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Wave dynamics on time-depending graph with Aharonov-Bohm ring
D. A. Eremin1, E. N. Grishanov1, D. S. Nikiforov2, I. Y. Popov2
department of Mathematics and IT, Ogarev Mordovia State University Bolshevistskaya Str. 68, Saransk, Russia
2 Department of Higher Mathematics, ITMO University, Kroverkskiy pr. 49, St. Petersburg, 197101, Russia
[email protected], [email protected], [email protected], [email protected]
Aharonov-Bohm ring (AB ring) is an element frequently used in nanosystems. The paper deals with wave dynamics on quantum graph consisting of AB ring coupled to a segment. It is assumed that the lengths of the edges vary in time. Variable replacement is made to come to the problem for stationary geometric graph. The obtained equation is solved using the expansion with respect to a complete system of eigenfunctions of the unperturbed self-adjoint operator for the stationary graph. The coefficients of the expansion are found as solutions of a system of differential equations numerically. The influence of the magnetic field is studied. The comparison with the case of stable geometric graph is made.
Keywords: quantum graph, Aharonov-Bohm ring.
Received: 15 June 2018 Revised: 18 July 2018
1. Introduction
Nanostructures in a magnetic field attract great attention of physicists due to its interesting behavior. Many of these properties, e.g., giant magnetoresistance, have found wide range of applications in nanotechnology, computer hardware, etc. One of the basic elements for nanostructures in a magnetic field is Aharonov-Bohm ring [1], i.e. a nano-sized conducting ring in a magnetic field orthogonal to the ring plane. The Aharonov-Bohm effect was observed and studied in many physical situations (see, e.g., [2-6]. For such structures, quantum graph model is actively used [7-10]. It is a rather effective mathematical model allowing one to describe the spectral and transport properties of many complex physical systems [11-15]. We consider a quantum graph with edge lengths varying in time. Although this model is very interesting from a physical point of view (see, e.g., [16]), there are only a few works devoted to this problem. One can mention papers concerning the time-dependent boundary conditions [17] or time dependent point-like interactions [18], but consideration of time-dependent graphs began recently [19-23]. Wave dynamics for time-dependent quantum graphs in a magnetic field was not studied previously. There are only works concerning concerning Aharonov-Bohm rings in fluctuating magnetic field (see, e.g., [24]). In the present paper, we construct and study a model of time-dependent quantum graph with loop in magnetic field. We investigate the dependence of the dynamics on a magnetic field.
2. Model
2.1. Graph description
We consider quantum graph r with a loop (ring) shown in Fig. 1. It is assumed that the magnetic field acts at the ring. Lengths of graph edges, ring (Lr (t)) and segment (Lg(t)), vary in time in accordance with the following relations:
where r, l are constants, L(t) is some twice continuously differentiable function. Obviously, Lr(t) and Le(t) should be positive.
PACS 03.65.-w, 02.30.Tb, 02.10.Db, 73.63.Nm
DOI 10.17586/2220-8054-2018-9-4-457-463
Lr (t) = 2 nrL(t), Le(t) = lL(t),
(1)
Fig. 1. Geometrical structure of the graph
2.2. Schrodinger operator on the graph
We start with the problem corresponding to the absence of a magnetic field. It means free Schrodinger operator acts on edges of the graph:
d ti2 82 ih— W(x, t) = - — —2 W(x, t), 0 < x < Lr(t), dt 2m dx2 (2)
d ti2 d2 (2)
ih - ^ (x,t) = - — ^ ^x^ 0 < x < L,(t).
We look for continuous solution satisfying the Kirchhoff conditions at the central vertex V0 and the Dirichlet conditions at the boundary one Vi:
W*(0,t) = W (0,t) = W (Lr (t),t),
Ve(Le (t),t) = 0, (3)
3 3 3
_ dx iMI- dx (Lr (t),i) = 0.
We replace variables in (2), (3) to obtain a problem with non-varying edges:
x
^ = L(t), (4)
ti = t,
Further, t1 is mentioned as t for simplicity. After the replacement, we come to the following equations:
d 1 d 2 L d
idtW = -L2dy-2 W + W, 0 < x < 2nr, (5)
• d T 1 d2 T L d T n ^ ^ idtW = -L2 df W + iLydfy W, 0 < x < 1
Here L = dL/dt. The boundary conditions remain the same as (3). The appearing factor L(t) in the last condition can be omitted due to its positivity.
Thus, we obtained the problem for stable geometric graph. To solve it, we will use eigenfunctions of the stationary Schrodinger operator. The corresponding problem for such quantum graph (with the length of segment
equals I and the radius of the ring equals r) has the form:
d2
d2
- dy2 0'(y)= ^(y), 0 < y < i,
d
2
-dy2 0r(y) = k2^r(y), 0 < y < 2nr,
&(0) = 0r (0) = 0r (2nr),
fa(t) = 0,
d Mv=0 + dy 0r \ y=0 - dy 0r\y=2nr =
The eigenfunctions are as follows
,(n)(y) _ sinfM1 - y)) ^ (y)_ Bn sin(kn) '
^(n)(y) _ COs(kn(y - nr)) ,
r Bn cos (knnr) '
the normalizing coefficient is
B2 _ 1 + nr n 2sin2(knl) cos2(nk„r) ' and kn is n-th root of characteristic equation
2 tan (nkr) _ cot kl.
Dealing with stationary geometric graph, we can expand solution into a series of the eigenfunctions which form a complete set as eigenfunctions of self-adjoint operator:
( Mv,t) \ _ v- u J ^(n)(y) x
£cn(tw :(n)(y) . (6)
(y,t) J V V ^rn)(y) ,
After the substitution of expansion (6) into equations (5), multiplication by and correspondingly,
summation of the both equations and integration of the obtained expression over graph r, we come to a system of ordinary differential equations for coefficients cn(t) of the expansion:
¿m(t)+ ik| Cm - Ç Cn L j y^ ^(m) dy _0. (7)
The system is truncated and solved numerically. We will discuss it in the next section, where the analogous procedure will be applied to the graph in a magnetic field.
2.3. Quantum graph in magnetic field
In the model, it is assumed that we have different operators acting at the loop and at the segment. While at the segment, we deal with the free Schrodinger operator, at the loop we consider the Landau operator, i.e. the Schrodinger operator with a magnetic field:
- d T , , h2 ( d $(t) \2 T , , r , ,
(x't)_2m Hx + mi) r(x,t)i 0 <x < Lr(t), (8)
d h2 d2 ih—^(x,t) _ - — dx^-*^(x,t), 0 < x < Le(t),
where the magnetic flux is $(t) _ nr2L2(t)B, $0 is the magnetic flux quantum and B is constant magnetic field. After variables replacement (4), one obtains:
ih (-LVÎy *r (V't) + dt*r (y,t^_
h2 1 d2 T , , h2 $ 1 d ^ , h2 $2 T , , - 2mL2¿y*^r(y,t) - imL¥0Ldy^r(y,t) +i^r(y,t), 0 < y < l0, (9)
L d T . . d T .A h2 1 d2 T . .
^ -Lydy+ dt^'^J _ -2mL2^^(y't).
To solve the equations, we use the expansion with respect to the complete system of eigenfunctions of the free stationary Schrodinger operator on the stationary graph:
i **(y,t) \ ^ UJ №(y) x
^Cn(t) Ifn) (y) . (10)
(y,t) J V V (y)
(9) and multiply the both sides by < we summarize both equations and integrate the expression over graph r. Finally, we come to a system of ordinary
We put expansion (10) into equations (9) and multiply the both sides by ^^ and ^m), correspondingly. Then.
0.2 0.4 0.6 o.a 1 1.2
Fig. 2. Wave function modulo. Left - at the ring, right - at the segment; t = 0.0 (Initial function) (arbitrary units)
Fig. 3. Wave function modulo for different values of B. Left - at the ring, right - at the segment; t = 1.6 (arbitrary units)
differential equations for coefficients Cn(t) of expansion (10):
2nr
h k
+• e» - e c L ( Mr ^+/
2nr , ,
+A nB e cni ^ ^+i
m ^ J dy
_ti_ n2r2L2B2 2m $0
o
2nr
E Cn J dy = 0.
(11)
3. Results and discussion
The initial conditions for system (11) are obtained from the initial condition for the wave function:
2nr I
Cn(0)=y Wr(y, 0)0n)dy + J **(y, 0)0((n)dy. (12)
0 0 We choose the initial value of the wave function in the following form:
Wr (y, 0) = 0, W£(y, 0) = (1 - cos 2ny)
Other parameters are chosen in the following way:
L(t) = a + b cos wt, a =1, b = 0.5, w = 1, ti = 2m = 1, l = r = 1.
System (11) is infinite. To solve it numerically, we make a truncation. We increase the number of equations up to the moment when the result becomes stable.
Results for B = 0,1, 5,10 are shown in Fig. 2-6. One can see that an increase of the magnetic field lead to greater localization of the solution at the segment and to stabilization of the wave function.
The stabilization of the wave function is observed. Comparing results for the graph having constant lengths of edges (L = 1) and for the time-dependent graph, one can see that in the stationary case, the magnetic field can stabilize the solution more quickly. Results for B = 5 are shown in Fig. 7-9.
0 2 4 6 a 0 0.2 0.4 0.6 O.a 1 12 1.4
Fig. 4. Wave function modulo for different values of B. Left - at the ring, right - at the segment; t = 3.24 (arbitrary units)
0 2 4 6 a 0 0.2 0.4 0.6 o.a 1 1.2 1.4
Fig. 5. Wave function modulo for different values of B. Left - at the ring, right - at the segment; t = 4.8 (arbitrary units)
E: = 0 1 1 1 1 ^ 1-5
0 2 4 6 a 0 0.2 0.4 0.6 o.a 1 1.2 1.4
Fig. 6. Wave function modulo for different values of B. Left - at the ring, right - at the segment; t = 6.4 (arbitrary units)
0 12 3 4 5 6 0 0 2 0.4 0.6 03
Fig. 7. Wave function modulo for B = 5. Left - at the ring, right - at the segment; thin curve -t = 0.0, dotted curve - t = 0.2, solid curve - t = 0.4 (arbitrary units)
0 12 3 4 5 6 0 02 0.4 0.6 03
Fig. 8. Wave function modulo for B = 5. Left - at the ring, right - at the segment; thin curve -t = 0.6, dotted curve - t = 0.8, solid curve - t = 1.0 (arbitrary units)
0 1 2 3 4 5 6 0 02 0.4 0.6 03
Fig. 9. Wave function modulo for B = 5. Left - at the ring, right - at the segment; thin curve -t = 1.2, dotted curve - t = 1.4, solid curve - t = 1.6 ... (arbitrary units)
Acknowledgements
This work was partially financially supported by the Government of the Russian Federation (grant 08-08), by grant 16-11-10330 of Russian Science Foundation.
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