superradiance transition and transport through nanostructureslkaplan/wnmp11.pdf · superradiance...

Superradiance Transition and Transport ThroughNanostructures

Lev Kaplan

Tulane University

March 5, 2011

In collaboration with:Luca Celardo (Brescia), Roman Sen’kov (CMU), Matthew Smith

(Tulane → AFRL), Suren Sorathia (Puebla), Vladimir Zelevinsky (MSU)

Lev Kaplan (Tulane University) Superradiance Transition WNMP 11 1 / 31

Outline

1 IntroductionMotivationSimple Model for Electron TransportNon-Hermitian Effective Hamiltonian

2 Superradiance Transition: Transmission and Resonance StructureAnalytic Results for TransmissionResonance Structure: Symmetric CaseResonance Structure: Asymmetric Case

3 Disorder and LocalizationComparison with RMT

4 Transport in Higher Dimensions

5 Conclusions and Outlook


Introduction Motivation

Motivation

Properties of open system depend on nature & degree of opening

Weak coupling: all internal states acquire small decay widths

Strong coupling: width segregation

“Superradiance transition”: analogy with Dicke superradiance inquantum optics [Sokolov & Zelevinsky]

Our objective:

Study relation between superradiance and mesoscopic transport

Start with simple paradigmatic model, then generalize


Introduction Simple Model for Electron Transport

Simple Model for Electron Transport

1D sequence of N wells and N + 1 potential barriers (~2 = 2m = 1)

E0 E

0

Δ Δ2extΔ1

ext

L L

V0

V0

Applications: semiconductor superlattices, 1D quantum dot arrays, ...[Tsu & Esaki; Pacher et al.; Morozov et al., ...]


Introduction Non-Hermitian Effective Hamiltonian

Non-Hermitian Effective Hamiltonian

Discrete quantum system with N intrinsic states |i〉, Hamiltonian H

Coupled to continuum of states |c ,E 〉 with amplitudes Aci (E )

Integrate out continuum

H → H(E ) = H + P(E )− i

2W (E )

Pij(E ) =∑c

P.V.

∫dE ′

Aci (E ′)Ac

j (E ′)∗

E − E ′Wij(E ) = 2π

∑c(open)

Aci (E )Ac

j (E )∗

Transmission: T ab(E ) = |Z ab(E )|2

where Z ab(E ) =N∑

i ,j=1

Aai

1

E −Hij(Ab

j )∗

E dependence may often be neglected over narrow energy window



Weak Tunneling Between Wells

In our case

H =

E0 + δ1 − i

2γ1 Ω 0 ... 0Ω E0 Ω ... 00 Ω E0 ... 0... ... ... ... ...

0 0 0 ... E0 + δ2 − i2γ2

where Ω =

2α2E0

V0(1 + αL/2)exp (−α∆)

[k =

√E0, α =

√V0 − E0

]

γ1,2 =8α3E0k

V 20 (1 + αL/2)

exp (−2α∆1,2) δ1,2 =k2 − α2

4αkγ1,2



Weak Tunneling Between Wells

H =

E0 + δ1 − i

2γ1 Ω 0 ... 0Ω E0 Ω ... 00 Ω E0 ... 0... ... ... ... ...

0 0 0 ... E0 + δ2 − i2γ2

Transmission between two leads:

T 12(E ) = T (E ) =

∣∣∣∣∣ (√γ1γ2/Ω)∏N

k=1(E − Ek)/Ω

∣∣∣∣∣2

where Ek are eigenvalues of H

For weak coupling, excellent agreement with brute force numerics



Transmission: Numerical Examples

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1T

(E)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)SR

Δ Δ

Δ Δ

Δext Δ

ext

ΔextΔ

ext

=1.5 =2

=2.15 =2.4


Superradiance Transition: Transmission and Resonance Structure Analytic Results for Transmission

Analytic Results for Transmission: Small N

Define asymmetry parameter q: γ1 = γ γ2 = γ/q

N = 1: H = −iγ/2− iγ/2q

Maximum transmission at E = 0: T (E = 0) ≡ T1 = 4q(q+1)2

Perfect transmission possible only in symmetric case q = 1

N = 2: H =

(− i

2γ Ω

Ω − i2γ/q

)[Volya & Zelevinsky; Alhassid,Weidenmuller, & Wobst]

T (E = 0) ≡ T2 =

∣∣∣∣ (γ/Ω)/√q

1 + (γ/Ω)2/4q

∣∣∣∣2Perfect transmission for any q, at critical coupling

( γΩ

)cr

= 2√q

Integrated Transmission

S =1

4Ω

∫T (E ) dE =

2πγ/Ω

(q + 1)[1 + (γ/Ω)2/4q]maximized at

( γΩ

)cr



Analytic Results for Transmission: General N

T (E ) =4γ1γ2P

2−[

1 + γ1γ2

(P2− − P2

+

)]2+ (γ1 + γ2)2P2

+

where

P±(E ) =1

N + 1

N∑n=1

(±1)n

E − EnEn = 2Ω cos

(πn

N + 1

)

For N 1 and −2Ω < E < 2Ω, let β = cos−1(E/2Ω):

P±(E ) ≈ 1

2Ω sin[(N + 1)β]

sin(Nβ)

− sin(β)

Near middle of band, |E | Ω, let E = En + r D = En + r2πΩ

N:

T (E ) ≈ 16γ1γ2

(4Ω + γ1γ2/Ω)2 sin2(πr) + 4(γ1 + γ2)2 cos2(πr)




Near middle of band, |E | Ω, let E = En + r D = En + r2πΩ

N:

T (E ) ≈ 16γ1γ2

(4Ω + γ1γ2/Ω)2 sin2(πr) + 4(γ1 + γ2)2 cos2(πr)

Oscillatory behavior of T (E ) independent of N1 Integer r : energies corresponding to Bloch poles

T (E = En) = T1 ≡4q

(q + 1)2

2 Half-integer r : energies midway between Bloch poles

T (E = En + D/2) ≈ T2 ≡∣∣∣∣ (γ/Ω)/

√q

1 + (γ/Ω)2/4q

∣∣∣∣2

Maximum and minimum T (E ) for general N expressed in terms ofT (E = 0) for one-well and two-well systems




Which of T1 = 4q/(q + 1)2 and T2 =∣∣∣ (γ/Ω)/

√q

1+(γ/Ω)2/4q

∣∣∣2 is maximum and

which is minimum? Depends on coupling γ/Ω!

For γ/Ω < 2:

T2 < T1 As γ increases, T2 and Tavg rise

For 2 < γ/Ω < 2q:

T2 > T1 At (γ/Ω)cr = 2√q, T2 and Tavg are maximized

For 2q < γ/Ω:

T2 < T1 As γ increases further, T2 and Tavg fall




Example for asymmetry parameter q = 10, and ten values of coupling γ:

-0.2 -0.1 0 0.1 0.2E

0

0.2

0.4

0.6

0.8

1

T(E

)

γ/Ω=0.1γ/Ω=1γ/Ω=1.8

-0.2 -0.1 0 0.1 0.2E

0

0.2

0.4

0.6

0.8

1

T(E

)

γ/Ω=3γ/Ω= 2 q

1/2

-0.2 -0.1 0 0.1 0.2E

0

0.2

0.4

0.6

0.8

1

T(E

)

γ/Ω=15γ/Ω=20γ/Ω=25

-0.2 -0.1 0 0.1 0.2E

0

0.2

0.4

0.6

0.8

1

T(E

)γ/Ω=100γ/Ω=400


Superradiance Transition: Transmission and Resonance Structure Resonance Structure: Symmetric Case

Resonance Structure: Symmetric Case (q = 1)

How are different transmission regimes reflected in resonance structure?

Evolution of resonance poles with increasing coupling (decreasing exteriorbarrier width) for N = 5:

2.3175 2.318 2.3185

Ei

1e-08

1e-06

0.0001

Γ i

Δ/Δext

=1

Δ/Δext

=2.4

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

2.317 2.318 2.319E

0

0.2

0.4

0.6

0.8

1

T(E

)

SR

Δ Δ

Δ Δ

Δext Δ

ext

ΔextΔ

ext

=1.5 =2

=2.15 =2.4

Superradiance transition!



Superradiance Transition: Symmetric Case (q = 1)

In General:

Small coupling γ: All N resonances acquire widths Γi ∼ γLarge γ: Only M widths continue to grow (where M = number ofopen channels), while remaining widths → 0

At what value of external coupling does superradiance transition (SR)occur?

Resonance overlap criterion: 〈Γ〉/D ≈ 1

Γ ∼ γ/N, D ∼ Ω/N ⇒ γ ≈ 2Ω

Coincides with maximum transmissioncriterion (for symmetric case)

In our model, this happens when ∆1,2 = ∆/2

E0 E

0

Δ Δ2extΔ1

ext

L L

V0

V0



Average of N − 2 Smallest Resonance Widths (Symmetric)

0 2 4 6 8 10γ/Ω

0

0.5

1

1.5<

Γ>/D

0 2 4 6 8 10γ/Ω

0

1

2

3

4

<Γ>

/DN=100

N=5



Integrated Transmission

S =1

4Ω

∫T (E ) dE

N→∞−−−−→ π(γ/Ω)

4 + (γ/Ω)2

Peaked at SR: γ/Ω = 2⇐⇒ ∆/∆ext = 2

1Δ/Δ

ext

0

0.1

0.2

0.3

0.4

S

N=3N=10N=50



Integrated Transmission: Enhancement at SR

Let S1 = integrated transmission at ∆ext = ∆ (or ∆ext = 0)

Let Smax = integrated transmissions at SR (∆ext = ∆/2)

Smax

S1∼ V0

4√

E0(V0 − E0)e√V0−E0∆

Weak tunneling√V0 − E0∆ 1 =⇒ Exponential enhancement

0.1 0.15 0.2 0.25 0.3 0.35Δ

012345

Log

(Sm

ax/S

1)

1000 1500 2000 2500 3000V

0

2

3

4

Log

(Sm

ax/S

1)Lev Kaplan (Tulane University) Superradiance Transition WNMP 11 18 / 31

Superradiance Transition: Transmission and Resonance Structure Resonance Structure: Asymmetric Case

Now Consider Asymmetric Coupling q > 1

Average of N − 2 smallest resonance widths

1 10 100γ/Ω

0

0.1

0.2

0.3

0.4<

Γ>/D

q=4q=10

2 2q 2q

Double SR transition at γ/Ω = 2 and γ/Ω = 2q

# Narrow resonances: N −→ N − 1 −→ N − 2


Superradiance Transition: Transmission and Resonance Structure Resonance Structure: Asymmetric Case

Integrated Transmission: Asymmetric Coupling

S =1

4Ω

∫T (E ) dE ≈ π(γ/2Ω)

(q + 1) [4 + (γ/Ω)2/q](q = asymmetry)

Peaked at γ/Ω = 2√q ⇐⇒ ∆ = ∆1 + ∆2 [Pacher & Gornik]

Geometric midpoint between SR transitions γ/Ω = 2 and γ/Ω = 2qExample for N = 100 and asymmetry q = 1, 10, 25

0 5 10 15 20 25 30γ/Ω

0

0.2

0.4

0.6

0.8

1

S

q=1q=10q=25


Disorder and Localization Introduce On-Site Disorder

Introduce On-Site Disorder

Random variation δL ∈ [−WL/4E0,+WL/4E0] in well widths=⇒ random variation δE0 ∈ [−W /2,+W /2] in on-site energies

Weak disorder

W < 2πΩ

Mean spacing (at center of energy band) D ≈ 2πΩ/N weaklyinfluenced by disorder

Maximum transmission at γ/Ω = 2√q as in clean case

Strong disorder

W > 2πΩ

Localized transport regime: T log-normally distributed

Mean spacing D ≈W /N

Maximum transmission at 〈Γ〉 ≈ D ⇐⇒ γ/N ∼W /N ⇐⇒ γ ∼W

Critical value of coupling proportional to disorder strength


Disorder and Localization Example of Weak Disorder

Example of Weak Disorder

Average transmission near center of energy band: −0.1 ≤ E/Ω ≤ 0.1

Disorder: W /Ω = 0.5

Vertical dashes: γ/Ω = 2√q

0.01 0.1 1 10 100

γ/Ω0

0.1

0.2

0.3

0.4

0.5

0.6

<T

>

q=1q=4q=25q=100

Notice: Maximum of each curve intersects q = 1 (symmetric) curve

At critical coupling, left & right tunneling probabilities equal!


Disorder and Localization Example of Strong Disorder

Example of Strong Disorder

Average lnT near center of energy band: −0.1 ≤ E/Ω ≤ 0.1

Asymmetry: q = 10

Disorder: W /Ω = 4, 10, 15

Arrows: γ/Ω = 1.2ND/Ω

1 10

γ/Ω

3

4

5

6

7

8

9

<ln

T>

- <

lnT

>0

Heff

W=4 H

eff W=10

Heff

W=15PB W=4PB W=10PB W=15


Disorder and Localization Comparison with RMT

Comparison with Random Matrix Theory

Tunneling probabilities on left and right: τa = 1− |〈Saa〉|2

Maximum transmission achieved when τ1 = τ2 [Beenakker]

In RMT, τ1,2 =4κ1,2

(1 + κ1,2)2where κ1,2 is effective coupling to each

channel [Celardo et al.]

Then τ1 = τ2 satisfied when

κ1 = κ2 [symmetric coupling]

κ1 = 1/κ2 ←− nontrivial case

Explicitly κ1 =πγ1|〈1|ψE=0〉|2

2Dκ2 =

πγ2|〈N|ψE=0〉|2

2D



Comparison with Random Matrix Theory

Explicitly κ1 =πγ1|〈1|ψE=0〉|2

2Dκ2 =

πγ2|〈N|ψE=0〉|22D

We know D = 2πΩ/N in middle of band

Now if eigenstates obey RMT, |〈1|ψE=0〉|2 = |〈1|ψE=0〉|2 =1

N

κ1 = 1/κ2 ⇐⇒ γ/Ω = 4√q

Off by factor of 2!



How to resolve discrepancy with RMT?

-2 -1 0 1 2E

0

0.005

0.01

0.015

0.02

<|Φ

1(E)|2 >

GOEW=0.5W=1W=10

1/N

Look at LDOS at edge of chain

For moderate disorder, factor of 2 enhancement at E = 0 comparedwith RMT

Accounting for this factor crucial for correctly computing transmission


Transport in Higher Dimensions Higher Dimensional Open Model

Higher Dimensional Open Model

D Dimensions: Lattice of N = M × L sites coupled to 2×M channels2D ANDERSON MODEL

γ1 γ2ΩE

0

Focus: G (E ) =M∑a=1

2M∑b=M+1

|Z ab(E )|2

Effective Hamiltonian:

Hii =

E0 + δE0

not coupled to leads

E0 + δE0 + δ1,2 − i2γ1,2

sites coupled to leads

Hij =

Ω nearest neighbors

0 otherwise


Transport in Higher Dimensions Conductance for Weak Disorder

Conductance as Function of Coupling Strength

Consider conductance G (0):

0.1 1 10 100

γ/Ω

0.01

0.1

1

10

<G

>

q=1q=25q=400

0.1 1 10 100

γ/Ω

0.01

0.1

1

10

<G

>

q=1q=25q=400

Q1D: 10× 100W /Ω =

√3/4

Theory: RMT(Melsen & Beenakker)

2D: 30× 30W /Ω = 2

G maximized at (γ/Ω)cr = 2√q


Transport in Higher Dimensions Increasing Disorder

Increasing Disorder W

Fix asymmetry q = 4:

1 10 1004

6

8

10

12

14

< L

n G

> -

< L

n G

>0

W=2W=16W=32W=60

0.1 1 10 100

γ/Ω

4

6

8

10

12

< L

n G

> -

< L

n G

>0

W=5W=20W=40W=60

2D: 20× 20

3D: 8× 8× 8

G maximized at (γ/Ω)cr ∼ ND ∼W


Conclusions and Outlook


SR transition in paradigmatic model of coherent quantum transport

Transport properties and resonance structure strongly affected bycoupling strength to environment

Symmetric coupling: Transmission peaked at SR transition

(γ/Ω)cr = 2√q

Typical resonance width is at maximum

Asymmetric coupling: Double SR transition

Transmission peaked at midpoint between two transitions (= localminimum of typical resonance width)

Results robust for weak to moderate disorder

For strong disorder, γcr ∼W

Results apply in Q1D, 2D, and 3D lattices

Outlook: Extension to include e-e interactions



Thank you!


superradiance transition and transport through nanostructureslkaplan/wnmp11.pdf · superradiance...

Documents