PT-symmetry breaking and laser-absorber modes in optical scattering systems
Using a scattering matrix formalism, we derive the general scattering properties of optical structures that are symmetric under a combination of parity and time-reversal (). We demonstrate the existence of a transition beween -symmetric scattering eigenstates, which are norm-preserving, and symmetry-broken pairs of eigenstates exhibiting net amplification and loss. The system proposed by Longhi Longhi (), which can act simultaneously as a laser and coherent perfect absorber, occurs at discrete points in the broken symmetry phase, when a pole and zero of the S-matrix coincide.
pacs:42.25.Bs, 42.25.Hz, 42.55.Ah
Schrödinger equations that violate time-reversal symmetry due to a non-Hermitian potential, but retain combined (parity-time) symmetry, have been extensively studied since the work of Bender et. al. Bender (); Bender2 (), who showed that such systems can exhibit real energy eigenvalues, suggesting a possible generalization of quantum mechanics. Moreover, -symmetric systems can display a spontaneous breaking of -symmetry, at which the reality of the eigenvalues is lost Bender (); Makris (). Although -symmetric quantum mechanics remains speculative as a fundamental theory, the idea has been fruitfully extended to wave optics Makris (); Musslimani (); Guo (); Ruter (). The classical electrodynamics of a medium with loss or gain breaks symmetry in the mathematical sense, although the underlying quantum electrodynamics is of course -symmetric. symmetry is maintained in optical systems by means of balanced gain and loss regions that transform into one another under parity; thus the combined operation, which also interchanges loss and gain, leaves the system invariant.
We show in this Letter that the scattering behavior of a general -symmetric system can exhibit one or multiple spontaneous symmetry-breaking transitions. This result applies to arbitrarily complex -symmetric scattering geometries, whereas earlier works on optical symmetry breaking were inherently restricted to waveguide (paraxial) geometries Makris (); Musslimani (); Guo (); Ruter () for which resonances in the propagation direction play no role. In addition, we elucidate the properties of certain singular solutions occurring in such systems, recently studied for special cases by several authors Mostafazadeh (); Schomerus (); Longhi (), where a pole and a zero of the scattering matrix (S-matrix) coincide at a specific real frequency. A real-frequency pole corresponds to the threshold for laser action, while a real-frequency zero implies the reverse process to lasing, in which a particular incoming mode is perfectly absorbed. A device exhibiting the latter phenomenon, which does not require -symmetry, has been termed a “coherent perfect absorber” (CPA) CPA (). A -symmetric scatterer, at these singular points, can function simultaneously as a CPA and a laser at threshold, as noted by Longhi Longhi (). The present work establishes the CPA-laser points as special solutions in a wider “phase” of -broken scattering eigenstates. We identify signatures of both the -breaking transition and the CPA-lasing points, for several exemplary and experimentally-feasible geometries.
S-matrix properties – Consider an optical cavity coupled to a discrete set of scattering channels, denoted . Incoming fields enter via the input channels, interact in the cavity, and exit via the output channels. For simplicity, we focus on the scalar wave equation, which directly describes one-dimensional (1D) and two-dimensional (2D) systems. A steady-state scattering solution for the electric field, , obeys
For amplifying or dissipative media, is complex. The frequency is real for physical processes but can be usefully continued to the complex plane. Henceforth, we set . Outside the cavity, has the form
Here and denote the input and output channel modes, whose exact form depends on the scattering geometry (e.g. plane waves, spherical/cylindrical waves, or waveguide modes). The complex input and output amplitudes, and , are related by the S-matrix:
For all values of , is a (complex) symmetric matrix reciprocity (); if and are real, it is also unitary.
We now review the properties of for a complex, -symmetric system Schomerus (). (Henceforth, will refer to any linear symmetry operation such that , including not only parity, but also rotations and inversion.) First, observe that the -operator maps incoming channel modes to outgoing ones:
where is the anti-linear complex conjugation operator. The generalized parity operator is a linear operator acting on scalar fields satisfying
where the system-dependent matrix mixes the channel functions, but never transforms between incoming and outgoing channels. Note that .
If the system is -symmetric, for any solution (2) there exists a valid solution at frequency :
This is the fundamental relation obeyed by -symmetric S-matrices, and we will now show that it has important implications for the eigenvalue spectrum.
Multiplying both sides of (7) by an eigenvector of with eigenvalue gives
Hence, the inverse of the complex conjugate of any eigenvalue of is an eigenvalue of . For real , this implies that , just as for S-matrices having pure symmetry, which are unitary. Unitarity imposes a stronger constraint: each eigenvalue is unimodular, so unitary S-matrices do not have poles or zeros for real . Both and symmetry imply that poles and zeros occur in complex conjugate pairs.
Symmetry-breaking transition – Eq. (7) is a weaker constraint than unitarity, and can be satisfied in two ways: either each eigenvalue is itself unimodular, or the eigenvalues form pairs with reciprocal moduli. These two possibilities correspond to symmetric and symmetry-broken scattering behavior. In 1D, there are just two scattering eigenvectors, so the entire S-matrix is either in the symmetric or broken-symmetry “phase”. (In higher dimensions, as we will see, the transition can occur in different eigenspaces of the S-matrix, so that the system can be in a mixed phase; initially we focus on 1D.) Let us denote the eigenvectors by . In the symmetric phase, each is itself -symmetric, i.e. , so the eigenstate exhibits no net amplification nor dissipation (). In the broken-symmetry phase, is not itself -symmetric but the pair satisfies by transforming into each other: , where . Each eigenstate in the pair spontaneously breaks symmetry; one exhibits amplification, and the other dissipation. Similar to transitions in Hamiltonian systems, such as the transition from real to complex eigenvalues of -symmetric Hamiltonians, the scattering transition can be induced by tuning the parameters of Berry ().
Having shown that a -breaking transition can take place in the scattering behavior of -symmetric systems, we turn now to some concrete examples to see how it occurs. First, consider an arbitrary 1D system with -symmetric ; its S-matrix is parameterized by
where ; are the reflection amplitudes from right and left; and is the (direction-independent) transmission amplitude. In general , but their relative phase must be or . Although depends on three real numbers, , its eigenvalues only depend on two, and , for we can scale out the phase of . One can show that the criterion for the eigenvalues of to be unimodular is:
For fixed , we write as a function of and a -breaking parameter . On varying and/or , violating (10) brings us into the broken-symmetry phase.
Fig. 1 shows how the transition occurs by varying , in a simple slab of total length with fixed in each half. The critical frequency can be shown to be
The resulting “phase diagram” is shown in Fig. 2(a). The discrete points in the broken-symmetry phase correspond to the CPA-laser solutions which we will soon discuss; these lie along a line given by the equation Mostafazadeh ()
The strong oscillations of in Fig. 1 occur as the system crosses this curve. Without fine-tuning , the system will not hit one of the CPA-laser solutions exactly; hence, varying does not bring to , though it can become quite large.
With different , more complicated phase diagrams are possible. Fig. 2(b) shows the effect of a real-index region inserted between the gain and loss regions. As we increase , internal resonances in the real-index region cause the trapping time to oscillate with ; hence the effective -breaking perturbation strength oscillates, and the system re-enters the symmetric phase periodically.
In these 1D structures, the symmetry-breaking transition can be probed by measuring the total intensity scattered by equal-intensity input beams of variable relative phase . In the symmetric phase, two particular values of correspond to unimodular S-matrix eigenstates; for other values of , both net amplification and dissipation can be observed in the total output , since the system is not unitary. In the broken-symmetry phase, where , it can be shown that
where . Thus all values of for balanced inputs give rise to amplification (Fig. 1). This signature of the transition should be easily observable.
Fig. 3 shows the behavior of the poles and zeros in the complex plane for the simple 2-layer structure. When , the poles and zeros of the -symmetric S-matrix are located symmetrically around the real axis, and can be labeled by the parity of the corresponding eigenvectors. As increases from zero, symmetry requires that the zeros and poles move symmetrically in the complex plane. Neighboring zeros (poles) approach each other pairwise and undergo an anti-crossing, causing strong parity mixing. The horizontal motion of the poles and zeros corresponds to the symmetric phase of the scatterer, and the vertical motion to the broken-symmetry phase. The anti-crossing corresponds to the phase boundary, as can be seen by comparing the points labeled A, B and C in Figs. 2 and 3.
CPA-Laser – The CPA-laser points occur when a pole and a zero of the S-matrix coincide on the real axis, as shown in Fig. 3. This corresponds to the singular case in which , implying , while their product remains unity. Physically, the scattering system behaves simultaneously as a laser at threshold and a CPA Longhi (). Fig. 3 also indicates a very interesting property of these solutions: only half the zero-pole pairs flow upwards and reach the real axis, while the other half go off to infinity and do not produce laser-absorber modes; thus, the CPA-laser lines have twice the free spectral range of the passive cavity resonances. For , they occur at frequencies , where is an integer, exactly half-way between alternate pairs of passive cavity resonances. This property of the CPA-laser should be straightforward to demonstrate experimentally.
In a general -symmetric system, the poles and zeros are related by complex conjugation, so any system that is a laser must be a CPA-laser. Even away from the CPA-laser singularities, the -symmetric cavity in the broken symmetry phase is a unique interferometric amplifier: for coherent input radiation not corresponding closely to the damped S-matrix eigenvector, amplification takes place—in particular, this typically occurs for one-sided illumination. However, coherent illumination from both sides, with appropriate relative phase and amplitude corresponding to the damped eigenvector, leads instead to strong absorption. The CPA-laser is the extreme case, where the gain/loss contrast is infinite.
The possibility of lasing in a system is not obvious, as naively one might argue that photons traversing the device should experience no net gain. The presence of spontaneous symmetry breaking invalidates this argument. Photons in amplifying modes spend more time in the gain region than the loss region; hence, as the gain/loss parameter is increased, it eventually becomes possible for one mode to overcome the outcoupling loss and lase. As a self-organized oscillator, the CPA-laser will automatically emit in the amplifying eigenstate.
Disk -scatterer – Consider any 2D -symmetric body in free space. Scattering in this system is naturally described using angular momentum channel functions,
For a region of linear dimensions and average index , only states with are significantly scattered, so we can truncate the infinite S-matrix to channels. As this S-matrix is tuned, multiple pairs of eigenstates can undergo -breaking, so the phases of the S-matrix are indexed by the number of pairs of -broken eigenstates. In Fig. 4, we show this behavior for a disk with balanced semicircular gain and loss regions, with the S-matrix calculated using the R-matrix method CPA (); wigner (). The symmetry-breaking is non-monotonic in , similar to the 1D example of Fig.2(b). The regions of strong -breaking are roughly associated with the resonances of the -symmetric disk, as symmetry-breaking is enhanced by the dwell time in the medium. This points to the possibility of microdisk based amplifier-absorbers.
Conclusion – We have shown that -symmetric optical scattering systems generically display spontaneous symmetry breaking, with a unimodular phase where the S-matrix eigenstates are norm-preserving, and a broken symmetry phase in which they are pairwise amplifying and damping with reciprocal eigenvalue moduli. This -breaking transition can be tested experimentally, using 1D heterojunction geometries (with realistic values of the gain/loss parameter) that are distinctly different from the paraxial geometries previously suggested for observing symmetry breaking in optics Makris (); Musslimani (); Guo (); Ruter (). In our analysis, we have neglected the role of the noise due to amplified spontaneous emission, which may be significant at the singular CPA-laser points Schomerus (). However this noise should not preclude observation of experimental signatures of CPA-lasing, e.g. the doubling of the free spectral range relative to the passive cavity. Moreover one may study the inteferometric amplifying behavior in the broken symmetry phase well below the CPA-laser points, where the noise will be substantially smaller.
We thank S. Longhi and A. Cerjan for useful discussions. This work was partially supported by NSF Grant No. DMR-0908437, and by seed funding from the Yale NSF-MRSEC (DMR-0520495).
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