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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