What is the Hamiltonian that describes the interaction of a spin qubit with an external magnetic field?
The Hamiltonian that describes the interaction of a spin qubit with an external magnetic field can be derived using the principles of quantum mechanics and the concept of Larmor precession. In this context, a spin qubit refers to a two-level quantum system, where the states are represented by the spin-up and spin-down states of a particle. The external magnetic field induces a coupling between the spin of the qubit and the magnetic field, leading to the Larmor precession of the spin.
To derive the Hamiltonian, let's consider a spin qubit with a magnetic moment given by μ, placed in an external magnetic field B. The interaction between the magnetic moment and the magnetic field can be described by the Zeeman interaction term. The Zeeman interaction energy is given by the dot product of the magnetic moment and the magnetic field, which can be written as:
E = -μ · B
Here, the negative sign arises from the convention of considering the energy of the spin-down state as lower than the energy of the spin-up state. The magnetic moment μ is proportional to the spin operator S, which can be written as:
μ = γS
where γ is the gyromagnetic ratio. The gyromagnetic ratio depends on the properties of the particle and the nature of the spin. For example, for an electron, γ is equal to the Bohr magneton divided by 2 times the electron mass.
Substituting the expression for μ into the Zeeman interaction energy, we obtain:
E = -γS · B
The Hamiltonian operator H is defined as the energy operator of the system. Therefore, the Hamiltonian that describes the interaction of the spin qubit with the external magnetic field can be written as:
H = -γS · B
In this Hamiltonian, S is the spin operator and B is the magnetic field vector. The dot product represents the interaction between the spin and the magnetic field.
The Hamiltonian H describes the energy of the spin qubit in the presence of the external magnetic field. It captures the Larmor precession of the spin qubit, which is the precession of the spin vector around the direction of the magnetic field. The frequency of the Larmor precession is determined by the magnitude of the magnetic field and the gyromagnetic ratio.
To illustrate this, let's consider the case of a spin-1/2 particle, such as an electron, in a uniform magnetic field B along the z-axis. In this case, the spin operator S can be written as:
S = (ħ/2)σ
where ħ is the reduced Planck constant and σ is the vector of Pauli matrices. Substituting this expression into the Hamiltonian, we obtain:
H = -γ(ħ/2)σ · B
Expanding the dot product, we find:
H = -γ(ħ/2)(σx Bx + σy By + σz Bz)
where σx, σy, and σz are the Pauli matrices and Bx, By, and Bz are the components of the magnetic field vector B.
The Hamiltonian that describes the interaction of a spin qubit with an external magnetic field is given by H = -γS · B, where S is the spin operator and B is the magnetic field vector. This Hamiltonian captures the energy of the spin qubit in the presence of the magnetic field and governs the Larmor precession of the spin qubit.
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