PT-OVS: Phase Diagram in Non-Hermitian Photonics

• PT-OVS is a framework for non-Hermitian two-mode photonic systems that organizes PT-symmetric and PT-broken phases via a full gain–loss parameter-space diagram.
• It reveals that PT breaking occurs not only in balanced gain–loss pairs but also in gain–gain and gain–lossless configurations by leveraging eigenvalue discriminants.
• The phase diagram provides actionable insights for designing optical devices such as beam splitters and non-reciprocal routers using exceptional-point principles.

PT-OVS, in the usage associated with the study of “Gain-gain and gain-lossless PT-symmetry broken from PT-phase diagram,” denotes a parity–time framework for non-Hermitian two-mode photonic systems in which PT-symmetric and PT-broken behavior are organized by a full gain–loss parameter-space diagram rather than by the standard balanced gain–loss construction alone. In this formulation, gain–gain, gain–lossless, gain–loss, loss–loss, loss–lossless, and lossless–lossless configurations are treated within one phase-space picture, and the boundary between PT-symmetry and PT-broken can be clearly defined in the full-parameter space including gain, lossless and loss (Zhang et al., 2022).

1. Formal setting and PT-symmetry condition

Parity-time symmetry in optics is formulated through a non-Hermitian Hamiltonian $H$ satisfying

$[H,\mathcal{PT}] = 0,$

even though $H$ itself is non-Hermitian. For one-dimensional systems with a complex potential $V(x)$, the PT condition is

$V(x)=V^*(-x).$

Under the paraxial-wave/Schrödinger analogy used in non-Hermitian optics, the complex refractive index $n(x)$ or complex permittivity $\epsilon(x)$ plays the role of the potential, and a typical optical PT condition becomes

$n(x)=n^*(-x),$

so that the real part is even and the imaginary part is odd.

The two-mode coupled system treated in the PT-OVS framework uses amplitudes $a_1(\xi)$ and $a_2(\xi)$, where $[H,\mathcal{PT}] = 0,$0 is either propagation distance or time. Its coupled-mode equations are

$[H,\mathcal{PT}] = 0,$1

with $[H,\mathcal{PT}] = 0,$2 the complex coupling coefficient and $[H,\mathcal{PT}] = 0,$3 the gain/loss rates. The sign convention is explicit: $[H,\mathcal{PT}] = 0,$4 denotes loss and $[H,\mathcal{PT}] = 0,$5 denotes gain. Writing

$[H,\mathcal{PT}] = 0,$6

the effective Hamiltonian is

$[H,\mathcal{PT}] = 0,$7

This formulation is broader than the standard optical PT configuration $[H,\mathcal{PT}] = 0,$8, $[H,\mathcal{PT}] = 0,$9. A plausible implication is that PT-OVS is best understood not as a single balanced gain–loss architecture, but as a generalized phase-space organization of non-Hermitian coupled photonic modes.

2. Eigenstructure and phase classification

The phase structure is controlled by the eigenvalues of the exponential evolution terms. Eliminating one variable yields

$H$0

The discriminant

$H$1

separates PT-symmetric and PT-broken behavior.

For $H$2, the system is in the PT-symmetric phase. The eigenvalues then have identical imaginary parts and opposite real parts:

$H$3

$H$4

The propagation is dominated by oscillatory energy exchange, with a uniform exponential factor multiplying the motion.

For $H$5, the system is in the PT-broken phase. The eigenvalues are purely imaginary:

$H$6

$H$7

In this regime one eigenmode experiences exponential amplification and the other exponential attenuation. In the language of the coupled structure, one supermode grows and one decays, so the modal symmetry is broken (Zhang et al., 2022).

The same model also yields the second-order equation

$H$8

whose solutions are exponentials $H$9 with the eigenvalues above. This connects the phase classification directly to observable propagation dynamics.

3. Full PT-phase diagram and exceptional-point lines

The boundary between PT-symmetric and PT-broken phases is obtained when the square-root term vanishes:

$V(x)$0

In normalized form,

$V(x)$1

These straight lines in the $V(x)$2 plane are the exceptional-point lines, where both eigenvalues and eigenvectors coalesce.

Between the exceptional-point lines, where $V(x)$3, the system is in the PT-symmetric phase. Outside them, where $V(x)$4, the system is PT-broken. On the exceptional-point lines themselves, the system is at the threshold of symmetry breaking. Because the threshold condition depends only on the difference $V(x)$5 and not on the individual signs alone, the same diagram encompasses gain–gain, gain–lossless, gain–loss, loss–loss, loss–lossless, and lossless–lossless cases (Zhang et al., 2022).

Regime	Sign structure	Phase condition
Gain–gain	$V(x)$6	PT-broken if $V(x)$7
Gain–lossless	one $V(x)$8, one $V(x)$9	PT-broken if $V(x)=V^*(-x).$0
Gain–loss	one negative, one positive	Same exceptional-point condition
Loss–loss	$V(x)=V^*(-x).$1	Same exceptional-point condition
Loss–lossless	one $V(x)=V^*(-x).$2, one $V(x)=V^*(-x).$3	Same exceptional-point condition
Lossless–lossless	$V(x)=V^*(-x).$4	Hermitian coupling at the origin

A recurring misconception in PT optics is that PT-breaking requires a balanced gain–loss pair. The phase diagram contradicts that restricted view: a large imbalance in gain between two gain channels, or between gain and zero-loss, is sufficient to produce PT-broken behavior with purely imaginary eigenvalues.

4. Transmission matrices, beam splitting, and the exceptional point

For coupled waveguides, the framework is expressed in terms of a transmission matrix $V(x)=V^*(-x).$5 that maps input fields to output fields after propagation distance $V(x)=V^*(-x).$6:

$V(x)=V^*(-x).$7

In the balanced gain–loss configuration $V(x)=V^*(-x).$8, $V(x)=V^*(-x).$9, the system is used as a beam splitter, and the form of $n(x)$0 changes across the phase boundary.

In the PT-symmetric phase, where $n(x)$1,

$n(x)$2

The corresponding transmission matrix is

$n(x)$3

The off-diagonal terms are proportional to $n(x)$4 and encode coherent coupling between channels, while the diagonal terms include gain/loss and phase accumulation. The factors $n(x)$5 make the oscillatory character explicit. For $n(x)$6, the matrix reduces to the familiar Hermitian $n(x)$7 directional-coupler form.

In the PT-broken phase, where $n(x)$8,

$n(x)$9

and

$\epsilon(x)$0

Here the evolution is governed by hyperbolic behavior, equivalently by exponentials, and one mode becomes dominant.

In the strongly broken limit $\epsilon(x)$1, the approximation

$\epsilon(x)$2

yields

$\epsilon(x)$3

The output is then dominated by the gain channel for any input state, and the structure acts as a non-reciprocal amplifier or unidirectional beam router.

At the exceptional point, where $\epsilon(x)$4, the coalesced eigenvalues satisfy

$\epsilon(x)$5

and the transfer matrix becomes

$\epsilon(x)$6

Both the PT-symmetric and PT-broken transmission matrices reduce to this form as $\epsilon(x)$7, providing a self-consistent description of the threshold (Zhang et al., 2022).

5. Gain–gain and gain–lossless PT-broken regimes

The defining extension of the PT-OVS picture is the explicit treatment of gain–gain and gain–lossless PT-broken phases. In the gain–gain case, both channels satisfy $\epsilon(x)$8, but PT-breaking still occurs when

$\epsilon(x)$9

In the gain–lossless case, with one neutral channel and one gain channel, the threshold becomes

$n(x)=n^*(-x),$0

These conditions show that a gain–loss pair is not required; what matters is the magnitude of the gain/loss imbalance relative to the coupling.

The dynamical distinction between PT-symmetric and PT-broken behavior is illustrated directly in coupled-waveguide examples. In the PT-symmetric gain–gain case, taking $n(x)=n^*(-x),$1 and $n(x)=n^*(-x),$2 gives $n(x)=n^*(-x),$3, so the system is PT-symmetric. Both waveguides exhibit periodic energy exchange with overall gain, and the power in each channel remains of the same order. In the PT-symmetric gain–lossless case, with $n(x)=n^*(-x),$4, $n(x)=n^*(-x),$5, and $n(x)=n^*(-x),$6, one again obtains oscillatory energy exchange, now with net gain contributed by the gain waveguide.

In the PT-broken gain–gain case, taking $n(x)=n^*(-x),$7, $n(x)=n^*(-x),$8, and $n(x)=n^*(-x),$9 gives $a_1(\xi)$0. Both modes increase exponentially because both waveguides are amplifying, but most of the energy concentrates in the waveguide with the higher gain. The intensity in the upper guide becomes about three times that in the lower over a certain distance, regardless of which port was initially excited. In the PT-broken gain–lossless case, with $a_1(\xi)$1, $a_1(\xi)$2, and $a_1(\xi)$3, both channels’ intensities increase exponentially when the gain channel is excited, while excitation from the lossless channel produces pronounced energy exchange at first and then exponential growth dominated by the gain waveguide (Zhang et al., 2022).

These examples establish that PT-broken behavior in gain–gain and gain–lossless systems is not merely formal. In gain–gain systems, both physical waveguides are amplifying media, yet the asymmetry in gain still selects a dominant supermode. In gain–lossless systems, the neutral waveguide mainly participates in transitional energy exchange, after which the gain waveguide dominates the transport.

6. Photonic significance and relation to earlier PT optics

The PT-OVS construction functions as a design framework for beam splitters, optical routers, and related non-Hermitian photonic devices. In the PT-symmetric region, where

$a_1(\xi)$4

the transmission matrix supports oscillatory, beam-splitting behavior. By choosing the propagation length $a_1(\xi)$5 appropriately, the output splitting ratio can be tuned through the PT-symmetric transfer matrix. In the PT-broken region, where

$a_1(\xi)$6

the same formalism yields unidirectional, amplifying routing, especially in the strong-broken limit where the output is effectively locked to the gain channel. Tuning close to the exceptional-point lines

$a_1(\xi)$7

provides access to the transition region in which the coalescence of eigenvalues and eigenvectors produces sharp changes in propagation.

Within the historical development of PT optics, earlier studies focused mainly on gain–loss systems and on loss–loss or passive PT realizations. The broader significance of PT-OVS is that it gives a full-parameter-space PT-phase diagram including gain, lossless, and loss modes, and shows that PT-like phase transitions and broken-symmetry behaviors occur in gain–gain and gain–lossless systems as well, with the same exceptional-point condition. This suggests that PT symmetry in photonics is more naturally viewed as a phase-diagram property of coupled non-Hermitian modes than as a property tied exclusively to balanced gain–loss pairs (Zhang et al., 2022).