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A NEW MODEL FOR QUANTUM DOT LIGHT EMITTING-ABSORBING DEVICES: PROOFS AND SUPPLEMENTS
1H. Neidhardt, 2L. Wilhelm, 3V. A. Zagrebnov 1,2WIAS Berlin, Mohrenstr. 39, 10117 Berlin, Germany
3Institut de Mathematiques de Marseille - UMR 7373 CMI - Technopole Chateau-Gombert 39, rue F. Joliot Curie 13453 Marseille Cedex 13, France
PACS 85.60.Jb, 88.40.H, 85.35.Be, 73.63.Kv, 42.50.Pq DOI 10.17586/2220-8054-2015-6-1-6-45
Motivated by the Jaynes-Cummings (JC) model, we consider here a quantum dot coupled simultaneously to a reservoir of photons and to two electric leads (free-fermion reservoirs). This new Jaynes-Cummings-Leads (JCL) model permits a fermion current through the dot to create a photon flux, which describes a light-emitting device. The same model is also used to describe the transformation of a photon flux into a current of fermions, i.e. a quantum dot light-absorbing device. The key tool to obtain these results is the abstract Landauer-Buttiker formula.
Keywords: Landauer-Buttiker formula, Jaynes-Cummings model, coupling to leads, light emission, solar cells. Received: 22 December 2014
1. Introduction
In the following the fermion current going through a quantum dot is analyzed as a function of
(1) the electro-chemical potentials on leads and of
(2) the contact with the external photon reservoir.
Although the latter is the canonical JC-interaction, the coupling of the JC model with leads needs certain precautions, if we wish to remain in the one-particle quantum mechanical Hamiltonian approach and scattering theory framework. To this end, we proposed a new Jaynes-Cummings-Leads (JCL-) model [19]. This model makes possible a photon flux into the resonator, created by a fermion current through the dot; i.e. it describes a light-emitting device, as well as to transform the external photon flux into a current of fermions, which corresponds to a quantum dot light-absorbing device.
The paper is an extended version of [19], which means that the JCL-model, as well as all theorems, corollaries of this article one can already find in [19], however, without any proofs. In the following article, we are going to close this gap and give full proofs of all statements. In doing so, we have added some statements absent in [19].
The paper is organized as follows. The JCL-model is introduced and discussed in Sections 2.1-2.7. For simplicity, we choose for the lead Hamiltonians the one-particle discrete Schrodinger operators, with constant one-site (electric) potentials on each of leads. In Section 2.5, we show that the our model fits into the framework of trace-class scattering. In Section 2.7, we verify
the important property that the coupled Hamiltonian has no singular continuous spectrum. Our main tool for analysis of different currents is an abstract Landauer-Buttiker-type formula applied in Sections 3.1 and 3.2 to the case of the JCL-model. This allows us to calculate the outgoing flux of photons induced by electric current via leads. This corresponds to a light-emitting device. We also found that pumping the JCL quantum dot with a photon flux from a resonator may induce a current of fermions into the leads. This reversing imitates a quantum light-absorbing cell device. These are the main properties of our model and the main application of the Landauer-Buttiker-type formula of Sections 3.1 and 3.2. They are presented in Sections 4 and 5, where we distinguish contact-induced and photon-induced fermion currents.
To describe the results of Sections 4 and 5, note that in our setup, the sample Hamiltonian is a two-level quantum dot decoupled from the one-mode photon resonator. Then, the unperturbed Hamiltonian H0 describes a collection of four totally decoupled sub-systems: a sample, a resonator and two leads. The perturbed Hamiltonian H is a fully coupled system, and the feature of our model is that it is totally (i.e. including the leads) embedded in the external electromagnetic field of the resonator. Hence, each electron can be interpreted as a fermion with internal harmonic degrees of freedom, or a Fermi-particle caring its individual photon cloud.
Similar to the "Black Box" system-leads (SL-) model {HSL,H0}, [1], [4], it turns out that the JCL-model also fits into the framework of the abstract Landauer-Buttiker formula, and in particular, is a trace-class scattering system {HJCL = H, HSL}. The current in the SL-model is called the contact-induced current Jcd. This current was the subject of numerous papers, see e.g. [1,5], or [4]. Note that the current Jei is due to the difference of the electro-chemical potentials between two leads, but it may be zero even if this difference is not null [12,13].
The fermion current in the JCL-model, takes into account the effect of the electron-photon interaction under the assumption that the leads are already coupled. This is called the photon-induced component Jli of the total current. To the best of our knowledge, the present paper is the first which rigorously studies this phenomenon. We show that the total free-fermion current Jel in the JCL-model decomposes into a sum of the contact- and the photon-induced currents: Jel := JC + Jff. An extreme case is when the contact-induced current is zero, but the photon-induced component is not, c.f. Section 5.1. In this case, the flux of photons Jph out of the quantum dot (sample) is also non-zero, i.e. the dot serves as a light emitting device, c.f. Section 5.2. In general the Jph = 0 only when the photon-induced component Jvp՝ = 0.
By choosing the parameters of the model in an suitable manner, one can get either a photon emitting or a photon absorbing system. Hence, the JCL-model can be used either as a light emission device or as a solar cell. Proofs of explicit formulas for fermion and photon currents Jf hl, Jph are contained in Sections 4 and 5.
Note that the JCL-model is called mirror symmetric if (roughly speaking) one can interchange left and right leads and the JCL-model remains unchanged. In Section 5 we discuss a surprising example of a mirror symmetric JCL-model such that the free-fermion current is zero but the model is photon emitting. This peculiarity is due to a specific choice of the photon-electron interaction, which produces fermions with internal harmonic degrees of freedom.
2. Jaynes-Cummings quantum dot coupled to leads 2.1. Jaynes-Cummings model
The starting point for the construction of our JCL-model is the quantum optics Jaynes-Cummings Hamiltonian HJC. Its simplest version is a two-level system (quantum dot) with the energy spacing e, defined by Hamiltonian hs on the Hilbert space hS = C2, see e.g. [16]. It is assumed that this system is "open" and interacts with the one-mode ш photon resonator with Hamiltonian hph.
Since mathematically hph coincides with a quantum harmonic oscillator, the Hilbert space of the resonator is the boson Fock space hph = F+(C) over C and
hph = ub*b. (2.1)
Here, b* and b are verifying the Canonical Commutation Relations (CCR) creation and annihilation operators with domains in F+(C) ~ ^2(N0). Operator (2.1) is self-adjoint on its domain:
dom (hph) = l (հօ,հւ,հշ,...) E f(N o) : u2|kn|2 < ж.
[ n£no )
Note that the canonical basis {фп := (0, 0,... ,kn = 1, 0,.. .)}nen0 in ^2(N0) consists of eigenvectors of operator (2.1): հք,1Փո = սա фп.
To model the two-level system with the energy spacing e, one fixes in C2 two ortho-normal
vectors {eg ,ef}, for example ef := ^^ and ef := ^^, which are eigenvectors of Hamiltonian
hS with eigenvalues {Af = 0, Af = e}. To this end, we set:
hS = e (0 , (2.2)
and we introduce two ladder operators:
a+ :=(° ^ and a- :=(! 0) . (2.3)
Then, one obtains hS = e a+a- as well as ef = a+ef, ef = a-ef and a-ef = 0. So, ef is the ground state of Hamiltonian hS. Note that the non-interacting Jaynes-Cummings Hamiltonian HJC resides in the space HJC = hS 0 hph = C2 0 F+(C) and it is defined as the matrix operator:
HJC := hf 0 IhPh + Ibs 0 hph . (2.4)
Here, կբե denotes the identity operator in the Fock space hph, whereas կ3 stays for the identity matrix in the space f)S.
With operators (2.3), the interaction VSb between quantum dot and photons (bosons) in the resonator is defined (in the rotating-wave approximation [16]) by the operator:
Vsb := gsb (a+ 0 b + a- 0 b*) . (2.5)
Operators (2.4) and (2.5) define the Jaynes-Cummings model Hamiltonian:
Hjc := HJC + Vsb , (2.6)
which is a self-adjoint operator on the common domain dom(HJC) Ո dom(VSb). The standard interpretation of HJC is that (2.6) describes an "open" two-level system interacting with an external one-mode electromagnetic field [16].
Since the one-mode resonator is able to absorb infinitely many bosons, this interpretation sounds reasonable, but one can see that the spectrum a(HJC) of the Jaynes-Cummings model is discrete. To this end, note that the so-called number operator NJC := a+a- 0 I^h + կ3 0 b*b commutes with HJC. Then, since for any u > 0:
HJn>o := {Coef 0 фп + Zief 0 Փո-jco.iec , НПСо := {Zoef 0 Фо}?оес ,
are eigenspaces of operator NJC, they reduce HJC, i.e. HJC : HJnC ^ HJnC. Note that HJC = 0n>0 HJnC, where each HJnC is invariant subspace of operator (2.6). Therefore, it has
the representation:
HJC = 0 HJC , n> 1 ,hJC = 0 . (2.7)
n£no
Here operators hJC are the restrictions of HJC, which act in each HJnC as follows:
hJCiZo es0 0 фп + Zi es 0 фп-i) (2
= [Zonw + ZigsbVn] es 0 фп + [Zi(e + (n - 1)ш) + Zogsb^n] es 0 фп-i .
Hence, the spectrum Ծ(HJC) = |JTl^o ծ (Нп)). By virtue of (2.8), the spectrum ծ(hJC) is defined for n ^ 1 by eigenvalues E(n) of two-by-two matrix hJC acting on the coefficient space {Zo, Zi}:
J (Zi) = (e +(nJ' agsnf) (Zo) = E(n) (Zi). (2.9)
Then, (2.7) and (2.9) imply that the spectrum of the Jaynes-Cummings model Hamiltonian HJC is pure point:
Ծ(Hյc) = ) = {0} U U n + 1 (e - ш) ± \J(e - ш)2/4 + gSbn\ .
nen l z j
This property evidently persists for any system Hamiltonian hs with discrete spectrum and linear interaction (2.5) with a finite mode photon resonator [16].
We resume the above observations concerning the Jaynes-Cummings model, which is our starting point, by following remarks:
(a) The standard Hamiltonian (2.6) describes instead of flux only oscillations of photons between resonator and quantum dot, i.e. the system hs is not "open" enough.
(b) Since one our aim is to model a light-emitting device, the system hs needs an external source of energy to pump it into the dot, which will be transformed by interaction (2.5) into the outgoing photon current by pumping the resonator.
(c) To reach this aim we extend the standard Jaynes-Cummings model to our JCL-model by attaching to the quantum dot hs (2.2) two leads, which are (infinite) reservoirs of free fermions. Manipulating with electro-chemical potentials of fermions in these reservoirs we can force one of them to inject fermions in the quantum dot, whereas another one to absorb the fermions out the quantum dot with the same rate. This current of fermions throughout the dot will pump the dot and induce a photon current according scenario (b).
(d) The most subtle point is to invent a leads-dot interaction Vls, which ensures the above mechanism and which is simple enough that one would still be able to treat this JCL-model using our extension of the Landauer-Buttiker formalism.
2.2. The JCL-model
First, let us make some general remarks and formulate certain indispensable conditions when one follows the modeling (d).
(1) Note that since the Landauer-Buttiker formalism [13] is essentially a scattering theory on a contact between two subsystems, it is developed only on a "one-particle" level. This allows one to study with this formalism only ideal (non-interacting) many-body systems. We impose this condition on many-body fermion systems (electrons) in two leads. Thus, only direct interaction between different components of the system: dot-photons VSb and electron-dot VlS are allowed.
(2) It is well-known that fermion reservoirs are technically simpler to treat than boson ones [13]. Moreover, in the framework of our model, it is also very natural since we study electric current, although produced by "non-interacting electrons". So, below we use fermions/electrons synonymously.
(3) In spite of the precautions formulated above, the first difficulty to consider in an ideal many-body system interacting with quantized electromagnetic field (photons) is induced indirect interaction. If electrons can emit and absorb photons, it is possible for one electron to emit a photon that another electron absorbs, thus creating an indirect photon-mediated electron-electron interaction. This interaction makes it impossible to develop the Landauer-Buttiker formula, which requires a non-interacting framework.
Assumption 2.1. To solve this difficulty, we forbid in our model the photon-mediated interaction. To this end, we assume that every electron (in leads and in dot) interacts with its own distinct copy of the electromagnetic field. So, considering electrons together with their photon fields as non-interacting "composed particles", allows us to apply the Landauer-Buttiker approach. Formally, it corresponds to the "one-electron" Hilbert space hel 0 hph, where hph is the Hilbert space of the individual photon field. The fermion description of composed-particles hel 0 hph corresponds to the antisymmetric Fock space F-(hel 0 hph).
The composed-particle assumption 2.1 allows us to use the Landauer-Buttiker formalism developed for ideal many-body fermion systems. Now, we come closer to the formal description of our JCL-model with two (infinite) leads and a one-mode quantum resonator.
Recall that the Hilbert space of the Jaynes-Cummings Hamiltonian with two energy levels is HJC = C2 0F+ (C). The boson Fock space is constructed from a one-dimensional Hilbert space, since we consider only photons of a single fixed frequency. We model the electrons in the leads as free fermions residing on discrete semi-infinite lattices. Thus:
hel = f (N) © C2 © f(N) = hel 0 hs 0 herl, (2.10)
is the one-particle Hilbert space for the electrons and for the dot. Here, hf, a E {l, r}, are the respective Hilbert spaces of the left and right lead, while hS = C2 is the Hilbert space of the quantum dot. We denote by:
{Onen, {$n }1j=o,
the canonical basis consisting of individual lattice sites of հ„, a E {l,r}, and of hS, respectively. With the Hilbert space for photons, hph = F+(C) ~ ^2(N0), we define the Hilbert space of the full system, i.e. quantum dot with leads and with the photon field, as follows:
H = hel 0 hph = (/2(N) 0 C2 0 £2(N)) 0 ^2(No). (2.11)
Remark 2.2. Note that the structure of full space (2.11) takes into account the condition 2.1 and produces composed fermions via the last tensor product. It also manifests that electrons in the dot as well as those in the leads are composed with photons. This is different than the picture imposed by the the Jaynes-Cummings model, when only the dot is composed with photons:
H = ^2(N) 0 C2 0 £2(No) 0 f(N) , HJC = C2 0 £2(No) , (2.12)
see (2.4), (2.5) and (2.6), where HJC = hS 0 hph. The next step is a choice of interactions between subsystems: dot-resonator-leads.
According to (2.10), the decoupled leads-dot Hamiltonian is the matrix operator:
(hf
h
0 0 (ui\
hS 0 on и = us , {ua E f(N)}ae{,r} , Us E C2
0 hf) \Ur)
where hea = -AD + va with a constant potential bias va e R, a e {l, r}, and hs can be any self-adjoint two-by-two matrix with eigenvalues {AfJ, Xs := Xs + e}, e > 0, and eigenvectors {es, es}, cf (2.2). Here, AD denotes the discrete Laplacian on £2(N) with homogeneous Dirichlet boundary conditions given by:
(ADf )(x) := f (x + 1) - 2f (x) + f (x - 1), x E N, dom(AD) := {f E ^2(No): f (0) := 0},
which is obviously a bounded self-adjoint operator. Notice that Ծ(AD) = [0,4].
We define the lead-dot interaction for coupling gel e R by the matrix operator acting in (2.10) as follows:
( 0 (•Л1 )5[ 0 Հ Vel = gel Mi )S$ 0 (;5[)^ , (2.13)
V 0 (;5S )5{ 0 )
where non-trivial off-diagonal entries are projection operators in the Hilbert space (2.10) with the scalar product u,v ^ (u,v) for u,v e hel. Here, {5s} is ortho-normal basis in hll, which in general may be different from {efj, e!(}. Hence, interaction (2.13) describes quantum tunneling between leads and the dot via contact sites of the leads, which are supports of 5i and 5i.
Then we define the Hamiltonian for the system of interacting leads and dot as hel := hol + vel. Here, both hol and hel are bounded self-adjoint operators on hel.
Recall that photon Hamiltonian in the one-mode resonator is defined by operator hph = шЬ*Ь with domain in the Fock space F+(C) ~ ^2(No), (2.1). We denote the canonical basis in £2(N o) by {Тп}пе^. Then for the spectrum of hph one obviously gets:
Ծ(Մհ) = Ծ№(Մհ) = U {ПШ}. (2.14)
We introduce the following decoupled Hamiltonian Ho, which describes the system when the leads are decoupled from the quantum dot and the electron does not interact with the photon field:
Ho := Hel + Hph, (2.15)
where
Hel := hol 0 Ip and Hph := V 0 hph. The operator Ho is self-adjoint on dom(Ho) = dom^e; 0hph). Recall that hol and hph are bounded self-adjoint operators. Hence, He and Hel are semi-bounded from below, which yields that Ho is semi-bounded from below.
The interaction of the photons and the electrons in the quantum dot is given by the coupling of the dipole moment of the electrons to the electromagnetic field in the rotating wave approximation. Namely:
Vph = gph ((•,el)el 0 Ь + (•, el)es0 0 Ь*) , (2.16)
for some coupling constant gph e R. The total Hamiltonian is given by:
H := Hel + Hph + Vph = Ho + Vel + Vph, (2.17)
where Hel := hel 0 I^ph and Vel := vel 0 I^ph.
In the following, we call S = {H, Ho} the Jaynes-Cummings-leads system, in short JCL-model, which we are going to analyze. In particular, we are interested in the electron and photon currents for that system. The analysis will be based on the abstract Landauer-Buttiker formula, cf. [1,13].
Lemma 2.3. H is bounded from below self-adjoint such that dom(H) = dom (Ho).
Proof. Let c ^ 2. Then,
||ЬХп||2 ^ ||Ь*Хп||2 = n +1 ^ c-in2 + c, n e No. Consider elements f e hS 0 hph Ո dom(Ihe; 0 hph) with the following:
f = EPie 0 Y, j e{0,1}, l e No,
j,l
which are dense in HJC := hS 0hph. Then, ||f ||2 = Ец Ш՜2 and ||(V ||2 = Е^вц^2.
We obtain the following:
H((;eS)eS 0 V)f ||2 ^ Е!вц|21ЬТ,|2 ^
j,l
Eli|2(c-il2 + c) = c-i ||(Ihe; 0 Ь*Ь) f ||2 + cr՝12 j,i
Similarly,
||((•,eS)e'S 0 Ь*)f ||2 ^ c-i||(V 0 Ь*Ь)f ||2 + c| If c ^ 2 is large enough, then we obtain that Vph is dominated by Hph with relative bound less than one. Hence, H is self-adjoint and dom(Ho) = dom(H). Since He and Vel are bounded and Hph is self-adjoint and bounded from below, it follows that H = He + Hph + Vel + Vph is bounded from below [17, Thm. V.4.1]. □
2.3. Time reversible symmetric systems
A system described by the Hamiltonian H is called time reversible symmetric if there is a conjugation Г defined on H such that ГН = HГ. Recall that Г is a conjugation if the conditions Г2 = I and (rf, rg) = (fg), f,g e H.
Let hpn, n e No, the subspace spanned by the eigenvector Тп in hph. We set:
Нпа := hea 0 hnh, n e No, a e {l,r}. (2.18)
Notice that
H = 0 H
)па
пе^^0,ае{1,т]
Definition 2.4. The JCL-model is called time reversible symmetric if there is a conjugation Г acting on H such that H and Ho are time reversible symmetric and the subspaces Hna, n e No,
a e {l,r}, reduces Г.
Example 2.5. Let y1 and yS1 be conjugations defined by the following:
la fa := fa := [ЮЩ}кеn, fa e hf, a e{l,r},
and
Yelf = Yel (fs (0)\ := [МЩ
Ysfs = Ys{fS (1)) = f(1)J
We set դel := yIl ф is Ф yIl. Further, we set:
YphФ := Ф = {^(n)}nеno, Ф e hph. Let Г := iel 0 iph. One easily checks that Г is a conjugation on H = he l 0 hph.
Lemma 2.6. Let yO, a e {S,l, r}, and Yph be given by Example 2.5.
(i) If the conditions yS es = es and ySS es — es are satisfied, then Ho is time reversible symmetric with respect to Г and, moreover, the subspaces H-n,a, n e No, a e {l,r}, reduces Г.
(ii) If in addition the conditions Ys^o = and yS1= ^f are satisfied, then JCL-model is time reversible symmetric.
Proof. (i) Obviously we have
Yf hel = heY, a E {l, r}, and Yphhph = hphYph.
If yS1 eS = eS and YSleS = e\ are satisfied, then YSlhes = hSSyS1 which yields Yelh0l = h0lYel and, hence, ГЯ0 = ГЯ0. Since YelhOl = ha and Yphhph = hph one gets ГHna = Hna which shows that Hna reduces Г.
(ii) Notice that y^i = ^f, a E {l,r}. If in addition the conditions Ys^o = and Ysl$S = are satisfied, then Yelvel = velYel is valid, which yields Yelhel = helYel. Hence, ГЯ = HГ. Together with (i), this proves that the JCL-model is time reversible symmetric. □
Choosing the following:
(1) ■ * (0) ' « ■ ^ Ъ է-' (219)
one satisfies the condition YsleS = eS and Ys eS = e\ as well as YseS = eS and Ys eS = eS. 2.4. Mirror symmetric systems
A unitary operator U acting on H is called a mirror symmetry if the following conditions are met:
UHna = Hna,, a, a' E{l,r}, a = a' are satisfied. In particular, this yields UHJC = HJC, HJC := hSl 0 hph.
Definition 2.7. The JCL-model is called mirror symmetric if there is a mirror symmetry commuting with H0 and H.
One can easily verify that if H0 is mirror symmetric, then
Hna, U = UHna, n E N0, a, a' E {l,r}, a = a',
where
Hna := hf 0 Ihph + 0 hpnh = Հ + nu, n E No, a, a' E {l,r}, a = a'.
In particular, this yields that va = va>. Moreover, one gets UHS = HsU where Hs := hSj 0 Цр^ +
Ihel 0 hph.
Notice that if H and H0 commute with the same mirror symmetry U, then also the operator Hc := hsl 0 Ihph + Ihel 0 hph also commutes with U, i.e, is mirror symmetric.
Example 2.8. Let S = {H,H0} be the JCL-model. Let vl = vr and let efi and e՛? as well as and ծք be given by (2.19). We set:
us eS := eS and u£sleS = -eS, (2.20)
as well as
uphYn = e-innYn, n E No. (2.21)
Obviously, Us := ussl 0uph defines a unitary operator on HJC. Straightforward computation shows that:
Us Hs = Hs Us and Us Vph = VphUs. (2.22)
Furthermore, we set:
usX := 8rn, and ufr8rn = 8ln, n E N, (2.23)
14 and
u
(0 0 uer)
0 us 0
\uer 0 0
We thus have the following:
Vel U
el
ffl\
fs
\fr)
( < fs, (us )4s >s[ ^
< fr, ur )*Հ >ss + <fi, ui y^ >5ss
V < fs, (usyss > 51 ,
(2.24)
Since 5s := ֊֊շ (ess + Պ) and 51 := ֊շ (ess - Գ) we get from (2.20)
(us )*5'0 = 5s and (uD^s = 5s.
Obviously, we then have
(utr ) + 5l = 51 (ufi)փ 51 = 51.
Inserting (2.25) and (2.26) into (2.24), we find
Vel u
el
Further, we have:
el
uelVel
fl fs
fr
fl fs
fr
/ <fs ,5s >5[ \
<fr,5Г >5s + <f],5[ >5s
<fs ,5s >5Г
(2.25)
(2.26)
(2.27)
( < fs,5s >5[ \ <fi,5[ >5s + <fr,5Г > 50
< fs, 5s >5r
(2.28)
Comparing (2.27) and (2.28), we get uelvel = veluel. Setting U := uel ® uph one immediately
proves that UH0 = H0U and UH = HU. Since UH symmetric.
= Hna,, it is satisfied that S is mirror We note that Example 2.8 S is also time-reversible symmetric.
2.5. Spectral properties of H: first part
In the following, our goal is to apply the Landauer-Buttiker formula to the JCL-model. By Lp(H), 1 Հ p Վ ж, we denote in the following the Schatten-v.Neumann ideals.
Proposition 2.9. If S = {H, Ho} is the JCL-model, then (H + i)-1 - (Ho + i)-1 e Li(H). In particular, the absolutely continuous parts Hac and Щс are unitarily equivalent.
Proof. We have
(H + i)-1 - (Ho + i)-1 = (Ho + i)-1V(H + i)
1
1
(Ho + i)-1 V(Ho + i)-1 - (Ho + i)-1V(Ho + i)-1V(H + i)
where V = H - Ho = Vel + Vph. Taking into account Lemma 2.3, it suffices to prove that (Ho + i)-1V(Ho + i)-1 e L1(H). Using the spectral decomposition of hph with respect to hph = 0neNo hnh, where hnh are the subspaces spanned by Yn, we obtain the following:
(Ho + i)-1 =0 (heol + nu + i)-1 0 I
n£n0
hpnh.
(2.29)
We have (Ho + i)-1V(Ho + i)-1 = (Ho + i)-1 V + Vph)(Ho + i)-1. Since vdl is a finite rank operator, we have ||vel||Ll < <x>. Furthermore, հՈ is obviously one-dimensional for any n G N0. Hence, ll^ph ||ll = 1. From (2.29) and Vei = vel 0 I^k., we obtain the following:
||(Ho + i)-'Vei(Ho + i)-1 ||l1 = E ll(h0l + nu + i)-1vei(h0l + nu + i)-1 ||լ
neno
^ E H(ho + nu + i)-2|| Hvel |li
nen0
Since hei is bounded, we get:
||(hfl + nu + i)-1|| = sup (V(A+nU^+I)-1 ^ c(n +1)-1, (2.30)
xea(hol)
for some c > 0. This immediately implies that ||(Ho + i)-1Vel(Ho + i)-1||Ll <
We are going to handle (Ho + i)-1Vph(Ho + i)-1. Letpn be the projection from hph onto hnh. We have the following:
(Ho + i)-1 (;eS)ef 0 b (Ho + i)-1
= E (հ^ + mu + i)-\,eS)ef (hf + nu + i)-1 0 f^bpt
m,neNo
= E (hf +(n - 1)u + i)-1 (•,eS0 )eSS (hf + nu + i)-1 0 /Л-^, Tn>
nen
From (2.30), we get the following:
(hef + (n - 1)u + i)-1 (•, eS)eSS (hef + nu + i)-1) 0 /ПГп^ Y> n G N, which yields:
^ c
2
n
Ll n(n + 1)
||(Ho + i)-1 (•, eS)eS 0 b (Ho + i)-1 ||լ ^ c2 £ ֊Ո+֊) < rc.
nen n\n ՝ լ)
Since
||(Ho + i)-1 (•, eS)eS 0 b* (Ho + i)-1^ = ||(Ho + i)-1 (•,eS0 )ef 0 b (Ho + i)-1^, one gets (Ho + i)-lVph(Ho + i)-1 G L1(H), which completes the proof. □
Thus, the JCL-model S = {H,Ho} is a L1-scattering system. Let us recall that hea =
D I П „Л „„ uel _ uel _ f)2t
-AD + va, a G {l, r}, on hf = he = i Lemma 2.10. Let a G {l,r}. We have the following:
°(hf) = Vac(hf) = [va, 4 + va]. The normalized generalized eigenfunctions of hea are given by:
ga(x, A) = n-2 (1 - (-A + 2 + va)2/4)-4 sin (arccos((-A + 2 + va)/2)x)
for x G N, A G (va, 4 + va).
Proof. We prove the absolute continuity of the spectrum by showing that:
{ga(x,A) | A g (-2,2)} is a complete set of generalized eigenfunctions. Note that it suffices to prove the lemma for
((Ad + 2)f )(x) = f (x + 1) + f (x - 1), f (0) = 0.
The lemma then follows by replacing A with -A + 2 + va. Let A e (-2,2) and
gAD (x, A) = n՜2 (1 - A2/4)-4 sin (arccos(A/2)x).
Note that gAD (0, A) = 0, when the boundary condition is satisfied. We substitute - =
arccos(A/2) e (0, n), i.e. A = 2 cos(-) and obtain the following:
sin(-(x + 1)) + sin(-(x - 1)) = 2sin(-x) cos(-),
when gAn (x, A) satisfies the eigenvalue equation. It is obvious that gAD (•, A) e ^2(No) for A e (-2, 2). To complete the proof of the lemma, it remains to show the ortho-normality and the completeness. For the ortho-normality, we have to show that
E gAD (x,X)gAn (x, v) = 5(A - v).
xeN
Let վ e Co°°((-2, 2)). We use the substitution - = arccos(v/2) and the relation
sin(arccos(y)) = (1 - y2)՜2
to obtain the following:
dv E gAD (x, A)gAD (x,v)^(v)
22
xen
, гж ^ sin(-) sin ( arccos(A/2)x) sin(-x)
2п I d^ E֊--\--—ф(2 cos(-))
Jo xeN (sin(-)) 2 (sin(arccos(A/2))) 2
(2n)-i fП d- E . (sin(-)) 1-г (>rccos(A/2)-M)x +
'o xen (sin(arccos(A/2))) 2
e—i(arccos(A/2)—^)x - ei(arccos(A/2)+^)x __ e-i(arccos(A/2)+^)x J^(2cos(-))
Observe that for the Dirichlet kernel:
E (eixy + e-ixy) - 1 = 2n 5(y),
xeno
when
r 2
dv E gAD (x, A)gAD (x, v)^(v) =
՜2 xen
fП d--(sin(-)) 2-r (5(arccos(A/2) - -) + 5(arccos(A/2) + -)) ^(2cos(-)) = ф(А).
Jo (sin(arccos(A/2))) 2
In the second equality we use that the summand containing 5(arccos(A/2) + -) is zero since both arccos(A/2) > 0 and - > 0. Thus, the generalized eigenfunctions are orthonormal. Finally, using once more the substitution - = arccos(v/2), we obtain the following:
J dv gAD (x,v)gAD (y,v) =
r2 , 1
\Л 2
2
dv - (v/2)2j 2 sin (arccos(v/2)x) sin ( arccos(v/2)y)
rn
2n-i / d- (sin(-))-i sin(-)sin(-x) sin(-y) = 5xy, Jo
for x,y e N, when the family of generalized eigenfunctions is also complete. □
From these two lemmas, we obtain the following corollary that gives us the spectral properties of H0.
Proposition 2.11. Let S = {H, H0} be the JCL-model. Then, a(H0) = aac(H0) Uapp(H0), where aac(H0) = (J [vl + nu,vl + 4 + nu] U [vr + nu, vr + 4 + nu]
nen0
and
&pp(Ho) = U A + nu : j = 0,1}.
nen0
The eigenvectors are given by g(m, n) = eԼ 0 Yn, m = 0,1, n E N0. The generalized eigenfunc-tions are given by ցռէ, A,n) = ga(^,A — nu) 0 Yn for A E aac(H0), n E N0, a E {l, r}.
Proof. It is well known (see e.g. [15]) that for two self-adjoint operators A and B with asc(A) =
asc(B) = 0, we have asc(A 0 1 + 1 0 B) = 0,
aac(A 0 1 + 1 0 B) = (aac(A) + a(B)) U (a(A) + aac(B))
and
app(A 0 1 + 1 0 B) = app(A) + app(B).
Furthermore, if фа(^а) and фв(^в) are (generalized) eigenfunctions of A and B, respectively, then фА(ХА) 0 фв(Хв) is a (generalized) eigenfunction of A 0 I + I 0 B for the (generalized) eigenvalue Aa + XB.
The lemma follows now with A = hf and B = hph using Lemmata 2.10 and (2.14) and the fact that hs has eigenvectors {ef, es} with eigenvalues {Af, A s = Af + e}. □
2.6. Spectral representation
For the convenience of the reader, we define here what we mean under a spectral representation of the absolutely continuous part K0^c of a self-adjoint operator K0 on a separable Hilbert space K. Let k be an auxiliary separable Hilbert space. We consider the Hilbert space L2(R, dA, k). By M, we define the multiplication operator induced by the independent variable A in L2(R, dA, k). Let Ф : Kac(Ko) —> L2(R, dA, k) be an isometry acting from Kac(Ko) into L2(R, dA, k) such that: Фdom(KQC) С dom(M) and
МФf = §Kacf, f E dom(K0ac).
Obviously, the orthogonal projection P := ФФ* commutes with M which yields the existence of a measurable family, {P(A)}agr, such that:
(Pf )(A) = P(A) f (A), f e L2(R, A, k).
We set L2(R, dA, k(A)) := PL2(R, A, k), k(A) := P(A)k, and call the triplet
n(Kac) := {L2(R,dA, k(A)), M, Ф}
a spectral representation of K0ac. If {L2(R, dA, k(A)), M, Ф} is a spectral representation of Kac, then Kac is unitarily equivalent M0 := M \ L2(R, dA, k(A)). Indeed, one has Ф^^* = M0. The function (A) := dom(k(A)), A E R, is called the spectral multiplicity function of K0ac. Notice that 0 Վ է0 (A) Վ ж for A E R.
For a E {l, r}, the generalized eigenfunctions of h^ define generalized Fourier transforms by Փէ : hs = h£a'ac(hi) ^ L2([va,va + 4]) and
(Фа fa )(A) = E ց» (X,A)fa (x), fa E hf. (2.31)
xeNo
We then set:
haW
(2.32)
C Л e [va,va + 4] 0 Л e R \ [va,va + 4].
One can easily verify that n(heJ;) = {L2(R,dW, ha(W)), М,фО;} is a spectral representation of ha = hea,ac, a = l,r, where we always assumed implicitly that (ф^f a)(Л) = 0 for Л e R \ [v a,v a + 4]. Setting:
hel (Л)
hel(W) := 0 С C2,
hel (Л)
Л R,
(2.33)
and introducing the map:
фЫ : hel,ac(hol)
hel
L2
if
,d\, hel(W)),
defined by:
ф-f = (f
where f :=
rr el,ac
fl fr
(2.34)
(2.35)
we obtain a spectral representation n(ho 'ac) = {L2(R, dЛ, h(W)), М,ф} of the absolutely continuous part heol'ac = he 0 he of Հ'. One easily verifies that 0 Վ Հ™լ (Л) Վ 2 for Л e R.
Introducing:
Л^ш := min{vl,vr} and Л^ := max{vi + 4,Vr + 4}, one easily verifies that ^ (Л) = 0 for Л e R \ [ЛЛ^].
(2.36)
Notice, if vr + 4 Վ vl, then
he1(Л)= fc, Л e [Vr,Vr + 4] U [Vl,Vl + 4], I {0}, otherwise
which shows that hel has a simple spectrum. In particular, it holds ^ (Л) = 1 for Л e [vr ,vr +
4] U [vl,vi + 4] and otherwise (Л) = 0.
Let us introduce the Hilbert space h := l2(No,
C2)
Regarding he1(Л - nu) as a subspace of hn, one regards:
®n£n0 hn, hn
C2, n N0.
h(Л) := ф hn(Л), hn(Л) := hel(Л - nu), л e r,
n£no
(2.37)
as a measurable family of subspaces in h. Notice that 0 ^ dim(h(W)) < ж, Л e R. We consider the Hilbert space L2(R, dW, h(W)).
Furthermore, we introduce the isometric map Ф : H(Hac) —> L2(R, dW, h(W)) defined by
(*f )(Л) = e (((ФttlflՈ)(fЛ - ^ , Л e R, V Л J n^0 Wrfr (n))(W - nu))՝
(2.38)
where
e (ft)) e e hei,ac(hei) 0 h
n£n0 VJr^ 4 n£n0
Ph = ЛЛ
n
n£n
(he 0 hpnh \
„el,ac (iel\
՝ 0 hn = Ш
hrl 0 hpnh
where hph = 0neN0 hnh and hnh is the subspace spanned by the eigenvectors Yn of hph. One easily verifies that Փ is an isometry acting from Hac(Hac) onto L2(R, dW, h(W)).
Lemma 2.12. The triplet {L2(R, dA, h(A)), M, Ф} forms a spectral representation of Щс, that is, n(Hoac) = {L2(R, dA, h(A)), M, Ф} where there is a constant d G No such that 0 ^ (A) ^
дel _дel , ,
2dmax for A G R where dmax := maX- min and Amax and Amin are given by (2.36).
Proof. It remains to be shown that Ф transforms H^c into the multiplication operator M. We have
(hflfl)(n) + nufl(n
Ho J = Հ37 I
neno
which yields the following:
Haf = n® \(hflfr)(n) + nufr(n)
(&Hacf)(\)= Ш ((Փք (hflfl)(n))(A - nu)+ nu^ffl(n))(A - nu)\ ( o f )(A) = ^ Wfl(hffr)(n))(A - nu) + nu^ffr(n))(A - nu))
= e (A(f\\(fA - ""D = (МФf )(A), A G R.
S0 Welfr("))(a - nu)) v յ)ՀԻ
which proves the desired property.
One easily checks that h(A) might only be only non-trivial if A - nu G Xmin, Amax]. Hence, we obtain that h(A) is non-trivial if the condition:
A- Ael A- Ael■
'՝ "max հ n ^ min
uu
is satisfied. Hence,
{A — Ael A — Ael 1
n G No : A Amax ^ n Վ A Amin\ , A G R.
uu
or
{Ael ֊ Ael 1
n G No : 0 ^ n Հ max - max I , A G R.
Hence 0 Վ H (A) Վ dmax for A G R. □
In the following we denote the orthogonal projection from h(A) onto hn(A) by Pn(A), A G R, cf (2.37). Since h(A) = ®nen0 hn(A) we have %) = Enen0 Pn(A), A G R. Further, we introduce the following subspaces:
hna(A) := ha(A - nu), A G R, n G No.
Notice that:
hn(A) = e hna(A), A G R, n G No.
ae{l,r}
By Pna(A) we denote the orthogonal projection from h(A) onto hna (A), A G R. Clearly, we have
Pn (A) = Eae{l,r} Pna (A), A G R.
Example 2.13. In general, the direct integral n(H^c) can be very complicated, in particular, the structure of h(A) given by (2.37) is difficult to analyze. However, there are interesting simple cases: (i) Let v = vl = vr and 4 Վ u. In this case we have hel(A) = C2 for [v,v + 4] and
h(A)
C2, A G [v + nu, v + nu + 4], n G N {0}, otherwise.
(ii) Let vr = 0, vl = 4, ш0 = 4. Then
h(A)
(herl (A)
ЫГ (A) hei(A)
C,
C2,
C2,
A e [0,4), A e [4,8), A e [8,12),
where
Ъаа> (A) =
heJ (A)
Ф ,
ha (A)
a, a' e {l,r}, a = a'.
Hence, dim(h(A)) = 2 for A ^ 4.
□
Let Z be a bounded operator acting on Hac(H0) and commuting with Hac. Since Z commutes with H0ac there is a measurable family {Z(A)}AeR of bounded operators acting on h(A) such that Z is unitarily equivalent to the multiplication operator induced by {Z(A)}agr in n(H0՝c). We then set:
ZmarK (A):= Pma (A)Z(A) \ h пк (A), A e R, m,n e No, a, к e{l,r}.
Let Zma пк := PmaZPn„ where Pma is the orthogonal projection from H onto Hma ^ Hac(H0), cf. (2.18). Clearly, the multiplication operator induced {Zman>t(A)}AeR in n(H^c) is unitarily equivalent to ZmaՈк.
Since, by Lemma 2.12, h(A) is a finite dimensional space, the operators Z(A) are finite dimensional ones and we can introduce the following quantity:
Vma-пк(A) := ^^апк(AfZma^(A)), A e R, m, n e No, a, к e{l,r}.
Lemma 2.14. Let H0 be the self-adjoint operator defined by (2.15) on H. Furthermore, let Z be a bounded operator on Hac(H0) commuting with Щc
(i) Let Г be a conjugation on H, cf. Section 2.3. If Г commutes with H0 and Pma, n e N0,
a e {l,r} and ГZГ = Z* holds, then omanH(A) = апкma (A), A e R.
(ii) Let U be a mirror symmetry on H. If U commutes with H0 and Z, then &тапк (A) = ama,(A), A e R, m,n e N0, a, a', к, К e {l, r}, a = a', к = К.
Proof. (i) Since Г commutes with H0 the conjugation Г is reduce by Hac(H0). So without loss of generality, we assume that Г acts on Hac(H0). We set Гпа := Г \ Hпа. Notice that:
Г = ф Гпа.
пе!^о,ае{1,г}
There is a measurable family {Г(A)}лeR of conjugations such that the multiplication operator induced by {Г(A)}лeR in n(Hac) is unitarily equivalent to Г. Moreover, since Г commutes with Pna, we see that the multiplication operator induced by the measurable family:
Гпа(A):=T(A) \ h^(A), A e R, m e N0, a e {l,r}, ..Using TZ Г = Z * we get Pma Zm
^Tm-a (A)ZmaՈк (A)ГՈк (A) = Zпк ma (A)*, A e R.
is unitarily equivalent to Г^. Using YZ Г = Z * we get Г ma Zmaux Гпк = ZՈк ma. Hence,
(2.39)
If X is a trace class operator, then ^(ГХГ) = tr(X). Using that we find
пк (A) = ЦГ
пк (A)Zm a пк
(A)*
пк
(A)) =
tr(ГՈк (A)ZmaՈк (A)*Гmа Гma ZmaՈк (A)ГՈк (A))
From (2.39), we obtain the following:
(X) = tr(Zn
к ma к ma m = Пк "m^
(X), X E R,
which proves (i).
(ii) Again, without loss of generality we can assume that U acts only Hac(H0). Since U commutes with H0, there is a measurable family {U(X)}AeR of unitary operators acting on h(X) such that the multiplication operator induced by {U(X)}AeR is unitarily equivalent to U. Since
UHna = Hna, we have U(X)hna (X) = hna, (X), X E R. Hence,
Vmn (X) = tr(U (X)ZmanK (XX)* Zman к (X)U (X)*) = tr(U (X)Zma,nк (X)*U (X)*U (X)Zma^ (X)U (X)*).
Hence,
(X) = tr(PnH,U(X)Z(X)*U(X)*Pmaf (X)U(X)Z(X)U(XyPnH, (X)). Since U commutes with Z, we find that:
Oma^ (X) = tr(PnH, Z(X)*Pma, (X)Z(X)PnH, (X)) = ^nH, (X) , X E R, which proves (ii). □
2.7. Spectral properties of H: second part
Since we have full information for the spectral properties of H0, we can use this to show that H has no singular continuous spectrum. Crucial for that is the following lemma: with the help of [6, Cor. IV.15.19], which establishes existence and completeness of wave operators and absence of singular continuous spectrum through a time-falloff method. We cite it as a Lemma for convenience, with slight simplifications that suffice for our purposes.
Lemma 2.15 ([6, Corollary IV.15.19]). Let {H0, H} be a scattering system and let Л be a closed countable set. Let F+ and F- be two self-adjoint operators such that F+ + F- = PHc0 and
s - lim eTltH° F±e±itH° = 0. If (H - i)-1 - (Ho - г)-1 E L^(H), (1 - PHc0)Y(Ho) E L^(H), and
d*
((Ho - г)-1 - (H - i)-1)e-itH0y(Ho)F±
< oo
for all y E C0°(R \ Л), then W±(H, H0) exist and are complete and osc(H) = osc(H0) = է. Furthermore, each eigenvalue of H and H0 in R \ Л is of finite multiplicity and these eigenvalues accumulate the most at points of Л or at
We already know that the wave operators exist and are complete since the resolvent difference is trace class. Hence, we need Lemma 2.15 only to prove the following proposition.
Proposition 2.16. The Hamiltonian H defined by (2.17) has no singular continuous spectrum, that
is, Osc(H) = է
Proof. At first we have to construct the operators F±. To this end, let F : L2(R) ^ L2(R) be the usual Fourier transform, i.e
(Ff )(թ) := f (թ) := — f e~iv"xf (x)dx, f E L2(R,dx), թ E R.
\/2n JR
Further, let П± be the orthogonal projection onto L2(R±) in L2(R). We set:
F± =
where Փ is given by (2.38). We immediately obtain F- + F+ = Pac(H0). We still have to show that:
s — Hm \\eTitHo<£*Fn±F*Фв±гШ° f || = 0
for f E Hac (H0). We prove the relation only for F+ since the proof for F- is essentially identical. We have the following:
(п+ГФегШо f)(x) = (2n)-2 xr+ (x) Լ ք(թ) = xr+ Ш(х + t)
with ф = F f. Now,
\\e-itHo $*Fn+F*$eltHo f \\2 = \\n+F *§eitHo f \\2 =
dx
ф(х +1)
dx
ф(х)
t
0.
Concerning the compactness condition, we already know that (H — i) 1 — (H0 — i) 1 E Li(H) С from Proposition 2.9. Let
Л := լ| [vl + nu, vr + nu,vl + 4 + nu, vr + 4 + nu},
which is closed and countable. We know from Corollary 2.11 that H0 has no singular continuous spectrum and the eigenvalues are of finite multiplicity. It follows then that (1 — Pac(H0))j(H0) is compact for every y E C^(R \ Л). The remaining assumption of Lemma 2.15 is as follows:
Լ dt |((H — i)-1 — (Ho — i)-1)y(Ho)e-itHoF±
<00.
■ ph
If we can prove this, then we immediately obtain that H has no singular continuous spectrum.
Now (H — i)-1 — (Ho — i)-1 = (H — i)-1(Vel + Vph)(Ho — i)-1. But (H — i)-1 is bounded,
ran(F±) С Hac(Ho) = (hel © Kl) 0 h and VphPac(H0) = 0. Also, Vel = vel 0 Ihph and
ker(vei)± С 01 © hs © .
Hence, it suffices to prove:
dt
Pa(Ho — i)-1Y(Ho)e-itHo F±
<,
a E [l, r}, where P? = pi 0 կբհ and p^ is the orthogonal projection onto he. In the following we treat only the case F+. The calculations for F- are completely analogous. We use that Փ maps HSc into the multiplication operator M induced by Л. Hence, we get the following:
PaY(H0)e-itHo <&*Ff | = )e-itHo <&*Ff | =
= (2n)-2 I dЛдa(1,Л — пш)%Л) f dx e-iX(x+t) f(x)2) 2'
JSa.n "'r+ /
4n£no
where supp (f) С R+, Y(Л) := (Л — i)-1j(Л), Л E R, and that Y(Л) E C0°°(R \ Л). We find:
[va + nu0 ,va + nu + 4]. Notice
I' dЛga(1,Л — nu)Y(Л) I' dxe-iX(x+t)f(x) =
J J r+
Ր
dЛ ga(1, Л)7(Л + nu) dx e-i(x+™)(x+t) f (x)
J r+
CVa+4
t
+
which yields:
P1aФ*Ф7 (Ho )e 0 Փ*7՜ f
1 , ^ Г va +4
(2n)-2 E
vnen0
d A ga(1, A)Yy(A + nuo) dx e-i(X+nM0)(x+t) f (x)
Since the support of y(A) is compact, we see that the sum EneN0 is finite. Changing the integrals, we get:
[ dA ga(1, A - nu)Y(A) f dx e-iX(x+t) f (x)
J San JR +
-in^0(x+t)
dx f(x)e
Integrating by parts m-times, we obtain:
+
Va+4
dA ga(1, A)y(A + nu)e
-iX(x+t)
j dA ga(1, A - nu)Y(A) f dx e-iX(x+t) f (x) =
■jSan ./r+
p-inw(x+t) rva+4 dm
■J dA e-iX(x+t) — (ga(1, A)j (A + nu))
(-i)m
dxf(xh , ,v
J+ (x + t)m
Hence,
[ dA ga(1, A - nu)Y(A) f dx e-iX(x+t) f (x) JSa.n ֊'R +
^ C
dx |f (x)I՜— «+ ^ л (x + t)m
which yields:
( dA ga(1,A - nu)Y(A) ( dxe-iX(x+t) f (x)
JSa.n ֊'R +
21 ^ cn
t(2m-1)
for m N where:
Cn :=
fVa+4
d
dm
d m
ga(1, A)y(A + nu)
Notice that Cn = 0 for sufficiently large n G N. Therefore
/ \ 1/2
PaY(Ho)e-itH0f I ^ E C2n
nen0
1
tm-1/2
f G L2(R+,dx),
which shows that
PfY(Ho)e-itH0F+ G L1(R+ ,dt) for m ^ 2.
□
3. Landauer-Buttiker formula and applications 3.1. Landauer-Buttiker formula
The abstract Landauer-Buttiker formula can be used to calculate currents through devices. Usually one considers a pair S = {K, Ko} be of self-adjoint operators where the unperturbed Hamiltonian Ko describes a totally decoupled system, that means, the inner system is closed and the leads are decoupled from it, while the perturbed Hamiltonian K describes the system where the leads are coupled to the inner system. An important component is system S = {K, Ko}, which represents a complete scattering or even a trace class scattering system.
In [1], an abstract Landauer-Buttiker formula was derived in the framework of a trace class scattering theory for semi-bounded self-adjoint operators which allows one to reproduce the results of [18] and [7] rigorously. In [13], the results of [1] were generalized to non-semi-bounded operators. Following [1], we consider a trace class scattering system S = {K, Ko}. We
2
+
+
a:
2
2
2
a
recall that S = {K,K0} is called a trace class scattering system if the resolvent difference of K and K0 belongs to the trace class. If S = {K, K0} is a trace class scattering system, then the wave operators W±(K,K0) exist and are complete. The scattering operator is defined by S(K, K0) := W+(K, K0)*W-(K, K0). The main components, besides the trace class scattering system S = {K, K0}, are the density and the charge operators p and Q, respectively.
The density operator p is a non-negative bounded self-adjoint operator commuting with K0. The charge Q is a bounded self-adjoint operator commuting also with K0. If K has no singular continuous spectrum, then the current related to the density operator p and the charge Q is defined as follows:
Հց = —г tr (W-(K, K)pW- (K, K0)* [K, Q]), (3.1)
where [K, Q] is the commutator of K and Q. In fact, the commutator [K, Q] might be not defined. In this case, the regularized definition:
JSpq = -г tr (W- (K, K0)(I + K2)pW- (K, K0)* — [K, Q] ֊) , (3.2)
is used, where it is assumed that (I + K^)p is a bounded operator. Since the condition (H — i)-1[H, Q](H + г)-1 e Li(H) is satisfied, definition (3.2) makes sense. By £i(H) is the ideal of trace class operators is denoted.
Let K0 be self-adjoint operator on the separable Hilbert space K. We call p be a density operator for K0 if p is a bounded non-negative self-adjoint operator commuting with K0. Since p commutes with K0, one sees that p leaves invariant the subspace Kac(K0). We then set
pac := p \ Kac (K0).
call pac the ас-density part of p.
A bounded self-adjoint operator, Q, commuting with K0, is called a charge. If Q is the charge, then:
Qac := Q \ Kac(K0),
is called its ac-charge component.
Let n(K0ac) = {L2(R,dA, k(A)), M, Ф} be a spectral representation of K^c. If p is a density operator, then there is a measurable family {pac(A)}AeR of bounded self-adjoint operators such that the multiplication operator:
(MPac f )(A) := paC(A) f (A), f e dom(MPac) := L2(R, dA, k(A)),
is unitarily equivalent to the ас-part pac, that is, MPac = ФpаcФ*. In particular, this yields that: ess-sup Aerllpac(A)||B(t(A) = \\pac\\B(Rac{Ko)). In the following, we call {pac(A)}aer the density matrix of pac.
Similarly, one obtains that if Q is a charge, then there is a measurable family {Qac(A)}AeR of bounded self-adjoint operators, such that the multiplication operator:
(MQac f )(A) := Qac(A) f (A),
f e dom(Qac) := {f e L2(R, dA, k(A)) : Qac(A) f (A) e L2(R, dA, k(A))},
is unitarily equivalent to Qac, i.e. MQac = ФQаcФ*. In particular, one has:
ess-sup agR ||Qac(A) llB(k(A)) = l\QacllB(Kac(Ko)). (3.3)
If Q is a charge, then the family {Qac(A)}AeR is called the charge matrix of the ac-component of Q.
Let S = {K, K0} be a trace scattering system. By {S(A)}agR, we denote the scattering matrix, which corresponds to the scattering operator S(K, K0) with respect to the spectral representation n(K0ac). The operator T := S(K,K0) — Pac(K0) is called the transmission operator.
By {T(A)}AeR, we denote the transmission matrix which is related to the transmission operator. Scattering and transmission matrices are related by S(A) = Tk(A) + T(A) for a.e. A e R. Notice that T(A) belongs for to the trace class a.e. A e R.
Theorem 3.1 ([13, Corollary 2.14]). Let S := {K, K0} be a trace class scattering system and let {S(A)}AeR be the scattering matrix of S with respect to the spectral representation П(Кас). Furthermore, let p and Q be density and charge operators and let {pac(A)}AeR and {Qac(A)}AeR be the density and charge matrices of the ac-components pac and charge Qac with respect to П(Кас), respectively. If (I + Ko2)p is bounded, then the current JpQ defined by (3.2) admits the representation:
JpQ = 2n Լ tr{pac(A)(Qac(A) - S*(A)Qac(A)S(A)))dA, (3.4)
where the integrand on the right side and the current JpQq satisfy the estimate:
\tr(pac(A)(Qac(A) - S*(A)Qac(A)S(A)))\ Վ (3.5)
^ipWiilwojht (A)||£l(k(A))||Q(A)||£(k(A)),
for a.e. A e R and
Jsp,q\ Հ CoII(H + г)-1 - (Ho + i)-1||Li(K), (3.6)
where Co := П||(l + H2)pHm.
In applications, not every charge Q is a bounded operator. We say the self-adjoint operator Q commuting with K0 is a p-tempered charge if Q(H0 - i)-p is a bounded operator for p e N0. As above, we can introduce Qac := Q \ dom(Q) Ո Kac(K0). It follows that QEKo (A) is a bounded operator for any bounded Borel set A. This yields that the corresponding charge matrix, {Qac(A)}AeR, is a measurable family of bounded self-adjoint operators such that:
ess-supa£r(1 + A2)p/2||Qac(A)||£(e(A)) < ж.
To generalize the current JpQq to tempered charges Q, one uses the fact that Q(A) := QEKo (A) is a charge for any bounded Borel set A. Hence, the current Jpq(a) is well-defined by (3.2) for any bounded Borel set A. Using Theorem 3.1 one gets that for p-tempered charges, the limit
JP, q := Am JP, q(a) (3.7)
exists, provided (H0 — i)p+2p is a bounded operator. This gives rise to the following corollary:
Corollary 3.2. Let the assumptions of the Theorem 3.1 be satisfied. If for some p e N0 the operator (H0 — i)p+2p is bounded and Q is a p-tempered charge for K0, then the current defined by (3.7) admits the representation (3.4), where the right hand side of (3.4) satisfies the estimate (3.5). Moreover, the current JpQq can be estimated in the following manner:
\JSp,q\ Հ Cpll(H + г)-1 - (Ho + i)-1||Li(K), (3.8)
where Cp := П||(l + H2)p+2/2pHmHQ(I + H2)-p/2Hm.
At first glance, the formula (3.4) is not very similar to the original Landauer-Buttiker formula of [7,18]. To make the formula more convenient, we recall that a standard application example for the Landauer-Buttiker formula is the so-called black-box model, cf. [1]. In this case, the Hilbert space K is given by:
N
K = © 0 Kj, 2 Վ N < ж. (3.9)
j=1
and Ko by:
N
Ko = Ks © 0 Kj, 2 Վ N < то. (3.10)
j=1
The Hilbert space KS is called the sample or dot and KS is the sample or dot Hamiltonian. The Hilbert spaces Kj are called reservoirs or leads and Kj are the reservoir or lead Hamiltonians. For simplicity, we assume that the reservoir Hamiltonians Kj are absolutely continuous and the sample Hamiltonian KS has a point spectrum. A typical choice for the density operator is:
N
P = fs(Ks) © 0 fj(Kj), (3.11)
j=1
where fS(•) and fj(•) are non-negative bounded Borel functions, and for the charge:
N
Q = gs(Hs) © 0g3(Hj), (3.12)
j=1
where gS(•) and gj(•) a bounded Borel functions. Making this choice the Landauer-Buttiker formula (3.4) takes the form:
1 N r
JpQ = 2П E (fj(X) - fk(X))g3((X)dX, (3.13)
jk=1 r
where
j(X) := tr(Tjk(X)*Tjk(X)), j,k =1,...,N, Xe R, (3.14)
are called the total transmission probability from reservoir k to reservoir j, cf. [1]. We call it the cross-section of the scattering process going from channel k to channel j at energy X e R. {Tjk (X)}agr is called the transmission matrix from channel k to channel j at energy X e R with respect to the spectral representation n(K0ac). We note that {Tjk(X)}agr corresponds to the transmission operator:
Tjk := PjT(K,Ko)Pk, T(K, Ko) := S(K,Ko) - Pac(Ko), (3.15)
acting from the reservoir k to reservoir j where T(K, Ko) is called the transmission operator. Let {T(X)}agr be the transmission matrix. Following [1], the current JpQ given by (3.13) is directed from the reservoirs into the sample.
The quantity ||T(X)||L2 = tr(T(X)*T(X)) is well-defined and is called the cross-section of the scattering system S at energy X e R. Notice that:
N
Հ X) = ||T ( X)||L2 = tr(T (X)*T (X)) = E j (X). Xe R,
j k=1
We point out that the channel cross-sections Ojk (X) admit the property:
NN
E j (X) = E (X), Xe R, (3.16)
j=1 j=1
which is a consequence of the unitarity of the scattering matrix. Moreover, if there is a conjugation J, such that KJ = JK and KoJ = JKo holds, that is, if the scattering system S is time reversible symmetric, then we have even more, namely, it holds that:
j (X) = ok3 (X), Xe R. (3.17)
Usually, the Landauer-Buttiker formula (3.13) is used to calculated the electron current entering the reservoir j from the sample. In this case one has to choose Q := Qf := — ePj where
Pj is the orthogonal projection form K onto Kj and e > 0 is the magnitude of the elementary charge. This is equivalent to choosing gj (A) = —e and gk (A) = 0 for k = j, A e R. In doing so, we get the Landauer-Buttiker formula simplifying to:
e N r
JQ = — ^֊ E Afj(A) — fk(A))j(A)dA. (3.18)
' j հ֊ j_-, j r
To restore the original Landauer-Buttiker formula, one sets:
fj(A) = f (A — и), A e R, (3.19)
where щ is the chemical potential of the reservoir Kj and f (•) is a bounded non-negative Borel function called the distribution function. This gives rise to the following formula:
e N r
Jp Q.1 = — ^֊ E ' (f (A — fj) — f (A — fk)) j (A)dA. (3.20)
Հ֊ k—1 R
In particular, if we choose one:
f (A) := fFD(A):=r+1eex, 0, A e R, (3.21)
where fFD(•) is the Fermi-Dirac distribution function, and inserting (3.21) into (3.20) we obtain:
e N r
Jp Q* = E (fFD (A — fj) — fFD (A — fk)) j (A)dA. (3.22)
' j A֊ k_լ J R
If we have only two reservoirs, then they are usually denoted by l (left) and r (right). Let j = l and k = r. Then,
Jp Qei = —֊ (fFD (A — fi) — fFD (A — fr ))&ir (A)dA. (3.23)
p'Q1 2֊ JR
One easily checks that JpQq Վ 0 if щ ^ fr. That means, the current is leaving the left reservoir and is entering the right one which is in accordance with physical expectations.
Example 3.3. Notice that sc := {hei, h01} is a £i scattering system. The Hamiltonian hei takes into account the effect of coupling of reservoirs or leads hi := l2(N) and hr := l2(N) to the sample hs = C2 which is also called the quantum dot. The Hamiltonians for the leads are given by: hea = — AD + va, a = l,r. The sample or quantum dot Hamiltonian is given by hesl. The wave operators are given by:
(hel, h0l) := s-lim elthde-ith<ePac(h0). (3.24)
The scattering operator is given by sc := w+(hel, h^^')*w-(hel, he). Let n(hg 'ac) be the spectral representation of h0l 'ac introduced in Section 2.6. If pel and qel are density and charge operators for h0l, then the Landauer-Buttiker formula takes the following form:
1
JSp*l,, զ1 = tr (peJc(A) (qeJc — Sc (A)*q£(A)Sc(A))) , (3.25)
where {sc(A)}AeR, {qel(A)}AeR and {pel(A)}AeR are the scattering, charge and density matrices with respect to n(h0l'ac), respectively. The condition that ((h0l)2 + Ihei)pel is a bounded operator is superfluous because hel is a bounded operator. For the same reason we have that every p-tempered charge qel is in fact a charge, that means, qel is a bounded self-adjoint operator.
The scattering system sc is a black-box model with reservoirs hel and he'. Choosing
pel = fl (hel) e fs (hes) 0 fr (hel),
where fa(-), a = l, r, are bounded Borel functions, and
qel = 9i(hf) © gs (heS) © gr (hfl), where ga(-), a e {l, r}, are locally bounded Borel functions, then from (3.13) it follows that:
JSplKe = 2П E Լ(fa(A) - fx(A))ga(A)ac(A)dA,
a, x£{l , r}
a=K
where {ac(A)}AeR is the channel cross-section from left to right and vice versa. Indeed, let {tc(A)}AeR be the transition matrix which corresponds to the transition operator tc := sc - I^i. Clearly, one has tc(A) = I^A) - sc(A), A e R. Let {pf(A)}AeR be the matrix which corresponds to the orthogonal projection pf from hel onto hf. Further, let tccrl(A) := pfl (A)tc(A)pfl and tcr := pfl (A)tc(A)pf. Notice that both quantities are in fact scalar functions. Accordingly, the channel cross-sections ծ^(A) and (A) at energy A e R are given by ac(A) := ծԼ(A) = К(A)\2 = \tcri(A)\2 = a'ci(A), A e R.
In particular, if gl(A) = 1 and gr = 0, then:
Jpe qi = 7^ f (fl(A) - fr(A))ac(A)dA, (3.26)
p ,ql 2n JR
and qfl := pfl. Following [1], Jpcel qel denotes the current entering the quantum dot from the left lead.
3.2. Application to the JCL-model
Let S = {H,H0} now be the JCL-model. Furthermore, let p and Q be the density operator and a charge for H0, respectively. Under these assumptions, the current Jsp q is defined by:
JpQ := -itr (W-(H, Ho)(I + H2)pW-(H,Ho)*[H,Q]^^J , (3.27)
and admits representation (3.4). If Q is a p-tempered charge and (H0-i)p+2p is a bounded operator, then the current JpQ is defined in accordance with (3.7) and the Landauer-Buttiker formula (3.4) is also valid.
We introduce the intermediate scattering system Sc := {H,Hc}, where:
Hc := hel 0 I¥h + Ihei 0 hph = Ho + Vel.
The Hamiltonian Hc describes the coupling of the leads to the quantum dot, but under the assumption that the photon interaction is not switched on.
Accordingly, Sph := {H,Hc} and Sc := {Hc,H0} are ^-scattering systems. The corresponding scattering operators are denoted by Sph and Sc, respectively. Let n(Hac) = {L2(R,dA, hc(A)), M, Ф^ of Haf be a spectral representation of Hc. The scattering matrix of the scattering system {H, Hc} with respect to n(Hac) is denoted by {Sph(A)}AeR. The scattering matrix of the scattering system {Hc, H0} with respect to n(Hac) = {L2(R, dA, h0(A)), M, Ф0} is denoted by {Sc(A)}a£r.
Since Sc is a ^-scattering system, the wave operators W±(Hc, H0) exist and are complete and since ФcW±(Hc, H0)Ф* commutes with M, there are measurable families {W±(A)}AeR of isometries acting from ho (A) onto hc(A) for a.e. A e R such that:
(ФсW±(Hc, Дэ)Ф* f )(A) = W±(A) f (A), A e R, f e L2(R, dA, ho(A)).
The families {W±(A)}AeR are called wave matrices.
Straightforward computation shows that Sph := W+(Hc, Ho)*SphW+(Hc, Ho) commutes with Ho. Hence, with respect to the spectral representation n(H3c), the operator Sph is unitarily
equivalent to a multiplication induced by a measurable family { Sph (A)}AeR of unitary operators in ho(A). Straightforward computation shows that:
Sph( A) = W+( A)* SPh (A)W+( A), (3.28)
for a.e. A e R. Roughly speaking, { Sph (A)}AeR is the scattering matrix of Sph with respect to the spectral representation П(ИЗС). Furthermore, let
pc := W-HcHo)pW-HcHo)* (3.29)
and
Qc := W+Hc, Ho)QW+(Hc, Ho)*. (3.30)
The operators pc and Qc are the density and tempered charge operators for the scattering system Sph. Indeed, one easily verifies that pc and Qc are commute with Hc. Moreover, pc is non-negative. Furthermore, if Q is a charge, then Qc is also a charge. This gives rise to the introduction of currents .J^Q := J^
JcP,q •= -itr (W-(Hc, Ho)pW-(Hc, Ho)*^֊ [Hc, Q]h֊) , (3.31)
and Tph — JSph and JP,Q •— Jpc,Qc,
JphQ := -itr (Ш-(И, Hc)pcW-(H, Hc)*֊ [H,Qc]֊) , (3.32)
which are well defined. If Q is p-tempered charge and (Ho — i)P+2p is a bounded operator, then one easily checks that Qc is a p-tempered charge and (Hc — i)P+2pc is a bounded operator. Hence the definition of the currents Jpc qc can be extended to this case and the Landauer-Buttiker formula (3.4) holds. '
Finally, we note that the corresponding matrices {pcac(A)}AeR and {Qcac(A)}AeR are related to the matrices {pac(A)}agr and {Qac(A)}agr by
pcac( A) = W-( A)pac( A)W-( A)* and Qcac( A) = W+( A)Qac( A)W+( A)* (3.33)
for a.e. A e R.
Proposition 3.4 (Current decomposition). Let S = {H, Ho} be the JCL-model. Furthermore, let p and Q be the density operator and a p-tempered charge, p e No, for Ho, respectively. If (Ho — i)P+2p is a bounded operator, then the decomposition,
JP,Q = Jp,Q + JpQ, (3.34)
holds where JrpQ and JppQ are given by (3.31) and (3.32).
In particular, let {Sc( A)}AeR, {pac(A)}AeR and {Qac(A)}AeR be scattering, density and charge matrices of Sc, p and Q with respect to n(Hc) and let {Sph(A)}AeR, {pcac(A)}AeR and {Qac( A)}AeR be the scattering, density and charge matrices of the scattering operator Sph, density operator pc, cf. (3.29), and charge operator Qc, cf. (3.30), with respect to the spectral representation n(Hac}. Then, the following representations:
rc 1 I 1__/ fW С Հ\\*/
JCPQ := 7rftr(pac( A)(Qac( A) — Sc( A)*Qac( A)Sc( A))dA, (3.35)
АП JR
JPphQ := 2Է Լ tr(pcac(A)(Qcac(A) — Sph(A)*Qcac(A)SPh(A)))dA, (3.36)
take place.
Proof. Since Sc and Sph are ^-scattering systems from Theorem 3.1 the representations (3.35) and (3.36) are easily follow. Taking into account (3.33), we get the following:
trPacWQacW — SphWQacWSphW)) =
tr(W-WpacW-^y(w+WQacWw+^) — SphW QcM)SphW)). Using Sc(Л) = W+^)*W-^) we find that:
tr(pcacW(Qcac (Л) — SphW*Q^WSphW))=tr(pacWx (3.37)
(ScW*QacWSc^) — W-^)*SphW*W+ WQacWW+^Sph^W-^))) .
Since [Hc,H0} and [H,Hc} are ^-scattering systems, the existence of the wave operators W±(H,Hc) and W±(Hc,H0) follows. Using the chain rule, we find W±(H,H0) = W±(H, Hc)W±(Hc, H0) which yields:
S = W+(H,H0)*W+(H,H0)
= W+(Hc, H)*W+(H, Hc)W-(H, Hc)W-(Hc, H0) = W+(Hc, H0)*SphW-(Hc, H0). Hence, the scattering matrix [S(Л)}лек of [H, H0} admits the representation
S(Л) = W+^)*Sph^)W-^), Л E R. (3.38)
Inserting (3.38) into (3.37), we get the following:
JPphQ = ֊ j tr(paCW(Sc(X)*Qac(X)Sc(Л) — S(ЛуQac(X)S(Л))Щ (3.39)
АП J r
Using (3.39), we obtain the following:
jrp,Q + JphQ = Լ tr(PacW(Qac (Л) — S (X)*Qac (Л)S (X)))dX. Finally, taking into account (3.4), we obtain (3.34). □
Remark 3.5.
(i) The current Jcp q is due to the coupling of the leads to the quantum dot and it is therefore called the contact induced current.
(ii) The current JpphQ is due to the interaction of photons with electrons and it is called the photon induced current. Notice the this current is calculated under the assumption that the leads are already in contact with the dot.
Corollary 3.6. Let the assumptions of Proposition 3.4 be satisfied. With respect to the spectral representation ЩЩ12) of Hac the photon induced current JppQ can be represented by:
1 r
JPp!Q = ^ tr( paC(X) (QacW — SphW* QacW Sph (Л))Щ, (3.40)
АП J r
where the measurable families [ Sph(Л) }Aer and [ pac(Л) }Aer are given by (3.28) and
SacW := ScWpacWScW* Л E R, (3.41)
respectively.
Proof. Using (3.33) and Sc^) = W+^)*W-^), we find: tr(pcacW(QcacW — Sph(X)*QCac (\)Sph(\))) =
tr (Sc WpacWScW* (QacW — W+Л)* SphW*W+^)Q ac^)W+^)* Sph^)W+^))) .
Taking into account the representations (3.28) and (3.41), we get the following:
trPUmicW - SPh(X)*Qcac(X)SPh(X))) =
tr(Sc(X)pac(X)Sc(X)*(Qac(X) — SPh(X)* Qac(A) Sph(X))), which immediately yields (3.40). □
Remark 3.7. In the following, we call { pac(X) }лек, cf. (3.41), the photon modified electron density matrix. Notice that { pac (X) }лек might be non-diagonal, even if the electron density matrix {Pac(X)}лек is diagonal.
4. Analysis of currents
""՝ 'pQ and JpQ
In the following, we analyze currents Jp Q and jpQ under the assumption that p and Q have the
tensor product structure:
p = pel 0 pph and Q = qel 0 qph, (4.1)
where pel and pph as well as qel and qph are density operators and (tempered) charges for h0l and hph, respectively. Since pph commutes with hph, which is discrete, the operator pphhas the form:
pph = £ Pph(n)(-, Tn)Tn, (4.2)
(n
neno
where pph(n) are non-negative numbers. Similarly, qph can be represented as:
ph
nen0
where qph(n) are real numbers.
qph = £ qph(n)(-, Tn)Tn, (4.3)
Lemma 4.1. Let S = {H, H0} be the JCL-model. Assume that p = 0 and Q have the structure (4.1) where pel is a density operator and qel is a charge for h0l.
(i) The operator (H0 — i)p+2 p, p e No, is bounded if and only if the condition:
sup pph(n)np+2 < TO, (4.4)
nen0
is satisfied.
(ii) The charge Q is p-tempered if and only if:
sup |qph(n)|n-p < to, (4.5)
nen
is valid
Proof. (i) The operator (H0 — i)p+2p admits the representation:
(Ho — i)p+2p = 0 pph(n)(h0l + nu — i)p+2pel.
peno
We have:
II(Ho — i)p+2p||l(h) = sup pph(n)|(h0l + nu — i)p+2pel\\z(hel) (4.6)
peno
peno
— / 7 ol .
n
= sup pph(n)np+2n-(p+2) (h0l + nu — i)p+2pel
l(hel)
Since lim^oo n (p+2) (h0 + nu — i)p+2pel = up+2|pel|£(hei), we obtain for sufficiently
l(h )
large n e N0 that:
up+2
U "„el.. ,
hel
\\pel |L(hel) ^ n-(p+2)||(h0l + nu — i)p+2pel|£(hel) .
Using that and (4.6), we immediately obtain (4.4). Conversely, from (4.6) and (4.4), we obtain that (Ho - i)p+2p is a bounded operator. (ii) As above, we have:
Q(Ho - i)-p = 0 qph(n)qel.
n£no
Hence:
\\Q(Ho - i)-ph(H) = sup \qph(n)\\\qel(hf + nu - i)-p\\mel).
n£n0
Since np\\(hf + nu - i)-p\\L(hel) = u,-p\\qel\\L(hel), we similarly obtain, as above, that (4.5)
holds. The converse is obvious. □
4.1. Contact induced current
Let us recall that Sc = {Hc, Ho} is a £i-scattering system. Straightforward computation shows that:
W±(Hc,Ho)= w±(hel,he0l) 0 I¥h, where w±(hel, hfl) is given by (3.24). Hence:
Sc = sc 0 Ihph, where sc := w+h^hf )*w- (hfl,hfl).
Proposition 4.2. Let S = {H,Ho} be the JCL-model. Assume that p and Q are given by (4.1) where pel and qel are density and charge operators for hfl and pph and qph for hph, respectively. If for some p e No the conditions (4.4) and (4.5) are satisfied, then the current Jp Q is well defined and admits the representation:
Jp,Q = J, Y := E qph(n)pph(n), (4.7)
n£no
where Jpe qel is defined by (3.2). In particular, if tr(pph) = 1 and qph = կPh, then JpQ = Jpcel qel.
Proof. First, we note that by lemma 4.1 the operator (Ho - i)p+2p is bounded and Q is p-tempered. Hence, the current JpQ is correctly defined and the Landauer-Buttiker formula (3.4) is valid.
With respect to the spectral representation n(Hc) of Lemma 2.12, the charge matrix {Qac(A) }agr of Qac = q^c 0 qՓ admits the representation:
Qac (A) = 0 qfc(A - nu)qph(n), A e R. (4.8)
n£n0
Since Sc = sc 0 Ihph, the scattering matrix {Sc(A)}AeR admits the representation:
Sc(A) = 0 sc(A - nu), A e R.
n£n0
Hence:
Qac(A) - Sc(A)*Qac(A)Sc(A) = (4.9)
0 qph(n) {qfc(A - nu) - sc(A - nu)*qfc(A - un)sc(A - nu)) .
n£n0
Moreover, the density matrix {pac(A)}AeR admits the representation:
pac(A)= 0 pph(n)pealc (A - nu) (4.10)
n£n0
Inserting (4.10) into (4.9) we find the following:
pac ( A) (Qac( A) — Sc( A)*Qac( A)Sc( A)) =
0 qph(n)pph(n)pealc(A — nu) (qealc(A — un) — Sc(A — nufCXA — un)8c(A — nu))
neno
Since y = Eneno qph(n)pph(n) is absolutely convergent by (4.4) and (4.5), we obtain that:
tr(pac(A) (Qac(A) — Sc( A)*Qac(A)SC(A))) = (4.11)
^ qph(n)pph(n)tr pc(A — nu) qc(A — un) — Sc(A — nu)*q^(A — un)sc(A — nu)))
neno
Clearly, we have:
tr (pac(A — nu) (qeJc(A — un) — Sc(A — nu)*qealc(A — un)Sc(A — nu))) 4II pac (A — nu) ||,£(hn(A))\\qt(A — nu) ||£(hn(A)), A e R. We insert (4.11) into the Landauer-Buttiker formula (3.35). Using (4.4) and (4.5) as well as
/ \\pac(A)|khn(A))\\qac( a)||£(hn(A))d A <
r
we see that we can interchange the integral and the sum. By doing so, we get:
Jp>Q = E qph(n)pph(n)֊ է tr {palc(A — nu)x
neno Հո jr
(qealc(A — un) — sc(A — nu)*qalc(A — un)sc(A — nu))) dA.
Using (3.25) we prove (4.7).
If tr(pph) = 1, then £no pph(n) = 1. Furthermore, if pph = I¥h, then qph(n) = 1. Hence, Y =1. □
4.2. Photon induced current
To calculate the current JppQ, we use representation (3.40). We then set:
Smhn (A) : = Pm(A) Sph (A) \ hn(A), A e R,
where { Sph (A)}AeR is defined by (3.28) and Pm(A) is the orthogonal projection from h(A), cf. (2.37), onto hm(A) : = hel(A — mu), A e R.
Proposition 4.3. Let S = {H, Ho} be the JCL-model. Assume that p and Q are given by (4.1) where pel and qel are density and charge operators for hal and pph and qph for hph, respectively. If for some p e No the conditions (4.4) and (4.5) are satisfied, then the current JppQ is well-defined and it admits the following representation:
1 г
JpQ = E pph(m) E qph(n)tt dA tr {SadA — mu) x (4.12)
meno neno ՝JR
(qalc( a—nu)5mn — Snm (A)*qalc( a—nu) Snm ( a)))
where { Salc(A) }AeR is the photon modified electron density defined, cf. (3.41), which takes the following form:
Salc(A) = Sc(A)pel(A)sc(A)*, A e R. (4.13)
Proof. By Lemma 4.1 we get that that the charge Q is p-tempered and (H0 — i)pp is a bounded operator. By Corollary 3.2, the current JpphQ := J^Q is well-defined.
Since (Qac(Л) — Sph(Л)* Qac(Л) Sph (Л)) is a trace class operator for Л e R, we get from (3.40) and (4.10) that:
Ц PacW (QacW — SphW* Qac (Л) Sph (Л))) =
£ pph(m)tr (pel(Л — mu) Рт(Л) (q^(\) — Sph(X)* QacW Sph (Л)) Pm(Л))
m£n0
Furthermore, we have:
Рт(Л) (Qac(ty — Sph(ty* Qac(Л) Sph (Л)) Рт(Л)
= qph(m) (де1(Л — mu) — Рт(Л) Sph(Л)* Qac(X) Sph (Л)) Рт(Л) = qph(m)qel(Л — mu) — £ qph(n) S^X)* qel(Л — nu) SПт(Л) ,
nGn0
for Л E R where S^W* := Рп(Л) Sph (Л)Рт(Л), Л E R. Notice that Engn0 is a sum with a finite number of summands. Hence:
tr (рас(Л) (Qac (Л) — SphW* Qac (Л) Sph (Л))) = £ Pph(m) £ qph(n)x
тем0 nGn0
tr (р?1(Л — mu) (qel(Л — mu)5mn — S^W* qe1(Л — nu) S^W )) We are going to show that
£ Pph(m) £ \qph(n)\l |tr (pel(Л — mu) x
mGn0 nGn0
el Sph el
{qe1(Л — mu)rn — ^т(Л)* qel(Л — nu) S^) Clearly, one has the following estimate:
dЛ < oo.
\tr (р?\Л — mu) (^(Л — mu^n — S^W qel(Л — nu) S^Л)))
2\\ р1(Л — mu) ll£(hm(A)) (\\qel (Л — mu)W&(hm(\))5nrn + \\qel (Л — nu)|£(hn(A}}
Furthermore, we get:
I „ \\ р1(Л — mu) \kMA))\\qel(Л — -т^щих^пт Հ
J AGr
f \\Pel(Л) \\լ(^))\№1 (Л)\\тт(х)^Л j agr
and
f \\ pe (Л — mu) \\l(hm(A)) Wq^'^ — 'Ո^ՀհՀՃ))^ Վ
r
Աճես*) \\р1(Л — (m — n)u) \\тт-п(Х)^Л j agr
If the conditions (4.4) and (4.5) are satisfied, then
£ pph(m)\qph(m)\ j \\ Г(Л) \Um(A))\\qel(Л)\\цЫа)^Л < ж.
rnGn jr
Furthermore, we obtain:
£ pph(m) £ |qph(n)l / В II pel(X — (m — n)u) ^„.„(л))^ ^
men0 nen0 •/Лек
(^max — vmia + 4)||p£||^ £ pph(m) £ |qph(n)| < TO,
men |m-n|^dmax
where dmax is introduced by Lemma 2.12. To prove the following:
£ pph(m) £ |qph(n)| < to,
men0 |m-n|^dmax
we again use (4.4) and (4.5). The last step allows us to interchange the integral and the sums, which immediately proves (4.12). □
Corollary 4.4. Let S = {H, H0} be the JCL-model. We assume that p and Q are given by (4.1), where pel and qel are density and charge operators for h0l and pph and qph for hph, respectively. If pel is an equilibrium state, i.e. pel = fel(h0l), then:
JpQ = £ </*(*)— l (pph(n)fel(X — nu) — pph(m)fel(X — mu)) x
tr ( Snhm (X)*qealc(X — nu) Snm (X)) dX. (4.14)
Proof. From (4.12), we obtain the following:
JpQ = £ qph(n) £ pph(m)2n[dXfel(X — mu)x
nen0 men0 jr
tr (qa[(X—nu)smrt — Spm (X)*q:lc(X—nu) Spm (X))
Hence,
JpQ = £ qph(n)2n f dX £ pph(m)fel(X — mu)x
nen0 jr men0
tr (qalc(X — nu)5mm — S£ (X^^X — nu) Sppm (X)) . This gives the following:
JpQ = £ qph(n)֊ / dX (pph(n)fel(X — nu)tr (q:lc(X — nu)) — (4.15)
.-m 2n Jr ՝֊ ՝֊ 7
nen0
£ pph(m)f el(X — mu)tr ( Sphm (X)*qalc(X — nu) Spm (X))).
Since
£ pph(m)fel(X — mu)tr {Sphm (X)*q£(X — nu) Sphm (X)) =
en0
£ (pph(m)fel(X — mu) — pph(n)fel(X — nu)) tr ( Spt (X)*a(X — nu) Sphm (X)) +
men0
pph(n)fel(X — nu) £ tr( Spm (X)*qealc(X — nu) Spm (X)) ,
men0
then inserting this into (4.15) we obtain (4.14). □
5. Electron and photon currents 5.1. Electron current
To calculate the electron current induced by contacts and photon contact, we make the following choice throughout this section. We set
QO := qea 0 qph, qea = -epO and qph := կհ, a e {l,r}, (5.1)
where pO denotes the orthogonal projection from hel onto hO. By e > 0, we denote the magnitude of the elementary charge. Since peO commutes with hf, one can easily verify that QO commutes with H0, which shows that QO is a charge. Following [1], the flux related to QO gives us the electron current JpQd entering the lead a from the sample. Notice QO = -ePa where Pa is
the orthogonal projection from H onto := hO 0 hph. Since qph = Ihph, condition (4.5) is immediately satisfied for any p ^ 0.
Let f (•) : R —> R be a non-negative bounded measurable function. We set:
pel = pf e peS e pfl, peO := f (hi - ща), a e{l,r}, (5.2)
and p = pel 0 pph. By щO the chemical potential of the lead a is denoted. In applications, one sets f (A) := fFD(A), A e R, where fFD(A) is the so-called Fermi-Dirac distribution given by (3.21). If в = ж, then fFD(A) := xR- (a), A e R. Notice that [pel,pel] = 0. For pph, we choose the Gibbs state:
pph := Ze-ehph, Z = tr(e-ehph) = . (5.3)
Hence, pph = (1 - е-вш)e-ehph. If в = ж, then pph := (•, Xo)Xo. Clearly, tr(pph) = 1. We note that pph(n) = (1 - е-13ш)е-пвш, n e N0, satisfies the condition (4.4) for any p ^ 0. Accordingly, p0 = pel 0 pph is the density operator for H0.
Definition 5.1. Let S = {H, H0} be the JCL-model. If Q := Qf, where QO is given by (5.1), and p := p0 := pel 0 pph, where pel and pph are given by (5.2) and (5.3), then Jf Qel := JS Qel
is called the electron current entering the lead a. The currents Jc nel and Jphnel are called the contact-induced and photon-induced electron currents.
5.1.1. Contact induced electron current. The following proposition immediately follows from Proposition 4.2.
Proposition 5.2. Let S = {H, H0} be the JCL-model. Then the contact induced electron current Jpo Qel, a e {l, r}, is given by Jcpo Qel = Jpcel qel. In particular, one has:
JcP0lQd = -֊ (f (A - nO) - f (A - fix)ac(A)dA, a, к e {l,r}, a = к, (5.4)
where {ac(A)}AeR is the channel cross-section from left to the right of the scattering system sc = {hel, h0l}, cf. Example 3.3.
Proof. Since tr(pph) = 1 it follows from Proposition 4.2 that Jp)o Qel = Jpcel qel. From (3.26), cf. Example 3.3, we find (5.4). □
If щ > fr and f (•) is decreasing, then Jc Qel < 0. Hence, the electron contact current
p0,Ql
is going from the left lead to the right which is in accordance with the physical expectations. In particular, this is valid for the Fermi-Dirac distribution.
Proposition 5.3. Let S = {H, H0} be the JCL-model. Further, let pel and pph be given by (5.2) and (5.3), respectively. If the charge QO is given by (5.1), then the following holds:
(E) If /it = цг, then JcpoQei = 0, a e {l, r}.
(S) If vt ^ Vr + 4, then J cpo Qei = 0, a e {l, r}, even if / = ц.
(C) If ef = 5f and ef = Sf, then Jpo Qei = 0, a e {l, r}, even if = .
Proof. (E) If /I = ir, then f (A — ) = f (A — /r). Applying formula (5.4) we obtain Jpo Qel = 0.
(S) If vt ^ vr + 4, then ho'ac has simple spectrum. Hence the scattering matrix {sc( A)}AeR of the scattering system sc = {hel, he} is a scalar function which immediately yields ac(A) = 0, A e R, which yields J Լ Qel = 0.
(C) In this case, the Hamiltonian hel decomposes into the direct sum of two non-interacting Hamiltonians. Hence, the scattering matrix of {sc( A)}AeR of the scattering system sc = {hel, he} is diagonal, which immediately yields J^ qbi = 0. □
5.1.2. Photon induced electron current. It is hopeless to analyze the properties of (4.12) if we make no assumptions concerning pel and the scattering operator sc. The simplest assumptions is that pel and sc commute. In this case, we get pel (A) = pel(A), A e R.
Lemma 5.4. Let S = {H, Ho} be the JCL-model. Furthermore, let pel be given by (5.2). If one of the cases (E), (S) or (C) of Proposition 5.3 is realized, then the pel and sc commute.
Proof. If (E) holds, then pel = f (hi1) which yields [pel, sc] = 0. If (S) is valid, then the scattering matrix {sc(A)}AeR is a scalar function which shows [pel,sc] = 0. Finally, if (C) is realized, then the scattering matrix {sc(A)}AeR diagonal. Since the pel is given by (5.2) we get [pel, sc] =0. □
We will now calculate the current J^Qei, see (4.12). Clearly, we have Pa(A) = EneN0 ped(A — nu) and I^) = Pl(A) + Pr(A), A eY We then set:
Pna(A) : = Pa(A)Pn(A) = Pn(A)Pa(A) = pel(A — nu), a e {l,r},
n e No, A e R. In the following we use the notation Tph (A) = Sph (A) — I^(A), A e R, where { Tph(A) }AeR is called the transition matrix and { Sph(A) }AeR is given by (3.28). We set:
Tpm (A):= Pka (A) Tph (A)PmK (A), A e R, a, к e{l,r}, k,m e No,
and:
t£m» (A) = tr( Tpm(A)* Tk!hm„(A)), A e R, (5.5)
which is the cross-section between the channels ka and mK.
Proposition 5.5. Let S = {H, Ho} be the JCL-model.
(i) If pel commutes with the scattering operator sc and qel, then:
Cq 1 = — E TT / СPph(n)f (A — la — nu) — Pph(m)f (A — /к — mu)) dpaniH(A) dA.
,-kvt j r
m,n£n0
k£{l,r}
(5.6)
(ii) If in addition S = {H, Ho} is time reversible symmetric, then
JpoQai = — E tT f (pph(n)f (A — ia — nu) — pph(m)f (A — la — mu)) a^ma, (A) dA,
a m,n£no Jr
(5.7)
a, a' e {l, r}, a = a'.
Proof. (i) Let us assume that:
qel = £ Як (hi),
Ke{l,r}
Notice that
qealc(X)= £ Як(X)pi(X), X e R. (5.8)
Ke{l,r}
Inserting (5.8) into (4.12) and using qph = կբե, we obtain the following:
1
JPph0Q = £ pph(m) £ —[dXфa(X — mu)gi(X — nu)x
men0 nen0 JR
ae{l,r} Ke{l,r}
tr [ped(X — mu) (pkl(X — nu)5mp — Spm (X)*pkl(X — nu) Spm (X)/ where, for simplicity, we have set:
Фa(X) := f (X — fj,a), X e R, n e N0, a e{l,r}. (5.9)
Accordingly, we get:
JPphoQ = £ pph(n)֊ f dX фк(X — nu)gi(X — nu)tr (pK(X — nu)) —
nen0 2n Jr V У
ke{l,r}
£ £ pph(m)֊ f dXфa(X — mu)gK(X — nu)x (5.10)
nen0 men0 2n r
ke{l,r} ae{l,r}
tr (pf(X — mu) Spm (X)*pkl(X — nu) Spm (X)pf(X — mu) Since the scattering matrix { Sph (X)}^R is unitary, we have:
pkl(X — nu) = £ pK(X — nu) Smhp (X)*ped(X — mu) (X)pZ(X — nu), (5.11)
men0 ae{l,r}
for n e N0 and к e {l, r}. Inserting (5.11) into (5.10), we find that:
1
JPph0Q = £ £ pph(n)2֊j dX фк(X — nu^(X — nu)x
nen0 men0 ՝JR
ке^^} ae{l,r}
tr (p^(X — nu) Sphm (X)*pal(X — mu) Smhp (X)pt(X — nu)) — 1 f
E E pph(m)— dXфa(X — mu)як(X — nu)x
pen0 men0 кe{l,r} ae{l,r}
tr p(X — mu) Spm (X)*pe(X — nu) Spt (X)pf(X — mu)) . Using the notation (5.5), we find the following:
JHq = £ £ pph(n)֊ I dX фк(X — nu^(X — nu) э^(X) —
pen0 men0 jR
ке^^} ae{l,r}
1
E E pph(m)— Լ dX фa(X — mu)як(X — nu) aPphHma (X)
pen0 men0 ке{^} ae{l,r}
By (3.16), we find that:
£ *rnhan« (Л)= £ Հլта (Л) Л E r.
rnGn0 rnGn0
aG{l,r} aG{l,r}
Using that, we obtain the following:
1
2П
Jp0,Q = £ ^ Lx (5.12)
rn,nGn0 a,kG{l,r}
(pph(n^x(Л — nu) — р^^фа^ — mu)) gH(Л — nu) аПт (Л) dЛ.
Setting да(Л) = — e and gK(Л) = 0, к = a, we obtain (5.6). (ii) Straightforward computation shows that:
£ f (pph(n)f (Л — Va — nu) — Pph(m)f (Л — Va — mu)) (Л) dЛ =
nrnGNh r
£ I' (Pph(m)f (Л — Va — mu) — pph(n)f (Л — Va — nu)) a^(Л) dX. n.rnGno r
Since a^ (Л) = a^^ (Л), Л E R, we obtain the following:
£ [ (pph(n)f (Л — Va — nu) — pph(m)f (Л — Va — mu)) օք^(Л) dЛ =
nrnGbh r
— £ I' (pph(n)f (Л — Va — nu) — Pph(m)f (Л — Va — mu)) ՅԼ^ (Л) dЛ n.rnGno r
which yields:
£ Լ (pph(n)f (Л — Va — nu) — Pph(m)f (Л — Va — mu)) օ?^ (Л) dЛ = 0.
Using that, we immediately obtain the representation (5.7) from (5.6). □
Corollary 5.6. Let S = [H, H0} be the JCL-model.
(i) If the cases cases (E), (S) or (C) of Proposition 5.3 are realized, then the representation (5.6) holds.
(ii) If the case (E) of Proposition 5.3 is realized and the system S = [H,H0} is time reversible symmetric, then
JphQi = — £ 2r! (Pph(n)f (Л — V — nu) — pph(m)f (Л — V — mu)) d£m<x, (ЛЩ (5.13)
" rn,nGn0 Jr
n E N0, a E [l,r} where v := v = Vr and a = a'.
(iii) If the case (E) of Proposition 5.3 is realized and the system S = [H, H0} is time reversible and mirror symmetric, then Jph Qel = 0.
р0, Q a
Proof. (i) The statement follows from Proposition 5.5(i) and Lemma 5.4.
(ii) Setting va = Va' formula (5.13) follows (5.7).
(iii) If S = [H,H0} is time reversible and mirror symmetric, we obtain from Lemma 2.14 (ii) that ծՈ^՛ (Л) = аПh՛т (Л), Л E R, n,m E N0, a, a' E [l,r}, a = a'. Using that, we obtain from (5.13) the following:
JPhQi = — £ ֊ I (Pph(n)f (Л — V — nu) — Pph(m)f (Л — V — mu)) a? nta (ЛЩ.
.-rt zn j r rn,nGn0
Interchanging m and n, we obtain:
= - E f f (pph(m)f (A - щ - mu) - pph(n)f (A - щ - nu)) na (A)dA.
m,'n&i0 П jr
Using that S is time reversible symmetric we get from Lemma 2.14 (i) that:
Jp0Q = - E 2֊ L(pph(m)f (A - щ - mu) - pph(n)f (A - щ - nu)) d? (A)dX.
m,n£N0
which shows that Jphnel = -Jphnel. Hence Jphnel = 0. □
We note that by Proposition 5.3, the contact induced current is zero, i.e. J po Qel = 0. Hence,
if the S is time reversible and mirror symmetric, then the total current is zero, i.e. Js Qel = 0.
p0,Qa
Remark 5.7. Let the case (E) of Proposition 5.3 be realized, that is, = щг. Moreover, we assume for simplicity that 0 =: vr Հ v := v;.
(i) If в = ж, then pph (n) = 80n, n e N0. From (5.6), we immediately obtain that Jp Qel = 0. That means, if the temperature is zero, then the photon-induced electron current is zero.
(ii) The photon-induced electron current might be zero even if в < ж. Indeed, let S = {H,H0} be time reversible symmetric and let the case (E) be realized. If u ^ v + 4 and hel(A) := hnl(A) = hel(A - nu), n e N0. Hence, one always has n = m in formula (5.13), which immediately yields J^Qd = 0.
(iii) The photon-induced electron current might be different than zero. Indeed, let S = {H,H0} be time reversible symmetric and let v = 2 and u = 4, then one sees that to calculate the
JphQei, one has to consider m = n + 1 in formula (5.13). Therefore, we find that:
p0,Q
JP0>Qf = E 2п X
n£N0
jR dA (pph(n)f (A - щ - nu) - pph(n + 1)f (A - щ - (n + 1)u)) дЩn+l)r (A).
If pph is given by (5.3) and f (A) = fFD(A), cf. (3.21), then one easily verifies that
д
—pph(x)fFD(A - щ - xu) < 0, х,щ,А e R. дх
Hence, pph(n)fFD(А - щ - nu) is decreasing in n e N0 for А,щ e R which yields {pph(n)f (А - щ - nu) - pph(n + 1)f (А - щ - (n + 1)u)) ^ 0. Therefore, J^Qel Վ 0 which means that the photon-induced current leaves the left-hand side and enters thel right-hand side. In fact, Jp^Qd = 0 implies that dnh(n+i)r (А) = 0 for n e N0 and А e R, which means that there is no scattering from the left-hand side to the right one and vice versa which can be excluded generically.
5.2. Photon current
The photon current is related to the charge by equation:
Q := Qph = -Ihel 0 n,
where n = dr(1) = b*b is the photon number operator on hph = F+(C), which is self-adjoint and commutes with hph. It follows that Qph is also self-adjoint and commutes with H0. It is not bounded, but since dom(n) = dom(hph), it is immediately obvious that Qph(H0 + в)-1 is bounded,
when N is a tempered charge. Its charge matrix with respect to the spectral representation n(Hac) of Lemma 2.12 is given by:
Qah (A) = — e nPn(A).
neno
We recall that Pn(A) is the orthogonal projection from h(A) onto hn(A) = hel(A — nu), A e R. We will now calculate the photon current or, as it is also known, the photon production rate.
5.2.1. Contact induced photon current. The following proposition is, in fact, in accordance with the physical intuition.
Proposition 5.8. Let S = {H, Ho} be the JCL-model. Then Jc QPh = 0.
Proof. We note that qeJc(A) = Ihei(A), A e R. Inserting this into (3.25), we obtain Jspcel qel = 0. Applying Proposition 4.2 we prove Jpo Qph = 0. □
The result reflects the fact that the lead contact does not contribute to the photon current, which is plausible from the physical point of view.
5.2.2. Photon current. From Proposition 5.8, we see that only the photon-induced photon current JPphoQVh contributes to the photon current JpoQPh. Since JpoQPh = JphoQVh, we call JphoQVh simply the photon current.
Using the notation Tpm (A) := Pn(A) Tph (A) \ hel(A — mu), A e R, m,n e No. We set: Tpm(A) = 'Tpm (A)sc(A — mu), A e R, m,n e No, (5.14)
and
Tpm(A):= Ppk(A)fpm(A) \ hl(A — mu), A e R, (5.15)
m,n e No, a, к e {l,r}, as well as ^ma(A) := tr(ftma(A)*Ttma(A)), A e R.
Proposition 5.9. Let S = {H, Ho} be the JCL-model.
(i) Then:
Հ^ = E (n — m)pph(m)T- / f (A — ia — mu)aprhnt,a(A)dA. (5.16)
m,neno a^E^r}
(ii) If pel commutes with sc, then:
1 Ր
Jpo,Qph = E (n — m)pph(m)֊ f (A — ia — mu) a^ma (A)dA. (5.17)
Tr
a^E^r}
(iii) If pel commutes with sc and S = {H, Ho} is time reversible symmetric, then:
1 r
JphQPh = У — dAx (5.18)
poQ TT Jr
m,nENo,n>m ^aE^r}
(n — m) (pph(m)f (A — /a — mu) — pph(n)f (A — /к — nu)) a^ma (A), where a' e {l, r} and a' = a.
Proof. (i) From (4.12) we get
J^Qph = - E npph (m) X
m,n£n0
1 г
-Լ dA tr( pfc(A - mu) (Pn(A)8mn - ^ (А)*САА - nu) S^ (А)
Hence:
Tph — _ V^ ™rph(™\1 f fT-fs^n-™,.Л fp (W- Sph (\
JPp0h,Qph = - E mpph(m)֊ f tr(Sac(A - mu) (.Pm(A) - S^A)* Pm(A) S^A))) dA+
,-в.т zn j r
mGn0
E npph(m)1 Լ tr f Salc(A - mu) SnKA)* Pn(А) Snm(A) ) dA.
m,n£n0 m=n
Using the relation Pm(A) = I^A) -J2nen0,m=n Pn(A), А e R, we obtain the following:
J^Qph = - E mpph(m)֊ ( tr f pfc(A - mu) f Sn!m(A)* Pn(A) Sn!m(A))) dA+
к\т ճւ П J r
m,n&n0
m=n
E npph(m)֊ f tr (pfc(A - mu) Snhm(A)* Pn(A) Snhm(A)) dA.
2n Jr v '
m=n
Since Tph (А) = Sph (А) - Ih(a), А e R, we find
J^QPh = - E (m - n)pph(m)֊ Լ tr f pt(A - mu) fphm(А)* fphm(А) ) dA.
m,n£N0
Using (4.13) and definition (5.14) one readily sees that:
JToQph = - E (m - n)pph(m)2- ( tr fpea[(A - mu)fm (А)*^(А)) dA .
m,nei% Հո JR
Since pealc = pfl e pfl where pO is given by (5.2), we find the following:
J^ = - E (m - n)pph(m)֊ jR f (А - ща - mu)tr ffpm (А)*(А)) dA,
m,n£n0
where we have used (5.15). Using dphma (А) = tr(Tp^ (А)*Tpn^ (А)), we prove (5.16).
(ii) If pealc commutes with sc, then Salc (А) = peJcc(А), А e R, which yields that one can replace dnhma(А) by э'Кma (А), А e R. Therefore, (5.17) holds.
(iii) Clearly, we have:
JPo,QPh = (5.19)
1j
E (n - m)pph(m)— f (А - ща - mu) Э^ma (A)dA +
:КГ_ JR
m,n£n0,n>m
E (n - m)pph(m)— f (А - ща - mu) эЩma (A)dA
2П
m,nEN0,n<m
Moreover, a straightforward computation shows the following:
£ (n — m)pph(m)֊ / f (X — fa — mu) d£ma (X)dX =
m,pen0,p<m
a,кe{l,r}
E (m — n)pph(n)֊ I f (X — цк — nu) 3^ (X)dX.
,-кчт JR
m,peN0,p>m a^e^r}
Since S = {H, H0} is time reversible symmetric, we find the following:
(n — m)pph(m)֊ ( f (X — fa — mu) 3*Լո>է (X)dX = (5.20)
j r
m,pen0 ,p<m
a,кe{l,r}
1 f
£ (m — n)pph(n)— f (X — fк — nu) ЗКma (X)dX.
,-кчт JR
m,peN0,p>m a,кe{l,r}
Inserting (5.20) into (5.19), we obtain (5.18). □
Corollary 5.10. Let S = {H, H0} be the JCL-model and let f = fFD. If case (E) of Proposition 5.3 is realized and S = {H,H0} is time reversible symmetric, then JpQ Qph
^ 0.
Proof. We set f := fl = fr. One has:
pph(m)f (X — f — mu) — pph(n)f (X — f — nu) =
e-meш(i — e-(p-m)^)fFD(X — f — mu)fFD(X — f — nu) ^ 0,
for n > m. From (5.18), we see that Jp Qph ^ 0. □
Remark 5.11. We will now comment the results. If J^q^ ^ 0, then system S is called light emitting. Similarly, if J1phQPh Վ 0, then we call it light absorbing. Of course if S is light emitting and absorbing, then JpJhQPh = 0.
(i) If в = to, then pph(m) = 50m, m e N0. Inserting this into (5.16), we get:
JToQPh = £ n 2П Լ f (X — fa)(jPphK 0a (X)dX > 0
pe n0 r
a^e^lr}
Hence, the system S is light emitting. (ii) Let us show S might be light emitting even if в < to. We consider the case (E) of Proposition 5.3. If S is time reversible symmetric, then it follows from Corollary 5.10 that the system is light emitting.
If the system S is time reversible and mirror symmetric, then J^Qd = 0, a e {l, r}, by Corollary 5.6(iii) . Since Jcpo Qel =0 by Proposition 5.3, we get that JSo Qel = 0 but the photon current is larger than zero. So our JCL-model is light emitting by a zero total electron current JS nel.
Let vr = 0, vl = 2 and u = 4. Hence S is not mirror symmetric. Then, we get from
Remark 5.7(iii) that Jphr.el = — Jphnel Հ 0. Hence, there is an electron current from the
v 7 p0,Qel p0,Qrl ^ '
left to the right lead. Notice that by Proposition 5.3 Jc nel = 0. Hence, JS nel Հ 0.
p0,Ql p0,Ql
(iii) To realize a light absorbing situation, we consider the case (S) of Proposition 5.3 and assume that S is time reversible symmetric. Notice that by Lemma 5.4, sc commutes with pel. We make the following choices:
vr = 0, vl ^ 4, u = v՛, vl = 0, Vr = u = v՛.
It follows out that with respect to the representation (5.18) one has only to m = n — 1, к = r and a = l. Hence,
Jph = Հր 1
JP0,Qph Հ-^ 2n
nGN П
X
Լ dЛ (pph(n — 1)f (Л — (n — 1)u) — pph(n)f (Л — (n + 1)u)) ^-Dr (Л)
Since, f (Л) = fFD(Л), we find:
Pph(n — 1)f (Л — (n — 1)u) — pph(n)f (Л — (n + 1)u) = Pph(n — 1)f (Л — (n — 1)u)f (Л — (n + 1)u) x (1 + ев(А-(п+1)и) — е-ви (1 + ев(А-Ш(п-1)))) ,
or
Pph(n — 1)f (Л — (n — 1)u) — Pph(n)f (Л — (n + 1)u) =
Pph(n — 1)f (Л — (n — 1)u)f (Л — (n + 1)u)(1 — е-вш )(1 — ев(х-шп)).
Since Л — nu ^ 0 we find pph(n — 1)f (Л — (n — 1)u) — pph(n)f (Л — (n +1)u) ^ 0 which yields JPph0,Qch Վ
To calculate JphQei, we use formula (5.7). Setting a = l, we obtain a' = r, which yields
р0 ,Qi
Հքք = — £ 2Пx
l rn.nGno '
Լ dЛ (pph(n)f (Л — Vr — nu) — Pph(m)f (Л — VI — mu)) (Л). One verifies that 300 (Л) = 0 and аП (Л) =0 for m = n + 1, n E N. Hence,
ph _ ^^ e
nGn
Լ dЛ (pph(n)f (Л — Vr — nu) — pph(n — 1)f (Л — Vl — (n + 1)u)) apnh(n+1)r (Л) Since Vr = u and Vl = 0, we find:
JP0&t = —h,2n x
JPh0,Qf = £ 2n x
l nGN П
l f (Л — (n + 1)u)pph(n — 1)(1 — е~вш) 3Pph(n+i)r(Л) dЛ Հ 0.
r
Hence, there is electron current flowing from the left to right induced by photons. We recall that Jc nel = 0.
P0,Qf
Acknowledgments
The first two authors would like to thank the University of Aalborg and the Centre de Physique Theorique - Luminy for hospitality and financial support. We thank Horia D. Cornean for discussions concerning the JCL-model and Igor Yu.Popov for the opportunity to present this complete version of our results. The work on this paper has been partially supported by the ERC grant no.267802 "AnaMultiScale".
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