An exceptional point is a special point in parameter space at which two (or more) eigenvalues and eigenvectors coincide. The discovery of exceptional points within mechanical and optical systems has uncovered peculiar effects in their vicinity. Here, we consider perhaps the simplest quantum model that exhibits an exceptional point and allows for an analytical treatment. In particular, we reexamine a two-level atom driven by a laser and suffering from losses. The same exceptional point arises in several non-Hermitian matrices, which determine various aspects of the dynamics of the system. There are consequences for some important observables, for example, the spectrum evolves from being a Lorentzian-like singlet to a Mollow triplet upon passing through the exceptional point. Our analysis supports the perspective that viewing certain quantum systems through the lens of exceptional points offers some desirable explanatory advantages.

Topics
Quantum mechanical calculations, Coherence, Atomic fluorescence spectroscopy, Optical spectroscopy, Spectral linewidths, Quantum mechanical systems and processes, Quantum mechanical operators, Open quantum systems, Density-matrix, Quantum optics

Modelling physical situations as eigenvalue problems requires physicists to frequently employ matrix diagonalization techniques on square matrices. All is well and good when dealing with diagonalizable matrices, but sometimes non-diagonalizable—or defective—matrices arise from seemingly reasonable physical theories.1,2

A defective N × N square matrix does not have N linearly independent eigenvectors, and therefore it lacks a complete basis of eigenvectors. For example, consider the following pair of 2 × 2 matrices:
(1)
where a and b are real numbers. Normal matrices, including Hermitian and unitary matrices, are not defective.3 In the case of M1, there are two distinct eigenvalues a and b that are associated with the linearly independent eigenvectors (1,0)T and (0,1)T⁠, respectively. However, the defective matrix M2 has the repeated eigenvalue a, and only one eigenvector (1,0)T⁠. When the diagonalization of some matrix M is impossible, the best one can do is to arrive at the Jordan normal form J=A−1MA⁠, which features square blocks (instead of just the eigenvalues of M) along the diagonal of J. In general, the procedure for finding the matrix A is quite involved, as described in Ref. 3. Such diagonalization problems typically arise in a course on linear algebra, where it is natural to connect the formal mathematical issue to real-world examples such as those encountered within physics.4,5

Defective matrices can arise in non-Hermitian physical systems.6–8 Closed systems, which are isolated from the external environment, may be described using Hermitian operators that guarantee diagonalizability. However, non-Hermicity allows for the conservation of probability to be violated, which is typically the case in open systems which interact with their environment.9,10 In these situations, energy or information may flow into or out of the system. This permits the possibility of a defective theory: by varying parameters in the governing non-Hermitian matrix, one may find a point of non-diagonalizability—an exceptional point—at which (at least two) eigenvalues and eigenvectors coalesce simultaneously.11,12 Accessible accounts of exceptional points within mechanical,13,14 electromagnetic,15,16 and acoustical17 setups have already been provided. In particular, Ref. 14 presents an illuminating discussion of Newton's classical equation for a damped harmonic oscillator from the standpoint of an exceptional point.

The response of classical systems in the neighborhood of their exceptional points has traditionally led to surprising behaviors.4,5 More recently, the nature of exceptional points in quantum systems has been investigated experimentally, mostly with superconducting qubits.18–20 These open systems can be modelled with master equations,9,10 which can be manipulated to provide non-Hermitian matrices that govern the system's dynamics. We will pursue this theoretical framework here, which sits comfortably alongside a course encompassing quantum optics or related fields. Notably, exceptional points have been expertly introduced at a more general level in Ref. 1, while excellent primers on the underlying topic of non-Hermitian physics are supplied by the foundational Refs. 11 and 12.

In what follows, we reconsider perhaps the simplest quantum system exhibiting an exceptional point: a two-level atom driven by a laser. This example arguably presents an ideal first encounter with spectral degeneracies within a quantum model. We discuss some remarkable phenomena occurring in the vicinity of the exceptional point. Most notably, the emission spectrum of the atom is reconstructed from being a simple singlet to an exotic triple-peaked structure after the system passes through its exceptional point, which is a particularly vivid manifestation of non-Hermitian physics.

The rest of this work is organized as follows. We start by laying out the closed system version of the model in Sec. II. We then introduce the open system description in Sec. III, which requires some manipulations with master equations at the level of Refs. 9 and 10. Armed with these theoretical foundations, we then reveal the exceptional points inherent to the model in Sec. IV and the differential equations necessary for studying the dynamics of the system in Sec. V. The consequences for the population dynamics (Sec. VI), first-order coherence (Sec. VII), second-order coherence (Sec. VIII), and optical spectrum (Sec. IX) are then discussed before we draw some conclusions in Sec. X.

Let us consider an atom with a two-dimensional Hilbert space composed of the ground state |0⟩ and the excited state |1⟩⁠. To move between these two states, we introduce the raising operator σ†=|1⟩⟨0| and the lowering operator σ=|0⟩⟨1|⁠, which satisfy the anticommutation relation {σ,σ†}=I⁠. These operators act so that σ†|0⟩=|1⟩ and σ|1⟩=|0⟩⁠. If the transition frequency between the two levels is ω0, the corresponding energy can be described with the term ℏω0|1⟩⟨1|⁠. If the atom is coupled to a laser of amplitude Ω, frequency ωL⁠, and phase θ, the driving energy can be modelled with the term ℏΩeiθe−iωLt|1⟩⟨0| and its Hermitian conjugate. The Hamiltonian operator may then (in the rotating wave approximation) be written as21 
(2)
In order to remove the explicit time dependence in Eq. (2), we move into the rotating frame of the driving laser using the unitary transformation U=exp(iωLtσ†σ)⁠, which leads to
(3)
where we have set ℏ=1 here and in all subsequent equations, and where the laser-atom detuning frequency Δ=ω0− ωL⁠. We also used the transformation equation Ĥ=UĤU†+ i(∂tU)U†⁠, along with the Baker–Campbell–Hausdorff formula eABe−A=B+[A,B]+1/2[A,[A,B]]+⋯ for two operators A and B.22 
The Hamiltonian operator of Eq. (3) is not diagonal in the |0⟩ and |1⟩ basis, so a suitable transformation needs to be found. We suppose that two new states |+⟩ and |−⟩ bring about the diagonal form of the Hamiltonian,
(4)
As can be checked by direct substitution into Eq. (4), the required superposition states are
(5)
(6)
where ϕ is defined through
(7)
The two energy levels, which are independent of the driving phase θ, are given by
(8)
where we have introduced the auxiliary frequency
(9)
Clearly there are no spectral degeneracies between the two states |+⟩ and |−⟩ in this completely closed system; hence, no exceptional points are present. This fact can also be seen by directly solving the Schrödinger equation i∂t|ψ⟩=Ĥ|ψ⟩ with the trial solution |ψ⟩=A|0⟩+B|1⟩⁠. The dynamical matrix describing the system in the {|0⟩,|1⟩} basis is then
(10)
whose eigenvalues are exactly those of Eq. (8). Crucially, the governing matrix of Eq. (10) is clearly seen to be Hermitian H=H†⁠, forbidding any exceptional point physics. This fully Hermitian theory also ensures that the norm ⟨ψ|ψ⟩=1 is both time-independent and probability preserving, as should be the case for a closed system.
In the style of Refs. 9 and 10 losses may be introduced into the model by upgrading the Hamiltonian dynamics of Sec. II. At the more comprehensive level of the density matrix ρ, which accounts for mixed states and not just the pure states as described by some state vector |ψ⟩⁠, the master equation reads as23,
(11)
where γ≥0 is the damping decay rate of the atom, and where the Lindblad form of this equation is derived in Ref. 24. The first term on the right-hand-side of Eq. (11) is the von Neumann equation and accounts for the dynamics of the closed system using the Hamiltonian operator from Eq. (3). Dissipation into the external environment is described by the second term on the right-hand-side of Eq. (11), which allows for non-unitary evolution.9,10 Since the atom is associated with the states |0⟩ and |1⟩ only, the density matrix is composed of just four elements,
(12)
where ρn,m=⟨n|ρ|m⟩ and n,m∈{0,1}⁠. The necessary condition ρ0,0+ρ1,1=1 ensures that probability is conserved. Using the master equation of Eq. (11) to compute the time evolution of the four matrix elements ρn,m leads to the following dynamical equation:
(13)
where the 4 × 4 Liouvillian matrix
(14)
which is non-Hermitian L≠L†⁠, unlike the Hermitian matrix of Eq. (10) that describes the closed version of the model. The non-Hermicity of Eq. (14) raises the possibility for exceptional point physics to arise thanks to the open nature of the system.
Before delving into the nature of the exceptional points and their consequences for the dynamics of the system, we lay the foundations for results needed later by considering the simplest response of the atom: its steady state behavior. Considering sufficiently large timescales t→+∞⁠, such that the time derivative ∂tρ=0⁠, the steady state density matrix is given by the solution of the linear homogeneous equation Lρ=0⁠. Since the determinant of Eq. (14) is zero, there are infinitely many solutions at first glance. Therefore, it is enough to solve just three of the simultaneous equations arising from the four-dimensional system Lρ=0 since the probability preservation condition ρ0,0+ρ1,1=1 uniquely determines the density matrix. After some algebra, we obtain the steady state density matrix elements,
(15)
(16)
(17)
which will be employed later on. Having warmed up by manipulating the density matrix of the atom in the steady state, we now turn to searching for the exceptional points contained within the governing Liouvillian matrix.

The behavior of the atom is fully described by the Hermitian matrix of Eq. (10) when it is considered to be a lossless system. With the help of the non-Hermitian Liouvillian matrix of Eq. (14), dissipation is accounted for, which increases the chance for exceptional points to occur in the open version of the system.

The four complex eigenvalues λn of the Liouvillian matrix may be found from Eq. (14) via the quartic equation
(18)
where the three real-valued coefficients an are given by
(19)
(20)
(21)
The solution of the effective cubic equation within Eq. (18) defines the dynamics, while the fourth solution λ4=0 corresponds to its steady state behavior. The three nontrivial eigenvalues are
(22)
(23)
(24)
where S and T are defined in terms of D via
(25)
The three physical parameters of the problem (the detuning frequency Δ, driving amplitude Ω, and loss rate γ) enter the eigenvalues via the quantities
(26)
(27)
Eigenvalue degeneracies occur when λ2=λ3⁠, so S = T. Equivalently, D = 0, which leads to the sextic equation
(28)
where the three real-valued coefficients bn are given by
(29)
(30)
(31)
The discriminant D of the bi-cubic equation appearing within Eq. (28) has the crucial proportionality
(32)
which determines the character of the bi-cubic solutions. Notably, the discriminant vanishes at a critical value of the detuning frequency,
(33)
The exact cubic solutions of Eq. (28) [not given here, but of the same formal structure as Eqs. (22)–(24)] allow one to find the values of Ω corresponding to eigenvalue degeneracies, which (when they arise along with eigenvector degeneracies) are exceptional points. The results are plotted in Fig. 1(a) as a function of Δ, which shows that for nonzero Δ, there are two exceptional points for a given detuning frequency (solid green and dashed orange lines) up to the critical detuning frequency Δc (vertical gray line) above which no spectral degeneracies exist.
Fig. 1.
(Color online) Exceptional points in a two-level atom. (a) The values of the driving amplitude Ω, in units of the damping rate γ, corresponding to an exceptional point in the Liouvillian matrix L [cf. Eq. (14)]. These values are found from the solutions of Eq. (28), which exist up to a critical detuning Δc as marked by the vertical grey line [cf. Eq. (33)]. (b) The real and imaginary parts of the three nontrivial Liouvillian eigenvalues iλ1,iλ2, and iλ3 as a function of Ω when Δ = 0 [cf. Eqs. (34)–(36)]. Vertical grey line: exceptional point ΩEP=γ/8. (c) Same as for panel (b), but for nonzero detuning, where Δ=γ/20 [cf. Eqs. (22)–(24)].
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(Color online) Exceptional points in a two-level atom. (a) The values of the driving amplitude Ω, in units of the damping rate γ, corresponding to an exceptional point in the Liouvillian matrix L [cf. Eq. (14)]. These values are found from the solutions of Eq. (28), which exist up to a critical detuning Δc as marked by the vertical grey line [cf. Eq. (33)]. (b) The real and imaginary parts of the three nontrivial Liouvillian eigenvalues iλ1,iλ2⁠, and iλ3 as a function of Ω when Δ = 0 [cf. Eqs. (34)–(36)]. Vertical grey line: exceptional point ΩEP=γ/8⁠. (c) Same as for panel (b), but for nonzero detuning, where Δ=γ/20 [cf. Eqs. (22)–(24)].

Fig. 1.
(Color online) Exceptional points in a two-level atom. (a) The values of the driving amplitude Ω, in units of the damping rate γ, corresponding to an exceptional point in the Liouvillian matrix L [cf. Eq. (14)]. These values are found from the solutions of Eq. (28), which exist up to a critical detuning Δc as marked by the vertical grey line [cf. Eq. (33)]. (b) The real and imaginary parts of the three nontrivial Liouvillian eigenvalues iλ1,iλ2, and iλ3 as a function of Ω when Δ = 0 [cf. Eqs. (34)–(36)]. Vertical grey line: exceptional point ΩEP=γ/8. (c) Same as for panel (b), but for nonzero detuning, where Δ=γ/20 [cf. Eqs. (22)–(24)].
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(Color online) Exceptional points in a two-level atom. (a) The values of the driving amplitude Ω, in units of the damping rate γ, corresponding to an exceptional point in the Liouvillian matrix L [cf. Eq. (14)]. These values are found from the solutions of Eq. (28), which exist up to a critical detuning Δc as marked by the vertical grey line [cf. Eq. (33)]. (b) The real and imaginary parts of the three nontrivial Liouvillian eigenvalues iλ1,iλ2⁠, and iλ3 as a function of Ω when Δ = 0 [cf. Eqs. (34)–(36)]. Vertical grey line: exceptional point ΩEP=γ/8⁠. (c) Same as for panel (b), but for nonzero detuning, where Δ=γ/20 [cf. Eqs. (22)–(24)].

Close modal
The preceding analysis is considerably simplified for the case of vanishing laser-atom detuning (Δ = 0). In this resonant case, the three eigenvalues λn of Eqs. (22)–(24) reduce to
(34)
(35)
(36)
where we have introduced the auxiliary frequency
(37)
The real and imaginary parts of the complex eigenvalues iλn are plotted as the magenta lines in Fig. 1(b), which showcases the presence of a single exceptional point (gray line) when the driving amplitude is equal to
(38)
The unnormalized eigenvectors Λ1,2,3 corresponding to the eigenvalues λ1,2,3 are given by
(39)
which confirm the existence of an exceptional point at ΩEP due to the simultaneous coalescence of both the eigenvectors Λ2 and Λ3 and the eigenvalues λ2 and λ3. This analysis demonstrates the presence of an exceptional point in a simple quantum model.

Let us now briefly return to the more complicated case of nonzero atom-laser detuning frequency, as is considered in the plots of Fig. 1(c). For the example situation with Δ=γ/20⁠, we use the full eigenvalues λn as defined through Eqs. (22)–(24) to plot the real and imaginary parts of iλn as the cyan lines in panel (c). These plots demonstrate two exceptional points (vertical gray lines), as predicted by the locator plot provided in Fig. 1(a).

Armed with the knowledge of the locations of the exceptional points in the two-level atom, we are nearly ready to consider the consequences of exceptional point physics for some typical observables, but before that we briefly provide the equations for the atomic dynamics in Sec. V, which sets up the remaining calculations.

The average value of an operator, for example, the mean population of the atom as described by ⟨σ†σ⟩⁠, can be obtained from the master equation of Eq. (11). The relevant equation of motion arises after applying the trace property ⟨O⟩=Tr(Oρ)⁠, which is valid for any operator O⁠, in the manner of Refs. 9 and 10. This process results in
(40)
where the 3-vector correlators Ψ and the drive term P are
(41)
while the 3 × 3 dynamical matrix is given by
(42)
The expectation values of the operators σ, σ†⁠, and σ†σ can then be found by solving the coupled equations of Eq. (40) in the style of Ref. 25 for example. The three complex eigenvalues of the dynamical matrix of Eq. (42) correspond to the three nonzero eigenvalues of the Liouvillian matrix of Eq. (14) (up to a factor of the imaginary unit i), such that the locations of the exceptional points are wholly unaffected. For example, in the simplest case of resonance (Δ = 0), one finds the following three complex eigenvalues:
(43)
(44)
(45)
where the frequency Ω̃ is defined in Eq. (37), and where the exceptional point is given by Eq. (38). A somewhat analogous treatment of the dynamics and exceptional point physics of a damped harmonic oscillator using purely classical equations is provided in Ref. 14, which acts as a complement to the analysis presented here. In what follows, we use the solutions of the coupled equations defined in Eq. (40) to show how exceptional points can drastically change the dynamical response of the atom. One can expect some surprises due to changing character of Eqs. (44) and (45) above and below ΩEP from complex quantities to wholly imaginary quantities.
In the steady state, the average values for the operators contained within Ψ [cf. Equation (41)] may then be found from Ψ=−H−1P⁠. The steady state population of the atom ⟨σ†σ⟩ and the steady state coherences ⟨σ⟩ and ⟨σ†⟩ then read [cf. Eqs. (15)–(17)]
(46)
(47)
(48)
The lower population bound of limt→+∞⟨σ†σ⟩=0 is reached at vanishing driving (⁠ Ω→0⁠) since the atom resides in its ground state. The upper bound of limt→+∞⟨σ†σ⟩=1/2 is met in the large driving limit (⁠ Ω→+∞⁠), which reveals that population inversion is prohibited in this system. We plot the steady state atom population in Fig. 2(a) as a function of the driving amplitude Ω, which shows the evolution of the steady state population between the two bounds 0 and 1/2. A smaller Δ implies a higher population since driving close to resonance excites the atom most efficiently.
Fig. 2.
(Color online) Population of a two-level atom. (a) Steady state population as a function of the driving amplitude Ω, in units of the damping rate γ [cf. Eq. (46)]. (b) Classical part (dashed green line) and quantum part (solid orange line) making up the steady state population when Δ = 0 [cf. Eqs. (50) and (51)]. Horizontal line: maximum of the classical field. (c) Dynamic population of the atom as a function of time, in units of γ−1 [cf. Eqs. (52)–(54)]. Increasing values of Ω are displayed with increasingly thick lines, along with the result at ΩEP (dashed red line) and vanishing Ω (dotted cyan line). In panel (c), Δ = 0, and the initial condition is ⟨σ†σ⟩=1 at t = 0.
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(Color online) Population of a two-level atom. (a) Steady state population as a function of the driving amplitude Ω, in units of the damping rate γ [cf. Eq. (46)]. (b) Classical part (dashed green line) and quantum part (solid orange line) making up the steady state population when Δ = 0 [cf. Eqs. (50) and (51)]. Horizontal line: maximum of the classical field. (c) Dynamic population of the atom as a function of time, in units of γ−1 [cf. Eqs. (52)–(54)]. Increasing values of Ω are displayed with increasingly thick lines, along with the result at ΩEP (dashed red line) and vanishing Ω (dotted cyan line). In panel (c), Δ = 0, and the initial condition is ⟨σ†σ⟩=1 at t = 0.

Fig. 2.
(Color online) Population of a two-level atom. (a) Steady state population as a function of the driving amplitude Ω, in units of the damping rate γ [cf. Eq. (46)]. (b) Classical part (dashed green line) and quantum part (solid orange line) making up the steady state population when Δ = 0 [cf. Eqs. (50) and (51)]. Horizontal line: maximum of the classical field. (c) Dynamic population of the atom as a function of time, in units of γ−1 [cf. Eqs. (52)–(54)]. Increasing values of Ω are displayed with increasingly thick lines, along with the result at ΩEP (dashed red line) and vanishing Ω (dotted cyan line). In panel (c), Δ = 0, and the initial condition is ⟨σ†σ⟩=1 at t = 0.
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(Color online) Population of a two-level atom. (a) Steady state population as a function of the driving amplitude Ω, in units of the damping rate γ [cf. Eq. (46)]. (b) Classical part (dashed green line) and quantum part (solid orange line) making up the steady state population when Δ = 0 [cf. Eqs. (50) and (51)]. Horizontal line: maximum of the classical field. (c) Dynamic population of the atom as a function of time, in units of γ−1 [cf. Eqs. (52)–(54)]. Increasing values of Ω are displayed with increasingly thick lines, along with the result at ΩEP (dashed red line) and vanishing Ω (dotted cyan line). In panel (c), Δ = 0, and the initial condition is ⟨σ†σ⟩=1 at t = 0.

Close modal
The underlying nature of the average atom population ⟨σ†σ⟩ can be probed by decomposing the operator σ into two parts σ=⟨σ⟩+υ⁠. Here, the mean-field quantity ⟨σ⟩ is a complex number accounting for the classical behavior, while the operator υ tracks the quantum field. The operator σ†σ can then be decomposed as σ†σ=⟨σ†⟩⟨σ⟩+υ†υ+⟨σ†⟩υ+ ⟨σ⟩υ†⁠. Since the quantum field υ has no mean field by construction ⟨υ⟩=0 and the mean population has no interference terms,
(49)
Therefore, the classical and quantum contributions to the mean population in the steady state can be deciphered as
(50)
(51)
We plot both contributions to Eq. (49) as a function of Ω in Fig. 2(b) for the simplest case of zero detuning. The classical field population described by the dashed green line and Eq. (50) has the lower bound of 0 (occurring with both weak and strong driving) and an upper bound of just 1/8, as is marked by the thin gray line. The quantum field associated with Eq. (51) varies more widely between 0 and 1/2 (solid orange line) and starts to dominate above Ω=γ/(22)⁠. It is the only contribution in the large driving limit Ω≫γ⁠, highlighting the quantum nature of the atom at saturation.
The full dynamical population of the atom follows from the time-dependent solution of Eq. (40). With zero detuning and with the initial population ⟨σ†σ⟩=1 at t = 0, the expressions for the atom populations are considerably simplified. Taking into account the exceptional point at ΩEP⁠, we find that the non-decaying part of the solution takes the form of trigonometric oscillatory functions when the driving is strong (⁠ Ω>ΩEP⁠), hyperbolic functions when the driving is weak (⁠ Ω<ΩEP⁠), and otherwise a linear in time function exactly at the exceptional point (⁠ Ω=ΩEP⁠),
(52)
(53)
(54)
where Ω̃ is defined in Eq. (37) and where
(55)
We plot the dynamical atom population in Fig. 2(c) as a function of time. Strikingly, only driving amplitudes above the exceptional point lead to the onset of population cycles (medium green line), which are increasingly pronounced for larger driving amplitudes (thick orange line). Therefore, within this system, by evolving from non-oscillatory to oscillatory atomic population dynamics, one can infer that an exceptional point (dashed red line) has been safely passed through. Within classical mechanics, a similar dynamical transition is seen to occur for a damped harmonic oscillator moving between its under-damped and over-damped regimes.14 
In the two limiting cases of vanishing driving and very strong driving,
(56)
(57)
The purely exponential decay (with the time constant 1/γ⁠) of the undriven atom is seen with the dotted cyan line in Fig. 2(c). The very strong driving result of Eq. (57) is not shown, but the expression reveals cosinusoidal oscillations (maximally bound between 1 and 0), which are exponentially damped toward a steady state population of 1/2 with the characteristic time constant of 4/3γ⁠.
The temporal stability of the light emitted from the atom can be described by its coherence in time, as judged by correlation functions.26,27 In particular, the first-order degree of coherence g(1)(τ) is concerned with the first power of the light field, while the second-order degree of coherence g(2)(τ) is associated with the second power of the light field. Several neat experiments to measure the degrees of coherence of photons have been proposed,28–30 which allows one to characterize the nature of the emitted light. There are three kinds of first-order coherence, which allows one to determine the degree to which the light is monochromatic or not.31 Fully coherent (and perfectly monochromatic) light occurs when |g(1)(τ)|=1⁠, light with partial coherence manifests itself when 0<|g(1)(τ)|<1⁠, and perfectly incoherent light arises when |g(1)(τ)|=0⁠. For example, a reasonable approximation to full coherence is the monochromatic light emitted from a laser, while a good approximation to full incoherence is the white light emanating from a thermal source. The degree of first-order coherence g(1)(τ) is defined via31 
(58)
where the time delay τ≥0⁠. The master equation of Eq. (11) yields the necessary equation of motion for the set of two-time correlators31 
(59)
(60)
where P and H are defined in Eqs. (41) and (42), respectively. The degree of first-order coherence follows from the solution of Eq. (59), subject to the steady state normalization given by the denominator in Eq. (58). We consider zero detuning so that the resulting analytic expressions are more compact. Depending on the value of Ω, we have the absolute values as follows:
(61)
(62)
(63)
where Ω̃ and Γ are defined in Eqs. (37) and (55), respectively. The asymptotic behaviors of |g(1)(τ)| at short and long time delays are captured by the expressions
(64)
(65)
At zero time delay, the correlator is exactly unity, representing full coherence, which comes directly from the definition of Eq. (58). With large time delays, there is a greater chance of randomness and so the correlator reduces to a small and constant amount [cf. Eq. (65)]. In between these extreme temporal delays, the exceptional point at ΩEP defines the crossover between regimes of either oscillating or non-oscillating coherences [cf. Eqs. (61) and (63)], similar to the population dynamics in Sec. VI. In the vanishing driving and very strong driving limits, the first-order coherence function reduces to
(66)
(67)
The weak driving expression describes perfect monochromatic light due to the coherent driving of the laser, as represented by the cyan line in Fig. 3(a). The very strong driving result of Eq. (67) is not shown in panel (a) but exhibits cosinusoidal oscillations between 0 and 1, which are quickly washed out to zero with longer delay times.
Fig. 3.
(Color online) Degrees of coherence and optical spectrum of an atom. (a) Absolute value of the first-order coherence g(1)(τ) as a function of the time delay τ, in units of the inverse damping rate γ−1 [cf. Eqs. (61)–(63)]. (b) Second-order coherence g(2)(τ) as a function of τ [cf. Eqs. (71)–(73)]. (c) Optical spectrum S(ω) as a function of the frequency ω of the emitted photons [cf. Eqs. (79)–(81)]. In this figure, we consider Δ = 0 and present several values of the driving amplitude Ω as marked by the legend in panel (a), as well as the results at the exceptional point (dashed red lines) and vanishing Ω (dotted cyan lines).
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(Color online) Degrees of coherence and optical spectrum of an atom. (a) Absolute value of the first-order coherence g(1)(τ) as a function of the time delay τ, in units of the inverse damping rate γ−1 [cf. Eqs. (61)–(63)]. (b) Second-order coherence g(2)(τ) as a function of τ [cf. Eqs. (71)–(73)]. (c) Optical spectrum S(ω) as a function of the frequency ω of the emitted photons [cf. Eqs. (79)–(81)]. In this figure, we consider Δ = 0 and present several values of the driving amplitude Ω as marked by the legend in panel (a), as well as the results at the exceptional point (dashed red lines) and vanishing Ω (dotted cyan lines).

Fig. 3.
(Color online) Degrees of coherence and optical spectrum of an atom. (a) Absolute value of the first-order coherence g(1)(τ) as a function of the time delay τ, in units of the inverse damping rate γ−1 [cf. Eqs. (61)–(63)]. (b) Second-order coherence g(2)(τ) as a function of τ [cf. Eqs. (71)–(73)]. (c) Optical spectrum S(ω) as a function of the frequency ω of the emitted photons [cf. Eqs. (79)–(81)]. In this figure, we consider Δ = 0 and present several values of the driving amplitude Ω as marked by the legend in panel (a), as well as the results at the exceptional point (dashed red lines) and vanishing Ω (dotted cyan lines).
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(Color online) Degrees of coherence and optical spectrum of an atom. (a) Absolute value of the first-order coherence g(1)(τ) as a function of the time delay τ, in units of the inverse damping rate γ−1 [cf. Eqs. (61)–(63)]. (b) Second-order coherence g(2)(τ) as a function of τ [cf. Eqs. (71)–(73)]. (c) Optical spectrum S(ω) as a function of the frequency ω of the emitted photons [cf. Eqs. (79)–(81)]. In this figure, we consider Δ = 0 and present several values of the driving amplitude Ω as marked by the legend in panel (a), as well as the results at the exceptional point (dashed red lines) and vanishing Ω (dotted cyan lines).

Close modal
Intensity correlations of the atom can be captured by the degree of second-order coherence g(2)(τ)⁠, which can be readily measured experimentally with photon counting setups.28–30 This second-order correlator may be defined by31,
(68)
for a time delay τ≥0 between two different emissions, and with the normalization taken in the steady state. This important quantity describes a classical bunching effect when g(2)(0)>g(2)(τ)⁠, where photon emissions are clumped together in time. Conversely, antibunching occurs when g(2)(0)<g(2)(τ)⁠. This is a decidedly non-classical phenomena where the emitted photons are well separated in time from one another.31 The edge case of g(2)(0)=g(2)(τ) suggests that photons are emitted with a random spacing in time. The master equation of Eq. (11) yields the necessary equation of motion for the set of two-time correlators,31 
(69)
(70)
where P and H are defined in Eqs. (41) and (42), respectively. The degree of second-order coherence follows from the solution of Eq. (69) subject to the steady state normalization of Eq. (68). We consider zero detuning so that the resulting analytic expressions are more manageable and list the solutions for the regions Ω>ΩEP, Ω=ΩEP and Ω<ΩEP in that order,
(71)
(72)
(73)
where Ω̃ and Γ are given by Eqs. (37) and (55), respectively. The asymptotics of the full solutions at short and long delay times are
(74)
(75)
The vanishing time delay result of zero describes non-classical, antibunching behavior. This perfect antibunching follows since the two-level atom cannot simultaneously emit two photons: a finite delay time is necessary for the atom to be re-excited from its ground state in order to re-emit another photon. The opposing limit implies randomly bunched light, which arises because, for large time delays, the field intensities at t and t+τ should be completely uncorrelated. Otherwise, for intermediate delay times, the full solutions of Eqs. (71)–(73) showcase the remarkable impact of the exceptional point residing at ΩEP⁠. The correlator g(2)(τ) either displays characteristic oscillations—or not—depending upon whether the driving amplitude has passed through the exceptional point ΩEP (Fig. 3(b)), since the nature of the eigenvalues change from being complex to wholly imaginary. In the two limiting cases of vanishingly weak driving and very strong driving, we find the exact expressions reduce to
(76)
(77)
The weak driving result is denoted by the cyan line in Fig. 3(b), and its lack of oscillations as it develops from 0 to 1 monotonically in time is a distinguishing feature. The strong driving result is not shown in panel (b), but the highly oscillatory expression displays extrema at 0 and 2, which will eventually damp out to unity with the time constant 4/3γ⁠.
The optical spectrum of the fluorescent light emitted by the atom S(ω) is defined in terms of the Fourier transform of the two-time correlator ⟨σ†(t)σ(t+τ)⟩⁠. The spectrum measures the intensity of the photons emitted by the atom at the frequency ω, and mathematically it reads31,
(78)
which has been normalized so that ∫0∞S(ω)dω=1⁠. This ensures that S(ω) can be treated as the probability of the system emitting a photon at the frequency ω. By inserting the solutions already found in Sec. VII for the first-order correlator g(1)(τ) into the integrand of Eq. (78), and after carrying out the time integral, the explicit form of the spectral line shape may be found.32 With zero atom-laser detuning, the optical spectrum may be written as
(79)
(80)
(81)
where δ(x) is Dirac's delta function. The first term, describing a weighted delta spectral peak or “Rayleigh scattering peak,” is common to all the three above-mentioned expressions and arises from the light elastically scattered by the atom from the driving laser.32 The second term is a standard Lorentzian term, which emerges due to the relaxation process from the excited state to the ground state, and it defines the central (and unshifted) spectral peak, with the characteristic broadening γ. Finally, the third term accounts for the spectral differences due to the size of the driving amplitude, including the possible presence of sidebands. Above the exceptional point [cf. Eq. (79)], the final spectral term is defined using the auxiliary functions S+>(ω) and S−>(ω)⁠, which describe the twin satellite peaks helping to form the celebrated spectral triplet discovered by Mollow.32 These sidebands are peaked around the so-called Mollow frequency Ω̃⁠, and together with the central Lorentzian form the inelastic spectral response. At the exceptional point [cf. Eq. (80)], the final spectral term presents as an unshifted Student's t-distribution (with ν = 3 degrees of freedom) such that the overall spectrum is a singlet. Furthermore, below the exceptional point [cf. Eq. (81)], the final term is the sum of two purely Lorentzian line shapes (with different weighting coefficients L+< and L−<⁠). These Lorentzians are centered at resonance such that passing through the exceptional point leads to an interesting spectral transition from a simple singlet structure to an intriguing triplet. The exact forms of the auxiliary functions needed to fully describe the spectrum, as given in Eqs. (79) and (81), are defined by
(82)
(83)
Equation (82) is composed of a weighted Lorentzian part (the first term), which is associated with spontaneous emission and a weighted dispersive part (the second term) due to interferences, which helps to define the Mollow triplet present for Ω>ΩEP⁠. Notably, in the extreme limit of vanishing driving, the spectrum in Eq. (81) collapses into the solitary delta peak,
(84)
a feature wholly coming from the Rayleigh scattering peak, which mathematically can be traced back to the steady state term in the g(1)(τ) solutions as given in Eq. (61). The evolution with Ω of the spectrum of the atom is plotted in Fig. 3(c). Most notably, the aforementioned Mollow triplet structure32 is striking for larger drivings (orange line), before a singlet spectrum emerges with weaker drivings near to the exceptional point case (dashed red line). Only a delta peak (dotted cyan line) survives with vanishing driving. These results demonstrate the critical significance of the exceptional point for the optical response of the atom and the vivid consequences for the emission spectrum.

Exceptional points are increasingly influential across modern physics, particularly in wave physics as manifested in mechanics13,14 and electromagnetics.15,16 Their impact on quantum objects can be treated within an open systems approach; here, we have reconsidered perhaps the simplest example: that of a two-level atom. We have shown the consequences of passing through an exceptional point on the response of the atom, including striking changes to its optical spectrum (which evolves from being a singlet to a beautiful triplet) and significant reconstructions of correlation functions (which transition from being non-oscillatory to oscillatory). We believe that looking at quantum systems from the viewpoint of exceptional points is rather instructive, since it (i) neatly categorizes distinct behavioral regimes and (ii) helps to explain the character of common observables via the properties of complex eigenvalues arising from non-Hermitian matrices. This perspective may become increasingly widespread with the continuing rise in popularity of non-Hermitian quantum physics.12 

C.A.D. is supported by the Royal Society via a University Research Fellowship (URF/R1/201158) and an International Exchanges grant (IES/R1/241078) with A. Cidrim (Universidade Federal de São Carlos, Brazil). V.A.S. acknowledges support from the Horizon Europe through the Marie Sklodowska-Curie Actions (2021-PF-01, Project No. 101065500, TeraExc).

The authors have no conflicts to disclose.

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