Experimental measurement of the quantum geometric tensor using coupled qubits in diamond

Abstract Geometry and topology are fundamental concepts, which underlie a wide range of fascinating physical phenomena such as topological states of matter and topological defects. In quantum mechanics, the geometry of quantum states is fully captured by the quantum geometric tensor. Using a qubit formed by an NV center in diamond, we perform the first experimental measurement of the complete quantum geometric tensor. Our approach builds on a strong connection between coherent Rabi oscillations upon parametric modulations and the quantum geometry of the underlying states. We then apply our method to a system of two interacting qubits, by exploiting the coupling between the NV center spin and a neighboring 13C nuclear spin. Our results establish coherent dynamical responses as a versatile probe for quantum geometry, and they pave the way for the detection of novel topological phenomena in solid state.


INTRODUCTION
The quantum geometric tensor (QGT) constitutes a central and ubiquitous concept in quantum mechanics, by providing a geometric structure to the Hilbert space [1][2][3][4][5]. The imaginary part of this tensor corresponds to the well-known Berry curvature [6,7], which acts as an effective 'electromagnetic' tensor in parameter space. This geometric quantity, which is formally associated with the parallel transport of wave functions [8], is responsible for striking observable phenomena such as the geometric phase [8], the anomalous Hall effect [9] and topological states of matter [10]. In contrast, the real part of the QGT constitutes the Fubini-Study metric [2,3,5], which defines a notion of distance (a Riemannian metric) in parameter space through the overlap of wavefunctions. This 'quantum metric', which is intimately related to quantum fluctuations and dissipative responses of the system [2,5,11,12], has been shown to play an important role in various contexts, including quantum phase transitions [13], open quantum systems [14], orbital magnetism [15,16], localization in insulators [11], semi-classical dynamics [17,18], excitonic Lamb-shifts in transition-metal dichalcogenides [19], superfluidity in flat bands [20] and topological matter [21,22]. In the context of quantum information, the quantum metric is equivalent to the quantum Fisher information, which is a witness for multipartite entanglement [23].
Various manifestations of the QGT have been observed in experiments, using very different physical platforms and probes. On the one hand, the local Berry curvature has been detected in ultracold atomic gases [24][25][26], coupled optical fibers [27], and solids [28,29]. On the other hand, a first manifestation of the quantum metric-the so-called Wannier-spread functional of Bloch bands [30]was recently measured in cold atoms [31], based on the proposal [32]; see [12,[33][34][35] for other proposals to detect quantum geometry. Nevertheless, direct and systematic measurement of the complete QGT has never been performed.
Here, we report on the first experimental measurement of the complete QGT, using a qubit formed by an NV center spin in diamond. Following C The Author(s) 2019. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. the proposal of [32], we exploit the relation between the QGT and the response of quantum systems upon parametric modulations to map out the full Fubini-Study metric as well as the local Berry curvature of the underlying quantum states. We then apply our method to a system of two interacting qubits, obtained by coupling the NV center spin to a nearby 13 C nuclear spin. Our results not only enforce the deep connections between outof-equilibrium dynamics and quantum geometry [36][37][38][39][40][41][42][43][44], but also reveal a universal tool for detection of geometric and topological properties in quantum systems.

DETECTING THE QGT THROUGH RABI OSCILLATION
We start by considering the Hamiltonian H (λ) of a generic discrete quantum system, which depends on a set of dimensionless parameters λ = (λ 1 , λ 2 , · · · , λ N ), where N is the dimension of parameter space. For a single qubit, the relevant parameter space corresponds to the twodimensional Bloch sphere. Defining the eigenstates and eigenvalues of this generic Hamiltonian, H (λ) |n(λ) = n (λ)|n(λ) , a geometric structure emerges upon projecting the dynamics onto a single (non-degenerate) band n (λ). The resulting quantum geometry is captured by the QGT, which is defined as [7] For simplicity, hereafter we denote ∂ μ ≡ ∂ λ μ . The real part Re (χ μν ) = g μν is the Fubini-Study metric, which introduces a notion of distance in parameter space, while the imaginary part I m (χ μν ) = −F μν /2 is related to the Berry curvature F μν responsible for the Berry phase. It is useful to express the QGT in the form so as to highlight the relation between this geometric quantity and the coupling matrix elements connecting the eigenstates |n(λ) and |m(λ) upon a parametric modulation [32],

EXPERIMENTAL SETUP
In our experiment, we first perform a full quantumgeometric measurement using a two-level system, as described by the general Hamiltonian where the angles (θ, ϕ) form the relevant parameter space (the Bloch sphere). Considering the lowenergy dressed state, the components of the QGT read g θθ = 1 4 , g ϕϕ = 1 4 sin 2 θ, g θϕ = 0, F θϕ = sin θ/2. These components fully characterize the underlying quantum geometry: the quantum metric g corresponds to the natural metric of a sphere S 2 , embedded in R 3 with fixed radius R = 1 2 , while the Berry curvature F θϕ corresponds to the 'magnetic' field of a fictitious Dirac monopole located at the center of that sphere [22].
The experimental setup is sketched in Fig.  1D. The two-level system in Eq. (4) is obtained from a single nitrogen-vacancy (NV) center in an electronic grade diamond. We apply a magnetic field B z = 509 G along the NV axis to lift the degeneracy of the states m s = ±1. A two-level system is supported by the spin sublevels m s = 0 and m s = −1. We first prepare the system in the eigenstate of the Hamiltonian H (θ 0 , ϕ 0 ), i.e. n(θ 0 , ϕ 0 ) = cos θ 0 2 | − 1 + sin θ 0 2 e i ϕ 0 |0 . This is achieved by first applying a 532 nm green laser pulse to initialize the NV center spin in the m s = 0 state. A subsequent microwave pulse H i (t) = sin(ω 0 t + ϕ 0 )σ x , applied over a duration t θ 0 = θ 0 , rotates the NV center spin around the axisn(ϕ 0 ) = (cos ϕ 0 , sin ϕ 0 , 0) by an angle θ 0 . The initial state preparation is verified by a spin-locking type experiment, which confirms that the NV spin is prepared in the eigenstate of H (θ 0 , ϕ 0 ) [see supplementary data].
The precise control over the AWG allows us to engineer the microwave driving field with accurate amplitude and phase modulation. This leads to implementation of the generic two-level system where In the experiment, we calibrate the driving amplitude in the Hamiltonian [Eq.  (C) elliptical parametric modulation for the measurement of the local Berry curvature (as indicated by color map) with θ t = θ 0 + a θ sin(ωt) and ϕ t = ϕ 0 ± a ϕ cos(ωt). (D) Experimental setup used for the quantumgeometric measurement, based on an NV center spin in diamond. A green laser pulse polarizes the NV center spin into the | m s = 0 state. The engineered microwave created from an arbitrary waveform generator (Tektronix AWG 70002A, 16GS/s) is amplified before being delivered to the sample and coherently drives the NV center spin. The NV center spin state is detected by an APD via spin-dependent fluorescence. (E) An example of parametric-modulation resonance measurement. The probability that the NV center spin remains in the initial eigenstate at time T = 400 ns as a function of the modulation frequency, for a linear parametric modulation θ t = θ 0 + a θ sin(ωt), ϕ t = ϕ 0 with (θ 0 , ϕ 0 ) = ( 5π 6 , 0) and a θ = 0. : The parametric modulation drives a coherent transition between the eigenstates of H (θ 0 , ϕ 0 ), which is detected by rotating the NV center spin around the axisn(ϕ 0 ) by an angle 2π − θ 0 . This rotation maps the eigenstates of H (θ 0 , ϕ 0 ) back to the NV center spin state |0 and | − 1 , which is then measured by spin-dependent fluorescence.

EXPERIMENTAL RESULTS
In the experiment, we implement two types of modulations [32]: (a) a 'linear' modulation θ t = θ 0 + a θ sin(ωt), ϕ t = ϕ 0 + a ϕ sin(ωt); (b) an 'elliptical' modulation θ t = θ 0 + a θ sin(ωt), ϕ t = ϕ 0 + a ϕ cos(ωt); see Fig. 1A-C. Setting a θ , a ϕ 1, the time-dependent Hamiltonian can be expressed as After preparing the NV center spin in the eigenstate |n(θ 0 , ϕ 0 ) of the Hamiltonian H (θ 0 , ϕ 0 ), we apply the engineered microwave driving field with parametric modulation [see Eq. (5)] and fix the time duration T. We sweep the parametric modulation frequency ω, and measure the probability p 0 (T) that the NV spin remains in the initial eigenstate |n(θ 0 , ϕ 0 ) . In Fig. 1E, we show an example of such a parametric-modulation resonance measurement; see supplementary data for the experimental data using other types of modulations. The results indicate that a coherent transition between the eigenstates becomes resonant when ω ∼ = A ≡ ω c . We then measure the resonant coherent oscillation upon parametric modulation with ω =ω c , as shown in Fig. 2A-C. The observed Rabi frequencies under resonant parametric modulations, which reveal the information about the coupling matrix elements connecting the eigenstates [see Eq. (3)] upon parametric modulation, are shown in Fig. 2D-F. The experimental results allow us to determine the quantum geometry of the prepared dressed states precisely.
As a central result, we show in Fig. 3 the experimental extraction of the full QGT, based on Rabi-oscillation measurements. This provides a first demonstration that coherent responses upon parametric modulations can be used as a powerful tool to access the complete geometry of a discrete quantum system. We point out that the present quantum-geometry measurement is based on coherent dynamic responses upon periodic driving, and in this sense, it does not rely on any adiabaticity constraints (i.e. small modulation velocity [36,37]). It should be noted, however, that this method uses small modulation amplitudes, and hence small  Rabi frequencies, which require systems exhibiting long coherence times. The agreement between the experiment results and the theoretical predictions can be improved by increasing the measurement time, which allows for better determination of the oscillation frequency. Furthermore, in contrast with the excitation-rate measurement of [31,32,41], the QGT is extracted from Rabi oscillations [39], where the initial state is recovered after each Rabi period; in principle, this allows for detection of geometry and topology through a non-destructive measurement. Besides, our quantum-geometry measurement can also be used to characterize the topology of the underlying system. For this analysis, we extend the Hamiltonian to the form where r is a tunable parameter. As for Eq. (4), the geometry of the Hamiltonian in Eq. (8) is that of a fictitious monopole located close to a sphere S 2 , whose position in parametric space depends on the additional parameter r . The topology of the system then relies on whether this fictitious monopole is located inside the sphere or not, as captured by the Chern number C = 1 2π ∫ S 2 F θϕ d θ d ϕ [13]. Figure 4 shows the Berry curvature measurement in two distinct topological phases. In the non-trivial regime, the Chern number can equally be determined from the metric 2 is the determinant of the QGT [22]. Altogether, these results indicate that topology can indeed be finely analyzed based on our geometric-detection scheme.

APPLICATION TO INTERACTING QUBITS
As a second application, we further extend our experiment to extract the QGT of an interacting two-qubit system. The interacting two-qubit system is formed by an NV center electron spin coupled to a 13 C nuclear spin located in the vicinity of the NV center. We determine the strength of the corresponding spinspin interactions using a pulsed optically detected magnetic resonance experiment; we obtain the interaction parameters: A x ≈ 2.79 MHz and A z ≈ 11.832 MHz [see Eq. (9) below]. By engineering microwave driving fields with designed frequency and phase, we obtain the following effective Hamiltonian where σ and τ are Pauli operators associated with the first and second qubits, respectively. Henceforth, we denote the eigenstates of the Hamiltonian in π/4 π/2 3π/4 π 0 π/4 π/2 3π/4 π Figure 5. Quantum geometry of an interacting two-qubit system. (A) The measured Fubini-Study metric, compared with the theoretical predictions: g θϕ (green curve), g ϕϕ (red curve) and g θθ (blue curve). (B) The measured local Berry curvature F θϕ is compared with the theoretical prediction (curve). The amplitude of the driving field [see Eq. (9)] is m w = 2.13 MHz. The Chern number estimated from the integral of the Berry curvature is C = 0.00 ± 0.01, which is in agreement with the prediction (C = 0) in this strongly interacting regime.
Eq. (9) as | 1 , | 2 , | 3 , | 4 , according to their ordered eigenenergies 1 < 2 < 3 < 4 . The competition between the local term ( mw ) and the spin-spin interaction in the Hamiltonian Eq. (9) leads to a rich topological phase diagram. In the regime mw the spin-spin interaction becomes less significant and we thus recover the topological properties of the two-level system, for which the Chern number is C = 1 in the eigenstate | 3 (see the measurements described in the previous section); note that the other eigenstates exhibit similar behaviors. The spin-spin interaction eventually dominates upon decreasing the value of the local parameter; below the critical value, mw < (c 1 ) mw , the Chern number of the eigenstate changes from C = 1 to C = 0, which can be seen as a drastic effect of the spin-spin interaction. This vanishing of the Chern number in the strongly interacting regime is clearly captured by our QGT measurement, as reported in Fig. 5. These results demonstrate the measurement of both the Fubini-Study metric and the Berry curvature deep in the interacting regime, and show excellent agreement with theoretical predictions [see supplementary data].
As previously noted, the QGT contains information regarding the entanglement properties of interacting systems, through the concept of quantum Fisher information [23]. As an interesting perspective, our detection method could be applied to more complex interacting systems in view of revealing their quantum fluctuations and entanglement properties.

CONCLUSION
To summarize, we have experimentally demonstrated a powerful connection between the quantum geometric tensor and the coherent dynamic response of a quantum system upon a parametric drive. Based on this fundamental relation, we first extracted the complete QGT, including all the components of the Fubini-Study metric and those of the local Berry curvature, by driving Rabi oscillations in a single qubit. These measurements clearly revealed the topological (monopole-type) structure associated with this simple setting. We point out that this method is readily applicable to observe other intriguing topological defects, such as tensor monopoles defined in 4D parameter spaces [22]. Furthermore, we have applied this detection method to an interacting two-qubit system, which suggests potential applications to many-body quantum systems with geometric features [12,32,45]. Altogether, our results demonstrate that coherent dynamic responses can serve as a powerful tool to access the geometric and topological properties of quantum systems and open a way to explore the fundamental role of the QGT in various scenarios, ranging from many-body systems to open quantum systems.

NOTE ADDED
Materials and methods are available as supplementary material. Two other experimental measurements of the QGT were reported after the completion of our work [46], in polaritons [47] and superconducting qubits [48].