Proposal for geometric generation of a biexciton in a quantum dot using a chirped pulse
PProposal for geometric generation of a biexciton in a quantum dot using a chirped pulse
H. Y. Hui and R. B. Liu
Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China (Dated: November 29, 2018)We propose to create a biexciton by a coherent optical process using a frequency-sweeping (chirped) laserpulse. In contrast to the two-photon Rabi flop scheme, the present method uses the state transfer throughavoided level crossing and is a geometric control. The proposed process is robust against pulse area uncertainty,detuning, and dephasing. The speed of the adiabatic operation is constrained by the biexciton binding energy.
PACS numbers: 78.67.Hc, 42.50.Dv, 32.80.Xx, 03.65.Ud
I. INTRODUCTION
Semiconductor quantum dots (QDs) have manifold uses inquantum information and computation. They have been uti-lized to generate single-photons [1, 2, 3, 4, 5, 6] with goodindistinguishability [7]. More recently it has been proposedand partially realized [2, 8, 9, 10, 11] that QDs in two-excitonstates, called biexcitons, can be used to generate pairs of en-tangled photons, by cascade emission of photons [12]. The en-tanglement of photon pairs in this scheme was noted [13] to beimperfect, because of the slight di ff erence in energy betweenthe two single-exciton levels [14, 15]. However, considerableimprovement has recently been made [16, 17], which suggeststhat the scheme should be of high experimental value in quan-tum optics, quantum computation [18] and quantum cryptog-raphy [19, 20], and can also be used to test foundations ofquantum mechanics [2]. Biexciton is also of interest in itselfbecause it serves as the physical basis for a 2-bit conditionalquantum logic gate [21].A number of works have already been done on the op-tical coherent control of the single-exciton states in, e.g.,InAs / GaAs QDs [22, 23, 24, 25, 26] and CdSe / ZnSe QDs [6].In the recent experiments of optical coherent control of biex-citon states, two approaches were used. The first one ap-plies two optical beams each in resonance with the | g (cid:105) → | X (cid:105) (ground state to single exciton) and | X (cid:105) → | XX (cid:105) (single- tobi-exciton) transitions [21, 27]. However, it was noted [28]that a better approach is to apply degenerate pulses with fre-quency equal to half the biexciton energy, such that the spon-taneously emitted photons have frequencies di ff erent fromthat of the excitation pulse. This has been followed by re-cent works [28, 29, 30]. Experiments have been done on bothInAs / GaAs and CdSe / ZnSe QDs, and the phenomenon of two-photon Rabi oscillation is the prime indicator of successfulcontrol in these experiments.In this paper, we propose to use a frequency-sweepingpulse [31] for a geometric generation of a biexciton in a quan-tum dot. The scheme is based on the adiabatic state transferfrom the ground state to the final biexciton state via avoidedenergy level crossing, in which the intermediate exciton is by-passed. The utilization of level anti-crossing follows the ideaof the STIRAP (stimulated Raman adiabatic passage) for adi-abatic state transfer in a Λ -type 3-level system [32, 33]. Buthere since both the exciton and biexciton transitions couple tothe same optical pulse, independent control of the two transi- tions as required in the STIRAP is not feasible. Instead, thefrequency sweeping [31] is proposed to realize the adiabaticstate evolution. The geometric scheme bears the robustnessagainst some uncertainty in the system parameters such as en-ergy levels and dipole magnitude and in laser pulse parameterssuch as amplitude, shape, and frequency, which is unavoidablein realistic experiments. Bypassing the intermediate single-exciton state minimizes the possibility of generating single-photon emission which, e.g., may contaminate an entangledphoton pair in quantum optics application. Constrained by thebiexciton binding energy, the adiabatic state transfer can becompleted in picosecond timescales for a typical CdSe quan-tum dot, and thus the e ff ect of the exciton dephasing can belargely avoided.This paper is organized as follows: In Sec. II we formu-late the problem and give the waveform of the pulse used; inSec. III we demonstrate numerically the creation of a biexci-ton which is robust against small uncertainty in all parameterscharacterizing the system and keeps the occupation of single-exciton state relatively low; in Sec. IV we show that dephas-ing, modeled in the Lindblad formalism [34], only slightlyreduce the e ffi ciency. II. MODEL AND MECHANISM
The biexciton system can be modeled by a four-level sys-tem: the ground state | g (cid:105) , the biexciton state | XX (cid:105) , and twointermediate single-exciton states with di ff erent linear polar-izations | X (cid:105) and | Y (cid:105) [14, 15]. Because the two pathways ofexcitation, | g (cid:105) → | X (cid:105) → | XX (cid:105) and | g (cid:105) → | Y (cid:105) → | XX (cid:105) , is inde-pendent and can be implemented independently by applyingdi ff erent polarizations of excitation in experiments, we onlyconsider | g (cid:105) → | X (cid:105) → | XX (cid:105) .The Hamiltonian is written as H = ( ω + δ ) | X (cid:105) (cid:104) X | + ω | XX (cid:105) (cid:104) XX | + (cid:2) Ω ( t ) ( | g (cid:105) (cid:104) X | + | X (cid:105) (cid:104) XX | ) + H . c . (cid:3) , (1)where we have defined ω such that 2 ω is the energy betweenground state and biexciton, and ω + δ is the energy of the singleexciton, with δ equal to half the biexciton binding energy ∆ E . Ω ( t ) is the time-dependent coupling cause by a laser pulse.We write the Hamiltonian in a frequency-modulated rotatingreference frame, with Ω ( t ) = Ω t exp ( i ( ω − ∆ ) t − i φ t ) (where a r X i v : . [ c ond - m a t . m e s - h a ll ] J un (1/d ) d W / d/ d
Wd fd
FIG. 1: (Color online) (a) Laser pulse amplitude and time-dependentfrequency as defined in Eq. (3b): Ω t and ˙ φ t , scaled with respect to δ . (b) The three eigenvalues of Hamiltonian in Eq. (2) under thepulse in Eq. (3b) (solid lines), and when Ω t = | g (cid:105) is occupied. The parameters here are A = . δ , α = . δ , µ = T = /δ . ∆ ≡ detuning) , as H = − ∆ − ˙ φ t Ω t Ω t δ Ω t Ω t ∆ + ˙ φ t . (2)Here the basis is { e − i [ ( ω − ∆ ) t − φ ] | g (cid:105) , | X (cid:105) , e i [ ( ω − ∆ ) t − φ ] | XX (cid:105)} .When Ω t =
0, the eigenvectors of H are the three basisstates, with eigenvalues {− ˙ φ t , δ , ˙ φ t } . We envisage that when˙ φ t sweeps from negative to positive, the eigenvector wouldchange from | g (cid:105) to | XX (cid:105) , following which the system wouldbe excited, by-passing the intermediate state adiabatically, if apulse, being su ffi ciently slow-varying, is applied to induce thelevel anti-crossing.In the following, we choose the following specific func-tional forms for Ω t and ˙ φ t [31] Ω t = A sech( α t ) , (3a)˙ φ t = µα tanh( α t ) , (3b) (a)(b)FIG. 2: (Color online) Final population of the biexciton state. Here α = . ≈ − and T = − ≈ δ =
10 meV, consistent with the bindingenergy of CdSe / ZnSe QDs. (a) Fixing δ =
10 meV and ∆ = A and µ (i.e. µα ) are varied. If A is too small, the process fails to be adi-abatic. Alternatively, if µα is too small, no anti-crossing phenomenacould be observed. (b) Fixing A = µα = ∆ and binding energy 2 δ are varied. The result is acceptable fordetuning within ± as shown in Fig. 1(a). Using these waveforms, the adiabaticeigenvalues of the Hamiltonian are computed and plotted inFig. 1(b). As expected, the eigenvectors | g (cid:105) and | XX (cid:105) ex-change with each other. To illustrate the level anti-crossing,the eigenvalues for the case of no interactions ( Ω t =
0) isalso plotted in dashed lines. We see that the single-excitonstate does not participate in the level crossing. Thus it canbe inferred that the occupation of | X (cid:105) would be kept low be-cause the third eigenvalue δ is separated from the remainingtwo eigenvalues; the larger δ , the lower would be the occupa-tion of | X (cid:105) .The square of components of eigenvectors for the middleeigenvalue are plotted in solid lines in Fig. 1(c). As this eigen-vector changes from initially | g (cid:105) to finally | XX (cid:105) , we expect thisto be followed by the actual physical system, if the pulse issu ffi ciently slow-varying.It should be pointed out that although we have chosenspecific waveforms in Eq. (3b) for the pulse shape, otherchoices are also possible, provided that “anti-crossing” simi-lar to Fig. 1 can be produced. For instance, Gaussian shape for Ω t and linear frequency sweep could also be used [31]. How-ever, waveforms in Eq. (3b) shows better adiabaticity, and isused in the simulation. III. SIMULATIONS
The adiabaticity can be kept in two ways, by increasingthe duration of process or the pulse amplitude A . The oc-cupation of intermediate state can also be suppressed by in-creasing the duration. However, long duration is an undesir-able parameter in experiment because of dephasing. In thefollowing we fix a duration of t ∈ [ − T , T ], and investigatethe adiabaticity of the state transfer as well as the interme-diate state population. We numerically solve the evolution Ψ ( t ) = a ( t ) | g (cid:105) + b ( t ) | X (cid:105) + c ( t ) | XX (cid:105) , with initial conditions Ψ (0) = | g (cid:105) . An example is given by the solid lines Fig. 1(c).We see that the actual evolution follows adiabatic approxi-mation closely. Note that with δ ≈
10 meV in the case ofCdSe / ZnSe QDs [15, 35], the duration 2 T ≈
10 ps, which ismuch shorter than the exciton dephasing time.To investigate on the dependence on the parameters, we plotthe final biexciton population | c ( T ) | as a function of A (cou-pling magnitude) and µα (frequency-sweeping amplitude) inFig. 2(a), δ (binding energy /
2) and ∆ (detuning) in Fig. 2(b).These plots have some foreseeable characteristics. The case µα ≈ A . This is demonstrated in the peaks andtroughs along µα = ∆ can be accepted. Uncertaintyin δ (as large as ± ff ect the e ffi cacy of thisprocess, either.From the figures we remark that in contrast to the processesof Rabi oscillation, this process is largely independent of thepulse area ( A and α ) and even the pulse shape. This is anexperimentally crucial feature, as the control over pulse areais often inexact under realistic conditions, which would makethe transferred population lower than expected as in the ordi-nary two-photon Rabi flop scheme.It is also of interest to investigate on the relation betweentime duration T ∼ α − in Eq. (3b). and the single-exciton in-termediate population max | b ( t ) | . In Fig. 3 we plot max | b ( t ) | as a function of α ; where for each α we use large enough A such that the transferred population ( | g (cid:105) → | XX (cid:105) ) is larger FIG. 3: µ = . δ =
10 meV are fixed, while A is adjustedaccordingly such that for each α the final biexciton population is atleast 0 .
99 and max | b ( t ) | is minimized. Roughly a linear correlationis demonstrated, until α is too large. (a)(b)FIG. 4: (Color online) (a) The numerical solution for the evolution ofstates in CdSe / ZnSe QDs, for the case of no dephasing (solid lines)and the case with dephasing (dots). (b) The terminal state popula-tions for di ff erent values of γ ij . The vertical dotted line indicates thecase of (a), 3 . − than 0 .
99 while max | b ( t ) | is minimized. We see a near-linearcorrelation. Physically we understand that when we limit theprocess to complete in shorter interval, the process becomessimply a transfer via real excitation of the intermediate state( | g (cid:105) → | X (cid:105) → | XX (cid:105) ). IV. EFFECTS OF DEPHASING
The analysis so far is less realistic in that we have ne-glected the e ff ect of dephasing present in QDs, which gen-erally drives pure state into mixed state. We thus consider therelaxation and dephasing of excitons and biexcitons, whichmay be caused by spontaneous emission and electron-phononscattering [36, 37]. At low-temperatures, we consider just thespontaneous emission as the limiting factor of the quantumoperation [38]. The spontaneous emission is modelled by anadditional Lindblad term in the master equation for densitymatrix ρ : ∂ t ρ = − i [ H , ρ ] + L ( ρ ) , (4)where the Lindblad super-operator L is defined by: L ( ρ ) = (cid:88) i j γ i j (cid:104) σ † i j ρσ i j − σ i j σ † i j ρ − ρσ i j σ † i j (cid:105) , (5)with { i , j } = {| XX (cid:105) , | X (cid:105)} , or {| X (cid:105) , | g (cid:105)} , signifying the transitionfrom i to j .In the case of CdSe / ZnSe QDs, in the elimination ofelectron-phonon interactions at low temperatures, it was de-termined to be γ i j ≈ . − [6]. Together with δ =
10 meVand the pulse shape of Eq. (3b), the evolution of di ff erent statepopulations are plotted in Fig. 4. It shows only a slight reduc-tion of the final population of | XX (cid:105) , while those of | X (cid:105) and | g (cid:105) increase. V. CONCLUSION
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