TY - JOUR

T1 - Spin and reoccupation noise in a single quantum dot beyond the fluctuation-dissipation theorem

AU - Wiegand, Julia

AU - Smirnov, Dmitry S.

AU - Hübner, Jens

AU - Glazov, Mikhail M.

AU - Oestreich, Michael

N1 - Funding information: Wiegand Julia 1 Smirnov Dmitry S. 2 Hübner Jens 1 * Glazov Mikhail M. 2 Oestreich Michael 1 † Institut für Festkörperphysik, 1 Leibniz Universität Hannover , Appelstraße 2, D-30167 Hannover, Germany 2 Ioffe Institute , Polytechnicheskaya 26, 194021 St. Petersburg, Russia * jhuebner@nano.uni-hannover.de † oest@nano.uni-hannover.de 14 February 2018 February 2018 97 8 081403 13 July 2017 14 November 2017 ©2018 American Physical Society 2018 American Physical Society We report on the nonequilibrium spin noise of a single InGaAs quantum dot charged by a single hole under strong driving by a linearly polarized probe light field. The spectral dependency of the spin noise power evidences a homogeneous broadening and negligible charge fluctuations in the environment of the unbiased quantum dot. Full analysis of the spin noise spectra beyond the fluctuation-dissipation theorem yields the heavy-hole spin dynamics as well as the trion spin dynamics. Moreover, the experiment reveals an additional much weaker noise contribution in the Kerr rotation noise spectra. This additional noise contribution has a maximum at the quantum dot resonance and shows a significantly longer correlation time. Magnetic-field-dependent measurements in combination with theoretical modeling prove that this additional noise contribution unveils a charge reoccupation noise which is intrinsic in naturally charged quantum dots. Deutsche Forschungsgemeinschaft 10.13039/501100001659 GRK 1991 OE 177/10-1 Russian Science Foundation 10.13039/501100006769 14-12-501067 The efficient optical spin manipulation of individual two-level systems opens fascinating perspectives for spin-photon interfaces, quantum cryptography, and quantum information processing [1,2] . One of the particularly promising solid-state candidates for such spin-photon quantum devices is a single spin localized in a semiconductor quantum dot (QD). Such a system not only provides very long coherence times and a large spin-photon coupling strength, but also provides controlled tuning of the emission wavelength and of the fine-structure splitting. Indeed efficient spin manipulation [3–5] , spin detection [6,7] , and generation of highly indistinguishable photon pairs [8–10] were already demonstrated in optimized quantum dot microcavity structures. The spin dynamics of semiconductor quantum optical systems can be optimally studied by spin noise spectroscopy (SNS) [11] which avoids nonresonant optical excitation. The quantum optical technique has been transferred to semiconductor physics during the past decade [12–17] , revealing not only charge and nuclear spin dynamics [18,19] , but also higher-order spin correlators [20,21] . SNS is mostly used as a weakly interacting nondestructive measurement technique. In this case the fluctuation-dissipation theorem relates the spin noise spectrum to the magnetic spin susceptibility of the system [22,23] and thus strongly limits the informative abilities of the equilibrium SNS. SN measurements beyond the fluctuation-dissipation theorem, i.e., under conditions out of thermal equilibrium, require a special theoretical analysis, but they can offer more information about the coupling and correlation between spin coherences, the response to resonant driving fields, and the charging dynamics [24–27] . Lately, the first SN of a single QD was reported [28] where the QD resonance was inhomogeneously broadened due to slow charge fluctuations of residual impurities in the surrounding of the QD [29] . In this Rapid Communication we employ SNS in view of spin-photon interfaces and investigate the spin dynamics of a coherent superposition of a single hole and the corresponding trion state of a homogeneously broadened strongly driven QD. The observed dynamics in the coupled QD microcavity system allow us to understand physical limits and challenges of optically driven charged QDs as solid-state qubits—one of these challenges being the intrinsic reoccupation noise. The studied sample comprises a single layer of self-assembled In(Ga)As QDs grown by molecular beam epitaxy (MBE) on a (001)-oriented GaAs substrate. The QD layer has a gradient in QD density from zero to about 100 dots / ? m 2 . A p -type doping of ? 10 14 / cm 3 ensures that a fraction of the QDs is occupied by a single hole [30] . The QDs are embedded in an asymmetric GaAs ? microcavity with 13 (top) and 30 (bottom) AlAs/GaAs Bragg mirror pairs and a Q -factor of about 350 determined from reflectivity measurements. The QD microcavity is operating in the weak-coupling regime, enhances the light-matter coupling, increases the ratio of SN to optical shot noise, and enables measurements in reflection geometry. The measurement setup is a low-temperature confocal microscope with two detection arms, one for photoluminescence (PL) analysis and one for SNS (see Ref. [28] and the Supplemental Material [31] ). The QD sample is cooled down to 4.2 K and, for PL measurements, excited above the QD barrier by a cw diode laser with a photon energy of 1.59 eV. The black solid line in Fig. 1 shows the PL spectrum of a single QD in a sample region of very low QD density. The measured linewidth of the PL does not reflect the natural linewidth of the QD but is limited by the optical excitation power and the spectrometer resolution of about 20 ? eV . All results below are measured on this specific QD, but control measurements on other QDs yield consistent results. The transition at higher photon energy is attributed to the neutral exciton ( X 0 ) and the transition at lower photon energy to the positively charged trion ( X 1 + ) (see the Supplemental Material [31] for details), which is in good agreement with the QD PL spectra in Ref. [32] . The assignment is confirmed below by spectrally resolved measurements of the SN power. 10.1103/PhysRevB.97.081403.f1 1 FIG. 1. PL spectrum of the QD showing trion ( X 1 + ) and exciton ( X 0 ) transition. The red data points depict the corresponding SN power measured as a function of laser detuning with respect to the trion resonance. The red line is a guide to the eye. The inset shows a typical SN spectrum (the black dots) at a detuning of ? = ? 97 ? eV with a Lorentzian fit (the red line). The SN of the QD is measured by a linearly polarized cw Ti:sapphire ring laser which is stabilized to a Fizeau wavelength meter and focused onto the sample to a spot with a diameter of 1 ? m . The fluctuations of the Kerr rotation angle due to spin fluctuations in the single QD are measured by a polarization bridge with a low-noise balanced photoreceiver. The resulting electric signal is amplified, digitized, and Fourier transformed in real time to obtain a SN power density frequency spectrum. The total SN power is obtained by integrating the SN frequency spectrum. A small longitudinal magnetic field ( B z = 31 mT ) is applied to increase the longitudinal spin-relaxation time T 1 and thereby improves the signal-to-noise ratio [28] . The SN spectrum is isolated from the background of optical shot and electrical noise by subtracting a spectrum acquired in a purely transverse magnetic field ( B x = 27 mT ). Here, the small transverse magnetic field efficiently suppresses the SN since the projection of the longitudinal spin component on the direction of detection is strongly reduced and the broad transverse spin component with the transversal spin-relaxation time T 2 ? T 1 is negligible in the measured frequency bandwidth. The red dots in Fig. 1 depict the integrated SN power measured as a function of probe laser detuning ? with respect to the trion resonance for a laser intensity of I = 1.1 ? W / ? m 2 . The SN power spectrum P SN ( ? ) consists of two distinct maxima located symmetrically around the QD resonance with a sharp dip at ? = 0 providing the evidence of the homogeneous broadening of the QD resonance [33] . Thus, processes usually leading to inhomogeneous broadening in unbiased QDs, such as charge fluctuations in the QD environment [28] , play a negligible role for this QD. The inset shows a typical SN frequency spectrum measured for a probe laser energy strongly negatively detuned from the optical resonance by ? = ? 97 ? eV . This SN spectrum has a Lorentzian line shape, and the corresponding half-width at half maximum (HWHM) yields the spin-correlation rate. In this case the QD excitation is negligible, and the HWHM is proportional to the inverse longitudinal heavy-hole spin-relaxation time T 1 h = 1 / ( 2 ? HWHM ) . We find from the Lorentzian fit T 1 h = 2.51 ( 11 ) ? s , which evidences efficient decoupling of hole and nuclear spins in the applied magnetic field [28,34] . Next, we examine the SN spectra at small detunings. The SN power has a minimum at zero detuning but does not reach exactly zero which seems, at first glance, inconsistent with the SN from a homogeneously broadened two-level system. A more detailed investigation of the SN frequency spectra at a laser intensity of 1.1 ? W / ? m 2 in the quasiresonant regime reveals: (a) an additional noise contribution and (b) the influence of strong coherent excitation of the trion. Figure 2(a) shows the SN spectrum at 9 ? eV detuning as an example. In addition to the main Lorentzian denoted ? (the red line) that dominates the SN spectrum at large laser detuning (the inset in Fig. 1 ) we observe a second Lorentzian contribution denoted ? (the blue line). The second Lorentzian has a significantly smaller width, i.e., a longer correlation time, and a significantly smaller maximal noise power compared to the ? contribution. The respective SN power spectra of the ? and ? contributions are depicted in Fig. 2(b) . The SN power spectrum of the ? contribution is in excellent agreement with the expected line shape for the SN of a single hole, (1) P SN , ? ( ? ) = A ? 2 ( ? 2 + ? 2 ) 2 , where the parameter ? describes the width of the SN power spectrum (cf. Fig. 2(b) ) and A determines the amplitude of the spectrum. Note, that strong coherent excitation of the trion results in a much larger ? than the intrinsic linewidth ? (see the Supplemental Material [31] and Ref. [35] ). Indeed, a fit to the data based on Eq. (1) , shown by the gray line in Fig. 2(b) , yields a value of 17.08 ( 58 ) ? eV for ? which is about one order of magnitude larger than the typical ? of self-assembled QDs, in agreement with the model below. 10.1103/PhysRevB.97.081403.f2 2 FIG. 2. (a) Typical SN spectrum for a probe laser detuning of ? = 9 ? eV . The spectrum consists of two Lorentzian contributions ? and ? . (b) SN power spectrum of the ? and ? contributions. The gray lines correspond to a global fit according to Eqs. (1) and (7) , respectively. (c) Correlation rates ? s ( ? contribution) and ? n ( ? contribution) as a function of detuning. The gray lines are Lorentzian fits. (d) Intensity dependence of the trion linewidth ? 1 . The red line is a fit according to Eq. (5) describing saturation broadening. The strong coherent driving of the system leads to the formation of new dressed states akin to trion-polaritons, representing a coherent superposition of the ground (hole) and excited (trion) states superimposed by the light [36] . The eigenfrequencies of these states depend on the intensity of the probe light and the detuning. This results into a renormalization of the SN power spectrum width ? (see the Supplemental Material [31] ) as shown in Fig. 2(b) . Therefore the observed ? contribution is related to the dressed-state spin noise. The red dots in Fig. 2(c) show the detuning dependence of the ? -correlation rate which, for large detunings, is associated with the relaxation of the heavy-hole spin. The rate strongly increases in the vicinity of the trion resonance which proves that trion excitation by the probe laser significantly affects the spin dynamics. Consistently, the correlation rates increase as well with higher laser intensities (not shown) [17] . The SN power and the correlation rate of the ? contribution closest to zero detuning could not be extracted from the SN spectrum since its SN power decreases towards zero and its Lorentzian width becomes much larger than the detection bandwidth of 1.8 MHz. The broadening of the SN power spectrum and the increase in the correlation rate of the ? contribution can be readily explained in the framework of a four-level system shown in Fig. 3(a) . Absorption of a ? + or ? ? cavity photon by the QD results in a transition from the hole | ± 3 / 2 ? spin state to the trion | ± 1 / 2 ? state, respectively, as shown by the red arrows. The generation rate is given by [37] (2) G = E 2 ? ? 2 + ? 2 , where E is the matrix element of the trion optical transition, which is proportional to the interband dipole matrix element and the electric-field amplitude in the cavity (see the Supplemental Material [31] ). The transition back to the ground state (the blue arrows in Fig. 3(a) ) can be induced either by stimulated photon emission or by trion recombination without light emission into the main cavity mode. In the weak-coupling regime the trion recombination rate R = G + ? 0 can be presented as the sum of the generation rate G , representing the probe-induced recombination and the spontaneous recombination rate ? 0 . In addition, nonradiative trion recombination can result in a transition of the hole from the QD into an outer state | out ? via the Auger recombination process with a rate ? a [35,38,39] . The recharging process returns a hole from | out ? back into the QD ground state with a rate ? r . We will show below that the generation and recombination processes are by a few orders of magnitude faster than the Auger recombination, QD recharging, and spin-relaxation processes so that the steady-state occupancy of the trion state n tr is determined by the balance of generation and recombination rates, (3) n tr = G G + R n , with n being the probability to find the QD in the charged state, i.e., occupied by a hole or a trion. The average spin-relaxation rate is the weighted sum of spin-relaxation rates of the hole in the ground-state 1 / T 1 h and the electron in the trion 1 / T 1 e [26] , (4) ? s = n h n 1 T 1 h + n tr n 1 T 1 e , where n h = n ? n tr is the probability that the QD is in the ground state. Taking into account the dependence of the generation rate on the detuning by Eq. (2) , one can see that the dependence of ? s on ? is described by a Lorentzian as shown by the gray line in Fig. 2(c) . The HWHM of the Lorentzian profile ? s ( ? ) is determined by the trion linewidth (see the Supplemental Material [31] ) and denoted as ? 1 in the following. Figure 2(d) shows the measured ? 1 as a function of the probe laser intensity. The measured intensity dependence is perfectly described by saturation broadening (see the Supplemental Material [31] ), (5) ? 1 ( I ) = ? 1 + I / I 0 , where I / I 0 = 2 E 2 / ( ? ? 0 ) . The fit of ? 1 according to Eq. (5) is shown as the red line in Fig. 2(d) and yields the intrinsic HWHM of the trion transition ? = 2.2 ( 23 ) ? eV and the saturation intensity I 0 = 0.07 ( 15 ) ? W / ? m 2 . The value of ? is in good agreement with the homogeneous linewidth found in Ref. [40] for positively charged QDs. 10.1103/PhysRevB.97.081403.f3 3 FIG. 3. (a) Sketch of the QD states and the relevant transitions. (b) Splitting of electron and hole spin states extracted from the ratio of ? and ? SN power spectra as a function of B z . At the same time, the strong increase in the spin-relaxation rate ? s in the vicinity of the trion resonance is related to the fast spin relaxation of the electron in the trion, i.e., the electron spin in the trion relaxes orders of magnitudes faster than the hole spin. The fit of ? s for different detunings and intensities yields the longitudinal electron-in-trion spin-relaxation time of T 1 e = 32.8 ( 32 ) ns (see the Supplemental Material [31] for details). Now we proceed to the detailed investigation of the ? component which is characterized by a different dependence on the detuning and a significantly lower correlation rate compared to the ? component. The SN power spectrum of the ? contribution has a maximum at the QD resonance which suggests one of the following origins (see the Supplemental Material [31] ): (i) spin splitting noise, which can be produced by nuclear spin fluctuations [18] , (ii) resonance frequency fluctuations due to charge noise [28] , and (iii) occupancy noise of the resident charge carrier in the QD. In order to have a deeper insight into the particular mechanism we performed measurements of the SN power spectra as a function of the longitudinal magnetic field. These measurements show that the noise power of the ? component increases proportional to B z which directly excludes spin splitting noise. The trion resonance frequency fluctuations can also be excluded because they are characterized by a different detuning dependence and a shorter correlation time (see the Supplemental Material [31] ). Hence, the ? contribution is related to the occupancy noise (ON) of the QD. At first glance, the fluctuations of the hole or trion occupancy ? n h or ? n tr alone cannot produce a noise of the Kerr rotation angle ? K because of the time-reversal symmetry [41] . However, the symmetry analysis (see the Supplemental Material [31] ) reveals possible contributions of the form ? K ? ? n h B z , ? n tr B z . Thereby the application of a longitudinal magnetic field opens up the possibility to study the charge dynamics in the system by means of SNS. Quantum dot occupancy noise arises when the average occupancy is smaller than unity. The correlation rate of the QD occupancy ? n is determined by the rates of the transition from the trion state to an outer state and by the reoccupation of the QD by a hole, see Fig. 3(a) . The ejection of the hole out of the QD can result from the Auger process. Reoccupation can either result from hole tunneling from a nearby acceptor or capture of a free hole. The origin of the measured occupancy noise is the finite Kerr rotation of an occupied QD as compared to the absence of any Kerr rotation for an empty QD in a longitudinal magnetic field at the trion resonance. The average Kerr rotation angle is (6) ? K ? ? 1 2 ? ? 2 ( ? 1 2 + ? 2 ) 2 n ? z , where ? z = ? z e ? ? z h is the difference between electron and hole spin splittings. The average QD occupancy fluctuation squared is given by ? ? n 2 ? = n ( 1 ? n ) which along with Eq. (6) determines the dependence of the occupancy noise power on the detuning, (7) P ON , ? ( ? ) = A ( ? 2 ? ? 2 ) 2 ( ? 2 + ? 2 ) 4 1 ? n 4 ? z 2 , cf. Eq. (1) . A fit to the SN power data of the ? contribution based on this equation is shown in Fig. 2(b) as the gray line. We note however, that the exact shape of this dependence can be sensitive to the details of the charge dynamics in the outer states (see the Supplemental Material [31] ). The comparison of the ? and ? SN power spectra as a function of B z allows by assuming 1 ? n ? 1 to estimate the spin splitting ? z , which is depicted in Fig. 3(b) . The linear dependence on B z additionally proves the correct identification of the origin of the ? contribution and yields a difference of the electron and hole g -factors | g e ? g h | ? 4 , which is reasonable considering the strong renormalization of the electron and hole g -factors in In(Ga)As QDs [42,43] . The blue dots in Fig. 2(c) depict the correlation rate of the reoccupation noise ? n , which decreases strongly for increasing laser detuning. This dependence has again a Lorentzian shape and is described by (8) ? n = n tr n ? a + ? r , where ? r is the reoccupation rate. Close to the optical resonance, ? n is dominated by the Auger recombination rate ? a . Taking into account the saturation intensity I 0 and trion linewidth ? determined from the fit of the ? contribution we extracted the Auger rate ? a = 2.93 ( 28 ) ? s ? 1 . This is in very good agreement with the value measured in Ref. [35] for similar QDs. For large detuning, the correlation time is not dominated by the Auger process anymore but by the reoccupation time which is slow in our sample. Our measurements of ? n ( ? ) yield an estimate for the reoccupation rate of ? r = 0.207 ( 96 ) ? s ? 1 . In conclusion, we have measured the nonequilibrium SN of a homogeneously broadened single QD inside a microcavity in the weak-coupling regime. The presented results extend SNS to the coherent single spin dynamics investigation far beyond the fluctuation-dissipation theorem which uncovers the hidden potential to study strongly nonequilibrium but yet coherent spin dynamics of the excited states and charge dynamics in the system. The SN of the strongly excited artificial atom shows two very distinct contributions. The dominant contribution is related to the optically driven heavy-hole trion transition which is potentially useful for spin-photon interfacing. This contribution shows the anticipated saturation broadening and a combined spin-relaxation time resulting from the ground state (hole) and the excited state (electron in the trion). The second contribution is much weaker but may be parasitic for spin-photon interfacing. This contribution results from the intrinsic loss of the heavy hole due to the small but finite nonradiative recombination rate and the subsequent reoccupation of the QD by a hole. The control of this contribution is left for future investigations and may represent an important challenge for applications (see the Supplemental Material [31] ). We demonstrate the ability of SNS to access single charge carrier and trion spin and charge dynamics as well as the optical properties of interband transitions under nonequilibrium conditions. We expect that this approach can become an essential powerful tool on hand for the characterization of future spin-photon interfaces and spin-based information processing devices. We thank K. Pierz (PTB) for providing the sample and acknowledge financial support by the joint research project Q.com-H (BMBF 16KIS00107), the German Science Foundation (DFG) (GRK 1991, OE 177/10-1), the Basis Foundation and RF President Grant No. SP-643.2015.5. Theory was developed under partial support of the Russian Science Foundation (Grant No. 14-12-501067).

PY - 2018/2/14

Y1 - 2018/2/14

N2 - We report on the nonequilibrium spin noise of a single InGaAs quantum dot charged by a single hole under strong driving by a linearly polarized probe light field. The spectral dependency of the spin noise power evidences a homogeneous broadening and negligible charge fluctuations in the environment of the unbiased quantum dot. Full analysis of the spin noise spectra beyond the fluctuation-dissipation theorem yields the heavy-hole spin dynamics as well as the trion spin dynamics. Moreover, the experiment reveals an additional much weaker noise contribution in the Kerr rotation noise spectra. This additional noise contribution has a maximum at the quantum dot resonance and shows a significantly longer correlation time. Magnetic-field-dependent measurements in combination with theoretical modeling prove that this additional noise contribution unveils a charge reoccupation noise which is intrinsic in naturally charged quantum dots.

AB - We report on the nonequilibrium spin noise of a single InGaAs quantum dot charged by a single hole under strong driving by a linearly polarized probe light field. The spectral dependency of the spin noise power evidences a homogeneous broadening and negligible charge fluctuations in the environment of the unbiased quantum dot. Full analysis of the spin noise spectra beyond the fluctuation-dissipation theorem yields the heavy-hole spin dynamics as well as the trion spin dynamics. Moreover, the experiment reveals an additional much weaker noise contribution in the Kerr rotation noise spectra. This additional noise contribution has a maximum at the quantum dot resonance and shows a significantly longer correlation time. Magnetic-field-dependent measurements in combination with theoretical modeling prove that this additional noise contribution unveils a charge reoccupation noise which is intrinsic in naturally charged quantum dots.

UR - http://www.scopus.com/inward/record.url?scp=85042180062&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.97.081403

DO - 10.1103/PhysRevB.97.081403

M3 - Article

AN - SCOPUS:85042180062

VL - 97

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 8

M1 - 081403

ER -