Control of the electromagnetic environment of a quantum emitter by shaping the vacuum field in a coupled-cavity system
Robert Johne, Ron Schutjens, Sartoon Fattah poor, Chao-Yuan Jin, Andrea Fiore
aa r X i v : . [ qu a n t - ph ] M a r Control of the electromagnetic environment of a quantum emitter by shaping thevacuum field in a coupled-cavity system
Robert Johne, Ron Schutjens, Sartoon Fattah poor, Chao-Yuan Jin, and Andrea Fiore Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany COBRA Research Institute, Eindhoven University of Technology,PO Box 513, NL-5600MB Eindhoven, The Netherlands (Dated: August 14, 2018)We propose a scheme for the ultrafast control of the emitter-field coupling rate in cavity quantumelectrodynamics. This is achieved by the control of the vacuum field seen by the emitter througha modulation of the optical modes in a coupled-cavity structure. The scheme allows the on/offswitching of the coupling rate without perturbing the emitter and without introducing frequencychirps on the emitted photons. It can be used to control the shape of single-photon pulses forhigh-fidelity quantum state transfer, to control Rabi oscillations and as a gain-modulation methodin lasers. We discuss two possible experimental implementations based on photonic crystal cavitiesand on microwave circuits.
Spontaneous emission (SE) is at the heart of quantumoptics and quantum photonics. Tremendous progress hasbeen achieved in optimizing the SE of quantum emitters(QEs) in atomic systems [1] and artificial atoms suchas quantum dots [2–4] and superconducting circuits [5]by placing them into resonators. These coupled cavity-QE systems have been established as nonclassical lightsources [6] and may serve as light-matter interfaces [1, 7–9].In cavities, the presence of an increased optical densityof states enhances the emission and absorption propertiesof QEs. Typically, the QE-cavity interaction is governedby a coupling constant g given by the dipole moment ofthe QE and the vacuum electric field associated to thecavity mode. This interaction constant is thus given byintrinsic properties of the system, which are difficult tomodify. However, g is a crucial parameter, since it deter-mines the relevant timescale of the interaction. Indeed,an QE in the excited state decays into the cavity modewith a decay time τ dec = (cid:16) g κ (cid:17) − [10] in the weak cou-pling regime ( g ≪ κ ), where κ is the cavity loss rate.On the other hand, in the strong coupling regime, wherethe coupling constant g exceeds the loss of the cavity κ , acoherent and reversible energy exchange between the cav-ity and the QE takes place with a characteristic timescale τ r ∝ /g .So far, the control of the QE-cavity interaction in thesolid state has been performed mostly by tuning theirspectral overlap [11–16] and in the large majority of ex-periments on timescales much longer than the interactiontime. Dynamic control is however needed for the controlof the photon waveform and the establishment of QE-photon entanglement. Such dynamic control has beendemonstrated recently by using a combined variation ofthe loss rate and the cavity field by ultrafast carrier injec-tion [17], and by ultrafast detuning in photonic crystaldiodes [18]. However, both these techniques produce atemporal variation of the cavity frequency, resulting in a frequency chirp of the emitted photons, which limits thefidelity of quantum state transfer [19].In this letter, we propose the concept of pure vacuumfield modulation as a method to control the QE-cavityinteraction in real time. We demonstrate that the vac-uum field of an optical mode in a given cavity can becompletely suppressed by varying the frequency of twocoupled lateral cavities, without producing any frequencychirp of the target mode. This enables the on/off switch-ing of the QE-cavity interaction rate g , which is fun-damentally different from control techniques based oncontrolling the QE-cavity detuning. As an example, wetheoretically demonstrate the shaping of a single photonpulse into a symmetric wavepacket as a prerequisite forhigh-fidelity quantum state transfer [7]. Finally we dis-cuss two experimentally feasible platforms to implementthe proposed scheme. The full and direct control of thelight-matter interaction constant g represents powerfultool to develop advanced applications in quantum infor-mation science, e.g. switching Rabi oscillations, and itcan serve as the basis for a new class of gain modulatedlasers.We first consider the coupling of three in-line cavitieswith a QE placed in the central cavity (called target cav-ity) with frequency ω t as shown in Fig.1. Furthermorewe assume that the frequency of the two outer cavities ω l,r = ω t ± ∆, named left and right control cavity in thefollowing, can be tuned at will under the assumption thatthe detuning ∆ is the same for both but with a differentsign. The Hamiltonian of the empty three-cavity systemreads ( ~ = 1) H cc = ( ω t + ∆) − iκ l η η ω t − iκ t η η ( ω t − ∆) − iκ r , (1)where κ r,l,t are the loss rate of the control cavitiesand target cavity, respectively, and η denotes the cou-pling rate between the adjacent cavities. Neglecting ( ) ( ) ( ) left control cavity right control cavitytarget cavityη ηκ l κ r κ t (a)(b) (c) -8 0 8 Δ/η | α t ( ) | ω / η Δ/η ω ω ω FIG. 1. (a) Illustration of the coupled cavity scheme; (b)Calculated eigenfrequencies ω i /η as a function of the dimen-sionless cavity detuning ∆ /η . (c) Calculated | α (1) t | , whichgoverns the modulation of the QE-cavity coupling constant g .The used parameters are κ t,r,l = 0 . η . the loss rates, the Hamiltonian can be exactly diagonal-ized, which yields three non-degenerate eigenvalues ω i ( i = 1 , , ω = ω t ω , = ω t ± p η + ∆ . (2)The most interesting eigenvalue, which we will considerin the following is ω = ω t because its frequency is inde-pendent from the detuning of the control cavities. Thecorresponding eigenvector reads ~α (1) = α (1) l α (1) t α (1) r = 1 q η ) − ∆ η . (3)Fig.1 (b) and (c) show the frequencies ω i versus detun-ing and the target cavity fraction | α (1) t | of the mode ω calculated by solving the eigenvalue problem for H cc . In-terestingly, by changing the detuning of the control cavi-ties, the target cavity fraction α (1) t of the eigenmode canbe changed from α (1) t (∆ = 0) = 0 to α (1) t (∆ ≫ η ) ≈ ω = ω t can be freely engineered to have no electric fieldcomponent in the target cavity or the full field inten-sity corresponding to a decoupled target cavity. This hasdrastic consequences for a QE with frequency ω e = ω t coupled to the target cavity with strength g < κ t . Dueto its spatial position it can couple to the mode in case α (1) t > α (1) t = 0 (i.e. whenthe three cavities are in resonance). Moreover, also the effective loss rate of the modescan be changed due to the mixing of the cavity com-ponents. An approximate diagonalization ( η ≫ κ t , κ c )of Eq.(1) including the loss terms yields similar eigen-vectors as given in Eq. (3). The effective loss of thecoupled eigenmode ω can be then written as κ = | α (1) t | κ t + | α (1) l | κ l + | α (1) r | κ r . Thus, if the loss ratesare equal, the tuning of the left and right control cavitiesallows for a pure g -modulation.In order to describe the interaction of the coupled cav-ity system with the QE, we write the full Hamiltonianincluding the QE with frequency ω e and the QE-targetcavity coupling g using the basis of the coupled modes: H eff = ω e − iγ g g g g ω − iκ g ω − iκ g ω − iκ , (4)where the coupling rate g i = α ( i ) t g ( i = 1 , ,
3) and α ( i ) t describes the target cavity fraction of each eigen-mode. Assuming that η ≫ g and ω e = ω t , the QE isspectrally decoupled from the modes ω , (as we willassume in the following). It is immediately obvious,that by changing the detuning of the control cavities∆ one can modulate directly the effective interactionstrength between the QE and the cavity mode α (1) t g ,without changing the frequency of the coupled cavitymode ( ω = ω t ). The SE rate of the QE (neglectingthe spontaneous decay into leaky modes given by γ ) canbe described in the uncoupled-cavity case by γ = g κ t [10]. This obviously changes in case of the coupled cavi-ties to γ eff = | α (1) t | g κ . One obtains for the ratio of theSE rates γ eff γ = | α (1) t | κ t ( | α (1) t | κ t + | α (1) l | κ l + | α (1) r | κ r ) , (5)which can be tuned from zero (all cavities in resonanceand | α (1) t | = 0) to one (∆ ≫ η and | α (1) t | ≈ → ∆( t )in such a way that the typically sharp rise of the emittedsingle photon wavepacket can be slowed down to exactlymatch the time inverted decay tail.We simulate the dynamics using a wavefunction ap-proach and replace the loss of the target cavity by thecoupling to a quasi-continuum of modes representing awaveguide coupled to the target cavity [19, 20]. TheHamiltonian of the system reads H = X i = t,r,l ω i a + i a i + N X k =1 ω k b + k b k − X i = r,l iη ( a + t a i − a + i a t )(6)+ ig ( a + t σ − σ + a t ) − r κ t ∆ ω k π N X k =1 ( a + t b k − b + k a t ) . The first term is the free evolution of the three cavitymodes with operators a i and the second term describesthe evolution of the quasi-continuum with mode frequen-cies ω k and operators b k . The remaining terms describethe cavity-cavity coupling, the QE-target cavity couplingand the target cavity-continuum coupling, respectively.By plugging the expansion of the wavefunction (limitingourself to only a single excitation in the system) | Ψ i = ( c e | e, , , i + c t | g, , , i + c l | g, , , i (7)+ c r | g, , , i ) ⊗ | vac i + | g, , , i ⊗ X k c ( k ) κ b + k | vac i in the time dependent Schr¨odinger equation − i∂ t | Ψ i = H eff | Ψ i using an effective Hamiltonian including the lossof the cavity modes and the decay of the QE we ob-tain the evolution equations for the state amplitudes ina frame rotating at ω t ˙ c e = gc t − γc e (8)˙ c t = − gc e − η ( c l + c r ) + κ ′ N X k =1 c ( k ) κ (9)˙ c l = ηc t − i ∆( t ) c l − κ l c l (10)˙ c r = ηc t + i ∆( t ) c r − κ r c r (11)˙ c ( k ) κ = − i ∆ k c ( k ) κ − κ ′ c t , (12)where κ ′ = q κ t ∆ ω k π is the target-cavity-continuum cou-pling and ∆ ω k is the spacing of the quasi-continuummodes. The QE-dynamics are given by the amplitude c e , while the target and the two control cavities are de-scribed by amplitudes c t and c l,r , respectively. Finallythe waveguide quasi-continuum is described by the am-plitudes c ( k ) κ .Starting from a inverted QE ( c e (0) = 1) we calculatethe evolution of the state amplitudes as well as the outputpulse, which is given by the inverse Fourier transformof the amplitudes of the continuum c ( k ) κ ( T ) at a giventime T much larger than the duration of the effectiveinteraction of the QE with the cavity system [19, 20].We explicitly take into account the losses of the controlcavities, while the main loss channel of the target cavityis the quasi continuum coupling. Furthermore the QE isweakly coupled to the cavity mode g < κ t .Figure 2(a) shows the detuning of the control cavi-ties for the three scenarios ∆ ≫ η , a time dependent Δ>η Δ(t) Δ=0Δ>η Δ(t) Δ=096 1230 tg tg i n t e n s i t y ( a . u . ) φ ( t ) Δ / η p o p u l a t i o n (a)(b) (c)(d) FIG. 2. (a) Detuning ∆ /η for three different cases: ∆ =const(blue), ∆ = ∆( t ) (black), and ∆ = 0 (red). (b) intracavity dy-namics for the three cases. Solid lines show the QE population | c e | and dashed lines show the cavity photon population | c t | .(c) Pulses emitted into the waveguide for ∆ = const (blue)and for ∆ ( t ) (black). The red dashed line is a Gaussian fitof the emitted symmetric photon pulse. (d) Phase φ ( t ) of theemitted photon pulse versus time for the symmetric photonpulse. The used parameters are { g, η, κ r,l } = { . , , } κ t . detuning ∆( t ) and ∆ = 0. The intracavity dynamicsfor the QE population | c e ( t ) | (solid lines) and the tar-get cavity photon population | c t ( t ) | (dashed lines) areshown in Fig. 2 (b) for the three detuning plotted inFig.2(a). As expected, even for a system with losses theabove analysis holds i.e. the QE is completely decou-pled in case of ∆ = 0 and it decays as it would decayin an uncoupled cavity for large detuning ∆ ≫ η . Dueto the active modulation of the QE-cavity coupling via atime dependent detuning ∆( t ) one can shape the emit-ted wavepacket in principle arbitrarily. Here, we showthe shaping into an symmetric pulse, which can be ab-sorbed with in principle unit fidelity by a similar systemwith time-inverted control cavity detuning. The outputpulse is shown on Fig.2(c) together with the natural pulseshape without active control in case ∆ ≫ η . A Gaus-sian fit of the shaped wavepacket reveals a nearly perfecttime-symmetric wavepacket. The phase of the shapedwavepacket is shown in Fig.2(d) and it is constant in time(apart from the small deviations for long time due to nu-merical error caused by small amplitudes), since there isno frequency tuning involved in contrast to recent pro-posals using ultrafast electrical control of the QE energies[19]. The fidelity F of the absorption of this symmetricphoton pulse by a similar system with time inverted oper-ation can be obtained by calculating the overlap integralof the incident pulse with the time inverted pulse [19].This yields for the present case F = 0 .
997 and can befurther improved by optimization of the dynamic tuning.The dynamic tuning of QE-cavity coupling requiresthat the coupled cavity modes follow the adiabatic eigen-states given by Eq.(2). Assuming a linear time depen-dence of the detuning ∆( t ) = βt , the problem can be de-scribed by a generalized Landau-Zener model [21, 22].Theresulting condition √ β ≪ η needs to be satisfied to en-sure adiabaticity. This sets an upper speed limit on thedynamic tuning. Including the QE-cavity coupling, thecondition for shaping photon pulses can be written as2 g /κ ≪ √ β ≪ η , which is very well satisfied for thesimulations shown in Fig.2. In case one opts for theswitching of Rabi oscillations the more stringent condi-tion reads κ < g ≪ √ β ≪ η . On the other hand, detri-mental off-resonant coupling of the QE to modes ω , incase of ∆ ≈ g / η , which should be small. However, there alsothe spontaneous decay of the QE in leaky modes γ even-tually may affect the performance. For both the lowerand upper tuning speed limits, a large cavity coupling η is desired and should be carefully engineered duringthe implementation. Note that while we focus here onthe adiabatic tuning, diabatic effects may open up addi-tional applications of the coupled-cavity system e.g. forunconventional beam splitters and for the generation ofentanglement [23].The proposed scheme can be in principle implementedin any coupled-cavity QED system. Here we discuss twopractical implementations, namely a semiconductor cav-ity consisting of quantum dots in a photonic crystal andand a microwave system using circuit QED.In the solid state various implementations of coupledcavity systems have been realized such as microdiscs[24, 25], nanowires [26, 27], ring resonators [28] and pho-tonic crystals [29–32]. Here we consider a coupled pho-tonic crystal cavity-quantum dot system. The desiredtuning can be implemented by modulation of two laserbeams impinging on the control cavities injecting free car-riers and thus providing the local refractive index change.The modulation speed is only limited by the free carrierlifetime in the material and can well exceed the SE timeof a weakly coupled quantum dot [17]. By applying elec-tric fields to reduce the free carrier lifetime down to a fewpicoseconds a modulation speed of about 100 GHz canbe reached.To be more precise, we consider a system consisting ofthree in-line coupled L3 photonic crystal cavities. Three-dimensional finite element simulations are used to deter-mine the local density of optical states (LDOS) D expe-rienced by an QE in the central target cavity dependingon its wavelength for detunings of the control cavities asshown in Fig.3(a-c). The refractive index change can betranslated into a frequency/wavelength change by usingthe expression ∆ ω/ω = − ∆ n/n . For ∆ = 0 the mode ω has no target cavity fraction and hence the dipolein the central cavity interacts only with the modes ω , (Fig.3(a)). For intermediate detuning all three coupledmodes have electric field distributions in the target cav-ity mode, each at its own frequency (Fig.3(b)). In caseof very large detunings the central cavity mode is com-pletely decoupled and only a single peak is visible in theLDOS spectrum (Fig.3(c)). The difference in the loss (i) (ii) (iii) D / D Δ (nm) y x (i)(ii)(iii) |α t(1) ( ∆ )| D λ (μm) λ (μm) D ( a . u . ) λ (μm) (a)(d) Δ= 0 nm Δ = 2.5 nm Δ = 11 nm (b) (c)(e)
FIG. 3. Spectra of the local density of optical states D experi-enced by a dipole in the target cavity for a detuning (a) ∆ = 0nm (b) ∆ = 2 . D/D calculated by finite element simulations (squares) incomparison to the results from the coupled oscillator model | α (1) t | (red line). (e) Electric field distributions of the modewith frequency ω for three different detunings labelled (i-iii)indicated in (d). rates of modes ω , arises from different diffractive out-of-plane losses [33]. The dependence of the LDOS atfrequency ω on the detuning is shown in Fig.3(d). Itis in good agreement with the simple coupled oscillatormodel D t = | α (1) t | D , where D is the LDOS of thedecoupled target cavity mode determined by the finiteelement calculations. Once again the LDOS experiencedby the QE can be fully controlled by tuning the controlcavities. The electric field profile of the mode ω is shownfor three different detunings in Fig.3 (e).An alternative implementation is possible in super-conducting circuits [5] due to the ultrahigh quality res-onators as well as long coherence times of qubits. Thedevelopment of tunable superconducting resonators [34–36] has been the basis for tunable couplers with unprece-dented level of control [37]. This approach is also basedon a coupled cavity system, where the frequency detun-ing of the of the resonators is controlled by a magneticflux applied to a superconducting quantum interferencedevice. Thus, circuit QED provides an ideal platform torealize the present proposal in the microwave regime.Summarizing, the present proposal illustrates that acoupled-cavity system can be used to fully and deter-ministically control the QE-cavity coupling without in-ducing a spectral detuning between the QE and the cav-ity mode. This enables active shaping of single photonpulses in ultrafast experiments. Finally we propose anexperimental design, which can be used to implement thetuning scheme. The present proposal provides a powerfultechnique to control the SE of QEs and paves the waytowards a fully controllable single photon source. Fur-thermore, the results show the potential of the coupledcavity approach for actively controlling the light-matterinteraction in the solid state enabling more advanced ap-plication in quantum information processing.The authors acknowledge enlightening discussions oncoupled cavities with Massimo Gurioli (Univ. Firenze)and assistance from Leonardo Midolo in the finite el-ement simulations. This work is part of the researchprogramme of the Foundation for Fundamental Researchon Matter (FOM), which is financially supported bythe Netherlands Organization for Scientific Research(NWO), and is also supported by the Dutch TechnologyFoundation STW, applied science division of NWO, theTechnology Program of the Ministry of Economic Affairsunder Project No. 10380. 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