9 To analyze the reflection properties of the lattice shown in Fig.3, let us expand the state vector | ψ ( t ) 〉 as where | c$_{n}$ ( t ) | 2 is the occupation probability of site | n 〉 and | c$_{a}$ ( t ) | 2 the occupation probability of the boundary impurity site | a 〉 . From Eqs.(3) and (63), it follows that the occupation amplitudes c$_{n}$ and c$_{a}$ satisfy the following coupled equations Plane wave solutions to Eqs.(65-67) with wave number (momentum) k (0 ≤ k ≤ π ), corresponding to eigenstates of H with energy E ( k ) = − 2 κ$_{0}$ cos k , are of the form c$_{n}$ ( t ) = ¯ c$_{n}$ ( k ) exp( − iE ( k ) t ), c$_{a}$ ( t ) = ¯ c$_{a}$ ( k ) exp( − iE ( k ) t ), where FIG. 4: (color online) Probability of particle reflection R ( k ) for (a) an amplifying impurity site [Im( E$_{a}$ ) > 0], and (b) an absorbing impurity site [Im( E$_{a}$ ) < 0] for κ$_{a}$/κ 0 = 1 and Re( E$_{a}$ ) = 0. Curves 1, 2 and 3 correspond to | Im( E$_{a}$ ) | /κ$_{a}$ = 1 / 3, 2/3 and 1, respectively. According to Eqs.(54) and (55), a spectral singularity occurs at k = π/ 2 (i.e., E 0 = − 2 κ 0 cos k = 0) for | Im( E$_{a}$ ) | /κ$_{a}$ = 1 (curve 3) . 0 0.5 GLYPH<c112> GLYPH<c112> 10 0 10 2 10 4 10 6 10 8 R k reflectance ( ) 0 0.5 GLYPH<c112> GLYPH<c112> 10 -6 10 -4 10 -2 10 0 waveGLYPH<.notdef>number k (a) (b) 2 3 1 1 2 3 | ψ ( t ) 〉 = c$_{a}$ ( t ) | a 〉 + ∞ ∑ n =1 c$_{n}$ ( t ) | n 〉 (64) i dc$_{n}$ dt = − κ$_{0}$ ( c$_{n}$$_{+1}$ + c$_{n}$$_{−}$$_{1}$ ) n ≥ 2 (65) i dc$_{1}$ dt = − κ$_{0}$c$_{2}$ − κ$_{a}$c$_{a}$ (66) i dc$_{a}$ dt = − κ$_{a}$c$_{1}$ + E$_{a}$c$_{a}$. (67) ¯ c$_{n}$ ( k ) = exp[ − ik ( n − 1)] + r ( k ) exp[ ik ( n − 1)] (68) ¯ c$_{a}$ ( k ) = κ$_{a}$ [exp( ik ) + r exp( − ik )] (69) and r ( k ) = − κ 2 a − κ 2 $_{0}$(2 cos k + E$_{a}$/κ$_{0}$ ) exp( ik ) κ 2 a κ 2 $_{0}$(2 cos k + E$_{a}$/κ$_{0}$ ) exp( − ik ) (70) is the spectral reflection coefficient (see, for instance, [37]). Note that c$_{n}$ ( t ) is given by the superposition of two traveling waves, a regressive wave exp[ − ikn − iE ( k ) t ] that propagates along decreasing values of n , and a pro- gressive wave exp[ ikn − iE ( k ) t ] that propagates in the opposite direction, i.e. which is reflected from the lat- tice boundary [49]. The probability of reflection from the lattice boundary is given by FIG. 5: (color online) Same as Fig.4, but for for κ$_{a}$/κ 0 = √$_{2 and Re(}$$_{E}$$_{a}$$_{) = 0.}$ Curves 1, 2 and 3 correspond to | Im( E$_{a}$ ) | /κ$_{a}$ = 2 / 3, 4/3 and 2, respectively. According to Eq.(57), there are two spectral singularities, corresponding to the energies E 0 = ± [4 κ 2 0 − Im $^{2}$( E$_{a}$ )] 1 / $^{2}$, which coalesce at | Im( E$_{a}$ ) | /κ$_{a}$ = 2 (curve 3). R k reflectance ( ) waveGLYPH<.notdef>number k (b) (a) 0 0.5 GLYPH<c112> GLYPH<c112> 10 -10 10 -5 10 0 1 2 3 0 0.5 GLYPH<c112> GLYPH<c112> 10 0 10 5 10 10 15 3 2 1 R ( k ) = | r ( k ) | 2 = [( κ 2 a − 2 κ 2 $_{0}$) cos k − κ$_{0}$ Re( E$_{a}$ )] 2 + [ κ 2 $_{a}$sin k + κ$_{0}$ Im( E$_{a}$ )] 2 [( κ 2 a 2 κ 2 $_{0}$) cos k κ$_{0}$ Re( E$_{a}$ )] 2 + [ κ 2 $_{a}$sin k − κ$_{0}$ Im( E$_{a}$ )] 2 (71) sorbing potential, and R ( k ) ≥ 1 for Im( E$_{a}$ ) > 0, i.e. for an amplifying potential. In particular, for Im( E$_{a}$ ) > 0 the reflection probability R ( k ) goes to infinity at wave numbers k = k$_{0}$ (with 0 ≤ k$_{0}$ ≤ π ) that satisfy simulta- In the Hermitian limit (Im( E$_{a}$ ) = 0), one has R ( k ) = 1, i.e. the incident wave is completely reflected from the lat- tice boundary. This is merely a consequence of conserva- tion of the particle probability in the scattering process. Conversely, for a complex-valued energy E$_{a}$ of the impu- rity site, one has R ( k ) ≤ 1 for Im( E$_{a}$ ) < 0, i.e. for an ab-