Download Dotan Davidovich research proposal

Adiabatic and Non-adiabatic Transport in Networks
Dotan Davidovich
Adviser: Doron Cohen
Physics Department, Ben Gurion University of the Negev
(Dated: October 27, 2011)
====== 1.
Background
Quantum networks and quantum random networks appear in many different work on various subjects, starting
from future quantum computation models[1] ,quantum internet[2] and even in the explanation of photosynthesis[3].
Of particular interest are the currents and counting statistics[4, 5] in driven networks. These were studied in the
adiabatic regime[6] for open and closed systems called ”pumping”[7–9] and ”stirring”[10–12] respectably. There are
still open question regarding these systems behaviour for non-adiabatic driving and beyond the 2-level landau-zener
dynamics[13].
====== 2.
Abstract
We would like to study a generalized Landau-Zener crossing problem [13]. In our setup a shuttle level crosses a band
of network levels. A particle that is located initially in the shuttle has the probability to be transferred into the
network. The questions that we would like to ask are the following:
(1) What is the survival probability 𝑃 (𝑡) of the particle in the shuttle?
(2) What is the final distribution of the occupation probabilities in real space?
(3) What is the final distribution of the probabilities in energy space?
(4) What is the statistics of the induced currents in the network.
We would like to know how the answers depend on the rate in which the energy of the shuttle is varied, ranging from
the adiabatic to the diabatic / sudden regimes.
With regard to (4) we are going to analyse the distribution of the circulated charge 𝑄. In the stochastic approximation
the distribution is bounded to the range |𝑄| < 1, while in the quantum adiabatic limit |𝑄| might be arbitrarily larger
due to a stirring effect. Consequently we would like to develop a phenomenology for the induced stirring in random
networks.
Modelling .–
𝑁 sites system, the sites are randomly scattered on an 𝐿 × 𝐿 sheet, the on site energies are randomly chosen from a
range of [0, Δ]. The system is build using several parameters:
𝐿
𝑟0 = √ = average distance.
𝑁
𝑐0 = typical coupling between neighbouring sites.
𝜉 = interaction range
𝜖𝑖 = on-site energy = random ∈ [0, Δ]
𝑟𝑖 = site’s location = random ∈ [0, 𝐿] × [0, 𝐿]
The hopping amplitude is determined by the distance between the sites in the following manner:
2
𝑐𝑖𝑗 = hopping amplitude = ±𝑐0 𝑒
𝑟0 −|𝑟𝑖 −𝑟𝑗 |
𝜉
(1)
The ± indicates that the sign is random but we also simulate only positive hopping amplitude.
Another optional definition of the model is 𝑁 site lattice with random on-site energies taken from [0, Δ], and only
nearest neighbours interaction. The interaction is defined as followed:
𝑐𝑖𝑗 = ±𝑐0 e−𝑟𝑎𝑛𝑑𝑜𝑚 ,
𝑟𝑎𝑛𝑑𝑜𝑚 ∈ [−𝜎, 𝜎]
(2)
Δ
in
We have two dimensionless parameters that describe the system ,𝜉/𝑟0 and Δ/𝑐0 in the first system, and 𝜎 and 𝑐0
the second. These are approximately the same systems because in the second model 𝜎 plays the roll of 𝜉/𝑟0 so we
get a random network. Participation number.– The main parameter we use to characterize the eigenstates of the
system is the participation number 𝑃 𝑁𝜓 which tells how many sites are involved at a certain state 𝜓 , this is defined
by:
𝑃 𝑁𝜓 = ∑︀
1
* 2 , 𝑖 = site number
(𝜓
𝑖 𝜓𝑖 )
𝑖
(3)
This gives 1 if the particle is in 1 site and 𝑁 if the particle is uniformly distributed along the system. The average
participation number for the eigenstates of the system is:
∑︀
𝑃 𝑁𝑛
𝑃 𝑁𝑛 = 𝑛
(4)
𝑁
In Fig.1 we show the dependence of 𝑃¯𝑁 on the dimensionless parameters of the model.
Time dependent scenario .– The particle starts at one of the sites with energy 𝜖0 = 𝑢(𝑡). The energy is raised
linearly at rate 𝑢˙ until the site’s energy is far above the band. We then have the final state |𝜓(𝑡𝑓 )⟩ which we
characterize with 𝑃 𝑁𝑖 and 𝑃 𝑁𝑛 , see Fig.3.
Survival probability - 𝒫(𝑡).– The survival probability is the probability to remain in the starting site. If the initial
condition was 𝑝0 (0) = 1, 𝑝𝑖 (0) = 0 for 𝑖 ̸= 0, then 𝒫(𝑡) = 𝑝0 (𝑡). In our simulation at moderate 𝑢˙ the particle will
”flow” to the band as seen in Fig.2.
====== 3.
Work plan
Survival probability at the starting site.– It is of interest to characterize the probability of the particle to stay
in the shuttle at the end of the process and how it depends on 𝑢˙ , the adiabatic and sudden regime are well studied
,here we want to know what is the survival probability 𝑃 (𝑡) of the particle in the shuttle , after the shuttle has passed
the band.
Final site-occupation distribution.– As the shuttle passes through the band the particle ”flows” to the other
sites. We would like to find out what is the probabilities of the particle to be at the different sites at the end of
the process and the distribution of these probabilities and how they depend on the shuttle rate. We can see a the
dependence in Fig.3.
Final energy distribution.– As the shuttle passes through the band the particle ”flows” to the other eigenstates.
We would like to find out what is the probabilities of the particle to be at the different eigenstates/energies at the
end of the process and the distribution of these probabilities and how they depend on the shuttle rate. We can see a
the dependence in Fig.3.
Induced currents.– We would like to know what are the statistics of the induced currents and of the circulated
charge in the network , and how they depend on the rate of the shuttle from the adiabatic to the diabatic/sudden
regimes.The circulated charge is the charge passed through some close path.
Optional scenarios.– The starting conditions can be changes so the particle starts at an eigenstate in the middle of
the band , and the level we change with time, partially ”scoops” it and partially ”scatters” it over other eigenstates.
3
Geometrical conductance.– The distribution of the geometrical conductance of a current created by the moving
of a scatterer through a network is described in Fig.2 [11] but the fitting parameters of this distribution are yet to be
explained. We want to find a theory to explain these fitting parameters.
====== 4.
Preliminary results
In our simulation the particle starts at one of the sites at very low energy 𝜖0 = 𝑢(𝑡) (far from the network band) ,
then the site’s energy is raised linearly at rate 𝑢˙ until it is far above the band levels. We would like to characterize
the system at the end of the process and it’s dependence on 𝑢.
˙ We start by solving the Schrdinger equation with
time-dependent Hamiltonian in order to get 𝜓(𝑡).
⎛ ⎞
⎛
⎞
1
𝜖0 𝑐01 𝑐02 · · ·
⎟
⎜
⎜
⎟
𝑐
𝜖
𝑐
·
·
·
10
1
12
𝜕𝜓
⎜0⎟
⎜
⎟
(5)
𝜖0 = 𝑢(𝑡) ,
𝜓(𝑡 = 0) = ⎜0⎟
= −𝑖𝐻𝜓 ,
𝐻 = ⎜𝑐20 𝑐21 𝜖2 · · ·⎟ ,
𝜕𝑡
⎝ ⎠
⎝
⎠
..
..
.. . .
.
.
··· .
.
Where 𝑢(𝑡) ∝ 𝑡.
As the shuttle level crosses the band the particle ”flows” to the band , this can be shown by looking at the
probability to be at each site as shown in Fig.2. At the end of the process we have an Hamiltonian with eigenstates
|𝑛⟩ and the particle state |𝜓(𝑡 = 𝑡𝑓 )⟩ described at the standard basis |𝑖⟩. We would like to find out what are the
spatial and energy distribution of the particle at the end, and how these depend on the rate of energy change.
First,we have to realize that there are oscillation between the different |𝑛⟩ states the particle has ”flown” to, so we
have to average over them. We Start by defining site and energy distributions:
𝑝𝑖 (𝑡) = |⟨𝑖|𝜓(𝑡)⟩|2 ,
𝑝𝑛 (𝑡) = |⟨𝑖|𝜓(𝑡)⟩|2
(6)
Where 𝑝𝑖 is time dependent and 𝑝𝑛 does not depend on time when the simulation ends 𝑡 > 𝑡𝑓 so we have to average
over 𝑝𝑖 . The time average over 𝑝𝑖 is:
2
𝑝¯𝑖 = 𝑝𝑖¯(𝑡) = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩|
(7)
This can be proofed as followed: We start from the particle state at a given time:
∑︁
|𝜓(𝑡)⟩ =
𝑒−𝑖𝐸𝑛 𝑡 |𝑛⟩⟨𝑛|𝜓(𝑡)⟩
(8)
𝑛
The probability to be at site 𝑖 in time 𝑡 is:
∑︁
𝑝𝑖 = |⟨𝑖|𝜓(𝑡)⟩|2 =
𝑒−𝑖(𝐸𝑛 −𝐸𝑚 )𝑡 ⟨𝑖|𝑛⟩⟨𝑚|𝑖⟩⟨𝑛|𝜓(𝑡)⟩⟨𝜓(𝑡)|𝑚⟩
𝑚,𝑛
=
∑︁
|⟨𝑖|𝑛⟩⟨𝑛|𝜓(𝑡)⟩|2 +
𝑛=𝑚
∑︁
𝑒−𝑖(𝐸𝑛 −𝐸𝑚 )𝑡 ⟨𝑖|𝑛⟩⟨𝑚|𝑖⟩⟨𝑛|𝜓(𝑡)⟩⟨𝜓(𝑡)|𝑚⟩
(9)
𝑛̸=𝑚
When we take the time average , the second term averages to zero so we are left with:
𝑝¯𝑖 = 𝑝𝑖¯(𝑡) = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩|
2
(10)
For presentation purposes we define the joint probability distribution:
2
𝑃𝑛,𝑖 = |⟨𝑖|𝑛⟩ ⟨𝑛|𝜓⟩|
It follows from the above definitions that:
∑︁
𝑝𝑛 =
𝑃𝑛,𝑖
(11)
(12)
𝑖
𝑝¯𝑖 =
∑︁
𝑛
𝑃𝑛,𝑖
(13)
4
We can use 𝑃𝑛,𝑖 to calculate the participation number for the sites and eigenstates at the end of the process at different
rates:
𝑃 𝑁𝑛 = ∑︀
1
2
𝑛 𝑝𝑛
,
1
𝑃 𝑁𝑖 = ∑︀
𝑖
𝑝¯2𝑖
(14)
The spatial and energy participation numbers are described at Fig.3, from which we choose an intermediate rate and
describe 𝑃𝑛,𝑖 at this rate in Fig.4.
Currents and counting statistics
The current operator describing the current in a single bond is obtained by introducing a flux and then taking it to
zero:
(︀
)︀
(︀
)︀
*
𝐼𝑖→𝑗 = 𝑖 |𝑖⟩𝐻𝑖𝑗 ⟨𝑗| − |𝑗⟩𝐻𝑖𝑗
⟨𝑖| = 𝑖𝑐𝑖𝑗 |𝑖⟩⟨𝑗| − |𝑗⟩⟨𝑖|
(15)
⟨𝐼𝑖→𝑗 (𝑡)⟩ = ⟨𝜓(𝑡)|𝐼𝑖𝑗 |𝜓(𝑡)⟩ = 𝑖(𝜓𝑖 𝑐𝑖𝑗 𝜓𝑗* − 𝜓𝑗 𝑐𝑖𝑗 𝜓𝑖* ) = −2𝑐𝑖𝑗 Im{𝜓𝑖 𝜓𝑗* }
From the currents we can define the counting operator:
∫︁ 𝑡
𝐼𝑖→𝑗 (𝑡′ )𝑑𝑡′
𝑄𝑖→𝑗 (𝑡) =
(16)
(17)
0
We would like to characterize the counting statistic in our system. If a stochastic picture were applicable, then 𝑄
would satisfy |𝑄| ≤ 1 But in the quantum adiabatic limit (for slow rate) |𝑄| might be arbitrarily large due to a stirring
effect.This is shown at Fig. 5.
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5
18
0.5
16
∆/c0
1
1.5
14
2
12
2.5
10
3
8
3.5
6
4
4
4.5
2
1
2
3
σ
4
FIG. 1: Average Participation number for different 𝜎 and Δ/𝑐0 . We would like our system to be at the ”green” zone, so we
don’t miss the geometrical aspects of the model. Later we use 𝜎 = 1 and Δ/𝑐0 = 1.
1
0.8
Pi
0.6
0.4
0.2
0
−30
−20
−10
0
10
20
30
u
FIG. 2: The Probability to be at any site as a function of time(parametrized by the starting site’s energy 𝑢) at an intermediate
rate, 𝑢=20.
˙
6
40
PN
i
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
rate
PN
n
15
10
5
0
0
10
20
30
40
50
rate
FIG. 3: Average Participation numbers at the end of the process for different rates for 𝜎 = 1 and Δ/𝑐0 = 1. one can identify
several regimes: slow, intermediate and fast. In the slow regime we have 𝑃 𝑁𝑛 ≈ 1. In the fast regime both 𝑃 𝑁𝑛 ≈ 1 and
𝑃 𝑁𝑖 ≈ 1, as expected from the sudden approximation.
|i> , Distance −−−>
<−−− energy ,|n>
5
0.025
10
15
0.02
20
25
0.015
30
0.01
35
40
0.005
45
10
20
30
40
FIG. 4: Probability distribution for sites and eigenstates sorted by distance from the starting site and state’s energy for 𝜎 = 1
and Δ/𝑐0 = 1. Intermediate rate (𝑢=20),
˙
averaged over 1000 realizations. The spatial dispersion is narrow, energy dispersion
is wide. For slow rates (𝑢˙ ≤ 1), the energies involved are only the low ones and the spatial dispersion is wide. For fast rate
(𝑢˙ ≥ 90), the particle mainly stays at the starting site.
7
1.8
1.6
Maximal Q
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
80
90
100
rate
FIG. 5: Maximal |𝑄| for different rates(𝜎 = 1 and Δ/𝑐0 = 1), one can recognize a low rate region there is no zero probability
to have |𝑄| ≥ 1. Red dots are maximal |𝑄| for different realizations with the same parameters.
200
#
150
100
50
0
−0.5
0
0.5
1
Q max
FIG. 6: Histogram for maximal |𝑄| for 𝑢˙ = 20 (10,000 values).
1.5