Exceptional points for parameter estimation in open quantum systems: Analysis of the Bloch equations

We suggest to employ the dissipative nature of open quantum systems for the purpose of parameter estimation: The dynamics of open quantum systems is typically described by a quantum dynamical semigroup generator ${\cal L}$. The eigenvalues of ${\cal L}$ are complex, reflecting unitary as well as dissipative dynamics. For certain values of parameters defining ${\cal L}$, non-hermitian degeneracies emerge, i.e. exceptional points ($EP$). The dynamical signature of these $EP$s corresponds to a unique time evolution. This unique feature can be employed experimentally to locate the $EP$s and thereby to determine the intrinsic system parameters with a high accuracy. This way we turn the disadvantage of the dissipation into an advantage. We demonstrate this method in the open system dynamics of a two-level system described by the Bloch equation, which has become the paradigm of diverse fields in physics, from NMR to quantum information and elementary particles.


Felix Bloch [1] pioneered the dynamical description of open quantum systems. Originally
Bloch's equations describe the relaxation and dephasing of a nuclear spin in a magnetic field.
Soon it became apparent that the treatment can be extended to a generic two-level-system (TLS), such as the dynamics of laser driven atoms in the optical regime [2]. The open twolevel-system has been used to model many different fields of physics. For example, the TLS or a q-bit is at the foundation of quantum information [3][4][5]. In particle physics the TLS algebra has been employed in studies of possible deviations from quantum mechanics (QM) in the context of neutrino oscillations [6], as well as quantum entanglement [7][8][9][10], associated with electron/positron collisions and entangled systems due to EPR-Bell correlations [11].
The Bloch equation describing the dynamics of the three components of the polarization is: In the rotating frame, the dynamics is governed by two relaxation parameters T 1 , T 2 , a pure dephasing rate 1 , a detuning parameter ∆ = ω s − ν and a driving term = µE, where ω s is the unperturbed frequency of the system and ν the driving frequency.
In the absence of dissipation the eigenvalues of the matrix are pure imaginary, and the dynamics is a free precession of the polarization vector characterized by the Rabi frequency: When dissipation is present the eigenvalues of the homogeneous part of Eq.
(1) become complex, reflecting a decaying oscillation dynamics leading asymptotically to a steady state.
The Bloch equation is the simplest example of a quantum Master equation. Bloch rederived the equation from first principles, employing the assumption of weak coupling between the system and bath [12,13]. These studies have paved the way for a general theory of quantum open systems. Davis [14] rigorously derived the weak coupling limit, resulting in a quantum Master equation which leads to a completely positive dynamical semigroup [15].
Based on a mathematical construction, Lindblad and Gorini, Kossakowski and Sudarshan (L-GKS) obtained the general structure of the generator L of a completely positive dynamical semigroup [16,17], whereρ = Lρ, andρ is the density operator of the system. In the Heisenberg representation the L-GKS generator becomes: whereX is an arbitrary operator. The hamiltonianĤ is hermitian andV is defined to operate in the Hilbert space of the system. The curly brackets denote an anti commutator.
For any finite Hilbert space we can define a vector space of operators using the scalar product Â ·B = tr Â †B . Typically a vector base of traceless operators with the addition of the identity is employed. With this vector space, Eq. (2) is expressed as a set of coupled linear differential equations: where Y is the vector of basis operators andM is the representation of the generator L in this vector space. The eigenvalues of the matrixM reflect the non-hermitian dynamics generated by L. In general they are complex with the steady state eigenvector having an eigenvalue of zero.
The solution for this equation is WhenM is diagonalizable, we can writeM = T ΛT −1 , for a non-singular matrix T and a diagonal matrix Λ, with the eigenvalues {λ i } on its diagonal. Then the exponential ofM is expressed as eM t = T e Λt T −1 , resulting in dynamics that follows the functional form of a sum of decaying oscillatory exponentials.
However, the quantum master equation, Eq. (3), is represented by a non-hermitian ma-trixM with complex eigenvalues. Therefore, for special values of the system parameters the spectrum of the non-hermitian matrix is incomplete, due to the coalescence of several eigenvectors, referred to as a non-hermitian degeneracy. The difference between hermitian degeneracy and non-hermitian degeneracy is essential: In the hermitian degeneracy, several different orthogonal eigenvectors are associated with the same eigenvalue, and the matrix is diagonalizable. On the other hand, in the case of non-hermitian degeneracy several orthogonal eigenvectors coalesce to a simple self-orthogonal eigenvector (see for example [18]). In such cases the matrixM is not diagonalizable.
The exponential of a non-diagonalizable matrixM can be expressed using its Jordan normal form:M = T JT −1 . Here, J is a Jordan-blocks matrix which has (at least) one non-diagonal Jordan block with the form J i = λ i I + N , where I is the identity matrix and N is a matrix with ones on its first upper off-diagonal. The exponential ofM is expressed as eM t = T e Jt T −1 , but here e Jt is not diagonal: The exponential of the block J i in e Jt will have the form: The matrix N is nilpotent and therefore the Taylor series of e N t is finite, thus it is a polynomial in the matrix N t. This gives rise to a polynomial behaviour of the solution in Eq. (4).
The point in the spectrum where the coalescence of the eigenvectors take place is known as an exceptional point (EP ). When two eigenvalues of the master equations coalesce, a second-order non-hermitian degeneracy is obtained. We refer to it as a second-order EP and denote it by EP2. Similarly, a third-order non-hermitian degeneracy is denoted by EP3.
Different manifestations of the EP phenomenon have been described in optics [19,20], in atomic physics [21][22][23], in electron-molecule collisions [24], superconductors [25], quantum phase transitions in a system of interacting bosons [26], electric field oscillations in microwave cavities [27], and in PT-symmetric waveguides [28]. It is not clear however what effect EP has on the dynamics of non-hamiltonian open quantum systems. In this letter we focus on the following issues: • What are the values of the external parameters which lead to non-hermitian degeneracy, i.e. exceptional points (EP )?
• What are the observable dynamical consequences of these EP ?
For particular external parameters, such as laser intensity and detuning, the dynamics is controlled by second order EP or by third order EP. In this case the analysis of the transient measured signal by harmonic inversion, which is used to retrieve the system parameters, becomes extremely sensitive to the external control parameters. Moreover, we will show that this extremely high sensitivity can be used to determine with high accuracy the physical parameter of the TLS interacting with an environment: The energy gap ω s , the transition dipole µ and the strength of the interaction with the bath Γ.

II. THE TWO-LEVEL-SYSTEM MASTER EQUATION
For the two level system, we define the vector space by the three traceless polarization operatorsŜ x ,Ŝ y ,Ŝ z and the identityÎ. In this case the L-GKS master equation Eq. (2) is The effective rotating-frame Hamiltonian of the system under a driving field with detuning ∆ and driving frequency is:Ĥ The two-level-system L-GKS equation for an operatorX with relaxation and pure dephasing becomes where κ ± are kinetic coefficients, κ + /κ − = exp(hω/k B T ), and γ is the pure dephasing rate. Eq. (7) can be merged with the Bloch's equation (1) where 1 (7) can be written in matrix notation as an inhomogeneous equation, using the with the matrix L defined as: where: The general solution for such an equation is: with S 0 = S(0) and S eq fulfills L S eq = (κ − − κ + ) e 3 .
We define M = L − ΓI, where I is the 3 × 3 identity matrix, and Γ = K − γ, to obtain with: The master equation Eq. (7) is a common form for TLS found in the literature [5,29,30].
Eq. (12) which determines the EP interpolates between two extreme cases. The first is associated with a cold bath generating only spontaneous emission, then Γ = K = κ − . The second is a hot singular bath dominated by pure dephasing, then Γ = −γ.
In strong laser fields a more careful derivation accounts for the shift in the rotating wave Hamiltonian eigenvalues due to external driving. This shift alters the kinetic coefficients in the Master equation. As a result, the kinetic coefficients γ become field dependent [31,32].
In the present letter we limit the study to weak driving fields and postpone to future study strong field modifications.

III. EXCEPTIONAL POINTS IN THE BLOCH EQUATIONS
The task is to find the eigenvalues of the generator matrix (12). We first define the variables: and: With these definitions the eigenvalues of Eq. (12) become: For real W (i.e. for Γ 2 X 2 + Y 3 ≥ 0) all eigenvalues are real. For Γ 2 X 2 + Y 3 < 0, W is complex, and two of the eigenvalues are complex (complex conjugate to each other).
We now look for non-hermitian degeneracies of the eigenvalues. This occurs when W vanishes. In such cases the second and third eigenvalues are degenerated, and an EP2 occurs. Figure 1 shows a map of EP2 as a function of and ∆ for fixed Γ = 0.1. The curve of the exceptional points is the curve for which W = 0.
A third order exceptional point, EP3, occurs when X = Y = 0, leading to the relation ∆ = ± 1 108 Γ, = 8 108 Γ. These triple-degeneracies EP3 occur twice, and have a cusp-like behaviour, emerging from the EP2-curves. They are marked with red asterisks in Figure 1. This is consistent with the analysis of Mailybaev [33] of non hermitian degenracies of a two-parameters family of 3×3 matrices. He observed two smooth curves of EP2 that emerge into a single point cusp point of EP3.

CEPTIONAL POINTS
As a dynamical signature we can follow the excited state population of a driven twolevel-system initially at the ground state. One can then tune the parameters ∆ and close to an exceptional point (EP). The population is completely determined by the polarization S z = Ŝ z (t). Thus we follow a time series of the polarization observable.
When the matrixM that generates the dynamics is diagonalizable, the polarization, as well as other correlation functions of quantum dynamics, follow the functional form of a sum of decaying oscillatory exponential signals, Cf. Eq. (4): where ω k are the frequencies, which are the eigenvalues ofM , d k are the associated amplitudes, and both ω k and d k can be complex. This analytic form is used in harmonic inversion methods to find the frequencies and amplitudes of the time series signal [34][35][36]. At exceptional points,M is not diagonalizable. due to a non-hermitian degeneracy. For this case the dynamics obtains a polynomial character, Cf. Eq. (5) and Ref. [18]: Here, ω (r k ) k denotes a frequency with multiplicity of r k + 1. Note that for non-degenerate frequencies, i.e. r k = 0, we have d k,0 = d k and ω S z (t) = d 1 e −iω 1 t + d 2 e −iω 2 t + d 3 e −iω 3 t , for a non-degenerate case, or S z (t) = d 1 e −iω t 1 + (d 2,0 + d 2,1 t) e −iω (1) 2 t , for EP2 (r k = 1), or S z (t) = (d 1,0 + d 1,1 t + d 1,2 t 2 ) e −iω (2) 1 t , for EP3 (r k = 2).
Fuchs et al. showed that since the standard harmonic inversion methods were designed for Eq. (16), applying them to a signal generated by Eq. (17) will lead to divergence of the amplitudes d k . To overcome this divergence, they developed an extended harmonic inversion method [37].
To test the procedure of identifying exceptional points we simulated the dynamics of a two-level system with varying field parameters (i.e. and ∆) generating a time series of polarization S z (n) = S z (nδt), with a time sampling interval δt. The harmonic inversion method was then applied to this time series to estimate the frequencies and the corresponding amplitudes. The suspected exceptional points were located by identifying possible degeneracies of the assigned frequencies ω k . A further test is the divergence of the amplitudes when Eq.
(16) is employed, and a better fit using Eq. (17). We applied this procedure for two domains in the parameters space, the details follow: A. Identifying an EP2 by varying ∆ The procedure for identifying an exceptional points was applied for fixed Γ = 0.1, = 0.01 and varying the laser detuning ∆ = ω s − ν. Figure 2   A more stringent test for the EP3 are based on the following criterions.
• All eigenvalues have to be degenerate.
• All three amplitudes should diverge.
A demonstration of this second test is shown in Figure 4: In this figure, we examined for each of the three amplitudes whether its magnitude is larger than a predetermined threshold value. For each amplitude, this binary criterion is represented by a single color in an RGBimage. The emergence of black points indicates where all three amplitudes are large.
A very fine resolution in the ∆-space was crucial in order to locate the exceptional points. The entire structure of two peaks in the amplitude can be found in a section with length of 10 −5 in ∆, while the "triangle" of exceptional points is located on an area of 2 × 10 −2 in ∆ (see Figure 1 above).
The procedure of identifying the EP3 by the degeneracies of the frequencies and diver-  11)). The next task is to invert the relations between the eigenvalues and the system parameters (Eq. (15)).
Extracting Γ is simple: The formulas for ∆ and are more complicated. To obtain and ∆ one has to invert   where Γ = iω (2) 1 as obtained from Eq. (18). This parameter estimation becomes extremely robust and sensitive. This is a consequence of the special non analytic character close to the EP3: When ν → ν EP 3 and E → E EP 3 then the three frequencies obtained by the standard harmonic inversion coalesce, leading to a branch point (Cf. Chapter 9 in Ref. [18]): where α k and β k are parameters. At the EP3 the first order derivatives of every one of ω k=1,2,3 with respect to the field parameters ν and E diverges. Since Γ = i 3 3 k=1 ω k it is clear that also ∂Γ/∂ν → ∞ and ∂Γ/∂E → ∞ as ν → ν EP 3 and E → E EP 3 . Therefore when ν → ν EP 3 and E → E EP 3 , Γ can be determined to any desired accuracy and following Eq.(20) also the other system parameters ω s and µ.

VI. DISCUSSION
Bloch's equation has become the template for the dynamics of open quantum systems. Such systems are characterised by decoherence with a dynamical signature of decaying oscillatory motion. It is therefore surprising that the existence of non hermitian degeneracies has been overlooked. Our finding of an intricate manifold of double degeneracies EP2 and triple degeneracies EP3 in the elementary TLS template suggests that such phenomena should be common in other systems. We expect that for any Markovian quantum dynamics described by the L-GKS generator [16,17] a manifold of exceptional points should emerge.
As found in the present study, non hermitian degeneracies EP have a subtle influence on the dynamics. The hallmark of EP dynamics is a polynomial component in the decay leading to non-Lorenzian lineshapes. We suggest an experimental procedure to identify the EP, using harmonic inversion of the polarization time series. This procedure can identify the merging of the eigenvalues of the generator of the dynamics. In addition the amplitudes of the reconstructed signal diverge. A confirmation is the ability of the modified method [37] to reconstruct the signal.
The most significant result of our work is that different fields of physics employing the TLS model, have a universal behaviour involving EP. This extreme sensitivity of harmonic inversion in the neighbourhood of an EP, allows us to determine the system parameters: ω sthe energy gap, µ-the dipole transition moment, and Γ-the decoherence rate. The most sensitive point is the third order branch point (EP3).