Dynamics of a quantum oscillator coupled with a three-level Lambda-type emitter
aa r X i v : . [ qu a n t - ph ] M a y Dynamics of a quantum oscillator coupled with a three-level Λ − type emitter Alexandra Mˆırzac and Mihai A. Macovei ∗ Institute of Applied Physics, Academiei str. 5, MD-2028 Chi¸sin˘au, Moldova (Dated: May 2, 2019)We investigate the quantum dynamics of a quantum oscillator coupled with the most upperstate of a three-level Λ − type system. The two transitions of the three-level emitter, possessingorthogonal dipole moments, are coherently pumped with a single or two electromagnetic field sources,respectively. We have found ranges for flexible lasing or cooling phenomena referring to the quantumoscillator’s degrees of freedom. This is due to asymmetrical decay rates and quantum interferenceeffects leading to population transfer among the relevant dressed states of the emitter’s subsystemwith which the quantum oscillator is coupled. As an appropriate system can be considered ananomechanical resonator coupled with the most excited state of the three-level emitter fixed onit. Alternatively, if the upper state of the Λ − type system possesses a permanent dipole then it cancouple with a cavity electromagnetic field mode which can be in the terahertz domain, for instance.In the latter case, we demonstrate an effective electromagnetic field source of terahertz photons. I. INTRODUCTION
Lasing and cooling effects are among the most studiedones due to their enormous potential applications in themicro- or nano-world [1–5]. Presently, quantum technolo-gies [6–8] require precise tools allowing a complete controlof the quantum interaction between light and matter and,of course, the above mentioned phenomena occurring ina wide range of systems. Particularly, certain quantumsystems offer additional control mechanisms via exter-nally applied coherent light sources and, therefore, cool-ing phenomenon was successfully demonstrated in few-level atomic systems [9–12], for instance. On the otherside, various optomechanical systems are intensively in-vestigated recently because of their extreme sensitivityto ultra-weak perturbations [13, 14]. Thereby, coolingor lasing in these systems are of fundamental interest aswell [4, 15–18]. Furthermore, artificially created atom-iclike systems such as quantum dots or quantum wellsare also suitable for modern applications and exhibit anadvantage with respect to engineering of their dipole mo-ments, transition frequencies, etc. [19–21]. In these cir-cumstances, ground-state cooling of a nanomechanicalresonator with a triple quantum dot via quantum inter-ference effects was demonstrated in [22], see also [23–25].Enhanced nanomechanical resonator’s phonon emissionvia multiple excited quantum dots was demonstrated aswell, in Ref. [26]. Moreover, among other applicationsof these systems or various optoelectronical schemes isthe generation of electromagnetic field in the terahertzdomain. The importance of the terahertz waves towardssensing, imaging, spectroscopy or data communicationsis highly recognized [27–29]. In this context, quantumsystems possessing permanent dipoles were shown to gen-erate terahertz light [30–34]. Additionally, they exhibitbare-state population inversion as well as multiple spec-tral lines and squeezing [35–37]. ∗ Electronic address: [email protected]
Thus, there is an increased interest for novel quantumsystems exhibiting lasing in a broad parameter range orcooling of micro- or nano-scale devices. From this pointof view, here, we investigate a laser pumped Λ − typethree-level system the upper state of which is being cou-pled with a quantum oscillator described by a quantizedsingle-mode boson field. More specifically, as a quan-tum oscillator can serve a vibrational mode of a nanome-chanical resonator containing the three-level emitter or,respectively, an electromagnetic cavity mode field if theupper state of the three-level sample, embedded in thecavity, possesses a permanent dipole. The frequency ofthe quantum oscillator is significant smaller than all otherfrequencies involved to describe the model, however, it isof the order of the generalized Rabi frequency character-izing the laser-pumped three-level qubit. In concordanceto the dressed-state picture of the three-level system, wehave identified two resonance conditions determining theoscillator’s quantum dynamics, namely, when the quan-tum oscillator’s frequency is close to the doubled gener-alized Rabi frequency or just to the generalized Rabi fre-quency, respectively. Correspondingly, we treat these twosituations separately. We have found steady-state lasingor cooling regimes in both situations for the quantumoscillator’s field mode, however, for asymmetrical spon-taneous decay rates corresponding to each three-levelqubit’s transition. The mechanisms responsible for theseeffects are completely different for the two situations. Inthe case when the doubled generalized Rabi frequency isclose to the oscillator’s one, the model is somehow similarto a two-level system interacting with a quantized fieldmode where the spontaneous decay pumps both levels.On the other side, if the oscillator’s frequency lies nearresonance with the generalized Rabi frequency, then thesample is close to an equidistant three-level system wherethe single-mode quantum oscillator interacts with bothqubit’s transitions. The latter situation includes single-or two-quanta processes accompanied by quantum inter-ference effects among the involved dressed-states lead-ing to deeper cooling regimes and flexible ranges for las-ing effects. This is different from other related schemes FIG. 1: (a) The schematic of the model: A laser pumpedthree-level Λ − type system the upper state of which, | i ,is coupled with a quantum oscillator mode of frequency ω .The oscillator can be described by a single mode of a nano-mechanical resonator containing the three-level emitter. Al-ternatively, if the upper state of the three-level system pos-sesses a permanent dipole then it can couple with an electro-magnetic cavity mode which can be in the terahertz ranges,for instance. Here, the pumping laser’s frequencies are equalto the average transition frequency of the three-level emitter( ω + ω ) /
2. Ω and Ω are the corresponding laser-qubitcoupling strengths, i.e. the Rabi frequencies, whereas γ ′ s arethe respective spontaneous decay rates. (b) The semi-classicallaser-qubit dressed-state picture where each bare-state levelis dynamically split in three dressed-states {| Ψ i , | Ψ i , | Ψ i} .Resonances occur at: (I) ω = 2Ω or (II) ω = Ω, respectively,where Ω is the generalized Rabi frequency. based on electromagnetically induced transparency pro-cesses [22–25]. In the case the model contains an elec-tromagnetic cavity mode, which describes the quantumoscillator, then its frequency can be in the terahertz do-main and, thus, we demonstrate an effective coherentelectromagnetic field source of such photons. While las-ing or cooling effects are available for two-level systems aswell [3–5], three-level ones may have an advantage in thesense that show improved results for the same parametersinvolved. This may help when there are only certain ac-cessible parameter ranges. Furthermore, certain realisticnovel systems are described by a three-level model. Forinstance, as a concrete Λ − type system may be taken alaser-pumped color center emitter embedded on a vibrat-ing membrane where strong coupling strengths can beachieved via vacuum dispersive forces [25]. Few coupledquantum dots are appropriate systems too [23, 38]. Also,as alternative systems can be asymmetrical real or ar-tificial few-level molecules possessing permanent dipoles, d αα = 0 [30–37, 39, 40]. If d ≫ { d , d } , then an elec-tromagnetic resonator mode can couple with the upperstate of the Λ − type system via its permanent dipole. The article is organized as follows. In Sec. II we de-scribe the analytical approach and the system of interest,while in Sec. III we analyze the obtained results. Thesummary is given in Sec. IV. II. THEORETICAL FRAMEWORK
The Hamiltonian describing a quantum oscillator offrequency ω coupled with a laser-pumped Λ − type three-level system, see Fig. 1(a), in a frame rotating at ( ω + ω ) /
2, is: H = ~ ωb † b + ~ ω S − S ) + ~ gS ( b + b † ) − ~ X α ∈{ , } Ω α ( S α + S α ) . (1)We have assumed here that as a pumping electromag-netic field source it can act a single laser of frequency ω L pumping both arms of the emitter or, respectively, twolasers fields { ω L , ω L } each driving separately the twotransitions of the Λ − type sample possessing orthogonaltransition dipoles. Additionally, we have also consideredthat ω L = ω L ≡ ( ω + ω ) /
2, see Fig. 1(a). Here ω αβ are the frequencies of | α i ↔ | β i three-level qubit’stransitions, { α, β ∈ , , } . The components enteringin the Hamiltonian (1) have the usual meaning, namely,the first and the second terms describe the free energiesof the quantum oscillator and the atomic subsystem, re-spectively, whereas the third one accounts for their mu-tual interaction via the most upper-state energy levelwith g being the respective coupling strength. The lastterm represents the atom-laser interaction and { Ω , Ω } are the corresponding Rabi frequencies associated with aparticular driven transition. Note that if the upper stateof the investigated model contains a permanent dipolethen the external coherent light sources interact withit as well. The corresponding Hamiltonian is: H pd = ~ S P i ∈{ , } G i cos ( ω Li t ), where G i = d E i / ~ with E i being the lasers amplitudes. However, the Hamiltonian H pd can be considered as rapidly oscillating, because ω Li ≫ G i , and being further neglected. Thus, the Hamil-tonian (1) and the analytical approach developed hereallow to treat concomitantly both situations, namely,when either a nanomechanical resonator or an electro-magnetic cavity is taken as a quantum oscillator. Finally,the three-level qubit’s operators, S αβ = | α ih β | , obey thecommutation relation [ S αβ , S β ′ α ′ ]= δ ββ ′ S αα ′ - δ α ′ α S β ′ β whereas those of the quantum oscillator’s: [ b, b † ] = 1and [ b, b ] = [ b † , b † ] = 0, respectively.In the Born-Markov approximations [41–43], the wholequantum dynamics of this complex model can be moni-tored via the following master equation:˙ ρ + i ~ [ H, ρ ] = − X α ∈{ , } γ α [ S α , S α ρ ] − γ [ S , S ρ ] − κ (1 + ¯ n )[ b † , bρ ] − κ ¯ n [ b, b † ρ ] + H.c.. (2)The right-hand side of Eq. (2) describes the emit-ter’s damping due to spontaneous emission as well asthe quantum oscillator’s damping effects with ¯ n =1 / [exp ( ~ ω/k B T ) −
1] being the mean oscillator’s quantanumber due to the environmental thermostat at temper-ature T . Here k B is the Boltzmann constant, γ ’s arethe corresponding decay rates of the three-level qubit,see Fig. 1(a), while κ describes the quantum oscillator’sleaking rate, respectively. The physics behind our modelcan be easier highlighted if we turn to the three-levelqubit-laser dressed-state picture given by the transfor-mation: | i = sin θ | Ψ i − cos θ √ (cid:0) | Ψ i + | Ψ i (cid:1) , | i = cos θ √ | Ψ i + 12 (1 + sin θ ) | Ψ i −
12 (1 − sin θ ) | Ψ i , | i = − cos θ √ | Ψ i + 12 (1 − sin θ ) | Ψ i −
12 (1 + sin θ ) | Ψ i , (3)where sin θ = ω / (2Ω) and cos θ = √ / Ω withΩ = p + ( ω / being the generalized Rabi fre-quency whereas Ω = Ω ≡ Ω . Applying the transfor-mation (3) to the Hamiltonian (1) one arrives at the cor-responding Hamiltonian’s expression in the dressed-statepicture, i.e., H = H + H d + H + H , where H = ~ ωb † b + ~ Ω R z ,H d = ~ g (cid:0) sin θR + cos θ ( R + R ) / (cid:1)(cid:0) b + b † (cid:1) ,H = ~ g cos θ (cid:0) R + R (cid:1)(cid:0) b + b † (cid:1) / ,H = − ~ g sin 2 θ √ (cid:0) R + R + H.c. (cid:1)(cid:0) b + b † (cid:1) , (4)with R z = R − R . Here the dressed-state three-levelqubit’s operators are: R αβ = | Ψ α ih Ψ β | and obeying thesame commutation relations as the old ones. In the in-teraction picture, characterized by the unitary operator U ( t ) = exp ( iH t/ ~ ) , (5) H d can be considered as a fast oscillating one and omit-ted from the dynamics, while the last two Hamiltonianstransforms as: H I = ¯ g (cid:0) R e i Ω t + H.c. (cid:1)(cid:0) b † e iωt + H.c. (cid:1) ,H I = − ˜ g (cid:0) ( R + R ) e i Ω t + H.c. (cid:1)(cid:0) b † e iωt + H.c. (cid:1) , (6)where ¯ g = ~ g cos θ/ , (7)whereas ˜ g = ~ g sin 2 θ/ (2 √ . (8) Analyzing the above Hamiltonians one can observe thatthe quantum dynamics of our model is determined bytwo resonances (see Fig. 1b), namely, ( I ) at2Ω = ω, (9)and ( II ) at Ω = ω. (10)Therefore, in what follows, we shall treat these two casesseparately. Thus, the Hamiltonian for the first situation,( I ), will be H = ¯ δb † b + ¯ g (cid:0) R b † + bR (cid:1) , (11)while for the second case, ( II ), is H = ˜ δb † b − ˜ g (cid:0) ( R + R ) b † + b ( R + R ) (cid:1) , (12)where, respectively, ¯ δ = ω −
2Ω whereas ˜ δ = ω − Ω. Addi-tionally, applying the dressed-state transformation (3) tothe corresponding damping part of the master equation(2), followed by the operation (5), one arrives at a masterequation, see Appendix A, which allows to obtain an ex-act system of equations describing the quantum dynam-ics of the examined system. Note that rapidly oscillatingcomponents in the above Hamiltonians, i.e. (11,12), aswell as in the final master equation (A1) were dropped,meaning that Ω ≫ { g, γ, γ , γ } .In what follows, we shall compare the two situations,i.e. ( I ) and ( II ), for the same parameters range anddiscuss the physics behind. III. RESULTS AND DISCUSSIONS
The equations of motion, for the first situation ( I ),describing the oscillator’s quantum dynamics (i.e., meanquanta number and its quantum statistics, qubit’s popu-lations etc.) can be obtained with the help of Eq. (A1):˙ P (0) n = i ¯ g ( P (5) n − P (3) n ) − κ ¯ n (cid:0) ( n + 1) P (0) n − nP (0) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (0) n − ( n + 1) × P (0) n +1 (cid:1) , ˙ P (1) n = i ¯ g ( P (5) n − P (3) n ) − κ ¯ n (cid:0) ( n + 1) P (1) n − nP (1) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (1) n − ( n + 1) × P (1) n +1 (cid:1) + γ (1)0 P (0) n − γ (1)1 P (1) n , ˙ P (2) n = i ¯ g ( P (5) n + P (3) n ) − κ ¯ n (cid:0) ( n + 1) P (2) n − nP (2) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (2) n − ( n + 1) × P (2) n +1 (cid:1) + γ (2)0 P (0) n − γ (2)1 P (1) n − γ (2)2 P (2) n , ˙ P (3) n = i ¯ δP (4) n − i ¯ gn ( P (1) n − P (2) n − P (1) n − − P (2) n − ) − κ (1 + ¯ n ) (cid:0) (2 n − P (3) n − n + 1) × P (3) n +1 + 2 P (5) n (cid:1) − κ ¯ n (cid:0) (2 n + 1) P (3) n − nP (3) n − (cid:1) − γ (3)3 P (3) n , ˙ P (4) n = i ¯ δP (3) n − κ (1 + ¯ n ) (cid:0) (2 n − P (4) n + 2 P (6) n − n + 1) P (4) n +1 (cid:1) − κ ¯ n (cid:0) (2 n + 1) P (4) n − nP (4) n − (cid:1) − γ (4)4 P (4) n , ˙ P (5) n = i ¯ δP (6) n + i ¯ g ( n + 1)( P (1) n + P (2) n − P (1) n +1 + P (2) n +1 ) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (5) n − n + 1) P (5) n +1 (cid:1) − κ ¯ n (cid:0) (2 n + 3) P (5) n − nP (5) n − − P (3) n (cid:1) − γ (5)5 P (5) n , ˙ P (6) n = i ¯ δP (5) n − κ ¯ n (cid:0) (2 n + 3) P (6) n − nP (6) n − − P (4) n (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (6) n − n + 1) P (6) n +1 (cid:1) − γ (6)6 P (6) n . (13)Here γ (1)0 = (cid:0) ( γ ( − ) + γ (+) ) sin θ + γ cos θ (1 + sin θ ) (cid:1) / γ (1)1 = 2 γ (0)0 + ( γ ( − ) + γ (+) ) sin θ/ γ cos θ (1 +sin θ ) / γ (2)0 = (cid:0) ( γ (+) − γ ( − ) ) sin θ − γ sin θ cos θ (cid:1) / γ (2)1 = 2(Γ ( − ) − Γ (+) ) + ( γ (+) − γ ( − ) ) sin θ/ − γ sin θ cos θ/ γ (2)2 = 2 (cid:0) γ (0)0 + Γ ( − ) + Γ (+) + γ cos θ (1 +sin θ ) / (cid:1) and γ (3)3 = ( γ + γ ) cos θ/ γ (0)0 + Γ ( − ) +Γ (+) + γ cos θ (1+sin θ ) / γ (4)4 = γ (5)5 = γ (6)6 = γ (3)3 .Further, γ ( ± ) = γ (1 ± sin θ ) + γ (1 ∓ sin θ ) , Γ ( ± ) = γ ( ± ) cos θ/ γ (1 ∓ sin θ ) / γ ( ± )0 = ± (cid:0) γ (1 ∓ sin θ ) − γ (1 ± sin θ ) (cid:1) sin θ cos θ/ γ (0)0 = ( γ + γ ) cos θ/ ρ (0) = ρ + ρ + ρ , ρ (1) = ρ + ρ , ρ (2) = ρ − ρ , ρ (3) = b † ρ − ρ b , ρ (4) = b † ρ + ρ b , ρ (5) = ρ b † − bρ , ρ (6) = ρ b † + bρ , where ρ αβ = h α | ρ | β i , and then pro-jecting on the Fock states | n i , i.e., P ( i ) n = h n | ρ ( i ) | n i , { i ∈ · · · } and n ∈ { , ∞} , see also [44]. Thus, theanalytical approach developed here allows us to obtainan exact system of equations describing the quantumdynamics of the composed system laser pumped spon-taneously damped qubit plus leaking phonon mode withinthe rotating wave, Born-Markov and secular approxima-tions, respectively, and to extract the variables of interestwith the help of the traced density operator over the cor-responding degrees of freedom.In order to solve the infinite system of Eq. (13), wetruncate it at a certain maximum value n = n max sothat a further increase of its value, i.e. n max , does notmodify the obtained results. Thus, the steady-state meanquanta’s number is expressed as: h b † b i = n max X n =0 nP (0) n , (14)with n max X n =0 P (0) n = 1 , (15)while its steady-state second-order correlation function is Ω (cid:144)H W L X b † b \ (cid:144) n Ω (cid:144)H W L g b H L H L (a) (b)FIG. 2: (a) The mean quanta number of the quantum os-cillator h b † b i / ¯ n and (b) its second-order correlation function g (2) b (0) versus ω / (2Ω ) for the situation (I). Here g/γ = 4, γ /γ = 0 . γ/γ = 0, κ/γ = 10 − , ω/γ = 50, Ω /γ = 20and ¯ n = 1. Ω (cid:144)H W L X b † b \ (cid:144) n Ω (cid:144)H W L g b H L H L (a) (b)FIG. 3: (a) The scaled mean quanta number of the quantumoscillator h b † b i / ¯ n and (b) the corresponding second-order cor-relation function g (2) b (0) against the scaled control parameter ω / (2Ω ) for the situation (I). Here g/γ = 4, γ /γ = 0 . γ/γ = 0, κ/γ = 10 − , ω/γ = 50, Ω /γ = 20 and ¯ n = 15. defined as usual [45], namely, g (2) b (0) = h b † b † bb ih b † b i = 1 h b † b i n max X n =0 n ( n − P (0) n . (16)Respectively, the steady-state mean value of the dressed-state inversion operator, h R z i = h R i − h R i , can beobtained as follows: h R z i = n max X n =0 P (2) n . (17)Figure (2) shows the steady-state behaviors of the meanquanta number and its quantum statistics based onEqs. (13) and Exps. (14,15,16). The maximum for h b † b i occurs around ¯ δ = 0, i.e., at the resonance whenthe quanta’s frequency ω equals the dressed-state split-ting frequency 2Ω due to pumping lasers. Importantlyhere, the quanta’s statistics is near Poissonian meaningthat we have obtained lasing regimes in our system, seeFigs. 2(a,b). Also, lasing is taking place if γ /γ ≪ h R i > h R i , that is, we have dressed-statepopulation inversion and this is the reason for lasing ef-fect, see Fig. 6(a). To avoid any confusion via lasing wemean generation of quantum oscillator’s quanta possess-ing Poissonian statistics, i.e., g (2) b (0) = 1. Respectively, Ω (cid:144)H W L X b † b \ (cid:144) n Ω (cid:144)H W L g b H L H L (a) (b)FIG. 4: (a) The mean quanta number of the quantum os-cillator h b † b i / ¯ n and (b) its second-order correlation function g (2) b (0) versus ω / (2Ω ) for the situation (II) with γ /γ ≪ Figure (3) depicts the cooling regimes in this system, un-der situation (I). This happens when γ /γ ≪ h R i < h R i leading to quanta’s absorption pro-cesses, see Fig. 6(b). The minimum in the mean quantanumber followed by an increased second-order correlationfunction g (2) b (0) occur around ¯ δ = 0, that is, at resonancecondition, see Figs. 3(a,b).Further, for the sake of comparison, we will keep thesame parameters and shall investigate the quantum dy-namics for the second situation, i.e. ( II ). The respectiveequations of motion describing the quantum oscillator’sdynamics as well as the quantum emitter’s one are givenin Appendix B, i.e., Eqs. (B1). Particularly, Fig. 4(a)shows the mean quanta’s number of the quantum oscil-lator in this case, whereas Fig. 4(b) depicts the corre-sponding behavior of the second-order quanta’s correla-tion function as a function of ω / (2Ω ) when γ /γ ≪ | i of the three-level emit-ter has a permanent dipole then it can couple with asingle cavity electromagnetic field mode of terahertz fre-quency, for instance. In this case, we have obtained a co-herent electromagnetic field source generating terahertzphotons. Regarding external applied field intensities I :For transition wavelengths of the order of 1 µm , sponta-neous decay rates within the range 10 − Hz, andthe corresponding THz interval for the Rabi frequenciesΩ ∼ − Hz, one obtains I within few to several kW/cm which correspond to moderate laser intensities.Respectively, Fig. 5(a) emphasizes the cooling regime inthe examined system, and for the second situation ( II ),occurring when γ /γ ≪
1. The second-order correlationfunction increases respectively, see Fig. 5(b), demonstrat-ing enhanced phonon-phonon or photon-photon correla-tions depending on the model we have in mind. Com- Ω (cid:144)H W L X b † b \ (cid:144) n Ω (cid:144)H W L g b H L H L (a) (b)FIG. 5: (a) The scaled mean quanta number of the quantumoscillator h b † b i / ¯ n and (b) its second-order correlation function g (2) b (0) versus ω / (2Ω ) for the situation (II) with γ /γ ≪ pared with Fig. (3) describing same things but for thefirst situation (I), the cooling is significantly enhancedin the second case ( II ) while keeping identical parame-ters, see Fig. (5) and Fig. (3). The steady-state meanvalue of dressed-state inversion operator h R z i , in thelasing regime, behave differently in this case, compareFig. 7(a) with Fig. 6(a). In the second situation (II), h R z i approaches zero values, while the mean quanta’snumber is large, although has a minimum, see Fig. 4(a).As we shall explain below, these behaviors are due toquantum interference effects. However, cooling occurs for h R i < h R i facilitating quanta’s absorption processes,see Fig. 7(b). Note that we have carefully checked theconvergence of our results with respect to various valuesfor n max .Although both situations ( I ) and ( II ) show coolingor lasing phenomena, the mechanisms behind them arecompletely different. If γ = γ and γ = 0, the firstsituation ( I ) resembles a two-level system {| Ψ i , | Ψ i} of frequency 2Ω interacting, respectively, with a quan-tum oscillator of frequency ω , with 2Ω ≈ ω , see also[3]. The spontaneous decay acts in both directions, i.e. | Ψ i ↔ | Ψ i , with a corresponding impact on coolingor lasing effects. The cross-correlation terms from theMaster Equation (A1) do not influence the quantum dy-namics in this case from the simply reason that they donot enter at all in the equations of motion (13). On theother side, the second situation ( II ) is close to an equidis-tant three-level system | Ψ i ↔ | Ψ i ↔ | Ψ i , where eachtransition being of frequency Ω interacts as well with thequantum oscillator possessing the frequency ω , however,with Ω ≈ ω . In this case transitions may take place viasingle oscillator’s quanta processes among the dressed-state | Ψ i ↔ | Ψ i ↔ | Ψ i or, respectively, involving two-quanta effects among the dressed-states | Ψ i ↔ | Ψ i .This also means that cross-correlation terms from theMaster Equation (A1) do influence the quantum dynam-ics in this case. This is clearly elucidated also if oneinspects the variables ρ ( i ) , { i ∈ · · · } , given in theAppendix B, since it contain single or two-quanta pro-cesses appearing concomitantly. The various decay pathsamong the dressed-states involved | Ψ i ↔ | Ψ i ↔ | Ψ i lead to quantum interference effects, see also Eq. (A1),although the dipole moments corresponding to the twobare transitions of the Λ − type sample are orthogonal toeach other. These cross-correlations [46–48] among thedressed-states contribute to a more flexible domain forlasing and deeper cooling regimes compared to the situ-ation ( I ) and for the same parameters involved. Thus,one can conclude that quantum interference effects viasingle- or two-quanta processes distinguish the situation( II ), described by the Hamiltonian (12), from the corre-sponding one characterized by the Hamiltonian (11), i.e.,the case ( I ). This is also the reason that the three-levelemitter’s population dynamics behave differently as wellin these two cases, compare Fig. (6) and Fig. (7). Noticethat when ω / → | Ψ i = (cid:0) | i − | i (cid:1) / √
2, whereas h b † b i / ¯ n = 1 and g (2) b (0) = 2, see Figs. (2-5), meaning that the quantumoscillator’s mode is in a thermal state and no cooling orlasing effects take place, respectively. Here, these phe-nomena occur for ω / = 0, when some populationresides on the higher upper state | i , which is distinctfrom other related schemes based, however, on coher-ent population trapping effects or electromagnetically in-duced transparency phenomenon [22–25]. Furthermore,we have observed that there are no cooling effects forboth cases described here, ( I ) or ( II ), if γ = γ while γ = 0. However, the phenomenon it will appear as youincreases γ while keeping γ = γ . Finally, the temper-atures ranges considered here are within several Kelvinsfor phonon cooling effects to few hundreds of Kelvins forcoherent THz photon generation, respectively. IV. SUMMARY
Summarizing, we have investigated a laser-pumpedthree-level Λ − type system the upper state of which isbeing coupled with a quantum oscillator characterizedby a single quantized leaking mode. We have identifiedtwo distinct situations leading to cooling or lasing ef-fects of the quantum oscillator’s degrees of freedom andhave described the mechanisms behind them. Particu-larly, we have demonstrated that the interplay betweensingle- or two-quanta processes accompanied by quantuminterference effects among the induced emitter’s dressed-states are responsible for flexible lasing or deeper cool-ing effects, respectively. This leads also to mutual in-fluences between the quantum oscillator’s dynamics andthe three-level emitter’s quantum dynamics, respectively.The coherent terahertz photons generation is identified asone of the possible application resulting from this study. Acknowledgments
We acknowledge the financial support via grant No.15.817.02.09F as well as the useful discussions with Vic-tor Ceban, Profirie Bardetski and Corneliu Gherman.
Appendix A: The master equation
Below, one can find the final Master Equation used toobtain the corresponding equations of motion describingthe quantum dynamics of both the quantum oscillator aswell as of the three-level Λ − type emitter, that is,˙ ρ + i ~ [ H, ρ ] = − γ [ R (+) , R (+) ρ ] − γ [ R ( − ) , R ( − ) ρ ] − sin θ γ (+) [ R , R ρ ] − sin θ γ ( − ) [ R , R ρ ] − γ (0)0 (cid:0) [ R , R ρ ] + [ R , R ρ ] (cid:1) − Γ (+) [ R , R ρ ] − Γ ( − ) [ R , R ρ ] − γ (+)0 (cid:0) [ R , R ρ ] + [ R , R ρ ] (cid:1) − γ ( − )0 (cid:0) [ R , R ρ ] + [ R , R ρ ] (cid:1) − γ θ × [ 12 ( R + R ) − R , (cid:0)
12 ( R + R ) − R (cid:1) ρ ] − γ θ (1 − sin θ ) [ R + R , ( R + R ) ρ ] − γ θ (1 + sin θ ) [ R + R , ( R + R ) ρ ] − κ (1 + ¯ n )[ b † , bρ ] − κ ¯ n [ b, b † ρ ] + H.c., (A1)where R ( ± ) = sin 2 θ √ R ∓ cos θ √ (1 ± sin θ ) R ± cos θ √ (1 ∓ sin θ ) R . The following terms: [ R , R ρ ], [ R , R ρ ][ R , R ρ ] and [ R , R ρ ] as well as their Hermitianconjugate parts characterize the cross-damping effects orquantum interference phenomena [46–48]. As an exercise,we present the equations of motion for the dressed-statepopulations of the three-level emitter in the absence ofthe quantum oscillator, that is g = 0, h ˙ R i = γ (+)11 h R i − γ (+)22 h R i + γ (+)33 h R i , h ˙ R i = γ ( − )11 h R i + γ ( − )33 h R i − γ ( − )22 h R i , h R i = 1 − h R i − h R i . (A2)Here, γ ( ± )11 = γ ( ± ) sin θ/ γ cos θ (1 ∓ sin θ ) / γ ( ± )22 =2 γ (0)0 + Γ ( ∓ ) / γ cos θ (1 ± sin θ ) / γ ( ± )33 = γ ( ± ) cos θ/ γ (1 ∓ sin θ ) /
8. One can observe that thecross-correlation terms from the Master Equation (A1)do not contribute to population quantum dynamics givenby Eqs. (A2). However, their influence will appear in thepresence of the quantum oscillator, i.e. when g = 0, andthis is clearly shown here, compare Fig. (6) and Fig. (7).The steady-state solutions of the above system of equa-tions are: h R i = (cid:0) γ (+)11 γ ( − )22 + γ ( − )11 γ (+)33 (cid:1) / (cid:0) γ (+)11 ( γ ( − )22 + γ ( − )33 )+ γ (+)22 ( γ ( − )11 + γ ( − )22 ) + γ (+)33 ( γ ( − )11 − γ ( − )33 ) (cid:1) , (A3)whereas the solution for h R i can be obtained fromExp. (A3) via an exchange of upper signs, i.e. ( ± ) → ( ∓ ). Ω (cid:144)H W L X R z \ - - - - Ω (cid:144)H W L (a) (b)FIG. 6: The mean dressed-state inversion operator h R z i = h R i − h R i as a function of ω / (2Ω ) obtained in thesteady-state for the first situation (I). (a) γ /γ ≪ γ /γ ≪
1. The solid lines are obtained with the full sys-tem of equations (13), while the dashed lines in the absenceof the quantum oscillator, i.e. with Exp. (A3). All otherparameters are as in Fig. (2) and Fig. (3), respectively. Ω (cid:144)H W L X R z \ - - - - Ω (cid:144)H W L (a) (b)FIG. 7: The same as in Fig. (6) but for the second case (II).The solid lines are obtained with the full system of equationsof motion (B1), while the dashed lines with Exp. (A3). Allother parameters are as in Fig. (4) and Fig. (5), respectively. Fig. (6) and Fig. (7) depict the steady-state values ofthe dressed-state inversion operator h R z i for the bothcases studied here, ( I ) and ( II ), and in the presence ofthe quantum oscillator (solid lines) as well as in its ab-sence (dashed curves), respectively. One can observe thatthere is a clear difference between the cases with g = 0and g = 0 in the lasing regimes, compare Fig. 6(a) andFig. 7(a). As it was described above, this distinction isdue to cross-correlation terms or quantum interferenceeffects arising in the second case (II). Correspondingly,in the cooling regimes the quantum oscillator’s influenceon the steady-state mean value of the qubit inversion op-erator is not quite significant, although still visible. Appendix B: The equations of motion when ω ≈ Ω ,i.e., for the case (II) Here, we shall present the equations of motion for thesecond situation ( II ) obtained with the help of the Mas-ter Equation (A1), that is,˙ P (0) n = i ˜ g ( P (3) n − P (5) n − P (9) n + P (7) n ) − κ ¯ n (cid:0) ( n + 1) P (0) n − nP (0) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (0) n − ( n + 1) P (0) n +1 (cid:1) , ˙ P (1) n = i ˜ g ( P (7) n − P (9) n ) − κ ¯ n (cid:0) ( n + 1) P (1) n − nP (1) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (1) n − ( n + 1) P (1) n +1 (cid:1) + ˜ γ (1)0 P (0) n − ˜ γ (1)1 P (1) n − ˜ γ (1)2 P (2) n , ˙ P (2) n = − i ˜ g ( P (9) n + P (7) n ) − κ ¯ n (cid:0) ( n + 1) P (2) n − nP (2) n − (cid:1) − κ (1 + ¯ n ) (cid:0) nP (2) n − ( n + 1) P (2) n +1 (cid:1) + ˜ γ (2)0 P (0) n + ˜ γ (2)1 P (1) n − ˜ γ (2)2 P (2) n , ˙ P (3) n = i ˜ δP (4) n − ˜ γ (3)3 P (3) n + ˜ γ (3)7 P (7) n + i ˜ g ( n (2 P (0) n − P (1) n − − P (2) n − ) − (2 n + 1) P (1) n ) − κ (1 + ¯ n ) (cid:0) (2 n − P (3) n − n + 1) P (3) n +1 + 2 P (9) n (cid:1) − κ ¯ n (cid:0) (2 n + 1) P (3) n − nP (3) n − (cid:1) , ˙ P (4) n = i ˜ δP (3) n − i ˜ gP (12) n − κ (1 + ¯ n ) (cid:0) (2 n − P (4) n + 2 P (10) n − n + 1) P (4) n +1 (cid:1) − κ ¯ n (cid:0) (2 n + 1) P (4) n − nP (4) n − (cid:1) − ˜ γ (4)4 P (4) n + ˜ γ (4)8 P (8) n , ˙ P (5) n = i ˜ δP (6) n + i ˜ g (cid:0) P (11) n + ( n + 1)( P (1) n +1 − P (2) n +1 ) − n + 1)( P (0) n − P (1) n ) (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (5) n − n + 1) P (5) n +1 (cid:1) − κ ¯ n (cid:0) (2 n + 3) P (5) n − nP (5) n − − P (7) n (cid:1) − ˜ γ (5)5 P (5) n + ˜ γ (5)9 P (9) n , ˙ P (6) n = i ˜ δP (5) n + i ˜ gP (12) n − κ ¯ n (cid:0) (2 n + 3) P (6) n − nP (6) n − − P (8) n (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (6) n − n + 1) × P (6) n +1 (cid:1) − ˜ γ (6)6 P (6) n + ˜ γ (6)10 P (10) n , ˙ P (7) n = i ˜ δP (8) n + i ˜ g (cid:0) P (13) n + n ( P (1) n − P (2) n ) − n ( P (0) n − − P (1) n − ) (cid:1) − κ ¯ n (cid:0) (2 n + 1) P (7) n − nP (7) n − (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n − P (7) n − n + 1) P (7) n +1 + 2 P (5) n (cid:1) + ˜ γ (7)3 P (3) n − ˜ γ (7)7 P (7) n , ˙ P (8) n = i ˜ δP (7) n + i ˜ gP (14) n − κ ¯ n (cid:0) (2 n + 1) P (8) n − nP (8) n − (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n − P (8) n − n + 1) P (8) n +1 + 2 P (6) n (cid:1) + ˜ γ (8)4 P (4) n − ˜ γ (8)8 P (8) n , ˙ P (9) n = i ˜ δP (10) n + i ˜ g (cid:0) n + 1)( P (0) n +1 − P (1) n +1 ) − ( n + 1) × ( P (1) n + P (2) n ) − P (15) n (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (9) n − n + 1) P (9) n +1 (cid:1) − κ ¯ n (cid:0) (2 n + 3) P (9) n − nP (9) n − − P (3) n (cid:1) + ˜ γ (9)5 P (5) n − ˜ γ (9)9 P (9) n , ˙ P (10) n = i ˜ δP (9) n − i ˜ gP (16) n − κ ¯ n (cid:0) (2 n + 3) P (10) n − nP (10) n − − P (4) n (cid:1) − κ (1 + ¯ n ) (cid:0) (2 n + 1) P (10) n − n + 1) × P (10) n +1 (cid:1) + ˜ γ (10)6 P (6) n − ˜ γ (10)10 P (10) n , ˙ P (11) n = 2 i ˜ δP (12) n + i ˜ g (cid:0) nP (5) n − ( n + 1) P (3) n (cid:1) − κ (1 + ¯ n ) × (cid:0) nP (11) n − ( n + 1) P (11) n +1 + P (15) n (cid:1) − κ ¯ n (cid:0) ( n + 1) × P (11) n − nP (11) n − − P (13) n (cid:1) − ˜ γ (11)11 P (11) n , ˙ P (12) n = 2 i ˜ δP (11) n + i ˜ g (cid:0) nP (6) n − ( n + 1) P (4) n (cid:1) − κ (1 + ¯ n ) × (cid:0) nP (12) n − ( n + 1) P (12) n +1 + P (16) n (cid:1) − κ ¯ n (cid:0) ( n + 1) × P (12) n − nP (12) n − − P (14) n (cid:1) − ˜ γ (12)12 P (12) n , ˙ P (13) n = 2 i ˜ δP (14) n + i ˜ g (cid:0) ( n − P (7) n − nP (3) n − (cid:1) − κ (1 + ¯ n ) × (cid:0) ( n − P (13) n − ( n + 1) P (13) n +1 + 2 P (11) n (cid:1) − κn ¯ n × (cid:0) P (13) n − P (13) n − (cid:1) − ˜ γ (13)13 P (13) n , ˙ P (14) n = 2 i ˜ δP (13) n + i ˜ g (cid:0) ( n − P (8) n − nP (4) n − (cid:1) − κ (1 + ¯ n ) × (cid:0) ( n − P (14) n − ( n + 1) P (14) n +1 + 2 P (12) n (cid:1) − κn ¯ n × (cid:0) P (14) n − P (14) n − (cid:1) − ˜ γ (14)14 P (14) n , ˙ P (15) n = 2 i ˜ δP (16) n + i ˜ g (cid:0) ( n + 1) P (5) n − ( n + 2) P (9) n (cid:1) − κ (1 + ¯ n )(1 + n ) (cid:0) P (15) n − P (15) n +1 (cid:1) − κ ¯ n × (cid:0) ( n + 2) P (15) n − nP (15) n − − P (11) n (cid:1) − ˜ γ (15)15 P (15) n , ˙ P (16) n = 2 i ˜ δP (15) n + i ˜ g (cid:0) ( n + 1) P (6) n +1 − ( n + 2) P (10) n (cid:1) − κ (1 + ¯ n )(1 + n ) (cid:0) P (16) n − P (16) n +1 (cid:1) − κ ¯ n × (cid:0) ( n + 2) P (16) n − nP (16) n − − P (12) n (cid:1) − ˜ γ (16)16 P (16) n . (B1)Here ˜ γ (1)0 = γ (1)0 , ˜ γ (1)1 = γ (1)1 , ˜ γ (1)2 = γ sin θ cos θ/ γ (2)0 = γ (2)0 , ˜ γ (2)1 = − γ (2)1 , ˜ γ (2)2 = γ (2)2 , ˜ γ (3)3 = γ cos θ (1 + 3 sin θ ) / γ cos θ (1 − θ ) / γ (+) + γ ( − ) ) sin θ/ γ (0)0 + Γ ( − ) + 9 γ cos θ/
16 + γ cos θ (cid:0) (1 + sin θ ) + (1 − sin θ ) / (cid:1) /
4, ˜ γ (3)7 = γ (+)0 + γ cos θ (1 − sin θ ) /
4, ˜ γ (4)4 = ˜ γ (3)3 , ˜ γ (4)8 = ˜ γ (3)7 , ˜ γ (5)5 = γ cos θ (1 − θ ) / γ cos θ (1 + 3 sin θ ) / γ (+) + γ ( − ) ) sin θ/ γ (0)0 + Γ (+) + 9 γ cos θ/
16 + γ cos θ (cid:0) (1 − sin θ ) + (1 + sin θ ) / (cid:1) /
4, ˜ γ (5)9 = γ ( − )0 + γ cos θ (1+sin θ ) /
4, ˜ γ (6)6 = ˜ γ (5)5 , ˜ γ (6)10 = ˜ γ (5)9 , ˜ γ (7)7 = ˜ γ (6)6 ,˜ γ (7)3 = ˜ γ (6)10 , ˜ γ (8)8 = ˜ γ (7)7 , ˜ γ (8)4 = ˜ γ (7)3 , ˜ γ (9)5 = γ (+)0 + γ cos θ (1 − sin θ ) /
4, ˜ γ (9)9 = ˜ γ (3)3 = ˜ γ (10)10 , ˜ γ (10)6 = ˜ γ (9)5 ,˜ γ (11)11 = ( γ + γ ) cos θ/ γ (0)0 +Γ ( − ) +Γ (+) + γ cos θ (1+sin θ ) /
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