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.

Claims

1. A spin-detection method comprising the following steps: a) placing a sample (E) containing spins (SE) in a static magnetic field (B.sub.0); b) magnetically coupling the sample to an electromagnetic resonator (REM) having a resonant frequency ω.sub.0/2π 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; and d) 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 $T_0=\frac{\hslash {\omega }_0}{k_B}$ where ℏ is the reduced Planck constant and k.sub.B is Boltzmann's constant.

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.Math.Γ.sub.1.sup.−1 and 10.Math.Γ.sub.1.sup.−1, and preferably between Γ.sub.1.sup.−1 and 5.Math.Γ.sub.1.sup.−1, where Γ.sub.1 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 Γ.sub.2* of the spins of the sample are chosen such that $\frac{{\Gamma }_P}{{\Gamma }_2^{*}}<1$ and preferably $\frac{{\Gamma }_P}{{\Gamma }_2^{*}}<0.1,$ where ${\Gamma }_P=\frac{4Q{g}^2}{{\omega }_0}.$

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.

Description

[0026] The appended drawings illustrate the invention:

[0027] FIG. 1, which has already been described, schematically illustrates an EPR-spectroscopy apparatus according to the prior art;

[0028] FIG. 2 schematically illustrates an EPR-spectroscopy apparatus for implementing a method according to the invention;

[0029] FIG. 3 illustrates the structure of a counter of microwave photons able to be used in the apparatus of FIG. 2;

[0030] FIG. 4a and

[0031] FIG. 4b illustrate experimental results.

[0032] 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.

[0033] 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 ω.sub.0 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.sub.p. The third length of waveguide GO3 forms a pair of half-wavelength (λ/2) resonators resonant at an angular frequency ω.sub.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 ω.sub.0 and of a pump photon at the angular frequency ω.sub.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.sub.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.sub.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 ω.sub.0.

[0034] 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.

[0035] 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.

[0036] 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.

[0037] 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.

[0038] 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).

[0039] This may be demonstrated in the following way.

[0040] 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 T.sub.1=(Γ.sub.1).sup.−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 T.sub.1—for example longer than or equal to 5T.sub.1 even 10T.sub.1. The probability that a spin relaxes by emitting a photon in a mode of the electromagnetic resonator is equal to p.sub.1=Γ.sub.P/Γ.sub.1. The photon counter is considered to have a bandwidth equal to Γ.sub.2*, 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.sub.P/Γ.sub.1.

[0041] The number of noise photons (i.e. of photons not originating from spins) is given by n Γ.sub.2*/Γ.sub.1+αΓ.sub.1.sup.−1 where, as explained above, n=1/(e.sup.ℏω.sup.0.sup./k.sup.B.sup.T−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.

[0042] 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.

[0043] 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 (p.sub.1(1−p.sub.1)N)}.

[0044] 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)}.

[0045] In total, the standard deviation of the detection noise is therefore equal to

[00001]$\sqrt{.Math.n.Math.{\Gamma }_2^{*}/{\Gamma }_1+\alpha {\Gamma }_1^{-1}+\left[p_1\left(1-p_1\right)+\eta \left(1-\eta \right)\right]N}.$

[0046] The signal-to-noise ratio of this method of incoherent detection by photon counting is therefore equal to:

[00002]$S{N}_{i,\mathrm{CP}}=\eta pN{p}_1/\sqrt{.Math.n.Math.{\Gamma }_2^{*}/{\Gamma }_1+\alpha {\Gamma }_1^{-1}+\left[p_1\left(1-p_1\right)+\eta \left(1-\eta \right)\right]N}$

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.

[0047] Specifically, in the ultimate limit where p=1 (maximum spin polarization), p.sub.1=1 (spins relax dominantly via the Purcell effect) and n˜0, this last condition corresponding to

[00003]$T\ll \frac{\hslash {\omega }_0}{k_B},$

the following is obtained:

SN.sub.i,CP=ηN/√{square root over (αΓ.sub.P.sup.−1+η(1−η)]N)},

whereas it will be recalled that:

SN.sub.e,h=2N√{square root over (Γ.sub.P/2Γ.sub.2.sup.+)}

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.sub.P).sup.−1 may be as low as necessary.

[0048] SN.sub.i,CP may therefore be arbitrarily high, even if N=1 and Γ.sub.P/2Γ.sub.2*>>1, provided that the efficiency of the detector is high, and that the dark count rate is low enough.

[0049] 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.

[0050] 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.sub.P/Γ.sub.1 in a bandwidth given by Γ.sub.2*. The corresponding noise power is given by nΓ.sub.2*/Γ.sub.1. The standard deviation is √{square root over (nΓ.sub.2*/Γ.sub.2)}.

[0051] The signal-to-noise ratio of this method of incoherent homodyne detection is therefore equal to:

SN.sub.i,h=pNΓ.sub.P/√{square root over (nΓ.sub.1Γ.sub.2*)}.

[0052] It may be seen that the ratio

SN.sub.e,h/SN.sub.i,h=2√{square root over (Γ.sub.1/Γ.sub.P)}

is always greater than 2, and even very much greater than 2 in situations where Γ.sub.1>>Γ.sub.P. Hence, this method is less suited to detection of low numbers of spins than the method of the invention.

[0053] In the case of counting the photons of a spin-echo signal, the number of photons detected is given by ηp.sup.2N.sup.2(Γ.sub.P/2Γ.sub.2*), which is the square of the amplitude of the signal multiplied by the efficiency η of the detector.

[0054] The duration of the echo is (Γ.sub.2*).sup.−1), and hence the number of dark counts is α(Γ.sub.2*).sup.−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.sub.P/2Γ.sub.2*))}.

[0055] The signal-to-noise ratio of the detection of a spin-echo signal by photon counting is therefore equal to

SN.sub.e,CP=ηp.sup.2N.sup.2(Γ.sub.P/2Γ.sub.2*)/√{square root over (α(Γ.sub.2*).sup.−1+ηp.sup.2N.sup.2(1+n)(Γ.sub.P/2Γ.sub.2*))}.

[0056] In the “ultimate” limit where p=1, Γ.sub.1≈Γ.sub.P and n≈0, the following is obtained:

[00004]$S{N}_{e,\mathrm{CP}}=\frac{\eta N^2\left(\frac{{\Gamma }_P}{2{\Gamma }_2^{*}}\right)}{\sqrt{{\alpha \left({\Gamma }_2^{*}\right)}^{-1}+\eta N^2\left(\frac{{\Gamma }_P}{2{\Gamma }_2^{*}}\right)}}<S{N}_{e,h}=2N\sqrt{{\Gamma }_P/2{\Gamma }_2^{*}}$

[0057] 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.

[0058] The ratio SN.sub.i,CP/SN.sub.e,CP is equal to

SN.sub.i,CPM/SN.sub.e,CPM=(1/N)√{square root over (Γ.sub.2*/Γ.sub.P)}.

[0059] 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.sub.c=√{square root over (Γ.sub.2*/Γ.sub.P)}

[0060] If N>N.sub.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.

[0061] 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

[0062] 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 (2Γ.sub.2*/Γ.sub.P)} and when Γ.sub.2*/Γ.sub.P>>1, and provided that

[00005]$T_0\le \frac{\hslash {\omega }_0}{k_B}.$

[0063] 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 ω.sub.0 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 ω.sub.0. In this experiment, Γ.sub.2*≅10.sup.5s.sup.−1, Γ.sub.P≅10s.sup.−1, and Γ.sub.1=Γ.sub.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.sup.−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.sub.P.sup.−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.

[0064] 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.

[0065] 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.

[0066] 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:

[0067] Firstly, it is necessary to count radio-frequency photons, which are less energetic than the microwave photons emitted by electron spins.

[0068] Secondly, the condition

[00006]$T_0\le \frac{\hslash {\omega }_0}{k_B},$

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

[0069] This makes application of the invention to the detection of nuclear spins more complex, but not fundamentally so.

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