Logical gates and quantum processors with trapped ions and cavities

1
Logical gates and quantum processors with
trapped ions and cavities

• MIGUEL ORSZAG
• FACULTAD DE FISICAPONTIFICIA 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
QUANTUM COMPUTING AND QUANTUM INFORMATION
Classical Information unit Bit
Quantum Information Unit Quantum Bit.QUBIT
Qubit Microscopic system limited by two quantum
states.
Superposition
Qudit Microscopic system limited by N Quantum
States.
4
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5
2.-Quantum Gates
1 qubit gate
6
2.-Quantum Gates
C-NOT
7
QUANTUM GATES C_NOT GATE
control
target
2 qubit gate
8
QUANTUM GATES
Controlled Not (C-NOT)
Qudit
a
b
a
b
9
Quantum Gates
Toffoli
3 qubit gate
10
QUANTUM GATES
Toffoli
a
control1
control2
b
3 qubit gate
target
c
The target Only changes if both Controls Jk1
11
Quantum Copying Machines

• 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.
Fidelity
2
12
Quantum Copying Machines
Universal Quantum Copying Machine UQCM
machine
original

• Ideal copying process

blanc

• Analysis of the copies

1.
2.
Hilbert-Schmidt NORM

• An ideal copy should be written as

13
QUANTUM COPYING MACHINE
Universal Quantum Copying MachineUQCM

• Result

WANTED STATE
UNWANTED STATE
14
QUANTUM COPYING MACHINE
IMPLEMENTED BY A CIRCUIT
15
QUANTUM COPYING MACHINES
Quantum Copying Machine implemented by a circuit
Preparation phase
16
QUANTUM COPYING MACHINES
Implemented by a circuit
Copying phase
17
QUANTUM COPYING MACHINES
Analysis of the copies
Universal, doesnot depend on the input state
UQCM
Duplicator
Triplicator
Universal,but restricted To real numbers(input
coeff)
18
QUANTUM PROCESSORS
CLASSICAL PROCESSOR
Apparatus capable of developing any function
within the DATA REGISTER, such funtion being
specified by a PROGRAM REGISTER.
QUANTUM PROCESSOR
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.
19
QUANTUM PROCESSORS
Limitations due to Quantum Mechanics.

• 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)

20
3.-A GENERAL QUANTUM PROCESSOR
Fixed Unitary Operator (processor)
State of the DATA
Program State
Residual State (independent of the data)
U
Unitary Operator ACTING ON THE DATA ONY
This Type of Processor is necessarily stochastic
21
Stochastic Processor for a qubit
rotation of a qubit
How to implement it???
22
A STOCHASTIC QUANTUM PROCESSOR
Data qubit
Program 1 qubit
Program 2 qubit
C_ NOT GATE
Toffoli Gate
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)
23
To understand the procedure, consider a single
Program Register. First we define the program and
the data
DATA STATE
PROGRAM STATE
24
Bad result is for 1in P1

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

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
QUANTUM PROCESSORS
Processor Performance I

• State of Data

• Operation to implement

31
PROCESADORES CUANTICOS
Processors performance I

• Program Register base

• Program State

32
QUANTUM PROCESSORS
Processors performance I

• State that inputs the machine

33
QUANTUM PROCESSORS
Processors performance I

• The output state is

34
QUANTUM PROCESSORS
Processors performance
35
QUANTUM PROCESSORS
Processors performance II
36
QUANTUM PROCESSORS
Processors performance II

• Program state

• Input state

37
QUANTUM PROCESSORS
Processors performance II

• Output State

38
QUANTUM PROCESSORS
Processors performance II
39
QUANTUM PROCESSORS
Processors performance II

• Output State

40
QUANTUM PROCESSORS
Processors performance II

• If we measure in

1.
Success probability
2.
Error probability
41
PROCESADORES CUANTICOS
Processors performance
42
PROCESADORES CUANTICOS
Processors performance III
43
QUANTUM PROCESSORS
Processors performance III

• Program State

• Input State

44
QUANTUM PROCESSORS
Processors performance III

• Output State

45
QUANTUM PROCESSORS
Processors performance III

• Output State

46
QUANTUM PROCESSORS
Hillery-Buzek (2002)
Qubits
P1/4
Base de Bell
47
QUANTUM PROCESSORS
Hillery-Buzek (2002)
Qudits
48
QUANTUM PROCESSORS
Efficiency of processors
49
QUANTUM PROCESSORS
Efficiency of the Vidal-Cirac 2002 processor
50
QUANTUM PROCESSORS
Efficiency of the Vidal-Cirac 2002 processor
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
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

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QUANTUM PROCESSORS
Effiviency of the processors Hillery-Buzek 2004
The processor is given by the unitary operator
Ajk is an operator in Hd, jñ, kñ are the
basis for Hp
The data and program states are
The required operation is
It is useful to take c1 z c0
56
QUANTUM PROCESSORS
Efficiemcy of Processors Hillery-Buzek 2004

• Case with only one program state, the operator A
  is

Summ módulo 2

• To improve the probability, one uses a vector
  program given by

ck1 z ck
In the two program case, the operator A
Summ módulo 4
57
QUANTUM PROCESSORS
Rendimiento de Procesadores Hillery-Buzek 2004

• To generalize this process, an N dimensional
  vector program is used

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
JAYNES CUMMINGS MDEL TWO LEVEL ATOM AND DIPOLE
APPROXIMATIONS
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
Normally in QO this last two terma are neglected
in the So-called RWA
62
a
Two level atom
b
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 n4. 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 n2, (typical n70 )
68
MICROMASER
69
MICROMASER
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, Qcavity108, Tcav2K, nth2
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
Thus, an operation like
Is not possible, with
INPUT QUBIT
initial state of cloner
final state of cloner
Blank copy
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
The quality of the copy is measured through the
FIDELITY
UNIVERSAL QUANTUM COPYING MACHINE
BASIS
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
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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 alGiovannetti 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,0gt--?(e,0gt 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
82
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83
Where
Is the frequency difference between these two
transitions.
84

The exchange
IS POSSIBLE
85
C-NOT GATE
N0 ATOMIC STATE IS NOT CHANGED N1 ATOMIC STATE
IS EXCHANGED
CONTROL
TARGET
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
g
State swapping.The excitation of atom 1 is
transferred to the cavity a
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 of
Marginally satisfies the earlier requirement.
The whole scheme should
With the flight time of 100
Take about 700
Which 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
93
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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.Braunst
ein,M.Hillery,D.Bruss, Phys.Rev.A,56,3446(1998) 7
.-C.Simon,G.Weihs,A.Zeilinger, Phys.Rev.Lett,84,29
93(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.Bru
ne et alPhys.Rev.A,78,1800(1995) 12.-V.Giovannetti
,D.Vitali,P.Tombesi,A.Eckert,Phys.Rev.A,52, 3554(1
995) 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
1.-INTRODUCTION
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 ions 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
How to implement it???
100
A STOCHASTIC QUANTUM PROCESSOR
Data qubit
Program 1 qubit
Program 2 qubit
C_ NOT GATE
Toffoli Gate
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)
101
To understand the procedure, consider a single
Program Register. First we define the program and
the data
PROGRAM STATE
DATA STATE
102
Bad result is for 1in P1

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

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
4.-Implementing quantum gates with a trapped Ion
3 level Trapped ion (harmonic trap) interacting
with two laser fields highly detuned from the
upper level (RAMAN SCHEME)
106
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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
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
hc
Integers(small)
Laser Frequencies
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
The Hamiltonians
C-NOT
Toffoli
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
X,Y vibrations
Control target
bus
123
Toffoli
X,Y,Z vibrations
-
Control 1
Control 2
target
bus
124
5.-DISCUSSION
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(200
5)
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
latest experiment on phonon lifetimes
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 n1 state can have a life
of 40ms!!!!!!!!!!!GREAT
L.Deslauriers,P.Haljan,J.P.Lee,K.A.Brickman,B.B.Bl
inov,M.J.Madsen, C.Monroe,PRA,70,043408(2004)
127
a word about measurement
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 its
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
a word about measurement in general
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.
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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.
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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.
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CONCLUSIONS
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 P3/4 We
analize the preparation process, discuss
decoherence and also propose various possible
measurement schemes on the program qubit space.
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