Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0001 Acta Polytechnica 62(1):1–7, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague CONSERVED QUANTITIES IN NON-HERMITIAN SYSTEMS VIA VECTORIZATION METHOD Kaustubh S. Agarwal, Jacob Muldoon, Yogesh N. Joglekar∗ Indiana University Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202 U.S.A. ∗ corresponding author: yojoglek@iupui.edu Abstract. Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time (PT ) symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a PT -symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful addition to the well-established methods for characterizing time-invariants in non-Hermitian systems. Keywords: Parity-time symmetry, pseudo-Hermiticity, conserved quantities. 1. Introduction Since the seminal discovery of Bender and coworkers in 1998 [1], non-Hermitian Hamiltonians H with real spectra have become a subject of intense scrutiny [2– 4]. The initial work on this subject focused on taking advantage of the reality of the spectrum to define a complex extension of quantum theory [5] where the traditional Dirac inner product is replaced by a Hamiltonian-dependent (CPT ) inner product. Soon it became clear that this process can be thought of as identifying positive definite operators η̂ ≥ 0 that intertwine with the Hamiltonian [6–8], i.e. η̂H = H†η̂, and that a non-unique complex extension of standard quantum theory is generated by each positive def- inite η̂ [9, 10]. These mathematical developments were instrumental to elucidating the role played by non-Hermitian, self-adjoint operators, biorthogonal bases, and non-unitary similarity transformations that change an orthonormal basis set into a non-orthogonal, but linearly independent basis set in physically realiz- able classical and quantum models [11]. A decade later, this mathematical approach gave way to experiments with the recognition that non- Hermitian Hamiltonians that are invariant under com- bined operations of parity and time-reversal (PT ) rep- resent open systems with balanced gain and loss [12– 15]. The spectrum of a PT -symmetric Hamiltonian HPT(γ) is purely real when the non-Hermiticity γ is small. With increasing γ, a level attraction and resulting degeneracy turns the spectrum into complex- conjugate pairs when the non-Hermiticity exceeds a nonzero threshold γPT [16]. This transition is called PT -symmetry breaking transition, and at the thresh- old γPT the algebraic multiplicity of the degenerate eigenvalue is larger than the geometric multiplicity, i.e. it is an exceptional point (EP) [17]. Fueled by this physical insight, the past decade has seen an explosion of experimental platforms, usu- ally in classical wave systems, where effective PT - symmetric Hamiltonians with balanced gain and loss have been realized. They include evanescently cou- pled waveguides [18], fiber loops [19], microring res- onators [20, 21], optical resonators [22], electrical cir- cuits [23–25], and mechanical oscillators [26]. The key characteristics of this transition, driven by the non- orthogonality of eigenstates, are also seen in systems with mode-selective losses [27–29]. In the past two years, these ideas have been further extended to mini- mal quantum systems, thereby leading to observation of PT -symmetric breaking and attendant phenom- ena in a single spin [30], a single superconducting transmon [31], ultracold atoms [32], and quantum photonics [33]. We remind the readers the effective Hamiltonian approach requires Dirac inner product, and is valid in both PT -symmetric and PT -broken regions. Apro- pos, the non-unitary time evolution generated by the effective HPT signals the fact that the system under consideration is open. In this context, ev- ery intertwining operator η̂ – positive definite or not – represents a time-invariant of the system. In other words, although the state norm ⟨ψ(t)|ψ(t)⟩ or the energy ⟨ψ(t)|HPT|ψ(t)⟩ of a state |ψ(t)⟩ = exp(−iHPTt)|ψ(0)⟩ of a PT -symmetric system are not conserved [8], the expectation values ⟨ψ(t)|η̂|ψ(t)⟩ remain constant with time. For a system with N degrees of freedom, a complete characterization of in- tertwining operators for a given system is carried out by solving the set of N2 simultaneous, linear equations, i.e. η̂HPT = H † PTη̂. (1) 1 https://doi.org/10.14311/AP.2022.62.0001 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en K. S. Agarwal, J. Muldoon, Y. N. Joglekar Acta Polytechnica In the past, several different avenues have been used to obtain these conserved quantities. They include spectral decomposition methods [8, 34], an explicit recursive construction to generate a tower of inter- twining operators [25, 35], sum-rules method [36], and the Stokes parametrization approach for a PT - symmetric dimer [37]. Here, we present yet another approach to the problem, and illustrate it with two simple examples. The plan of the paper is as fol- lows. In Section 2, we present the eigenvalue-equation approach for intertwining operators and the details of the vectorization scheme. This method is valid for any finite dimensional PT -symmetric Hamilto- nian. In Section 3, we present results of such analysis for a quantum PT -symmetric dimer with static or time-periodic gain and loss. Corresponding results for a classical PT -symmetric dimer are presented in Sec- tion 4. We conclude the paper with a brief discussion in Section 5. 2. Intertwining operators as an eigenvalue problem For a PT -symmetric system undergoing coherent but non-unitary dynamics with static Hamiltonian HPT, the expectation value of an operator η̂ satisfies the following linear-in-η̂ first-order differential equation ∂t⟨ψ(t)|η̂|ψ(t)⟩ = −i⟨ψ(t)|η̂HPT − H † PTη̂|ψ(t)⟩. (2) This equation is reminiscent of the Gorini Kossakowski Sudarshan Lindblad (GKSL) equation [38, 39] (hence- forth referred to as the Lindblad equation) that de- scribes the dynamics of the reduced density matrix of a quantum system coupled to a much larger envi- ronment [40–42]. Interpreting η̂ as an N × N matrix, all η̂s that satisfy Eq. (2) can be obtained from the corresponding eigenvalue problem Ekη̂k = −i(η̂kHPT − H † PTη̂k) ≡ Lη̂k, (3) for 1 ≤ k ≤ N2. We vectorize the matrix η̂ into an N2-sized column vector |ηv ⟩ by stacking its columns, i.e. [η̂]pq → ηvp+(q−1)N [43]. Under this vectoriza- tion, the Hilbert-Schmidt trace inner product carries over to the Dirac inner product, Tr(η̂†1η̂2) = ⟨η v 1 |ηv2 ⟩ where ⟨ηv1 | is the Hermitian-conjugate row vector ob- tained from the column vector |ηv1 ⟩. Using the identity Aη̂B → (BT ⊗ A)|ηv ⟩, the eigenvalue problem Eq. (3) becomes det(L − E1N 2 ) = 0 where the N2 × N2 “Li- ouvillian” matrix is given by L = −i [ HTPT ⊗ 1N − 1N ⊗ H † PT ] , (4) and 1m is the m × m identity matrix. Thus, the intertwining operators are distinct eigenvectors |ηvm⟩ with zero eigenvalue in Eq. (3). The N2 eigenvalues of the Liouvillian L are simply related to N eigenvalues ϵm of the HPT as Epq = −i(ϵp − ϵ∗q ). (5) Since the spectrum of HPT is either real (ϵp = ϵ∗p) or complex conjugates (ϵp = ϵ∗q for some pair), there are N zero eigenvalues of L when HPT has no symmetry- driven degeneracies; the number of zero eigenvalues grows to N2 if the Hamiltonian is proportional to the identity matrix [34]. This analysis also provides a transparent way to construct corresponding inter- twining operators via the spectral decomposition of HPT [8]. Note that when E = 0, due to the linearity of the intertwining relation, Eq. (1), without loss of generality, we can choose the N intertwining operators η̂m to be Hermitian. So what is the advantage of this approach? For one, it gives us N(N − 1) other, (generally non-Hermitian) operators whose expectation value in any arbitrary state evolves simply exponentially in time. When Epq is purely imaginary, it leads to the non-Hermitian η̂pq whose expectation value in any state remains constant in magnitude; on the other hand, if Epq is purely real, one can choose a Hermitian η̂pq whose expectation value exponentially grows or decays with time. This analysis of constants of motion is valid for systems with a static, PT -symmetric Hamiltonian. It can be suitably generalized to time-periodic, PT - symmetric Hamiltonians via the Floquet formal- ism [25, 28, 32, 44, 45]. When HPT(t) = HPT(t + T) is periodic in time, the long-time dynamics of the system is governed by the Floquet time-evolution op- erator [46] GF (T) = Te −i ∫ T 0 HPT(t′)dt′, (6) where T stands for the time ordered product that takes into account non-commuting nature of the Hamiltonians at different times. The (stroboscopic) dynamics of the system at times tm = mT is then given by |ψ(tm)⟩ = GmF |ψ(0)⟩, and the corresponding, Hermitian, conserved operators η̂ = η̂† are determined by [25, 34] G † F η̂GF = η̂. (7) Vectorization of Eq. (7) implies the conserved quanti- ties are given by eigenvectors of the “Floquet Liouville time-evolution” matrix G = GTF ⊗ G † F (8) with unit eigenvalue. Since GF (T) inherits the PT symmetry of the time-periodic Hamiltonian, the eigen- values κm of GF (T) either lie on a circle (|κp| = const.; PT -symmetric phase) or occur along a radial line in pairs with constant geometric mean (|κpκq | = const.; PT -broken phase). Therefore, it is straightforward to see that among the N2 eigenvalues λpq ≡ κpκ∗q of G, there are N unit eigenvalues, giving rise to N conserved quantities. As in the case with the static Hamiltonian, the remaining N(N − 1) eigenvectors give operators that vary exponentially with the strobo- scopic time tm irrespective of the initial state |ψ(0)⟩. 2 vol. 62 no. 1/2022 Conserved quantities in non-Hermitian systems If λpq is real, we can choose them to be Hermitian, as in the case of a static Hamiltonian. We now demonstrate these ideas with two concrete examples. 3. Quantum PT -symmetric dimer We first consider the prototypical PT -symmetric dimer (N = 2) with a Hamiltonian given by H1(t) = Jσx + iγf(t)σz = HT1 ̸= H † 1. (9) We call this model “quantum” because it arises natu- rally in minimal quantum systems undergoing Lind- blad evolution when we confine ourselves to trajecto- ries that undergo no quantum jumps [31], as well as in wave systems [18–22]. Here J > 0 denotes coupling between the two degrees of freedom and γ > 0 is the strength of the gain-loss term. H1 is PT -symmetric with the parity operator P = σx and time-reversal operator T = ∗ (complex conjugation). The eigenval- ues ϵ1,2 = ± √ J2 − γ2 ≡ ±∆(γ) of the Hamiltonian H1(γ) remain real when γ < γPT = J and become purely imaginary when γ exceeds the threshold. In the static case, f(t) = 1, using Eq. 1, it is easy to show that η̂1 = P = σx is the first in- tertwining operator [34, 35], and the recursive con- struction gives the second intertwining operator as η̂2 = η̂1H1/J = 1 + (γ/J)σy . However, the corre- sponding 4 × 4 Liouvillian matrix L, Eq. (4), has two nonzero eigenvalues that are given by E± = ±2i∆. The corresponding eigen-operators are given by η̂± = 1 J2 [ (γ ± i∆)2 −i(γ ± i∆) +i(γ ± i∆) 1 ] . (10) Note that the 2 × 2 matrices η̂± have rank 1, and thus are not invertible. In the PT -symmetric region (∆ ∈ R), the operators η̂± are not Hermitian, whereas in the PT broken region (∆ ∈ iR), they are Hermitian. Next we consider the time-periodic case, i.e. f(t) = f(t + T) where f(t) = sgn(t) for |t| < T/2 denotes a square wave. This piecewise constant gain and loss means that the Hamiltonian switches from H1+ = Jσx + iγσz for 0 ≤ t < T/2 to H1− = T H1+T = Jσx −iγσz for T/2 ≤ t < T . The non-unitary Floquet time-evolution operator can be explicitly evaluated as [47] GF (T) = e−iH1−T /2e−iH1+T /2, (11) = G012 + iGxσx + Gyσy, (12) where G0 = [J2 cos(∆T) − γ2]/∆2, Gx = −J sin(∆T)/∆ and Gy = −Jγ[1 − cos(∆T)]/∆2 are coefficients that remain real irrespective of where ∆(γ) is real or purely imaginary. When γ → 0, this repro- duces the expected result GF (T ) = exp(−iJσxT) and in the limit T → 0, the time-evolution operator re- duces to 12 as expected. On the other hand, as ∆ → 0, the power series for GF (T) terminates at second order in T in a sharp contrast to the static case, where it terminates at first order in time. The eigenvalues of GF , Eq. (12), are κ1,2 = G0 ± i √ G2x − G2y. (13) Thus the EP contours separating the PT -symmetric phase (|κ1| = |κ2|) from the PT -broken phase (|κ1| ≠ |κ2|) are given by Gx = ±Gy [47]. It is easy to check that η̂1 = σx satisfies G † F η̂1GF = η̂1 and is a strobo- scopically conserved quantity. The second conserved operator is obtained from the symmetrized or anti- symmetrized version of the recursive construction [34], i.e. η̂2 = { (η̂1GF + G † F η̂1)/2, −i(η̂1GF − G † F η̂1)/2. (14) In the present case, the symmetrized version re- turns η̂1 while the antisymmetrized version gives the second, linearly independent conserved operator as η̂2 = Gx12 + Gyσz . Following the procedure out- lined in Section 2 gives us two unity eigenvalues of G, Eq. (8), with corresponding conserved operators. The remaining two eigenvalues are complex conju- gates with unit length in the PT -symmetric region, i.e. λ3 = λ∗4 = eiϕ with eigen-operators η̂+ = η̂ † − that are Hermitian conjugates of each other. In the PT -broken region, the two complex eigenvalues with equal phase satisfy |λ3λ4| = 1. Figure 1 shows expectation values normalized to their initial values, ηα(t) ≡ ⟨ψ(t)|η̂α|ψ(t)⟩ ⟨ψ(0)|η̂α|ψ(0)⟩ (15) calculated with initial state |ψ(0)⟩ = | + x⟩ as a func- tion of dimensionless time t/T. The system param- eters are γ = 0.5J, JT = 1, and | + x⟩ is the eigen- state of σx with eigenvalue +1. Thus, the system is in the PT -symmetric region. Figure 1a shows that η1(t) is conserved in this evolution at all times, not just stroboscopically at tm = mT. On the other hand η2(t), shown in Figure 1b, has a periodic be- havior with a period ∼ 30T (not shown). Although η2(t) varies with time, it is stroboscopically conserved, η2(tm) = 1. The dotted red line shows ℜλt2 = 1. Figure 1c shows that the real part of η+(t), with eigenvalue λ3 = −0.44 + 0.9i, also shows periodic vari- ation. The dotted black line shows ℜλt3, and the fact that ℜη+(tm) matches it stroboscopically confirms the simple sinousoidal variation of this eigen-operator. Figure 1d shows corresponding results for the fourth operator η̂− = η̂ † + with eigenvalue λ4 = −0.44 − 0.9i. We conclude this section with transformation prop- erties of GF (T) and the conserved operators η̂. When the periodic Hamiltonian is Hermitian, i.e. H0(t) = H † 0 (t) = H0(t + T), shifting the zero of time to t0 leads to a unitary transformation, GF (T + t0, t0) = U(t0)GF (T)U†(t0), (16) U(t0) = Te −i ∫ t0 0 H0(t′)dt′. (17) 3 K. S. Agarwal, J. Muldoon, Y. N. Joglekar Acta Polytechnica Figure 1. Conserved quantities for a Floquet, quantum PT -symmetric dimer. System parameters are γ = 0.5J, JT = 1, |ψ(0)⟩ = | + x⟩, and ηα(t) denote normalized expectation values. (a) η̂1 = σx is an eigen-operator of G with eigenvalue λ1 = 1; η1(t) is constant. (b) η̂2 = Gx12 + Gyσz is the second eigen-operator of G with λ2 = 1; η2(t) oscillates with time, but is stroboscopically constant at t/T = n; the dotted red line shows ℜλt2 = 1. (c) η̂+ is a non-Hermitian eigen-operator with unit-length eigenvalue λ3 = −0.44 + 0.9i. The real part of its normalized expectation value stroboscopically matches ℜλt3 shown in dotted black. (d) Corresponding result for η̂− = η̂ † + with eigenvalue λ4 = λ∗3 . Therefore the conserved operators are also unitar- ily transformed. However, in our case, Eq. (16) be- comes a similarity transformation, GF (T + t0, t0) = SGF (T)S−1 where S = T exp(−i ∫ t0 0 HPT(t ′)dt′) does not satisfy S†S = 1 = SS†. Under this transformation, the conserved operators change as η̂ → S−1†η̂S−1. This non-unitary transformation of the conserved quantities under a shift of zero of time suggests that they are not related to “symmetries” of the open system with balanced gain and loss. 4. Classical PT -symmetric dimer We now consider a different example characterized by a non-Hermitian Hamiltonian with purely imaginary entries. We call such a system “classical” because having HPT = −H∗PT ensures that the non-unitary time evolution operator exp(−iHPTt) is purely real, and therefore |ψ(t)⟩ remains real if |ψ(0)⟩ is. Such classical Hamiltonian arises naturally in describing the energy density dynamics in mechanical or electrical cir- cuits [23–26, 28], where |ψ(t)⟩ encodes time-dependent positions, velocities, voltages, currents, etc. and is obviously real. As its simplest model, we consider a dimer governed by the Hamiltonian H2(t) = Jσy + iγf(t)σz = −H∗2 . (18) On one level, the Hamiltonian H2(t), Eq. (18), is “just a change of basis” from H1(t), Eq. (9); H2(t) = exp(−iπσz/4)H1(t) exp(+iπσz/4). However, since H2(t) models effective, classical systems where the entire complex state space is physically accessible, it is necessary to treat it differently. A physical real- ization of H2(t) is found in a single LC circuit whose inductance L(t) and capacitance C(t) are varied such that its characteristic frequency J = 1/ √ L(t)C(t) remains constant [25]. Hamiltonian H2(t) is PT -symmetric with PT = σx∗. In the static case (f(t) = 1), the two, Hermi- tian intertwining operators are given by η̂1 = σy and η̂2 = η̂1H2/J = 12 − (γ/J)σx. In addition, the vec- torization approach gives two, rank-1 eigen-operators η̂± = 1 J2 [ (γ ± i∆)2 −(γ ± i∆) −(γ ± i∆) 1 ] , (19) with eigenvalues E± = ±2i∆. As we discussed in Section 3, these operators are not Hermitian in the PT -symmetric phase, and become Hermitian in the PT -broken phase. For the Floquet case, we choose a gain-loss term that is nonzero only at discrete times. This is accomplished by choosing the dimensionless function f(t) as f(t) = T [δ(t) − δ(t − T/2)] = f(t + T). (20) The resulting Floquet time-evolution operator GF (T) can be analytically calculated [25]. Since the Hamilto- nian H2(t) is Hermitian at all times except tk = kT/2, the evolution is mostly unitary, punctuated by non- unitary contributions that occur due to δ-functions at 4 vol. 62 no. 1/2022 Conserved quantities in non-Hermitian systems times tk. The result is GF (T) = e+γT σze−iJ T σy /2e−γT σze−iJ T σy /2 = G012 + Gxσx + iGyσy + Gzσz, (21) where the four real coefficients Gk are given by G0 = cos2(JT/2) − sin2(JT/2) cosh(2γT), (22) Gx = − sin(JT) sinh(2γT)/2, (23) Gy = − sin(JT)[1 + cosh(2γT)]/2. (24) Gz = − sin2(JT/2) sinh(2γT). (25) As is expected, the purely real GF (T) reduces to exp(−iJTσy ) in the Hermitian limit γ → 0. The EP conoturs, on the other hand, are determined by the constraint G2x + G2y − G2z = 0, which reduces to cos(JT/2) = tanh(γT) [25]. Two linearly independent Floquet intertwining oper- ators obtained by solving Eq. (7) are given by η̂1 = σy and η̂2 = −i(η̂1GF −G † F η̂1)/2. The latter simplifies to η̂2 = Gy12 + Gzσx − Gxσz . We leave it for the reader to check that, as in the case of Floquet quantum PT dimer problem, the symmetrized version of the recur- sive procedure, Eq. (14), does not lead to a result that is linearly independent of η̂1. Following the recipe in Section 2, we supplement these analytical results with symbolic or numerical results for four eigenvalues λk and four eigen-operators η̂1, η̂2, η̂± of G, Eq. (8). Figure 2 shows the behavior of normalized expec- tation values ηα(t) calculated with |ψ(0)⟩ = | + x⟩ as a function of time. The system parameters are γ = 0.5J and JT = 1, and therefore the system is in the PT -symmetric region. Note that since |ψ(t)⟩ is purely real, ⟨ψ(t)|η̂1|ψ(t)⟩ = 0 independent of time [25]. On the other hand η2(t), shown in Fig- ure 2a, has a periodic behavior. Although η2(t) varies with time, it is stroboscopically conserved, η2(tm) = 1. Figure 2b shows that the real part of η+(t), with unit-magnitude eigenvalue λ3 = −0.65 + 0.756i, also varies periodically. The dotted black line shows ℜλt3, and the fact that ℜη+(tm) matches it stroboscopically confirms the simple sinousoidal variation of this eigen- operator. Since the system is in the PT -symmetric phase, η̂− = η̂ † +, and therefore ℜη−(t) = ℜη+(t). Figure 2c shows the corresponding imaginary parts ℑη+(t) = −ℑη−(t) for the eigen-operator with the complex conjugate eigenvalue λ4 = λ∗3. We note that in the PT -broken regime, the non-unit-modulus eigen- values are not complex conjugates of each other, and therefore the corresponding eigen-operators will not satisfy the relations shown in Figures 2b-c. 5. Conclusions In this article, we have presented a new method to obtain intertwining operators or conserved quantities in PT -symmetric systems with static or time-periodic Hamiltonians. In this approach, these operators ap- pear as zero-E eigenmodes of the static Liouvillian L or as λ = 1 eigenmodes of the Floquet G. For Figure 2. Conserved quantities for a classical PT - symmetric dimer with γ = 0.5J, JT = 1, |ψ(0)⟩ = | +x⟩. Since |ψ(t)⟩ is purely real, the expectation value of η̂1 = σy is always zero. (a) η̂2 = Gy12 + Gxσz − Gzσx is the second eigen-operator of G with λ2 = 1. η2(t) oscillates with time, but is stroboscopically constant at t/T = n; the dotted red line shows ℜλt2 = 1. (b) Since the system is in the PT -symmetric phase, ℜη+(t) = ℜη−(t) (solid black) shows periodic behavior with values that stroboscopically match ℜλt3, shown in dotted black. (c) Corresponding imaginary parts, ℑη−(t) = −ℑη+(t) (dot-dashed black) show similar, stroboscopically matching behavior. an N-dimensional system, in addition to the N con- stants of motion, this approach also leads to N(N − 1) operators whose expectation values in any arbitrary state undergo simple exponential-in-time change. We have demonstrated these concepts with two simple, physically motivated examples of a PT -symmetric dimer with different, periodic gain-loss profiles. We have deliberating stayed away from continuum models because extending this approach or the recursive con- struction [34, 35] to infinite dimensions will probably be plagued by challenges regarding domains of result- ing, increasingly higher-order differential operators. The definition of an intertwining operator via Eq. (1) can be generalized to obtain conserved observables for Hamiltonians that posses other antilinear symmetries, such as anti-PT symmetry [48–50] or anyonic-PT symmetry [51, 52]. The recursive procedure to gener- ate a tower of such operators [34], and the vectoriza- tion method presented in Section 2 remains valid for arbitrary antilinear symmetry. Thus, this approach can be used to investigate constants of motion in such systems as well. 5 K. S. Agarwal, J. Muldoon, Y. N. Joglekar Acta Polytechnica References [1] C. M. Bender, S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. 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