How does the spin qubit evolve over time under the influence of the Hamiltonian for Larmor precession?
The evolution of a spin qubit under the influence of the Hamiltonian for Larmor precession is a fundamental concept in the field of quantum information. To understand this evolution, let us first define what a spin qubit is and how it behaves.
A spin qubit is a two-level quantum system that can be represented by a two-dimensional vector space. The two levels, often denoted as |0⟩ and |1⟩, correspond to the spin-up and spin-down states of a spin-1/2 particle, such as an electron or a nucleus. The state of the qubit can be described by a superposition of these two basis states, where the coefficients of the superposition represent the probability amplitudes of finding the qubit in each state.
The Hamiltonian for Larmor precession describes the evolution of the spin qubit in the presence of a magnetic field. It is given by:
H = -γB0σz,
where γ is the gyromagnetic ratio, B0 is the magnitude of the magnetic field along the z-axis, and σz is the Pauli matrix that acts on the spin qubit. This Hamiltonian represents the interaction of the spin qubit with the magnetic field, causing the spin to precess around the z-axis at a frequency determined by the gyromagnetic ratio.
To understand the time evolution of the spin qubit under this Hamiltonian, we need to solve the time-dependent Schrödinger equation:
iħ∂ψ/∂t = Hψ,
where ħ is the reduced Planck's constant, ψ is the state vector of the qubit, and t is time. By solving this equation, we can obtain the time evolution of the qubit state.
Let us consider an initial state of the spin qubit given by |ψ(0)⟩ = α|0⟩ + β|1⟩, where α and β are complex probability amplitudes. By substituting this initial state into the Schrödinger equation and solving for ψ(t), we find:
|ψ(t)⟩ = e^(-iγB0σzt/ħ)(α|0⟩ + β|1⟩),
where e^(-iγB0σzt/ħ) represents the time evolution operator. This equation shows how the spin qubit evolves over time under the influence of the Hamiltonian for Larmor precession.
The time evolution operator can be expanded using the Baker-Campbell-Hausdorff formula to obtain:
e^(-iγB0σzt/ħ) = e^(-iγB0t/ħ)cos(γB0t/ħ) – iσzsin(γB0t/ħ).
This equation reveals that the time evolution of the spin qubit involves both a rotation around the z-axis and a phase factor. The rotation angle around the z-axis is given by γB0t/ħ, which depends on the strength of the magnetic field, the gyromagnetic ratio, and the time. The phase factor depends on the initial state of the qubit and the time.
For example, if the initial state of the spin qubit is |0⟩, then the time evolution of the qubit state can be written as:
|ψ(t)⟩ = e^(-iγB0t/ħ)|0⟩ = cos(γB0t/ħ)|0⟩ – i sin(γB0t/ħ)|1⟩.
This equation shows that the spin qubit remains in the |0⟩ state, but acquires a phase factor determined by the magnetic field strength and the time.
The spin qubit evolves over time under the influence of the Hamiltonian for Larmor precession by undergoing a rotation around the z-axis and acquiring a phase factor. The rotation angle and the phase factor depend on the strength of the magnetic field, the gyromagnetic ratio, the initial state of the qubit, and the time.
Other recent questions and answers regarding Examination review:
- How can the time evolution of the qubit state be computed using the eigenvalues of the Hamiltonian for Larmor precession?
- What is the relationship between the angular momentum and the Hamiltonian for Larmor precession?
- What is the Hamiltonian that describes the interaction of a spin qubit with an external magnetic field?
- How is a quantum gate or unitary transformation on a qubit state performed using the Bloch sphere?