A quantum dot single-photon source with on-the-fly all-optical polarization control and timed emission
AA quantum dot single-photon source with on-the-fly all-opticalpolarization control and timed emission
Dirk Heinze, Artur Zrenner, and Stefan Schumacher
1, 2 Physics Department and Center for Optoelectronics and Photonics Paderborn (CeOPP),Universit¨at Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA (Dated: October 14, 2018)
Sources of single photons are key elementsin the study of basic quantum optical conceptsand applications in quantum information science.Among the different sources available, semicon-ductor quantum dots excel with their straightforward integrability in semiconductor based on-chip solutions and the potential that photonemission can be triggered on demand. Usually,the photon emission event is part of a cascadedbiexciton-exciton emission scheme. Importantproperties of the emitted photon such as polariza-tion and time of emission are either probabilisticin nature or pre-determined by electronic prop-erties of the system. In this work, we study thedirect two-photon emission from the biexciton. We show that emission through this higher-ordertransition provides a much more versatile ap-proach to generate a single photon. In the schemewe propose, the two-photon emission from thebiexciton is enabled by a laser field (or laser pulse)driving the system into a virtual state inside theband gap. From this intermediate virtual state,the single photon of interest is then spontaneouslyemitted. Its properties are determined by thedriving laser pulse, enabling all-optical on-the-flycontrol of polarization state, frequency, and timeof emission of the photon.
Semiconductor quantum dots have proven theirpromise as basic building blocks in various applicationsin the field of semiconductor based quantum optics andquantum communication. These semiconductor nanos-tructures have been used as well-controlled on demandquantum emitters for single photons as well as forlasing at the single-photon level and to generate po-larization entangled pairs of photons.
It has beendemonstrated that the interaction of the quantum dot’selectronic excitations with optical fields and the emis-sion characteristics of photons can be tailored to a largeextend by use of optical cavities. Even on-chip solu-tions of quantum-dot cavity systems with build-in elec-trically pumped microlaser sources have recently beendemonstrated. If a semiconductor quantum dot is excited from itselectronic ground state, the lowest excited configurationsare the exciton states with one electron-hole pair in thesystem. Through further excitation from either of theexcitons, the biexciton state with two electron-hole pairscan be excited (cf. Fig. 1). These excited states are rel- atively long-lived with lifetimes typically on the order ofa nanosecond such that optical transitions can be stud-ied in detail and also photon emission can be utilizedefficiently. Most previous studies have focused on theemission of one or two subsequent (cascaded) photonsfrom the biexciton to exciton or exciton to ground-statetransition, respectively.
In contrast to this cascadedemission, recently it was noted that semiconductor quan-tum dots can also efficiently couple to an optical lightfield through a direct two-photon transition from groundstate to biexciton and vice versa. Both of these statesare spin-zero states such that a direct two-photon transi-tion is allowed and efficient. Fully stimulated coherenttwo-photon excitation has been demonstrated in bothdegenerate and two-color scenarios. On the otherextreme, a fully spontaneous two-photon emission wasreported and explored. In this Letter we analyze a mixed scenario for thetwo-photon emission from the biexciton: one photon isstimulated, the other one spontaneously emitted as il-lustrated in Fig. 1a. In analogy to a partially stimu-lated down-conversion process, the first photon is trig-gered/stimulated by an external pump laser field. Thisfield is off-resonant to all one-photon active transitionsand drives the system between the biexciton state and Quantum Dot& Cavity γ pure ! ω H ! ω V b | B !| G ! | X V !| X H ! biexcitoncavityground statesingle-photonemissionvirtualstate a | B !| G ! | X ! ↔| G !| B ! ↔| X ! pump FIG. 1:
Single-photon generation through two-photonemission from a quantum-dot biexciton. a
The firstphoton is triggered by a laser field in a stimulated emissionprocess into a virtual level inside the bandgap. The spon-taneously emitted second photon is channeled into a cavitymode and has properties such as polarization and frequencycomplementary to the stimulating laser light field. b Illus-tration of the theoretical model for the quantum-dot cavitysystem analyzed. Details are given in the text. a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec de t un i ng f r o m t w o - pho t on r e s onan c e Δ ( m e V ) -0.5-20 0 4010 20 30 50 Photon density x 10 -3 -3 x 10 ab cde BiexcitondensityExcitondensityGround statedensity no r m . pu l s e i n t en s i t y Pulse envelope
FIG. 2:
All-optical control of the single-photon emission event. a
Computed time-resolved photon density inside thecavity for different detunings of the laser pulse from the two-photon resonance condition. The temporal profile of the laserpulse is shown in b and biexciton, exciton, and ground state densities in c - e , respectively. a virtual level inside the system’s band gap. However,the first photon only gets actually emitted if a secondphoton is spontaneously emitted (in our setup into acavity mode) and bridges the remaining energy gap tothe ground state. Given the first photon’s stimulatednature, its properties are determined by the pump. Fol-lowing from fundamental energy and spin conservation,the second photon has complementary properties such aspolarization and frequency. Therefore changes in the pa-rameters of the pump laser allow for all-optical controland on-the-fly changes to the properties of the emittedsingle photon.Here we present a detailed theoretical analysis for aquantum-dot cavity system. We show numerically that asingle-photon source as discussed above can be realizedfor a wide range of realistic system parameters. Our cal-culations show that the properties of the emitted photon(as a true quantum object) can indeed be all-opticallycontrolled with the classical pump laser field. Further-more, we show that for excitation with a picosecondpump, the time of the emission event can be chosen tobe inside the short time window marked by the presenceof the pump pulse.The microscopic many-particle theory we use in ouranalysis is based on the quantum-dot cavity model illus-trated in Fig. 1b. Included are the relevant electronicconfigurations of the quantum dot. These are groundstate | G (cid:105) , excitons | X H (cid:105) and | X V (cid:105) , and biexciton | B (cid:105) .The electronic system is coupled to the photons in twocavity modes with orthogonal polarizations and frequen-cies ω H,V . In addition to the quantized light fields inthe cavity modes an off-resonant coherent laser field isincluded to trigger the photon emission. For a given ini-tial state and given external laser field, we evaluate the dynamical evolution of all occupations and coherences inthe system. In particular, we extract detailed informa-tion about the photon emission from the system. Photonemission from the cavity and loss of electronic coherenceson the relevant timescales are included. More details onthe theoretical approach are given in the Methods Sec-tion below.To study the scheme outlined above and illustrated inFig. 1a, initially we prepare the quantum dot in the biex-citon state with no photons in the cavity. Recent studieshave shown the robust initialization of the biexciton.
Then a picosecond light pulse is applied, driving the sys-tem between the biexciton and a virtual state inside theband gap. When the pulse frequency ω L is tuned suchthat the two-photon resonance condition from ground tobiexciton state is fulfilled, E B − E G = (cid:126) ( ω L + ω H,V ), en-ergy conservation allows spontaneous emission of a sin-gle photon into the cavity mode. To increase the prob-ability of the emission event to occur during the pres-ence of the pulse, first we use a high-quality cavity with κ = (cid:126) /
10 ps − (quality factor Q ≈ (cid:126) ω = 1 . g = (cid:126) /
10 ps − ≈ µ eV. For theseparameters, Fig. 2a shows the computed time-resolvedphoton density inside the V-polarized cavity mode fordifferent detuning of the V-polarized pump pulse fromthe two-photon resonance condition. The envelope ofthe 5 ps pulse with peak Rabi energy of 1 . ≈ κ . We ob-serve a slight field-induced shift of the emission in Fig. 1afrom the ideal two-photon resonance condition at ∆ = 0.In addition to the main emission feature, weak oscilla-tions are visible as vertical stripes in Fig. 2a from theemission from the biexciton-exciton transition into theoff-resonant cavity mode.The dynamics of the biexciton, exciton, and groundstate densities are depicted in Fig. 2c-e. Clearly visible isthe adiabatic following of these densities during the pres-ence of the off-resonant pump pulse. However, only forthe frequency window close to the two-photon resonancecondition, the ground state and biexciton densities arechanged by a sizeable amount after the pump pulse haspassed. We stress that not a significant amount of den-sity is generated in the exciton state(s) as the emissionis strongly dominated by the direct two-photon channel.We note that the absolute probability for the photonemission to occur and with it the potential brightness ofthe source can be further optimized by increasing pumpintensity and pulse length along with other system pa-rameters such as biexciton binding energy, cavity fre-quency, coupling strength and cavity quality. We furthernote that for higher pumping intensities and strongercouplings the system dynamics can also become morecomplex. True benchmarking of the potential perfor-mance will be left for a future study possibly along withan experimental demonstration. We note that for thesystem parameters chosen here, by increasing the pulselength by a factor of 10 to 50 ps, we find an almost linearincrease of the emitted photon density with pulse lengthwhile only generating an insignificant amount of excitondensity.In Fig. 3 we demonstrate that with the polarizationstate of the pump field, the polarization state of theemitted photon can be controlled. The pump electricfield amplitude and polarization state is parametrized as (cid:126)E = E · (cos ( p ) (cid:126)σ + +sin ( p ) (cid:126)σ − ), here with the real-valuedparameter p with p = 0 ( p = π/
2) corresponding to thelimiting case of circularly polarized light σ + ( σ − ). Pa-rameters are the same as in Fig. 2. The two limiting casesare the circular polarization states σ + and σ − : accordingto spin selection rules, when the pump is + circularly po-larized, the emitted photon is − circularly polarized andvice versa. Figure 3c shows the time-averaged (averagedfrom 12 to 45 ps) proportions of photon densities in thecircularly polarized states. With changing the pump po-larization parameter p the expected (almost sinusoidal)change in the polarization state is found. The achievablecontrast is slightly reduced by the spontaneous biexcitondecay through the exciton states. These results give un-ambiguous evidence that the emitted photon stems froma spontaneous emission into the cavity mode and can notresult from photons pumped into the system through thelaser source. By changing the polarization state of the pu m p po l a r i z a t i on t i m e ( p s ) σσ + - - σ + σ - σ + σ p r opo r t i on o f pho t on den s i t i e s pump polarization+ ρσ polarized photon density σ polarized photon density+ - a bc pu m p po l a r i z a t i on t i m e ( p s ) - σ + σ - ρ -3 x10 -3 x1001 FIG. 3:
Optical polarization control of the single-photon emission.
Pump frequency is tuned to the maxi-mum emission at ∆ = − .
54 meV in Fig. 2. Shown are thetime-resolved a left- and b right-circularly polarized photondensities ρ + and ρ − inside the cavity for different polarizationstates of the pump pulse triggering the emission. c Time-averaged proportions of ρ + and ρ − . pump anywhere in between the + and − circular polar-ization state (in general elliptically polarized), any polar-ization state can be realized for the emitted single photonof interest. Therefore, the classical laser field of the pumpcan be used to control the polarization state of a singlephoton as a true quantum object.In a scenario where also frequency filtering is appliedduring photon detection, the contrast for this polariza-tion control could reach near 100 % (no background pho-tons are emitted in the spectral range of interest). Toachieve efficient polarization control, the cavity modesmust be degenerate (as can be realized in a micro pillarcavity for example), however, no frequency fine-tuning ofcavity modes for example through temperature is neededas the stimulating laser can be tuned as needed.In Fig. 4 we demonstrate that the scheme proposedhere does not rely on the strong coupling or the highquality of the cavity mode used above. We present resultsobtained with a much lower coupling strength and in alow-quality cavity with g/κ = 0 .
04 with g = 1 /
50 ps − ≈ µ eV and κ = 1 / − ( Q ≈ . Photon Photon PhotonPulse Pulse Pulse
Photon density abc
FIG. 4:
Controlling the time of emission with the ar-rival time of the pump.
A lower-quality cavity with lowercoupling strength with g/κ = 0 .
04 is used. a The 5 ps pumppulse is centred at 15 ps. For a broad range of pump fre-quencies, a built-up of photon density and with it photonemission from the cavity is observed. In b we demonstratethat the timing of the photon emission can be controlled byvarying the arrival time of the pump. For a pump detuning of∆ = − .
54 meV, the normalized photon density is shown forpump arrival at 25 ps, 35 ps, and 45 ps, respectively. c Pumpenvelope corresponding to panel b . space such that frequency can be controlled in a widerspectral range. Applying a chirp in frequency and/or po-larization state of the pump to control the complex tem-poral dynamics of the single-photon emission event couldalso be of great interest. An experimental realization ofthe proposed scheme would be highly desirable. We envi-sion that the scheme we propose bears great promise forrealization of the next generation of versatile quantum-dot based single-photon sources. I. METHODS
We include the relevant electronic configurations of thequantum dot in our theory. These are ground state | G (cid:105) ,excitons | X H (cid:105) and | X V (cid:105) , and biexciton | B (cid:105) . The elec-tronic system is then coupled to the photons in two cav-ity modes with orthogonal polarizations and frequencies ω H,V . In addition to the quantized light fields in the cav-ity modes an off-resonant coherent laser field is includedto trigger the emission. In rotating wave approximation,the many-particle system Hamiltonian then reads: H = E G | G (cid:105)(cid:104) G | + E B | B (cid:105)(cid:104) B | + (cid:88) i = H,V E i | X i (cid:105)(cid:104) X i | + (cid:88) i = H,V (cid:16) (cid:126) ω i b † i b i − (cid:2) g (cid:0) | G (cid:105)(cid:104) X i | b † i + | X i (cid:105)(cid:104) B | b † i (cid:1) + h . c . (cid:3)(cid:17) + (cid:88) i = H,V (cid:16)(cid:0) | G (cid:105)(cid:104) X i | Ω ∗ i ( t ) + | X i (cid:105)(cid:104) B | Ω ∗ i ( t ) (cid:1) + h . c . (cid:17) . Here, b † i ( b i ) denote creation (annihilation) operators ofphotons in the cavity modes H and V and Ω i ( t ) givesthe Rabi energy of the time dependent coherent laserfield projected onto the respective transitions with i -polarization. As for the validity of our hybrid theory in-cluding both classical and quantized light fields, we notethat it is important that in all our evaluations below, thelaser field is off-resonant to the cavity modes by severalmeV. The coupling strength of the electronic system tocavity modes is given by g . The time-evolution of thesystem density operator ρ s obeys the following equationof motion: ∂∂t ρ s = − i (cid:126) [ H , ρ s ] + L cavity ( ρ s ) + L pure ( ρ s ) . (1)Coupling of the system to the environment is in-cluded through the two dissipative terms L cavity ( ρ s ) and L pure ( ρ s ). The finite lifetime (cid:126) /κ of the photons insidethe cavity is taken into account through the Lindbladterm L cavity ( ρ s ) = κ (cid:88) i = H,V (2 b i ρ s b † i − b † i b i ρ s − ρ s b † i b i ) . (2)A pure dephasing of coherences between electronic con-figurations is included through L pure ( ρ s ) = − (cid:88) χ,χ (cid:48) ,χ (cid:54) = χ (cid:48) γ pure χχ (cid:48) | χ (cid:105)(cid:104) χ | ρ s | χ (cid:48) (cid:105)(cid:104) χ (cid:48) | , (3) with χ, χ (cid:48) ∈ { G, X H , X V , B } . We use the same value γ pure χχ (cid:48) = γ = (cid:126) /
200 ps − ≈ µ eV for all electronic co-herences, which is a realistic value for pure dephasingof excitonic coherences at low temperature. We notethat our results are qualitatively robust even with a puredephasing much faster than assumed here. Fine struc-ture splitting between exciton levels – typically of theorder of several tens of µ eV – does only cause minorquantitative changes to the results and with E H = E V is set to zero for simplicity. Degeneracy is assumed forthe cavity modes, ω H = ω V , which is needed for effi-cient polarization control. A reasonable biexciton bind-ing energy of 3 meV is assumed and the cavity modesare tuned 5 meV to the red of the biexciton to excitontransitions. In this work we assume the system to beinitially in the biexciton configuration with no photonsinside the cavity. For this initial condition, the systemdynamics is obtained by explicitly solving Eq. (1) in thefinite-dimensional Fock-space spanned by the degrees offreedom of our system. Expectation values of any opera-tor ˆ A are computed by taking the trace with the systemdensity operator, (cid:104) ˆ A (cid:105) = tr { ˆ Aρ s } .We note that for simplicity some secondary effects thatcould slightly change the results have not been consid-ered: phonon-assisted transitions into cavity mode orlaser field are expected to be weak for the detuning of sev-eral meV used; the radiative loss into other off-resonantcavity modes is expected to play a small quantitative roleon the timescales studied. II. ACKNOWLEDGMENTS
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