Basic concepts
Coordinate system
In the case of the nitrogen-vacancy (NV) center, the z-axis is aligned parallel to the N-V axis, and typically, the static magnetic field is also aligned with this axis. While there is freedom in choosing the orientation of the x-y plane, it is essential to ensure that the relative positions of the nuclear spins (as reflected in the corresponding hyperfine tensor) and the orientations of the driving fields are properly aligned.
Simphony provides a default NV model to which one can add carbon-13 nuclear spins using lattice indices. To identify the choice of the coordinate system, see the documentation of the default NV model.
Spin-Hamiltonian
The central spin system can be described by the following Hamiltonian:
where
\(\boldsymbol{S_i} = (S_{x,i}, S_{y,i}, S_{z,i})\) are the spin operators,
\(\gamma_i\) are the gyromagnetic ratios,
\(\boldsymbol{B}_\text{static}\) is the static magnetic field,
\(\boldsymbol{B}_\text{drive}(t)\) is the time-dependent driving magnetic field,
\(\Delta_i\) are the zero-field splitting or nuclear quadrupole parameters,
\(\boldsymbol{A_{ij}}\) describes the interaction (e.g., hyperfine or dipolar) between the i-th and j-th spin.
The first term represents the Zeeman interaction with static and driving fields. The second term accounts for zero-field splitting (for spin-1) and nuclear quadrupole effects. The last term describes interactions between spins, such as hyperfine coupling between an electron spin and nearby nuclear spins.
The goal of Simphony is to make it easy to create such spin models, either by using the default one or by building a custom model, and to set the time dependence of the driving magnetic field(s) through a sequence of pulses to coherently control the coupled spin system.
Quantum numbers and qubit subspace
The spins in the central spin register are typically spin-1/2 or spin-1. For quantum computing applications, it is necessary to define a qubit subspace within the Hilbert space of a spin-1 system. Similarly, for spin-1/2 systems, the ordering of basis states must be specified to correctly define the computational basis for the qubits.
Units
Frequencies are expressed in MHz, while time is measured in \(\mu\text{s}\). Energies are given in terms of frequency, which corresponds to the unit system \(h = 1\) in the Hamiltonian. The strength of magnetic fields is measured in Tesla (T).
Bases and frames
By applying pulses to the spin system, nontrivial time evolution occurs. The goal is to generate gates by using appropriate pulse sequences. A common question is in which basis and reference frame the time evolution realizes the desired ideal gate.
- Bases:
product: Basis states of each spin and their tensor product states.eigen: Eigenstates of the full spin Hamiltonian, which include the effects of non-secular terms such as hyperfine interactions.
- Frames:
lab: The laboratory frame governed by the Schrödinger equation.rotating: A frame rotating with respect to the lab frame, often chosen to simplify systems with time-dependent Hamiltonians.
In Simphony, the product basis and the rotating frame are used as defaults.
Pulse segments
Simphony is designed to simulate pulse sequences that consist of microwave and radio-frequency pulses, usually acting alternately. During simulation, Simphony determines the simulation segments, where each segment is separated by pulse boundaries—that is, whenever a pulse starts or ends, a new simulation segment begins—and identifies which pulses are active within each segment. The simulation then proceeds through these time segments, discretizing them and computing the evolution by exponentiating the corresponding Hamiltonian for each piece.
In certain cases, the computation can be significantly accelerated. For example, if there is no active pulse, or if there is a single pulse with a fixed frequency and constant envelope. In the latter case, Simphony applies an optimized method that performs the time evolution only for a single sinusoidal wave and then computes the full time evolution for the entire pulse from this result.
Rotating frame
The rotating frame is often introduced to simplify the interpretation of gates implemented by pulse sequences. Currently, Simphony performs all computations in the lab frame, but the results can be analyzed in a rotating frame, which is commonly used when defining and interpreting quantum gates.
In Simphony, the operator corresponding to the rotating frame is:
where \(f_i\) is the rotation frequency associated with spin \(i\), and \(\sigma_{z,i}\) denotes the Pauli-Z operator acting on the corresponding qubit subspace of spin \(i\).
Virtual rotations
Virtual rotations are phase shifts applied in software rather than by applying a physical pulse. These rotations do not require additional time and are commonly used to adjust the effective phase of subsequent pulses in a pulse sequence.
In Simphony, virtual rotations are represented as ideal Z-rotations applied instantly in the qubit subspace of the corresponding spin.
Tensor product convention
In multi-spin systems, the ordering of operators in the tensor product follows a fixed convention. In Simphony, the rightmost operator acts on the first spin in the register, consistent with the standard Kronecker product ordering used in quantum computing frameworks.