Detecting spins with a microwave photon counter
Emanuele Albertinale, Léo Balembois, Eric Billaud, Vishal Ranjan, Daniel Flanigan, Thomas Schenkel, Daniel Estève, Denis Vion, Patrice Bertet, Emmanuel Flurin
DDetecting spins with a microwave photon counter
Emanuele Albertinale , Léo Balembois , Eric Billaud , Vishal Ranjan , Daniel Flanigan ,Thomas Schenkel , Daniel Estève , Denis Vion , Patrice Bertet , Emmanuel Flurin ∗ Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette Cedex, France National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK Accelerator Technology and Applied Physics Division,Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: February 3, 2021)Quantum emitters respond to resonant illumination by radiating electromagnetic fields. A com-ponent of these fields is phase-coherent with the driving tone, while another one is incoherent, con-sisting of spontaneously emitted photons and forming the fluorescence signal. Atoms and moleculesare routinely detected by their fluorescence at optical frequencies, with important applications inquantum technology [1, 2] and microscopy [3–6]. Spins, on the other hand, are usually detectedby their coherent response at radio- or microwave frequencies, either in continuous-wave or pulsedmagnetic resonance [7]. Indeed, fluorescence detection of spins is hampered by their low sponta-neous emission rate and by the lack of single-photon detectors in this frequency range. Here, usingsuperconducting quantum devices, we demonstrate the detection of a small ensemble of donor spinsin silicon by their fluorescence at microwave frequency and millikelvin temperatures. We enhancethe spin radiative decay rate by coupling them to a high-quality-factor and small-mode-volumesuperconducting resonator [8], and we connect the device output to a newly-developed microwavesingle photon counter [9] based on a superconducting qubit. We discuss the potential of fluorescencedetection as a novel method for magnetic resonance spectroscopy of small numbers of spins.
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Microwave measurements at the quantum limit haverecently become possible thanks to the development ofsuperconducting parametric amplifiers that linearly am-plify a signal at cryogenic temperatures with minimaladded noise [10]. These advances enable efficient mea-surement of the field quadratures X and Y of a givenmicrowave mode, as needed for qubit readout in circuitquantum electrodynamics [11]. Even then, the signal-to-noise ratio remains ultimately limited by vacuum fluc-tuations enforced by Heisenberg uncertainty relations,imposing that the quadrature standard deviations sat-isfy δX = δY = 1 / . As a result, a linear amplifier isnot well suited for detecting fluorescence signals consist-ing of a few incoherent photons emitted randomly overmany modes. In contrast, such signals are ideally de-tected by a photon counter: because it measures in theenergy eigenbasis, it is in principle noiseless when thefield is in vacuum and only clicks for photons incomingwithin its detection bandwidth [12].Operational Single Microwave Photon Detectors (SM-PDs) have been developed only recently based on cavityand circuit quantum electrodynamics [9, 13–18]. Here,we report the first use of such a SMPD for sensing appli-cations, to detect spin fluorescence at microwave frequen-cies. Consider an ensemble of N electron spins 1/2, res-onantly coupled with a spin-photon coupling constant g to a resonator of frequency ω and linewidth κ (Fig. 1a).After being excited by a π pulse, the spins will relax ex-ponentially into their ground state with a characteristictime T . If their dominant relaxation channel is radiative(the so-called Purcell regime [8]), they will do so by spon- taneously emitting N microwave photons at the Purcellrate Γ P = 4 g /κ = T − . Whereas a linear amplifier canonly detect these incoherent photons as a slight increaseof noise above the background [19, 20], a single-photoncounter with sufficient bandwidth is expected to detecteach of them as a click occurring at a random time andrevealing an individual spin-flip event.For our demonstration, we use the electronic spins ofan ensemble of bismuth donors implanted about
100 nm below the surface of a silicon chip enriched in the nuclear-spin-free silicon 28 isotope. These spins couple mag-netically to the inductor of a superconducting LC res-onator with frequency ω / π = 6 .
94 GHz patterned inaluminium on the surface of the chip (see Fig. 1b). Ap-plying a static magnetic field B ≈
17 mT parallel to theinductor tunes the lowest transition frequency of the bis-muth donors in resonance with the resonator. At thisfield, the energy loss rate κ i in the resonator is 3.5 timeshigher than the energy leak rate κ c in the measuring line,yielding a total resonator bandwidth κ/ π = 0 .
68 MHz with κ = κ i + κ c . As the electron spin of donors in siliconhave hour-long spin-lattice relaxation times at low tem-peratures [21], they easily reach the Purcell regime whencoupled to micron-scale superconducting resonators [8].For our sample parameters, we measure a spin relaxationtime T = 300 ± ms (see Sec. 3 in Methods) domi-nated by the radiative contribution. In our experiment[see Fig.1(a)], the resonator coupled to the spins has asingle input-output port connected through a circulator(and coaxial cables) to both the line used to drive thespins and to a SMPD. a r X i v : . [ qu a n t - ph ] F e b B (mT) ω / π ( G H z ) Si Bie - η d = 0.66 α = 0.6 ms -1 ge . m m B SMPD
Buffer WastePump
R D M a)b) c) φ φ/φ ω b + ω p = ω q + ω w ω b ( φ ) ~ ω ω ω b / π ( G H z ) π ω g ω κ c η d = 0.53 α = 1.5 ms -1 κ i Control and readout
Figure 1:
Principle of spin detection with a photon counter. ( a ) Schematics of the experiment. Each spin in theensemble is coupled with a strength g to a resonator of angular frequency ω and internal loss rate κ i , itself coupled with arate κ c to an input-output line. This line allows to drive the spins with microwave pulses through a circulator and to route thephotons emitted by the spins while relaxing radiatively towards a Single Microwave Photon Detector (SMPD). ( b ) Spin deviceschematics. The spins are bismuth donors implanted in a silicon substrate isotopically enriched in the Si isotope, on top ofwhich a superconducting LC resonator is patterned. A magnetic field B is applied parallel to the substrate to tune the lowestbismuth donor frequency in resonance with ω at B ∼ mT. ( c ) SMPD device and operation schematics. The SMPD relieson a transmon qubit at frequency ω q / π = 6 .
13 GHz , coupled to three ports, namely buffer, pump, and waste. The buffer portconsists of a resonator at frequency ω b ( φ ) that can be tuned to ω by applying a flux φ to a SQUID loop inserted in it (seeinset). The waste port consists of a resonator at a fixed frequency ω w / π = 7 .
63 GHz . Incoming photons at the buffer areconverted into excitation of the transmon qubit and into a photon in the waste resonator by a four-wave mixing process enabledby a pump tone sent via the pump port at frequency ω p = ω q + ω w − ω b . The SMPD operation consists of a three-step cycleof total duration . µ s which can be repeated continuously. The Reset (R) step is achieved by driving the waste port andthe pump. The Detection step (D) is achieved by switching off the waste port, and keeping the pump on. The Measurementstep (M) is performed by switching off the pump, and sending a microwave pulse on the buffer port to perform dispersive qubitreadout. The SMPD efficiency η d and dark count rate α are indicated in the inset for relevant values of φ in the inset. This SMPD (see Fig. 1c) consists of a superconductingcircuit with a transmon qubit [22] of frequency ω q capac-itively coupled to two coplanar waveguide resonators: a’buffer’ resonator whose frequency ω b can be tuned to ω by applying a magnetic flux to an embedded supercon-ducting quantum interference device (SQUID) [23], and a’waste’ resonator with fixed frequency ω w . As describedin Ref. [9], the detection of a photon (here at frequency ω b ) relies on the irreversible excitation of the transmonwhen driven by a non-resonant pump tone at frequency ω p = ω q + ω w − ω b . The SMPD is cycled continuously,each cycle consisting of three steps (see Fig. 1c). First,a reset step ( R ), during which the qubit is set to itsground state by turning on the pump (violet pulse) whileapplying to the waste resonator a weak resonant coher-ent tone (green pulse). Second, a detection step ( D ) thatstarts when the microwave at ω w is switched off, while thepump is kept on: a photon possibly entering the buffergets mixed with the pump through a four-wave mixingprocess that triggers both the excitation of the transmonand the creation of a photon in the waste; this photon islost in the
50 Ω port of the waste, which guarantees theirreversibility of the detection and the mapping of theincoming photon into a transmon excitation. The thirdstep ( M ) is the measurement of the transmon state us-ing the dispersive shift [24] of the buffer resonator (orangepulse). At ω b = ω , the probability to detect a click whenone photon reaches the detector during the detection win-dow D (intrinsic photon detector efficiency) is measuredto be η d = 0 . ± . (mainly limited by the transmonenergy relaxation, see Methods), and the rate of falsepositive detection, referred to as the the dark count rate,is α = 1 .
53 clicks / ms . The detection duty cycle (step D duration over total cycle duration) is η duty = 0 . , witha complete cycle lasting . µ s . Note that the detec-tor evidently saturates for signals having more than 1photon every µ s , approximately. The detector band-width ∆ ω/ π ≈ is larger than the spin resonatorlinewidth κ/ π , implying that there is no filtering of thephotons emitted by the spins.As a first experiment, we measure the spontaneousemission of the spin ensemble: a π -pulse inverts the spinpopulation, so that (cid:104) S z (cid:105) = N/ , and (cid:104) S x (cid:105) = (cid:104) S y (cid:105) =0 , S x,y,z = (cid:80) Ni =1 S ( i ) x,y,z being the sum of the individ-ual dimensionless spin operators S ( i ) (Fig. 2a). Since (cid:104) S x,y (cid:105) = 0 , the coherent part of the output field also sat-isfies (cid:104) X (cid:105) = (cid:104) Y (cid:105) = 0 . On the other hand, by energyconservation, a flux of incoherent photons is emitted at arate (cid:104) X + Y (cid:105)− / proportional to − ∂ t (cid:104) S z (cid:105) (see Fig. 2a),forming the spin fluorescence signal and triggering countsin the SMPD. This signal decays back to within the spinrelaxation time T . Note that the π pulse perturbs theSMPD during a dead-time of ∼ µ s , after which it canbe used normally.One measurement record consists of × consecu-tive SMPD detection cycles, spanning a total measure-ment time of . Each cycle yields one binary outcome c ( t i ) , t i being the time around which cycle i is centered.Figure 2a displays an example of a measurement recordat early ( to . ) and at late times ( to . ): dueto the emission by the spins, more counts are observedin the to . interval than in to . . Repeatingthe measurement 500 times and histogramming the num-ber of counts, we obtain the average count rate (cid:104) ˙ c ( t d ) (cid:105) as a function of the delay t d after the π pulse. Figure2b shows this rate with and without π -pulse applied.Without pulse, a constant (cid:104) ˙ c ( t d ) (cid:105) = 1 .
53 counts / ms isrecorded, which corresponds to the dark count rate α of the SMPD. With π -pulse, (cid:104) ˙ c ( t d ) (cid:105) shows an excess of .
85 counts / ms exponentially decaying towards , with afitted time constant of
309 ms . Because this time is the p ( C )
900 10502.4 c ( c oun t s / m s ) No pulse a) b)c) C li cks d (s)0.1 1.11 A p π π pulse Integration window C XS z S x t d (s)C spins δ C X + Y - Figure 2:
Detection of spin relaxation by photon count-ing. ( a ) Pulse sequence and spin dynamics. The appliedpulse amplitude A p , transverse and longitudinal magnetiza-tions (cid:104) S x (cid:105) and (cid:104) S z (cid:105) , output field quadrature (cid:104) X (cid:105) , and photoncount rate (cid:104) ˙ c (cid:105) are shown as a function of the delay t d after the π pulse. Solid curves are sketches of the expected dynamics.After a transient during the pulse, (cid:104) S x (cid:105) quickly goes to be-cause of the spin ensemble inhomogeneous broadening; (cid:104) S z (cid:105) on the other hand is inverted and relaxes towards equilibriumin a characteristic time T = 0 . s. During this time, (cid:104) X (cid:105) isalso , whereas a flux of spontaneous photons (the spin fluo-rescence) (cid:104) X + Y (cid:105) − / is emitted. The lower panel showsone typical experimental time trace of the detected clicks as afunction of t d . ( b ) Average count rate (cid:104) ˙ c (cid:105) ( t d ) measured in 19ms time bins in the case where a π pulse is (magenta) or is not(blue) applied to the spins. An exponential fit for t d > . (solid line) leads to the characteristic time T = 309 ms . Theobserved excess rate at short times t d <
50 ms (inset) hasbeen investigated and attributed to fast-relaxing two-levels-systems. ( c ) Measured probability distribution of the numberof counts C integrated from . to
585 ms , obtained for500 repetitions of the experiment when a π pulse is eitherapplied (magenta) or not (blue) to the spins. Solid lines rep-resent Poissonian fits. same as the independently measured spin relaxation time T (see Methods), we conclude that the SMPD detectsthe photons spontaneously emitted by the spins upon re-laxation. Note that at short t d we moreover observe anextra excess rate of . / ms decaying with a
20 ms time constant (see inset of Fig. 2b), which we attributeto the radiative relaxation of spurious two-level systemspresent at sample interfaces.To analyse the photo-counting statistics of the fluo-rescence signal, we integrate the number of counts C = (cid:80) i c ( t i ) over a window of duration t w = 540 ms. Theprobability histogram p ( C ) is shown in Fig. 2c. With a) t (ms) 1 X + Y -
10 t i = τ t i (ms) b)c) C li cks Blindtime Blindtime p ( c ) τ τ A p XS z S x τ < t i <1 ms c c c e Figure 3:
Detection of spin echo by photon counting. ( a ) Pulse sequence and spin dynamics. A Hahn echo pulse se-quence (solid orange line) consisting of a π/ pulse, a delay τ ,and a π pulse is applied to the spin ensemble, causing a revivalof the spin transverse magnetisation (cid:104) S x (cid:105) at time τ , and theemission of a coherent microwave echo on the X quadrature.After the π/ pulse, the spin population (cid:104) S z (cid:105) slowly decaysby spontaneous emission with a characteristic time T , within addition a small drop during the echo. Being coherent, theecho can be detected by homodyne detection or by a SMPD.Solid curves are sketches of the expected dynamics. An ex-ample of an experimental sequence is shown in the bottom,with a click detected at the echo time. ( b ) Average number ofcounts (cid:104) c (cid:105) in µ s bins, as a function of the time t from thebeginning of the echo sequence, averaged over 83 sequences.Blue-shadowed areas represent the µ s blind time of thedetector after each strong spin pulse. The increased countprobability at t i = 2 τ is the spin-echo. ( c ) Average probabil-ity p ( c ) of having one or no count in the time bin centred atecho time t i = 2 τ (orange) or in one of the subsequent bins τ < t i < ms (blue). The difference between click probabili-ties (dashed lines) lead to a signal c e of 0.3 photons re-emittedcoherently. and without π pulse, an average of (cid:104) C ( π ) (cid:105) = 1050 and (cid:104) C (0) (cid:105) = 900 counts are detected, the difference defin-ing the spin signal C spin = 150 photons. The ratio ofthis signal to the total number of excited spins N de-fines an overall detection efficiency η = C spin /N . For aPoissonian distribution, one expects the width of p ( C ) to be δC = √ αt w and δC = (cid:112) αt w + η (1 − η ) N with-out and with π pulse, respectively. In our case, because αt w (cid:29) ηN , both distributions have approximately thesame width δC (cid:39) counts, dominated by the darkcounts fluctuations contribution.It is interesting to note that the signal-to-noise ratio ηN/ (cid:112) αt w + η (1 − η ) N can in principle become arbitrar- ily large for an ideal SPD for which α ∼ and η ∼ ,even for N approaching . This reflects the fact thatin the Purcell regime, N spins once excited will emit N photons over a timescale of a few T , and that anideal SMPD will detect them all noiselessly. This is inmarked difference with previous experiments using su-perconducting qubits for ESR spectroscopy, which reliedon either Free-Induction-Decay collection by a tunableresonator [25] or measurement of the spin ensemble mag-netisation by a flux-qubit [26, 27]. SMPD detection ofspin fluorescence in the Purcell regime thus appears as aparticularly promising method for detecting small num-bers of spins. In our experiment, the SNR is equal to . , already exceeding the SNR of echo-based detectionas discussed in the following, despite the imperfections ofthe present SMPD.Using additional spin measurements by homodyne de-tection [7] combined with a numerical simulation of theexperiment, we estimate that N = (13 . ± . × spins are excited by the π pulse (Sup. Mat.). The over-all detection efficiency is thus η = 0 . ± . . Writingthis efficiency as η = η d η duty η int η col , with η int = 0 . afactor due to the finite integration window, we deducea collection efficiency η col = 0 . ± . between thespins and the detector. This is due in part to the spinresonator internal losses which contribute for a factor κ c / ( κ i + κ c ) = 0 . , and in part to losses in the microwavecircuitry joining the two devices.We now turn to another method of spin detection bya SMPD, during the emission of a spin echo at the endof a Hahn echo sequence π/ − τ − π − τ − echo (see Fig.3a). After the first π/ pulse, which brings all the spinsalong the x axis at t = 0 , spins lose phase coherence ina time ∼ T E due to the spread of their Larmor frequen-cies, so that (cid:104) S x ( t ) (cid:105) ∼ ( N/ − t/T E . In our experiment, T E ∼ κ − because the spin excitation bandwidth is setby the cavity and not by the much larger spin ensem-ble inhomogeneous linewidth (Fig. 4). Phase coherenceis transiently restored around t = 2 τ by the refocusing π pulse, yielding (cid:104) S x ( t ) (cid:105) ∼ ( N/ − | t − τ | /T E . The oscil-lating transverse magnetization generates a short phase-coherent microwave pulse of duration T E in the detec-tion line, called the spin echo, with the photon statis-tics of a coherent state. In the limit N Γ P T E (cid:28) , itsamplitude can be shown to be (cid:104) X e (cid:105) ∼ N (cid:112) η col Γ P T E / ,corresponding to an average photon number (cid:104) X e (cid:105) [28]much smaller than the number N of spins. Spin echoesare usually detected by linear amplification and phase-coherent demodulation [7, 28], with a signal-to-noise ra-tio (cid:104) X e (cid:105) /δX = 2 (cid:104) X e (cid:105) ultimately limited by the vacuumfluctuations [28]. Here, we show that spin echoes canalso be detected by a microwave SMPD, as photon echoesat optical frequencies [29]. Note that the signal-to-noiseratio upper-bound that an ideal SMPD could reach islimited by photon shot noise during the echo and equalto (cid:104) X e (cid:105) / (cid:112) (cid:104) X e (cid:105) = (cid:104) X e (cid:105) , i.e. half the one of phase-coherent detection.In our demonstration of microwave photon echo de-tection, the echo duration is shorter than the detectioncycle, and only one photon at most can be detected atthe echo time τ ; we thus center the detection step D ofthe SMPD at τ . We also chose τ = 350 µ s , larger thanthe detector dead time. A typical photo-counting trace isvisible in Fig. 3a, showing in particular one click at theexpected echo time. Repeating several echo sequencesyields (cid:104) c ( t i ) (cid:105) (see Fig. 3b), clearly showing an excess ofcounts for t i = 2 τ .The click probability histogram is shown in Fig. 3 atand out of the echo time. The average number of detectedphotons during the spin-echo, c echo = (cid:104) c (2 τ ) (cid:105) − (cid:104) c ( t i > τ ) (cid:105) = 0 . , is as expected much lower than C spin , thenumber of photons detected in the spontaneous emissionexperiment of Fig. 2. The standard deviation δc echo =0 . during the echo (Fig. 3) yields a signal-to-noise ratio c echo /δc echo = 0 . , significantly lower than the one ob-tained with the spontaneous emission method, althoughboth measurements were performed with the same rep-etition time ∼ T and thus also the same initial spinpolarization.We finally demonstrate that SMPD detection can beused to perform usual spin characterisation measure-ments. First, spin spectroscopy is performed by vary-ing the magnetic field B around the resonance valueand using the three different detection methods alreadymentioned: homodyne detection with echo, SMPD fluo-rescence detection, and SMPD detection of the echo. Asseen in Fig. 4d, all three methods give similar spectra.Second, we observe Rabi nutations in the fluorescencesignal. In Fig. 4b, the spin signal C spin is plotted as afunction of the spin driving pulse duration τ . Oscilla-tions are observed, with a frequency linearly dependenton the pulse amplitude, reflecting the Rabi oscillationsof (cid:104) S z (cid:105) since C spin = η [1 + 2 (cid:104) S z (cid:105) ] / . This Rabi nuta-tions can also be measured with the microwave photonechoes method, by varying the duration τ of the refocus-ing pulse; the SMPD signal (cid:104) c (2 τ ) (cid:105) shows the expectedRabi oscillations (Fig. 4c). The oscillation contrast inFigs. 4b and c diminishes with pulse duration τ due tothe spread of Rabi frequencies in the ensemble. This in-homogeneity has a different impact on the spontaneousemission signal and on the echo signal [30], which is quan-titatively reproduced by simulations (see Methods), asseen in Figs 4b-c. Finally, the spin coherence time is mea-sured by microwave photon echo detection. In Fig. 4d, (cid:104) c (2 τ ) (cid:105) is plotted as a function of τ . An exponential fit tothe data yields T = 2 . ms, in agreement with the valuemeasured using homodyne detection (Sup. Mat.). Over-all, this demonstrates that SMPD detection can be usedto perform standard ESR spectroscopy measurements.Our results represent the first use of a SMPD for quan-tum sensing. Beyond the fundamental interest of such aproof of principle, we conclude by discussing the poten- a) b)d)c) B (mT) C s p i n s c ( τ ) X Echo JPAEcho SMPDSpont SMPD τ (µs) A τ A Ω / π C s p i n s ( τ ) τ (µs)0.50.0 0 12 π /2 τ A c ( τ ) ( τ ) τ (ms) τ τ π /2 π c ( τ ) ( τ ) ,, Figure 4:
Spin ensemble characterisation by photoncounting. ( a ) Spin ensemble resonance lineshape when scan-ning the magnetic field B measured with the three spin de-tection methods: homodyne echo detection (red curve), echodetection by photon counting (orange curve) and spontaneousemission detection by photon counting (blue curve). ( b ) Mea-sured (magenta dots) and simulated (solid line) average spinsignal (cid:104) C spin (cid:105) as a function of the duration T of a microwavepulse exciting the spins. ( c ) Measured (orange dots) and sim-ulated (solid line) average number of clicks (cid:104) c (2 τ ) (cid:105) detected atecho time, as a function of the duration T of the second pulseof the Hahn echo sequence. The extracted Rabi frequency de-pendence on the pulse amplitude A (inset of panel b) matchesthe one obtained through the spontaneous emission signal.( d )Measured (orange dots) average number of photons de-tected at echo time (cid:104) c (2 τ ) (cid:105) as a function of the time delay τ between the pulses of the Hahn echo sequence. An exponen-tial fit (solid line) yields a coherence time T = 2 . . tial of SMPDs for spin detection. Whereas the SNR ofecho detection by a SMPD (Fig. 3) and by homodynedetection are comparable, spin fluorescence detection bya SMPD (Fig. 2) on the other hand presents several fea-tures that make it a truly interesting method for ESRspectroscopy. Indeed, the SNR can reach much highervalues than in echo-detection, since the signal (numberof emitted photons) can be as high as the total number ofexcited spins, whereas the noise is entirely dominated bySMPD non-idealities, which are likely to be improved infuture devices [31–33]. We therefore expect that the de-velopment of better SMPDs with lower dark count ratesand higher efficiency will push further ultra-sensitive spindetection, possibly down to a single spin. This perspec-tive is all the more interesting that the method appliesequally well to spins with short coherence times such asencountered in real-world spin systems, making practicalsingle-spin ESR spectroscopy a possible future perspec-tive. Acknowledgements
We acknowledge technical support from P. Sénat, D.Duet, P.-F. Orfila and S. Delprat, and are grateful forfruitful discussions within the Quantronics group. Thisproject has received funding from the European UnionsHorizon 2020 research and innovation program underMarie Sklodowska-Curie Grant Agreement No. 765267(QuSCO). E.F. acknowledges support from the ANRgrant DARKWADOR:ANR-19-CE47-0004. We acknowl-edge support from the Agence Nationale de la Recherche(ANR) through the Chaire Industrielle NASNIQ undercontract ANR-17-CHIN-0001 cofunded by Atos, and ofthe Région Ile-de-France through the DIM SIRTEQ(REIMIC project).
Author contributions
E.A., P.B. and E.F. designed the experiment. T.S. pro-vided the bismuth-implanted isotopically purified siliconsample, on which V.R. fabricated the Al resonator. E.A.designed and fabricated the SMPD with the help of D.V.and E.F.. E.A., V.R., E.F. performed the measurements,with help from L.B, D.F. and P.B.. E.A., P.B. and E.F.analysed the data. E.A., E.B. and V.R. performed thesimulations. E.A., P.B. and E.F. wrote the manuscript.D.F., D.V., D.E. and E.F. contributed useful input to themanuscript. ∗ Electronic address: emmanuel.fl[email protected][1] H. J. Kimble, M. Dagenais, and L. Mandel, “Photon An-tibunching in Resonance Fluorescence,”
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METHODS
1. SETUP
The experimental setup used to drive the spin ensembleand operate the SMPD, with six input lines (labeled 1-6) and one output line (labeled 7), is shown in Fig. 5.We first discuss the room-temperature part (microwaveand dc signals generation), and then the low-temperaturepart (cabling inside the dilution refrigerator).
Room temperature setup
The room-temperature setup includes four microwavesources and two 4-channel arbitrary waveform genera-tors (AWG 5014 from Tektronix). All microwave pulsesneeded in the experiment are generated by mixing theoutput of a source with AWG channel outputs used todrive the I and Q ports of an I/Q mixer at an intermedi-ate frequency indicated in Fig. 5. The pulses are used todrive the spins (at the spin resonator frequency ω ) andto operate the SMPD.SMPD operation requires:- a dc flux-bias of the SQUID in the buffer resonator,in order to tune ω b in resonance with ω . This is achievedwith a dc current source (Yokogawa 7651) connected toan on-chip antenna near the SQUID (line 4 in Fig. 5).- microwave pulses at the pump frequency ω p to satisfythe 4-wave mixing condition ω p = ω q + ω w − ω b - microwave pulses to readout the qubit state via thequbit-state-dependent dispersive shift of the buffer res-onator. They are at frequency ω b + χ qb , the buffer res-onator frequency with qubit in the e state.- microwave pulses at the waste frequency ω w to resetthe qubit.Moreover, qubit readout pulses are amplified by a flux-pumped Josephson Parametric Amplifier (JPA) in degen-erate mode [34]. The JPA needs dc flux biasing to adjustthe JPA frequency; it is provided by an on-chip antennanear the JPA SQUID array, fed by a constant voltagesource biasing a resistor at room-temperature (line 1 inFig. 5). The JPA also requires flux-pumping to achievegain. The pump tone is generated by frequency-doublingthe same source used to generate the readout pulses (line2 in Fig. 5), followed by mixing with an intermediate fre-quency (see Fig. 5). The relative phase between signaland pump is adjusted with a phase shifter for maximumgain on the signal-bearing quadrature.The same source (Keysight MWG, shown in yellow inFig. 5) is used for driving the spins, qubit state read-out, JPA pumping, and as local oscillator for signal de-modulation yielding the quadratures of the qubit readoutpulses. Spin driving pulses and qubit state readout pulsesare sent via the same line (line 3 in Fig. 5). Spin driv-ing pulses require much larger powers than qubit read- out pulses. Therefore, in the room-temperature setup,the line was split before recombination, and in one ofthe branches an amplifier was inserted in-between twomicrowave switches.A second source (Vaunix Labbrick, shown in green inFig. 5) is used for the qubit reset pulses at ω w (line 5). Athird source (Keysight, shown in purple in Fig. 5) is usedfor SMPD pumping. Pump pulses are generated throughI/Q mixing and amplification of the generator output.The signal is then passed through a
70 MHz band-pass fil-ter to prevent spurious wave mixing caused by side-bandresonances and LO leakage, before reaching the cryostatinput on line 6. A fourth source (Vaunix Labbrick, shownin blue in Fig. 5) is used for SMPD tuning and charac-terization.
Low-temperature setup
Line 3 is heavily attenuated at low-temperatures inorder to minimize spurious excitations of the transmonqubit in the SMPD and therefore dark counts (see Fig. 5).It is then connected to the spin resonator input via a dou-ble circulator. The reflected signal is routed by the samecirculator towards the SMPD input (buffer resonator),and the signal reflected on the SMPD is finally routed to-wards the input of the JPA and the detection chain. Twodouble circulators isolate the SMPD from the JPA, tominimize noise reaching the SMPD and potentially caus-ing spurious qubit excitations and dark counts. The JPAoutput (reflected signal) is routed to a High-Electron-Mobility-Transistor (HEMT) amplifier from Low-NoiseFactory anchored at the 4K stage of the cryostat, andthen to output line 7. Infrared filters are inserted onall the lines leading to the SMPD to minimize out-of-equilibrium quasi-particle generation leading to spuriousqubit excitations and dark counts. To minimize heatingof the low-temperature stage by the strong pump tone ofthe SMPD, the necessary attenuation of the pump line at mK is achieved with a dB directional coupler thatroutes most of the pump power towards the mK stagewhere it is dissipated.Using the same line both for spin excitation and SMPDreadout raises potential issues that are now discussed.First, the spin excitation pulse also leads to a large fieldbuild-up in the buffer resonator (since ω b = ω ), whichexcites the qubit and perturbs the proper functioning ofthe SMPD during a time that we quantify to be µ s (detector dead-time). Then, one may also wonder aboutspurious excitation of the spins caused by the repeatedqubit readout pulses. This is avoided, because qubitreadout is performed at ω b + χ qb , which is thus shiftedfrom ω by χ qb / π = − . MHz. I R spin resonator I R I R -
50 Ohm - - - - I R - I R bias tee JPA I R - - - I R HEMT
AWG 5014 φ RFLOI Q
10 MHz M
I QLORF 2F QRF ILO SG
20 MHz
DAQ
50 MHz - + I LORF
AWG 5014
80 MHz 120 MHz
RFLO I RF LOI QA1 A2A2
Keysight MWGKeysight MWGVaunix LabbrickVaunix LabbrickYokogawa
A3 A4 A5
1. Spin pulse 2. SPD reset 3. Detection4. Readout A1: AML-0120-L4401A2: HVA-500M-20-B A3: AMP0-2G20-30-3A4: MiniCirc ZVA183S+A5: MiniCirc ZVE8G+A-tekH60 circulatorPulsarQS2 couplerMarki I/Q mixeror SSB mixer Clear Mw splitter50 Ohm loadWaveline S11330 fast switch Marki ADA-0416 freqency doubler φ Phase shifter k Ω Figure 5:
Schematic of the setup.
2. FABRICATIONSpin sample
The bismuth donors are implanted in a nm epilayerof Si -enriched silicon. The implantation profile rangesfrom 50 to 150nm depth, with a peak concentration of × donors / cm (see refs [28, 35] for more details).The spin resonator consists of an interdigitated capacitor,shunted by a µ m -wide, µ m -long inductive wire, witha design similar to the one used in [28]. It is depositedon top of the silicon sample by evaporation of a
50 nm -thick aluminium film through a resist mask patternedusing e-beam lithography, followed by liftoff. The chipis then placed inside a 3D copper cavity into which apin protrudes, controlling the capacitive coupling to theinput line [28]. A superconducting coil applies an in-plane magnetic field B to tune the spin frequency. Single microwave photon detector
The single photon detector circuit is based on the de-sign by Lescannne et al. [9]. It is fabricated using wetetching of a
60 nm aluminium layer evaporated on a high-resistivity intrinsic silicon substrate. Before metal de-position, the substrate is pre-cleaned with a SC1 pro-cess. The wafer is first immersed for
10 min at ° C in a bath of 5 parts H O to 1 part H O (30 %) to 1part NH OH (29 %) , then is immersed for in HF( ) solution to remove the surface oxide. The substrateis then loaded in an electron-beam evaporator within
10 minutes , after which a
60 nm aluminium layer is de-posited. Patterning of the circuit is achieved by elec-tron beam lithography of a UV3 resist mask, followedby wet etching of the aluminium using a TMAH-baseddeveloper (Microposit CD26). The Josephson junctionsare evaporated using the Dolan bridge technique and re-contacted to the main circuit through aluminium ban-dage patches [36]. Finally, circuit gaps are isotropicallytrenched with a SF6-based reactive ion etch, which hasshown to decrease the internal losses of superconductingresonators [37, 38]. The resulting
10 mm by chip isglued and wired to a Printed-Circuit-Board, placed in acopper box, magnetically shielded, and attached to thecold stage of the dilution refrigerator.
3. CHARACTERISATIONElectron spin resonance spectroscopy by homodynemeasurements
Prior to the experiments reported in the main text, thespin ensemble is characterised by pulsed electron spin res-onance spectroscopy, comparable to previous work [28]. Interestingly this can be done in the same cooldown asthe SMPD measurements reported in the main text, be-cause of the fact that the SMPD setup also includes aJosephson Parametric Amplifier (JPA) for qubit statereadout. To switch from single photon detection to ho-modyne spin measurements, we simply tune the bufferresonator frequency ω b at a frequency far from ω , andtune the JPA at resonance with ω . In that way, thespin-echo signal simply reflects off the SMPD withouttriggering any qubit excitation, and gets amplified bythe JPA, exactly as was achieved in similar experiments[28]. Output signal demodulation then yields the spin-echo quadrature and its integral A e .We measure the spin relaxation time T at B = 17 mTwith an inversion recovery sequence, in which a π pulseis first applied, followed after a duration τ by a Hahn-echo detection sequence. The echo area A e is shown asa function of τ in Fig. 6a, together with an exponentialfit yielding T = 300 ± ms. We also measure the spincoherence time by measuring the echo amplitude as afunction of the delay τ between the π/ pulse and theecho (see Fig. 6b). An exponential fit yields T = 2 . ms.Rabi nutations are obtained by measuring the echo area A e as a function of the refocusing pulse amplitude A (seeFig. 6c). Finally, the bismuth donor spin spectrum is ob-tained by recording the echo amplitude A e as a functionof the field B (see Fig. 6d).The data in Fig. 6 are modelled using a simulation tooldescribed elsewhere [30]. It computes the evolution ofthe spin ensemble under the application of driving pulsesat the resonator input. The spread in spin Larmor fre-quency (due to strain-induced inhomogeneous broaden-ing [39]) and in spin-photon coupling (due to the spa-tial inhomogeneity of the B field generated by the res-onator) are taken into account by describing the spin asan ensemble of packets, with coupling constant density ρ ( g ) and frequency density ρ spin ( ω ) . The evolution ofeach packet is computed independently under the drivepulses, and the echo response is obtained by summingthe packet contributions. Purcell relaxation is also takeninto account [30].Here, we make two extra simplifying assumption. Be-cause the inhomogeneous broadening is much larger thanthe cavity linewidth κ , and that the signal originates es-sentially from spins within this linewidth, we considerthe spin density to be constant, ρ spin ( ω ) = ρ spin . More-over, we model the coupling constant inhomogeneityas a Gaussian centered on ¯ g and width δg , ρ ( g ) = δg √ π e − ( g − ¯ g δg . We adjust the values of ¯ g and δg to get good agreement with the relaxation and Rabi nu-tation data in Fig. 6b and d, yielding ¯ g / π = 290 Hz,and δg / π = 25 Hz. Because the spin density ρ spin onlyrescales the signal amplitude in Fig. 6, its determinationrequires other measurements that are described below,enabling us to infer the number of excited spins and the1 τ (ms) π /2 B (mT) a) b)d)c) π τ τ A π /2 A τ (s) π /2 ππ τ -550 08000.05 A e ( a . u . ) A e ( a . u . ) A e ( a . u . ) A e ( a . u . ) A e Figure 6:
ESR spectroscopy of the spin ensemble. ( a )Measured (blue dots) and simulated (solid line) integratedecho as a function of the delay τ between the inversion π -pulse and the Hahn echo sequence. An exponential fit (notshown) yield a characteristic decay time T = 300 ± ms.( b ) Measured (blue dots) and simulated (solid line) integratedecho as a function of the delay τ between π/ and π pulsesof the Hahn echo sequence. An exponential fit (not showed)yields a characteristic decay time T = 2 . . ( c ) Measured(blue dots) and simulated (solid line) integrated echo as afunction of the amplitude A of the π pulse of the Hahn echosequence revealing Rabi oscillations. ( d ) Measured integratedecho (blue dots) as a function of the in-plane magnetic field B used to tune the spin ensemble frequency. overall photon detection efficiency. Single photon detector characterisation and tuning
The single microwave photon detector consists of atransmon qubit whose ground g and first excited state e encode the detector click. The transmon frequency is ω q / π = 6 .
14 GHz and its anharmonicity is −
200 MHz .It is capacitively coupled to a tunable buffer resonator(maximum frequency ω maxb / π = 7 .
09 GHz ) and a wasteresonator ( ω w / π = 7 .
62 GHz ) with dispersive shifts χ qb / π = − . and χ qw / π = − . respec-tively. The buffer resonator is coupled to the exter-nal microwave line via a capacitance (energy dampingrate κ b = 13 . × s − at the working point ω b / π =6 .
946 GHz ), while the waste resonator is coupled througha Purcell filter to a
50 Ω -terminated line (energy dampingrate κ w = 2 . × s − ). A pump (a.u.) a) b)d)c) η d ω photon /2 π (GHz) -0.4 0.400.5-0.80.0 0 0.46.6726.664 A pump (a.u.) ω p / π ( G H z ) ω p / π ( G H z ) PePe
Re[r] I m [ r ] Figure 7:
Single microwave photon detector character-isation. ( a ) Measured (blue dots) and fitted (orange solidline) complex reflection coefficient r of the buffer resonatorat working point ω b / π = 6 .
946 GHz . The fitting functiontakes into account flux noise of the SQUID enabling the tun-ing the resonator. ( b ), ( d ) Probability P e of finding the qubitin its excited state (color scale) as function of the amplitude A pump and frequency ω pump / π of the pump activating theparametric process of photo-detection. When no photon isimpinging, the buffer resonator is close to its vacuum state(b) no parametric process is activated and the qubit is mostlyin its ground state P e ≈ ; in contrast, when photons areinjected (d) the parametric process is activated at pump fre-quencies for which the conservation of energy is respected.The quadratic dependence of the pump activation frequencyon the pump amplitude is due to the Stark shift of the qubitfrequency for increasing pump power. ( c ) Measured (bluedots) efficiency of detection η d at ω b = 6 .
946 GHz , as a func-tion of the input photon frequency ω photon / π . From the fit(orange solid line), obtained with a model of two coupled cav-ities, we extract a bandwidth ∆ det / π ≈ . . Detector tuning
In order to perform spin detection the SMPD must betuned in resonance with the spin-emitted photons. Thisis achieved by changing the magnetic flux threading asuperconducting quantum interference device (SQUID)embedded in the input resonator of the detector, whichallows a tunability range of about
200 MHz (see fig. 7aand Fig1 of the main text). The photon detector will benow characterised in the vicinity of this working point.2 a) b) -3 -2 -1 -0.004 p ( V ) -3 -2 -1 -0.004 p ( V ) V (Volt) V (Volt) c) d) p e t D (us)0.02 0 60 η d t D (us)0.40.8 t D t D Figure 8:
SMPD performance
Probability p ( V ) of measur-ing the average quadrature voltage V when probing the bufferresonator for qubit readout, when a pulse is applied ( b ) or not( a ) to the qubit prior to the measurement. Dashed line in-dicates the readout threshold, chosen to minimise the ratio α/η d , each measure falling on the left (resp. right) is associ-ated to the qubit being in its ground (resp. excited) state. ( c )Measured (blue dots) and fitted (orange solid line) probability p e of finding the qubit in its excited state as function of time T after the reset sequence, showing out-of-equilibrium qubit ex-cited population reaching thermal equilibrium on a timescale ≈ T . Black dashed line at t D = 5 µ s marks the point atwhich detector is operated. ( d ) Measured (blue dots) and fit-ted (orange solid line) detector efficiency η d as a function ofthe duration of the detection step t D . Fit model takes intoaccount bandwidth-limited detection efficiency for short de-tection windows and T -decay effect for increasing t d . Blackdashed line at t D = 5 µ s marks the point at which detector isoperated, to optimise the photo-detected echo signal. Qubit readout
The qubit readout is based on standard dispersivereadout through the buffer resonator. A tone is sent atthe buffer resonance frequency pulled by the qubit dis-persive shift ω b + χ qb (thus avoiding to drive the spinresonator at ω = ω b ). The qubit state is encoded in thephase of the reflected tone which is subsequently ampli-fied, demodulated and numerically integrated. The read-out performances are evaluated by histograming this re-flected signal conditioned on the application of a π pulsein Fig. 8a and b. We observe two Gaussian distributionsseparated by σ corresponding to the qubit being in theground and excited state, the spurious tail between theGaussian corresponds to relaxation events of the qubitduring the readout time. A threshold enables the dis-crimination of the qubit state. It is chosen to minimise the ratio p ( e | /p ( e | π ) between the false positive and truepositive, therefore optimising the dark count rate withrespect to the efficiency. We measure a ground state fi-delity of − p ( e |
0) = 99 . and an excited state fidelityof p ( e | π ) = 71% Efficiency calibration
The first characterisation of the device as a photondetector consists in tuning the 4-wave mixing processby sending a weak coherent tone at the detector fre-quency while scanning the pump frequency and ampli-tude. When the 4-wave mixing matching condition issatisfied the qubit is left in its excited state, as shown inFig. 8d. The chosen working point is the one maximis-ing the probability of the qubit excitation while minimis-ing the residual qubit excitation due to pump heating orspurious process. Note that the frequency drift for thematching condition corresponds to the qubit AC-starkshift induced by the pump tone increasing amplitude.The detector efficiency is measured by sending a weakcoherent tone onto the buffer resonator with ¯ n photonson average, and measuring the probability p e of findingthe qubit in its excited state. This measurement is com-pared to the excited state probability p darke in the ab-sence of incoming pulse for the same detection window.The detector efficiency for a given detection window t d is then defined as η d = (p e − p darke ) / ¯ n . The photon num-ber ¯ n is independently calibrated through measurementsof qubit dephasing and AC-Stark shift [40] taking intoaccount the flux-noise causing extra broadening of thebuffer resonance. Fig. 7c shows the measured efficiencyas a function of the photon frequency in the vicinity of ω b / π = 6 .
946 GHz . The efficiency for the detection win-dow t d = 5 µ s used in the main text is η d = 0 . ± . . Weunderstand quantitatively the infidelity budget of the de-tector. One source of inefficiency is caused by the readoutexcited state fidelity which limits the detection efficiencyto . , the other source of infidelity is due to the T de-cay of the qubit during the detection window which limitsthe efficiency to . . The product of these two figuresis close to the measured value of η d = 0 . , showing thatqubit relaxation is the dominant limiting factor. Detector bandwidth
As long as the qubit lies in its ground state, the de-tector response (Fig. 7c) can be modelled by consideringthat the buffer and waste resonator are coupled with aconstant G = − ξ p √ χ qb χ qw due to the 4-wave parametricprocess involving the qubit, where ξ p is the pump am-plitude in units of square root of photons and χ qb ( χ qw ) the dispersive coupling of the buffer (waste) resonator tothe qubit. One can write down the system of coupled3equations for the buffer and waste intra-resonator fields α and β : ˙ α = − iδ b α − i G β − κ b √ κ b α in (1) ˙ β = − iδ w β − i G ∗ α − κ w √ κ w β in (2)where δ b and δ w are the buffer and waste frequencies inthe frame rotating at the probing frequency and α in , β in are the respective input field amplitudes. Now using therelation between the intra-resonator fields and the inputand output flux √ κ b α = α in + α out , from the equilibriumsolution of the coupled system we can extract the trans-mission coefficient | S | = | β out /α in | . Assuming zeroinput flux on the waste this leads to: | S | = (cid:12)(cid:12)(cid:12)(cid:12) ξ p √ κ b κ w χ b χ w − δ b δ w + 2 iδ b κ w + 2 iδ w κ b + κ b κ w + χ b χ w ξ (cid:12)(cid:12)(cid:12)(cid:12) This expression can be directly related to the detectorefficiency when varying the input photon frequency, Fig-ure 7c show a fit of this expression to experimental datawith only ξ p and a scale factor as free parameters. Fromthe curve we extract a bandwidth of ∆ det / π ≈ . . Dark Counts
A key figure of merit of the detector is its dark countrate. We characterise this quantity by applying a re-set pulse to the qubit through the waste resonator andby keeping the pump tone turned on while no photonpulse is sent to the buffer resonator. By varying the du-ration of the pump tone, we observe an increasing ex-cited state population as shown in Fig.8c. The residualqubit population rises with a slope of . − from aninital value of . × − . The qubit reaches a finitepopulation of . × − after a few characteristic time T = 8 . µ s . Note that the qubit is initialised well belowits thermal population by the reset process. This finitepopulation can be divided in distinct contributions. Inthe absence of the pump, we measure a residual excitedstate population of the qubit of . × − tone whichis attributed to out-of-equilibrium quasi-particles in thesuperconducting film [41]. By detuning the pump fromthe matching condition, the heating effect of the pumpalone can be evaluated. We measure a negligible riseof the excited population, smaller than − , comparedto the population in absence of pumping. Therefore,most of the excess qubit population δp e ∼ . × − can be attributed to the finite temperature of the bufferline. Such a finite thermal occupancy n th , buffer triggersthe detector over its full bandwidth ∆ det and leads todark counts that are integrated over the qubit lifetime T . The expected rise of qubit population is thus given -0.60.2 0 100 T (us) a) X ( V )
70 100-0.0030.004 -1.4e4-1e4 S z T (us)
680 720T (us) b) -2.6e4-1e4 S z T (us) c) T (us) = 4 π = 12000 s z s z e Figure 9:
Spin ensemble simulations. ( a ) Measured (bluedots) and simulated (orange solid line) electromagnetic fieldamplitude at the output of the spin cavity as a function ofthe time T from the π/ pulse of an echo sequence. The echoappears as a slight increase of the field amplitude at twicethe separation between the π/ and π pulse (inset). The spinspectral density ρ spin is the only free parameter of the sim-ulation, accordance with the experimental data is achievedfor ρ spin = 14 . − . ( b ) Simulated time evolutionof (cid:104) S z (cid:105) during an echo sequence, using the same pulse pa-rameters as in the experiment. ( c ) Simulation of the timeevolution of (cid:104) S z (cid:105) during a π -pulse with the same parametersof the experiment of photo-detected incoherent relaxation. Aspin density ρ spin = 12 spins kHz − was adjusted so that theratio between the variation of (cid:104) S z (cid:105) at echo time (in panelb) and upon the π pulse excitation (panel c) reproduces theexperimental ratio C spin / ( η duty c e ) . by δp e = η d ∆ det T n th , buffer which gives a thermal occu-pancy of the buffer line of n th , buffer ∼ . × − thatcorresponds to a residual temperature for the microwaveline of
27 mK .
4. ESTIMATION OF THE NUMBER OF SPINS
The goal of this section is to explain how we deter-mine the number of spins N excited in the spontaneousemission detection experiment, enabling us to quantifythe overall photon detection efficiency η . We use twoindependent methods, using two different datasets.The first method was already used in previouswork [35, 42]. It relies on measurements performed withhomodyne detection, without any use of the SMPD. Wemeasure a complete spin-echo sequence (including the4control pulses), and we use simulations to fit the data.The ratio of spin-echo to control pulse amplitude allowsfitting the spin density ρ spin needed to account for thedata as the only adjustable parameter.Results are shown in Fig. 9a. Quantitative agreementis obtained for ρ spin = 14 . − . We can thendetermine N , the total number of spins excited by a π pulse in the spontaneous emission detection experimentof Fig.2 in the main text, by running a second dedicatedsimulation with spin density ρ spin . We obtain N = 14 . × .The second method uses a comparison between thetwo different types of measurements performed with theSMPD. On the one hand, the number of photons C spin detected in the spontaneous emission experiment scaleslinearly with N . On the other hand, the echo signal c e scales like N , as explained in the main text, indicativeof the phase-coherent character of echo emission. There-fore, the ratio C spin /c e can give access to N . Note thatbecause both datasets are obtained with the same SMPDand setup, C spin /c e is independent of η , as required for aproper evaluation of N .To make this reasoning quantitative, we again resort tosimulations of both the echo sequence and the π pulse.From each simulation, we extract the number of spinsinvolved by computing the change in the total magneti- zation (cid:104) S z (cid:105) . The ratio of these two numbers should beequal to the experimentally determined C spin / ( η duty c e ) (the η duty correction is due to the fact that echo detec-tion is gated and therefore is insensitive to the detectorduty cycle); we use the spin density ρ spin as the onlyadjustable parameter to reach the agreement.For the experimental value of C spin we use the datashown in Fig. 2 of the main text. For c e we use differ-ent pulse parameters than shown in Fig. 3 of the maintext (same parameters for the π/ pulse, but lower am-plitude and . µ s duration for the π pulse), to get a lowervalue of c e and minimize the risk of SMPD saturation.The experimental ratio is then C spin / ( η duty c e ) = 3 · .Figures Fig. 9b and 9c show the evolution of (cid:104) S z (cid:105) atearly times in the case of a Hahn echo sequence and ofa π -pulse respectively, with the pulse parameters usedin the experiment. The correct ratio is reproduced for ρ spin = 12 spins kHz − . This yields N = 1 . × .The two methods are in agreement within an estimateduncertainty σ spin ≈ on the number of spins takingpart to the process. We take the average of the twovalues N = (13 . ± . × as the reference value forthe efficiency estimation. From this, an overall collectionefficiency η = 1 . × −2