### 碩士論文

### Department of Physics College of Science

### National Taiwan University Master Thesis

### 使用約瑟夫森分支放大器的量子測量之研究

### Study of Quantum Measurement by a Superconducting Josephson Bifurcation Amplifier

### 柯百謙 Bai-Cian Ke

### 指導教授：管希聖 博士

### Advisor: Hsi-Sheng Goan, Ph.D.

### 中華民國 98 年 7 月

### July, 2009

## 誌 謝

此篇論文可以順利完成，首先要感謝指導教授管希聖博士。儘管繁重的教 學與研究工作，老師依然無時無刻不關心學生的學習狀況。每當學生在學習上 遇到瓶頸或是對未來感到迷惘，總會放下手邊的工作與學生長談。撰寫論文期 間，老師用一種驚人的方式，一個字一個字的修改學生的論文。老師的認真負 責與對學術的堅持更是學生心中永遠的榜樣。口試期間，還要感謝陳岳男教授 與周忠憲教授的詳細審閱以及寶貴的指正與建議，使論文得以更臻完備，在此 獻上最深的敬意。

我還要感謝 501 室的所有同學與學長。陳柏文學長常常解決我的疑惑，甚 至主動找我討論，提點我該注意的地方。大黃學長的電腦功力也深深令我佩 服。超級好人劉彥甫學長相當的幽默天真，犧牲自己以活絡實驗室的氣氛，也 充分展現紳士風度，參加前女友的婚禮。成功的阿宅吳致盛，常識異常地豐 富，爛梗大家都知道，像他用的這麼自然的應該沒幾個。以及一起奮鬥的黃胖 胖，每次開會結束都一起在陽台喝飲料、分享心事。帥氣的冷智群，唱歌好 聽，同時也是玩桌上遊戲的好夥伴。魔術師公館小明是個難以形容的神奇人 物，在此由衷地祝福他街頭藝人之路可以一帆風順。

此外，我還要感謝成功大學的蔡錦俊教授與陳家駒教授。他們在我大學時 代不斷的督促我好好用功，讓我在學業上甚至人生的道路上，受益良多且深 遠。還有超低溫物理實驗室的怪獸學長，他是我心中最真心的朋友以及最好的 典範。不僅以學長的角色教導做實驗應有的態度，也以朋友的角色陪我渡過許 多難關。當然也感謝實驗室的所有同學，給我一個快樂的大學生活。

最後最感謝我的父母與洪雅琪小姐。我的父母在整個求學期間都默默地為 我付出，不斷的支持我、鼓勵我，讓我可以無後顧之憂地完成學業。洪雅琪小 姐在英文方面可以說是對我諄諄教誨。雖然被我技巧性地騙來組了一個不符合 他程度的讀書會，但在發覺事實真相後，還是很有耐心從音標開始教我。除此 之外，更是陪我渡過了最低潮的日子，將她的快樂分享給我，忍受我亂發脾

的付出。

要感謝的人很多，掛一漏萬，若有遺漏在此也一併獻上內心最深的謝意。

### 摘 要

最近，一種新型的放大器，稱為約瑟夫森分支放大器（JBA），用以測量 超導量子位元（qubit），已經被提議和建造出來。JBA 解決了建構在傳統超導 Josephson junction 量子位元測量裝置的散熱問題，此惱人的散熱問題是由此 裝置的電壓切換到 normal state 所引起 。本論文旨在模擬使用 JBA 測量量 子位元的過程，並提供對理解量子測量問題所必需的相關知識。我們一開始回 顧一些基本的超導量子電路元件，並介紹兩種不同類型的量子位元：flux qubit 和 charge qubit。由於 Josephson junction 的非線性電感，JBA 的數 學模型可由驅動非線性振盪器所描述，此數學模型被稱為 Duffing 振子。因 此，我們著重於量子 Duffing 振子的性質和介紹 JBA 的運作原理 。測量量子 位元的過程本身是一個開放量子系統的問題。為了來描述它的行為，我們推導 了 驅 動 Duffing 振 子 和 量 子 位 元 系 統 的 縮 減 密 度 矩 陣 的 quantum master equation。我們區分了熱環境和測量裝置對系統的影響，並使用 Floquet formalism 處理時間上的週期性問題。並在最後提出一些 Duffing 振子和量子 位元測量的模擬結果。

Recently, a new type of amplifier, called the Josephson bifurcation amplifier (JBA), to read out the state of a superconducting quantum bit (qubit), has been proposed and constructed. This JBA has solved the annoying dissipation problem of voltage switching to the normal state in traditional superconduct- ing Josephson junction based qubit measurement devices. This thesis aims to model the qubit readout process by the JBA, and to provide the essential in- put toward the understanding of the quantum measurement problem. We first review some basic elements of superconducting quantum circuit, and introduce two different types of qubits: flux qubits and charge qubits. Due to the non- linear inductance of a Josephson junction, the mathematical model of the JBA can be linked to a driven non-linear oscillator, known as the Duffing oscillator.

So we focus on the properties of the quantum Duffing oscillator and present the operation principles of the JBA. The qubit readout process is itself an open quantum system problem. To describe its dynamics, we derive the quantum master equation for the reduced density matrix of the combined driven quan- tum Duffing oscillator and qubit system. We distinguish the influence of the thermal environment on the combined system from that of the measurement device, and use the Floquet formalism to tackle the time-periodical driven problem. Simulation results of the Duffing oscillator and qubit measurement will be presented.

Table of Contents vi

List of Figures viii

1 Introduction 1

2 Introduction to superconducting quantum bits 4

2.1 Josephson junctions . . . 6

2.1.1 The Josephson effect . . . 6

2.1.2 A Josephson junction with a nonlinear inductance . . . 8

2.1.3 The current-biased Josephson junction . . . 9

2.2 The Cooper-pair box and the SQUID . . . 11

2.2.1 The single cooper-pair box device . . . 11

2.2.2 The SQUID device . . . 12

2.3 Charge qubits and flux qubits . . . 15

2.3.1 Charge qubits . . . 15

2.3.2 Advanced charge qubits . . . 17

2.3.3 Flux qubits . . . 18

2.3.4 Advanced flux qubits . . . 20

2.4 The quantronium . . . 21

2.5 The Josephson bifurcation amplifier . . . 22

2.5.1 The quantronium with a JBA readout . . . 23

3 The Floquet formalism 27 3.1 The Flouqet theory . . . 28

3.1.1 General form of the solution . . . 28

3.1.2 Some properties of quasienergy and QES . . . 29

3.2 The extended Hilbert space . . . 31

3.2.1 Operators in the extended Hilbert space . . . 32

3.2.2 The Floquet Hamiltonian . . . 34

3.3 Driven two-level systems and oscillators in the Floquet picture . . . . 35

3.3.1 Driven two-level systems . . . 35

3.3.2 Driven oscillators . . . 37 vi

3.3.3 The rotating wave approximation . . . 39

3.4 Time evolution operators . . . 40

3.5 Conclusions . . . 42

4 Quantum dissipation 43 4.1 The Density Matrix . . . 43

4.1.1 Pure states and mixed states . . . 44

4.1.2 Ensemble average . . . 46

4.2 Derivation of the Master equation . . . 46

4.2.1 Equations of motion of the density matrix of closed systems . 46 4.2.2 Integro-differential form of the equation of motion for the den- sity matrix . . . 47

4.2.3 The Born approximation . . . 51

4.2.4 The Markovian approximation and bath correlation functions 51 4.3 Master equations of driven systems . . . 53

4.3.1 The derivation of master equations . . . 53

4.3.2 Microscopic models of dissipation . . . 55

5 The quantum Duffing oscillator 59 5.1 Hamiltonian of quantum Duffing oscillator . . . 59

5.2 The Floquet-Born-Markovian master equation . . . 60

5.2.1 The driven weak-coupling master equation . . . 60

5.2.2 Complete set property of Floquet states . . . 61

5.2.3 The Floquet master equation . . . 61

5.2.4 The rotating wave approximation . . . 63

5.2.5 Dynamics of the quantum Duffing oscillator . . . 63

5.2.6 Expectation value of x(t) . . . 65

5.3 Numerical simulation . . . 65

5.3.1 Amplitude response . . . 66

5.3.2 Varying temperatures and the nonlinearity coefficients . . . . 68

5.3.3 Varying driving amplitudes . . . 68

5.3.4 Expansion in x space . . . 69

5.4 Driven quantum Duffing oscillator coupled to a qubit . . . 72

5.4.1 The JBA response . . . 72

5.4.2 Behaviors of the qubit . . . 73

6 Conclusions 78

Bibliography 81

A Classical Duffing oscillation 85

2.1 The current-biased josephson junciton and its equivlent circuit. . . . 9 2.2 The ”tilted-washboard” effective potential versus phase difference of a

current-biased Josephson junction. . . 10 2.3 The single Cooper pair box. One side of a small superconducting island

is connected via a Josephson tunnel junction to a large superconducting reservoir, and another side is coupled capacitively to a voltage source. 11 2.4 The superconducting quantum interference device, SQUID, and its

equivalent circuit. . . 12 2.5 The dc-SQUID. A superconducting loop with two Josephson junctions

replaces the single junction in the current-biased Josephson junction circuit. . . 14 2.6 (a) The energy spectrum of a charge qubit versus gate voltage. (b) The

lowest two energy levels near V_{g} = 0.5, the part of (a) circumscribed
by dashed lines. . . 17
2.7 The single Cooper pair transistor. A superconducting loop with two

Josephson junctions replaces the single junction in a SCB for a tunable
E_{J}. . . 18
2.8 The equivalent circuit of a transmon. . . 19
2.9 The three-junction SQUID. . . 21
2.10 The circuit diagram of the quantronium with preparation, tuning, read-

out blocks. . . 22 2.11 Quantronium circuit with JBA readout port. A JBA readout port

replaces the voltage-switching measurement in the original design of
the quantronium. . . 24
3.1 The quasienergy spectrum, _{α} versus ω. Solid lines: f = 0.001 and

α = 0.001. Dashed lines: f = 0 and α = 0.001. . . 39 5.1 Quasienergy spectrum and response amplitude as a function of the

driving frequency. Every avoided crossing in the quasienergy spectrum
corresponds to a N-photon excitation. Parameters are k_{B}T = 0.1~ω0,
α = 0.1α_{0}, f = 0.1f_{0} and γ = 0.005ω_{0}. . . 67

viii

5.2 (a)Response amplitude for different values of temperature T , k_{B}T =
0.1, 0.5, and 1.0~ω0, with α = 0.1α_{0}. (b)Response amplitude for dif-
ferent value of the nonlinearity α, α = 0.095, 0.1, and 0.105α_{0} with
k_{B}T = 0.1~ω0. The remaining parameters are f = 0.1f_{0}and γ = 0.005ω_{0}. 69
5.3 (a) A 3D diagram of the response amplitude versus the driving fre-

quency and the driving amplitude. The two arrows label a shift of the
critical area. (b) A response amplitude profile versus the driving ampli-
tude f with ω_{ex}= 1.4ω_{0}. The remaining parameters are k_{B}T = 0.1~ω0,
α = 0.1α0 and γ = 0.005ω0. . . 70
5.4 The amplitude distribution function. The two peaks concentrate to

one when the driving frequency is away from the critical region. From
the upper left to the upper right and then from the lower left to the
lower right, ω_{I} = 1.155, 1.16, 1.165, 1.17, 1.18, 1.195ω_{0}. The remaining
parameters are k_{B}T = 0.1~ω0, α = 0.1α_{0}, f = 0.1f_{0} and γ = 0.005ω_{0}. 71
5.5 The time evolution diagrams of the response amplitude when qubit’s

states are (a) |0i and (b) |1i with ω_{I} = 0.024. (c)(d)(e) The ampli-
tude distribution function with qubit’s state |0i (dotted line) or |1i
(solid line). The driving frequency ωex equals 1.17ω0. The remaining
parameters are k_{B}T = 0.1~ω0, α = 0.1α_{0}, f = 0.1f_{0} and γ = 0.005ω_{0}. 74
5.6 The σ_{z} expectation value of the qubit. The JBA’s environment in-

fluences the qubit through the JBA. The remaining parameters are
ω_{q} = 0.01ω_{0}, ω_{qx} = 0.01, ω_{ex} = 1.16, k_{B}T = 0.1~ω0, α = 0.1α_{0},
f = 0.1f_{0}. The initial state of the qubit is |1i. . . 76
5.7 (a) The amplitude response. The qubit decays from |1i to |0i caused by

its environment, which making the response amplitude can’t maintain a higher level. (b) The amplitude distribution function. Solid line:

t=6000. Dashed line: t=4000. γ = 0.005, γq = 0.0002, ωq = 0.1, and
ω_{I} = 0.024. . . 77
A.1 The response amplitude profile. (a)The arrows indicate where the re-

sponse amplitude must jump up to a bigger or a lower response am-
plitude. ω_{0} = 1, k = 0.02 and F = 0.02 (b)The response curve is for
different values of the driving amplitude strength. ω_{0} = 1, k = 0.02
and, from the bottom to the top, F = 0.003, 0.006, 0.01, 0.015 and
0.02. . . 89
A.2 _{0} determines the direction of the response amplitude’s turning . . . . 89

### Introduction

The quantum information science has developed for decades. In the beginning, it didn’t draw much attention, because no quantum algorithms that had practical use and outperformed their classical counterparts were found. Not only it has great dif- ficulties to realize a quantum computer, but also researchers even thought that the calculating speed of a quantum computer is much slower than the speed of a classical computer. This situation remained until Shor’s algorithm [1, 2] and Grover’s algo- rithm [3–5] were proposed. Shor’s algorithm makes it possible to efficiently factorize large semi-prime integers and Grover’s algorithm enables searches within a large un- sorted database. Those two problems are impossible solved or very time-consuming for a classical computer. People recognized a problem difficult for a classical com- puter to solve may be easy to solve for a quantum computer. Because of those key motivations, people pay more and more attention on the field of quantum information science.

”Is it possible to realize a quantum computer?”, many people may ask this ques- tion. In fact, it is still a very long distance for people to realize a practical quantum

1

computer. However, regardless whether a quantum computer can be built ultimately, people will still benefit much on the road to the final goal of implement a quantum computer. Researchers have been trying to find methods to control quantum systems precisely, and to develop controllable quantum systems to construct universal quan- tum gates [6], which can be used to implement arbitrary unitary operations. Those methods and devices developed may be used in other purposes. Although few-qubit controls and manipulations are still a challenge, it is believed that one day the quan- tum computer will be realized.

There are three stages in quantum computation : preparation, manipulation, and readout. In this thesis, we focus on the readout process. At the end of quantum- state manipulation, we need to read out the final results. Or even in the middle of manipulation, we read out the qubit’s state for the purpose of error correction.

Many traditional schemes to read out the states of superconducting Josephson junc- tion qubits, such as the phase qubit, quantronium etc., are involved with the voltage switching of a readout Josephson junction to the dissipative normal state under the direct measurement. A new type of amplifier, the Josephson bifurcaiton amplifier (JBA) [7], to read out the states of a qubit, constructed by I. Siddiqi et al. in 2004, has solved the annoying dissipation problem. The mathematical model of the JBA is a driven non-linear oscillator, known as the Duffing oscillator in classical physics [8,9].

This thesis investigates the quantum Duffing oscillator and some basic super- conducting quantum information devices. First we review some basic elements of quantum circuits: Josephson junctions, superconducting Cooper-pair boxes (SCB’s), and superconducting quantum interference devices (SQUID’s). The property of be- having like a nonlinear inductance makes the Josephson junction play a crucial role

in a quantum circuit. The discrete Cooper-pair number in SCB and the magnetic flux quantum number make, respectively, the SCB and the SQUID ideal candidates as qubits. Then we introduce basic types of quantum bits, flux qubits and charge qubits, and a special kind of qubit in charge-phase regime, called the quantronium.

After that, we present an introduction of the working principle of a JBA and how a JBA can be modeled as a driven quantum Duffing oscillation.

Second, we describe a mathematical technique, the Floquet formalism, usually
used to deal with time-periodic problems. Analogous to the Bloch theory, the princi-
ple of the Floquet formalism is to expanse ,besides the space domain, the time domain
function by a time-periodic basis, e^{inωt}. Next, we introduce the concept of a mas-
ter equation. In an open system, the Schr¨odinger equation is no longer sufficient to
describe the dynamics of the system of interest. The density matrix and the master
equation is thus required. Then, we present the master equation for a driven system,
which differ form the ordinary master equation, that is usually in the Lindblad form.

There are time-ordering operators inside the time-dependent master equation, which could be very troublesome.

Finally, we present a master equation with Floquet states as a basis. In the Floquet picture, the problem of the time-ordering operators is readily solved. Consequently, the dynamics of the driven quantum Duffing oscillator can be described more easily by using this improved master equation. Then we describe how to use a JBA to measure a qubit, present some numerical results and discuss the dynamical behavior of the combined system of JBA and the measured qubit.

### Introduction to superconducting quantum bits

A bit is the most fundamental unit of classical computation and information. A bit in
a classic computer has only two possible states, either 0 or 1. Besides |0i or |1i, a quan-
tum bit, or qubit, can have superposition of states, α |0i + β |1i with |α|^{2}+ |β|^{2} = 1.

Furthermore, there are many useful quantum effects , such as quantized energy levels and entanglement, in qubits. Researchers try to take advantage of quantum effects and hope to ultimately create quantum computers to solve time-consumsing prob- lems or problems which are impossible to be solved in classical computers, such as factoring large numbers and simulating large quantum systems.

A quantum superconducting Josephson-junction circuit may contain a large num- bers of energy levels, while for qubit operations only two levels are required. Moreover, these two qubit levels must be well decoupled from the other levels. Typically, that means that a qubit should involve a low-lying pair of levels, well separated from the spectrum of higher levels, and not being close to resonance with any other levels.

4

There are three stages in quantum computation : preparation, manipulation, and readout. Although any quantum two-state system can be considered as a qubit, to be able to be isolated from other energy levels and environment and to be prepared, manipulated, and read out determine whether it is a good qubit or bad one.

Relaxation and decoherence caused by coupling to environment make most physi- cal systems behave like classical systems, except microscopic systems, such as atoms.

However, superconducting circuits maintain quantum properties with macroscopic or mesoscopic size. The size is not the only difference between atoms and super- conducting circuits. Parameter-controlling in superconducting circuits is easier than in atoms, and coupling between two superconducting circuits can be turned on and turned off at will. Well-designed superconducting circuits may have better coherence time than atoms do, providing more time for quantum computing. Preparing initial states and measuring final states are also easier in superconducting circuits. Because of those advantages, although there are still many obstacles in the way to practical application, studying superconducting circuits is one of the main streams in quantum information processing. [10–14]

In this chapter, I will start from the basics of a Josephson junction, an important element of superconducting circuits, and introduce two fundamental types of super- conducting qubits and some advanced types of qubits.

### 2.1 Josephson junctions

The fundamental structure of a Josephson junction consists of a sandwich of two su- perconductors separated by an insulating layer, typically fabricated from oxidation of the superconductors, and thin enough to allow tunneling of discrete charges through the barrier. That is why a Josephson junction is also called a superconducting tunnel junction or a Josephson tunnel junction.

For the purposes of creating a two-level system which is isolated from and not by external excitation resonant with other energy levels, the harmonic system is not suitable, in which all of energy gaps are the same. A nonlinear system is required.

A Josephson junction [15, 16] is the electronic circuit element that has nonlinear and non-dissipative properties at arbitrarily low temperatures. Because of the properties of nonlinearity, when the driving frequency ω is detuned from the natural oscillation frequency ω0, the system is very sensitive between two possible oscillation states that differ in amplitude and phase. So, a Josephson junction is an important element not only of creating a qubit but also of quantum readout measurement.

### 2.1.1 The Josephson effect

As stated above, a junction consists two strongly superconducting electrodes con- nected by a weak link. The weak link can be an insulating layer as Josephson origi- nally proposed, or a normal metal layer made weakly superconductive by the so-called proximity effect, or simply a short, narrow constriction in otherwise continuous su- perconducting material [16]. According to quantum mechanics, the electrons would tunnel through the weak link or barrier layer. There are two effects of pair tunneling, DC and AC Josephson effects [15].

DC Josephson effect: a dc current flows across the junction in the absence of any electric or magnetic field. The relationship between the phase difference δ and the current I of superconducting pairs across the junction is

I_{J} = I_{c}sin δ . (2.1)

The critical current I_{c}is the maximum zero-voltage superconducting current that can
pass through the junction above which the superconducting state will become normal
state. It is proportional to the transfer interaction. Because no voltage apply, the
phase difference δ is a constant. For finite voltage situations involving the ac Joseph-
son effect, a more complete description is required.

AC Josephson effect: when a dc voltage is applied across the junction, an ac current flows across the junction. The phase difference δ is no longer a constant. The relationship between voltage and phase difference is

˙δ = −2eV/~ (2.2)

or

δ(t) = −2e

~ Z t

0

V dt + δ(0) . (2.3)

and the superconducting current is

I_{J} = I_{c}sin (δ(0) − 2eV t/~) . (2.4)

Furthermore, considering more general cases, we can apply a time-dependent voltage, and write down the function in some significant symbols,

I_{J}(t) = I_{c}sinΦJ(t)

ϕ_{o} = I_{c}sin δ(t) , (2.5)

where the generalized flux is defined by Φ_{J} =Rt

−∞V (t^{0})dt and ϕ_{0} = ~/2e is the re-
duced flux quantum, or ϕ_{0} = Φ_{0}/2π, where Φ_{0}, h/2e, is the magnetic flux quantum.

Actually, phase difference is not a gauge-invariant quantity; for a given physical
situation, there is not only one unique value of phase difference. Hence it cannot
in general determine the current I_{J}, which is a well-defined gauge-invariant physical
quantity. The phase difference mentioned before is not the real phase difference
between two superconductor [16], defined by

δ ≡ δ^{0} −2π
Φ_{0}

Z

A · dl . (2.6)

where δ^{0} is the real phase difference and the integration over the vector potential A
is from one electrode of the weak link to the other. Thus, the difficulty is cured. In
addition to curing the conceptual problem, the introduction of the gauge-invariant
phase difference is the key to working out the effects in a magnetic field, which cannot
be treated without introducing the vector potential A.

### 2.1.2 A Josephson junction with a nonlinear inductance

At first, let’s take a short review of a conventional inductance.

L = Φ/I or I = φ/L ,

where L is the inductance, Φ is the magnetic flux, and I is the current.

We thus expand Eq. (2.5)
I_{J}(t) = 1

LΦ_{J}(t) − 1

6L_{J}ϕ^{2}_{0}Φ^{3}_{J}(t) + O[Φ^{5}_{J}(t)] (2.7)
or simplely

IJ(t) = Icsin δ(t) = Ic

δ(t) − δ^{3}(t)
3! + ...

. (2.8)

**I**

**J** **C** **R**

Figure 2.1 The current-biased josephson junciton and its equivlent circuit.

By comparing the functions of a Josephson junction and a conventional inductance, it is very easy to find that besides the linear term in the relation of current and magnetic flux, there are additional nonlinear high-order terms in a Josephson junction. A Josephson junction, therefore, can be considered having a nonlinear inductance.

### 2.1.3 The current-biased Josephson junction

A Josephson junction schematically shown in Fig. 2.1 as a sandwich structure can be modeled as a parallel circuit which consists of a nonlinear inductance, a resistance, and a capacitance.

According to Kirchhoff’s rule and some relationships, I = C ˙V = C ¨δ, δ = ^{2e}

~Φ,
and I_{j} = I_{c}sin δ, the equation of the circuit is

~

2eC ¨δ + ~

2eR˙δ + I_{c}sin δ = I_{e}, (2.9)
where C is the capacitance, R is the resistance, and V is the voltage across the
capacitance. Then, it is useful to define some meaningful parameters, E_{C} ≡ ^{(2e)}_{2C}^{2} and

### δ

**U**

Figure 2.2 The ”tilted-washboard” effective potential versus phase differ- ence of a current-biased Josephson junction.

E_{J} ≡ _{2e}^{~}I_{c}. The kinetic energy of the quasi-partical of phase δ is

K( ˙δ) = ~^{2}δ˙^{2}

4E_{C} , (2.10)

the potential energy of it is

U (δ) = E_{J}(1 − cos δ) − ~

2eI_{e}δ , (2.11)

and the Hamiltonian has the form

H = E_{C}n^{2}− E_{J}cos δ − ~

2eI_{e}δ . (2.12)

The relationship of potential versus phase is shown in Fig. (2.2). It is obvious that nonlinear inductance, cos δ, makes potential oscillate and bias current makes it slope.

When current bias is applied, the pendulum potential becomes tilted. By the way, a current-biased Josephson junction can be considered as a qubit, because the potential is cosine function, making energy gaps different.

**V** _{g}

_{g}

**C** _{g}

_{g}

Figure 2.3 The single Cooper pair box. One side of a small superconducting island is connected via a Josephson tunnel junction to a large superconduct- ing reservoir, and another side is coupled capacitively to a voltage source.

### 2.2 The Cooper-pair box and the SQUID

### 2.2.1 The single cooper-pair box device

There is a small superconducting island in a superconducting Cooper-pair box (SCB) device as shown in Fig. 2.3. One side of the island is connected via a Josephson tunnel junction to a large superconducting reservoir, and the other side is coupled capacitively to a voltage source. Cooper pairs can only transfer to the island one by one in the device. The number of electrons on the island is controlled by the bias voltage.

The Hamiltonian of the cooper-pair box is

H = Eˆ C(ˆn − ng)^{2}− EJcos ˆδ , (2.13)

where ng = CgVg/2e is the offset Cooper pair number caused by the gate voltage Vg

through gate capacitance C_{g}, and n is the number of extra Cooper pairs between the
two capacitances, the gate capacitance and the capacitance in the Josephson junc-
tion. Therefore, the first term, E_{C}(ˆn − n_{g})^{2}, represents the electrostatic energy of

**J** **C** **L** **R**

### ⊕ Φ

A_{1} A_{2}

Figure 2.4 The superconducting quantum interference device, SQUID, and its equivalent circuit.

the island, where E_{C} = (2e)^{2}/2 (C + C_{g}). Due to the nonlinear inductance of the
Josephson junction, the second term, E_{J}cos ˆδ, appears.

### 2.2.2 The SQUID device

A superconducting quantum interference device (SQUID) is a device involved with quantum interference.

A rf-SQUID, shown in Fig. 2.4, consists of a superconducting loop interrupted by a tunnel junction. a external magnetic flux is sent through the loop, inducing quantum interference.

According to the Meissner effect, we have
J (r) = |ψ(r)|^{2} q~

m^{∗}∇θ(r) − q^{2}
m^{∗}cA(r)

, (2.14)

where A is the vector potential and q ≡ −2e for a Cooper pair. Inside a supercon- ductor, the current vanlishs,

∇θ(r) = −2e

~c

A(r) . (2.15)

Choosing a contour inside the superconducting loop, with Eq. (2.6) we can get

Φ_{t}=
I

A · dl = Z A2

A1

A · dl + Z A1

A2

A · dl

= −2e

~c Z A2

A1

∇θ(r) · dl + Z A1

A2

A · dl

= 2e

~cδ , (2.16)

where Φt is total magnetic flux and Eq. (2.6) has been used.

With magnetic flux Φ = Φt − Φe where Φe is external magnetic flux and the
inductance energy ^{Φ}_{2L}^{2}, the Hamiltonian of a rf-SQUID is given by

H = Eˆ _{C}nˆ^{2} − E_{J}cos ˆδ + E_{L}(ˆδ − δ_{e})^{2}

2 , (2.17)

where δ_{e} = ^{2e}

~Φ_{e}. The first term E_{C}nˆ^{2} is electrostatic energy of the capacitance in
the Josephson junction, and the second term is related to the Josephson energy. The
last term corresponds to the inductance energy of the loop, and E_{L} = _{4π}^{Φ}2^{2}^{0}L. In the
next part, another device, a dc-SQUID, and a very important concept related to it
will be introduced.

A dc-SQUID is a device which consists of two tunnel junctions in a supercon- ducting loop and is biased by an external current. It is similar to a current-biased Josephson junction with a two-junction loop, as shown in Fig. 2.5, instead of a single junction.

Two superconducting phases, δ_{1,2}, is involved, and according to Eq. (2.5), the
external current is

I_{c1}sin δ1 − I_{c2}sin δ2 = I_{e}. (2.18)

### Φ **I**

## ⊕

Figure 2.5 The dc-SQUID. A superconducting loop with two Josephson junctions replaces the single junction in the current-biased Josephson junc- tion circuit.

It is convenient to define some new variables,

δ± = δ_{1}± δ_{2}

2 , (2.19)

and in a symmetric case, which the two Josephson junction are the same I_{c1} = I_{c2},
Eq. (2.18) reduces to the form

2I_{c}cos (δ_{e}/2) sin δ− = I_{e} . (2.20)

Comparing Eq. (2.20) with Eq. (2.5), we can find that 2Iccos (δe/2) is the effective
critical current. Most importantly, it can be tuned by the external magnetic flux
and consequently the effective Josephson energy, EJ = _{2e}^{~}2Iccos (δe/2) is tunable too.

The Hamiltonian can be written by generalizing Eqs. (2.12),(2.17) for the phases δ±.

H = E_{C}ˆn^{2}_{+}+ E_{C}nˆ^{2}_{−}− 2E_{J}cos ˆδ_{+}cos ˆδ−+ E_{L}

2ˆδ_{+}− δ_{e}2

2 + ~

2eI_{e}δˆ− , (2.21)
where ˆn+ and ˆn− are the conjugate momentum of ˆδ+ and ˆδ−. According to quan-
tum mechanics-just like the familiar position and momentum operators ˆx and ˆpx-the

operators ˆδ and Cooper-pair number operator ˆn on the capacitor are canonically conjugate, as expressed by the commutator braket, [ˆδ, ˆn] = i.

### 2.3 Charge qubits and flux qubits

### 2.3.1 Charge qubits

A superconducting Josephson junction qubit in which the charging energy is much large than the Josephson coupling, EC EJ, is called a charge qubit. In this regime, a convenient basis is formed by the charge states, and the phase terms can be consid- ered as perturbation. This is why this kind of qubits are called charge qubits. The necessary of one-qubit and two-qubit gates can be performed by controlling applied gate voltages and magnetic fields. Different designs will be presented that not only in complexity, but also in flexibility of manipulations.

In this subsection, the simplest charge qubit, cooper-pair box, Fig. 2.3, is pre- sented in details. This example illustrates how charge qubits provide two energy states, which satisfy the requirements of qubits.

In charge regime, at first we expand all operators in the basis of the charge states {|ni}. The Hamiltonian of a cooper-pair box, Eq. (2.13), is

H = Eˆ _{C}(ˆn − n_{g})^{2}− E_{J}cos ˆδ .

Then by using the properties of orthonomal and complete set, hn | ˆn | n^{0}i = δ_{n,n}^{0} and
I =P

n|nihn|, the first term is rewritten as X

n

E_{C}(n − n_{g})^{2}|nihn| (2.22)

and by using the commutator relation, h ˆδ, ˆni = i ,

⇒ h ˆδ^{m}, ˆn
i

= imˆδ^{m−1} , m > 0 ,

⇒ h

ˆ
n, e^{iˆ}^{δ}i

=

ˆn,X

m

iˆδm

m!

= e^{iˆ}^{δ} . (2.23)

The commutator relation Eq. (2.23) is similar to the commutator relation of number
operator ˆa^{+}ˆa and the creation operator ˆa^{+}, [ˆa^{+}a, ˆˆ a^{+}] = ˆa^{+}. So, e^{iˆ}^{δ} and e^{−iˆ}^{δ} can be
presented in charge basis,

e^{iˆ}^{δ}=X

n

|n + 1ihn| , e^{−iˆ}^{δ} =X

n

|nihn + 1| , (2.24)

and the second term of Eq. (2.13) is 1

2E_{J}X

n

(|nihn + 1| + |n + 1ihn|) . (2.25)

By combining Eq. (2.22) and Eq. (2.25), in this basis the Hamiltonian reads H =ˆ X

n

E_{C}(n − n_{g})^{2}|nihn| −1

2E_{J}(|nihn + 1| + |n + 1ihn|)

. (2.26)

The energy spectrum of Eq. (2.26) is shown in Fig. 2.6a.

Under suitable conditions, when charge number on a gate capacitor n_{g} controlled
by gate voltage V_{g} equals half integers, the lowest two energy states are well-isolated
from other states, shown in Fig. 2.6b. Because of that, near n_{g} = 1/2, the Hamilto-
nian can be reduced to

H = −ˆ 1

2(σ_{z}+ ∆σ_{x}) , (2.27)

where = E_{C}(1 − 2n_{g}), and ∆ = E_{J}. The qubit eigenenergies are then given by the
equation

E_{1,2} = ∓1
2

q

E_{C}^{2} (1 − 2n_{g})^{2} + E_{J}^{2} . (2.28)

**n**

**n**

_{g}-2 -1 0 1 2 3 0.5

**n**

**n**

_{g}(a) (b)

0.5

Figure 2.6 (a) The energy spectrum of a charge qubit versus gate voltage.

(b) The lowest two energy levels near V_{g} = 0.5, the part of (a) circumscribed
by dashed lines.

So, under suitable conditions charge qubits provide physical realizations of qubits
with two charge states differing by one cooper-pair charge on a small island. For
quantum computation, it is required to have the ability to rotate a state on the Bloch
sphere to any position at will, and consequently σ_{z} and σ_{x} rotation are necessary. In
a cooper-pair box, pure σ_{x} rotation is acquirable, as n_{g} = 1/2, but pure σ_{z} rotation
is not, since E_{J} is fixed. In previous section, an important concept is mentioned. A
two-junction loop can substitute for the single Josephson junction, creating a SQUID-
controlled qubit, Fig. 2.7. Thus, the effective Josephson energy E_{J} is tunable and
pure σ_{z} rotations can be performed.

### 2.3.2 Advanced charge qubits

Operated in E_{J}/E_{C} 1 regime, basic charge qubits have good anharmonicity to form
two-level systems but their energy bands shown in Fig. 2.6 have slopes, making them
very sensitive to low-frequency charge noise. The magnitudes of charge dispersion and

**V** _{g} **I**

_{g}

Figure 2.7 The single Cooper pair transistor. A superconducting loop with
two Josephson junctions replaces the single junction in a SCB for a tunable
E_{J}.

anharmonicity are both determined by the ratio E_{J}/E_{C}. The low value of the ratio
of E_{J}/E_{C} brings not only good manipulations of qubits but also serious decoherence.

Many researchers keep trying to find solutions for this problem. A famous example
is the transmon [17], Fig. 2.8. The fundamental idea of the transmon is to shunt
the Josephson junction of a small Cooper-pair box with a large external capacitor
to increase the charging energy E_{C} and to increase the gate capacitor to the same
size. This make the charge dispersion reduces exponentially in E_{J}/E_{C}, while the
anharmonicity only decreases algebraically with a slow power law in E_{J}/E_{C}.

### 2.3.3 Flux qubits

In the previous section, we describe the quantum dynamics of low-capacitance Joseph-
son devices where the charging energy dominates over the Josephson energy, E_{C}
E_{J}, and the relevant quantum degree of freedom is the charge on superconducting

**C**

_{B}**C**

_{r}**L**

_{r}**V**

_{g}**C**

_{in}**C**

_{g}**C**

_{J }**E**

_{J }### Φ

### island

Figure 2.8 The equivalent circuit of a transmon.

island. We now talk about another quantum regime, the phase regime, E_{J} E_{C},
in which the flux states are the better basis. This kind of qubits are called flux qubits.

A rf-SQUID is the simplest example of a flux qubit. The Hamiltonian, Eq. (2.17), is

H = Eˆ _{C}ˆn^{2}− E_{J}cos ˆδ + E_{L}(ˆδ − ˆδ_{e})^{2}

2 ,

and in the phase regime, the potential energy is given by

U (δ) = −E_{J}cos δ + E_{L}(δ − δe)^{2}

2 . (2.29)

The potential energy is cosine function added a second power function. δe in a flux
qubit play as the same role as ng do in a charge qubit. The lowest area can be
approximated to a double-well. When δe equals π or odd π, a symmetric double-well
potential energy appears. It is similar to that of n_{g} equal 1/2 in a charge qubit.

Because of the tunneling through center barrier, the lowest two energy level split
with a gap ∆, which depends on the height of the barrier. When δ_{e} doesn’t equal π

or odd π, the potential energy becomes unsymmetric, the probability of the lowest
energy pair is not half in each well. This situation is like when n_{g} is near 1/2, in a
charge qubit, the probability is not the same in |0i and |1i. The Hamiltonian of a
flux qubit can be truncated to the lowest two energy states in a simple form of

H = −ˆ 1

2(σ_{z}+ ∆σ_{x}) , (2.30)

where ∆ depends on E_{J} and is given by

= 4π s

6 E_{J}
E_{L} − 1

EJ

Φ_{e}
Φ_{0} − 1

2

. (2.31)

In this form, the pure operator X-rotation can be performed by setting Φ/Φ_{e} = 1/2,
but the pure Z-rotation can not. In order to solve this problem, we can replace the
single junction with a two-junciton loop that introduces an additional external flux
Φ˜_{e} as another control variable. Therefore, the effective Josephson energy becomes
tunable.

### 2.3.4 Advanced flux qubits

The main idea in a SQUID is to create a double-well potential, requiring large enough inductance. This implies that the qubit contains a large qubit loop, making itself influenced by magnetic fluctuations of environment seriously. One way to overcoming this difficulty is using a three-junction device pointed out by Mooij et al. [18]. In a three-junction-loop qubit, as shown in Fig. 2.9, ELis not the only element to creating a double-well potential. The loop, therefore, can be much small than a rf-SQUID and the qubit is relatively free from charge and magnetic environment fluctuations.

### ⊕ Φ

Figure 2.9 The three-junction SQUID.

### 2.4 The quantronium

With device parameters locating between charge qubits and flux qubits, the quantro-
nium [19, 20] is a very special kind of qubits. Neither ˆn nor ˆδ is a good quantum
number since the quantronium is operated in E_{J} ∼= EC regime. The circuit of a
quantronium is shown in Fig. 2.10. the island connected to two Josephson junc-
tions and a voltage is applied to it through a capacitance. The two small Josephson
junctions and a large Josephson junction with a higher critical current E_{J 0} ≈ 20E_{J}
form a closed loop, and an external magnetic flux is applied to it. The two small
junctions define the superconducting island of the box, and the phase ˆγ of the large
junciton, so-called read-out junciton, coupled to the qubit. A readout pulse current
I_{b}(t), with a peak value approaching the large junciton’s critical current, is applied to
the parallel combination of the large junction and the small junctions. If the state of
the qubit is |1i, the supercurrent adds the readout pulse will make the large junction
switch to a finite voltage state. If the state of the qubit is |0i, the large junciton will
stay in the superconducting zero voltage state.

When the qubit operates in the charge-flux regime, E_{J} ∼= EC, no matter charge
or flux noise decoherence can be reduced to higher order, because the slope of energy

### ~

**U**
**C**_{g}

**1 E****J**

**2**

**1 E****J**

**2**

preparation “quantronium” circuit readout

**E**** _{J0}**
δ γ

Φ **2C**

**2C**

**I**_{b}**(t)**

tuning

Figure 2.10 The circuit diagram of the quantronium with preparation, tun- ing, readout blocks.

levels in the charge degree of freedom and in the flux degree of freedom are both flatter
than simple charge qubits and flux qubits. Moreover, if the qubit is maintained at
the double degeneracy point, n_{g} = 1/2 and Φ = 0, the influence of both flux and
charge noise sources vanishes to first order.

### 2.5 The Josephson bifurcation amplifier

In this section, we will introduce briefly the Josephson bifurcation amplifier designed to measure the states of charge qubits. In order to measure the state of a charge qubit, the number of Cooper-pairs, we need a very sensitive device and the accuracy of it must much bigger than 2e. Before the Josephson bifurcation amplifier, there are many measurement devices such as the single-electron transistor (SET) or the read- out junction of quantronium, but in those devices, the dissipation problem is usually serious, because they are involved with switching to finite voltage states. This flaw was conquered by the introduction of the Josephson bifurcation amplifier (JBA). I.

Siddiqi et al. [7] constructed a new type of amplifiers based on the transition of a rf-driven Josephson junction between two distinct oscillation states near a dynami- cal bifurcation point. The main advantages of JBA are speed, high secsitivity, low backaction, and the most special character is the absence of on-chip dissipation. The measurement of quantronium with JBA [21] was published by I. Siddiqi et al. some years later and quantum nondemolition readout using a JBA [22, 23] was also pub- lished by I. Siddiqi et al. soon.

The central element of a JBA is a Josephson junction whose critical current I_{0}
can be modulated by an input signal, i.e. states of a qubit. Another sinusoidal signal
drives this Joseson junction. An output port is connected to this circuit to measure
the reflected component of the drive signal. Simply, a JBA is a driven Josephson
junction with a tunable critical current. The anharmonic potential of the Joseph-
son junction and the sinusoidal driving make up a famous mathematical model, the
Duffing oscillator, which have two distinct possible oscillation states that differ in
amplitude and phase.

### 2.5.1 The quantronium with a JBA readout

Figure 2.11 is a quantronium circuit with preparation and readout ports [22]. The
middle part is a quantronium qubit. The two parallel Josephson junctions have
capacitances C_{J}/2 and Josephson energies E_{J}(1 ± d)/2, where d is the asymmetry
factor quantifying the difference between the two junctions (0 ≤ d ≤ 1), and E_{J} =
ϕ_{0}I_{0}, where I_{0} is the sum of critical currents of the junctions. The island is biased
by a voltage source V_{g0} in series with a gate capacitance C_{g}. The Hamiltonian of the

### ~

**U**
**C**_{g}

**1 E****J**

**2**

**1 E****J**

**2**

preparation “quantronium” circuit JBA readout

**E**** _{J0}**
δ γ

Φ **2C**

**2C**

tuning

### ~

**R**
**U(t)**

**φ**

**0**

**φ**

**1**

Figure 2.11 Quantronium circuit with JBA readout port. A JBA readout port replaces the voltage-switching measurement in the original design of the quantronium.

quantronium is

H = E_{CP} (ˆn − n_{g})^{2}− E_{J}

cosδ

2cos ˆθ − d sinδ 2sin ˆθ

, (2.32)

where E_{CP} = (2e)^{2}/2(C_{g} + C_{J}), ˆθ is the superconducting operator (”conjugate” to
ˆ

n—i.e., [ˆθ, ˆn] = i), and δ is the superconducting phase across the series combination
of the two small junctions. n_{g} and δ can be tuned by biased voltage and external
flux, respectively, and the energy spectrum is sufficiently anharmonic, i.e. the gaps
between any two energy levels are strongly unequal. This suggests that the first two
energy states can form a qubit.

For the purpose of measurement, a JBA readout device is coupled to the quantro-

nium. The total Hamiltonian with the external flux Φ = 0 is
Htot = ECP(ˆn − ng)^{2}− EJ

"

cos ˆθ ⊗ cos ˆδ

2 − d sin ˆθ ⊗ sin δˆ 2

#

+
Qˆ^{2}

2C − E_{J 0}cos ˆδ − U (t)

R ϕ_{0}δ ,ˆ (2.33)

where E_{J 0} is the Josephson energy of the readout junction and U (t) is a time-
dependent driving potential. To study a measurement problem, it is very convenient
to rewrite the Hamiltonian in the following form,

H_{tot} = H_{S}+ H_{I}+ H_{P} , (2.34)

where H_{S} is the Hamiltonian of system, i.e. the quantronium, H_{P} is the Hamiltonian
of the probe, i.e. the JBA, and H_{I} is their interaction Hamiltonian. So Eq. (2.33) is
rewritten as

H_{tot} = H_{S}+ H_{I} + H_{P}

= E_{CP}(ˆn − n_{g})^{2}− E_{J}cos ˆθ

− E_{J}
(

cos ˆθ ⊗

"

cos ˆδ 2 − 1

#

− d sin ˆθ ⊗ sin δˆ 2

)

+
Qˆ^{2}

2C − EJ 0cos ˆδ − U (t)

R ϕ0δ .ˆ (2.35)

To approximate a two-level system, the Hamiltonian at the optimum n_{g} = 1/2 is
truncated to

H_{S} = −~ω01

2 σ_{z} ,
H_{I} = −

(
ασ_{z}⊗

"

cos δˆ 2− 1

#

− βσ_{y} ⊗ sinδˆ
2

) ,

H_{P} = Qˆ^{2}

2C − E_{J 0}cos ˆδ − U (t)

R ϕ_{0}δ ,ˆ (2.36)

where α = E_{J}(h0| cos ˆθ|0i − h1| cos ˆθ|1i)/2 and β = idE_{J}(h0| sin ˆθ|1i − h1| sin ˆθ|0i)/2.

With d = 0, the requirement [H_{S}, H_{I}] = 0 of a quantum non-demolition (QND) is

fulfilled. The qubit in this case is coupled to the JBA through only a σ_{z} operator.

If cos ˆδ in H_{P} is expanded to the order of ˆδ^{4}, we then have a system of a nonlinear
driven quantum Duffing oscillator. In the following chapters, we will discuss the
properties of a driven quantum Duffing oscillator in order to understand the behavior
of the JBA. The time-dependent driving, U (t) in Eq. (2.36), is a generally a periodic
in time function. So, we will describe in the next chapter a formalism, known as the
Floquet formalism, to deal with the periodic in time problem.

### The Floquet formalism

Originally, the Flouqet theory is a mathematical theory dealing with differential equa-
tions. In 1965, Jon H. Shirley introduced this method to solve the Schr¨odinger
equation with periodic in time [24]. By using the method of separation variables,
a time-independent Schr¨odinger equation becomes an eigenvalue-eigenfunction equa-
tion, ˆH |ψ_{α}i = E_{α}|ψ_{α}i. After getting the eigenenergies and eigenvectors, the time
evolution of states is easily solved, |Ψ_{α}(t)i = P C_{α}e^{−iE}^{α}^{t}|ψ_{α}i. However, there is no
well-defined eigenenergy and eigenvector with a time-dependent Hamiltionian, which
means there is no stationary state, and a time-ordered integral form, ˆT [e^{R Hdt}], is
always involved in the time evolution of states. According to the Flouqet theorem,
if a time-periodic system is expanded in a time-space Hilber space, the time-periodic
Schr¨odinger equation becomes an eigenvalue-eigenfunction equation, too. Therefore,
the knowledge of the time-independent Schr¨odinger equation can be used in time-
dependent one. Besides, there are many advantages for different cases using the
Floquet theory [25, 26].

In this chapter, we try to introduce the basic concepts of the Flouqet theory and 27

show two examples, a two-level system and a nonlinear oscillator, with periodic in time driving. First, a general form of the solution in the Floquet theory is introduced and determined. Besides, any operator with either time-dependent or time-independent terms, is presented in the Floquet picture as a time-indenpendent operator. As a re- sult, the time-dependent Hamiltonian can be transformed to a time-independent ma- trix in the Floquet picture. Therefore, how to solve the time-dependent Schr¨odinger equation becomes a pure eigenvalue-eigenvector question. Finally, the time evolution of states can be obtained.

### 3.1 The Flouqet theory

### 3.1.1 General form of the solution

Suppose there is a Schr¨odinger equation with a periodic Hamiltonian,

id

dt|Ψ(t)i = ˆH (t) |Ψ(t)i , (3.1)

where H is a Hermitian matrix of period functions of t with a period τ , ˆH (t − τ ) =
H (t) . The general form of the solution of a differential equation with periodicˆ
coefficients is given by Floquet’s theorem. So, the Floquet theorem asserts that the
solutions of the Schr¨odinger equation (3.1) in a time-periodic potential with a period
τ , can be described as a linear combination of the qusienergy states (QES) |ψ_{α}(t)i,
That is |Ψ (t)i which satisfies Eq. (3.1) can be written as

|Ψ (t)i =X

α

C_{α}|ψ_{α}(t)i , |ψ_{α}(t)i = e^{−i}^{α}^{t}|φ_{α}(t)i , (3.2)

where |φ_{α}(t)i is a periodic state, |φ_{α}(t + τ )i = |φ_{α}(t)i, and is a real parameter
called the quasienergy. The Floquet theorem for time-periodic problems is similar to
the Bloch theorem for a space-periodic problems in Solid-state physics. The role of

the quasienergy in the Floquet theorem is therefore similar to that of the quasimo- mentum in the Bloch theorem.

Now, the next goal is to find a equation for the QES |ψ_{α}(t)i. Defining
F ≡ ˆˆ H − id

dt (3.3)

and substituting Eq. (3.2) into Eq. (3.1) reveals that F (t) |ψˆ α(t)i = 0

⇒ F (t)eˆ ^{−i}^{α}^{t}|φ_{α}(t)i = 0

⇒ e^{−i}^{α}^{t}H (t) |φˆ α(t)i − e^{−i}^{α}^{t}id

dt |φα(t)i − |φα(t)i ide^{−i}^{α}^{t}
dt = 0

⇒ F (t) |φˆ _{α}(t)i = _{α}|φ_{α}(t)i . (3.4)

Equation (3.4) seems like an eigenvalue-eigenfunction equation. If a suitable basis
can be found, it is just required to solve an eigenvalue-eigenfunction equation instead
of a time-dependent Schr¨odinger equation. In this case, using the eigenvalue _{α}, the
eigenvector |φ_{α}(t)i and Eq. (3.2), we can get the evolution of all states.

### 3.1.2 Some properties of quasienergy and QES

If _{α} is one of the quasienergies and |φ_{α}(t)i is the corresponding quasieigenvector,
considering the follow transform,

^{0}_{α,m}≡ _{α}+ mω , (3.5)

|φ^{0}_{α}(t)i ≡ e^{imω}|φ_{α}(t)i , (3.6)
where ω = 2π/τ . This can be checked by substituting them into Eq. (3.2). We
then can find that the QES |ψ_{α}(t)i is unchanged upon this transformation, and so is

|Ψ (t)i. This means if _{α} is one of the quasienergies, so is _{α}+ mω, and the Floquet

states are physically equivalent if their quasienergies differ by mω.

Because of the time-periodic properties, the Floquet states |φ_{α}(t)i can be ex-
panded in Fourier series,

|φ_{α}(t)i =

∞

X

n=−∞

|φ_{α,n}i e^{−inωt} , (3.7)

with the Fourier components of the Floquet states

|φ_{α,n}i = 1
τ

Z τ 0

dte^{inωt}|φ_{α}(t)i , (3.8)

and the QES

|ψ_{α}(t)i = e^{−i}^{α}^{t}

∞

X

n=−∞

|φ_{α,n}i e^{−inωt} . (3.9)

Finally, the total state |Ψ (t)i, Eq. (3.2),

|Ψ (t)i =X

n,α

Cαe^{−i(}^{α}^{+nω)t}|φα,ni . (3.10)

Thus a state can be considered as a superposition of stationary states with energies
equal to (_{α}+ nω). This is why we call _{α} quasienergy.

The time-periodic Hamiltonian ˆH (t) can also be expanded in Fourier series, H (t) =ˆ

∞

X

n=−∞

Hˆne^{−inωt}. (3.11)

and thus

F (t) =ˆ

∞

X

n=−∞

Hˆ_{n}e^{−inωt}− id

dt . (3.12)

For the Hermitian oprator ˆF , one can introduce the composite Hilbert space which contains time-periodic wave fuction. The eigenvectors of ˆF satisfy the orthonormality condition

hφ_{α}(t)|φ_{β}(t)i = δ_{α,β} , (3.13)

and form a complete set

X

α

|φ_{α}(t)ihφ_{α}(t)| = I . (3.14)

After using Eq. (3.12) and Eq. (3.7), another form of Eq.(3.4) is given, X

n^{0},n

Hˆ_{n}^{0}e^{−in}^{0}^{ωt}− id
dt

e^{−inωt}|φ_{α,n}i = _{α}

∞

X

n=−∞

e^{−inωt}|φ_{α,n}i ,

⇒ X

n^{0},n

ˆH_{n}^{0}e^{−i(n}^{0}^{+n)ωt}− nωe^{−inωt}

|φ_{α,n}i = _{α}

∞

X

n=−∞

e^{−inωt}|φ_{α,n}i . (3.15)

For the mth component, X

n^{0},n

δm,n^{0}+nHˆn^{0} − nωδm,n

|φα,ni = α|φα,mi

⇒ X

n

ˆH_{m−n}− nωδ_{m,n}

|φ_{α,n}i = _{α}|φ_{α,m}i . (3.16)

Now, we can define ˆH_{n}^{0}−n≡ ˆH_{n}^{0}_{,n} and get
X

n

ˆH_{m,n}− nωδ_{m,n}

|φ_{α,n}i = _{α}|φ_{α,m}i . (3.17)

### 3.2 The extended Hilbert space

The above equation is formally equivalent to a time-independent Schr¨odinger equation and ˆF is the Hermitian operator which determines the quasienergies and the Floquet states. The corresponding Hilbert space is the direct product T ⊗ R of the original Hilbert space R and the Hilbert space of the time-periodic functions T . The inner product in T is defined by

hn|mi = 1 τ

Z τ 0

dtn^{∗}(t)m(t) . (3.18)

The most simple basis {|ni} of orthonormalized vectors for this space is the set of vectors defined by ht|ni = exp[−inωt], and the spatial orthonormalized basis {|βi} is

chosen arbitrarily for convenience, hαm | βni = hα | βi1

τ Z τ

0

e^{imωt}e^{−inωt}dt = δ_{αβ}δ_{nm}, (3.19)
and a time-periodic function is given by

|Φ (t)i = X

αm

f_{αm}e^{−imωt}|αi =X

αm

f_{αm}|αmi , (3.20)

where

f_{αm} =
Z τ

0

e^{imωt}f_{α}dt . (3.21)

### 3.2.1 Operators in the extended Hilbert space

In matrix form, any time-indepandent operator A in the extended Hilbert space T ⊗R is written as

I ⊗ A =

...

A 0 0 0 0

0 A 0 0 0

· · · 0 0 A 0 0 · · ·

0 0 0 A 0

0 0 0 0 A

...

...

n = 2

n = 1

n = 0

n = −1 n = −2

...

. (3.22)

For a time-periodic operator with exp[−iωt] term, B(t) = Ae^{−iωt}, the matrix element
of time doman is given by

m
e^{−iωt}

n = 1 τ

Z τ 0

e^{imωt}e^{−iωt}e^{−inωt}dt = δ_{m,n+1} . (3.23)
For a time-periodic operator C(t), C(t) = Ae^{iωt}, the matrix element of time doman
is given by

m
e^{iωt}

n = 1 τ

Z τ 0

e^{imωt}e^{iωt}e^{−inωt}dt = δ_{m,n−1}. (3.24)

B(t) and C(t) are written respectively as

B(t) =

... 0 1 0 0 0 0 0 1 0 0

· · · 0 0 0 1 0 · · · 0 0 0 0 1 0 0 0 0 0

...

⊗A and C(t) =

... 0 0 0 0 0 1 0 0 0 0

· · · 0 1 0 0 0 · · · 0 0 1 0 0 0 0 0 1 0

...

⊗A .

(3.25)
A simple concept here is that a e^{iωt} term would ”shift up” the spatial part a ”block”,
a e^{−iωt} term would ”shift down” the spatial part a ”block”. Furthermore a e^{2iωt} term

‘shifts up’ two block and so on.

Besides those operators, there is still another term, id/dt in operator ˆF . Through the similar method, the matrix element of id/dt is given by

m

id dt

n

= 1 τ

Z τ 0

e^{imωt}id

dte^{−inωt}dt = nωδm,n , (3.26)
and in matrix form id/dt is written as

id dt =

...

2ω 0 0 0 0

0 ω 0 0 0

· · · 0 0 0ω 0 0 · · ·

0 0 0 −ω 0

0 0 0 0 −2ω

...

(3.27)