Exceptional Point and Cross-Relaxation Effect in a Hybrid Quantum System

Exceptional points (EPs) are exotic degeneracies of non-Hermitian systems, where the eigenvalues and the corresponding eigenvectors simultaneously coalesce in parameter space, and these degeneracies are sensitive to tiny perturbations on the system. Here, we report an experimental observation of the EP in a hybrid quantum system consisting of dense nitrogen (P1) centers in diamond coupled to a coplanar-waveguide resonator. These P1 centers can be divided into three subensembles of spins and cross relaxation occurs among them. As a new method to demonstrate this EP, we pump a given spin subensemble with a drive field to tune the magnon-photon coupling in a wide range. We observe the EP in the middle spin subensemble coupled to the resonator mode, irrespective of which spin subensemble is actually driven. This robustness of the EP against pumping reveals the key role of the cross relaxation in P1 centers. It offers a novel way to convincingly prove the existence of the cross-relaxation effect via the EP.

Popular Summary

Exceptional points (EPs) are exotic degeneracies of non-Hermitian systems, where the eigenvalues and the corresponding eigenvectors simultaneously coalesce in parameter space. Experimentally, there are various hybrid quantum systems fabricated on chips, such as a spin ensemble coupled to a resonator. To demonstrate the EPs in these systems, one needs to explore a method to precisely tune the coupling between the spin ensemble and the resonator. In this work, we experimentally implement such an EP by tuning the coupling via the generation of the spin excitations.

The hybrid quantum system that we study is composed of paramagnetic dense nitrogen centers in diamond coupled to a coplanar-waveguide resonator. These impurity centers can be divided into three subensembles of spins. We find that only for the middle subensemble can the EP occur in the hybrid quantum system, while the EP does not occur for the other two subensembles, due to the effects of the cross relaxation among these spin subensembles. It is known that an EP can enhance the sensitivity to perturbation on the system. However, our observation of the EP for the middle subensemble does not show any appreciable differences when pumping any of the three spin subensembles. This robustness of the EP against driving reveals the key role of the cross relaxation, which induces the same spin excitations in each subensemble. Further study on monitoring the evolution of this non-Hermitian system can reveal the intriguing time-dependent responses of the system on the drive tone. It will shed new light on the hybrid quantum systems and stimulate further works in this new direction.

Figure 1

(a) A schematic of the hybrid system. The diamond sample with P1 centers is glued on a coplanar-waveguide resonator. (b) A P1 center that involves the substitution of a carbon atom by a nitrogen atom. (c) The energy levels and allowed state transitions in the P1 center.

Figure 2

The experimental realization of the exceptional point. (a) The transmission spectrum measured with a probe tone of power $-120$ dBm, corresponding to an average photon number $\overline{n}=404$. (b) The Rabi splitting due to the coupling between the magnons of the $s=0$ subensemble and the resonator photons versus the power of the drive tone. Here, the static magnetic field is tuned to have the magnons of the $s=0$ subensemble in resonance with the resonator mode: ${\omega }_0/2\pi ={\omega }_c/2\pi =3.093$ GHz. Note that the decay rate of the resonator is nearly independent of the power of the drive tone [cf. Fig. 8 in Appendix pp3]. (c) The effective magnon-photon coupling (gray circles) extracted from the Rabi splittings in (b). The fitting curve is obtained using Eq. (6), with parameters $g/2\pi =$ 17.2 MHz, $\kappa /2\pi =0.6$ MHz, and $\gamma /2\pi =11.9$ MHz. (d) The transmission spectrum simulated with Eq. (1), where the fitting curve in (c) is used for the effective magnon-photon coupling.

Figure 3

The peak positions and line widths of the two polariton modes. (a) The data shown by squares and triangles are the peak positions of the two polariton modes extracted from Fig. 2 and the solid curves are the real parts of the two eigenvalues calculated using Eq. (5). (b) The data denoted by squares and triangles are the line widths of the two polariton modes extracted from Fig. 2 and the solid curves are the imaginary parts of the two eigenvalues calculated using Eq. (5). When $P_d>93.7$ dBm, only one peak is measured. We use both Eqs. (1) and (5) to deduce the line widths of the two modes in this region.

Figure 4

The effect of the cross relaxation. (a),(b) The Rabi splittings related to the $s=0$ subensemble, where the drive tone is applied in resonance with the magnons of the $s=+$ and $-$ subensembles, respectively. The dashed curves are simulated as the real part of Eq. (5). (c),(d) The Rabi splittings related to the $s=+$ and $-$ subensembles, where the drive tone is in resonance with the magnons of the $s=+$ and $-$ subensembles, respectively. The dashed curves are simulated as the real part of Eq. (4), but with ${\omega }_c$, ${\omega }_0$ and $g_{\mathrm{eff}}$ replaced by ${\tilde{\omega }}_c$, ${\omega }_{\pm }$ and $g_{\mathrm{eff},\pm }$ respectively. In (c), the magnons of the $s=+$ subensemble are tuned to have frequency ${\omega }_{+}/2\pi =3.106$ GHz via the static magnetic field. The magnons of the $s=0$ and $-$ subensembles have frequencies ${\omega }_0/2\pi =3.012$ GHz and ${\omega }_{-}/2\pi =2.918$ GHz. The frequency of the resonator mode is found to be ${\omega }_c/2\pi =3.095$ GHz, which is slightly blue shifted from ${\omega }_c/2\pi =$ 3.093 GHz in Fig. 2 due to the static magnetic field (see Appendix pp3). In (d), the magnons of the $s=-$ subensemble are tuned to have frequency ${\omega }_{-}/2\pi =3.080$ GHz via the static magnetic field. The magnons of the $s=0$ and $+$ subensembles have frequencies ${\omega }_0/2\pi =3.175$ GHz and ${\omega }_{+}/2\pi =3.269$ GHz. The frequency of the resonator mode is found to be ${\omega }_c/2\pi =3.090$ GHz, which is slightly red shifted from ${\omega }_c/2\pi =3.093$ GHz in Fig. 2 due to the static magnetic field. The other parameters used are the same as in Fig. 2.

Figure 5

A schematic of the experimental setup, where the hybrid system in Fig. 1 is placed in a cryogenic chamber.

Figure 6

The effective-coupling dependence of the parameters of the eigenvector. (a) The normalized amplitudes $A_{1,2}/\sqrt{1+A_{1,2}^2}$ and (b) the relative phases ${\varphi }_{1,2}/\pi$ of the two eigenvectors versus the effective coupling strength $g_{\mathrm{eff}}/2\pi$. Here, we choose $\kappa /2\pi =0.6$ MHz and $\gamma /2\pi =11.9$ MHz, as in the main text.

Figure 7

The transmission amplitude versus the duration time of the drive tone. The $s=0$ subensemble is tuned to be in resonance with the resonator mode. We resonantly pump the $s=0$ subensemble with a drive tone and measure the transmission amplitude at 3.090 GHz. To measure the transmission spectra in our experiment, the duration time of the drive tone is chosen to be 3000 s before each measurement of the transmission amplitude is implemented.

Figure 8

(a) The dependence of the resonator-mode bare frequency on the static magnetic field. The transmission spectrum is obtained in the same way as in Fig. 2, but using a strong probe tone. (b) The measured line width of the resonator mode versus the drive power.

Figure 9

The effective magnon-photon coupling versus the driving power. (a) The data for $g_{\mathrm{eff}}$ (circles) extracted from the Rabi splitting in Fig. 4 (see the main text), where the $s=0$ subensemble is in resonance with the resonator mode and the $s=+$ subensemble is resonantly pumped by a drive tone. The solid curve is calculated for $g_{\mathrm{eff,+}}$ using Eq. (D9) with ${\mathrm{\Delta }}_c/2\pi =-94$ MHz. (b) The data for $g_{\mathrm{eff}}$ (circles) extracted from the Rabi splitting in Fig. 4, where the $s=0$ subensemble is in resonance with the resonator mode and the $s=-$ subensemble is resonantly pumped by a drive tone. The solid curve is calculated for $g_{\mathrm{eff,-}}$ using Eq. (D9) with ${\mathrm{\Delta }}_c/2\pi =94$ MHz. The other parameters are the same as in Fig. 2.

Figure 10

The transmission spectra without the EP. (a),(b) The Rabi splittings related to the $s=+$ subensemble, where the drive tone is in resonance with the $s=0$ and $s=-$ subensembles, respectively. (c),(d) The Rabi splittings related to the $s=-$ subensemble, where the drive tone is in resonance with the $s=0$ and $s=+$ subensembles, respectively. The dashed curves are simulated as the real parts of the two eigenvalues, as in Figs. 4 and 4.