Interplay between geometric and dynamic phases in a single-spin system

We use a combination of microwave fields and free precession to drive the spin of a nitrogen-vacancy (NV) center in diamond on different trajectories on the Bloch sphere and investigate the physical significance of the frame-dependent decomposition of the total phase into geometric and dynamic parts. The experiments are performed on a two-level subspace of the spin-1 ground state of the NV, where the Aharonov-Anandan geometric phase manifests itself as a global phase, and we use the third level of the NV ground-state triplet to detect it. We show that while the geometric Aharonov-Anandan phase retains its connection to the solid angle swept out by the evolving spin, it is generally accompanied by a dynamic phase that suppresses the geometric dependence of the system dynamics. These results offer insights into the physical significance of frame-dependent geometric phases.

Figure 1

(a) The NV center in diamond, formed from a substitutional nitrogen adjacent to a lattice vacancy in the diamond matrix. (b) Optical transitions and ground-state structure of the NV, depicting Zeeman-separated $m_S$ energy levels and transition frequencies. (c) A variable-detuning spin-echo sequence nested inside a resonant Ramsey sequence, showing microwaves applied to each two-level subspace and laser preparation and readout pulses. The time between spin-echo pulses ${\tau }_{\text{SE}}$ is varied and time delays either side of the spin echo sequence are added to ensure the total time between the Ramsey pulses ${\tau }_R$ remains fixed. The resonant microwave Rabi frequency for each transition is about $500\mathrm{kHz}$. (d) Ramsey fringes for different spin-echo times ${\tau }_{\text{SE}}$ from the $\{0,-1\}$ interferometer while only the detuning $\mathrm{\Delta }$ of the $m_S=0\to m_S=-1$ microwaves are changed. The frequency of the Ramsey fringes is consistent with $\mathrm{\Delta }{\tau }_{\text{SE}}/4$, albeit slightly offset due to the low Rabi frequency effectively amplitude modulating the interference fringes. This shows that spin-echo results in a global phase in the $\{0,+1\}$ subspace that can be read out using interferometry on the $m_S=0\to m_S=-1$ transition.

Figure 2

Pulse sequences applied to the NV spin to drive cyclic trajectories and measured spin populations for multiple circuits. (a) Sequence 1: A resonant Ramsey sequence applied to the $\{0,-1\}$ subspace reads out the phase accumulated by the $\mathcal{C}$ pulse applied to the $\{0,+1\}$ system. The path traversed by the spin in the Bloch sphere consists of a closed circle subtending a cone with solid angle $\mathrm{\Theta }$. (b) Sequence 2, both the resonant interferometry sequence and $\mathcal{C}$ pulse act on the $\{0,-1\}$ system. The path the spin takes in the resonant frame is noncyclic, with initial and final azimuthal spins separated by a relative phase $\mathrm{\Delta }\varphi$. Transformation to the detuned frame of reference yields a cyclic evolution, though the geometric and dynamic phases both depend on the absolute phase of the $\mathcal{C}$ pulse, which is uncontrolled. [(c) and (d)] Experimental results for varying the $\mathcal{C}$ pulse detuning and cumulative rotation number $N$ for Sequences 1 and 2, respectively. Error bars derive from photon counting statistics and lines are fits of the form ${cos}^2\left(N{\varphi }_T\right)$ with fixed Rabi frequency.

Figure 3

Calculated paths taken by the Bloch vector of the NV spin under the application of Sequence 2, observed in the resonant frame (left), rotating at the two-level splitting (equivalently, the resonant microwave frequency) and (right) in the detuned frame, rotating at $\omega \pm \mathrm{\Delta }$, where $\mathrm{\Delta }$ is the detuning of the $\mathcal{C}$ pulse. Different colors represent different phases of the microwave field sampled when the $\mathcal{C}$ pulse is applied. In the resonant frame, where the NV spin Bloch vector is stationary, the $\mathcal{C}$ pulse drives noncyclic trajectories, so that the initial and final spins are separated azimuthally by $\mathrm{\Delta }\varphi$. In the detuned frame, the same trajectories are closed, subtending solid angles proportional to the Aharonov-Anandan phase.

Figure 4

Composite cyclic evolution pulse sequence, Bloch sphere trajectories, free-evolution fringes and results. (a) Sequence 3 consists of two $\mathcal{C}/2$ pulses acting on the $\{0,+1\}$ subsystem with detuning $\mathrm{\Delta }$, separated in time by $\tau =1/\mathrm{\Delta }$. A resonant Ramsey sequence applied to the $\{0,-1\}$ subspace reads out the total phase accumulated over the three geometric stages ${\mathcal{C}}_1$, ${\mathcal{C}}_{\tau }$, and ${\mathcal{C}}_2$ traced by the spin vector on the Bloch sphere (note inverted $z$ axis) (b). (c) We first verify that the spin vector precesses at a rate $\mathrm{\Delta }$ by measuring interference fringes resulting from application of only the $\{0,+1\}$ pulses of Sequence 3 and varying the time between pulses. Gray points depict the fitted period, dashed lines denote $1/\mathrm{\Delta }$. (d) Applying Sequence 3 to the NV spin yields data consistent with application of a single $\mathcal{C}$ pulse, indicating the the geometric nature of the free precession stage ${\mathcal{C}}_{\tau }$ is suppressed by an accompanying dynamic phase. Error bars derive from standard error from three repeated measurements, solid lines denote fits.

Figure 5

(a) Sequence 4 is identical to Sequence 3 but with a full $\mathcal{C}$ pulse applied after a time ${\tau }_1=(1-\eta )\tau$, followed by further free evolution ${\tau }_2=\eta \tau$ such that ${\tau }_1+{\tau }_2=1/\mathrm{\Delta }$, with $\eta$ a parameter that determines where in the free evolution the $\mathcal{C}$ pulse is applied. (b) Paths taken by the spin vector under each geometric manipulation step depicted on the Bloch sphere, with the frame rotation rate set by the microwave field driving $\mathcal{C}$ pulses, note the inverted $z$ axis. ${\mathcal{C}}_i$ denotes evolution under the microwave fields and ${\mathcal{F}}_i$ denotes free-evolution steps. (c) Experimental results for various values of $\eta$, which determines where in the free-evolution step the complete $\mathcal{C}$ pulse is applied. The output of the complete interferometry sequence is identical to two $\mathcal{C}$ pulses applied consecutively.