Partially-PT-symmetric optical potentials with all-real spectra and soliton families in multi-dimensions
aa r X i v : . [ n li n . PS ] D ec Partially- PT -symmetric optical potentials with all-real spectra and soliton families inmulti-dimensions Jianke Yang
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA
Multi-dimensional complex optical potentials with partial parity-time ( PT ) symmetry are pro-posed. The usual PT symmetry requires that the potential is invariant under complex conjugationand simultaneous reflection in all spatial directions. However, we show that if the potential is onlypartially PT -symmetric, i.e., it is invariant under complex conjugation and reflection in a single spatial direction, then it can also possess all-real spectra and continuous families of solitons. Theseresults are established analytically and corroborated numerically. In optics, light propagation is often modeled bySchr¨odinger-type equations [1]. If the medium containsgain and loss, the optical potential of the Schr¨odingerequation would be complex. A surprising finding inrecent years is that, if this complex potential satisfiesparity-time ( PT ) symmetry, then the linear spectrumcan still be all-real, thus admitting stationary light trans-mission [2–7]. Here PT symmetry means that the poten-tial is invariant under complex conjugation and simulta-neous reflection in all spatial directions. In one dimen-sion (1D), PT -symmetry condition is V ∗ ( x ) = V ( − x );in 2D, this condition is V ∗ ( x, y ) = V ( − x, − y ); and soon. Besides all-real spectra, PT -symmetric potentialshave been found to support continuous families of opti-cal solitons [8–12]. But if the complex potential is not PT -symmetric, then the linear spectrum is often non-real, and soliton families often do not exist [13]. Otherfindings on PT systems can be found in [14–29].In this Letter, we show that in multi-dimensions, if thecomplex potential is not PT -symmetric but is partially- PT -symmetric, then such potentials can still admit all-real spectra and continuous families of solitons. Here par-tial PT -symmetry means that the potential is invariantunder complex conjugation and reflection in a single spa-tial direction (rather than in all spatial directions simul-taneously). For example, in 2D, partially- PT -symmetricpotentials are such that either V ∗ ( x, y ) = V ( − x, y ) or V ∗ ( x, y ) = V ( x, − y ). Partially- PT -symmetric poten-tials constitute another large class of complex potentialswith all-real spectra and soliton families, and they mayfind interesting applications in optics. For simplicity, weconsider the 2D case throughout the Letter, but similarresults hold for three and higher dimensions too.The model for nonlinear propagation of light beams incomplex optical potentials is taken asiΨ z + ∇ Ψ + V ( x, y )Ψ + σ | Ψ | Ψ = 0 , (1)where z is the propagation direction, ( x, y ) is the trans-verse plane, ∇ = ∂ xx + ∂ yy , and σ = ± V ( x, y ) is assumedto possess the partial PT symmetry V ∗ ( x, y ) = V ( − x, y ) . (2)The real part of this potential is symmetric in x , and its imaginary part anti-symmetric in x . No symmetry isassumed in the y direction.First, we show that the spectrum of this partially- PT -symmetric potential can be all-real. Eigenvalues of thispotential are defined by the Schr¨odinger equation( ∇ + V ) ψ = λψ, (3)where λ is the eigenvalue and ψ the eigenfunction.We start by considering separable potentials, where V ( x, y ) = V ( x ) + V ( y ) . For these potentials, the partial PT symmetry condition(2) implies that V ∗ ( x ) = V ( − x ) , V ∗ ( y ) = V ( y ) . Thus the function V ( x ) is PT -symmetric and V ( y )strictly real. Eigenvalues of this separable potential are λ = Λ + Λ , and the corresponding eigenfunctions are ψ ( x, y ) =Ψ ( x )Ψ ( y ), where[ ∂ xx + V ( x )] Ψ ( x ) = Λ Ψ ( x ) , [ ∂ yy + V ( y )] Ψ ( y ) = Λ Ψ ( y ) . Since V ( x ) is PT -symmetric, its eigenvalues Λ can beall-real. Since V ( y ) is strictly real, its eigenvalues Λ are all-real as well. Thus eigenvalues λ of the separablepotential V ( x, y ) can be all-real.Next we consider separable potentials perturbed by lo-calized potentials, V ( x, y ) = V ( x, y ) + ǫV p ( x, y ) , (4)where V is separable, V p localized, ǫ a small real param-eter, and both V , V p satisfy the partial- PT -symmetrycondition (2). Since V p is localized, continuous eigenval-ues of the perturbed potential V are the same as those ofthe separable potential V and are thus all-real. We nowshow that discrete eigenvalues of V are also real.Suppose λ is a simple discrete real eigenvalue ofthe separable potential V . Since V is partially- PT -symmetric, the eigenfunction ψ of λ is partially- PT -symmetric as well, i.e., ψ ∗ ( x, y ) = ψ ( − x, y ). Underperturbation ǫV p , the perturbed eigenvalue and eigen-function can be expanded into the following perturbationseries, λ = λ + ǫλ + ǫ λ + . . . ,ψ = ψ + ǫψ + ǫ ψ + . . . . Substituting these expansions and the perturbed poten-tial (4) into Eq. (3), at O ( ǫ ) we get Lψ = ( λ − V p ) ψ , (5)where L ≡ ∇ + V − λ . Since λ is a simple eigenvalue,the kernel of the adjoint operator L ∗ then contains asingle eigenfunction ψ ∗ . Then in order for Eq. (5) to besolvable, the solvability condition is that its right handside be orthogonal to ψ ∗ , which yields λ = h ψ ∗ , V p ψ ih ψ ∗ , ψ i , (6)where the inner product is defined as h f, g i = Z ∞−∞ Z ∞−∞ f ∗ ( x, y ) g ( x, y ) dxdy. Since λ is simple, it is easy to show that h ψ ∗ , ψ i 6 = 0.A key consequence of partial PT symmetry is that, iffunctions f and g are both partially- PT -symmetric, thentheir inner product h f, g i is real, because h f, g i ∗ = h f ∗ , g ∗ i = h f ( − x, y ) , g ( − x, y ) i = h f, g i . Since ψ and V p are partially- PT -symmetric, the innerproducts in Eq. (6) then are real, thus λ is real.Pursuing this perturbation calculation to higher or-ders, we can show that λ n is real for all n ≥
1, thus theeigenvalue λ remains real under perturbations ǫV p .For general partially- PT -symmetric potentials, we usenumerical methods to establish that their spectra canbe all-real. To illustrate, we take the complex potential V ( x, y ) to be V ( x, y ) = 3 (cid:16) e − ( x − x ) − ( y − y ) + e − ( x + x ) − ( y − y ) (cid:17) +2 (cid:16) e − ( x − x ) − ( y + y ) + e − ( x + x ) − ( y + y ) (cid:17) + iβ h (cid:16) e − ( x − x ) − ( y − y ) − e − ( x + x ) − ( y − y ) (cid:17) + (cid:16) e − ( x − x ) − ( y + y ) − e − ( x + x ) − ( y + y ) (cid:17)i , (7)where we set x = y = 1 .
5, and β is a real constant.This potential is not PT -symmetric, but is partially- PT -symmetric with symmetry (2). For β = 0 .
1, thispotential is displayed in Fig. 1 (top row). It is seenthat Re( V ) is symmetric in x , Im( V ) anti-symmetric in x , and both Re( V ), Im( V ) are asymmetric in y . Thespectrum of this potential is plotted in Fig. 1(c). It xy (a) −6 0 6−606 xy (b) −6 0 6−606 −0.1 0 0.1 −2 0 2−101 Re( λ ) I m ( λ ) (c) β =0.1 −2 0 2−101 Re( λ ) I m ( λ ) (d) β =0.3 FIG. 1: (a,b) Real and imaginary parts of the partially- PT -symmetric potential (7) for β = 0 .
1; (c, d) spectrum of thispotential for β = 0 . is seen that this spectrum contains three discrete eigen-values and the continuous spectrum, which are all-real.Thus we have numerically established that partially- PT -symmetric potentials can have all-real spectra. For thesereal eigenvalues, their eigenfunctions respect the partial PT symmetry of the potential.For potential (7) with varying β , we have found thatits spectrum is all-real as long as | β | is below a thresh-old value of 0 . PT symmetry. This phase transition is illustrated in Fig.1(d), where the spectrum at β = 0 . PT -symmetricpotentials [2, 5, 6, 8]. We see that it arises in partially- PT -symmetric potentials too.Next we examine whether these partially- PT -symmetric potentials support continuous families of soli-tons. These solitons are special solutions of Eq. (1) inthe form of Ψ( x, y, t ) = ψ ( x, y ) e iµz , (8)where µ is a real propagation constant, and ψ ( x, y ) sat-isfies the equation ∇ ψ + V ( x, y ) ψ + σ | ψ | ψ = µψ (9)and vanishes when ( x, y ) goes to infinity. In 1D, non- PT -symmetric potentials cannot admit soliton families [13].However, in higher dimensions, we will show analyticallyand numerically that partially- PT -symmetric potentialsdo support continuous families of solitons.First, we show analytically that, from each real dis-crete eigenvalue of the partially- PT -symmetric potential,a continuous family of solitons bifurcates out under eachof the focusing and defocusing nonlinearities. Suppose µ is a discrete simple real eigenvalue of the potential and ψ is its eigenfunction, i.e., Lψ = 0, where L ≡ ∇ + V − µ .Then we seek solitons with the following perturbation ex-pansion ψ ( x, y ; µ ) = ǫ / (cid:2) c ψ + ǫψ + ǫ ψ + . . . (cid:3) , where ǫ ≡ | µ − µ | ≪
1, and c is a certain non-zeroconstant. Substituting this expansion into Eq. (9), the O ( ǫ / ) equation is automatically satisfied. At O ( ǫ / ),we get the equation for ψ as Lψ = c (cid:0) ρψ − σ | c | | ψ | ψ (cid:1) , where ρ = sgn( µ − µ ). The solvability condition of this ψ equation is that its right hand side be orthogonal tothe adjoint homogeneous solution ψ ∗ . This conditionyields an equation for c as | c | = ρ h ψ ∗ , ψ i σ h ψ ∗ , | ψ | ψ i . (10)For the real eigenvalue µ , its eigenfunction ψ pos-sesses partial PT symmetry. Thus the two inner productsin the above equation are both real. Then for a certainsign of ρ , i.e., when µ is on a certain side of µ , the rightside of Eq. (10) is positive, hence this equation is solv-able for the constant c . Since the soliton in Eq. (9)is phase-invariant, we can take c to be positive withoutany loss of generality.Pursuing this perturbation calculation to higher or-ders, we can find that this perturbation solution can beconstructed to all orders for any small ǫ , thus a con-tinuous family of solitons bifurcates out from the lineareigenmode ( µ , ψ ). In this construction process, par-tial PT symmetry of the potential is critical. For in-stance, in the absence of this partial PT symmetry (and PT symmetry), it is generally impossible to guarantee thereality of inner products in Eq. (10), which makes thisequation unsolvable for c .Next we corroborate these analytical results numeri-cally. The partial- PT potential (7) with β = 0 . σ = 1). Then power curves of soliton familiesbifurcated from the first and second eigenmodes of thepotential are displayed in Fig. 2. Here the power P isdefined as R R | ψ | dxdy . Interestingly, these two powercurves are connected through a fold bifurcation, mean-ing that solitons bifurcated from these two eigenmodesbelong to the same solution family, and the power of thissolution family has an upper bound.Profiles of solitons on this power curve are also dis-played in Fig. 2. Here the amplitude fields of solitons atpoints ‘b,c’ of the power curve (with µ = 1 .
3) are plot-ted on the right column of the figure. It is seen that the µ P a bcd xy amplitude at “b” −6 0 6−606 xy phase at “b” −6 0 6−606 −101 xy amplitude at “c” −6 0 6−606 FIG. 2: Upper left: power diagram of the soliton family inpotential (7) with β = 0 . σ = 1 (blue segments are stableand red unstable); upper and lower right: amplitude fields ofsolitons ( | ψ | ) at points ‘b, c’ of the power curve; lower left:phase field of the soliton at point ‘b’. soliton at point ‘b’ has higher amplitude, obviously be-cause it is on the upper power branch. The phase fieldsof these two solitons are similar, thus only the phase fieldat point ‘b’ is shown. Note that these solitons share thesame partial PT symmetry of the complex potential (7).Lastly, we examine linear stability of this soliton fam-ily. For this purpose, we perturb these solitons by normalmodesΨ( x, y, z ) = e iµz h ψ ( x, y ) + f ( x, y ) e λz + g ∗ ( x, y ) e λ ∗ z i , where f, g ≪