## Extending the time of coherent optical response in ensemble of singly-charged InGaAs quantum dots

### Coherent optical response in (In,Ga)As QDs

We study n-doped (In,Ga)As quantum dots structures embedded into the microcavity with GaAs/AlAs distributed Bragg reflectors (for details see the “Methods” section). The QD emission is represented by the photoluminescence (PL) spectrum in Fig. 1a which was measured from the edge of the sample in order to avoid the cavity impact (blue dotted line). The PL maximum at the photon energy of 1.4355 eV corresponds to the radiative recombination of excitons from the lowest confined energy state, while a weak shoulder at higher energies around 1.45 eV is apparently related to the emission from the first excited exciton states. The width of the PL line reflects the magnitude of inhomogeneous broadening for the optical transitions with the full width at the half maxima (FWHM) of 10 meV. The corresponding transmission spectrum with a band centered at 1.434 eV and FWHM of 1.4 meV is shown by the red dashed line in Fig. 1a. Using a microcavity with a quality factor *Q* ~ 1000 facilitates the efficient generation of non-linear coherent optical signal due to the significant increase of light–matter interaction^{24,26,27}.

Transient four-wave mixing experiments with heterodyne detection are performed at a temperature *T* = 2 K and a magnetic field is applied in the plane of the sample (see the “Methods” section and Fig. 1c). The time-resolved electric field amplitude of the four-wave mixing signal is shown in Fig. 1c by the blue line for *τ*_{12} = 33.3 ps and *τ*_{23} = 100 ps, where *τ*_{ij} is the time delay between pulses *i* and *j* in the sequence. Two- and three-pulse echoes are observed at times *t* = 2*τ*_{12} (2PE) and *t* = 2*τ*_{12} + *τ*_{23} (3PE), respectively. They are well described by Gaussian peaks with the FWHM of about 10 ps which is mainly determined by the spectral width of the excitation pulses^{16}.

In what follows we use the magnitude of the electric field amplitude at the PE peak maximum ∣*P*_{PE}∣ to characterize the strength of the photon echo signal. Note that the data in Fig. 1c correspond to a single scan where the areas of the excitation pulses are set below *π*/2 which results in simultaneous appearance of both 2PE and 3PE signals. In the next sections we present the two-pulse PE data for excitation with areas of pulses 1 and 2 approximately equal to *π*/2 and *π*, respectively. As for the three-pulse PE data we use a sequence of three *π*/2 pulses. The pulse energy of \({{{{{{{\mathcal{P}}}}}}}}=5\) pJ corresponds to the pulse area of about *π*. We note that the areas of excitation pulses do not change the temporal dynamics of the 2PE and 3PE signals as function of *τ*_{12} and *τ*_{23}, which is the main task of our study. They influence the amplitude of echoes and their ratio. Under optimal conditions the fluence of a PE pulse is estimated to be about 0.5 fJ.

In order to address various spin configurations, we use different linear polarization schemes in the excitation and detection paths. The direction of polarization is assigned with respect to the magnetic field direction, i.e. H and V polarizations are parallel and perpendicular to **B**, respectively. The polarization scheme is labeled as *A**B**D* or *A**B**C**D* for two- or three- pulse echoes. Here, the first two (*A**B*) or three (*A**B**C*) letters indicate the linear polarizations of the optical pulses in the excitation sequence and the last letter (*D*) corresponds to the polarization direction in the detection, e.g. the data in Fig. 1c are taken in the HVVH polarization configuration.

### Photon echo from trions in QDs

In order to observe long-lived spin-dependent echoes it is necessary to address trion *X*^{−} (charged exciton) complexes, which correspond to the elementary optical excitation in a charged QD. The energy spectrum in the charged QD can be well described by a four-level energy scheme with Kramers doublets in the ground and excited states at *B* = 0, which are determined by the spin of the resident electron *S* = 1/2 and the angular momentum of the heavy hole *J* = 3/2, as shown in Fig. 2a. In contrast to the exciton in a neutral QD, this four-level scheme allows establishing optically induced long-lived spin coherence in the ground state^{17}.

Although the photon energies for resonant excitation of trion and exciton (*X*) complexes are different in one and the same QD, it is not possible to perform selective excitation of only charged QDs by proper choice of the photon energy. This is due to the strong degree of inhomogeneous broadening for optical transitions in the QD ensemble, which is considerably larger than the energy difference between the *X* and *X*^{−} resonances. It is, however, possible to distinguish between exciton and trion contributions using polarimetric measurement of photon echo signal^{28,29}. Figure 2b shows polar plots of two-pulse PE magnitude measured at *τ*_{12} = 66 ps using HRH and HRV polarization schemes. The diagrams are obtained by rotation of the polarization direction of the second pulse with linear R-polarization which is defined by angle *φ*_{2} with respect to the H-polarization. In both polarization schemes, the signal is represented by rosettes with fourth harmonic periodicity when the angle *φ*_{2} is scanned. Such behavior corresponds to PE response from trions where the PE is linearly polarized with the angle *φ*_{PE} = 2*φ*_{2} and the PE amplitude is independent of *φ*_{2}^{29}. In case of the neutral exciton the polar plot is different because the PE signal is co-polarized with the second pulse (*φ*_{PE} = *φ*_{2}) and it amplitude follows \(| \cos {\varphi }_{2}|\).

We note that the small increase of the PE amplitude by about 15% in HHH as compared to HVH remains the same under rotation of the sample around *z*-axis which excludes an anisotropy of dipole matrix elements in *x**y*-plane as possible origin of asymmetry (see the blue pattern in Fig. 2b). The difference could be provided by a weak contribution from neutral excitons. This is because in HRH configuration the PE from trions is the four-lobe pattern \(\propto | \cos^{2}{\varphi }_{2}|\) while for excitons it corresponds to a two-lobe pattern \(\propto {\cos }^{2}{\varphi }_{2}\). Finally, we conclude that independent of the polarization scheme the main contribution to the coherent optical response with a photon energy of 1.434 eV in the studied sample is attributed to trions. This demonstration is very important for proper interpretation of the results because long-lived spin-dependent echoes can be observed only in charged QDs. Moreover it has large impact for applications in quantum memory protocols where high efficiency is required.

We evaluate the optical coherence time *T*_{2} and the population lifetime *T*_{1} of trions in QDs from the decay of PE amplitude of the two- and three-pulse echoes, respectively. The data measured at *B* = 0 in co-polarized configuration (HHH for 2PE and HHHH for 3PE) are shown in Fig. 2c. In the case of 2PE, the amplitude is scanned as a function of 2*τ*_{12} (blue dots), while for 3PE the dependence on *τ*_{23} is shown (green triangles). The exponential fit of two-pulse echo \(| {P}_{{{{{{{{\rm{2PE}}}}}}}}}| \propto \exp (-2{\tau }_{12}/{T}_{2})\) gives *T*_{2} = 0.45 ns which is in agreement with previous studies in (In,Ga)As/GaAs QDs^{16,22,24}. The decay of 3PE has a more complex structure. At short delay times, its magnitude decays exponentially with a time constant of *T*_{1} = 0.27 ns which we attribute to the trion lifetime *τ*_{r}. However, the signal does not decay to zero and shows a small offset with a magnitude of about 5% of the initial amplitude at long delay times *t* > 1 ns. This weak signal is governed by the dynamics of population grating in the ground state of the QDs ensemble and can be provided by many different reasons, which are out of the scope of this paper. We note that *T*_{2} ≈ 2*T*_{1} indicates that the loss of optical coherence under resonant excitation of trions is governed by their radiative recombination.

### Long-lived spin-dependent photon echo in QDs

Application of the transverse magnetic field (**B**∣∣**x**) leads to Zeeman splitting of the Kramers doublets in the ground resident electron and optically excited trion states. The electron spin states with spin projections *S*_{x} = ±1/2 are split by *ℏ**ω*_{e} = *g*_{e}*μ*_{B}*B*, while the trion states with angular momentum projections *J*_{x} = ±3/2 are split by *ℏ**ω*_{h} = *g*_{h}*μ*_{B}*B*. Here, *ω*_{e} and *ω*_{h} are the Larmor precession frequencies of electron and heavy hole spins, *g*_{e} and *g*_{h} are the electron and hole *g* factor, and *μ*_{B} is the Bohr magneton. Optical transitions between all four states are allowed using light with H or V linear polarization, as shown in Fig. 2a. The energy structure can be considered as composed of two Λ schemes sharing common ground states. The magnetic field induces the asymmetry between these two Λ schemes allowing one to transfer optical coherence induced by the first optical pulse into the spin coherence by application of the second optical pulse^{15,17}.

The first pulse creates two independent coherent superpositions between the ground and excited states (optical coherences). For an H-polarized pulse the optical coherences correspond to the density matrix elements *ρ*_{13} and *ρ*_{24} (see Fig. 2a and Supplementary Note 1). The second pulse creates the populations *ρ*_{ii} (*i* = 1, 2, 3, 4) for H-polarization or accomplishes transfer of optical coherences into the spin coherences of trions (*ρ*_{34}) and electrons (*ρ*_{12}) when the second pulse is V-polarized. Due to inhomogeneous broadening of the optical resonance frequencies *ω*_{0} optical excitation with a sequence of two pulses leads to appearance of population (co-polarized HH-sequence) or spin (cross-polarized HV-sequence) gratings in the spectral domain with the period of 1/*τ*_{12}. For the HH-sequence at zero magnetic field the population gratings in the left and right arms of the optical scheme are equal, i.e. *ρ*_{11} = *ρ*_{22} and *ρ*_{33} = *ρ*_{44} (see Fig. 2a). However, in magnetic field due to the Zeeman splitting of electrons and holes, a spin grating for the component directed along magnetic field appears. For the HV-sequence the spin grating is given by the *y**z* components.

Thus, a sequence of two-linearly polarized pulses can be used to initialize a spin grating in the ground and excited states. The addressed spin components depend on the polarization of the exciting pulses. For linearly co-polarized HH sequence the spin components along the magnetic field direction (*x*-axis) are addressed (see Supplementary Eq. (35)

$${S}_{x}=-{J}_{x}\propto \sin \left(\frac{{\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}}{2}{\tau }_{12}\right)\exp \left(-\frac{{\tau }_{12}}{{T}_{2}}\right)\cos \left({\omega }_{0}{\tau }_{12}\right).$$

(1)

In case of cross-polarized HV sequence the spin grating is produced in the plane perpendicular to the magnetic field direction (see Supplementary Eqs. (36) and (37)

$${S}_{y}+i{S}_{z}={J}_{y}-i{J}_{z} \propto i\exp \left(i\frac{{\omega }_{{{{{{{{\rm{e}}}}}}}}}\,-\,{\omega }_{{{{{{{{\rm{h}}}}}}}}}}{2}{\tau }_{12}\right)\\ \times \exp \left(-\frac{{\tau }_{12}}{{T}_{2}}\right)\cos \left({\omega }_{0}{\tau }_{12}\right).$$

(2)

The evolution of spin gratings for trions and resident electrons is governed by their population and spin dynamics in magnetic field. The hole spin grating lifetime is limited by the trion lifetime. The electron spin grating in the ground state is responsible for the long-lived spin-dependent echo which appears if the third pulse is applied. The latter transforms the spin grating back into the optical coherence, leading to the appearance of the photon echo after the rephasing time *τ*_{12}^{15}. The decay of LSPE as a function of *τ*_{23} is governed by the spin dynamics of resident electrons. HHHH and HVVH polarization schemes give access to longitudinal *T*_{1,e} and transverse \({T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }\) spin relaxation times, respectively.

In the studied (In,Ga)As/GaAs QDs the value of *g*_{h} = 0.18 is of the same order of magnitude as the electronic *g*-factor *g*_{e} = −0.52^{30}. Therefore, it should be taken into account in contrast to previous studies where the Zeeman splitting in the trion state was neglected. In addition, it should be noted that the PE signal depends sensitively on the orientation of crystallographic axes with respect to the magnetic field direction due to the strongly anisotropic in-plane *g*-factor of the hole in semiconductor quantum wells and dots^{30,31}. In our studies, the sample was oriented with the [110] crystallographic axis parallel to **B** which corresponds to the case when the H- and V-polarized optical transitions have the photon energies of *ℏ**ω*_{0} ± *ℏ*(*ω*_{e} − *ω*_{h}) and *ℏ**ω*_{0} ± *ℏ*(*ω*_{e} + *ω*_{h}), respectively.

The three-pulse PE amplitude as a function of delay time *τ*_{23} and magnetic field *B* are shown in Fig. 3. In full accord with our expectations, we observe that application of a moderate magnetic field *B* < 1 T drastically changes the dynamics of three-pulse PE. In HHHH polarization scheme the large offset emerges which decays on a timescale significantly longer than the repetition period of laser pulses, i.e. *T*_{1,e} ≫ 10 ns. The short decay, which is also present at *B* = 0, with the time constant *T*_{1} = 0.26 ns is associated to the trion lifetime. In the HVVH polarization scheme, long-lived oscillatory signal appears which is attributed to the Larmor spin precession of resident electrons and decays exponentially with \({T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }\). At shorter delays, the signal behavior is more complex due to the superposition of spin-dependent signals from trions and resident electrons.

Further insight can be obtained from the magnetic field dependence of LSPE signal which is measured at the long delay *τ*_{23} = 2.033 ns when the contribution from trions in three-pulse PE is negligible (see Fig. 3b). The delay time *τ*_{12} is set to 100 ps which is shorter than the optical coherence *T*_{2}. At zero magnetic field, the PE is absent in the HVVH polarization scheme and shows only very weak amplitude in HHHH configuration. An increase of magnetic field leads to the appearance of LSPE in both polarization configurations. For HHHH we observe a slow oscillation which is governed by Larmor precession of both electron and hole spins during *τ*_{12} when the spin grating is initialized by the sequence of two pulses. In the HVVH scheme the LSPE oscillates much faster because it is mainly determined by the Larmor precession of resident electron spins during *τ*_{23}, which is roughly 20 times longer than *τ*_{12}.

In order to describe the experimental results quantitatively, we extended the theory from Langer et al.^{15} by taking into account both electron and heavy-hole Zeeman splitting (for details see Supplementary Note 1). We analytically solve the Lindblad equation for the (4 × 4) density matrix to describe the temporal evolution between the first and second pulses for 0 < *t* < *τ*_{12} and after the third pulse for *t* > *τ*_{12} + *τ*_{23}. The spin dynamics of trions and electrons in external magnetic field for *τ*_{12} < *t* < *τ*_{12} + *τ*_{23} is described by the Bloch equations. The three-pulse PE amplitude in HHHH scheme is given by

$${P}_{{{{{{{{\rm{HHHH}}}}}}}}} \propto \;{{{{{{{{\rm{e}}}}}}}}}^{-\frac{2{\tau }_{12}}{{T}_{2}}}\left[2{{{{{{{{\rm{e}}}}}}}}}^{-\frac{{\tau }_{23}}{{\tau }_{{{{{{{{\rm{r}}}}}}}}}}}{\cos }^{2}\left(\frac{{\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}}{2}{\tau }_{12}\right)+{{{{{{{{\rm{e}}}}}}}}}^{-\frac{{\tau }_{23}}{{T}_{{{{{{{{\rm{T}}}}}}}}}}}{\sin }^{2}\left(\frac{{\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}}{2}{\tau }_{12}\right)\right.\\ \left.\;+\,{{{{{{{{\rm{e}}}}}}}}}^{-\frac{{\tau }_{23}}{{T}_{{{{{{{{\rm{1,e}}}}}}}}}}}{\sin }^{2}\left(\frac{{\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}}{2}{\tau }_{12}\right)\right]$$

(3)

Here \({T}_{{{{{{{{\rm{T}}}}}}}}}^{-1}={\tau }_{{{{{{{{\rm{r}}}}}}}}}^{-1}+{T}_{{{{{{{{\rm{h}}}}}}}}}^{-1}\) is the spin lifetime of the trion. For moderate magnetic fields *B* ≤ 1 T we can assume that the spin relaxation time of hole in QDs *T*_{h} is significantly longer than *τ*_{r} and, therefore, in our case *T*_{T} = *τ*_{r}^{32}. The first and second terms on the right hand side correspond to the trion contribution, while the last term is due to the LSPE from resident electrons.

For HVVH polarization we obtain

$${P}_{{{{{{{{\rm{HVVH}}}}}}}}} \propto {{{{{{{{\rm{e}}}}}}}}}^{-\frac{2{\tau }_{12}}{{T}_{2}}}\left[{{{{{{{{\rm{e}}}}}}}}}^{-\frac{{\tau }_{23}}{{T}_{{{{{{{{\rm{T}}}}}}}}}}}{r}_{{{{{{{{\rm{h}}}}}}}}}\cos ({\omega }_{{{{{{{{\rm{h}}}}}}}}}{\tau }_{23}-({\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}){\tau }_{12}-{\phi }_{{{{{{{{\rm{h}}}}}}}}})\right.\\ \left.\;+\,{{{{{{{{\rm{e}}}}}}}}}^{-\frac{{\tau }_{23}}{{T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }}}{r}_{{{{{{{{\rm{e}}}}}}}}}\cos ({\omega }_{{{{{{{{\rm{e}}}}}}}}}{\tau }_{23}+({\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}){\tau }_{12}-{\phi }_{{{{{{{{\rm{e}}}}}}}}})\right]$$

(4)

where for simplicity we introduce the following parameters: phases *ϕ*_{e}, *ϕ*_{h} and amplitudes *r*_{e} and *r*_{h}. The subscript e, h corresponds to the electron or trion contributions which are given by the first and second terms on right-hand side in Eq. (4), respectively. The parameters are given by Supplementary Eqs. (55)-(57). They are determined by the Larmor precession frequencies *ω*_{e} and *ω*_{h}, delay time *τ*_{12}, trion lifetime *τ*_{r}. The *g*-factors of electrons and holes are known from previous studies^{30,33}. Therefore, the only unknown parameter is the spin dephasing time of resident electrons \({T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }\). Note that if the *g*-factors of electrons and holes are unknown they can be used as additional fitting parameters in the description below.

In order to determine \({T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }(B)\), we fit the transient signals in HVVH polarization for different magnetic fields as shown exemplary for the transient at *B* = 0.1 T by the solid red line in Fig. 3a. For the LSPE when *τ*_{23} ≫ *τ*_{r} = *T*_{2}/2 only the second term in Eq. (4) remains, which simplifies the fitting procedure. Three parameters of the LSPE signal, i.e. decay rate \(1/{T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }\), amplitude *r*_{e}, and phase *ϕ*_{e}, were extracted from the fit which are plotted as blue dots in Fig. 4 as a function of the magnetic field. It follows from Fig. 4a that the spin dephasing rate increases linearly with the increase of *B*. Such behavior is well established in ensembles of QDs and it is related to the fluctuations of electron *g*-factor value in different QDs^{32}. It can be described as

$$\hslash /{T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }=\hslash /{T}_{{{{{{{{\rm{2,e}}}}}}}}}+{{\Delta }}{g}_{{{{{{{{\rm{e}}}}}}}}}{\mu }_{{{{{{{{\rm{B}}}}}}}}}B,$$

(5)

where *T*_{2,e} is the transverse spin relaxation time and Δ*g*_{e} is the inhomogeneous broadening of the electron *g*-factor. The linear fit with this expression shown in Fig. 4a by the red dashed line gives *T*_{2,e} = 4.3 ns and Δ*g*_{e} = 4 × 10^{−3}.

The parameter *ϕ*_{e} in Fig. 4b starts from −0.8 rad in magnetic fields below 0.1 T and approaches zero in fields above 0.8 T. The amplitude *r*_{e} in Fig. 4c gradually rises with an increase of *B* up to 0.4 T and remains the same in larger magnetic fields. We calculate the magnetic field dependence of amplitude and phase of LSPE using Supplementary Eqs. (56) and (57), respectively, using *g*_{e} = −0.516, *g*_{h} = 0.18, *T*_{T} = *τ*_{r} = 0.26 ns and *τ*_{12} = 33.3 ps. The resulting curves are shown by red solid lines in Fig. 4 and are in excellent agreement with the experimental data. We note that in the limit of large magnetic fields, which corresponds to the condition of ∣(*ω*_{e}−*ω*_{h})∣*τ*_{12} ≫ 1, the amplitude of LSPE saturates (*r*_{e} → 1) and the phase of the signal approaches zero (*ϕ*_{e} → 0) which gives the simple expression \({P}_{{{{{{{{\rm{HVVH}}}}}}}}}\propto \cos [{\omega }_{{{{{{{{\rm{e}}}}}}}}}{\tau }_{23}+({\omega }_{{{{{{{{\rm{e}}}}}}}}}-{\omega }_{{{{{{{{\rm{h}}}}}}}}}){\tau }_{12}]\) for a long-lived signal at *τ*_{23} ≫ *τ*_{r}. We emphasize that this expression takes into account the non-zero *g*-factor of the hole *g*_{h} which plays an important role in the formation of the LSPE signal.

After evaluation of \({T}_{{{{{{{{\rm{2,e}}}}}}}}}^{* }(B)\), we can reproduce the LSPE signals as a function of *τ*_{23} and *B* using Eqs. (3) and (4) which are shown by red curves in Fig. 3 in both HHHH and HVVH polarization configurations. Here, the longitudinal spin relaxation time *T*_{1,e} is set to 23 ns. In general this relaxation process can be neglected because *T*_{1,e} strongly exceeds *τ*_{23}. Excellent agreement is obtained at all time delays and magnetic fields. We note that the small discrepancies in HHHH polarization configuration at the magnetic fields around 0 and 1 T are attributed to the presence of a weak background signal possibly due to a population grating in the ground states as previously discussed for the case of Fig. 2c. Nevertheless, importantly the HVVH configuration which corresponds to fully coherent transformation between optical and spin coherence is free from any background.