METHOD FOR DETECTING SPINS BY PHOTON COUNTING

Abstract

A method of detecting spins in a sample, includes exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse for flipping the spins, and detecting a noise signal produced by the return of the spins to equilibrium by means of a device for counting radio-frequency or microwave photons.

Description

The invention relates to a method for detecting spins by photon counting. It mainly, but not exclusively, applies to EPR spectroscopy (EPR standing for Electron Paramagnetic Resonance).

EPR spectroscopy exploits the ability of unpaired electrons, which are found in certain chemical species (free radicals and salts and complexes of transition metals) to absorb and re-emit the energy of electromagnetic radiation, typically microwave radiation, when they are placed in a magnetic field.

The absorption or emission spectrum of the electromagnetic radiation provides information on the chemical environment of the unpaired electron or of the nucleus, respectively.

FIG. 1 schematically illustrates an EPR-spectroscopy apparatus according to the prior art, see for example (Bienfait 2016), (Probst 2017) and (Ranjan 2020).

A sample E containing a set of N electronic spins SE is placed in a cryogenic enclosure CRY, in the magnetic field B₀ generated by a magnet or superconducting coil A. The strength of the magnetic field determines the resonant angular frequency ω₀ of the electronic spins:

ω₀=−γB₀

where γ is the gyromagnetic ratio of the electron. The frequency f_L=ω₀/2π is called the Larmor frequency. Typically, it is located in the microwave spectral region because the gyromagnetic ratio of a free electron is equal to about 28 GHz/T.

The sample E is furthermore magnetically coupled to an electromagnetic resonator REM tuned to the Larmor frequency and modeled by a parallel LC circuit. In the case of (Bienfait 2016), (Probst 2017) and (Ranjan 2020), the resonator is a planar structure comprising two interdigital electrodes that form a capacitor and that are connected at their center by an inductive conductive line, in proximity to which line the sensitive region is located. The plane of the resonator is parallel to the magnetic field B₀. Other types of resonators may be used, for example conductive cavities surrounding the sample on all sides.

The coupling constant of the sample to the resonator is designated “g”, and its quality factor is designated Q. Typically, the coupling constant is sufficiently high for the Purcell effect to dominate the dynamics of relaxation of the spins: Γ₁≈Γ_P where Γ₁ is the energy relaxation rate of the spins and Γ_P is the Purcell factor, which is given by

Γ_P=4g²/κ

where κ=ω₀/Q is the rate of dissipation of energy in the resonator.

A signal generator GS applies, to the resonator REM, a sequence of microwave pulses IEX at the Larmor frequency. These so-called excitation pulses excite the spins of the sample coupled to the resonator.

The most widely used sequence of excitation pulses is the so-called “spin-echo” sequence. It comprises a first impulse so-called “π/2” pulse, which flips the spins—which are initially aligned with the magnetic field B₀—into a plane perpendicular thereto. The spins precess around the magnetic field B₀ and, on doing so, emit a first electromagnetic signal at the Larmor frequency. This signal is called the FID signal (FID standing for Free Induction Decay) because its intensity decreases exponentially due to spin decoherence, with a time constant T₂*=(Γ₂*)⁻¹. The decoherence rate Γ₂* depends on the properties of the sample and on the uniformity of the magnetic field B₀. After a certain time—shorter than time taken for the spins to lose their coherence T₂=(Γ₂)⁻¹—a so-called “π” pulse is applied. This second pulse inverts the orientation of the spins and causes the emission of a second so-called echo electromagnetic signal.

The electromagnetic signals emitted by the spins are converted into an electronic response signal RS by the resonator. A gyrator G makes it possible to separate the excitation pulses from the response signal RS. In particular, it is the echo signal that is used to detect the spins.

The response signal RS is delivered, via a suitable transmission line LT (typically a coaxial cable), to an electronic detecting system SED.

For example, in the case of (Bienfait 2016), (Probst 2017) and (Ranjan 2020), the electronic detecting system comprises a Josephson parametric amplifier JPA pumped at an angular frequency ω_p≈2ω₀; the amplified signal is subsequently amplified by a HEMT amplifier HA, then mixed in a mixer ML with a signal SOL from a local oscillator OL at the angular frequency ω₀, and the in-phase component I and quadrature component Q of the baseband signal resulting from the mixing are detected (homodyne detection).

It is possible to demonstrate—see for example (Bienfait 2016)—that, in the case where the quality factor of the resonator is limited by the coupling to the detection antenna (overdamped regime)—the amplitude of a spin-echo signal is equal to

X _e =pN√{square root over (Γ_P/2Γ₂*)}

where “p” is the polarization at equilibrium of the set of spins, which depends on the temperature according to Maxwell-Boltzmann statistics, and N is the number of spins in the sample.

The noise level is given by

δX=√{square root over (n)}/2

with n=n_eq+n_amp where n_eq is the thermal noise of the microwave field, n_eq=1+2n with n=1/(e^ℏω0/kBT−1) expressing the average number of photons per mode at the temperature T (k_B: Boltzmann's constant) and n_amp is the noise added by the amplifier. The signal-to-noise ratio of homodyne echo detection is therefore equal to

SN _e,h=2pN√{square root over (Γ_P/2nΓ₂*)}.

where the subscripts “e” and “h” stand for “spin echo” and “homodyne detection”, respectively.

Even in the ideal case where p=1 (full polarization of the sample) and n=1 (detector at the quantum noise limit and very low temperature), to obtain a signal-to-noise ratio equal to 1 it is therefore necessary for the number of spins in the sample to satisfy N≥√{square root over (2Γ₂*/Γ_P)}.

However, in general Γ₂*/Γ_P>>1, and therefore N>>1.

The authors of (Probst 2017) obtained, by means of an apparatus of the type shown in FIG. 1, a detection sensitivity of 65 spins/√{square root over (Hz)} and an acquisition rate (limited by Γ_P) of 10 Hz, which allowed them to detect the signal generated by a sample containing only 200 spins. To do this, they worked at a temperature of 10 mK, such that and p≈1 and n_eq≈1, maximized the quality factor of the electromagnetic resonator and its coupling constant to the sample and used an amplifier at the quantum noise limit, so that n_amp≈0.

Other techniques for detecting spin have been used in the prior art, but they have not allowed such high sensitivities to be achieved.

In particular, other excitation sequences may be used. For example, it is possible to use only a “π/2” pulse and to detect the FID signal directly, without inducing an echo.

In (Kubo 2012), an FID signal was detected not by conventional electronic techniques, such as homodyne or heterodyne detection, but by counting microwave photons by means of a transmon, which is a type of superconducting qubit (a qubit being a two-level quantum system). In this method, both the sample and the detecting qubit are arranged in a microwave cavity the resonant frequency of which must be modified dynamically in order to allow the electron spins to first be excited with a “π/2” pulse, then the FID signal to be detected by the qubit. This technique is complex to implement and has only allowed a sensitivity of the order of 10⁵ spins/√{square root over (Hz)} to be achieved, several orders of magnitude worse than the—earlier—result of (Probst 2017).

The use of a “π” pulse alone induces the emission of an incoherent signal (“noise”) caused by the return of spins to equilibrium. Such a signal has been observed—in Nuclear Magnetic Resonance—by (McCoy 1989), but considered of no practical interest due to its very low sensitivity.

The invention aims to improve the sensitivity of the detection of spins, and in particular to make possible measurements on samples containing a very low number of spins, for example 10 or fewer, or even just one.

According to the invention, this aim is achieved by virtue of detection, by means of a counter of radio-frequency or microwave photons, of the incoherent signal emitted by the spins excited by a spin-inverting pulse (“π” pulse). One subject of the invention is therefore a spin-detection method comprising the following steps:

a) placing a sample containing spins in a static magnetic field;b) magnetically coupling the sample to an electromagnetic resonator having a resonant frequency ω₀/2π equal to the Larmor frequency of the spins in the static magnetic field, the coupling constant and the quality factor of the resonator being sufficiently high for the coupling to the resonator to dominate the dynamics of relaxation of the spins;c) exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse at said Larmor frequency; andd) detecting an electromagnetic signal emitted by the spins of the sample in a mode of the electromagnetic resonator in response to said pulse by means of a device for counting radio-frequency or microwave photons;characterized in that the radio-frequency or microwave electromagnetic pulse at the Larmor frequency is a spin-flipping pulse, whereby the detected signal is a noise signal produced by the return of the spins to equilibrium.

By “radio-frequency”, what is meant is frequencies comprised between 1 MHz and 1 GHz and by “microwave” what is meant is frequencies comprised between 1 GHz and 100 GHz.

The appended drawings illustrate the invention:

FIG. 1, which has already been described, schematically illustrates an EPR-spectroscopy apparatus according to the prior art;

FIG. 2 schematically illustrates an EPR-spectroscopy apparatus for implementing a method according to the invention;

FIG. 3 illustrates the structure of a counter of microwave photons able to be used in the apparatus of FIG. 2;

FIG. 4a and

FIG. 4b illustrate experimental results.

The method of the invention may be implemented by means of an apparatus of the type illustrated in FIG. 2, which differs from that of FIG. 1 essentially in that its electronic system SED′ for detecting the microwave signal emitted by the spins of the sample E is based on a device CP for counting microwave photons.

The photon-counting device CP may be a superconducting qubit, in particular a transmon, such as described in (Lescanne 2019) and illustrated in FIG. 3. This device comprises a Josephson junction JJ in transmon configuration connected to three lengths of planar waveguide. The first length of waveguide GO1 forms a half-wavelength (λ/2) resonator resonant at the angular frequency ω₀ of the photons to be detected. The second length of waveguide GO2 is intended to direct, to the Josephson junction, a so-called pump signal, at an angular frequency ω_p. The third length of waveguide GO3 forms a pair of half-wavelength (λ/2) resonators resonant at an angular frequency ω_w referred to as the “waste” angular frequency, and is coupled, via a so-called “Purcell” band-pass filter, to a cold environment (on the scale of millikelvin) having a characteristic impedance of 50 ohm. The Josephson junction in transmon configuration behaves as a two-level system. When it is in its ground state, the simultaneous arrival of a photon at the angular frequency ω₀ and of a pump photon at the angular frequency ω_P causes the Josephson junction to transition to its excited state, the remaining energy being dissipated to the cold environment in the form of a photon at the angular frequency ω_w. The state of the transmon is read by probing one of the two resonators to which the qubit is coupled (the waste mode for example) with a microwave pulse at its resonant frequency; because of the dispersive coupling to the qubit, the phase of the pulse reflected by the resonator allows the state of the qubit and therefore whether or not a photon is present to be deduced. The device is reset by injecting a photon of angular frequency ω_w into GO3, said photon combining with a pump photon in the Josephson junction to return the latter to its ground state, excess energy being removed via a photon at the wavelength ω₀.

Other types of devices allow microwave or even radio-frequency photons to be counted. For example, (Walsh 2017) proposes a bolometer-type detector that uses a Josephson junction to detect heating of a graphene sheet induced by a single photon.

Furthermore, the electronic system GS for generating signals of the apparatus of FIG. 2 is configured to generate, instead of spin-echo sequences, single inverting or “π” pulses IS.

More generally, inverting pulses, which flip the spins by π rad, may be replaced by pulses that flip by a non-zero angle φ that may be less than or equal to π rad (“flipping” pulses). The case where φ=π rad (inversion) is preferred because it maximizes the intensity of the signal emitted by the spins.

The spins of the sample, which are excited by an inverting or flipping pulse, return to equilibrium by spontaneously, and therefore incoherently, emitting photons at the Larmor frequency, forming what is called “spin noise”. The spontaneous emission is highly accelerated by the Purcell effect, and hence almost all of these photons are emitted in a mode of the electromagnetic resonator and are coupled to the transmission line LT, which guides their propagation to the photon-counting device CP. In FIG. 2, the reference RS′ designates the response signal propagating along the transmission line LT. It will be noted that, contrary to the spin-echo signal RS of FIG. 1, it consists entirely of noise.

Whereas, as discussed above with reference to (McCoy 1989), the detection of spin noise via conventional electronic techniques (homodyne or heterodyne demodulation) is not very sensitive, the present inventors have discovered that, unexpectedly, detection of spin noise by photon counting makes it possible to achieve a higher sensitivity than the prior art (homodyne detection of a spin-echo signal).

This may be demonstrated in the following way.

If N is the number of spins in the sample and p (comprised between 0 and 1, and in practice close to 1) is the polarization, the number of excited spins is equal to pN. These spins relax with a time constant T₁=(Γ₁)⁻¹. It is possible to consider that all the spins will have relaxed at the end of an acquisition window sufficiently long with respect to T₁—for example longer than or equal to 5T₁ even 10T₁. The probability that a spin relaxes by emitting a photon in a mode of the electromagnetic resonator is equal to p₁=Γ_P/Γ₁. The photon counter is considered to have a bandwidth equal to Γ₂*, which allows it, in principle, to collect all the photons emitted by the spins, and a quantum efficiency η. The number of photons detected by the counter is therefore equal to ηpNΓ_P/Γ₁.

The number of noise photons (i.e. of photons not originating from spins) is given by n Γ₂*/Γ₁+αΓ₁⁻¹ where, as explained above, n=1/(e^ℏω0/kBT−1) is the average number of photons per mode at the temperature T and α is the dark count rate, i.e. the count rate in the absence of photons.

The noise level corresponds to the standard deviation of the number of noise photons detected, which, assuming that the dark photons have a Poisson distribution, is the square root thereof.

Another source of noise results from the fact that the number of detected photons originating from spins itself varies, because the number of photons emitted by the spins is a random variable of standard deviation √{square root over (p₁(1−p₁)N)}.

Furthermore, since detection efficiency is finite, the number of photons detected is also a random variable, of standard deviation √{square root over (η(1−η)N)}.

In total, the standard deviation of the detection noise is therefore equal to

$\sqrt{\langle n\rangle {\Gamma }_2^{*}/{\Gamma }_1+\alpha {\Gamma }_1^{-1}+\left[p_1\left(1-p_1\right)+\eta \left(1-\eta \right)\right]N}.$

The signal-to-noise ratio of this method of incoherent detection by photon counting is therefore equal to:

$S{N}_{i,\mathrm{CP}}=\eta pN{p}_1/\sqrt{\langle n\rangle {\Gamma }_2^{*}/{\Gamma }_1+\alpha {\Gamma }_1^{-1}+\left[p_1\left(1-p_1\right)+\eta \left(1-\eta \right)\right]N}$

where the subscript “i” stands for “incoherent” (and, therefore, spin noise) and “CP” stands for detection by photon counting. Herein lies the fundamental difference with the conventional method of homodyne detection. Whereas signal-to-noise ratio in homodyne detection is intrinsically limited by vacuum fluctuations that mean that with photon counting there is a parameter regime in which the signal-to-noise ratio may be arbitrarily high.

Specifically, in the ultimate limit where p=1 (maximum spin polarization), p₁=1 (spins relax dominantly via the Purcell effect) and n˜0, this last condition corresponding to

$T\ll \frac{\hslash {\omega }_0}{k_B},$

the following is obtained:

SN _i,CP =ηN/√{square root over (αΓ_P⁻¹+η(1−η)]N)},

whereas it will be recalled that:

SN _e,h=2N√{square root over (Γ_P/2Γ₂⁺)}

in homodyne detection. However, there is no theoretical limit on the value that the efficiency of the detector or the dark count rate may reach, i.e. η may be as close to 1 as desired, and α(Γ_P)⁻¹ may be as low as necessary.

SN_i,CP may therefore be arbitrarily high, even if N=1 and Γ_P/2Γ₂*>>1, provided that the efficiency of the detector is high, and that the dark count rate is low enough.

It is interesting to also calculate, for the purposes of comparison, the signal-to-noise ratio achievable by homodyne detection of spin noise and by counting the photons of a spin-echo signal.

In the case of homodyne detection of spin noise, the total power emitted by the spins is given by the number of photons emitted in the detection window, which is equal to pNΓ_P/Γ₁ in a bandwidth given by Γ₂*. The corresponding noise power is given by nΓ₂*/Γ₁. The standard deviation is √{square root over (nΓ₂*/Γ₂)}.

The signal-to-noise ratio of this method of incoherent homodyne detection is therefore equal to:

SN _i,h =pNΓ _P/√{square root over (nΓ₁Γ₂*)}.

It may be seen that the ratio

SN _e,h /SN _i,h=2√{square root over (Γ₁/Γ_P)}

is always greater than 2, and even very much greater than 2 in situations where Γ₁>>Γ_P. Hence, this method is less suited to detection of low numbers of spins than the method of the invention.

In the case of counting the photons of a spin-echo signal, the number of photons detected is given by ηp²N²(Γ_P/2Γ₂*), which is the square of the amplitude of the signal multiplied by the efficiency η of the detector.

The duration of the echo is (Γ₂*)⁻¹), and hence the number of dark counts is α(Γ₂*)⁻¹. The noise level corresponds to the standard deviation, i.e. to the square root, of this number of counts. Furthermore, it is necessary to take into account the shot noise due to the echo itself, which is a coherent state of the field and therefore has a standard deviation given by pN√{square root over ((1+n)η(Γ_P/2Γ₂*))}.

The signal-to-noise ratio of the detection of a spin-echo signal by photon counting is therefore equal to

SN _e,CP=ηp²N²(Γ_P/2Γ₂*)/√{square root over (α(Γ₂*)⁻¹+ηp²N²(1+n)(Γ_P/2Γ₂*))}.

In the “ultimate” limit where p=1, Γ₁≈Γ_P and n≈0, the following is obtained:

$S{N}_{e,\mathrm{CP}}=\frac{\eta N^2\left(\frac{{\Gamma }_P}{2{\Gamma }_2^{*}}\right)}{\sqrt{{\alpha \left({\Gamma }_2^{*}\right)}^{-1}+\eta N^2\left(\frac{{\Gamma }_P}{2{\Gamma }_2^{*}}\right)}}<S{N}_{e,h}=2N\sqrt{{\Gamma }_P/2{\Gamma }_2^{*}}$

So there is in principle no advantage in terms of signal-to-noise ratio in detecting an echo by photon counting rather than by coherent homodyne detection.

The ratio SN_i,CP/SN_e,CP is equal to

SN _i,CPM /SN _e,CPM=(1/N)√{square root over (Γ₂*/Γ_P)}.

It may therefore be seen that the method of the invention is advantageous with respect to the detection by counting an echo signal when the number of spins of the sample is less than

N _c=√{square root over (Γ₂*/Γ_P)}

If N>N_c, the method of detection by spin-echo and photon counting may therefore be more sensitive than the method according to the invention. However, in this case it will generally be preferable to employ conventional homodyne detection.

In conclusion, it may be seen that none of these techniques allows a signal-to-noise ratio as high as that provided by the invention to be achieved in the case of samples

It will be clear from the foregoing that the method of the invention is particularly advantageous when the number N of spins of the sample is of the order of or less than √{square root over (2Γ₂*/Γ_P)} and when Γ₂*/Γ_P>>1, and provided that

$T_0\le \frac{\hslash {\omega }_0}{k_B}.$

The technical result of the invention has been validated experimentally by detecting the microwave signal emitted by a set of N≅200 donors (bismuth atoms) in silicon coupled to a resonator at the frequency ω₀ by a device for counting microwave photons that was similar to the one described in the reference (Lescanne 2019) and that was tuned to the frequency ω₀. In this experiment, Γ₂*≅10⁵s⁻¹, Γ_P≅10s⁻¹, and Γ₁=Γ_P. The signal of the spins was detected according to the two modalities envisioned in this patent. FIG. 4a illustrates the results of an experiment in which spin-noise photons were counted, with a counter such that η≅0.2 and α=1.5 ms⁻¹. It may be seen that the number of photons detected per unit time just after a n pulse applied to the spins decreased exponentially with a time constant Γ_P⁻¹=100 ms, before reaching a constant value corresponding to the dark count rate. It was a question of spontaneous emission by the spins via the Purcell effect. The total number of counts detected during each sequence, read from the graph of FIG. 4a by subtracting the baseline corresponding to the dark count (about 1.5 counts/ms) and integrating the number of remaining counts, is about 50, which is close to the expected value ηN.

The detection of spins via the echo method detected by photon counting has been graphed in FIG. 4b, which shows the probability of counting one photon per unit time, said probability being obtained by averaging over a number of acquisition sequences. The two microwave pulses intended to generate the echo signal may be seen at the times t=0 and t=0.35 ms. The spin-echo is detected at the time t=0.7 ms, as expected. The amplitude of the signal is about 0.3 counts per echo sequence, which is close to the expected theoretical value of 0.4 counts on average.

The invention has been described with reference to its application to the detection of electronic spins, and more particularly to EPR spectroscopy (EPR standing for Electron Paramagnetic Resonance), but it is not limited thereto. In particular, it may be applied to the detection of nuclear spins and more particularly to NMR spectroscopy (NMR standing for Nuclear Magnetic Resonance). This is important, because while few molecular species have unpaired electrons detectable by EPR, very many nuclei—and in particular the most common thereof, the proton—have nuclear spin and are therefore detectable by NMR.

Extension of the technique of the invention to the detection of nuclear spins poses no difficulty in principle. However, since the gyromagnetic ratios of atomic nuclei are about three orders of magnitude lower than the gyromagnetic ratio of the electron, the Larmor frequencies used in NMR are typically much lower than those encountered in EPR (a few MHz or tens of MHz, instead of several GHz), despite the use of stronger magnetic fields. This has two consequences:

Firstly, it is necessary to count radio-frequency photons, which are less energetic than the microwave photons emitted by electron spins.

Secondly, the condition

$T_0\le \frac{\hslash {\omega }_0}{k_B},$

which must preferentially be met to obtain a high sensitivity, requires even greater cooling.

This makes application of the invention to the detection of nuclear spins more complex, but not fundamentally so.

BIBLIOGRAPHY

(McCoy 1989) “Nuclear spin noise at room temperature”, M. A. McCoy and R. R. Ernst, Chemical Physics Letters 159, 587 (1989).

(Kubo 2012) “Electron spin resonance detected by a superconducting qubit”, Y. Kubo et al. Phys. Rev. B 86, 06514 (2012)

(Bienfait 2016) “Reaching the quantum limit of sensitivity in electron spin resonance” A. Bienfait, J. J. Pla, Y. Kubo, M. Stern, X. Zhou, C. C. Lo, C. D. Weis, T. Schenkel, M. L. W. Thewalt, D. Vion, D. Esteve, B. Julsgaard, K. Moelmer, J J L Morton, P. Bertet, Nature Nanotechnology 11, 253 (2016).

(Probst 2017) “Inductive-detection electron-spin resonance spectroscopy with 65 spins/√Hz sensitivity” S. Probst, A. Bienfait, P. Campagne-Ibarcq, J. J. Pla, B. Albanese, J. F. Da Silva Barbosa, T. Schenkel, D. Vion, D. Esteve, K. Moelmer, J. J. L. Morton, R. Heeres, P. Bertet, Appl. Phys. Lett. 111, 202604 (2017).

(Walsh 2017) “Graphene-Based Josephson-Junction Single-Photon Detector” Walsh, Evan D., et al. Physical Review Applied, vol. 8, no. 2, August 2017.

(Lescanne 2019) “Detecting itinerant microwave photons with engineered non-linear dissipation” R. Lescanne, S. Deléglise, E. Albertinale, U. Réglade, T. Capelle, E. Ivanov, T. Jacqmin, Z. Leghtas, E. Flurin, arxiv:1902:05102.

(Ranjan 2020) “Pulsed electron spin resonance spectroscopy in the Purcell regime” V. Ranjan, S. Probst, B. Albanes, A. Doll, O. Jacquit, E. Flurin, R. Heeres, D. Vion, D. Esteve, J. J. M. Morton, P. Bertet, J. Mag. Res. 310 (2020).

Claims

1. A spin-detection method comprising the following steps: a) placing a sample (E) containing spins (SE) in a static magnetic field (B0);b) magnetically coupling the sample to an electromagnetic resonator (REM) having a resonant frequency ω0/2π it equal to the Larmor frequency of the spins in the static magnetic field, the coupling constant and the quality factor of the resonator being sufficiently high for the coupling to the resonator to dominate the dynamics of relaxation of the spins;c) exciting the spins of the sample by means of a radio-frequency or microwave electromagnetic pulse (IS) at said Larmor frequency; andd) detecting an electromagnetic signal (RS′) emitted by the spins of the sample in a mode of the electromagnetic resonator in response to said pulse by means of a device (CP) for counting radio-frequency or microwave photons;wherein the radio-frequency or microwave electromagnetic pulse at the Larmor frequency is a spin-flipping pulse, whereby the detected signal is a noise signal produced by the return of the spins to equilibrium.

2. The method as claimed in claim 1, wherein the device for counting radio-frequency or microwave photons is spaced apart from the electromagnetic resonator and connected thereto via a waveguide or a transmission line (LT).

3. The method as claimed in claim 1, wherein, at least in steps c) and d), the sample is kept at a temperature lower, preferably by at least a factor of 10, than

4. The method as claimed in claim 1, wherein, in step d), the electromagnetic signal is detected during an acquisition window of duration comprised between 0.5·Γ1−1 and 10·Γ1−1, and preferably between Γ1−1 and 5·Γ1−1, where Γ1 is the relaxation rate of the spins of the sample coupled to the electromagnetic resonator.

5. The method as claimed in claim 1, wherein the coupling constant g between the spins of the sample and the electromagnetic resonator, the quality factor of the electromagnetic resonator at the Larmor frequency and the decoherence rate Γ2* of the spins of the sample are chosen such that

6. The method as claimed in claim 1, wherein the spin-flipping pulse is a spin-inverting pulse.

7. The method as claimed in claim 1, wherein the device for counting radio-frequency or microwave photons is a qubit.

8. The method as claimed in claim 6, wherein the device for counting radio-frequency or microwave photons is a transmon.

9. The method as claimed in claim 1, wherein the spins of the sample are electron spins.