System and method for determining a perturbation energy of a quantum state of a many-body system

1. A method for determining a perturbation energy of a quantum state of a many-body system, comprising:

constructing a wave function that approximates the quantum state by adjusting parameters of the wave function to minimize an expectation value of a zeroth-order Hamiltonian, wherein:

the zeroth-order Hamiltonian explicitly depends on a finite mass of each of a plurality of interacting quantum particles that form the many-body system;

the quantum state has a non-zero total angular momentum;

the wave function is expressed as a linear combination of explicitly correlated Gaussian basis functions; and

each of the explicitly correlated Gaussian basis functions includes a preexponential angular factor;

calculating a theoretical estimate of the perturbation energy based on the wave function and a perturbation Hamiltonian that explicitly depends on the finite mass of each of the plurality of interacting quantum particles; and

outputting the theoretical estimate of the perturbation energy for use in configuring an apparatus such that the apparatus is capable of performing a measurement of a transition energy of the many-body system, the transition energy being based on the perturbation energy, the measurement of the transition energy having an experimental uncertainty that is equal to or less than the theoretical estimate of the perturbation energy.

2. The method of claim 1 , wherein said adjusting the parameters includes iteratively adjusting (i) one or more variational parameters of the explicitly correlated Gaussian basis functions and (ii) one or more weights of the linear combination.

3. The method of claim 2 , wherein said adjusting the one or more variational parameters includes adjusting one or more integer parameters of the preexponential angular factor of each of the explicitly correlated Gaussian basis functions.

4. The method of claim 2 , wherein said adjusting the one or more variational parameters includes adjusting one or more complex variational parameters of each of the explicitly correlated Gaussian basis functions.

5. The method of claim 1 , wherein:

the zeroth-order Hamiltonian represents a non-relativistic energy of the many-body system; and

the perturbation Hamiltonian represents one or more relativistic corrections to the non-relativistic energy.

6. The method of claim 5 , the perturbation Hamiltonian including one or more of:

a mass-velocity term;

a Darwin term;

an orbit-orbit interaction term;

a spin-orbit interaction term; and

a spin-spin interaction term.

7. The method of claim 1 , wherein:

only one of the plurality of interacting quantum particles represents an atomic nucleus; and

each of a remaining plurality of the interacting quantum particles represents an electron.

8. The method of claim 1 , wherein:

each of a first plurality of the interacting quantum particles represents an atomic nucleus; and

each of a remaining second plurality of the interacting quantum particles represents an electron.

9. The method of claim 1 , the zeroth-order Hamiltonian including an external-field term that represents an interaction of the many-body system with an external field.

10. The method of claim 1 ,

further comprising adding the theoretical estimate of the perturbation energy to the expectation value of the zeroth-order Hamiltonian to obtain a total energy of the quantum state;

wherein said outputting the theoretical estimate of the perturbation energy includes outputting the total energy of the quantum state.

11. A system for determining a perturbation energy of a quantum state of a many-body system, comprising:

a processor;

a memory in communication with the processor, the memory storing machine-readable instructions that, when executed by the processor, control the system to:

construct a wave function that approximates the quantum state by adjusting parameters of the wave function to minimize an expectation value of a zeroth-order Hamiltonian that explicitly depends on a finite mass of each of a plurality of interacting quantum particles that form the many-body system, the quantum state having a non-zero total angular momentum, the wave function being expressed as a linear combination of explicitly correlated Gaussian basis functions, each of the explicitly correlated Gaussian basis functions including a preexponential angular factor,

calculate a theoretical estimate of the perturbation energy based on the wave function and a perturbation Hamiltonian that explicitly depends on the finite mass of each of the plurality of interacting quantum particles, and

output the theoretical estimate of the perturbation energy for use in configuring an apparatus such that the apparatus is capable of performing a measurement of a transition energy of the many-body system, the transition energy being based on the perturbation energy, the measurement of the transition energy having an experimental uncertainty that is equal to or less than the theoretical estimate of the perturbation energy.

12. The system of claim 11 , wherein the machine-readable instructions that, when executed by the processor, control the system to adjust the parameters of the wave function include machine-readable instructions that, when executed by the processor, control the system to iteratively adjust (i) one or more variational parameters of the explicitly correlated Gaussian basis functions and (ii) one or more weights of the linear combination.

13. The system of claim 12 , wherein the machine-readable instructions that, when executed by the processor, control the system to adjust the one or more variational parameters include machine-readable instructions that, when executed by the processor, control the system to adjust one or more integer parameters of the preexponential angular factor of each of the explicitly correlated Gaussian basis functions.

14. The system of claim 12 , wherein the machine-readable instructions that, when executed by the processor, control the system to adjust the one or more variational parameters include machine-readable instructions that, when executed by the processor, control the system to adjust one or more complex variational parameters of each of the explicitly correlated Gaussian basis functions.

15. The system of claim 12 , wherein:

the zeroth-order Hamiltonian represents a non-relativistic energy of the many-body system; and

the perturbation Hamiltonian represents one or more relativistic corrections to the non-relativistic energy.

16. The system of claim 15 , the perturbation Hamiltonian including one or more of:

a mass-velocity term;

a Darwin term;

an orbit-orbit interaction term;

a spin-orbit interaction term; and

a spin-spin interaction term.

17. The system of claim 11 , wherein:

only one of the plurality of interacting quantum particles represents an atomic nucleus; and

each of a remaining plurality of the interacting quantum particles represents an electron.

18. The system of claim 11 , wherein:

each of a first plurality of the interacting quantum particles represents an atomic nucleus; and

each of a remaining second plurality of the interacting quantum particles as an electron.

19. The system of claim 11 , the zeroth-order Hamiltonian including an external-field term that represents an interaction of the many-body system with an external field.

20. The system of claim 11 , wherein:

the memory stores additional machine-readable instructions that, when executed by the processor, control the system to add the theoretical estimate of the perturbation energy to the expectation value of the zeroth-order Hamiltonian to obtain a total energy of the quantum state; and

the machine-readable instructions that, when executed by the processor, control the system to output the theoretical estimate of the perturbation energy include machine-readable instructions that, when executed by the processor, control the system to output the total energy of the quantum state.