One of the primary difficulties in manipulating quantum information is the inevitable loss, or “decoherence,” of the information stored in the quantum system due to interaction with its environment. Researchers have developed a variety of techniques to reduce the rate of decoherence in quantum information systems or to limit its effect, enabling more reliable operations on the quantum state. We demonstrate two all-optical approaches to overcoming sources of decoherence of the electron spin qubit of the nitrogen-vacancy (NV) center in diamond: geometric-phase-based operations and “superadiabatic” operations. By using optical techniques to address defects in diamond, we advance the possibility of manipulating individual spin qubits integrated into photonic networks or spin arrays.

## STIRAP

Both the geometric phase and the superadiabatic implementations employ a quantum control technique for three-level lambda systems known as stimulated Raman adiabatic passage, or STIRAP. When two lower states are coupled to an upper state (a “lambda” configuration), a coherent state transfer from one lower state to the other can be performed by driving both transitions to the upper state simultaneously and modulating the relative speeds of those transitions (see the first figure below, at left). In the NV center, STIRAP can be performed by choosing two of the ground states and one excited state. By applying optical laser fields resonant with both ground-to-excited state transitions and smoothly tuning the relative strength and phase of the two lasers, the state of the NV center can be coherently transferred from one ground state to the other without stimulating a transition to the excited state. As described below, the coherent control provided by STIRAP can be used to generate geometric phase or modified to perform superadiabatic protocols in the qubit basis formed by the two chosen ground states.

## Geometric Phase

Geometric phase, also known as Berry phase, is a phase difference that is acquired over the course of a system’s evolution through a complete cycle in parameter space. Simple examples of classical geometric phase can be found: a vector at the north pole of a sphere that undergoes parallel transport to the equator, then follows the equator for a quarter rotation, and finally moves back to the pole will acquire a 90-degree relative phase as a result of the path it follows. Likewise, a quantum geometric phase can accumulate between two qubit states due to a cycle of the system within any parameter space – for example, the Bloch sphere. An important characteristic of geometric phase is that the degree of phase acquired depends only on the area enclosed by the path of the cycle, not by dynamic factors such as the duration of the cycle or its particular shape.

To demonstrate these properties, we use STIRAP to drive the state of the NV center from one pole of the Bloch sphere to the other along a semicircle. Once reaching the other pole, the phase relation between the resonant lasers is changed, and the system is returned to its original state along another semicircle. This cycle encloses a “slice” of the Bloch sphere, and thus those states accumulate a global geometric phase (see above, center). This phase is then measured relative to the third ground state of the NV center. By changing the size of the slice, we are able to control the amount of phase between the states (see above, right; a top-down view of the Bloch sphere).

Since the geometric phase acquired depends only on the area enclosed by the cycle, geometric phases are inherently resistant to some of the noise that is endemic to experimental instrumentation, making it a good candidate approach for improving the reliability of quantum information processing. To test the robustness of the protocol to noise, we intentionally introduce additional noise to the STIRAP loop. Our results show that the geometric phase operations are particularly resilient to noise parallel to the cycle, and function even in the presence of high degrees of noise, which is an essential feature for any extended quantum information protocols (see above).

## Superadiabatic Protocols

Adiabatic processes like STIRAP can be used on their own to evolve quantum systems from one state to another, since they maintain coherent control as long as the transfer is performed slowly relative to the energetic differences between states. However, the long evolution times required make decoherence more problematic. To mitigate decoherence, a variety of control schemes have been developed to reproduce the results of an adiabatic process in a shorter time. We employ one of these schemes, known as “superadiabatic transitionless driving” or SATD, to accelerate STIRAP. We engineer nonadiabatic dynamics to depart from the ideal adiabatic evolution path, but meet it at the target endpoint (see below, left).

For optically-driven STIRAP in the NV center, one of the primary sources of decoherence is occupation of the excited state, which spontaneously decays in around 11 nanoseconds. Performing SATD entails modifying the STIRAP pulse shapes in a manner that intentionally occupies the excited state, but removes the population before much of it decays. A variant of SATD, MOD-SATD, behaves the same way but also minimizes the population required to be in the excited state (see below, right). We are able to create the proper pulse shapes through the sub-nanosecond control of an amplitude electro-optic modulator.

We show that both the SATD and MOD-SATD protocols operate with significantly higher fidelities than a simple adiabatic procedure does, and are substantially faster at reaching a target fidelity (below, left). Furthermore, we show that both protocols are robust against experimental imperfections. Interestingly, more drastic pulse correction is required than what is theoretically optimal in the absence of decoherence; simulation shows that this deviation can be reproduced by taking decay from the excited state into account (below, right).

To learn more about our studies, please refer to:

“Optical manipulation of the Berry phase in a solid-state spin qubit,” C. G. Yale*, F. J. Heremans*, B. B. Zhou*, A. Auer, G. Burkard, and D. D. Awschalom, *Nature Photonics* **10**, 184 (2016)

Related article: "Quantum optics: Robust light-controlled qubits," *Nature Photonics ***10**, 147 (2016)

“Accelerated quantum control using superadiabatic dynamics in a solid-state lambda system,” B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. C. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, Nature Physics 13, 4 (2017).