Coherent frequency up-conversion of microwaves to the optical telecommunications band in an Er:YSO crystal

0.61 MB·English

by  Xavier Fernandez-Gonzalvo

Coherent frequency up-conversion of microwaves to the optical telecommunications
band in an Er:YSO crystal
Xavier Fernandez-Gonzalvo,1 Yu-Hui Chen,1 Chunming Yin,2 Sven Rogge,2 and Jevon J. Longdell1
1The Dodd-Walls Centre for Photonic and Quantum Technologies & Department of Physics,
University of Otago, 730 Cumberland Street, Dunedin, New Zealand.
2Centre of Excellence for Quantum Computation and Communication Technology,
School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia
(Dated: June 22, 2015)
The ability to convert quantum states from microwave photons to optical photons is important
for hybrid system approaches to quantum information processing. In this paper we report the up-
conversionofamicrowavesignalintotheopticaltelecommunicationswavelengthbandusingerbium
dopants in a yttrium orthosilicate crystal via stimulated Raman scattering. The microwaves were
5 appliedtothesampleusinga3Dcopperloop-gapresonatorandthecouplingandsignalopticalfields
1 weresinglepassed. Theconversionefficiencywaslow,inagreementwithatheoreticalanalysis,but
0 can be significantly enhanced with an optical resonator.
2
n PACSnumbers:
u
J
I. INTRODUCTION for its 4I15/2 ↔ 4I13/2 optical transition [32], and the
9
wavelength of this transition is in the telecommunica-
1
Superconducting qubits are a rapidly advancing part tionsband, wherepropagationlossesinopticalfibresare
] of quantum information science. The ability to reach minimized. Because Er3+ is a Kramer’s ion (odd num-
h deep into the strong coupling regime of cavity QED us- ber of 4f electrons), for the nuclear spin free isotopes
p ingmicrowaveshasrevolutionizedquantumopticsinthe (allbut167Er),thegroundstateisdoublydegenerate. It
-
t microwaveregime[1–4],andallowsthecouplingbetween also has rather large effective g values [33, 34], such that
n
superconducting qubits and a broad range of microwave microwavefrequencysplittingscanbeachievedwithonly
a
u frequencyquantumsystems[5,6]. Distributionandstor- modest magnetic fields.
q age of microwave quantum states, however, present diffi- In the present letter we report the up-conversion of
[ cult challenges. A way around this problem would be to

convertquantumstatesofmicrowavephotonsintooptical
2
4v pphrooptoangastaionndovficqeuvaenrtsuam. Tshtaistewsobuledtwaelelonwsluopnegrcdoinstdaunccte- 4I13/2 34 Po Ω Ωξ
1 ing qubit nodes using optical fibres, and it would also μ
0 allow for quantum memories for light to be used [7–11], t
2 whicharecurrentlymoredevelopedthantheirmicrowave Ω
0 ξ
counterparts[12–15]. Quantumfrequencyconversionhas Ω
1. been achieved between optical frequencies [16–20], and S
0 2
recently between microwave frequencies [21]. However,
5
1 sofar,quantumfrequencyconversionfromthemicrowave 4I15/2 Ωμ D B
: to the optical domain remains an unsolved challenge. 1 α D
v There are a number of approaches being investigated 1 2
i
X for the up-conversion process. Opto-mechanical ap- B
proaches [22–26] currently have the highest reported ef-
r
a ficiencies and can achieve MHz bandwidths. In such ap- FIG. 1: (Color online) Left: the 4I and 4I levels in
15/2 13/2
proachesbothanopticalandamicrowaveresonatorsare Er:YSO are Zeeman split under the presence of an external
parametrically coupled through a micro-mechanical res- magnetic field B(cid:126). The Raman heterodyne signal is produced
onator. In order to have quiet frequency conversion this when a microwave field Ωµ and an optical field Ωξ drive two
transitionsinathreelevelatom. Acoherenceisproducedon
rather low frequency intermediate mechanical resonator
thethirdtransitionwhichgeneratesanopticalsignalfieldΩ .
needsto be cooledtoitsquantumgroundstate, andthis S
Thiscanbedetectedasabeatnoteontheopticaldrivefield,
iscurrentlychallenging. Anotherapproachistousecon-
i.e. amodulationintheopticaloutputpowerP atthesame
o
ventionalnon-linearopticalmaterialstomakeresonantly
frequency as the microwave field. Right: depiction of the ex-
enhanced modulators [27–29]. perimental setup. A copper made loop-gap resonator holds
Two recent proposals [30, 31] have suggested using an Er:YSO sample inside. Light is coupled in and out using
rare earth doped solids, with a particular focus on er- apairofprisms. Microwavesarecoupledviatwostraightan-
biumdopedyttriumorthosilicate(Er:YSO).Er:YSOhas tennas. Themagneticfieldisappliedintheverticaldirection,
many attractive features for frequency up-conversion: it parallel to the D1-D2 plane of the crystal, at an angle α as
has narrow inhomogeneous and homogeneous linewidths measured from D1.
2
a microwave signal into the optical telecommunications ersarecoupledwithapairofstraightantennasinsidethe
band using rare earth ion dopants in a crystal, by per- cavityspace. Theinputlight,at1536nm,iscoupledinto
forming microwave Raman heterodyne spectroscopy in and out of the sample with the aid of a pair of coupling
Er:YSO.Ramanheterodynespectroscopywithradiofre- prisms, and fibre coupled collimators. The input fibre
quency (ca. 0-200MHz) is a commonly used technique is a single mode fibre, while for the output one we use a
for nuclear spins in rare earth dopants [35–39]. It has multi-modefibreforeaseofcoupling. Asuperconducting
also been demonstrated in the microwave regime in ruby magnet generates a magnetic field perpendicular to the
[40, 41] and metalloprotiens [42]. These systems, how- longitudinal direction of the sample (i.e. in the D1-D2
ever, are not as attractive for the realization of quantum plane) between 0 and 300mT. The angle α, measured
frequency conversion because they exhibit much broader from D to B(cid:126), can be varied by rotating the sample. A
1
optical lines. more detailed explanation of the complete experimental
setup can be found in Appendix C.
ThestrengthforeachoftheopticaltransitionsinFig.1
II. FREQUENCY UP-CONVERSION isgivenbytheproductoftheelectronictransitiondipole
moment and the overlap of the two spin states. This
Raman heterodyne spectroscopy uses the three wave overlap is calculated by diagonalizing the spin Hamilto-
mixing that occurs from three energy levels in a ∆ con- nian [33] an taking the inner product of the respective
figuration, as shown in Fig. 1. To enhance the efficiency eigenstates. The orientation of the magnetic field has to
of the process we use a microwave resonator for the low- be chosen carefully, so as to maximize the difference be-
est frequency field. The 4I ground state of Er:YSO tween the quantisation axes for the ground and excited
15/2
is Zeeman split under the presence of an applied mag- states and thus allow ∆ transitions. For the situation in
netic field B(cid:126), making the |1(cid:105)↔|2(cid:105) transition resonant which B(cid:126) is contained in the D1-D2 plane the calculated
with the microwave cavity. When the input microwave angle that maximizes the overlap between states |2(cid:105) and
field Ωµ is applied it generates a coherence between lev- |3(cid:105) is αM =29◦.
els |1(cid:105) and |2(cid:105). Simultaneously, the optical coupling field Afterthefrequencyconversionprocess,theACcompo-
Ω drives a second coherence between levels |2(cid:105) and |3(cid:105). nentoftheheterodynesignaldetectedinthephotodiode
ξ
The presence of these two coherences generates a third is high-pass filtered and amplified, and sent into an RF
one between levels |1(cid:105) and |3(cid:105), which gives an output spectrum analyser. An inconvenient consequence of us-
signal field Ω at a frequency equal to the sum of the ing a multi-mode fibre for the output light is that there
S
frequencies of the microwave and the coupling fields. As is loss in the modulation due to dephasing of the differ-
long as the sample is small compared to the wavelength entpropagationmodes. Inoursetupthislossistypically
ofthemicrowavefieldthesignalfieldwillbegeneratedin from 3 to 10dB, and it depends on the arrangement of
the same spatial mode as the coupling beam. The signal thefibre. Fromthepowerdetectedbythespectrumanal-
field can then be readily detected in a photodiode as a yser we can work out the optical power of the generated
heterodyne beat note on the coupling beam. signal sideband PS.
B. Raman heterodyne spectroscopy
A. Experimental realisation
The crystalline structure of YSO belongs to the C6 Our Raman heterodyne spectroscopy results are pre-
2h sented in the color plot of Fig. 2. The power of the gen-
symmetry group, with two crystalographically inequiv-
erated signal field is measured as we scan the magnetic
alent sites where erbium can replace yttrium. In this
field and the coupling laser frequency f . On the left
work we focused on ‘Site 1’ with a transition wavelength ξ
side,inwhite,weplotanopticalabsorptionspectrumfor
of λ = 1536.478nm [43]. YSO has three orthogonal op-
tical1extinction axes D , D and b. We use a cylindrical |B(cid:126)| = 0 (dotted line) and for |B(cid:126)| = 0.178T (solid line).
1 2
Er:YSO sample of 4.95mm diameter by 12mm length, Note that due to various etalon effects the background
with an erbium number concentration of 0.001%. The of these measurements is not constant. In the optical
optical b axis of the crystal is aligned along the length absorption spectrum for |B(cid:126)| = 0.178T the four optical
of the cylinder, and so the D -D plane is parallel to transitions can be observed. The strong ones (|1(cid:105) ↔ |3(cid:105)
1 2
the end faces. The sample sits inside a copper three- and|2(cid:105)↔|4(cid:105))appearaspeaksaround∆fξ =±1.6GHz,
dimensional loop-gap microwave resonator, with a res- while the weak ones (|1(cid:105)↔|4(cid:105) and |2(cid:105)↔|3(cid:105)) appear as
onant frequency of 4.9GHz and a linewidth of 16MHz smaller shoulders at about ∆fξ = ±3.4GHz. The ratio
(quality factor Q (cid:39)300). This kind of resonator pro- between the absorption level of the weak and the strong
vides very good filling factors (∼0.8) and makes optical linesisclosetotheexpectedvalueforα(cid:39)αM. Fromthe
coupling to the sample a simple task, since two optical absorptionmeasurementswecanalsoextractaninhomo-
windowscanbeopenedintheendcapsatanullpointof geneousbroadeningoftheopticaltransitionof∼2.5GHz
the surface currents, thus not affecting the properties of FWHM.ComparingtheRamanheterodynespectroscopy
thecavityverymuch. Inputandoutputmicrowavepow- data with the absorption spectrum at |B(cid:126)| = 0.178T we
3
10−12
6
4 10−13
2
∆ f (GHz)ξ − 02 1100−−1154SP (W)
−4
−6 10−16
−8
10−17
200
Hz) 100 x10
(kR 0
P
E
∆ f −100 x10
−200
0 0.05 0.1 0.15 0.2 0.25
B (T)
FIG. 2: (Color online) Top: Raman heterodyne spectroscopy on Er:YSO, showing frequency conversion from microwave to
opticaltelecomfrequencies. Thestrengthofthemagneticfieldisplottedinthehorizontalaxis,andthecouplinglaserdetuning
isplottedontheverticalaxis. Thecolorscaleindicatesthepoweroftheoutputsignalfield. Ontheleft,thewhitedottedline
represents the optical absorption spectrum for |B(cid:126)| = 0. The solid white line corresponds to the optical absorption spectrum
for|B(cid:126)|=0.178T.Bottom: EPRspectrumoftheEr:YSOsample. Theregionsawayfromthemainpeakshavebeenmagnified
for clarity. In both plots the presence of double peaks along the horizontal axis is due to misalignment in the magnetic field,
breaking the magnetic degeneracy of the two inequivalent orientations of Er+3 in YSO.
see that the main four peaks in the color plot (in red) agrees with our numerical simulations. From this EPR
coincide with the absorption on each of the lines, as it is shift we can extract a cavity-atoms cooperativity factor
to be expected. It can also be seen that the peak signal of the order of 6×10−2.
is slightly higher for the lowest frequency peaks, which Comparing the Raman heterodyne and the EPR spec-
can be explained using hole burning arguments. tra we can see that most of the features present in the
Raman heterodyne spectroscopy data are also replicated
in the EPR data. The EPR peak present at B ≈ 0.03T
C. Electron paramagnetic resonance we assign to the Er atoms in Site 2 of YSO. Because
the optical transition for these atoms is at a different
wavelength we don’t see a signal in Raman heterodyne
Beneath the Raman heterodyne spectroscopy data is
spectroscopy.
the electron paramagnetic resonance (EPR) spectrum of
oursample. TotaketheseEPRmeasurementsweapplya
frequency modulated (FM) microwave signal into the in-
put port of the microwave cavity and monitor the trans- III. CHARACTERIZATION OF THE
CONVERSION PROCESS
mitted intensity using a lock-in amplifier. In this way
we are able monitor the resonant frequency shift of our
cavity (∆f ) as is done in Pound frequency locking In this section we characterise the conversion process
EPR
[44]. Asthespintransitionsaresweptthroughresonance byexaminingthedependencyoftheoutputsignalpower
withthecavitytheypulltheresonatorfrequencyfirstone withtheinputmicrowaveandcouplingpowers. Wecom-
waythentheother, resultingindispersiveshapedpeaks. pare these measurements with a numerical model of our
The collection of vertical lines in the Raman heterodyne experiment, which we also use to find out the various
spectrum and the smaller peaks in the EPR spectrum losses in our setup. Finally we estimate the efficiency of
are due to the 167Er isotope, which has non-zero nuclear theconversionprocessanddiscussseveralwaysbywhich
spin (I = 7/2) and therefore exhibits hyperfine splitting it can be increased.
even for |B(cid:126)|=0.
The EPR data presented in Fig. 2 shows a maximum
frequency shift of around 180kHz. The measurements A. Scaling with the input powers
shown in the figure are taken for an input microwave
power of 0dBm, which is enough to start to saturate the Figure 3 shows, in red, the dependence of the signal
microwave transition in the absence of the optical field. field power with the input microwave power P and the
µ
For saturation-free measurements, at lower microwave detected coupling laser power P . The laser coupling
ξ
powers,wegetamaximumshiftofaround260kHz,which powerismeasuredattheoutputofthesystem,andisnot
4
P thelossofheterodynesignalinthemulti-modefibreis
0.6 µ
equivalenttoalossinmicrowavepower. Thismulti-mode
0.5 10−12 (W)S fiinbgreonlotshseisgmeoemaseutrriecdaltoarbraenbgeetmweenento3fatnhdefi1b0rdeB. Tdoeptehnedse-
0.4 P
W) 10−14 two numbers we have to add the effects of lowering the
P (pS0.3 −10 P0µ (dB1m0) 20 tuenmknpoewrantubrueticnanthbeeceoxapxeicatlemditcorobweaovnetwheiroesr,dewrhoifchafaerwe
0.2
dB. For the optical losses we can measure the loss from
0.1 the sample to the detector at room temperature, and it
is around 5dB. It is hard to make an estimation of how
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 lowering the temperature will modify this number. We
P (mW)
ξ could observe a total decrease in transmitted power of
-10dB through our complete optical setup after the sys-
FIG.3: (Coloronline)Signalpowerasafunctionofdetected
tem was cold, but this number takes into account both,
couplinglaserpower(mainfigure)andinputmicrowavepower
inputandoutputlosses,whichwecannotmeasuresepa-
(inset) in red, along with the corresponding theoretical pre-
rately with the system inside the cryostat. All in all, we
dictions in blue. The faster-than-linear growth of P versus
S
consider our fitted loss parameters to be in reasonable
P shows optical cooling of the spins via optical pumping.
ξ
agreement with our observations.
corrected for optical losses between the sample and the
power meter. In blue we plot the expectations for these C. Conversion efficiency
measurementsbasedonourtheoreticalmodel,brieflydis-
cussed below. The dependency of P with P follows
S µ By comparing the input and the signal field pow-
the expected pattern for a three wave mixing process: it
ers we can calculate a number conversion efficiency
ionucrrecaasseesalrinoueanrdlyPuµn=til20itdrBemac.heTshaesseamtueraastuiornempeonitnst,arine ηTnhi=s ePPffiSµc·ieffnµξc,ywηheraecfcoµuinsttshfeoirntphuetfmraicctrioownaovfemfriecqruoewnacvye.
taken for P =1.8mW. n
ξ photons converted into optical telecom photons. For a
The dependency of P with P , however, doesn’t fol-
S ξ coupling power of ∼2mW and making the appropriate
lowaquadraticrelationforlargelaserpowers. Thefaster
corrections for ζ and ζ−1 we get a conversion efficiency
thanquadraticrateatwhichthesignalincreaseswiththe µ ξ
pump laser power is due to optical pumping improving of O(cid:0)10−12(cid:1). In order to get closer to the target of
thepopulationdifferencebetweenthetwoI sublevels, unity conversion efficiency the most important improve-
15/2
lowering the effective spin temperature. This fact is par- mentwillbetoaddadoublyresonantopticalcavity(for
ticularlyconvenientsince,foralownoiseconversionpro- the coupling and signal fields), which will improve the
cess, the spins temperature will need to be small com- efficiency by a factor proportional to the finesse of the
pared with the frequency of the input microwave field. cavity squared F2, where F can be as high as O(cid:0)105(cid:1).
These measurements are taken for an input microwave On top of this effect, cavity enhancement of the cou-
power P =0dBm. pling field should increase the effectiveness of the opti-
µ
cal cooling of the spins, additionally increasing the ef-
ficiency of the conversion process. There are also nu-
merous other improvements that can be made. A more
B. Numerical model and propagation losses
homogeneous magnetic field is very desirable, since it
would reduce the microwave inhomogeneous linewidth.
To model the experiment and plot the blue lines in
The optical depth used in this experiment is also rather
Fig.3,wetreateacherbiumatomasathreelevelsystem
low (0.02mm−1). Much larger optical depths have been
and use standard master equation techniques. This is
observedinEr:YSOwithoutthepenaltyofbroaderinho-
described in detail in Appendix B. The optical and spin
mogeneous lines [45]. Astonishingly narrow absorption
dephasing times are not know precisely and are allowed
lines with good optical depth have also been reported
to vary, as is the spin lifetime. In the fitting process
for Erbium dopants in isotopically pure yttrium lithium
we also introduce two free parameters ζ and ζ−1 which
µ ξ fluoride [46]. The Q-factor of ∼300 for our microwave
take into account the propagation losses of P from the
µ resonator is also rather modest – much higher Q-factors
setup input to the microwave cavity and the inverse loss
for copper resonators have been reported [47]
of P from the photodiode detector to the sample. The
ξ
fitted values for these loss and inverse loss parameters
areζ =13.1dBandζ−1 =-6.4dB.Itishardtocompare
µ ξ
thesetwonumberswiththemeasuredlossesofthesetup, IV. CONCLUSIONS
since losses are hard to quantify at low temperatures for
practicalreasons. Themeasuredζ atroomtemperature In summary, we have presented a novel way to con-
µ
is about 8dB. In the regime where P is proportional to vert microwave signals into the optical telecommunica-
S
5
tionsband,bymeansofacryogenicallycooledrareearth Tofurthersupportthisclaimthereisthefollowingex-
sample inside a three-dimensional microwave cavity. We perimental observation, represented in Fig. A.1. When
have matched our Raman heterodyne spectroscopy ex- measuring the Raman heterodyne signal with the spec-
perimental results with a theoretical counterpart. Fi- trumanalyserthenoisebackgroundisn’tflat. Insteadwe
nally, although the efficiency of this initial demonstra- can observe a small peak at the signal generator’s out-
tion is low, there are many ways to improve it, the most putfrequency. Thisispick-upnoisecomingdirectlyfrom
significantofwhichisenhancingtheeffectofthetwoop- the signal generator’s oscillator to the analyser (and the
tical fields with an optical resonator. Among possible amplifier preceding it), and it is present even when the
designs for this optical resonator is the Fabry-Perot res- outputofthesignalgeneratorisshutdown(buttheoscil-
onator as proposed in [30] or a whispering gallery mode lator inside the signal generator is still on) or the optical
type resonator as investigated in [28]. detectorisblocked. Whenmeasuringtheheterodynesig-
nalpowerversustheinputmicrowavepower,forhighmi-
crowavepowerstheheterodynesignalismuchbiggerthan
ACKNOWLEDGEMENTS thepick-upnoise,andthelateronecanbeneglected. For
low microwave powers, however, the detected peak be-
comes smaller than the pick-up noise alone. This means
WewouldliketoacknowledgetheMarsdenFund(Con-
that the detected signal and the pick-up noise interfere
tractNo. UOO1221)oftheRoyalSocietyofNewZealand
destructively,whichinturnmeanstheyarecoherentwith
andtheARCCentreofExcellenceforQuantumCompu-
each other. It is safe to assume that the pick-up noise
tation and Communication Technology (CE110001027)
will be coherent with the signal coming out of the signal
for their support. S.R. acknowledges a Future Fellow-
generator since they come from the same source, so we
ship (FT100100589).
can conclude that the converted signal is coherent with
the input microwave signal, and therefore so is the con-
version process.
Appendix A: A note on coherence
While we haven’t performed any direct measurement Appendix B: Theoretical model
ofphasepreservationintheup-conversionprocesswecan
be certain that this process is a coherent one. The ob-
To model the experiment and plot the blue lines in
servedheterodynepeaksareafewtensofHzwide. Both
Fig. 3, we treat each erbium atom as a three level atom
the optical and the microwave transitions in Er:YSO are
driven by a microwave field which connects the two low-
muchwiderthanthat,hencetheonlyexplanationisthat
est states |1(cid:105) and |2(cid:105), and an optical pump field which
thissignalisindeedgeneratedinaRamanscatteringpro-
connects states |2(cid:105) and |3(cid:105). In the interaction picture we
cess, which is inherently coherent. In other words: the
have the following Hamiltonian for a single atom:
only process in our sample that can generate such a nar-
row signal is a coherent process. In addition, we are H =δ σ +δ σ +Ω ((cid:126)r)(σ +σ )+Ω ((cid:126)r)(σ +σ ),
2 22 3 33 µ 12 21 ξ 23 32
performing heterodyne detection, so our measurements (B1)
areonlysensitivetothatlightwhichiscoherentwiththe where δ is the detuning from the microwave cavity fre-
2
pump beam. quency, δ is the detuning from the coupling laser fre-
3
quency and σ = |i(cid:105)(cid:104)j|. Because in the center of the
ij
loop-gap resonator the magnetic field is rather uniform
we can take the microwave Rabi frequency to be a con-
−105
stant Ω ((cid:126)r)⇒Ω . We take the optical pump field as a
pick−up µ µ
plane wave propagating along the z axis so the result-
−110 Low P
µ ing Rabi frequency, which satisfies Ωξ((cid:126)r)≡Ωξ·eink32z,
m) −115 isrepresentedbyatravellingwavealongthezaxis,where
B
P (dsa −120 ktr3a2nissitthioenw. avevectorforlightresonantwiththe|2(cid:105)↔|3(cid:105)
The dynamics of each of the atoms is governed by the
−125
master equation
−130
−200 −100 0 100 200
ρ˙ =−i[ρ,H]+L ρ , (B2)
∆f (Hz) loss
where ρ is the density operator of a single atom and
FIG. A.1: (Color online) Heterodyne signal as measured by Lloss is the loss Lindblad superoperator. Contributing
thespectrumanalyser,showinginterferencebetweenthegen- to L are the collapse operators [48] describing: the
loss
eratedsignalandthepick-upnoisefromthesignalgenerator. spontaneous emissions from state |3(cid:105) to states |1(cid:105) and
√ √
The sidebands are attributed to sidebands in the laser fre- |2(cid:105) ( γ σ , γ σ ); spin lattice relaxation between
31 31 32 32 √ √
quency due to mechanical vibrations in the laser box. states |1(cid:105) and |2(cid:105) ((N + 1) γ σ , N γ σ ); and
b 21 12 b 21 21
6
the dephasings for the microwave and optical transitions andthespininhomogeneouslinewidthistakenfromEPR
√ √
( γ σ , γ σ ). Here N = (e(cid:126)ω/(kT)−1)−1 is the results (∆ =2π×13MHz).
2d 22 3d 33 b µ
meannumberofbathquantaatthemicrowavefrequency. The dephasing rates and the spin lattice relaxation
Foroursituation,witha5GHzmicrowavefrequencyand time are not known for this temperature and magnetic
a 4.2K temperature, N ≈17. field, so they are allowed to vary and the values that
b
The steady state coherence on the |3(cid:105) ↔ |1(cid:105) transi- gavethebestfittothedataarechosen. Thesevaluesare
tion is given by ρ (δ ,δ ), and it can be obtained from γ =2.8×106s−1,γ =1.7×106s−1andγ =27.4s−1
31 2 3 3d 2d 21
the steady state solution of Eq. (B2). Then, the total (1 ms lifetime).
polarization at a given position z and at the |3(cid:105) ↔ |1(cid:105) We also introduce two free parameters ζ and ζ−1
µ ξ
frequency will be given by whichtakeintoaccountthepropagationlossesofP from
µ
the setup input to the microwave cavity and the inverse
(cid:90)(cid:90)
loss of P from the photodiode detector to the sample.
P(z)=Nd dδ dδ g(δ ,δ )ρ (δ ,δ ,z)+c.c. , ξ
13 2 3 2 3 31 2 3 The fitted values for these loss and inverse loss parame-
(B3) ters are ζ =13.1dB and ζ−1 =-6.4dB, which are well
µ ξ
where N is the density of atoms, d is the electric within the experimental expectations.
ij
dipole moment of the transition between |i(cid:105) and |j(cid:105), and
g(δ ,δ ) describes the distribution of the microwave (δ )
2 3 2
and optical (δ ) detunings due to inhomogeneous broad- Appendix C: Experimental setup
3
(cid:82)(cid:82)
ening, normalized so that dδ dδ g(δ ,δ )=1. In our
2 3 2 3
calculations g(δ ,δ ) is assumed to be a two-dimensional
2 3
Gaussian function with standard deviations ∆ and ∆ .
µ o Sync
Lock-In
This P((cid:126)r) acts as a source term and generates a side-
Amp.MDet. Amplifier
band signal via Signal
Generator MMM
∂E (z) iµ ω c
S = 0 31 P(z) , (B4) SM
∂z 2n
where ω is the angular frequency of the |3(cid:105)↔|1(cid:105) tran- PC
31 Spectrum
sition, n is the refractive index of the sample, and µ
0 Analyser
and c are the magnetic permeability and the speed of BiasMT
Tunable
light in the vacuum. In the optically thin limit this dif-
Laser
ferentialequationistrivialtosolveandtheresultcanbe
rearranged to give:
α L d I (cid:32)(cid:82)Leinkµzdz(cid:33) FIG.C.1: (Coloronline)Experimentalsetup. Thepartdrawn
|E (z =L)|= 31 · 23· ·Re 0 ·E , inblueisusedforEPRmeasurementsonly. PC:polarization
S 2 d13 πΩξ L ξ controller. SM: single mode fibre. MM: multi-mode fibre.
(B5)
where α31 is the absorption coefficient for the |1(cid:105)↔|3(cid:105) The setup for our experiment is depicted in Fig. C.1.
transition, L is the length of the sample, Eξ is The pump beam is generated by a fibre coupled external
the √amplitude of the coupling laser beam, and cavity diode laser at 1536nm, and then amplified by an
(cid:82)(cid:82)
I = 2π∆o dδ2dδ3g(δ2,0)ρ31(δ2,δ3,z =0). The first erbium doped fibre amplifier. A polarization controller
and the second terms on the right hand side of Eq. (B5) is used to maximize the output heterodyne signal. The
can be easily obtained from experimental absorption pump beam travels through a single mode fibre and is
measurements for the different transitions. The fourth then collimated using a fibre coupled GRIN lens colli-
term is a phase-matching factor due to the fact that the mator. With the help of a couple of right angle prisms
driving laser has a propagating phase of eink32z, while the beam is sent in and out of the microwave resonator,
the side-band signal has a propagation phase of eink31z passing through the Er:YSO sample on its way. The
(where kij is the wave vectors for the |i(cid:105) ↔ |j(cid:105) transi- output light, consistent of the pump beam and the up-
tion),anditcanbecalculatedveryaccurately. Thethird converted signal is then coupled into a multi-mode fibre
term including I is calculated numerically by solving the using a second GRIN lens collimator. At the output of
master equation as explained above. themulti-modefibrethelightiscollimatedintoaHama-
The spontaneous emission rates for the two optical matsu G7096-03 photodiode detector. A bias tee and a
transitions used to model the experiment are γ = battery serve the double purpose of biasing the photodi-
31
60 s−1 and γ = 30 s−1. These are calculated from ode and separating the AC from the DC components of
32
theknown11msexcitedstatelifetimeandthebranching thedetectedsignal. TheACcomponentisthenhighpass
ratios expected from the spin Hamiltonians. filteredandanalysedwithaFieldFoxN9916Avectornet-
Theopticalinhomogeneouslinewidthistakenfromthe work analyser working as a spectrum analyser. To drive
optical absorption measurements (∆ = 2π × 1GHz), the microwave cavity we use an R&S SMP 22 microwave
o
7
signal generator. Microwaves are coupled in and out of The part drawn in blue in Fig. C.1 corresponds to the
thecavityusingacoupleofstraightantennas. Thecavity EPR setup, consisting of an amplitude detector and an
itselfhasaresonantfrequencyof4.9GHz,aloadedQfac- SRS SR830 lock-in amplifier. In order to do EPR ex-
tor of around 300 and a room temperature transmission periments we add FM modulation to the microwave sig-
(|S |2) of around 6dB. nal at 3kHz. When this FM signal passes through the
21
Theresonatorandthecouplingopticssitinsideanen- cavity it gets converted into an AM signal with a modu-
casing stainless steel tube filled with about a mbar of lation amplitude proportional to the slope of the cavity
helium, that acts as a thermal exchange gas. This tube transmission curve. Using and amplitude detector we
is then inserted into a liquid helium bath cryostat. In can measure this modulation amplitude with a lock-in
this way we can avoid optical distortions created by the amplifier, which needs to be synchronized with the FM
boiling helium and thermal shocks that could be detri- modulation at the signal generator.
mental to the various optical components. Surrounding The laser source, the signal generator, the lock-in am-
theendoftheencasingtubeisasuperconductingmagnet plifier, the variable current source powering the magnet
poweredbyavariablecurrentsource,whichcangenerate and the spectrum analyser are all remotely controlled
magnetic fields of up to 0.3T. from a computer (not shown in the figure).
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