Robust mode conversion in NV centers using exceptional points

We show that microwave-driven NV centers can function as robust mode switches by utilizing a special degeneracy called an exceptional point (EP). While previous theoretical and experimental work on EP-based mode switches applies only to pure states, we develop here a general theory for switching between mixed states, statistical ensembles of different pure states, resulting from the interaction with the environment. Our theory is general and applicable to all leading platforms for quantum information processing and quantum technologies. However, our numerical simulations use empirical parameters of NV centers. We provide guidelines for coping with the main challenges for experimental realization of this protocol: decoherence and mixed-state preparation.

Figure 1

(a) Ground states of the NV center. Two microwave fields drive transitions between the $|{}^3{A}_2,0\rangle$ and $|{}^3{A}_2,\pm 1\rangle$ states, with Rabi frequencies ${\mathrm{\Omega }}_{1,2}$, detunings ${\mathrm{\Delta }}_{1,2}$, and left-/right-circular polarizations ${\sigma }_{\pm }$. Interaction with the environment causes pure dephasing at a rate ${\gamma }^{(1,2)}$ (straight dashed arrows) and downward or upward transitions at rates ${\kappa }_u^{(1,2)}$ or ${\kappa }_d^{(1,2)}$, respectively (wiggly dashed arrows). The evolution of the system is governed by the Lindbladian operator $\widehat{\mathcal{L}}$ [Eq. (3)]. (b) Exceptional points in the Lindbladian. A filled contour plot of the maximal condition number of the eigenvalues of $\widehat{\mathcal{L}}$ [Eq. (7)]. We scan ${\mathrm{\Delta }}_1$ and ${\mathrm{\Omega }}_1$ while fixing all other parameters: ${\mathrm{\Omega }}_2=400,{\mathrm{\Delta }}_2=1400,{\kappa }_u^{(1,2)}={\kappa }_d^{(1,2)}=1,{\gamma }^{\left(1\right)}=900$, and ${\gamma }^{\left(2\right)}=1500$ (in kHz). The condition number diverges when modes coalesce at exceptional points (EPs) (scale on the right).

Figure 2

Real and imaginary parts of the eigenvalues of the Lindbladian operator near the isolated EP. We scan ${\mathrm{\Delta }}_1$ and ${\mathrm{\Omega }}_1$ while holding all other parameters fixed (given in Fig. 1). In this regime, the Lindbladian has four simple eigenvalues (${\lambda }_{3,4,5,6}$) and two complex-conjugate pairs (${\lambda }_{1,2}$ and ${\lambda }_{7,8}$) that coalesce at the EP2. When changing the parameters in a loop around the EP2 (details in Appendix pp4), the simple eigenvalues return to the starting point (dashed lines) while the remaining eigenstates swap (arrowed lines), i.e., schematically, $|1\rangle \leftrightarrow |2\rangle$ and $|7\rangle \leftrightarrow |8\rangle$. The red-rimmed points (associated with ${\lambda }_{1,8}$) are transferred into cyan-rimmed points (${\lambda }_{2,7}$) and vice versa.

Figure 3

Nonreciprocal mode switches in NV centers. We initialize the system in either ${\widehat{\rho }}_{\mathrm{in}}^{\left(1\right)}$ or ${\widehat{\rho }}_{\mathrm{in}}^{\left(2\right)}$ and let it evolve as the parameters are varied along the path from 2. Panels (a), (b), (d), and (e) show the moduli of the off-diagonal density-matrix elements at the input and output. (Diagonal entries are set to zero for clarity.) (a), (b) The initial state ${\widehat{\rho }}_{\mathrm{in}}^{\left(1\right)}$ is adiabatically transported into ${\widehat{\rho }}_{\mathrm{out}}^{\left(1\right)}\propto {\widehat{\rho }}_{\mathrm{in}}^{\left(2\right)}$ after a clockwise loop around the EP. (d), (e) The initial state ${\widehat{\rho }}_{\mathrm{in}}^{\left(2\right)}$ is adiabatically transported into ${\widehat{\rho }}_{\mathrm{out}}^{\left(2\right)}\propto {\widehat{\rho }}_{\mathrm{in}}^{\left(1\right)}$ after a counterclockwise loop. (c), (f) Normalized projections of the initial and final states on the basis states at $t=0$.

Figure 4

Higher-order EPs, found by searching the four-dimensional parameter space spanned by the two Rabi frequencies ${\mathrm{\Omega }}_{1,2}$ and frequency offsets ${\mathrm{\Delta }}_{1,2}$ [see Fig. 1]. (a) A fourth-order degeneracy is found at ${\mathrm{\Delta }}_1=-80.8,{\mathrm{\Omega }}_1=278.2,{\mathrm{\Delta }}_2=44.3,{\mathrm{\Omega }}_2=-445$ (in units of kHz). (b) The plots show real and imaginary parts of the eigenvalues (top and bottom plots) as a function of ${\mathrm{\Omega }}_1$, at three values of ${\mathrm{\Delta }}_1$, above (A) at (B) and below the EP (C). The plots demonstrate that two pairs of second-order EPs (EP2s) merge into an EP4. (c) A fifth-order degeneracy is found at ${\mathrm{\Delta }}_1=-62,{\mathrm{\Omega }}_1=314,{\mathrm{\Delta }}_2=58,{\mathrm{\Omega }}_2=436$. (d) Similar to (b), the plots show the real and imaginary parts of the eigenvalues along the three cuts (labeled A–C) in (c). The plots demonstrate that two pairs of EP2s merge with an EP3 to form the fifth-order EP.

Figure 5

Dynamically encircling an isolated EP. The middle panels (a)–(d) show the evolution of the adiabatic coefficients $|a_i{\left(t\right)|}^2={|{\vec{S}}_i^L\left(t\right)·\vec{S}\left(t\right)|}^2$ [Eq. (D2)] during the loop. The side panels in (a)–(d) show the normalized projections of the initial and final states onto the instantaneous states at the beginning of the loop, $|{\vec{S}}_i^L\left(0\right)·{\vec{S}}_{\mathrm{in}/\mathrm{out}}^{(1,2)}{|}^2$. (a) When initialized in state ${\vec{S}}_{\mathrm{in}}^{\left(1\right)}$ and evolved in a clockwise manner, the coefficients $a_1$ and $a_8$ are predominant throughout the evolution, which implies that the state evolves adiabatically, hence, the final state is the coalescing partner state ${\vec{S}}_{\mathrm{in}}^{\left(2\right)}$. (b) When reversing the direction of the loop, the system does not evolve adiabatically and the final state is ${\vec{S}}_5^R$. (c), (d) Show the situation when the system is initialized in ${\vec{S}}_{\mathrm{in}}^{\left(2\right)}$, where only a counterclockwise loop enables adiabatic mode swapping. The parameters are the same as in Fig. 3 from the main text.