1.

INTRODUCTION

Semiconductor quantum dots (QDs) have revealed very promising and conclusive demonstrations of the quantum control of the exciton coherence by optical means. The possibility to drive the QD in the Rabi-flopping regime with resonant ultra-short laser pulses has been shown, yielding a precise determination of the quantum phase between the ground and excited states [1], [2]. However, neutral quantum dots exhibit richer physics than a two-level system, as the exciton states present a level-degeneracy lifting induced by electron/hole anisotropic exchange interaction (few tens of µeV); this fine structure splitting (FSS) naturally builds up a V-shaped system, offering the extra possibilities to write and read the coherent superposition of the two excited states. The resulting quantum beats associated to the FSS have been already reported, showing a coherence time lasting over few hundreds of picosecond [3].

Resonant or quasi-resonant excitation allows one to transfer an optical coherence to the V-shaped system with respect to the optical selection rules. The two exciton states denoted |X> and |Y> are related to optical dipoles mainly aligned respectively along [110] ≡ X and 11̄0 ≡ Y crystallographic axes[4], [5]. As a consequence, a resonant optical field with a polarization having both X and Y components imprints a determinate initial phase in between the |X> and |Y> states [6].

We propose here a more sophisticated experimental protocol allowing the quantum-beat phase manipulation, not only by means of light polarization control, but also by applying a longitudinal magnetic field. The key role of the magnetic field consists in changing the symmetry of the linear dipoles into elliptic ones (see Fig. 1): the optical selection rules evolution results in a competition between the longitudinal magnetic field and the effective in-plane electron/hole exchange field, providing a magnetic field tuning of the exciton quantum beats phase.

Figure 1:

Scheme of the V-shape system, as a as a function of the Zeeman splitting δ_z. When δ_z = 0, the optical dipoles are linearly polarized and parallel respectively to the X and Y-axes. Their energies are split by the FSS δ₁. In the limit δ_z >>δ₁, the optical dipoles become circularly polarized, whereas in the intermediate regime defined by δ_z ∼ δ₁ and delimited by the dashed circle, the dipoles show elliptic symmetry with principle axes still aligned with X and Y directions. Their energy separation simply writes . Ω_x(t) and Ω_y(t) denote the optical pulse couplings with respect to the excitons dipole symmetry at zero magnetic field

2.

SAMPLE AND EXPERIMENTAL SETUP

The studied sample contains three layers of neutral InGaAs/GaAs QDs, with a nominal density of∼5×10¹⁰ cm−2. The QDs were embedded inside a λ-planar GaAs cavity made of 24 AlGaAs/GaAs Bragg pairs for the bottom mirror and 12 for the top mirror. The role of the cavity is to increase the detection sensitivity of the signal[7] and its Q-factor is close to ∼2000. To measure the exciton dynamics, we use a reflection- type pump-probe setup, as depicted in Fig. 2. A picosecond mode-locked Ti:Sa laser is split into three beams (pump, probe, and reference beams) with energy matching the optical mode of the cavity centered at 1.337 eV. The pump and probe beams are focused on the sample with a microscope objective (N.A. = 0.5) giving a spot diameter of ∼1 µm. In this way, we estimate that only N_QD ∼ 300 QDs spectrally in resonance with the cavity mode contribute to the reflectivity signal. The standard deviation of the probed QDs sample √N_QD/N_QD ≈ 5% is small enough to catch the essential aspects of the exciton dynamics with average values [8]. New experimental strategies are required to increase the signal-to-noise ratio of the small number of probed QDs. In this sense, we developed an optical heterodyne amplification of the reflectivity signal [9], [10]. The probe and reference beams are passed through acousto-optic modulators leading to an optical frequency shift of the first diffraction order: the probe optical frequency becomes ω_pr = ω_L +ω₁ and the reference ω_ref = ω_L +ω₂, with ω₁ = 110 MHz and ω₂ = 112 MHz. We note here that ω₁, ω₂ are several orders of magnitude less than the cavity-mode linewidth. Then, the reflected part of the probe pulse having interacted with the QDs and the reference beam are temporally overlapped onto the avalanche photodiode, giving rise to the two-wave mixing signal, i.e., the heterodyne signal beating at |ω₁ −ω₂|= 2 MHz in our case.

Figure 2:

Scheme of the pump/probe setup with heterodyne amplification of the reflectivity signal. AOM: acousto-optics modulator; HWP: half-wave plate; NPCBS: non-polarizing cube beam splitter; APD: avalanche photodiode; Δt: pump-probe delay.

To go further in describing the advantage of the heterodyne detection, we can express the whole intensity contributions received by the APD:

where E^⃗_p,r (t), E^⃗_ref(t) are the electric field associated respectively to the reflected pump and reference pulse envelopes. In the same way, E^⃗_QD(t) refers to the envelope of the probe pulse field reflected by the QDs, of which exact derivation is given further, and E^⃗_{pr, r} stands as the inherent reflected probe component that has not interacted with the QDs. <···> finally denotes the detector time averaging. As an optical chopper modulates the pump intensity at a frequency ω_p = 1.7 kHz, a double stage lock-in amplifier is then able to isolate with a high pump-signal rejection, where E^⃗_ref plays both the role of an amplification and a polarization analysis of the QDs reflectivity signal. The average probe power is set at 0.1 µW, whereas the reference and pump powers are close to 5 µW.

The polarization configurations used in this work are described in Figs. 3(a) and 3(b) for the three beams: the pump linear polarization makes an angle ξ with the X axes, while the probe polarization is aligned with one of the eigenstate directions and the reference could be either parallel or perpendicular to the probe polarization direction. Figure 3(c) provides a first insight into the experimental results. This preliminary experiment consists in measuring the decay of the reflectivity signal associated to both |X> and |Y> exciton eigenstates, which are addressed selectively by an X (i.e., ξ = 0) and Y (i.e., ξ = π/2) linearly polarized pump. In the same way, the population dynamics of the |X> (|Y>) exciton is recorded with the corresponding probe polarization along X (Y), and finally the reference is set parallel to the probe in order to amplify the probe reflectivity change. The heterodyne signals are shown in Fig. 3(c) and the dynamics for the |Y> state has been artificially set negative for clarity. Remarkably both dynamics present almost the same amplitude and are well fitted with the same mono-exponential decay of 710±10 ps, giving with good approximation the radiative lifetime, assuming the exciton relaxation rate is negligible over the radiative time scale as already confirmed in Refs. [11], [12]. From this experimental result, we conclude that the two optical exciton transitions have nearly the same oscillator strength; this assumption will deeply simplify the derivation of the reflected probe pulse onto the QDs.

Figure 3:

Description of the experimental polarization configurations. (a) The linear polarization of the probe and reference fields are parallel and aligned with the crystallographic axes X, whereas they are set perpendicular in (b). (c) Dynamics of the reflectivity signal: the pump polarization is parallel to an eigenstate direction X or Y, and the polarization of the probe and reference are specified in the legend. The signal detected with orthogonal reference and probe polarizations has been multiplied by five.

Lastly, to verify that the polarization configurations mentioned here permit one to selectively study the exciton eigenstate population dynamics, the reference linear polarization is now rotated by π/2 and set perpendicular to the probe. The heterodyne signal is shown by the continuous gray line in Fig. 3(c) and appears to be vanishingly small, confirming that the reflected probe acquires no significant orthogonal polarization, when the probe polarization is aligned with one of the exciton eigenstate directions.

3.

MODELIZATION OF THE PHOTOINDUCED HETERODYNE SIGNAL

This pump-probe experiment offers a lot of possible polarization configurations giving each different aspect of the exciton dynamics, as three independent beams are involved. Before presenting a complete experimental study, this section is devoted to modelizing the photo-induced heterodyne signal, in various polarization configurations. First, we will pay attention to writing the exciton eigenstates basis in the presence of a longitudinal magnetic field. Then, the reflected probe pulsed E^⃗_QD(t) will be calculated, and its mixing with the reference having different linear polarization direction and giving rise to the optical heterodyne beats will be discussed.

3.1

Exciton eigenstates symmetry in the presence of a Faraday magnetic field

When applying a magnetic field along the growth direction, the Hamiltonian describing the exciton states is written as [13]

where |± 1> are written in Fig. 1, δ₁ is the FSS value, and δ_z is the Zeeman splitting given by g_zµ_BB_z with g_z being the exciton longitudinal g factor. Diagonalization of the Hamiltonian leads to new eigenstates denoted |EL+> and |EL−>, with their corresponding energies:

α is the angle defined as 1/2 Arctan (δ_z/δ₁). When the condition δ_z>>δ₁ is fulfilled, α tends to π/4 and the circular symmetry of the exciton states is restored. It is now clear that for the intermediate regime of Zeeman splitting, the states present an elliptic symmetry. When regarding more precisely the interaction with the resonant linearly polarized pump pulse, the Hamiltonian is given by

where . The optical coupling coefficients in expressed in the {|0>,|EL−>,|EL+>} basis are complex functions of (ξ, α), and differ for each optical transition |0>→|EL+> and |0>→|EL−>. The key point is that their arguments are different, yielding the possibility to control the initial phase (i.e., just after the passage of the pump pulse) between |EL+> and |EL−> by changing α (i.e., the magnetic field) when fixing ξ. At this stage, it is easy to check on Eq. (2) that, for ξ = 0 or ξ = π/2, both optical couplings have the same arguments and are simultaneously either purely imaginary or real, preventing the phase manipulation by applying the magnetic field.

The more intriguing case is for ξ = π/4. When this pump polarization configuration is employed, the argument of the optical coupling to |EL+> state, Arg[sinα −i cosα], is α − π/2, whereas the other one, Arg[cosα +i sinα], is α, so that the difference of both arguments becomes α independent. This also means that the ξ = π/4 configuration renders inoperative the magnetic field control of the exciton quantum beats phase. From this very preliminary analysis, we conclude that low- symmetry pump polarization direction ξ is required to validate the magnetic field induced phase control protocol (i.e., for ξ different from 0, π/4, and π/2).

3.2

Expressions of the reflectivity signals

We propose here to express the probe electric field reflected by the QDs; this latter should be a function of ξ, α, and the pump-probe delay Δt but also depends on the probe polarization, while the heterodyne signal depends in addition on the reference polarization. First we consider the density operator evolution given by the Liouville equation:

We use the eingenstates basis {|0>,|EL−>,|EL+>}. Before the pulse arrival, the density operator is written as Diag[1,0,0] and becomes after pump pulse ρ(0⁺) = W(∞).ρ(0−).W^†(∞), where the unitary pump operator [14] is and exists in the interval of the pump pulse duration τ_p∼2 ps. The corresponding spectral width of the laser 1/τ_p is on the one hand much larger than the exciton splitting energy, and on the other hand comparable with the cavity-mode linewidth. ρ(0⁺) naturally serves as the initial condition to solve Eq. (3). Then we find for the population terms

and for the coherence terms

where θ = ∫Ω_p (t)dt is the Rabi area of the pump pulse, γ = 1/τ_r +1/T₂, T₂ is the coherence time of the excitons beatings, and .

While ρ(t) describes the free evolution of the system after the pump pulse, we note σ(t) the density matrix taking into account the interaction between the QDs and the probe field for a given pump-probe delay Δt. As the pulse duration τ_p is much shorter than the interband coherence time, σ(t) is governed by the coherent evolution equation:

where for a probe polarization direction along X. The solution of Eq. (4) is calculated in the first order in the probe field and can be expressed by

with the conditions −τ_p/2 <t’ −Δt <τ_p/2 and σ_ii(Δt) = ρ_ii(Δt) indicating that the probe field does not change significantly the population terms under the experimental precautions |Ω_pr(t)|<<|Ω_p(t)| and |Ω_pr(t)|τ_p <<1. Only the second terms in Eq. (5) mixes the interactions with the pump and probe fields and participates to the photo-induced signal, whereas the first term solely describes the pump-induced resonant fluorescence. As the reference beam is polarized along the crystallographic axes, it is very convenient to express σ in the {|0>,|X>,|Y>} basis, by applying the change-of-basis operator P according to σ̂ * = P · σ̂ · P^† with

Due to this operation, the coherence elements in σ result in a mixing with population and coherence terms of σ. Then the probe-induced stimulated QDs emission—or in the classical point of view, the reflected part of the probe by the QDs—is now calculated by first expressing the quantum average dipole with j ={X,Y}. In the classical far-field approximations, the QD radiated field is proportional to the oscillating dipole, and the two orthogonal components are

and

with , where µ₀ is the magnetic permeability and the complex coefficient β takes into account the cavity effect on the probe reflection, and contains the phase factor due to the wave propagation from the QDs to the photo-detector. Depending on the reference polarization, the experiment is then sensitive selectively to the QD fields polarized along the X or Y direction. At this stage, it is easy to verify on Eq. (7) that for (ξ = 0, α = 0) (i.e., the pump parallel to the probe polarization direction along X and no applied magnetic field), the Y component of the QDs emitted field is zero giving no heterodyne signal when the reference polarization is along Y as demonstrated in Fig. 3(c). As a lock-in amplifier detection is used, one needs to calculate the amplitude of the harmonic beating at |ω₁ −ω₂|, noted R. We will consider a perfect alignment of the reference and probe polarization directions with the crystallographic axes. In our measurement, the probe polarization is kept constant along X and only the pump and reference polarization change, so that we write the signal R(Δt,ξ,X) when the reference polarization is aligned with the X direction. Then we find, omitting the amplitude pre-factor,

At this stage of the analysis, it is important to comment more on the expression R(∆t,ξ,X) written in (8), in which we can clearly distinguish two components: one decaying with the radiative lifetime refers to the population term, and the other one having a γ decay corresponds to the coherent term oscillating with a period h/δ. The relative amplitude of both components depends only of (ξ,α). Regarding more particularly the coherent term in Eq. (8), this latter can be rewritten under the form defining a phase and an amplitude associated to the quantum beats, which are plotted respectively in Figs. 4(a) and 4(b) as a function of the ratio δ_z/δ₁ for different angles ξ.

Figure 4:

(a) Evolution of the exciton quantum beats phase with the ratio δ_z/δ₁ for different pump polarization directions ξ. (b) Evolution of the beats amplitude versus δ_z/δ₁ for different values of the parameter ξ.

Figure 4(a) confirms the qualitative arguments discussed previously in Sec. IIIA, where the phase remains independent of the applied magnetic field for high-symmetry pump polarization, i.e., ξ = 0,π/4,π/2. However, for low- symmetry pump polarization, the phase becomes sensitive to the magnetic field and saturates to −π/2+2ξ for 0 <ξ < π/2 and −3π/2+2ξ for π/2 <ξ <π. For example, for ξ = π/8[π/2] (3π/8[π/2]), the phase of the quantum beats saturates to−π/4(π/4). The amplitude related to the coherent beating shown on Fig. 4(b) increases with the magnetic field up to the saturation and looks largely independent of the angle ξ. When no magnetic field is applied, the coherent signal amplitude vanishes whatever the angle ξ is, and only the population signal remains [see Fig. 3(c)].

4.

RESULTS AND DISCUSSION

We present in this section the experimental results showing the reflectivity signals gathered in Fig. 5 and obtained when the probe and reference polarization directions are the same and parallel to the X direction. The purpose of this work is to compare the phase of the exciton quantum beats when changing the magnitude of the longitudinal magnetic field; that is why we choose to plot the reflectivity signals in the phase domain ϕ defined for each magnetic field as rather than in the time domain ∆t. Our choice renders the phase evolution more clear. This has required first to measure the FSS δ₁ and the g_z values.

Figure 5:

Reflectivity signals obtained when the probe and reference polarization are set parallel. (a) The pump polarization angle is ξ = 0 and ξ = π/2. (b) ξ = π/4. (c) ξ = 7π/12. (d) Evolution of the exciton splitting energy with the magnetic field, extracted from the period of oscillation. The red points come from Fig. 5(a) and ξ = π/2. (e) Evolution of 1/T2 as a function of the applied magnetic field. The blue dashed fitting curve takes into account the inhomogeneities of the FSS and exciton Landé-g factor g_z, whereas for the black dotted curve, only the g-factor dispersion is considered with the following numerical values: Γg = 0.62 and .

Figure 5(a) shows the experimental results for ξ = 0 and ξ = π/2. When the reference polarization is set parallel to the probe one, we have demonstrated that the coherence signal is a modulation of the decaying population signal [see expression of R(∆t,ξ,X) in Eq. (8)], which is confirmed experimentally. The period of oscillations as a function of the magnetic field can be easily fitted and finally be converted into exciton splitting energy in Fig. 5(d). Therefore, we deduce from the splitting evolution that the average values describing the probed QDs sample δ₁ and g_z are respectively 15 µeV and 2.8. The magnetic field value satisfying the condition δ₁ = δ_z(= g_zµ_BB_z) is then B_z ∼ 100 mT.

The continuous lines in Fig. 5(a) represent the fitting curves, which match very well the experimental data for all the magnetic field values. During the fitting procedure using the expression of R(∆t,ξ,X), τ_r, δ₁, and g_z are maintained fixed as constant parameters, whereas only two free parameters are used: the coherence time T2 and a global renormalization pre-factor. In particular, it is remarkable that the amplitudes of both the population and coherence components that are deeply correlated are very well reproduced simultaneously. As expected, the amplitude of the oscillating part tends to zero for δ₁ >δ_z and saturates in the regime δ₁<<δ_z, as predicted in Fig. 4(b). Finally, the experimental results in Fig. 5(a) show in very good agreement with the modelization that the ξ = 0 and ξ = π/2 reflectivity signals oscillate in phase opposition whatever the magnetic field amplitude is, as demonstrated in Fig. 4(a).

Figures 5(b) and 5(c) show the reflectivity dynamics evolution with the magnetic field, for a pump polarization corresponding to ξ = π/4 and ξ = 7π/12. Once again, as the probe and reference polarizations are set parallel, the quantum beats are in addition to the mono-exponential population signal. However, this latter contribution was removed from the experimental data after the fitting procedure in Figs. 5(b) and 5(c) in order to center the oscillations and make the magnetic field dependence of the phase more clear.

Figure 5(b) shows the two main features in good agreement with the theoretical model: first, the amplitude of the oscillating component grows up when increasing the magnetic field, and secondly no phase shift is observable when changing the magnetic-field magnitude, as expected from the ξ = π/4 experimental configuration. A dashed line joining the intersections of the curves with their respective zero-baselines centered at ϕ = π appears then vertical, and is a guide for the eyes to note the absence of the magnetic-field induced phase shift.

The ξ = 7π/12 case in Fig. 5(c) is the more interesting experimental configuration, as it differs from the high-symmetric pump polarization directions in respect of the exciton dipole polarization, and provides the evidence of the magnetic-field control of the quantum beats phase. The reflectivity signal is plotted over a half-period of oscillation (i.e., 0<ϕ < π). We select only two data for clarity: one measured at low magnetic field (B_z = 53 mT corresponding to α_min ≈ 0.26), and the other one at high magnetic field (B_z = 343 mT corresponding to α_max ≈0.65, close to its maximum limit of π/4). In both cases, the theoretical model fits the data satisfactorily. For the higher magnetic field the curve intersects the zero baseline for ϕ ≈ π/3. A vertical line is added on the plot, as a cursor pointing at this specific angle. The second vertical line points to the zero of the second curve. It is clear that the distance between these two lines reveals the magnetic-field induced phase shift in the exciton quantum beats. The corresponding amplitude of the phase shift is simply written as |ψ_x (α_max,7π/12)− |ψ_x (α_min,7π/12)| and has a value of ≈ 0.38 rad (close to π/8).

Until now, the inhomogeneities inherent to the probed QDs sample have been overlooked. In this regard, the coherence rate 1/T₂ evolution with the magnetic field gives quantitative details on the dispersion of the δ₁ and g_z related distributions. Figure 5(e) shows the 1/T₂ evolution as a function of the applied magnetic field, extracted from the ξ = π/2 experiments. Very similar trends are found with other pump linear polarizations ξ. The dependence looks linear for the highest magnetic fields and deviates substantially from the linear behavior for B_z < 150 mT. These observations are then compatible with the following description: when B_z >150mT, the dispersion of the exciton Landé g factor dominates the quantum beats damping providing a linear dependence of 1/T₂ versus B_z, with a slope close to ∼µ_BΓ_g/2ℏ, where Γ_g is the FWHM of a Lorentzian distribution associated to g_z; for the lowest magnetic fields, i.e., B_z < 150 mT, both δ₁ and g_z dispersions combine. To go further quantitatively, the dispersions Γ_g and Γ_δ1 can be simultaneously evaluated, under the reasonable assumptions of uncorrelated distributions, by averaging numerically the coherent part of the reflectivity signal over both distributions for each magnetic field, as follows:

where is the Lorentzian distribution centered on x̄ and with a dispersion is the expected coherence time at zero magnetic field. Equation (11) is then numerically approximated by , where ᾱ and δ̄ are the quantities used in the fitting procedure as constant parameters for a fixed magnetic field and evaluated with the average values δ₁ and g_z; is the inhomogeneous coherence time taking into account the whole dispersed parameters. It is then possible to find the best set of parameters , Γ_g, and Γ_δ1, which makes the dependence with the magnetic field as close as possible to the measured one. The blue dashed curve in Fig. 4(e) shows the evolution for ps, Γ_g = 0.62, and Γ_δ1 = 5.9 µeV, and presents the same trends as the experimental results. More precisely, the fast increasing of the coherence rate below B_z = 150 mT is well reproduced. Then we can conclude that, for our sample, δ₁ is closed to 15±3 µeV and g_z = 2.8±0.31.

5.

CONCLUSION

We have demonstrated the possibility to tune the exciton quantum beats phase by applying a longitudinal magnetic field [15]. The model describes the experiments with a very good accuracy. The magnitude of the FSS δ₁, as well as the pump polarization direction ξ defines the range where the phase becomes sensitive to the magnetic field. In our sample, δ₁ being only 15 µeV, the magnetic field reaching the saturation of the phase is then restricted to ∼100 mT [see Fig. 2(a)]. However, a relatively small value of the FSS, indicating moderate anisotropic QDs seems to ensure similar oscillator strength of the |X> and |Y> exciton states. Finally, heterodyne amplification of the reflectivity should provide an efficient tool to study exciton dynamics at the level of a single quantum dot.