Wave dynamics on time-depending graph with Aharonov Bohm ring

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1 NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 218, 9 (4), P Wave dynamics on time-depending graph with Aharonov Bohm ring D. A. Eremin 1, E. N. Grishanov 1, D. S. Nikiforov 2, I. Y. Popov 2 1 Department of Mathematics and IT, Ogarev Mordovia State University Boshevistskaya Str. 68, Saransk, Russia 2 Department of Higher Mathematics, ITMO University, Kroverkskiy pr. 49, St. Petersburg, 19711, Russia ereminda@mai.ru, evgenyg@mai.ru, dmitrii.nikiforov@gmai.com, popov1955@gmai.com PACS w, 2.3.Tb, 2.1.Db, Nm DOI / Aharonov Bohm ring (AB ring) is an eement frequenty used in nanosystems. The paper deas with wave dynamics on quantum graph consisting of AB ring couped to a segment. It is assumed that the engths of the edges vary in time. Variabe repacement is made to come to the probem for stationary geometric graph. The obtained equation is soved using the expansion with respect to a compete system of eigenfunctions of the unperturbed sef-adjoint operator for the stationary graph. The coefficients of the expansion are found as soutions of a system of differentia equations numericay. The infuence of the magnetic fied is studied. The comparison with the case of stabe geometric graph is made. Keywords: quantum graph, Aharonov Bohm ring. Received: 15 June 218 Revised: 18 Juy Introduction Nanostructures in a magnetic fied attract great attention of physicists due to its interesting behavior. Many of these properties, e.g., giant magnetoresistance, have found wide range of appications in nanotechnoogy, computer hardware, etc. One of the basic eements for nanostructures in a magnetic fied is Aharonov Bohm ring [1], i.e. a nano-sized conducting ring in a magnetic fied orthogona to the ring pane. The Aharonov Bohm effect was observed and studied in many physica situations (see, e.g., [2 6]. For such structures, quantum graph mode is activey used [7 1]. It is a rather effective mathematica mode aowing one to describe the spectra and transport properties of many compex physica systems [11 15]. We consider a quantum graph with edge engths varying in time. Athough this mode is very interesting from a physica point of view (see, e.g., [16]), there are ony a few works devoted to this probem. One can mention papers concerning the time-dependent boundary conditions [17] or time dependent point-ike interactions [18], but consideration of time-dependent graphs began recenty [19 23]. Wave dynamics for time-dependent quantum graphs in a magnetic fied was not studied previousy. There are ony works concerning concerning Aharonov Bohm rings in fuctuating magnetic fied (see, e.g., [24]). In the present paper, we construct and study a mode of time-dependent quantum graph with oop in magnetic fied. We investigate the dependence of the dynamics on a magnetic fied. 2. Mode 2.1. Graph description We consider quantum graph Γ with a oop (ring) shown in Fig. 1. It is assumed that the magnetic fied acts at the ring. Lengths of graph edges, ring (L r (t)) and segment (L (t)), vary in time in accordance with the foowing reations: { Lr (t) = 2πrL(t), (1) L (t) = L(t), where r, are constants, L(t) is some twice continuousy differentiabe function. shoud be positive. Obviousy, L r (t) and L (t)

2 458 D. A. Eremin, E. N. Grishanov, D. S. Nikiforov, I. Y. Popov FIG. 1. Geometrica structure of the graph 2.2. Schrödinger operator on the graph We start with the probem corresponding to the absence of a magnetic fied. It means free Schrödinger operator acts on edges of the graph: i t Ψ r(x, t) = 2 2 2m x 2 Ψ r(x, t), x L r (t), i (2) t Ψ (x, t) = 2 2 2m x 2 Ψ (x, t), x L (t). We ook for continuous soution satisfying the Kirchhoff conditions at the centra vertex V and the Dirichet conditions at the boundary one V 1 : Ψ (, t) = Ψ r (, t) = Ψ r (L r (t), t), Ψ (L (t), t) =, x Ψ (, t) + x Ψ r(, t) x Ψ r(l r (t), t) =. We repace variabes in (2), (3) to obtain a probem with non-varying edges: y = x L(t), t 1 = t, (3) (4) Further, t 1 is mentioned as t for simpicity. After the repacement, we come to the foowing equations: i t Ψ r = 1 2 L 2 y 2 Ψ r + i L L y y Ψ r, x 2πr, i t Ψ = 1 L 2 2 y 2 Ψ + i L L y y Ψ, x. (5) Here L = dl/dt. The boundary conditions remain the same as (3). The appearing factor L(t) in the ast condition can be omitted due to its positivity. Thus, we obtained the probem for stabe geometric graph. To sove it, we wi use eigenfunctions of the stationary Schrödinger operator. The corresponding probem for such quantum graph (with the ength of segment equas and the radius of the ring equas r) has the form: d2 dy 2 φ = k 2 φ, y, d2 dy 2 φ r = k 2 φ r, y 2πr, φ () = φ r () = φ r (2πr), φ () =, d y φ y= + d dy φ r y= d dy φ r y=2πr =.

3 Wave dynamics on time-depending graph with Aharonov Bohm ring 459 The eigenfunctions are as foows the normaizing coefficient is = sin(k n( y)), B n sin(k n ) r = cos (k n(y πr)) B n cos (k n πr), B 2 n = and k n is n-th root of characteristic equation 1 2 sin 2 (k n ) + πr cos 2 (πk n r), 2 tan (πkr) = cot k. Deaing with stationary geometric graph, we can expand soution into a series of the eigenfunctions which form a compete set as eigenfunctions of sef-adjoint operator: ψ (y, t) c n (t). (6) ψ r (y, t) = n After the substitution of expansion (6) into equations (5), mutipication by φ (m) r and φ (m), correspondingy, summation of the both equations and integration of the obtained expression over graph Γ, we come to a system of ordinary differentia equations for coefficients c n (t) of the expansion: ċ m (t) + i k2 m L 2 c m n c n L L Γ r y φ(n) y φ(m) dy =. (7) The system is truncated and soved numericay. We wi discuss it in the next section, where the anaogous procedure wi be appied to the graph in a magnetic fied Quantum graph in magnetic fied In the mode, it is assumed that we have different operators acting at the oop and at the segment. Whie at the segment, we dea with the free Schrödinger operator, at the oop we consider the Landau operator, i.e. the Schrödinger operator with a magnetic fied: i ( t Ψ r(x, t) = 2 i 2m x + Φ(t) ) 2 Ψ r (x, t), x L r (t), L(t)Φ i (8) t Ψ (x, t) = 2 2 2m x 2 Ψ (x, t), x L (t), where the magnetic fux is Φ(t) = πr 2 L 2 (t)b, Φ is the magnetic fux quantum and B is constant magnetic fied. After variabes repacement (4), one obtains: ( i L ) L y y Ψ r(y, t) + t Ψ r(y, t) = m L 2 y 2 Ψ r(y, t) i 2 Φ 1 m LΦ L y Ψ r(y, t) + i 2 Φ 2 m L 2 Φ 2 Ψ r (y, t), y, (9) ( i L ) L y y Ψ (y, t) + t Ψ (y, t) = m L 2 y 2 Ψ (y, t). To sove the equations, we use the expansion with respect to the compete system of eigenfunctions of the free stationary Schrödinger operator on the stationary graph: Ψ (y, t) C n (t). (1) Ψ r (y, t) = n We put expansion (1) into equations (9) and mutipy the both sides by φ (m) r and φ (m), correspondingy. Then, we summarize both equations and integrate the expression over graph Γ. Finay, we come to a system of ordinary r

4 46 D. A. Eremin, E. N. Grishanov, D. S. Nikiforov, I. Y. Popov FIG. 2. Wave function moduo. function) (arbitrary units) segment; t =. (Initia FIG. 3. Wave function moduo for different vaues of B. segment; t = 1.6 (arbitrary units) differentia equations for coefficients C n (t) of expansion (1): Ċ m + i km 2 2m L 2 C m 2πr L C n y φ(n) r L y φ(m) r dy + n + πrb m Φ n C n 2πr r y φ(m) r dy + i π 2 r 2 L 2 B 2 2m Φ 2 n C n 2πr y φ(n) y φ(m) dy φ (m) dy =. (11) 3. Resuts and discussion The initia conditions for system (11) are obtained from the initia condition for the wave function: C n () = 2πr Ψ r (y, ) r dy + We choose the initia vaue of the wave function in the foowing form: Other parameters are chosen in the foowing way: Ψ r (y, ) =, 2 Ψ (y, ) = (1 cos 2πy) 3. L(t) = a + b cos ωt, a = 1, b =.5, ω = 1, = 2m = 1, = r = 1. Ψ (y, ) dy. (12) System (11) is infinite. To sove it numericay, we make a truncation. We increase the number of equations up to the moment when the resut becomes stabe. Resuts for B =, 1, 5, 1 are shown in Fig One can see that an increase of the magnetic fied ead to greater ocaization of the soution at the segment and to stabiization of the wave function. The stabiization of the wave function is observed. Comparing resuts for the graph having constant engths of edges (L = 1) and for the time-dependent graph, one can see that in the stationary case, the magnetic fied can stabiize the soution more quicky. Resuts for B = 5 are shown in Fig. 7 9.

5 Wave dynamics on time-depending graph with Aharonov Bohm ring 461 FIG. 4. Wave function moduo for different vaues of B. segment; t = 3.24 (arbitrary units) FIG. 5. Wave function moduo for different vaues of B. segment; t = 4.8 (arbitrary units) FIG. 6. Wave function moduo for different vaues of B. segment; t = 6.4 (arbitrary units)

6 462 D. A. Eremin, E. N. Grishanov, D. S. Nikiforov, I. Y. Popov FIG. 7. Wave function moduo for B = 5. segment; thin curve t =., dotted curve t =.2, soid curve t =.4 (arbitrary units) FIG. 8. Wave function moduo for B = 5. segment; thin curve t =.6, dotted curve t =.8, soid curve t = 1. (arbitrary units) FIG. 9. Wave function moduo for B = 5. segment; thin curve t = 1.2, dotted curve t = 1.4, soid curve t = (arbitrary units)

7 Wave dynamics on time-depending graph with Aharonov Bohm ring 463 Acknowedgements This work was partiay financiay supported by the Government of the Russian Federation (grant 8-8), by grant of Russian Science Foundation. References [1] Aharonov Y., Bohm D. Significance of eectromagnetic potentias in the quantum theory. Phys. Rev., 1959, 115, P [2] Léna C. Eigenvaues variations for Aharonov-Bohm operators. J. Math. Phys, 215, 56, P [3] Fischer A.M., CampoV.L., Jr., Portnoi M.E., Romer R.A. Exciton Storage in a Nanoscae AharonovBohm Ring with Eectric Fied Tuning. Phys. Rev. Lett., 29, 12, P [4] Entin-Wohman O., Imry Y., Aharony A. Effects of externa radiation on biased Aharonov-Bohm rings. Phys. Rev. B, 24, 7, P [5] Sheykh I. A., Gakin N. G., Bagraev N. T. Conductance of a gated AharonovBohm ring touching a quantum wire, Phys. Rev. B, 26, 74, P [6] Nichee F., Komijani Y., Henne S., Ger C., Wegscheider W., Reuter D., Wieck A. D., Ihn T., Enssin K. AharonovBohm rings with strong spinorbit interaction: the roe of sampe-specific properties. New J. Phys., 213, 15, P [7] Kurasov P., Enerbäck. Aharonov-Bohm ring touching a quantum wire: ow to mode it and to sove the inverse probem. Rep. Math. Phys., 211, 68, P [8] Kokoreva M. A., Marguis V. A., Pyataev M. A. Eectron transport in a two-termina AharonovBohm ring with impurities. Physica E, 211, 43, P [9] Kokoreva M. A., Pyataev M. A. Spectra and transport properties of one-dimensiona nanoring superattice.. Int. J. Mod. Phys. B, 213, 27(2), P [1] Grishanov E.N., Eremin D.A., Ivanov D.A., Popov I.Y., Smirnov P.I. Periodic chain of disks in a magnetic fied: buk states and edge states. Nanosystems: Physics, Chemistry, Mathematics., 215, 6(5), P [11] N. I. Gerasimenko, B. S. Pavov, Scattering probems on noncompact graphs. Theoret. Math. Phys., 1988, 74, P [12] P. Exner, P. Keating, P. Kuchment, T. Sumada, A. Tepyaev,Anaysis on graph and its appications. Proc. Symp. Pure Math. Providence, RI, 28, 77. [13] P. Ducos, P. Exner, O. Turek. On the spectrum of a bent chain graph. J. Phys. A: Math. Theor., 28, 41, P /1-18. [14] I.Y.Popov, A.N.Skorynina, I.V.Binova. On the existence of point spectrum for branching strips quantum graph. J. Math. Phys., 214, 55, P. 3354/1-2. [15] I. Y. Popov, P. I. Smirnov, Spectra probem for branching chain quantum graph. Phys. Lett., 213, A 377, P [16] J.V. Jose, R. Gordery. Study of a quantum Fermi-acceeration mode. Phys. Rev. Lett., 1986, 56, P. 29. [17] A.J. Makowski, S.T. Dembinski. Exacty sovabe modes with time-dependent boundary conditions. Phys. Lett., 1991, A,154(5-6), P [18] C. Cacciapuoti, A. Mantie, A. Posiicano. Time dependent deta-prime interactions in dimension one. Nanosystems: Phys. Chem. Math., 216, 7(2), P [19] Z.A. Sobirov, D.U. Matrasuov, Sh. Ataev and H. Yusupov In Compex Phenomena in Nanoscae Systems. Eds. G. Casati, D. Matrasuov. Berin, Springer, 29. [2] D.U. Matrasuov, J.R. Yusupov, K.K. Sabirov, Z.A. Sobirov. Time-dependent quantum graph. Nanosystems: Phys. Chem. Math., 215, 6(2), P [21] O. Karpova, K. Sabirov, D. Otajanov, A. Ruzmetov, A.A. Saidov. Absorbing boundary conditions for Schrödinger equation in a timedependent interva. Nanosystems: Phys. Chem. Math., 217, 8(1), P [22] Eremin D.A., Grishanov E.N., Kostrov O.G., Nikiforov D.S., Popov I.Y. Time dependent quantum graph with oop. Nanosystems: Physics, Chemistry, Mathematics, 217, 8, P [23] Popov I.Y., Nikiforov D.S. Cassica and quantum wave dynamics on time-dependent geometric graph. Chinese Journa of Physics. 218, 56(2), P [24] Marquardt F., Bruder C. Aharonov-Bohm ring with fuctuating fux. Phys. Rev. B, 22, 65, P