CC BY
62
A model of electron transport through a boson cavity
A.A. Boitsev1, J. Brasche2, H. Neidhardt3, I.Y. Popov1
xITMO University, Kronverkskiy, 49, St. Petersburg, 197101, Russia
2Institut für Mathematik, TU Clausthal, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany
3Weierstrass Institute, Mohrenstr. 39, D-10117 Berlin, Germany
boitsevanton@gmail.com, johannes.brasche@tu-clausthal.de, hagen.neidhardt@wias-berlin.de, popov1955@gmail.com
We propose a model describing electronic transport through a boson cavity. We use the Jaynes-Cummings model dealing with a two-level quantum dot coupled to a quantized electro-magnetic field and two semi-infinite wires. The mathematical background of our model is given by the theory of self-adjoint extensions of symmetric operators. Using the boundary triplets approach, the gamma-field and the Weyl function were calculated. In addition, we obtained the scattering matrix for the model system.
Keywords: boundary triplets, extensions, scattering, solvable model.
Received: 21 February 2018 Revised: 1 March 2018
1. Introduction
To study the steady state current flowing through a quantum device in the framework of Landauer-Buttiker approach, it is necessary to consider a quantum system as an inner system (quantum dot) with left and right leads attached to it, i.e. free-fermion reservoirs with two different electro-chemical potentials. The goal was to calculate the steady electron current going from one lead through the dot the other one. This current is directly related to the transmission coefficients of some natural scattering system related to this particle transport problem. This, approach was justified a few years ago [1-3]. It, effectively, reduces the problem to investigation of a one-particle Hamiltonian.
In the present paper we consider the Jaynes-Cummings model [4,5]. The Jaynes-Cummings model serves to determine how quantization of the radiation field affects the predictions for the evolution of the state of a two-level system in comparison with semi-classical theory of light-atom interaction. It is often applied to description of the interaction between an atom and a laser field [6]. We used a version suggested in [7,8]. More precisely, the authors of [7,8] use the discrete Hamiltonian. In contrast, we use the continuous Hamiltonian. Namely, we have a two-level quantum dot coupled, from one side, to the standard Jaynes-Cummings one-mode photon resonator, and from the other side to two semi-infinite leads. To construct the model, we use the operator extensions theory approach (see, e.g., [9-11] and references in [12]) in the framework of boundary triplet approach (see, e.g., [13-16]). Detailed mathematical background of the model is developed in [17].
2. Preliminaries 2.1. Linear relations
A linear relation © in H is a closed linear subspace of H © H. The set of all linear relations in H is denoted by C(H). We denote also by C(H) the set of all closed linear (not necessarily densely defined) operators in H. Identifying each operator T e C(H) with its graph gr (T), we regard C(H) as a subset of C(H).
The role of the set C(H) in extension theory becomes clear from Proposition 2.3. However, it's role in the operator theory is substantially motivated by the following circumstances: in contrast to C (H), the set C(H) is
closed with respect to taking inverse and adjoint relations © 1 and ©*. The latter is given by: © 1 = {{g,f} : if,g}e ©} and:
PACS 85.60.Jb, 88.40.H, 85.35.Be, 73.63.Kv, 42.50.Pq
DOI 10.17586/2220-8054-2018-9-2-171-178
A linear relation Ö is called symmetric if ö c Ö* and self-adjoint if Ö = ©*.
2.2. Boundary triplets and proper extensions
Let us briefly recall some basic facts on boundary triplets. Let S be a densely defined closed symmetric operator with equal deficiency indices n±(S) := dim(N±j), Nz := ker(S* — z), z G C±, acting on some separable Hilbert space H.
Definition 2.1.
(i) A closed extension S of S is called proper if dom(S) c dom(S) c dom(S*).
(ii) Two proper extensions S', S are called disjoint if dom (S") n dom (S) = dom (S) and transversal if in addition dom (S') + dom (S) = dom (S*).
we denote by Exts the set of all proper extensions of S completed by the non-proper extensions S and S* is denoted. For instance, any self-adjoint or maximal dissipative (accumulative) extension is proper.
Definition 2.2 ([18]). A triplet n = {H, r0, rj, where H is an auxiliary Hilbert space and r0, r : dom (S*) ^ H are linear mappings, is called a boundary triplet for S* if the "abstract Green's identity":
(S*f,g) — (f, S*g) = (rif, rog) — (rof,rig), f, g G dom (S*). (1)
is satisfied and the mapping r := (r0, r1)T : dom (S*) ^ H © H is surjective, i.e. ran (r) = H © H.
A boundary triplet n = {H, r0, r1} for S* always exists whenever n+(S) = n_(S). Note also that n±(S) = dim(H) and ker (r0) n ker (r1) = dom (S).
With any boundary triplet n one associates two canonical self-adjoint extensions Sj := S* \ ker(r), j g {0,1}. Conversely, for any extension S0 — S* g Exts, there exists a (non-unique) boundary triplet n — {h, r0, ri} for S* such that S0 := S* \ ker (r0).
Using the concept of boundary triplets one can parametrize all proper extensions of A in the following way.
Proposition 2.3 ( [19]). Let n = {H, r0, r1} be a boundary triplet for S*. Then the mapping
Exts 3 S ^ rdom (S) = {(r0f, r1f)T : f G dom (S)} =: © G C(H) (2)
establishes a bijective correspondence between the sets Exts and C(H). We write S = SQ if S corresponds to © by (2). Moreover, the following holds:
(i) SQ = Sq» , in particular, SQ = SQ if and only if ©* = ©.
(ii) SQ is symmetric (self-adjoint) if and only if © is symmetric (self-adjoint).
(iii) The extensions SQ and S0 are disjoint (transversal) if and only if there is a closed (bounded) operator B such that © = gr (B). In this case (2) takes the form
Sq := Sgr(B) = S* \ ker (r — B^). (3)
In particular, Sj := S* \ ker (r) = Sq3 , j G {0,1}, where ©0 := ^^ j and ©1 := ^^ j = gr (O) where
O denotes the zero operator in H. Note also that C(H) contains the trivial linear relations {0} x {0} and H x H parameterizing the extensions S and S*, respectively, for any boundary triplet n.
2.3. Gamma field and Weyl function
It is well known that Weyl function is an important tool in the direct and inverse spectral theory of Sturm-Liouville operators. In [19], the concept of Weyl function was generalized to the case of an arbitrary symmetric operator S with n+(S) = n_(S) < to. Following [19], we briefly recall basic facts on Weyl functions and 7-fields associated with a boundary triplet n.
Definition 2.4 ( [19]). Let n = {H, r0, r1} be a boundary triplet for S* and S0 = S* \ ker(r0). The operator valued functions y(-) : p(S0) ^ [H, H] and M(•) : p(S0) ^ [H] defined by:
Y(z) := (r0 \ Nz)-1 M(z):=r17(z), z G KS0), (4)
are called the 7-field and the Weyl function, respectively, corresponding to the boundary triplet n.
Clearly, the Weyl function can equivalently be defined by:
M(z)r0fz =rfz, fz G Nz, z G p(S0).
The Y-field y(-) and the Weyl function M(•) in (4) are well defined. Moreover, both y(-) and M(•) are holomorphic on p(S0) and the following relations:
Y(z) = (I + (z - z)(So - z)-1)y(C), z, z e p(So), (5)
and
M(z) - M(Z)* = (z - Z)y(Z)*Y(z), Z, z e p(So), (6)
hold. Identity (6) yields that M(•) is [H]-valued Nevanlinna function (M(•) e R[H]), i.e. M(•) is [H]-valued holomorphic function on C± satisfying:
M(z) = M(z)* and Im(M(z)) > o, z e C+ u C_.
Im(z)
It follows also from (6) that 0 e p(Im(M(z))) for all z e C±.
2.4. Krein-type formula for resolvents
Let n = {H, r0, r1} be a boundary triplet for S*, M(•) and y(-) the corresponding Weyl function and Y-field, respectively. For any proper (not necessarily self-adjoint) extension S0 e ExtS with non-empty resolvent set p(S0) the following Krein-type formula holds (cf. [19]):
(SQ - z)-1 - (So - z)-1 = y(z)(© - M(z))-1Y*(z), z e p(So) n p(S0). (7)
Formula (7) extends the known Krein formula for canonical resolvents to the case of any S0 e ExtS with p(S0) = 0. Moreover, due to relations (2), (3) and (4) formula (7) is related to the boundary triplet n. We emphasize, that this relation makes it possible to apply the Krein-type formula (7) to boundary value problems (see, e.g., [14] and references in [12]).
2.5. Scattering and Weyl function
Let S be a densely defined closed symmetric operator with finite equal deficiency indices n±(S) and n = {H, r0, r1} is a boundary triplet for S*, Let S0 = S* \ kerr0 and S0 is a self-adjoint extension corresponding to © e (7(H). As dimH is finite, by (7) (S0 - z)-1 - (S0 - z)-1 is a finite rank operator and the system {S0, S0} is a so-called complete scattering system, i.e. the wave operators:
W(S0,So) = s - lim eitSee-itSoPac(S0),
exists and they are complete, i.e. their ranges coincide with the absolutely continuous subspace Hac(S0) of S0 (see, e.g. [20], [21], [22]). By Pac(S0) we denote the orthogonal projection on absolutely continuous subspace Hac(S0) of S0. The scattering operator S(S0, S0) of a scattering system {S0, S0} is defined as:
S(S0 ,So) = W+(S0,So)*W-(S0,So).
If we regard the scattering operator as an operator in Hac(S0) then it becomes unitary and commutes with absolutely continuous part Sac = S0 \ Hac(S0)ndom (S0). of S0 and thus it is unitarily equivalent to a multiplication operator induced by a family {S0(z)} of unitary operators in a spectral representation of Sac ([20], Proposition 9.57). This family is called a scattering matrix of a scattering system S(S0, S0).
Since the dimension dimH is finite, then the Weyl function M(•) corresponding to boundary triplet n = {H, r0, r1} is a matrix-valued Nevanlinna function. By Fatous theorem ([23]), the limit M(A+i0) = lim M(A+
e^0+0
¿e) exists for almost all A e R. We denote the set of real point where the limit exists by £M. We will use the notation HM(a) = ran(M(A)), A e £M. By PM(A) we will denote the orthogonal projection on HM(a). We will also
use the notation Ns(z) — (© - M(z)) , z e C \ R, where © e C*(H) is a self-adjoint relation corresponding to S0. This function is well defined and the limit N0(A + ¿0) = (© - M(A + ¿0))-1 exists almost for every A e R. This set we will denote as .
Theorem 2.5. ( [13]) Let S be a densely defined symmetric operator with finite deficiency indices in separable Hilbert space H, let n be a boundary triplet corresponding to S* with corresponding Weyl function M (•), S0 is a self-adjoint extension of S, S0 = S* \ kerr0, © e (5(H), then in L2(R, dA, HM(A)) the scattering matrix of the complete scattering system {S0, S0} is given by:
60(A) = Ihm(x) (A + ¿0)) Ne (A + i0)V^(M (A + ¿0))
for A e £M n .
3. Scattering
3.1. Model description
A rigorous construction of a proposed model can be found in [17]. We will show here only the main
results. We consider the Hilbert space H = L2(R_) © L2(R+) where R_ = (—to, 0) and R+ = (0, to). On the
d2
subspaces Hi := L2(R_) and Hr := L2(R+) we consider the closed symmetric operators H; = —+ v; and
dx2
d2
Hr = — + vr. We set H := H; © Hr = L2(R) and H := H; © Hr. Operator H can be regarded as the
dx2
symmetric operator:
d2 . . f v; x G R_
A = —r^- + v(x), v(x) := < ^
dx2 |vr x G R+
with domain dom(A) = W02,2(R) := {f G W2'2(R) : f (0) = f'(0) = 0}. The operator A is symmetric and has deficiency indices n±(A) = 2. For simplicity we assume that 0 < vr < v;. It can be checked (see [17]) that the triplet nA = {HA, rA, rA} with:
HHl C
HA := © = © rAf := K) IM , rA
jA__^ _ ^ rA , .= (f(-0)\ rA .= (-f'(-0)'
Hnr C . f (+°)J ' 1 ^ f'(+°)y
defines a boundary triplet for A*. The Weyl function MA(z) of the boundary triplet nA is given by: ,„-4, N fmH' (z) 0 \ (¿Vz — v; 0 \ , „ N
MA(z)=(v 0() mHr(z)J = ( 0 l ¿Vz-^J , z G P(A0),
where A0 := A* [ ker (rA). Notice that A0 has Dirichlet boundary conditions.
We will view the point zero as a quantum dot or quantum cavity. In particular, the Hilbert space HA = C2 is considered to be the state space of the quantum dot and the self-adjoint operator B as the Hamiltonian of the dot, where AB := A* [ ker (rA — B^) is a self-adjoint extension of A. The Hamiltonian B describes a two level system to which we are going to couple bosons. The state space of the bosons is the Hilbert space T = 12(N0), N0 := N U {0}. The boson operator T is given by:
T£ = T{£fc}fc£No = {k£fc}fc£No ,
S = {6}fc£No G dom(T) := {{£fc}fceNo G ^(N0) : {k£fc}fc£No G M^)}.
The Hamiltonian T describes a system of bosons which do not interact mutually. The number of bosons is not fixed and varies from zero to infinity. The Hilbert space T has a natural basis given by ek = {¿fcj}j£No. Let us introduce the creation and annihilation operator b* and b, respectively, defined by:
b*ek = %/k + 1efc+1, k G N0, and bek = Vkefc_1, k G N0,
where e_1 =0. One easily checks that T = b*b.
In order to couple these two systems let us consider the closed symmetric operator:
S := A << It + IH ® T (8)
in the Hilbert space K := H < T. Setting the following:
K; := Hi < T, S; := H; <g> I% + < T,
Kr := Hr < T, Sr := Hr < IT + IHr < T,
we obtain:
K = K; © Kr and S = S; © Sr. The corresponding Weyl function MS(•) is given by:
MS(z) = (MS0(z) MST (z)) , z G p(S0), S0 = SD © SD,
IiD < It + Ih < T anc
conditions and:
where SD = H;D < IT + IH < T and SD = HrD < IT + IHl < T - the extensions with the Dirichlet boundary
mHl (z - T) - Re(mHl (i - T ))
M(z) =-i- v --^, z G C+,
( ) Im(mH (i - T)) '
^ mHr(z - T) - Re(mH (i - T))
MSr (z) =-i-, „ , -rr---, z € C+.
( ) Im(mHr (i - T)) , ±
Moreover, we have:
and:
where:
'^Im(mH< (i - T)) 0 \ /Z-1/2 0 N
0 ^Im(mHr (i - T))) = ^ 0 Z-1/2v
/Re(mHl (i - T)) 0 \_fzi 0\
Re(mHr (i - T)W \0 Z,
r
Z := V y/Tz + (T + v)2 + T +
V,
Zr := V VTz + (T + vr)2 + T + vr.
3.2. Scattering matrix
We want to show that the limit lim MS (A + ¿e) exists. The candidate for this limit is the function MS (A), A e R.
Lemma 3.1. The operator-function MS(A), A e R is a bounded operator. Consider:
MSi (A) = mHl (A - T) - Re(mH (¿ - T)) = ^ mH (A - n) - Re(mH (j - n)) ( ) Im(mH (¿ - T)) ^ Im(mH (¿ - n)) .
We then rewrite as follows:
mHi (A - n) - Re(mHi (¿ - n)) _ ¿VA - n - vi + V1 + (n + vi)2 + n l
Im(mHi ^ - n)) 1 ( 1—/ ^ ^-x -1
" ' ' ' ' ° ' n + vi
— (V/1 + (n + V)2 + n + v^
and consider this sequence when n > A - vi, then the numerator can be rewritten as:
(n + Vi j2 + n + vi - /n - A + V; =
1 /l + (n + Vi)2 - (n + Vi - 2A)
2 -2\J/1+ (n + V;)2 + n + V; + Vn - A + V;
Notice that:
VV/1 + (n + Vi)2 + n + Vi
Vl + (n + Vi)2 + n + Vi + Vn - A + Vi
has the finite limit, so it is bounded and we have to examine:
Vl + (n + vi)2 - (n + vi - 2A) =
1 + 4nA - 4A2 + 4v; A
V^l + (n + Vi)2 + (n + Vi - 2A)
which also has the finite limit and so bounded. Thus, the sequence is bounded for n > A - v;, so it is bounded for n € No. Analogously, we deal with MSr.
Lemma 3.2. lim MS(A + ie) = MS(A), A € R Consider:
mHl (A - n) - (mHl (A + ie - n))
|MSl (A) - MSl (A + ie) II =
®
Im(mHl (i - n))
0
We obtain:
mHi (A — n) — (mHi (A + ie — n))
Im(mHi (i — n))
i^A — n — V; — iVA + ie — n — V;
y/TT(n + V;)2 + n +V
a/A — n — V; + a/ A + ie — n — V;
e
^ ^/l + (n + V)2 + n + V;)
V^/l + (n + V; )2 + n + V;
-1
A — n — V;
The norm in the right hand side is bounded for integer n, so:
mHl (A — n) — (mHl (A + ie — n)) Im(mHl (i — n))
uniformly with respect to n. Analogously we deal with MSr. Let us show that the operator:
(0 — MS (A + ie))_1 exists. It is sufficient to show that (Im(MS(A)))_1 exists.
Lemma 3.3. (Im(MS(A)))_1 exists for A = n + v; and A = n + vr.
Consider:
^ 0, e ^ +0,
Im(MS (A)) _ Im(mH (A — T)) _ m Im(iVA — n — V;) _ Re(VA — n — V;)
Im(M (A))_T^Im(mH(i — n))
Im(mH (i — T ))
Im(mHi (i — n))
and Re(^A - n - ) = 0 when n > A - and Re(^A - n - ) = \/A - n - when n < A - If A = n +
then:
n<A-vi
Im(MSi (A))-1 _ 0 ^ (i — n))
. — n — V;
Analogously, we deal with MSr.
For scattering matrix description, we need to calculate 0m(M (A + 0i)) = VMM (A)).
Lemma 3.4. Operator \J Im(M (A)) has the following form: (
■ - 1 Va — n — o Im( mHi ( i — n))
VIm(M(A)) _
\
n<A-Vr /T-
VA — n — V, ^ ^ 0
^ Im(mHr (i — n)) ^
n=0 v v n>A-vr /
Im(MSi (A)) _ Im
Now, as far as mH = i\Jz - we obtain:
-Hi
(A — T ) — Re(mHi (i — T )) \ Im(mH (A — T ))
Im(mH (i — T)) y Im(mHi (i — T)) '
Im(mHi (A — T)) _ ^ Im(mHi (A — n)) _ ^ Im(iVA — n — v;)
m
Im(mHi(i — T)) ^^ Im(mHi(i — n)) ^^ Im(mHi(i — n))
As far as when n < A — v ; we have Im(i\/A — n — v ;) = \/A — n — v ; ) and when n > A — v ; we have Im(i\/A — n — v ;) = 0, we obtain that:
n<A-v i
9-
a/A — n — V;
Im(mH i (A — T))
Im(mHi (i — T )) ^ Im(mHi (i — n))'
0
0
Then,
Im(MS (A)) =
/n<A-Vl p;-
m vA - n - vl ^ 0
^ Im(mHl (i — n)) ^
n=0 v v " n>A-vi
n<A-vr 0)
a/ A — n — vr Im(mHr (i — n))
® ® 0
n>A-vr y
and
Im(M S (A))
n<A-Vl r;-
/T\ yA — n — vl ^ m 0
^ Im(mH (i — n)) ^
n=0 v v n>A-vi
\
Let O be a matrix of the form:
n<A-vr ®
a/A — n — Vr Im(mHr (i — n))
® ® 0
n>A-vr /
O =
® / an Yn
n=AYn ßnV 0
\
/-"P\ I Kn nn
nSv
Then © - M(A) is a block-diagonal matrix with the first block of the form
(
®
n=0
( VA — 2n — vl — Re(mHl (i — 2n))
V
Im(mHl (i — 2n))
Yn
ßn —
¿VA — 2n — 1 — vl — Re(mHl (i — 2n — 1)) Im(mHl (i — 2n — 1)) /
and the second block of the form:
®
( VA — 2n — vl — Re(mH (i — 2n)) n Im(mHr (i — 2n))
nn
\
«VA — 2n — 1 — vl — Re(mH (i — 2n — 1))
Im(mHr (i — 2n — 1))
7
Let:
, , » , VA — 2n — vl — Re(mHl (i — 2n))\ Ai(n) = (an — ---- J X
Im(mHl (i — 2n))
ßn —
VA — 2n — 1 — vl — Re(mHl (i — 2n — 1))
Im(mHl (i — 2n — 1))
7n7 n
and:
A2(n) = Kn —
VA — 2n — vl — Re(mHr (i — 2n)) Im(mHr (i — 2n)) ¿Va — 2n — 1 — vl — Re(mHr (i — 2n — 1)) Im(mHr (i — 2n — 1))
nn^n.
Theorem 3.5. The operator (© — MS(A + ie)) 1 exists. It has the block structure with the first block of the form:
t ¿VA — 2n — 1 — vl — Re(mHl (i — 2n — 1)) \
~ 1 V" Im(mH (i — 2n — 1)) —Yn
_ ¿VA — 2n — vl — Re(mH (i — 2n))
^ —Y n
®
Ai(n)
Im(mHl (i — 2n))
7
and the second block of the form:
~~ 1
n)
\
®
n=0
A2(n)
( VA — 2n — 1 — vl — Re(mHr (i — 2n — 1)) n Im(mHr (i — 2n — 1))
nn
— /
VA — 2n — vl — Re(mHr (i — 2n)) Im(mHr (i — 2n)) /
0
0
0
0
0
Y
n
n
n
^n -
X
—
n
n
—
n
The expression for the scattering matrix is given in Theorem 2.5. 4. Conclusion
An extended Jaynes-Cummings model using the ideas of [8] is constructed. It is based on the theory of self-adjoint extensions of symmetric operators. The theory gives one a rigorous mathematical methodology for introducing coupling between bosons (photons), two-level quantum dot (quantum resonator) and two semi-infinite wires. Boundary triplets, gamma-field and Krein Q-functions are found. We obtain the Weyl function. This allows us to determine the scattering matrix. Mathematical details of the model are thoroughly investigated in [17].
Acknowledgements
This work was partially financially supported by grant 16-11-10330 of Russian Science Foundation, by DFG Grants BR 1686/3-1, NE 1439/3-1. A.A.B. and I.Y.P. thank Professor J. Brasche and Clausthal University of Technology for kind hospitality.
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