Passive $\mathcal{PT}$-symmetry breaking transitions without exceptional points in dissipative photonic systems
Yogesh N. Joglekar Andrew K. Harter






Corresponding author: yojoglek@iupui.edu
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Department of Physics, Indiana University-Purdue University Indianapolis (IUPUI), Indianapolis 46202, Indiana, USA



Over the past decade, parity-time (𝒫𝒯)-symmetric Hamiltonians have been experimentally realized in classical, optical settings with balanced gain and loss, or in quantum systems with localized loss. In both realizations, the 𝒫𝒯-symmetry breaking transition occurs at the exceptional point of the non-Hermitian Hamiltonian, where its eigenvalues and the corresponding eigenvectors both coincide. Here, we show that in lossy systems, the 𝒫𝒯 transition is a phenomenon that broadly occurs without an attendant exceptional point, and is driven by the potential asymmetry between the neutral and the lossy regions. With experimentally realizable quantum models in mind, we investigate dimer and trimer waveguide configurations with one lossy waveguide. We validate the tight-binding model results by using the beam propagation method analysis. Our results pave a robust way toward studying the interplay between passive 𝒫𝒯 transitions and quantum effects in dissipative photonic configurations. 


(080.1238) Array waveguide devices; (270.5585) Quantum information and processing.
Passive 𝒫𝒯-symmetry breaking transitions without exceptional points in dissipative photonic systems
    Andrew K. Harter
    December 30, 2023
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§ INTRODUCTION


A fundamental principle of traditional quantum theory is that the observables of a system are Hermitian operators <cit.>. This self-adjoint character of observables is defined with respect to a global (Hamiltonian-independent) Dirac inner product. In particular, the Hamiltonian of a closed quantum system is Hermitian. It determines the energy levels of the system and therefore the experimentally observable transition frequencies. Thus, it came as a great surprise when Carl Bender and co-workers discovered a broad class of non-Hermitian, continuum Hamiltonians with purely real spectra <cit.>. The salient feature of such Hamiltonians is the presence of complex potentials V(x) that are invariant under the combined parity (𝒫:x→ -x) and time-reversal (𝒯=*) operations, i.e. V^*(-x)=V(x). Initial efforts on this subject focused on developing a self-consistent complex extension of quantum mechanics via a redefinition of the inner product that is used to define the adjoint of an observable <cit.>. This line of inquiry led to significant mathematical developments in understanding the properties pseudo-Hermitian operators <cit.>. But it did not elucidate a simple physical picture for complex potentials that are a hallmark of 𝒫𝒯-symmetric Hamiltonians. 

Experimental progress on the 𝒫𝒯-symmetric systems started with two realizations <cit.>. First, Schrodinger equation is isomorphic with paraxial approximation to the Maxwell's equation where the local index of refraction n(x)=n_R(x)+in_I(x) plays the role of the potential V(x). Second, it is easy to engineer a lossy index of refraction n_I(x)<0, and not too difficult to engineer a gain either, i.e. n_I(x)>0. These dual realizations provided a transparent, physical insight into the meaning of complex, 𝒫𝒯-symmetric potentials: they represent balanced, spatially separated loss and gain <cit.>. Since then, over the past decade coupled photonic systems described by non-Hermitian, 𝒫𝒯-symmetric effective Hamiltonians have been extensively investigated <cit.>. Light propagation in such systems shows non-trivial functionalities, such as unidirectional invisibility <cit.>, that are absent in their no-gain, no-loss counterparts. We emphasize that these realizations are essentially classical. Gain at few-photons level is random due to spontaneous emission <cit.>; in contrast, loss is linear down to single-photon level. Thus, engineering a truly quantum 𝒫𝒯-symmetric systems is fundamentally difficult. Therefore, in spite of a few theoretical proposals <cit.>, there are no experimental realizations of such systems that show quantum correlations present. 

By recognizing that a two-state loss-gain Hamiltonian is the same as a two-state loss-neutral Hamiltonian apart from an “identity-shift" along the imaginary axis, the language of 𝒫𝒯-symmetry and 𝒫𝒯 transitions has been adopted to purely dissipative, classical systems as well <cit.>. Indeed, the first ever observation of 𝒫𝒯-symmetry breaking was in two coupled waveguides, one with loss γ and the other without <cit.>. As the loss strength was increased from zero, the net transmission first decreased from unity, reached a minimum, and then increased as γ was increased beyond a threshold, signaling the passive 𝒫𝒯 symmetry breaking transition. This mapping provided a clear way forward to define passive 𝒫𝒯-symmetry breaking phenomenon in truly quantum, effectively two-level, dissipative systems. It led to the first observations of 𝒫𝒯-symmetry breaking in the quantum domain with quantum-correlated single  photons <cit.> and ultracold atoms <cit.>. 

In this paper, we extend the notion of passive 𝒫𝒯 transition to lossy Hamiltonians that do not map onto a 𝒫𝒯-symmetric Hamiltonian. By using tight-binding models and beam propagation method (BPM) analysis of experimentally realistic setups, we demonstrate that such transitions – driven by the emergence of a slowly decaying eigenmode – occur without the presence of exceptional points. 




§ AVOIDED LEVEL CROSSING IN GAIN-LOSS SYSTEMS


The prototypical effective Hamiltonian for experimentally realized, classical gain-loss systems is given by H_PT(γ)=-J σ_x+iγσ_z≠ H^†_PT where σ_k are the standard Pauli matrices. This Hamiltonian is invariant under combined operations of 𝒫=σ_x and 𝒯=*. With ħ=1, the parameters of the Hamiltonian H_PT have units of s^-1. In optical settings, the time is proportional to the distance z traveled along the waveguide; therefore, the inter-site coupling J is inversely proportional to the coupling length L_c, i.e. J=π c/(n_0L_c) where c/n_0 is the constant speed of light in the waveguide with local index n_0. Similarly, the loss γ and the potential offset δ (measured in s^-1) are linearly proportional to the inverse penetration depth and the propagation-constant offset (measured in m^-1) respectively. The eigenvalues of H_PT are given by ±√(J^2-γ^2), are real for γ≤ J, and become complex conjugate pair for γ>J. The transition from the 𝒫𝒯-symmetric phase, i.e. purely real eigenvalues, to 𝒫𝒯-symmetry broken phase, i.e. some complex-conjugate eigenvalues, occurs at the threshold γ_PT=J. At this point, eigenvalues and eigenvectors of H_PT(γ), both, become degenerate. Thus the 𝒫𝒯-symmetry breaking transition point is the same as the second-order exceptional point of the Hamiltonian H_PT <cit.>. 

What happens when this Hamiltonian is perturbed by an antisymmetric real potential? Without loss of generality, such potential can be implemented by iγ→ iγ+δ. The eigenvalues of the perturbed Hamiltonian H(γ,δ) immediately become complex,

    λ_±(γ,δ)=±√( J^2-γ^2+δ^2-2iγδ).

Figure <ref> shows that when δ>0, for γ≪ J, the imaginary parts  λ_± are quite small and grow linearly with γ. This behavior changes to 
a steep dλ_±/dγ at γ=J, signaling the enhanced sensitivity in the neighborhood of the exceptional point <cit.>. However, the eigenvalues are always complex and therefore, the system is never in the 𝒫𝒯-symmetric phase when δ≠ 0. Thus, the notion of 𝒫𝒯 symmetry breaking transition cannot be extended to perturbed gain-loss Hamiltonians H(γ,δ). 




In truly gain-loss systems, the 𝒫𝒯-symmetry breaking, defined by real-to-complex spectrum, occurs at the exceptional point, defined by coincidence of both eigenvalues and eigenvectors. Note that similar avoided level crossings <cit.> also occur in coupled lasers with static cavity losses and pump-current controlled gains, and show surprising phenomena such as pump-induced laser death <cit.>, loss-induced revival of lasing <cit.>, and laser self-termination <cit.>.





§ PT-SYMMETRY BREAKING IN DISSIPATIVE SYSTEMS


The prototypical Hamiltonian for experimentally realized neutral-loss systems is given by 

    H_D(γ)=-Jσ_x-iγ(1-σ_z)=-Jσ_x-2iγ|2⟩⟨ 2|

In the quantum context, this represents a two-state system with loss only in the second state. In the classical context, this represents two evanescently coupled waveguides, where there is absorption only in the second waveguide <cit.>. Note that H_D(γ)=H_PT(γ)-iγ 1_2; however due to the imaginary shift that is not invariant under time-reversal operation, the dissipative Hamiltonian does not commute with the antilinear 𝒫𝒯 operator, i.e. [𝒫𝒯,H_D]≠ 0. The eigenvalues of Eq.(<ref>) are given by λ_D±=-iγ±√(J^2-γ^2) and always have positive decay rates Γ_±=-λ_D± for the two eigenmodes of the lossy Hamiltonian H_D. When γ≤ J, the two decay rates are equal and they increase linearly with γ, i.e. dΓ_±/dγ>0. Traditionally, this region is called the 𝒫𝒯-symmetric phase. When γ=J, the corresponding eigenmodes become degenerate. However, when γ>J, the decay rate for one of the eigenmodes starts to becomes smaller,

    Γ_+=γ-√(γ^2-J^2) J^2/2γ,

while the second mode decays faster with Γ_-→ 2γ. Traditionally, this region is called the 𝒫𝒯-broken phase. We define the emergence of a slowly decaying mode, i.e. 

    dΓ/dγ=0 atγ=γ_PT,

as the defining characteristic of the passive 𝒫𝒯 transition in loss-neutral systems <cit.>. Indeed, the increased total transmission with increasing loss seen in Ref. <cit.> is due to this mode. We note that this criterion, defined by the sign of dΓ/dγ changing from positive to negative for one of the modes, is not equivalent to defining the transition by the presence of an exceptional point, where the two decaying eigenmodes coalesce. For the identity-shifted Hamiltonian H_D(γ), Eq.(<ref>), they happen to coincide. The latter criterion is only meaningful for the identity-shifted cases, whereas Eq.(<ref>) is physically motivated and has straightforward experimental consequences <cit.>. 

In the following sections we elucidate the consequences of this difference between dissipative 𝒫𝒯 systems and gain-loss systems for dimer and trimer models, while keeping evanescently coupled photonic waveguides in mind as their experimental realizations.




§ DISSIPATIVE 𝒫𝒯 DIMER AND TRIMER CASES


Starting from the dissipative dimer Hamiltonian H_D, Eq.(<ref>), we now consider its Hermitian, on-site perturbation, 

    H_2(γ,δ)=([       0      -J;      -J -2iγ-2δ ]).

Note that H_2(γ,δ) represents two waveguides with evanescent coupling J, with a loss potential 2γ in the second waveguide and an extra on-site potential -2δ, that is generated by an offset in the real part of the index of refraction vis a vis the first waveguide. 




It is straightforward to analytically obtain the eigenvalues λ_2D± of H_2(γ,δ), similar to those in Eq.(<ref>), and the corresponding eigenvectors. Analyzing their behavior in the neighborhood of γ/J=1, δ=0 requires further care, as it entails developing the corresponding Puiseux series in fractional powers of the distance from the exceptional point <cit.>. In this paper, we primarily focus on numerical results across the entire parameter space instead of analytical results in the vicinity of the exceptional point. Figure <ref> shows the evolution of the decay rates Γ_±(γ) for the two eigenmodes as a function of the on-site perturbation. When δ=0, the exceptional point and the passive 𝒫𝒯-symmetry breaking threshold, defined by the emergence of the slowly decaying mode, coincide. When δ/J=0.1, the decay rates for the two modes are always different. Initially, both increase with γ until near γ∼ J where one takes off and the other starts to decrease, thus indicating the emergence of the slowly decaying mode. When the perturbation is increased further, δ/J=0.8, a similar behavior is observed, but on a weaker scale. The inset in Fig. <ref> shows the numerically obtained derivative of the decay-rate of the slow mode, dΓ_+/dγ. When δ=0, we see that it is unity for γ<J and diverges to -∞ as γ→ J^+. For δ>0, the inset provides a quick determination of the 𝒫𝒯 transition threshold γ_PT. These results are even in δ, and the emergence of a slowly decaying mode is a robust feature for all offsets δ≠ 0. We would like to point out that in the dissipative case the avoided-level-crossing location is instrumental to determining the passive 𝒫𝒯 transition point. In contrast, the counter-intuitive phenomena in coupled lasers do not occur at the avoided-level-crossing location, but at the location where an amplifying mode emerges <cit.>. 




There is no exceptional point for the Hamiltonian H_2(γ,δ≠ 0), and yet the passive 𝒫𝒯 transition phenomenon, defined by the emergence of a slow mode, is robust for δ≠ 0. To show this, we obtain a key experimental signature <cit.>, the total transmission T(γ,δ) as a function of the on-site potential δ. Starting from an initial injection into the first,  neutral site, i.e. |ψ(0)⟩=|1⟩, the total transmission at time t (or equivalently at distance z=ct/n_0 traveled along the waveguide) is given by T≡⟨ψ(t)|ψ(t)⟩ where |ψ(t)⟩=exp(-iH_2t)|ψ(0)⟩. Keeping in mind Ref. <cit.>, we obtain the net transmission at a single-coupling length, i.e. z=L_c, or equivalently, t=π/J. Figure <ref> shows that T(γ,δ) has a robust upturn feature for all δ, from δ=0 to a large value of δ/J∼ 1; it means that the key signature used in the passive 𝒫𝒯 symmetry breaking experiments <cit.> does not probe the exceptional point, but rather, the slowly decaying mode. The results in Fig. <ref> bolster the rationale for using Eq.(<ref>) as the definition of passive 𝒫𝒯 symmetry breaking transition with or without exceptional points. 
 
Let us now consider a dissipative trimer with only one lossy waveguide. The nearest-neighbor tunneling Hamiltonian for a trimer with open boundary conditions is given by H_0=-J( |1⟩⟨ 2|+|2⟩⟨ 3| + h.c.). With loss and a different on-site potential in the central waveguide, the trimer Hamiltonian becomes

    H_tc=H_0-(iγ+δ) |2⟩⟨ 2|.

Eq.(<ref>) has one zero eigenvalue with an antisymmetric eigenmode that does not couple to the center, lossy site. The other two eigenvalues are given by [-(iγ+δ)±√(8-(γ-iδ)^2)]/2. Figure <ref> shows the decay rates for the three modes as a function of the loss γ and on-site potential δ in the central waveguide. The zero decay rate, Γ_1=0, denotes the antisymmetric eigenmode. When δ=0, we see that the Hamiltonian H_tc has an exceptional point at γ_PT=2√(2)J. For γ<γ_PT both decay rates Γ_2,Γ_3 increase with loss strength, and the system is in the 𝒫𝒯-symmetric phase. It is characterized by a net transmission that decreases when γ is increased. When γ>γ_PT, the decay rate Γ_2 starts to decrease and, as a result, the total transmission is increased with increasing γ <cit.>. When δ≠ 0 these features remain robust but the Hamiltonian H_tc does not have an exceptional point. 




If the loss is in one of the outer waveguides, the trimer Hamiltonian is given by

    H_te=H_0-(iγ+δ)|1⟩⟨1|.

Figure <ref> shows the eigenmode decay rates Γ_k as a function of the loss strength γ for various on-site potentials δ. When δ=0, the trimer has two modes with equal decay rates, Γ_1=Γ_2, and a third one with the fastest decay rate Γ_3. Near γ/J∼ 1.4, the former two change over to slowly-decaying modes, signifying the 𝒫𝒯 transition. As δ>0 is increased, the two degenerate decay rates split, and faster of the two, Γ_2, approaches the third one, Γ_3. Results in the last panel, corresponding to δ/J=0.75, hint at the existence of an exceptional point in the vicinity of these parameters. In general, similar results are found for multiple losses and/or on-site potentials for the trimer model. The emergence of slowly decaying modes is commonplace, even when the existence of exceptional points is not. 




Lastly, we consider a trimer with periodic boundary conditions, with the Hamiltonian

    H_t(γ,δ,J')=H_tc-J'(|1⟩⟨ 3|+|3⟩⟨ 1|),

where 0≤ J'≤ J denotes the tunable coupling between the first and the third waveguides. This ring configuration can be easily realized via femtosecond-laser direct-written waveguides in glass <cit.>. We leave as an exercise for the reader to verify the following results. The antisymmetric mode continues to be an eigenmode of the Hamiltonian H_t with real eigenvalue +J'. The other two decaying eigenmodes have eigenvalues given by 

    λ_t±=-1/2(iγ+δ+J')±1/2√(8J^2+[(δ-J')+iγ]^2).

It follows from Eq.(<ref>) that the exceptional point at δ=0 shown in Figure <ref> survives along the line δ=J'. When δ≠ J', it is also straightforward to check that a slow mode emerges at a loss strength γ_PT∼ 2√(2)J, even though the Hamiltonian H_t does not have an exceptional point.  

It is worthwhile to mention that these trimer models cannot be identity-shifted to a 𝒫𝒯-symmetric model that supports a purely-real to complex-conjugate spectrum transition. That is, they are not isomorphic with any balanced gain-loss trimer models. They stand on their own and display the passive 𝒫𝒯 transition without an exceptional point over a wide range of Hamiltonian parameters. In the following section, we verify these results via beam propagation method (BPM) analysis of three planar coupled waveguides. 







§ BEAM PROPAGATION METHOD ANALYSIS


The results presented in Sec. <ref> are based on a tight-binding Hamiltonian where the spatial extent of a “site" is ignored. In a coupled-waveguides realizations of a lattice model, the electric field strength across the waveguide is not constant, and its spatial variation in the direction transverse to the waveguide needs to be taken into account. We verify our tight-binding results by obtaining the time-evolution of the wavefunction ψ(x,t) in a planar waveguide trimer with realistic parameters <cit.>. We remind the reader that ψ(x,t) is the envelope of the actual electric field, i.e., E(x,t)=exp[ik_0z-ic(k_0/n_0)t]ψ(x,t) where k_0 is the wavenumber of the rapidly varying part of the electric field, c is the speed of light, and n_0 is the real index of refraction of the cladding. Under paraxial approximation, the envelope ψ(x,t) obeys the continuum Schrodinger equation for a particle with an effective mass m=k_0 n_0^2/c in a potential V(x)=ck_0[1-n(x)^2/n_0^2]. The position-dependent index of refraction n(x) differs from that of a cladding only within each waveguide. For a small refractive-index contrast Δ n/n_0∼ 10^-4≪ 1, the potential becomes V_p=2ck_0Δ n_p/n_0 within waveguide p. We model the losses by adding negative imaginary parts to the index contrast Δ n_p and generate different on-site potentials by choosing different real parts while satisfying the small-contrast constraint. (For more details regarding these calculations, see Refs. <cit.>.) 

Figure <ref> a,b show the resultant intensity plots I(x,z)=|ψ(x,z=ct/n_0)|^2 dx for a dissipative trimer with an exceptional point, δ=0. The initial state is a symmetric combination of three single-waveguide eigenmodes with weight ratios 1:9:1. The symmetric combination ensures that the initial state is decoupled from the antisymmetric eigenmode with zero decay rate. Figure <ref>a shows the rapid decay of the initial pulse when the loss strength γ/J=2.75 is just below the threshold γ_PT=2√(2)J. When the loss strength is increased four-fold to γ/J=10, Fig. <ref>b, the same pulse travels farther signaling the emergence of the slowly decaying mode in the 𝒫𝒯 broken regime. Figure <ref> c,d show corresponding results for a trimer with no exceptional point and a very large on-site potential δ/J=1. We use such a large offset because the BPM simulations are virtually indistinguishable from the δ=0 case for δ≲ J/2. The eigenvalue flow in Fig. <ref> shows that in this case, after an initial linear increase, the decay rate of the slow-mode Γ_2 does not change appreciably at large γ≥γ_PT. Figure <ref>c shows that when γ/J=2.75 the initial pulse has a partial oscillation back into the central, lossy waveguide. When the loss strength is increased to γ/J=10, Fig. <ref>d, the pulse travels farther indicating a 𝒫𝒯-broken regime. These representative BPM results are consistent with tight-binding model findings, and show that in dissipative systems, a 𝒫𝒯 transition can occur with or without exceptional points. 




§ DISCUSSION

In this work, we have proposed a meaningful extension of the passive 𝒫𝒯 symmetry breaking phenomenon to dissipative Hamiltonians that are not identity-shifted from a balanced gain-loss Hamiltonian. We have shown that passive 𝒫𝒯 transitions, defined by the emergence of a slowly decaying mode, are ubiquitous in dissipative photonic systems and are not contingent on the existence of exceptional points. This is in sharp contrast with classical systems with balanced gain and loss, where 𝒫𝒯-symmetry breaking and exceptional points go hand in hand, and the concept of 𝒫𝒯 symmetry breaking cannot be meaningfully extended to unbalanced Hamiltonians. Our results may also be applicable to dissipative metamaterial systems, where non-trivial phenomena occur in the presence of balanced gain and loss <cit.>. 

Due to their exceptional tunability, relatively easy scalability, and the ability to implement one and (post-selected) two-qubit operations, integrated photonic systems at a quantum level are of great interest for quantum simulations and quantum computing. By introducing mode-dependent, tunable losses, such systems can be further tailored for true quantum simulations of non-Hermitian Hamiltonians in general and (dissipative) 𝒫𝒯-symmetric Hamiltonians in particular. Such simulators will permit the investigation of quantum attributes, such as entropy, entanglement metrics, many-particle correlations, etc. across the passive 𝒫𝒯 transition. Carrying out such studies in balanced gain-loss systems remains an open question and, most likely, is fundamentally impossible <cit.>. 



§ ACKNOWLEDGMENT
 
This work was supported by NSF grant no. DMR-1054020.



99


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