arXiv:1802.09855v2 [quant-ph] 3 Jul 2018 colo hsc n srnm,Cri nvriy Cardiff University, Cardiff Astronomy, and Physics wells of School quantum Muljarov triple A. E. and and Tanimu A. double in states Resonant h S ssa ai h S fasml ytm usu- the system, applying simple of a advantage of RSs The the solvable. basis analytically a ally [9,10]. as sys- (RSE) uses expansion RSE modified has The resonant-state power- [8] a called a physics into method of nuclear electromagnetics ful basis in RSs in developed a suggested find recently as first been to idea, used order This be the tem. in inside can revealed functions expansion, therefore of [6,7] for and set RSs complete system, of a quantum studies form can Early (such they well). system that the quantum of de- a out have states leaking as RSs time, the in that general, exponentially showing In cay eigenvalues, RSs. energy define complex bound- strictly these conditions However, system. ary the bound- outside wave solv- conditions outgoing to ary with [5]). equivalent Schr¨odinger e.g. equation actually the (see, is ma- ing textbooks singularities scattering many in the these described of Finding also singularities as the [4] study trix to [3]. is investigated nances be to of still are aspects systems fundamental quantum many in grow- resonances, RSs this in of spite in nanos- interest In formed semiconductor ing potentials. are quantum of of states field types electronic various the different due in where mechanics progress tructures, quantum rapid in a particular, interest to in years, significant recent gravity. attracted in and have physics phenomena quantum mechan- Resonant to classical electrodynamics from and ranging ics physics, in of math- resonances fields with a dealing different quan- are in which People describe, exhibit. resonances systems They tum natural works [2]. way, pioneering Siegert rigorous the and ematically since [1] century, me- Gamow quantum a of in almost known for been chanics have (RSs) states Resonant Introduction 1 wl eisre yteeditor) the by inserted be (will No. manuscript EPJ ehp,amr rdtoa a fdaigwt reso- with dealing of way traditional more a Perhaps, PACS. the study, revisited. this also of is part as potentials analyzed therefore, subgro is functions; transmission, different delta quantum Dirac between the transitions as such their observables, and res bo states wave normal onant outgoing and generalized anti-bound with bound, Schr¨odinger’s equation includes This vestigated. Abstract. 2018 4, July version: Revised / date Received: wells 36.eSltoso aeeutos on tts–03.65. – states bound equations: wave of Solutions 03.65.Ge h ulsto eoatsae ndul n rpeqatmw quantum triple and double in states resonant of set full The F43A ntdKingdom United 3AA, CF24 motn smttc loigepii nltcexpres- some analytic and explicit solutions allowing exact asymptotics out quan- working important triple first and systems, We double well in [13]. tum problem as transmission state such and bound system, states the revisit quantum of a density determin- of local and the properties to- functions spectral of all the set RSs, ing complete normal a and forming anti-bound gether of bound, spectrum includes and full RSs double The systems. for well/barrier Schr¨odinger equation quantum 1D triple the of a solutions is potentials. work such present in The RSs triple of barriers. and study double or thorough 1D investigated wells as been systems quantum not simple Dirac such have in knowledge, even however, our far These, so of RSs. the best the of the by resonances to described potential, internal are the the which across by system, determined particles mainly of are trans- scattering quantum as and and such mission barriers phenomena, through Rather, tunneling states. are lattice. quantum bound systems crystal to quantum 1D limited of a not properties of spectral structure model- the [12] band However, solu- potential electronic periodic Kronig-Penney the the famous ing as the potentials well as such of [11], in tions literature states the Bound known sequence functions. are a delta multiple-quantum RSE with Dirac we the a it of work, approximating in of potential, this use simplification well/barrier further well-known In its a potentials. for take complicated generating RSs for more un- of as treating better basis well for as analytic wells, properties, an quantum their triple of and derstanding systems, double quantum-mechanical as such (1D) one-dimensional ple per- of analytically. case treated in be obvious cannot becomes which systems turbations various to RSE elkonpolmo on ttsi delta-like in states bound of problem known well nti ok eivsiaetefl pcrmo eigen- of spectrum full the investigate we work, this In sim- of RSs the study to is paper this of aim The naycniin.Tetasomto fres- of transformation The conditions. undary h unu elptnil r oee by modeled are potentials well quantum The . p swl sterl fec ugopin subgroup each of role the as well as ups nn ttswihaealegnouin of eigensolutions all are which states onant zOe ytm 32.gQuantum 73.21.Fg – systems Open Yz l/are tutrsi in- is structures ell/barrier 2 A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells sions. Then we demonstrate how the bound states appear wave outside the system, even though their energies are or disappear in the spectrum transforming into anti-bound real and negative. All other RSs have pn 6= 0 and κn < 0 states as the parameters of the potential change. Then we which results in complex eigen energies and wave func- extend our consideration to the full spectrum of RSs and tions which oscillate and grow exponentially in the exte- (ipn−κn)|x| discuss the physical meaning of the normal RSs, also pay- rior: ψn(x) ∝ e → ∞, according to Eq. (4). ing attention to the their evolution and transformation As a consequence of this exponential growth, the wave into/from bound and anti-bound states [4,14,15,16]. Fi- functions of the RSs are not orthogonal and not normaliz- nally, we investigate the role of the RSs in the quantum able in the usual way. RSs instead require a proper general transmission. orthonormality condition which would include the stan- dard one as a special case, valid for bound states. For a one-dimensional system, this general orthonormality of 2 Resonant states of one-dimensional RSs is given [2,6,9] by quantum systems xR δnm = ψn(x)ψm(x)dx Z In general, RSs of a quantum-mechanical system are eigen xL ψ (x )ψ (x )+ ψ (x )ψ (x ) solutions of the Schr¨odinger equation − n L m L n R m R , (6) i(kn + km) Hˆ (r)ψn(r)= Enψn(r) , (1) where δnm is the Kronecker delta, and xL and xR are satisfying the outgoing wave boundary conditions (BCs). two arbitrary points outside the system, one to the left of it (xL) and one to the right (xR). For bound states, Here Hˆ (r) is the Hamiltonian of a single particle, ψn(r) it can be easily seen, by taking the limits xR,L → ±∞ and En are, respectively, its eigen wave function and eigen energy, and r is a three-dimensional coordinate. Having in and noting that the second term vanishes due to the van- ishing wave function, that Eq. (6) leads to the standard mind application to e.g. planar semiconductor heterostruc- ∞ tures, we reduce our consideration in this work to a non- orthonormality: δnm = −∞ ψn(x)ψm(x)dx. For exponen- relativistic 1D Schr¨odinger’s problem. For brevity of no- tially growing wave functionsR the divergence of the inte- tations, we make use of the units ~ = 1 and m = 1/2, gral at |xR,L| → ∞ is exactly compensated by the second where m is the particle mass (e.g. the electron effective term. Furthermore, as the normalization does not depend mass in a semiconductor). It is also convenient to intro- on xL and xR, it is usually convenient to take these points duce the eigen wave number kn of the particle associated exactly at the boundaries of the system. with a given RS and use it instead of the energy En which is linked to it via the non-relativistic parabolic dispersion relation 3 Double well 2 En = k . (2) n We model a symmetric double quantum well structure by A 1D time-independent Schr¨odinger equation then takes a superposition of two Dirac delta functions, the form: V (x)= −γδ(x − a) − γδ(x + a) , (7) 2 d 2 − + V (x) ψn(x)= k ψn(x) , (3) dx2 n where 2a is the distance between the wells and γ is the strength of the potential which has the meaning of the where V (x) is the potential of the particle, which is chosen depth of each quantum well multiplied by its width. Fig- in such a way that it vanishes outside the system. ure 1 sketches this potential along with a realistic coupled In 1D, the outgoing wave BCs for RSs reduce to quantum well structure it models. While potentials mod- eled by delta functions sometimes fail to catch interesting ikn|x| ψn(x) ∝ e for |x| → ∞ , (4) physical phenomena, such as the band crossing [17], an ob- vious advantage of this model is its simplicity and explicit which are also known as Siegert BCs [2]. Solving Eq. (3) analytical solvability. The solution for this potential, in with the BCs Eq. (4) inevitably leads to the fact that the terms of bound states, has been covered in depth in many energies En are generally complex, texbooks, see e.g. [11]. The first few resonant states in dou- ble barrier structures (γ < 0) were found in [3]. We revisit κ 2 2 κ2 κ En = (pn + i n) = (pn − n)+2ipn n , (5) this problem again, in order to study the full spectrum of RSs and their properties, which has not been done in the where pn and κn are, respectively, the real and the imag- κ literature. This is also of practical importance, as the full inary part of the eigen wave number: kn = pn + i n. For set of RSs can be further used as a basis for the RSE. bound states pn = 0 and κn > 0, so that the energy is κ2 real negative En = − n < 0, and the general Eq.(4) re- duces to the standard BC of the wave function vanishing 3.1 Exact solution −κn|x| away from the system: ψn(x) ∝ e → 0 at |x| → ∞. For anti-bound states [15], if they exist in the spectrum, A general solution of Eq.(3) with the potential Eq.(7) has pn = 0 and κn < 0, corresponding to a purely growing the form (for brevity of notations, we drop in this and the A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells 3
Expressing the ratio A/C from Eqs.(11) and (12) and combining the results obtain A ik(eika ∓ e−ika) eika ± e−ika = = , (15) C (ik + γ)eika eika After rearrangement this yields a transcendental secular equation 2ik 1+ = ∓e2ika (16) γ
determining all the RS eigenvalues kn. Note that the upper (lower) sign corresponds to even (odd) RSs.
Fig. 1. A sketch of the potential of a symmetric double well structure (red line) approximated by two delta-functions (grey 3.2 Bound and anti-bound states arrows). To find bound and anti-bound states of the system, we make a substitution k = iκ in Eq. (16) and solve the latter following section the index n labeling RSs): for real values of κ. Then the eigen energy E = −κ2 takes real negative values. For bound states, κ should be Aeikx + Be−ikx x > a, positive, as required by the evanescent form of the wave ψ(x)= C eikx + C e−ikx |x| 6 a, (8) function outside the system. For anti-bound states instead 1 2 Deikx + Fe−ikx x< −a, the wave function has a pure exponential growth to the exterior which is provided by κ < 0. with constant coefficients standing at the exponentials. While the secular equation Eq. (16) apparently depends Applying the outgoing wave BCs leads to B = D = 0. on two parameters, γ and a, this parametric space reduces Furthermore, using the mirror symmetry of the potential, to a single parameter V (−x) = V (x), splits all the solutions into two groups: α = γa (17) even and odd states, having the property which can be treated as the effective system size or the ψ(−x)= ±ψ(x) . (9) effective strengths of the potential. Concentrating on the dependence of the eigen states on the system size (e.g. From this we obtain F = ±A and C1 = ±C2 = C. Then keeping the strength γ fixed), it is convenient to introduce the wave function takes the form a dimensionless wave number q = 2κ/γ. Then Eq. (16)
ikx takes the form Ae x > a, −q±α q± =1 ± e , (18) ψ(x)= C eikx ± e−ikx |x| 6 a, (10) −ikx where index + (−) labels even (odd) parity states. The ±Ae x< −a. full solution of Eq. (18) found numerically with the help of the Newton-Raphson method implemented in MATLAB The wave function ψ(x) must be continuous at any point is shown in Fig. 2 for positive values of α and q. It demon- ′ but its derivative ψ (x) is discontinuous at x = ±a. The strates the dependence of the imaginary wave vector for break in the derivative can be evaluated by integrating two bound (even and odd) states in the system as function Eq. (3) across the delta-function potential wells. This yields of the effective width α. four boundary conditions determining the relation between At large distances between the wells (α ≫ 1) the two the coefficients A and C, as well as the eigenvalues k. How- states are quasi-degenerate, ever, as the symmetry of the potential has been already −α taken into account leading to Eq. (10), only one pair of q± ≈ 1 ± e , (19) BCs (e.g. at x = a) provides a unique information: illustrating the fact that each isolated delta-like quantum ′ ′ ψ (a +0+) − ψ (a − 0+)= −γψ(a) , (11) well accommodates only one bound state with q = 1. As clear from Fig. 2, Eq. (19) is a good approximation of the ψ(a +0 ) − ψ(a − 0 )=0 , (12) + + full solution Eq. (18) for α > 3. As α increases, the split- ting between the levels becomes exponentially small, re- where 0+ is a positive infinitesimal. The other pair of BCs (at x = −a) is then fulfilled automatically. Substituting flecting the vanishing tunnel coupling between the wells. In the opposite limit of small width a or small wave the wave function Eq. (10) into the BCs Eqs. (11) and (12), κ obtain number (i.e. small binding energy), one can obtain a simple analytic approximation, based on the Taylor ex- ikAeika − ikC(eika ∓ e−ika)= −γAeika , (13) pansion of the exponential in Eq. (18), Aeika − C(eika ± e−ika)=0 , (14) e−qα ≈ 1 − qα + q2α2/2 , (20) 4 A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells
2.0 1.5 barriers wells ground state exact 1.0 (even) Eq. (19) s q 1.5 n Eq. (21) RS, Im(s) eve 0.5 d bound od
1.0 0.0 RS, Re(s) antibound -0.5 0.5 excited state (odd) effective wave number number wave effective effective wave number effective -1.0
0.0 -1.5 0 1 2 3 4 5 -2 -1 0 1 2 3 4 effective system width α effective potential strength α Fig. 2. Exact (black solid lines) and approximate (blue and red Fig. 3. Effective wave number s = −2ika as function of the dashed and dotted lines) effective wave numbers q = −2ik/γ = effective potential strength α = γa plotted for even and odd 2κ/γ of the two bound states in a double quantum well system eigen states using the explicit functional dependence Eq. (23). modeled by a double delta function potential, as functions of The plot demonstrates transitions from bound to anti-bound the effective width of the system α = γa. states and then to normal RSs. Regions for potential wells (α > 0) and barriers (α < 0) are indicated, as well as for bound (s > 0) and anti-bound (s < 0) states. Blue and green valid for |qα| = |κa| ≪ 1. For the even parity state, it lines show the real and imaginary parts of the wave numbers is sufficient to use the expansion Eq.(20) up to 1st order, s for the pair of the lowest energy normal RSs formed from a while the same level of approximation for the odd parity pair of degenerate anti-bound states at α ≈−0.27. state requires also the 2nd order to be taken into account. Then approximate solutions of Eq. (18) take the form: dependence α(s) given by Eq. (23). Its plot is presented in 2 α − 1 Fig. 3. Since a> 0, the region of positive s corresponds to q+ ≈ (even), q− ≈ 2 (odd). (21) α +1 α2 bound states. We see two bound states for α> 1 and only They are shown in Fig. 2 by dashed lines matching the one for 0 <α< 1. The odd state transforms at α = 1 exact solution (solid lines) at small α (for the even state) from bound to anti-bound, as negative s corresponds to or at small q (for the odd state). growing exponentials outside the systems, see Eq.(10) for The analytic approximation Eq. (21) also allows us to k = is/2a and s < 0. Another anti-bound state forms find a condition for bound states to exist in the system, from the even bound state at α = 0 when the wells switch which requires that q > 0. Indeed, when a bound state into barriers. At the same time, as α changes its sign from disappears from the spectrum, its binding energy van- positive to negative, the odd anti-bound state goes away to ishes, meaning that q → 0. This makes the approximation infinity as s → −∞ and then comes back at small negative Eq. (21) valid, so that it precisely determines the critical values of α as an even anti-bound state, coexisting with values of the system parameters when the bound state dis- the other even anti-bound state up to α ≈−0.27. At that point the two anti-bound states merge, now transforming appears. While the ground state exists for any α> 0 (q+ is always positive), the excited (odd) bound state exists into a pair of normal RSs, which then evolve as α decreases only for further. Both RSs have the same imaginary part and the α> 1 (22) opposite real part of k, shown in Fig. 3 by blue and greens lines, respectively. and disappears at α = 1 (when q− is vanishing), as the width of the system becomes insufficient to accommodate it, given the tunnel coupling between the wells. However, 3.3 Resonant states a quantum state itself cannot disappear from the system completely. Instead, it transforms into an anti-bound state We now consider all possible solutions of Eq. (16) in the which can be observed for α< 1. complex k-plane, generating bound, anti-bound and nor- To see it more clearly and also to investigate the de- mal RSs, as shown in Fig. 4 for the case of a double well pendence on the potential strength (e.g. keeping the width and a double barrier structure. For the wells (γa = 3), one 2a fixed), we introduce another dimensionless imaginary can see two bound states and an infinite countable number wave number s =2κa, so that Eq. (16) takes the form of normal RSs with nonzero real and imaginary parts of s k. Furthermore, these normal RSs all have complex wave α = . (23) 1 ± e−s functions which cannot be made real by redefining the nor- malization constant, unlike bound or anti-bound states. The function s(α) can be easily plotted without solving These RSs appear in pairs: Each RS with the eigen wave the transcendental equation, due to the explicit functional number k and the wave function ψ has a counterpart with A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells 5
2 Double barrier: Double well: even even 1 odd odd )
0 ka Im(
-1
-2 -30 -20 -10 0 10 20 30 Re ( ka ) Fig. 5. As Fig. 1 but for a triple well potential.
Fig. 4. Complex eigen wave numbers kn of the RSs in a double delta-potential structure, with two wells (γa = 3, shiny balls) the help of the secular equation Eq. (16), we find the nor- and two barrier (γa = −3, open circles). Even and odd parity malization constants in Eq. (10): states are shown, respectively, in black and red. γ −1 1 A = C 1+ , C = . 2ik 2 ±[a − (γ +2ik)−1] the eigen wave number −k∗ and the wave function ψ∗, p (24) so that the spectra of RSs shown in Fig.4 possess a mir- The normalized wave functions of a double well or a dou- ror symmetry about the imaginary axis, which is a general ble barrier system are now ready to use in the RSE which property of an open system, not related to its spatial sym- can be applied for various perturbations. This is however metry. For the barriers (γa = −3), there are only normal outside the scope of the present paper and will be pre- RSs seen in the spectrum, as this potential strength is sented elsewhere. too big for anti-bound states to exist, see Fig.3. In both spectra, normal RSs of even and odd parity appear in al- ternating order and are almost equally spaced for large k 4 Triple well dominated by the real part. The reason for that is that these states have the same nature as Fabry-P´erot modes We now add a third well (barrier) positioned at x = b, in an optical system, with a half integer multiple of their somewhere between the two equal wells (barriers): −a < De-Broglie wavelength λ =2π/k approximately matching b
2.5 for the region inside it. Equating coefficients at the same exact exponentials in Eq. (27), obtain D2 = ±C1 = ±C and Eq. (38) C = ±D = ±D, where we have introduced constants C 2.0 ground state 2 1 q Eq. (21) and D for brevity of notations. Then the wave function Eq. (41) takes a simplified form: 1.5 Eq. (42) Eq. (19) Aeikx x > a, ikx −ikx 6 6 1.0 Ce ± De 0 x a, ε = 0.5 ψ(x)= ikx −ikx (28) De ± Ce −a 6 x< 0, −ikx 0.5
±Ae x< −a. number wave effective odd excited even excited The existence of the third delta function in the potential 0.0 Eq. (25) leads to a new break in the derivative of the wave 0 1 2 3 4 5 6 7 8 9 10 function at x = 0, and to two more BCs: effective distance α
ψ′(ε) − ψ′(−ε)= −βψ(0) , (29) Fig. 6. As Fig. 2 but for a triple quantum well system modeled by a triple-delta potential with b = 0 and ε = 1/2. ψ(+ε) − ψ(−ε)=0 , (30) in addition to the pair of BCs given by Eqs. (11) and (12). 4.2 Bound and anti-bound states Using Eq. (30) for an odd parity state [the lower sign in Eq. (28)] results in a condition C = D meaning that Repeating the procedure used in Sec. 3.2, we first intro- ψ(0) = 0, as should be for any anti-symmetric state. This duce a purely imaginary wave number k = iκ, expressed makes however the odd state insensitive to the potential in terms of a real valued κ and then use an effective di- well or barrier if the latter is placed exactly in the center of mensionless wave number q = 2κ/γ, in order to study the system, thus keeping ψ′(x) continuous at x = 0. The the dependence of the bound state on the system size. odd parity solution of the Schr¨odinger equation with the In addition to the effective width/strength α defined by potential Eq. (25) and b = 0 is thus the same as for the Eq. (17), we introduce a relative strength of the middle double delta potential Eq. (7) and is described in detail well/barrier: in Sec.3. We therefore concentrate below on even parity β ε = . (36) states. γ For even parity states, Eq. (30) is automatically ful- filled due to Eq. (9), but Eq. (29) brings in a unique in- Equation (35) then takes the form formation about the middle well/barrier: 2ik(C − D) = q + ε −β(C + D), or q =1+ e−qα (37) q − ε D 1+2ik/β σ = = − . (31) [compare with Eq. (18) for +]. The exact numerical solu- C 1 − 2ik/β tion of Eq.(37) for ε =0.5 is shown in Fig. 6 by black solid lines displaying two even parity bound states, as well as At the same time, Eqs. (11) and (12) now give the odd parity state which is the same as in Fig. 2. While the ground state having the highest value of q exists for ika ika −ika ika ikAe − ik(Ce − De )= −γAe , (32) any size of the system (i.e. for all α> 0), the 2nd excited Aeika − (Ceika + De−ika)=0 , (33) (even) state disappears in this case at α = 5. To understand this behavior, we again use the Tay- which result, after having combined them with Eq. (31), lor expansion Eq. (20) up to 2nd order, obtaining from in two different expressions for the ratio A/C, Eq. (37) an approximation for even states:
A ik(eika − σe−ika) eika + σe−ika 2 − ε(α − 1) = = , (34) q ≈ . (38) C (ik + γ)eika eika 1+ α(1 − αε/2) determining the secular transcendental equation for even- Obviously, for ε = 0, Eq. (38) is equivalent to the ap- parity states: proximation Eq. (21) for the even parity values q+. The approximation Eq. (38) is shown for ε = 0.5 in Fig. 6 by 2ik 1 − 2ik/β dashed blue lines, demonstrating a good agreement with 1+ = e2ika . (35) the full solution for α → 0 (ground state) and for q → 0 γ 1+2ik/β (2nd excited states). The last limit allows us to obtain the following inequality for ε and α: In the limit β → 0, Eq.(35) reduces back to the secular equation (16) for the even-parity (ground) state of the 2 double quantum well. α> 1+ , (39) ε A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells 7
1.0 15 barriers wells ε = 8 ε exact 10 = 32 ground state Eq. (38) ε = 2 s q Eq. (21) 5 ε Eq. (41) = 0.5 Eq. (19) bound
0.5 0 antibound ε = 0.5 -5
ε = -4 ε effective wave number number wave effective excited state wave number effective -10 = 2 ε = 8 ε = 32 0.0 -15 0 1 2 3 4 5 -1 0 1 2 3 4 5 6 effective distance α effective potential strength α Fig. 7. As Fig. 6 but for ε = −4. Fig. 8. As Fig. 3 but for a triple-delta potential with b = 0 and different ε as labeled. All curves are obtained by plotting the inverse of the function Eq. (44). Note that only even parity showing under which conditions an even bound state ex- solutions are displayed, with an excited state branch shown by ists. When both γ > 0 and β > 0, this inequality refers a dashed curve for each ε. to the 2nd excited state in a triple well. In particular, for the example in Fig. 6, the even excited state disappears at α =1+2/ε = 5. which has an explicit solution for α(s): If, however, there is a barrier in the middle of two 2s wells, i.e. γ > 0 but β < 0, there is a maximum of two α = , bound states in the spectrum, one even (the ground state) 1+ ε + e−s ± (1 − ε)2 +2(1+3ε)e−s + e−2s and one odd (the excited state), and the same Eq. (39) p (44) now becomes a condition for the ground state to exist. [compare with Eq. (23)]. Again, the advantage of repre- Indeed, if the barrier is high enough, namely if β < −2γ, senting the solution in the form of Eq. (44) is that it can the ground state also disappears from the spectrum at be displayed without solving the secular equation. The the system size smaller than that determined by Eq. (39). plots of it are presented in Fig. 8, showing the evolution of This case presents an interesting situation when a one- bound and anti-bound states with the effective potential dimensional symmetric potential well structure cannot ac- strength α. We see that as α decreases the bound states commodate any bound states. An illustration for ε = −4 transform into anti-bound states and then to normal RSs is provided in Fig. 7 showing that the ground state disap- (not shown in Fig. 8), as in the case of a double barrier, pears at α = 1/2, in agreement with Eq.(39) and Fig.9 see Fig. 3. below. Finally, by fixing α (i.e. the product of the potential To analyze the behavior at large system sizes, we take strength γ and the width a) the dependence q(ε) or s(ε) the limit α → ∞, which makes the exponential term in on the relative potential strength ε, given by Eq. (36), can Eq. (37) small. This results in a quadratic equation for q: be extracted. Expressing ε from Eq. (37) obtain
q2 − q(1 + ε + e−qα)+ ε(1 − e−qα)=0 (40) 1 − q + e−qα s 1 − s/α + e−s ε = q = . (45) 1 − q − e−qα α 1 − s/α − e−s giving solutions Taking the inverse of this function, we find the dependence 1+ ε −α q0 ≈ 1+ e (41) s(ε) [or q(ε)] which is displayed in Fig. 9, showing the evo- 1 − ε lution of states with the the potential ratio ε continuously for the ground and changing between positive and negative values, thus cov- ering also an important case of mixed potentials (with a ε −εα q2 ≈ ε + e (42) barrier in the middle). Interestingly the two even states 1 − ε displayed show a sort of avoided crossing which is getting for the 2nd excited state. These approximate values are sharper with increased potential strength/width α, owing also plotted in Figs. 6 and 7 showing an agreement with to a smaller tunnel coupling between the wells. the full solution. To study the dependence on the quantum well strength γ (i.e. α for a fixed a), we introduce, as in Sec. 3.2, the 4.3 Solutions for an asymmetric case effective wave number s =2κa. Then Eq.(35) becomes Let us now consider a more general case of an arbitrary s s/α + ε position of the middle well/barrier at x = b, with −a < =1+ e−s (43) α s/α − ε b 15 barriers wells α = 2 bound 3 Double well: Triple well α 10 = 8 even b = 0, even s 2 odd b = 3 a/5 α = 0.5 5 ) 1 0 ka Im ( Im 0 -5 α = 0.5 effective wave number effective -1 -10 α = 8 α = 2 antibound -15 -2 -5 0 5 10 15 -30 -20 -10 0 10 20 30 relative potential strength ε Re ( ka ) Fig. 9. As Fig. 8 but for the relative potential strength depen- dence given by Eq. (45), for different values of α as labeled. Fig. 10. As Fig. 4, for double (γ = 3/a), triple symmetric between 6 amplitudes in the wave function Eq. (26), one (b = 0, γ = 3/a, β = 6/a) and triple asymmetric (b = 3a/5, γ = 3/a, β = 6/a) quantum well structures. needs to satisfy 3 pairs of BCs, describing the continuity of the wave function and discontinuity of its first derivative at x = −a, b and a. We skip details of this derivation, Finally, expressing η from Eq.(46), we find an ex- which can be made in a similar way to Secs.3.2 and 4.2. plicit dependence of the wave numbers k on the middle We present a resulting secular equation for k, which can well/barrier strength β: be written compactly as 1 − ξ2 ξ2(1 − η) − 2ξ cos(2kb)+1+ η =0 , (46) βa = s , (52) 1 − 2ξc + ξ2 after introducing where ξ = e−s/(1 − s/α), in accordance with Eq. (47). e2ika 2ik The function k(β) can be obtained by simply inverting ξ = and η = . (47) the function β(k) given by Eq. (52). For the symmetric 1+2ik/γ β quantum well structure (b = 0), using c = 1 in Eq.(52) leads to We again first study the dependence of the full solution 1+ ξ βa = s , (53) of Eq. (46) for bound states on the potential strength γ, 1 − ξ for fixed a, b and β. Introducing α = γa and s = −2ika, as before, and solving the quadratic equation (46) for ξ, which is exactly the same as Eq. (45). obtain two branches of the solution: s 4.4 Resonant states α±(s)= −s , (48) 1 − e /ξ±(s) RSs for both symmetric and asymmetric triple well struc- where tures are shown in Fig. 10. The spectrum of RSs for the c ± c2 + η2 − 1 ξ±(s)= , (49) symmetric structure is quite similar to that of the dou- p1 − η ble well which is also shown for comparison (the same as s demonstrated in Fig. 4). Note that odd RSs remain the c(s) = cosh(sb/a) and η(s)= − . (50) βa same for both systems. For the triple well, we now see two bound states, in accordance with our analysis in Sec. 4.2. It is instructive to see that for b = 0, the two branches Indeed, for α = 3 and ε = 2 the inequality Eq. (39) is become 1+ η fulfilled allowing the second excited state to exist. ξ− = 1 and ξ+ = (51) The spectrum of RSs for an asymmetric triple well 1 − η structure with b = 3a/5, γ = 3/a and β = 6/a is quite corresponding to the odd and even parity states and coin- different. First of all, being shifted from the center of the ciding with the solution for a symmetric triple well struc- structure, the middle quantum well mixes even and odd ture given by Eq. (16) with the lower sign used and by RSs. As a result, stronger deviations from the double well Eq. (35), respectively. Taking further the limit η → ∞ spectrum of RSs is seen. Choosing the ratio 2a/(a − b) (corresponding to β → 0), obtain solutions for a dou- equal to an integer, as in the present case, the third well in ble well structure: ξ∓ = ±1 which, after substitution into the middle splits the structure into two resonators having Eq. (49), give exactly Eq. (23). commensurable widths aL and aR. In our case, aL =8a/5 A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells 9 and aR =2a/5, thus splitting the full width of the system in 4:1 proportion. Therefore resonances accommodated in 2 the right (narrower) subsystem can be enhanced, owing 1.0 to an additional constructive interference of wave, by the left (wider) subsystem and the full-width structure. As a result, one can see a quasi-periodic modulation in the Bound only spectrum with the period of about π/aR, which is five + 1 RS times larger than the separation between the RS wave 0.5 + 3 RSs numbers, which is approximately π/(2a), see Sec. 3.3. 100 RSs Analytic 5 Role of the resonant states in the |t(k)| probability Transmission transmission 0.0 0 2 4 6 8 10 12 In this section, we study the role of RSs in observables, Wave vector k (1/ a) such as the local density of states or the scattering matrix. Fig. 11. The probability of transmission trough a double Below we consider, as an example, the transmission of a Dirac delta well structure with γ = 3/a, evaluated using the particle through a quantum system consisting of two Dirac analytic transmission ta(k) (thick red curve) and its spectral delta wells. A particle traveling in free space is described representation t(k) calculated for different number of RSs taken by a wave function in the form of plane wave with a wave into account in the Mittag-Leffler expansion: bound states only number k. We first calculate analytically its transmission (magenta), bound states and 1 pair of RSs (dark blue), bound states and 3 pairs of RSs (dark green), and 100 RSs in total, amplitude ta(k) as a function of the real wave number k of the particle. This transmission can be found by choosing including the bound states (black dashed curve). appropriate BCs outside the system, namely by allowing the system to be excited with an incoming wave. To do delta-function potential Eq. (7). Using the explicit form of so, we keep in Eq. (8) the term with an incoming wave the wave functions Eq. (10), their normalization Eq. (24), which now has a non-vanishing amplitude D 6= 0, while and the secular equation (16), one can write the transmis- requiring that B = 0. The BCs at x = ±a, given by sion, with the help of Eqs. (55) and (56), in the form of an Eqs. (11) and (12), are the same as for the RSs. Applying infinite series over its poles: them and solving a set of algebraic equations, we find the R transmission amplitude t(k)= ke−2ika n , (57) k − k Xn n A 4k2 ta(k)= = 2 4ika . (54) where D 4k(k − iγ) − γ (1 − e ) 2 ikna Rn = ± . (58) Now taking the analytic continuation of this function into γ[(γ +2ikn)a − 1] the complex k plane, it is easy to see that ta(k) has sim- This result can be compared with the analytic transmis- ple poles at k = kn, where kn are the wave numbers of sion ta(k), given by Eq. (54), which is done in Fig. 11. all possible RSs (including bound, anti-bound and normal Using the ML expansion Eqs. (57) and (58), we also RSs), which satisfy the secular Eq. (16). study in Fig. 11 the role of different RSs in the trans- For an arbitrary one-dimensional potential with com- mission. We first note that in this representation, bound pact support, i.e. vanishing (or constant) outside the sys- states play a small but non-negligible role, producing some tem area |x| 6 a, the transmission amplitude is given by background contribution. The maxima of the transmis- sion reaching the value of 1 for this symmetric quantum −2ika t(k)=2ike Gk(a, −a) , (55) structure can be described by only taking into account ′ in the summation Eq. (57) the corresponding normal RSs. see e.g. [18]. Here, Gk(x, x ) is the Green’s function of the Adding the very first pair of normal RSs already describes Schr¨odinger equation (3) for a given fixed wave number ′ quite well the first peak in the transmission. The agree- k. For the coordinates x and x within the system, the ment is further improved by adding more RSs. With three Green’s function is vanishing on an infinitely large circle pairs of RSs, the first peak of the transmission is fully re- in the complex k plane, and therefore, one can apply to it produced, but the other two are described only qualita- the Mittag-Leffler (ML) theorem which yields [6,9,13] tively. To correct this and to describe other peaks, more ′ RSs in Eq.(57) are needed. Taking all of them into ac- ′ ψn(x)ψn(x ) Gk(x, x )= , (56) count, the correct transmission is fully reproduced. 2k (k − k ) Xn n n where ψn(x) are the RS wave functions normalized ac- 6 Conclusion cording to Eq. (6). For illustration, we apply the general result given by We have studied the full set of resonant states of a one- Eqs. (55) and (56) to the particular case of the double dimensional Schr¨odinger problem with double and triple 10 A. Tanimu, E. A. Muljarov: Resonant states in double and triple quantum wells quantum wells or barriers approximated by Dirac delta functions. This full set includes bound, anti-bound and normal resonant states. We have revisited the problem of finding bound states in delta-well potentials and have worked out simple analytic expressions for important lim- iting cases and compared them with the full numerical solution. The latter is in turn presented here as universal dependencies containing the minimal number of parame- ters. 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