Methods of constructing and designing RF pulses and exciting or inverting two-level systems

1. A method of performing broadband excitation or inversion of a two-level spin system, comprising:

providing a two-level system comprising at least one qubit (spin) and at least one harmonic oscillator (spring);

defining a radio frequency (RF) bandwidth for the two-level system;

bounding a radio frequency (RF) amplitude or total energy for an RF pulse;

determining a desired terminal magnetization profile or flip angle for the at least one spin;

determining a desired terminal magnetization profile or flip angle for the at least one spring;

mapping an endpoint of a trajectory of the spins to an endpoint of a trajectory of the springs, wherein the endpoints of the spins and springs correspond to the desired terminal magnetization profile or flip angle;

employing a single control or two controls simultaneously in both an x direction and a y direction;

calculating a converging solution for an RF pulse; or

steering the spring and the spin to a desired terminal magnetization profile or flip angle by applying the calculated converging solution for the RF pulse to the two-level system.

2. The method of claim 1 , wherein the calculating a converging solution comprises generating RF pulse parameters, providing a first control design comprising a minimum-energy broadband pulse, or providing a second control design comprising an amplitude-limited broadband pulse.

3. The method of claim 1 , wherein

(i) the two-level spin system is selected from the group consisting of a logical qubit spin system, a nuclear spin system, a photon spin system, an electron spin system, an atomic spin system, and a dot spin system;

(ii) a dynamic connection between nonlinear spin and linear spring systems are calculated under optimal forcing to design an RF pulse based on the design of a control to steer linear harmonic oscillators;

(iii) a condition of a one-to-one correspondence to a spin trajectory is satisfied;

(iv) the RF pulse conditions result in the spins and springs having coinciding magnetization on the same axis and excitation or inversion of at least 99% of the spins and springs;

(v) the RF pulse compensates for a distribution of spin and spring frequencies;

(vi) the RF pulse is an excitation pulse, a reverse excitation pulse; or an inversion pulse; or

(vii) the RF pulse results in a spin flip angle selected from π, π/2, or π/4.

4. The method of claim 1 , wherein the RF pulse is computed using linear systems to force the spring.

5. The method of claim 1 , wherein the RF pulse is an excitation pulse or an inversion pulse for a nonlinear Bloch system.

6. A method of constructing an RF pulse comprising:

(i) obtaining an energy parameter;

(ii) obtaining a bandwidth parameter;

(iii) obtaining a desired flip angle (magnetization profile), wherein the flip angle is between 0 and 180 degrees;

(iv) converting the desired flip angle from a spatial coordinate system to a linear coordinate system;

(iv) inputting the energy parameter, the bandwidth parameter, and the linear coordinates into the system; or

(v) analytically deriving parameters for the RF pulse,

wherein the RF pulse produces a desired magnetization profile of a final spin or spring state.

7. The method of claim 6 , wherein

the bandwidth parameter corresponds to a range of frequencies of a sample;

the method can be performed on-line; or

the method uses analytical approaches or optimization-free algorithms.

8. The method of claim 6 , comprising:

providing a two-level system comprising at least one qubit (spin) and at least one harmonic oscillator (spring) and mapping the spins to the springs;

providing a first control design comprising a minimum-energy broadband pulse; or

providing a second control design comprising an amplitude-limited broadband pulse.

9. The method of claim 6 , wherein the RF pulse

(i) produces a desired distribution of final spin or spring states or a desired magnetization profile when applied to a two-level spin system;

(ii) achieves at least 99% broadband excitation or inversion; and

(iii) is not a hyperbolic secant pulse.

10. The method of claim 6 , wherein the RF pulse

(i) compensates for a distribution of spin and spring frequencies; or

(ii) satisfies experimental requirements selected from a bound on an RF pulse amplitude or total energy of the RF pulse.

11. The method of claim 6 , wherein the bandwidth parameter is between −40 kHz and 40 kHz.

12. The method of claim 6 , wherein the energy parameter comprises a maximum allowable power in Watts or dbm.

13. The method of claim 6 , wherein flip angle is between 0 and 360° or selected from a π, a π/2, or a π/4 flip angle.

14. A method of designing a broadband radiofrequency (RF) pulse comprising:

(i) providing a two-level spin system comprising at least one qubit (spin) and at least one harmonic oscillator (spring);

(i) employing at least a first control design and optionally, a second control design simultaneously in both an x and a y direction; or

(ii) achieving a desired flip angle of a qubit, wherein

the flip angle of a qubit is controlled by steering the spring between specific states.

15. The method of claim 14 , wherein the qubit is selected from the group consisting of a logical qubit, a nuclear spin, a photon, an electron, an atomic spin, and a dot spin.

16. The method of claim 14 , wherein the qubit is a nuclear spin and a broadband RF pulse compensates for a distribution of spin and spring frequencies.

17. The method of claim 14 , further comprising numerical optimization, resulting in an RF pulse or an RF pulse sequence that places bounds on a radio-frequency (RF) amplitude or a total energy of the RF pulse.

18. The method of claim 14 , wherein

the first control design comprises a minimum-energy broadband pulse; and

the second control design comprises an amplitude-limited broadband pulse;

the flip angle can be between about 0° and 360°; or

the RF pulse performs at least 99%, exact, or asymptotically exact excitation or inversion over a defined bandwidth, optionally with a bounded amplitude.

19. The method of claim 14 , wherein the RF pulse has a bang-bang pulse shape.

20. The method of claim 14 , wherein an excitation or inversion of the spins can be adjusted by selecting different amplitude bounds and RF pulse durations.