PHYSICAL REVIEW B 86, 064514 (2012)

Electron spin resonance detected by a superconducting qubit
Y. Kubo,1 I. Diniz,2 C. Grezes,1 T. Umeda,3 J. Isoya,3 H. Sumiya,4 T. Yamamoto,5 H. Abe,5 S. Onoda,5 T. Ohshima,5
V. Jacques,6 A. Dréau,6 J.-F. Roch,6 A. Auffeves,2 D. Vion,1 D. Esteve,1 and P. Bertet1
1

Quantronics Group, Service de Physique de l’Etat Condensé (CNRS URA 2464), CEA – Saclay, 91191 Gif-sur-Yvette, France
2
Institut Néel, Centre National de la Recherche Scientifique, BP 166, 38042 Grenoble, France
3
Research Center for Knowledge Communities, University of Tsukuba, Tsukuba 305-8550, Japan
4
Sumitomo Electric Industries Ltd., Itami 664-001, Japan
5
Japan Atomic Energy Agency, Takasaki 370-1292, Japan
6
Laboratoire de Physique Quantique et Moléculaire (CNRS, UMR 8537), Ecole Normale Supérieure de Cachan, 94235 Cachan, France(Received 25 May 2012; published 10 August 2012)
A new method for detecting the magnetic resonance of electronic spins at low temperature is demonstrated.
It consists in measuring the signal emitted by the spins with a superconducting qubit that acts as a singlemicrowave-photon detector, resulting in an enhanced sensitivity. We implement such an electron-spin resonancespectrometer using a hybrid quantum circuit in which a transmon qubit is coupled to a spin ensemble consistingof nitrogen-vacancy (NV) centers in diamond. With this setup we measure the NV center absorption spectrum at
30 mK at an excitation level of ∼ 15 μB out of an ensemble of 1011 spins.
DOI: 10.1103/PhysRevB.86.064514

PACS number(s): 03.65.−w, 85.25.Cp, 76.30.Mi

I. INTRODUCTION

Electron spin resonance (ESR) spectroscopy at low temperatures is often complicated by the long spin-lattice energy
relaxation time, which can reach minutes at sub-Kelvintemperatures. For continuous-wave (CW) ESR spectroscopy,
this implies working at low powers to avoid saturating thespins; in pulsed ESR this imposes low repetition rates. In bothcases a higher sensitivity for detecting the signal absorbed
or emitted by the spins would be desirable. Recently, tools
borrowed from superconducting quantum electronics havebeen applied to high-sensitivity ESR spectroscopy. High-Q
superconducting coplanar waveguide resonators have beenused for CW-ESR at millikelvin temperatures with a cryogenic
low-noise high electron mobility transistor (HEMT) ampliﬁer
followed by homodyne detection,1–4 and for pulsed ESR at
kelvin temperatures with a commercial ESR spectrometer,5yielding promising results in terms of sensitivity. Here wego one step further and use an on-chip single microwave
photon detector6 based on a superconducting qubit7 to realize
a high-sensitivity low-temperature ESR spectrometer.
The principle of our experiment is compared to moreconventional ESR techniques in Fig. 1. In conventional ESR
spectroscopy, a microwave pulse is applied to an ensembleof spins close to their resonance frequency ωs through a
low-Q cavity of frequency ωc with which the spins are tuned
in resonance (ωs = ωc ) by a magnetic ﬁeld. After being
excited by the pulse, the spins reradiate coherently part of
the absorbed energy through the cavity into the detection
waveguide, giving rise to a free induction decay (FID) signal
measured by homodyne detection, which yields the spin
absorption spectrum after Fourier transform.8 What limits thesensitivity of a typical commercial
CW spectrometer operating√
at 300 K to ∼ 1010 spins/ Hz for a line width of 0.1 mT and
an integration time of 1 s is the overall noise temperature of
the detection chain. In this work, we replace the detection
chain by a superconducting qubit and its readout circuitry.
This results in an increased sensitivity since a superconducting
1098-0121/2012/86(6)/064514(6)qubit is a nearly ideal single microwave photon detector6 at
its resonance frequency ωge . In order to transfer part of theexcitation of the spins to the superconducting qubit, the ESR
resonator is made frequency tunable and with a high-quality
factor. Spectroscopy is performed by ﬁrst exciting the spinswith a weak microwave pulse, collecting the radiated FID
signal with the resonator tuned at ωs , then transferring thissignal to the qubit at ωge , and measuring its ﬁnal state.
Repeating this experimental sequence yields the probability
Pe to ﬁnd the qubit in its excited state, which reproduces
the spin absorption spectrum. The sensitivity of such an ESR
spectrometer is set by the efﬁciency at which signal photons
can be transferred from the spins to the resonator, then to thequbit, and by the ﬁdelity with which the qubit state can bemeasured.

II. DEVICE AND EXPERIMENTAL SETUP

We implement this method using a recently reported hybrid
quantum circuit9 that includes an ensemble of electronic spins
magnetically coupled to a superconducting resonator, itself
electrically coupled to a superconducting qubit, as sketched inFig. 2(a). The spins are negatively charged nitrogen-vacancy(NV) color centers in diamond, whose structure and energy
levels are summarized in Figs. 2(b) and 2(c). The ground stateof NV centers has a spin one with splitting ω± /2π  2.88 GHz
between states mS = 0 and mS = ±1 at zero magnetic ﬁeld.10
Each of the two mS = 0 to mS = ±1 transitions is further
split into three peaks separated by 2.2 MHz due to thehyperﬁne (HF) coupling to the 14 N nuclear spin.11 In theexperiment a static magnetic ﬁeld BNV = 1.1 mT12 is appliedalong the [1,1,1] crystallographic axis to lift the degeneracy
between states mS = ±1. Centers having their N -V axis
along [1,1,1] (called ensemble I in the following) undergoa different Zeeman shift from those along the three other1,1,1 axes (ensemble III), resulting in two different ESR
frequencies ω+I /2π = 2.91 GHz and ω+III /2π = 2.89 GHz

064514-1©2012 American Physical Society

Y. KUBO et al.

(a)

PHYSICAL REVIEW B 86, 064514 (2012)(a)ωc = ωs

1

FID

ωs

10 μB

Q
R

2

(b)

ωc 2 ω

C

S

Pe 3

time

10

(b)

BNV

BNV // [111]

(c)

mI

ms= +1

0
+1
-1

ge

III

ωs
1

N I=1

3

V

ω+III

S=1

ms=0

Pe

ωp

I

ms= -1ωge

10 μB
FIG. 1. (Color online) Comparison between conventional pulsed
electron spin resonance (ESR) and qubit-detected ESR. (a) Conventional ESR: A spin ensemble is placed in a cavity with a frequency ωcand driven with a microwave pulse resonant with the cavity. When the
ESR frequency ωs matches ωc , the spins absorb the microwave pulse,and emit immediately after a free-induction decay (FID) signal into
the waveguide connected to the cavity. (b) Qubit-detected ESR: The
cavity is now frequency tunable and embeds both the spin ensemble
and a superconducting qubit with a frequency ωge . In a ﬁrst step(1), the spins are probed by a spectroscopy pulse with a frequency
ωp , which excites them if its frequency matches ωs . In a second
step (2), the cavity frequency is tuned to ωc = ωs , receives the FID
signal from the spins, and is afterwards tuned to transfer this signal
to the superconducting qubit at ωge . Finally (3) the qubit excitedstate probability Pe (ωp ) is measured, mapping the spins absorption
spectrum. Very low excitation powers can be used given the highsensitivity of the method.

for the mS = 0 to mS = +1 transition on which we will
exclusively focus in the following.
The diamond crystal used is of the high-pressure hightemperature (HPHT) I b type and has a NV center concentration of ∼ 3 ppm and a residual nitrogen concentrationof ∼ 20 ppm. In our setup, it is glued on top of the ESR
cavity C, a coplanar waveguide superconducting resonator13of quality factor Q ∼ 104 made of a niobium thin ﬁlm sputteredon a silicon substrate. The spin ensemble S detected in theexperiment consists of the ∼ 1011 NV centers that lie within
the mode volume of C, thus within a few microns of the
diamond surface. The cavity frequency ωc () can be tuned
on a nanosecond timescale by a ﬂux  applied throughthe loop of a superconducting quantum interference device(SQUID) inserted into C.1,14–16 The superconducting qubit
is a Cooper-pair box of the transmon type17,18 with resonancefrequency ωge between its ground state |g and excited state |e.
It is coupled to an additional resonator R which is nonlinear
and used to read-out the qubit state. As explained in detailin Ref. 19, this read-out is performed in a single shot bymeasuring the phase of a microwave pulse reﬂected on R. This
phase takes two different values depending on the qubit state,
and repeating ∼ 104 times the same experimental sequence
yields the qubit excited state probability Pe .

ω+I
0
±1

FIG. 2. (Color online) (a) Sketch of the implementation of qubitbased ESR. The spin ensemble S consists of nitrogen-vacancy (NV)centers in a diamond crystal. They are coupled to the frequencytunable coplanar waveguide resonator C used as the ESR cavity. C
also embeds the ESR-detector qubit Q, a superconducting qubit of thetransmon type whose state can be read-out with another resonator R.
Microwave pulses for spin spectroscopy as well as for qubit read-out
are sent via an input port coupled both to C and to R. (b) and (c)Sketch and energy levels of NV centers in diamond. In our setup a DC
magnetic ﬁeld BNV is applied along the [1,1,1] direction, resulting indifferent Zeeman splittings for centers having the N -V axis parallelto BNV (ensemble I, in red) and those having their axis along thethree other 1,1,1 axes (ensemble III, in blue). The ESR frequencies
ω±I,I I I are further split in three resonance lines due to the hyperﬁneinteraction with the spin-1 14 N nuclear spin.11

III. SINGLE PHOTON STORAGE AND RETRIEVALAs reported in an earlier work9 (see also Ref. 21), it ispossible with this circuit to coherently exchange a single
quantum of excitation between the spin ensemble S andthe qubit Q via the tunable cavity C. To demonstrate that,the qubit is prepared in |e; its excitation is transferred to thecavity by sweeping adiabatically ωc () through ωge , which isthen tuned suddenly in resonance with the spins at ωK (whereK = +I, + I I I ) for some interaction time τ . The excitation
left in the cavity is ﬁnally transferred back into the qubit,
which is then read-out. As shown in Fig. 3, the resulting
qubit excited state probability Pe (τ ) is found to oscillate,revealing the conversion of a single microwave photon into
an elementary collective excitation of the spin ensemble. For
well-deﬁned interaction times τs,K (see Fig. 3), the excitationin the qubit is swapped into the spin ensemble;9 at a later time
it is recovered in the qubit with a ﬁdelity ∼ 0.1.
A quantum-mechanical description of this experiment22,23
is useful in the discussion of the ESR results presented
below. Each of the NK effective spins of the ensemble atωK is modelled as an effective harmonic oscillator with†frequency ωjK and annihilation (creation) operator bjK (bjK ),an approximation valid in the low-excitation limit as is the casethroughout this article. The spin ensemble and cavity are then†
described by Hamiltoniansh̄ωjK bjK bjK and h̄ωc ()a † a,a (a † ) being the cavity annihilation (creation) operator. The
coupling between the resonator and the spin ensemble is†
described by a Hamiltonian HK = −ih̄ gjK (bjK a + H.c.),064514-2

ELECTRON SPIN RESONANCE DETECTED BY A . . .

PHYSICAL REVIEW B 86, 064514 (2012)0.3

τs,III

Q
C

τ

S

0

100
200
300
Interaction time τ (ns)

2

3
R
aSWAP

(b)

+III
0

τs,I
0

ωp

τS

C
Q

0.1

ω+I

0.1

1

0.2

aSWAP

ω+III

0.2

(a)

R

400

FIG. 3. (Color online) Single photon transfer between qubit andspins (Ref. 9). (inset) Pulse sequence used to excite the qubit in state|e and transfer its excitation to S via C back and forth. (panel) Swap
oscillations for the two spin frequencies ω+I and ω+III . Swap times
τs,I and τs,III are indicated by arrows. Note that the two curves are
shifted vertically for clarity with corresponding scales on the left (forensemble I) and the right axes (ensemble III).

with gjK the coupling constant of spin jK to the resonator. This†
Hamiltoniancan be rewritten as HK = −ih̄gK (bK a + H.c.)2 1/2
with gK = ( gjK ) the spin ensemble resonator collectivecoupling constant and bK = (1/gK ) gjK bjK the annihilation
operator of the collective spin excitation coupled to the cavity.
This super-radiant mode has a spatial proﬁle given by thespatial dependence of the coefﬁcients gjK , which reproduces
the proﬁle of the magnetic ﬁeld inside the cavity mode. Notealso that bK involves all the spins belonging to group K,
even if they have different frequencies due to slightly different
magnetic environment in the crystal, also including the threepossible states of the 14 N nuclear spin causing the hyperﬁne
structure; as a result this mode is coupled to NK − 1 darkmodes that act as a bath.16,22–24 Using these notations, one
describes the oscillations shown in Fig. 3 as occurring between
states |1c ,0K  and |0c ,1K , where |1c  = a † |0c  is the usual†
Fock state with one photon in the cavity, and |1K  = bK |0K 
is the ﬁrst excited state of the super-radiant mode; dampingof these oscillations is due to inhomogeneous broadening, andcan be interpreted as damping of state |1K  into the bath ofdark states.16
IV. ELECTRON SPIN RESONANCE
PROTOCOL AND DISCUSSION

The ESR protocol is shown in Fig. 4. It consists of anexperimental sequence similar to the one used for single
photon storage (see Fig. 3), but with the spin ensemble initially
excited at several photons level instead of the qubit preparedin its excited state. More precisely, a low-power microwave
pulse of duration t = 2 μs and varying frequency ωp isapplied to the spins while ωc () is far detuned; the resulting
excitation is transferred ﬁrst into the cavity by tuning ωc ()suddenly in resonance with ωK for the swap time τs,K , theninto the qubit by an adiabatic swap interaction; the qubit state is
ﬁnally measured. Provided the average number of microwave
photons emitted by the spins into the ESR cavity stays much+I

Density ρ(ω) (a.u.)

Probability Pe

0.3

πProbability Pe

S

0.2

0.1

0

2.88

2.89
2.90
2.91
Frequency ω /2π (GHz)2.92

FIG. 4. (Color online) (a) Experimental pulse sequence used for
qubit-detected ESR: (1) The spins are ﬁrst weakly excited by a 2 μs
microwave pulse with a frequency ωp ; (2) the resulting spin excitationis transferred to the cavity C by a fast ﬂux pulse, which brings ωc inresonance with ωK (K = +I, + I I I ) for a swap time τs,K , and thento the qubit Q by an adiabatic swap (aSWAP). (3) The qubit excitedstate probability Pe is ﬁnally measured. (b) Measured (open circles)
and calculated (solid line) Pe (ωp ) for spin ensemble +III (left) and +I(right). The spin density ρ(ω) used in the calculation is shown as adashed line.

lower than 1 to avoid saturating the qubit, the resulting
excited state probability Pe (ωp ) is expected to reproduce thespin ensemble absorption spectrum. Experimental results ofFig. 4(b) indeed display the characteristic HF structure of NV
centers consisting in three peaks separated by 2.2 MHz for bothspin ensembles + I and + III. This validates the concept ofelectron spin resonance detected by a superconducting qubit.
Note that Pe (ωp )  1 showing that qubit saturation is avoided
as wanted.
We now discuss the sensitivity of this qubit-based ESR
spectrometer. The qubit state is detected in a single shot witha ﬁdelity of  0.7 at the end of an experimental sequencethat lasts typically 50 μs, yielding a 1% imprecision on theprobability Pe in one second. To translate this sensitivity in amagnetic moment unit, one needs to know with what efﬁciency
the excitation of the spin ensemble is actually transferred tothe qubit. The transfer of one microwave photon from thecavity to the qubit is performed with an efﬁciency of orderunity (in our experiment it is around 0.7 limited by losses in
the cavity and qubit), so the limiting factor is the efﬁciencyof the transfer of the spin ensemble excitation to the cavity
during their resonant interaction. At ﬁrst sight one might thinkthat since the spin ensemble and cavity are in the strong
coupling limit, one excitation of the spin ensemble should also
be converted into a microwave photon with an efﬁciency oforder 1, similar to what happened in the coherent oscillations
shown in Fig. 3. This reasoning is not correct here because
the collective spin mode bωp excited by the spectroscopy pulse
does not necessarily match perfectly the super-radiant modebK . Indeed, although the spatial matching of the two modes
is excellent since the spectroscopy pulse is applied through064514-3

Y. KUBO et al.

PHYSICAL REVIEW B 86, 064514 (2012)

the cavity, this is not the case for spectral matching: only
spins having a resonance frequency within the spectroscopy
pulse bandwidth δ/2π = 150 kHz around ωp contribute tobωp , whereas all spins within the hyperﬁne line (total width
/2π ∼ 5 MHz) contribute to the super-radiantmode bK . As√
a result, one expects an overlap of order δ/ between thebωp and bK modes, implying that there should be ∼ /δ =20 times less excitations in the bK mode (and thus also inthe cavity mode after the swap interaction) than in the bωp
mode.
This argument can be made rigorous and quantitative.
The collective spin mode excited by the√spectroscopy pulse
 2

is deﬁned as bωp = [ gjK αωp (ωjK )bjK ]/gjK |αωp (ωjK )|2 ,with αωp (ω) the pulse Fourier transform. The quantity of interest is then the correlation function β(ωp ,τs ) = a(τs )bω† p (0)giving the probability amplitude for an excitation created in bωp
by the spectroscopy pulse to be transferred into a photon inside
the cavity mode after an interaction time τs . This function canbe computed numerically given a certain spin distribution ρ(ω)
using the formulas derived in Appendix A. In our experiment,
the line width of each hyperﬁne peak w+I /2π = 1.6 MHz
and w+I I I /2π = 2.4 MHz, and coupling constants g+I /2π =2.9 MHz and g+I I I /2π = 3.8 MHz have been determined fromother measurements,9 so that a direct comparison with theory
without any adjustable parameter is possible as shown in Fig. 4.
The agreement is quantitative [note that we have also includedin the distributions ρ(ω) additional ESR frequencies caused by
the hyperﬁne interaction of the NV center with neighboring
13
C nuclei with the 1.1% natural abundance as expected]. From
this calculation, we deduce that the average excitation of thespin ensemble at resonance in the data shown in Fig. 4 is ∼ 15,
in agreement with the qualitative argument presented above.
In the present state of the experiment, the qubit-based ESR
spectrometer therefore measures the spectrum of an ensembleof 1011 NV centers at an excitation level of order 15 μB , in a
total integration time of one minute.
Thanks to this very low excitation level, the experimental
sequence can be safely repeated at 20 kHz, despite the NV
centers energy relaxation time reaching minutes at 30 mK.3,20
More precisely, two factors contribute to make this experimentpossible: i) at the end of each experimental sequence theexcitation of the bωp mode quickly decays into the bath of darkmodes, allowing the next experimental sequence to start withbωp in its ground state and thus keeping the average number of
excitations transferred to the qubit well below 1 as needed to
avoid saturation, ii) the low excitation rate ensures on the other
hand that the ensemble of 1011 spins stays far from saturationeven after repeating the sequence for hours. The upper limit ofpossible repetition rate is determined by the qubit relaxation
time, 2 μs in our experiment. Although the experiment was
indeed performed at 20 kHz, it could probably have been
pushed to 100 kHz, resulting in a sensitivity increased by afactor ∼ 2.
We ﬁnally note that our calculation reproduces a puzzlingfeature of the data that was not discussed yet: the middle
peak of the Pe (ωp ) curve has a lower amplitude than thetwo other peaks, both for the + I and the + III curves as
seen in Fig. 4, although the spin density ρ(ω) used in thecalculation is a simple sum of three Lorentzians with the same
amplitude. Our ESR protocol thus appears to slightly distort

the absorption spectrum. This phenomenon originates fromthe ωp dependence of the energy transfer efﬁciency from thespins into the cavity, caused by the fact that gK ≈  in our
sample. It could probably be corrected in future experiments
either by increasing gK or by transferring the spin excitation
to the cavity with an adiabatic passage.
Besides detecting a large ensemble of N = 1011 electronic
spins at near single-excitation level, it is interesting to discuss
what is the minimal number of spins Nmin that could be
detected with a similar experimental protocol in order tocompare it to the sensitivity of existing conventional spectrometers. For that we will change perspective in the following
discussion, and assume that the spins being measured canactually be excited at saturation. We suppose that the N
spins, of inhomogeneous line width , have been
√ excited bya hard π/2 pulse. In the weak coupling limit g N  κ  (κ = ωc /Q being the cavity damping rate), the spins emit in
the cavity n̄ = g 2 N 2 /(42 ) photons (see Appendix B). Taking
a conservative estimate for the minimal average excitation thatcan be detected by a superconducting qubit within one second
to be 0.05, a spin-cavity coupling√ constant g/2π = 10 Hz,
one obtains Nmin = 105 spins/ Hz for a 0.1 mT line widthcorresponding to /2π = 2.8 MHz. This ﬁgure is ﬁve ordersof magnitude better than a commercial spectrometer at 300 K,
and two orders√of magnitude better than the record sensitivityof 106 spins/ Hz for a 0.01 mT line width that was recently
reported with a surface loop-gap resonator operated at 10 K26
and a coplanar waveguide resonator at 4 K.5 Note however thatin order to operate such a qubit-based spectrometer in practice,
one would need i) to use a repetition rate around 10 kHz,
which requires a reasonably short spin-lattice relaxation timeor some way to repump rapidly the spins into their ground state(optically25 or electrically, for instance) and ii) to ﬁnd a qubit
design that withstands large magnetic ﬁelds usually needed for
ESR. Interesting alternative possibilities could be to physicallyseparate the qubit-detector from the spins, which would allow
more easily the application of large magnetic ﬁelds to the spins
without perturbing the qubit, or be to continuously monitor
the qubit state with a parametric ampliﬁer as demonstrated
in recent experiments27 instead of pulsing the qubit state
detection as done here.
In conclusion we have discussed an ESR spectrometer inwhich the signal coming from the spins is detected by asuperconducting qubit acting as a single-microwave-photon
detector. We have implemented this idea on an ensemble of∼ 1011 NV centers coupled to a transmon qubit, measuring
their absorption spectrum at an excitation level of ∼ 15 μB ,with a well-resolved hyperﬁne structure. Estimates indicate√that this spectrometer would be able to detect 105 spins/ Hz
with a 0.1 mT line width, a gain of two orders of magnitude in sensitivity compared to the best reported values
for conventional spectrometers. Our work thus demonstrates
the potential of superconducting circuits for electron spinresonance spectroscopy.

ACKNOWLEDGMENTS

We acknowledge useful discussions with B. Julsgaard,
K. Moelmer, J. J. L. Morton, G. A. Briggs, and within the064514-4

ELECTRON SPIN RESONANCE DETECTED BY A . . .

PHYSICAL REVIEW B 86, 064514 (2012)

Quantronics group, and technical support from P. Sénat, P.-F.
Orﬁla, T. David, J.-C. Tack, P. Pari, P. Forget, and M. de
Combarieu. We acknowledge support from European project
Solid, ANR projects Masquelspec and QINVC, C’Nano,
Capes, and Fondation Nanosciences de Grenoble.

=k αωp (ωk )gk2



gk t1 (−iω)
iγ0 + (ω − ωp )

(A4)αωp (ωj ) gj22
t1 (−iω)k αωp (ωk )gk=,
iγ0 + (ω − ωp )  α (ω ) 2 g 2
j

j

ωp

j

j

APPENDIX A

We now explain in more detail how the theory curves
in Fig. 4 are calculated. As explained in the main text, theenergy transfer efﬁciency from the resonator to the qubit
is of order unity and can be well modeled by an ideal
adiabatic passage. In this context the quantity of interest is theresonator population after the interaction with the spins. The
calculations are performed in the Holstein-Primakoff approximation, in which the spins and the resonator are describedby harmonic oscillators. The total system Hamiltonian is††
H /h̄ = ωc ()a † a + ωj bj bj + igj (bj a − bj a † ), gj being
the coupling constant of spin j with the resonator. We needto calculate the probability that the excitation created at t = 0
in the spins to be transferred to the cavity after a time t,
this probability is the square modulus of 0|a(t)bω† p |0. The
spins excitation is created by a microwave pulse of central
frequency ωp with a pulse envelope in frequency describedby αωp (ω) = α(ω − ωp ), a typical envelope is a Lorentzianfunction with FWHM δ. We can deﬁne an operator bω† p that
describes the excitation induced by this pulse as1†αωp (ωk )gk bk , (A1)bω† p = 2 2k|αωp (ωj )| gj
jthis comes simply from the standard atom-ﬁeld interaction
for a classical light source such as the one used in theexperiment. As shown in Ref. 22 the quantity 0|a(t)bω† p |0
can be calculated by considering an effective non-Hermitian
Hamiltonian⎞⎛ig1 ig2 . . .
ω̃0⎟⎜ −ig1 ω̃1⎟.
(A2)
Heff /h̄ = ⎜ω̃2⎠⎝ −ig2..
..
.
.
with complex angular frequencies ω̃0 = ωc () − iκ/2 andω̃k = ωk − iγ0 /2; here, γ0 is the spontaneous emission
rate of each spin (that we take here to be zero since
NV centers at low temperature have negligible energy relaxation). Indeed, introducing the vector X(t) of coordinates [a(t)a † (0), . . . ,bj (t)a † (0), . . .] it can be shown thatdX/dt = −(i/h̄)Heff X. The formal solution to this equationis thenX(t) = L−1 [(s + iHeff /h̄)−1 X(0)] ,where 2 t1 (−iω) = i/[ω − ω0 + iκ/2 − W (ω)] with W (ω) =j gj /[ω − ωj + iγ0 /2]. Note that we evaluated tωp (s) fors = −iω, this is sufﬁcient to perform the Laplace transforminversion as there are no singularities in the imaginary axisof tωp . We deﬁne the spin density ρ(ω) encompassing thecoupling strength, which is possibly different for each spin, g2
as ρ(ω) = j g2j δ(ω − ωj ). Using this deﬁnition in theKequation above we havetω p =(A5)

The spectral width of the microwave pulse is, in our case,much smaller than any scale that characterizes our distribution
ρ(ω). This allows the rewriting of the convolution above as(α ∗ ρ)(ωp )= A ρ(ωp ) ,2(|α| ∗ ρ)(ωp )where the constant A = √α(ω)dω
|α(ω)|2 dω(A6)

is purely characterized

by the pulse envelope√ √ with no dependence on ωp , yielding forexample A = δ π/2 for a Lorentzian envelope. This meansthat if we consider that the spins are distributed at a typicalrange  the equation above gives a rigorous justiﬁcation of therule of thumb that says that the efﬁciencyof the spin-resonator√transfer is given by the overlap δ/.
Finally to generate the theoretical curve in Fig. 4, we
perform a numerical inversion of the Laplace transform foreach ωp and take |0|a(t)bω† p |0|2 at t = τS,I I I or t = τS,I .
APPENDIX B

We now make explicit the calculation of the sensitivityof our qubit-based ESR spectrometer in the weak coupling
limit gK  κ  . The Hamiltonian coupling the spins tothe cavity ﬁeld is H = h̄g(S− a † + H.c.), where S− = σi,− ,σi,− being the lowering operator of spin i. In the absence ofdriving ﬁeld, the equation for the intracavity mean ﬁeld is theneasily obtained asκda= − a − igS− .
dt
2

(A3)

which gives 0|a(t)bω† p |0 = xG † · X(t) = L−1 [tωp (s)] with
xG = (1,0,0, . . .) and L[f (s)] = e−st f (t)dt (s being a complex number). The initial condition X(0) is the one producedby bω† p given in Eq. (1), thusαωp (ωk )gk[(s + iHeff )−1 ]0,ktωp (−iω) =  k
2 2
j αωp (ωj ) gj

(α ∗ ρ)(ωp )gK t1 (−iω)
.
iγ0 + (ω − ωp ) (|α|2 ∗ ρ)(ωp )Right after a π/2 pulse on the spins, |S− | = N/2.
Neglecting the back-action of the cavity ﬁeld on the spins(which is justiﬁed in the weak coupling limit), we simply getthat S−  = (N/2)e−t . From that one shows that a(t) =−igN (e−κt/2 − e−t )/[κ − 2], which in the limit κ  yields a maximum photon number in the cavity of n̄ =g 2 N 2 /(42 ).

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