使用約瑟夫森分支放大器的量子測量之研究

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_BT = 0.1~ω0, α = 0.1α₀, f = 0.1f₀ and γ = 0.005ω₀. . . 67

viii

5.2 (a)Response amplitude for different values of temperature T , k_BT = 0.1, 0.5, and 1.0~ω0, with α = 0.1α₀. (b)Response amplitude for dif- ferent value of the nonlinearity α, α = 0.095, 0.1, and 0.105α₀ with k_BT = 0.1~ω0. The remaining parameters are f = 0.1f₀and γ = 0.005ω₀. 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= 1.4ω₀. The remaining parameters are k_BT = 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, 1.195ω₀. The remaining parameters are k_BT = 0.1~ω0, α = 0.1α₀, f = 0.1f₀ and γ = 0.005ω₀. 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_BT = 0.1~ω0, α = 0.1α₀, f = 0.1f₀ and γ = 0.005ω₀. 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 = 0.01ω₀, ω_qx = 0.01, ω_ex = 1.16, k_BT = 0.1~ω0, α = 0.1α₀, f = 0.1f₀. 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. ω₀ = 1, k = 0.02 and F = 0.02 (b)The response curve is for different values of the driving amplitude strength. ω₀ = 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_csin δ . (2.1)

The critical current I_cis 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_csin (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_csinΦJ(t)

ϕ_o = I_csin δ(t) , (2.5)

where the generalized flux is defined by Φ_J =Rt

−∞V (t⁰)dt and ϕ₀ = ~/2e is the re- duced flux quantum, or ϕ₀ = Φ₀/2π, where Φ₀, 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 ≡ δ⁰ −2π Φ₀

Z

A · dl . (2.6)

where δ⁰ 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_Jϕ²₀Φ³_J(t) + O[Φ⁵_J(t)] (2.7) or simplely

IJ(t) = Icsin δ(t) = Ic



δ(t) − δ³(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_csin δ, the equation of the circuit is

~

2eC ¨δ + ~

2eR˙δ + I_csin 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)_2C² 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) = ~²δ˙²

4E_C , (2.10)

the potential energy of it is

U (δ) = E_J(1 − cos δ) − ~

2eI_eδ , (2.11)

and the Hamiltonian has the form

H = E_Cn²− E_Jcos δ − ~

2eI_eδ . (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

C _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

A₁ A₂

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_Jcos 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) − q² 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_2L², the Hamiltonian of a rf-SQUID is given by

H = Eˆ _Cnˆ² − E_Jcos ˆδ + E_L(ˆδ − δ_e)²

2 , (2.17)

where δ_e = ^2e

~Φ_e. The first term E_Cnˆ² 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^Φ2²⁰L. 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_c1sin δ1 − I_c2sin 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± = δ₁± δ₂

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_ccos (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_Cˆn²₊+ E_Cnˆ²₋− 2E_Jcos ˆδ₊cos ˆδ−+ E_L



2ˆδ₊− δ_e2

2 + ~

2eI_eδˆ− , (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_Jcos ˆδ .

Then by using the properties of orthonomal and complete set, hn | ˆn | n⁰i = δ_n,n⁰ 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^iˆδi

=



ˆn,X

m

 iˆδm

m!



= e^iˆδ . (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^iˆδ and e^−iˆδ can be presented in charge basis,

e^iˆδ=X

n

|n + 1ihn| , e^−iˆδ =X

n

|nihn + 1| , (2.24)

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

2E_JX

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_C² (1 − 2n_g)² + E_J² . (2.28)

n

-2 -1 0 1 2 3 0.5

n

(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

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

C

L

V

C

C

C

E

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 _Cˆn²− E_Jcos ˆδ + E_L(ˆδ − ˆδ_e)²

2 ,

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

U (δ) = −E_Jcos 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 Φ₀ − 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 EJ

2

1 EJ

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 I₀ 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 = ϕ₀I₀, where I₀ 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 EJ

2

1 EJ

2

preparation “quantronium” circuit JBA readout

E_J0 δ γ

Φ 2C

2C

tuning

~

R U(t)

φ

φ

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

#

+ Qˆ²

2C − E_{J 0}cos ˆδ − U (t)

R ϕ₀δ ,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_Jcos ˆθ

− E_J (

cos ˆθ ⊗

"

cos ˆδ 2 − 1

#

− d sin ˆθ ⊗ sin δˆ 2

)

+ Qˆ²

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 = Qˆ²

2C − E_{J 0}cos ˆδ − U (t)

R ϕ₀δ ,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 ˆδ⁴, 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_αi = E_α|ψ_αi. After getting the eigenenergies and eigenvectors, the time evolution of states is easily solved, |Ψ_α(t)i = P C_αe^−iEαt|ψ_αi. 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^−iαt|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 ^−iαt|φ_α(t)i = 0

⇒ e^−iαtH (t) |φˆ α(t)i − e^−iαtid

dt |φα(t)i − |φα(t)i ide^−iαt 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)

|φ⁰_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,ni e^−inωt , (3.7)

with the Fourier components of the Floquet states

|φ_α,ni = 1 τ

Z τ 0

dte^inωt|φ_α(t)i , (3.8)

and the QES

|ψ_α(t)i = e^−iαt

∞

X

n=−∞

|φ_α,ni 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_ne^−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

n⁰,n



Hˆ_n⁰e^−in0ωt− id dt



e^−inωt|φ_α,ni = _α

∞

X

n=−∞

e^−inωt|φ_α,ni ,

⇒ X

n⁰,n

 ˆH_n⁰e^{−i(n0+n)ωt}− nωe^−inωt

|φ_α,ni = _α

∞

X

n=−∞

e^−inωt|φ_α,ni . (3.15)

For the mth component, X

n⁰,n



δm,n⁰+nHˆn⁰ − nωδm,n



|φα,ni = α|φα,mi

⇒ X

n

 ˆH_m−n− nωδ_m,n

|φ_α,ni = _α|φ_α,mi . (3.16)

Now, we can define ˆH_n⁰−n≡ ˆH_n⁰_,n and get X

n

 ˆH_m,n− nωδ_m,n

|φ_α,ni = _α|φ_α,mi . (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^imωte^−inωtdt = δ_αβδ_nm, (3.19) and a time-periodic function is given by

|Φ (t)i = X

αm

f_αme^−imωt|αi =X

αm

f_αm|αmi , (3.20)

where

f_αm = Z τ

0

e^imωtf_αdt . (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^imωte^−iωte^−inωtdt = 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^imωte^iωte^−inωtdt = 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^imωtid

dte^−inωtdt = 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)