Fundamental limits for non-destructive measurement of a single spin by Faraday rotation
FFundamental limits for non-destructive measurement of a single spin by Faradayrotation
D. Scalbert
Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Universit´e de Montpellier, Montpellier, FR-34095, France (Dated: December 7, 2018)Faraday rotation being a dispersive effect, is commonly considered as the method of choice fornon-destructive detection of spin states. Nevertheless Faraday rotation is inevitably accompaniedby spin-flips induced by Raman scattering, which compromises non-destructive detection. Here, wederive an explicit general relation relating the Faraday rotation and the spin-flip Raman scatteringcross-sections, from which precise criteria for non-destructive detection are established. It is shownthat, even in ideal conditions, non-destructive measurement of a single spin can be achieved only inanisotropic media, or within an optical cavity.
Introduction.
Encoding information into the spin stateof a single electron, nucleus, or atom, and reading this in-formation non-destructively constitute some of the mainchallenges of quantum computing and spintronics. Thesechallenges have motivated strong experimental efforts to-wards electrical and optical detection of single spin statesin semiconductors [1]. Optical detection has been demon-strated by polarized photoluminescence or polarization-dependent absorption [2–4], but these methods are de-structive. Dispersive methods like non-resonant Kerr orFaraday rotation can be in principle non-destructive andopen a way to quantum non-demolition measurementsof a single spin state [5]. Nevertheless it is known thateven for dispersive measurements, the probe laser mayeventually flip the targeted spin, compromising the non-destructive measurement. This limitation is of funda-mental nature, since spin Faraday rotation is inevitablylinked to spin-flip Raman scattering [6–8]. This issuehas been also addressed in details in the context of spinnoise spectroscopy, where the signal can be interpretedeither as Faraday or as Raman noise [9, 10]. To overcomethis problem new schemes for quantum non-demolitionmeasurements have been proposed [11–14]. In practicethe conditions for non-destructive measurements can beeven more challenging to realize due to non-ideal exper-imental conditions, such as light scattering in the sam-ple substrate, low detector quantum efficiency, mixingof spin states etc. Since the first detections of a singlespin by Kerr or Faraday effect [15, 16], and thanks tostrong experimental efforts, and technological progressnon-destructive measurements with these methods arewithin reach [17, 18].Until now the fundamental limits imposed by spin-flipRaman scattering (SFRS) on such measurements havenot been established quantitatively. This Letter intendsto fill this gap through the derivation of an explicit andgeneral relation between the SFRS, and the spin Fara-day rotation (SFR) cross-sections. The notion of cross-section is a very general and convenient way to character-ize the probability of scattering, capture, or absorption,of light or particles when they interact with atoms, de-fects or impurities in solids. As shown in [19] it is alsoadapted to characterize the Faraday rotation induced by a spin polarization. Instead, in general the Faraday ro-tation is characterized by the Verdet constant, which isthe proportionality factor between the rotation angle onone side, and the interaction length times the magneticfield intensity on the other side. This definition is in-convenient for SFR because the rotation angle is propor-tional to the spin polarization density rather than themagnetic field. A spin polarization density can be cre-ated without applied magnetic field by optical pumpingor can appear locally because of a spontaneous spin fluc-tuation. This property is exploited for example in spinnoise spectroscopy [20–22]. In these situations it is bet-ter to introduce the proportionality factor σ F between therotation angle θ F and spin polarization density J z suchthat θ F = σ F J z (cid:96) , where (cid:96) is the interaction length [19]. σ F has the dimension of a surface and can be consideredas a cross-section for SFR.As we show below there exists a direct and generalrelation between the spin-flip Raman scattering cross-section σ R , and σ F . This relation shows that, withoutan optical cavity, a quantum non-demolition of a singlespin by Faraday rotation is not always feasible even inan ideal experiment. General relations between cross-sections.
We considera transparent dielectric material with spins embeddedin it. In these conditions the electric induction can bewritten as [23] D = (cid:15) (cid:48) E + i E × ( G J ) . (1) (cid:15) (cid:48) is the real part of the dielectric tensor, and E is theelectric field. In general G is a second rank tensor, whichbecomes a scalar in optically isotropic media such asatomic vapors or cubic semiconductors. J stands for thespin density. We will consider below dielectric materi-als with point group symmetry C v (relevant for crystalswith wurtzite structure or for quantum dots with smallin-plane asymmetry), or with cubic symmetry. With thehigh-symmetry axis along z , G takes the form G = G G
00 0 G , (2)In the following we will only consider a light beampropagating along z , and linearly polarized along x . The a r X i v : . [ c ond - m a t . o t h e r] D ec field amplitude in the dielectric is then given by E ( r , t ) = E ˆ x exp[ i ( k · r − ωt )] with k = k ˆ z , and the light intensityin the dielectric medium is I = (cid:15) E cn , where n is therefractive index.To calculate the Faraday rotation angle θ F one firstconsiders that J = J ˆ z is time-independent. The solutionof the wave equation is well known and gives θ F = ωG J(cid:96) cn(cid:15) (3)From the definition of the Faraday rotation cross-section [19] one obtains σ F = ωG cn(cid:15) (4)Let us now calculate the spin-flip Raman scattering(SFRS) cross-section σ R for non-polarized spins. For thispurpose one can consider a small volume v whose dimen-sions are much smaller than the optical wavelength, andthat contains N non-interacting spins [24]. SFRS oc-curs because of spin fluctuations. By definition of σ R the power of light scattered by the N spins in v is givenby P s = I σ R N . We have thus to calculate P s . In thevolume v the amplitude of the spin-dependent dipole mo-ment induced by the incident field is given by p ( r , t ) = i − G E J z ( r , t ) v G E J y ( r , t ) v e i ( k · r − ωt ) . (5)The total power emitted by this time-fluctuating dipoleis given by P s = 23 14 π(cid:15) ( c/n ) < | ¨ p ( r, t ) | >, (6)where < ... > denotes time average. Since the spin fluc-tuations are much slower than the variations of the elec-tromagnetic field one can neglect the time-derivatives of J y,z . Thus we get P s = E nω π(cid:15) c (cid:2) G < ( J z ( r , t ) v ) + G < ( J y ( r , t ) v ) > (cid:3) (7)For N independent and randomly oriented spins we have < ( J y,z ( r , t ) v ) > = 13 N s ( s + 1) , (8)where s is the value of the individual spins. Hence, weobtain the SFRS cross-section σ R = ( G + G ) ω π(cid:15) c s ( s + 1) . (9)By comparing Eqs. (4) and (9) we obtain one of the mainresult of this paper, which relates σ R and σ F σ R = 8 π η ) s ( s + 1) (cid:16) nσ F λ (cid:17) (10) with η = ( G /G ) , and λ = 2 πc/ω is the light wave-length in vacuum.It may be useful to give the expression of the differ-ential cross section for forward (or backward) scattering.Taking into account the radiation pattern of the dipoleand that only the y component of the Raman dipole [25]participates to forward-scattering. Starting from Eq. (7)one easily get the differential cross section (cid:18) dσ R d Ω (cid:19) = 13 s ( s + 1) (cid:16) nσ F λ (cid:17) , (11)in agreement with Eq. (7) from reference [8] for s = 1 / x and interacting with a single spin s situated at the po-sition of the beam waist z = 0, and in a pure spin-upor spin-down state s z = ± s (see Fig. (1)). At the beamwaist w the intensity of the field decreases with the dis-tance ρ from the beam axis as I ( ρ ) = I exp( − ( ρ/w ) ).Since the spin is in a pure state there is no fluctua-tions along z , but only quantum spin fluctuations in the(x,y) plane. Among these, only spin fluctuations in the y direction contribute to SFRS. Hence, in Eq. (7) onlythe second term of the right-hand side must be kept,but with spin fluctuations evaluated for spin-up or spin-down state. Using s z = ± s and (cid:104) s x (cid:105) = (cid:104) s y (cid:105) one finds (cid:104) s y (cid:105) = s/
2. Inserting this value in Eq. (7) one obtainsthe following expression for the SFRS cross-section for apure spin-up or spin-down state σ R,pure = 4 π ηs (cid:16) nσ F λ (cid:17) . (12)Since the Raman dipole is along z there is no forwardSFRS in this case (see Fig. (1)), in the sense that thereis no frequency-shifted scattered light in this direction.However, the static spin component along z induces adipole parallel to y . The emitted light is not frequency-shifted and is cross-polarized with respect to the incidentfield. It corresponds to spin-Rayleigh scattering, and isat the origin of the Faraday effect. The relevant dipoleassociated with a pure spin state s z = ± s is according toEq. (5) p y = ∓ iG E Je − iωt , and the emitted field alongthe z -axis can be expressed as E y ( z, t ) = ± iσ F s nλ | z | E e i ( kz − ωt ) , (13)where k = nω/c is the wavevector inside the sample.Besides, the field of the gaussian beam along the z axisis given by E x ( z, t ) = E nπw λ (cid:114) (cid:16) zz c (cid:17) e i ( kz − ωt − ψ ( r )) . (14)where z c is the Rayleigh length, and ψ ( z ) is the Gouyphase shift. For small rotation the Faraday rotation an-gle is given by Re [ E y ( z, t ) /E x ( z, t )]. Close to the spin FIG. 1. Up: Schematics of the geometry for dispersive mea-surement of a single spin by Faraday rotation. The orientedlines represent a focused gaussian beam propagating from leftto right, and the arrow represents a spin at the beam waist(z=0), polarized along the z -axis. The green (resp. red) cir-cles represent the radiation pattern for Rayleigh scattering(resp. SFRS). In an isotropic medium and for a spin one-half the Rayleigh and Raman scattering intensities are ex-actly equal. Bottom: Illustration of the effect of the Gouyphase shift on the light polarization of the gaussian beam.At z < x .For 0 < z (cid:28) z c the light is elliptically polarized, while for z (cid:29) z c the light becomes linearly polarized with the polar-ization rotated by the angle θ F with respect to the incominglight polarization. ψ ( z ) ≈
0, hence the Rayleigh field and the incident fieldare in phase quadrature. They become in phase only for z (cid:29) z c where ψ ( z ) → π/
2, resulting in a rotation of thepolarization plane (see Fig. (1)) θ F ± = ∓ σ F πw s = ∓ θ F . (15)As expected SFR is proportional to the ratio of σ F to the beam cross-section. Note however that the ro-tation angle is not constant across the beam but gen-erally increases off-axis. Also SFR is inevitably accom-panied by SFRS. Thus there is a finite probability ofspin-flips induced by light, which limits the possibility ofnon-destructive measurement.The total power of elastically scattered light is givenby P el. = G I ω π(cid:15) c s . (16)As for Raman scattering one gets the correspondingcross-section σ el. = 8 π s (cid:16) nσ F λ (cid:17) , (17)so that σ R,pure σ el. = η s . (18) Condition for non-destructive measurement of a sin-gle spin.
We will consider the case of a spin one-halfonly. For a non-destructive measurement one demands
FIG. 2. Principle of a single-shot measurement of a singlespin via the Faraday effect using a pair of undistinguish-able photons and two-photons detectors D y ↑ and D y ↓ . D x is an optional one-photon detector. One probe-photon is sentthrough the sample and interacts with the spin. The experi-ment is repeated until a probe-photon is transmitted throughthe PBS (and no photon detected on D x ), and interferes witha reference-photon. A first click on D y ↑ or D y ↓ prepares thespin in the spin-up or spin-down state. Subsequent clicks atthe same detector validate the non-destructive measurement. that the acquisition time be short enough to avoid anyspin-flip induced by inelastic light scattering, but longenough to determine the spin state | s z (cid:105) = | ± (cid:105) from aFaraday effect based measurement. This can be done bymeasuring the light intensity after a linear polarizer aver-aged over a large number of photons (see Supplementarymaterial). But the fundamental detection limit is morerigorously determined by looking at the level of the singlespin-single photon interaction, which amounts to detectthe single-photon Faraday effect [26, 27]. This interac-tion generally leads to a spin-photon entangled state suchas | Ψ (cid:105) = 1 √ (cid:18) | − θ F (cid:105)| + 12 (cid:105) + | θ F (cid:105)| − (cid:105) (cid:19) , (19)where |± θ F (cid:105) = cos( θ F ) | x (cid:105)± sin( θ F ) | y (cid:105) is the photon state.Hence, a phase-sensitive detection of the y -component ofthe photon polarization state projects the spin in a purestate | ± (cid:105) . If the spin relaxation time is long enough,and if non-destructive measurement is achieved, the out-come of subsequent measurements should always be thesame. Figure (2) illustrates a possible phase-sensitive de-tection setup, using pairs of undistinguishable photons.One photon from each pair (probe photon) is sent thoughthe sample and interacts with the spin, while the otherphoton (reference photon), which is phase-stabilized withrespect to the first one, is sent on the reference path. Af-ter interaction the probe photon state is projected onthe x and y polarization states by the polarizing beamsplitter (PBS). y -polarized probe photons then interferewith reference photons at the 50:50 beam splitter (BS). Aclick, corresponding to a two-photons detection [28], willoccur either at detector D y ↑ or at detector D y ↓ . Thisprepares the spin in a known pure state. As long as thespin state is conserved, following clicks will always occurat the same detector. At least one click is necessary todetect the spin state. The average number of clicks for n photons incident on the sample is (assuming no loss) n click = n |(cid:104) y | Ψ (cid:105)| (cid:39) n θ for small rotation angles. Be-sides, the number of spin-flips due to SFRS in the sametime interval is n sf = n σ R,pol /πw . A non-destructivemeasurement requires both n sf (cid:28) n click ≥
1. UsingEqs. (15) and (12) we get8 η (cid:16) nπw λ (cid:17) (cid:28) . (20)Taking into account that diffraction imposes πw ≥ λ/n ,we finally obtain the following condition for a non-destructive measurement η (cid:28) . (21)which can be satisfied only in anisotropic media. Thisis consistent with the fact that for isotropic media SFRSoccurs with the same probability as spin-Rayleigh scat-tering (see Eq. (18)).We assume now that the spin is placed inside a planarmicrocavity. The detailed calculation of the SFRS cross-section is complicated by the modification of the opticalmodes in which the incident field can be scattered. Wewill limit ourselves to a qualitative discussion. On onehand the Faraday rotation is amplified by the qualityfactor of the cavity σ F,cav (cid:39) Qσ F [19, 29]. On the otherhand the field intensity in the cavity is, for a micro-cavityof thickness comparable to the optical wavelength insidethe cavity, amplified by a comparable factor. The Ramandipole for a spin polarized along the cavity axis is alsoparallel to this axis, hence most of the field is scatteredin directions parallel to the plane of the cavity for whichthe optical modes density is not modified. Therefore,in a first approximation, the SFRS cross-section is alsoamplified by the Q -factor σ R,cav (cid:39) Qσ R . The relationbetween Faraday and Raman cross-sections becomes σ R,cav (cid:39) π ηsQ (cid:16) nσ F,cav λ (cid:17) (22)Hence, the condition for non-perturbative measurementsbecomes ηQ (cid:28) , (23)which means that a non-destructive measurement is alsopossible in isotropic media provided the spin is placed inan optical cavity. Application to semiconductors.
To finish we apply thecalculation to direct band gap semiconductors of cubicor wurtzite structure, with localized spin states, such asD X or negatively charged QDs. We note that the cal-culation applies also to QDs of C v symmetry, in whichcase G may take two different values depending on theincident light polarization with respect to the principalaxis of the G tensor. We assume that the optical responseis dominated by the excitonic transitions towards theselocalized states. As is evident from Eq. (5) G and G are associated respectively to the z and y -components ofthe Raman dipole, contributed for respectively by onlylight-hole transitions and by both light-hole and heavy-hole transitions. Using the well-known selection rules forthese transitions we obtain G ∝
13 1 E lh − hν (24) G ∝
12 1 E hh − hν −
16 1 E lh − hν , (25)where E hh and E lh are respectively the energies of theheavy-hole and light-hole excitonic transitions, and hν isthe incident photon energy. Finally, η = 4 (cid:104) E lh − E hh E hh − hν + 2 (cid:105) . (26)Hence, the condition given by Eq. (21) will be fulfilled forlarge enough splitting between light-hole and heavy-holeas compared to the detuning from the heavy-hole exci-tonic resonance, which can be easily realized in quantumdots. Conclusion.
In conclusion we have derived a general re-lation which connects the SFRS and SFR cross-sections,valid in conditions of weak absorption. Using this rela-tion, criteria for non-destructive measurement of a sin-gle spin state by Faraday rotation are deduced. Thesecriteria show that non-destructive measurements requireeither a high enough optically anisotropy, or the use ofan optical cavity. These criteria may serve as quantita-tive guidelines to select the experimental conditions fornon-destructive measurements. The above criteria can beeasily adapted for real experiments in order to take intoaccount losses due to light scattering in the substrate andlimited quantum efficiency of the detector for example.
Acknowledgements.
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