High Speed Homodyne Detector for Gaussian-Modulated Coherent-State Quantum Key Distribution

Yuemeng Chi

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto

High Speed Homodyne Detector for Gaussian-Modulated Coherent-State Quantum Key

Distribution

Yuemeng Chi

Master of Applied Science

Graduate Department of Electrical and Computer Engineering

University of Toronto

2009

We developed a high speed homodyne detector in the telecommunication wavelength region for a Gaussian-modulated coherent-state quantum key distribution experiment. We are able to achieve a 100 MHz bandwidth, ultra-low electronic noise and pulse-resolved homodyne detector. The bandwidth of this homodyne detector has reached the same order of magnitude of the best homodyne detectors reported. By overcoming photodiode response functions mismatch, choosing proper laser sources, ensuring the homodyne detector linearity and stabilizing the homodyne detection system, we demonstrate that the homodyne detector has a 10 dB shot-noise-to-electronic-noise ratio in the time domain at a local oscillator of 5.4108 photons/pulse at a laser repetition rate of 10 MHz. With this homodyne detector, we expect to increase our GMCS QKD experiment speed by 100 times, which will improve the key generation rate by 1-2 orders of magnitude.

ii Dedication

First and foremost I owe my deepest gratitude to my supervisors, Professor Hoi-

Kwong Lo and Professor Li Qian, who have supported me thoughout my research with their patience and knowledge. I attribute my master work to their encouragement and eﬀort. Without their support, this thesis would not have been completed. It is an honor for me to work with them during my last two years’ study. One simply could not wish for a better or friendlier supervisor.

I am also deeply appreciate the help and advice received from Doctor Bing Qi, who essentially teaches me everything in an optical and electrical lab. I am grateful to Doctor

Bing Qi for his knowledge and many useful discussions that motivated me.

Special thanks are extended to Professor J. Stewart Aitchison, Professor Lacra Pavel, and Professor Joyce Poon for their time, advice, and willingness to serve on the commit- tee.

I would like to show my gratitude to Professor Alex Lvovsky at University of Calgray for generously providing the homodyne detector electronic design and printed circuit board layout. I also thank Professor SunHyun Youn and Nitin Jain for their kind help in building the homodyne detector circuit and sharing a lot of construction experience.

I would also extend my thanks to Liang Tian, who has helped a lot in tuning the homodyne detector circuit. I also would like to acknowledge Professor Namdar Saniei and Doctor Wen Zhu for helpful discussions. It is a pleasure to thank a friendly and cheerful group of fellow students, Viacheslav Burenkov, Wei Cui, Chi-Hang Fred Fung,

Junbo Han, Wolfram Helwig, Kenny Ho, Dongpeng Kang, Xiongfeng Ma, Jason Ng,

Wing-Chau Ng, Chris Sapiano, Peyman Sarraﬁ, Gigi Wong, Fei Ye, Jiawen Zhang, Lijun

Zhang, and Eric Zhu for their support and friendship. I also thank 3GMetalWorx Inc. for providing a professional shielding metal box for the homodyne detector circuit.

Furthermore, I would like to thank Ms. Diane Silva and Ms. Linda Liu for their eﬃcient and professional administrative work.

iii Finally and most importantly, this thesis would not have been possible without the endless love and support from my family. This thesis is dedicated to my husband and my parents.

iv Contents

1 Introduction 1

1.1 Background ...... 1

1.2 Motivation: high speed homodyne detector ...... 6

1.3 Objective ...... 7

1.4 Organization ...... 8

2 Review of GMCS QKD and Homodyne Detection 9

2.1 Gaussian-modulated coherent-state quantum key distribution ...... 9

2.1.1 Protocol ...... 9

2.1.2 State of the art ...... 12

2.2 Homodyne detection ...... 14

2.2.1 Introduction ...... 14

2.2.2 State of the art ...... 19

2.3 Summary ...... 21

3 GMCS QKD over 20 km Fiber 22

3.1 Experimental setup ...... 22

3.2 Secure key rate formula ...... 24

3.3 Results ...... 27

3.4 Discussions ...... 33

3.5 Summary ...... 36

v 4 High Speed Homodyne Detector 37

4.1 Requirement of a homodyne detector in GMCS QKD ...... 37

4.2 High speed homodyne detector design ...... 38

4.2.1 Homodyne detector optical setup in ﬁber ...... 38

4.2.2 Homodyne detector electrical circuit ...... 39

4.3 Challenges ...... 40

4.3.1 Low electronic noise ...... 41

4.3.2 Diﬀerent photodiode response functions ...... 42

4.3.3 Linearity ...... 45

4.3.4 Laser source ...... 47

4.3.5 Optical stability ...... 52

4.4 Summary ...... 54

5 Performance of the Homodyne Detector 55

5.1 Experimental plan ...... 56

5.2 Measurement with CW light ...... 56

5.2.1 Experimental setup ...... 57

5.2.2 Noise measurement in the time domain ...... 58

5.2.3 Noise measurement in the frequency domain ...... 60

5.3 Measurement with pulsed light ...... 62

5.3.1 Pulsed laser source ...... 62

5.3.2 Noise measurement in the time domain ...... 63

5.3.3 Noise measurement in the frequency domain ...... 73

5.4 Conclusions and discussions ...... 77

6 Conclusion and Future Work 81

6.1 Signiﬁcance and contribution ...... 81

6.2 Future work ...... 83

vi List of Tables

1.1 Secure key rate (bit) per pulse for GMCS [1], decoy state [2] and DPS

[3] protocols over 5-km telecommunication ﬁber. Secure key rates for de-

coy state and DPS protocols are simulation results based on experimental

conditions...... 5

3.1 Parameters used in the key rate simulation , from Ref. [1]. The length of

the ﬁber is 5 km...... 31

3.2 20-km GMCS QKD parameters and results (e: experimental result; c:

calculated result) ...... 33

3.3 Secure key rate in our GMCS QKD experiment and in Ref. [4] over 20-km

ﬁber. rep. :repetition ...... 33

4.1 Speciﬁcations of FGA04 InGaAs photodiode (typical values)...... 39

4.2 Speciﬁcations of the two lasers used in photodiode linearity test ...... 49

5.1 Parameters in the key rate simulation (given in Ref. [1]). Here we assume

²A and Nleak are the same for high-speed and low-speed GMCS QKD experiments...... 71

6.1 HD peformance comparison between our group and Ref. [5] ...... 82

vii List of Figures

1.1 One-time-pad scheme ...... 2

1.2 Alice prepares four photons with arbitrary polarizations. Eve taps them

from the channel and uses her basis (horizontal and vertical basis) to

measure them. The polarization of those photons will collapse to Eve’s

basis. Thus, Eve cannot perfectly duplicate those states...... 3

2.1 GMCS QKD protocol...... 11

2.2 Schematic of a homodyne detection. BS: beam splitter; SIG: signal; LO:

local oscillator; PD: photodiode; φ: introducing a phase between the signal

and the LO; Black line: optical path; Blue line: electrical path; Dashed

box: homodyne detector ...... 15

2.3 LO and signal states in the phasor space...... 16

2.4 Scheme of a photocurrent subtraction. PD: photodiode ...... 19

3.1 Schematic of the GMCS QKD system. L: 1550 nm CW ﬁber laser, PC1−5:

polarization controllers; PBS1−3: polarization beam splitters or combiners;

AM0−1: amplitude modulators; PM1−2: phase modulators; SW1−2:optical

switches; AOM+(AOM−): upshift(downshift) acousto-optic modulators;

VOA1−2: variable optical attenuators; ISO: isolator; C: ﬁber coupler; HOM: homodyne detector [1, 6, 7] ...... 23

viii 3.2 Quadrature variances prepared by Alice, quadrature variances measured

by Bob, and equivalent input noise χ. Quadrature variance prepared by

Alice, and equivalent input noise χ are referred to the input. Noise on

Bob’s side is referred to the output...... 25

3.3 QKD experimental results. The equivalent input noise has been deter-

mined experimentally to be χ = 6.13[6, 7]...... 28

3.4 Determine δ by using a high modulation variance VA 40000 and a weak LO (105 photon/pulse). The result is δ = 0.0049 ...... 29

3.5 Noise of the balanced HD as a function of LO power. With a LO of

1.2107 photons/pulse, the electronic noise is 6.8 dB below the shot noise

(plot with raw data obtained from [1]) ...... 30

3.6 Secure key rate as a function of the electronic noise (in shot noise unit)

under the “general model”. Parameters in this simulation are in Table 3.1 . 31

3.7 The leakage from LO to signal. LO: local oscillator; SIG: signal; LE:

leakage; PBS: polarization beam splitter; LO is 5-6 orders of magnitude

higher than signal. The leakage from signal to LO is negligible. Arrowed

lines indicate the polarization of the beam ...... 32

3.8 Key rate simulation when ultra-low loss ﬁber and standard ﬁber are used

in the GMCS QKD experiment. Parameters are based on Table 3.2 under

the “realistic model” when β = 0.898 ...... 34

3.9 Key rate simulation with excess noise and in the absence of excess noise.

Parameters are based on Table 3.2 under the “realistic model” when β =

0.898 ...... 35

4.1 Homodyne detection setup in the telecommunication wavelength. LO:

local oscillator; OVD: optical variable delay; FC: 50:50 ﬁber coupler; VOA:

variable optical attenuator; PD: photodiode; AMP: electronic ampliﬁers. . 39

ix 4.2 A simpliﬁed homodyne detector circuit. PD: photodiode (Thorlabs,FGA04);

OPA847: operational ampliﬁer (Texas Instrument) ...... 40

4.3 Photo of the circuit board of the homodyne detector. Two FGA04 photo-

diodes are in the upper left corner...... 42

4.4 Customized metal box for shielding, constructed by 3GMetalWorx Inc..

Dimensions (cm): 8.126.674.22; Thickness (cm): 0.0406 ...... 43

4.5 Photodiode impulse responses test. (a)Photodiode impulse responses test

setup. Laser: Picoquant PDL 800-B, pulse width: 30-50 ps; RLoad = 50 Ohm. Laser pulses are sent to the photodiode and the output electrical

voltage V0 is measured by an oscilloscope. (b) V0 as a function of time for ten photodiodes. We acknowledge Liang Tian’s testing work in this

experiment...... 44

4.6 Residual signals at the output of the HD, after balancing the two signals

to the photodiodes. Laser: PriTel Mode-lock femsto-second ﬁber laser;

pulse width: 0.7 ps - 1.2 ps; LO power = 13.8 µW. Horizontal scale: 20

ns/div; Vertical scale: 10 mV/div...... 45

4.7 Photodiode impulse response with (a) pulse width: 0.7-1.2 ps; horizontal

scale: 2 ns/div; vertical scale: 50 mV/div (b) pulse width: 50 ns; edge

time: 15 ns; horizontal scale: 20 ns/div; vertical scale: 2 mV/div . . . . . 46

4.8 (a) Circuit linearity test circuit. It is the ampliﬁcation part of the circuit

in Fig. 4.2. OPA847: electronic ampliﬁer; (b) Circuit linearity test setup.

Electrical pulses (generated by a function generator, repetition rate: 10

MHz; pulse width: 50 ns) is sent to the circuit in (a). The output is

measured by an oscilloscope...... 47

4.9 Output peak voltage as a function of the input peak voltage for both

positive and negative signals. The slope (HD electronic ampliﬁers voltage

gain) is 22. The straight line is the linear ﬁt...... 48

x 4.10 (a) Photodiode linearity test circuit. PD: photodiode; (b) Photodiode lin-

earity test setup. Laser I or II pulses are sent to one of the photodiodes.

An oscilloscope is used to measure the output peak voltage. PD: photo-

diode; Switch is connected with port 1,2 to test the photodiode linearity,

while connected with port 3 to measure the optical power...... 50

4.11 Photodiode output peak voltage as a function of the photon number per

pulse with laser I ( 1 ps pulse width) as a source...... 51

4.12 Photodiode output peak voltage as a function of the photon number per

pulse for laser II ( 50 ns pulse width) as a source ...... 52

5.1 Experimental setup with CW LO (NP Photonics). CW L: 1550 nm CW

ﬁber laser; VOA1-2: variable optical attenuators; LO: local oscillator; FC:

50:50 ﬁber coupler; OVD: optical variable delay; PD: photodiodes; AMP:

HD electronic ampliﬁers; OSC: oscilloscope (Lecroy); RFSA: RF spectrum

analyzer (HP 8564E); Dashed line box: homodyne detector...... 57

5.2 Calibration setup to get the balanced condition. Pulsed L: 1550 nm Pri-

Tel pulsed femto second laser; VOA1-2: variable optical attenuators; LO:

local oscillator; FC: 50:50 ﬁber coupler; OVD: optical variable delay; PD:

photodiodes; AMP: HD electronic amplifers; OSC: oscilloscope (Lecroy);

RFSA: RF spectrum analyzer (HP 8564E); Dashed line box: homodyne

detector...... 58

5.3 Oscilloscope graph for HD noise measurement at a CW LO power of 0.187

mW. Horizontal scale: 5 µs/div. Vertical scale: 2 mV/div...... 59

5.4 Voltage variance as a function of the CW LO power in the time domain.

The red line has a slope of 1...... 60

xi 5.5 RF spectrum analyzer background noise spectrum (lowest curve), HD

electronic noise spectrum (second lowest curve) and HD noise spectra at

CW LO powers of 6.4400, 4.1400, 2.5380, 1.5960, 1.0180, 0.6460, 0.4140,

0.2528,0.1592,0.1014,0.0640,0.0410 and 0.0252 mW (from the highest to

the third lowest curve). Resolution bandwidth: 1 MHz ...... 61

5.6 Noise spectrum from DC to 10 MHz at an LO power of 0.3 mW. Resolution

bandwidth: 1 MHz ...... 62

5.7 HD noise power and shot noise power as a function of CW LO in the

frequency domain. The red line has a slope of 1...... 63

5.8 Experimental setup of HD noise measurement in the time domain. CW

L: 1550 nm CW ﬁber laser; VOA1-2: variable optical attenuators; LO:

local oscillator; FC: 50:50 ﬁber coupler; OVD: optical variable delay; PD:

photodiodes; AMP: HD electronic ampliﬁers; OSC: oscilloscope (Lecroy);

Dashed line box: homodyne detector...... 64

5.9 HD output waveform under the balanced condition at an LO power of

0.786 mW. Horizontal scale: 50 ns/div. Vertical scale: 50 mV/div. The

square box on this graph indicates one cycle...... 64

5.10 One measurement frame at an LO power of 0.4 mW. Horizontal scale: 10

mV/div, Vertical scale: 5 µs/div...... 66

5.11 Data processing procedure (LO = 0.77 mW). (a) 500 original curves (one

frame) S; (b) Background curve (i.e., the average curve) of S; (c) 500

processed curves (one frame) T ...... 67

5.12 HD noise variance and shot noise variance as a function of the pulsed LO

power in the time domain. The red line has a slope of 1...... 69

5.13 Correlation coeﬃcient between nth (X) and n + mth (Y) pulses at an LO

power of 0.77 mW ...... 70

xii 5.14 Quadrature value of nth pulse X(n) and that of n+1 pulse Y (n) = X(n+1)

at an LO power of 0.77 mW ...... 71

5.15 Key rate simulation as a function of the transmission distance with a

high-speed HD (running at 10 MHz repetition rate) and a low-speed HD

(running at 100 kHz repetition rate) ...... 72

5.16 Experimental setup in the frequency domain. CW L: 1550 nm CW ﬁber

laser; VOA1-2: variable optical attenuators; AM: amplitude modulator;

PC: polarization controller; LO: local oscillator; FC: 50:50 ﬁber coupler;

OVD: optical variable delay; PD: photodiode; AMP: HD electronic am-

plifers; RFSA: RF spectrum analyzer (HP 8564E); Dashed line box: ho-

modyne detector...... 73

5.17 Noise spectrum at an LO power of 0.786 mW. Frequency range: DC to

100 MHz. Resolution bandwidth: 100 kHz...... 74

5.18 Noise spectrum at an LO power of 24.56 µW when (a)two photodiodes are

illuminated; (b)one photodiode is blocked. Resolution bandwidth: 100 kHz 75

5.19 Noise spectra at diﬀerent LO powers. Frequency span: 5 to 6 MHz. Reso-

lution bandwidth: 10 kHz. LO powers are 0.0029, 0.0072, 0.0142, 0.0292,

0.0458, 0.0721, 0.1136, 0.1784, 0.2920, 0.4580, 0.7180, 1.1340, 1.7760,

2.9200 mW from the lowest curve to the highest curve, respectively. . . . . 76

5.20 Noise power as a function of the LO power in the frequency domain. The

red line has a slope of 1...... 77

6.1 A self-diﬀerencing scheme in homodyne detection. Red pulses are optical

pulses and blue pulses are electrical pulses...... 84

xiii Chapter 1

Introduction

In the information era, cryptography has become extremely important for governments, the military, businesses and the public. It allows people to achieve secure communications between diﬀerent places without worries of information leakage. Quantum cryptography

(quantum key distribution) has been proved to be unconditionally secure and has been studied widely in the past decade[8, 9]. To improve the quantum information transmission speed, the construction of a high speed homodyne detector (HD) for Gaussian- modulated coherent-state (GMCS) quantum key distribution (QKD) experiments will be investigated in this thesis. In this chapter, background knowledge on GMCS QKD and homodyne detection will be presented.

1.1 Background

How to achieve a secure communication has been an important question for thousands of years. In a secure communication, Alice and Bob would like to communicate in a secure manner in the presence of an eavesdropper, traditionally called Eve. Many encryption techniques have been invented, however, their securities are not completely guaranteed.

For example, one famous encryption scheme called RSA, invented by Rivest, Shamir and

Adleman in 1978 [10], is widely used in payment card chip technology. The RSA idea

1 Chapter 1. Introduction 2

Figure 1.1: One-time-pad scheme employs that for certain functions f(x), it is easy to compute f(x) given x, but diﬃcult to recover x given the value of f(x). For instance, one can easily compute 97 83 =

8051, while it requires a lot of eﬀort to ﬁnd out the prime factors of 8051. Security of RSA is based on the computational complexity assumption of factorization of large integers. However, this assumption can be violated by fast factoring using a quantum computer[11], hence the security is threatened.

As a signiﬁcant development to realize secure communication, the one-time-pad scheme was ﬁrst proposed by G. Vernam in 1917 [12] and has since been proved secure by C.

Shannon [13]. In this scheme, shown in Fig. 1.1, Alice and Bob share a binary key string (pad) with a length equal to the original message. Alice ﬁrst performs an XOR operation (i.e. addition modulo two) between every bit of the original message and every corresponding bit of the key to generate an encrypted message. She then transmits this encrypted message to Bob. Bob decodes the original message by performing an XOR operation between the encrypted message and the same key as Alice used to encrypt the message. The key has to be as long as the message, and cannot be reused. Whenever

Alice wants to send a new message, she needs to ﬁnd a way to distribute the key to Bob without leaking any information about the key to Eve. This is called the key distribu- Chapter 1. Introduction 3

Figure 1.2: Alice prepares four photons with arbitrary polarizations. Eve taps them from the channel and uses her basis (horizontal and vertical basis) to measure them. The polarization of those photons will collapse to Eve’s basis. Thus, Eve cannot perfectly duplicate those states. tion problem. Therefore, the problem of secure communication with the one-time-pad encryption becomes the problem of key distribution. Solving the key distribution problem is challenging because classical key information (i.e. the key string is represented by classical bits) can be tapped and duplicated during transmission through a classical communication channel (e.g. a ﬁber-optic link) without being detected by Alice and Bob.

The key distribution problem is solved by the proposal of quantum key distribution

(QKD)[8, 9, 14, 15]. The security of QKD is based on fundamental quantum mechanical principles, such as the quantum no-cloning theorem [16, 17, 18] , which states that an arbitrary quantum state cannot be perfectly duplicated, and the Heisenberg uncertainty principle which states that two non-commuting observables cannot be precisely known together. For example, if Alice uses the polarization of a single photon to encode key information and sends it to Bob, Eve may intercept this single photon from the channel and measure the polarization of the single photon in an orthogonal basis. However, the polarization of the single photon will collapse to one of the basis states chosen by Eve and will not be able to be perfectly duplicated (shown in Fig. 1.2). Another example

h is the Heisenberg uncertainty principle ∆x∆p C 4π , stating that the position (x) and Chapter 1. Introduction 4 momentum (p) of a particle (or wave) cannot be precisely known at the same time. If

Eve is trying to measure the position, she will inevitably introduce more uncertainty to the momentum. If Alice and Bob monitor the uncertainty of both x and p, they will be able to detect Eve if the added noise on x or p is above a threshold. Due to the diﬃculty of encoding information in the position and momentum of a particle (or wave), a scheme of encoding information in the amplitude and phase quadrature of the electric ﬁeld is developed [19].

QKD has been studied intensively for more than one decade [8, 9, 14, 15, 20, 21, 22].

There are many impressive progresses in both theory and experiment [14, 19, 23, 24,

25, 26]. QKD systems have been implemented with single photon source [14, 15, 27], in which properties of single photons (polarization, phase, etc) are used to encode key information, and coherent states source [19, 28], in which information is encoded in the measurable quantities of a coherent state, such as the amplitude and phase quadratures of a faint laser pulse (weak coherent state).

In the best-known BB84 QKD protocol [14], a perfect single photon source is used to guarantee the security. Unfortunately, a single photon state is difficult to generate in practice, despite tremendous efforts [29, 30, 31]. In practical systems, highly attenuated coherent laser sources that have a non-negligible chance to emit single-photon pulses are used. However, with such highly attenuated laser sources, unconditional security cannot be directly applied because of their finite probability of emitting multi-photon pulses.

To solve this problem, special techniques based on currently available highly attenuated coherent laser sources, such as decoy state protocol, have been implemented to improve the key rate [23, 32, 33, 34, 35, 36, 37, 38].

To avoid the requirement for a true single-photon source, coherent states QKD has been proposed in the past few years as a promising alternative to the commonly utilized single photon QKD. QKD based on Gaussian-modulated coherent-state (GMCS) protocol has recently attracted increasing interests [19, 39, 40, 41, 42]. This protocol will be Chapter 1. Introduction 5 discussed in detail in Section 2.1.1. There are several advantages of this protocol over single photon QKD. [43, 44].

1. The coherent state can be produced easily by a practical laser source, while the

perfect single photon source is still unavailable.

2. The homodyne detector in the GMCS QKD can be constructed using high eﬃcient

PIN photodiodes [39], while the eﬃciency of the single photon detector in the

telecommunication wavelength region is low.

3. More than one bit of information could be transmitted by sending one coherent

state and hence GMCS protocol could achieve a high key rate.

GMCS QKD has a distinct advantage in its potential application in achieving a higher key rate over a short distance [19]. A comparison of the key rate per pulse for GMCS, decoy state and diﬀerential phase shift (DPS, another example of single photon QKD)

QKD is shown in Table 1.1. More detailed comparison between GMCS QKD and single photon QKD can be found in Ref. [45].

As shown in Table 1.1, the secure key rate per pulse over a short distance in the

GMCS QKD experiment is two orders of magnitude higher than those in single photon

QKD experiments (decoy state and DPS QKD). However, the current experimental implementation of GMCS QKD is below 1 MHz repetition rate [1, 43], much lower than

Table 1.1: Secure key rate (bit) per pulse for GMCS [1], decoy state [2] and DPS [3] protocols over 5-km telecommunication ﬁber. Secure key rates for decoy state and DPS protocols are simulation results based on experimental conditions. Secure key rate per pulse

GMCS [1] 0.3

Decoy state [2] B0.002

DPS [3] B0.001 Chapter 1. Introduction 6 single photon QKD at a few GHz repetition rate [4, 46], thus the secure key rate per second in a GMCS QKD experiment is still lower than that of a high speed single photon

QKD experiment. Development of a high speed GMCS QKD system will be an interest- ing research direction in future experimental QKD implementation. Since the repetition rate of GMCS QKD experiments is fundamentally limited by the speed of the homodyne detector (HD, to be shown in Sections 1.2 and 2.1.2), a high speed HD that meets the requirements of GMCS QKD experiments should be developed as the ﬁrst step to achieve this long term goal.

1.2 Motivation: high speed homodyne detector

In GMCS QKD experiments, key information is encoded in electric ﬁeld quadratures of weak coherent states by Alice. On Bob’s side, he will measure the quadrature values using a homodyne detection technique. Homodyne detection is extremely useful since it can be used to characterize the electric ﬁeld of an optical signal, even if it is a non-classical signal, such as squeezed state. This unique characteristic makes homodyne detection a popular implementation in quantum applications [47, 48, 49].

Homodyne detection has been used for many years in optical communication [49, 50].

There are also commercialized fast (up to 40 GHz) balanced receivers that can be used in homodyne detection [51, 52]. Unfortunately, they are not suitable for GMCS QKD experiments. In classical optical communications, usually strong light is sent to the signal port. For example, -20 dBm signal is used in Ref. [53] and 0 dBm signal is used in Ref. [54]. However, in GMCS QKD experiments, the signal pulse is very weak, less than -95 dBm in Ref. [1] and -90 dBm in Ref. [43] . Therefore, the homodyne detector

(HD) used in GMCS QKD experiments should be very sensitive, which requires ultra- low electronic noise of the HD. In addition to the low noise requirement, in GMCS QKD experiments, HDs need to be able to measure the individual pulse in the time domain Chapter 1. Introduction 7 since key bits are encoded in every signal pulse. Owing to these requirements, most research groups have to develop their own HDs for quantum information and quantum optics implementations [40, 43, 47, 48, 55]. To date, only a few groups have been able to construct broad bandwidth HDs ( 100−250 MHz) for quantum measurements [5, 47, 56,

57], however, none is demonstrated at 1550 nm wavelength. High-speed HDs operating in the telecommunication wavelength region that can be used in quantum optics and quantum information are still lacking. In this thesis, my goal is to develop a high speed

HD in the telecommunication wavelength that will be suitable for high speed GMCS

QKD experiments.

1.3 Objective

The long term objective is to experimentally develop a high speed QKD system based on

GMCS protocol. This will be achieved by developing high speed modulations, detections, data acquisitions, classical data processing algorithms, etc (to be discussed in Section

2.1.2). As the ﬁrst stage to achieve this objective, the goal of my thesis is to enable a fully-ﬁber based high speed GMCS QKD system by constructing a fast detection system: homodyne detector (HD) in the telecommunication wavelength region.

My research is carried out with the following objectives.

• Establish the requirements and challenges of an HD for a high-speed GMCS QKD

system.

• Provide both electrical and optical designs of a high speed HD to meet those re-

quirements.

• Construct and test the HD, and outperform the previous demonstrations reported

in the literature. Chapter 1. Introduction 8

• Evaluate HD performance in real QKD and suggest directions for future high speed

GMCS QKD experimental implementation.

1.4 Organization

The thesis is organized as follows: I brieﬂy review the development of GMCS QKD and homodyne detection in Chapter 2, from principle to the state of the art. In Chapter 3,

I will discuss an example of using a custom-designed low speed HD in a 20-km GMCS

QKD implementation over an existing system. The main work of this thesis is discussed in Chapter 4 and Chapter 5. In Chapter 4, requirements of a high speed HD that can be used in high speed GMCS QKD experiments are presented. Challenges and solutions of constructing a high speed HD are also discussed in this chapter. In Chapter 5, I will present the testing results of our constructed HD in both time and frequency domains, with CW and pulsed light, respectively. Our results show that this HD has a bandwidth of 100 MHz, which will enable GMCS QKD experiments at a repetition rate of 10

MHz. The expected key rate of GMCS QKD experiments with this HD will be increased by 1-2 orders of magnitude, which is comparable to the high speed QKD based on single photon protocols. In Chapter 6, I will summarize the thesis, discuss the signiﬁcance of the work and suggest future work for developing a high speed GMCS QKD system. Chapter 2

Review of GMCS QKD and

Homodyne Detection

2.1 Gaussian-modulated coherent-state quantum key

distribution

Over the past few years, quantum key distribution (QKD) using Gaussian-modulated coherent-state (GMCS) protocol has drawn a lot of attention [19, 39, 40, 41, 42]. The uncertainty principe, which states that the amplitude quadrature (x) and the phase quadrature (p) of a coherent state cannot be precisely known simultaneously, is the foun- dation of GMCS QKD. Any eavesdropping of one quadrature will introduce additional noise to the other quadrature. Alice and Bob are able to detect Eve if they monitor the variances of x and p together. In this section, I will give an overview of the GMCS QKD protocol and the state of the art of this topic.

2.1.1 Protocol

In the classical electromagnetism, a light ﬁeld can be characterized by x cos ωt + p sin ωt, where ω is the angular frequency, and x and p are amplitude and phase quadratures.

9 Chapter 2. Review of GMCS QKD and Homodyne Detection 10

When the light intensity is very weak and the quantum eﬀect is considered, the measurement of one quadrature will introduce more uncertainty to the other quadrature. In the

GMCS QKD, Alice encodes x and p on each bit and Bob randomly chooses x or p to decode information. Eve does not know which basis Bob will select to perform measurement. If Eve eavesdrops in a random basis, according to the uncertainty principle, her measurement of x (p) will disturb the variable p (x), i.e. Eve cannot simultaneously reduce measurement error on both quadratures. Therefore, the presence of Eve inevitably introduces additional noise in Bob’s measurements, which we call excess noise. After the key transmission, by comparing some of Bob’s measurement results with the x, p values prepared by Alice, Bob can estimate the excess noise, which is attributed to Eve. Hence,

Bob can estimate the amount of information leaked to Eve. Alice and Bob then distill the secure information by a classical data processing. Note that, the diﬀerence between

GMCS QKD and classical optical communication is that weak (quantum) signal is used in GMCS QKD. If the excess noise in GMCS QKD is above a threshold, no secure key can be transmitted. An example of using a strong signal in GMCS QKD will be shown in Fig. 3.4 of Section 3.3, in which no positive secure key is generated.

The protocol of the GMCS QKD is described as follows (shown in Fig. 2.1) [19, 43,

39, 45, 58].

1. Alice generates two random sets of continuous variables x and p with a Gaussian

distribution that has a zero average (variance = VAN0, N0 is the shot noise unit). For a continuous Gaussian noise channel, a Gaussian distribution of the input will

yield an optimal channel capacity [59]. In the GMCS QKD, Alice encodes random

bits (key information) by modulating the amplitude quadrature (x) and the phase

quadrature (p) of weak coherent states Sx + ipe (typically less than 100 photons

in each pulse) with her Gaussian distributed sets x, p. Experimentally this is

realized by modulating the intensity and the phase of each pulse. On the receiver’s

side, Bob measures either x or p quadrature of the weak coherent states randomly Chapter 2. Review of GMCS QKD and Homodyne Detection 11

Figure 2.1: GMCS QKD protocol.

by using a homodyne detection. Note that, the signal cannot be ampliﬁed before

being detected, since quantum coherent states cannot be perfectly cloned without

paying penalty (adding excess noise).

2. Through an authenticated public channel (Eve can listen to the information but

cannot modify), Bob informs Alice about the quadratures he picked. Alice then

discards the quadratures that were not measured by Bob. At this stage, Alice

shares a set of correlated Gaussian variables (called the ”raw key”) with Bob.

3. Alice and Bob then publicly compare a random sample of their raw key to evaluate

the transmission eﬃciency of the quantum channel ηG (including channel eﬃciency

G and Bob’s system eﬃciency η), and the excess noise ² of the QKD system. Excess

noise is the noise above the vacuum noise level associated with channel losses [39].

It reﬂects possible leakage information to Eve. Based on the parameters, Alice and

Bob can evaluate their mutual information IAB and the information obtained by

1 Eve IBE [19, 60]. This stage is a classical data processing process and can be

1Here, we assume the secure key is made of Bob’s data. Bob will publish some of his measurements Chapter 2. Review of GMCS QKD and Homodyne Detection 12

divided into reconciliation (correcting the errors while minimizing the information

revealed to Eve. In this thesis, we only consider reverse reconciliation) and privacy

ampliﬁcation (making the key secret). If the reconciliation is perfect, a secure key

of length IAB − IBE will be distilled after the classical data processing.

2.1.2 State of the art

The GMCS QKD protocol was developed in 2002 [28]. The security of the GMCS QKD was ﬁrst proven against individual attacks with direct [28] or reverse [19, 60] reconciliation schemes. Security proofs were then given against general individual attacks [60] and general collective attacks [43, 61, 62]. Until recently, three groups have independently claimed they proved the unconditional security [63, 64, 65].

The ﬁrst GMCS QKD experiment based on homodyne detection was demonstrated in free space [19]. This experiment was carried out with 780 nm optical light. Hence it did not establish the feasibility of implementation in ﬁber communication networks, which are widely used for long-distance communication.

Fiber-based GMCS QKD systems over a practical distance remain challenging. So far only two groups have implementations of the GMCS QKD over a practical distance operated fully on ﬁber-based components [1, 43, 66]. To reduce excess noise arising from the leakage from the local oscillator (LO), in Ref. [43], Mach-Zehnder interferometers

(MZIs) with largely imbalanced path lengths (80 m) were employed to separate the signal and the leakage in the time domain (time-multiplexing scheme). However, in practice it is quite challenging to stabilize a MZI with a large length imbalance.

One recent paper [66] by the same group reported a field test of a GMCS QKD prototype integrated into a preinstalled quantum cryptography telecommunication net- work. In that paper, a polarization-time multiplexing scheme is employed. The system and Alice will modify her data according to Bob’s results. This process is called reverse reconciliation since this flow has a reverse direction from the key transmitting flow. Chapter 2. Review of GMCS QKD and Homodyne Detection 13 is running at 500 kHz laser repetition rate.

In contrast, our group (Experimentalists: Dr. Bing Qi and Lei-Lei Huang, under the supervision of Profs Li Qian and Hoi-Kwong Lo) has developed a polarization-frequency- multiplexing scheme to eﬀectively suppress the leakage of the LO with balanced MZIs.

[1]. The system design will be described in Section 3.1. In this experiment, the laser repetition rate is 100 kHz.

In parallel with GMCS QKD based on homodyne detection, a heterodyne detection scheme, in which Bob measures both x and p quadratures simultaneously, was proposed

[67] (Note that, local oscillator and signal are at the same frequency. The heterodyne detection used here is different from that defined in classical communication in which local oscillator and signal are at different frequencies). The security proof of the GMCS

QKD based on heterodyne detection is given in Refs. [63, 68]. With this scheme, there is no need to choose a random quadrature on Bob’s side. The experimental implementation of GMCS QKD based on heterodyne detection has been realized in 2005 [69].

Although the GMCS QKD has a potential application in transmitting at high key rates, so far all reported experimental systems are running below 1 MHz. There are several limitations for a practical GMCS QKD system in achieving high speeds.

• The repetition rate of a GMCS QKD system is essentially limited by the homodyne

detector (HD) bandwidth. In QKD, each pulse quadrature encoded with Gaussian

random numbers has to be measured individually in the time domain. This will

require the response time for HD to be shorter than the inverse of the laser repeti-

tion rate. In other words, the bandwidth of the HD in the frequency domain should

be greater than the repetition rate. Intuitively, one should increase the bandwidth

of the HD to obtain a high repetition rate. However, the electronic noise is pro-

portional to the HD bandwidth [1, 40]. A wider bandwidth will result in greater

electronic noise and lead to a lower secure key rate [1]. Chapter 2. Review of GMCS QKD and Homodyne Detection 14

• The repetition rate is also limited by the speeds of computer-driven data acquisition

systems, which are typically lower than a few MHz. In Ref. [1] , a data acquisition

card (NI, PCI-6115) with a sampling rate of 10 M samples/s is employed to acquire

data. If one wants to get suﬃcient data points for each pulse (like 20), the maximum

repetition rate can only be a few hundreds of kHz.

• Speed of the classical reconciliation data processing will limit the GMCS QKD

speed.

Owing to the above limitations, there is no reported GMCS QKD experiment above

1 MHz repetition rate. For a GMCS QKD key transmission demonstration, if data processing is not performed in the real time, a fast oscilloscope (40 G Samples/s) can be used to acquire and store raw data. Considering that the HD bandwidth is a fundamental limitation, we will develop a fast HD that can enable a high speed GMCS QKD system in this thesis.

2.2 Homodyne detection

2.2.1 Introduction

Homodyne detection is a well-established technique for measuring the amplitude and phase quadrature of a weak optical signal [49, 70]. Fig. 2.2 shows a schematic of homodyne detection. The signal is mixed at a beam splitter (with a 50/50 splitting ratio) with a strong local oscillator (LO)[49, 71, 72] with a deﬁned optical phase (φ in Fig. 2.2).

This phase is introduced by a phase modulator at the LO beam in Fig. 2.2. The output ports of the beam splitter are attached to two photodiodes. The photocurrent diﬀerence

(after the subtraction shown in Fig. 2.2) is finally amplified by an electronic amplifier.

I follow Ref. [72] to derive the output of a homodyne detection. The electric ﬁelds of the signal ES(t) and the LO EL(t) are Chapter 2. Review of GMCS QKD and Homodyne Detection 15

Figure 2.2: Schematic of a homodyne detection. BS: beam splitter; SIG: signal; LO: local oscillator; PD: photodiode; φ: introducing a phase between the signal and the LO;

Black line: optical path; Blue line: electrical path; Dashed box: homodyne detector

ES(t) = ES + δXS(t) + iδPS(t), (2.1) and

iφ EL(t) = [EL + δXL(t) + iδPL(t)]e . (2.2)

where ES and EL are real time-independent terms, δXS(t) and δPS(t) (δXL(t) and

2 δPL(t) ) are real and describe changes of amplitude and phase quadratures of the signal (LO) ﬁeld. Re[ES(t)] = ES + δXS(t) is the amplitude quadrature of the signal, while Im[ES(t)] = δPS(t) is the phase quadrature of the signal. Here the spatial mode distribution and the fast oscillating term eiωt are neglected. Fig. 2.3 shows signal and

LO states in the phasor space.φ in Fig. 2.3(same as φ in Fig. 2.2) is the relative phase diﬀerence between the signal and the LO. For the case where the LO beam is far more intense than the signal beam EL Q ES [71, 72], the electric ﬁelds after the beam splitter

2Here, I use upper cases X and P to represent electric ﬁeld quadratures, in order to distinguish with the Gaussian distributed quadratures x, p used in QKD. Chapter 2. Review of GMCS QKD and Homodyne Detection 16

Figure 2.3: LO and signal states in the phasor space.

E1(t) and E2(t) are,

1 E1 = º (EL + ES), (2.3) 2 and 1 E2 = º (EL − ES). (2.4) 2

2 2 The photocurrents from the two photodiodes are proportional to SE1S and SE2S respectively, 1 SE S2 = [SE (t)S2 + E (t)E(t) + E (t)E(t) + SE (t)S2] (2.5) 1 2 L L S S L S

1 SE S2 = [SE (t)S2 − E (t)E(t) − E (t)E(t) + SE (t)S2] (2.6) 2 2 L L S S L S

The ﬁrst term in Eq. (2.5) can be written as

1 1 I = SE (t)S2 = [E + δX (t) + iδP (t)][E + δX (t) − iδP (t)] 1 2 L 2 L L L L L L 1 [SE S2 + 2E δX (t)], (2.7) 2 L L L

2 2 where the small terms δXL(t) and δPL(t) are neglected. Chapter 2. Review of GMCS QKD and Homodyne Detection 17

The second and third terms I2 and I3 are

1 1 I = E (t)E(t) = [E + δX (t) + iδP (t)]eiφ[E + δX (t) − iδP (t)] 2 2 L S 2 L L L S S S 1 [E E eiφ + E eiφδX (t) + E eiφ(−iδP (t))], (2.8) 2 L S L S L S and

1 1 I = E (t)E(t) = [E + δX (t) + iδP (t)]e−iφ[E + δX (t) − iδP (t)] 3 2 S L 2 S S S L L L 1 [E E e−iφ + E e−iφδX (t) + E e−iφ(iδP (t))], (2.9) 2 S L L S L S

Here, ESδX(ESδP ) is neglected compared to ELδX(ELδP ). All higher order terms in δX(δP ) are neglected.

Their sum will be

1 I + I = [2E δX (t) cos φ + 2E δP (t) sin φ + 2E E cos φ]. (2.10) 2 3 2 L S L S L S

1 2 Because EL Q ES, I4 = 2 SES(t)S is neglected compared to I1.

Combining all the terms from I1 to I4, we have,

1 SE (t)S2 SE S2 + 2E δX (t) + 2E [(E + δX (t)) cos φ + δP (t) sin φ]. (2.11) 1 2 L L L L S S S

Similarly, the results for the other photodiode can be found. We can also get the diﬀer- ential current i−(t) from the two photodiodes.

2 2 i−(t) = SE1S − SE2S = 2EL[(ES + δXS(t)) cos φ + δPS(t) sin φ] (2.12)

2 The variance of this current, < i−(t) A, is obtained by

2 2 2 2 2 2 < i−(t) A 4EL(δXS cos φ + δPS sin φ). (2.13)

Here, since δX and δP are independent electric ﬁeld ﬂuctuations, the variance of cross terms, such as < δXδP A, cancels out in the averaging process.

In particular, if the phase diﬀerence φ is 0 or π~2, we will obtain Chapter 2. Review of GMCS QKD and Homodyne Detection 18

¢ ¨ ¨ 2ELδXS(t), φ = 0; i (t) 2E [(E + δX (t)) cos φ + δP (t) sin φ] = ¦ (2.14) − L S S S ¨ ¨ 2E δP (t), φ = π~2. ¤¨ L S

As a result, the current difference i−(t) will be proportional to either the X or the P quadrature of the signal field depending on the phase difference between the signal and the LO (0 or π~2). Because i−(t) is also proportional to the LO field, the LO should be much stronger (typically 6 orders of magnitude stronger) than the signal to facilitate detection of a weak signal. In GMCS QKD, Bob uses homodyne detection and measures either X or P quadrature of the signal field by randomly introducing a phase difference of 0 or π~2.

The variance of the current will be ¢ ¨ 2 2 ¨ 4ELδXS, φ = 0; < i2 (t) A ¦ (2.15) − ¨ ¨ 4E2 δP 2, φ = π~2. ¤¨ L S

2 < i−(t) A is proportional to the variance of the amplitude quadrature (when φ = 0) or the phase quadrature (when φ = π~2). The variance of the electric ﬁeld ﬂuctuation is called shot noise [71].

In a practical HD, two reverse-biased PIN-photodiodes (shown in Fig. 2.4) are employed to produce photocurrents from the optical signals, followed by a subtraction operation [48]. If pulsed light hits the two photodiodes, a positive pulse will be generated by

PD1 and a negative pulse will be generated by PD2. The intensity difference are obtained after subtraction as shown in Fig. 2.4. Following this structure, electronic amplifiers (not shown) are used to amplify this differential signal.

In a homodyne detection system, in addition to the shot noise due to electric ﬁeld

fluctuation, electronic amplifier noise, which is affected by the gain and bandwidth, also contributes to HD noise. Electronic noise is assumed to be independent of the intensity of the LO if the HD circuit is operating in its linear region [73]. Shot noise and electronic Chapter 2. Review of GMCS QKD and Homodyne Detection 19

Figure 2.4: Scheme of a photocurrent subtraction. PD: photodiode noise are independent of each other. Thus their variances can be added, and will be called the total noise of HD or HD noise in this thesis.

A simple experiment to measure the shot noise is to send vacuum to the signal port of a beam splitter, measure the total noise of HD and subtract electronic noise from the total noise of HD. Shot noise can only be detected when it exceeds HD electronic noise, which requires a strong LO ﬁeld. In most homodyne detection demonstrations with a pulsed LO, the photon number in each pulse is typically 107-108 [1, 43, 39, 40, 48]. To verify an HD to be shot-noise limited, we should have (1) the shot noise exceeds the electronic noise; (2) the total noise of HD, y, is related to the LO power by an equation of the form, y = ax + b, where a, b are constants. The shot noise (the total noise - the HD electronic noise) is linearly dependent on the LO power.

2.2.2 State of the art

Homodyne detection plays an important role in quantum optics [73] and quantum information [19]. HD was originally designed for measurement in the frequency domain to Chapter 2. Review of GMCS QKD and Homodyne Detection 20 evaluate field quadrature noise [57]. Such measurements are performed in the frequency domain by observing a certain spectral component of the photocurrent difference signal using an electronic spectrum analyzer. Frequency domain measurements do not give the electric field quadrature values in the time domain. With the development of quantum information, time domain homodyne detection has become increasingly important. How- ever, constructing an HD for use in the time domain (the signal and the LO are both pulsed light) is technically challenging, because

• The electronics must be fast enough to ensure temporal separation of responses to

individual laser pulses to avoid an overlap between consecutive pulses.

• Precise subtraction of the two photocurrents in the absence of the signal is necessary

in the time domain. A residual signal (incomplete subtraction of positive and

negative signals) makes our measurement diﬃcult and may also saturate the HD

ampliﬁers (to be discussed in Section 4.3.2).

• The HD should provide a flat amplification profile in the frequency domain to

ensure a broad bandwidth. This frequency span should be at least from DC to the

repetition rate.

Quantum tomography experiments are one of the most important applications of time- domain homodyne detection [5, 74]. In quantum tomography measurements, differential photocurrent is observed in the real time and integrated over the desired temporal mode to obtain a single value of a field quadrature. Homodyne detection is also actively used in quantum information experiments. In GMCS QKD experiment, homodyne detection is used by the receiver (Bob) to measure the field quadrature encoded with Gaussian random numbers [19, 43]. In QKD based on the BB84 protocol, homodyne detection may also be used to recover the phase information encoded on each pulse [75].

The ﬁrst time domain HD were performed below 1 kHz and achieved a shot-noise- to-electronic-noise ratio of 9 dB [76]. Hansen et al. [48] built an HD working at a Chapter 2. Review of GMCS QKD and Homodyne Detection 21 repetition rate of 204 kHz and yielded a 14 dB shot-noise-to-electronic-noise ratio. So far only four other groups in the world have been able to achieve a bandwidth of 100-250

MHz, which allows for tens of MHz repetition rate. Zavatta et al. [47] developed an

HD working at a repetition rate of 82 MHz with a shot-noise-to-electronic-noise ratio of

7 dB at 786 nm wavelength. During the course of my M. A. Sc. study, other groups are also developing high speed HDs. Jain et al. [77] have demonstrated HD working at

76 MHz laser repetition rate at 800 nm wavelength. Okubo et al. [5] have developed pulse-resolved HD at a laser repetition rate of 76 MHz with a bandwidth of more than

250 MHz at 1064 nm wavelength. Haderka et al. [57] have reported HD at a repetition rate of 53.8 MHz at 800 nm wavelength. High speed HD with such a broad bandwidth and a high repetition rate in the telecom wavelength region is lacking [39, 40, 55, 78]. We will target on constructing a high speed and pulse-resolved HD in the telecommunication wavelength region.

2.3 Summary

In this chapter, the GMCS QKD protocol and the homodyne detection principle have been introduced. I also provided an overview of the state of the art of GMCS QKD and homodyne detection implementations. QKD based on GMCS protocol opens a door to very high secret key generation rates [79]. However, its current speed is fundamentally limited by the bandwidth of the homodyne detector. To achieve a long term objective of developing a high speed ﬁber-based GMCS QKD, my master thesis goal is to construct a high speed HD at 1550 nm wavelength. Chapter 3

GMCS QKD over 20 km Fiber

In this chapter, a 20-km ﬁber based GMCS QKD experiment using an existing system will be presented as a real example of implementing HD in QKD. Comparing this result with

QKD experiments based on single photon protocols, GMCS QKD has a great advantage in generating high key rate over a short distance. Due to the limitation of homodyne detector bandwidth, the repetition rate of GMCS QKD is still low. To develop a high speed GMCS QKD system, building a high speed HD is in demand.

The 20-km QKD experimental results have been published on Lasers and Electro- optics (CLEO/QELS 2008)[6] and SPIE Optics + Photonics 2008[7], which I am a co- author. This chapter is mainly based on the two conference papers. This work follows a 5-km GMCS QKD experiment that has been published on Phys. Rev. A, 76, 052323

(2007). The system is developed by Dr. Bing Qi and Lei-Lei Huang, under the supervision of Profs. Li Qian and Hoi-Kwong Lo.

3.1 Experimental setup

The schematic is shown in Fig. 3.1. In this setup, a 1550 nm continuous-wave (CW) ﬁber laser (NP Photonics) is employed as the source. Alice uses an amplitude modulator (AM0 in Fig. 3.1) to generate 200-ns laser pulses at a repetition rate of 100 kHz. A polarization

22 Chapter 3. GMCS QKD over 20 km Fiber 23

beam splitter (PBS1) is used to split the pulses into a weak signal and a strong local oscillator (LO). The splitting ratio can be controlled by a polarization controller (PC1).

In the signal arm (upper arm) of Alice’s setup in Fig. 3.1, coherent state SxA + ipAe is modulated by the second amplitude modulator (AM1) and a phase modulator (PM1).

AM1 and PM1 are driven by Arbitrary Waveform Generators (AWG) which contain random amplitude and phase data produced from xA, pA (Gaussian distributed with average zero, variance = VAN0, N0 is the shot noise unit). The frequency of the LO will be upshifted by 55 MHz by an acoustic-optic modulator (AOM+). Signal and LO will be combined by a second polarization beam splitter (PBS2) and transmitted through a 20-km telecommunication ﬁber.

Figure 3.1: Schematic of the GMCS QKD system. L: 1550 nm CW ﬁber laser, PC1−5: polarization controllers; PBS1−3: polarization beam splitters or combiners; AM0−1: amplitude modulators; PM1−2: phase modulators; SW1−2:optical switches; AOM+(AOM−): upshift(downshift) acousto-optic modulators; VOA1−2: variable optical attenuators; ISO: isolator; C: ﬁber coupler; HOM: homodyne detector [1, 6, 7]

On Bob’s side, PBS3 is used to separate signal (lower arm) and LO (upper arm). The frequency of the LO will be downshifted by 55 MHz by a second acoustic-optic modulator

(AOM−). Bob randomly chooses either x or p with his phase modulator (PM2) driven by a third AWG which contains a binary random ﬁle for choosing x or p. To decode the

Gaussian random numbers encoded in the signal pulses, Bob combines the signal and

LO by a fiber coupler (C in Fig. 3.1) and performs a homoydne detection with a low Chapter 3. GMCS QKD over 20 km Fiber 24 speed homodyne detector (HD). Follow the design in Ref. [48], the HD is constructed by a pair of photodiodes and a low noise charge sensitive amplifier. A fiber isolator has been placed in the signal arm of Bob’s Mach-Zehnder Interferometer (MZI) to reduce the noise from multiple reflections of LO in Bob’s system. A 12-bit data acquisition card

(NI, PCI-6115) at a sampling rate of 10M samples/s is employed to measure the output of the HD.

For the speciﬁc HD used in this experiment, LO has to contain 3 106 photon per pulse so that the shot noise can exceed the electronic noise (shown in Fig. 3.5, where the HD noise is 3 dB above the electronic noise). In a GMCS QKD experiment, the signal contains less than 100 photons per pulse [1, 43], which is much weaker than that of the LO. For a practical system, there will be some leakage from the LO arm to the signal arm. Any leakage photon from LO to signal will interfere with LO and introduce excess noise. To overcome this problem, a polarization-frequency-multiplexing scheme is used to separate leakage from signal (see Ref. [1] for detailed description). In this scheme, polarization beam splitters are used to split and combine the signal and the LO.

Acoustic-optic modulators (AOM+ and AOM-) are used to up-shift and down-shift the frequency of LO by 55 MHz on Alice and Bob’s sides.

3.2 Secure key rate formula

In this section, I will present the QKD secure key rate formulas based on GMCS protocol.

A detailed derivation of the key rate formula and the security analysis against individual attack can be found in Ref. [19, 39, 45, 60]. The mutual information shared between

Alice and Bob is IAB. The maximum information of Bob’s key available to Eve is limited by IBE [13, 59] based on reverse reconciliation. The secret key information that Alice and Bob can distill is ∆I (to be deﬁned later).

Before giving the key rate formula, let me ﬁrst deﬁne the notations that will be Chapter 3. GMCS QKD over 20 km Fiber 25

Figure 3.2: Quadrature variances prepared by Alice, quadrature variances measured by

Bob, and equivalent input noise χ. Quadrature variance prepared by Alice, and equivalent input noise χ are referred to the input. Noise on Bob’s side is referred to the output.

employed in the formulas. Here all noises/variances are in units of the shot noise N0.

Alice’s modulation variance is VA (variance of x or p quadrature modulated by Alice), and V = VA + 1 is the quadrature variance of the coherent state prepared by Alice (1 is the shot noise of a coherent state, see Fig. 2.1 the left plot in Alice’s box). The channel efficiency (transmission) is G , and the total efficiency of Bob’s device (optical loss and detector efficiency) is η. χ is the equivalent noise measured at the input, which is composed of quantum noise of channel χvac, noise outside Bob’s system ²A and noise contributed by Bob’s devices NBob. Fig. 3.2 shows the quadrature variances prepared by Alice, quadrature variances measured by Bob and equivalent input noise χ.

The mutual information between Alice and Bob IAB is determined by the Shannon entropy [59]. According to Refs. [19, 39],

1 I = log [(V + χ)~(1 + χ)], (3.1) AB 2 2 where 1 − ηG χ = χ + ² = + ², (3.2) vac ηG which can be separated into “vacuum noise” χvac (noise associated with the channel loss and detection eﬃciency of Bob’s system) and “excess noise” ² (noise due to the Chapter 3. GMCS QKD over 20 km Fiber 26 imperfections in a non-ideal QKD system).

There are two models that can be used to estimate Eve’s information. The “general model” assumes that losses and noise in Bob’s system can be controlled by Eve [19]. In contrast, in a “realistic model”, we assume that Eve has no control over Bob’s system.

Under the “general model”, the mutual information between Bob and Eve IBE is

1 I = log [(ηG)2(V + χ)(V −1 + χ)]. (3.3) BE 2 2 If a reverse reconciliation algorithm[19] is adopted and the key is generated from

Bob’s data, the secure key rate is [19]

∆I = βIAB − IBE. (3.4) where β is the reconciliation eﬃciency (β B 1). With a perfect reconciliation β = 1, the maximum secure key with a length of IAB − IBE can be obtained.

Under the “realistic model”, part of the excess noise (called ²A, due to imperfections outside Bob’s system) might originate from Eve’s attack, while the other part of the noise is attributed to Bob’s devices over which Eve has no control (called NBob). The total excess noise ² can be written as [19]

² = ²A + NBob~ηG, (3.5)

where ²A refers to the input and NBob is noise measured at the output. From Eqs. (3.2) and (3.5) , the equivalent input noise is

1 − ηG N χ = + ² + Bob . (3.6) ηG A ηG

With a reverse reconciliation scheme, the mutual information shared by Bob and Eve under the “realistic model” is

1 ηGV + 1 + ηG² I = log [ A ]. (3.7) BE 2 −1 2 η~(1 − G + G²A + GV ) + 1 − η + NBob

The secure key is again be calculated by Eq. (3.4) given a reconciliation eﬃciency β. Chapter 3. GMCS QKD over 20 km Fiber 27

3.3 Results

We perform a QKD experiment with an LO of 1.2107 photons/pulse and a signal with a modulation variance of 10. The channel efficiency G and the total efficiency of Bob’s device have been calibrated carefully to be G=0.405 and η = 0.44 (including optical loss in Bob’s system 0.61 and the efficiency of the HD 0.72). Data are transmitted by frames. Each frame contains 7000 points (Gaussian random numbers). Among them, Bob performs x quadrature measurements on 3531 points and p quadrature measurements on

3469 points. The same random patterns are used repeatedly in our experiment.

In GMCS QKD systems, the phase between the signal and the LO should be only dependent on the phase information encoded by Alice. However, in practice, the zero point of the phase diﬀerence φ0 (the phase diﬀerence when Alice encodes phase 0) will drift with time. The GMCS QKD protocol is very sensitive to this phase drift because a small phase drift would reduce the secure key rate dramatically [1]. A novel phase remapping scheme is proposed to remove the excess noise due to the phase drift φ0: once

Alice and Bob know the value of φ0, Alice can simply modify her data to incorporate this phase drift. The security analysis of GMCS QKD still holds (see Ref. [1] for detailed description).

The correlated variables shared by Alice and Bob after phase-remapping are shown in

Fig. 3.3. The equivalent input noise has been determined experimentally to be χ = 6.13.

Excess noise can be determined by Eq. (3.2) to be 1.51.

To estimate the secure key rate under the “realistic model”, we separate ² into ²A

(noise outside Bob’s system) and NBob (noise inside Bob’s system) = Nele (HD electronic noise) + Nleak (leakage noise). Under this model, Eve has no control over Bob’s devices.

1. ²A is the excess noise due to imperfections outside Bob’s system, which includes the phase noise of the laser source, imperfect amplitude and phase modulations,

the phase noise of the interferometers, etc. Chapter 3. GMCS QKD over 20 km Fiber 28

Figure 3.3: QKD experimental results. The equivalent input noise has been determined experimentally to be χ = 6.13[6, 7].

Following [39], we assume that ²A is proportional to the modulation variance VA

and can be described by ²A = VAδ. To estimate the value of the coeﬃcient δ,

Alice uses a large modulation variance (VA 40000) to encode her information and employs a weak LO (105 photon/pulse, to reduce the leakage). With the same

process as QKD, the equivalent input noise χ and excess noise ² can be achieved

from the experimental results. Assuming all other excess noise in Eq. (3.2) except

1 ²A are negligible, i. e. χ VAδ , δ can be determined from experimental results

of the equivalent input noise χ and the modulation variance VA. Note that, large modulation is not used for real QKD since excess noise is so strong that no secure

key can be distributed between Alice and Bob.

Figure 3.4 shows the correlated variables of Alice and Bob when a large modulation

1 χ in Fig. 3.4 (with a large modulation VA) is 32 times larger than that of Fig. 3.3 (in real QKD when weak signal is used) Chapter 3. GMCS QKD over 20 km Fiber 29

Figure 3.4: Determine δ by using a high modulation variance VA 40000 and a weak LO (105 photon/pulse). The result is δ = 0.0049

VA is used. The calculated δ is = 0.0049. Therefore, for a modulation variance of

VA = 10, the expected excess noise outside Bob’s system is ²A = δVA =0.049.

2. From Eq. (3.2) and the estimated ²A above, the noise from Bob’s sytem NBob is 0.26.

There are two main sources of NBob, the electronic noise from the HD (Nele) and

the noise associated with the leakage from LO to signal (Nleak).

(a) As shown in Fig. 3.5, the total noise of HD (variance of HD output voltages)

is related to the LO power by a form of y = ax + b (y: total noise of the

HD, x: LO power, a, b are constants) when vacuum state is detected (i. e.

vacuum is sent to HD signal port). This test is extremely important since it

veriﬁes the HD noise is indeed the sum of the shot noise and electronic noise.

The HD has a 6.8 dB shot-noise-to-electronic-noise ratio when the QKD is Chapter 3. GMCS QKD over 20 km Fiber 30

Figure 3.5: Noise of the balanced HD as a function of LO power. With a LO of 1.2107 photons/pulse, the electronic noise is 6.8 dB below the shot noise (plot with raw data obtained from [1])

7 performed at an LO of 1.2 10 photons/pulse. The corresponding Nele is therefore 10−0.68 = 0.21 (in shot noise unit).

According to above noise analysis, the electronic noise of the HD will con-

tribute to the excess noise and ultimately aﬀect the secure key rate, which

is shown in the key rate simulation in Fig. 3.6 under the “general model”

[19] (based on Eqs. (3.1),(3.3) and (3.4) in Section 2.1.1). Parameters in this

simulation are obtained from Ref. [1] and shown in Table 3.1. As shown in

Fig. 3.6, the key rate drops as the electronic noise increases. Positive secure

key can be achieved as the electronic noise is below 0.13 (in shot noise unit),

i.e. 8.86 shot-noise-to-electronic-noise ratio2. Therefore, we will need a 10-

2In the “general model”, electronic noise from Bob’s homodyne detector can be controlled by Eve. Under the “realistic model”, we are able to tolerate a stronger electronic noise if we assume Eve has no control over Bob’s system. Chapter 3. GMCS QKD over 20 km Fiber 31

0.25

0.2

0.15

0.1 Key rate (bit/pulse)

0.05

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Electronic noise (in shot noise unit)

Figure 3.6: Secure key rate as a function of the electronic noise (in shot noise unit) under the “general model”. Parameters in this simulation are in Table 3.1 .

times shot-noise limited HD (shot noise is 10 dB above the electronic noise)

in GMCS QKD experiments.

For the particular HD used in the 20-km GMCS QKD experiment, the band-

width is about 1 MHz, which limits the pulse repetition rate at 100 kHz. To

enhance the speed, a broadband HD should be used in the GMCS QKD. How-

ever, the electronic noise scales with the HD bandwidth. A broad bandwidth

will include more electronic noise which also deteriorates QKD key rate by

increasing Nele. Therefore, we have to make a tradeoﬀ between the repetition rate and the electronic noise.

(b) Noise due to leakage photon from LO to signal is another excess noise source

Table 3.1: Parameters used in the key rate simulation , from Ref. [1]. The length of the

ﬁber is 5 km.

VA G η ²A Nleak β 16.9 0.758 0.44 0.056 0.02 0.898 Chapter 3. GMCS QKD over 20 km Fiber 32

Figure 3.7: The leakage from LO to signal. LO: local oscillator; SIG: signal; LE: leakage;

PBS: polarization beam splitter; LO is 5-6 orders of magnitude higher than signal. The leakage from signal to LO is negligible. Arrowed lines indicate the polarization of the beam

in Bob’s system. Nleak can be calculated by subtracting Nele from NBob, i.e.

Nleak = 0.05 (in shot noise unit). As shown in Fig. 3.7, polarization drift of the signal and the LO before they enter the polarization beam splitter (Fig.

3.7, drifting from states shown in solid lines to states shown in dotted lines)

will induce leakage photons from the LO to the signal beam. These leakage

photons will interfere with LO and contribute to excess noise.

Leakage photon number will be larger at a stronger LO power. To suppress

Nleak, a weak LO power is preferred. However, shot noise is scaled with LO

power and it is the noise unit in our experiment. At a lower LO power, Nele (in shot noise unit) due to HD electronic noise will become larger. Therefore,

a tradeoﬀ between Nele and Nleak has to be made. In future GMCS QKD experiments, a more systematic investigation on this issue should be performed.

The experimental results are summarized in Table 3.2. Using Eqs. (3.1),(3.4) and Chapter 3. GMCS QKD over 20 km Fiber 33

Table 3.2: 20-km GMCS QKD parameters and results (e: experimental result; c: calculated result)

rea rea VA G η χ ² ²A NBob Rβ=1 Rβ=0.898 10(e) 0.405 (e) 0.44 (e) 6.13 (e) 1.51(c) 0.049(e) 0.26(c) 0.12 (c) 0.05(c)

(3.7), the secure key rate is 0.05 bit/pulse 3 (β= 0.898) over 20 km ﬁber under the

“realistic model”.

3.4 Discussions

Table 3.3 compares our key rate with that of Ref. [4] based on decoy BB84 protocol.

Both systems are implemented over 20 km telecom ﬁber.

Table 3.3: Secure key rate in our GMCS QKD experiment and in Ref. [4] over 20-km

ﬁber. rep. :repetition Our experiment Ref. [4]

Key rate (bit/pulse) 0.05 0.00098

Key rate (bit/s) 5 k (100 kHz rep. rate) 1.02 M (1.036 GHz rep. rate)

For the key rate per pulse, our GMCS QKD experiment has an obvious advantage, because (1) HD with high-efficiency PIN photodiodes (HD efficiency= 72 %) is used in our experiment while low-efficiency single photon detector (10 %) is used in Ref. [4] ; (2) more than one bit can be obtained in each pulse based on GMCS protocol. However, the key rate per second of Ref. [4] is two orders of magnitude higher than that of our GMCS

QKD experiment, due to its fast repetition rate at GHz. Therefore, at a low repetition rate of 100 kHz in our experiment, the advantage of GMCS QKD cannot be thoroughly demonstrated.

3The secure key rate is less than 1 bit/pulse. After the key transmission, Alice and Bob will perform reverse reconciliation and privacy ampliﬁcation, which will sacriﬁce a lot of bits. Chapter 3. GMCS QKD over 20 km Fiber 34

0 10

−1 10 Low loss fiber

−2 10

−3 Standard fiber 10

−4

Key rate (bit/pulse) 10

−5 10

−6 10 0 10 20 30 40 50 60 70 Distance (km)

Figure 3.8: Key rate simulation when ultra-low loss ﬁber and standard ﬁber are used in the GMCS QKD experiment. Parameters are based on Table 3.2 under the “realistic model” when β = 0.898

To increase the secure key rate of the GMCS QKD, the subsequent improvements can be made in future experiments.

• Implement ultra-low loss ﬁber

Key rate is a function of the channel transmittance G, which is dependent on the

ﬁber loss. A high transmittance (low loss) channel will yield a high secure key.

Currently ultra-low loss ﬁber with a loss of 0.164 dB/km has been developed by

Corning and is already used in quantum cryptography experiment [80]. If we use

this ultra-low loss ﬁber in our GMCS QKD system, we can expect to improve the

secure key rate. Fig. 3.8 shows the secure key rate as a function of the transmission

distance for ultra-low loss ﬁber and standard ﬁber (loss 0.21 dB/km) respectively. Chapter 3. GMCS QKD over 20 km Fiber 35

0 10

Without excess noise

With excess noise

−1 10 Key rate (bit/pulse)

−2 10 0 5 10 15 20 Distance (km)

Figure 3.9: Key rate simulation with excess noise and in the absence of excess noise.

Parameters are based on Table 3.2 under the “realistic model” when β = 0.898

Although ultra-low loss ﬁber can increase the transmission distance by 10 km, the

key rate over a short distance does not improve too much (less than one order).

Furthermore, this ultra-low loss ﬁber is 3 times more expensive than that of the

standard ﬁber (based on their quotation).

• Reduce excess noise

In the GMCS QKD, Eve can obtain information by monitoring excess noise. Min-

imizing the excess noise can also improve the secure key rate. However, for a

practical system, the phase noise, polarization noise, electronic noise of Bob’s ho-

modyne detector and losses in Bob’s system will contribute excess noise [39]. To

see the improvement achieved by eliminating the excess noise, key rate is simulated

with excess noise and in the absence of excess noise (although it is hard to realize Chapter 3. GMCS QKD over 20 km Fiber 36

in a practical system) in Fig. 3.9. GMCS QKD key rate can not be improved by

orders even though a very unrealistic situation of no excess noise is assumed.

• Increase the bandwidth of our homodyne detector

Although the key rate per pulse of the GMCS QKD is 2 orders of magnitude higher

than that of single photon QKD, the repetition rate at 100 kHz in our experiment

is far lower than that of Ref. [4] at GHz. Thus, GMCS QKD key rate per second

is 2 orders of magnitude lower. If the repetition rate of GMCS QKD experiments

can be successfully increased by 100-1000 times, key rate per second of GMCS

QKD experiments will achieve Mbits/s, comparable with Ref. [4]. Constructing

a high speed HD is an eﬀective way to improve the secure key rate of GMCS QKD

experiments by a few orders.

3.5 Summary

In summary, as an example of using HD in a real QKD implementation, we demonstrated a GMCS QKD experiment over 20 km ﬁber based on our existing system. Under the

“realistic model” with β = 0.898 [19], our key rate over a 20-km ﬁber is 0.05 bit/pulse.

To greatly enhance the secure key rate in GMCS QKD experiments, several improvements, such as using ultra-low loss ﬁber, minimizing the excess noise, and increasing the repetition rate, are discussed. Comparing their predicted improvements, increasing the repetition rate of GMCS QKD experiments will be the most eﬀective way of enhancing the key rate by a few orders, which will require a high speed homodyne detector in the region of telecommunication wavelength. We will discuss our high speed HD construction and performance in the subsequent two chapters. Chapter 4

High Speed Homodyne Detector

As discussed in Section 2.1.2, the speed of GMCS QKD is mainly limited by the homodyne detector (HD) bandwidth. In this chapter, high speed HD requirements will be analyzed.

As the ﬁrst stage to enable a high speed GMCS QKD system, designs and challenges in constructing a high speed HD will be discussed from both electrical and optical sides.

Much of the work reported in Chapter 4 and 5 was done by me, under the daily supervision of Dr. Bing Qi and in frequent discussions with Profs. Li Qian and Hoi-

Kwong Lo. HD circuit design and printed circuit broad layout are provided by Prof.

Alex Lvovsky’s group at University of Calgary.

4.1 Requirement of a homodyne detector in GMCS

QKD

As discussed in Section 3.3, HD with a high shot-noise-to-electronic-noise ratio is preferred for the GMCS QKD experiment. Note that, theoretically the shot-noise-to- electronic-noise ratio can always be improved by increasing the LO power. However, for a practical HD, the photodiodes and electronic ampliﬁer will be saturated if the LO power is suﬃciently high ( 109 LO photon/pulse reported by [5, 43]). Hence our proposed goal

37 Chapter 4. High Speed Homodyne Detector 38 is to construct a 10-times shot-noise limited HD (with a shot-noise-to-electronic-noise ratio of 10 dB) at an LO of 108 − 109 photons/pulse.

The repetition rate of the GMCS QKD experiment described in Chapter 3 is at 100 kHz, which is fundamentally limited by the HD bandwidth. Considering that extending the bandwidth of the HDs to 100 MHz is feasible (demonstrated by [47], but not in the telecommunication wavelength), my goal is to construct an HD with a bandwidth of

100 MHz, which can improve the speed of GMCS QKD experiments by 100 times.

4.2 High speed homodyne detector design

A full homodyne detection system includes both optical and electrical designs (see Fig.

2.2 in Chapter 2). On the optical side, the LO and the signal mix at the beam splitter and the two output beams need to be carefully balanced before being sent to photodiodes.

On the electrical side, HD circuit will perform a subtraction of the two electrical signals generated by photodiodes and amplify this diﬀerential signal. In this section, a particular homodyne detection design used in the telecommunication wavelength region will be presented.

4.2.1 Homodyne detector optical setup in ﬁber

Based on telecommunication components, the optical setup of the homodyne detection can be designed as shown in Fig. 4.1 (left dashed line box) , which is a practical implementation of Fig. 2.2 at 1550 nm wavelength. In this setup, the signal and the LO beams with the same frequencies will interfere at a two-by-two ﬁber coupler (FC in Fig. 4.1) with a splitting ratio of 50:50. A variable optical attenuator (VOA) and an optical variable delay (OVD) are placed in the output paths of the ﬁber coupler, for adjusting losses and the lengths of the two paths accurately. Finally, the balanced signals will be sent to two photodiodes. To avoid disturbance from the environment, such as atmospheric Chapter 4. High Speed Homodyne Detector 39

Figure 4.1: Homodyne detection setup in the telecommunication wavelength. LO: local oscillator; OVD: optical variable delay; FC: 50:50 ﬁber coupler; VOA: variable optical attenuator; PD: photodiode; AMP: electronic ampliﬁers. turbulence, I used an enclosure to isolate the system of Fig.4.1.

4.2.2 Homodyne detector electrical circuit

The HD circuit design and the printed circuit board layout we use are from Prof. Alex

Lvovsky’s group at the University of Calgary. They have demonstrated quantum tomography experiments in characterizing the coherent or squeezed states at 800 nm wavelength

[56, 81]. Although in this thesis we are primarily interested in the telecommunication wavelength region (rather than 800 nm), their circuit design should be useful, because after the conversion of optical signals to photocurrents by the photodiodes, the subsequent ampliﬁcation will not be dependent on the wavelength. In practice, HD circuits with photodiodes working in the telecommunication wavelength region have to be tuned carefully to obtain an optimal condition for high speed GMCS QKD experiment, which we shall discuss in Section 4.3.

Table 4.1: Speciﬁcations of FGA04 InGaAs photodiode (typical values). Bandwidth Responsivity Capacitance Noise equivalent power Dark current º (GHz) (A/W) (pF) (W/ Hz) (nA)

2 0.9 1.0 1.510−15 0.5 Chapter 4. High Speed Homodyne Detector 40

Figure 4.2: A simpliﬁed homodyne detector circuit. PD: photodiode (Thorlabs,FGA04);

OPA847: operational ampliﬁer (Texas Instrument)

A simpliﬁed HD circuit (illustrated by the right dashed line box in Fig. 4.1) is shown in Fig. 4.2. Owing to high responsivity and high speed, two InGaAs photodiodes from

Thorlabs (FGA04) are employed in the HD circuit. Table 4.1 shows their speciﬁcations.

The two photodiodes are reversely biased and followed by two OPA847 operational am- pliﬁers. All components are soldered on a printed circuit board, which is eventually shielded by a custom-designed metal box.

Given the optical and the electrical designs of a high speed HD, we will discuss the construction challenges in the next section.

4.3 Challenges

Constructing a broadband HD used for quantum detection is quite challenging and only a few groups in the world have successfully demonstrated HDs with 100 MHz bandwidth

[5,30 - 32]. In this section, I will discuss those challenges and our corresponding solutions. Chapter 4. High Speed Homodyne Detector 41

4.3.1 Low electronic noise

As discussed in Section 3.3, a high shot-noise-to-electronic-noise ratio of an HD at a particular LO power will correspond to a low electronic noise and yield a high secure key rate, hence, an HD with ultra-low electronic noise is desired in GMCS QKD experiments.

In a practical HD circuit, there are several sources that will contribute to electronic noise [72],

1. Dark noise of photodiodes, which is only the noise due to photodiode dark current.

4kB TB 2. Johnson noise (thermal noise) of circuit. It is given by R , where kB is the Boltzmann constant, T is temperature, B is bandwidth and R is the impedance of

the circuit.

3. Electronic ampliﬁer noise. It is in the form of 2eGBF , where e is the elementary

charge, G is the voltage gain, B is the bandwidth and F is called excess noise factor

(Note that, the excess noise is diﬀerent from that in GMCS QKD).

Compared to Johnson noise and electronic ampliﬁer noise, photodiode dark noise is usually neglected [72]. Since the gain for a HD is huge, the electronic ampliﬁer noise is

7 orders of magnitude higher than that of Johnson noise 1. Therefore, most of the HD electronic noise is contributed by electronic ampliﬁer noise.

The HD circuit is built on a printed circuit board with discrete surface mounted components (shown in Fig. 4.3). The two photodiodes are placed closely in the upper left corner of the board. To minimize the parasitic capacitance, pins of the photodiodes are cut as short as possible and mounted very close to the board.

Shielding of electromagnetic waves is also very important in reducing the impact of the environmental noise on the circuit [83]. In discussion with engineers from 3GMetalWorx

1If T = 300 K, R = 50 Ohm, G = 2200 (see Section 4.3.3, we have converted the trans-impedance gain into voltage gain), and F is of the order of 1 [82], the ratio of Johnson noise to electronic ampliﬁer noise is 10−7. Chapter 4. High Speed Homodyne Detector 42

Figure 4.3: Photo of the circuit board of the homodyne detector. Two FGA04 photodiodes are in the upper left corner.

Inc., our group designed a customized metal box to shield HD circuit board, as shown in Fig. 4.4. In this shielding design, HD power and output are placed in separated compartments in order to minimize their radiations. On top of the board, separated metal walls are used to prevent the crosstalk among the photodiodes, the ampliﬁers and the output wire. Finally, two covers for the separated walls and the box are implemented to completely shield the HD detector.

4.3.2 Diﬀerent photodiode response functions

The HD requires two photodiodes with almost identical response functions, however, practical fabrication process cannot yield identical photodiodes. Figure 4.5 shows the impulse response measurement circuit and results for ten photodiodes (see the ﬁgure caption for measurement descriptions). Hence two photodiodes selected for the HD will inevitably exhibit response mismatch in the time domain, which we call photodiode mismatch. The residual signal after subtraction and ampliﬁcation cannot be neglected.

Fig. 4.6 shows the residual signal at the output of HD, after balancing the two optical signals to the photodiodes. Chapter 4. High Speed Homodyne Detector 43

Figure 4.4: Customized metal box for shielding, constructed by 3GMetalWorx Inc.. Di- mensions (cm): 8.126.674.22; Thickness (cm): 0.0406

A strong LO power is preferred to achieve a large shot-noise-to-electronic-noise ratio, since shot noise scales with the LO power. Owing to the large gain of the HD electronic ampliﬁers ( 2.2 104 V/A, to be shown in Section 4.3.3), residual signal induced by photodiode mismatch will exceed HD output saturation level when the LO power is suﬃciently high. However, if a low LO power is chosen to avoid saturation, shot noise may not be able to exceed electronic noise. Hence, minimizing the residual signal is necessary in achieving a high shot-noise-to-electronic-noise ratio. In the context of GMCS QKD, the residual signal should be less than 10−4 of the signal of one arm [39]2.

To overcome this problem, (1) a wide pulse with long rising/falling time is used to minimize the mismatch, since both positive and negative electrical pulses generated by photodiodes will be smooth with long rising/falling time. Fig. 4.7 compares the impulse

2This is a rough estimation. Assuming that each arm has 108 photons/pulse, shot noise will be 104 photon/pulse (square root of the arm signal), due to the photon number poissonian distribution of a coherent pulse. Since we want to observe shot noise without saturating the ampliﬁers, the residual signal should not be larger than the shot noise. Chapter 4. High Speed Homodyne Detector 44

Figure 4.5: Photodiode impulse responses test. (a)Photodiode impulse responses test setup. Laser: Picoquant PDL 800-B, pulse width: 30-50 ps; RLoad = 50 Ohm. Laser pulses are sent to the photodiode and the output electrical voltage V0 is measured by an oscilloscope. (b) V0 as a function of time for ten photodiodes. We acknowledge Liang Tian’s testing work in this experiment.

responses when a narrow or a wide pulse is sent to the photodiode 3. However, a wide pulse width will limit the repetition rate; (2) data processing is used to remove this mismatch, which will be discussed in Chapter 5.

3The measurement procedure is similar to that was done in Fig. 4.5. See Section 4.3.4 for the detailed experimental setup and procedure. Chapter 4. High Speed Homodyne Detector 45

Figure 4.6: Residual signals at the output of the HD, after balancing the two signals to the photodiodes. Laser: PriTel Mode-lock femsto-second ﬁber laser; pulse width: 0.7 ps

- 1.2 ps; LO power = 13.8 µW. Horizontal scale: 20 ns/div; Vertical scale: 10 mV/div.

4.3.3 Linearity

In the GMCS QKD, random bits are encoded to signal pulses. The HD is used to measure the quadrature of individual pulse in the time domain. It is important to ensure the HD is operating in its linear region to minimize the measurement error. (If the HD output is not linearly dependent on the input, the measurement of the pulse quadrature will not be accurate).

There are two linearities one should consider in building a HD: (1) the photodiodes must work in their linear regions at the required LO power; (2) the HD electronic ampli-

ﬁers should operate in the linear region over the desired input range. In this section, I

ﬁrst investigate the linearity of HD electronic ampliﬁers. The linearity of the photodiodes will be discussed in detail in the next section.

I use the circuit and the setup shown in Fig.4.8 to test the linearity of HD electronic Chapter 4. High Speed Homodyne Detector 46

Figure 4.7: Photodiode impulse response with (a) pulse width: 0.7-1.2 ps; horizontal scale: 2 ns/div; vertical scale: 50 mV/div (b) pulse width: 50 ns; edge time: 15 ns; horizontal scale: 20 ns/div; vertical scale: 2 mV/div ampliﬁers. The experimental procedure is:

1. Positive and negative electrical pulses (10 MHz, 50 ns width) are generated by a

function generator (Agilent) and then sent to the HD electronic ampliﬁers input,

as shown in Fig. 4.8 (a). Chapter 4. High Speed Homodyne Detector 47

Figure 4.8: (a) Circuit linearity test circuit. It is the ampliﬁcation part of the circuit in

Fig. 4.2. OPA847: electronic ampliﬁer; (b) Circuit linearity test setup. Electrical pulses

(generated by a function generator, repetition rate: 10 MHz; pulse width: 50 ns) is sent to the circuit in (a). The output is measured by an oscilloscope.

2. Output pulse peak voltage is measured by an oscilloscope at the HD electronic

ampliﬁers output.

3. Measurements in step 1 and 2 are repeated with diﬀerent input electrical signals.

Output pulse peak voltage (absolute value) is plotted as a function of the input pulse peak voltage (absolute value) for both positive and negative signals in Fig.4.9. HD electronic amplifier gains are almost the same for both positive and negative signals in the amplifiers linear region (deviating 1 % from the linear fit). According to Fig.4.9, the voltage gain (slope) is 22. Because the resistor connected to the first amplifier input in Fig. 4.8 (a) has a 1000 Ohm resistance, the trans-impedance gain of the HD electronic amplifiers is 2.2 104 V/A 4.

4.3.4 Laser source

A pulsed laser source is required in GMCS QKD experiments. Here we will discuss the choice of the pulse repetition rate and pulse width. The linewidth and phase noise of

4With an input voltage of V , the input current across the 1000 Ohm resistor is V/1000. Since the output voltage is 22 V , the trans-impedance gain will be the output voltage divided by the input current 22V /(V /1000) = 2.2104 V/A. Chapter 4. High Speed Homodyne Detector 48

2000

1800 Positive signal input 1600 Negative signal input 1400

1200

1000

800

600

400 Output peak voltage (mV) 200

0 0 10 20 30 40 50 60 70 Input peak voltage (mV)

Figure 4.9: Output peak voltage as a function of the input peak voltage for both positive and negative signals. The slope (HD electronic ampliﬁers voltage gain) is 22. The straight line is the linear ﬁt. the laser source are not considered here since they will contribute to ²A of the excess noise ². In GMCS QKD, excess noise is assumed to be controlled by Eve and insecure information can be removed by future classical data processing.

• Pulse repetition rate

Pulse repetition rate is fundamentally limited by the bandwidth of the HD. Com-

mercial photodiode can go up to 100 GHz. However, due to the ultra-low noise

requirement of GMCS QKD experiments, HD ampliﬁcation circuit has a maximum

bandwidth of 100-250 MHz so far [5, 56], since HD electronic ampliﬁer noise scales

with its bandwidth.

The residual signal due to incomplete subtraction of the positive and negative pho-

tocurrents also limits the repetition rate. As an optical pulse strikes the photodiode,

the photocurrent will be generated. If the incident optical pulse width is too nar- Chapter 4. High Speed Homodyne Detector 49

row, the photodiode may not be fast enough to respond and signiﬁcant oscillations

will occur at the falling edge of the electrical pulse. Since two photodiodes are

employed in the HD circuit, in addition to photodiode intrinsic mismatch (such as

responsivity, and response time), diﬀerent oscillation shapes from the two photodi-

odes will result in a considerable residual signal. Hence, wide optical pulse is used

to minimize the oscillations, however, this will limit the repetition rate.

• Pulse peak power and width

In the GMCS QKD, the photon number in each LO pulse is of the order of 107 −109,

to ensure (a) the shot noise exceeds the electronic noise ; (2) the electronic circuit

is not saturated. Here I will discuss photodiode linearities in this required power

range.

The circuit and the setup shown in Fig. 4.10 are used in this test. Two types of

pulsed laser sources (shown in Table 4.2) running at 10 MHz repetition rate are

used to test photodiode linearity.

The experimental procedure is:

1. I send pulsed light (laser I or laser II) to only one photodiode by connecting

the switch with port 1 in Fig. 4.10 (b) and use an oscilloscope to measure

the electrical pulse peak voltage across the 10 Ohm resistor in Fig. 4.10 (a).

Optical power of the pulsed light is measured by connecting the switch with

port 3 in Fig. 4.10 (b).

2. Step 1 is repeated by sending pulsed light to the other photodiode when the

Table 4.2: Speciﬁcations of the two lasers used in photodiode linearity test Laser number Type Pulse Width

I PriTel mode-lock femto second 0.7-1.2 ps

II CW modulated by amplitude modulator 50 ns Chapter 4. High Speed Homodyne Detector 50

Figure 4.10: (a) Photodiode linearity test circuit. PD: photodiode; (b) Photodiode linearity test setup. Laser I or II pulses are sent to one of the photodiodes. An oscilloscope is used to measure the output peak voltage. PD: photodiode; Switch is connected with port 1,2 to test the photodiode linearity, while connected with port 3 to measure the optical power.

switch is connected with port 2.

Figure 4.7 (a) and (b) (in Section 4.3.2) show the oscilloscope waveforms of the

electrical output pulses with laser I or II as the source, respectively. Comparing

the two graphs, the electrical pulse peak voltage measured by the setup (Fig. 4.10

(a)) with laser I as a source is much higher than that with laser II. It is natural

because laser I has a narrow width (0.7-1.2 ps). Since the photodiode bandwidth is

2 GHz, the output pulse with a source of laser I has a width of 500 ps. The areas

(equivalent to energy) under two pulses are almost equal.

As shown in Fig. 4.11 (laser I) and Fig.4.12 (laser II), output electrical pulse peak

voltage across the 10 Ohm resistor (in Fig. 4.10 (a)) is plotted as a function of the

photon number in each optical pulse for both positive and negative photodiodes,

respectively. Chapter 4. High Speed Homodyne Detector 51

250

Positive photodiode 200 Negative photodiode

150

100

50 Output pulse amplitude (mV)

0 0 5 10 15 7 Photon numbers per pulse x 10

Figure 4.11: Photodiode output peak voltage as a function of the photon number per pulse with laser I ( 1 ps pulse width) as a source.

With laser I as a source (Fig.4.11), both photodiodes start to saturate when input

photon number/pulse exceeds 108. With laser II as a source (Fig. 4.12), photo-

diodes output voltage deviates 4 % from their linear ﬁts at up to 109 incident

photon per pulse. Power of laser II is limited to the non-tunable CW laser power.

One may ﬁnd the photodiodes saturate at a lower optical input power level with

laser I as the source. In fact, the photodiodes are saturated by the high peak power

of laser I. At a photon number of 108 per incident pulse,

– Laser I: peak power = 108 hν~0.7ps = 18 W

– Laser II: peak power = 108 hν~50ns = 0.26 mW

To choose an appropriate laser source, one needs to guarantee that the photodiodes

are working in their linear regions to avoid saturation, which can be realized by

broadening the optical pulse. Chapter 4. High Speed Homodyne Detector 52

20 Positive photodiode Negative photodiode

5 Output pulse amplitude (mV)

0 0 2 4 6 8 10 8 Photon numbers per pulse x 10

Figure 4.12: Photodiode output peak voltage as a function of the photon number per pulse for laser II ( 50 ns pulse width) as a source

Considering diﬀerent photodiode response functions, the linearity of the HD electronic ampliﬁer and that of the photodiode, a 50-ns-wide pulse with a rising/falling time of 15 ns generated by externally modulating CW laser at a repetition rate of 10 MHz is chosen as the laser source.

4.3.5 Optical stability

The two beams between ﬁber coupler output ports and photodiodes (see Fig. 4.1) should be well balanced in both lengths and intensities. However, owing to imperfect stability of the optical setup, such as atmospheric turbulence and polarization dependent loss, intensities of the two beams may drift. Here, I will give a simple example to show how

HD output is inﬂuenced by small intensity drifts of the two beams. Note that, the optical source is chosen to be 50-ns-wide pulses generated by modulating CW light in this analysis. Chapter 4. High Speed Homodyne Detector 53

1. Suppose the LO (in front of the ﬁber coupler) has 2 108 photons in each pulse.

The optical peak power for each balanced beam is 0.26 mW. ( 108 hν/50 ns =

0.26 mW, where hν=1.310−19 J is single photon energy.)

2. Assuming at the beginning the two beams are perfectly balanced, during the mea-

surement, intensity of one beam has 1 % drift (for example, due to atmospheric

turbulence) while the other does not change. The residual pulse due to incomplete

cancelation of positive and negative signals will have a peak power of 2.6 µ W.

3. Considering the photodiode responsivity of 0.9 A/W and the HD electronic ampli-

ﬁers trans-impedance gain of 2.2 104 V/A, the residual signal at the HD output

will be 51 mV (2.6 µW 0.9 A/W 2.2 104 V/A). This is comparable with the

shot noise voltage that will be shown in Fig. 5.11 (c) in Section 5.3.2.

If the intensities of the two beams drift, HD output voltage will signiﬁcantly drift by several tens of mV or even hundreds of mV, which might exceed the oscilloscope range (when an optimal oscilloscope setting is chosen) and introduce diﬃculty in the measurement.

Several sources may introduce the time-varying losses: (1) polarization dependent components on the two beams, such as the ﬁber coupler with a polarization dependent splitting ratio and the optical variable delay with a polarization dependent loss. If the light polarization drifts with time, intensities of the two beams will also change and result in time-varying HD output voltage; (2) Atmospheric turbulence (which may cause

fiber bending) and temperature fluctuation (which may change the refractive index and increase reflection) are also possible sources of fiber beam losses.

To minimize intensity ﬂuctuations of the two beams caused by environmental disturbance, an enclosure is used to isolate the HD setup (Fig. 4.1), which could keep the homodyne detection system stable during one measurement time requiring 50 µs (see

Sections 5.2.2 and 5.3.2). Chapter 4. High Speed Homodyne Detector 54

In summary, several challenges in building a broad bandwidth and shot-noise limited

HD are discussed here. (1)HD electronic ampliﬁers and photodiodes should work in their linear regions. (2) A proper laser source has to be chosen considering HD bandwidth, components’ linearities and photodiode mismatch. (3) HD package should be carefully designed to improve the system stability.

4.4 Summary

In this chapter, HD requirements for a high speed GMCS QKD system are proposed:

(1) a broad bandwidth of 100 MHz; (2) 10 dB shot-noise-to-electronic-noise-ratio in the time domain with a pulsed LO over a range of 108-109 photons/pulse. Based on the

HD circuit design from Prof. Lvovsky’s group, challenges in building a high speed HD in the telecommunication wavelength region, from both electrical side to optical side, are discussed. By choosing an optimal laser source, carefully stabilizing electrical and optical components of the HD, and ensuring the linearities of photodiodes and electronic ampliﬁers, we constructed a high speed HD required by GMCS QKD experiments. Chapter 5

Performance of the Homodyne

Detector

In Chapter 4, requirements and challenges of constructing a shot-noise limited homodyne detector (HD) for high speed GMCS QKD have been discussed. Based on limitations from diﬀerent photodiode response functions, linearities of photodiodes and HD electronic ampliﬁers (see Section 4.3), I propose the HD electronic design and optical setup in

Section 4.2.

As discussed in Section 2.2.1, shot noise is proportional to the local oscillator (LO) power and electronic noise is independent of the LO power. To prove the noise generated by the HD is indeed the sum of shot noise and electronic noise, I need to verify that the output noise variance, y, is related to the LO power, x, by an equation of the form, y = ax + b where a and b are constants. On the other hand, as discussed in Section

3.3, shot-noise-to-electronic-noise-ratio ( shot noise variance divided by electronic noise variance) at a speciﬁc LO power can be used to determine the magnitude of the electronic noise (Nele, in shot noise unit) in QKD. Therefore, it is necessary to distinguish the two noises by measuring the total noise of HD as a function of the LO power. In this chapter, the HD performance is tested by this measurement.

55 Chapter 5. Performance of the Homodyne Detector 56

5.1 Experimental plan

To verify that the HD is shot-noise limited and to determine the shot-noise-to-electronic- noise ratio, experiments are carried out as follows.

• Measure HD noise in the time domain (by an oscilloscope) and the frequency do-

main (by an RF spectrum analyzer) with diﬀerent continuous wave (CW) local

oscillators.

• Measure HD noise in the time domain and in the frequency domain with diﬀerent

pulsed local oscillators. The laser design has been discussed in Section 4.3.4.

Although a pulse-resolved shot-noise limited HD is our ultimate goal to implement in a high speed GMCS QKD (see Section 2.1.2), it is technically challenging due to the requirements of high subtraction (i.e. small residual signal), ultra-low noise and ﬂat gain over a broad bandwidth (see Section 2.2.2 and 4.3). As the ﬁrst stage of verifying the

HD to be shot-noise limited, I test HD noise with CW LO, to reduce the residual signal due to diﬀerent response functions of the two photodiodes.

To illustrate feasibility of using this HD in future high speed GMCS QKD experiments, HD noise is also tested with a pulsed LO and individual measurement of each pulse will be shown. To have a thorough understanding of HD performance, measurements are performed in both the time and the frequency domains. Finally, I will predict the performance of a GMCS QKD experiment that uses our high speed HD.

5.2 Measurement with CW light

As discussed in Section 4.3.2, diﬀerent response functions of two photodiodes of the HD will cause incomplete subtraction of the positive and the negative photocurrents and result in a residual signal (like in Fig. 4.6). To avoid the residual signal and have a Chapter 5. Performance of the Homodyne Detector 57

Figure 5.1: Experimental setup with CW LO (NP Photonics). CW L: 1550 nm CW

fiber laser; VOA1-2: variable optical attenuators; LO: local oscillator; FC: 50:50 fiber coupler; OVD: optical variable delay; PD: photodiodes; AMP: HD electronic amplifiers;

OSC: oscilloscope (Lecroy); RFSA: RF spectrum analyzer (HP 8564E); Dashed line box: homodyne detector. complete subtraction of the positive and negative signals, the total noise of HD is ﬁrst measured as a function of the CW LO power.

5.2.1 Experimental setup

In the setup shown in Fig 5.1, a CW laser (NP Photonics) is used as an LO and is sent to one port of the ﬁber coupler. The other input port is left unconnected, and I treat it as vacuum input by default. At the output, an oscilloscope and an RF spectrum analyzer connected by an RF splitter are used to measure the HD noise. To eliminate high frequency noises which are beyond the desired bandwidth 1, a low pass ﬁlter with

107 MHz bandwidth is applied at the HD output (it is omitted in our setup diagram). In the subsequent discussions, LO power is deﬁned to be the optical power measured before it enters the ﬁber coupler.

1It may come from the electromagnetic oscillation of the feedback loop in the HD circuit. For this particular HD, the oscillation is over 600-800 MHz Chapter 5. Performance of the Homodyne Detector 58

5.2.2 Noise measurement in the time domain

HD noise is measured as a function of the LO power in the time domain. The experimental procedure is as follows.

1. The ﬁrst step is to calibrate the system by balancing the two output paths of the

ﬁber coupler and their losses.

Figure 5.2: Calibration setup to get the balanced condition. Pulsed L: 1550 nm PriTel pulsed femto second laser; VOA1-2: variable optical attenuators; LO: local oscillator; FC:

50:50 ﬁber coupler; OVD: optical variable delay; PD: photodiodes; AMP: HD electronic amplifers; OSC: oscilloscope (Lecroy); RFSA: RF spectrum analyzer (HP 8564E); Dashed line box: homodyne detector.

As shown in Fig. 5.2, a pulsed laser (PriTel femto second pulsed laser) is sent

to the ﬁber coupler as the LO. I change the delay time and attenuation until the

HD output voltage exhibits a minimum peak-to-peak value (shown in Fig. 4.6 in

Section 4.3.2). Most of the positive and negative electrical signals generated by

the two photodiodes cancel out under this condition (The degree of the subtraction

is quantiﬁed by the common mode rejection ratio, which will be discussed and

compared with other groups in Section 5.3.3).

2. I replace the pulsed laser by the CW source. With the setup shown in Fig. 5.1,

the output voltage is measured in a 50 µs window, with a sampling rate of 5 G Chapter 5. Performance of the Homodyne Detector 59

samples/s. Fig. 5.3 shows the waveform of the oscilloscope (raw data). The total

noise of HD is obtained by calculating the voltage variance in this time window.

Figure 5.3: Oscilloscope graph for HD noise measurement at a CW LO power of 0.187 mW. Horizontal scale: 5 µs/div. Vertical scale: 2 mV/div.

3. With a diﬀerent LO power, the measurement in Step 2 is repeated.

4. Oscilloscope noise is obtained by measuring output voltage variance in the absence

of any input to the oscilloscope channel.

Figure 5.4 shows HD noise and shot noise as a function of the LO power. Shot noise is linearly dependent on the CW LO power. HD detector is 10-times shot-noise limited at a CW LO power of 2.8 mW. Note that, the oscilloscope noise is included in the electronic noise (In fact, it is 5% of the electronic noise). Chapter 5. Performance of the Homodyne Detector 60

1 10

10 dB 0 10 2 Electronic noise

−1 10

HD noise Shot noise Noise variance (mV) −2 10

−3 10 −3 −2 −1 0 1 10 10 10 10 10 LO power (mW)

Figure 5.4: Voltage variance as a function of the CW LO power in the time domain. The red line has a slope of 1.

5.2.3 Noise measurement in the frequency domain

In the frequency domain, HD noise is measured as a function of frequency at diﬀerent

LO powers. The experimental procedure is as follows.

1. Same as Step 1 in the time domain measurement, the two output beams of the

ﬁber coupler are carefully balanced by using the PriTel pulsed LO.

2. CW laser output is used as the LO of the HD. With the setup in Fig. 5.1, HD noise

spectrum is measured by an RF spectrum analyzer from DC to 150 MHz.

3. Measurement in step 2 is repeated at diﬀerent LO powers.

4. Electronic noise is measured by blocking the optical inputs to photodiodes. The

power supply of the HD electronic ampliﬁers (not shown in Fig. 5.1) is on.

5. Background noise is measured by blocking two photodiodes and turning oﬀ the

power supply of the HD electronic ampliﬁers. Chapter 5. Performance of the Homodyne Detector 61

Figure 5.5: RF spectrum analyzer background noise spectrum (lowest curve), HD electronic noise spectrum (second lowest curve) and HD noise spectra at CW LO powers of

6.4400, 4.1400, 2.5380, 1.5960, 1.0180, 0.6460, 0.4140, 0.2528,0.1592,0.1014,0.0640,0.0410 and 0.0252 mW (from the highest to the third lowest curve). Resolution bandwidth: 1

MHz

HD noise power as a function of the frequency at diﬀerent LO powers is shown in Fig.

5.5. It indicates that (1) the HD has a bandwidth of 100 MHz (when power drops by

3 dB); (2) the noise power increases as the LO power increases.

Noise power is calculated by integrating the area under each curve. To eliminate the strong noise centered at DC (shown in Fig. 5.6, might be contributed by pink noise, instrument noise and small incomplete subtraction of positive and negative signals), the spectral noise power (in Fig. 5.5) is integrated from 5 MHz to 110 MHz, since the noise below 5 MHz is not mainly contributed by HD electronic noise and shot noise.

Figure 5.7 shows the shot noise is linearly dependent on the LO power in the frequency domain. HD is 10-times shot-noise limited at an LO power of 2.6 mW, which has a good agreement with that in the time domain, which is 2.8 mW. Here, the RF spectrum Chapter 5. Performance of the Homodyne Detector 62

−10

−20

−30

−40

−50 Noise power (dBm/MHz) −60

−70 0 2 4 6 8 10 Frequency (MHz)

Figure 5.6: Noise spectrum from DC to 10 MHz at an LO power of 0.3 mW. Resolution bandwidth: 1 MHz analyzer noise (6% of the electronic noise) is included in the electronic noise.

5.3 Measurement with pulsed light

To measure electric ﬁeld quadratures of individual pulse in GMCS QKD, a broad bandwidth and shot-noise limited HD working with pulsed LO is required. Performance of the HD with pulsed LO will be discussed in this section.

5.3.1 Pulsed laser source

As discussed in Section 4.3.4, I choose 50-ns-wide (with 15 ns rising/falling time) pulsed laser by externally modulating CW light with an amplitude modulator controlled by a function generator at a repetition rate of 10 MHz. Chapter 5. Performance of the Homodyne Detector 63

−3 10

HD noise Shot noise −4 10

10 dB −5 10 Electronic noise Noise power (mW)

−6 10

−7 10 −2 −1 0 1 10 10 10 10 LO power (mW)

Figure 5.7: HD noise power and shot noise power as a function of CW LO in the frequency domain. The red line has a slope of 1.

5.3.2 Noise measurement in the time domain

Experimental setup

The experimental setup in the time domain is shown in Fig. 5.8. Pulsed LO is sent to one input port of the ﬁber coupler. The other input port is left unconnected, and I treat it as vacuum input by default. An oscilloscope is used to measure the noise from the HD output.

Measurement

The experimental procedure is:

1. I carefully balance the two output beams of the ﬁber coupler by changing the delay

time and attenuation until a minimum peak-to-peak output voltage is achieved (20

mV in this particular case shown in Fig. 5.9). Under this condition, most of the Chapter 5. Performance of the Homodyne Detector 64

Figure 5.8: Experimental setup of HD noise measurement in the time domain. CW L:

1550 nm CW ﬁber laser; VOA1-2: variable optical attenuators; LO: local oscillator; FC:

50:50 ﬁber coupler; OVD: optical variable delay; PD: photodiodes; AMP: HD electronic ampliﬁers; OSC: oscilloscope (Lecroy); Dashed line box: homodyne detector.

Figure 5.9: HD output waveform under the balanced condition at an LO power of 0.786 mW. Horizontal scale: 50 ns/div. Vertical scale: 50 mV/div. The square box on this graph indicates one cycle.

positive and negative signals cancel out. To quantify how good the subtraction is,

the common mode rejection ratio (CMRR) can be estimated. Chapter 5. Performance of the Homodyne Detector 65

CMRR is used to characterize the subtraction ability of the HD while it rejects

DC and preserves AC of the two input signals. Ideally, one expects the HD output

Vo = A(V1 − V2), where V1 and V2 (voltage or current) are input signals. However,

for a practical HD, the output Vo can be described as A V = A (V − V ) + 2 (V + V ) (5.1) o 1 1 2 2 1 2

where A1 is the rejection mode gain and A2 is the common mode gain (typically

smaller than A1). The CMRR is deﬁned as

A1 2 A1 CMRR = 10 log10( ) = 20 log10 S S. (5.2) A2 A2

If the input and output signals are represented in power (i.e. V1, V2, and Vo are

power inputs and output), CMRR is deﬁned as 10 log S A1 S. 10 A2

To estimate CMRR experimentally, several approximations are made.

• If only one photodiode is illuminated and the other is blocked (i.e. V1 x 0,V2 =

0), output Vo1 is A V = A V + 2 V A V (5.3) o1 1 1 2 1 1 1

where A1 Q A2 is assumed considering that the rejection mode gain is far larger than the common mode gain for a balanced receiver.

• When both photodiodes are both illuminated (i.e. V1 = V2), output Vo2 is A A V = A (V − V ) + 2 (V + V ) 2 2V = A V (5.4) o2 1 1 2 2 1 2 2 1 2 1

where V1 = V2 (i.e. perfectly balanced signal inputs) is assumed.

If V is the same in the two cases, ratio of S A1 S can be obtained from S Vo1 S. 1 A2 Vo2

2 Under this particular condition in Fig. 5.9, Vo1 = 15 V and Vo2 20 mV (by read-

15 ing the peak-to-peak voltage in Fig. 5.9). CMRR can be obtained by 20 log10 S 0.02 S

2It is estimated from the average optical power shining on one photodiode 0.786~2 = 0.393 mW, pulse 4 duty cycle = 50 %, photodiode responsivity η = 0.90 A/W, and the HD circuit gain GTI =2.2 10 V/A. Output HD voltage when only one photodiode is illuminated will be 0.393 mW/50% η GTI = 15 V Chapter 5. Performance of the Homodyne Detector 66

= 58 dB. Note that, Vo1 cannot be measured directly by blocking one photodiode, since this high voltage will saturate the HD circuit.

2. 50 µs is chosen as one frame to measure the HD output voltages (shown in Fig.

5.10), while there are 500 pulse cycles in each frame. The main reason to make

measurement within a very short time is to avoid the intensity drift introduced by

the system instability and environmental inﬂuences (see Section 4.3.5).

3. Step 2 is repeated under diﬀerent LO powers.

4. Oscilloscope noise is measured in the absence of signal input.

Figure 5.10: One measurement frame at an LO power of 0.4 mW. Horizontal scale: 10 mV/div, Vertical scale: 5 µs/div.

Data processing

The following data processing is applied to the original waveform in Fig. 5.10. Chapter 5. Performance of the Homodyne Detector 67

1. I break each frame into 500 segments of equal length, named S(i).(i = 1, 2, ...500)

Each segment S(i) corresponds to 100 ns, which is the period of the pulsed LO.

These segments, having the same time span, are folded onto each other, as shown

in Fig. 5.11 (a).

Figure 5.11: Data processing procedure (LO = 0.77 mW). (a) 500 original curves (one frame) S; (b) Background curve (i.e., the average curve) of S; (c) 500 processed curves

(one frame) T . Chapter 5. Performance of the Homodyne Detector 68

2. As shown in Fig. 5.11 (a), S(i) exhibits a DC oﬀset in which the voltages are not

zero at the non-pulsed region (0-15 ns and 70-100 ns), caused by the HD circuit and

residual signals (peaks at 20 ns and 65 ns) due to diﬀerent response functions of

the two photodiode (see Section 4.3.2). I calculate the average voltage of all curves

in S(i) as the background (shown in Fig. 5.11 (b) ). Processed pulses are obtained

by subtracting the background from S(i), named T (i) (shown in Fig. 5.11 (c)).

3. For each processed curve T (i), noise is calculated by adding all voltages within the

pulsed region (from 15 ns to 70 ns), named N(i), by a similar approach in Ref. [5].

4. Variance of N(i) at each LO power is obtained as the HD noise variance.

5. The oscilloscope noise is measured in the same way in the absence of input to the

oscilloscope channel.

Results

HD noise variance and shot noise variance are plotted as a function of the LO power in

Fig. 5.12. Shot noise is linearly dependent on the LO power. The HD is 10-times shot- noise limited at an LO power of 5.4108 photons/pulse (0.7 mW) in the time domain.

Here, the oscilloscope noise (13 % of the electronic noise) is included in the electronic noise. The results with pulsed LO should not be directly compared to those obtained with CW LO, because we adopt a data processing technique that average noises within a 55 ns time window in each cycle.

In GMCS QKD, Bob needs to measure the electric field quadrature of each signal pulse individually. In the test of shot noise when vacuum is sent to the signal port, one needs to verify that each pulse yields only one quadrature value. As a simple way to check, the correlation coefficient (CC) between adjacent pulses is evaluated. At a specific

LO power of 0.77 mW, I perform HD noise measurement 10 times and combine those noises to obtain a larger sample size NLarge(j)(j = 1, 2, ..., 5000). Two arrays X(k) and Chapter 5. Performance of the Homodyne Detector 69

0 10

10 dB

) −1

2 10

Electronic noise

−2

Noise variance (V 10 HD noise Shot noise

−3 10 −3 −2 −1 0 1 10 10 10 10 10 LO power (mW)

Figure 5.12: HD noise variance and shot noise variance as a function of the pulsed LO power in the time domain. The red line has a slope of 1.

Y (k) (k = 1, 2, ..., 4900) are assigned by

X(k) = NLarge(k),Y (k) = NLarge(k + m)(m = 0, 1, 2, ..., 10). (5.5)

CC between X and Y is deﬁned as

E(XY ) − E(X)E(Y ) CC = » » . (5.6) E(X2) − E2(X) E(Y 2) − E2(Y ) Figure 5.13 depicts the CC between two measured quadrature sequences X and Y at an LO power of 0.77 mW as a function of m. The value of CC when m = 1 is 0.034, which is comparable with other HDs developed by Ref. [5] (0.04) and Ref. [57] (0.07).

The small correlation between two adjacent pulses could be contributed by HD electronic noise correlation. Fig. 5.14 shows a direct representation of the sets X and Y when m =

1. X and Y are very independent. These results demonstrate the realization of individual measurement of each pulse. Chapter 5. Performance of the Homodyne Detector 70

1.2

0.8

0.6

0.4

0.2 Correlation coefficient 0

−0.2 0 2 4 6 8 10 m

Figure 5.13: Correlation coeﬃcient between nth (X) and n + mth (Y) pulses at an LO power of 0.77 mW

HD noise measurements in the time domain with pulsed LO can be used to predict the future high speed QKD performance. Using the GMCS QKD experimental parameters in Ref. [1] (shown in Table 5.1), under the “realistic model”, GMCS QKD key rates as a function of the transmission distance with this new constructed high-speed HD and existing low-speed HD are simulated in Fig. 5.15. Note that, parameters used in this simulation are based on a low-speed GMCS QKD system and may not be valid for a high-speed experiment. This simulation is used to predict the order of magnitude of the key rate of a GMCS QKD experiment using our high-speed HD. In this simulation, an

LO photon number of 108 per pulse, at which the low-speed HD is shot-noise limited (see

Fig. 3.5 in Section 3.3), is selected for a low speed GMCS QKD system. For comparison, the same LO level is chosen for a high speed GMCS QKD system. The key rate when our high-speed HD shows 10-times shot-noise limited (at an LO of 5.4108 photons/pulse) is also simulated for a high speed GMCS QKD experiment. Chapter 5. Performance of the Homodyne Detector 71

0 Y (V)

−1

−2

−3 −3 −2 −1 0 1 2 3 X (V)

Figure 5.14: Quadrature value of nth pulse X(n) and that of n+1 pulse Y (n) = X(n+1) at an LO power of 0.77 mW

Key rates can be increased by about 2 orders of magnitude by employing our high speed HD in the GMCS QKD experiment (comparing the blue solid, the green dashed- dotted curve with the red dashed curve). This is due to the 10 MHz repetition rate permitted by 100 MHz broad bandwidth of the HD. This simulation results show GMCS

QKD key rate with the high-speed HD is expected to achieve a few Mbits/s over a short distance (0-15 km), comparable to the world-fastest key rates of 1.02 Mbits/s achieved

Table 5.1: Parameters in the key rate simulation (given in Ref. [1]). Here we assume ²A and Nleak are the same for high-speed and low-speed GMCS QKD experiments.

VA η χ ²A Nleak 16.9 0.44 2.25 0.056 0.02 Chapter 5. Performance of the Homodyne Detector 72

7 10

6 10

5 10

4 10

3 10

Key rate (bit/s) High speed (10 MHz, LO = 5.4*108 photons/pulse) 2 10 High speed (10 MHz, LO = 108 photons/pulse) Low speed (100 kHz, LO = 108 photons/pulse) 1 10

0 10 0 5 10 15 20 25 30 Distance (km)

Figure 5.15: Key rate simulation as a function of the transmission distance with a high- speed HD (running at 10 MHz repetition rate) and a low-speed HD (running at 100 kHz repetition rate)

by Ref. [4] over 20 km and 1.34 Mbits/s achieved by Ref. [46] over 10 km using single photon QKD protocols. If one looks at the blue solid and green dashed-dotted curves obtained by using the fast HD in the simulation, a stronger LO power can be used to achieve a higher key rate.

In summary, this fast HD has been demonstrated to be 10-times shot-noise limited at a pulsed LO level of 5.4 108 photons/pulse in the time domain. Meanwhile, individual measurements of each pulse are demonstrated. This HD meets the requirements discussed in Section 4.1 and is expected to improve the key rate by 2 orders of magnitude in future

GMCS QKD experiments. Chapter 5. Performance of the Homodyne Detector 73

5.3.3 Noise measurement in the frequency domain

Experimental setup

The experimental setup in the frequency domain is shown in Fig. 5.16. An RF spectrum analyzer (HP 8564E) is used to measure the HD output noise.

Figure 5.16: Experimental setup in the frequency domain. CW L: 1550 nm CW ﬁber laser; VOA1-2: variable optical attenuators; AM: amplitude modulator; PC: polarization controller; LO: local oscillator; FC: 50:50 ﬁber coupler; OVD: optical variable delay; PD: photodiode; AMP: HD electronic amplifers; RFSA: RF spectrum analyzer (HP 8564E);

Dashed line box: homodyne detector.

Measurement

The experimental procedure is:

1. Same as Step 1 in the time domain measurement in Section 5.3.2 , the system has

to be calibrated to a well-balanced condition.

2. Noise power is measured from DC to 100 MHz ﬁrst (shown in Fig. 5.17). There

are peaks at DC, 10 MHz, and higher harmonics of the repetition rate. The DC

component is mainly due to the pink noise and DC oﬀset (see the time domain

data processing step 2 in Section 5.3.2). Those 10 MHz harmonic frequencies are

mainly contributed by residual signal power due to the incomplete subtraction of

the positive and negative signals. Chapter 5. Performance of the Homodyne Detector 74

−10

−20

−30

−40

−50

Noise power (dBm/100 kHz) −60

−70 0 20 40 60 80 100 Frequency (MHz)

Figure 5.17: Noise spectrum at an LO power of 0.786 mW. Frequency range: DC to 100

MHz. Resolution bandwidth: 100 kHz.

3. CMRR measurement in the frequency domain

Direct measurement of CMRR in the frequency domain can be obtained by measur-

ing the noise power at the repetition rate (10 MHz) when (1) both two photodiodes

have incident optical powers (shown in Fig. 5.18 (a), this peak at 10 MHz is min-

imized by carefully balancing the two channels); and (2) only one photodiode has

incident optical power while the other photodiode is blocked (shown in Fig. 5.18

(b)). Note that, CMRR can not be directly measured at high LO powers. Ow-

ing to the huge trans-impedance gain (2.2104 V/A, see Section 4.3.3), the HD is

easily saturated when only one photodiode is illuminated. In this experiment, the

maximum LO to avoid saturation is 49.4µW and I choose an LO power of 24.56µW

in the particular measurement shown in Fig. 5.18.

4. Noise spectrum measurement

In Fig. 5.17, the peaks at 10 MHz and its harmonics are not mainly contributed

by the shot noise. To measure shot noise, a frequency span from 5 MHz to 6 MHz Chapter 5. Performance of the Homodyne Detector 75

Figure 5.18: Noise spectrum at an LO power of 24.56 µW when (a)two photodiodes are illuminated; (b)one photodiode is blocked. Resolution bandwidth: 100 kHz

(far from the peaks) is chosen. This noise measurement is repeated at diﬀerent LO

powers.

5. Spectrum analyzer noise is measured in the same way of step 4 in the absence of

input signal. Chapter 5. Performance of the Homodyne Detector 76

Figure 5.19: Noise spectra at diﬀerent LO powers. Frequency span: 5 to 6 MHz. Reso- lution bandwidth: 10 kHz. LO powers are 0.0029, 0.0072, 0.0142, 0.0292, 0.0458, 0.0721,

0.1136, 0.1784, 0.2920, 0.4580, 0.7180, 1.1340, 1.7760, 2.9200 mW from the lowest curve to the highest curve, respectively.

Results

From Fig. 5.18, the CMRR is obtained to be 54.33 dB by ﬁnding the ratio of powers at

10 MHz of the two cases, which is better than 45 dB in Ref. [5] measured at low LO powers (the exact power is not given).

Noise spectra at diﬀerent LO powers are shown in Fig.5.19. I integrate the spectral noise under each curve over this 1 MHz frequency span. In Fig. 5.20, the noise power is plotted as a function of the LO power. Shot noise is linearly dependent on the LO power. The HD is 10-times shot-noise limited at an LO power of 1.54 mW (1.2109 photons/pulse). The noise of the RF spectrum analyzer (6% of the electronic noise) is included in the electronic noise. Chapter 5. Performance of the Homodyne Detector 77

−5 10

HD noise Shot noise

−6 10

10 dB

−7 10 Electronic noise Noise power (mW)

−8 10

−9 10 −3 −2 −1 0 1 10 10 10 10 10 LO power (mW)

Figure 5.20: Noise power as a function of the LO power in the frequency domain. The red line has a slope of 1.

With pulsed LO, HD noise measurements are performed in the time and the frequency domains. In the time domain, to remove DC oﬀset and residual signals, we performed data processing (see Section 5.3.2), which applies a window of 55 ns in the time domain.

In the frequency domain, a 1-MHz frequency span is chosen to measure the spectral noise. Diﬀerent schemes employed in the two domains make it diﬃcult to compare the shot-noise-to-electronic-noise ratio with the pulsed LO.

5.4 Conclusions and discussions

In this chapter,the HD performance is tested by measuring HD noise as a function of the LO power in both time and frequency domains. The main results are summarized as follows.

1. The HD has a bandwidth of 100 MHz. It is of the same order as the broad- Chapter 5. Performance of the Homodyne Detector 78

est bandwidth HDs working at non-telecommunication wavelengths reported by

other groups [5, 30-32]. In the telecommunication wavelength region, no group has

reported HDs with broader bandwidths for use in quantum measurements.

2. With CW LO, shot noise is veriﬁed to be linearly dependent on the LO power in

the time/frequency domains. Shot noise can be above electronic noise by 10 dB at

2.6-2.8 mW LO powers. Shot-noise-to-electronic-noise-ratios are in good agreement

in the two domains. The measurement with CW light is just a preliminary test of

the HD and will not be used in future real GMCS QKD experiments.

3. With pulsed LO running at 10 MHz, shot noise is tested to be linearly dependent

on the LO power in the time/frequency domains. By using this HD, future high

speed GMCS QKD is able to operate at a repetition rate of 10 MHz, which is 1-2

orders of magnitude higher than current reported GMCS QKD systems [1, 43]. The

expected key generation rate (a few Mbits/s) is of the same order of magnitude as

the world-fastest single photon QKD experiments [4, 46].

In summary, for the ﬁrst time, a 100 MHz broad bandwidth HD in the telecommunication wavelength region is reported. At a pulse repetition rate of 10 MHz, a 10-times shot-noise limited HD performance can be achieved at an LO of 5.4 108 photons per pulse. With this high-speed HD, future GMCS QKD key rate is expected to achieve a few Mbits/s over a short distance.

In future, HD can be improved in the following ways.

1. Photodiode pair that is fabricated under the same condition [51] can be used to

eliminate the residual signal due to diﬀerent photodiode response functions. This

will be helpful to improve the subtraction of the HD and achieve a better shot-

noise-to-electronic-noise-ratio by using a higher LO power without saturating the

HD. With a well-matched photodiode pair, we expect to use an LO pulse containing Chapter 5. Performance of the Homodyne Detector 79

1010 − 1011 photons, which might improve the current shot-noise-to-electronic-noise

ratio by 1-2 orders of magnitude. 3

2. Electronic noise of a HD will contribute to the excess noise and reduce key rate. In

future, to reduce electronic noise, we could cool the ampliﬁers and the photodiode.

Furthermore, we can choose ampliﬁers with lower gains, since the electronic noise

scales with gain (see Section 4.3.1). To ensure the shot noise can be detected, a

stronger LO power will be required, because HD output is proportional to EL (see Section 2.2.1). To increase the LO power without saturating the HD circuit, the

problem of the residual signal due to diﬀerent photodiode response functions should

be solved ﬁrst.

3. Instead of the data processing in Section 5.3.2, ﬁltering DC component by a high

pass ﬁlter and harmonic frequencies at laser repetition rate by notch ﬁlters can be

used to remove the residual signal and HD circuit DC oﬀset (shown in Fig. 5.11 (b)),

which is suggested by Prof. Lvovsky. This will not improve the HD performance

directly but will avoid the trouble of implementing the data processing.

4. HDs based on integrated circuits (IC) have an advantage over those based on dis-

crete circuits. The main advantages of an IC are: (1) components are built in

a compact way and the parasitic impedance can be reduced; (2) a good quality

control and repeatable performance can be achieved. However, developing the HD

based on IC is also challenging: (1) designing the interface between commercial

photodiodes and IC might be diﬃcult; (2) a custom IC may lack ﬂexibility for

corrections and incremental improvements. Although there are challenges, it still

3This can be roughly estimated from Fig. 4.9 in Section 4.3.3 and Fig. 5.11 in Section 5.3.2 (c). The linear region of the HD electronic ampliﬁer is at least 2V obtained from Fig. 4.9. If the residual signal is neglected using a well-matched photodiode pair, shot noise magnitude can be 2V without saturating the HD, which is 2 orders of magnitude greater than 20mV in Fig. 5.11 (c) in Section 5.3.2 at an LO of 6108 photons/pulse (0.77 mW). Therefore, we expect to use 6 1010 photons/pulse without saturating the HD by employing a perfectly matched photodiode pair. Chapter 5. Performance of the Homodyne Detector 80

might be a promising direction for further improving HD performance. Chapter 6

Conclusion and Future Work

6.1 Signiﬁcance and contribution

Pulse-resolved balanced homodyne detection has attracted a great attention for its application in quantum information and quantum optics [73]. A broadband and pulse-resolved

HD in the telecommunication wavelength region used in GMCS QKD has been developed and tested in this thesis. Table 6.1 compares the performance of our HD and the HD from Hirano’s group, which has the broadest bandwidth in quantum optics and quantum information applications. The bandwidth of our HD has achieved the same order of magnitude as that in Ref. [5]. HD subtraction performance (indicated by CMRR) of our HD is 9 dB higher (Note that, authors of Ref. [5] measured the CMRR using a low

LO power but they did not specify their LO power. The comparison might not be made with the same LO power).

In Ref. [5], squeezed state is measured in both time and frequency domains. During their measurement, pulses are identical when a particular phase between LO and signal is

ﬁxed. Therefore, measurement in the time domain is not the only way to characterize the squeezed state. Measuring the noise in the frequency domain can also used to determine the uncertainty of squeezed states.

81 Chapter 6. Conclusion and Future Work 82

In contrast to the squeezed state measurement, in GMCS QKD experiments, signal pulses should be measured individually since they are modulated with Gaussian distributed keys. Hence it is very important for us to realize the pulse measurement in the time domain, which is demonstrated in the HD noise correlation measurement. Our experiment shows the correlation coeﬃcient of adjacent quadratures is 0.034, which is on the same order of magnitude as Ref. [5] (0.04) and Ref. [57] (0.07).

With our 100 MHz bandwidth and pulse-resolved HD, we expect to perform GMCS

QKD experiments at a repetition rate of 10 MHz, which will improve the GMCS QKD key rate per second by 1-2 orders and will be of the same order as the key rate achieved by current high speed single photon QKD experiments [4, 46].

For the 20-km GMCS QKD work based on an existing system with a low speed

HD (developed by Dr. Bing Qi and Lei-Lei Huang), two conference papers have been accepted by the Conference on Lasers and Electro-optics (CLEO/QELS 2008) and SPIE

Optics + Photonics 2008 [6, 7]. High speed HD work (done by me, under the daily supervision of Dr. Bing Qi, and in frequent discussions with Profs. Li Qian and Hoi-

Kwong Lo) has been presented in QuantumWorks 2009 (poster) [84]. A journal paper is under preparation. Collaborated work on high speed random number generators, in which I worked on preliminary random number tests with NIST Statistical Test Suite, has been accepted by 9th Asian Conference on Quantum Information Science [85].

Table 6.1: HD peformance comparison between our group and Ref. [5] Our HD Ref. [5]

Wavelength 1550 nm 1064 nm

Bandwidth 100 MHz 250 MHz

CMRR 54 dB (LO = 24.56 µW) 45 dB (low LO power) Chapter 6. Conclusion and Future Work 83

6.2 Future work

In this thesis, we developed a high speed shot-noise limited HD in the telecommunication wavelength region. There are a few practical problems we can try to solve, such as diﬀerent photodiode response functions. For a fully telecommunication ﬁber-based high speed GMCS QKD, an optical system and a control system should be investigated in future.

1. Diﬀerent photodiode response functions

In high speed HD construction, a very practical problem is the residual signal

due to diﬀerent response functions of the photodiodes. To overcome this problem,

there are a few feasible methods. First, a well-matched photodiode pair that two

photodiodes are fabricated under idenetical condition can be used to minimize the

residual signal. Second, an HD based on one photodiode can be studied, which we

call a self-diﬀerencing scheme.

This self-diﬀerencing scheme is ﬁrst introduced by Ref. [86] to construct high speed

single photon detectors. If we use this idea in HD, the schematic is shown in Fig.

6.1,

(a) The signal and local oscillator interfere at the left 50/50 optical splitter.

(b) We can introduce a time delay between the resulting two pulses to separate

them in the time domain.

should expect two separate optical pulses (shown as the red pulses). We call

them pulse 1 and pulse 2.

(d) We use one photodiode to detect both optical pulses, converting them into

two electrical pulses (shown as the lower right two blue pulses). Chapter 6. Conclusion and Future Work 84

Figure 6.1: A self-diﬀerencing scheme in homodyne detection. Red pulses are optical pulses and blue pulses are electrical pulses.

(e) The two electric pulses are then splitted into four electric pulses by the electric

splitter. We call them pulse 1’, 2’, 1”, 2”.

(f) We introduce another time delay which is identical to the optical delay in-

troduced in Step (b) by an electric delay. The purpose is to align the second

pulse in the upper path (pulse 2’) and the ﬁrst pulse in the lower path (pulse

1”) in the time domain.

(g) We perform a subtraction between the two paths. We expect to receive a large

negative pulse (pulse 1’) and a large positive pulse (pulse 2”). In the middle

of these two strong pulses we may observe the desired result (pulse 1” - pulse

2’).

(h) Output signal will be ampliﬁed by electronic ampliﬁers (not shown).

The advantages of this scheme are

(a) All the components (except the electronic ampliﬁers) are passive with a high Chapter 6. Conclusion and Future Work 85

speed and low noise.

(b) We use only one photodiode. Therefore we can eliminate most of photodiode

mismatch discussed in Section 4.3.2.

But there are still a few challenges in this scheme,

(a) All the passive devices should not introduce much distortion to the shapes of

electrical pulses.

(b) With a huge gain of subsequent electronic ampliﬁer, the pulses 1’ and 2” will

saturate the electronic ampliﬁers. One possible solution is to let electronic

ampliﬁers work in a gating mode (i.e., it is operating when the incoming signal

is the subtraction of pulses 1” and 2’). An alternative method is to carefully

choose the repetition rate to overlap pulse 1’ and pulse 2” from the previous

cycle in the time domain. This will also double the eﬀective repetition rate.

This novel self-diﬀerencing scheme is worth trying to solve the photodiode mismatch

problem.

2. Local oscillator power

The optimal LO power has to be investigated systematically. The excess noises

from Bob’s system have two sources: HD electronic noise Nele and noise associate

with the leakage photon from LO to signal Nleak. To suppress Nele, a strong LO should be chosen in order to improve the shot-noise-to-electronic-noise ratio. How-

ever, controlling the leakage photon from the LO beam to the signal beam is more

technically challenging with a strong LO. Therefore, a tradeoﬀ between the two

noises has to be made when we choose a proper LO power. Chapter 6. Conclusion and Future Work 86

3. Optical scheme

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leakage interference of LO and leakage from LO to signal [1, 43, 44]. It is necessary

to conduct more research to select an optimal scheme for future high speed GMCS

QKD.

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