Download Master Thesis

Master in Quantum Science and Technology
Quantum field theories
in
superconducting circuits
Laura García-Álvarez
Advisor:
Prof. Enrique Solano
Department of Physical Chemistry
Facultad of Science and Technology
University of the Basque Country UPV/EHU
Leioa, September 20th, 2013
x
Agradecimientos
Quizá esta parte final de la tesis sea la más difícil de escribir, la que te obliga a reflexionar
sobre el camino recorrido hasta llegar aquí, a pensar en quiénes han hecho eso posible. Sé que
necesariamente mis palabras representarán algo incompleto, y que mi incapacidad de expresarme hará que lo siguiente sólo sea un espejismo de lo que realmente ha significado para mí
este año y el agradecimiento que tengo a las personas que me han ayudado.
Siempre que me vienen a la cabeza las palabras “tesis de máster”, me evocan no sólo las
horas de trabajo y de cálculos, sino todas las inseguridades, las reflexiones y las decisiones que
ha supuesto este año para mí. Como mi primer trabajo de investigación, esta tesis representa
algo más que un trabajo científico, es un importante primer paso hacia lo que en los próximos
años he decidido que forme parte de mi vida: las a veces interminables horas de cálculos
infructuosos, las discusiones en la pizarra delante de lo que parece un problema irresoluble, y,
sobre todo, la satisfacción al estudiar y desarrollar algo nuevo.
Hay muchas personas que han estado presentes y participando de un modo u otro en
esta “tesis de máster”. Por ello quiero agradecer al Prof. Enrique Solano, que me ha ofrecido
la oportunidad de participar en este proyecto, me ha apoyado y asesorado siempre que lo
he necesitado. Muchas gracias al Dr. Guillermo Romero, por guiarme en esta tesis, por sus
consejos, por sus clases sobre circuitos superconductores, sus discusiones en la pizarra y su
paciencia conmigo y con mis preguntas. Gracias también al resto de miembros del grupo de
investigación QUTIS, por estar dispuestos a ayudar con cualquier consulta que tuviera, y al
Prof. Íñigo Eguskiza, por sus tutorías y el tiempo que me ha dedicado.
No menos importantes son otras personas, que, aunque no hayan ayudado en los detalles
científicos, han sido imprescindibles para que haya terminado esta tesis. Quiero agradecer a
mis compañeros de clase, con los que compartido este año de trabajo y las incertidumbres
sobre el futuro, por nuestros debates no sólo de física y su buen humor y amistad. Mi familia,
con su confianza ciega en mis decisiones y su apoyo incondicional, mis amigos y las personas
más cercanas, que son los que me han acompañado silenciosamente en esta carrera de fondo,
a ellos, en especial, gracias.
3
Contents
Contents
4
1 Introduction
6
2 Circuit quantum electrodynamics
2.1 Conditions for macroscopic quantum coherence . . . . .
2.2 Josephson effect . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Basic features of superconductivity . . . . . . . .
2.2.2 Flux quantization . . . . . . . . . . . . . . . . . .
2.2.3 Josephson effect . . . . . . . . . . . . . . . . . . .
2.3 Quantum description of superconducting circuits . . . .
2.3.1 Variables of superconducting circuits . . . . . . .
2.3.2 Lagrangian of a circuit . . . . . . . . . . . . . . .
2.4 Superconducting qubits . . . . . . . . . . . . . . . . . .
2.5 Transmission line resonators . . . . . . . . . . . . . . . .
2.6 Galvanic coupling . . . . . . . . . . . . . . . . . . . . . .
2.6.1 The model . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Qubit potential . . . . . . . . . . . . . . . . . . .
2.7 New design of coupling between a qubit and a resonator
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3 Quantum field theories
3.1 Basic concepts of quantum field theory . . . . . . . .
3.1.1 Field theory methods . . . . . . . . . . . . . .
3.1.2 Renormalization . . . . . . . . . . . . . . . .
3.1.3 Beyond perturbative methods . . . . . . . . .
3.2 Quantum electrodynamics . . . . . . . . . . . . . . .
3.3 Quantum field theories in one dimension . . . . . . .
3.3.1 Quantum electrodynamics renormalization . .
3.3.2 Simplified model of quantum electrodynamics
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4 Quantum field theories in superconducting circuits
4.1 Model of quantum field theory . . . . . . . . . . . . . . . . . . . . .
4.1.1 Jordan-Wigner transformation . . . . . . . . . . . . . . . . .
4.1.2 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . .
4.1.3 Symmetric form of the Hamiltonian . . . . . . . . . . . . . .
4.2 Experimental proposal . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Experimental setup for simulating interacting quantum fields
4.2.2 Hamiltonian of the superconducting device . . . . . . . . . .
4.3 Digital quantum simulation of quantum field theories . . . . . . . . .
4.4 Generation of interaction terms . . . . . . . . . . . . . . . . . . . . .
4.4.1 The Mølmer-Sørensen multi-qubit gate . . . . . . . . . . . . .
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4.4.2
4.4.3
4.4.4
The sequence of gates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiqubit entangling gate . . . . . . . . . . . . . . . . . . . . . . . . .
Codification of information . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Conclusions
53
Bibliography
54
5
Chapter 1
Introduction
The twentieth century has brought a revolution in the world of science and technology, with
the development of quantum mechanics and the theory of computation among the greatest
advances. The advancement of technology gave rise to digital computers, whose power growth
has been described successfully by the Moore’s law. Nevertheless, computer hardware improvement will eventually reach the limits of miniaturization at atomic levels, where quantum effects
become important. Here, the theory of quantum computation, which is based on the idea of
using quantum mechanical phenomena to perform computations, could provide a solution to
the eventual failure of Moore’s law.
The fields of quantum information and quantum computation rely on fundamental ideas
of quantum mechanics, computer science and information theory, and study the information
processing tasks that can be accomplished by using quantum mechanical systems. The promising idea of encoding the information in quantum bits or qubits, and carrying out operations
through quantum gates for the sake of solving unreachable problems in classical computation,
encounters with several challenges such as the complete control over single quantum systems,
the identification of well-defined qubits, and the possibility of performing gate operations before the quantum system losses its coherence. The computational limitation of most powerful
classical computers, such as exponential growth of computing time when the complexity of the
system increases, motivates the construction of such a quantum computer.
Nowadays, the construction of a quantum computer of sufficient size for a exponential gain
in efficiency with respect to classical computations is not available. However, the efforts in this
direction have given rise to other alternative methods of computation more accessible with
current technology known as quantum simulations. Quantum simulation [1, 2] is a high-impact
discipline that consists of the intentional artificial reproduction of the quantum dynamics
of a system, which is usually difficult to study, in other controllable quantum system. The
development of quantum technologies allows the implementation of quantum simulators on
various platforms: trapped ions, ultracold atoms, quantum dots, photonic systems, optical
networks, and superconducting circuits, among others.
One of the main problems that one can address in the field of quantum information is the
study of quantum field theories (QFTs) [3]. QFT is among the deepest descriptions of nature,
being considered a fundamental tool to describe the microscopic universe. This theory combines
the principles of quantum mechanics, the concept of field, and special relativity. In this sense,
QFTs has an interdisciplinary character that describes physical phenomena in different fields,
such as high energy physics, condensed matter and astrophysics.
Traditional computations of QFT are based on perturbation theory through Feynman diagrams [3], or in numerical simulations through lattice gauge theories [4]. Although perturbation
theory is suitable for weak coupling, Feynman diagrams are not a reliable tool of computation
when achieving the so-called strong coupling regime. In this case, we need to perform complex
numerical simulations with lattice gauge theories, which provide us of incomplete information.
In order to study non-perturbative phenomena of QFT, quantum simulators might be an important alternative tool of calculation. In this sense, new methods of computation have been
6
proposed in the field of quantum information [5, 6, 7, 8, 9, 10, 11], which theoretically would
calculate exponentially faster than classical computers [11].
Up to now some proposals of simulating QFT are unfeasible with current technologies,
since they are based on lattice gauge theories and need a high number of controllable quits. In
addition, other feasible proposals are designed for ion traps where the degrees of freedom are
the finite number of vibration modes and the internal energy levels of ions.
Unlike other technologies superconducting circuits naturally provide a continuum of boson
electromagnetic modes, which have not yet been used as a tool in QFT simulations. Other
advantages of this platform are its scalability, operating speeds of a few nanoseconds, the fact
that its design allows tuning parameters through electromagnetic fields, and that the frequency
range in which they work spans the interval from 1 to 10 GHz, similar to mobile devices. In this
sense, any key result in this field would be a potential breakthrough in quantum communication
protocols. The aim of this work is to profit from the advantages of superconducting circuits [12,
13, 14, 15, 16, 17, 18, 19, 20, 21] and propose the quantum simulation of fermionic field modes
interacting via a continuum of bosonic modes.
In this thesis, we first introduce in chapter 2 the basic theory of superconducting circuits
with a detailed computation of the galvanic coupling between a flux qubit and a microwave
resonator, which is usually employed to reach higher values of coupling. In this chapter, we
have also introduced and analyzed a new design of flux qubit that allows to switch between
transversal and longitudinal coupling with the resonator and to tune the strength coupling
without renormalizing the qubit energy.
Chapter 3 is devoted to review some basic features of QFT, and in particular of quantum
electrodynamics (QED), which is the model studied in our proposal. We consider simplified
models of QED in one spatial dimension and analyze their advantages and drawbacks. We
propose and study a simulatable model that conserves some features of QED, such as particle
creation and annihilation.
In chapter 4, we explain our proposal of simulating the QFT model described in the previous chapter. We will profit from the continuum of bosonic modes provided by transmission
lines in superconducting devices. Here, we describe a digital method of simulation suitable
for a superconducting circuit including flux qubits, and in particular, the new configuration
introduced in chapter 2 may be used.
Finally, in chapter 5, we expose our concluding remarks and mention the aspects that will
need further study in the frame of quantum simulations of QFT with superconducting circuits.
7
Chapter 2
Circuit Quantum Electrodynamics
Many different physical architectures have been proposed as quantum information processors.
Among them, superconducting circuits are a solid state system where we can observe quantum
effects though they are macroscopic objects. Usually, quantization is only relevant in atom
sized objects, since the thermal motion of a large number of atoms masks its quantum properties. However, certain phenomena such as superconductivity, enable macroscopic variables as
voltages to exhibit quantum behaviour.
The phenomenon of zero resistivity and perfect diamagnetism, known as superconductivity,
is explained by describing a macroscopic wavefunction. At low temperatures, the electrons
near the Fermi surface interact attractively due to the exchange of lattice vibrations and form
Cooper pairs. The many-body wavefunction for the electrons is a coherent state in which a
macroscopic number of pairs are all in the same state, that is, all electron pairs adopt the
same phase like a unique wave, and the same energy. This macroscopic quantum coherence in
superconductors is only evident when the devices are cold enough and small enough to avoid
the decoherence effects.
2.1
Conditions for macroscopic quantum coherence
Macroscopic variables of superconducting circuits, such as voltages or currents, show quantum
behaviour under suitable conditions [22]. Evidence for quantum superposition states in this
platform can be seen when the size and the temperature of the systems prevents themselves
from the interaction with the environment, that is, from acquiring classical behaviour.
Ultra-low dissipation
The first requirement for a superconducting circuit to show quantum effects is the condition of
very low dissipation, that is, a negligible resistance in the experiments. The latter is achieved
by using metallic parts which exhibit a superconducting phase at the operating temperatures
of the experiment, and also at the qubit transition frequency ωq , which satisfies the condition
~ωq ∆, where ∆ is the energy gap of the superconducting material. Typical choices in
experiments are aluminium or niobium, which are low temperature superconductors.
Ultra-low noise
The energy of the thermal fluctuations in the experiment must be much less than the superconducting qubit transition energy, kB T ~ωq . Not only the superconducting device must be
cooled, but also the wires of the control and readout ports connected to it in order to avoid
noise. The energy splitting of the two lowest states in a superconducting qubit is in the 1 − 10
GHz range, thus the cryogenic temperature of the experiment must be around 20 mK.
8
2.2
2.2.1
Josephson effect
Basic features of superconductivity
The phenomenon of superconductivity mentioned above appears when the material is cooled
below a critical temperature Tc . In this case, the physical device stops showing any electric
resistance and expels the magnetic fields. At this point, the electron density of states acquire a
small gap 2∆ separating the occupied and unoccupied states. In conventional superconductors,
the relation between the energy gap and the critical temperature reads: 2∆ ' 3.5kB Tc , that
is, the higher the Tc , the greater the gap.
The explanation to superconducting properties from a microscopic point of view came
with the BCS theory of superconductivity, which is based in the fact that each electron at the
Fermi surface suffers from an effective attractive interaction and becomes bound in a Cooper
pair. However, the most exciting developments of the ensuing decade came as a result of the
Ginzburg-Landau theory [23], where the behavior of the whole ensemble of superconducting
electrons is described by a many-particle wavefunction:
p
Ψ(r, t) = Ψ0 (r, t)eiθ(r,t) = ns (r, t)eiθ(r,t) .
(2.1)
The local density of superconducting electrons is given by ns (r, t) = |Ψ0 (r, t)|2 . All the
Cooper pairs are described by the same phase θ(r, t), and therefore, macroscopic variables can
depend on this quantity. Its physical meaning can be analyzed by considering the continuity
equation for the probability of a charged particle in an electromagnetic field:
∂
1
(Ψ∗ Ψ) +
∇ · [~(Ψ∗ ∇Ψ − Ψ∇Ψ∗ ) − 2qiA(Ψ∗ Ψ)] = 0 ,
∂t
2mi
∂
(Ψ∗ Ψ) + ∇ · Jρ = 0 ,
(2.2)
∂t
which, in this case, become in a continuity equation for the condensate density ns . Therefore
Jρ also describes a flow of particles, and can be related with a current density Js if we multiply
by the charge of a superconducting electron, q:
Js = qJρ = qns vs .
(2.3)
By replacing the macroscopic wavefunction of Eq. (2.1) into the expression of Jρ that
appears in Eq. (2.2), we obtain:
ns
Jρ =
(~∇θ − qA) .
(2.4)
m
If we compare the above expression with Eq. (2.3), we obtain the velocity of superconducting
electrons:
1
(2.5)
vs = (~∇θ − qA) ,
m
where we can interpret ~∇θ as the canonical momentum, p, of the superconducting electrons.
2.2.2
Flux quantization
h
The flux enclosed by a superconductor must be quantized in units of the flux quantum Φ0 = 2e
.
In order to prove this statement, a closed curve C in the bulk of a superconducting ring is
considered, as shown in Fig. 2.1.
In the curve C, far from the surface, the density current Js and the magnetic field B vanish.
This condition implies that the integral of Js around the contour C must also vanish:
I
I
ns
Js · dl = q
(~∇θ − qA) · dl = 0 ,
m
C
I
I C
~∇θ(r, t) · dl =
qA · dl .
(2.6)
C
C
9
C
Figure 2.1: A superconducting ring pierced by a magnetic flux. Along the contour C, which
contains superconducting and nonsuperconducting regions, both current and magnetic field
vanish.
In order to evaluate the integral on the right hand side of Eq. (2.6), we have to realize that
the real function θ is not well defined, since there exist multiple phase values giving the same
result for the macroscopic wavefunction:
p
p
Ψ(r, t) = ns (r, t)eiθ(r,t) = ns (r, t)ei(θ0 (r,t)+2πn) , ∀n ∈ Z ,
(2.7)
where θ0 (r, t) is the principal value of the phase, ranging in the interval (−π, π].
Then, the integral around the path C is computed by taking into account that the integrand
is the gradient of a scalar function, and that the path is in a multiply connected region (see
Fig. 2.1):
I
C
~∇θ(r, t) · dl = ~ lim [θ(r2 , t) − θ(r1 ] = 2~πn .
r2 →r1
Now, we evaluate the integral on the left hand side by using the Stoke’s theorem:
I
Z
Z
qA · dl = q (∇ × A) = q B · dS .
C
(2.8)
(2.9)
S
If we replace the charge of the superconducting electron by its value, q = 2e, we finally
obtain the statement of magnetic flux quantization:
ΦB =
where the flux quantum is defined as Φ0 =
2.2.3
h
n = nΦ0 ,
2e
(2.10)
h
2e .
Josephson effect
The Josephson effect, discovered by Brian D. Josephson in 1962 [24], is a coherent phenomenon
which predicts a superconducting current flow between two superconductors separated by an
insulating layer that is thin enough for electron pairs to tunnel. In what follows, we analyze
the main features of a Josephson junction shown in Fig. 2.2. Besides exhibiting interesting
macroscopic quantum mechanical properties, this junction is a basic component in circuit
QED technology.
In deriving the Josephson equations, the arguments followed by Feynman [25] will be introduced. We define the wavefunctions of the superconducting states in the left and the right
superconductor as Ψ1 and Ψ2 , respectively. We consider the simplest case with the same homogeneous and superconducting material on both sides, which lead us to describe the dynamics
10
2
1
U2
U1
Figure 2.2: Josephson junction, two superconductors separated by a thin insulating layer.
with the following coupled Schrödinger equations:
∂Ψ1
∂t
∂Ψ2
i~
∂t
i~
= U1 Ψ1 + aΨ2
= U2 Ψ2 + aΨ1 ,
(2.11)
where U1 and U2 are the ground state energies of each superconductor, and a is a parameter
characterizing the overlap between the wavefunctions Ψ1 and Ψ2 . If a = 0, the dynamics is
described by uncoupled equations, one for each superconductor.
We know that the wavefunctions take the form shown in Eq. (2.1):
p
Ψ1 =
n (r, t)eiθ1 (r,t)
p 1
Ψ2 =
n2 (r, t)eiθ2 (r,t) ,
(2.12)
where n1 and n2 are the density of superconducting electrons and θ1 and θ2 are the phases of
each superconductor.
Now, by substituting Eq. (2.12) into Eq. (2.11), we obtain:
√
√
√
ṅ1
i~
√ + i n1 θ̇1 eiθ1 = U1 n1 eiθ1 + a n2 eiθ2
2 n1
√
√
√
ṅ2
i~
(2.13)
√ + i n2 θ̇2 eiθ2 = U2 n2 eiθ2 + a n1 eiθ1 ,
2 n2
where the dot notation indicates time derivates.
In each case, real and imaginary parts are equaled, thus obtaining the set of equations:
√
~ṅ1
= a n2 sin(θ2 − θ1 )
√
2 n1
√
√
√
−~ n1 θ̇1 = U1 n1 + a n2 cos(θ2 − θ1 )
√
~ṅ2
= −a n1 sin(θ2 − θ1 )
√
2 n2
√
√
√
−~ n2 θ̇2 = U2 n2 + a n1 cos(θ2 − θ1 ) ,
(2.14)
(2.15)
(2.16)
(2.17)
which can be compacted in the governing equations for the Josephson effect. Equations (2.14)
and (2.16) can be written to represent a current flow from the left side superconductor to the
right side one:
2a √
ṅ1 = −ṅ2 =
n1 n2 sin(θ2 − θ1 ) .
(2.18)
~
This first Josephson equation establishes a relation between the superconducting current
density and the phase difference ϕ = θ2 − θ1 across the junction. In the following we exploit
11
the fact that superconductors on both sides are identical, hence n1 ∼ n2 ∼ n. In this case, and
considering the absence of any scalar and vector potentials, the supercurrent density varies
sinusoidally:
2qa
n sin ϕ
~
= Jc sin ϕ ,
q ṅ =
Js
(2.19)
where Jc is the critical Josephson current density, determined by the coupling a between the
superconductors.
We must note that in order to keep the density of superconducting electrons in the electrodes constant, the junction has to be attached to a current source avoiding the charge and
discharge of the two electrodes, thereby keeping n constant. We reinterpret then Eq. (2.18)
as the current that would begin to flow if there were no current source. In spite of n1 and n2
remaining constant, if we were included the new currents, Js is still described by Eq. (2.19).
Now, Eqs. (2.15) and (2.17) can be reduced to the second Josephson equation, which is a
differential equation for the temporal change of the phase difference ϕ:
θ̇2 − θ̇1 = ϕ̇ =
1
(U1 − U2 ) .
~
(2.20)
We consider now the above equations if a potential difference between the two superconductors is present, that is, if U1 − U2 = qV . This means that the voltage is related with the
phase difference in the following way:
Z
q
ϕ(t) = ϕ0 +
V (t)dt ,
(2.21)
~
where ϕ0 is the value of ϕ at t = 0.
The Josephson current in Eq. (2.19) depends on the potential. Without any voltage, there
is a DC supercurrent across the junction given by Js = Jc sin ϕ0 . In the presence of a constant
voltage, an AC current appears:
2π
Js = Jc sin ϕ0 +
Vt ,
(2.22)
Φ0
where we have used the expression of the flux quantum, Φ0 =
h
2e .
The Josephson frequency is:
ν
2π/Φ0 V
1
MHz
=
=
,
= 483.597898(19)
V
2πV
Φ0
µV
(2.23)
and depends only on fundamental constants.
From the two Josephson equations, Eq. (2.19) and Eq. (2.21), and the relation for an
˙ we deduce that the Josephson junction turns out to be
inductance in a circuit V = LI,
equivalent to a non-linear inductor with:
LJ =
h
2πqIC cos ϕ
,
(2.24)
where IC is the critical current through the junction.
Energy operator for a Josephson element
The energy operator of the Josephson element can be deduced from the discreteness of charge
that tunnels across the barrier. The charge QJ having flown through the Josephson element is
an integer N times the charge q = 2e of a Cooper pair:
QJ (t) = 2eN (t) .
12
(2.25)
This is a direct consequence of the translational invariance of the Josephson potential
energy.
We treat N as an operator whose eigenstates correspond to macroscopic states of the circuit
with a well defined number of Cooper pairs, such that:
X
N̂ =
N |N ihN | .
(2.26)
N
The tunnelling of Cooper pairs through the barrier can be translated in a coupling between
the eigenstates of the operator N , which can be expressed in the following Hamiltonian:
HJ = −
EJ
2
NX
=∞
N =−∞
(|N ihN + 1| + |N + 1ihN |) ,
(2.27)
with EJ = Φ0 IC /2π stands for the Josephson energy.
The above Hamiltonian can be written in terms of the phase difference across the junction,
ϕ, which we treat now as an operator. This representation is derived by the introduction of a
new basis:
NX
=∞
|ϕi =
eiN ϕ |N i .
(2.28)
N =−∞
We notice that under the change ϕ → ϕ + 2π the state |ϕi remains unaltered. Conversely,
we have that:
Z 2π
1
|N i =
dϕ e−iN ϕ |ϕi ,
(2.29)
2π 0
which allows us to rewrite the Hamiltonian of Eq. (2.27) in this new basis:
HJ = −
EJ 1
2 2π
2π
Z
dϕ (eiϕ + e−iϕ )|ϕihϕ| .
(2.30)
0
We introduce the operator eiϕ̂ which acts eiϕ̂ |N i = |N − 1i, and is defined by:
e
iϕ̂
1
=
2π
Z
0
2π
dϕ eiϕ |ϕihϕ| .
(2.31)
This way, the Hamiltonian in Eq. (2.27) reads:
HJ = −EJ cos ϕ̂ .
(2.32)
The operator ϕ̂ is the phase conjugate of the number operator N̂ , whose eigenvalues include
all the integers, even the negative ones. This relation is reflected by the fact of:
[ϕ̂, N̂ ] = i .
(2.33)
However, we have to take care with the operator ϕ̂ due to the phase character reflected in
the fact of |ϕi remaining unaffected under the change ϕ → ϕ + 2π. In this sense, only periodic
functions of ϕ̂ have a non-ambiguous meaning.
The quantum version of the Josephson equations, Eq. (2.19) and Eq. (2.21), is obtained if
we take into account Eq. (2.33) and consider that N̂ couples linearly with the voltage operator
V̂ through the Cooper pair charge 2e.
Hereafter, I will refer to the Josephson element as a non-linear inductance, while the term
Josephson junction is reserved to the physical realization of the junction, which is modelled by
a capacitor, C, and the non linear inductance, LJ , in parallel, as shown in Fig. 2.3.
13
(a)
(b)
C

LJ


Figure 2.3: (a) Model of the physical realization of the Josephson junction, which is usually
denoted as a cross (b).
martes 17 de septiembre de 13
2.3
Quantum description of superconducting circuits
Superconducting quantum circuits are integrated devices whose size allows to describe them by
using a lumped element approximation or transmission line equations. Despite the similitudes
between conventional electronics and quantum circuits, there are still important conceptual
differences. In conventional circuits the collective electronic degrees of freedom such as currents
and voltages are classical variables, whereas in quantum circuits, these degrees of freedom are
treated by quantum operators which may not commute. In the latter, macroscopic quantum
effects involving the collective behaviour of many particles are present.
The problem of writing systematically the Hamiltonian of a given a superconducting circuit
made up of capacitors, inductors and Josephson junctions has been introduced in Refs. [26, 27].
The choice of the variables of the circuit as well as the procedure to write the Lagrangian and
the Hamiltonian will be described below.
2.3.1
Variables of superconducting circuits
A superconducting circuit is formed by a set of branches, which are electrical components
connected at nodes. The variables of the circuit can be associated to both branches or nodes.
We will start by describing the former.
The branches can be capacitors, inductors or Josephson junctions, and after choosing the
convention on the direction of each branch, we can associate a current flowing through it and
a voltage across it as shown in Fig. 2.4.
(t)
vb
b

(t)
ib
martes 17 de septiembre de 13
Figure 2.4: Scheme of a branch connected at two nodes, and convention for the voltage and
current associated to it.
14
The voltage and the current of a branch b are given by:
Z end of b
~ · dS
~ ,
vb =
E
beginning of b
I
1
~ · dS
~ .
B
ib =
µ0 around b
(2.34)
The previous values are used to define the dynamical branch variables appearing in the
Hamiltonian of the circuit, that is, the branch fluxes and branch charges:
Z t
φb (t) =
vb (t0 ) dt0 ,
−∞
t
Z
Qb (t) =
ib (t0 ) dt0 .
(2.35)
−∞
We notice that if the branch is a capacitive element, vb = f (Qb ), and if it is a inductive
element, ib = g(φb ), neglecting the mutual inductances. The energy can be computed by taking
into account if the branch is a capacitive or inductive element, and the power flowing into the
branch. It is given by vb ib = vb Q̇b = ib φ̇b , therefore the energy of a branch is:
Z Qb
h(Qb ) =
f (Q) dQ ,
0
Z φb
h(φb ) =
g(φ) dφ .
(2.36)
0
On the other hand, the node variables of the circuit are electrical quantities which depend
on the particular description of the topology of the circuit, and can be expressed as a linear
combination of the branch variables. In order to write the Hamiltonian we have to find a set
of independent variables. The procedure used to eliminate redundant branches is also valid to
relate the node and branch representations.
First, a node of the circuit is chosen as the ground node, while the remaining nodes are
called active nodes. A spanning tree is built from the ground node by choosing a set of branches
such that every active node of the network is related to the ground node by only one path.
The remaining branches that have not been chosen in the spanning tree are referred as
closure branches. These branches participate in closed loops ` of the circuit, and are related
with the set of independent branches by the constraints imposed by flux quantization, that is:
X
φb = φ̃` ,
(2.37)
all b around `
where φ̃` is the static flux threading the loop `.
Now, we can define the flux of a node n by the time integral of the voltage measured from
the ground node to it along the path chosen by the previous procedure. In terms of the branch
variables, it is expressed:
X
φn =
Snb φb where Snb = 1, −1 or 0 .
(2.38)
b
The value of Snb depends on the convention on the direction of each branch of the path.
The flux of each branch b can be obtained conversely from the fluxes of their two end nodes n
and n0 , and the static flux of the a closed loop in the case of closure branches. We distinguish
if the branch belongs to the set of branches of the spanning tree, where the flux is given by:
φb = φn − φn0 ,
(2.39)
or, on the contrary, if the branch is a closure one associated to a loop `:
φb = φn − φn0 + φ̃` .
15
(2.40)
2.3.2
Lagrangian of a circuit
The Lagrangian can be derived by associating to the capacitive and inductive branches the
kinetic and the potential terms, respectively. In this sense, in the branch representation we can
associate:
1 2
C φ̇ ,
2 b
1 2
φ .
L →
2L b
C
→
(2.41)
Once the Lagrangian is obtained, we can define the conjugate momenta of branch fluxes
by the usual relation:
∂L
qb =
,
(2.42)
∂ φ̇b
and write the Hamiltonian via the Legendre transformation:
X
H=
φ̇b qb − L .
(2.43)
b
The commutator of the branch fluxes and their conjugate branch charges is:
[φb , qb ] = i~ .
(2.44)
The Josephson junctions (JJs) are other fundamental elements in superconducting circuits.
As depicted in Fig. 2.3, a Josephson junction consists of a non-linear inductor in parallel with
a capacitance. The energy term associated with the non linear inductor has been derived in
section 2.2.3 and it is given by Eq. (2.32), thus in order to describe the complete Josephson
junction we just include the capacitive kinetic term in the Lagrangian term associated:
JJ
→
1
CJ φ̇2b − EJ cos ϕ .
2
(2.45)
The phase difference ϕ is related with the branch flux of the Josephson junction via the
second Josephson equation and Eq. (2.35):
ϕ̇ =
ϕ =
2e
V ,
~
2e
φb mod 2π .
~
(2.46)
The Lagrangian of any circuit composed by capacitances, inductors and Josephson junctions
can be written systematically by taking into account the energy operator of each element, given
by Eq. (2.41) and Eq. (2.45). We must notice that in the Lagrangian the potential energy has
a minus sign, and we have to add this factor to the potential terms, that is, the inductors and
the non-linear inductor of the Josephson junction.
The above derivation has been performed in the branch representation, but we can change
to the node representation by using the relations of Eq. (2.39) and Eq. (2.40).
2.4
Superconducting qubits
A qubit is a fundamental element in the field of quantum information consisting of a quantum
two-level system. In quantum simulations, qubits can play the role of the atoms where only two
energy levels are relevant. We can take advantage of superconducting circuits, and in particular
of Josephson junctions, to build a two-level system that will be called superconducting qubit.
The Josephson junction is a non-linear inductor in parallel with a capacitance. The nonlinearity means that the energy levels are not regularly spaced. In this sense, considering a
16
specific circuit configuration we can obtain two low-lying states that are sufficiently separated
from higher excited states, such that a two-level approximations holds.
When a circuit is composed by Josephson junctions the terms that appear in the Hamiltonian have the form:
1 qb2
φb
H =
− EJ cos
,
2 CJ
ϕ0
φb
,
(2.47)
H = EC Nb2 − EJ cos
ϕ0
where we define the parameter of EC = 2e2 /2CJ , which represents the electrostatic energy,
namely the energy used to store a Cooper pair in the capacitor. This energy EC along with
EJ , the Josephson energy, are the two characteristic parameters of the Josephson junctions.
Superconducting qubits can be classfied according to the ratio EJ /EC , and to the variables
by which they are controlled. Three basic kinds of qubits corresponds to this classification:
charge qubits, flux qubits, and phase qubits.
Charge qubit
The charge qubit, or Cooper-pair box, consists of a small superconducting island placed
between the barrier of the Josephson junction and a plate of a capacitor, as depicted in
Fig. 2.5(a,b). This device operates in the charge regime where EJ /EC < 1, and uses the
Cooper pairs on the isolated area. Applying a voltage Vg , it induces a charge difference between the two sides of the Josephson junction, thus the qubit can be controlled by electric
fields.
(a)
(b)
J
E
EJ
Cg

EJ
g
V
(c)
Vg
(d)
E
J
E
J
Cg
e
↵E
J
EJ

Ie
Figure 2.5: Schemes of (a) a charge qubit or Cooper-pair box, (b) a split Cooper-pair box, (c)
a flux qubit consisting of three Josephson junctions, and (d) a phase qubit.
martes 17 de septiembre de 13
The main source of noise in this sort of qubit are charge fluctuations caused by charges
trapped in the substrate and oxide layers of Josephson junctions.
Flux qubit
The flux qubit, depicted in Fig. 2.5(c), is a superconducting ring interrupted by one or three
Josephson junctions. This qubit operates in the flux regime EJ /EC > 1, and couples to external
magnetic fields through the magnetic flux flowing across the closed loop. This flux induces two
circulating supercurrent states, either clockwise or anticlockwise, which define the qubit.
17
The dominant source of noise are the flux fluctuations, in fact, due to the high sensitivity
to external magnetic fields of a one-junction flux qubit, flux qubits are usually built with three
Josephson junctions.
Phase qubit
The phase qubit consists of a single Josephson junction connected to a current source as shown
in Fig. 2.5(d). This qubit profits from the phase difference ϕ of the Josephson junction in the
flux regime EJ /EC 1, and is affected by flux fluctuations mainly.
Transmon and fluxonium qubits
New proposals and improved versions of the basic qubits have been developed with the aim of
avoid the flux and charge noises.
A modified type of charge qubit is the named transmon [28], depicted in Fig. 2.6(a), where
additional components have been added. The transmon consists of a split Cooper pair box,
shunted by a section of transmission line, which can be well approximated as a lumped-element
capacitor. This qubit operates in the flux regime EJ /EC > 1 and reduces the charge noise.
(b)
(a)

50
⌦
g
C
in
C

L
r
V

Cin

Lr
EJ

C
r C
s
g
C

Cr
E
J
EJA

L
JA

CJ Lr

Figure 2.6: Schemes of (a) a transmon qubit, and (b) a fluxonium.
Other improved qubit is the fluxonium [29], shown in Fig. 2.6(b), where a small junction
is shunted to a series array of large-area tunnel junctions, which provide a large inductance.
The charge and flux noise can be avoided with an appropriate choice of the parameters.
2.5
Transmission line resonators
In circuit QED, where the interaction between light and matter is reproduced, the qubits are
used as the substitute of two-level atoms and the role of the photons is played by transmission
line resonators.
In the context of electrical circuits, photons are understood as the quantized excitations
of an electromagnetic resonator, and can be confined in one dimension and travel along a
transmission lines. The schematic representation of this element is shown in Fig. 2.7(a), where
we can appreciate a central superconducting wire placed between two lateral ground planes.
The central superconductor carries the signal and has two gaps in two points placed an integer
number of half-wavelengths apart. These gaps are capacitance points that act like mirrors due
to their large impedance, and impose the boundary conditions that generate the resonator with
a discrete set of modes of standing waves of the current and voltage.
The analysis of the circuit diagram associated to the transmission line resonator, shown in
Fig. 2.7(b), can be performed in terms of node variables. The Lagrangian associated reads:
L=
XC
n
2
2
ψ˙n −
X 1
(ψn+1 − ψn )2 ,
2L
n
n
18
(2.48)
(a)
~
E
(b)
L0
C0
L0
 
C0
C0
L0
C0
L0
C0
g
Figure 2.7: (a) Schematic representation of a transmission line resonator, with a central superconducting wire with two gaps running between two ground plates. The electric field is
confined between the central wire and the ground plates. In (b), the effective circuit diagram
of the transmission line resonator formed by capacitances and inductances.
with ψn the node flux, Cn and Ln the capacitance and inductance, respectively.
In order to analyze this system, we will take the continuum limit. This limit is characterized
by a thin discretization of the lattice spacing a → 0, under the assumption that the inductance
and capacitance per unit length are defined as l(xn ) = Ln /a, and c(xn = Cn /a, respectively.
The Lagrangian of the system reads:
!
X
Cn 2
1
ψn − ψn+1 2
L=
a
ψ̇ −
.
(2.49)
2a n 2 Lan
a
n
In the continuum limit, we shall consider the following notation ψn → ψ(xn , t). Moreover,
if we suppose that ψ(xn , t) does not experience abrupt changes at microscopic scale, we can
think that in the continuum ψ(x, t) is a smooth function of variable x. We notice that since
the distance between two neighbour points is given by ∆xn = a, then in the limit a → 0:
Z
X
(2.50)
f (xn )∆xn → dx .
lim∆xn →0
Moreover, we must consider the following relations
∂ψ(xn , t) 1 2 ∂ 2 ψ(xn , t)
+ a
,,
∂xn
2
∂x2n
∂ψ(xn , t) 1 2 ∂ 2 ψ(xn , t)
ψ(xn−1 , t) ≈ ψ(xn , t) − a
+ a
,
∂xn
2
∂x2n
ψn+1 − ψn ≈ a∂xn ψ(xn , t) .
ψ(xn+1 , t) ≈ ψ(xn , t) + a
Hence, the Lagrangian of the system is
!
Z
c(x) ψ(x, t) 2
1
ψ(x, t) 2
L = dx
−
,
2
∂t
2l(x)
∂x
19
(2.51)
(2.52)
where we have considered the general case of position-dependent inductance and capacitance
per unit length, that is, an inhomogeneous resonator.
We can compute the corresponding equation of motion of this Lagrangian:
d
1 ∂ψ(x, t)
= c(x)ψ̈(x, t) ,
(2.53)
dx l(x) ∂x
and solve it by introducing an ansatz where we decompose the flux in the sum of each flux
amplitude, ψn , times the corresponding normal spatial mode, un (x):
X
ψn (t)un (x) .
(2.54)
ψ(x, t) =
n
The flux amplitude of the mode n is given by:
Z
1 `
ψn (t) =
c(x)ψ(x, t)un (x) dx ,
A 0
(2.55)
where A is a normalization constant, and ` is the length of the transmission line resonator.
The substitution of Eq. (2.54) in the equation of motion leads us to the differential equation:
d
1 ∂u(x)
= −ωn2 c(x)un (x) ,
(2.56)
dx l(x) ∂x
n
which is a Sturm-Liouville problem if we consider the boundary conditions ∂u
∂x x=0,` = 0.
These conditions come from the fact of the current, given by I = ∂x ψ(x,t)
, vanishes at the
L0
resonator edges.
The eigenfunctions un (x) form an orthogonal basis, where we have to consider the normalization constant. In this sense, the orthogonality relation reads:
Z `
c(x)un (x)um (x) dx = Cr δnm ,
(2.57)
0
R`
where Cr = 0 c(x)dx is the total resonator capacitance.
The Lagrangian of Eq. (2.52) can be rewritten as a sum of eigenmodes by taking advantage
of the mode decomposition of ψ(x, t) along with the orthogonality relation and Eq. (2.56):
X Cr 2
L=
ψ˙n − ωn2 ψn2 .
(2.58)
2
n
The Hamiltonian can be obtained by defining the conjugate momentum of the flux ψn , that
is the charge qn = Cr ψ˙n , and by introducing the operators of the flux and the charge in terms
of the creation and annihilation operators, a† and a:
r
~
(a† + an ) ,
ψn =
2ωn Cr n
r
~ωn Cr †
qn = i
(an − an ) ,
(2.59)
2
where an and a†n are annihilation and creation bosonic operators satisfying the commutation
relation [an , a†n ] = δnm .
Finally, the Hamiltonian reads:
X
1
†
H=
~ωn an an +
,
(2.60)
2
n
which is a sum of harmonic of different frequencies ωn , not necessarily equally spaced due to
the inhomogeneity of the resonator.
20
2.6
Galvanic coupling
The galvanic coupling between a flux qubit and a resonator is used for reaching a significantly stronger coupling [30], even reaching the ultrastrong coupling regime of light-matter
interaction [31, 32, 33, 34]. Here, the theory proposed by J. Bourassa et al. [35] including
nonlinear effects coming from the Josephson junctions belonging to the flux qubit structure
will be revised.
2.6.1
The model
Let us consider the lumped circuit element depicted in Fig. 2.8, which describes a flux qubit
positioned in the centre of a transmission line resonator of length 2`. We proceed by analyzing
this setup via the circuit network theory developed in section (2.3).
According to our circuit array, we have the following relation between branch and node
fluxes:
ψj = φj ∀j .
(2.61)
So we can write the Lagrangian for the left and the right parts since we can rewrite all flux
branches in terms of node fluxes. Thus the Lagrangians of the transmission lines are:
(l)
Ltl
j
j−1
X
C ˙ 2 X 1
=
ψm −
(ψm+1 − ψm )2 ,
2
2L
(r)
Ltl =
m=1
N
X
m=k
C ˙ 2
ψm −
2
m=1
N
−1
X
(2.62)
1
(ψm+1 − ψm )2 ,
2L
m=k
(2.63)
where L is the inductance and C the capacitance of each lumped resonator. For the central
part of the circuit, where the flux qubit is located, the relations between branch and node
fluxes are:
φa = Φ 2 − Φ1 ,
φd = Φ 1 − Φ3 ,
φb = Φ 4 − Φ3 .
(2.64)
Furthermore, because of the flux quantization around the closed loop we have φa + φc −
φb + φd + φx = N Φ0 , thus φc = Φ4 − Φ2 − φx taking N = 0, that is, there is no trapped flux.
c
2
j+1
1
2
j
3
4
a
b
11
1
2
3
d
j
k
k
N
1
N
3
N
g
Figure 2.8: Following the circuit network theory, we choose a spanning tree depicted by the
arrows where we define the flux branches. In addition, each node of the network is defined by
node fluxes.
21
Then the Lagrangian for the central part reads:
1
1
(Φ1 − ψj )2 −
(ψk − Φ3 )2
2L
2L
CJ,1
1
2
(Φ2 − Φ1 )
(Φ̇2 − Φ̇1 ) + EJ,1 cos
2
ϕ0
CJ,2
1
2
(Φ4 − Φ2 ) − 2πf
(Φ̇4 − Φ̇2 ) + EJ,2 cos
2
ϕ0
CJ,3
1
2
(Φ4 − Φ3 )
(Φ̇4 − Φ̇3 ) + EJ,3 cos
2
ϕ0
CJ,4
1
2
(Φ1 − Φ3 ) ,
(Φ̇1 − Φ̇3 ) + EJ,4 cos
2
ϕ0
Lc = −
+
+
+
+
(2.65)
where we introduce the frustration f = φx /Φ0 with Φ0 = h/2e is the flux quantum.
We take the continuum limit of the transmission lines, and the total Lagrangian is:
Z +` 0
1
C (x)
2
(∂t ψ)2 −
(∂
ψ)
L =
dx
x
2
2L0 (x)
−`
1
1
−
(Φ1 − ψj )2 −
(ψk − Φ3 )2
2L
2L
CJ,1
1
2
+
(Φ̇2 − Φ̇1 ) + EJ,1 cos
(Φ2 − Φ1 )
2
ϕ0
CJ,2
1
2
(Φ̇4 − Φ̇2 ) + EJ,2 cos
(Φ4 − Φ2 ) − 2πf
+
2
ϕ0
CJ,3
1
2
+
(Φ̇4 − Φ̇3 ) + EJ,3 cos
(Φ4 − Φ3 )
2
ϕ0
CJ,4
1
2
(Φ̇1 − Φ̇3 ) + EJ,4 cos
(Φ1 − Φ3 ) ,
(2.66)
+
2
ϕ0
where C 0 (x) and L0 (x) are the position-dependent capacitance and inductance per unit length
of the coplanar waveguide resonator (CWR), respectively.
The node flux description allows us to derive the current conservation at each node of the
network by considering the equations of motion. These equations defined by the flux Φ1 are
given by the Euler-Lagrange equations:
d ∂L
∂L
,
(2.67)
=
dt ∂ φ̇j
∂φj
in our case:
EJ,1
1
0 = CJ,1 (Φ̈2 − Φ̈1 ) +
sin
(Φ2 − Φ1 )
ϕ0
ϕ0
EJ,4
1
− CJ,4 (Φ̈1 − Φ̈3 ) −
sin
(Φ1 − Φ3 )
ϕ0
ϕ0
1
−
(Φ1 − ψj ) .
(2.68)
L
Now we impose the boundary conditions where the resonator is in contact with the central
part of the circuit, that is, Φ1 = ψj+1 and Φ3 = ψk−1 . So that the above equation becomes:
EJ,1
1
0 = CJ,1 (Φ̈2 − ψ̈j+1 ) +
sin
(Φ2 − ψj+1 )
ϕ0
ϕ0
EJ,4
1
− CJ,4 (ψ̈j+1 − ψ̈k−1 ) −
sin
(ψj+1 − ψk−1 )
ϕ0
ϕ0
1
−
(ψj+1 − ψj ) .
(2.69)
L
22
In the continuum limit, this equation reads:
EJ,1
1
−
sin
(Φ2 − ψ(0 , t))
0 = CJ,1 (Φ̈2 − ψ̈(0 , t)) +
ϕ0
ϕ0
EJ,4
1
−
+
−
+
− CJ,4 (ψ̈(0 , t) − ψ̈(0 , t)) −
sin
(ψ(0 , t) − ψ(0 , t))
ϕ0
ϕ0
1 ∂ψ(x, t) .
−
L0 ∂x x=0−
−
(2.70)
The above equation is nothing but the Kirchhoff’s law at node A, see Fig. 2.8. A direct
calculation of the Euler-Lagrange equations for remaining nodes leads to:
1 ∂ψ(x, t) 1 ∂ψ(x, t) − 0
=− 0
,
(2.71)
L
∂x x=0−
L
∂x x=0+
which indicates the current conservation for the coplanar waveguide resonator (CWR).
Now, we will find the eigenmodes of the transmission line resonator by assuming the two
following approximations: (i) the qubit does not affect these eigenmodes since most of the
current flows through the resonator, and (ii) the Josephson energy of the fourth Josephson
junction is much larger than the Coulomb energy, EJ EC . The former approximation is
justified if the total inductance of the qubit loop is larger than the Josephson inductance of
the coupling zone, LJ,4 , that is:
3
X
LJ,k LJ,4 ,
(2.72)
k
ϕ20 /EJ,k ,
with LJ,k =
the inductance of each Josephson junction k.
Now, the problem of the resonator intersected by the fourth Josephson junction can be
analyzed by considering the two approximations. Since the qubit loop acts as a small perturbation and does not affect the resonator, the two terms that correspond to Josephson junctions
of this loop can be neglected in Eq. (2.70), which will read:
EJ,4
1
1 ∂ψ(x, t) −
+
−
+
CJ,4 (ψ̈(0 , t) − ψ̈(0 , t)) = −
. (2.73)
sin
(ψ(0 , t) − ψ(0 , t)) − 0
ϕ0
ϕ0
L
∂x x=0−
This equation can be simplified even further by considering the second approximation. The
phase difference ∆ψ/ϕ0 = (ψ(0− , t) − ψ(0+ , t))/ϕ0 , that relates the fluxes to left and right
of the qubit position, can be considered small enough to expand in series due to the small
fluctuations of the phase. This fact can be explained from the ratio between the characteristic
energies of the fourth junction, EJ /EC 1, which allows us to approximate the cosine of the
potential to a harmonic oscillator. Therefore, the previous equation may be linearized:
EJ,4
1 ∂ψ(x, t) −
+
−
+
,
CJ,4 (ψ̈(0 , t) − ψ̈(0 , t)) = − 2 (ψ(0 , t) − ψ(0 , t)) − 0
L
∂x x=0−
ϕ0
1
1 ∂ψ(x, t) −
+
−
+
CJ,4 (ψ̈(0 , t) − ψ̈(0 , t)) = −
(ψ(0 , t) − ψ(0 , t)) − 0
,
(2.74)
LJ,4
L
∂x x=0−
E
1
where we have used ϕJ,4
.
2 = L
J,4
0
The boundary conditions are provided by the equations of motion for the flux at the
discontinuities of the transmission line. First, the current, which is given by I = ∂x ψ(x,t)
,
L0
vanishes at the two edges of the resonator. This condition of charge conservation may be
expressed:
∂ψ(x, t) ∂ψ(x, t) =
=0.
(2.75)
∂x x=+`
∂x x=−`
23
The second boundary condition is derived from the fact of the current that flows into the
Josephson junction at point x = 0 on the right side has to exit on the left side. This condition
is related to the phase drop ∆ψ, and can be expressed from Eq. (2.71) and Eq. (2.74) in the
following way:
1 ∂ψ(x, t) ∆ψ
1 ∂ψ(x, t) =− 0
= CJ,4 ∆ψ̈ +
.
(2.76)
− 0
L
∂x
L
∂x
LJ,4
x=0−
x=0+
Now, to find the eigenmodes of the linearized transmission line resonator we consider the
Lagrangian of the resonator:
Z +` 0
C (x)
1
2
2
L=
dx
,
(2.77)
(∂t ψ) −
(∂x ψ)
2
2L0 (x)
−`
and we obtain the equations of motion for the node fluxes in the remaining nodes of the
resonator:
1 ∂ 2 ψ(x, t)
∂ 2 ψ(x, t)
=
.
(2.78)
L0 C 0 ∂x2
∂t2
These equations are solved following the treatment of the transmission line resonator described in section 2.5. First, we use an ansatz where ψ(x, t) is decomposed over normal modes
un (x):
X
ψn (t)un (x) ,
(2.79)
ψ(x, t) =
n
with ψn is the flux amplitude of the eigenmode n.
When the spectral decomposition of ψ(x, t) with separation of variables is inserted in
Eq. (2.78), we obtain:
1 ∂ 2 ψn
1
∂ 2 un
=
= −ωn ,
(2.80)
ψn ∂t2
L0 C 0 un ∂x2
that is,
∂ 2 ψn
+ ω 2 ψn = 0 ,
∂t2
∂ 2 un
+ kn2 un = 0 ,
∂x2
(2.81)
where kn2 = L0 C 0 ωn2 is the linear dispersion relation of each mode.
From here, the use of the subscript n is avoided without loss of generality. The first boundary
condition given by Eq. (2.75) allows us to write:

 A cos(k(x + `)) if x < 0
.
(2.82)
u(x) =

B cos(k(x − `)) if x > 0
The second boundary condition of Eq. (2.76) leads to:
∂u(x) ∂u(x) = Ak sin(k`) = −Bk sin(k`) =
,
∂x x=0−
∂x x=0+
(2.83)
and it may be fulfilled in two different ways, we can choose either k = πN
` , where N is an
integer, or A = −B. The former option describes even normal modes that do not couple to the
Josephson junction because no current is flowing through it, and the second option provides
odd normal modes which, on the contrary, do couple to the junction. Then, we choose the odd
normal modes and relate the current flowing through the Josephson junction with the phase
24
difference ∆ψ. Let us insert the ansatz for ψ(x, t) in Eq. (2.76), this lead us with the following
equation:
ψ(t)
1
1
(u(0− ) − u(0+ )) ,
Ak sin(k`) = −ω 2 ψ(t)CJ,4 (u(0− ) − u(0+ )) + ψ(t)
0
L
LJ,4
(2.84)
where we have used the equation of motion of the time dependent part ψ(t).
Now, by making use of the function for the normal modes in Eq. (2.82) we obtain the
transcendental eigenvalue equation for the wave vectors kn :
2L0
ωn2
kn = cot(kn `) +
1− 2 ,
(2.85)
LJ,4
ωp
√
0 0
where ωp
n = kn / L C are the resonance frequencies for the odd modes of the system, and
ωp = 1/ LJ,4 CJ,4 is the plasma frequency of the fourth junction. The orthogonality relation
of these eigenmodes can be computed by first considering the case of m 6= n:
Z
`
un (x)um (x) dx = Am An
`
2kn sin(kn `) cos(km `) − 2km cos(kn `) sin(km `)
,
2
kn2 − km
(2.86)
which reduces to:
Z `
0
un (x)um (x) dx = −CJ,4 (un (0− ) − un (0+ ))(um (0− ) − um (0+ )) = −CJ,4 δun δum , (2.87)
C
`
where we have used the transcendental equation for the eigenmodes, Eq. (2.85), and we have
renamed the spatial eigenmodes difference as un (0− ) − un (0+ ) = δun .
The remaining case of m = n can also be computed from the integral and reduced by using
Eq. (2.85), leading to:
Z
`
un (x)un (x) dx = A2n ` +
`
δu2n L0
kn2 2LJ,4
ω2
1 − n2 .
ωp
(2.88)
Hence, the generalized orthogonality equation can be expressed by combining the previous
results:
Z `
0
C
un (x)um (x) dx + CJ,4 δun δum = δnm ηn ,
(2.89)
`
where
ηn =
A2n `
δu2 L0
+ 2n
kn 2LJ,4
ωn2
1+ 2
ωp
(2.90)
depends on the mode n and can be regarded as the effective mass of each one. This equation
can be used to fix the normalization constant Am if we equal it to the total capacitance of the
circuit, that is:
Z `
C0
un (x)um (x) dx + CJ,4 δun δum = δnm CΣ ,
(2.91)
`
2C 0 `
with CΣ =
+ CJ,4 .
The transmission line resonator can be described by a sum of harmonic oscillators by using
the spectral decomposition of ψ(x, t) in Eq. (2.79) in the odd spatial modes. This expression is substituted in the Lagrangian of Eq. (2.77), which is simplified with the help of the
orthogonality relation and becomes:
L=
X CΣ
n
2
CΣ 2 2
ψ˙n2 −
ω ψ .
2 n n
25
(2.92)
−1.1
−1.15
−1.2
E(EJ )
−1.25
−1.3
−1.35
0.48
0.49
0.5
f
0.51
0.52
Figure 2.9: The energy spectrum of the flux qubit for values of α = 0.75 and EJ /EC = 32.
The corresponding Hamiltonian can be found by a Legendre transformation:
H=
X π2
CΣ 2 2
n
+
ω ψ ,
2CΣ
2 n n
n
(2.93)
where the charge πn = CΣ ψ˙n as been defined as the conjugate momentum to the flux ψn . This
harmonic oscillator can be quantized as usual by imposing the canonical commutation relations
[ψn , πm ] = i~δnm . These coordinates can be expressed by introducing the ladder operators a†n
and an as:
r
~
ψn =
(a† + an ) ,
2ωn CΣ n
r
~ωn CΣ †
πn = i
(an − an ) ,
(2.94)
2
where [an , a†m ] = δnm .
It follows that the linearized transmission line resonator intersected by a Josephson junction
is described by a sum of harmonic oscillators of frequency ωn :
X
1
†
H=
~ωn a a +
.
(2.95)
2
n
2.6.2
Qubit potential
The node fluxes description is well suited for deriving the correct boundary conditions coming
from the first Kirchhoff’s law. However, to obtain the effective flux qubit potential we need to
consider the branch variable description, and to impose the matching conditions coming from
the boundary conditions. In the central part containing the three-junctions array we impose a
flux configuration with two equal junctions EJ,1 = EJ,3 = EJ , and a smaller one EJ,2 = αEJ .
The Lagrangian for this part reads in terms of branch variables:
U
= −[cos ϕa + cos ϕb + α cos(ϕb − ϕa + 2πf − ∆ψ/ϕ0 )],
EJ
where the phases and branch fluxes are related by ϕj = φj /ϕ0 , with the reduced flux quantum
ϕ0 = Φ0 /2π. The eigenvalues of the Hamiltonian associated to this potential can be computed
numerically, and they are shown in Fig. 2.9.
26
2.7
New design of coupling between a qubit and a resonator
We introduce a new design for coupling between a qubit and a transmission line. The configuration of Josephson junctions is shown in Fig. 2.10. Compared to existing configurations [36],
our qubit design adds two additional loops in order to achieve a tunable qubit-resonator coupling strength, without affecting the qubit energy [37]. Mathematically, this occurs when the
qubit energy is independent of the fluxes defining the interaction, which can be modulated in
a continuous way.
f5
f3
f1
f2
f4
Figure 2.10: Scheme for ultrastrong coupling between a qubit and a transmission line where
the coupling can be modulated without affecting the qubit energy.
The branch variable description of the circuit depicted in Fig. 2.10 is accomplished first
by taking into account the flux quantization around all the closed loops. The branch fluxes
are renormalized such that ϕj = φj /ϕ0 , with the reduced flux quantum ϕ0 = Φ0 /2π and
Φ0 = h/2e the flux quantum. The flux of each loop must be equal to an integer number of flux
quantum Φ0 . The above condition leads to the following equations:
ϕ2 − ϕ3 − ϕ1 + 2πf1 = 2πN1 ,
ϕ3 − ϕr − ϕ4 + 2πf2 = 2πN2 ,
ϕ6 − ϕ2 + ϕ1 − 2πf3 = 2πN3 ,
ϕ4 − ϕ5 + 2πf4 = 2πN4 ,
ϕ7 − ϕ6 + 2πf5 = 2πN5 ,
(2.96)
where we have also introduced the frustration of each loop, fj = φx, j/Φ0 with φx,j the flux
flowing through each loop j.
Hereafter, we will assume no trapped flux in the loops, that is, Ni = 0. The junctions are
defined such that in the qubit are involved EJ,1 = EJ,2 = EJ and EJ,3 = αEJ . In addition, we
assume that EJ,4 = EJ,5 = EJ,6 = EJ,7 = βEJ and f4 = −f5 .
With these conditions the qubit potential reads:
−
Uq
= cos ϕ1 + cos ϕ2 + α cos ϕ3 + β(cos ϕ4 + cos ϕ5 + cos ϕ6 + cos ϕ7 ) ,
EJ
(2.97)
and, from the constraints imposed by the flux quantization we rewrite the potential:
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
+ β[cos(ϕ2 − ϕ1 − ϕr + 2π(f1 + f2 )) + cos(ϕ2 − ϕ1 − ϕr + 2π(f1 + f2 + f4 ))
+ cos(ϕ2 − ϕ1 + 2πf3 ) + cos(ϕ2 − ϕ1 + 2π(f3 + f4 ))] .
27
(2.98)
Now, by defining:
ϕ̄ = ϕ2 − ϕ1 − ϕr + 2π(f1 + f2 ) ,
θ̄ = ϕ2 − ϕ1 + 2πf3 ,
(2.99)
we can develop the following expression for a phase θ:
cos θ + cos(θ + 2πf4 ) = 2 cos(πf4 ) cos(θ + πf4 ) ,
(2.100)
and we can rewrite the Josephson potential in terms of these variables and the previous
trigonometrycal relation obtaining:
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
+ 2β cos(πf4 )(cos(ϕ̄ + πf4 ) + cos(θ̄ + πf4 )) ,
(2.101)
that is:
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
f4
+ 2β cos(πf4 ) cos ϕ2 − ϕ1 − ϕr + 2π f1 + f2 +
2
f4
+ cos ϕ2 − ϕ1 + 2π f3 +
.
2
(2.102)
This expression can be simplified even further if we consider that the phase amplitude
of the resonator is smaller than the unity, |ϕr | 1. Before approximating the potential, we
rewrite it in terms of ϕ = ϕ2 − ϕ1 − ϕr + 2π(f1 + f2 + f4 /2) and θ = ϕ2 − ϕ1 + 2π(f3 + f4 /2):
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
+ 2β cos(πf4 )(cos ϕ cos ϕr + sin ϕ sin ϕr + cos θ) .
(2.103)
Now, we approximate |ϕr | 1:
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
ϕ2
+ 2β cos(πf4 ) cos ϕ 1 − r + . . . + sin ϕ (ϕr + . . . ) + cos θ .
2
(2.104)
The target of this design is a qubit-resonator tuneable strength coupling that does not affect
the qubit energy. We notice that we can set the external fluxes such that f3 = f1 + f2 + 1/2,
which implies that θ = ϕ + π, and the potential reads:
−
Uq
EJ
= cos ϕ1 + cos ϕ2 + α cos(ϕ2 − ϕ1 + 2πf1 )
+ 2β cos(πf4 )ϕr sin ϕ + O(ϕ2r ) ,
(2.105)
where the term that renormalizes the qubit energy has disappear, making the qubit energy to
be independent from the flux f4 that tunes the coupling strength.
The effective Hamiltonian of the new setup must be calculated in order to study the behaviour of the coupling strengths as a function of f1 , f2 and f4 . Let us calculate the kinetic
energy, where only one mode of the resonator has been taken into account:
T =
1
1X
CJ,j ϕ20 ϕ˙j 2 + Cr ψ̇n2 ,
2
2
j
28
(2.106)
−1.65
−1.75
−1.85
E(EJ )
−1.95
−2.05
−2.15
0.46
0.48
0.5
f1
0.52
0.54
Figure 2.11: The energy spectrum of the new qubit depicted in Fig. 2.10 as a function of the
external flux f1 , with α = 2, β = 0.1, f4 = 0, and EJ /EC = 32.
where CJ,j and Cr are the capacitances the junctions and the resonator, respectively, and ψn is
the flux variable of the mode n of the resonator. We assume that CJ,1 = CJ,2 = CJ , CJ,3 = αCJ
and CJ,4 = CJ,5 = CJ,6 = CJ,7 = βCJ , which lead us to:
T =
1
ϕ2
ϕ20
CJ ϕ˙1 2 + ϕ˙2 2 + αϕ˙3 2 + 0 CJ β ϕ˙4 2 + ϕ˙5 2 + ϕ˙6 2 + ϕ˙7 2 + Cr ψ̇n2 .
2
2
2
(2.107)
The variables of the resonator ϕr and ψn are related in the following way:
1
ϕr =
ϕ0
~
2Cr ωn
1
2
(a†n + an )[un (x + ∆x) − un (x)] = ϕn δn ,
(2.108)
with ψn = ϕ0 ϕn = 1/ϕ0 (~/2Cr ωn )1/2 (a†n + an ) and δn = un (x + ∆x) − un (x).
Now, if we rewrite the kinetic energy by using the constraints of Eq. (2.96) and assume
that the external fluxes does not depend on time, we can achieve the following expression:
T
ϕ20
CJ (1 + α + 4β) ϕ˙1 2 + ϕ˙2 2 − (2α + 8β)ϕ˙1 ϕ˙2
2
Cr
2
2
2
− 2ϕ0 CJ (ϕ˙2 − ϕ˙1 ) δn ϕ˙n + βCJ δn +
ψ˙n .
2
=
(2.109)
In order to obtain the Hamiltonian, we consider the fluxes φj = ϕ0 ϕj and ψn , and calculate
their conjugate momenta:
Q1 =
Q2 =
qn =
∂L
∂T
=
= CJ (1 + α + 4β)φ˙1 − CJ (α + 4β)φ˙2 + 2CJ βδn ψ˙n ,
∂ φ˙1
∂ φ˙1
∂T
∂L
=
= −CJ (α + 4β)φ˙1 + CJ (1 + α + 4β)φ˙2 − 2CJ βδn ψ˙n ,
˙
∂ φ2
∂ φ˙2
∂L
∂T
=
= 2CJ β φ˙1 − φ˙2 δn + (Cr + 2βCJ δn2 )ψ˙n ,
∂ ψ˙n
∂ ψ˙n
(2.110)
which can be related with the time derivatives of fluxes through the capacitance matrix:

 
 ˙ 
φ1
Q1
CJ (1 + α + 4β) −CJ (α + 4β)
2CJ βδn
~˙ .
~





Q2
−CJ (α + 4β) CJ (1 + α + 4β)
−2CJ βδn
Q=
=
φ˙2  = M φ
qn
2CJ βδn
−2CJ βδn
Cr + 2βCJ δn2
ψ˙n
29
(a)
(b)
1
c(1)
x
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
f2
0
0.5
0.505
0.51
0.515
0.52
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.5
0
0.505
0.51
0.515
0.52
f1
(d)
1
0.8
0.8
1
(2)
cz
0.8
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
c(2)
x
f2
f2
f1
(c)
1
c(1)
z
0.8
f2
0
0.5
0.505
0.51
0.515
0.52
0
0.5
0
0.505
0.51
0.515
0.52
f1
f1
Figure 2.12: Following Eq. (2.113), we plot coupling strengths of first and second order of the
(1)
(1)
(2)
(2)
new flux qubit: (a) cx , (b) cz , (c) cx , and (d) cz as a function of external fluxes f1 and
f2 . The plots corresponds to values of α = 2, β = 0.1, f4 = 0, and EJ /EC = 32.
The Lagrangian and the Hamiltonian in a matrix form can be written:
1 ~˙ T ~˙
Mφ − U ,
L= φ
2
1 ~ T −1 ~
H= Q
M Q+U ,
2
(2.111)
with the matrix M :


CJ (1 + α + 4β) −CJ (α + 4β)
2CJ βδn
−2CJ βδn  .
M =  −CJ (α + 4β) CJ (1 + α + 4β)
2CJ βδn
−2CJ βδn
Cr + 2βCJ δn2
The above expression for the Hamiltonian in Eq. (2.111) is provided because M is a symmetric
matrix, M t = M .
30
The inverse matrix, M −1 , is computed giving the followng results:
M −1 (1, 1) = M −1 (2, 2) =
M −1 (1, 2) = M −1 (2, 1) =
M −1 (1, 3) = M −1 (3, 1) =
=
M −1 (3, 3) =
Cr (1 + α + 4β) + 2CJ β(1 + α + 2β)δn2
,
CJ (Cr (1 + 2α + 8β) + 2CJ β(1 + 2α + 4β)δn2 )
Cr (α + 4β) + 2CJ β(α + 2β)δn2
,
CJ (Cr (1 + 2α + 8β) + 2CJ β(1 + 2α + 4β)δn2 )
2βδn
−
Cr (1 + 2α + 8β) + 2CJ β(1 + 2α + 4β)δn2
−M −1 (2, 3) = −M −1 (3, 2) ,
1 + 2α + 8β
.
(2.112)
Cr (1 + 2α + 8β) + 2CJ β(1 + 2α + 4β)δn2
The Hamiltonian can be expressed in terms of Pauli matrices if we truncate the Hilbert
space to dimension two, corresponding to the subspace generated by the two eigenvectors
associated with the two lowest eigenvalues. We are allowed to perform this approximation
due to the anharmonicity of the energy distribution, in which the pair of lowest states are
sufficiently separated from the others.
Hence, the interaction part of the Hamiltonian in Eq. (2.111) can be written in terms of
Pauli matrices in the following general form:
Hint =
X
ϕnr
n=1,2...
3
X
(n)
cj σj ,
(2.113)
j=0
where σ0 stands for the identity operator, and the σj with j = 1, 2, 3 correspond to the
Pauli matrices. The constants cj are the different coupling strengths which are calculated by
a numerical evaluation of the Hamiltonian, and shown in Fig. 2.12.
The computations have been performed for a resonator of impedance Z ∼ 50 Ω, and
capacitance Cr = 850 fF, coupled galvanically to the qubit through a Josephson junction of
capacitance CJ = 10 fF. The frequency of the first mode of the resonator is approximately
ωr /2π = 7 GHz, which leads to a phase drop of ϕr = 0.1218,
p by substituting the value of the
spatial mode slip δn = 0.42831 in the relation ϕr = δn /ϕ0 ~/2ωr (Cr + CJ ).
31
Chapter 3
Quantum Field Theories
This chapter is devoted to revise briefly basic concepts and formalism of quantum field theory.
In particular, we will consider quantum electrodynamics which represents the early triumphs
of quantum field theory with extremely accurate predictions in quantities. A simplified model
of QED in one spatial dimension will be introduced and explained, with an analysis of their
problems of renormalization. An alternative Hamiltonian will be proposed in order to study a
simplified version of QED, and their extensions to a full-fledged QFT will be also analyzed.
3.1
Basic concepts of quantum field theory
The attempts of developing a relativistic quantum mechanics where an agreement between
quantum mechanics and special relativity was achieved, along with the need for a description of classical fields that accounts for their quantum-mechanical aspects, paved the way for
developing quantum field theories.
Special relativity tells us that two observers in two different inertial frames must agree on
the predicted results of experiments, which in the case of quantum mechanics, requires the
observers to agree on the value of the wave function at a particular spacetime point.
The main problem is that in quantum mechanics space and time are treated differently,
while the former is the eigenvalue of an operator which indicates the particle position, the
latter is just a parameter. In order to combine quantum mechanics and relativity we must put
space and time at the same level, thus we demote position from its stage as an operator and
set it as a variable on operators. In this sense, we assign an operator φ̂(x) to each point x in
space, thus conforming a set of operators called quantum field. The time dependence of the
operators is shown in the Heisenberg picture:
φ(x, t) = ei
Ht
~
φ(x, 0)e−i
Ht
~
.
(3.1)
Therefore, both position and time are now labels on operators. This fact can be interpreted
as a feature of the quantization of systems of fields, instead of the quantization of systems of
particles given by quantum mechanics, where the position x̂ is an operator associated with
physically observable quantities of the discrete system of particles.
A classical field is described by a function φ(x, t) with x a three-dimensional spatial coordinate, and t the time. The field can be a scalar, a vector or a tensor, and usually takes on
distinct values at each point in spacetime, and consequently, it is said to have infinite degrees
of freedom. There are physical phenomena described by classical fields, for instance classical
electrodynamics with the electric and magnetic fields, which do not include certain aspects
involving discrete particles, in this case photons. In this sense, the corresponding quantum
theory, quantum electrodynamics, accommodates the observations of quantum mechanics.
The procedure followed to achieve a quantum field theory is to reinterpret the dynamical
variables of the classical field theory as operators that obey canonical commutation relations,
that is, quantize the classical theory. In this way, we also ensure to include the features of
32
special relativity, since classical field theory is suited to relativistic dynamics due to the Lorentz
invariance of their expressions.
The implications of these new descriptions by Lorentz invariant quantum fields are theories
where the number of particles might be not conserved. The possibility of describing phenomena
where the number of particles can change over time is a particular feature of QFT, in contrast
with quantum mechanics where we have a fixed number of particles, with each particle having
a small number of degrees of freedom.
Lagrangian formulation
The Lagrangian formalism of classical field theory is inherited by quantum field theory. A
classical field is described by a function φ(x, t) that can be a scalar, vector, or tensor, where
x is a three-dimensional coordinate in space and t is the time. A fundamental quantity in
classical field theory is the action functional, S, defined as the time integral of the Lagrangian,
L. The Lagrangian can be expressed as well as the spatial integral of the Lagrangian density,
L which is a function of the field strength φi (x) and its derivatives ∂µ φi . Therefore, the action
reads:
Z
Z
S = L dt = L(φi , ∂µ φi ) d4 x ,
(3.2)
with (t, x) = (x0 , x1 , x2 , x3 ) = xµ .
The principle of least action is a variational principle of classical mechanics that can be
extended to QFT. This principle states that the ‘path’ taken by a system in the configuration
space between times t1 and t2 is the one for which the action is stationary to first order:
Z
∂L
∂L
∂L
4
0 = δS = d x
δφi − ∂µ
δφi + ∂µ
δφi
.
(3.3)
∂φi
∂(∂µ φi )
∂(∂µ φi )
The last term vanishes if we consider the Gauss’s theorem and the surface integral over the
fourth-dimensional region of integration. We assume δφ = 0 in the initial and final configurations at time t1 and t2 , respectively, as well as on the spatial boundary of the region. Thereby,
we obtain the Euler-Lagrange equations of motion for the system:
∂L
∂L
∂µ
−
=0.
(3.4)
∂(∂µ φi )
∂φi
We notice that all expressions above are explicitly Lorentz invariant if L is a Lorentz scalar,
that is, the fields fulfill the same equations after performing a boost or rotation to a different
frame of reference.
Hamiltonian formulation
The construction of the Hamiltonian in the case of classical fields follows analogous rules to the
case of discrete systems. Recall that, in the last case, given a Lagrangian L(qi , q̇i ) as a function
of the coordinates qi and their time derivatives q̇i , the Hamiltonian is obtained by a Legendre
transformation H = pi q̇i − L where the conjugate momenta are given by pi = ∂L/∂ q̇i . In our
case, the role of each dynamical variable qi (t) is played by φi (x, t), with x a continuous index.
Therefore, we can define:
∂L
(3.5)
πi (x) ≡
∂ φ̇i (x)
and
H = πi φ̇i − L ,
(3.6)
as the momentum density conjugate to φ(x),
R and the Hamiltonian density, respectively. The
Hamiltonian itself is given by the integral d3 x H.
33
Canonical quantization
We can go from classical to quantum mechanics via canonical quantization. This means that
we promote the fields and their corresponding momenta, φi (x) and πi (x), to operators, with
commutation relations:
[φ(x, t), φ(x0 , t)] = 0 ,
[π(x, t), π(x0 , t)] = 0 ,
[φ(x, t), π(x0 , t)] = i~δ (3) (x − x0 ) .
(3.7)
As we can notice, in the Heisenberg picture, these operators should be taken at equal times.
The field operators are related with creation, annihilation and count of particles, and it
is convenient to introduce the ladder operators ak and a†k when we are treating with systems
where the number of particles is not conserved. The canonical commutation relations of these
ladder operators depend on whether the fields are related with bosons or fermions, leading to
bosonic commutation relations or fermifermioniconic anticommutation relations, respectively.
3.1.1
Field theory methods
The discussion of quantum fields in interaction leads to the description of realistic phenomena,
such as scattering, or creation and annihilation of particles. This situation requires solving
extremely complex field equations, which is a problem usually addressed by the use of perturbation theory.
In the perturbative approach to quantum field theory, the Hamiltonian is divided into that
of the free fields plus the interaction terms, where the last are treated as a perturbation when
the coupling strength is weak enough. In this case, the interaction term is expanded in a series
in powers of the coupling, where each term can be thought of as forces between particles being
mediated by other particles.
The study of the equations of motion of the interacting fields leads to a perturbation series
solution suitable for scattering processes. This solution is known as the S-matrix expansion and
contains the complete information about all scattering processes, such that one can extract
the transition probability for a certain process to any perturbative order.
When the interaction Lagrangian density does not depend on derivatives of the fields, the
choice of the interaction picture is particularly successful, since the interaction fields and freefields satisfy the same equations of motion and the same commutation relations. Consequently,
the results easily derivable for free-fields can be translated to interacting fields.
The S-matrix determines the final state |φ(∞)i into which the initial state |φ(−∞)i = |ii
evolves after a collision process, |φ(∞)i = S|φ(−∞)i = S|ii. Thus, the probability amplitude
of a transition process between the initial state and a specific final state |f i is given by:
hf |φ(−∞)i = hf |S|ii .
(3.8)
The S-matrix can be expressed in terms of the interaction Hamiltonian density in the
interaction picture as follows:
Z
Z
∞
X
(−i)n
S =
. . . d4 x1 d4 x2 . . . d4 xn T{HI (x1 )HI (x2 ) . . . HI (xn )}
n!
n=0
Z
4
= T exp[−i d xHI (x)] ,
(3.9)
where T is the time-ordering symbol indicating that the operators are written in chronological
order, which in the exponential indicates that each term of its Taylor series is time-ordered.
The S-matrix can be expanded as a sum of normal products where all annihilation operators
stand to the right of all creation operators by using the Wick’s theorem [3]. This theorem relates
34
time-ordered products of a certain number of field operators with the normal product of these
fields and all their possible contractions, where a contraction of two fields is defined as the
vacuum expectation value of their time-ordered product.
The action of normal-ordered operators consists of absorbing certain number of particles
and then emitting other ones, with no intermediate emission and reabsorption of particles. The
non-vanishing contractions correspond to Feynman propagators, and represent virtual particles
being emitted and reabsorbed in intermediate states.
One can associate each normal product of the perturbative contributions with a particular
transition from an initial state to a final one, leading to a graphic representation of different
processes through vertices and lines attached to the vertices. These diagrams, called Feynman
diagrams, are a powerful and intuitive tool used to organize the perturbative expansion of the
S-matrix by associating a number of loops with the power of the coupling.
3.1.2
Renormalization
In the perturbation theory of QFT, the calculations of physical quantities lead to divergent
integrals that must be treated in order to obtain physical predictions. The infinities arising in
the formalism can be removed by techniques of regularization, such as the cut-off method or
the dimensional regularization, which consist of modifying the divergent integrals into finite
ones that can be evaluated. The regularized integrals depend on the formalism employed, but
we achieve a finite and well-defined theory in all orders of perturbation, which is a first step
towards a valid and finite theory.
The second step, called renormalization, originates from the recognition that the noninteracting or bare particles appearing in the formulation of the perturbation theory are different from the real physical particles which interact. The renormalization takes into account the
modification in the particles due to the interaction, and relates the properties of the physical
particles to those of the corresponding bare particles, leading to predictions of the theory in
terms of the former.
The last step consists of removing regularization and restoring the original theory, which
leads to finite physical predictions independent of the method of regularization used. The
requirement introduced by the renormalization, that physical properties are equal to the measured values, allows to absorb the infinities in the relations between bare and physical particles,
and to express the observable predictions of the theory in terms of the observed properties of
the particles.
The procedure of renormalization is not valid for all field theories, but only field theories
whose predictions, given in terms of a finite number of parameters, remain finite when the
regularization is removed are called renormalizable.
Power counting is a very simple and useful tool to rapidly determine which theories should
be expected to be renormalizable. Physical quantities in QFT have dimensions of mass to some
power if we consider that ~ = c = 1. In this case, length and time has the same dimensions
inverse to energy, [E] = [x]−1 = [t]−1 = M with M the mass dimension. The analysis of the
divergences of the Feynman diagrams lead to a criterion for non-renormalizability based on
the dimensions of the coupling constants of the Lagrangian, which states that if any of the
coupling constants has a negative mass dimension, the theory is non-renormalizable, and if all
the coupling constants are positive or zero the theory could be renormalizable.
3.1.3
Beyond perturbative methods
Calculations of quantum field theory quantities are performed through Feynman diagrams and
perturbative expansions of the interactions as it has been shown in previous sections. The
validity of most of these computations is related with the weakness of the coupling constant,
that is, if the coupling constant is insufficiently small, the perturbation theory does not generate
correct results.
35
The values of the coupling constant for which the perturbative methods become unreliable
are known as the strong-coupling regime. Many phenomena in QFT cannot be understood
in perturbation theory and require other techniques to be calculated, such as lattice gauge
theories [4].
In general, the numerical simulations of QFTs is computationally hard, with the computing
time growing exponentially with the system size. Nevertheless, a quantum simulator [1, 2] could
provide a remarkable efficient way to simulate these theories [5, 6, 7, 9, 10, 11] exponentially
faster than classical computers [11].
3.2
Quantum electrodynamics
The relativistic quantum field theory of electrodynamics is called quantum electrodynamics [3].
In essence, it describes how light and matter interact. In addition, it is the first theory where full
agreement between quantum mechanics and special relativity is achieved. This gauge theory
is described by a set of equations, namely, Maxwell’s equations and the Dirac equation, whose
form is determined essentially by relativistic invariance.
Opposite to classical electrodynamics, which is described by the electric and magnetic fields
and does not include certain aspects of electromagnetism involving discrete particles, QED
accommodates the observations of quantum mechanics of processes mediated by photons.
The interacting field theory of QED describes basic processes, such as electron-positron
creation
R 3 and annihilation. Formally, this theory is described by the QED total Lagrangian
L = d x LQED , where the Lagrangian density LQED is (~ = c = 1):
LQED = LDirac + LMaxwell + Lint
1
= ψ̄(iγ µ ∂µ − m)ψ − (Fµν )2 − eψ̄γ µ ψAµ .
4
(3.10)
Here, Aµ , and ψ and ψ̄ = ψ † γ 0 are bosonic and fermionic fields [3], respectively, Fµν =
∂µ Aν − ∂ν Aµ is the electromagnetic field tensor, e = −|e| is the electron charge, and γ µ are
the Dirac matrices satisfying the Dirac algebra {γ µ , γ ν } = 2g µν .
Dirac Lagrangian
The first part of the Lagrangian corresponds to the Dirac Lagrangian describing spin-1/2
particles, which is a Lorentz invariant expression from which one can construct the known
Dirac Hamiltonian of one-particle quantum mechanics. From the Lagrangian density we obtain
π = ∂L
= iψ † , and then, the Hamiltonian density reads:
∂ ψ̇
HDirac = iψ † ψ̇ − ψ̄(iγ µ ∂µ − m)ψ = ψ̄(−iγ · ∇ + m)ψ .
(3.11)
Since the Dirac equation is a factorization of the Klein-Gordon equation, every solution of
the Dirac equation also satisfies the Klein-Gordon equation, which means that the Dirac field
can be written as a linear combination of plane waves:
ψ(x) = u(p)eipx + v(p)e−ipx ,
(3.12)
where p0 = ω = (p2 +m2 )1/2 , and u(p) and v(p) are four-component constant spinors. Plugging
the previous equation into the Dirac equation, we obtain:
(p
/ + m)u(p)eipx + (−p/ + m)v(p)e−ipx = 0 .
(3.13)
Therefore, we require that:
(p
/ + m)u(p) = 0 ,
(−p
/ + m)v(p) = 0 ,
36
(3.14)
where we have introduced the Feynman notation p/ ≡ γ µ pµ .
Each of these equations has two linearly independent solutions, thus we introduce the index
r = 1, 2 and denote the spinors as ur (p) and vr (p). The solutions involving ur (p) and vr (p)
are referred to as positive and negative solutions respectively, and in the quantum theory are
interpreted in terms of particles and antiparticles. The two-fold degeneracies of the positive
and negative solutions for a given momentum p are related with the possible spin orientations.
The general solution of the Dirac equation when quantized is the Dirac field, described as:
Z
d3 p
1 X
−ip·x
†
ip·x
p
ψ(x) =
a
(p)u
(p)e
+
b
(p)v
(p)e
,
(3.15)
r
r
r
r
(2π)3 2ωp r
Z
d3 p
1 X
−ip·x
†
ip·x
p
ψ̄(x) =
b
(p)v̄
(p)e
+
a
(p)ū
(p)e
,
(3.16)
r
r
r
r
(2π)3 2ωp r
with the creation and annihilation operators obeying the anticommutation rules {ar (p), a†s (p0 )} =
{br (p), b†s (p0 )} = (2π)3 δ (3) (p − p0 )δrs .
The Hamiltonian associated with this Lagrangian can be written:
Z
d3 p X †
†
HDirac =
ω
a
(p)a
(p)
+
b
(p)b
(p)
,
(3.17)
p
r
r
r
r
(2π)3 r
where we have used the inner products between spinors, and dropped an infinite negative
constant term that comes from anticommuting br (p) and b†r (p).
Maxwell Lagrangian
The second part of the complete Lagrangian of QED corresponds to the Maxwell Lagrangian
describing radiation, from which we obtain the Maxwell’s equations of electromagnetism. The
complete Maxwell Lagrangian density contains sources jµ :
1
1
LMaxwell = − (Fµν )2 − jµ Aµ = − (∂µ Aν − ∂ν Aµ ) − jµ Aµ ,
4
4
(3.18)
where Aµ = (φ, A) is a four-vector potential whose components are the scalar and vector
potential of the theory of electromagnetism, and j µ = (ρ, j) a four-vector containing the charge
and current densities. The quantization can be accomplished in different gauges with equivalent
results, where the four components of the potential are treated as quantum fields with certain
bosonic commutation relations.
All observable quantities can be expressed in terms of the electric E and magnetic B fields,
which are related with the potentials through the following equations (c = 1):
B=∇×A ,
∂A
E = −∇φ −
.
∂t
(3.19)
These relations do not determine the vector and scalar potentials uniquely, but for an
arbitrary function χ(x, t) the gauge transformation:
Aµ (x) → A0µ (x) = Aµ (x) + ∂µ χ(x)
(3.20)
leaves the fields unaltered. The predictions for observable quantities must be invariant under
gauge transformations, hence it is a fundamental requirement of any theory formulated in
terms of the potentials Aµ (x) to be gauge-invariant.
The equations of motion from the Lagrangian density below are two Maxwell’s equations:
∂µ F µν = j ν ,
37
(3.21)
which in terms of the electric field E and the magnetic field B can be written in LorentzHeaviside units and c = 1:
∇·E = ρ ,
∇×B = j+
∂E
.
∂t
(3.22)
The two remaining Maxwell’s equations:
∇·B = 0 ,
∂B
= 0,
∇×E+
∂t
(3.23)
can be codified in the following expression:
µνρσ ∂ ρ F µν = 0 ,
where µνρσ is the four-dimensional Levi-Civita tensor, defined by:

+1 if µ 6= ν 6= ρ 6= σ , symmetric,





−1 if µ 6= ν 6= ρ 6= σ , antisymmetric,
µνρσ =





0 otherwise.
(3.24)
(3.25)
A symmetric permutation of indices means that an even number of indices are exchanged,
while an antisymmetric permutation means that an odd number of indices are exchanged.
Although Eq. (3.24) is not obtained from the Maxwell Lagrangian applying the principle of
least action, it can be shown that they are an identity known as the Bianchi identity.
The quantization of the fields in the Coulomb gauge (∇ · A = 0) allows us to express the
potential in terms of creation and annihilation operators as follows:
Z
A(x) =
2
1 X
d3 k
−ik·x
†
ik·x
√
ε
(k)
a
(k)e
+
a
(k)e
,
r
r
r
(2π)3 2ωk
(3.26)
r=1
where ωk = c|k|, and εr (k) are the unit polarization vectors, which in Coulomb gauge are
transversal to k. The quantized states of the electromagnetic field are photons, and ar (k) and
a†r (k) are their creation and annihilation operators satisfying bosonic commutation relations.
The sum respect to r is over two polarization states of the photons for all momenta k.
The Hamiltonian associated with Lagrangian (3.18), in the case of no charges are present,
is composed by harmonic oscillators:
Z
d3 k X
HMaxwell =
ωk a†r (k)ar (k) ,
(3.27)
(2π)3 r
where we have dropped an infinite energy term coming from the commuting of ar (k) and a†r (k).
Interaction Lagrangian
We consider now the interaction of electrons with the electromagnetic field described by the last
part of the QED Lagrangian in Eq. (3.10). The electromagnetic interaction can be introduced
into the free-fermion Lagrangian density, LDirac :
LDirac = ψ̄(iγ µ ∂µ − m)ψ ,
38
(3.28)
by considering a local symmetry of the fields:
ψ(x)
ψ̄(x)
→
→
ψ 0 (x) = eieχ(x) ψ(x) ,
ψ̄ 0 (x) = e−ieχ(x) ψ̄(x) .
(3.29)
When we take as basic requirement the invariance with respect to local phase transformations of the matter field, we must insist that the derivatives be transformed into covariant
derivatives, that transform under the phase in exactly the same way as the field itself, i.e.
Dµ ψ → eieχ(x) Dµ ψ .
(3.30)
The above expression holds true if we introduce a gauge field Aµ as follows:
∂µ ≡
∂
→ Dµ = ∂µ + ieAµ (x) ,
∂xµ
(3.31)
with e = −|e| the electron charge, and Aµ transforming as in Eq. (3.20). This substitution is
usually referred to as the minimal substitution.
The resulting theory describes the interaction of electrons with an electromagnetic field,
specified by the potentials Aµ , through the following Lagrangian density achieved after performing the substitution of the derivatives in the Dirac Lagrangian:
L = LDirac + Lint = ψ̄(iγ µ ∂µ − m)ψ − eψ̄γ µ ψAµ .
(3.32)
The gauge invariance of the theory of QED requires invariance when simultaneously transforming the Dirac fields according to the local phase transformation of Eq.(3.29), and the
electromagnetic potentials according to the gauge transformation of Eq. (3.20).
To obtain the complete Lagrangian density for QED, it must be added the Maxwell Lagrangian of the radiation field, that is, of the electromagnetic field in absence of charges.
3.3
Quantum field theories in one dimension
From the beginnings of QFT studies, theoretical models in one spatial dimension have been
developed with the aim of shedding light on the general ideas of the theory. 1 + 1 dimensional
models such as the Thirring model [38], the Schwinger model [39] or the Luttinger-Schwinger
model [40], have proven useful to acquire intuition about QFT.
In this sense, we consider the QED Lagrangian density in one dimension and analyze its
features in the following sections.
3.3.1
Quantum electrodynamics renormalization
The theory of QED in 1 + 1 dimensions is an attempt of simplification in order to treat
and simulate the interaction terms easily. Nevertheless, this naive treatment lead us to some
problems related to the renormalization of the theory. In this sense, we perform a simple
exercise of dimensional analysis to find out the mass dimensions of the coupling constant and
determine the renormalizability of the theory based on the criterion described in section 3.1.2.
In the unit system where ~ = c = 1, we consider that the action must be dimensionless,
therefore the position and the Lagrangian density have the following dimensions:
[x] = M −1 ,
Z
L =
dd x L , M −d [L] = M → [L] = M d+1 ,
with M , the mass dimension.
39
(3.33)
Now, based on the previous results for the dimensions of a Lagrangian density and using
the terms of Eq. (3.10), we can deduce in the case of QED in d spatial dimensions that:
[∂] = M ,
d
ψ̄i∂/ψ = M [ψ]2 → [ψ]d = M 2 ,
(3.34)
and that:
d+1
1
[− F 2 ] = M d+1 → [F ] = M 2 ,
4
d−1
[F ] = [∂A] = M [A] → [A] = M 2 ,
(3.35)
and finally, the interaction term where the coupling constant appears has the followingg dimensions:
d−1
1−3d
eψ̄Aψ = [e]M d M 2
→ [e] = M 2 .
(3.36)
We deduce from this last result that in QED, the coupling constant e is a dimensionless
constant in the case of d = 3, that is, when we treat the interaction in three spatial dimensions.
In one spatial dimension, the coupling constant has negative mass dimension, M −1 , hence
according to the criterion of non-renormalizability, the theory is non-renormalizable, that is,
all scattering amplitudes are divergent at a sufficiently high order in perturbation theory.
3.3.2
Simplified model of quantum electrodynamics
The problem we are addressing in this section is to find a simplified version of QED that
may be scaled towards quantum simulations of full-fledged QFT in quantum simulators. We
consider a simplified version of QED which may describe Yukawa interactions, and the coupling
of fermions to the Higgs field. In particular, the model that we have in mind assumes the
following conditions: (i) 1+1 dimensions; (ii) scalar fermions and bosons; (iii) variable coupling
strength, and it is described by the Hamiltonian (~ = c = 1)
Z
Z
Z
†
†
†
(3.37)
H = dpωp (bp bp + dp dp ) + dkωk ak ak + dx g(x)ψ † (x)ψ(x)A(x) .
Here, A and ψ are bosonic and fermionic fields, respectively, bp (b†p ) and dp (d†p ) are fermionic
and antifermionic annihilation (creation) operators, and ak (a†k ) is the annihilation (creation)
bosonic operator.
For the sake of clarity, we consider a simplified version of Eq. (3.37) thus introducing an
additional condition for the model: (iv) one fermionic and antifermionic field mode interacting
through a continuum of bosonic modes. More fermionic modes, including spinors, and polarization in the bosonic modes, can be treated in a similar way. The Hamiltonian with one fermionic
and antifermionic modes and a continuum of bosonic modes is:
Z
H = ωb† b + ωd† d + dkωk a†k ak
Z
† −ikx
ikx
† −ipx
ipx
ipx
† −ipx
+
dxdk gk (ak e + ak e
)(b e
+ de )(be + d e
) .
(3.38)
The fermionic and antifermionic annihilation operators obey the anticommutation relations
{b, b† } = {d, d† } = I, and the bosonic operators satisfying the conmutation relation [ak , a†k0 ] =
δ(k − k 0 ).
The model above suffers from the lack of the gauge symmetry U (1), therefore it is not
a QED theory, but a simplified version. Despite this drawback, the Hamiltonian introduced
in Eq. (3.37) conserves features of QED if we consider the interaction, where g(x)ψ † (x)ψ(x)
represents a charge density and A(x) stands for the scalar potential of electromagnetism, it
40
describes the action of an electrostatic potential. The model in Eq. (3.38) will allow to describe
pair creation and annihilation processes and self interactions of the fields. This model may also
describe Yukawa interactions and the coupling of scalar fermions to the Higgs field.
It is noteworthy to mention that the model (3.38) does not represent a full-fledged theory
of QFT. This is because its number of degrees of freedom is finite in the fermionic modes, and
a basic feature of QFT is that the fields have infinite degrees of freedom. We achieve the quasicontinuum in the bosonic modes, but when we scale the system towards a complete simulation
of QFT by increasing the number of qubits, the number of degrees of freedom remains finite
in the fermionic modes.
41
Chapter 4
Quantum field theories in
superconducting circuits
Quantum field theories [3] are among the deepest descriptions of physical nature and, consequently, there are different approaches to study them, as Feynman diagrams [3] or lattice gauge
theories [4]. The former is only valid at weak coupling, while in the region of strong coupling,
where perturbative methods become unreliable and interesting non-perturbative phenomena
appear, lattice field theory can be used to obtain certain quantities.
Due to the complexity of the numerical simulations of QFTs, new methods of computation
have been proposed in the field of quantum information [5, 6, 7, 9, 10, 11], which theoretically
would calculate exponentially faster than classical computers [11].
Here, we propose the quantum simulation of fermionic field modes interacting via a continuum of bosonic modes with superconducting circuits [12, 13, 14, 15, 16, 17, 18, 19, 20, 21],
which are among the best adapted quantum technologies in terms of quantum control and scalability. An important feature of superconducting devices is that, unlike any other experimental
setup, they offer naturally a strong coupling to the continuum of bosonic modes. Therefore, this
system is a specially suited platform to realize quantum simulations of interacting fermionic
and bosonic quantum field theories.
4.1
Model of quantum field theory
A fundamental interaction appearing in nature is that of fermions with bosons. Some of the
basic processes that take place in this physical scenario, such as electron-positron creation
and annihilation, are well described by the interacting field theory called quantum electrodynamics [3]. In this proposal we consider an analogous but simplified model of QED that may
describe also Yukawa interactions, and the coupling of fermions to the Higgs field. This model
is described in subsection 3.3.2 and has been studied by assuming: (i) 1+1 dimensions; (ii)
scalar fermions and bosons; (iii) variable coupling. The Hamiltonian is that of Eq. (3.38) and
reads (~ = c = 1):
Z
†
†
H = ωb b + ωd d + dkωk a†k ak
Z
† −ikx
ikx
† −ipx
ipx
ipx
† −ipx
+
dxdk gk (ak e + ak e
)(b e
+ de )(be + d e
) ,
(4.1)
with fermionic and antifermionic annihilation (creation) operators b(b† ) and d(d† ), respectively,
and ak (a†k ), the annihilation (creation) bosonic operator.
Our purpose is to provide a method for the quantum simulation of interacting QFTs by
exploiting the state-of-the-art in superconducting circuits [12, 13, 14, 15, 16, 17, 18, 19, 20],
leading to a scalable analog/digital quantum simulation.
42
4.1.1
Jordan-Wigner transformation
The Jordan-Wigner transformation [41, 42] will be used in our derivation to relate the fermionic
and antifermionic operators with tensor products of Pauli matrices, which are operators that
we can simulate in the superconducting circuit.
This transformation is based on a mapping between fermionic operators and spin-1/2 operators. We can first relate the spin raising and lowering operators, S + and S − , with the creation
and annihilation operators f † and f in the following way:
S+ = f
S− = f † ,
(4.2)
with |0i ≡ |+i and |1i ≡ |−i the equivalence between the spin and the number states.
Nevertheless, the first problem arise when we consider spins on different sites, because they
commute while fermion operators anticommute, that is:
Si+ Sj+ = Sj+ Si+
and
Si− Sj− = Sj− Si− ,
fi fj = −fj fi
and
fi† fj† = −fj† fi† .
(4.3)
It is necessary to cancel the extra minus sign to obtain a faithful spin representation when
we treat with different labels of the fermionic operators. In one dimension, the following nonlinear transformation solves the problem:
X †
Sl+ = fl exp(iπ
fj fj )
j<l
Sl− =
fl† exp(−iπ
X
fj† fj ) ,
(4.4)
j<l
where the phase value, 1 or −1, depends on whether the number of occupied modes is even
or odd, respectively. This sign assignation is particularly easy to perform in one dimension
because the modes can be ordered in a line and the sign is related to the number of particles
occupying the lower modes.
The spins obey the algebra [S + , S − ] = 2Sz . Since the spin operators and the Pauli matrices
are related such that S + = σ + , S − = σ − , and Sz = 12 σ z , we can derive the relation:
σ z = 1 − 2f † f .
Now, by using the fact of f † f taking the values 0 or 1, we can identify e±iπf
and then, the inverse transformations are:
Y
fl† = σl− (1 − 2fl† fl )
(4.5)
†f
= 1 − 2f † f ,
j<l
Y
fl = σl+ (1 − 2fl† fl ) ,
(4.6)
j<l
which can be rewritten by considering Eq. (4.5) in:
z
fl† = IN ⊗ IN −1 ⊗ · · · ⊗ σl− ⊗ σl−1
⊗ · · · ⊗ σ1z
z
fl = IN ⊗ IN −1 ⊗ · · · ⊗ σl+ ⊗ σl−1
⊗ · · · ⊗ σ1z ,
where the operators have been expressed in a tensor product.
43
(4.7)
4.1.2
Interaction Hamiltonian
The Hamiltonian of Eq. (3.38) that we want to simulate in superconducting circuits must be
reformulated in terms of Pauli matrices in order to relate the interaction with an effective
Hamiltonian describing the superconducting circuit.
We apply the Jordan-Wigner transformation derived in section 4.1.1, by assigning to the
fermionic and antifermionic operators the following tensor products of Pauli matrices:
b† = I ⊗ σ −
b = I ⊗ σ+
,
,
d † = σ − ⊗ σz
d = σ + ⊗ σz
(4.8)
With this transformation and using the fact that:
1
σ ± = (σx ± iσy ) ,
2
σi σj = ijk σk + δij ,
the interaction part of the Hamiltonian reads after some computations:
Z
1
1
Hint =
dxdk gk (ak eikx + a†k e−ikx )(I ⊗ I + σz ⊗ I − I ⊗ σz )
2
2
1
1
(σx ⊗ σx − σy ⊗ σy ) cos(2px) − (σx ⊗ σy + σy ⊗ σx ) sin(2px) .
+
2
2
4.1.3
(4.9)
(4.10)
Symmetric form of the Hamiltonian
The interaction Hamiltonian in Eq. (4.10) can be represented in a symmetric form. It is useful
a reorganization of the Hamiltonian in terms of cosines, therefore we split the integrals over
x in two parts, one with x < 0 and the other with x > 0, which allows us to rewrite some
terms as cosines and sines. There are three kinds of terms in the interaction, which can be
symmetrized as follows:
Z ∞
Z
H1 =
dx dkgk A(ak eikx + a†k e−ikx )
Z−∞
Z
∞
=
dx dkgk 2A cos(kx)(ak + a†k ) ,
Z0 ∞
Z
H2 =
dx dkgk A cos(2px)(ak eikx + a†k e−ikx )
Z−∞
Z
∞
=
dx dkgk 2A cos(kx) cos(2px)(ak + a†k ) ,
Z0 ∞
Z
H3 =
dx dkgk A sin(2px)(ak eikx + a†k e−ikx )
−∞
Z ∞ Z
=
dx dkgk 2A sin(kx) sin(2px)i(ak − a†k ) .
(4.11)
0
Now, by drawing on the following trigonometric expressions:
1
cos(kx) cos(px) = (cos(kx + 2px) + cos(kx − 2px)) ,
2
1
sin(kx) sin(px) = − (cos(kx + 2px) − cos(kx − 2px)) ,
2
44
(4.12)
the interaction part of the Hamiltonian can be expressed as:
Z ∞ Z ∞
dk gk (ak + a†k ) cos(kx)(2I ⊗ I + σz ⊗ I − I ⊗ σz )
dx
Hint =
0
+
+
−∞
1
(ak + a†k )(cos(kx + 2px) + cos(kx − 2px))(σx ⊗ σx − σy ⊗ σy )
2
i
†
(ak − ak )(cos(kx + 2px) − cos(kx − 2px))(σx ⊗ σy + σy ⊗ σx ) .
2
(4.13)
This form of writing the interaction terms has a natural correspondence with the Hamiltonian that will be obtained in Eq. (4.14) associated with the dynamics of the system, such that
the Hamiltonians Hj that will appear on each exponential of Eq. (4.17), namely the Hamiltonians that we have to simulate, will have a cosine dependence. This gives an additional value
to our proposal provided by the physics of superconducting coplanar transmission lines.
4.2
Experimental proposal
A distinctive feature of superconducting circuits, unlike other proposals, is that they provide naturally the continuum of bosonic modes. In fact, superconducting circuits are to our
knowledge the only physical scenario where matter qubits can be coupled strongly to a onedimensional continuum of bosonic modes. For instance, the reflection of a propagating microwave photon with almost 100% probability has been reported [43]. Other implementations
have access only to a discrete and small number of bosonic modes. Here we will present a
proof-of-principle architecture for the quantum simulation of QFTs using state-of-the-art superconducting circuits technology. The fermionic modes will be encoded in superconducting
qubits, in a scalable approach that may lead to a full-fledged quantum simulation of QFTs [1, 2].
4.2.1
Experimental setup for simulating interacting quantum fields
To profit from the natural continuum of bosonic modes provided by superconducting circuits [12, 13, 14, 15, 16, 17, 18, 19, 20, 21], that involves the interaction between superconducting devices made of Josephson junctions and coplanar transmission lines, and as a
proof-of-principle proposal of the dynamics governed by Eq. (4.10), let us consider the setup
depicted in Fig. 4.1a involving two one-dimensional coplanar waveguides (CPW). It consists of
a transmission line supporting a continuum of modes of the electromagnetic field (open line),
that interacts with three superconducting qubits. In addition, there is a cavity with a single
bosonic mode that interacts with two of the qubits. Notice that two superconducting elements
interact simultaneously with both coplanar waveguides. Furthermore, we consider a symmetric
arrangement of qubits for both CPWs. In this setup, we consider the couplings among each
qubit and the CPWs as well as the superconducting qubit energies to be tunable. In particular,
we will require the ability to switch the interaction on and off, as well as to establish situations
where qubits and CPWs could exchange excitations, or the field supported by the CPWs could
experience displacements depending on the qubit state. These conditions can be accomplished
considering specific architectures in superconducting circuits [36, 37, 21]. An extension to many
fermionic modes can be accomplished by adding more superconducting qubits, as can be seen
in Fig. 4.2.
45
(a)
(b)
UMS ( ⇡/2, 0)
UMS (⇡/2, 0)
qubit 1
UC
↵(f✏j )
qubit 2
ancilla qubit
UC
↵(fqj )
Figure 4.1: (a) Schematic representation of the proposal for simulating interacting quantum
fields. An open transmission line supporting the continuum of bosonic modes (terminated
by arrows at each extreme) interacts with three superconducting qubits, while the second
one-dimensional waveguide (terminated by capacitors at each end) supports a single mode
of the electromagnetic field and interacts with two superconducting qubits. Each qubit can
be individually addressed through on-chip flux lines producing fluxes f to tune the coupling
strength, and fq to tune the qubit energy as well as the kind of coupling (transversal or
longitudinal). (b) Sequence of multiple and single gates acting on superconducting qubits to
generate a single Trotter step.
4.2.2
Hamiltonian of the superconducting device
In our framework, the coupling of qubits to the cavity and to the open line (bosonic continuum)
ol + H c ,
should be described by the interaction Hamiltonian, Hint = Hint
int
Hint = ~
3
X
α(fj )
Z
dkgk cos(kxj )(ak + a†k )
+ ~
j=1
cjµ (fqj )σµj
µ=x,z
j=1
3
X
X
Z
β
dkgk cos(kxj )i(a†k
−
ak )σyj
+~
2
X
j=1
gj (c + c† )
X
cjµ (fqj )σµj , (4.14)
µ=x,z
where σx and σz are Pauli operators, ak (a†k ) and ωk stand for the annihilation (creation)
operator and the frequency associated with the kth mode of the the open line, respectively,
whereas the operator c (c† ) annihilates (creates) excitations in the cavity. In addition, xj stands
for the jth qubit position, and the coefficients α can be tuned over the range [0, αmax ] via some
external parameter fj acting on the jth superconducting qubit. Moreover, the coefficients
cµ can be tuned at convenience via an external parameter fqj , such that the switching from
transversal (cx ) to longitudinal (cz ) coupling, and vice versa, is allowed. The same parameter
fqj will allow to tune the qubits energy. The coefficient β corresponds to the capacitive coupling
of the resonator and the flux qubits, which become relevant when the inductive coupling is
switched off, and cannot be tuned.
For practical purposes, the open line corresponds to a large cavity supporting a set of
modes, such that their frequency spacing is small enough to consider them as a continuum.
Notice that the cosine dependence in Eq. (4.14) emerges from the cavity boundary conditions.
It is noteworthy to mention that, writing the Hamiltonian (4.10) in a symmetric form with
respect to the x coordinate will also lead to a cosine dependence. In this sense, our setup
becomes a suitable scenario for simulating the spatial dependence.
Previous proposals for simulating quantum fields [8, 5, 9, 6, 7, 10, 11] lack from versatility
and the access to the continuum of bosonic modes.
46
As a matter of fact, the schematic architecture in Fig. 4.1a could be reached using on-chip
superconducting circuits as the one developed in Refs. [18, 19], where an array of twelve interacting cavities has been shown experimentally. The superconducting elements can be transmon [44] or flux qubit [45]. In particular, our discussion will be focused on the later, considering
a new configuration depicted and analyzed in section 2.7.
It has been shown in Refs. [36, 37] and in section 2.7, that a particular arrangement of
the flux qubit architecture has a three-fold purpose. First, it allows us an independent control
on the qubit energy and the qubit-cavity coupling strength. Second, it has the advantage
of switchable coupling in both strength and the crossover from longitudinal to transversal.
Third, it allows us to neglect higher-order regimes in the light-matter interaction, leading to a
linear theory. In addition, our proposal for simulating QFTs could be optimized by using the
gradiometric flux qubit [46], that forbids crosstalk among external fluxes applied to the loop,
thus allowing an independent control of the qubit gap and its symmetry point.
4.3
Digital quantum simulation of quantum field theories
The digital methods for quantum simulation allows us to simulate systems whose dynamics
differ from the ones of the experimental setups. This feature is a key point in order to simulate
a model of QFT. Our proposal allows us to reproduce the full Hamiltonian dynamics described
by the Schrödinger equation
i~∂t |ψi = H|ψi ,
(4.15)
R
where the Hamiltonian H = dV H(x, y, z) with dV = dxdydz, and H(x, y, z) is the Hamiltonian density. In our simplified model of a 1-dimensional
scalar QED theory, the interaction
R
†
Hamiltonian in the Schrödinger picture is Hint = g dxψ (x)A(x)ψ(x), with ψ(x) and A(x),
the fermionic and bosonic fields, respectively, and H = ψ † (x)A(x)ψ(x) the Hamiltonian density. Notice that in our proposal, the fermionic modes associated with ψ(x) are discretized,
whereas we can achieve a continuum of bosonic modes through coplanar waveguides.
The evolution of any initial state |ψ0 i according to the Schrödinger equation (4.15) reads
|ψ(t)i = e−iHt |ψ0 i ≈ e−i
P
x,j
Hj (x)t
|ψ0 i ,
PN
(4.16)
where the Hamiltonian density is a sum of N terms, H(x) = j=1 Hj (x).
Our approach consists of a discretization
of
R
Pthe spatial coordinates x that allows us to
write the Hamiltonian H as H = dxH(x) ≈ x H(x), thus been achieved an expansion of
the Hamiltonian in a sum of terms associated with different spatial points. Taking into account
that in general [Hj (x), Hj 0 (x0 )] 6= 0, |ψ(t)i can be approximated with Trotter techniques as
follows:
P
|ψ(t)i = e−iHt |ψ0 i ≈ e−i x H(x)t |ψ0 i
−iH1 (x) nt −iH2 (x) nt
−iHN (x) nt
≈
e
e
...e
t
t
t
× e−iH1 (x+δ) n e−iH2 (x+δ) n . . . e−iHN (x+δ) n
n
−iH1 (x+2δ) nt −iH2 (x+2δ) nt
−iHN (x+2δ) nt
× e
e
...e
...
,
(4.17)
where n is the number of Trotter steps and δ the spacing between adjacent spatial points.
In this digital approach we have used the Trotter formula in a model where the space has
been discretized. The Trotter expansion consists of dividing the time t into n time intervals
of length t/n, and applying sequentially l times the evolution operator of each term of the
Hamiltonian for each time interval. In this case the evolution operators are associated with the
different terms of the Hamiltonian density in different spatial points.
47
4.4
Generation of interaction terms
Let us discuss how the Hamiltonian of Eq. (4.14) allows to generate the interactions needed to
simulate the dynamics governed by the Hamiltonian of Eq. (3.38).
In this case, the Hamiltonian of Eq. (3.38) presents just four kinds of interactions if we
analyze its reduced form of Eq. (4.13). We can observe single and two qubit gates coupled to
the continuum in H1 and H2 , respectively:
Z
H1 = σi dxdk gk f1 (k, x)(ak + a†k ) ,
Z
H2 = (σi ⊗ σj ) dxdk gk f2 (k, x)(ak + a†k ) ,
(4.18)
with σq = {σx , σy , σz } for q = 1, 2, 3, and fi (k, x) are functions that depend on k and x.
There is also a term coupling two qubit gates with the momentum operators associated
with the continuum bosonic modes, H3 and interactions that involve only bosonic modes H4 :
Z
H3 = (σi ⊗ σj ) dxdk gk if3 (k, x)(a†k − ak ) ,
Z
H4 =
dxdk gk f4 (k, x)(ak + a†k ) .
(4.19)
In Fig. 4.1b, we show the set of quantum gates for a single Trotter step [2, 9] that can be
generated by the superconducting circuit setup. Each gate corresponds to the evolution under
the Hamiltonian (4.14) for different values of the parameters fj , fqj .
In this sense, the simulator should provide the possibility of generating multi-qubit gates,
and coupling spin operators to a continuum of bosons. In addition, a high level of quantum
control will be required in order to apply sequentially a set of quantum gates that will provide
the fermion-boson interactions in a digital quantum simulation context [9].
In the following sections, we will introduce the gates that can be performed in the experimental setup and their action over the qubits.
4.4.1
The Mølmer-Sørensen multi-qubit gate
The Mølmer-Sørensen (MS) gate [48] is a multi-qubit entangling gate involved in the generation
of the interaction terms of the Hamiltonian of this proposal of quantum simulations of quantum
field theories. This gate, UMS , can be parametrized as:
θ
2
UMS (θ, φ) = exp −i (cos φSx + sin φSy ) ,
(4.20)
4
P
where Sx,y = i σix,y is extended to as many qubits as fermionic modes are involved, in the
simplest case, two qubits.
The main control parameter of the gate is the phase θ, when taken the value θ = π/2 makes
the MS gate maximally entangling, generating GHZ states [48]. The other phase φ allows to
choose between a σ x -type or σ y -type MS gate, φ = 0 and φ = π/2, respectively.
Other remarkable property of this gate is the periodicity respect to the first parameter,
that is, UMS (θ, φ) = UMS (θ + 2πn, φ) for n ∈ Z.
The inverse MS gate, with negative values of θ, can be accomplished by MS gates with
positive values of θ, with the following correspondences:

if n odd

 UMS (π − θ, φ)
UMS (−θ, φ) ≡
,
(4.21)

 UMS (π − θ, φ) Qn σ̃j
if n even
j=1
48
with σ̃jx = cos φσjx + sin φσjy , and n the number of qubits participating in the MS gate.
4.4.2
The sequence of gates
In general, each Hamiltonian Hj in Eq. (4.17) involves a product of fermionic operators coupled
to an integralR in the bosonic modes. For example, one possible interaction could be H =
g(bi d†j + dj b†i ) dk cos(kx)(ak + a†k ). The Jordan-Wigner transformation has allowed to write
the above interaction as the exponential of a tensorial product of Pauli matrices with an
integral in the bosonic modes. To generate this exponential, we apply the following sequence
of quantum gates [9, 47].
= UMS (−π/2, 0)Uσz (φ)UMS (π/2, 0)
h π
i
h π
i
= exp i Sx σ1x exp [iφσ1z ] exp −i Sx σ1x
π 4 i
h 4
π z
Sx σ1 + sin
Sx σ1y ,
(4.22)
= exp iφ cos
2
2
P
with the operator Sx = ni=2 σix acting on the second qubit if we consider the system with two
flux qubits and the ancilla, or acting on all the remaining qubits of the system if we consider
a scaled up proposal with n qubits and the ancilla.
In order to calculate the interaction one can use the following identities:
 Qn
x
π  i=2
Qn σi x for n − 1 = 4k, k ∈ N,
− i=2 σi for n − 1 = 4k − 2, k ∈ N, ,
cos
(4.23)
Sx =

2
0
for n even
U
and
sin
π
2
Sx
 Qn
x
for
 Q
i=2 σi
n
− i=2 σix for
=

0
for
n − 1 = 4k − 3, k ∈ N,
n − 1 = 4k − 1, k ∈ N,
n odd
,
(4.24)
which lead us to the following result in the case of n qubits and the ancilla:
U
= UMS (−π/2, 0)Uσz (φ)UMS (π/2, 0)
Z
z
x
x
= exp [−iφ(σ ⊗ σ ⊗ σ ⊗ ...) dk cos(kx)(ak + a†k )] ,
(4.25)
where UMS is the Mølmer-Sørensen gate studied above, and the central gate reads
Z
z
Uσz (φ) = exp[iφσ1 dk cos(kx)(ak + a†k )],
(4.26)
which adds the interaction with the open transmission line.
In the simplest case of three qubits including the ancilla, the gates that act in the first
two qubits are, from right to left, one Mølmer-Sørensen interaction UMS (π/2, 0), one local
R
gate Uc = exp[iφσ1z dk cos(kx)(ak + a†k )] that will couple the spin operators to the bosonic
continuum, and an inverse Mølmer-Sørensen interaction UMS (−π/2, 0). The application of
these three operations will
gate coupled with an integral in the bosonic
R generate the two-qubit
†
modes, H2 = (σi ⊗ σj ) dxdk gk f2 (k, x)(ak + ak ).
R
We will consider also operators of the form Uc = exp[iφσ1z dk cos(kx)i(a†k − ak )], where
the capacitive coupling between the flux qubit and the resonator has been taken into account
in order to simulate the coupling of two qubit gates with
associated
R the momentum operators
†
with the continuum of bosonic modes, H3 = (σi ⊗ σj ) dxdk gk if3 (k, x)(ak − ak ).
49
Figure 4.2: Scheme for the implementation of a set of N fermionic modes coupled to a continuum of bosonic modes. Each fermionic mode is encoded in a nonlocal spin operator distributed
among N superconducting qubits.
The same scheme of gates can be applied on more qubits in order to scale the system
simulating interactions involving a larger number of fermions. The gate Uc will be used independently on each qubit to generate single qubit gates coupled to the bosonic continuum. Besides, the ancilla qubit will allow to generate the gates involving just the bosonic
modes. This will
R be achieved coupling the ancilla qubit to the continuum with an interaction
Uc = exp[iφσaz dk cos(kx)(ak +a†k )], where σaz is the Pauli operator associated with the ancilla
qubit and preparing this qubit into an eigenstate of σaz .
The spatial dependence in the Hamiltonian is simulated just introducing phases in the
bosonic fields, A(x), by letting them to evolve freely when all coupling strengths are turned
off (α(fj ) = 0 in Eq. (4.14)). For example, if we have the state at time t to be
R
†
|ψi = U2 e−iδN̂ eiδN̂ e−ig dk cos(kx0 )(ak +ak )F e−iδN̂
× eiδN̂ U1 |ψ0 i ,
(4.27)
R
where N̂ = dkωk a†k ak is the free energy term for the bosonic field, and F stands for
the fermionic part, the gates U1 and U2 can be used to cancel the terms eiδN̂ and e−iδN̂ ,
respectively. Thus, we are generating effectively interaction terms at positions x0 + δ and
−x0 + δ. Notice that this involves the application of a block of physical gates UB (δ) =
U2 UMS (−π/2, 0)Uc UMS (π/2, 0)U1 . It means that to approach the continuum in space, we have
to apply several blocks |ψi ≈ UB (δ)UB (2δ) · · · UB (jδ) · · · |ψ0 i.
4.4.3
Multiqubit entangling gate
The way of scaling this formalism to a larger number of fermionic modes is to consider more
superconducting elements coupled both to the cavity and to the open transmission line along
the same lines as in the previous formalism, as depicted in Fig. 4.2. Considering N superconducting qubits, N fermionic modes can be encoded. Accordingly, our proposal can efficiently
implement a large set of fermionic modes interacting with the bosonic continuum. This will
represent a significant step forward towards full-fledged quantum simulation of QFTs in controllable superconducting circuits.
50
Let us consider the Hamiltonian describing the interaction of N qubits and the cavity
c . Furthermore, we assume identical superconducting qubits positioned along the
mode, Hint
CPW cavity such that all couplings are identical, and we set the parameters fqj and fj such
that α(fj ) = 0, and cz = 1 (cx = 0). In this case, the dynamics will produce a Mølmer-Sørensen
gate, up to local rotations on the superconducting qubits, if the ratio between the coupling
strength g and the resonator frequency ω0 satisfies the condition g/ω0 1. This comes out
because the dynamics is driven by the effective second-order Hamiltonian Heff = (g 2 /ω0 )Sx2 .
The inverse MS gate UMS (−π/2, 0) can be implemented by considering the relation:
UMS (−π/2, 0) = UMS (7π/8, 0) ,
(4.28)
and by letting the system evolve until the desired time. In addition, the local gate Uc (φ)
acting on the first qubit can be applied via the cavity, letting the remaining qubit to interact
dispersively with the resonator, as well as turning off the interaction between the qubits and
the open CPW.
4.4.4
Codification of information
The information will be encoded in the two levels of each flux qubit, by the association of
fermionic and antifermionic creation and annihilation operators onto the four internal levels of
the two flux qubits in the simplest case.
We recall the Jordan-Wigner mapping performed in section 4.1.1, which is given by:
b† = I ⊗ σ −
b = I ⊗ σ+
d† = σ − ⊗ σz
,
d = σ + ⊗ σz ,
,
(4.29)
and associate the fermionic operators to the following operators acting in the two superconducting qubit states as follows:
b† = | ↑↓ih↑↑ | + | ↓↓ih↓↑ | ,
b = | ↑↑ih↑↓ | + | ↓↑ih↓↓ | ,
d† = | ↓↑ih↑↑ | − | ↓↓ih↑↓ | ,
d = | ↑↑ih↓↑ | − | ↑↓ih↓↓ | ,
(4.30)
where the states | ↑i and | ↓i correspond to the two levels of a qubit.
With this mapping, the vacuum state corresponds to the state |0i = | ↑↑i, the state with one
fermion is |f i = | ↑↓i, and the state with one antifermion is |f¯i = | ↓↑i. We notice that creating
a state of one fermion and one antifermion is achieved by applying both creation operators b†
and d† , and the result will depend on the order of the operators, since they anticommute. That
is, b† d† |0i = −d† b† |0i = | ↓↓i.
Some features of the model related with QED may be analyzed and simulated in the
experiment. Fermion self-interaction is appreciated when one measures the evolution of a single
fermion and the probability |hf, 0, 0|U (t)|f, 0, 0i|2 at time t.
Pair creation and annihilation can be also observed by measuring the transition probabilities
between a state with no fermions into a state with a fermion and an antifermion, and vice versa.
In the approach to a full-fledged quantum field theory, we scale the formalism to a larger
number of fermionic modes by adding more superconducting qubits between the resonator and
the open transmission line. In this case, an analogous mapping between the fermionic operators
and tensor products of Pauli matrices is performed by the Jordan-Wigner transformation:
z
b†l = IN ⊗ IN −1 ⊗ ... ⊗ σl− ⊗ σl−1
⊗ ... ⊗ σ1z ,
−
z
d†m = IN ⊗ IN −1 ⊗ ... ⊗ σm
⊗ σm−1
⊗ ... ⊗ σ1z ,
51
(4.31)
where l = 1, 2, ..., N/2, m = N/2+1, ..., N , with N the total number of fermion plus antifermion
modes and Ij the identity operator.
An analogous encoding of information by considering all the levels of the superconducting
qubits can be performed. In this case, the fermion-antifermion pair creation and annihilation
can be observed for each fermionic mode k, as well as the fermion self interaction.
Finally, considerations related with renormalization are needed so that our proposal can
be used to make predictions of QFT phenomena. In this sense, one might associate meassured
quantities in the simulator with observable quantities of the simulated system.
In conclusion, we have proposed a protocol for simulating interacting bosonic and fermionic
fields in superconducting circuits, which provide the continuum of bosonic modes in a natural
way. This is a significant advantage over other proposals for simulating quantum field theories
in trapped ions or cold atoms. In the latter, the number of discrete field modes grows slowly
with the size of the system. Our protocol establishes a path to performing future quantum
simulations that overcome classical computers. This new approach may represent a significant
step towards the full fledged quantum simulation of quantum field theories in perturbative and
nonperturbative regimes.
52
Chapter 5
Conclusions
In this thesis, we have first introduced in chapter 2 the basic theory of superconducting circuits
with a detailed analysis of the galvanic coupling between a flux qubit and a resonator, usually
employed to reach higher values of coupling.
We have proposed a new design of flux qubit with five superconducting loops that allows
to switch between transversal and longitudinal coupling with the resonator, and to tune the
strength coupling without renormalizing the qubit energy. Such advantages make it an ideal
candidate for the proposal introduced in chapter 4.
Basic features of QFT have been reviewed in chapter 3, and, particularly, the Lagrangian
of quantum electrodynamics has been introduced. We have considered a simplified model of
QED in one spatial dimension, but the first and naive approximation turns out to be nonrenormalizable. We have proposed an alternative model that suffers from the lack of the QED
characteristic gauge symmetry U (1), but which can reproduce some of its typical phenomena,
such as particle creation and annihilation and self-interaction processes.
In chapter 4, we proposed the simulation of a simplified model of QED with interacting fermionic and bosonic modes in superconducting circuits. We have considered a proof-ofprinciple proposal involving two one-dimensional coplanar waveguides and three flux qubits.
It consists of a transmission line supporting a continuum of modes of the electromagnetic field
(open line), that interacts, inductively and capacitively, with three superconducting qubits. In
addition, there is a cavity with a single bosonic mode that interacts with two of the qubits.
A digital method of simulation of the model suitable for superconducting devices, involving Mølmer-Sørensen gates and single qubit rotations, has been described. The multiqubit
entangling gate in the case of the scaled system with N qubits is also analyzed.
The codification of information related with the characteristic phenomena of QED, such as
pair creation and annihilation and self-interaction processes, has also been described through
the mapping between the two-qubit states and the states with one fermion, one antifermion,
and fermion and antifermion. Considering the scaled system with more than one fermionic and
antifermionic mode, the association can be accomplished in a similar way.
In this work, we have profited from the advantages of superconducting circuits to propose
the quantum simulation of fermionic field modes interacting via a continuum of bosonic modes,
which is a significant step forward towards quantum simulations of full-fledged QFTs.
An in-depth analysis of some problems that have emerged in this work is needed. In particular, the introduction of the capacitive coupling between the flux qubit and the open line
has to be analyzed in all cases in order to accurately determine when it becomes relevant.
This work has pointed to several open lines of research, among which we can highlight the
simulation of a continuum of fermionic modes that would be considered, with the continuum
of bosonic modes, a simulation of full-fledged QFT, with infinite degrees of freedom. The
problem of renormalization must be also analyzed in order to make further predictions of QFT
phenomena, and the question of which quantities can be extracted from the simulation should
be also studied in the light of new proposals of embedding quantum simulations.
53
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