Quadratic Zeeman Spectral Diusion of Thulium Ion Population in a Yttrium Gallium Garnet Crystal Jacob H. Davidson1Antariksha Das1yNir Alfasi1Rufus L. Cone2Charles W. Thiel2and Wolfgang Tittel1 3 4

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Quadratic Zeeman Spectral Diﬀusion of Thulium Ion Population in a Yttrium

Gallium Garnet Crystal

Jacob H. Davidson,1, ∗Antariksha Das,1, †Nir Alfasi,1Rufus L. Cone,2Charles W. Thiel,2and Wolfgang Tittel1, 3, 4

1QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands

2Department of Physics, Montana State University, Bozeman, Montana 59717, USA

3Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland

4Schaﬀhausen Institute of Technology - SIT, 1211 Geneva 4, Switzerland

(Dated: October 12, 2022)

The creation of well understood structures using spectral hole burning is an important task in the

use of technologies based on rare earth ion doped crystals. We apply a series of diﬀerent techniques

to model and improve the frequency dependent population change in the atomic level structure of

Thulium Yttrium Gallium Garnet (Tm:YGG). In particular we demonstrate that at zero applied

magnetic ﬁeld, numerical solutions to frequency dependent three-level rate equations show good

agreement with spectral hole burning results. This allows predicting spectral structures given a

speciﬁc hole burning sequence, the underpinning spectroscopic material properties, and the relevant

laser parameters. This enables us to largely eliminate power dependent hole broadening through the

use of adiabatic hole-burning pulses. Though this system of rate equations shows good agreement

at zero ﬁeld, the addition of a magnetic ﬁeld results in unexpected spectral diﬀusion proportional to

the induced Tm ion magnetic dipole moment and average magnetic ﬁeld strength, which, through

the quadratic Zeeman eﬀect, dominates the optical spectrum over long time scales. Our results

allow optimization of the preparation process for spectral structures in a large variety of rare earth

ion doped materials for quantum memories and other applications.

I. INTRODUCTION

Rare-earth ion doped crystals (REICs) are interesting

materials due to their long-lived excited states and their

exceptionally long optical coherence times at cryogenic

temperature [1, 2]. In particular, along with the pos-

sibility for spectral tailoring of their inhomogeneously

broadened 4fN-4fNtransitions, this makes them prime

candidates for a number of applications in classical and

quantum optics. Examples include laser stabilization,

RF spectrum analysis, narrow band spectral ﬁltering,

and quantum information storage and processing [2–7].

Thulium-doped Yttrium Gallium Garnet (Y3Ga5O12,

Tm:YGG) is one such material. Its 3H6↔3H4transition

at 795 nm wavelength features an optical coherence time

of more than 1 ms [8–10], which is one of the longest

among all studied REICs. In combination with the ac-

cessibility of this transition—within the range of com-

mercial diode lasers—this makes it a natural candidate

for applications.

The quality of created features and the resulting con-

sequences for associated applications, are dependent on

the spectroscopic properties of the dopant ions and their

numerous interactions with other atomic components in

their local crystalline environment [11, 12], the details of

the optical pumping process, and the spectral and tempo-

ral proﬁle of the applied laser pulses[13, 14]. Deep under-

standing of the relation between spectroscopic properties,

∗These authors contributed equally to this work. Present Address:

National Institute of Standards and Technology (NIST), Boulder,

Colorado 80305, USA

†These authors contributed equally to this work.

optical control ﬁelds, and spectral diﬀusion dynamics has

resulted in improvements of this process in a number

of other rare-earth-doped materials including Tm:YAG,

Eu:YSO, and Pr:YSO [15–17]. However, this important

connection has thus far not been made for Tm:YGG.

In this paper we track the evolution of population

within the electronic levels of Tm3+ ions in YGG (see

Fig. 2 for simpliﬁed level scheme) by semi-continuous

monitoring of spectral holes for many sequences of ap-

plied spectral hole burning pulses. The characteristic

shapes and sizes of these spectral features are matched

to a rate equation model that encompasses the ground

(3H6), excited ( 3H4), and bottleneck (3F4) levels in this

material with associated lifetimes and branching ratios.

At zero magnetic ﬁeld we see good agreement between

our numerical model and measured results across many

diﬀerent pump sequences of varying duration, power, and

spectral shape. With the addition of an external mag-

netic ﬁeld the agreement with our numerical model dis-

appears as spectral diﬀusion from local host spins begins

to dominate the shape of all spectral features over long

timescales. We characterize the nature of this unexpected

behavior and expand our model accordingly by adding a

spectral diﬀusion term to account for a quadratic Zeeman

interaction with present noisy magnetic ﬁelds [18, 19].

The letter is structured as follows: In section II we

describe the experimental setup used to collect our mea-

surements. In section III we detail the atomic level struc-

ture in Tm:YGG and introduce spectral hole burning, the

workhorse of our investigations, to select a known set of

atomic population. In section IV we introduce and apply

a rate equation model which shows good agreement to the

measured spectral hole features. In section IV A we detail

the use of adiabatic pulses to shape spectral holes at zero

arXiv:2210.05005v1 [quant-ph] 10 Oct 2022

magnetic ﬁeld with the goal of creating high-resolution

features. Section V shows un-controlled changes to cre-

ated spectral holes in the presence of magnetic ﬁelds and

connects these noise eﬀects to the quadratic Zeeman ef-

fect. Section VI extends this quadratic Zeeman connec-

tion to the characterization of spectral diﬀusion results

that diﬀerentiates the measured results from those pre-

dicted by our model over longer timescales.

II. EXPERIMENTAL SETUP

To measure spectral holes, from population storage

in various atomic levels, over diﬀerent timescales in

Tm:YGG we use the setup detailed in Fig. 1. A

CW diode laser tuned to the ion transition frequency at

795.325nm [20] is locked to a reference cavity resulting

in a linewidth of roughly 5kHz [21]. To craft short pulses

of high extinction ratio, its continuous wave emission is

directed to a free space AOM. The sinusoidal driving sig-

nal of the AOM is mixed with a signal modulated by an

arbitrary function generator, which allows programmable

control of the transmitted pulse amplitude for the ﬁrst

order light.

After the amplitude control, the pulsed light is di-

rected to a ﬁber coupled phase modulator driven using

arbitrary waveforms for serrodyne frequency shifting and

more complex chirped pulse shapes as detailed in section

IV A. The optical signals are then sent through a polar-

ization controller to a 1% Tm:YGG crystal grown by Sci-

entiﬁc Materials Corp. and housed in a pulse tube cooled

cryostat at 500-700mK. A superconducting solenoid cen-

tered on the crystal applies a homogeneous magnetic ﬁeld

from 0-2T(using about 1mA/mT of current) along the

crystal’s <111>axis. Signals transmitted through the

crystal are directed to a ﬁber-coupled photo diode and

recorded for subsequent analysis

Experimental control is handled on a number of diﬀer-

ent time scales via custom Python scripts that ensures

signals are created at the correct moment [22]. For se-

quencing on timescales of longer than a second, the built-

in Python timing functions are used to adjust the ex-

periment. On all timescale shorter than seconds, timing

is handled by pre-programming a pulse generator that

produces correctly timed trigger signals for the various

devices. Waveforms for the arbitrary voltage signals are

generated by custom scripts and uploaded to the respec-

tive devices for arbitrary control of instantaneous pulse

frequencies and amplitudes.

III. TM:YGG SITE AND LEVEL STRUCTURE

Garnet crystals such as YGG have cubic crystal struc-

ture with O10

hspace group symmetry, which yields six

Tm3+ ion substitution sites, each with D2point group

symmetry[9, 23–25]. The magnetic and optical behav-

ior of Tm3+ ions in each of these sites is identical but

AWG1

trigger

Pulse Generator

AWG2

trigger

AOM

RF Synth

29 dB

40 dB

0th

1st

ECDL 795nm

QWP

HWP

+PDH

+FALC

QWP

Scope

trigger

Mkr

T= 600mK, B = 0 - 2 T

Tm:YGG

FIG. 1. Schematic of the experimental setup. A PC programs

a sequence on a pulse generator (SpinCore Pulseblaster) with

nanosecond timing resolution that produces a set of trig-

ger pulses for all devices. Waveforms written to an arbi-

trary waveform generator (AWG1,Tektronix AWG 70002A)

voltage channel, are subsequently ampliﬁed (SHF S126 A),

and drive an electro optic phase modulator (PM) to gen-

erate side bands on the laser light at arbitrary frequencies.

The AWG marker channels drive a set of home-built electri-

cal switches which gate the drive signal of an acousto-optic

modulator (AOM) to create short pulses from the laser light.

This gated AOM signal is mixed with fast arbitrary voltage

pulses(AWG2,Tektronix AFG 3102), ampliﬁed (Mini-Circuits

ZHL-5W-1+), and sent to an acousto-optic modulator (AOM

Brimrose 400MHz) to synthesize controllable-amplitude laser

pulses with rise times as short as 2.5ns. We use a single light

source (Toptica DL Pro 795nm) for diﬀerent tasks (optical

pumping, pulse generation, etc). The laser frequency is set

via a wavemeter (Bristol 871) and locked to a thermally and

acoustically isolated high ﬁnesse optical cavity (Stable Laser

Systems) via the Pound-Drever-Hall method and a fast feed-

back loop acting on the laser current and piezo voltage (Top-

tica PDH and FALC Modules). Transmitted signals from the

crystal are directed to a variable-bandwidth photo detector

(NewFocus 2051) and displayed on an oscilloscope (Lecroy

Waverunner 8100A) conﬁgured by the PC and synchronized

with the experimental sequence by a trigger signal.

the crystal structure leads to eﬀective in-equivalence be-

tween the sites due to six diﬀerent orientations that the

ion and its entire local environment can take within the

lattice [26]. However, for a few speciﬁc directions relative

to the crystalline axes, ions at in-equivalent sites can be

cast into classes that share the same projections of ap-

plied electromagnetic ﬁelds ( ~

E,~

B) onto their local site

axes.

Given the orientation of our magnetic ﬁeld, parallel to

the crystalline ~

B|| <111>axis, we cast the ions into

two diﬀerent classes as depicted in Fig. 2 a. The ions

in each separate class experience diﬀerent magnetic ﬁeld

projections onto the axes in their local frame. One class

features magnetic ﬁeld projections along the ion’s local

X and Z axes (blue), and the other projections along

the local Y and Z axes (red). This diﬀerence becomes

evident from a simple spectral hole burning experiment,

which we use to introduce the remaining results of the

paper.

Frequency Oset (MHz)

Absorption (cm-1)

Energy

H

H

B = 0

Magnetic Field (B)

+1/2

-1/2

 g

+1/2

-1/2

 e

F

τ1 =1ms

τ1 =64ms

1-ζ

<001>

<010>

<100>

FIG. 2. a. Depiction of the six ion substitution sites relative

to the cubic crystal cell. For a magnetic ﬁeld along the crys-

talline <111>axis the sites are cast into two classes featuring

diﬀerent ﬁeld projection, shown in red and blue. Small black

arrows indicate the local site X,Y,Z, axes for the red site in

the <100> <010>plane. b. Energy level structure of the

Tm ion transition of interest with previously measured life-

times (τ), branching ratios (ζ), and ground and excited split-

tings (∆g, ∆e). c. A pair of hole-burning spectra with the

main spectral hole pictured in the center. Each hole, shown

in an associated color to part a., is burned with a polariza-

tion that selects one of the two classes of ions and produces

anti-holes with diﬀerent splittings (indicated by the colored

arrows). Additional modulations of the optical depth out-

side the hole originate from a slight orientation oﬀset during

crystal cutting and polishing.

In spectral hole burning, a long optical pump pulse

excites atomic population in a narrow spectral window

within an inhomogeneously broadened absorption line.

Excited ions subsequently decay — either back into the

original state, or into another energy level that often be-

longs to the electronic ground state manifold. Scanning

a weak laser beam over a spectral interval centered on

the frequency of the original pump pulse reveals sections

of decreased and increased absorption — so-called spec-

tral holes and anti-holes. Spectral holes occur at fre-

quencies of reduced ground-state population, i.e. with

oﬀset ∆ = 0 (for the central hole), ∆ = De(for the

side hole), and anti-holes can be observed whenever the

ground-state population is increased, which happens at

∆ = Dgand ∆ = Dg±De. Here, Dgand Deare ground

and excited state splittings. Consult Ref.[13, 27, 28] for

more details on spectral hole burning.

In our case, the ground and excited state splittings de-

pend on the magnitude and direction of the applied mag-

netic ﬁeld — which vary for each class of Tm ions [26].

At 7.5 mT we recorded a pair of hole burning spectra,

shown in Figure 2 c, for orthogonal pumping polariza-

tions. This allowed us to selectively address ions in either

of the two classes. We found two sets of anti-holes, each

of which split according to the diﬀerent ﬁeld projections

experienced by the two classes of Tm3+ ions. This ability

to select out a single class of ions becomes important in

section VI as spectral diﬀusion depends on the magnetic

projection on each speciﬁc class of ions.

IV. MODELING RESULTS USING

THREE-LEVEL RATE EQUATIONS

To further understand the eﬀects of a given hole burn-

ing process on our REIC ensemble we turn to solutions

of the Maxwell-Bloch equations that describe the inter-

action of light with one or many atomic systems. These

diﬀerential equations can be quite diﬃcult to solve, given

the complexity that there does not exist a single ﬁxed

Rabi frequency to drive all ions [29]. In the limit of exci-

tation pulse lengths much shorter or much longer than T2

the rate equation approach has been shown to be an ef-

fective model [30]. Thus for the case of Tm:YGG, due to

the long T2and low optical depth we reduce the Maxwell-

Bloch equations to a set of rate equations that describe

the conserved total atomic population and how it ﬂows

through the diﬀerent available levels as a function of time

[14].

Note that in the case of narrow band excitation of an

inhomogeneously broadened transition where only a cer-

tain portion of atoms are driven, frequency dependence

must be added. Following [14] and [15, 31] we describe

the dynamics of our atomic ensemble with equations 1-3.

∂ng(t)

∂t =R(∆)(ne−ng) + 1−ζ

ne+1

nb(1)

∂ne(t)

∂t =R(∆)(ng−ne)−1

ne(2)

∂nb(t)

∂t =ζ

ne−1

nb(3)

This system of coupled diﬀerential equations describes

the relative change of atomic population, n(g,e,b)(t, ∆) in

three ion levels as a function of time, and frequency oﬀset,

∆. The branching ratio ζdetermines how much popula-

tion decays from the excited state |eithrough the bottle-

neck state |bibefore reaching the ground state |giwith

respective level lifetimes Teand Tb. To drive the system,

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QuadraticZeemanSpectralDiusionofThuliumIonPopulationinaYttriumGalliumGarnetCrystalJacobH.Davidson,1,AntarikshaDas,1,yNirAlfasi,1RufusL.Cone,2CharlesW.Thiel,2andWolfgangTittel1,3,41QuTechandKavliInstituteofNanoscience,DelftUniversityofTechnology,2628CJDelft,TheNetherlands2DepartmentofPhysics,Montan...

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Quadratic Zeeman Spectral Diusion of Thulium Ion Population in a Yttrium Gallium Garnet Crystal Jacob H. Davidson1Antariksha Das1yNir Alfasi1Rufus L. Cone2Charles W. Thiel2and Wolfgang Tittel1 3 4.pdf

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Quadratic Zeeman Spectral Diusion of Thulium Ion Population in a Yttrium Gallium Garnet Crystal Jacob H. Davidson1Antariksha Das1yNir Alfasi1Rufus L. Cone2Charles W. Thiel2and Wolfgang Tittel1 3 4

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