Spectroscopy of rare earth ion doped solids for quantum memory applications

Thesis

Reference

Spectroscopy of rare earth ion doped solids for quantum memory applications

HASTINGS-SIMON, Sara Rose

HASTINGS-SIMON, Sara Rose. Spectroscopy of rare earth ion doped solids for quantum memory applications. Thèse de doctorat : Univ. Genève, 2008, no. Sc. 3948

URN : urn:nbn:ch:unige-5517

DOI : 10.13097/archive-ouverte/unige:551

Available at:

http://archive-ouverte.unige.ch/unige:551

Disclaimer: layout of this document may differ from the published version.

Universit´e de Gen`eve Facult´e des Sciences

Groupe de Physique Appliqu´ee Professeur N. Gisin

Spectroscopy of Rare Earth Ion Doped Solids

for

Quantum Memory Applications

Th` ese

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es sciences, mention physique

par

Sara Rose Hastings-Simon

n´ee Hastings de

Californie (´Etats-Unis)

Th`ese N° 3948

Gen`eve

Atelier de reproduction de la Section de physique 2008

This thesis is dedicated to the memory of my grandmother Celia Hastings.

Abstract

This thesis concerns the study of rare earth ion doped solids for quantum memory applications. We characterize different systems in the context of their usefulness for a quantum memory protocol (CRIB) based on a modified photon echo.

In chapter one we present the motivation for this work, explaining the need for quantum memories to expand quantum communication to longer ”real world”

distances via quantum repeaters, and how a solid state based quantum memory could be realized in these systems. A background on rare earth ion doped solids is presented which introduces the basic properties we study here. Chapter two describes the equipment and experimental techniques such as spectral hole burning and photon echoes that are employed in this work.

We concentrate our measurements on Er and Nd ions doped into different solids.

These ions are attractive for the specific wavelengths of the transitions which are accessible with stable diode lasers.

In Nd³⁺:YVO₄ we perform spectral hole burning spectroscopy. In our search for a Λ system in this material we demonstrate narrow spectral holes that arise from population trapping in the ground state Zeeman levels. We perform preliminary spectral tailoring in the system using the Zeeman levels.

In Er³⁺:Y₂SiO₅ we also make spectral hole burning measurements motivated by the search for a Λ system. We concentrate our efforts on the time dynamics of the hole and anti-hole which are created with the ground state Zeeman levels. In addition we find a very long lifetime hole that was not predicted.

For the modified photon echo memory one must be able to control the absorption frequencies of the ions in a reversible way. The linear Stark effect is an attractive candidate to provide this controlled broadening and we measure the Stark shifts and broadening in Er doped materials.

The optical coherence time in Er doped silicate fiber is investigated with spectral hole burning and photon echo techniques. We measure very narrow holes for an amorphous system. Er doped fiber is naturally suited to integration in a quantum communication system as standard telecom fiber works at the Erbium transition wavelength.

Finally, using photon echoes as a model for the CRIB protocol, we examine the preservation of information stored with this method, and the possibility of using such a protocol to store multiple temporal modes.

Contents

Abstract vi

I English version 1

1 Introduction 3

1.1 General Introduction . . . 3

1.2 Quantum Memory . . . 4

1.2.1 Introduction . . . 4

1.2.2 Controlled Reversible Inhomogeneous Broadening . . . 5

1.2.3 Quantum Repeaters . . . 6

1.3 Rare Earth Ion Doped Solids . . . 6

1.3.1 Introduction . . . 6

1.3.2 Level Structure . . . 6

1.3.3 Homogeneous Linewidth . . . 7

1.3.4 Inhomogeneous Linewidth . . . 9

1.3.5 Zeeman Levels . . . 9

1.3.6 Stark Effect . . . 9

2 Equipment and Experimental Techniques 11 2.1 Lasers . . . 11

2.2 Modulators . . . 11

2.3 Detectors . . . 12

2.4 Cryostat . . . 13

2.5 Spectral Hole Burning . . . 14

2.5.1 Introduction . . . 14

2.5.2 Homogeneous Linewidth . . . 16

2.5.3 Spectral Hole Burning in Lambda Systems . . . 16

2.6 Photon Echoes . . . 18

2.6.1 Two pulse photon echoes . . . 18

2.6.2 Three pulse photon echoes . . . 19

3 Spectral Hole Burning Spectroscopy in Nd³⁺:YVO₄ 20

3.1 Introduction . . . 20

3.2 Experimental Setup/Methods . . . 21

3.3 Results and Discussion . . . 22

3.3.1 Spectral Hole Burning Mechanism . . . 22

3.3.2 Homogeneous Linewidth . . . 23

3.3.3 Hole Burning Spectrum . . . 24

3.3.4 Zeeman Level Lifetime . . . 26

3.3.5 Spectral Tailoring . . . 27

3.4 Conclusion . . . 28

4 Population Storage in Zeeman states in Er³⁺:Y2SiO5 30 4.1 Introduction . . . 30

4.2 Experimental Setup . . . 30

4.3 Results and Discussion . . . 31

4.3.1 Spectral Hole Burning Spectrum . . . 31

4.3.2 Hole and Anti-hole dynamics . . . 33

4.3.3 Longlived holes . . . 34

4.4 Conclusions . . . 36

5 Linear Stark effect in Er doped materials 37 5.1 Introduction . . . 37

5.2 In Erbium doped LiNbO3 crystals with channel waveguides . . . 38

5.3 In Erbium doped silicate fiber . . . 39

5.4 Conclusion . . . 42

6 Coherence Lifetime Measurements in Er³⁺ doped Fiber 43 6.1 Introduction . . . 43

6.2 Spectral Hole Burning Measurements . . . 44

6.3 Two Pulse Photon Echo Measurements . . . 45

6.4 Three Pulse Photon Echo Measurements . . . 46

6.5 Discussion . . . 46

6.6 Conclusions . . . 50

7 Interference Effects in Photon Echoes 51 7.0.1 Introduction . . . 51

7.0.2 Time Bin Interference . . . 52

7.0.3 Multi Mode Storage and Interference . . . 52

7.0.4 Interference of Free Induction Decay . . . 54

7.0.5 Conclusion . . . 55

8 Conclusions 56

II Version fran¸caise 59

Abstract 61

9 Introduction G´en´erale 62

10 M´emoire quantique 64

10.1 Introduction . . . 64

10.2 CRIB . . . 65

10.3 R´ep´eteur quantique . . . 66

10.4 Solides dop´es aux terres rares . . . 66

10.4.1 Introduction . . . 66

10.4.2 Structure des niveaux . . . 66

10.4.3 Largeur homog`ene . . . 68

10.4.4 Largeur inhomog`ene . . . 69

10.4.5 Niveaux de Zeeman . . . 69

10.4.6 Effet Stark . . . 69

Bibliography / Bibliographie 72

III Appendix / Annexe 79

11 Second Harmonic Generation Phase Matching as a Measurement Method for Index of Refraction Difference 81

Acknowledgements / Remerciements 91

IV Published articles / Articles publi´ es 93

Publication list / Liste des publications 95

Part I

English version

Chapter 1 Introduction

1.1 General Introduction

The first proposal for quantum cryptography was made in 1984 by Charles Ben- nett and Gilles Brassard [1], and various demonstrations followed [2]. All quantum cryptography schemes have the inherent requirement that photons (for example sin- gle photons or coherent states) are sent between the two people who wish to share a secret key. For distances up to ∼ 500 km this can be done in optical fibers at telecom wavelengths where the attenuation is quite small (0.35 and 0.20 dB/km for 1300 and 1550 nm, respectively). However when the distances become larger than this it is no longer possible to communicate effectively, for example over a distance of 1000 km the transmission would be 10⁻²⁰, so even with a repetition rate of 100 GHz this would translate into only 1 bit transmitted every 10⁹ seconds (or roughly 1 bit per 32 years), and additional losses and noise such as detector noise would reduce this number further. In classical communication these losses are overcome by amplifiers along the transmission lines, but the no-cloning theorem means that it is not possible to use such an amplifier for the quantum bits. One could imagine to have stations along the way where the message is decoded, then recoded and sent along, but this introduces additional points of in the transmission line where the information is available, thus reducing security.

Instead for longer distances one can use a quantum repeater scheme [3]. The basic idea is to split the long distance into a set of smaller distances and perform entanglement swapping across each smaller distance, until you get the information from one end to the other. However, even this process involves small probabilities, and a single lost photon will prevent the entanglement from propagating from the sender to the receiver, meaning the whole process must be started again. This can be overcome by establishing entanglement for each link independently and only pro- ceeding to the next entanglement swapping step when all the photons have arrived.

This requires memories that can store the photons at each entanglement swapping station until its partner photon has also arrived. It is these memories, in the view of this process, that are the focus of the work in this thesis.

The work in this thesis is motivated by the goal of constructing a quantum mem- ory for single photons, to implement a quantum repeater for quantum cryptography

as described above. Thus it is this motivation which guides the choice of materials and properties studied here.

The thesis consists of three parts, chapter 1 and 2 introduce the motivation and basic physical principles of this work, along with the main equipment and experimen- tal techniques used. Chapters 3 - 7 describe the experiments that were performed to study the material properties and demonstrate the usefulness of a photon echo type quantum memory. The published and submitted papers can be found in part IV. Supporting materials and an additional experiment which is not related to the topic of this thesis can be found in part III.

1.2 Quantum Memory

1.2.1 Introduction

In quantum cryptography one needs to transmit qubits, or ”quantum bits” from one station to another [2]. Photons, or so called flying qubits are well suited for this application as, unlike atoms, or other qubits, they can be sent relatively easily via standard telecom fiber. However, they do not make good stationary qubits as they will not simply stay in one place like a crystal nor can you store them in a trap like an atom. The simplest solution one can imagine is to put a photon into a cavity or optical fiber loop. However problems come up, with the cavity it is not possible to make a high-Q cavity with fast switching capabilities (so either you loose the photon out of one of the mirrors or some intracavitary switching element absorbs it) [4].

With a fiber loop you have the same kind of problems with absorption as if you had just sent the photon the whole distance to begin with (plus the switching problem as well) [5]. Just as a reminder, we can not store qubits the way classical bits are stored (by reading the value and writing them down in some classical memory), because the measurement will change the system and the cryptography wont work at all. Hence the need for a way to store and retrieve the qubit that preserves its wavefunction.

There are various proposals for ways to realize this quantum memory. A natural first approach is to try to use individual quantum systems, such as trapped ions or atoms [6] or single defect centers in diamond. The main challenge in such imple- mentations is to achieve strong enough coupling between the light and the system used as a memory. One possible solution is the use of high-finesse cavities. Another attractive possibility is to use ensembles instead of individual systems. In this way the light-matter interaction is enhanced. Approaches using atomic gases are well developed, in particular using electromagnetically induced transparency [7, 8, 9, 10]

and off-resonant Raman interactions [11]. However these approaches typically allow one to store only photons with rather limited bandwidth (of order MHz). More- over the achievable storage times are fundamentally limited by the atomic motion.

It is thus of interest to look for alternative ensemble-based protocols, and in par- ticular for implementations in the solid state, where motional effects play no role.

A well-known approach towards the realization of classical memories for light in solid state systems are photon echoes [12, 13]. Photon echoes in rare-earth doped

materials allow the storage of classical light fields with large bandwidth. However standard photon echoes are limited in efficiency and do not allow the realization of high-fidelity quantum memories due to unavoidable fluorescence. Here we focus on a modified photon echo approach which overcomes these limitations.

This modified photon echo, is based on controlled reversible inhomogeneous broadening (CRIB) of a narrow, spectrally isolated absorption line [14, 15, 16].

This protocol requires an atomic ensemble with a large optical depth, a long optical coherence time, and the ability to inhomogeneously broaden a single absorption line in the ensemble in a controlled and reversible way.

1.2.2 Controlled Reversible Inhomogeneous Broadening

The CRIB proposal consists of a series of steps, a simple version is described here.

In the first step a narrow homogeneous absorption line is prepared from a broad, natural inhomogeneous absorption profile. This is done by using an intense laser to optically pump ions absorbing at all but one specific frequency into long lived auxiliary states (for example the ground state Zeeman or hyperfine levels). Next this narrow absorption line is broadened in frequency in a way that is reversible. This can be done, for example, by using the linear Stark effect and applying a position dependent electric field (electric field gradient) with a controllable voltage. At this point the light you want to store is sent into the crystal and absorbed. Just as in a standard photon echo, the dipole moments of the ions will begin to precess. If we consider the ion at the center of the broadening as a reference then those with a detuning ∆ (-∆) will acquire a phase ∆τ (-∆τ) after a time τ.

When it is time to read out the memory you simply reverse the polarity of the field. Ions that previously had a detuning of ∆ (-∆) will now be detuned by -∆

(∆) and will acquire a phase -∆τ (∆τ) after a time τ. After a time equal to the waiting time the dipole moments will line up once again, creating a macroscopic dipole moment, and the light will be re-emitted.

In the simple case this emission is in the forward direction and a portion of the light is reabsorbed by the crystal, the crystal length can be chosen to optimize the emitted light with the optimal value at αL∼ 2. There are more complex variations to the simple case that would lead to longer storage times and a higher emission percentage. During the waiting time it is possible to transfer the excitation into a ground state with a longer coherence time before transferring it back for the readout.

This would allow for longer storage times. In addition you could apply a position dependent phase shift (via two counter-propagating π pulses, or directly via an electric field) which would turn the CRIB into a true time reversal process such that the light would be emitted in the backwards direction, and without reabsorption.

Details of these additional options can be found in reference [16].

The first demonstration of this protocol has been realized in Eu³⁺:Y₂SiO₅ [15], and further work in Pr³⁺:Y2SiO5 has been done [17]. We are interested in exploring Kramers ions which may have advantages over the non-Kramers ions used previously.

1.2.3 Quantum Repeaters

A CRIB based memory could be a very useful ingredient for a quantum repeater architecture. A quantum repeater is a protocol for generating long distance entan- glement [3]. The protocol works by dividing the long distance into certain number of elementary links. Then entangled states are generated and stored independently for each link. Long distance entanglement is then generated via entanglement swap- ping [18, 19] between neighboring links in a hierarchical manner. The independent generation makes such a protocol much more efficient than direct transmission. The capability of storing the generated entanglement at each link is essential for profit- ing from a quantum repeater. Quantum memories based on CRIB are particularly attractive because they are naturally suited for storage and retrieval of multiple temporal modes analogously to a photon echo based memory where storage and retrieval of multiple temporal mode has been demonstrated. Such a memory would lead to a significant further enhancement of the rate at which entanglement can be generated, a factor of N for a memory with N modes [20]. The rate could be further enhanced via frequency multiplexing within a single memory.

1.3 Rare Earth Ion Doped Solids

1.3.1 Introduction

Rare Earth elements (RE) is the name given to the 15 elements in the Lanthanoid group, from lanthanum to lutecium (atomic number 57-71) along with scandium and yttrium, all of which are naturally occurring on the earth. RE elements have in common a partially filled 4f shell. The transitions within the 4f shell are shielded from the environment by the outer-lying filled 5s and 5p shells. This shielding contributes to the long coherence times at low temperatures (and corresponding narrow homogeneous linewidth) for the 4f→4f transitions. When RE are doped into inorganic crystals they occur most often as triply charged ions (RE³⁺), substituting for a host ion at one or more well defined sites, though double charged ions (such as Eu²⁺ and Sm²⁺) are also common. The energy level structure depends on the symmetry properties of this site.

1.3.2 Level Structure

Free RE ions have simple electronic energy levels which can be calculated directly from the Hamiltonian. When placed into a crystal host, interactions between the RE ions and the host material leads to a perturbation to levels. In general one starts with the energy levels of the free ion and treats the interactions as perturbations.

The relevant interactions and their energies are shown in Table 1.1 [21], and their origins are described below.

Configuration splitting (central electrostatic field) refers to the energy levels from different orbital in the ion, for example going from 4f^N to 4f^N−15d. The non-central electrostatic field is the description for interactions between different electrons in

Interaction Mechanism Energy(cm⁻¹) Configuration splitting 10⁵

Non-central electrostatic field 10⁴ Spin-orbit interaction 10³ Crystal field interaction 10²

Hyperfine splitting 10⁻⁴−10⁻¹ Superhyperfine splitting 10⁻⁴−10⁻³

Table 1.1: Energy level scales of rare earth ions in crystals [21].

the many electron atom.

Thespin orbit interaction, also called fine coupling, is the interaction between the spin of the electron and the magnetic field it sees from orbiting around the nucleus.

This gives rise to a magnetic dipole-dipole type interaction. The levels are labelled as^2S+1L_J where L is the orbital angular momentum (denoted with letters S, P, D,...

corresponding to 0, 1, 2,...), S is the spin of the electron, and J is the total angular momentum. It is transitions between these levels we study in this thesis, for example the ⁴I_15/2 →⁴I_13/2 transition in Er³⁺ doped solids. For a free ion the energies of the levels depend only on the total J so they are 2J+1 degenerate.

Thecrystal field levels arise from the electrostatic interaction between the electron and the electric field in the crystal. For ions with an odd number of electrons in the RE³⁺ state, so called Kramers’ Ions (both Er³⁺and Nd³⁺), the interaction normally splits the levels into J+1/2 doubly degenerate levels, so called Kramers’ doublets, unless the RE site symmetry is high, in which case more degeneracy remains. In general the Kramers’ doublets can be described by a fictitious spin 1/2. The de- generacy can be lifted with the application of a magnetic field via the first order Zeeman effect, splitting the Kramers’ doublets into two spin states (ms =±1/2).

The Hyperfine interaction, or magnetic hyperfine interaction, is present in first order for isotopes with a non-zero nuclear spin. The interaction between the nuclear spin moment of the ion and the electronic angular momentum leads to a further splitting of the crystal field levels for such ions [22]. Hyperfine groundstate levels are of interest for their particularly long coherence times and lifetimes [23, 24], however, they have not been investigated in this work.

A second, smaller nuclear spin angular momentum interaction, theSuperhyperfine interaction, can be observed for certain host crystals. In this case it is the nuclear spin of the host ion which interacts with the electron angular momentum of the RE ion. The energy splittings that arise are smaller than for the hyperfine interaction, but we have seen signatures of this interaction in various spectral hole burning (SHB) and photon echo (PE) measurements performed in this thesis, for instance between Al and Er ions.

1.3.3 Homogeneous Linewidth

The linewidth of a transition is related to the coherence time T₂ as

Γh= 1

πT₂ (1.1)

The upper limit for the coherence time is given by twice the state lifetime as

1

T2 = _2T¹₁ + _T¹∗

2, where T₂^∗ is the pure dephasing. There are some materials where coherence times approaching this lifetime limited time has been observed such as Eu³⁺:Y₂SiO₅ with a linewidth of 122 Hz (corresponding to a T₂ of 2.6 ms) [25, 24]

and Er³⁺:Y₂SiO₅ with a linewidth of 50 Hz [26] (corresponding to a T₂ of 6.3 ms).

In general there are dephasing processes which shorten the coherence time below the 2T1limit, but steps can be taken to reduce their impact. Spin flips in neighboring ions (other RE³⁺ions or host ions) lead to fluctuating magnetic fields at the optically active ion which broadens the transition [27, 28]. This effect can be lessened by using lower RE³⁺ dopant concentrations and hosts with a small or zero nuclear magnetic moment, thus reducing the number of neighbor ions that can contribute. In addition one can apply a small magnetic field to increase the energy splitting (via the Zeeman effect) between the spin states to reduce the spin flips [28]. Kramers’ ions, such as those studied in this thesis, interact more strongly due to their large magnetic moments. However the levels can more easily be split by larger energies owing to the large magnetic moment, a process which is more difficult for non-Kramers’

ions or host nuclear spins. Interactions with lattice phonons (absorption, emission or scattering) also increase the width of the transition [29]. These effects can be reduced by decreasing the temperature and thus the number of phonons present in the system.

Spectral Diffusion

Spectral diffusion is the broadening of the homogeneous linewidth of an ensemble of ions with the same transition frequency. The central frequency of the ions is initially the same, but interactions with the local environment change the frequencies. If these interactions are different for different ions, and they have a time dependent character then over time the central frequencies of the ions will drift, or ”diffuse”

and as a result the linewidth of the ensemble will broaden [30, 31].

In rare earth ion doped solids one typical mechanism for this diffusion is dipole- dipole magnetic interactions between neighboring dopant ions and between the ion and spins in the host material. As explained above, spin diffusion is often driven by spin flips in the material. In order to reduce the magnitude of this effect one can apply a magnetic field which splits the Zeeman levels by more than k_BT and greatly reduces the spectral diffusion [28]. Note that the relevant width for quantum memory applications is the width of the spectral hole measured via SHB, which can be strongly affected by spectral diffusion.

The homogeneous linewidth measured by SHB is the linewidth some ms after the hole is burned. Over this timescale spectral diffusion can widen the hole [28], making the Γ_h measure larger than that predicted by equation 10.1.

1.3.4 Inhomogeneous Linewidth

The homogeneous linewidth is ideally the same for each ion in the crystal, but the central frequency of a given transition is slightly different for each ion. This difference is due to strains and imperfections in the crystal which lead to different crystal fields at different positions in the crystal [32]. The absorption profile for a given sample is given by the sum of the absorption for each of the absorbers, leading to an inhomogeneous broadening of the absorption. The inhomogeneous width dominates over the homogenous width for cryogenic temperatures. The width of this profile, the inhomogeneous linewidth, depends on the type of ions, the host and the doping concentration.

A large relative inhomogeneous linewidth (ie Γ_inh/Γ_h À 1) would allow for fre- quency multiplexing in memory applications. Ratios on the order of 10⁸ have been reported for Er³⁺ doped materials [26], and in general are relatively high in rare earth ion doped crystals, in the range of 10⁵-10⁶, making them an attractive system for classical and quantum information processing.

1.3.5 Zeeman Levels

The degeneracy in the energy levels for Kramers’ ions can be lifted via the Zeeman effect under the application of a magnetic field. Using an effective spin 1/2 model one can describe the Zeeman splitting for an applied magnetic field of B~ as,

∆ω=µ_B~g·B~ (1.2)

Where~gis the effective g vector, a phenomenological vector that can be measured, and µ_B is the Bohr magneton (14 GHz/Tesla). In general g is a tensor, in a crystal

~g is fixed for each ion by the symmetries of the crystal field [22]. In an amorphous material the direction of~g varies over the sphere for different ions. Typical values for the magnitude of g in rare earth doped solids are order of 1-10 [33, 34] resulting in Zeeman splitting on the order of 10-100 GHz/Tesla.

1.3.6 Stark Effect

In the presence of an external DC electric field, the energy levels of an atom with a permanent electric dipole moment are shifted linearly with the field. This phe- nomenon is known as the linear DC Stark effect [35]. For example, if the electric dipole moments are different for different electronic levels of a given atom this shift leads to a shift in the associated optical transition frequency between the electronic levels.

Following from the inversion symmetry of a free ion one can see that a free RE ion has no permanent dipole moment, and thus exhibits no linear Stark effect.

However, a linear DC Stark effect can be observed in certain RE doped solids, where the dipole moment in the ion is induced by local electric fields (crystal field interaction). In order to induce a dipole moment in the RE ion it must sit at a site with an inversionless crystal field (a non centrosymmetric site) [36, 37]. A

non-centrosymmetric site is necessary but not sufficient. In addition one must have mixing of the f wavefunctions with opposite parity ones which is provided by the crystal field, and moreover there are a number of specific point symmetries such as ones with a symmetry plane perpendicular to the main rotation axis where dipole moments are also forbidden.

For the linear Stark effect the transition frequency, ω, of an ion is shifted by the electric field according to the formula,

∆ω= ∆µ_eχE

~ cosθ. (1.3)

Here ∆µe is the difference between the permanent electric dipole moments of the two states connected by the transition, E is the applied DC electric field, χ = (²+ 2)/3 is the Lorentz correction factor,² is the dielectric constant of the sample, and θ is the angle between the vectors∆µ~ _e and E.~

There are also higher order Stark shifts that can be observed for ions without a permanent dipole moment. For the second order, or quadratic, Stark effect a dipole moment arises from the interaction with the electric field via a polarization effect.

The shift in frequency is due to the Stark interaction between the induced dipole moment and the applied field, therefore quadratic in the applied field. In general the quadratic Stark effect is much smaller than the linear effect for ions in crystals that have a permanent electric dipole moment.

In a crystal, because of the ordered structure of the crystallin lattice, the dipole moments are aligned along a set of one or more well defined directions. The sym- metry of the crystal along with the site symmetry at the rare-earth-ion position determines the number of directions. Ions with ∆µ~ _e aligned along the same direc- tion experience the same shift in their resonance frequency with an applied electric field (as observed here in Er³⁺:LiNbO₃). In crystals with a centrosymmetric crystal symmetry the dipole moments are aligned along two opposite directions such that an applied field splits the resonance frequency into two, the pseudo Stark shift. In contrast, in an amorphous material, such as an optical fiber, the dipole moments are randomly orientated, such that the projection of ∆µ~ _e along the direction of an applied electric field varies continuously for different ions in the solid. This gives rise to a broadening of the transition with an applied electric field.

Chapter 2

Equipment and Experimental Techniques

2.1 Lasers

All of the experiments in this thesis are based on the interaction of ions with coherent radiation from a laser. These interactions are on resonance and thus require a laser at the frequency of the transitions in the ions (∼1530 nm and 879 nm for the materials studied here).

Aside from light at the correct frequency, spectral hole burning and photon echo techniques place different requirements on the laser. For spectral hole burning it is necessary to have a laser with a well defined, stable, wavelength over tens of ms.

Power needs are modest, with incident powers in the range of µW-mW sufficient.

In contrast, to produce photon echoes, the wavelength stability requirement is less stringent and much higher powers (tens to hundreds of mW) are needed.

Two different types of lasers were used for the experiments, an external cavity diode lasers (for 1530 nm) and a Ti:sapphire laser (for 879 nm, though diode lasers are also available at this wavelength). One of the advantages of diode lasers is their inherent stability, the cavity reduces the bandwidth to as low as hundreds of kHz without the need for active stabilization. However there is a gap in the wavelengths reachable by diodes, notably around the 606nm transition frequencies of Pr. This necessitates the use of dye lasers, which are difficult to stabilize, one of the main motivations for us to work with Nd and Er.

2.2 Modulators

To create optical pulses for the experiments we modulated the light from continuous wave lasers. Two types of modulators were used, acousto optic modulators (AOM) and electro optic modulators (EOM). Different modulators were used depending on the type of pulses that were needed (length, and attenuation), it is also possible to use multiple modulators in series to increase the attenuation ratio.

An AOM uses diffraction off sound waves in a crystal to modulate light. A piezo-

electric transducer is attached to one end of a crystal, and an oscillating electrical signal (RF wave) at about 50-500 MHz drives the transducer which produces sound waves in the crystal. These sound waves change the index of refraction in the crys- tal, and incoming light scatters off the resulting periodic index modulation. The angle of the diffracted beam, θ, is given by sin (θ) = m(_2Λ^λ ), where m is the order of diffraction,λis the wavelength of the light and Λ is the wavelength of the soundwave in the material.

Typically the first diffracted beam is used as the signal beam, the light is ”off”

when there is no RF wave sent to the transducer and there is no light diffracted. The intensity in the signal beam can be changed by changing the sound wave intensity.

The signal beam acquires a frequency shift due to the deflection off the sound wave.

The magnitude of the frequency shift is given by the frequency of the RF wave. The frequency shift can be used to modulate the frequency of the beam by scanning the frequency of the RF wave around some central frequency. In this case the angle of deflection changes, but the AOM can be used in ”double pass” configuration where the first deflected beam is sent back into the modulator and then deflected a second time, which cancels the angular deflection.

AOMs have very good extinction ratios, values up to 50-60 dB can be achieved in double pass formation. The pulse duration can not be shorter than the time it takes for the sound wave to cross the focused beam which is generally on the order of tens of nanoseconds. In addition a lower limit on the pulse duration is imposed by the frequency of the RF wave, f, as 1/f.

In order to create shorter pulses to perform photon echo experiments EOMs must be used. One example of an EOM is simply a Mach-Zehnder interferometer where a phase change can be made in one arm. By varying the phase it is possible to make the light interfere constructively or destructively at the output, and thus modulate the output. The phase is changed by applying an electric field which changes the index of refraction of the material the light is propagating through (for example Lithium Niobate) thus changing the phase at the output. Pulses created by EOMs can be much shorter (as fast as tens of picoseconds), but the suppression is typically much less than with AOMs (15-30 dB).

2.3 Detectors

To detect the light in the experiments photodiodes were used. For the majority of the experiments, where detected light levels were in the range of nW-mW, standard photodiodes provide sufficient sensitivity. In the case of the few photon, free in- duction decay experiment a single photon avalanche diode that can measure light at the single photon level was used. Standard photodiodes are simply semiconduc- tor diodes where an incoming photon excites an electron producing a photocurrent.

This photocurrent, which is proportional to the intensity on the detector can be am- plified and measured on an oscilloscope. Single photon avalanche diodes are based on photodiodes but operate reverse biased at voltages above the breakdown voltage such that a single incident photon launches an avalanche and can be detected. Such detectors provide very high sensitivity but can not, in general, resolve photon num-

ber (ie their output is either 0 or more than 0), making them useful for very low light levels only.

2.4 Cryostat

All of the measurements on RE-ion-doped crystals in this thesis were performed at cryogenic temperatures below 10 K where the optical coherence time is much longer. There are a variety of different implementations but in general Helium is used to cool the sample to temperatures as low as hundreds of mK. When higher temperatures are desired a small heater is used to stabilize the temperature up to 10 K. Magnets (permanent or coils) can be placed on the outside of the cryostat to produce small magnetic fields, or superconducting coils can be placed inside the cold head to produce larger fields. The different types of cryostats used for this work are described below.

Bath Cryostat

A bath cryostat is perhaps the most simple cryostat. The sample is in a chamber that is filled with liquid helium which cools the sample to 4K, and there are no problems with thermalization as the liquid is in direct contact with the sample. Windows on the cryostat allow optical access to the sample. One problem with the sample being in a bath of helium is small bubbles that can form in the liquid, disturbing the optical path. To avoid this one can reduce the pressure in the liquid chamber below 50 mbar (corresponding to 2.7 K) where the helium becomes a superfluid. In general adjusting the pressure (by pumping on the liquid chamber) allows for the temperature to be controlled and lowered down to 1.5 K.

Cold Finger Cryostat

This is basically identical to the bath cryostat, with the difference that the sample is in thermal contact with a cold finger rather than in the bath. The cold finger can be cooled in the same way by pumping on the helium liquid down to 1.7 K.

Because the sample is not in the bath one avoids the problem of bubbles but extra care (we found the most effective is to attach the sample and sample holder with a thermal silver paste) must be taken to ensure good thermal contact between the sample and the cold finger. In addition one uses a radiation shield, a metal shield with small holes, to allow optical access to the sample but limit the absorption of thermal radiation from the 300 K outside through the windows.

Pulse Tube Cooler

A pulse tube cooler, in contrast to the cryostats described above, is a closed system cryostat where helium is continuously recycled in the closed system. This allows a pulse tube cooler to be installed anywhere there is an electrical plug without need for a helium recuperation system (or without venting off used helium into the air). It is

limited in that it can cool down to temperatures around 2.2 K. The system is made up of a compressor that is connected by tubing to a cold head which is in vacuum.

The compressor compresses the helium gas which is sent to the heat exchanger where it is expanded, drawing heat from the cold head. The expanded gas is then returned to the compressor and the cycle starts again. The sample is attached to the cold head via a cold finger and is accessed via windows (and protected with a radiation shield.) One drawback to this type of cryostat, aside from its limited cooling power is the vibrations introduced into the system from the rotary valve which creates the pressure wave.

Dilution Refrigerator

A dilution refrigerator allows one to reach temperatures much colder than with a standard cryostat (tens of mK). It uses a mixture of two helium isotopes (helium- 3, a rare isotope with 1 neutron and helium-4, the most common isotope with 2 neutrons). At low temperatures this mixture undergoes a spontaneous phase separation to form a mixture of two phases, one that is rich in helium-3, one poor in helium-3. One removes helium-3 from the poor phase, and as a result helium-3 flows from the rich phase to replace the missing atoms. In order to cross the barrier energy is needed, and it is taken in the form of heat from the sample holder, thus cooling the sample. Temperatures of a few mK can be achieved via this method.

In the fridge we used there are no windows to provide optical access to the sample.

Instead, thermalized fibers bring light to and from the sample under test.

2.5 Spectral Hole Burning

2.5.1 Introduction

Spectral Hole Burning (SHB) spectroscopy is a technique that makes it possible to study the properties (linewidth, lifetime, etc.) of a single homogenous absorption line within an inhomogeneously broadened ensemble of ions doped in a crystal [38, 39, 40]. In the case where there are more than two energy levels, SHB can be used to do optical pumping between the levels [23, 41, 42, 43] and study the evolution of the system. In addition preparing a narrow absorption line is a key step in various quantum memory and quantum computing proposals such as CRIB. Hole burning can be achieved by storing population in the excited state (transient SHB) or in a longer lived metastable state [40].

In transient SHB spectroscopy [39] of ions with only one ground state and one excited state an intense laser is used to excite ions from the ground state to the excited state. The population difference can then be probed by attenuating the laser and scanning the frequency around the burning frequency. The ions in the excited state lead to a hole in the inhomogeneous absorption profile manifested as an increase in transmitted light at the corresponding transition wavelength. The width of the hole is limited by the homogenous linewidth of the transition, spectral diffusion [30, 31], and power broadening [44].

a

b

Laser

PBS AOM

Cryostat

time

time I

l

l/4

l/2

Figure 2.1: a. Experimental setup for spectral hole burning, here the AOM is used in double pass to create pulses which are focused onto the sample in the cryostat and detected in transmission. A mirror and detector before the cryostat monitor incident light to correct for intensity fluctuations. b. Intensity and wavelength of laser pulses for the spectral hole burning sequence.

For ions with more ground states the spectrum becomes more complex, as de- scribed below. In some systems it is also possible to produce persistent spectral holes of many hours or longer. This process, which can only be observed in certain materials involves physical photon induced changes to the ion or host. [45, 46]

In SHB experiments one measures the transmitted intensity after a crystal (or fiber) of length L. The transmitted intensity at position L, I(L) is given by

I(L) = I(0)e^−αL (2.1)

where I(0) is the incident intensity and α is the absorption coefficient. The absorption coefficient at a frequency ω is linearly proportional to the population difference,α=σ(N1-N2), where σ is the cross section of the transition and N1 (N2) is the number of ions in the ground (excited) state at ω. Therefore SHB measure- ments can be used to extract information about the dynamics of the population. In particular if there is no population in the excited state the absorption coefficient is linearly proportional to the ground state population.

A typical SHB experimental setup is shown in figure 2.1a. It is necessary to use a laser with a linewidth smaller than that of the ion being studied in order to measure the linewidth of the ions via this method. This is generally difficult since Γ_h is in the range of 1-100 kHz. Pulses can be created with an AOM or by acting directly on the current of the laser. Similarly the frequency of the laser during the probe pulse can be scanned directly by acting on the current of the laser diode, or indirectly by scanning the driving frequency for the AOM. In the second case the AOM must be used in so called double pass mode (as shown in figure 2.1a), as the direction of the beam shifts with the change in driving frequency. A pickoff mirror can be used before the cryostat to monitor and correct intensity fluctuations.

The typical pulse sequence (2.1b) is an intense, fixed frequency burning pulse, followed by one or multiple attenuated read pulses where the frequency is scanned over the initial burning frequency. The length of the burning pulse is typically on the order of tens or hundreds of milliseconds, in general there is no need to burn for longer than the longest lifetime in the system. The readout pulse must be much lower in intensity to avoid burning during the scan which would change the population.

The transmitted light as a function of frequency is measured with a detector and an oscilloscope.

2.5.2 Homogeneous Linewidth

The corresponding homogenous linewidth, Γ_h, of a transition can be measured di- rectly via spectral hole burning and the spectral hole linewidth, Γ_holeas Γ_h = Γ_hole/2 in the limit of negligible laser line width and power broadening. Typically one fits the spectral hole with a Lorentzian or Gaussian and extracts the width of the hole from the fitting parameters.

Power broadening is the process by which the hole is widened due to saturation in the center of the spectral hole as compared with the side while the spectral hole is being created. By measuring the hole width for a set of different pump powers one can extract the homogenous linewidth in the limit of zero power broadening [44], which is given by

Γ_holem = Γ_hole0(1 + r

1 + I

I_s) (2.2)

Where I is the intensity of the burning pulse, I_s is the saturation intensity, Γ_holem is the measured spectral hole linewidth and Γhole0 is the corresponding hole linewidth with zero power broadening.

Note also that the homogeneous linewidth measured by SHB for the measure- ments presented in this thesis is the linewidth some ms after the hole is burned.

Over this timescale spectral diffusion can widen the hole [30, 31, 28]. Thus linewidths measured via SHB are upper limits on the spectral diffusion in the system.

2.5.3 Spectral Hole Burning in Lambda Systems

In the case of an ion with two ground states and two excited states where ∆E_g,

∆E_e ¿ Γ_inh., for instance corresponding to Zeeman levels, the spectrum is more complex as the burning frequency can be in resonance with transitions from either of the two ground states to either of the two excited states due to the large inho- mogeneous broadening. These four alternatives correspond to four different classes of ions within the inhomogeneous profile. Ions can be optically pumped via the excited state into the other ground state, leading to an increase in absorption from this other ground state (an anti-hole). Thus in addition to the central hole one expects additional holes as well as anti-holes at spacings given by the Zeeman level spacings. Figure 2.2a shows the hole burning spectrum for one class of ions, the complete spectrum for the four classes is shown in figure 2.2b. If the ground state

gGm_BB g_Em_BB

1 2 3 4

1 2 3 4

a b

gEmBB

gGm_BB gGm_BB -g_Em_BB

3 4 1

2

pump pump pump pump pump

g+

g-

e-

e+

Figure 2.2: a. Four level hole burning spectrum (Zeeman levels) for the class of ions where the laser is in resonance with the |g₋i → |e₋i transition. The g_G (g_E) is the Land´e g factor for the ground (excited) state, µ_B is the Bohr magneton and B is the applied magnetic field. The pump transition (solid line) and possible probe transitions (dashed lines) between the four levels are shown in the energy diagram and labelled on the corresponding transmission spectrum of holes and anti-holes.

b. Four level hole burning spectrum with inhomogeneous broadening. The four different classes of ions within the inhomogeneous profile that are in resonance with the pump beam are shown, along with the resulting transmission spectrum of holes and anti-holes (here the probe transitions are not shown for clarity).

levels are long lived this can lead to population trapping and longer lived spectral holes as compared to the transient hole which is limited to the excited state lifetime.

In addition to Zeeman levels one can construct such a system out of hyperfine levels, or even superhyperfine levels. If the number of levels increase the spectra quickly becomes very complicated, such as for Pr³⁺ doped solids [39, 47].

The structure of the observed spectrum and its time evolution depends strongly on the lifetimes of the various states. In order to have efficient optical pumping and trapping in the ground state level it is necessary to have a relaxation time between the ground state levels that is much longer than the optical T₁lifetime of the excited state. In addition one must take into account the branching ratio for ions in the excited state to relax into the other ground state than the one they started in. The product of this ratio and the T₁ lifetime defines an effective excited state lifetime that is the time it takes for an ion in the ground state to be transferred to the other ground state (assuming the pumping rate is much larger that the other rates.) If the ground state relaxation time is longer than this effective excited state lifetime then a single frequency pump laser will transfer population from one ground state to the other where it will be stored for the ground state lifetime. The signatures of this efficient storage in the ground state Zeeman levels are the appearance of anti- holes at the ground state Zeeman splitting, and a central hole with a decay time longer than the excited state T₁ lifetime. The anti-holes arise from the additional population in the ground state (and thus have a lifetime given by the ground state relaxation lifetime), and the long lived central hole is present because the ions do not repopulate their original ground state within the T₁ lifetime.

If, however, the effective excited state lifetime is longer or of the same order as

echo p/2 p

t12

t12

echo p/2

t23

t12

p/2

t12

p/2

w

u

v

|g

|e

u w₀

w+

w_-

v u

w0

w₊ w_-

v

pulse p

a b

c

Figure 2.3: a. Sequence of pulses for two pulse photon echo. b. A Bloch sphere picture of the two photon echo process. The excited ions dephase in the u-v plane, a π pulse changes the phase of the ions by 180^◦, the ions rephase. c. Sequence of pulses for three pulse photon echo.

the ground state lifetime, or the branching ratio is very low, the ions in the second ground state will decay back into the first ground state and also be pumped by the laser. In this case one will observe spectral holes instead of anti-holes due to a decrease in population of both ground states. And in addition the central hole will be transient and will decay with the T₁ lifetime.

The signatures of population trapping can be observed in SHB experiments, and in addition the dynamics of the trapping can be investigated [41]. By varying the delay between the burning pulse and readout pulse one can determine the lifetime of the central hole. In the case of Zeeman level trapping one can determine the effective g-factors for the ground and excited states by varying the magnetic field and measuring the splitting between the central hole and side holes/anti-holes.

2.6 Photon Echoes

2.6.1 Two pulse photon echoes

Photon echoes, which are the optical analogy to spin echoes [48] were first observed in ruby [49]. In two pulse photon echoes the first pulse is ideally aπ/2 pulse which, in the Bloch sphere picture, excites the ions to the u-v plane where they have the same phase and thus a macroscopic dipole moment along direction v. Ions with a different frequency will precess at different speeds and directions in the u-v plane with respect to the central frequency such that the macroscopic dipole moment spreads out and the radiation decays. In order to rephase the ions one then applies a second pulse, ideally aπ pulse which rotates the ions by 180^◦ around the u axis. At this point the ions will begin to rephase. At a time t₁₂ where t₁₂ is equal to the time between the first and second pulses the ions will have rephased and now oscillate with the same phase, creating a macroscopic dipole moment and the emission of light [50]. This process is shown in figure 2.3a and b.

Any decay of the coherence during the waiting time t₁₂ will lead to a reduction

in the emitted echo. The intensity of the two pulse echo is given by

I ∼e^(−4t12/T2) (2.3)

In order to determine the T2 for the material the echo can be measured as a function of time and the decay due to dephasing is just a single exponential, in the absence of other drcoherence mechanisms acting during this time, such as spectral diffusion.

There is another effect which leads to emission directly after excitation when the ions have a macroscopic dipole moment. The so called free induction decay [51], FID, is the emission immediately following excitation arising from the macroscopic dipole moment present before the ions begin to dephase.

2.6.2 Three pulse photon echoes

In contrast to two pulse photon echo measurements, which measure the so called instantaneous homogenous linewidth, three pulse photon echoes can be used to study the effect of spectral diffusion on the decoherence time [30, 31]. The values can be compared directly to those obtained via SHB. One can think of a three pulse echo as the analog to a two pulse echo where the second π rephasing pulse is split into two π/2 pulses with a variable delayt23 between the two, as shown in figure 2.3c.

In a three pulse photon echo the first pulse creates an atomic coherence, but the second pulse at a time t₁₂ later transfers the coherence into a frequency-dependent population grating in the ground and excited states. The third pulse after a delay of t23 scatters off the grating, forming an echo a time t12 after the third pulse [52].

The intensity of the three pulse photon echo is given by

I ∼e^(−4t12/T2)e^(−2t23/T1) (2.4) In absence of spectral diffusion, the grating decays with the radiative lifetime T₁. Spectral diffusion during the waiting time before the third pulse smears the population grating and decreases the echo intensity. By measuring the echo intensity as a function of t₂₃ one can measure the spectral diffusion.

Chapter 3

Spectral Hole Burning

Spectroscopy in Nd ³⁺ :YVO ₄

3.1 Introduction

The CRIB protocol places strict demands for the ion and host material. For a memory to be efficient the optical depth must be very large [53, 54, 20], such that the input is absorbed with high probability, as any light not absorbed leads directly to a loss in efficiency. In addition the excited state must have a sufficiently long coherence time, either for the duration of the storage, or if there are appropriate long lived ground states then a coherence time long enough to transfer the excitation onto the ground state. There must be a linear Stark shift which is sufficiently large to provide the required broadening for a reasonable applied electric field. A level structure with (at least) two long lived states (such as ground states) that can be efficiently coupled to an excited state via optical transitions, a so called lambda- system, for the first step (optical pumping) of the protocol is also required. For the storage of single photons a third ground state level is necessary. The levels must be accessible with optical frequencies, and a stable laser at this frequency is required.

Three-level lambda systems with efficient optical pumping have been so far demon- strated in Praseodymium [23, 42, 43], Europium [55, 56] and Thulium [57, 58] doped crystals. Population storage in Zeeman sublevels of Nd³⁺:LF₃has also been observed in a different context [41]. Praseodymium-doped crystals have excellent optical and hyperfine coherence times [42, 43]. However, these systems have a small separation between hyperfine ground states (a few MHz). This is a strong limitation for ap- plications that require a large spectral bandwidth. Moreover, the transition used in Pr doped crystals is at a wavelength of 606 nm, where only dye lasers of limited spectral resolution are available commercially. These must be stabilized, a difficult process, for use in spectral hole burning applications.

We have studied the spectroscopy of Nd³⁺:YVO₄ using SHB techniques in view of the material requirements for quantum memory protocols. Nd³⁺:YVO₄, and Nd³⁺

in general has a few properties which makes it an attractive candidate quantum information processing applications. Firstly, the oscillator strength of the ⁴I9/2 →

4F_3/2in Nd³⁺:YVO₄is one of the largest of all the rare earth ions and hosts, making it

c-axis k c a

b

Laser AOM

PBS

Cryostat

time

time I

l

l/4

l/2

0.9 mm

Figure 3.1: a. Experimental setup, the AOM is used in double pass to create pulses which are focused onto the sample in the cryostat and detected in transmission. A mirror and detector before the cryostat monitor incident light to correct for intensity fluctuations. b. Intensity and wavelength of laser pulses for the spectral hole burning sequence. c. Orientation of the Nd³⁺:YVO₄ crystal, the light propagates through the crystal perpendicular to the crystal c-axis.

possible to achieve the high optical depths required. Secondly, Nd³⁺ is an attractive ion because the 879 nm is within reach of diodes, thus stable external cavity diode lasers can be used.

3.2 Experimental Setup/Methods

The YVO₄ crystal is uniaxial, and Nd³⁺ ions substitute for Y³⁺ ions in sites of D_2d point symmetry. As described in chapter 1, with this symmetry the ground state ⁴I9/2 splits into five Kramers doublets, only the lowest being populated at liquid helium temperatures, the⁴F_3/2 excited state splits into two Kramers doublets.

Under an applied magnetic field the ground and excited state doublets each split into two Zeeman levels. The optical lifetime of the ⁴F3/2 state has been measured by stimulated photon echo to be 40 µs [26].

The YVO₄single crystals doped with Nd³⁺(doping concentration 0.001 % (Nd/Y nominal molar ratio)) used for this experiment come from E. Cavalli (Dipartimento di Chimica Generale ed Inorganica, Universit di Parma, Parma, Italy) and M. Bet- tinelli (Dipartimento Scientifico e Tecnologico, Univ. Verona, and INSTM, UdR Verona, Italy) and were grown by spontaneous nucleation from a Pb₂V₂O₇ flux [59].

The experiments were performed on the 879 nm, ⁴I9/2 → ⁴F3/2 transition in a crys- tal with a thickness of 0.9 mm along the direction of light propagation, oriented as shown in figure 3.1c. Light at 879 nm from a Microlase Ti:sapphire laser (MBR-110) in continuous mode with an output power of ∼100 mW is modulated by an AOM in a double pass setup. By modulating both the amplitude and frequency of the RF wave used to drive the AOM, created by an arbitrary waveform generator, we can create pulses and scan the frequency of the light. A λ/2 plate is used to control the polarization of the light on the sample.

We made measurements in two different bath cryostats in the lab of S. Kr¨oll

-6 -4 -2 0 2 4 6 -4.0

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

-alpha*L

Frequency [GHz]

-La

Figure 3.2: Absorption as a function of wavelength over the inhomogeneities broad- ened absorption spectrum of the transmission. The inhomogeneous broadening is roughly 2.1 GHz and the absorption is strongly polarization dependent with an αL of -3.7 for light polarized parallel to the crystal c axis (filled circles) and -0.6 for light polarized perpendicular to the crystal c axis (open circles).

(Atomic Physics, Lund Institute of Technology, Lund, Sweden), Cryovac model 100 with external Helmholtz coils which can produce a magnetic field of up to 15 mT at the sample and a Oxford instruments Spectromag with a superconducting magnetic that can produce a field up to 8 T at the sample. The light was focused directly onto the sample in the cryostat and the transmitted light was focused onto a detector (Thorlabs, model PDB150A). A pickoff mirror before the cryostat directed a small percentage of the incident light onto an identical detector to normalize the signal to laser and AOM intensity fluctuations. The experimental setup is shown in figure 3.1.

3.3 Results and Discussion

3.3.1 Spectral Hole Burning Mechanism

The inhomogeneous absorption spectrum was measured in transmission by slowly scanning the frequency of the Ti:sapphire laser source (see figure 3.2). The inho- mogeneous broadening is roughly 2.1 GHz and the absorption is strongly polarized with an αL of -3.7 for light polarized parallel to the crystal c axis and -0.6 for light polarized perpendicular to the crystal c axis. The following measurements were performed with the light polarized parallel to the crystal c axis.

For the spectral hole burning measurements we used the AOM to create an intense burning pulse. We then probed the spectral hole by scanning the frequency of the AOM around the initial burning frequency using a 1 ms delayed and attenuated read pulse (see Figure 3.1b).

We measured the area of the central hole as a function of the applied magnetic field (B parallel to c) at a temperature of 4.2 K as shown in figure 3.3. Without an applied magnetic field no or very weak spectral holes were observed. However, under

0 4 8 12 0.0

0.4 0.8 1.2

Hole Area (au)

B field (mT)

Figure 3.3: Area of the central hole 1 ms after spectral hole burning as a function of the applied magnetic field. Here B was parallel to the c axis with T = 4.2 K the application of a low magnetic field (a few mTesla) the spectral hole grew very large and saturated at fields of about 10 mTesla. In general, a spectral hole at the burning frequency can be observed due to population remaining in the excited state or due to ions which are trapped in other ground state levels. In our experiment, where the waiting time between the burning and scan pulses was much longer than the excited state lifetime (1 ms as compared to 40 µs), there clearly cannot be any contribution to the spectral hole from ions in the excited state. Therefore the presence of a spectral hole shows that population is trapped in other ground state levels. The strong magnetic field dependence of the hole suggests that population trapping is occurring in ground state Zeeman levels.

There were also weak holes observed at zero applied magnetic field. These could have several origins. For instance, two isotopes of Nd, ¹⁴³Nd and ¹⁴⁵Nd, naturally occurring at 12.2% and 8.3% respectively, have a nuclear spin and thus a hyperfine structure in which population could be trapped. In addition, interactions between the nuclear moment of the host ion Vanadium and the electronic magnetic moment of Nd³⁺ (superhyperfine interaction) can give rise to additional energy levels where trapping can occur [60].

3.3.2 Homogeneous Linewidth

The homogeneous linewidth, Γ_h, of a transition can be measured directly via spectral hole burning and the spectral hole linewidth, Γhole as Γh = Γhole/2 in the limit of negligible laser line width and power broadening.

The homogeneous linewidth has been previously measured to be 15 kHz at mag- netic fields higher than 1.5 Tesla using two pulse photon echoes [26]. However, spec- tral diffusion effects can broaden the hole considerably compared with line width values measured via two pulse photon echo, where the width is measured at a much shorter timescale [28]. The relevant width for a CRIB-based memory is the width of the spectral hole that can be prepared, as the storage time in the excited state is limited to the inverse of the width of the prepared spectral hole. [54].

Spectral holes measured at T=2.1 K, for fields of B = 15 mT and 300 mT parallel