Optically induced rotation of a quantum dot exciton spin
We demonstrate control over the spin state of a semiconductor quantum dot exciton using a polarized picosecond laser pulse slightly detuned from a biexciton resonance. The control pulse follows an earlier pulse, which generates an exciton and initializes its spin state as a coherent superposition of its two non-degenerate eigenstates. The control pulse preferentially couples one component of the exciton state to the biexciton state, thereby rotating the exciton’s spin direction. We detect the rotation by measuring the polarization of the exciton spectral line as a function of the time-difference between the two pulses. We show experimentally and theoretically how the angle of rotation depends on the detuning of the second pulse from the biexciton resonance.
pacs:03.67.Lx, 42.25.Ja, 42.50.Dv, 78.67.Hc
Coherent manipulation of the quantum states of a physical system is a critical step towards novel applications in quantum information processing (QIP). Semiconductor quantum dots (QDs) exhibit atomic-like energy spectrum and are compatible with modern micro- and optoelectronics. They are considered as excellent candidates for forming the building blocks for these future technologies Loss and DiVincenzo (1998), and they form the best interface between quantum light and matter Li et al. (2003); Zrenner et al. (2003); Simon et al. (2010). Physical realizations of quantum bits (qubits) and logic gates require controlled two level systems. The spins of QD-confined charge carriers have been proposed for this task Loss and DiVincenzo (1998); Economou et al. (2006) and recent experiments successfully demonstrated their state preparation and control. Preparation was achieved either by electron-hole pair photogeneration followed by electrical field induced separation Kroutvar et al. (2004); Young et al. (2007); Ramsay et al. (2008), or by optical pumping Atatüre et al. (2006); Gerardot et al. (2008); Press et al. (2008). Control of the initiated state was then demonstrated either by the AC-Stark effect, induced by an optical pulse Press et al. (2008); Berezovsky et al. (2008), or by the accumulation of a geometric phase Wu et al. (2007); Greilich et al. (2009); Kim et al. (2010). These impressive achievements require a series of optical pulses and the presence of a strong fixed magnetic field.
A QD confined electron - hole pair (exciton) is the fundamental optical excitation in QDs, essential for interfacing light qubits with matter spin qubits. It was also proposed for QIP realizations Troiani et al. (2000); Biolatti et al. (2000). The spin state of the optically active (bright) exciton Bonadeo et al. (1998); Flissikowski et al. (2001); Boyle et al. (2008); Kosaka et al. (2008), and that of the dark exciton Poem et al. (2010) have been directly accessed optically. Moreover, we have recently shown that unlike carriers’ spin, the exciton spin can be initiated at any desired state by a single optical pulse Benny et al. (2011). Partial control of the exciton’s spin was recently demonstrated Boyer de la Giroday et al. (2010), albeit the time required for a single operation was comparable to the exciton’s radiative lifetime (1 ns).
Here we demonstrate, for the first time, control over the spin of the bright exciton using a single, few picosecond long, laser pulse. The pulse duration, which is two orders of magnitude shorter than the exciton lifetime, permits many coherent operations. Moreover, the same control can be applied to the spin of the dark exciton, whose lifetime is three orders of magnitude longer McFarlane et al. (2009).
Here, we demonstrate this control using a circularly polarized laser pulse, slightly detuned from a resonance of an excited biexciton state. The pulse selectively couples one of the exciton spin states to the biexciton, while leaving the other state unaffected. The method is conceptually similar to that used recently on ensembles of charged QDs Wu et al. (2007); Greilich et al. (2009). The experiment is conducted as follows: At first, a right (R) or left (L) circularly polarized laser pulse tuned into an excitonic resonance of a single semiconductor quantum dot generates an excited bright exciton. The spin state of this excited exciton has a total angular momentum projection of J=1 () or -1 (), respectively. The excited exciton then non-radiatively relaxes to the ground-state. As discussed below, the spin is preserved during this fast relaxation Benny et al. (2011). This results in the formation of a coherent superposition of the two non-degenerate ground exciton eigenstates Gammon et al. (1996); Kulakovskii et al. (1999),
which by themselves correspond to excitations by horizontally (H) or vertically (V) linearly polarized pulses, respectively Benny et al. (2011). Since the eigenstates are non-degenerate, they evolve at different paces, and the exciton spin precesses in time between the and the states Bonadeo et al. (1998); Flissikowski et al. (2001); Boyle et al. (2008); Benny et al. (2011) [Fig. 1(a)]. Resonant excitation by an R- polarized pulse results in the following spin state precession:
where is the energy difference between the exciton eigenstates. Then, a second, delayed, circularly polarized pulse is tuned into (or slightly detuned from) an excited biexciton resonance. As illustrated in Fig. 1(b), this particular resonant level includes two states, in which the two electron spins are parallel to each other and they are anti-parallel to the two holes’ spins. The total angular momentum projection of these two states is J=2. We note that the biexciton resonance used here, where both the electrons and the holes are in triplet configurations, is different than that used in Ref. Benny et al., 2011, where the electrons are in a singlet configuration.
Since an R (L) polarized pulse carries with it angular momentum of 1 (-1), when tuned into the biexciton resonance, it couples only the J=1 (-1) exciton state to the J=2 (-2) biexciton state. The duration of the laser pulse is much shorter than the period of the exciton’s precession. Hence during the pulse, the coupled exciton - biexciton states can be safely viewed as an isolated two-level system Economou et al. (2006). There is an analytical solution for the dynamics of this system for the case of hyperbolic-secant pulse shape Economou et al. (2006); Rosen and Zener (1932); Takagahara (2010). The coupling that such a pulse induces between the relevant exciton and biexciton states is given by Takagahara (2010), , where is the laser frequency, is the pulse bandwidth, and is the Rabi frequency. If the exciton spin state just before the second pulse is given by Eq. (Optically induced rotation of a quantum dot exciton spin), the state after an R-polarized hyperbolic-secant pulse is given by Economou et al. (2006); Takagahara (2010),
where is the Gaussian hypergeometric function (also denoted as ), and , where is the detuning from the resonance frequency . Using known properties of hypergeometric functions Takagahara (2010), the probability to populate the biexciton if the time-difference between the two pulses is , reads:
A second, non-detuned () -pulse (), coincident in time with the first pulse () transfers the entire excitonic population to the biexciton state Economou et al. (2006). In general, the absorption of the second pulse depends on the direction of the precessing exciton spin relative to the polarization of the light pulse Benny et al. (2011). Since the intensity of the photoluminescence (PL) emission from the biexciton spectral lines is a measurement of the pulse absorption, the emission intensity oscillates as the delay between the two pulses increases. These oscillations provide an experimental way to measure the excitonic spin by projecting it onto directions determined by the polarization of the second pulse Benny et al. (2011). The second pulse affects also the non-transferred excitonic population. This is because a circularly polarized pulse couples only one component of the exciton spin state. Thus, the pulse affects the relative amplitude and phase between the excitonic spin eigenstates. The change in relative phase can be interpreted as a “rotation” of the exciton Bloch sphere about the - axis, and the change in the relative amplitude as “squashing” of the Bloch sphere from the pole towards the pole. The exciton state after such an operation is generally expressed, up to normalization, by the first two terms on the right hand side of Eq. (Optically induced rotation of a quantum dot exciton spin). In order to detect it, one needs to measure the spin direction of the exciton after the second pulse. We show below that the excitonic spin projection on the - axis of the spin Bloch sphere is readily available by measuring the net polarization of the PL from the exciton lines.
The idea is schematically described in Fig.1(a): As long as the exciton’s spin is oriented along the Bloch sphere’s equator, its projection on the - axis is zero, and both eigenstates of the exciton are equally populated. In contrast, if the spin is forced, by the second pulse, to move in a trajectory which leaves the equator, then the populations of the two eigenstates are no longer equal. Once the second pulse is turned off, the exciton spin again precesses around the - eigenstates axis, and the eigenstates population difference created during the pulse is kept constant. Since the PL emission is proportional to the probability of population, variations in the population result in measurable changes in the PL from the exciton’s spectral lines. Clearly, the normalized difference between the emission intensities of the two cross-linearly polarized exciton lines is a direct experimental measurement of the spin projection on the - axis of the Bloch sphere. The energy difference between the cross linearly polarized components is larger than their spectral widths. Therefore, their intensities can be simultaneously measured using a circular polarizer in front of the monochromator.
The excitonic PL emission in our experiment is not temporally resolved. Therefore, it also contains contribution from the biexciton population [Eq. (Optically induced rotation of a quantum dot exciton spin)], which decays incoherently into excitonic population. However, due to the polarization selection rules [Fig. 1(b)], these incoherent excitons equally populate the or eigenstates, and therefore do not contribute to the eigenstates population difference. The difference can thus be calculated directly from the exciton state immediately after the second pulse, Eq. (Optically induced rotation of a quantum dot exciton spin). With Eq. (Optically induced rotation of a quantum dot exciton spin) and using properties of hypergeometric functions Takagahara (2010) one obtains,
being the Gamma function. Eq. (5) shows that the oscillations in have the same frequency as those of the biexcitonic population, Eq. (Optically induced rotation of a quantum dot exciton spin), but they are shifted in phase by /2. The amplitude of the oscillations, Eq. (6), depends on the pulse intensity (), its bandwidth () and its detuning (). In particular, the sign of the amplitude is given by the sign of the detuning, and on resonance the amplitude vanishes. By using Eqs. (Optically induced rotation of a quantum dot exciton spin)-(6), the angle of the induced rotation is given by
A larger-than- pulse permits any rotation angle.
The studied sample contains one layer of strain-induced InGaAs QDs embedded in the center of a one optical wavelength (in matter) microcavity that enhances the PL collection efficiency. For the optical measurements the sample was placed inside a tube immersed in liquid Helium, maintaining sample temperature of 4.2K. A spatially isolated single QD from a low-density area of the sample was excited with two dye lasers, synchronously pumped by a frequency doubled Nd:YVO passively mode-locked pulsed laser. The dye laser’s pulse temporal full-width-at-half-maximum (FWHM) was measured to be 9 ps. An in-situ microscope objective was used both to focus the exciting beams onto the sample, and to collect the light emitted from it. The collected light was projected upon a desired polarization, dispersed by a 1 meter monochromator, and then detected by either an electrically-cooled charge-coupled-device array detector or by a single channel, single photon, silicon avalanche photodetector. The system provides spectral resolution of about 10 eV. One dye laser was tuned to an excited exciton resonance, 29 meV above the ground-state exciton emission line. Excitation in this resonance results in a high () degree of linear polarization memory of the exciton emission line. Moreover, the width of this resonance, which is greater than 500 eV, is much larger than its polarization splitting of 60 eV Benny et al. (2011). Therefore, relaxation to the ground exciton state occurs before any appreciable dephasing or rotation of the spin state can occur, and the spin state of the excited exciton, determined by the polarization of the exciting laser, is preserved during this fast relaxation Poem et al. (2010); Benny et al. (2011). The other dye laser was tuned into (or slightly detuned from) an excited biexciton resonance, at 33.7 meV above the exciton doublet. Using detailed polarization sensitive spectroscopic studies Benny et al. (2011) and a many-carrier theoretical model Poem et al. (2007) we unambiguously identified this resonance as the J=2 biexciton states described in Fig. 1(b).
In Fig. 2 we present the R polarized PL intensity from the QD vs. the PL energy and the time difference () between the first pulse, tuned into the exciton resonance, and the second one, detuned by -63 eV from the biexciton resonance. Both lasers were R polarized. The higher (lower) energy doublet is the emission from the ground state exciton (biexciton). The oscillations of the PL from the biexciton reflect the precession of the exciton spin state, as initialized by the first pulse Benny et al. (2011). This behavior is described by Eq. (Optically induced rotation of a quantum dot exciton spin). The oscillations of the PL from the two excitonic components, reflect the variations of the exciton spin projection on the eigenstates axis, induced by the detuned second pulse, as described by Eq. (5). The line at 1.28293 eV is due to the positively charged exciton. Its PL does not oscillate with .
Fig. 3(a) presents a PL-excitation spectrum of the biexciton line, for 30 ps difference between the first and second pulse. The intensity of the second pulse was tuned slightly below population inversion at resonance excitation. At this particular intensity the emission intensity from both the exciton and biexciton lines is optimized. This allows simultaneous testing of Eqs. (Optically induced rotation of a quantum dot exciton spin) and (5). The dashed line represents the calculated biexciton population [Eq. (Optically induced rotation of a quantum dot exciton spin)], for a 9 ps FWHM hyperbolic-secant pulse (=145 eV). The deviation of the measured intensity from the theoretical line at low energies is due to a near-by J=0 excited biexciton resonance Benny et al. (2011). We note that the width of the J=2 biexciton resonance is completely determined by the spectral width of the laser. This indicates that any dephasing and relaxation processes are significantly slower than the pulse duration. Indeed, in cw excitation the spectral width of this resonance is significantly narrower. This implies that the J=2 biexciton remains coherent during the second pulse. In Fig. 3(b) we present the intensity of the PL from the biexciton lines as a function of . Black (blue) line presents off (almost on) biexcitonic resonant excitation. In both cases, the evolution is cosinusoidal, as in Eq. (Optically induced rotation of a quantum dot exciton spin). In Fig. 3(c) we present the intensity of the PL from the exciton lines as a function of , for both off and almost on resonant excitation as in (b). Solid (dash) line denotes the PL from the H- (V-) polarized component of the excitonic doublet. In Fig. 3(d) we present the difference between the PL intensities from these two cross-linearly polarized components, normalized by their sum at negative delay time (before the second pulse). The oscillations are sinusoidal in time, as in Eq. (5). Their amplitude depends on the detuning from resonance. below (above) resonance, the amplitude is negative (positive) and on resonance it vanishes, as expected from Eq. (6). This dependence is summarized in Fig. 4, which presents the amplitudes of the measured oscillations in the exciton polarization, vs. the normalized detuning, . The lines present the amplitude calculated by Eq. (6), for =0.35 (0.7- pulse), slightly below inversion. The figure describes co- (blue) and cross- (red) circularly polarized laser pulses. The measured and calculated spin rotation angles [Eq. (7)] vs. the detuning are presented in the inset to Fig. 4. The agreement between the measured rotations and the theoretically calculated ones (which assume no dephasing) indicates a close to unity rotation fidelity.
In summary, we demonstrated that the polarization of the QD exciton spin can be rotated by a single, short optical pulse. We showed that the rotation can be directly detected by monitoring the intensities of the two components of the excitonic spectral line.
The support of the US-Israel binational science foundation (BSF), the Israeli science foundation (ISF), the ministry of science and technology (MOST) and that of the Technion’s RBNI are gratefully acknowledged.
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