Quantum Entanglement and Superconducting Qubits

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Quantum Entanglement and

Superconducting Qubits

Kvantmekanisk sammanflätning och supraledande qubits

Wai Ho, TANG

Faculty of Health, Science and Technology Physics, Bachelor Degree Project

15 ECTS credits

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Conventional computing based on classical technologies is approaching its limits. There-fore scientists are starting to consider the applications of quantum mechanics as a means for constructing more powerful computers. After proposing theoretical methods, many experimental setups have been designed to achieve quantum computing in reality. This thesis gives some background information on the subject of quantum computing. We rst review the concept of quantum entanglement, which plays a key role in quantum com-puting, and then we discuss the physics of the SQUIDs-cavity method proposed by Yang et al., and give the denitions of quantum gates which are the elements that are needed to construct quantum circuits. Finally we give an overview of recent developments of SQUIDs-cavity systems and quantum circuits after Yang et al.'s proposal in 2003. These new developments help to take a step towards the constructions of higher levels of quan-tum technologies, e.g. quanquan-tum algorithms and quanquan-tum circuits.

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1 Introduction

After a nearly century-long development of quantum mechanics, there are still a lot of mysteries to be solved, and potential techniques to be used. On the other hand, due to the demand of more powerful technologies (for example, super-computers for solving more complicated prob-lems), the engineering eld develops dramatically fast.

To obtain new technical developments, quantum properties attracted interest in a lot of re-search elds: computation, cryptography, teleportation ([7], [12], [15], [19] and [24]). All these areas are tightly related to the theory of quantum entanglement.

One of the proposed methods is to use superconducting quantum interference devices (SQUIDs) and a cavity. A comprehensive review of this method has been given in [26]. By placing two uncoupled SQUIDs in a cavity they can be treated as identical, the two SQUIDs are essentially in communication by performing the same processes to both SQUIDs. By using appropriated procedures (section III of [26]), including excitation of states and adjusting the resonance be-tween the cavity eld and the SQUIDs, the entangled state of the composite system will follow a specic path of evolution. Therefore, quantum gates can be constructed (see section IV of [26]). Such quantum gates are the basic elements for constructing quantum circuits, algorithms and computations [16].

In order to understand the principles behind these ideas, it is important to know what the denition and properties of quantum entanglement are. On the other hand, the physics of SQUIDs, cavity elds and their interactions are required for understanding the concrete exper-imental setup, two-SQUIDs-in-cavity.

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recent years. Section 4 provides a concluding discussion about my work and its applications. Finally, the mathematical details of canonically conjugate relations between magnetic uxes and electric charges and the quantization of cavity electromagnetic elds are given in appen-dices A and B, respectively.

2 Description of Quantum Systems

In quantum mechanics the state of a physical system is described by a density matrix; for a simple system, a state vector in Hilbert space is enough for the description. The dimension of the state space is the number of eigenstates of the Hamiltonian of the system. For a simple system which can be described by a state vector, an observation or measurement of the system is done by acting with the corresponding observable operator on the state vector. Through the measurement the state of the system collapses to an eigenstate of the observable operator. Otherwise, for a system described by a density operator, each of the constituent systems is changed under measurement as above, so the density operator collapses.

This is very dierent from the situation in classical physics in which any observables of a system can be dened as functions of the basic dynamical variables, and any measurements do not disturb others. However, the view that no disturbance comes from measurement violates a fundamental law in nature - the uncertainty principle, which states that two observables cannot be known exactly simultaneously if the corresponding observable operators do not commute. This can be understood by the nature of state collapse of measurement. A typical example is given by the position operator ˆx and the momentum operator ˆp. Their commutation relation [ˆx, ˆp] = i¯h 6= 0 implies the non-existence of any simultaneous eigenvector. The state of the system cannot be measured for both operators at the same time, so the product of uncertainties, ∆xand ∆p, cannot be zero.

2.1 Single-particle System

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2.1.1 State Vector Representation i. Bra-Ket Notation and Superposition

For a simple system, the state of the system can be represented by a state vector, also called ket vector, in Hilbert space, |ψi. Each ket has its corresponding state vector, also called bra vector, in the dual space, hψ|. Importantly, the rule of superposition applies to the states of the system: If |αi and |βi are two possible states of the system, then

|γi = cα|αi + cβ|βi (2.1)

also is a possible state of the system. Its bra vector in the dual space is given by

hγ| = c∗_αhα| + c∗_βhβ| , (2.2) where ci and c∗i denote a complex number and its complex conjugate, respectively.

ii. Inner Product: Normalization and Orthogonality

Due to the condition of normalization, which follows from the probability interpretation of the wave function, the inner product of a ket with itself is required to be 1, i.e.

hψ|ψi = 1. (2.3) A state | ˜ψi with arbitrary modulus always can be normalized to

|ψnormalizedi =

1

h ˜ψ| ˜ψi| ˜ψi . (2.4) If the inner product of two non-zero state |ϕi and |ψi is zero, i.e.

hϕ|ψi = 0 , (2.5) then we say that |ϕi and |ψi are orthogonal to each other.

iii. Basis Kets

As in linear algebra, due to the rule of superposition, it is convenient to choose a set of basis kets {|eii , i = 1, 2, 3, ..., N } to represent any state of system in a Hilbert space with N dimensions.

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z-axis direction. However, the set of eigenstates of spin-up and -down along x-axis direction are given by |+xi = 1 √ 2(|+zi + |−zi) , (2.6a) |−xi = 1 √ 2(|+zi − |−zi) , (2.6b) and give another valid choice (the coecient _√1

2 comes from the normalization). The eigenstates

are orthogonal to each other, which means that the inner product between two eigenstates is given by

hei|eji = δij , (2.7)

where δij is the Kronecker delta.

iv. Probability Interpretation and Projection Operator

Given a set of basis kets {|eii , i = 1, 2, 3, ..., N }, an arbitrary state can be expressed as a

superposition of basis kets:

|ψi =

N

X

i=1

ci|eii , (2.8)

where the coecients ci can be found by

ci = hei|ψi . (2.9)

The modulus squared of the coecient ci is interpreted as the probability pi of a system

col-lapsing from state |ψi to state |eiiafter measurement, i.e.

pi = |ci|2 = c∗ici = |hei|ψi|2 = hψ|eii hei|ψi . (2.10)

For convenience, it is useful to interpret |eii hei| in the last line of eq.(2.10) as the projection

operator ˆPi, which projects an arbitrary state |ψi to the selected state |eii. Then the eq.(2.10)

can be rewritten as

pi = hψ| ˆPi|ψi , (2.11)

whereby it is interpreted as the expectation value of ˆPi in state |eii (see vi. below). The

pro-jection operator is the foundation for the density matrix formalism. v. Completeness and Identity Operator

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be expressed as a superposition of basis kets like eq.(2.8). Hence, due to the completeness of the set of basis kets, the sum of the probabilities of an arbitrary state collapsing into |eiiwhere

i runs from 1, 2, ... to N is equal to 1, i.e.

Hence, from the last equality, it follows that

N X i=1 ˆ Pi = N X i=1

|eii hei| = ˆI (the identity operator). (2.14)

Also,

ˆ

PiPˆj = δijPˆi (2.15)

for i, j = 1, 2, · · · , N.

vi. Measurement on Operator and Expectation Value

In quantum mechanics, a measurement for a system described by the state vector |ψi is done by an observable operator acting on |ψi. If |ψi = |ai is an eigenstate of ˆA with eigenvalues a, i.e.

ˆ

A |ai = a |ai , (2.16) where ˆA is the operator for the observable A, a is the result of the measurement of A.

In general, an arbitrary state |ψi of a system is not an eigenstate of ˆA(one of the |aii) but it can

be expanded as a superposition of the eigenstates by the completeness of the set of eigenstates: |ψi =

N

X

i=1

ci|aii . (2.17)

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The system will collapse to the state |aii and give the measurement result ai with probability

pi = |ci|2. Assuming we have a huge amount of copies of a system, each of them can be measured

and will give a result according to the probability distribution. The expectation value hAi of the measurement is given by

hAi = N X i=1 piai = N X i=1 |ci|2ai = N X i=1 hψ|aii hai|ψi ai = N X i=1

hψ|aii hai| ˆA|ψi = hψ| ˆA|ψi . (2.19)

It should be noted that here there is no restriction on |ψi and ˆA. Any operator ˆA can be rewritten as

ˆ A = N X i,j=1 aij|eii hej| , (2.20)

where {|eii} is a basis, but not necessarily the basis of eigenstates of ˆA, and where

aij ≡ hei| ˆA|eji . (2.21)

If the basis set is chosen as the eigenstates of ˆA, {|aii}, the operator can be simply written as

ˆ A = N X i=1 ai|aii hai| . (2.22)

vii. Matrix Representation

Given a complete set of orthonormal basis kets {|eii}, an ordinary eigenvalue problem in

quan-tum mechanics can be rewritten as follows: ˆ

A |ψi = a |ψi , (2.23) where ˆA can be replaced by eq.(2.20), we obtain:

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Thus nally we have

N

X

j=1

aijcj = aci , (2.28)

where i runs from 1 to N. Obviously, these N equations can be expressed in matrix from. Then the calculation is more concrete and the language of linear algebra can be used directly. 2.1.2 Density Operator Representation

i. Density Operator

Not every quantum system can be described by a state vector in Hilbert space, especially this is not possible for quantum statistical systems. In such cases, a density operator or density matrix is introduced to tackle this problem; it is dened as follows:

ˆ ρ = N X i=1 pi|φii hφi| (2.29)

This describes an ensemble which is composed of systems |φii with probability pi. Note that

by denition the states of the subsystems are not necessarily orthonormal to each other. Direct from the denition, it is easy to obtain some important properties of a density operator; rst, T r[ ˆρ] = N X i=1 pi = 1 , (2.30)

and second, 0 ≤ pi ≤ 1 implies that

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where ψ is an arbitrary state vector. The trace property is a consequence of the completeness. The second property implements the fact the the pi are probabilities; it gives the positivity

condition of the density matrix. Conversely, if an operator ˆDsatises the above two properties, then ˆD can represent the density operator of some system.

ii. Basis Vectors

Once a set of basis vectors {|aii}is chosen, the density operator ˆρ can be represented in matrix

The set of basis vectors is still not necessarily orthonormal, but of course the measurable results must be consistent among all choices of basis vectors set. The properties will be more obvious if an orthonormal set, or even the set of eigenvectors of the Hamiltonian which can diagonalize the matrices, is chosen.

Moreover, it is important to note that due to the positivity condition of ˆρ, there always exists a spectral decomposition in which an orthonormal set {|ji} can diagonalize the density matrix, i.e. ˆ ρ = N X j=1 λj|ji hj| , (2.33)

where the coecients λj are real, non-negative, eigenvalues of the density matrix. Because the

trace of matrices related by similarity transformation is unchanged, the diagonaliztion from spectral decomposition reduces the complexity of derivation of theorems and calculations. iii. Measurement, Operator and Expectation value

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Here in the second line the completeness of the basis set is used. h ˆAφii ≡ hφi| ˆA|φii is the

expectation value of Â in state φi, and h Âρî is the expectation value of Â in system given by

the density matrix ˆρ.

iv. Pure State versus Mixed State

Considering the denition of density operator, it is similar to the projection operator ˆPi = |ii hi|.

Recalling the eq.(2.15), it is interesting to compute the square of the operator ˆρ: ρ2 =

N

X

i,j=1

pipj|φii hφi|φji hφj| . (2.39)

Now consider the trace of ˆρ2:

T r[ ˆρ2] = N X m,n,i,j=1 pipjham|φii hφi|ani han|φji hφj|ami (2.40) = N X i,j=1 pipjhφj|φii hφi|φji (2.41) = N X i,j=1 pipj|hφi|φji|2 (2.42) ≤ 1 (and ≥ 0). (2.43) Moreover, the equality T r[ˆρ2_{] = 1} _{implies that only one p}

i is non-zero and equal to one. In

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state vector |Ψi. Otherwise, T r[ˆρ2_{] < 1} _{implies that the system must be the sum of projection}

operators for at least two states, regardless of what basis set is chosen: ρ = PN ≥2

i pi|ψii hψi|,

then the system is in a mixed state. iv. Example

Supposing a system contains one electron, which is spinning along x-axis positively with prob-ability 3/4 and spinning negatively with probprob-ability 1/4. Hence, the density operator is given by ρx = 3 4|+xi h+x| + 1 4|−xi h−x| = 3/4 0 0 1/4 ! . (2.44) Let us check that the basic properties of a density matrix are satised:

T r[ρx] = 3 4+ 1 4 = 1 , (2.45) T r[ρ2_x] = 3 4 2 + 1 4 2 = 5 8 < 1 . (2.46) The second line shows that the system is in mixed state. If the density operator is expressed in the basis vectors of {|+zi , |−zi} (spinning along z-axis), it reads

ρz = 3 4 1/√2 1/√2 ! 1/√2 1/√2 + 1 4 1/√2 −1/√2 ! 1/√2 −1/√2 = 1/2 1/4 1/4 1/2 ! . (2.47) It is not dicult to check that the trace of ρz and ρ2z are same as above.

We know that the spin operator along the z-axis expressed in terms of the basis vectors of {|+zi , |−zi} is ˆ Sz = 1 0 0 −1 ! , (2.48)

so the expectation value of Sz in this example is

hSzi = T r[ˆρzSˆz] = T r " 1/2 1/4 1/4 1/2 ! 1 0 0 −1 !# = T r 1/2 −1/4 1/4 −1/2 ! = 0 , (2.49) which is expected because the composition states are spinning along the x-direction.

2.2 Multi-particle System

2.2.1 Description of Composite Systems

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on), then the state of each constituent system |ψii is an element of the Hilbert space of the

corresponding constituent system Hi where i = A, B, C, · · ·. The Hilbert space describing the

whole system is equal to the tensor product of the Hilbert space of all composite systems, i.e. H = HA⊗ HB⊗ HC⊗ · · · . (2.50)

The state vector describing the whole system is an element of the whole Hilbert space,

|Ψi ∈ H (2.51) and generally related to the state vectors of the constituent systems as

|Ψi = NA X i=1 NB X j=1 NC X k=1 · · · αijk···|ψA,ii ⊗ |ψB,ji ⊗ |ψC,ki ⊗ · · · (2.52) = NA X i=1 NB X j=1 NC X k=1

· · · αijk···|i, j, k, · · ·i , (2.53)

where Ni is the dimension of Hi, and the coecients satisfy the normalization condition:

X

i,j,k,···

|αijk···|2 = 1 . (2.54)

For example, the state vector |Ψi of a system composed of one spin-1/2 particle {|↑i , |↓i} and one spin-1 particle {|−1i , |0i , |+1i} is generally written in the form

(2.55) |Ψi = α↑,−1|↑, −1i + α↑,0|↑, 0i + α↑,+1|↑, +1i

+ α↓,−1|↓, −1i + α↓,0|↓, 0i + α↓,+1|↓, +1i .

If there are two composite systems in a whole system, by Schmidt decomposition (see e.g. page 65 in [3]) there always exists an orthonormal basis |imi of A and an orthonormal basis |jmi of

B such that |Ψi = D X m=1 λm|im, jmi with D ≤ min (NA, NB) . (2.56) 2.2.2 Entanglement

i. Separability versus Entanglement in Pure State

Consider the special case that the state vector of the whole system |Ψi can be written as the tensor product of the state vectors of the constituent systems:

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A state vector |Ψi with this property is called separable. If a measurement ˆA is applied only on the constituent system A, the state of system A will collapse to one of the eigenstates of the measurement |ai with eigenvalue a, but the states of other composite systems remain unchanged:

( ˆ_{A ⊗ Î}B⊗ ÎC ⊗ · · ·) |Ψi = ( Â ⊗ ÎB⊗ ÎC⊗ · · ·)(|ψAi ⊗ |ψBi ⊗ |ψCi ⊗ · · ·) (2.58)

= ( Â |ψAi) ⊗ (ÎB|ψBi) ⊗ (ÎC|ψCi) ⊗ · · · (2.59)

= (a |ai) ⊗ |ψBi ⊗ |ψCi ⊗ · · · . (2.60)

Thus there is essentially no communication between the constituent systems.

Otherwise, if the state |Ψi cannot be separated, then it is called entangled. A measurement on one composite system will then not only aect the constituent system, but also the others. For example, if the state of a 2 × 3 system (as given in eq.(2.55)) is

|Ψi = √1

2|↑, −1i + 1 √

2|↓, +1i , (2.61) and a projection operator ˆPA,↑ = |↑i h↑| is applied on system A, then one gets

( ˆPA,↑⊗ ˆIB) |Ψi =

|↑i h↑| ⊗ ˆIB) 1 √ 2|↑, −1i + 1 √ 2|↓, +1i (2.62) = √1 2|↑, −1i . (2.63) This means the even though no measurement is done on system B, its state is aected, too. This amounts to saying that the two constituent systems are in communication. This is an important idea and it has lead to the development of quantum computation and teleportation ([7], [12], [15], [19] and [24]).

ii. Separable versus Entanglement in Mixed State

The above concept can be extended to mixed states in similar form. If the density operator of the whole system ˆρ can be written as the tensor product of density operators of all constituent systems ˆρi in the form:

ˆ

ρ = ˆρA⊗ ˆρB⊗ ˆρC ⊗ · · · , (2.64)

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pm, i.e.

ˆ ρ =X

m

pmρˆA,m⊗ ˆρB,m⊗ ˆρC,m⊗ · · · , (2.65)

then it is more general and is called entangled. iii. Von Neumann Entropy

As mentioned, the entanglement between subsystems is essentially a communication between systems. It is then of interest to quantify how good the communication is, i.e. how much infor-mation can be exchanged under favourable conditions. In classical contexts, the Gibbs entropy in thermal and statistical physics and the Shannon entropy in information theory achieve this. In quantum mechanics an extension of this concept - von Neumann entropy, or entanglement entropy, obtain this objective.

For a single system described by the density matrix ˆρ, the von Neumann entropy is dened by S = −T r[ ˆρ ln( ˆρ)] = −

N

X

i=1

λiln(λi), (2.66)

where λi is the eigenvalues of the density matrix ˆρ. The last equality is similar to Shannon

entropy. Some properties of S are stated below: (i) Growth and pure state:

S(ρ) ≥ 0 (2.67) where equality holds for pure states ρ;

(ii) Maximality: S(ρ) is maximal and equal to (ln N) where N is the dimension of the Hilbert space of ρ if ρ is a maximally mixed state ˆI/N;

(iii) Concavity: S k X i=1 αiρi ! ≥ k X i=1 αiS(ρi) with k X i=1 αi = 1 ; (2.68) (iv) Subadditivity: S(ρAB) ≤ S(ρA) + S(ρB) (2.69)

where the reduced entropy SX(ρ)counts the trace for system X only. The equality is satised

if A and B are independent systems, i.e. ρAB = ρA⊗ ρB;

(v) Composite system in pure state: For a composite system AB in pure state, S(ρA) = S(ρB);

(vi) Strong subadditivity:

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for any composite system ABC.

Note that for the case of a composite system ρijk···, ρi is the reduced density operator for system

i, which is obtained by taking the trace over all other systems (e.g. i = A): ρA=TrBC···[ρABC···] =

X

j,k,···

(hψB,j| ⊗ hψC,k| ⊗ · · ·) ρABC···(|ψB,ji ⊗ |ψC,ki ⊗ · · ·) . (2.71)

A typical example of entanglement is the Bell state in the two-electron problem, given by |Ψi = √1 2(|↑↑i + |↓↓i) (2.72) or ˆ ρ = 1 2      1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1      . (2.73)

It is a pure state for the whole composite system AB so that S(ρAB) = 0, but it is maximally

entangled between the systems A and B that are related by S(ρA) = S(ρB). The reduced

density operator for A is

ρA =TrB[ρAB] (2.74)

=TrB[

1

2(|↑↑i h↑↑| + |↓↓i h↑↑| + |↑↑i h↓↓| + |↓↓i h↓↓|)] (2.75) = h↑B|

1

2(|↑↑i h↑↑| + |↓↓i h↑↑| + |↑↑i h↓↓| + |↓↓i h↓↓|) |↑Bi (2.76) + h↓B|

1

2(|↑↑i h↑↑| + |↓↓i h↑↑| + |↑↑i h↓↓| + |↓↓i h↓↓|) |↓Bi = 1 2|↑Ai h↑A| + 1 2|↓Ai h↓A| (2.77) = 1 2 1 0 0 1 ! . (2.78)

The von Neumann entropy for the subsystem A is SA= − 1 2 ln 1 2 − 1 2 ln 1 2 = ln (2) . (2.79) By symmetry between the systems A and B, ρB = ρA and SA = SB. The value ln(2) proves

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Now that we have introduced the formalism and concepts of quantum entanglement, we are nally in the position to discuss the system of our main interest; in the following sections, entanglement in SQUIDs-cavity systems will be reviewed and illustrated by examples, mainly based on the article [26].

3 Principles of Entanglement in SQUIDs-cavity System

In [26], Yang et al. proposed how Λ-type energy levels - a concept that will be explained below - achieved by using two superconducting quantum interference devices (SQUIDs) in a cavity eld, or a microwave pulse, could be used to construct three dierent gates which are important in computation: the controlled phase-shift gate, the controlled-NOT gate, and the SWAP gate. Moreover, the entanglement of two non-interacting SQUIDs in a cavity is able to perform a transfer of information by using the operations of gates. In this section, the Hamiltonian for a SQUID-cavity system employed in [26] is studied in detail, and then some supplementary material about gates is provided.

3.1 Hamiltonian of the System

Section II of [26] describes what happens when a SQUID is placed in a cavity with a magnetic eld. There are three components in the Hamiltonian of the whole system: the SQUID energy, the cavity eld energy, and the interaction energy between the cavity eld and the SQUID. 3.1.1 Superconducting Quantum Interference Device (SQUID)

A one-junction superconducting quantum interference device, also called a radio frequency (rf) SQUID, is essentially a superconducting ring of inductance L with a Josephson junction (for a schematic view, see Figure 1a). The junction is characterized by three parameters: the critical current Ic, the shunt capacitance C, and the shunt resistance R (Figure 1b). The cross mark

labels the Josephson junction.

In general, the current owing through the junction is given by

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Figure 1: (a) A schematic view of a rf SQUID, it is a superconducting ring of inductance L interrupted by a Josephson junction marked as a cross. (b) A schematic view of the Josephson junction, which is characterized by the critical current Ic, the shunt capacitance C, and the shunt resistance R. Adopted

from Figure 1 on page 61 of [5].

where g(θ) is a periodic function of the gauge-invariant phase dierence θ of the quantum mechanical wave function across the junction (Figure 2): g(θ +2π) = g(θ). Below for simplicity we will assume this form of g(θ):

g(θ) = sin θ . (3.2) From the basis laws of electrodynamics it follows that the equation of motion of the rf SQUID is given by (eq.(1) on page 61 of [5])

C ¨Φ + 1 R

˙

Φ = −∂U

∂Φ . (3.3) Here Φ is the total magnetic ux threading through the SQUID ring, including the ux due to the externally applied magnetic eld and the magnetic eld induced by the supercurrent in the ring:

Φ = Φext+ Φind . (3.4)

U is the potential energy of the SQUID. When a current I ows through a loop which carries a back electromotive force Eback = −L ˙I, and an amount of charge dQ = Idt passes through

during a small time dt, then a work dW is done which is given by (eq.(7.29) of [8])

dW = −EbackdQ = L ˙IIdt = LIdI (3.5)

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Figure 2: The electrons on the two sides of the junction are quantum mechanically described by a wave function. The electron probability density |ψ|2 _{is continuous across the junction, but the wave}

function ψ is not. The gauge-invariant phase dierence θ describes the phase shift of the wave function across the junction.

of the current is counted) Φ = LI. This results in three alternative expressions:

dW = LIdI (3.6a) = 1

LΦdΦ (3.6b) = ΦdI (3.6c) = IdΦ . (3.6d) The magnetic energy stored in the superconducting ring (inductor) can be obtained by eq.(3.6b):

UL = Z dW = Z Φind 0 1 LΦ 0 dΦ0 = Φ 2 ind 2L = (Φ − Φext)2 2L . (3.7) Note that UL only counts the eect of the supercurrent, but not the external magnetic eld.

However, the potential energy of the junction is obtained by eq.(3.6d) due to the uxoid quan-tization condition (a relation between the phase dierence θ and total magnetic ux threading the ring Φ; below eq.(1) on page 61 of [5]):

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is called the ux quantum. The integer n = 0, ±1, ±2, · · · can be omitted by the periodicity of the function g(θ), i.e. we can without loss of generality assume that

θ = 2π Φ Φ0

. (3.10)

The potential energy of the junction, called the Josephson coupling energy, is thus given by Ujunc = Z Φ 0 Isd Φ0 (3.11) = Z Φ 0 Icsin θ d Φ0 (3.12) = Z Φ 0 Icsin 2πΦ 0 Φ0 d Φ0 (3.13) = IcΦ0 2π 1 − cos 2π Φ Φ0 . (3.14) The rst term is a constant and can therefore be omitted. Hence, the total potential or magnetic energy of the SQUID is given by:

U (Φ) = UL+ Ujunc = (Φ − Φext)2 2L − EJcos 2π Φ Φ0 , (3.15) where EJ = IcΦ0 2π . (3.16) This form of the total potential energy of the system gives us the explanation for how the Λ-type energy levels are formed. In the U-Φ graph (Figure 3), it is a combination of global quadratic curve and rapid oscillation. Φext does nothing but adjusting the position of Φ at the

minimum of the quadratic trend. L and EJ (only Iccan be tuned) control the curvature of the

quadratic trend and the amplitude of the oscillation, respectively. In case of large L or large EJ, the oscillation dominates and therefore a W -type well appears. This results in the following

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Figure 3: This diagram shows the graph of the function y = (x − 0.5)2_{+ 0.5 cos (2πx)}_{. The variables}

y and x play the roles of U and Φ, respectively. The graph is a superposition of a quadratic curve and an oscillation. By adjusting the coecients, the curve looks like a W -type well with the property that the ground state |0i is in one well, the rst excited state |1i is in the other well, and the second excited state |2i is above the barrier between the two wells. The transitions between |2i and |1i or |0i are allowed, but the direct transition between |0i and |1i is forbidden due to the barrier. The transition lines look like the Greek letter "Λ", so this kind of energy levels is called Λ-type energy levels.

Comparing the equation of motion eq.(3.3) of the rf SQUID to the one for a classical mechanical damping system, i.e. to

m¨x + b ˙x = −∂U

∂x , (3.17) one sees that C, 1/R and Φ play the roles of mass m, damping coecient b and coordinate x, respectively. Therefore it is expected that there is a corresponding "kinetic" term in the total Hamiltonian of the SQUID, whose form is similar to that in mechanical system, i.e. to

p2

2m, where p is the momentum, which is the canonical conjugate variable of x. By considering the classical electromagnetic theory (LC circuit) in Lagrangian and Hamiltonian formulation and comparing it with the theory of mechanics (see appendix A), one see that Φ and Q, the total charge on the capacitor, are conjugate variables, analogously as position and momentum in mechanics; i.e. in a quantum mechanical description we have to impose the commutation relation

[ ˆΦ, ˆQ] = i¯h . (3.18) Hence the "kinetic" term for SQUID is Q2

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are indeed the magnetic energy of the superconducting ring and of the Josephson junction. Hence, the Hamiltonian of the SQUID ring is

ˆ Hs= ˆ Q2 2C + ( ˆΦ − Φext)2 2L − EJcos(2π ˆ Φ Φ0 ). (3.19) The rst term can be interpreted as the electric energy stored in the capacitor in the SQUID. 3.1.2 Cavity Field

The Hamiltonian of the electromagnetic eld can be fully understood only with the help of quantum electrodynamics (QED). Here we are dealing with a eld in a cavity, so cavity QED is to be employed. Cavity QED is simpler than the general situation because of the presence of a boundary condition. In free space, the frequency (or momentum) of the electromagnetic eld has a continuous value. In contrast, in a closed cavity only certain discrete frequencies of the electromagnetic eld can exist, analogically to e.g. the vibration modes of a two-end-xed string. The similarity between the string and the electromagnetic eld is important for understanding the quantized cavity electromagnetic eld. Performing canonical quantization to the Hamiltonian of the vibrating string or solid, the Hamlitonian can be transformed to a form similar to the Hamiltonian of a collection of uncoupled simple harmonic oscillators (SHO). The quanta in the vibration string or solid are called phonons, and are characterized by their normal frequency and direction.

Now, performing a similar canonical quantization of the cavity electromagnetic eld, the canon-ical relation is no longer between the coordinates and the momentum, but between the compo-nents of the electric eld and the magnetic eld in mode space (eqs.(2.94)-(2.100) of [4]). This is similar to the relation (3.18) between the magnetic ux and the electric charge. By dening ladder operators from the components of the electric eld and the magnetic eld in mode space (eqs.(2.106)-(2.109) of [4]), the Hamiltonian of the cavity eld can be written in the SHO-like form (eqs.(2.101), (2.102), (2.112) and (2.113) of [4])

ˆ Hcavity = X i ¯ hωi ˆ a†_iˆai+ 1 2 , (3.20) where the summation is over all normal modes. From now on, only a single normal mode ωi of

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where ˆa†

i and ˆai are the creation and annihilation operators of the normal mode ωi of the eld,

respectively. Replacing the components of the electric eld and the magnetic eld in mode space (eqs.(2.94) and (2.95) of [4]) by ladder operators (eqs.(2.106) and (2.107) of [4]), the magnetic eld can be rewritten as (eq.(6) of [26] explicitly, or rewriting eq.(2.111) of [4])

ˆ Bi(r, t) = s ¯ hωi 2µ0 h ˆ ai(t) + ˆa † i(t) i Bi(r), (3.22)

where Bi(r)is the component of the magnetic eld of normal mode ωi; the separation of r and

t is a direct result of the fact that we consider a closed cavity: we deal with a standing wave. A detailed mathematical derivation of the expression (3.22) is provided in appendix B.

3.1.3 Interaction Energy

Once the SQUID with its supercurrent is placed inside the cavity eld, a coupling energy arises through the interaction between the magnetic ux threading the ring produced by the supercurrent (Φ − Φext) and the ux produced by the cavity eld, which is given by

ˆ Φc= Z S ˆ B(r, t) · dS . (3.23) In eq.(3.23), the label i for the mode is omitted, and S is a surface bound by the SQUID ring. The magnetic ux is dened as in classical electrodynamics, but in the quantum me-chanical description the variables are replaced by their corresponding operators. The coupling Hamiltonian is then given by

ˆ

HI = λc ˆΦ − Φext ˆΦc, (3.24)

where the coupling parameter is λc = −1/L. The minus sign indicates that the coupling

energy is negative when the cavity magnetic eld and the one generated by the supercurrent threading the ring are parallel to each other. This term is similar to the coupling energy of spins. Altogether, combining the results for the SQUID, the cavity and the interaction we obtain, the total Hamiltonian of the whole system as

ˆ

H = ˆHs+ ˆHc,i+ ˆHI (3.25)

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3.2 Quantum Gates

The idea of quantum computation mainly involves four levels (page 65 of [16]). The basis is the qubit (state of system), which provides the storing ability in the same way as the classical bits - 0 and 1. The qubits can be changed through operations called quantum gates. Having dierent quantum gates, a quantum circuit can be built. A collection of quantum circuits can execute a quantum algorithm, which is a part of quantum computation.

Only the rst two levels are discussed in [26]. Section II of [26] focuses on the construction of qubits, i.e. the Hamiltonian. Sections II to III focus on how to cause the state transitions by using microwaves pulse. In section IV, two uncoupled SQUIDs in a cavity form a two-qubit system, while the physical process from section III is employed in specic procedures to achieve three individual gate operations: the controlled phase-shift gate, the controlled-NOT gate and the SWAP gate.

In order to achieve the operation, one can arrange that for each SQUID the coupling induced by cavity modes between the three lowest levels |0i, |1i, and |2i and any other levels is negligible. Assuming that the cavity eld is initially in the vacuum state, one can then describe the situation by an eective Hamiltonian H0 _{for the two SQUIDs which is given by eq.(25) in [26].}

The entanglement between the two SQUID qubits is generated via the ARA process that is shown in Figure 4; the middle step involves a microwave pulse. The details of the ARA process is given in section VI of [26].

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In the following I will give the denitions of those three gates and a brief review of how they are achieved.

3.2.1 Controlled Phase-shift Gate

The controlled phase-shift gate operates on a single-qubit state, on which it acts as |0i → |0i (unchanged) but |1i → eiθ_|1i_{with a phase shift θ ∈ [0, 2π]. In matrix form with respect to the}

standard basis {|0i , |1i}, this means:

UCP S,θ =

1 0 0 eiθ

!

. (3.26) The controlled phase-shift gate is obtained by a 3-step process of type ARA-H0_{-ARA; in more}

detail: in the two-SQUID qubit system, one qubit a acts as the control, the other qubit b as the target.

- Step 1: ARA process resulting in the transformation |1ia→ −i |2ia

- Step 2: time evolution according to the Hamiltonian H0_{, for an adapted time period}

- Step 3: ARA process resulting in the transformation |2ia→ i |1ia

In terms of basis states |0i and |1i for both a and b, these steps amount to

|0i_a|0i_b → |0i_a|0i_b → |0i_a|0i_b → |1i_a|1i_b (3.27) |0i_a|1i_b → |0i_a|1i_b → |0i_a|1i_b → |0i_a|1i_b (3.28) |1i_a|0i_b → − i |2i_a|0i_b → − i |2i_a|0i_b → |1i_a|0i_b (3.29) |1i_a|1i_b → − i |2i_a|1i_b → i |2i_a|1i_b → − |1i_a|1i_b , (3.30) showing that one has realized a controlled phase-shift gate for two SQUID-qubit system. The phase-shift, e.g. −1 in the last line, can be adjusted by the time length of the microwave pulse during the ARA process.

3.2.2 Controlled-NOT Gate

The controlled-NOT, or CNOT, gate is a two-qubit (|a, bi) gate. It can be achieved by com-bining controlled phase-shift gate with two ARA processes that amount to single-SQUID qubit rotatios of the individual SQUID qubits. During the operation, if the control qubit a is |0ia,

then the target qubit b remains unchanged. However, if a is |1ia, then the state of b will be

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{|0, 0i , |0, 1i , |1, 0i , |1, 1i}: UCN OT =      1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0      . (3.31)

This clearly satises U2

CN OT = I, and UCN OT is thereby unitary. A more general possibility is

to replace the ipped part by an arbitrary unitary matrix u, so that

UCN OT,gen =      1 0 0 0 0 1 0 0 0 0 u11 u12 0 0 u21 u22      , (3.32) with a 2 × 2-matrix u = u11 u12 u21 u22 ! (3.33) that satises u†= u−1 . (3.34) 3.2.3 SWAP Gate

The SWAP gate does what its name indicates: swapping the states of two qubits with each other. Briey,

ˆ

USW AP|α, βi = |β, αi . (3.35)

In matrix form with basis {|0, 0i , |0, 1i , |1, 0i , |1, 1i}:

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In principle, the SWAP gate can be obtained by combining three CNOT gates, but simpler by a 5-step process of type: ARA-H0_-ARA-H0_{-ARA. This amounts to}

|0i_a|0i_b → |0i_a|0i_b → |0i_a|0i_b → |0i_a|0i_b → |0i_a|0i_b → |0i_a|0i_b (3.37) |0i_a|1i_b → − i |0i_a|2i_b → i |2i_a|0i_b → − i |2i_a|0i_b → − i |2i_a|0i_b → |1i_a|0i_b

(3.38) |1i_a|0i_b → − i |2i_a|0i_b → i |0i_a|2i_b → |0i_a|1i_b → |0i_a|1i_b → |0i_a|1i_b

(3.39) |1i_a|1i_b → − |2i_a|2i_b → |2i_a|2i_b → i |2i_a|1i_b → − i |2i_a|1i_b → |1i_a|1i_b .

(3.40) In prinicple, any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations ([2], [18] and [20]), similar to the roles of resistor, capacitor and inductor in classical circuits. Hence, the use of ARA processes with the constructions of those three gates above is comprehensively studied in [26].

3.3 Recent Developments

Section 2 of this thesis provides many mathematical details, formalisms and they properties for the dierent quantum mechanical systems, which is useful for understanding the details of the cited paper, especially the recent developments below.

3.3.1 SQUIDs-cavity System

After Yang et al. in 2003 [26] proposed the possibility of achieving quantum entanglement and quantum gates by using SQUIDs-cavity system, Yang et al. also discussed further aspects of this system in 2004 [27]. They showed that quantum information transfer and entanglement with two-SQUID qubits has a high delity, and also that the coupling between SQUID and cavity is strong when the cavity electromagnetic wave is in the microwave range.

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e.g. in atoms and trapped ions in cavities.

Zhu et al. proposed another scheme for quantum information processing, with multi-SQUID qubits coupled through a cavity mode, in which quantum manipulations are insensitive to the state of the cavity. Also high-delity quantum information processing is possible because of the gate dependence on just global geometric features and the insensitivity to the state of the cavity modes.

3.3.2 Quantum Circuits

As mentioned, quantum computing involves at least ve main levels: qubits, gates, circuits, algorithms and computation. Yang et al. [26] only proposed the constructions of quantum qubits by SQUIDs-cavity system and three individual quantum gates, but without discussing that how a quantum circuit can be constructed by the proposed gates and other gates.

In the following years, many studies explored the potential of applying superconducting systems to quantum information processing. Liu et al. [14] analyzed the optical selection rules of the microwave-assisted transitions in a ux qubit superconducting quantum circuit. Blais et al. [1] have studied more types of two-qubit gates and the selection rules at the charge degeneracy point.

Most recently, Xiang et al. [25] gave an overview for many hybrid quantum circuits which com-bine elements from atomic physics, quantum optics, condensed matter physics and nanoscience in order to merge the best features of these constituents, such as long coherence times, fast op-erations, and scalability. Not only superconducting qubits but also other elements e.g. atoms, the spins, cavities and resonators are included. The quantum circuits combine gates made of various dierent kinds of qubits.

4 Conclusions

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on Yang et al.'s paper [26].

I have also described the basic properties of Hamiltonians of a SQUID, cavity, and their cou-pling in detail. The SQUID is composed of a superconducting ring and a Josephson junction, so that the structure of Λ-type energy levels is formed. The cavity can be understood only in the context of quantum electrodynamics; it plays the role of a medium of state transitions among the Λ-type energy levels through the coupling between the SQUIDs and the cavity elds. By using microwave pulses to excite the SQUIDs in cavity in specic ways, an entangled system and quantum gates can be constructed. Three individual quantum gates (the controlled phase-shift gate, the controlled-NOT gate, and the SWAP gate) are illustrated.

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Appendices

A Canonical Relation between Flux ˆΦ and Charge ˆ

Q

A.1 Canonically Conjugate Variables

The concept of canonical or conjugate variables arises in Lagrangian and Hamiltonian mechan-ics. Consider the Lagrangian of an N-dimensional system:

L(qj; ˙qj; t) = L(q1, q2, · · · , qN; ˙q1, ˙q2, · · · , ˙qN; t) = T − V , (A.1)

where T and V are the kinetic energy and the potential energy of the system, respectively; qi

and ˙qi =

dqi

dt are the i-th generalized coordinate and generalized velocity, respectively. The equations of motion in Lagrangian mechanics are

d dt ∂L ∂ ˙qi − ∂L ∂qi = 0 (i = 1, 2, · · · , N). (A.2) For each generalized velocity, there is one corresponding conjugate momentum dened as

pi =

∂L ∂ ˙qi

. (A.3)

The independent variable set (qj; ˙qj; t)changes to (qj; pj; t)by applying a Legendre

transforma-tion to the Lagrangian:

H(qj; pj; t) = N

X

i=1

piq˙i− L(qj; ˙qj; t) . (A.4)

The new quantity H(qj; pj; t) is called the Hamiltonian, which for systems with conservative

forces also is the total energy of the system. The generalized coordinate qi and its canonical

conjugate momentum pi satisfy the canonical equations of Hamilton, which are the equations

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This concept can be extended to the statement that any two variables A and B are canonically conjugate if they satisfy

˙ B = −∂H ∂A , (A.6a) ˙ A = ∂H ∂B (A.6b)

The details are described e.g. in the chapter 8 of [6]. It follows in particular that the corre-sponding operators in a quantum mechanical description satisfy the commutation relation

[ ˆA, ˆB] = i¯h . (A.7)

A.2 Relation between Φ and Q

To see how the magnetic ux Φ and charge Q are canonically conjugate, it is convenient to consider a circuit composed of an inductor of inductance L and a capacitor of capacitance C, i.e. a LC circuit (page 4 and 5 of [21] and section VI of [23]). Assuming that there is a potential dierence VC across the capacitor (and thus also across the inductor), there are a charge Q and

electric energy EC is stored in the capacitor:

Q = CVC , (A.8) and EC = 1 2CV 2 C = Q2 2C . (A.9) If an external voltage is applied, there is then a current I owing through the circuit (and the inductor), and hence a magnetic ux Φ threading the inductor and magnetic energy EL stored

in the inductor: VL= −VC = −L ˙I , (A.10) Φ = LI , (A.11) and EL= 1 2LI 2 ₌ Φ2 2L . (A.12) The Hamiltonian, or total energy, is given by

H = EC + EL =

Q2

2C + Φ2

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It follows that −∂H ∂Φ = − Φ L = −I = ˙Q , (A.14) and ∂H ∂Q = Q C = VC = L ˙I = ˙Φ , (A.15) so Φ and Q are canonically conjugate variables. Their commutation relation in quantum theory can be directly written down:

[ ˆΦ, ˆQ] = i¯h . (A.16)

B Field Quantization

B.1 Expressions for the Classical Electromagnetic Field

Consider the electric eld E and the magnetic eld B in a cavity. Because there are no charge or current sources, Maxwell's equations governing the electromagnetic elds are the same as in vacuum: ∇ · E = 0 , (B.1a) ∇ · B = 0 , (B.1b) ∇ × E = −∂B ∂t , (B.1c) and ∇ × B = 1 c2 ∂E ∂t , (B.1d) where c is the speed of light in vacuum. The elds can be expressed in terms of a scalar potential V and a vector potential A, as

E = −∇V −∂A

∂t , (B.2) and

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eq.(B.1) are clearly satised. However, the fourth equation gives a wave equation for A: ∇ × B = 1 c2 ∂E ∂t , (B.4) ∇ × (∇ × A) = 1 c2 ∂ ∂t −∇V − ∂A ∂t , (B.5) ∇ (∇ · A) − ∇2_{A = −}1 c2 ∂2_A ∂t2 , (B.6) ∇2_{A =} 1 c2 ∂2A ∂t2 (wave equation). (B.7)

The general solution of this wave equation gives A as (eqs.(2.5) and (2.6) of [22]) A (r, t) =X

n,λ

αn,λun,λ(r) exp (−iωnt) + c.c. (B.8)

where c.c. stands for complex conjugate. The separation of r and t is a feature of a standing wave in a cavity. The allowed frequencies of an electromagnetic wave are discrete in a cavity due to the boundary conditions of the cavity, so there is a summation over discrete values of n rather than an integration over a continuum. The Fourier coecients αn,λ are constant for a

free eld. The set of vector mode functions un,λ(r) satises the following equations:

∇2₊ωn2 c2 un,λ(r) = 0 (B.9) and ∇ · un,λ(r) = 0 . (B.10)

For a rectangular cavity of volume Lx× Ly× Lz, the modes are

un,λ = ˆ e(λ)n pLxLyLz exp (ikn· r) , (B.11) where ˆe(λ)

n is the unit polarization vector. Because an electromagnetic wave is a transverse wave,

ˆ

e(λ)n is perpendicular to the propagation wave vector kn, so there are only two possibilities for

λ, say 1 and 2, in a 3-dimensional cavity, and ˆe(1) _{⊥ ˆ}_e(2)_{, i.e. the three vectors ˆe}(λ)

n and k are

mutually perpendicular. The propagation wave vector

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determines the mode of the electromagnetic wave. According to the denitions, the explicit forms of E and B are obtained:

E = −∂A ∂t =

X

n,λ

iωnαn,λexp (−iωnt) un,λ(r) + c.c. (B.14)

and B = ∇ × A =X n,λ αn,λknexp (−iωnt) ∇ × un,λ(r) kn + c.c. . (B.15) The factor kn in the denominator in B is needed in order to keep the dimension of the whole

bracket to be the same as that of un,λ(r). Using the identity

∇ × [f (r)v(r)] = f (r) [∇ × v(r)] + [∇f (r)] × v(r) , (B.16) the bracket in B can be simplied:

∇ × un,λ(r) = ∇ × " ˆ e(λ)n pLxLyLz exp (ikn· r) # (B.17) = ˆe (λ) n pLxLyLz

∇ exp (ikn· r) (because ˆe(λ)n is a constant vector) (B.18)

= i exp (ikn· r) pLxLyLz kn× ˆe(λ)n , (B.19) and hence ∇ × un,λ(r) kn = i exp (ikn· r) pLxLyLz h ˆ_k n× ˆe(λ)n i = iˆkn× un,λ(r) . (B.20)

The last line shows that the bracket is essentially the same as un,λ(r), only diering with a

constant i and the direction via the cross product: ˆ kn× ê(1)n = ê (2) n , (B.21) and ˆ kn× ê(2)n = −ê (1) n . (B.22)

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is given by (eq.(8.5) of [8]) H = Z V ε₀ 2E 2 ₊ 1 2µ0 B2 dr (B.23) =X n,λ ε0ω2n 2 αn,λα ∗ n,λ+ α ∗ n,λαn,λ + k2 n 2µ0 αn,λα∗n,λ+ α ∗ n,λαn,λ , (B.24) where the un,λ satisfy the following orthogonality relations:

Z V u∗_n,λ(r) un0_,λ0(r) dr = δ_n,n0δ_λ,λ0 , (B.25) Z V un,λ(r) un0_,λ0(r) dr = 0 , (B.26) Z V _{∇ × u}∗ n,λ(r) kn ∇ × u_n0_,λ0(r) k0 n dr = δn,n0δ_λ,λ0 , (B.27) and Z V ∇ × un,λ(r) kn ∇ × un0_,λ0(r) k0 n dr = 0 , (B.28) where the integration is over the whole cavity space. To simplify the expression for H, the vacuum dispersion relation is used:

ω2_n= c2k_n2 (B.29) ⇒ k2 n= ω2 n c2 = ε0µ0ω 2 n , (B.30) so H =X n,λ ε0ωn2 2 αn,λα ∗ n,λ+ α ∗ n,λαn,λ + k2 n 2µ0 αn,λα∗n,λ+ α ∗ n,λαn,λ (B.31) =X n,λ ε0ω2n αn,λα∗n,λ+ α ∗ n,λαn,λ . (B.32)

Expressing αn,λ and α∗n,λthrough a coordinate qn,λ and its canonical momentum pn,λ according

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B.2 Quantizing the Cavity Field

Now, in order to quantize the system, promoting qn,λ and pn,λ to operators ˆqn,λ and ˆpn,λ,

respectively, a canonical commutation relation is imposed:

[ˆqn,λ, ˆpn0_,λ0] = i¯hδ_n,n0δ_λ,λ0 . (B.36)

Hence, αn,λand α∗n,λare replaced by the operators ˆαn,λand ˆα †

n,λin eq.(B.33). Analogously as in

the standard quantization of the harmonic oscillator, we dene the dimensionless annihilation operator ˆan,λ and creation operator ˆa

† n,λ: ˆ an,λ= r 2ε0ωn ¯ h αˆn,λ= r ω_n 2¯h ˆ qn,λ+ i ωpˆn,λ , (B.37) and ˆ a†_n,λ= r 2ε0ωn ¯ h αˆ † n,λ= r ωn 2¯h ˆ qn,λ− i ωpˆn,λ . (B.38) They have the following canonical commutation relations:

h ˆ an,λ, â † n0_,λ0 i = δn,n0δ_λ,λ0 , (B.39) and [ân,λ, ân0_,λ0] = 0 = h ˆ a†_n,λ, â†_n0_,λ0 i . (B.40) Then the Hamiltonian for the quantized electromagnetic eld can be rewritten as

ˆ H =X n,λ ¯ hωn ˆ a†_n,λˆan,λ+ 1 2 =X n,λ ¯ hωn ˆ Nn,λ+ 1 2 , (B.41) where ˆNn,λ is the number operator for the mode {n, λ}. The eigenstates of the system are then

|{Nn,λ}i =

Y

n,λ

|Nn,λi , (B.42)

where Nn,λ is the number of quanta, called photons, of mode {n, λ}, and satises

ˆ

Nn,λ|Nn,λi = Nn,λ|Nn,λi . (B.43)

The energy eigenvalue is

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The quantized vector potential, electric eld and magnetic eld are given by ˆ A (r, t) =X n,λ r ¯ h 2ε0ωn h ˆ an,λexp (−iωnt) un,λ(r) + â † n,λexp (iωnt) u ∗ n,λ(r) i , (B.45) ˆ E (r, t) = iX n,λ r ¯hωn 2ε0 h ˆ an,λexp (−iωnt) un,λ(r) − â † n,λexp (iωnt) u ∗ n,λ(r) i (B.46) and ˆ B (r, t) =X n,λ s ¯ hk2 n 2ε0ωn ˆ an,λexp (−iωnt) ∇ × un,λ(r) kn + â†_n,λexp (iωnt) ∇ × u∗ n,λ(r) kn (B.47) =X n,λ r ¯ hωnµ0 2 ˆ an,λexp (−iωnt) ∇ × un,λ(r) kn + â†_n,λexp (iωnt) ∇ × u∗_n,λ(r) kn . (B.48) More details can be found in chapter 2.1 of [22] and chapter 2.1 of [10].

In order to match the formula (B.48) with eq.(3.22), the set of vector mode functions is chosen to be real functions wn(r): ˆ B (r, t) =X n,λ r ¯ hωnµ0 2 h ˆ an,λexp (−iωnt) + â † n,λexp (iωnt) i ∇ × w_n,λ(r) kn (B.49) =X n,λ s ¯ hωn 2µ0 h ˆ an,λexp (−iωnt) + â † n,λexp (iωnt) iµ₀∇ × w_n,λ(r) kn (B.50) =X n,λ s ¯ hωn 2µ0 h ˆ an,λ(t) + â † n,λ(t) i Bn,λ(r) . (B.51)

By using real vector mode functions, before quantization, the electromagnetic eld can be expressed in mode space as

E =X n en(t) wn(r) (B.52) and B =X n bn(t) ∇ × wn(r) kn , (B.53) where en(t)and bn(t)are real coecients. They are called mode coecients of the electric and

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Quantum Entanglement and Superconducting Qubits