Optical remote control of a charge qubit
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Optical remote control of a single charge qubit
T. S. Santana ∗ and F. A. G. Almeida † Departamento de Física, Universidade Federal de Sergipe, 49100-000, Brazil (Dated: November 5, 2018)Both the electron transport-based qubits, implemented through double quantum dots, and thesources of indistinguishable single photons like self-assembled quantum dots are strong candidatesfor the implementation of quantum technologies, such as quantum computers and quantum re-peaters. Here, we demonstrate a reliable way of coupling these two types of qubits, uncoveringthe possibility of controlling and reading out the population of the double quantum dot via opticalexcitation. It is also shown that, in spite of the decoherence mechanisms affecting the qubits, theentanglement between them is achievable and, consequently, the implementation of the suggestedsystem in quantum technologies is feasible.
I. INTRODUCTION
The entanglement between two quantum systems wasnoticed by Einstein, Podolsky, and Rosen when they dis-cussed the validity of the quantum mechanics in its earlytimes . At that period, it was pointed out that the mea-surement in one of the systems could lead to instanta-neous information about the state of the other quantumsystem, even at a distance , although the informationcannot travel faster than the speed of light. Actually, theentanglement between two or more qubits is a crucial in-gredient for quantum computation, quantum informationand quantum communication . However, in nature, thephysical qubits always interact with some kind of reser-voirs, which mitigates the quantum correlation. In thiscase, some distillation protocol has to be applied in orderto recover the entangled state .Semiconductor devices became strong candidates forthe realization of quantum technologies, such as quan-tum computers , quantum memories and quantumrepeaters . The implementations are based on their op-tical properties, like in the case of self-assembled quan-tum dots (QDs) , on the electron transport or on theelectron spin, as in the case of double QDs defined bylithography techniques .If the noise originated from the density charge fluc-tuation in the vicinity of the QDs is irrelevant for itsdynamics, the coherence time of the charge qubit basedon the electron position in a double QD can be in theorder of 200 ns with a relaxation time to the groundstate of about 100 µ s – much greater than the typi-cal lifetime of excitonic states in optically active QDs,which is typically between hundreds of picoseconds anda few nanoseconds – allowing several qubit opera-tions to be performed before the quantum features ofthe system is completely lost. Moreover, high coher-ence and indistinguishability of successive single pho-tons have been reported for different kinds of solid-statephotonic devices , which have already been utilizedtowards the implementation of non-universal quantumcomputers .In this work, we suggest the construction of a bipar-tite system composed by a single-charge qubit and an optically active two-level system emitting single pho-tons, which may be achieved from the application oflithography techniques in optically active photonic de-vices such as QDs in semiconductor chips and directband-gap two-dimensional transition dichalcogenide ma-terials like WSe and MoSe . The interaction betweenthe charge qubit and the excited state of the optical QDvia Coulomb energy enables the optical remote controlof the first. Moreover, it is possible to obtain an en-tangled state between the emitter and the charge qubit,suggesting a new physical system for quantum communi-cation purposes, which requires the conversion betweenstationary and flying qubits and the guiding of the pho-tons emitted .This paper is organized as follows. In Sec. II, wepresent the physical description of the system, while theresults are exposed in Sec. III. In Sec. IV, we draw outour conclusions. II. MODEL
The system consists of two QDs distributed along ahemisphere of radius R with a single-photon emitter inits center, as depicted in Fig. 1 (a). The emitter QDis modeled as an optically driven two-level system in thedipole approximation , where the Rabi frequency is Ω– in this work only continuous wave excitation is consid-ered – the detuning between the optical transition andthe excitation field is ∆ and the radiative decay rate isΓ. When the emitter is in the ground state, there is nointeraction between itself and the single electron occupy-ing one of the traps. On the other hand, if the centralQD in driven to the excited state, the Coulomb interac-tion between the trapped electron and the exciton maybe significant, depending on the position of the electrontrap relative to the radiating dipole. Allowing the elec-tron to tunnel between the traps with a tunneling rateequal to J and assuming that the ionization energies forall the electron traps are equal, the Hamiltonian of thesystem under the rotating wave approximation is H ~ = (∆ + u ) σ † σd † d + (∆ + u ) σ † σd † d + Ω( t )2 σ x − J (cid:16) d † d + d † d (cid:17) , (1)where σ † ( σ ) is the creation (annihilation) operator forthe emitter, acting on the Hilbert subspace composed bythe state | g i and | e i , σ x = σ † + σ is the Pauli matrix,and d † n ( d n ) is the creation (annihilation) operator for the n -th electron trap, acting on the Hilbert subspace withstates | i and | i , as depicted in Fig. 1 (b). Consider-ing that the average length of the radiating dipole δr ismuch smaller than the distance R , the Coulomb energyis ~ u n = U e ( R ) δr cos( θ n ) /R , where U e ( r ) is the Coulombenergy of two electrons separated by a distance r and θ n is the angle between the n -th trap position vector fromthe emitter and the vector δ~r . Since we chose the secondelectron trap positioned at θ = π/ (a) Figure 1. (a) Sketch of the system composed by an emittersurrounded by two electron traps away by the same distance R . (b) The optically active QD is driven from its groundstate | g i to its excited state | e i through an excitation laserwith frequency detuned by ∆ from the optical transition. Thecoupling energy between the laser field and the exciton dipolemoment is ~ Ω and the emitter suffers radiative decay with arate equal to Γ. The single electron of the system can hopbetween the traps with a tunneling rate equal to J , changingthe Coulomb energy ~ u m . The radiative decay of the emitter is described by theLindblad superoperator, given by L ( σ ) ρ = Γ2 (cid:0) σρσ † − σ † σρ − ρσ † σ (cid:1) , (2)and the master equation of the system is dρdt = − i ~ [ H, ρ ] + L ( σ ) ρ, (3)where ρ is the density matrix of the system.For the initial state of the system, we consider that theemitter is in its ground state | g i , while the probability offinding the electron in any of the traps is 50%. All the off-diagonal elements of the density matrix are zero at t = 0.The trajectory of every element of the density matrix wasobtained by numerically solving the system of differential equations [Eq. 3] using the fourth-order Runge-Kuttamethod with error tolerance of 10 − . The steady statedensity matrix ρ s is determined from the point where t ≫ / Γ and dρ/dt ≈ . The exciton-phonon coupling may decrease theefficiency of the remote preparation of the charge qubit,as well as the entanglement between the qubits, but, forthe sake of simplicity, it was neglected here. III. RESULTS
It was observed that, when the parameters are ad-justed to have | u − u | > Ω > J and the detuning ∆tuned to compensate the energy shift caused by the pres-ence of the electron in one of the traps (∆ /u = − /u = − P i = h i | ρ s | i i being the probability of finding thesystem in the state | i i when ρ = ρ s , for Γ = 0 . / Ω sat = 3, J/ Ω sat = 1 / u / Ω sat = 9, u / Ω sat = 0and ∆ /u = −
1, where Ω sat = Γ / √ . t . . . . . P r o b a b ili t y P g P e P P Figure 2. Temporal evolution of the states population of thesystem composed by an optically active two-level system withstates | g i and | e i , and a charge qubit based on the electronposition states | i and | i . The parameters used were Γ = 0 . / Ω sat = 3, J/ Ω sat = 0 . u / Ω sat = 9, u = 0 and∆ /u = − In Fig. 3, the probability of finding the electron inthe state | i is obtained as a function of the Rabi fre-quency Ω and the tunneling rate J for Γ = 0 . /u = −
1, and u / Ω sat equal to 1 (a), 3 (b), 6 (c), and9 (d). If the Coulomb interaction is not high enough, theoptical remote control is inefficient and happens only fora reduced set of values for Ω and J [Fig. 3 (a)]. As thedifference | u − u | is increased [Fig. 3 (b) and (c)], theoptical control over the charge qubit becomes feasible,with P ≈ J . For high valuesof | u − u | [Fig. 3 (d)], the ability of controlling thecharge qubit saturates and the map of P as a functionof the Rabi frequency and the tunneling rate suffers onlysmall variations. Figure 3. Probability of finding the electron in the first trapafter the system reaches the steady state as a function ofthe Rabi frequency Ω and the tunneling rate J , both relativeto the saturation Rabi frequency Ω sat , for u = 0, Γ = 0 . u / Ω sat = 1, (b) u / Ω sat = 3, (c) u / Ω sat = 6,(d) u / Ω sat = 9 and ∆ = − u . The efficiency of the trapoccupancy control increases with the ratio Ω /J and tends tozero as J → In the absence of the Coulomb interaction or for u ≈ u , the two qubits would evolve without the influenceof each other, as can be noticed from the Hamiltonianof the system [Eq. (1)]. However, as the difference be-tween these two variables increases, the temporal evo-lution of the emitting dipole becomes dependent on thecharge qubit and vice versa. In this situation, the reso-nant frequency of the emitter will be shifted by u or u depending on the electron position state. If the Rabi fre-quency Ω is greater than the saturation Rabi frequencyΩ sat , the probability of finding the emitter in its excitedstate is significant (1 / ≫ Ω sat ), therefore it hasgreater influence on the evolution of the charge qubit. Fora small tunneling rate ( J < Ω , | u − u | ), the dynamics ofthe charge qubit is dominated by its interaction with theemitter. Consequently, for | u − u | much greater than Ωand J , it can be remotely controlled through the param-eters determining the dynamics of the emitting dipole,such as Rabi frequency, detuning and radiative decay.Taking advantage of the correlation between the twoqubits, the photons emitted from the dipole can be usedto monitor the electron tunneling between the traps.This is possible because the average photon countingrate , given by h N i = Γ h e | ρ s | e i , is sensitive to the de-tuning ∆ and the two possible energy shifts lead to twowell separated Lorentzian peaks. The number of pho-tons helps to identify the resonant frequencies, which op-timizes the charge qubit control, while the frequency of the photons tells which trap is occupied by the electron(Fig. 4). Figure 4. (a) Probability of finding the electron in the firsttrap (red solid line) and second trap (blue solid line) as func-tion of the laser detuning relative to the optical transition ofthe QD; (b) Expected number of emitted photons h N i as afunction of the detuning ∆ with the individual contribution ofthe optical transition shifted by u ( u ) represented by the red(blue) area. The parameters used were u / Ω sat = 9, u = 0,Ω / Ω sat = 3, J/ Ω sat = 0 . . The superposition of the electron position states canalso be created by inducing the electron to occupy one ofthe traps and, in the sequence, eliminating the excitationfield. In this case, the population of the electron positionstates are expected to coherently oscillate with frequencydetermined by J . The evolution of the trap occupancyand, consequently the tunneling rate J , can be monitoredby applying a relatively small excitation field (Ω ≪ J )on resonance with one of the optical transitions. Witha spectral distance between the transitions much greaterthan the linewidth of the Lorentzian peaks, the photonscattering will only happen when the corresponding trapis occupied. A second weak excitation field on resonancewith the other energy shift may also be used to comple-ment the monitoring of the electron.In order to quantify the entanglement between theemitting dipole and the electron position, we analyze thenegativity N defined as N = X λ< | λ | , (4)where λ are the eigenvalues of the partially transposeddensity matrix . The negativity varies from 0 for sep-arable states until 1 / /J , the states of the emitter have similar popu-lations and modest coherence elements, while the electrontends to occupy one of the traps. In this case, the outputfrom the electron position measurement has no informa-tion on the emitter’s state and the two two-level systemsare not entangled [Fig. 5 (a)]. A moderate entanglementis obtained by trading-off between the certainty of theelectron position and the photon coherence through thedecrease of the excitation power. The negativity indi-cates that a Rabi frequency between Ω sat and 3Ω sat anda tunneling rate from Ω sat / sat / | u − u | much greater than Ω sat . In this situ-ation, the dynamics of the bipartite system is dominatedby the photon emission with the electron occupying thefirst trap (for ∆ = − u ), and by the entangled state ofthe type | ψ i = a | g, i + b | e, i , where a and b are complexconstants. The fully entangled fraction, defined as F ( ρ s ) = max ψ h ψ | ρ s | ψ i , (5)where | ψ i are all the maximally entangled states of thesystem, gives a measure of how the mixed steady state ρ s approaches a Bell state . For Ω / Ω sat = 1 . J/ Ω sat = 0 . u / Ω sat = 9 [white lines in Fig. 5 (a)],we have F = 0 .
47 with | ψ i = ( | g, i − | e, i ) / √ N ≈ .
09, which is the maximal value ofthis map and it is greater than the values expected forthe thermal states of a gas-type system , for example.In Figs. 5 (b) and 5(c), we can observe the real and theimaginary parts of the steady-state density matrix ρ s ,respectively, for the parameters already specified. / Ω sat J / Ω s a t (a) 0 . . . . . N | g, i| g, i | e, i| e, ih g, |h g, |h e, |h e, | (b) − . . . . ℜ ( ρ ) | g, i| g, i | e, i| e, ih g, |h g, |h e, |h e, | (c) − . . . ℑ ( ρ ) Figure 5. (a) Negativity as a function of the ratios Ω / Ω sat and J/ Ω sat with u = 9Ω sat , u = 0 GHz, Γ = 0 . − u . Real (b) and imaginary (c) parts of the steady-state density matrix ρ s for Ω / Ω sat = 1 . J/ Ω sat = 0 . N ≈ .
09 [dashed white line in (a)].
The radiative decay of the emitter degrades the quan-tum correlation between the qubits, however, the entan-gled state can be distilled if some copies of the systemare available . Yet, although they are not maximally entangled, the probability of finding the dipole in its ex-cited state and the electron in the second trap is verysmall (0 . IV. CONCLUSION
In this work, we demonstrated how to remotely controland monitor a single-charge qubit using optical excitationvia the Coulomb interaction with an excitonic state of anoptically active QD. It was shown that the states of thecharge qubit can be manipulated by varying the inten-sity and the frequency of the excitation field. Moreover,the control of the electron position state was analyzed forseveral combinations of the system variables, from whereit was concluded that the efficiency tends to unit whenΩ ≫ J and u ≫ Ω sat . In this case, the electron posi-tion state can be identified through measurements on theamount of photons and their energies. Although the sec-ond electron trap was located to give u ≈
0, the resultspresented here are also valid when u has nonzero val-ues, since it differs from u enough to resolve the shiftedoptical transitions, as in Fig. 4 (b).The entanglement between the optically active qubitand the charge qubit was also investigated and it ispresent in spite of the radiative decay. When the effi-ciency of the charge qubit preparation tends to unit, theposition of the electron is independent of the opticallyactive qubit and no entanglement is observed. Whenthe Rabi frequency is diminished, the number of pho-tons decreases, the certainty about the electron positionbecomes smaller, but, in contrast, the entanglement be-tween the qubits competes with the photon emission dy-namics. Because the scattered photons carry informa-tion about the charge qubit, this system is a candidatefor physical implementations in the field of the quantumcommunication.A feasible implementation of this system is using solid-state devices, where the phonon-exciton interaction inthe optically active qubit may diminish the ability to re-motely control the charge qubits, as well as the quantumcorrelation between them. It can happen because thisdephasing mechanism would decrease the spectral reso-lution of the two transition energies originated from theCoulomb interaction with the electron. However, we doexpect these results to still approach reality, given thetypical ratio between the quantity of photons emitted atthe zero-phonon line and those belonging to the phononsideband . ACKNOWLEDGMENTS
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