via a Ramsey Field A1 interacts with the cavity Ca(initially in )through a Rotation, so State swapping.The excitation of atom 1 is transferred to the cavity a g

88. ATOM 2 IT CONTAINS THE INFORMATION TO BE CLONED This state can be prepared in the same fashion as the atom 1, for example with a Ramsey Field. Then we apply a Classical pulse, as described before, generating a C-NOT gate,nothing happens with 0 photons C-not

89. A3 and A4 are the atoms carrying the two copies(IDENTICAL)

90. FINAL STATE

91. DISCUSSION Experimental numbers(Haroche et al) An interaction time ofMarginally satisfies the earlier requirement. With the flight time of 100 The whole scheme should Take about 700Which is reasonable in a cavity with a Relaxation time of 16ms.They achieved a resolution required to Distinguish between 1 or 0 photons

92. Discussion of the dispersive C-NOT Gate We have solved numerically the Hamiltonian We introduced the exponentials to simulate numerically the flight time and duration of the pulse Scrodingers Eq was solved for the state

94. BIBLIOGRAPHY 1.-W.K.Wooters and W.H.Zurek,Nature,London,299,802(1982) 2.-V.Buzek,M.Hillery,Phys.Rev.A 54,1844(1996) 3.-D.Bruss et al, Phys.Rev.A 57,2368(1998) 4.-N.Gisin,S.Massar, Phys.Rev.Lett,794,153(1997) 5.-D.Bruss et al, Phys.Rev.Lett,81,2598(1998) 6.-V.Buzek,S.L.Braunstein,M.Hillery,D.Bruss, Phys.Rev.A,56,3446(1998) 7.-C.Simon,G.Weihs,A.Zeilinger, Phys.Rev.Lett,84,2993(2000) 9.-P.Milman,H.Olivier,J.M.Raimond, Phys.Rev.A,67,012314(20003) 10.-M.Paternostro,M.S.Kim,G.M.Palma,J.of Mod.Opt,50,2075(2003) 11.-M.Brune et alPhys.Rev.A,78,1800(1995) 12.-V.Giovannetti,D.Vitali,P.Tombesi,A.Eckert,Phys.Rev.A,52, 3554(1995) 13.-M.Orszag,J.Gonzalez,S.Dagach,sub Phys.Rev.A 14.- M.Orszag,J.Gonzalez,Open Sys and Info Dyn,11,1(2004)

95. Pontificia Universidad Católica de Chile A SINGLE ION STOCHASTIC QUANTUM PROCESSOR MIGUEL ORSZAG PAUL BLACKBURN

96. OUTLINE OF THE TALK 1.-INTRODUCTION 2.-QUANTUM GATES C-NOT,TOFFOLI 3.- A STOCHASTIC QUANTUM PROGRAMMABLE PROCESSOR (General, using quantum gates) 4.-IMPLEMENTING THE GATES VIA TRAPPED ION 5.-DISCUSSION Actual implementation of the processor with a trapped ion, decoherence and measurements

97. We first show how to realize the rotation of a qubit via a quantum processor, using two and three qubit gates. Next we discuss these C-NOT and TOFFOLI gates, implemented by making use the two and three dimensional Center of mass vibrational qubits using a single three level ion. Control and coupling of the ion´s internal electronic states is achieved via far detuned lasers exciting a Raman transition scheme.

98. Finally we put things together and come up with the proposal

99. Stochastic Processor for a qubit We want: How to implement it???

100. A STOCHASTIC QUANTUM PROCESSOR 1.-G.Vidal,L.Masanes,J.I.Cirac,PRL,788,047905(2002) 2.-M.A.Nielsen,I.L.Chuang,PRL,79,321(1997) 3.-M.Hillary,V.Buzek,M.Ziman,PARA,65,022301(2002) Data qubit Program 1 qubit Program 2 qubit C_ NOT GATE Toffoli Gate

101. To understand the procedure, consider a single Program Register. First we define the program and the data PROGRAM STATE DATA STATE

102. + In this case, a measurement on the program register will cause a collapse On the data qubit with the outcome. If we measure 0 in the program, we get the good answer, If we measure 1, wqe get the wrong answer With a probability 1/2 + Bad result is for 1in P1

103. To improve upon this scheme, we introduce a Toffoli Gate, as in the Figure. The data line and the first program qubit are unchanged, however,If the output of the program register is Indicating a failure, the Toffoli gate effectively acts like a C-Not Gate Between the data line and a second program qubit There is again a probability 1/2 of getting this time THEREFORE, WE HAVE INCREASED THE SUCCESS PROBABILITY to 3/4 (and so on…) Bad result only in the 11 case

104. One can generalize the argument, including MORE GENERALIZED TOFFOLI GATES, and having a success probability of Where N is the number of Program qubits or generalized Toffoli Gates (also the number of measurements) The price we pay is the increase of gates and number of measurements

105. 3 level Trapped ion (harmonic trap) interacting with two laser fields highly detuned from the upper level (RAMAN SCHEME)

107. FOR LARGE DETUNING WE PROCEED TO PERFORM AN ADIABATIC ELIMINATION OF THE UPPER LEVEL 2

108. Defining Go to the Heisenberg picture

109. Using a second trasnformation to eliminate The fast rotating terms

110. We get the following equations in the second picture Please notice that all the fast terms are gone

111. Under the assumption of large detunings We obtain a solution for By setting And similarly for the 2-3 transition

112. Upon inserting these adiabatic Solutions for the atomic operators An d In terms of And replacing them in the Hamiltonian We get the effective Hamiltonian after the Adiabatic elimination of level 2

113. Now we go to the interaction picture, with And transform to the new Hamiltonian

114. And replace the x, y, z operators by Width of ground State of oscillators

115. Called usually the Lamb Dicke parameter. The square of this quantity represents the ratio Between the recoil energy and the vibrational Energy in the i direction. Experimentally it ranges between 0.1 and 0.2.

116. Interaction Picture Laser Frequencies Integers(small) +hc

117. Stationary Terms We look for terms such that

118. Laser frequencies Stark shifted Atomic levels Please notice that by careful tuning of the lasers we can select a given Hamiltonian

119. C-NOT Toffoli The Hamiltonians To get the different effective Hamiltonians, we have to get the right laser frequencies

120. The Temporal Evolution C-NOT

121. The Temporal Evolution Toffoli - + - +

122. C-NOT Control target X,Y vibrations bus

123. Toffoli Control 1Control 2targetbus X,Y,Z vibrations -

124. The actual operation of the processor consists in three phases 1)Preparation,2)Processing and 3)Measurement In the first phase, the ion is Prepared in the ground state and is loaded with the data and program states. In the Processing phase, the lasers are switched on with the detunings and spatial orientations required for the required Pulse periods. In the detection phase, the y and z vibrational state of the ion is Measured, thus indicating whether the desired operation was applied successfully on the data state. The actual loading of the vibrational state could be done in a separate trap on an auxilliary ion. The vibrational state of this auxilliary ion can be transferred to our ion, following for example, the proposal of Paternostro et al M.Paternostro,M.S.Kim,P.L.Knight,PRA,71,022311(2005)

125. The feasibility of the proposal depends strongly on the Decoherence time which is of the order of 1-10ms. The characteristic time required for an operation like H3T, For a Lamb Dicke parameter =0.3 and Corresponds, for (full cycle) a time of 0.53 ms. In principle this time could be decreased since As long as the laser power is not beyond Watts/cm2 which would photoionize the atom

126. Alternatively,one may attempt to increase the motional state Decoherence time of the trap. C.Monroe et al have recently done experiments where the heating rate is very low (Cd+ ions)where they report Which means that an n=1 state can have a life of 40ms!!!!!!!!!!!GREAT…… L.Deslauriers,P.Haljan,J.P.Lee,K.A.Brickman,B.B.Blinov,M.J.Madsen, C.Monroe,PRA,70,043408(2004 )

127. Measurement or reconstruction of the quantum mechanical state of a trapped ion Where the information on the vibrational CM motion of a trapped ion can be Transferred to it´s electronis dynamic by irradiating a long living electronic transition by laser light and probing a strong transition for resonance Fluorescence. Was first suggested by S.Wallentowitz,W.Vogel, PRL,75,2932(1995) Other reconstruction schemes, applying coherent displacements of different Magnitudes was suggested both theoretically and experimentally by D.Liebfried, D.M.Meekhof,B.E.King,C.Monroe,W.M.Itano, D.J.Wineland,PRL,77,4281,(1996) QND measurements of vibrational populations in ionic traps. This scheme allows The production of of Fock states, associated with the CM motion and it is based On the fact that the Rabi Frequency between two internal states of the ion Induced by a resonant carrier field depends on the vibronic number. L.Davidovich,M.Orszag,N.Zagury,PRA,54,5118(1996) R.L.Matos Filho,W.Vogel,PRL,76,4520(1996) Direct Measurement of the Wigner Function L.G.Lutterbach,L.Davidovich,PRL 78,2547(1997)

128. A generalization of the QND measurement of the vibronic states (which was originally suggested for one dimensional vibrations, was extended to higher dimensions, quite appropriate in the present scheme, was done by W.Kaige et al Quantum Non-Demolition Measurements and Quantum State Manipulation in two dimensional Trapped Ion In Modern Challenges in Quantum optics, M.Orszag,J.C.Retamal, Edts, Springer Verlag,2001

129. Quantum non-demolition measurements are design to avoid The back-action of the measurement on the detected Observable. For example, in the optical domain, we find experiments Using Kerr effect in a solid or liquid medium. The signal field to be measured interacts non-linearly With a probe field, whose phase changes by a quantity Which depends linearly on the number of photons in the signal beam. In the Haroche group, they developed a QND method To measure the number of photons stored in a high Q cavity Which is sensitive to a very small number of photons.

130. The method is based on the detection of a dispersive phase shift produced by the field on the wave function of Non-resonant atoms which cross the cavity. This shift which is proportional to the photon number in the Cavity,is measured by atomic interferometry, using Ramsey fields. Since the atoms are non-resonant with the cavity, No photon is exchanged between them and the Cavity and the measurement is indeed a QND one. However, the information aquired by detecting A sequence of atoms modifies the field step by step,until It eventually collapses into a Fock state, which a priory is Unpredictable. A repetition of the measurement, for the same initial state Of the field will yield a distribution of Fock states, which reproduces the initial distribution of the field.

131. In a similar way, it is possible to realize a QND Measurement of the vibrational population distribution For an ion in a Paul trap. As in the cavity QED case, a Fock state is generated in the process.

137. We are proposing a scheme for implementing a single qubit Stochastic Quantum Processor using a single cold Trapped ion. The Processor implements an arbitrary rotation around the z-axis of the Bloch sphere of the data qubit, GIVEN TWO PROGRAM QUBITS. The operation is applied succesfully with a probability P=3/4 We analize the preparation process, discuss decoherence and also propose various possible measurement schemes on the program qubit space. CONCLUSIONS