Optical Heterodyne Detection of Cross-Phase

LMU

MPQ

Optical Heterodyne Detection of Cross-Phase Modulation Induced by a Single Atom Optische Heterodyn-Detektion einer Kreuzphasenmodulation induziert durch ein einzelnes Atom Jonas Neumeier

Master's Thesis

Max Planck Institute of Quantum Optics, Garching Quantum Dynamics Group directed by Prof. Dr. G. Rempe Faculty of Physics, Ludwig-Maximilians University, Munich

April 2018

Cover Illustration

The cover picture shows playful simulations with eld quadra-

tures changing over time in an optical phase space. By changing phase and amplitude of a light eld in a controlled way, it is possible to "draw" these artful forms.

Betreuer der Masterarbeit

Prof. Dr. Matthias Kling

i

&

Dr. Tatjana Wilk

Abstract Controlled interactions between individual photons are of fundamental interest as well as of technological signicance for applications such as quantum information devices and quantum sensors. Cavity quantum electrodynamics in the regime of strong light-matter coupling provides a promising platform for implementing suitable nonlinear interactions. We realize this with a single rubidium atom trapped at the centre of a high-nesse Fabry-Pérot cavity. In an N-type atomic energy-level scheme, one of two distinct transitions of the atom is strongly coupled to a cavity mode, and the other transition is o-resonant with another cavity mode (dispersive regime). Both modes are driven by light elds at wavelengths and

795 nm

(signal).

780 nm (probe)

A strong coherent control laser causes electromagnetically

induced transparency (EIT) together with the probe, and induces coupling between the two modes. A large cross-phase modulation is expected: single photons present in one mode induce a signicant phase shift on photons in the other mode. It can be understood as the result of a signal photon number dependent Stark shift changing the refractive index seen by the probe. This thesis reports on the successful implementation of a balanced optical heterodyne detection setup to the experiment, making phase and amplitude measurements at the single photon level possible. The detection setup combines analog downmixing with a eld-programmable gate array-assisted data acquisition, oering direct digitization of both quadratures. The mentioned cross-phase modulation is observed: a phase shift of

29°

induced by a single signal photon is measured.

ii

Contents Abstract

ii

1

Introduction

1

2

Theory

5

2.1

2.2

3

Cavity QED in the Strong-Coupling Regime

. . . . . . . . . . . . .

5

2.1.1

The Jaynes-Cummings Model

2.1.2

Driving and Dissipation

2.1.3

The Strong-Coupling Regime

. . . . . . . . . . . . . . . . .

9

2.1.4

Two Cavity Modes Coupled to an N-Type Atom . . . . . . .

11

Optical Heterodyne Detection

. . . . . . . . . . . . . . . . .

5

. . . . . . . . . . . . . . . . . . . .

8

. . . . . . . . . . . . . . . . . . . . .

16

2.2.1

Beat Notes and Visibility . . . . . . . . . . . . . . . . . . . .

17

2.2.2

Balanced Optical Heterodyning

. . . . . . . . . . . . . . . .

18

2.2.3

Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2.4

Recovering Amplitude and Phase via Demodulation . . . . .

25

2.2.5

Unbalanced Eciencies . . . . . . . . . . . . . . . . . . . . .

27

Experimental Setup

29

3.1

Balanced Optical Heterodyne Detection Setup . . . . . . . . . . . .

29

3.1.1

Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.1.2

Electronic Demodulation . . . . . . . . . . . . . . . . . . . .

33

Characterization of the Heterodyne Detection Setup . . . . . . . . .

36

3.2

3.3

3.4

3.2.1

Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.2.2

Frequency Response

. . . . . . . . . . . . . . . . . . . . . .

37

3.2.3

Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.2.4

Malus's Law . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Experiments at the Cavity . . . . . . . . . . . . . . . . . . . . . . .

44

3.3.1

Experimental Apparatus . . . . . . . . . . . . . . . . . . . .

44

3.3.2

Experimental Sequence, Data Acquisition . . . . . . . . . . .

46

Characterization of the Integrated Heterodyne Setup

. . . . . . . .

49

3.4.1

Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

3.4.2

Phase Stability

54

3.4.3

Optical Path Length Dierence

3.4.4

Signal-to-Noise

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

. . . . . . . . . . . . . . . . . . . . . . . . .

59

iii

Contents

4

5

Measurement of Cross-Phase Modulation

62

4.1

First Test with Empty Cavity

. . . . . . . . . . . . . . . . . . . . .

62

4.2

Cross-Phase Modulation

. . . . . . . . . . . . . . . . . . . . . . . .

63

Conclusion and Perspectives

70

Bibliography

72

Acknowledgements

78

iv

1 Introduction In the

20th

century quantum mechanics was discovered leading to a rst quan-

tum revolution. We used quantum mechanics to understand what already exists in nature at the smallest scales. With enormous success we were able to explain the individual behaviour of subatomic particles as well as atoms. We could make use of the knowledge for applications such as transistors and lasers [1] which have become indispensable in today's society, but we could not manipulate individual quantum particles. Nowadays it is believed that we are in the midst of a "second quantum revolution" [2] due to the prospect of new quantum technology which is thought to have a huge impact on our society [3]. The new technological advancement is based on the successful manipulation of individual quantum particles, such as single atoms and photons, which, not so long ago, was thought to remain in the realm of abstract theory [4].

We are now actively applying quantum mechanics to design

highly "unnatural" quantum states, states based on superposition and entanglement that are not likely to exist anywhere else in the universe. In addition to, e.g., explaining the structure of atoms, we can create articial atoms such as quantum dots which we can engineer to have desired electronic and optical properties. The now possible creation of designed states of coherent or entangled light and matter allows us to integrate them into complex systems with novel properties that have wide application in the development of quantum computers [5], quantum communication systems [6], quantum sensors and metrology [7, 8]. As the potential has been widely recognized [3], governments in many countries are now directing increasing funding towards quantum research to accelerate the development, and companies are trying to design and produce applications that take full advantage of the new technologies. For example, the Future and Emerging Technologies Flagship on Quantum Technologies is an existing highly-nanced program by the European Commission to move advanced quantum technologies from the laboratory to industry while advancing at the same time the fundamental science basis. Among many challenging goals of quantum research is the realization of controlled strong interactions between individual photons. Besides being of fundamental interest, strong photon-photon interactions enable unique applications such as single-photon switches, transistors, and all-optical deterministic quantum logic [9].

1

1 Introduction

Physicists have known for over half a century that light elds can interact inside nonlinear optical media.

However, the nonlinearity of conventional materials is

vanishing at the light powers associated with single photons. Therefore, new systems have to be designed. The regime where individual photons interact strongly is referred to as quantum nonlinear optics. We use cavity quantum electrodynamics (CQED) in the regime of strong lightmatter coupling which provides a promising platform for implementing nonlinear interactions between individual photons. Quantum electrodynamics (QED) is a very successful theory of light-matter interaction, treating both the fundamental components of matter and the electromagnetic eld quantum-mechanically. While an atom in free space interacts with a continuum of modes of the electromagnetic eld, inside a cavity or resonator only modes fullling a resonance condition are present, oering a much simpler theoretical treatment [10], and interesting enhancement eects. This is the basic idea of CQED. Spontaneous emission of an atom is commonly understood to be induced by vacuum uctuations of the electromagnetic elds. Two mirrors forming a cavity may increase the vacuum uctuations of a single eld mode in such a way that the coupling strength of the atom-cavity exceeds the interaction of the atom as well as the cavity with the free-space modes of the environment. In other words, the rate of coherent energy exchange between atom and cavity mode may become signicantly bigger than the dissipation rates of the system. This regime is referred to as the strong-coupling regime where the nonlinearity of quantum emitters is enhanced. It has been used, e.g., to realize photon blockades [11, 12], where photons of the same light eld inuence each other. A system which enables interaction between two dierent light elds is of great interest as well, and is used, e.g., for mutual switching [13, 14]. In this work, we focus on a phase shift of a light eld induced by another light eld.

Since the

1990s lots of active research has been going on towards the realization at very low light powers. Applications such as quantum phase gates [15], and quantum nondemolition measurements [16, 17] are one of the prospects. A possible mechanism to achieve this phase shift is electromagnetically induced transparency (EIT) with ensembles [1820], but here the single-photon phase shifts measured to date are quite weak, on the order of

10 µrad [21, 22].

However, the combination of Rydberg

states with EIT is very appealing [2325], and has led to impressive experimental results, e.g. the generation of a

π

phase shift with a single-photon pulse [26].

Another promising strategy is to couple a cavity to a single atom [15, 27, 28], an atomic ensemble [29], or a quantum dot [30].

2

1 Introduction

Here, we use a single four-level atom in an N-type conguration where one transition is strongly coupled to one cavity mode (780 nm), and the other separate transition is o-resonantly coupled to a dierent longitudinal mode of the same cavity (795 nm). A coherent control laser causes EIT together with the drive of the strongly coupled mode, and induces coupling between the two modes. In this system, a large cross-phase modulation is expected: single photons present in one mode induce a signicant phase shift on photons in the other mode. Studies using similar systems worked with huge detunings [13], or with just one of the two transitions mode-coupled [31, 32].

Many atoms were then needed to

observe notable eects. Our system provides therefore a unique platform. One of the challenges of this thesis project was to measure the just mentioned cross-phase modulation. Using conventional detectors (e.g., a single photon counter module), the phase information of a coherent optical light beam is erased, only yielding the optical power. As a result, interesting physics remains hidden. This is due to the inherently slow electronics of a detector relative to the fast oscillation of an electromagnetic eld in the visible spectrum. In optical heterodyne detection, using a reference beam (local oscillator), the frequency of the light eld is simply shifted into the accessible radio frequency domain, which makes it possible to preserve and resolve phase information with a normal photodetector. Moreover, the resulting electronic signal is proportional to the amplitude of the light eld in contrast to conventional light detection. Heterodyning stems back from the early days of radiotelegraphy in the beginning th of the 20 century [33], and is nowadays still most common in the radio frequency regime. However, the idea has been successfully transferred to optical frequencies with the invention of lasers [34]. Using a balanced optical heterodyne detection scheme, it is possible to reach very high sensitivity, making it feasible to investigate the CQED system at the single photon level. A signal-to-noise ratio only limited by the shot noise of the probe can be achieved. This interferometer-like setup was built and successfully implemented during the course of this thesis. Moreover, the setup was complemented by analog downmixing of the detector signal to yield the so called quadratures, together with direct digitization of these. The thesis is structured as follows: In Chapter 2, we provide important background and theory. We begin the chapter by presenting an introduction to CQED in the strong-coupling regime, starting with a simple two-level atom single cavitymode system. After that we include dissipation to the environment and external driving of the system with lasers. The strong-coupling regime is explained, and simulations of the system's spectral response are shown.

3

Then, we extend the

1 Introduction

two-level atom and one cavity mode scheme to a four-level atom and two cavity modes scheme. In the focus are interactions between single photons of two light elds at dierent wavelengths which lead to a cross-phase modulation.

Simula-

tions underpin this short introduction. Furthermore, we present balanced optical heterodyne detection. We start with an explanation of beat notes and visibility, being fundamental to heterodyning. We introduce the quantum theory of balanced optical heterodyne detection and arrive at an expression of the measured output voltage of a photodetector. Furthermore, we discuss noise and the signal-to-noise ratio of heterodyning, and the extraction of amplitude and phase information of a detected light eld. Finally, we conduct a classical analysis of the eect of unbalanced eciencies of a balanced detector scheme. In Chapter 3, we present the setup behind balanced optical heterodyning. We show the experimental apparatus, the experimental sequence and data acquisition. The heterodyne setup is characterized: visibility, frequency response of the detector, and noise characteristics are measured. Moreover, the proportionality of the measured signal to the electric eld amplitude of the detected light is experimentally conrmed. The analog downmixing yields two orthogonal quadratures which can be used to calculate the phase and the amplitude of the detected light eld. The determination of these quadratures is prone to errors. Therefore, a section is devoted to their characterization and correction. Furthermore, we demonstrate phase stability of the setup to guarantee a viable phase measurement. In addition, an important eect, which needs to be taken into account when recording a phase spectrum, and which is caused by an optical path length dierence between local oscillator and the probed light eld, is discussed. Finally, we show the signal-tonoise dependence on the local oscillator power. Experimental results are presented in Chapter 4. First, a known physical system, an empty cavity, was used to demonstrate the functionality of the heterodyne detection setup.

An amplitude and a phase spectrum were obtained to yield the

decay rate of the empty cavity. Second, we demonstrate a measurement of a phase shift on the probe light induced by a single intracavity signal photon on average using a single four-level atom in an N-type conguration. Chapter 5 concludes the thesis with a brief summary and perspectives. We present ideas on how to continue with the N-type system, and suggest dierent experiments which are believed to be possible now due to the new balanced optical heterodyne detection setup.

4

2 Theory In this chapter, a theoretical description of the atom-cavity system is given, and the concept of the balanced optical heterodyne detection is introduced.

2.1 Cavity QED in the Strong-Coupling Regime In this section, we present the basic principles of cavity quantum electrodynamics (CQED) in the strong-coupling regime at the single photon level.

First, we

briey introduce the Jaynes-Cummings (JC) model, where a two-level atom is interacting with a single mode of the electromagnetic eld inside a closed cavity. The anharmonic ladder of eigenstates of the coupled system proves the system to be a suitable platform for nonlinear quantum optics. Second, we include in the description external driving of the system with a laser, and dissipation.

Third,

we introduce the strong-coupling regime and its necessary conditions, a hallmark being the observation of normal mode splitting. In addition, we show simulations of the system's spectral response. Finally, we extend the two-level atom and single cavity mode scheme to a four-level atom and double cavity-modes scheme which is the platform we will investigate experimentally.

2.1.1 The Jaynes-Cummings Model As mentioned we are interested in the interaction between a single atom and a single mode of the electromagnetic eld. A cavity is the tool to isolate a single mode of the vacuum by creating boundary conditions. lar frequency

ωc

ωc = mcπ/L,

with

m ∈ N,

and the vacuum speed of light

be described quantum mechanically: the Fock states of photons

Only modes with angu-

t the cavity of length L according to the resonance condition

|ni,

c.

A single mode can

representing the number

n

in the mode, are a possible orthogonal basis of the Hilbert space. The annihilation operator a ˆ and creation operator a ˆ† lowers and raises, respec-

tively, the number of photons in a given mode by one. The associated harmonic eigenenergy ladder makes the cavity a linear optical element (see Figure 2.1(a)). 1 The eigenenergies are given by En = }ωc (n + ), where } is the reduced Planck 2 constant.

5

2.1 Cavity QED in the Strong-Coupling Regime

›

› |2,–›

|2,+

2√2g

|2 ωc

ωc

›

›

|e ωa

ωc

ωc

›

|0

(a)

› ›

|1,+ |1,–

2g

|1

›

|g

(b)

›

|0,g

(c)

Figure 2.1: (a) Harmonic energy ladder of a cavity mode with excitation energy

}ωc ,

where the rungs

n

count the number of photons. (b) A two-level

system with ground state

}ωa

|gi,

excited state

modelling a single atom.

|ei,

and transition energy

(c) Anharmonic Janyes-Cummings

energy ladder of the coupled system.

√ 2 ng

is the oscillatory energy

exchange rate between atom and cavity mode. Note:

In the illustration,

depicted (ωc

}

was set to 1, and the resonant case is

= ωa ).

The second ingredient we need for our system is a single atom, which is approximated by a two-level system.

In contrast to the intrinsically linear cavity,

the atom already saturates after one excitation by a photon at the atomic res-

ωa . We can assign a two dimensional state |gi and the excited state |ei, the two

onance angular frequency

Hilbert space

spanned by the ground

eigenstates of

the Hamiltonian operator, to describe the system. Possible associated operators are the atomic raising and lowering operators σ ˆ + = |eihg| and σ ˆ − = |gihe|, respectively, driving the transitions between both eigenstates, and the Pauli spin matrix σ ˆz = |gihg| − |eihe| = σ ˆ−σ ˆ+ − σ ˆ+σ ˆ − . The expectation value of the latter is an observable physical quantity and relates to the probability of nding the atom in state

|gi

or

|ei

upon a corresponding measurement.

The atom is schematically

represented in Figure 2.1(b). When the atom is placed inside the cavity, it can be well described by the Jaynes-

6


Cummings (JC) model [10]. In general, the Hamiltonian for the coupled system is:

ˆ =H â + H ˆc + H ˆ ac , H where

â H

is the Hamiltonian describing the atom,

(2.1)

ˆc H

is the Hamiltonian of the

ˆ ac the atom-eld interaction Hamiltonian. cavity eld, and H

Applying the rotating

wave- and dipole approximation, we arrive at the Jaynes-Cummings Hamiltonian:

ˆ JC = }ωa σ H ˆ+σ ˆ − + }ωc a ˆ† a ˆ + }g(ˆ σ−a ˆ† + σ ˆ+a ˆ).

(2.2)

The zero-point energy of the eld mode was neglected as we are only interested in the dynamics of the system.

The coupling constant

g

quanties the interac-

tion strength between the electric eld and the atom, and relates to the coherent exchange rate of an excitation. It is given by:

s g= with dipole moment

d,

mode volume

V,

ωc d2 , 2}V 0

(2.3)

and vacuum permitivity

0 .

Solving

ˆ JC H

gives rise to the eigenstates, which are commonly referred to as dressed-states with ground state

|0, gi

and excited states

|n, +i, |n, −i,

where

n ≥ 1

is the overall

number of excitations in the system:

|n, +i = sin (θ) |n, gi + cos (θ) |n − 1, ei |n, −i = cos (θ) |n, gi − sin (θ) |n − 1, ei. The mixing angle

θ

(2.4a) (2.4b)

depends on detuning between the atomic transition frequency

and the cavity frequency

δ = ωa − ωc : √ 2 ng p √ tan(θ) = δ + (2 ng)2 + δ 2

(2.5)

The eigenenergies are:

En,±

q √ 1 2 2 = }ωc n + } (ωc + ωa ) ± (2 ng) + δ . 2

(2.6)

In case of zero detuning, the dressed-states simplify to

1 |n, ±i = √ (|n, gi ± |n − 1, ei). 2 They form a nonlinear ladder of doublets with energy splitting

(2.7)

√ ∆E = } n2g

known as the Jaynes-Cummings ladder (Figure 2.1(c)). For a detailed derivation of the JC model, the reader is referred to textbooks, e.g. by Scully [35] or Walls [36].

7


2.1.2 Driving and Dissipation In order to be more realistic, driving and dissipation of the system have to be included in addition to the theory presented above. A driving term representing a coherent laser beam at angular frequency

ωd

is added to the Hamiltonian

ˆ JC . H

The laser is either shone on the cavity in which case the driving term takes the ˆ dc = }ηc e−iωd t a ˆ da = form H ˆ† + h.c., or it is incident on the atom and is given by H −iωd t + }ηa e σ ˆ + h.c., where ηc and ηc are the respective driving strengths. It is useful to rewrite the new Hamiltonian in the rotating frame:

ˆ = }∆da σ H ˆ+σ ˆ − + }∆dc a ˆ† a ˆ + }g(ˆ σ−a ˆ† + σ ˆ+a ˆ) + }ηa (ˆ σ− + σ ˆ + ) + }ηc (ˆ a+a ˆ† ), where

∆da = ωd − ωa (∆dc = ωd − ωc )

(2.8)

is the atom (cavity) detuning with respect

to the driving laser. Now, we include dissipation processes by coupling the system to the environment which is modelled as a thermal bath. They comprise leakage of photons out of the cavity mode quantied by the decay rate

κ, and spontaneous emission of the atom γ . A common approach is the usage

into free space quantied by the decay rate

of the Master equation in Lindblad form which takes into account the dissipative system-environment interaction. A short summary of this approach is given here. In [37], a detailed description can be found. A density matrix is used to consider system and environment jointly.

The Lindblad Master equation describes the

evolution of the system density matrix

ρ which represents a statistical ensemble of

quantum states with respective probabilities. The Lindblad Master equation takes the form:

i X ihˆ † †ˆ †ˆ ˆ ˆ ˆ ˆ 2Ci ρ(t)Ci − ρ(t)Ci Ci − Ci Ci ρ(t) . ρ(t) ˙ = − H, ρ(t) + } i

(2.9)

The rst term models the coherent evolution of the system and the second the coupling to the environment. with dissipation rates

γi .

√ ˆ Cî = γ i O i

are generalized dissipation operators

The system-environment interaction is irreversible and

unidirectional (Markov approximation). The equation of motion can be written in terms of the Lindblad superoperator

L:

ρ(t) ˙ = Lρ(t),

(2.10)

ρ(t) = eLt ρ(0).

(2.11)

and is formally solved by:

The system relaxes to a steady state

Lρss = 0.

The time dependent as well as

the steady state expectation values for an arbitrary experimental observable given by:

8

ˆ O

are


ˆ ˆ Lt ρ(0) hO(t)i =tr Oe ˆ ˆ hOss i =tr Oρss .

(2.12) (2.13)

2.1.3 The Strong-Coupling Regime Spontaneous emission of an atom is commonly understood to be induced by vacuum uctuations of the electromagnetic elds. A cavity may increase the vacuum uctuations of a single eld mode such that the coupling strength between atom and cavity exceeds the coupling of the atom and cavity to the environment. In other words, the rate of coherent energy exchange between atom and cavity mode becomes much bigger than the dissipation rates of the system. We refer to it as the strong-coupling regime of CQED. Experimentally, a signature of this regime is that the energy splitting of the rst doublet of the Janyes-Cummings ladder (see Figure 2.1(c)) becomes spectroscopically resolvable. Now, to gain more insights, we show simulations taking into acount the results presented in Subsection 2.1.2. The associated generalized dissipation operators are √ √ − ˆ and Cˆc = κˆ a. The Lindblad superoperator is constructed according Câ = γ σ to Equation 2.9. With the Master equation we nd numerical solutions for dierent parameters and observables via the Quantum Toolbox in Python (QuTiP) [38]. For the intracavity electromagnetic eld amplitude and phase spectra of the system, we express the detunings relative to the cavity frequency (cavity frame of reference) by substituting

ωa

by

ωc − ∆ca

in Equation 2.8 (∆ca

= ωc − ωa

is the

atom-cavity detuning). We will not consider the atom driving term now, so the employed Hamiltonian takes the form:

ˆ = }(∆dc − ∆ca )ˆ H σ+σ ˆ − + }∆dc a ˆ† a ˆ + }g(ˆ σ−a ˆ† + σ ˆ+a ˆ) + }ηc (ˆ a+a ˆ† ), where we scan

∆dc .

(2.14)

We yield the intracavity amplitude and phase spectra by rst

calculating the steady state expectation values for the quadrature operators of the eld:

Then we calculate the

1 † a +a ˆ) qˆ = (ˆ 2 i † pˆ = (ˆ a −a ˆ). 2 amplitude |α| via: p |α| = hˆ q i2ss + hˆ pi2ss

(2.15)

(2.16)

(2.17)

and the phase via:

θ = arctan

9

hˆ qi ss

hˆ piss

.

(2.18)

Ϭ͘ϭϮϱ

Ϭ͘Ϭϴ

Ϭ͘ϭϬϬ

Ϭ͘Ϭϲ ͮ ͮ;Ă͘Ƶ͘Ϳ

ͮ ͮ;Ă͘Ƶ͘Ϳ


Ϭ͘Ϭϳϱ Ϭ͘ϬϱϬ

Ϭ͘Ϭϰ

Ϭ͘ϬϮϱ

Ϭ͘ϬϮ

Ϭ͘ϬϬϬ

Ϭ͘ϬϬ

ϮϬ

Ϭ

dc/Ϯ ;D,ǌͿ

ϮϬ

ϮϬ

Ϭ

dc/Ϯ ;D,ǌͿ

ϮϬ

Figure 2.2: (a) Tuning the frequency of the drive with respect to the cavity yields the amplitude spectrum for the uncoupled (blue) and coupled (orange)

g/2π = 10 MHz: the blue curve with ∆ca /2π = 10 MHz and the orange curve with ∆ca /2π = −3 MHz. For all shown curves we chose κ/2π = 3 MHz, γ/2π = 3 MHz.

system. (b) The coupled system under detuned conditions for

The results are shown in Figure 2.2 and 2.3. In Figure 2.2(a), the amplitude spectrum in blue colour shows the case of vanishing coupling between atom and cavity (g

= 0), thus the root Lorentzian of the empty cavity with a linewidth √ square 1 3κ/2π . The maximum amplitude is 2ηc /κ. For the coupled (FWHM in MHz) of system (with ∆ca = 0), this resonance splits into two new modes (orange curve) at frequencies ±g/2π = ±10 MHz, called the normal modes. The linewidths of the √ 3(γ + κ)/(4π), and the maximum amplitude is 2ηc /(κ + γ). The resonances are curves in Figure 2.2(b) illustrate the asymmetric amplitude spectrum in the atomcavity detuned case (∆ca

6= 0):

The minimum and maxima shift in frequency and

the height of the resonances is unequal. Interestingly, the location of the minimum depends only on the transition frequency of the bare atom. As we nd ourselves in the cavity-frame of reference, an atom-cavity detuning means a change of the

2

atomic resonance only . In the atom-frame of reference (not shown here), the position of the minimum does not shift, however the position of the maxima do change. This atom dependent feature (the minimum) corresponds to an antiresonance of

1 The

FWHM (in MHz) of the Lorentzian of an intensity spectrum of the empty cavity is κ/2π . we can tune the atomic resonance via the intensity of the trapping laser which induces an ac-Stark shift.

2 Experimentally,

10


the system [39]. For an experimental observation, it is much more accessible in a phase measurement due to noise and its much more pronounced expression in a phase spectrum. Figure 2.3 depicts simulated phase spectra under various conditions. The red and green curves show atomic transition frequency detunings, the orange curve zero detuning. At the antiresonance, the phase undergoes a reverse phase shift compared to the shift at the normal modes. The blue curve illustrates the case of the empty cavity with its inverse tangent nature.

An ac-Stark shift

on the atom can be used as an all-optical way to control the corresponding phase modulation of the light as it changes the transition frequency of the bare atom.

Figure 2.3: Phase spectra of the system with an overall phase shift of

π in all cases.

The antiresonance depends only on the properties of the atom. The coupling strength was chosen

g/2π = 10 MHz

for the red, orange and

green curves. The blue curve shows a vanishing coupling between atom and cavity.

∆ca

is the atom-cavity detuning.

2.1.4 Two Cavity Modes Coupled to an N-Type Atom Typically, two electromagnetic eld modes do not interact without a medium. However, already a single four-level atom in N-type conguration (Figure 2.4) can enable strong interactions between two light elds that couple strongly to the outer separate transitions of the "N". A strong coherent laser couples the separate transitions. This system is capable of mediating strong nonlinear interactions between two dierent wavelengths. Recently, the experimental realization of such system

11


with two modes resonantly coupled to two transitions has been successful in our group as shown in [14]. To get a full description of this system, including theoretical and technical aspects, we refer to [14]. Here we continue to use and investigate the system, but in the dispersive regime where one atomic transition is stronglycoupled to one mode, and the other separate transition is o-resonant with the other mode. Observation of a large cross-phase modulation can be expected. The basic principles are explained in the following.

A ωa

gp

› ›

ωp

Δ42

|4

|3

Δc

Δp

B

ωc

ωs

Ωc

gs

ωb

› |2 ›

|1

Figure 2.4: Representation of the N-type system with central parameters which are explained in the text. The 'N' in N-type simply sketches the applied level scheme.

Let us have a look at the scheme in Figure 2.4. conguration with ground states

|1i

and

|2i,

A four-level atom in N-type

and excited states

|3i

and

|4i,

and

two cavity modes A and B constitute the system. The atomic transition from an

|ii (energy }ωi ) to a nal state |f i (}ωj ) at frequency ωf i is expressed by the operator σ ˆf i = |f ihi| with i, f = 1, 2, 3, 4. The excited P states j = 3, 4 are prone to spontaneous decay quantied by the rates γj = k γjk that is the sum of decays too ground states k=1,2. Latter states are (meta-) stable (γk = 0) except for a dephasing of rate rate γd between the two states. The cavity modes A (resonance angular frequency ωa , eld decay κa ) and B (ωb , κb ) have associated photon creation (annihilation) operators a ˆ† (a ˆ) and ˆb† (ˆb), respectively. Mode A (B) is excited by the probe eld (signal eld) at frequency ωp (ωs ) and driving strength ηp (ηs ). The coherent control laser beam with frequency ωc joins the separate probe and signal systems by driving the transition between |2i and |3i at Rabi frequency Ωc . We dene the single-photon detunings ∆p3 = ωp − ω31 and ∆c = ωc − ω32 . The frequency detuning of the atomic transition from |2i to |4i initial state

12


from mode B is

|1i

to

|3i

∆42 = ωb −ω42 , and the frequency detuning of the atomic transition ∆31 = ωa − ω31 . Furthermore, we dene the frequencies

from mode A is

of the the probe and the signal drives relative to the frequency of their respective cavity mode:

∆p = ωp − ωa

and

∆s = ωs − ωb .

Now we formulate the full Hamiltonian of the system:

ˆ =H â + H ˆc + H ˆ int + H ˆ d, H

(2.19)

ˆ a and H ˆ c is the unperturbed atom and cavity Hamiltonian, respectively. H ˆ Hint represents the interaction between atomic transitions and cavity modes, and ˆ d contains the external driving elds (probe, signal, and control beam). After H choosing the zero point of energy at level |2i, making use of the dipole and rotating

where

wave approximation, and the transformation to a rotating frame of reference, we nd the full Hamiltonian to be:

† † † ˆ = (∆p − ∆c + ∆31 )ˆ H σ11 σ ˆ11 − ∆c σ ˆ33 σ ˆ33 + (∆s + ∆42 )ˆ σ44 σ ˆ44 † † − ∆p a â ˆ − ∆sˆb ˆb † † ˆ + gp (ˆ a† σ ˆ13 + σ ˆ13 a ˆ) + gs (ˆb† σ ˆ24 + σ ˆ24 b) + (ηp a ˆ† + ηsˆb† + Ωc σ ˆ † + h.c.).

(2.20)

23

For simplicity, we chose

∆31 = ∆42 = ∆c = 0

} = 1.

Calculating the system's energy structure for

reveals a landscape of quadruplets for each combination of

probe and signal photon numbers (np and energy splittings

∆E>

ns ,

respectively).

The corresponding

(dierence of the outer levels of a quadruplet) and

∆E
any feasible electronic circuit for direct detection (