key: cord-0628934-ay0dsczn
authors: Ohlsson, Tommy; Zhou, Shun
title: Transition Probabilities for Flavor Eigenstates of Non-Hermitian Hamiltonians in the PT-Broken Phase
date: 2020-02-13
journal: nan
DOI: nan
sha: 86fc089cac3a5df23b1c0023cc9f964442e7919c
doc_id: 628934
cord_uid: ay0dsczn

We investigate the transition probabilities for the"flavor"eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time $t to +infty$. After making a connection between the PT-broken phase and the neutral-meson system in particle physics, we observe that the infinite-time behavior of the transition probabilities can be attributed to the negative decay width of one eigenstate of the non-Hermitian Hamiltonian. We also present some brief remarks on the situation at the so-called exceptional point, where both the eigenvalues and eigenvectors of the Hamiltonian coalesce.

exceptional point, where the transition between these two phases occurs. Then, in Sec. 3, the transition amplitudes and probabilities in the PT-broken phase will be introduced and studied, where the connection between the PT-broken phase and the neutral-meson system is also performed. Finally, in Sec. 4, we summarize the main results and draw our conclusions.

For a general discussion about the properties of PT-symmetric non-Hermitian Hamiltonians and their applications, one should be referred to the excellent review by Bender [1] and references therein. Particularly, in this work, we focus on the simple two-level system, for which the Hamiltonian is diagonalizable and space-time independent. The space-reflection operator P is defined as [13] 

and the time-reversal operator T is taken to be just the complex conjugation K, namely, T OT −1 = O * for any operators O in the Hilbert space. The most general form of the Hamiltonian for the two-level system is given by

where {a, b, c, d} are arbitrary complex constants. The PT symmetry of the Hamiltonian system requires that [PT , H] = 0, so we have (PT H) Ψ = 0 1 1 0

where Ψ stands for any vectors in the Hilbert space that the operators are acting on. From Eqs. (3) and (4), one can recognize that the PT symmetry of the system implies that a = d * and b = c * . Therefore, the most general non-Hermitian Hamiltonian H actually contains only four degrees of freedom (in terms of the number of real parameters), when it respects the PT symmetry. This is equal to the number of free parameters in the two-level system with the Hermitian Hamiltonian, where a and d are real while b = c * . For later convenience, we adopt the following parametrization of the most general PTsymmetric non-Hermitian Hamiltonian, viz.,

where all parameters {ρ, ϕ} and {σ, φ} are real and time-independent. For a recent study on time-dependent parameters in H, see Refs. [14, 15] . With the Hamiltonian in Eq. (5) , one can immediately figure out the eigenvalues of the system and their corresponding eigenvectors. More explicitly, the characteristic equation for this system is given by

where λ denotes the eigenvalues and 1 2 is the 2 × 2 identity matrix. From Eq. (6) , it is straightforward to find out two eigenvalues λ ± as λ ± = ρ cos ϕ ± σ 2 − ρ 2 sin 2 ϕ .

Under the condition that ρ 2 sin 2 ϕ < σ 2 is satisfied, the two eigenvalues are real. If this condition is not satisfied, namely, ρ 2 sin 2 ϕ ≥ σ 2 , we obtain either (i) two complex eigenvalues (if ρ 2 sin 2 ϕ > σ 2 holds)

which are complex conjugates to each other, or (ii) a degenerate real eigenvalue λ ± = λ 0 = ρ cos ϕ with multiplicity 2, since ρ 2 sin 2 ϕ = σ 2 holds. In Fig. 1 , we present the eigenvalues as a function of sin ϕ for different choices of the ratio of the parameters ρ and σ. The eigenvalues are displayed in the PT-symmetric phase (ρ 2 sin 2 ϕ < σ 2 : two real eigenvalues, see Subsec. 2.1) and the PT-broken phase (ρ 2 sin 2 ϕ > σ 2 : two complex-conjugate eigenvalues, see Subsec. 2.2) as well as the exceptional points are indicated (ρ 2 sin 2 ϕ = σ 2 : one degenerate real eigenvalue, see Subsec. 2.3). Some helpful comments on the eigenvalues and their corresponding eigenvectors of the non-Hermitian Hamiltonian H in Eq. (5) are in order.

As mentioned before, the two eigenvalues in Eq. (7) are real if the condition ρ 2 sin 2 ϕ < σ 2 is fulfilled. This is usually called the PT-symmetric phase of the system. In this case, we write the two eigenvectors corresponding to E ± = λ ± as |u + = (a + , b + ) T and |u − = (a − , b − ) T , where a ± and b ± are all complex numbers, and solve the equations

Using Eqs. (5), (7) , and (9), we obtain the following solutions

where cos α ≡ (ρ sin ϕ)/σ and sin α = (σ 2 − ρ 2 sin 2 ϕ) 1/2 /σ have been defined. 1 For clarity, the parameter space of {ρ, ϕ} and {σ, φ} will be constrained as follows. First, ρ and σ are moduli of the matrix elements of the Hamiltonian in Eq. (5) such that ρ ≥ 0 and σ ≥ 0 hold. Figure 1 : Illustration for the real (solid curves) and imaginary (dotted curves) parts of the normalized eigenvalues λ ± /σ = ξ 1 − sin 2 ϕ ± 1 − ξ 2 sin 2 ϕ as functions of sin ϕ for three different choices of the ratio of the parameters ρ and σ, i.e., ξ ≡ ρ/σ = 2 (red curves), 3 (orange curves), 4 (yellow curves), in the PT-symmetric phase (none or less shaded areas) and the PT-broken phase (shaded areas). The corresponding exceptional points are marked by black points ('•') and the black thin dashed curve shows the trajectory of the exceptional points.

Second, E ± are conventionally identified with the energy eigenvalues, so we assume both of them to be non-negative. Such a requirement implies E − = ρ cos ϕ ≥ 0 for σ 2 = ρ 2 sin 2 ϕ, i.e., ϕ ∈ [−π/2, π/2]. Furthermore, we must have ρ 2 ≥ σ 2 ≥ ρ 2 sin 2 ϕ. Otherwise, for σ > ρ, we get E − = ρ cos ϕ − σ 2 − ρ 2 sin 2 ϕ < ρ cos ϕ − ρ 2 − ρ 2 sin 2 ϕ = 0. In summary, we assume ρ ≥ σ > ρ sin ϕ > 0 with ϕ ∈ [0, π/2] and φ ∈ [0, 2π). In this parameter space, we have α ∈ [0, π/2]. Certainly, one can also choose a different range of ϕ such that the allowed region of α will be different. However, a different choice of the parameter space will essentially not affect our discussion.

In accordance with the general solutions in Eq. (10), we choose the following forms of the corresponding eigenvectors

where N ± are two arbitrary normalization constants and α ± ≡ α ± φ have been introduced. In order to fix the normalization constants, we compute the explicit PT -inner products of these two eigenvectors, i.e.,

where the superscript "T" denotes matrix transpose. Demanding the conventional normalization conditions [1] , namely,

one can determine the constants N ± up to an overall phase. The normalization conditions in the first identity in Eq. (16) lead to |N ± | 2 = 1/(2 sin α ∓ ), while the orthogonality conditions in the second identity in Eq. (16) give rise to φ = 0 or φ = π. Therefore, we conclude that the PT symmetry together with the orthogonality of the two eigenvectors under the PT -inner product justifies the particular form of the non-Hermitian Hamiltonian in Eq. (5) with φ = 0 or φ = π. Thus, in the subsequent discussion about the PT-symmetric phase, we will concentrate on this particular form for a PT-symmetric non-Hermitian Hamiltonian, i.e.,

and fix the normalization constants as N + = 1/ √ 2 sin α and N − = i/ √ 2 sin α. The eigenvectors are given in Eq. (11) , but now with α ± = α for φ = 0, and can be rewritten as

2 sin α e +iπ/4 · e −iα/2 e −iπ/4 · e +iα/2 , |u − = i √ 2 sin α e +iπ/4 · e +iα/2 e −iπ/4 · e −iα/2 .

In our choice of the phase convention for N ± , we can easily verify that PT |u ± = ±|u ± , indicating that |u ± are also the eigenvectors of the PT operator.

On the other hand, we can apply the bi-orthogonal formalism [16] to the non-Hermitian Hamiltonian in Eq. (17) . First, we have to find out the eigenvectors of H † , which possesses the same real eigenvalues E ± of H, namely,

One salient feature of the Hamiltonian in Eq. (17) is that the identity PH † P −1 = H holds. Therefore, we can multiply Eq. (19) on both sides by the parity operator P from the left and then observe that

implying that P|v ± ∝ |u ± . Without loss of generality, we simply choose |v ± = P|u ± and thus obtain the metric operator η [5] , i.e.,

where v ± | ≡ |v ± † and u ± | ≡ |u ± † . Then, one can verify that det η = csc 2 α − cot 2 α = 1 > 0 and the inverse of η is given by

By construction, the relation ηHη −1 = H † holds, so one can easily prove that there exists a charge-conjugation operator C defined as [17] 

satisfying the commutation relation

This non-Hermitian Hamiltonian system respects both the C and PT symmetries, and thus, the CPT symmetry. Since the PT -inner product is actually not positive-definite (due to det P = −1 < 0), it is necessary to introduce the η-and CPT -inner products for the definitions of the transition amplitudes and probabilities [18] . More explicitly,

• The η-inner product for any two state vectors |ψ and |χ reads

• The CPT -inner product for any two state vectors |ψ and |χ reads

where PC = η from the definition of the C operator in Eq. (23) has been used in the last step in Eq. (26) .

Therefore, the η-and CPT -inner products are equivalent and we can use either of them to calculate the transition amplitudes and probabilities between two quantum states. These calculations have been performed in Ref. [13] .

Under the condition σ 2 < ρ 2 sin 2 ϕ, one can check that [PT , H] = 0 remains to be valid for the most general form of the Hamiltonian H in Eq. (5) . This should be the case as we have derived the general form of the PT-symmetric Hamiltonian for arbitrary values of the parameters. However, as we have shown in Eq. (8), the Hamiltonian (5) has two complex eigenvalues E ± = λ ± , which in this PT-broken phase are labeled by primes in order to avoid any confusion with the ones in the PT-symmetric phase.

Following the same procedure as in the PT-symmetric phase to calculate the eigenvectors |u ± ≡ (a ± , b ± ) T corresponding to the eigenvalues E ± , we need to solve the equations

Introducing cosh α ≡ (ρ sin ϕ)/σ and sinh α = (ρ 2 sin 2 ϕ − σ 2 ) 1/2 /σ, where the identity cosh 2 α − sinh 2 α = 1 can be easily verified, we obtain the solutions to Eq. (27) as

Making a comparison between Eq. (10) in the PT-symmetric phase and Eq. (28) in the PTbroken phase, one observes the connection between these two cases by simply identifying α = −iα. Using Eq. (28), we explicitly rewrite the eigenvectors as

and try to fix the two normalization constants N ± by examining the PT -inner products of these two eigenvectors. As in Eqs. (12)- (15), we compute the PT -inner products, namely,

At first sight, it seems that one can choose proper values of N ± to guarantee the orthogonality conditions u ± |u ∓ PT = 0. However, as one can see from Eq. (31), this is only possible if both α = 0 and sin φ = 0 hold, or equivalently, at the so-called exceptional point with ρ 2 sin 2 ϕ = σ 2 . Therefore, for ρ 2 sin 2 ϕ > σ 2 under consideration, we have to determine the normalization constants N ± by requiring u ± |u ± PT = 0 and u ± |u ∓ PT = +1. These requirements differ significantly from those in the PT-symmetric phase. From Eq. (30) with u ± |u ± PT = 0, we immediately get φ = 0 (or φ = π), which is also consistent with our previous convention in the PT-symmetric phase. In addition, from Eq. (31) with φ = 0 and u ± |u ∓ PT = +1, we obtain N + = e −iπ/4 / √ 2 sinh α and N − = e +iπ/4 / √ 2 sinh α , and thus, using Eq. (29), we find the two eigenvectors as

It is helpful to make some comments on the further connection between the PT-symmetric and PT-broken phases. In the latter case, we have two eigenvalues E ± = ρ cos ϕ ± iσ sinh α , in which the replacement of α = −iα leads to the two eigenvalues E ± in the former case. At the same time, if we replace α by −iα everywhere in Eqs. (32) and (33), the eigenvectors |u ± will reduce to |u ± in Eq. (18) . Nevertheless, given the eigenvectors |u ± in Eqs. (32) and (33), one can check that PT |u ± = |u ∓ and H|u ± = E ± |u ± , indicating that the energy eigenstates |u ± are not eigenstates of the PT operator. This is the reason why this scenario is called the PT-broken phase. However, this is not in contradiction with the fact that [PT , H] = 0. Since PT is an anti-linear operator and E + = E * − , one should note that PT E ± |u ± = E ∓ PT |u ± . More explicitly, we have

implying [PT , H] = 0. This is quite different from the PT-symmetric phase, in which the two eigenvalues E ± are real. Now, we apply the bi-orthogonal formalism to the non-Hermitian Hamiltonian system in the PT-broken phase. As before, we have to find out the eigenvectors of H † , namely,

The identity PH † P −1 = H is still applicable, so we multiply Eq. (35) on both sides from the left by the P operator and then obtain

indicating P|v ± ∝ |u ± . Identifying |v ± = P|u ± , we can immediately compute the metric operator η , i.e., η ≡ s=± |v +s v −s | = P |u + u − | + |u − u + | P = 0 1 1 0 (37) and its inverse

Note that η = η −1 = P with det η = −1 < 0, so it is not positive-definite. In this case, it is impossible to find a Hermitian matrix to convert the non-Hermitian Hamiltonian into a Hermitian one via a similarity transformation. Furthermore, the C operator is given by C = P −1 η = η −1 P = 1 2 , which turns out to be the trivial 2 × 2 identity matrix. Similar to the PT-symmetric phase, we can define the η -inner product as well as the CPT -inner product as follows

• The η -inner product for any two state vectors |ψ and |χ reads ψ|χ η ≡ ψ|η |χ = |ψ † · η · |χ .

(39)

• The CPT -inner product for any two state vectors |ψ and |χ reads

where PC = η = P has been used in the last step.

Therefore, the two inner products are equivalent and we will not distinguish between them. In addition, since the C operator is trivial, these two inner products are also identical with the PT -inner product. However, as we have mentioned, the metric operator η = P is no longer positive-definite, and thus, the norm ψ|ψ η cannot be guaranteed to be positive. In fact, for the energy eigenstates |u + and |u − , we have u ± |u ± PT = u ± |u ± η = 0 and u ± |u ∓ PT = u ± |u ∓ η = +1. The identity η Hη −1 = H † , which now coincides with PHP −1 = H † , indeed leads to a unitary time evolution of the energy eigenstates.

Finally, let us give a brief discussion about the exceptional point (EP) at ρ 2 sin 2 ϕ = σ 2 . The EP can be identified as either the limiting case of α → 0 in the PT-symmetric phase or that of α → 0 in the PT-broken phase. In either limit, the energy eigenvalues become degenerate E ± (or E ± ) → E 0 = ρ cos ϕ. Moreover, for the eigenvectors |u ± in Eq. (18) and |u ± in Eqs. (32) and (33), the normalization constants N ± ∝ 1/ √ 2 sin α and N ± ∝ 1/ √ 2 sinh α are divergent in the respective limits of α → 0 and α → 0. However, this is an artificial divergence, since N ± (or N ± ) in the limit of α → 0 (or α → 0) cannot be determined from the PT -inner products of the relevant eigenvectors. The proper normalization can be taken as u 0 |u 0 = 1, with u 0 | ≡ |u 0 † , so we have

corresponding to the degenerate eigenvalue E 0 at the EP. The rich physics at the EPs and their practical applications have been briefly summarized in Refs. [19, 20] .

Since the time evolution of |u 0 is governed by the Schrödinger equation, we have |u 0 (t) = e −iE 0 t |u 0 , implying that only an overall phase factor will develop and no transitions between any two quantum states are expected. This is also true for the flavor eigenstates |u a = (1, 0) T and |u b = (0, 1) T , which are linear superpositions of the energy eigenstates. between two flavor eigenstates in the PT-broken phase in this section. In this scenario, the Schrödinger equation for the time evolution of the energy eigenstates is

and thus, we have

where the auxiliary parameters ω ≡ ρ cos ϕ and γ ≡ ρ 2 sin 2 ϕ − σ 2 have been defined. For reference, we list below the correspondences between the three new parameters {ω, γ, α } and the three original ones {ρ, σ, ϕ}, where φ = 0 has been assumed as in the previous section

In the following, we adopt the new set of parameters {ω, γ, α }, which can be converted back to the original one by using Eq. (46). To demonstrate the unitary time evolution, we calculate the norms of the time-evolved energy eigenstates to find u + (t)|u + (t) PT = |u + (t) † · P · |u + (t) = 0 ,

Similarly, one can also verify that u ± (t)|u ∓ (t) PT = +1, which is time-independent as it should be. Next, we introduce the flavor eigenstates in which basis the explicit form of the non-Hermitian Hamiltonian is specified. Recall the diagonalization of the Hamiltonian, i.e.,

where we have written A −1 = (|w + , |w − ) with |w ± being two column vectors. Obviously, we can identify |w ± with |u ± in Eqs. (32) and (33), since H|u ± = E ± |u ± . Hence, it is easy to derive

where one can note that A −1 = A T and A † = A . Furthermore, it is straightforward to verify that the flavor eigenstates are given by One can also prove that the norms u a (t)|u a (t)

Then, we proceed to compute the amplitudes and probabilities for the transitions between two flavor eigenstates. After some calculations, the transition amplitudes are found to be

while the corresponding transition probabilities are defined as P αβ ≡ |A αβ | 2 (for α, β running over a, b) and explicitly calculated as

One can observe non-conservation of the total probability, i.e., P aa +P ab = 1 or P ba +P bb = 1. Moreover, all probabilities in Eqs. (57)-(60) go to infinity for t → +∞, rendering them to be physically meaningless. However, the metric operator η = P is not positive-definite, so we should not expect the sum of transition probabilities to be conserved. One may instead compute the differences between the probabilities, i.e., P aa − P ab = − sinh 2 (α + γt) − sinh 2 (γt) / sinh 2 α = − sinh(α + 2γt)/ sinh α , (61) P ba − P bb = + sinh 2 (α − γt) − sinh 2 (γt) / sinh 2 α = + sinh(α − 2γt)/ sinh α , (62) which are unfortunately time-dependent. As a remedy for this problem, following the same strategy as in Ref. [13] , we construct the CPT flavor eigenstates |ũ a and |ũ b as follows

where the expressions for the two original flavor eigenstates |u a and |u b in Eqs. (51) and (52) have been used. Since the C operator is trivial in the PT-broken phase, we can easily prove that CPT |ũ a = PT |ũ a = +|ũ a and CPT |ũ b = PT |ũ b = −|ũ b . Therefore, the newly-constructed flavor eigenstates are eigenstates of both the CPT and PT operators. With these CPT flavor eigenstates, we repeat the calculations of the transition amplitudes and probabilities, and then obtain the amplitudesà αβ ≡ ũ β |u α (t) as

and the probabilitiesP αβ ≡ |Ã αβ | 2 as P aa =P ab = 1 2 (P aa + P ab ) = 1 2 sinh 2 α sinh 2 (γt) + sinh 2 (α + γt) ,

P ba =P bb = 1 2 (P ba + P bb ) = 1 2 sinh 2 α sinh 2 (γt) + sinh 2 (α − γt) .

Although these probabilities still become infinite in the limit of t → +∞, one can check that P aa −P ab = 0 andP ba −P bb = 0, where the time dependence is completely canceled out. It is interesting to note that there is no interference between the two amplitudes A aa and A ab when squaring the modified amplitudesà aa andà ab to calculateP aa andP ab , leading to a simple average of the probabilities in Eq. (69). The main reason can be traced back to the amplitudes in Eqs. (53) and (54), where A aa is purely imaginary, whereas A ab is real up to the same phase factor e −iωt . Similar observations can be made forP ba andP bb in Eq. (70). Therefore, it seems more reasonable to define the CPT flavor eigenstates as the final states in the sense of the time-independence of the probability differences.

The non-Hermitian Hamiltonian with complex eigenvalues has been known in particle physics for a long time. As a concrete example, the mixing and oscillation of the neutral-meson system {|P 0 , |P 0 }, such as K 0 -K 0 , D 0 -D 0 , and B 0 -B 0 , can be described by an effective non-Hermitian Hamiltonian [21] [22] [23] 

where both M and Γ are 2 × 2 Hermitian matrices. In order to make a distinction between the neutral-meson system and the PT-broken phase under consideration, we have set all 2 × 2 matrices in the former case in a sans-serif typeface. Without imposing either CPT or CP invariance, 2 the time-evolved neutral-meson states can be written as [11] |P 0 (t) = g

where z = 0 corresponds to the case of either CPT or CP invariance and the relevant time-evolution functions are given by

Note that for i = 1, 2, M i stand for the masses of the energy eigenstates |P i , while Γ i for the corresponding total decay widths. The masses and decay widths are related to the matrix elements of the effective Hamiltonian with the eigenvalues {E 1 , E 2 } via

where κ ≡ [(M 22 − iΓ 22 /2) − (M 11 − iΓ 11 /2)] /(2pq) and

The complex parameter z can be expressed as follows

, and ∆Γ ≡ Γ 2 − Γ 1 . Now, it is evident that z = 0 corresponds to M 11 = M 22 and Γ 11 = Γ 22 , as implied by the CPT theorem for local quantum field theories.

It is straightforward to calculate the transition amplitudes for |P 0 → |P 0 and |P 0 → |P 0 , namely,

where P 0 | ≡ |P 0 † and P 0 | ≡ |P 0 † have been defined. Accordingly, the corresponding transition probabilities turn out to be

with Γ ≡ (Γ 1 + Γ 2 )/2. Since the decay widths Γ 1 and Γ 2 are positive, the transition probabilities P P 0 P 0 (t) and P ∆m ≡ M 2 − M 1 = 0, δΓ ≡ Γ 11 − Γ 22 = −4ρ sin ϕ, and ∆Γ ≡ Γ 2 − Γ 1 = 4 ρ 2 sin 2 ϕ − σ 2 , namely,

From the previous discussion, one can recognize that E 1 = E + = ω+iγ and E 2 = E − = ω−iγ with ω = ρ cos ϕ and γ = ρ 2 sin 2 ϕ − σ 2 , and thus, M 1 = M 2 = ω and Γ 1 = −Γ 2 = −2γ.

It is interesting to observe that z = 0 and q/p = 1 are valid in the PT-broken phase, which cannot be simultaneously true for the neutral-meson system. At this point, it is helpful to give some remarks on the CPT and CP symmetries in the neutral-meson system, and the CPT and PT symmetries in the PT-broken phase. Following the convention in Ref. [23] , one can write down the discrete space-time symmetry transformations for the neutral-meson system as

implying that CP|P 0 = |P 0 and CP|P 0 = |P 0 . Observe that the time-reversal transformation will interchange the initial and final states, which form separately complete bases, and it is same in both the neutral-meson system and the PT-broken phase, i.e., T = T . In the matrix representation, we consider the two flavor eigenstates as |P 0 = (1, 0) T and |P 0 = (0, 1) T and their Hermitian conjugated states P 0 | = |P 0 † = (1, 0) and P 0 | = |P 0 † = (0, 1). It is then straightforward to obtain

where one can observe that the matrix forms of the CP and P operators are exactly the same. It is not difficult to verify that the CPT or CP invariance in the neutral-meson system guarantees M 11 = M 22 and Γ 11 = Γ 22 , while CP or T invariance leads to (M 12 ) = (Γ 12 ) = 0. However, as we have seen, the relations M 11 = M 22 , Γ 11 = Γ 22 , and (M 12 ) = (Γ 12 ) = 0 hold in the PT-broken phase.

Since the transition probabilities for the flavor eigenstates in the neutral-meson system have been calculated, we can apply them directly to the PT-broken phase. Using Eq. (81) as well as ∆m = 0 and Γ = (Γ 1 + Γ 2 )/2 = 0, we find that

and similarly using Eq. (82), we obtain

Comparing the above results with the ones in Eqs. (57) and (58), we realize the exchange between the expressions of P aa and P ab . Such an observation can be understood by noticing the fact that the PT -inner product and the ordinary inner product (i.e., the T -inner product) differ by the parity operator P that causes the exchange of the final flavor eigenstates. With the above comparative study, we can observe that the transition probabilities P aa and P ab calculated using the ordinary inner product become infinite in the limit of t → +∞ as well, as in the case of the PT -inner product. In analogy to the neutral-meson system, this observation can be attributed to the fact that the total decay width Γ 1 = −2γ < 0 of one energy eigenstate is negative. Therefore, it remains to be explored whether the PT-broken phase can be practically applied to a realistic dynamical system beyond particle physics or not.

The basic properties of non-Hermitian Hamiltonians in both the PT-symmetric and -broken phases are interesting and their practical applications have recently received a lot of attention. In this work, we have focused on the flavor transitions in the two-level quantum system with PT-symmetric non-Hermitian Hamiltonians. Extending our previous investigation on the PT-symmetric phase with two real eigenvalues, we have considered the PT-broken phase, in which the two eigenvalues are complex conjugates to each other.

First, after solving the eigenvalues and eigenvectors of the non-Hermitian Hamiltonian in the PT-broken phase, we have explicitly constructed the charge-conjugation operator C and the metric operator η , for which the identities C = 1 2 and η = P are valid. Second, using the PT -inner product, we have calculated the transition amplitudes and probabilities for the flavor eigenstates, i.e., |u α → |u β for α, β = a, b. After introducing the CPT flavor eigenstates CPT |ũ a = +|ũ a and CPT |ũ b = −|ũ b as the final states, we have found that the differenceP aa −P ab between, instead of the sumP aa +P ab of, the corresponding transition probabilities, vanishes and is time-independent. However, the probabilities themselves in the PT-broken phase have been found to be infinite in the limit of t → +∞, which is totally different from the corresponding result in the PT-symmetric phase. Third, in analogy to the neutral-meson system, we have also calculated the transition probabilities using the ordinary inner product, which is equivalent to the T -inner product, and observed that the infinitetime behavior of the probabilities originates from the negative decay width Γ 1 = −2γ < 0 of one energy eigenstate. For this reason, one might have to find practical applications of the PT-broken phase in dynamical systems beyond particle physics.

The results presented in this work indicate that the PT-broken phase has very different properties compared with the PT-symmetric phase and this deserves further exploration. As shown in Refs. [24] [25] [26] , the microcavity sensors prepared at the exceptional point will be much more sensitive to small perturbations, which can be implemented to realize a one-particle detection. In a similar way, the practical applications of the non-Hermitian Hamiltonian in the PT-broken phase may be accomplished only after coupling it to another system. This is the case for the neutral-meson system, where the weak interaction is switched on in order for the neutral mesons to decay. As we have mentioned, realistic applications may be lying beyond particle physics due to the negative decay width. We leave all these important points for future works.

Making sense of non-Hermitian Hamiltonians

Non-Hermitian physics and PT symmetry

Generalized PT symmetry and real spectra

Complex Extension of Quantum Mechanics

Pseudo-Hermitian Representation of Quantum Mechanics

The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics

PT symmetry as a necessary and sufficient condition for unitary time evolution

Some results on the dynamics and transition probabilities for non selfadjoint hamiltonians

Transition probabilities for non self-adjoint Hamiltonians in infinite dimensional Hilbert spaces

Flavor structures of charged fermions and massive neutrinos

Review of Particle Physics

Non-Hermitian neutrino oscillations in matter with PT symmetric Hamiltonians

Transition Probabilities in the Two-Level System with PT-Symmetric Non-Hermitian Hamiltonians

Analytically solvable 2 × 2 PT-symmetry dynamics from su(1,1)-symmetry problems

Non-Hermitian Hamiltonians and no-go theorems in quantum information

Interference phenomena of neutral K mesons

The C operator in PT-symmetric quantum theories

Appropriate inner product for PT-symmetric Hamiltonians

Physics of nonhermitian degeneracies

The physics of exceptional points

Remarks on Possible Noninvariance under Time Reversal and Charge Conjugation

Analysis of CP-Noninvariant Interactions and the K 0 1 , K 0 2 System

CP Violation

Enhancing the Sensitity of Frequency and Energy Splitting Detection by Using Exceptional Points: Applications to Microcavity Sensors for Single-Particle Detection

Sensors operating at exceptional points: General theory

Quantum Noise Theory of Exceptional Point Amplifying Sensors

S.Z. is also greatly indebted to his family, friends and colleagues for all their support and encouragement during his isolated stay in Huanggang, Hubei, China, where the new corona virus is particularly prevalent and the present work is completed.

P 0 P 0 (t) will vanish in the limit of t → +∞. Since the effective Hamiltonian in Eq. (71) takes the most general form, it can also be applied to the PT-symmetric non-Hermitian Hamiltonian in the PT-broken phase. Comparing the Hamiltonians in these two cases, i.e., Eqs. (17) and (71), we identify the following relations where it should be noted that ρ 2 sin 2 ϕ > σ 2 and (1 − ρ 2 sin 2 ϕ/σ 2 ) 1/2 = +i ρ 2 sin 2 ϕ − σ 2 /σ has been utilized. The parameter z in Eq. (78) is determined by δm ≡ M 11 − M 22 = 0,