Emission-frequency separated high quality single-photon sources enabled by phonons
Michael Cosacchi, Florian Ungar, Moritz Cygorek, Alexei Vagov, Vollrath Martin Axt
EEmission-frequency separated high quality single-photon sources enabled by phonons
M. Cosacchi, F. Ungar, M. Cygorek, A. Vagov,
1, 3 and V. M. Axt Theoretische Physik III, Universit¨at Bayreuth, 95440 Bayreuth, Germany Department of Physics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 ITMO University, St. Petersburg, 197101, Russia
We demonstrate theoretically that the single-photon purity of photons emitted from a quantumdot exciton prepared by phonon-assisted off-resonant excitation can be significantly higher in a widerange of parameters than that obtained by resonant preparation for otherwise identical conditions.Despite the off-resonant excitation the brightness stays on a high level. These surprising findingsexploit that the phonon-assisted preparation is a two-step process where phonons first lead to arelaxation between laser-dressed states while high exciton occupations are reached only with a delayto the laser pulse maximum by adiabatically undressing the dot states. Due to this delay, possiblesubsequent processes, in particular multi-photon excitations, appear at a time when the laser pulse isalmost gone. The resulting suppression of reexcitation processes increases the single-photon purity.Due to the spectral separation of the signal photons from the laser frequencies this enables theemission of high quality single photons not disturbed by a laser background while taking advantageof the robustness of the phonon assisted scheme.
On-demand single-photon sources continue to gain at-tention as key building blocks in quantum technologicalapplications, ranging from novel metrology over quantumcommunication to quantum computing. Semiconductorquantum dots (QDs) have proven to be suitable single-photon emitters [1–8] that due to their high compatibil-ity with existing semiconductor technology are promisingcandidates for device applications. In contrast to atomicsystems, these nanoscale structures are prone to the influ-ence of the surrounding solid state crystal matrix. Lon-gitudinal acoustic (LA) phonons are the main source ofdecoherence of excitons in semiconductor QDs even atcryogenic temperatures of a few Kelvin [9–13]. Never-theless, phonon-assisted off-resonant QD excitations havebeen shown to provide a robust alternative to resonantexciton preparation schemes [14–18]. In this letter, wedemonstrate theoretically that, quite unexpectedly, thecoupling to LA phonons combined with off-resonant driv-ing can be extremely beneficial for a single-photon sourcebased on a QD-cavity system, allowing for the generationof high-quality single-photons that are easily detectabledue to their spectral separation from the laser pulses usedfor the excitation of the QD.Placing a QD in a cavity strongly enhances the photonemission by enlarging the effective dot-cavity couplingand by setting a preferable emission axis. When excitingthe QD exciton resonantly, the frequencies of the exci-tation and the signal are identical - separating the twois a formidable experimental challenge. In fact, spectralseparability is achievable, e.g., by wetting layer excita-tion or by exciting the biexciton via the two-photon res-onance and subsequently exploiting the biexciton-excitoncascade [8, 19]. But while the former introduces a timejitter that reduces the on-demand character of the photonsource, the latter is sensitive to small fluctuations of exci-tation parameters such as the laser energy and the pulsearea. Both problems are overcome by an off-resonant laser pulsesphonons cavity quantum dot lossesradiative decay
FIG. 1. Sketch of the system under consideration. A two-level QD with a ground state | G (cid:105) and an exciton state | X (cid:105) iscoupled to a lossy single-mode microcavity. The | G (cid:105) → | X (cid:105) transition is driven by external laser pulses and the excitonstate is coupled to LA phonons in a pure-dephasing manner.Finally, the dot can decay radiatively. excitation of the quantum dot, which is thus extremelyadvantageous. Indeed, it has recently been shown thatthe robustness of off-resonant excitation schemes pavesthe way to excite two spatially separated QDs with dif-ferent transition energies simultaneously with the samelaser pulse, which is a milestone towards the scalabilityof complex quantum networks [20].The quality of a QD-cavity system as an on-demandsingle-photon source is typically quantified by several keyfigures of merit, such as the single-photon purity P andthe brightness B . While the former measures whether in-deed a single photon is emitted by the source, the lattercharacterizes its total photon yield [5]. When P = B = 1,the source emits a single photon with a probability ofunity at every excitation pulse via the cavity. The single-photon purity (SPP) can be extracted from a HanburyBrown-Twiss coincidence experiment [3, 7, 8, 21–24],which gives a conditional probability to detect a secondphoton when a first one has already been detected. Sup-pressing this probability is possible, e.g., by parametricdown-conversion, which enhances the SPP, albeit at thecost of a severely reduced brightness of the photon source a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l [25]. Maximizing both SPP and brightness is of utmostimportance to create efficient single-photon emitters.Simultaneously large P and B in a QD-cavity systemcan be achieved by exciting the dot resonantly by ultra-short laser pulses [3, 4, 7]. However, shortening the pulseduration is equivalent to widening it spectrally. Thedetrimental influence of exciting higher-lying states, es-pecially the biexciton state of the QD by short pulses isdiscussed in Ref. [26]. In view of the various advantagesof phonon-assisted off-resonant excitations listed above,the question arises how photonic characteristics such asSPP and brightness perform under off-resonant schemes.In short, we want to explore whether all of the advantagesof phonon-assisted off-resonant schemes come at the costof severely reduced photonic properties.It is expected that driving a QD off-resonantly ismuch less efficient. For longer and stronger pulses theresulting quantum state of a QD-cavity system con-tains an admixture of multi-photon states, which reducesthe SPP. Phonon-induced dephasing is expected to de-grade the quantum state even further. But paradoxi-cally quite the opposite can take place: a combinationof off-resonant driving with the phonon-induced relax-ation between laser-dressed QD states leads eventuallyto high exciton occupations in a subsequent adiabaticundressing process [27]. In this letter, we demonstratethat the delay of the exciton creation caused by the un-dressing suppresses the probability for multi-photon gen-eration. Therefore, comparing off-resonant and resonantexcitation with otherwise same conditions may, quite un-expectedly, yield enhanced SPPs in the off-resonant case.The best values predicted in this letter are even compa-rable to the best values obtained so far within resonantschemes addressing the exciton.We model the QD-cavity system as a laser-driven two-level system with a ground state | G (cid:105) and an excited state | X (cid:105) , H DL = − (cid:126) ∆ ω LX | X (cid:105)(cid:104) X | − (cid:126) f ( t ) ( | X (cid:105)(cid:104) G | + | G (cid:105)(cid:104) X | ),coupled to a single-mode microcavity (cf. Fig. 1), H C = (cid:126) ∆ ω CL a † a + (cid:126) g (cid:0) a † | G (cid:105)(cid:104) X | + a | X (cid:105)(cid:104) G | (cid:1) , which is on res-onance with the QD exciton. Here, ∆ ω LX and ∆ ω CL are the laser-exciton and cavity-laser detuning, respec-tively, and a is the photon annihilation operator in thecavity, which is coupled to the dot by the coupling con-stant g . A train of Gaussian pulses is assumed repre-sented by the laser envelope function f ( t ). The exci-tation can leave the system either via radiative decayor cavity losses modeled by Lindblad rates γ and κ , re-spectively. Finally, the exciton is coupled to a contin-uum of LA phonons in a pure-dephasing manner [28], H Ph = (cid:126) (cid:80) q ω q b † q b q + (cid:126) (cid:80) q (cid:0) γ X q b † q + γ X ∗ q b q (cid:1) | X (cid:105)(cid:104) X | . b q annihilates a phonon in the mode q coupled to the dotby the coupling constant γ X q . Full details of the modeland of our numerical approach are given in the supple-mental material [29]. It is worthwhile to note that we usepath-integral methods for our simulations that allow usto perform all simulations without approximation to the model [29, 36–38].For the calculations, standard GaAs parameters areused [39] for a QD of 6 nm diameter (for details onthe phonon coupling consider the supplement [29]). Ifnot stated otherwise, the excitation pulse full width athalf maximum is set to 7 ps, the cavity mode is reso-nant with the QD transition, the dot-cavity coupling is (cid:126) g = 50 µ eV, the radiative decay rate is (cid:126) γ = 20 µ eV, andthe cavity loss rate is (cid:126) κ = 50 µ eV. This corresponds to aPurcell factor of F P = g / ( γκ ) = 2 .
5. The initial phonondistribution is assumed to be thermal with a temperatureof T = 4 . P and the brightness B , are obtained from path-integral simulations of the two-time photonic correlationfunction G (2) ( t, τ ) = (cid:104) a † ( t ) a † ( t + τ ) a ( t + τ ) a ( t ) (cid:105) and thetime dependent photon occupation (cid:104) a † a (cid:105) ( t ), respectively.In order to express the SPP in terms of G (2) ( t, τ ) one firstneeds to take the average over the first time argument t ,i.e., G (2) ( τ ) = (cid:82) ∞−∞ dt G (2) ( t, τ ), which yields a functionwith the delay time τ of the coincidence measurementas its single argument. The probability p of detecting asecond photon during the same excitation pulse after afirst one has already been emitted thus can be obtainedby p = (cid:82) T Pulse / − T Pulse / dτ G (2) ( τ ) (cid:82) T Pulse / T Pulse / dτ G (2) ( τ ) , (1)where T Pulse is the separation of the pulses in the pulsetrain. The SPP is then defined as P = 1 − p . Note that −∞ < P ≤
1, where the lack of a lower bound is due tothe possibility of bunching instead of anti-bunching.In this work, the brightness of the source is mod-eled as the integrated leakage of the average photonnumber during the duration of one pulse, i.e., B = κ (cid:82) T Pulse / − T Pulse / dt (cid:104) a † a (cid:105) ( t ). Due to the definition, this quan-tity formally ranges in 0 ≤ B < ∞ without an upperbound since in principal infinitely many photons can ex-ist in a single electromagnetic field mode.In Fig. 2a the brightness simulated without phononsis shown as a function of the detuning ∆ ω LX betweenthe central laser frequency and the transition frequencyconnecting the ground and the exciton state of the QDas well as the pulse area Θ. An oscillatory behavior asa function of the pulse area with maxima at odd multi-ples of π is observed (cf. Fig. 2a). This is a consequenceof the well-known Rabi rotation of the exciton occupa-tion since the exciton feeds the cavity photons, which inturn are measured by the brightness. As a function ofthe detuning, the regions of high brightness are confinedto a fairly small range around resonance. The inclusionof phonons drastically changes this picture (cf. Fig. 2b).Through off-resonant excitation with detunings that canbe bridged by the emission of LA phonons, a nonvanin-shing brightness can be obtained in a previously dark without phonons with phonons051015 P u l s e a r e a Θ ( π ) a B r i g h t n e ss B b -0.4 0.0 0.4 0.8 1.2Detuning ∆ ω LX (meV)051015 c -0.4 0.0 0.4 0.8 1.2 80859095100 S i n g l e - ph o t o npu r i t y P ( % ) d FIG. 2. Brightness B (panels a, b) and SPP P (panels c, d)as a function of the laser-exciton detuning ∆ ω LX and the exci-tation pulse area Θ of a pulse in the pulse train. The left col-umn (a, c) is the result of a phonon-free calculation, the rightcolumn (b, d) includes the coupling to a continuum of LAphonons. Blue circle: resonant π -pulse excitation. Red circle:maximal SPP (with phonons). Red square: optimal SPP andbrightness for off-resonant excitation (with phonons). region. Note that the asymmetry with respect to thesign of the detuning is due to the low temperature of T = 4 . P at ∆ ω LX (cid:38) . P max = 98 .
8% (red circle) is even larger than90 .
7% obtained for the resonantly driven system (bluecircle). Combined with an appreciably large B , this indi-cates a possibility to have a good quality single-photonsource in the off-resonant excitation regime. Note that B = 0 .
46 observed at the point of P max (cf. red circlein Fig. 2b) is not much smaller than the maximal value of 0 .
67 achieved in the resonantly driven case (cf. bluecircle in Fig. 2b). It is also noteworthy that it is possibleto obtain a significantly larger brightness at the cost ofa slight decrease in the SSP. For example, if we choosea trade-off by maximizing the sum of the squares of thetwo figures of merit in the off-resonant regime, we obtain B = 0 .
53 and P = 98 .
1% (red square). This value for P is close to typical experimental values obtained for res-onant excitation of the quantum dot exciton (98 .
8% [4],99 .
1% [7]) even though the pulse lengths in Refs. [4, 7]have been slightly shorter [40].To explain the mechanism behind this observation, oneneeds to consider the dynamics of the QD-cavity states.In Fig. 3, the time dependent occupations in the resonantand the off-resonant case (cf. the blue and red circles inFig. 2, respectively) are compared. The considered statesare product states between the QD states and a photonstate with photon number n . After resonant π -pulse ex-citation (cf. Fig. 3a), the exciton state | X, (cid:105) withoutphotons is occupied (blue curve). The cavity couplingrotates the dot back to its ground state and produces onephoton in the cavity (orange curve). Because the driv-ing is still nonzero at this point, the dot is reexcited toproduce an occupation of the state | X, (cid:105) (green curve),which is shown in the inset of Fig. 3a. Finally, the cav-ity coupling leads to an occupation of the ground statewith two photons | G, (cid:105) (red curve), such that the SPPis diminished.In contrast to the π -pulse induced rotation of the Blochvector, the off-resonant excitation scheme exploits an ef-fect called adiabatic undressing [27]. Switching on thelaser transforms the dot states to a new energy eigen-basis commonly known as laser-dressed states, the gapbetween which can be bridged by LA phonons with typ-ical energies of a few meV. At low temperatures, thelower dressed state becomes occupied via phonon emis-sion. However, the phonon-induced relaxation is onlyefficient when both dressed states have roughly equal ex-citon components. Thus, the exciton state exhibits typi-cally occupations of the order of 50% after the relaxationis completed [27]. When the laser is turned off adiabati-cally, the lower dressed state is subsequently transformedto the exciton state in the original basis provided the de-tuning is positive (otherwise the ground state is reached[27]). This adiabatic undressing of the dot states there-fore boosts the exciton occupation only at the end of thepulse (cf. the blue curve in Fig. 3b). This in turn meansthat during the phase of phonon relaxation no photon canbe put into the cavity efficiently (cf. the orange curve inFig. 3b). When finally the adiabatic undressing-inducedexciton boost occurs, the occupation of | G, (cid:105) follows (cf.Fig. 3b). Since the excitation pulse is basically gone bythen, the reexcitation of the QD is strongly suppressed(green curve), such that effectively no second photon canbe put into the cavity (red curve). This implies a farhigher SPP than in the resonant counterpart, as is ob- O cc up a t i o n t − t p (ps)0.00.20.40.60.81.0 0 20 40 60 a O cc up a t i o n t − t p (ps)0.00.20.40.60.81.0 0 20 40 60 a O cc up a t i o n t − t p (ps)0.00.20.40.60.81.0 0 20 40 60 a O cc up a t i o n t − t p (ps)0.00.20.40.60.81.0 0 20 40 60 a O cc up a t i o n t − t p (ps) | X, i| G, i| X, i| G, i Driving0.00.20.40.60.81.0 0 20 40 60 a × − × − × − × − × − N o r m a li ze dd r i v i n g t − t p (ps)0 20 40 60 0.00.20.40.60.81.0 b N o r m a li ze dd r i v i n g t − t p (ps)0 20 40 60 0.00.20.40.60.81.0 b N o r m a li ze dd r i v i n g t − t p (ps)0 20 40 60 0.00.20.40.60.81.0 b N o r m a li ze dd r i v i n g t − t p (ps)0 20 40 60 0.00.20.40.60.81.0 b N o r m a li ze dd r i v i n g t − t p (ps) | X, i| G, i| X, i| G, i Driving0 20 40 60 0.00.20.40.60.81.0 b × − × − × − × − × − FIG. 3. Time-dependent occupations: a) after resonant π -pulse excitation (cf. blue circle in Fig. 2) and b) in the off-resonantphonon-assisted case (cf. red circle in Fig. 2). The occupations of the states | X, (cid:105) , | G, (cid:105) , | X, (cid:105) , and | G, (cid:105) are shown ascolored filled curves. The Gaussian envelope of the laser driving pulse normalized to its maximum value centered at t p is shownas a black dashed line. The insets show the same curves, respectively, on a zoomed-in scale for the occupations.FIG. 4. The difference between the SPP after off-resonantphonon-assisted excitation P off-res and after resonant π -pulse rotation P res is shown for two different pulse lengths(FWHM), namely: a) 7 ps and b) 14 ps, as a function of ra-diative decay (cid:126) γ and cavity losses (cid:126) κ . The cavity qualityfactor Q = ω c /κ is obtained via the cavity losses assuming acavity single-mode energy of (cid:126) ω c = 1 . . π and ∆ ω LX = 1 . served in Fig. 2d. In summary, the delay of the excitonoccupation caused by the two-step procedure of first re-laxing to a dressed state via phonon emission and thenreaching the exciton by adiabatic undressing is responsi-ble for the enhancement of the SPP.To quantify the robustness of the phonon-induced SPP enhancement against variations of other system param-eters, the difference between the SPP after off-resonantexcitation and after the resonant one is shown as a func-tion of the radiative decay γ and the cavity loss rate κ inFig. 4. A positive value (reddish shade) indicates a set ofparameters where the SPP is enhanced for off-resonantexcitation. We find such an enhancement for a wide pa-rameter regime in κ and γ that is experimentally wellaccessible. Also, changing the pulse length from 7 ps inFig. 4a to 14 ps in Fig. 4b does not change the phonon-induced SPP enhancement qualitatively. The reason whythe SPP for off-resonant excitation falls below the reso-nant one in the bad cavity limit and/or in the limit ofhigh radiative losses is that relaxation processes limit thetime available for the adiabatic undressing which even-tually becomes incomplete.In conclusion, we have presented a seemingly paradox-ical scheme for the phonon-assisted operation of a QD-cavity system as a single-photon source, where the ex-citation is spectrally separated from the generated pho-tons. Two factors that would separately lead to a qualitydegradation - off-resonant driving and dot-phonon cou-pling - in combination result in a huge boost in criti-cal characteristics of a single-photon source. We havedemonstrated that the achievable single-photon puritycan be noticeably higher than for resonant excitationwhile the brightness is still at an acceptable level. 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