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					        2-Qubit Quantum Information Processing by
 Zeeman-Perturbed Nuclear Quadrupole Resonance
Patrick Xian
Troy Borneman


1. Introduction

This lab introduces the essential concepts of quantum information processing (QIP) by Zeeman-perturbed
nuclear quadrupole resonance (NQR). The standard model for QIP with spin systems is based on using the
interactions between distinct two-level systems (qubits) to perform computations. However, if a single
quantum system has an energy structure with more than two levels, we may extend our computational
model to include logically defined qubits. For example, spin-3/2 nuclei contain four distinct energy levels
which may be logically mapped to the computational basis states of 2 qubits: 00, 01, 10, and 11. If the
energy seperation between any two levels is unique, the energy level structure appears as if it was gener-
ated by the Hamiltonian of two interacting spins-1/2 and we can perform universal computation. For
atomic nuclei with spin greater than 1/2, the presence of a quadrupolar interaction in addition to the usual
dipolar interaction leads to the desired energy level structure.


2. Physics of the Nuclear Quadrupole Interaction


  Nuclear Spin

An atomic nucleus is fundamentally a composite particle, composed of protons and neutrons bound
together by the Strong nuclear force. Quantum mechanically we describe the state of an atomic nucleus by
considering the structure of the various discrete energy configurations the nucleus may take. Nuclear
physics is concerned with the transitions and dynamics that result from the interaction of this energy level
structure with the surrounding environment. For the purposes of magnetic resonance, however, we con-
sider only magnetic interactions with the nuclear spin degree of freedom, and take the nucleus to be a
point particle undergoing no internal nuclear transitions. The spin of a nucleus is a complex result of the
charge distribution of the protons in the nucleus and is best understood as describing the number of ways
the nucleus may uniquely interact with a magnetic field. For example, a spin-1/2 nucleus may be consid-
ered to be either spin-up or spin-down, and interacts accordingly with an applied magnetic field. The two
possible spin configurations of a spin-1/2 particle make it ideally suited to define a binary quantum bit (a
physical qubit). A spin-3/2 nucleus, on the other hand has four possible spin configurations, allowing us to
conveniently define 2 logical qubits.


  Physical Origin of the Quadrupolar Interaction

The quadrupolar interaction results from the quadrupole moment of a spin-3/2 nucleus being immersed in
an electronic environment with non-vanishing electric field gradient (EFG). The Hamiltonian governing
thus interaction may be written as:
2        USEQIP NQR.nb




[H=Q· E=Q· ·(- V)]
The quadrupole moment, Q, is a manifestation of the non-spherical charge distribution of the nucleus. V is
a tensor that represents the electric potential generated from the electronic environment and is calculated
                                                                                                                 2
                                                                                                                     V
by solving the Laplace equation. We are interested in the components of V, which are given by Vii =              xi 2
                                                                                                                         ,

xi        x, y, z ) . The nature of V depends on the orbital structure of the surrounding electron cloud and
whether any symmetries exist. For example, if the electrons exist only in an s-orbital (spherical symme-
try), the solution of the Laplace equation is trivial - Vxx = Vyy = Vzz = 0 - and the EFG vanishes so that
we see no quadrupolar interaction despite a nonzero quadrupole moment.


        Hamiltonian and Energy Level Structure

The Hamiltonian characterizing the nuclear quadrupole interaction is:
                        2
Hq             6
                 q
                     3 Iz        I2         2
                                           Ix      2
                                                  Iy
In this equation          q   determines the stength of the quadrupolar interaction in units of angular frequency ( =
1 by convention here), Ix,y,z are the spin-3/2 angular momentum operators (see the next section for details),
and I 2          2
                Ix    2
                     Iy        2
                              Iz is the total angular momentum operator. The parameter    represents any assymetries
in the electronic field gradient and is given by:
            Vx x V y y
               Vz z

For our purposes we will set = 0, such that the electronic environment is considered to be cylindrically
symmetric. Note that this is not an approximation as one can easily find an appropriate sample for which
this symmetry condition is satisfied. The sample we will use for this lab, sodium chlorate (NaClO3),
exhibits this symmetry.
The simplified Hamiltonian characterizing the axially symmetric nuclear quadrupole interaction is now:
                        2
Hq             6
                 q
                     3 Iz        I2
To determine the energy level structure requires calculating the expectation value of the Hamiltonian.
From elementary quantum mechanics we know that this expectation value is given as:

    6
     q
          3 m2       I I        1
In this equation, I is the total angular momentum quantum number (3/2), and m is the magnetic quantum
number that can take the values -3/2, -1/2, 1/2, 3/2. Thus, we see that the energy of the system may take 4
different values, giving us an energy level diagram that has four levels. However, notice that due to the
magnetic quantum number being squared, there are only 2 distinct values that may be achieved. In this
case, we say that the +/- 1/2 and +/- 3/2 energy levels are doubly degenerate.
PROBLEM: What are the 4 possible values of the energy expectation value? Note that you should get
only two numbers that each occur twice due to the double degeneracy of the quadrupole Hamiltonian.
                                                                                          USEQIP NQR.nb    3




Answer: For I = 3/2 and m = +/- 1/2 you should get - q /2
                      For I = 3/2 and m = +/- 3/2 you should get   q /2


We can now determine the energy level diagram for this Hamiltonian to see that we have a well-defined
binary two-level system which naturally maps into a qubit.

                                                                          3
                                                          mz
                                                                          2


                                                                          1
                                                          mz
                                                                          2
     Spin-3/2 Angular Momentum Operators and Matrix Representation of Hamiltonian

In order to define the matrix representation of the quadrupolar Hamiltonian we must choose both a basis
and definition of the spin-3/2 angular momentum operators. The natural basis to choose is the 4 possible
values of the magnetic quantum number, which define the Iz operator:
                        1
             Iz      3, 0, 0, 0 , 0, 1, 0, 0 , 0, 0,           1, 0 , 0, 0, 0,   3   ;
                 2
             MatrixForm Iz
                  3
                       0      0   0
                  2
                       1
                  0           0   0
                       2
                              1
                  0 0             0
                              2
                                  3
                  0 0         0
                                  2


In order to properly define Ix, Iy, and It (the total angular momentum) we must consider the raising and
lowering operators and their commutation relations with the Iz operator. The raising and lowering opera-
tors simply act to move the system up or down one level in the energy level diagram and their commuta-
tion relations with the Iz operator are independent of spin.

I            Ix        i Iy

I            Ix        i Iy

    Iz , I             I
    I ,I                2 Iz
From these relations we can now calculate the matrix representations of Ix, Iy, and It
4    USEQIP NQR.nb




                        1
           Ix                        0,                  3 , 0, 0 ,                  3 , 0, 2, 0 , 0, 2, 0,       3   , 0, 0,         3 ,0   ;
                        2
                        1
           Iy                        0,                          3 , 0, 0 ,                 3 , 0,   2 , 0 , 0, 2 , 0,            3    , 0, 0,   3 ,0   ;
                        2
                                15                                               15                     15                    15
           It                            , 0, 0, 0 , 0,                               , 0, 0 , 0, 0,         , 0 , 0, 0, 0,            ;
                  4                                                              4                       4                    4
           MatrixForm Ix
           MatrixForm Iy
           MatrixForm It

                                 3
                0                                 0              0
                                 2
                    3
                                 0               1               0
                2
                                                                     3
                0                1               0
                                                                  2
                                                     3
                0                0                               0
                                                     2

                                                 3
                    0                                            0       0
                                             2
                        3
                                         0                               0
                    2
                                                                             3
                    0                                            0
                                                                         2
                                                                     3
                    0                    0                               0
                                                                 2

            15
                            0        0                   0
                4
                            15
            0                        0                   0
                            4
                                     15
            0               0                            0
                                         4
                                                     15
            0               0        0
                                                         4


PROBLEM: Check that the above matrices satisfy the usual commutation relations of the Pauli operators
                                1
    I ,I                        2
                                                             I
Now we can define the matrix form of the quadrupole Hamiltonian and trivially see that the eigenvectors
and eigenvalues are the same as for the energy level diagram given above:
                                                 wQ
           Hq wQ_        3 Iz.Iz     :                                               It ;
                      6
           MatrixForm Hq wQ
            wQ
                             0                   0           0
                2
                                wQ
            0                                    0           0
                                 2
                                                 wQ
            0                0                               0
                                                     2
                                                             wQ
            0                0                   0
                                                              2
                                                                                                 USEQIP NQR.nb   5




3. Pure Quadrupole Resonance (1 Logical Qubit)

We can see from the matrix representation and energy level structure of the quadrupole Hamiltonian that
the system dynamics break down into two identical two-level subsystems which do not mix. The two
subsystems correspond to the collection of states (3/2,1/2) and (-3/2,-1/2), respectively. Each of these
subsystems behaves as an isolated spin-1/2, thus allowing the dynamics to be represented as a direct sum
of SU(2) algebras, which are spanned by the usual spin-1/2 Pauli matrices.
                                     SU 2     0
H   SU(2)   SU(2)
                                       0    SU 2

Based on this observation we may define a new set of operators to describe the dynamics of the system:
                1
      Iz1                   1, 0, 0, 0 , 0,         1, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0        ;
                2
                1
      Iz2                   0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 1, 0 , 0, 0, 0,         1        ;
                2
                1
      Ix1                   0, 1, 0, 0 , 1, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0            ;
                2
                1
      Ix2                   0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 1 , 0, 0, 1, 0            ;
                2
                1
      Iy1                   0,        , 0, 0 ,     , 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0       ;
                2
                1
      Iy2      0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0,                        , 0, 0, , 0         ;
           2
      MatrixForm Iz1 Iz2
      MatrixForm Ix1 Ix2
      MatrixForm Iy1 Iy2
        1
            0           0       0
        2
                1
        0               0       0
                2
                        1
        0   0                   0
                        2
                                 1
        0   0           0
                                 2

            1
        0           0 0
            2
        1
            0 0 0
        2
                            1
        0 0 0
                            2
                    1
        0 0                 0
                    2


        0               0       0
                2

            0           0       0
        2

        0   0           0
                                 2

        0   0                   0
                        2


PROBLEM: Check that the above matrices satisfy the usual commutation relations of the spin-1/2 Pauli
operators in their respective subspaces
Based on these new operators we may now rewrite the quadrupole Hamiltonian as:
6   USEQIP NQR.nb




                1      2
Hq         q   Iz     Iz

    Addition of an RF Hamiltonian

In order to perform quantum information processing on this degenerate two-level system (qubit) we need
something more than a good definition of the eigenstates and energies of the system; we need a means to
drive the system between different energy states. Our ability to dynamically alter the state of the system
comes from the application of a time-dependent alternating radio-frequency (RF) field. The applied RF
field interacts with the dipole moment of the nuclei to drive transitions between states whose magnetic
quantum number differs by one. This condition is properly stated as: the selection rule of dipole allowed
transitions is | m| = 1. Thus, the RF field drives transitions between the states 3/2 1/2, -3/2 -1/2, and
1/2 -1/2. Note, however, that due to the energy structure of the quadrupole Hamiltonian, the states 1/2
and -1/2 are identical, so there is no notion of transition between them. We see, based on the selection
rules, that the RF field causes transitions only within the seperate spin-1/2 subsystems, without mixing
them.
So far we have only one coordinate system in this problem: the principle axis system (PAS) of the
quadrupole interaction. In this coordinate system we define the z-axis to be the axis about which the
charge distribution is symmetric. The Iz operator may be imagined as describing the spin angular momen-
tum in this 'z'-axis direction. In the laboratory, however, the coordinate system is defined by the physical
orientation of the coils used to generate the RF field. The two coordinate systems defined, respectively, by
the PAS of the quadrupole interaction and by the coils used in the laboratory, do not generally coincide.
This discrepancy must be accounted for.


    Change of Coordinates

In the laboratory coordinate system, the Hamiltonian describing the RF field is:
                                         1     2
Hrf      2     1    t cos   rf   t     Ix    Ix

This Hamiltonian represents an RF field of frequency rf and phase applied with a time-varying
strength 1 t . The tilde above the Ix operators represents the fact that the coordinate system being used is
different than in the description of the quadrupole Hamiltonian. In order to calculate the effect of this RF
Hamiltonian has on our system we must write both interactions in the same coordinate system. Because
  1 t is much smaller than q it is proper to rewrite the RF Hamiltonian in the coordinate system defined by

the PAS of the quadrupole interaction, with an angle between them:
                                                1     2              1     2
Hrf      2     1    t cos   rf   t     cos     Ix    Ix     sin     Iz    Iz
We now have two components to the RF Hamiltonian we must consider. The component Ix1 + Ix2 causes
a desired time-dependent mixing (transition) of the two states of the system. The component Iz1 + Iz2
causes an undesired time-dependent shift of the energy levels and must be appropriately dealt with.
Because 1 t is much smaller than q it is fine to simply drop this term from the Hamiltonian since it's
effect will be very small. It's important to note that this approximation is not entirely necessary; we can
perform calculations that include this term, but they will be significantly more complicated and yield very
little new information or insight into the system dynamics. Thus, the final RF Hamiltonian we will use is:
                                                                                                                        USEQIP NQR.nb   7




                                                                                 1         2
Hrf        2       1    t cos       cos                 rf    t                 Ix        Ix

  Rotating Wave Approximation

Currently, we have an RF Hamiltonian that represents a linearly polarized field applied along the x-axis of
the laboratory. Since the nuclear spin will be precessing about the quadrupolar z-axis with a frequency q
we would like to rewrite this Hamiltonian in terms of components that rotate in the same and opposite
senses, repsectively, as the nuclear spin:
                                      1             i             t
Hrf        2       1    t cos         2
                                               e             rf            ei    rf t            1
                                                                                                Ix    2
                                                                                                     Ix
The first exponential represents a field rotating in the opposite sense of the precession while the second
exponential represents a field rotating in the same sense as the precession. At this point we take a rotating
wave approximation and drop the first exponential since the frequency at which it rotates relative to the
precession frequency is very large and will introduce negligable transitions. Thus, the final RF Hamilto-
nian after the rotating wave approximation can be written:

Hrf            1       t cos ei       rf   t             1
                                                        Ix             2
                                                                      Ix
Which may be rewritten as:
                                                                            1         2                        1    2
Hrf            1       t cos     cos               rf    t                 Ix        Ix        sin   rf   t   Iy   Iy

  Rotating Frame Transformation

It is undesirable to have a time-dependence to the RF Hamiltonian as it makes calculating the dynamics
very difficult. In order to remove the time-dependence we go into a reference frame that rotates at the
same frequency as the RF. When we do this, we end up with a full quadrupole/RF Hamiltonian that looks
like:
                                  1             2                                          1     2
Hsys               q        rf   Iz            Iz                     1   t cos           Ix    Ix
Note that in this equation we have set the value of to be zero. This is because the simple 'hard' pulses we
apply in the lab are of constant amplitude and phase. We can always adjust the phase of our detection
scheme to compensate for a single phase offset, so we are justified in simply setting this phase to zero for
the time being. Note also that if we apply the RF at a frequency that is equal to the quadrupolar precession
frequency, then the total system Hamiltonian has the simple form
                                   1            2
Hsys               1   t cos      Ix           Ix
The two terms 1(t) and cos( ) dictate how fast the spins nutate around the axis of the RF field and
together are called the 'Rabi frequency'. The result of the laboratory and quadrupole PAS coordinate
systems not being aligned then acts to change the rate at which we can perform rotations. This may seem
undesirable, but it actually allows us another degree of freedom in the experiment, which leads to better
control. In the following sections we will include the RF field strength and the angle between coordinate
systems together into the Rabi frequency parameter:

  R    t                1   t cos
8    USEQIP NQR.nb




    Thermal Equilibrium

We have now fully described how a single logically defined quadrupolar qubit responds to the RF control
field. We must now determine what the initial state of the system is in order to calculate what we would
ideally expect to see for a simple one-pulse experiment. All physical systems naturally trend toward the
lowest energy configuration they can. In this case, our ground state is the +/- 1/2 energy level. In reality,
though, there is thermal energy in the environment that can cause excitations to a higher energy state. In
our experiment we have a crystal that contains roughly 1017 identical nuclei that each are in a specific
energy state at thermal equilibrium. The overall populations of the two energy levels and, therefore, the
initial state is given by the Boltzmann distribution
              1        Hsys kB T
    in        Z
                   e
If we series expand the exponential and assume that kBT is much greater than the energy scale of the
system Hamiltonian, then we can write the initial state as being:
                   q             1    2
    in        4 kB T
                            I   Iz   Iz
The I in the above equation stands for the 4x4 identity matrix, which we can drop from the description
since it does not evolve. Also, we may ignore the prefactor since it is simply a scaling factor that is carried
along throughout the calculation. The initial state we will use in calculations is then simply
               1         2
    in        Iz        Iz

    Exciting a Detectable Signel

We detect signal from the spin system using the same RF coil that is used for pulsing. As previously
mentioned, at thermal equilibrium the spins are rotating around the z-axis of the quadrupole PAS with a
frequency q. What is detected by the coil, by nuclear magnetic induction, is an alternating field in the xy-
plane. Initially, then, we will see no detectable signal from the spins in thermal equilibrium. We must use
our RF control fields to change the state of the spin system into something which may be detected. In
terms of quantum mechanics, the observable of our system is given as:

O            I1        I2        1
                                Ix    2
                                     Ix      1
                                          i Iy    2
                                                 Iy
In our experiment we are performing a spatially averaged ensemble measurement over the extent of the
crystal sample. Thus, what we observe is the expectation value of the above operator, which is defined as:

         O             Tr       O
We can see from this equation, when we plug in = in, that the expectation value is zero. What we need
then is to create a density matrix, , with components of Ix1 + Ix2 and Iy1 + Iy2 in it. In order to do this
we can simply apply a RF pulse of sufficient length and amplitude. The equation of motion for the density
matrix is the Liouville - von Neumann equation:
d
dt
                  i H,
                                                                                               USEQIP NQR.nb   9




The solution to this equation is
                                           †
   t      U t                  U t
The unitary propagator, U, is defined by the time-dependence of the system Hamiltonian

U t           ei         H t dt

PROBLEM: Calculate the effect of the RF pulse using the Hamiltonian above proportional to Ix1+Ix2
and the initial density matrix given by Iz1+Iz2. The solution is
                                         1          2                            1         2
   t      cos                  Rt       Iz         Iz          sin       Rt     Iy        Iy
We see, then, that we must set the product of the Rabi frequency, R, and the pulse time, t, equal to /2 in
order to maximize the detected signal


  The Free-Induction Decay

The signal we detect experimentally is the evolution of the initial state, , of the system - either thermal
equilibrium or after RF pulsing - under the internal Hamiltonian. For example, let's assume we have
applied a /2 pulse to the thermal equilibrium state such that
          1           2
  in     Ix          Ix
The unitary propagator determining the time evolution of in is given by the matrix exponential of the
quadrupole Hamiltonian
       FIDprop wQ_, t_ : MatrixExp                                   wQ t Iz1   Iz2   ;
       MatrixForm FIDprop wQ, t
          1
                  t wQ
          2                0        0              0
                           t wQ
              0            2        0              0
                                    t wQ
              0            0        2              0
                                               1
                                                       t wQ
              0            0        0          2




There is also an exponential damping of the FID signal due to relaxation processes such as T2. We may
phenomenologically describe relaxation effects as an exponential damping of the time-domain signal.
       FIDrelax T2_, t_ : Exp                            t    T2 ;

We first detect the in-phase portion of our signal (Ix1+Ix2), then the quadruture portion (Iy1+Iy2), then
Fourier tranform the result to obtain a spectrum
10    USEQIP NQR.nb




        Plot FIDrelax 0.0005, t
           Tr   Ix1     Ix2 .FIDprop 2   5000, t . Ix1     Ix2 .FIDprop 2   5000, t ¾ ,
           t, 0, 0.001 , PlotRange       1

         1.0



         0.5




                      0.0002   0.0004    0.0006   0.0008    0.0010


         0.5



         1.0

        Plot FIDrelax 0.0005, t
           Tr   Iy1     Iy2 .FIDprop 2   5000, t . Ix1     Ix2 .FIDprop 2   5000, t ¾ ,
           t, 0, 0.001 , PlotRange       1

         1.0



         0.5




                      0.0002   0.0004    0.0006   0.0008    0.0010


         0.5



         1.0



     Experiment: Sodium Chlorate Crystal Structure

In this experiment, we work with the chlorine-35 nuclei in sodium chlorate NaClO3 ) single crystal. The
direction of the principal z-axis of a NaClO3 molecule lies along the Na-Cl bond, which happens to be a 3-
fold rotation axis of the molecule such that Vxx=Vyy and the assymmetry parameter, , vanishes. From
solid state physics, a single crystal is a uniform construction of a large number of identical unit cells, each
of which also maintains the physical properties of the bulk. The unit cell of NaClO3 contains 4 magneti-
cally active Chlorine molecules whose individual PAS's do not coincide. The picture below is adapted-
(with modification) from JPSJ, 18, 1614(1963).
                                                                                          USEQIP NQR.nb   11




The four molecules each respond differently to an applied RF field, dependent upon their respective
orientations to the laboratory frame. This variation in response is captured in the angular dependence of
the Rabi oscillation frequency. In order to properly calculate the dynamics of the system, we must consider
how the Rabi frequency changes as a function of orientation for each molecule.
12   USEQIP NQR.nb




       s13 Table 0, i, 4                        ; s14               Table 0, i, 4                       ; s23          Table 0, i, 4                        ; s24            Table 0, i, 4                   ;
       sin Table 0, i, 4                        ;
       NaClBAngle
            1                                                                                            1
                       Sin            Cos               Sin                    Cos              ,                 Sin               Cos                 Sin                  Cos             ,
               3                                                                                         3
            1                                                                                                1
                           Sin          Cos                     Sin                Cos              ,                 Sin           Cos                 Sin                  Cos                 ;
               3                                                                                             3
                                                                 w0
                                                                                                                                    2
       For i       1, i          5, i      , w13                              6 Cos              2           4    3 Cos                         . Cos                    NaClBAngle                  i   ;
                                                                    4
                   w0
                                                                                            2
         w14                     6 Cos              2           4       3 Cos                           . Cos               NaClBAngle                  i        ;
                   4
                   w0
                                                                                        2
         w23                6 Cos               2           4       3 Cos                       . Cos                   NaClBAngle                  i        ;
                   4
                   w0
                                                                                        2
         w24                6 Cos               2           4       3 Cos                       . Cos                   NaClBAngle                  i        ;
                    4
         s13       i         w13; s14               i               w14; s23            i               w23; s24            i        w24;
                                                                2
         sin       i              1        Cos                          . Cos               NaClBAngle                  i       ;

       Manipulate
                       1                                                                                          1
        Plot                 Sin               Cos                      Sin             Cos              ,              Sin                 Cos                  Sin                  Cos            ,
                       3                                                                                          3
            1                                                                                                1
                           Sin          Cos                     Sin                Cos              ,                 Sin           Cos                 Sin                  Cos                 ,
               3                                                                                             3
                                                                                                                                                        3
            , 0,           , PlotRange                          0,        ,        1, 1         , GridLines                     0,          ,       ,            ,       , Automatic ,
                                                                                                                                        4       2           4
         GridLinesStyle          Directive Gray, Dashed ,
                                   3
         Ticks              0, , ,    ,   , Automatic , AxesLabel                                                               ,    Rabi       ,
                              4 2   4

         LabelStyle               Directive Bold, Medium, FontFamily                                                    "Helvetica"                     ,            ,       , 0, 2
                                                                                                                                                                         2
                                           1
       Manipulate Plot                              Sin                   Cos               Sin                   Cos           ,
                                            3
            1                                                                                                1
                       Sin              Cos                     Sin                Cos              ,                  Sin           Cos                    Sin                  Cos             ,
               3                                                                                             3
            1
                       Sin            Cos               Sin                     Cos                 ,        , 0, 2         , PlotRange                          0, 2            ,      1, 1         ,
               3
                                                            3                  5        3            7
         GridLines                    0,       ,        ,               , ,         ,           ,            ,2        , Automatic ,
                        4 2    4       4    2   4
         GridLinesStyle Directive Gray, Dashed ,
                          3       5    3    7
         Ticks    0, , ,     , ,     ,    ,    ,2                                                                , Automatic , AxesLabel                                     ,       Rabi    ,
                     4 2   4       4    2    4
                                                                                                                                                                         54.73
         LabelStyle               Directive Bold, Medium, FontFamily                                                    "Helvetica"                     ,            ,                      , 0,
                                                                                                                                                                             180
                                                                                          USEQIP NQR.nb   13




              Rabi
           1.0


           0.5


           0.0
                                                3
                          4          2           4

           0.5


           1.0




              Rabi
           1.0


           0.5


           0.0
                               3           5    3     7
                                                           2
                     4    2     4          4     2     4

           0.5


           1.0




  Experiment: Rabi Oscillations

A Rabi oscillation experiment is used to demonstrate control over a two-level system. We plot the magni-
tude of the FID signal observed after a single RF pulse of varying length. Ideally, we will see an exponen-
tially decaying oscillation of a single frequency, corresponding to a periodic transfer of population
between two energy states. However, since we have 4 molecules contributing signal to the measured FIDs
and each of these molecules generally has a different Rabi oscillation frequency, what we actually see is
the superposition of these frequencies. There are only a few special orientations of the crystal for which
the Rabi oscillation frequencies are the same and we see a simple exponentially decaying oscillation.
Generally, we will see an interference pattern of oscillations, as shown below
14    USEQIP NQR.nb




 6


 4


 2



             200         400         600          800   1000

 2


 4




4. Zeeman-Perturbed Quadrupole Resonance (2 Logical Qubits)


     The Addition of a Small Static Magnetic Field

Thus far we have been dealing solely with pure nuclear quadrupole resonance. In the absence of any
applied fields, the energy level structure of the interaction appears as a doubly degenerate two-level sys-
tem. Another way to say this is that the Hamiltonian and dynamics may be represented as a direct sum of
SU(2) algebras. This structure defines two degenerate logical qubits which do not interact and behave
identically so that the overall behavior is that of a single logically defined qubit. When we apply a weak (~
50 Gauss) external magnetic field along the laboratory z-axis the Zeeman interaction breaks the degener-
acy of the two qubits and the energy structure of the system now has four distinct levels:
The Hamiltonian and dynamics may now be represented as a direct product of SU (2) algebras, which
behaves like two physical two - level systems that directly interact. The combined Hamiltonian of the
Zeeman interaction and the quadrupole interaction is:
                      2
Hq z         6
              q
                   3 Iz        I2          o Iz

Here 0 is the Larmor frequency which determines the interaction strength of the nuclear dipole moment
with the applied magnetic field. The Iz operator for the Zeeman interaction has a tilde to denote the differ-
ence between the laboratory and quadrupole PAS coordinate systems. As we did previously with RF
fields, since the quadrupole interaction is much stronger than the Zeeman interaction - hence 'Zeeman
perturbed NQR' - we transform the Zeeman interaction into the PAS coordinate system of the quadrupole
interaction:
                      2
Hq z         6
              q
                   3 Iz        I2          o   cos Iz     sin Ix
In this equation is the overall angle between the two coordinate systems. Due to the presence of the Ix
operator, the eigenstates of the quadrupole interaction, which we used previously, are no longer the true
eigenstates of the system. The Ix operator causes significant mixing of the +1/2 and -1/2 states. Using first-
order degenerate perturbation theory we may define the new eigenstates to be

                   cos              12         sin         12
and
                                                                                                        USEQIP NQR.nb   15




                      sin             12                cos                 12
The angle         is the mixing parameter and is given by

                               f 1
       arctan                  f 1

where f depends on and is given by


f             1        4 tan2
The matrix form of the combined quadrupole and Zeeman Hamiltonian demonstrates that the degeneracy
of the quadrupole interaction is broken and a direct sum representation is no longer appropriate
                                          wQ
       Hqz wQ_, w0_, _ :                           3 Iz.Iz       It    Cos       w0 Iz;
                            6
       MatrixForm Hqz wQ, w0,
         wQ       3
                      w0 Cos                       0                       0                    0
          2       2
                                     wQ        1
                       0                           w0 Cos                  0                    0
                                     2         2
                                                                 wQ    1
                       0                           0                       w0 Cos               0
                                                                  2    2
                                                                                      wQ   3
                       0                           0                       0                   w0 Cos
                                                                                      2    2


The matrix form of this Hamiltonian looks very similar to that of two J-coupled spins-1/2. In fact, the
Hamiltonian may be rewritten in terms of two interacting spin-1/2 particles:
         1                 A    1                  B         J    A    B
H        2        A        z    2        B         z        2     z    z

PROBLEM: Determine the values of A, B, and J that allow the Zeeman perturbed quadrupole Hamilto-
nian to be written in the form of two coupled spins-1/2


    Mapping to 2 Logical Qubits

The energy level diagram of Hqz is:

                                                                                3
                                                                  mz
                                                                                2
                                                                               3
                                                                      mz
                                                                               2
16    USEQIP NQR.nb




We now have four levels corresponding to each of the eigenstates of the Hamiltonian. The splitting
between each of the levels is given by the difference in eigenenergies associated with each state. The solid
lines indicate strongly allowed transitions that emerge directly from the dipole selection rules. The dotted
lines indicate the weakly allowed transitions. These transitions are only allowed due to the mixing of the
+1/2 and -1/2 states due to the Ix component originally present in the Hamiltonian before applying perturba-
tion theory. Each of the transitions is unique and directly accessible, so we have universal control over the
system and can map the four energy levels to the computational basis states - 00, 01, 10, and 11 - in many
different ways. A convenient choice for experiments is shown in the above diagram.


     Control Through RF Hamiltonian

As with the case of pure quadrupole resonance, we will induce transitions between the computational basis
states through the application of a time-varying RF field. If apply a /2 RF pulse to the Zeeman-perturbed
quadrupole system, then examine the FID and it's Fourier transform (spectrum) we expect to see 4 peaks
associated with each of the transitions shown above. The positions of the four peaks depends on the
frequency separation of the energy levels of the system, which again depends on the orientation of the
molecule with respect to the laboratory frame. Also, note that we will generally see more than 4 peaks
since each of the 4 Chlorine molecules in our crystal is oriented differently. Due to the varying response of
each of the Chlorines and their respective transitions to the applied RF field, we will see anywhere from 2
to 16 peaks. The angular dependence of the transition frequencies for a single molecule is shown below


                                                                                        13         14


                                                                                             24   23




The intensity of each of the peaks is given by the transition probability for each transition. Fermi's golden
rule tells us that, to first order, the transition probabilities are given by the squared magnitude of the RF
perturbation matrix element connecting the two levels.
     USEQIP NQR.nb   17




13           14


     24   23

				
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