Deviation density matrix

HP xenon has also been used to enhance the polarization of a two-qubit NMR quantum computer using the C-enriched chloroform.Using the SPINOE transfer mechanism, this approach led to a polarization enhancement of the chloroform that was approximately 10 times the thermal values for H and Temporal spin-labeling methods along with measurements of the deviation density matrix were used to observe the formation of a pure spin state. The authors then demonstrated their approach by implementing a 2-qubit Grover s search algorithm. 

Fig. 30. (Top) Time evolution of the (filled squares) and (open circles) signals of CHCls after being dissolved in hyperpolarized e for 4min at —40°C showing the initial enhancements of + 18 for and -11 for Inset shows the deviation density matrix at time ti. (Bottom) Deviation density matrix (the diagonal is shifted to obtain a zero average for the nonground states) of the effective pure state. (Courtesy of Isaac Chuang. Reprinted from ref. 345 with permission. Copyright 2001, American Institute of Physics.)...

Fig. 30 shows the and signal enhancements as a function of time, and the deviation density matrix of the pure state under both thermal and HP xenon-enhanced conditions. 

From Equation (2.5.14), the deviation density matrix in thermal equilibrium under a... 

Applying Equation (2.6.14) with Op = n/2 leads to the same result as Equation (2.6.12), that is, the deviation density matrix has changed from 4 to ly with the application of the pulse. Writing the complete density matrices before and after the pulse, one obtains ... 

Putting Op = 7T in Equation (2.6.14), one obtains for the deviation density matrix the result... 

We can summarize the results of this section saying that the effects of RF pulses can be described in the rotating frame in terms of the rotation operators (usually around the X, y, —x,and —y axes) applied to the deviation density matrix starting from thermal equilibrium. The evolution of the system after or between the RF pulses is described as a free precession aroimd the z-axis, with a frequency that depends on the frequency offset and is therefore different for nuclei experiencing distinct local fields (due to chemical shifts, for example). 

The small terms involving J were neglected in this expression. The deviation density matrix is ... 

And the deviation density matrix corresponding to an ensemble of identical nuclei at thermal equilibrium is ... 

It can be easily shown from direct matrix products that Rx —n/2)I Rx n/2) = ly. This result shows thus that, starting from thermal equilibrium, where the deviation density matrix is given by Equation (2.9.3), one can generate single-quantum coherences in the density operator by applying a RF pulse with proper phase and duration. Therefore, a transverse magnetization can be excited and detected in the same way as the spin 1 /2 case. These considerations show then that, in the absence of quadrupolar interaction, the action of RF pulses in the spin 3/2 case is completely analogous to the case of spin 1 /2. 

On the other hand, after a selective pulse applied to a given transition connecting the levels specified by m and i — 1, the deviation density matrix contains coherences only between those levels. For I = 2i 12 and considering a selective nll)-x pulse applied to the central transition (1/2 —y -1/2), the deviation matrix is proportional to a fictitious spin-1/2 operator ly, whose form is analogous to the one given in Equation (2.9.12). 

As described in the text, the time evolution of any ensemble opeiatai can be determined using the deviation density matrix. Under action of the Zeeman Hamiltonian, the deviation density operato" evolves in time as ... 

The deviation density matrix in this case is tha-efore time-independent, as expected since we are dealing with thermal equilibrium. 

Let us consider a more general situation with arbitrary initial deviation density matrix Apo evolving in time under the Zeeman Hamiltonian. The calculation of the ensemble averages of the expectation values of the spin operator p is given below ... 

Conventional NMR deals with a large ensemble of spins. It means that the state of the system is in a statistical mixture, which is obviously inadequate for QIP. However, the NMR ability for manipulating spins states worked out by Cory et al. [24] and Chuang et al. [23] resulted in elegant methods for creating the so called effectively pure or pseudo-pure states. Behind the idea of the pseudo-pure states is the fact that NMR experiments are only sensitive to the traceless deviation density matrix. Thus, we might search for transformations that, applied to the thermal equilibrium density matrix, produce a deviation density matrix with the same form as a pure state density matrix. Once such state is created, all remaining unitary transformations will act only on such a deviation density matrix, which will transform as a true pure state. 

Figure 4.8 Experimental deviation density matrices (a), (b) and (c) were obtained after the application of the operations Uq,Ui, and U2 to the equilibrium density matrix, (d) Represents the average deviation density matrix. Adapted with permission from Reference [27] (Copyright 2007 American Chemical Society).

Figure 4.8 shows experimental results for the deviation density matrix obtained after applying each operation for a 2-qubit system Uq, U, and U2 as well as the average state (see also Problems P4.3 and P4.4). The deviation density matrices were obtained using the quantum state tomography process, which will be described in the next section. As it can be seen, the final averaged deviation density matrix is very similar to that of the pure state 100). 

Starting from the equilibrium state, the CNOT gates do not introduce any off-diagonal elements into the deviation density matrix, so we can regard (4.3.21) as two subsets of pseudo-pure states, the state 00)o with label state 0) and the state ll)i with state label... 

The first QST procedure dedicated to quadrupolar systems was developed by Kampermann and Veeman for spin 3/2 nuclei [8]. It is a direct adaptation from the method used for spin 1/2, but uses transition selective pulses instead of spin selective pulses. Similar to the spin 1 /2 method, the transition selective pulses are used to bring an specific set of populations and coherences to the reading position of the deviation density matrix (single quantum coherences) and after that the NMR spectrum is acquired. The transformations (transition selective pulses) that are applied to the deviation density matrix Ap in order to bring the unknown elements to the reading positions are. 

To obtain the off-diagonal elements of the deviation density matrix, njl transition selective pulses with proper phases are applied to the system prior to the readout pulse. The effect of the application of such pulses is to bring the off-diagonal elements of the density matrix to the main diagonal, as illustrated in (4.4.6), where only the main diagonal elements are displayed for simplicity,... 

Figure 4.12 Deviation density matrix truth table for CNOT gates implemented in a quadrupolar spin 3/2 system. Adapted with permission from Reference [31] (Copyright 2007 Elsevier).

To illustrate the Bloch sphere representation of an NMR system, let us consider the density matrix of an ensemble of spins 1/2 nuclei. Because the high temperature deviation density matrix is proportional to 1, the effect of an RF pulse is to induce rotations that transform the initial density matrix into a linear combination of the spin operator components,... 

P4.7 - Using the CNOT operations described in Section 4.3.3, derive the expression for the pseudo-pure deviation density matrix (4.3.21) starting from (4.3.19). 

Let us take the high temperature deviation density matrix for this three spin system as follow ... 

Let us first consider that the density matrix during the acquisition periods can be represented by the general deviation density matrix of Equation (4.4.2) ... 

Figure 6.3 (Quantum correlation feedback transfer demonstrated by Nelson and co-workers in 2000. The system of three spins starts at the equilibrium thermal state, with the deviation density matrix proportional to + /jf + A sequence of pulses creates a correlated (pseudo-entangled) between B and C ...

To analyze the NMR spectra of the rotated state, we have to build the deviation density matrix ... 

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