Phase-dependent quantum interferences with three-level artificial atoms
aa r X i v : . [ qu a n t - ph ] M a r Phase-dependent quantum interferences with three-level artificialatoms
Victor Ceban ∗ Institute of Applied Physics, Academy of Sciences of Moldova,Academiei str. 5, MD-2028 Chi¸sin˘au, Moldova
Abstract
The phase dependence of the cavity quantum dynamics in a driven equidistant three-level ladder-type system found in a quantum well structure with perpendicular transition dipoles is investigatedin the good cavity limit. The pumping laser phases are directly transferred to the superposedamplitudes of the cavity-quantum-well interaction. Their phase difference may be tuned in orderto obtain destructive quantum interferences. Therefore, the cavity field vanishes although theemitter continues to be pumped.
PACS numbers: 42.50.-p, 42.50.Ar, 42.50.Pq ∗ Electronic address: [email protected] . INTRODUCTION The confinement of quantum systems in a specific superposition of states may lead to var-ious quantum processes. In the realm of quantum optics, a particular interest is focused onthis type of phenomena, namely, quantum interference phenomena allow to explain and ob-serve various quantum effects [1–5], while etanglement processes [6–9] and shape-preservinglocalized light structures [10, 11] play a major role for the quantum computation and com-munication. A powerful tool in the control and manipulation of these effects originatesfrom an additional degree of freedom of the system given by its phase dependence. Forexample, quantum interference effects influence the collective fluorescence of a driven sam-ple of emitters, which becomes sensitive to phase dependence. Thus, the phase differenceof the two lasers pumping a collection of three-level emitters may decrease and cancel itsfluorescence when quantum interferences appear from a coherently driven source [12]. Thesuperflourescent behaviour of a sample of four-level emitters is modified by the vacuum in-duced quantum interferences and may be enhanced by varying the phase difference of thepumping lasers [13]. Moreover, for a well-chosen phase the sample may be trapped in itsexcited state and thus decoupled from the surrounding environment. The phase dependentcomplete or partial cancellation of the spontaneous emission is reached when a single four-level emitter is considered [14]. The spontaneous emission properties may also be controlledvia the phase difference of the pumping laser and a squeezed surrounding reservoir for athree-level ladder-type emitter [15]. In a different scenario, phase dependent systems maybe used to study the phase itself, e.g. , the carrier-envelope phase of a few-cycle laser pulsemay be determined via the behaviour of the populations of a qubit system [16].A more challenging goal has been the realization of quantum effects in systems made ofartificial atoms such as quantum wells (QWs), as these systems possess additional degreesof freedom, which leads to stronger decoherent phenomena [17]. The particular interest inthis type of artificial atoms for the current realm is the possibility to tailor their energeticstates via the layer thicknesses and materials used for the QW [18]. Quantum interferencephenomena as gain without inversion have been experimentally obtained for pumped three-level ladder-type coupled triple wells [19], while electromagnetically induced transparencyhas been observed in three-level QW systems with Λ-type transitions [20] as well as ladder-type intersubband transitions [21, 22]. A direct detection of ac Stark splitting, i.e. , dressed-2tate splitting, has been experimentally achieved in [21] for Ξ-type QWs. This type of QWs isparticularly interesting as it may be engineered as an equidistant three-level emitter [17, 18],an emitter difficult to implement with real atoms.In this paper, a pumped ladder-type three-level QW placed in a cavity is investigated.The QW architecture has equidistant energy levels and orthogonal transition dipoles. Eachtransition is resonantly driven by lasers with different phases. The energy level distribu-tion allows the optical cavity to couple with each of the QW transitions. Under the laserdriving, the QW exciton is prepared in a superposition of states, which leads to quantuminterference of the indistinguishable amplitudes of the cavity interaction with the differentexciton transitions. Strong destructive interferences may be achieved if the cavity is tunedto the most or less energetic dressed-state transition of the pumped QW. Therefore, thecavity field may be emptied for a well-chosen laser phase difference as the laser phases aretransferred to the interactional amplitudes. In this case, the pumped QW spontaneouslydecays in all directions except the cavity. Furthermore, this behaviour of the interferingQW-cavity system is associated with a quantum switch, where the income laser signals mayswitch the cavity field on and off by varying their phase difference.This article is organized as follows. In Sec. 2 the studied model is described, one presentsthe system Hamiltonian, the applied approximations and the master equation solving tech-nique. The results on the quantum interferences effect are discussed in Sec. 3. The summaryis given in Sec. 4.
II. THE MODEL
The model consists of a three-level equidistant ladder-type QW placed in an optical cavity.The QW is driven by two intense lasers and has perpendicular transition dipoles, whichallows to set each laser to pump a separate transition. The QW is described by its bare-states | i i , { i = 1 , , } and their corresponding energies ~ ω i . The atomic operators are defined as S ij = | i ih j | , { i, j = 1 , , } and obey the commutation rule [ S α,β , S β ′ ,α ′ ] = δ β,β ′ S α,α ′ − δ α ′ ,α S β ′ ,β . The most energetic level | i may spontaneously decay to the intermediate level | i with a rate γ , while the last one decays to the ground level | i with a rate γ . Thelaser pumping of the QW is expressed by semi-classical interactions with Rabi frequencyΩ (Ω ) corresponding to the laser of frequency ω L ( ω L ) and phase φ ( φ ) driving the3ower (upper) transition. The QW-cavity quantum interaction is described by the couplingconstant g ( g ) corresponding to the interaction of the optical resonator with the lower(upper) QW transition. The cavity field is defined by its frequency ω c and the bosoniccreation (annihilation) operators a † ( a ) that commute as [ a, a † ] = 1. The cavity is dumpedby a vacuum reservoir at a rate κ . The system Hamiltonian is defined as: H = ~ ω c a † a + ~ X i =1 ω i S ii + i ~ g ( a † S − S a ) + i ~ g ( a † S − S a )+ ~ Ω ( S e − i ( ω L t + φ ) + S e i ( ω L t + φ ) )+ ~ Ω ( S e − i ( ω L t + φ ) + S e i ( ω L t + φ ) ) . (1)where the first two terms are the free cavity and QW terms, the next two terms representthe QW-laser semi-classical interaction, while the last two terms describe the QW-cavityquantum interaction. The system dynamics is described by the master equation of thedensity operator ρ , namely: ∂ρ∂t = − i ~ [ H, ρ ] + κ L ( a ) + γ L ( S ) + γ L ( S ) , (2)where the Liouville superoperator is defined as L ( O ) = 2 O ρ O † − O † O ρ − ρ O † O for a givenoperator O . The second term of the equation describes the cavity damping, while the lasttwo terms represents the QW spontaneous emission.In the interaction picture, the Hamiltonian is brought to an easy diagonalizable form ofthe QW-lasers subsystem terms and is defined as: H = ~ ( ω c − ω L ) a † a + ~ Ω ( S + S ) + ~ Ω ( S + S )+ i ~ g ( a † S e − iφ − e iφ S a ) + i ~ g ( a † S e − iφ − e iφ S a ) . (3)Here, the lasers are considered to be resonant with the QW transitions and therefore ω L = ω L = ω L . Next, one adopts the semi-classical dressed-state transformation according tothe dynamical Stark splitting effect of the QW under the laser pumping [21]. In analogywith the Mollow triplet of a two-level emitter, the fluorescence spectra of the driven QWpossess sidebands that are symmetrical to the central bar-state frequency peak. However,in the case of the equidistant three-level emitter one has four degenerate sidebands due toits degenerate bare-state central peak. The new Hermitian base is defined considering thepumped QW subsystem eigenfunctions. The new atomic wavefunction basis vectors, i.e.,4he dressed-states, are defined as [23]: | i = − √ θ |−i − sin θ | i + 1 √ θ | + i , | i = 1 √ |−i + 1 √ | + i , | i = − √ θ |−i + cos θ | i + 1 √ θ | + i , (4)where θ = tan − (Ω / Ω ), Ω = p Ω + Ω . Once, the dressed-state transformation is applied,one tunes the cavity in resonance with the sideband transitions of the dressed-QW, i.e., ω c = ω L ± Ω or ω c = ω L ± g , ≪ Ω. In what follows, the cavity is set in resonance to the mostenergetic sideband, but note that the further discussions and results are also valid for thecase when the cavity is tuned to the less energetic transition, where a similar behaviour isobserved. The Hamiltonian within the secular approximation is brought to the form: H = ig ( a † R − + e iψ − e − iψ R + − a ) , (5)where g = ( g e − iφ sin θ − g e − iφ cos θ ) / ψ = arg( g ). The new set of atomic dressed-state operators is defined as R ij = | i ih j | , { i, j } ∈ {− , , + } and R z = R ++ − R −− . The newoperators obey the same commutation relations as the previous ones. The master equationis defined in the new basis as: ∂ρ∂t = − i ~ [ H, ρ ] + κ L ( a ) + γ a ( L ( R − ) + L ( R +0 ))+ γ b ( L ( R − ) + L ( R )) + γ c ( L ( R z ) + L ( R + − ) + L ( R − + ))) , (6)where γ a = γ (cos θ ) / γ b = γ (sin θ ) / γ c = ( γ sin θ + γ cos θ ) / γ , ≪ Ω.The master equation is numerically solved via projecting it in the system state basis[25]. The solving technique was adapted to the case when a three-level emitter and phasedependent lasers are considered. A first projection in the QW dressed-states leads to asystem of linear differential coupled equations defined by the variables: ρ (0) = ρ −− + ρ + ρ ++ , ρ (1) = ρ ++ + ρ −− , ρ (2) = ρ ++ − ρ −− , ρ (3) = ( a † ρ + − e iψ + e − iψ ρ − + a ) /
2, and ρ (4) = ( ρ + − a † e iψ +5 − iψ aρ − + ) /
2, where ρ ij = h i | ρ | j i , { i, j ∈ − , , + } are the QW reduced density matrixelements. The equations are next projected in the cavity field Fock states basis {| n i , n ∈ N } ,leading to the following set of equations:˙ P (0) n = − | g | ( P (4) n − P (3) n ) + κ ( n + 1) P (0) n +1 − κnP (0) n , ˙ P (1) n = − | g | ( P (4) n − P (3) n ) + κ ( n + 1) P (1) n +1 − ( κn + α/ P (1) n + γ cos θP (0) n , ˙ P (2) n = − | g | ( P (4) n + P (3) n ) + κ ( n + 1) P (2) n +1 − ( κn + β/ P (2) n , ˙ P (3) n = | g | n ( P (1) n − − P (1) n + P (2) n − + P (2) n ) / − κP (4) n + κ ( n + 1) P (3) n +1 − ( κ ( n − /
2) + ζ ) P (3) n , ˙ P (4) n = | g | ( n + 1)( P (2) n +1 + P (2) n − P (1) n +1 + P (1) n ) / κ ( n + 1) P (4) n +1 − ( κ ( n + 1 /
2) + ζ ) P (4) n , (7)where ζ = [ γ (2 + cos θ ) + 3 γ sin θ ] / α = γ sin θ + 2 γ cos θ , β = γ + γ sin θ , and P ( i ) n = h n | ρ ( i ) | n i .This system of equations (7) is numerically solved, considering the probability conserva-tion of the density matrix elements, i.e. , Tr[ ρ ] = 1, and their asymptotic behaviour thatallows the system to be truncated at a certain maximum n max of considered Fock states.The parameters of interest are estimated from the system variables and will be presentedand discussed in the next Section. One observes the cavity behaviour via the mean photonnumber h n i and the second-order photon-photon correlation function g (2) (0) defined by thediagonal elements of the QW’s reduced density matrix, deduced from the system (7) asfollows: h n i = h a † a i = ∞ X n =0 iP (0) i ≃ n max X n =0 iP (0) i , (8) g (2) (0) = h a † a † aa ih a † a i = 1 h n i ∞ X n =0 i ( i − P (0) i ≃ h n i n max X n =0 i ( i − P (0) i . (9) III. RESULTS AND DISCUSSIONS
The cavity field behaviour shows a good evidence of quantum interferences, as presentedin Fig. 1. For a certain configuration of laser phases and Rabi frequencies ratio, the meanphoton number is strongly decreased down to the zero value. This minimum describes a6
IG. 1: (a): The cavity mean photon number h n i and (b): the second-order photon-photon corre-lation function g (2) (0) as functions of the laser phase φ and Rabi frequencies ratio Ω / Ω . Here g /γ = 6, g /γ = 4, γ /γ = 2, κ/γ = 10 − , and φ = π/ complete cancellation of the cavity field and corresponds to the case when the two indistin-guishable amplitudes of the QW-cavity interaction are equal and in-phase. Therefore, whenthe cavity interacts equally with both of the QW transitions, the interaction amplitudescancel each other due to their destructive superposition. This destructive quantum inter-ference effect is also reflected in the behaviour of the field second-order correlation functiondescribing the photon distribution. When the cavity mean photon number is cancelled, g (2) (0) →
2, asymptotically describing a thermal distribution. The cavity is in equilibriumwith the surrounding electromagnetic vacuum, when maximum interference effect is reached.The phase difference of the input lasers plays a crucial role in the control of the quan-tum interference. The interaction amplitudes phases are related to the laser phases assuggested by the expression of the coupling constant g of the Hamiltonian form of equa-tion (5) at cavity-QW resonance and within the secular approximation. Therefore, adestructive superposition is obtained when the interaction amplitudes are in-phase, i.e., φ = φ + 2 πm, m ∈ Z , as shown in Fig. 1. At this condition, the system behaves simi-larly to the case when no laser phase was considered [23], where the field cancels simply for g /g = Ω / Ω .The possibility to control and turn-off the cavity field via quantum interferences suggests a7otential application of the studied QW-cavity system for quantum network circuits [26, 27].The model is sensitive to phase and intensity variations of the input lasers and acts asquantum switch, where the cavity field is turned on or off. Both input parameters arelargely confined in experimental conditions. Moreover, artificial-atom-based systems couldbe relevant candidates for on-chip quantum circuits [28]. IV. SUMMARY
The model of a pumped equidistant three-level ladder-type quantum well placed in anoptical cavity has been investigated in the good cavity limit. The emitter has perpendiculartransition dipoles and the cavity couples to both of the QW transitions. Two intense laserswith different phases are used to resonantly drive the emitter and each laser couples semi-classically to a different transition. It has been shown that the laser phases are transferred tothe QW-cavity interaction amplitudes. Therefore, the superposition of the indistinguishableamplitudes is phase dependent, so that the resulting destructive quantum interferences effecton the cavity field becomes sensitive to the phase difference of the input lasers.
Acknowledgement
The author is thankful to M. A. Macovei for fruitful discussions related to this study. Heacknowledges the financial support from the Academy of Sciences of Moldova via grant No.15.817.02.09F. [1] G. S. Agarwal,
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