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- Photonics Research
- Vol. 6, Issue 8, A51 (2018)

Abstract

1. INTRODUCTION

A fundamental principle of traditional quantum theory is that the observables of a system are Hermitian operators ^{[1]}. 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 ^{[2,3]}. The salient feature of such Hamiltonians is the presence of complex potentials ^{[4,5]}. This line of inquiry led to significant mathematical developments in understanding the properties of pseudo-Hermitian operators ^{[6–11]}. But it did not elucidate a simple physical picture for complex potentials that are a hallmark of parity-time (PT)-symmetric Hamiltonians.

Experimental progress on the PT-symmetric systems started with two realizations ^{[12,13]}. First, the Schrödinger equation is isomorphic, with paraxial approximation to the Maxwell’s equation, where the local index of refraction ^{[14]}. Since then, over the past decade, coupled photonic systems described by non-Hermitian, PT-symmetric effective Hamiltonians have been extensively investigated ^{[15–22]}. Light propagation in such systems shows nontrivial functionalities, such as unidirectional invisibility ^{[23,24]}, that are absent in their no-gain, no-loss counterparts. We emphasize that these realizations are essentially classical. Gain at the few-photons level is random due to spontaneous emission ^{[25,26]}; in contrast, loss is linear down to the single-photon level. Thus, engineering a truly quantum PT-symmetric system is fundamentally difficult. Therefore, in spite of a few theoretical proposals ^{[27]}, 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 ^{[28]}. Indeed, the first ever observation of PT-symmetry breaking was in two coupled waveguides, one with loss ^{[29]}. As the loss strength was increased from zero, the net transmission first decreased from unity, reached a minimum, and then increased as ^{[30]} and ultracold atoms ^{[31]}.

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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.

2. AVOIDED LEVEL CROSSING IN GAIN-LOSS SYSTEMS

The prototypical effective Hamiltonian for experimentally realized, classical gain-loss systems is given by ^{[32]}.

What happens when this Hamiltonian is perturbed by an antisymmetric real potential? Without loss of generality, such potentials can be implemented by ^{[33,34]}. However, the eigenvalues are always complex, and therefore, the system is never in the PT-symmetric phase when

Figure 1.Imaginary parts of the spectrum Eq. (

In true 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 ^{[35–37]} also occur in coupled lasers with static cavity losses and pump-current controlled gains and show surprising phenomena, such as pump-induced laser death ^{[38,19]}, loss-induced revival of lasing ^{[20]}, and laser self-termination ^{[39,40]}.

3. PT-SYMMETRY BREAKING IN DISSIPATIVE SYSTEMS

The prototypical Hamiltonian for experimentally realized neutral-loss systems is given by 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 ^{[29]}. Note that ^{[29–31]}. Indeed, the increased total transmission with increasing loss seen in Ref. ^{[29]} is due to this mode. We note that this criterion, defined by the sign of ^{[29–31]}.

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.

4. DISSIPATIVE PT DIMER AND TRIMER CASES

Starting from the dissipative dimer Hamiltonian

It is straightforward to analytically obtain the eigenvalues ^{[32–34]}. 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 ^{[39,40]}.

Figure 2.Decay rates

There is no exceptional point for the Hamiltonian ^{[29]}, the total transmission ^{[29]}, we obtain the net transmission at a single-coupling length, i.e., ^{[29]} does not probe the exceptional point, but rather, the slowly decaying mode. The results in Fig.

Figure 3.Net transmission

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 ^{[29–31]}. When

Figure 4.PT transition in three coupled waveguides with lossy center waveguide, Eq. (

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

Figure 5.PT transition in three coupled waveguides with first lossy waveguide, Eq. (

Lastly, we consider a trimer with periodic boundary conditions, with the Hamiltonian where ^{[41]}. We leave as an exercise for the reader to verify the following results. The antisymmetric mode continues to be an eigenmode of the Hamiltonian

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 BPM analysis of three planar coupled waveguides.

5. BPM ANALYSIS

The results presented in Section ^{[42]}. We remind the reader that ^{[42–44]}.)

Figures

Figure 6.BPM results for the intensity

6. 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 nontrivial phenomena occur in the presence of balanced gain and loss ^{[45,46]}.

Due to their exceptional tunability, relatively easy scalability, and the ability to implement one- and (postselected) 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, and many-particle correlations, across the passive PT transition. Carrying out such studies in balanced gain-loss systems remains an open question and, most likely, is fundamentally impossible ^{[47]}.

References

[1] J. J. Sakurai**. Modern Quantum Mechanics(1996)**.

[5] C. M. Bender. Making sense of non-Hermitian Hamiltonians**. Rep. Prog. Phys., 70, 947-1018(2007)**.

[11] M. Znojil. Three-Hilbert-space formulation of quantum mechanics**. SIGMA, 5, 001(2009)**.

[31] J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, L. Luo. Observation of parity-time symmetry breaking transitions in dissipative Floquet system of ultracold atoms**(2016)**.

[32] T. Kato**. Perturbation Theory for Linear Operators(1976)**.

[47] S. Scheel, A. Szameit. PT-symmetric photonic quantum systems with gain and loss do not exist**(2018)**.

Yogesh N. Joglekar, Andrew K. Harter. Passive parity-time-symmetry-breaking transitions without exceptional points in dissipative photonic systems [Invited][J]. Photonics Research, 2018, 6(8): A51

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