Robust mode conversion in NV centers using exceptional points

TL;DR

This paper demonstrates how NV centers can be used as topological mode switches by exploiting exceptional points, enabling non-reciprocal mode switching in open quantum systems at room temperature.

Contribution

It extends topological mode switching theory from pure to mixed states, showing practical implementation in NV centers with potential quantum information applications.

Findings

Successful simulation of topological and non-reciprocal mode switching

Quantum correlations decrease significantly at room temperature

Framework applicable to open quantum systems with mixed states

Abstract

We show that microwave-driven NV centers can function as topological mode switches by utilizing a special degeneracy called an exceptional point (EP). By tuning the intensities and frequencies of the driving fields, we find an EP---where two normal modes of the system coalesce---and, then, use it to simulate the dynamics and demonstrate topological and non-reciprocal mode switching. By comparing density matrices of the input and output states, we find that the quantum correlations decrease by three orders of magnitude at room temperature, and discuss ways for improving this result. This work extends the theory of topological mode switches (originally derived for pure states) to mixed states and is, therefore, applicable to general open quantum systems. Our theory enables exploring new phenomena (e.g., high-order EPs in low-dimensional systems) and presents a crucial step towards incorporating topological mode switches in quantum-information applications.

Figures

Equations

$${\Gamma }_{\ell }(t)\equiv {\int }_0^t\mathrm{Re}\left[{\lambda }_{\ell }(t^{\prime })-{\lambda }_j(t^{\prime })\right]d{t}^{\prime }>0,\forall j\ne \ell ,$$

$$\hat{H}={\Delta }_1|1⟩⟨1|+{\Delta }_2|-1⟩⟨-1|+$$

$$-\hslash {\Omega }_1\left(|1⟩⟨0|+|0⟩⟨1|\right)-\hslash {\Omega }_2\left(|-1⟩⟨0|+|0⟩⟨-1|\right),$$

$$\dot{\hat{X}}\equiv \hat{\mathcal{L}}[\hat{X}]=\frac{i}{\hslash }[\hat{H},\hat{X}]+\sum _j{\Gamma }_j\left(2{\hat{L}}_j^{†}\hat{X}{\hat{L}}_j-\{{\hat{L}}_j^{†}{\hat{L}}_j,\hat{X}\}\right).$$

$$\text{mbox}puredephasing:{\hat{L}}_{1,2}=|\pm 1⟩⟨\pm 1|-|0⟩⟨0|$$

$$\text{mbox}up/downjumps:{\hat{L}}_{3,4}=|\pm 1⟩⟨0|,{\hat{L}}_{5,6}=|0⟩⟨\pm 1|$$

$$\dot{\vec{S}}=\hat{M}(\vec{S}-{\vec{S}}_{\mathrm{eq}}).$$

$$\hat{M}{\vec{S}}_i^R={\lambda }_i{\vec{S}}_i^R,{\hat{M}}^T{\vec{S}}_i^L={\lambda }_i{\vec{S}}_i^L,$$

$$N({\lambda }_i)\equiv \frac{1}{|\cos ({\theta }_i)|}=\frac{|{\vec{S}}_i^L||{\vec{S}}_i^R|}{|{\vec{S}}_i^L\cdot {\vec{S}}_i^R|},$$

$$\hat{\rho }=\frac{1}{3}\left(\hat{1}+\sqrt{3}\sum _{i=1}^8\widehat{{\sigma }_i}{S}_i\right),$$

$$\hat{M}(s)=\sum _{j=1}^n{\lambda }_j(s)|{\psi }_j^R(s)⟩⟨{\psi }_j^L(s)|,$$

$$\epsilon \frac{\partial }{\partial s}|\psi (s)⟩=\hat{M}(s)|\psi (s)⟩.$$

$$|\psi (s)⟩=\sum _j{a}_j(s)\exp \left(-\frac{1}{\epsilon }{\int }_{s_0}^s{\lambda }_j(u)du\right)$$

$${\partial }_s{a}_{\ell }(s)+⟨{\psi }_{\ell }^L(s)|{\partial }_s{\psi }_{\ell }^R(s)⟩a_{\ell }(s)=$$

$$\displaystyle-\sum_{j\neq\ell}\langle{\psi^{L}_{\ell}(s)}|{\partial_{s}\psi^{R}_{j}(s)}\rangle a_{j}(s)\scalebox{0.9}{$\exp\left(-\frac{1}{\varepsilon}\int_{s_{0}}^{s}\left[\lambda_{j}(u)-\lambda_{\ell}(u)\right]du\right)$}$$

$${\Gamma }_{\ell j}(s)={\int }_{s_0}^s\mathrm{Re}\left[{\lambda }_j(u)-{\lambda }_{\ell }(u)\right]du>0$$

$$\displaystyle\hat{\sigma}_{1}=\left(\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&0\end{array}\right)\quad\hat{\sigma}_{2}=\left(\begin{array}[]{ccc}0&-i&0\\ i&0&0\\ 0&0&0\end{array}\right)$$

$$\displaystyle\hat{\sigma}_{3}=\left(\begin{array}[]{ccc}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right)\quad\hat{\sigma}_{4}=\left(\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 1&0&0\end{array}\right)$$

$$\displaystyle\hat{\sigma}_{5}=\left(\begin{array}[]{ccc}0&0&-i\\ 0&0&0\\ i&0&0\end{array}\right)\quad\hat{\sigma}_{6}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&1&0\end{array}\right)$$

$$\displaystyle\hat{\sigma}_{7}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&-i\\ 0&i&0\end{array}\right)\quad\hat{\sigma}_{8}=\frac{1}{\sqrt{3}}\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&-2\end{array}\right)$$

$$\displaystyle\scalebox{0.8}{$\left(\begin{array}[]{cccccccc}-\gamma_{11}&\Delta_{1}&0&0&0&0&\Omega_{2}&0\\ -\Delta_{1}&-\gamma_{22}&-2\Omega_{1}&0&0&\Omega_{2}&0&0\\ 0&2\Omega_{1}&-\gamma_{33}&0&\Omega_{2}&0&0&\gamma_{38}\\ 0&0&0&-\gamma_{44}&\Delta_{2}&0&-\Omega_{1}&0\\ 0&0&-\Omega_{2}&-\Delta_{2}&-\gamma_{55}&\Omega_{1}&0&-\sqrt{3}\Omega_{2}\\ 0&-\Omega_{2}&0&0&-\Omega_{1}&-\gamma_{66}&-(\Delta_{1}-\Delta_{2})&0\\ -\Omega_{2}&0&0&\Omega_{1}&0&(\Delta_{1}-\Delta_{2})&-\gamma_{77}&0\\ 0&0&\gamma_{83}&0&\sqrt{3}\Omega_{2}&0&0&-\gamma_{88}\end{array}\right)$},$$

$${\gamma }_{11}={\gamma }_{22}\equiv \left(\frac{{\gamma }_2}{8}+\frac{{\gamma }_1+{\kappa }_d^{(1)}+{\kappa }_u^{(1)}+{\kappa }_u^{(2)}}{2}\right)$$

$${\gamma }_{33}\equiv \frac{{\kappa }_u^{(2)}}{2}+{\kappa }_d^{(1)}+{\kappa }_u^{(1)}$$

$${\gamma }_{38}\equiv \left({\kappa }_d^{(1)}-{\kappa }_d^{(2)}-{\kappa }_u^{(1)}-\frac{{\kappa }_u^{(2)}}{3}\right)\frac{1}{\sqrt{3}}$$

$${\gamma }_{44}={\gamma }_{55}\equiv \frac{{\gamma }_1}{8}+\frac{{\gamma }_2+{\kappa }_u^{(1)}+{\kappa }_d^{(2)}+{\kappa }_u^{(2)}}{2}$$

$${\gamma }_{66}={\gamma }_{77}\equiv \frac{{\gamma }_1+{\gamma }_2}{8}+\frac{{\kappa }_d^{(1)}+{\kappa }_d^{(2)}}{2}$$

$${\gamma }_{83}\equiv -\frac{\sqrt{3}}{2}{\kappa }_u^{(2)}$$

$${\gamma }_{88}\equiv \frac{{\kappa }_u^{(2)}}{2}+{\kappa }_d^{(2)}.$$

$$\displaystyle-(\hat{M}^{-1}\vec{S}_{\mathrm{eq}})^{T}=\left(\begin{array}[]{cccccccc}0&0&\tfrac{\Delta\kappa_{2}}{3}+\tfrac{2\Delta\kappa_{1}}{3}&0&0&0&0&\tfrac{\Delta\kappa_{2}}{\sqrt{3}}\end{array}\right),$$

$$\Delta {\kappa }_i\equiv {\kappa }_d^{(i)}-{\kappa }_u^{(i)}.$$

$${\Delta }_1(t)={\Delta }_{\mathrm{EP}}+R_{\Delta }\cos (2\pi t/T+\varphi ),$$

$${\Omega }_1(t)={\Omega }_{\mathrm{EP}}+R_{\Omega }\sin (2\pi t/T+\varphi ).$$

$$\vec{S}(t)=\sum _{i=1}^8{a}_i(t){\vec{S}}_i^R(t)$$

Full text

A new class of adiabatic protocols enables robust mode conversion in open systems that possess a special degeneracy called an exceptional point (EP)—where multiple modes of the system coalesce Kato (2013); Strang (2006); Trefethen and Embree (2005). EP-based mode switches have intriguing physical properties, such as topological protection and non-reciprocity Uzdin et al. (2011); Berry and Uzdin (2011); Milburn et al. (2015); Hassan et al. (2017a, b), which were demonstrated experimentally in optical waveguides and optomechanics Doppler et al. (2016); Xu et al. (2016); Yoon et al. (2018); Zhang et al. (2018a, b); Zhang and Chan (2018) and theoretically proposed for several additional systems Chatzidimitriou and Kriezis (2018); Kaprálová-Žďánská and Moiseyev (2014); Gilary et al. (2013); Ke et al. (2016). Realizing robust non-reciprocal mode switching in quantum systems has far reaching consequences in quantum information processing and quantum control, as well as in quantum technology. Here we show how to realize EP-based mode switches in atomic and atom-like systems. While previous work on EP-based mode switches applies only to pure states, the theoretical description of atom-like systems typically requires mixed states—statistical ensembles of different pure states, which arise due to interactions with the environment. To bridge this gap, we develop a theory of mode switching between mixed states. Our protocol applies, most generally, to three-level systems in the V-configuration, and we perform numerical simulations using empirical parameters of nitrogen-vacancy (NV) centers—defects in diamond with exceedingly long coherence lifetimes and established optical and microwave mechanisms for initialization, manipulation and readout of their spin state Taylor et al. (2008); Maze et al. (2008); Balasubramanian et al. (2008); Jelezko and Wrachtrup (2006). Our theory enables exploring new phenomena (e.g., high-order EPs in low-dimensional systems) and presents a crucial step towards incorporating EP-based mode switches in quantum-information applications.

Robust mode switches are based on the adiabatic theorem, which describes the evolution of slowly varying closed systems. The theorem states that when preparing a system in a particular eigenmode, it remains in that mode during the evolution (given the conditions specified in Refs. 23; 24). Dynamic closed systems are described by Hermitian Hamiltonians that depend on a set of “control parameters.” When changing the parameters slowly along closed loops in parameter space, the theorem implies that the initial and final states are the same (up to a phase Pancharatnam (1956); Longuet-Higgins et al. (1958); Berry (1984)). The understanding that topological operations (i.e., executing closed control paths) may have outcomes that are robust against noise has lead to important discoveries in multiple areas of physics Xiao et al. (2010); Nayak et al. (2008); Lu et al. (2014); Mead (1992).

However, most physical systems exchange energy or particles with their environment. Open systems can be described by effective non-Hermitian operators, and their adiabatic transport properties are drastically different from closed ones. First, unlike Hermitian operators, non-Hermitian operators may have EPs, where multiple eigenmodes have the same eigenvalue and eigenvector Kato (2013); Strang (2006); Trefethen and Embree (2005). Near EPs, the eigenvalues are a multivalued functions of the system’s parameters and, consequently, when changing the control parameters along any closed loop that encircles an EP, the eigenmodes at the initial and final points may differ Dembowski et al. (2001). For example, when the EP is formed by the coalescence of two modes (hereafter called an EP2), the eigenmodes swap; i.e., $|{1}\rangle\!\!\rightarrow\!\!|{2}\rangle$ and $|{2}\rangle\!\!\rightarrow\!\!|{1}\rangle$. Secondly, in non-Hermitian systems, the adiabatic theorem holds only for “least decaying states” Nenciu and Rasche (1992); Sarandy and Lidar (2005); Miniatura et al. (1990), which are eigenmodes $\ell$ with eigenvalues $\lambda_{\ell}$ whose accumulated decay rate is positive:

[TABLE]

for all $t$ during the evolution (see appendix). When only two modes are involved, the accumulated decay rates, $\Gamma_{1,2}(t)$, have opposite signs, which implies that while one of the states can evolve adiabatically, the other one cannot. Therefore, when encircling an EP2 along a certain path, either $|{1}\rangle\!\!\rightarrow\!\!|{2}\rangle$ or $|{2}\rangle\!\!\rightarrow\!\!|{1}\rangle$ Uzdin et al. (2011); Berry and Uzdin (2011). For any given loop, when reversing the direction of the path, the sign of the accumulated decay rate is reversed [since the order of integration limits in Eq. (1) is exchanged], and this is the source for “non-reciprocity” of EP-based switches Doppler et al. (2016); Xu et al. (2016).

The performance of the switch is robust since it is related to a topological property of the system’s energy surfaces—the existence of an EP, which is a branch point in the system’s energy surfaces Doppler et al. (2016); Xu et al. (2016). In the adiabatic limit, the output of the switch depends only on whether or not the loop encircles an EP. Typically, small perturbations in the operational details only slightly distort the loop or move the location of the EP, but do not lift the degeneracy. The possibility of creating a non-reciprocal switches between pure states has attracted considerable attention Uzdin et al. (2011); Berry and Uzdin (2011); Milburn et al. (2015); Hassan et al. (2017a, b); Doppler et al. (2016); Xu et al. (2016); Yoon et al. (2018); Zhang et al. (2018a, b); Zhang and Chan (2018); Chatzidimitriou and Kriezis (2018); Kaprálová-Žďánská and Moiseyev (2014); Gilary et al. (2013); Ke et al. (2016) and, here, we generalize this concept for mixed states. Our protocol includes ($i$) finding an isolated EP in the eigenvalue spectrum of the NV center (Fig. 1), ($ii$) initializing the system in special superposition states (Fig. 2), and ($iii$) changing the control parameters in a loop around the EP (Fig. 3).

Figure 1(a) shows the electronic ground-state manifold of the negatively charged NV center (also called NV-). It consists of three spin-triplet states, $|{{}^{3}\!A_{2},m_{s}}\rangle$, where $m_{s}=0,\pm 1$ denotes the spin projection along the NV axis and $A_{2}$ marks the orbital symmetry Doherty et al. (2013); Chu and Lukin (2015). In the absence of external magnetic fields, the energy levels of $|{{}^{3}\!A_{2},0}\rangle$ and $|{{}^{3}\!A_{2},\pm 1}\rangle$ are split by $2.87$ GHz. We introduce right- and left-circularly polarized transverse microwave fields to selectively drive the $|{{}^{3}\!A_{2},0}\rangle\leftrightarrow|{{}^{3}\!A_{2},\pm 1}\rangle$ transitions111Alternatively, by applying a longitudinal magnetic field to lift the degeneracy of the $|{{}^{3}\!A_{2},\pm 1}\rangle$ states Togan et al. (2011), one can excite these states selectively with a linearly polarized microwave field. Alegre et al. (2007); London et al. (2014); Mrózek et al. (2015) (solid arrows). Using a semiclassical description Haken (1985) (where the electrons are treated quantum mechanically and the fields are treated classically) and employing the rotating-wave approximation Scully and Zubairy (1999) (which is valid when the driving fields are nearly resonant and relatively weak), the Hamiltonian of the driven NV center in the ground-state manifold is Scully and Zubairy (1999)

[TABLE]

where $\Omega_{i}\equiv\frac{\mathcal{E}_{i}\mu_{i}}{\hbar}$ denotes the Rabi frequency of each microwave field (with $\mathcal{E}_{i}$ the field amplitude, $\mu_{i}$ the transition dipole moment, and $i=1,2$) and $\Delta_{i}\equiv E_{i}-E_{0}-\hbar\omega_{i}$ denotes the single-photon detuning, i.e., the frequency offset from each atomic transition (with $E_{0,i}$ the energies of states $|{0,\pm 1}\rangle$ and $\omega_{i}$ the microwave frequencies).

The electronic state of the system is described by a density matrix, whose evolution is governed by the Lindblad master equation Breuer and Petruccione (2002); Scully and Zubairy (1999). In the Heisenberg picture, the Lindblad equation of motion for any observable $\hat{X}$ is

[TABLE]

The first term represents Hamiltonian evolution and the remaining terms describe incoherent processes due to the interaction with the environment. We consider two types of processes Breuer and Petruccione (2002); Am-Shallem et al. (2016): ($i$) pure dephasing at a rate $\gamma^{(1,2)}$ (straight dashed arrows) and ($ii$) downwards or upwards jumps at rates $\kappa_{d}^{(1,2)}$ or $\kappa_{u}^{(1,2)}$ respectively (wiggly dashed arrows). At thermal equilibrium, the ratio of upwards and downwards transitions is given by the Boltzmann factor $\kappa_{u}^{(1,2)}/\kappa_{d}^{(1,2)}=\exp(\hbar\omega_{1,2}/k_{B}T)$, where $k_{B}T$ is the thermal energy Am-Shallem et al. (2016). We assume that the system is at room temperature ($\approx 300K$), where upwards and downwards transition rates are almost equal222It is possible to enhance the rate of downward transitions using green light Doherty et al. (2013). This flexibility allows searching a wider parameter range, as will be explored in future work.. The incoherent processes are described by Mathisen and Larson (2018):

[TABLE]

with $\Gamma^{(1,2)}=\gamma^{(1,2)},\Gamma^{(3,4)}=\kappa_{d}^{(1,2)}$ and $\Gamma^{(5,6)}=\kappa_{u}^{(1,2)}$.

In order to find EPs in the eigenvalue spectrum of $\hat{\mathcal{L}}$, we rewrite the superoperator equation [Eq. (3)] in a form that is more convenient for theoretical investigation Mathisen and Larson (2018); Am-Shallem et al. (2015a). We introduce a basis of traceless orthogonal matrices, which spans the space of density matrices Byrd and Khaneja (2003); Goyal et al. (2016); Kimura (2003). Specifically, we choose the eight Gell–Mann matrices ($\hat{\sigma}_{1},\ldots,\hat{\sigma}_{8}$, defined in the appendix), which generalize Pauli matrices for three-level systems Gell-Mann (2010). By applying Eq. (3) to each Gell–Mann matrix ($\hat{\sigma}_{i}$) and taking the expectation value of the resulting equation, we obtain

[TABLE]

Here, $\vec{S}$ is an eight-dimensional vector, whose entries are $S_{i}=\mathrm{Tr}[\hat{\rho}\cdot\hat{\sigma_{i}}]$ and $\hat{\rho}$ is the density matrix. The real parts of the eigenvalues of $\hat{M}$ are the relaxation rates of the eigenmodes, while $\vec{S}_{\mathrm{eq}}$ is the steady state. Explicit expressions for $\hat{M}$ and $\vec{S}_{\mathrm{eq}}$ are given in the appendix [Eq. (B37) and Eq. (B40) respectively]. We consider here room temperatures, where $\vec{S}_{\mathrm{eq}}\approx 0$, since the rates of incoherent upwards and downwards transitions are equal. Left and right eigenvectors and eigenvalues of $\hat{M}$ satisfy

[TABLE]

where the superscript $T$ denotes matrix transposition. The matrix $\hat{M}$ is non-Hermitian and, hence, can have EPs. At an EP, the left and right eigenvectors of the degenerate eigenmode are orthogonal Moiseyev (2011) (i.e., $\vec{S}_{i}^{L}\cdot\vec{S}_{i}^{R}=0$). Therefore, the condition number, defined as the secant of the angle between left and right eigenvectors Trefethen and Bau III (1997),

[TABLE]

diverges at the EP.

Figure 1(b) reveals the location of EPs in the eigenvalue spectrum of $\hat{\mathcal{L}}$. We scan the parameters of the right-circularly polarized field ($\Omega_{1}$ and $\Delta_{1}$) while holding all other parameters fixed (see caption). At each point in parameter space, we compute the condition numbers of the eigenvalues of $\hat{M}$ and, then, plot the maximal condition number attained. The dark regions in the figure mark the location of the EPs. We determine the order of the degeneracy by plotting the eigenvectors at selected points along the dark lines. We find three lines of EP2s, which intersect at two points of EP3s (similar to Refs. 54; 45) and an isolated EP2 at ($\Delta_{1}^{\mathrm{EP}},\Omega_{1}^{\mathrm{EP}})\approx(-80,225)$ kHz. The same system can be used for finding fourth- and fifth-order EPs, as we show in the appendix.

Next, we demonstrate swapping of the instantaneous eigenvalues along loops that encircle the isolated EP2. Figure 2 shows surfaces of the real and imaginary parts of the eigenvalues of $\hat{M}$ [Eq. (B37)] as a function of $\Delta_{1}$ and $\Omega_{1}$. In the shown parameter regime, $\hat{M}$ has four simple eigenvalues ($\lambda_{3,4,5,6}$) and two complex-conjugate pairs of eigenvalues ($\lambda_{1,2}$ and $\lambda_{7,8}$) that coalesce at the isolated EP2. We choose a loop that encircles the EP2 and, then, compute the eigenvalues at each point along the path. (The path details are given in the appendix). As expected, the simple eigenvalues return to the starting point after the loop (dashed lines), while the remaining eigenstates swap (arrowed lines). That is, $|{1}\rangle\leftrightarrow|{2}\rangle$ and $|{7}\rangle\leftrightarrow|{8}\rangle$; the red-rimmed points (associated with $\lambda_{1,8}$) are transferred into cyan-rimmed points (associated with $\lambda_{2,7}$) after the cycle and vice versa.

Having shown that the eigenmodes of $\hat{M}$ swap when encircling the EP2, let us adopt a more intuitive picture and transform the Gell–man vectors into density matrices. For three-level systems, the transformation is Kimura (2003),

[TABLE]

where $\hat{\sigma}_{i}$ are the Gell–Mann matrices (see appendix). In order to conserve the probabilistic interpretation of $\hat{\rho}$, $\vec{S}$ must be real Byrd and Khaneja (2003); Goyal et al. (2016); Kimura (2003). Consequently, complex eigenmodes (such as $\vec{S}_{1,2}^{R}$) cannot be used as inputs for the switch and, instead, we initialize it in symmetric superpositions of complex-conjugate vectors, i.e., $\vec{S}^{(1)}_{\mathrm{in}}\propto\vec{S}^{R}_{1}+\vec{S}_{8}^{R}$ and $\vec{S}^{(2)}_{\mathrm{in}}\propto\vec{S}^{R}_{2}+\vec{S}_{7}^{R}$. Positivity of $\hat{\rho}$ implies that the length of $\vec{S}$ needs to be smaller than a critical value Kimura (2003). To satisfy this condition, we normalize $\vec{S}^{(1)}_{\mathrm{in}}$ and $\vec{S}^{(2)}_{\mathrm{in}}$ accordingly. Last, we use Eq. (8) to transform the Bloch vectors, $\vec{S}^{(1,2)}_{\mathrm{in}}$, into density matrices, $\hat{\rho}^{(1,2)}_{\mathrm{in}}$.

Next, we simulate the evolution of the system. We initialize the system in either $\hat{\rho}^{(1)}_{\mathrm{in}}$ or $\hat{\rho}^{(2)}_{\mathrm{in}}$ and solve Eq. (5) via the standard Runge-Kutta method Trefethen and Bau III (1997). Figure 3 summarizes our main result: The initial state $\hat{\rho}^{(1)}_{\mathrm{in}}$ is adiabatically transported into $\hat{\rho}^{(2)}_{\mathrm{in}}$ when the parameters are varied in a clockwise manner around the EP [panels (a–b)]. This happens because $\hat{\rho}^{(1)}_{\mathrm{in}}$ is the least-decaying eigenstate for a clockwise loop, which guarantees that the probability for quantum jumps into different eigenstates is negligible. Conversely, $\hat{\rho}^{(2)}_{\mathrm{in}}$ is adiabatically transported into $\hat{\rho}^{(1)}_{\mathrm{in}}$ only after a counterclockwise loop around the EP [panels (d–e)]. More details about the dynamics are given in the appendix (Fig. 5). Panels (c,f) show the projections of the initial and final Gell–Mann vectors on the eigenvectors at the beginning of the loop, reaffirming that $\vec{S}_{\mathrm{in}}^{(1)}\!\!\rightarrow\!\!\vec{S}_{\mathrm{in}}^{(2)}$ after a clockwise loop while $\vec{S}_{\mathrm{in}}^{(2)}\!\!\rightarrow\!\!\vec{S}_{\mathrm{in}}^{(1)}$ after a counterclockwise loop.

In order for the system to evolve adiabatically, the sweep rates of $\Delta_{1}$ and $\Omega_{1}$ need to be small compared to the “energy gap” Bohm (2012) (i.e., the distance between the complex eigenvalues). It implies that the cycle needs to be long compared to the dephasing time (e.g., we chose $T=15/\gamma^{(1)}$). Consequently, the final states, $\vec{S}_{\mathrm{out}}^{(1,2)}$, approach zero [since $\vec{S}_{\mathrm{eq}}\approx 0$]. From Eq. (8), one learns that when $|\vec{S}|\ll 1$, $\hat{\rho}$ is nearly diagonal, and it is hard to distinguish the final states from diagonal matrices. For visual clarity, in Fig. 3, we set the diagonal elements of $\hat{\rho}$ to zero, and show only the moduli of the off-diagonal elements. Since these terms are very small [$\mathcal{O}(10^{-4})$], it is challenging to distinguish between the final states. Possible approaches for fighting decoherence are discussed in the concluding paragraph.

Another challenge arises from the fact that the input states, $\hat{\rho}^{(1,2)}_{\mathrm{in}}$, are mixed. Experimentally, preparing pure states is a standard procedure Doherty et al. (2013) but the preparation of mixed states is more challenging. One approach for overcoming this problem is to prepare the system in pure states in the vicinity of $\hat{\rho}^{(1,2)}_{\mathrm{in}}$. The problem of finding the nearest pure state to a given density matrix is equivalent to finding the best rank-one approximation for that matrix, which can be solved by exploiting the singular value decomposition Markovsky (2008); Abdi and Williams (2010). Unfortunately, when applying this algorithm to $\hat{\rho}^{(1,2)}_{\mathrm{in}}$, one obtains pure states that have significant population in undesired states333The pure states have significant population in $\vec{S}^{R}_{5}$, which is the steady state since $|\mathrm{Re}[\lambda_{5}]|<|\mathrm{Re}[\lambda_{j}]|$ for all $j\neq 5$ [see Fig. 2(a)]. Since the cycle is much longer than the coherence lifetime, the final state is almost precisely $\vec{S}^{R}_{5}$, and the swapping of the partial populations of $\vec{S}_{\mathrm{in}}^{(1)}$ and $\vec{S}_{\mathrm{in}}^{(2)}$ becomes unmeasurable., which deteriorate the performance of the switch. Alternatively, one could use quantum control to find protocols for preparing arbitrary mixed states Bartana et al. (1997); Jirari and Pötz (2005); Roloff et al. (2009); Yan et al. (1993). However, experimental implementation of such protocols may be very difficult. These direction will be explored in future work.

To summarize, we presented a protocol for achieving robust mode conversion with NV centers by using EPs. Our work generalizes the existing theory of EP-based switches in two aspects:

1. By including mixed states, which are necessary for describing most existing platforms for quantum information processing, and 2. By treating multilevel systems, generalizing previous work that focused on two-level systems. We find that EP-based mode switches in Lindbladian systems require at least three electronic levels, and that one could force high-order EPs by carefully tuning several control parameters [see  Fig. 4]. Our analysis raises two challenges for experimental demonstration of this protocol: mixed-state preparation and decoherence. The former challenge is technical, and several ways for addressing it are mentioned above. The latter—fighting decoherence—is more fundamental. Quite generally, there is an incompatibility between adiabaticity (which implies slow evolution) and maintaining coherence (which requires, in turn, fast evolution) Miniatura et al. (1990). Adiabaticity requires that the parameter sweep rate should be smaller than the energy gap Born and Fock (1928) which, in our case, is on the order of the decoherence rate. Unfortunately, it implies that whenever the evolution period is long enough to enable adiabatic evolution, the signal degrades substantially. This shortcoming can be remedied by using modified protocols. For example, one could introduce additional lasers that actively restore the lost coherence during the evolution. Such “cycling transitions” exist in NV centers, and require cold-temperature conditions Togan et al. (2011). Another possibility is to use the notion of PT symmetry, which implies that under some appropriate conditions, a PT-symmetric non-Hermitian system evolves without dissipation Bender et al. (1999); Rüter et al. (2010). PT-symmetric evolution was recently observed in NV centers Wu et al. (2019), and this approach can be extended to design a PT-symmetric mode switch. Such protocols are expected to significantly improve the performance of the switch, and will be addressed in future work.

Acknowledgment

The authors would like to thank Ronnie Kosloff, Saar Rahav, Alexei Mailybaev, Yaniv Kurman, Sigal Wolf, and Idan Meirzada for helpful discussions. AP is supported by an Aly Kaufman Fellowship at the Technion and by The Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel. NM acknowledges the financial support of I-Core: The Israeli Excellence Center “Circle of Light,” and of the Israel Science Foundation (Grant No. 1530/15). NB acknowledges support from the European Union (ERC StG, MetaboliQs), the CIFAR Azrieli Global Scholars program, the Ministry of Science and Technology, Israel, and the Israel Science Foundation (Grant No. 750/14).

I Appendix

I.1 Adiabatic theorem for non-Hermitian systems

In this appendix, we sketch the proof of the adiabatic theorem for non-Hermitian systems following Nenciu and Rasche (1992). Let us investigate the conditions for adiabatic evolution of the normal modes of a non-Hermitian time-dependent operator, $\hat{M}(\varepsilon t)$ (with $\varepsilon>0$). We introduce a new variable, $s=\varepsilon t$, and consider the limit of $\varepsilon\rightarrow 0$ while $s$ is held fixed and finite. By invoking the normal-mode expansion of $\hat{M}(s)$, one can write

[TABLE]

where the time-dependent right- and lef-eigenvectors and corresponding eigenvalues are defined in Eq. (6) in the main text. The normal modes satisfy the biorthogonality condition $\langle{\psi^{L}_{i}(s)}|{\psi^{R}_{j}(s)}\rangle=\delta_{ij}$. Let $|{\psi(s)}\rangle$ be a solution of the differential equation

[TABLE]

Substituting the following ansatz

[TABLE]

into Eq. (A2) and using the biorthogonality condition, one obtains

[TABLE]

From Eq. (A4), one can easily read the conditions for adiabatic evolution. For Hermitian systems [where $\hat{M}(s)=\hat{M}^{\dagger}(s)$], the eigenvalues [$\lambda_{j}(s)$] are real and the right-hand side of Eq. (A4) contains only rapidly oscillating terms, which average to zero in the limit of $\varepsilon\rightarrow 0$. Conversely, when $\hat{M}(s)$ is non-Hermitian, the right-hand side of Eq. (A4) can be neglected only when

[TABLE]

for all $j\neq\ell$. To summarize, the condition for adiabatic evolution comes from requiring that the probability to jump from state $\ell$ to $j$ is small. In contrast to Hermitian systems, where the probability for quantum jumps oscillates in time, in non-Hermitian systems, it also contains exponentially growing or decaying factors, and can be small only if $\Gamma_{\ell j}(t)>0$ for all $j\neq\ell$.

I.2 Vectorizing the Lindblad master equation

In this appendix we derive Eq. (5) from the main text and present explicit expressions for the dynamical matrix $\hat{M}$ and the steady-state Gell–Mann vector $\vec{S}_{\mathrm{eq}}$. A similar formulation was introduced in Mathisen and Larson (2018) to analyze the stimulated Raman adiabatic passage (STIRAP) method Vitanov et al. (2001). To this end, we introduce the set of Gell–Mann matrices Gell-Mann (2010):

[TABLE]

We apply the Lindblad master equation [Eq. (3)] to each Gell-Mann operator ($\hat{\sigma_{i}}$) and compute the expectation value, obtaining Eq. (5) from the main text, with

[TABLE]

where we introduced the notation

[TABLE]

The steady-state vector is given by

[TABLE]

where we define the difference between upwards and downwards jumps as

[TABLE]

When the upwards and downwards rates are equal (i.e., in the high-temperature limit), the steady-state vector vanishes and the Gell–Mann vector will approach the origin at asymptotically large evolution times.

I.3 High-order EPs in a three-level system

In the main text, we show the existence of second- and third-order EPs in the ground-state manifold of the NV center. Here, we show that it is also possible to find high-order EPs in this system. For example, by searching the four-dimensional space spanned by $\Omega_{1,2}$ and $\Delta_{1,2}$ (see text), one can induce fourth- and fifth-order EPs, as shown in Fig. 4. More generally, the density matrix of an $N$-level system has $N^{2}-1$ real degrees of freedom. So one could potentially find eighth-order EPs in this system, but that would require searching a higher-dimensional parameter space. In order to find a real degenerate eigenvalue of degree $M$, one needs $M-1$ real parameters [to satisfy $\lambda_{1}(\vec{p})=\ldots=\lambda_{M}(\vec{p})$]. In order to find a complex degenerate eigenvalue of degree $M$, one needs $2(M-1)$ real parameters. We find high-order EPs by using the algorithm from Mailybaev (2006), which exploits versal deformation theory for finding EPs of a given order.

I.4 Dynamically encircling an isolated EP

In this appendix, we provide additional information about the evolution of the system during the loop in parameter space when encircling the EP in both directions (i.e., clockwise and counterclockwise). We choose an elliptic path of the form

[TABLE]

In Fig. 2, we use $\Delta_{\mathrm{EP}}=-80,\Omega_{\mathrm{EP}}=295,R_{\Delta}=100$, and $R_{\Omega}=30$ (all in kHz), while in Fig. 3, the radii are $R_{\Delta}=260$ and $R_{\Omega}=125$. The phase is $\phi=0.39\pi$ and the period is $T=15/\gamma^{(1)}$.

When initialized in $\vec{S}^{(1)}_{\mathrm{in}}$, the system evolves adiabatically only when the EP is encircled in a clockwise manner; the opposite is true for the second initial state, $\vec{S}^{(2)}_{\mathrm{in}}$. At each moment along the evolution, the instantaneous normal modes form a complete basis of the Hilbert space. That is, the state vector at time $t$ can be written in the form

[TABLE]

where $a_{i}(t)$ are called “the adiabatic coefficients”. We compute them be projecting the state vector, $\vec{S}(t)$, onto the instantaneous basis states, $\vec{S}_{i}^{L}(t)$. The system evolves adiabatically if it stays in the same instantaneous eigenstates throughout its evolution. It is important to emphasize that the instantaneous eigenstates themselves swap at the end of the loop; that is, if the system starts and ends with predominant coefficients $a_{1}$ and $a_{8}$, it means that the state swapped because $\vec{S}_{1}^{R}(t=T)=\vec{S}_{2}^{R}(t=0)$ and $\vec{S}_{8}^{R}(t=T)=\vec{S}_{7}^{R}(t=0)$. Figure 5 shows the evolution of the adiabatic coefficients during the loop [middle panels in (a–d)] and the projections of the input and output states on the instantaneous eigenstates (i.e., the normal modes) at the beginning of the evolution. More details are given in the caption.