=Paper=
{{Paper
|id=Vol-1482/733
|storemode=property
|title=Computer simulation of excitation conductivity by qubit model
|pdfUrl=https://ceur-ws.org/Vol-1482/733.pdf
|volume=Vol-1482
}}
==Computer simulation of excitation conductivity by qubit model==
https://ceur-ws.org/Vol-1482/733.pdf
Суперкомпьютерные дни в России 2015 // Russian Supercomputing Days 2015 // RussianSCDays.org
Computer simulation of excitation conductivity by
qubit model∗
Yuri I. Ozhigov, Nikita A. Skovoroda
Lomonosov Moscow State University, Faculty of Computational Mathematics and
Cybernetics, Moscow, Russia
We describe the computational model of quantum energy transfer in JCH like chains
based on qubit representation of quantum states. The chain consists of optical cavi-
ties, each of which contains one two level atom interacting with field inside the cavity.
Photons can jump between cavities that creates the effect of conductivity in the chain.
Mathematical model of the conductivity is based on the qubit representation of quan-
tum states in which any qubit has the fixed physical sense. This allows to include the
dephasing noise, input and output intensity and analyse the contra intuitive quantum
effects like dephasing assisted transport (DAT) and quantum bottleneck. The pro-
gram realization of the model is based on the package Mathematica and utilizes such
advantages of object oriented programming as module structure of the program. We
represent the results of numerical simulation: the optimal output and input intensi-
ties, DAT and bottleneck.
1. Introduction and background
Computer simulation is the important tool for the investigation of open quantum systems
(see [1]), because their dynamics rarely has analytical representation. The description of intrigu-
ing purely quantum effects, like DAT or bottleneck, requires sufficiently large dimensionality of
Hilbert space of quantum states. The extremal manifestation of this complexity are the fast
quantum algorithms, which can outperform their classical counterparts for the mathematical
tasks of the search type (Grover search algorithm, Shor integer factoring). The simulation on
classical computers thus requires rigid economy of computing resource that are of our disposal:
the memory.
It is convenient to measure the memory in terms of bits; for the quantum states it will be
qubit memory. For the classical simulation with n qubits we must store 2n bits, and this value
grows so fast that for the simulation on personal computers we are limited by the value n = 12,
for supercomputers the values n ≈ 30 are only accessible now and up to n = 60 in the foreseeable
future. The supercomputer simulation is effective if only we can select carefully very limited
computational task; to find the statement of such a task we should fulfil preliminary simulation
on the more flexible personal computers, for which the preparation and tuning of the program
is substantially easier.
In addition, even the usage of a supercomputer for the simulation of quantum dynamics
requires the peculiar algorithms substantially optimizing the standard technique of linear algebra
that can be found only in the framework of one ten of qubits, where the Hamiltonian matrix is
in principle visible.
In this work we propose the qubit model, in which any qubit has the fixed physical sense.
For example, if we encode the state of the single two level atom in the optical cavity in three
qubits, we can reserve the first qubit for its state: |0i1 — ground, and |1i2 — excited, and the
rest two qubits — for the state of the field, such that |00i2,3 , |01i2,3 , |10i2,3 , |11i2,3 encode
vacuum state, one, two and three photon states correspondingly. This approach makes possible
to measure exactly the memory that we reserve for quantum states and manipulate it as it is
convenient for the different forms of the model.
We will describe the qubit form of Jaynes-Cummings-Hubbard model (see [22, 23]), for the
∗
The work is supported by Russian Foundation for Basic Researches, Grant 15-01-06132 A.
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chain of optical cavities, connected by optical fibres. Each cavity contains one atom with two
energy levels: ground and excited that can interact with the field inside the cavity. Photons can
also jumps from one cavity to the neighbouring with the fixed amplitude that creates the energy
transport along the chain.
The cavity stores the photons of the fixed frequency ωc close to the atomic frequency that
creates the condition of valuable coupling between atoms and field. Fock states of the photons
can be also used as the logical states for the quantum computer; this scheme has been proposed
in the works [25, 26] and elaborated in the works [28–35]. The states of field can be effectively
stored in the states of atomic ensembles (see, for example, [27]) that could help the initialization
of a quantum computer.
We measure the effectiveness of the transport by the conductivity of excitations, that is
the time in which the excitation travels from the beginning of the chain and reaches the end.
We fix this time by the additional qubit called a sink, which interaction with the last cavity
is irreversible. We also consider the noise, which affects conductivity in the form of dephasing
that brings the interesting and practically important effect of increasing conductivity with the
growth of the noise intensity: dephasing assisted transport — DAT (see the works [11,12,18–21],
in which DAT is investigated in more details; in [36] one can find the treatment of conductivity
in noiseless chains).
The conventional mathematical representation of such open quantum systems is Markov
master equation in the form of Kossakowski-Lindblad (see [2] and [3]), which is applicable to
the systems, which environment has no memory (however, in some important processes the
environment has the memory, the main example is the process of life; the consideration of non
Markovian processes can be found, for example, in [4], see also [5]). The influence of environment
is typically treated as the source of decoherence, though it can create the interesting quantum
effects, like quantum Zeno effect, in which the measurements helps to conserve the needed
quantum state (see the works [6, 7]). The special noise can even suppress decoherence ( [8]; see
also the work [9] — the noise smoothing by a unitary control).
The influence of noise is very important for the conductivity in JCH chains — in artificial
devices as well as in living organisms (see [10–12]).
The conductivity of a single excitation can be formulated in the form of quantum walks
( [10, 13–16]).
DAT effect was investigated in details in the works [11, 12, 18–21] and others; this interest
has arisen due its role in the light harvesting Fenna-Matthews-Olson (FMO) complex in green
sulfur bacteria (see [17]).
Here we use excitation conductivity as the field of application of programming complex
based on qubit approach. In the next section we explain the mathematical model, then we
describe the computer program, and show the result of simulation made on working stations.
In the last section we briefly discuss the results and perspectives.
2. Quantum effects in the energy transfer
We give a definition, starting with the most detailed model ”excitations, photons, phonons.”
We consider a linear chain consisting of identical optical cavities with two level atoms inside.
Each cavity holds the photons with frequency ωc . Fock state with n photons inside i-th cavity we
denote by |nif ii . Inside each i-th cavity there is one two-level atom, which eigenstates: ground
and excited we denote by |0iat i , |1iat i correspondingly. The difference between frequencies of
these states (detuning) d = ωc − ωa is small in comparison with each of them: d ≪ ωc that
allows to use rotating wave approximation for the atom-photon interaction at the large enough
time frame, and Jaynes -Cummings model (JC). In addition, each atom is placed in a bath of
thermal phonons that have dephasing effect on its excitation. In this paper all phonons have
the same frequency ωp , for which DAT effect is maximal. In fact, this frequency is equal to
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the difference between eigen frequencies of the Hamiltonian of JCH model (see below) that are
responsible for the transfer of excitations (see the work [12]).
The transfer of energy from the cavity to cavity occurs through the flight of photons between
the cavities where − h1 δi,j is the amplitude of the flight from the cavity j to the cavity i per unit
of time, so that the amplitudes of the opposing flights are mutually conjugate, that is δi,j = δ̄j,i .
The Hamiltonian of our model is thus obtained by adding to the Jaynes-Cummings Hubbard
Hamiltonian HJCH the term of exciton-phonon interaction Hint ep :
H = HJCH + Hint ep ,
P P P + P
HJCH = hωa σi+ σi + hωc a+
i ai + (γai σi + γ̄ai σi+ ) + i6=j (δi,j a+ +
i aj + δ̄j, iaj ai ),
i i i
P
Hint ep = g (b+ +
i + bi )σi σi .
i
(1)
Here a+ i , a i denote operators of creation and annihilation of a photon in i-th cavity, b + +
i , b i - denote
creation and annihilation of a phonon in i-th cavity, σi+ , σi - denote creation and annihilation of
the excited state of the corresponding atom (excitation), γ is the amplitude of photon emitting
by excited atom in the unit of time. We assume that the photon jump is possible between
the neighbouring cavities only where it occurs with the same amplitude so that all δi,j = 0 for
|i − j| =
6 1, and for |i − j| = 1, δi,j = δ. Constant g is the intensity of exciton-phonon interactions
(square root of Huang-Phys factors).
In the simplified model, which we call ”excitation-phonon” photons are ignored, and exci-
tations are transmitted from one atom to the other; its Hamiltonian Hep has the form
X X
Hep = He + Hint ep , He = hωa σi+ σi + (µi,j σi+ σj + µ̄j,i σj+ σi ), (2)
i i6=j
The inflow of the energy can be viewed either as a constant occurrence of the excitation on
the first node, taking the energy income from the outer bath, or simply set the initial state of
the first atom as excited, regardless of photons. In the first case the bath inflows can create
excitation irreversibly, or we can simply consider the initial state of a strongly excited field in
the first cavity, which interacts with the first atom.
There are two ways for the simulation of dephasing as well. Either we consider it as the
interaction with the explicit phonons, or as the irreversible process with only excitations and
without explicit phonons.
Combining all of the above methods of studying the conductivity, we get all kinds of partic-
ular computing models. In any case, for the irreversible models master equation of Kossakowski-
Lindblad should be applied:
i · δt
− H
Uδt = e h̄ ,
X 1
ρ̃t+δt = δt (Li ρL∗i − (L∗i Li ρ + ρL∗i Li )), (3)
2
i
∗
ρt+δt = Uδt ρ̃t+δt Uδt .
which contains the unitary dynamics Uδt as the particular case. Here Lindblad opera-
tors Li describe the irreversible part of the process. For example, the irreversible runoff
of excitations to the sink from the last node (end) is expressed by the operator Lsink =
gsink |0iend |1isink h0|sink h1|end , where the positive coefficient gsink expresses the intensity of runoff,
dephasing in i-th node resulted from the interaction with the implicit phonons is reflected by
the operator Ldeph, i = σi+ σi , photon leak through the walls of the i-th cavity — by Ldet, i = ai ,
etc.
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We consider the resulting quantum effects of excitation conductivity. We determine the
conductivity by the degree of filling sink in a certain time.
1). Quantum bottleneck. Reduction of conductivity with increasing intensity of the runoff.
Graph of conductivity depending on the rate of the runoff is not monotonically increasing as
would be in the case of classical conductivity, but has a local maximum at a finite rate gsink 0 .
The nature of this counter-intuitive effect is purely quantum.
It can be explained by considering the dynamics of the population of the initial state ac-
cording to the time when the runoff intensity is large. If there is no runoff, the dynamics will be
similar to the graph of the cosine, which has a maximum point at the initial time. The runoff
constantly destroys the excitation at the end of the chain that makes the density matrix close to
the initial state in which a reduction of the excitonic population at the first atom decreases with
time very weak. This process competes with the obvious classic excitonic population decrease
with increasing runoff.
If the runoff is strong enough, it constantly holds the density matrix (and hence population
of excitations in the first atom) in the initial state in which there is practically no reduction of
excitonic population at all that results in ” freezing ” of the population, like in quantum Zeno
effect, in which the effect of ” freezing ” comes from frequent measurements of the quantum
state. In the case of the classical conductivity instead of cosine fall of excitonic population we
would have a linear decrease, and there would be no bottleneck.
2). Dephasing assisted transport — DAT. A plot of the conductivity on the intensity
of the noise (the number of phonons of the resonant frequency, or intensity of the interaction
between excitations and phonons, expressed in coefficients of Lindblad for dephasing) has a local
maximum at a non-zero point. This means that there is some non-zero noise intensity at which
the conductivity is maximal. There are quantum processes, connected with the conductivity,
the effectiveness of which is enhanced with the increase of noise, for example, mixing at random
walks on graphs ( [37]). Effect of DAT stands out among them due a special role it plays in
biology.
DAT as a purely quantum effect. Dephasing is direct suppression of the off-diagonal elements
of the density matrix. Quantum dynamics is obtained by adding the amplitude states, the
evolution of which leads to the same point of the classical space. Dephasing shifts the phase
of the amplitude to a certain area. If this area would be narrower than the natural spread
of the phase of states in the absence of dephasing, this influence will make the interference
more constructive, thus increasing conductivity. This effect can be illustrated by the example of
road traffic. If possibilities of cars are the same and allow rapid acceleration and braking, then
synchronous mode of motion (phase conservation) will support high bandwidth of the road. But
if the possibilities of vehicles are very different, then to increase bandwidth it is needed to the
contrary, smooth mode of braking and acceleration, i.e. dephasing.
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Figure 1. Dynamics of the population of the initial state with a large runoff intensity. Runoff shifts the
density matrix to the initial state, the population of excitation in the first atom (for two-atom chain this
is ρin,in ) is shifted to the maximum point, and remains almost constant for a long time, preventing the
sink filling.
3. Qubit-based model
3.1. Evolution equations
Kossakowski–Lindlbad equation in diagonal form is used for calculating non-unitary evolu-
tion:
NX
2 −1
∂ρ 1
ih̄ = [H, ρ] + i γi (Ai ρA∗i − (A∗i Ai ρ + ρA∗i Ai )) (4)
∂t 2
i
Unitary evolution is optimized by diagonalizing the Hamiltonian.
In both cases, the evolution operator is determined by a Hamiltonian H with a set of non-
√
normalized matrices Li = γi Ai .
For general (non-unitary) case, the evolution is computed by the equations (3) directy.
These evolution equations, as well as the measurement implementation, are independent of
the used model. Various models could be created by constructing a Hamiltonian H and a set of
Lindblad matrices Li according to that model rules and semantics.
3.2. Qubit-based model construction
In order to allow simple operator-based constructing of Hamiltonians and Lindblad matrices
we build the models by binding one a several qubits (a qubit group) to a specific state parameter,
for example to the excitation of the first atom (see [38]). The model construction methods
work with virtually huge matrices: for example for a chain of one-levelled excitons of a length
100, it would have a virtual Hamiltonian of size 2100 ∗ 2100 (one qubit per each exciton). If
excitations were four-levelled, that would make two qubits for each excitation. These matrices
are not actually constructed in memory, an energy-limiting projection is used which selects only
those states that have the energy between the specified lower and upper bounds. The resulting
Hamiltonian and Lindblad matrices which are used in the actual calculation have much smaller
sizes due to this projection.
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3.3. A chain of excitation-photon-phonon cavities
A general chain model was constructed. This model implements a chain of cavities, each
of which has similar multi-level excitation, photon field, and phonons. It has energy limiting
support and supports attaching an input source and an output sink.
Figure 2. The exciton-photon-phonon cavities chain schematics.
There are three types of qubit groups in the chain:
• bi is the phonon state for atom i, 1 — the corresponding phonon is present, 0 — the
corresponding phonon is absent;
• ai is the exciton state for atom i, 1 — excited state, 0 — not excited;
• pi is the photon state for atom i, 1 — there is a photon, 0 — no photon.
The chain model has the following parameters:
• Natoms — the number of atoms,
• k — photon tunnelling rate between neighbouring cavities,
• µ — photon–atom interaction strength,
• g — phonon–atom interaction strength,
• ωa — frequency of one excitation (atomic transition frequency),
• ωp — frequency of one photon (cavity frequency),
• ωg — frequency of one phonon,
• in — input rate, replenishment coefficient for the first excitation a1 ,
• out — output rate, sink coefficient for transferring excitation from aNatoms to s.
Hamiltonian of the system equals to Jaynes–Cummings–Hubbard Hamiltonian:
X X X X
− − − − ∗ + −
H= ωp p +
i pi + ωa a +
i ai + ω b b+
i bi + (kp+
i+1 pi + k pi pi+1 ) +
i i i i
X X (5)
− + ∗ + − ∗ − + −
(µpi ai + µ pi ai ) + (g + g ) ((bi + b+i )ai ai ).
i i
The sink s is attached to the last exciton in the chain. Input and output is performed using
Lindblad operators:
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Lin = in ∗ p+
1
(6)
Lout = out ∗ s+ p−
Natoms
In all the follow-up graphs and results, it is taken ωa = ωp = 0.1 and ωg = 0.01.
3.4. Dephasing models
Two dephasing models were implemented:
• Unitary-based: with explicit phonons as shown
P on 2. + + −
Hamiltonian of the system has the (g + g ∗ ) ((b−
i + bi )ai ai ) part.
i
• Lindblad-based: there are no explicit photons in the system. Hamiltonian of the system
−
does not have the corresponding part, Lindblad-like dephasing operators Di = p+i pi are
used instead [11].
With the addition of Lindblad-like dephasing operators, evaluation equation 3 takes the
following form:
X 1 X 1
ρ̃t+δt = δt (Li ρL∗i − (L∗i Li ρ + ρL∗i Li )) + δt di (Di∗ Di ρDi∗ Di − (Di∗ Di ρ + ρDi∗ Di )),
2 2
i i
∗
ρt+δt = Uδt ρ̃t+δt Uδt .
(7)
Here di are the coefficients of the corresponding dephasing operators. In this model, all of
di are equal to g.
4. Implementation details
The above-described model was implemented using the Wolfram Mathematica computer
algebra system, and was split into several modules, and the most important of them are
QuantumSystem, QuantumExperiment, QuantumM easures, and the models modules (one
module for each model). For this work the corresponding model module was called
QuantumM odelJCHqU ni. Those four modules would be referenced as System, Experiment,
M easures, and M odelJCH below.
Figure 3 shows the relationships between the modules in the system, with Experiment.nb
being the main notebook file with all the parameters and output. Dotted lines represent internal
dependencies, solid lines represent the data flow.
Some micro-optimization details are discarded to simplify the description.
4.1. The model representation and the M odelJCH module
As shown above, the model could be fully described using a Hamiltonian H and set of
Lindblad-like dephasing operators with corresponding coefficients: {(Li , li )} and {(Dj , dj )}.
The model is internally represented as a list of three elements:
• The description of the model and its parameters. For M odelJCH, stores the number of
possible particles in the system. Not used in computation, could be of any type;
• Hamiltonian H of the system — a square two-dimensional list of complex numbers;
• A list of Lindblad-based operators, each of which is by itself a list of three elements:
– The type of the operator, being either ”Lindblad” or ”Dephasing” — a string;
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Figure 3. The relationships between modules.
– The coefficient li or di , depending on the type — a complex number. This one is
optional, if the list has only two items, the coefficient is taken as being equal to 1;
– The matrix Li or Di , depending on the type — a square two-dimensional list of
complex numbers with the same size as H.
The M odelJCH module constructs such an list using the supplied parameters of the model
and the helper functions from the System module.
4.2. The Experiment module: QuantumExperiment function
The Experiment module contains everything related to the evolution logic, the most used
exported function is QuantumExperiment.
QuantumExperiment method accepts one mandatory argument system — the model repre-
sentation in the form mentioned in takes the model representation as described in 4.1, and some
named optional arguments (OptionsP attern): initialState, evolution, maxT , dT , plotStep,
measure, criteria, return.
initialState describes the initial state of the system, it is either a state vector number from
the basis (from 1 to N , where N ∗ N is the size of the Hamiltonian), a state vector of length N ,
or a density matrix (two-dimensional list of size N ∗ N ). There is no future difference between
passing in a number of some state vector or the corresponding state vector itself (for example,
2 corresponds to {0, 1, 0, ..., 0}).
evolution is the string that selects the numeric method for evolution, one of ”unitary”,
”eigen”, ”measurement”, and ”lindblad”, or a method that returns an evolution function itself.
Those methods will be described below.
maxT
maxT is the total time of emulation, dT is the step time (δt). In an ordinary case,
dT
is the maximum number emulation steps. plotStep is the number of emulation steps between
the measurements. For an optimized evolution method that implements an accurate analytical
solution (at the moment only the ”eigen” method for the unitary evolution), plotStep has the
effect of dT multiplier, where the effective dTe is equal to dT ∗ plotStep, and the maximum
maxT
number of emulation steps is equal to .
dT ∗ plotStep
measure is either a string ”diagonal” or ”entropy” or a measurement function, that accepts
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either one (current density matrix R) or two (current density matrix R and a density matrix
Rprev of the previous step) arguments.
criteria accepts a function of one argument — the current density matrix R. If criteria is
specified, the evolution stops on the step when the passed function returns T rue.
return is a string that describes what the method should return: ”measures” — the list of
measurements results, ”plot” — either two- or three-dimensional plot of the measurements results
(depends on whether the measurement returns a scalar or a vector), ”last” — the final density
matrix R, ”lastM easure” — the measurement result on the final density matrix, ”time” — the
total evolution time (for use with criteria), or ”time, lastM easure” — a list of two elements:
the total evolution time and the measurement result on the final matrix.
4.3. The Experiment module: the emulation methods
The evolution, system, dT , and initialState arguments are passed to the internal
QuantumEvolution method.
This method is designed to return an iteration function that performs the next step of the
evolution and returns a density matrix.
If evolution is not a string, but is a function — the method returns
evolution[system, dT, initialState].
For all the built-in function, the output matrix is normalized with M = (M + M ∗ )/2 and
M = M/T race[M ] to partially eliminate the effect of inaccurate number representation.
evolution = ”unitary” returns the most simple method Rt+δt = U · Rt · U ∗ , where U =
i · δt
− H
e h̄ . Lindblad-based part is discarded.
evolution = ”eigen” returns a method similar to ”unitary”, but does not use an iterative
algorithm and instead computes everything from the start. If the initialState is not a density
matrix, this method could be expressed as:
Vt = ER · DiagonalM atrix[EU n ] · ES
(8)
Rt = Vt · Vt∗
i · δt
− λj
Where Vt is the current state vector, EU is a vector of EUj = e h̄ , {λj } is the vector of
eigenvalues of H, ES is the set of eigenvectors of H, V0 is the initial state vector, ER = V0 ·ES − 1,
n is the number of step from 1.
The evolution of a density matrix has a similar, but has more computational operations.
evolution = ”measurement” implements the following equations:
X
Rat = Rt−δt + Rt−δt ∗ C + C · Rt + XAj · Rt−δt · XBj
j (9)
∗
Rt = U · Rat · U
Matrices Aj , Bj , and C could be easily computed from the sets {Li , li } and {Di , di }:
Ali = Li
Bli = dT · li · L∗i
Adi = Di∗ · Di
(10)
Bdi = dT · di · Di∗ · Di
1 X X
C = − dT ( li · L∗i · Li + dj · Dj∗ · Dj )
2
i j
evolution = ”lindblad” implements the following equations:
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X
∗
St = St−δt + ANt−δt · ( XAj · Rt−δt · XBj ) · ANt−δt
j
ANt = AN d · ANt−δt (11)
APt = AP d · APt−δt
Rt = APt · St · APt∗
Where
AN0 = AP0 = IdentityM atrix[N ],
−i · δt
H+C
AP d = e h̄ , (12)
−1
AN d = AP d ,
S0 = R0 .
−i · t
H+C
This results in ANt = e h̄ , APt = ANt−1 , St = ANt · Rt · ANt∗ .
There is also a ”lindblad−eigen” method that calculates ANt and APt using the eigenvector
−i · t
space of the matrix H + C each time instead of an iterative algorithm.
h̄
4.4. The M easures module
This model implements the two pre-defined measures, ”diagonal” and ”entropy”, as well as
a set of some helper functions. P
The entropy is calculated as λi · log(λi ) where λi is the i-th eigenvalue of the supplied
i
density matrix ρ.
Some of the helper functions are made to select the diagonal of the partial trace (to observe
the behaviour of a subsystem), to calculate the partial entropy of a subsystem or the mutual
entropy (for example the mutual entropy between subsystems or the mutual entropy between the
photons and the excitons in the system), to measure the distance between two density matrices
(to calculate the error rate or to plot a simple graph of how close did we get to the target state
over time), to calculate the distance of partial trace matrices (ignoring some qubits).
4.5. The System module
The System module also contains methods to construct density matrices from the state
vector or the basis state number (used by the Experiment module), and some helper functions
to ease the creation of Hamiltonians and Lindblad/Dephasing matrices in the model.
For example, QuantumHamiltonian is intended to ease the creation of models that are
based on qubit representation: this method accepts the total number of qubits in the system
and a list of rules, each one of which consists of:
• The positions of the qubits that this rule deals with,
• The initial state,
• The target state,
• The amplitude of this rule.
For example, two rules {{1, 2}, {0, 1}, {1, 0}, k} and {{1, 2}, {1, 0}, {0, 1}, k ∗ } could be used
− + −
to represent the interaction k · a+ ∗
2 a1 + k · a1 a2 , and one rule {{5}, {1}, {1}, ω} could be used
−
to represent ω · a+
5 a5 .
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A similar method QuantumLindblad constructs the Lindblad part of the model — the set
of matrices Li .
There are also helper methods that ease the implementation of limiting the total energy of
the system (and to pass the energy limit as the parameter to the model), for example the system
of 10 excitons with energy limited to 1 has only 10 basis states and Hamiltonian of size 10 ∗ 10,
not 1024∗1024. Based on those helper methods, QuantumHamiltonian and QuantumLindblad
implement an additional three-argument version, with the third argument being the definition
of the subspace to which the model is limited. That argument has to be set in order to directly
construct the Hamiltonian and Lindblad matrices for the reduced system — that is considerably
faster than constructing the matrices for the whole system and then selecting a subspace.
5. Computation results
For all numeric experiments with input enabled the total energy was not explicitly limited,
except for the natural limits imposed by the model. For numeric experiments with input disabled,
the initial state has one photon and the total energy is limited to 1 (because it could not raise
higher than that value).
5.1. Quantum bottleneck
Figure 4 shows the dependency of time taken to reach a target sink value on the input and
output rates. It could be seen, that though the transfer speed increases with the initial increase
of input and/or output rates, further increase of those rates has the effect of slowing down the
transfer.
The graph has a global minimum at one point — that is the optimal (in, out) values (which
are not necessarily equal to each other). Even with optimizing the input rate, the quantum
bottleneck effect is observed. Also, one can observe the quantum bottleneck effect over the
input rate. For any fixed in value there is an optimal out value and nice versa, and those are
not constant. The graph (with the exact optimal values) also depends on the parameters of the
model: µ and k, and also slightly depends on the desired target sink value, but has a similar
shape.
The exact form of those dependencies is out of the scope of this paper and will be described
separately.
Figure 4. The dependency of time taken to reach target sink value 0.9995 over in and out rates. µ =
0.8, k = 0.5
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5.2. Dephasing-assisted transport
µ = 0.2, k = 0.8, time = 150 µ = 0.8, k = 0.2, time = 60
Figure 5. The state of sink at the fixed time. Two atoms, no input (initial state has a photon in the
first cavity). Dependency over the output rate out and the dephasing coefficient g.
For high enough times, the absolute maximum over all possible output rates is reached when
g = 0. But non-zero values of g can greatly improve the conductivity in cases when the output
rate does not match with the optimal output rate. That is shown on figure 5. It could also be
seen that the effect does not depend on g being equal to some exact value in some cases, and
there is a broad range of possible values for g that improves the conductivity.
For very low times (or low target sink values, depending on the stop criterion) these graphs
are different and could show other short-lived effects, but we are inspecting target sink values
close to 1.0.
In the first part of figure 5 (µ = 0.2, k = 0.8) it could be seen that non-zero values for g
improve the conductivity for the case out < outOpt. In the second part (µ = 0.8, k = 0.2) it
could be seen that non-zero values for g improve the conductivity for the case out > outOpt.
The exact range of possible output rate values for which the conductivity could be improved
by adding dephasing depends on the parameters of the chain.
5.3. Unitary dephasing model
µ = 0.2, k = 0.8, time = 150 µ = 0.8, k = 0.2, time = 60
Figure 6. The state of sink at the fixed time. Two atoms, no input (initial state has a photon in the
first cavity). Dependency over the output rate out and the dephasing coefficient g.
For the unitary dephasing model, the main observed effects are the same as for the Lindblad-
based dephasing model in 5.2 — non-zero values of g can improve the conductivity in cases of
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inoptimal values of the output rate, though the overall effect of unitary dephasing (with the
supplied parameters) is milder.
Figure 6 depicts the same experiment as figure 5, but with the unitary dephasing model.
It is also worth noting that setups that were optimal to reach low target sink values (for
example 0.3 or 0.4) are not always optimal to reach target sink values close to 1.
6. Conclusions
A qubit-based approach to modelling the evolution of a quantum system is described.
A description of a reusable computational model is provided, mentioning some common
optimizations while still keeping the model generic — an optimized system in Hilbert space
of dimension 10 with an energy limit of 1 and a system in Hilbert space of dimension 210
system with the photon pumping are defined by the exact same code, and the main part of the
computational system is not bound to any specific system layout at all.
Quantum bottleneck effect was explained and reproduced over both input and output rates.
Two models of dephasing showed similar results in the long term (when the numeric exper-
iments were run long enough for the sink to reach a value close to 1).
For both dephasing models non-zero dephasing rates did not give a positive effect on the
conductivity in the case of optimal output rate, but gave a large positive effect in some cases
when the output rate was not optimal and when the conductivity is capped by the quantum
bottleneck effect.
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