Luis Foa Torres
Adriatic sea, near Trieste. Photo by Luis Foa Torres © 2005.
Mono-parametric quantum charge pumping in an open ring (a tale of interferences in the Fermi sea)
CEA CEA -- Grenoble. Grenoble. December 5th, 2005.
Luis Foa Torres
Outline of the talk Introduction Driven systems, Quantum Charge Pumping Adiabatic parametric scattering theory. Beyond the adiabatic theory Floquet Theory: some basics Transmittances and time-averaged currents in a single particle picture
Pumping in an open ring with a single parameter Conclusions and Perspectives
Driven systems
Luis Foa Torres
What is the influence of excitations by electromagnetic fields and time-dependent gate voltages on the electron transport?
Photon assisted tunneling Driven systems Present theoretical and experimental interests
Adiabatic and non adiabatic pumping of electrons Also used in the context of the tunneling time problem
… Theory for driven quantum transport is less developed as compared with the time-independent case. Calculations are usually restricted to the lowest order in the amplitude of the driving field.
Luis Foa Torres
Classical Pumps: the Archimedean screw
Archimedes, 3rd century BC.
Pumping in the Quantum domain Quantum Pumping Generation of a directed current (dc) at zero bias potential. Necessary condition for pumping.
T→ ≠ T←
Left-right symmetry breaking.
Luis Foa Torres
Pumping in the Quantum domain Typical Setup
M. Switkes et al Science 283, 1905 (1999).
Luis Foa Torres
Pumping in the Quantum domain Typical Setup
• Ballistic quantum dot. • Cyclic two-parameter deformation of the dot.
M. Switkes et al Science 283, 1905 (1999).
Different operational regimes according to the magnitude of
ω0 φ
Modulation frequency. Phase difference.
ω 0 ⎧ > 1, " non − adiabatic " QP
Dwell time in the dot.
Luis Foa Torres
Parametrical theory for adiabatic pumping P. W. Brouwer, PRB 44, 10135R (1998). Parametric pumping theory: S-matrix approach at low-frequency (first order in ).
S ( X1 (t ), X 2 (t ))
Main outcome of this theory: The average pumped current is proportional to the area in parameter space.
X2
Hence, at least two out of phase time-dependent parameters are needed to achieve pumping.
X1
Luis Foa Torres
Why does the parametric scattering theory require at least two parameters? What is the difference with a situation with only one parameter? Is it possible to operate a pump with a single parameter? What is the minimal number of parameters needed to obtain QP? what symmetries should they break?
Let us think beyond the adiabatic parametric scattering matrix theory… this means using either strong driving amplitudes or high frequencies…
Antecedentes teóricos, posibles enfoques.
Luis Foa Torres
Floquet Theory (a brief summary) Spatially periodic systems
periodically time-dependent systems
Bloch theorem
Floquet theorem
assures the existence of quasi-momenta associated with the Bloch functions.
assures the existence of quasi-energies and Blochtype states (Floquet's states).
∂⎞ ⎛ ˆ H ( x, t ) − ih ⎟ψ ( x, t ) = 0. Hˆ (t ) = Hˆ (t + T ) ⎜⎝ ∂t ⎠ There is a complete setψ ( x, t ) = exp(−iε t / h)φ ( x, t ), SepDynScales. α α α
of solutions of the form:
quasi-energies
Floquet's states or modes
φα ( x, t + T ) = φα ( x, t ) J. Shirley, PR 138, B979 (1965); Ya B. Zel’dovich, Sov.Phys. JETP 24, 1006 (1967). H. Sambe, PRA 7, 2203 (1973).
Luis Foa Torres
Floquet Theory (a brief summary) There are solutions of the TDSE of the form:
Hˆ (t ) = Hˆ (t + T )
ψ α ( x, t ) = exp(−iε α t / h)φα ( x, t ), φα ( x, t + T ) = φα ( x, t )
Floquet space (Sambe space)
R ⊗T
ν,n = ν ⊗ n
Hˆ F ( x, t )φα ( x, t ) = ε α φα ( x, t ), ∂ ˆ ˆ H F ( x , t ) = H ( x , t ) − ih γ , m Hˆ F ν , n = Hˆ γ( m,ν − n ) + nhω0δ γ ,ν δ n ,m ∂t Time- evolution operator
``Well, in our country,'' said Alice, still panting a little, ``you'd generally get to somewhere else if you ran very fast for a long time as we've been doing.'' Lewis Carroll, “Through the Looking Glass”.
U γ ,ν (t ,0) = ∑ γ , n e n
− iH F t / h
ν ,0 e
inω0 t
.
Luis Foa Torres
Beyond the adiabatic scattering theory Average Current Single electron picture
I L (t ) = i
I=
N L = ∑ c +j c j
e [H (t ), N L ], h
1
j∈L
T
dt I ∫ T
L
(t )
0
I=
e
[
∞
]
(n) (n) d ε T ( ε ) − T ∑ ∫ RL LR (ε ) f (ε )
2πh n = −∞
S. Camalet et al, PRB 70, 155326 (2004). S. Kohler, J. Lehmann & P. Hänggi, Phys. Reports 2005.
Transmission probability (n) RL
T
(ε ) = 4Γ
(n) R
{G }
R F ( R , n );( L , 0 )
2
ΓL( 0 )
Gˆ FR = (ε Iˆ − Hˆ F ) −1
Luis Foa Torres
Example 2: simple two-parameter pump
Main difference between a situation with one and two driving parameters: Possibility of having the analog of a magnetic flux in Floquet space.
Words on symmetry breaking and QP, phase rigidity and reciprocity, dynamical symm-breaking.
This provides the directional asymmetry needed for quantum pumping. Dynamical LR symmetry breaking. LFT, Physical Review B 72, 245339 (2005); condmat/0511223
Directional asymmetry between emission and absorption -> linear effect.
Luis Foa Torres
Pumping in an AB ring using a single parameter Ingredients
x
¾ Static magnetic field
Provides symmetry breaking, directional asymmetry.
¾ Time-dependent potential
LFT, Physical Review B 72, 245339 (2005); cond-mat/0511223
Provides additional inelastic processes removing phase-rigidity.
diferencias con Hänggi
Interplay between spatial interference and PAT, explain analytical results.
X
φ / φ0
Results a)
Luis Foa Torres
1.0
|I|
0.5
Fermi energy close to the resonant point, different driving amplitudes.
1E-6
0
2
∝
ω0
1.0
1E-14 1E-3
0.01
h ω0
0.1
1
Γ
A similar behavior is observed as a function of the driving amplitude. LFT, Physical Review B 72, 245339 (2005); condmat/0511223.
| I | [arb. units]
Small Fermi energy
1E-4
εF / E0
b)
1E-10
1E-5
1.0
0
0.5
0.0
2.0
1
I [arb.units]
| I | [e/(hV)]
0.01
0
-1 0.0
0
0.5
φ / φ0
1.0
1
εF / E0
The current can be reversed by changing the magnetic flux.
2
Conclusions and Perspectives
Luis Foa Torres
Time independent scheme for time-dependent transport in the presence of cyclic potentials. Quantum pumping in a ring with a single parameter, interplay of PAT and spatial interference. These ingredients have the same footing when the problem is formulated in Floquet space.
© Background photo by Luis Foa Torres, 2004.
Possibility of reversing the current by changing the magnetic flux.
Quantum pumping as an interference effect in Floquet space.
