Multiphoton blockades in pulsed regimes beyond the stationary limits
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Multiphoton blockades in pulsed regimes beyond the stationary limits
G. H. Hovsepyan, ∗ A. R. Shahinyan, † and G. Yu. Kryuchkyan
1, 2, ‡ Institute for Physical Researches, National Academy of Sciences,Ashtarak-2, 0203, Ashtarak, Armenia Yerevan State University, Alex Manoogian 1, 0025, Yerevan, Armenia
We demonstrate multiphoton blockades (PB) in the pulsed regime by using Kerr nonlinear dissi-pative resonator driven by a sequence of Gaussian pulses. It is shown that the results obtained forsingle-photon, two-photon and three-photon blockades in the pulsed excitation regime differ consid-erably from analogous results obtained for the case of continuous-wave (cw) driving. We stronglydemonstrate that for the case of cw pumping of the Kerr-nonlinear resonator there are fundamen-tal limits on populations of lower photonic number-states (with n = 0 , , , PACS numbers:
I. INTRODUCTION
The realization of quantum devices with photon-photon interaction has become an interesting and impor-tant research topic in quantum optics and quantum tech-nologies. The key element to obtain the mentioned in-teractions is strong optical nonlinearity that gives rise tomany important quantum effects at level of few-photons,including photon blockade[1–3].In the photon blockade (PB) the optical response toa single photon is modulated by the presence or absenceof the other photons. Particularly, the capture of a sin-gle photon into the system affects the probability that asecond photon is admitted. Thus, the photon blockade isthe analog of the Coulomb blockade for electrons in con-densed matter devices, where single electron transport isblocked by a strong Coulomb interaction in a confinedstructure. A simple consequence of photon blockade isthe antibunching of photons in emission in analogy tothe photon antibunching of resonance fluorescence on atwo-level atom [4, 5].One of the basic conditions for PB is that the photon-photon interaction strength should be larger regardingdecay rate of the system. In this respect, strong non-linearities on the few-photon level can be produced byinteraction between photons and an atom in a cavity [6–9], in systems with interacting photons or polaritons inarrays of cavities coupled to atoms or qubits [10–13], inoptomechanical systems and Kerr type nonlinear cavities[14–16].Photon blockade was first observed in an optical cav-ity coupled to a single trapped atom [3]. The PB ∗ [email protected] † anna [email protected] ‡ [email protected] has been predicted in cavity quantum electrodynamics(QED) [17], and recently in circuit QED with a singlesuperconducting artificial atom coupled to a microwavetransmission line resonator [18, 19]. PB was also exper-imentally demonstrated with a photonic crystal cavitywith a strongly coupled quantum dot [20], and was alsopredicted in quantum optomechanical systems [21, 22].An analogous phenomenon of phonon blockade was pre-dicted for an artificial superconducting atom coupled toa nanomechanical resonator [23], as well as the polaritonblockade effect due to polariton-polariton interactionshas been considered in [24]. Recently, PB was consideredin dispersive qubit-field interactions in a superconductivecoplanar waveguide cavity [25] and with time-modulatedinput [26].In most of the cited works PB was considered in non-linear optical systems driven by a continuous-wave laser.However, photon sources on demand require productionof photon pulses at fixed time-intervals. In this paper,we investigate PB in the pulsed regime, considering Kerr-type oscillatory dissipative system driven by a sequenceof classical Gaussian pulses separated by time intervals.This approach allows to expand the mechanism of PB forthe case of producing few-photon pulses at deterministictime intervals. Thus, we present comparative analysis ofone-photon, two-photon and three-photon blockades inthe pulsed regimes in addition to the results obtained forthe case of PB in cw driving field [16].In this way, we demonstrate, that contrarily to com-mon expectations, the results obtained in the pulsedregime considerably differ from those derived for themonochromatic driving [16]. More importantly, we seethat PB in Kerr-type systems under pulsed excitationcan be controlled by shape of pulses. In this spirit, weemphasize the idea of improving the degree of quantumeffects in open systems, as well as obtaining qualitativelynew quantum effects by applying of the sequence of tai-lored pulses. This approach was recently exploited forformation of high degree continuous-variable entangle-ment in the nondegenerate optical parametric oscillator[27, 28], and for investigation of quantum interferencein mesoscopic domain [29–31]. Recently, generation ofnonclassical states in cavity QED with pulsed input hasbeen investigated in [26] and production of a superposi-tion states in the periodically pulsed nonlinear resonatorhas been demonstrated in [30], [32]. Dynamics of period-ically driven nonlinear oscillator has been also studied inthe papers [33]- [37].We clarify the multiphoton photon blockade in Kerrnonlinear resonator under pulsed excitation mainly byconsidering photon-number effects and by analysingphase-space properties of resonator mode. However, wehave not discussed properties of the field transmissionwhen the photons are blockaded. Thus, we focus on anal-ysis of the mean photon number, the probability distribu-tions of photons, the second-order correlation functionsof photons and the Wigner functions in phase space. Inthis pulsed regime the ensemble-averaged mean photonnumbers, the populations of photon-number states andthe Wigner functions are nonstationary and exhibit a pe-riodic time dependent behavior, i.e. repeat the periodic-ity of the pump laser in an over transient regime. Besidesthis the results on production of photon Fock states atPB region depend on the parameters of Gaussian pulsessuch as the amplitude, the duration of pulses and thetime-interval between them, dissipation rates and Kerr-interaction coupling. Thus, we investigate production ofphotonic states for an arbitrary interaction time-intervalsincluding also time-intervals exceeding the characteristictime of dissipative processes, t ≫ γ − . However, the re-sults obtained are nonstationary that is conditioned byinteraction of the system with the sequence of Gaussiane.m. pulses.For comparison, the paper includes the investigation ofthe Kerr nonlinear dissipative resonator under cw driv-ing on the framework of the potential solution of theFokker-Planck equation for the quasiprobability distribu-tion function in complex P-representation. In this way,we obtain that contrarily to the pulsed regime for thecase of cw pumping there are fundamental limits on thepopulations of lower photonic-number-states.The paper is arranged as follows. In Sec. II we describeperiodically pulsed Kerr nonlinear resonator and discussthe case of cw driving in the exact quantum treatment ofPB. In Sec. III we consider one-photon, two-photon andthree-photon blockades in the pulsed regime on base ofthe populations of photon-number states and the second-order correlation functions. In Sec. IV we analyse theproduction of multi-photon states on base of the Wignerfunctions. We summarize our results in Sec. V. II. KERR NONLINEAR RESONATOR: PULSEDREGIME AND CW LIMIT
The Hamiltonian of Kerr nonlinear resonator underpulsed excitation in the rotating wave approximationreads as: H = ~ ∆ a + a + ~ χ ( a + ) a + ~ f ( t )(Ω a + + Ω ∗ a ) . (1)Here, time dependent coupling constant Ω f ( t ) that isproportional to the amplitude of the driving field consistsof the Gaussian pulses with the duration T which areseparated by time intervals τf ( t ) = X e − ( t − t − nτ ) /T , (2)while a + , a are the creation and annihilation operators, χ is the nonlinearity strength, and ∆ = ω − ω is thedetuning between the mean frequency of the driving fieldand the frequency of the oscillator, (see Fig. 1).This model seems experimentally feasible and can berealized in several physical systems. Particularly, the ef-fective Hamiltonian (1) describes a qubit off-resonantlycoupled to a driven cavity. In fact, it is well known thatthe Hamiltonian of two-level atom interacting with cav-ity mode in the dispersive approximation, if the two-levelsystem remains in its ground state, can be reduced to theeffective Hamiltonian (1). This model also describes ananomechanical oscillator with a † and a raising and low-ering operators related to the position and momentumoperators of a mode quantum motion. An important im-plementation of Kerr-type resonator has been recentlyachieved in the context of superconducting devices basedon the nonlinearity of the Josephson junction.We have included dissipation and decoherence in Kerrnonlinear resonator on the basis of the master equation: dρdt = − i ~ [ H, ρ ] + X i =1 , (cid:18) L i ρL † i − L † i L i ρ − ρL † i L i (cid:19) , (3)where L = p ( N + 1) γa and L = √ N γa + are theLindblad operators, γ is a dissipation rate, and N de-notes the mean number of quanta of a heat bath. Tostudy the pure quantum effects, we focus on the cases ofvery low reservoir temperatures, which, however, oughtto be still larger than the characteristic temperature T ≫ T cr = ~ γ/k B . This restriction implies that dis-sipative effects can be described self-consistently in theframe of the Linblad Eq. (3). In our numerical calcu-lation we choose the mean number of reservoir photons N = 0 . | n i which are spaced in energy E n = E + ~ ω n + ~ χn ( n −
1) with n = 0 , , ... . The lev-els form an anharmonic ladder with anharmonicity thatis given by E − E = 2 ~ χ . FIG. 1: Schematic representation of Kerr nonlinear resonatorunder a sequence of Gaussian pulses. In this scheme a emitterinvolving third-order susceptibility coupled to a resonator.FIG. 2: Schematic energy-level diagram with one-photon,two-photon and k -photon resonant transitions between statesof Kerr nonlinear resonator. Selective excitations leadingto multiphoton blockades are realized for the main frequen-cies of driving field: k ~ ω k = E k that are calculated as ω k = ω + χ ( k − Below, we concentrate on quantum regimes for theparameters leading to resolved oscillatory energy lev-els. In this case we can consider near the resonant se-lective transitions between lower photon-number states | m i → | n i . In this way, k -photon blockades for the pos-itive k = 0 , , , , ... are realized in both resonant andnear the resonant transitions between oscillatory initialand final states | i → | k i . According to the formula E n = E + ~ ω n + ~ χn ( n −
1) the resonant frequencies ofthese transitions equal to k ~ ω k = E k and can be derivedin the following form ω k = ω + χ ( k −
1) (see, Fig. 2).Thus, for one-photon transition k = 1 and E = ~ ω ,the resonance frequency is ω = ω , for two-photon tran-sition k = 2 and E = 2 ~ ω +2 χ , the other resonance fre-quency is ω = ω + χ , while for k = 3 , E = 3 ~ ω + 6 χ ,the resonance frequency is ω = ω + 2 χ .Considering the pulsed regimes of Kerr nonlinear reser-voir we assume that the spectral pulse-width, i.e. thespectral widths of pulses, should be smaller than the non-linear shifts of the oscillatory energy levels. It means thatthe duration of pulses should be larger than 1 /χ . Thus,for strong nonlinearities χ/γ >
1, we arrive to the fol-lowing inequalities for the duration of Gaussian pulses1 /γ > T > /χ .We solve the master equation Eq. (3) numerically based on quantum state diffusion method. The appli-cations of this method for studies of NDO can be foundin [29]-[32]. In the calculations, a finite basis of numberstates | n i is kept large enough (where n max is typically50) so that the highest energy states are never populatedappreciably. In the following the distribution of oscil-latory excitation states P ( n ) = h n | ρ | n i as well as theWigner functions W ( r, θ ) = X n,m ρ nm ( t ) W mn ( r, θ ) (4)in terms of the matrix elements ρ nm = h n | ρ | m i of thedensity operator in the Fock state representation willbe calculated. Here ( r, θ ) are the polar coordinates inthe complex phase-space plane, x = r cos θ , y = r sin θ ,whereas the coefficients W mn ( r, θ ) are the Fourier trans-form of matrix elements of the Wigner characteristicfunction.In this way, we calculate below nonstationary represen-tation of a single emitter coupled to an optical resonatorpopulations P , P , P , P of photon-number states | n i , n = 0 , , ,
3, as well as the mean photon number, theWigner functions and the second-order correlation func-tions.
A. Limits to the state populations for the case ofcw driving
In this subsection we shortly discuss selective excita-tion of a Kerr nonlinear resonator under cw driving. Inthis case the Hamiltonian (1) reads as: H = ~ ∆ a † a + ~ χ ( a † ) a + ~ (Ω a † + Ω ∗ a ) , (5)i.e. describes a standard anharmonic oscillator drivenby a monochromatic field in the rotating wave approx-imation. In this simplest case, an analytical resultsfor a dissipative driven nonlinear oscillator in steadystate have been obtained in terms of the solution of theFokker-Planck equation for the quasiprobability distribu-tion function P ( α, α ∗ ) in complex P-representation [38].This approach based on the method of potential equa-tions leads to the analytic solution for the quasiprobabil-ity distribution function P ( α, α ∗ ) within the frameworkof an exact nonlinear treatment of quantum fluctuations.In this way, the photon number probability distributionfunction p ( n ) = h n | ρ | n i can be expressed in terms ofcomplex P -representation as follows: p ( n ) = 1 n Z Z C dαdα † ( α ) n ( α † ) n exp ( − αα † ) P ( α, α † ) , (6)where C is an appropriate integration contour for each ofthe variables α and α ∗ , in the individual complex planes.After integration the mean photon number and the prob-ability distribution of photon-number states are repre- FIG. 3: Populations of photon number depending on the de-tuning. The parameters are: Ω /γ = 20, χ/γ = 15. χ/γχ/γ (1)(0) (2) ( ) (b)(a) P P
FIG. 4: Populations of photon numbers depending on theparameter of nonlinearity: the curves of P , P populations(a), the curves of P , P populations (b). The parametersare: Ω /γ = 20, ∆ /γ = 0. sented in the following form [39–41]: h a † a i = Ω (∆ + χ ) + ( γ/ F ( λ + 1 , λ ∗ + 1 , z ) F ( λ, λ ∗ , | ε | | ) , (7)where ε = Ω /χ , λ = ( γ + i ∆) /iχ , and F = F is thehypergeometric function: F ( a, b, z ) = ∞ X k =0 z k Γ( a )Γ( b ) k !Γ( k + a )Γ( k + b ) , (8) P ( n ) = | ε | n Γ( λ )Γ( λ ∗ ) n ! F ( λ, λ ∗ , | ε | | ) ∞ X k =0 | ε | k k !Γ( k + n + λ )Γ( k + n + λ ∗ ) . (9)In Fig. 3 we present the typical results for the pop-ulations P , P , P , P of the photon-number states | n i versus the detuning that are calculated on the Eq. (9).These results show the selective excitations of the Fockstates | i , | i , | i . As we see, the population P depictedin Fig. 3(b) displays the maximum 0.5 at ∆ = 0, thatcorresponds to the resonance transition | i → | i at thefrequency ω = ω . The other maximum P = 0 .
46 at∆ /γ = −
15 corresponds to the population of | i state through the Raman process with the energy conversa-tion E + 2 ω = ω k + E . This process involves theexcitation of Fock state | i , (in the transition | i → | i at the frequency of pump field ω = ω + χ ), and thedecay | i → | i at frequency ω k = ω + 2 χ . Note, thatthe population of vacuum state is zero in this spectralrange (see, Fig. 3(a)). The population P depicted inFig. 3(c) displays the maximum 0.24 at ∆ = −
15, thatcorresponds to the resonance transition | i → | i at thefrequency ω = ω + χ . As we see, the population P isvery small for all values of the detuning. Nevertheless,the population display maximum at ∆ /γ = −
30. Thispeak corresponds to three quanta excitation of | i stateat the frequency ω = ω + 2 χ .In general, from the analytic results and numeri-cal analysis we can conclude that the populations arestrongly limited. We demonstrate this fact in Fig. 4 byshowing the dependence of populations from the param-eter of nonlinearity. We observe in Fig. 4(a) that bothpopulations of vacuum and single-photon states mono-tonically increase with increasing of the nonlinearity pa-rameter. However, the population P is limited by thevalue 0.5. This effect of selective excitation can be in-terpreted as the single-photon Fock state blockading thegeneration of two or more photons. However, limit to P population is restricted possibility of observation PB incw regime. Note, that this result is in accordance withthe numerical results observed on framework of the mas-ter equation [16]. The behavior of the populations oftwo-photon and three-photon number states depicted inFig. 4(b) differ from P and P . We observe that thesepopulations display peaked structures and the maximumare realized for the definite parameters of nonlinearity.One of the questions that we will try to answer below iswhether or not the populations P n of | n i states for Kerrnonlinear dissipative resonator driven by a sequence ofGaussian pulses can exceed the cw limits. III. PB IN THE PULSED REGIMES
Now we present the results of this paper concerning se-lective excitations of photon-number states and hence theobservation of an effective PB due to pulsed excitation.In this sense, we note the main peculiarity of our paper.We investigate photon-number aspects of PB in nonsta-tionary regimes, for an arbitrary interaction time inter-vals, that also can exceed the characteristic time of dissi-pative processes, t ≫ γ − . These nonstationary regimesare conditioned by the specific form of excitation.If the Kerr nonlinear resonator is driven by a se-quence of pulses, its behavior is modified and essentiallydepends from the duration of pulses as well as time-intervals between them. In this way, we consider threeimportant regimes leading to one-photon, two-photonand three-photon blockades. In these regimes the pro-duction of single-photon, two-photon- and three-photonstates blockade the generation of the other photons. Each FIG. 5: The regime of single-photon blockade. The meanphoton number versus time-interval that repeats the Gaussianpulses and also indicates decay effects, for k = 1 (a); the time-evolution of the second-order photon correlation function ver-sus the dimensionless time, for k = 1 (b),(c); the second-orderphoton correlation function, for time-intervals at the peak val-ues of the mean photon numbers, versus the tuning parameter(d). The parameters are: χ/γ = 15, the maximum amplitudeof the driving field Ω /γ = 6, the mean number of reservoirphotons N = 0 . τ = 5 . γ − , T = 0 . γ − . of these regimes are realized for the appropriate choos-ing parameters, the detuning ∆, the nonlinearity param-eter χ , the pump amplitude Ω and the parameters ofpulses. Note, that in the case of pulsed excitation theensemble-averaged mean oscillatory excitation numbersand the populations of oscillatory states are nonstation-ary and exhibit a periodic time dependent behavior, i.e.repeat the periodicity of the pump laser in an over tran-sient regime. Below we demonstrate that the concrete PBregimes can be effectively prepared if the pulsed excita-tion is tuned to the corresponding resonance transitions.For convenience, we shall refer to k as a tuning contin-uous parameter instead of the detuning ∆. Thus, bychoosing the concrete value of tuning parameter k in theexpression ω k = ω + χ ( k −
1) we determine the detuning.The other conditions to improve the effectiveness ofPB concern to choosing the controlling parameters of thepulses. We assume the regimes involving short pulsesseparated by long time intervals that allow to increasethe weights of photon-state populations.
A. One-photon blockade
The analysis of the master equation Eq. (3) showssingle-photon blockade in Fig. 5 and Fig. 6. In thisregime, the probability distribution of excitation num-bers P ( n ) displays maximum, approximately equals tounity for single-photon state, n = 1, for the tuning pa- P P P P P ho t on nu m b e r s t a ti s ti c s Tuning parametr
FIG. 6: The regime of single-photon blockade. The popula-tions P ( n ) of | n i Fock states , for n = 0 , , ,
3, at the maximalpeak values of the mean photon numbers, versus the tuningparameter. The parameters are: χ/γ = 15, the maximumamplitude of the driving field Ω /γ = 6, the mean number ofreservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . rameter k = 1 or the detuning ∆ = 0. We also ana-lyze the populations of photonic states for the definitetime-intervals corresponding to the maximal peak val-ues of the mean photon numbers according to Fig. 5(a).The values of populations at these peaks versus the tun-ing parameter are depicted in Fig. 6. As we see, forthe parameters: χ/γ = 15, Ω /γ = 6, τ = 5 . γ − , and T = 0 . γ − the maximum of | i Fock state populationreaches to P = 0 .
82 approximately for k = 1, i.e. forthe near to resonance frequency ω = ω . In this casethe populations of the other states are very small, par-ticularly, P of vacuum state is approximately equals to0.14 and P of | i Fock state is 0.04.It is interesting to compare these results with the solu-tions obtained for the Kerr nonlinear resonator with cwdriving field. It has been shown numerically [16] that inthe case of cw-excitation the maximum rate of the pop-ulation P reach only to 0.5 for k = 1 and approximatelyequals to the population of vacuum state P . The limit0.5 for P has been also obtained by analytical calcu-lations in the previous section. Thus, the population P = 0 .
82 calculated in the pulsed regime essentiallyexceeds the analogous result in cw regime. This resultindicates the efficiency of the single-photon blockade inthe pulsed regime. Indeed, we can conclude that suchhigh population of single-photon state effectively block-ades the entering of the other photons in the resonator.To observe photon blockade we also turn to calculationof the normalized second-order correlation function forzero delay time g (2) that is defined as: g (2) ( t ) = h a † ( t ) a † ( t ) a ( t ) a ( t ) i ( h a † ( t ) a ( t ) i ) . (10) Correlation function Mean exitation number (d) t Correlation function Mean exitation number (e) C o rr e l a ti on f un c ti on t (c) P t (b) M ea n e x c it a ti on nu m b e r t (a) M ea n e x c it a ti on nu m b e r t (f) C o rr e l a ti on f un c ti on Tuning parameter
FIG. 7: The regimes of single-photon and two-photon block-ades. The mean photon numbers versus dimensionless time-interval for k = 2 (a) and k = 1 (b) that repeat the Gaussianpulses and involve also decay effects; time-evolution of thepopulation P of | i Fock state, for k = 2, versus dimen-sionless time (c); the second-order photon correlation func-tion versus the dimensionless time, for k = 2 (d), (e); thesecond-order photon correlation function for time-intervals atthe peak values of the mean photon numbers versus the tun-ing parameter (f). The parameters are: χ/γ = 30, the max-imum amplitude of pump field Ω /γ = 12, the mean numberof reservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . The results are depicted in Fig. 5(b, c, d). The non-stationary correlation function versus dimensionless timeintervals is shown in Fig. 5(b) with the curve of meanphoton number, while the result describing the photoncorrelation for short time intervals is presented in Fig.5(c). As we see, for short time intervals t = (0 − γ − only thermal cavity photons are involved and the timedependence of g (2) ( t ) in this area describes photon cor-relation of thermal bath at the averaged level g (2) = 2.During the pulses, if P reach to the maximum, the prob-ability of generation of a second photon is suppressed. Inthis case oscillatory mode acquired sub-poissonian statis-tics with the second-order correlation function g (2) < P P P P P ho t on nu m b e r s t a ti s ti c s Tuning parameter
FIG. 8: The regimes of single-photon and two-photon block-ades. The populations P ( n ) for n = 0 , , ,
3, for time-intervals at the peak values of the mean photon numbers ver-sus the tuning parameter. The parameters are: χ/γ = 30, themaximum amplitude of pump field Ω /γ = 12, the mean num-ber of reservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . shows the transition from photon bunching to antibunch-ing. The second-order correlation function versus thetuning parameter is plotted in Fig. 5(d). As calculationsshow, g (2) = 0 . k = 1 in accordance with the re-sult that for | i Fock state the normalized second-ordercorrelation function is zero. The correlation function dis-plays the peak at k = 1 .
6. This result corresponds to thephoton bunching with the level of mean photon number h n i = 0 .
38. Then the second-order correlation functionincreases with increasing of the tuning parameter k , how-ever, for negligible levels of the mean photon number. B. Two-photon and three-photon blockades
In this subsection we analyse the multiphoton blockadeon the base of photon-number effects. We will demon-strate how to chose the frequency of the driving field,the parameters of Gaussian pulses and the parameter ofnonlinearity in order to realize effective populations oftwo-photon and three-photon states that indicate mul-tiphoton blockades. It is obvious that for multiphotonblockade the regimes of photon state excitations withhigh levels of photon number in compared with the caseof single-photon state excitation should be considered.Such regimes imply a larger peak strength of drivingfield or/and a comparative small nonlinear parameter.We present below the results for two possible operationalregimes.The typical results indicating the two-photon blockadeare depicted in Fig. 7 and Fig. 8. The time evolutionof the mean photon number versus dimensionless time isdepicted in Fig. 7(a) for the tuning parameter k = 2,(for the detuning ∆ /γ = − χ/γ ), corresponding to two-photon resonant excitation; the analogous result for one-photon excitation, k = 1, is shown in Fig. 7(b). Thisbehavior repeats the periodicity of the driving pulses atthe over transient regime. The analogous time-evolutionis observed for the population P at k = 2 (see, Fig. 7(c)).This regime is favorable for the selective excitation oftwo-photon number state, if the detuning ∆ /γ = − χ/γ .In this case the population P = 0 .
61 takes place dueto two-photon excitation at the frequency ω = ω + χ , for the tuning parameter k = 2 (see, Fig. 8), whilethe population of one-photon state has a comparativesmall value P = 0 .
28 . The peculiarity of this regimeis that both one-photon blockade (for k = 1) and two-photon blockade (for k = 2) are effectively realized forthe same parameters: χ/γ = 15, Ω /γ = 6, τ = 5 . γ − , T = 0 . γ − . It should be mentioned that population P comes over cw limit for the population of | i Fock stateand (that it is remarkable) for the population of | i statein cw regime.In Fig. 7(e) the normalized second-order correlationfunction versus dimensionless time is shown. The time-evolution involves photon correlation of thermal bath fortime-interval that corresponds to resonator that involvesonly photon of thermal bath. Then, the peaks of g (2) takeplace at time-intervals at the front of pulses. Here quan-tum statistics is formed, but the mean photon numberis negligible, therefore the peak in the correlation func-tion is observed. At definite time intervals correspondingto peaked values of the mean photon number the cor-relation function equals to g (2) = 0 .
6. These values ofthe correlation function are investigated in depending onthe detuning (the parameter of tuning) and are shown inFig. 7(f). As we see, for k = 1 the effective population ofsingle-photon number state is observed, P = 0 .
79 (seeFig. 8), and photon antibunching is realized g (2) = 0 . k = 2, for which the maximal P = 0 .
61 population isrealized, the correlation function equals to g (2) = 0 . h n i = 1 .
52. Note, for the com-parison that for | i pure photon-number state the nor-malized second-order photon correlation function equalsto 0.5. In this regime the photon bunching, g (2) = 2 . k = 1 .
6. In the vicinity of k = 3 the large levelof g (2) is explained by the small level of the mean photonnumber (see, Fig. 9(a)).In the end of this subsection we shortly discuss three-photon blockade. For observation of the selective exci-tation of three-photon states we need the operationalregimes with more large levels of photon number thanhas been used for the previous cases. The typical resultsfor this regime are plotted in Fig. 9 and Fig. 10. Time-evolution of the mean photon number versus dimension-less time is plotted in Fig. 9(a). This result shows thecomparative large peak value of the mean photon num-ber for the parameters χ/γ = 11 and Ω /γ = 12. In thiscase, Fig. 10 demonstrates large population P = 0 .
48 of (d) T h i r d o r d e r c o rr e l a ti on f un c ti on Tuning parameter (a) M ea n e x c it a ti on nu m b e r t (b) M ea n e x c it a ti on nu m b e r Tuning parameter (c) C o rr e l a ti on f un c ti on Tuning parameter
FIG. 9: The regime of three-photon blockades. The meanphoton number versus dimensionless time, for k = 3 (a); themaximal peak values of the mean photon numbers depend-ing on the tuning parameter (b); the second-order correlationfunction at the peak values of the mean photon numbers ver-sus the tuning parameter (c); the third-order correlation func-tion at the peak values of the mean photon numbers versusthe tuning parameter (d). The parameters are: χ/γ = 11, themaximum amplitude of pump field Ω /γ = 12, the mean num-ber of reservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . | i Fock state for the tuning parameter k = 3 .
28. Thus,maximal population P takes place for near to the reso-nance frequency ω = ω + 2 . χ .The maximal values of the mean photon numbers fortime-interval during pulses versus the tuning parameterare depicted in Fig. 9(b). This curve displays two-peakstructure of the mean photon number. The first peakoccurs at k = 2 which corresponds approximately tothe mean photon number h n i = P + 2 P + 3 P =1.3in accordance with Fig. 8. The second characteristicdouble peak is at k = 3 .
28 and approximately equals to h n i = P + 2 P + 3 P =0.14+0.56+ 1.44.It is interesting to analyze selective excitation of three-photon state in the framework of the second-order andthird-order correlation functions. The results of calcu-lations are depicted in Fig. 9(c) and Fig. 9(d). As wesee, for the case of the maximal population P = 0 . | i Fock state, if k = 3 .
28 and the mean photonnumber is h n i = 2 , the second-order correlation func-tion is g (2) = 0 .
75, (note, that for the pure | i state g (2) = 2 / g (3) = 0 .
32. It should be mentioned, that the lastresult is in accordance with the result for pure | i Fockstate, g (3) = 2 / P P P P P ho t on - nu m b e r s t a ti s ti c s Tuning parameter
FIG. 10: The regime of three-photon blockades. The popu-lations P ( n ) for n = 0 , , ,
3, for time-intervals at the peakvalues of the mean photon numbers versus the tuning parame-ter. The parameters are: χ/γ = 11, the maximum amplitudeof pump field Ω /γ = 12, the mean number of reservoir pho-tons N = 0 . τ = 5 . γ − , T = 0 . γ − . three-photon states of Kerr nonlinear resonator in thepulsed regime in comparison with the analogous resultsobserved in the case of cw excitation (see, [16] and theresults depicted in Fig. 3 and Fig. 4). This observationindicates on the efficiency of multiphoton blockades forthe pulsed regimes. IV. PRODUCTION OF MULTI-PHOTONSTATES IN THE PULSED REGIMES
Recently, it has been demonstrated that formation ofsingle-photon states can be realized in a cavity stronglycoupled with emitter operated in the pulsed regime. Inthis way the photon-number effects and phase-space char-acteristics of photonic states can be controlled by choos-ing the parameters of driving pulses [26], [30]. In this sec-tion, we investigate production of two-photon and three-photon light states considering mainly the selectivity ofthe excitation of these states by driving pulses and theirphase-space properties.At first, we turn to the regime of single-photon block-ade shown in Fig. 5 and Fig. 6 considering maximal val-ues of the mean photon number for time-interval duringpulses depending on the tuning parameter. The resultof calculations are depicted in Fig. 11(a) where we plotthe dependence of the mean photon number for differentdetunings. This curve shows a large spectral width ofgeneration with the maximum at ∆ = 0. The peakedintensity is approximately h n i = P + 2 P =0.82+ 0.18with the populations have been obtained in Fig. 6.Now, we consider production of two-photon states byselective two-photon excitations in Kerr nonlinear res- (b) M ea n e x c it a ti on nu m b e r Tuning parametr (a) M ea n e x c it a ti on nu m b e r Tuning parametr
FIG. 11: The maximum values of the mean photon numbersdepending on the tuning parameter. The parameters are: χ/γ = 15, Ω /γ = 6, τ = 5 . γ − , T = 0 . γ − (a); χ/γ = 30,Ω /γ = 12, τ = 5 . γ − , T = 0 . γ − (b). The mean number ofreservoir photons N = 0 . | i state (a) ; the Wignerfunction of cavity mode for time-interval corresponding tothe maximal population of Fock state | i (b). The nega-tive regions of the Wigner functions are indicated in black.The parameters are: χ/γ = 15, the maximum amplitude ofpump field Ω /γ = 12, the mean number of reservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . onator. In order to demonstrate the process we turn tothe regime of two-photon blockade (see, the results de-picted in Fig. 7 and Fig. 8). For these parameters thepopulation of two-photon number state reaches P = 0 . /γ = − χ/γ . The curve of the maximalvalues of the mean photon numbers for these parametersare plotted in Fig. 11(b) in dependence from the detun-ing values. As we see, the mean photon number displaystwo-peak structure. The first peak occurs at k = 1 whichcorresponds approximately to the mean photon number h n i = P + 2 P =0.82+0.18. The second characteristicdouble peak is at k = 2 and approximately equals to h n i = P + 2 P =0.3+ 1.22.To obtain a complete description of the system we turnto calculation of the Wigner function in phase-space. Thenumerical results are given for the regime used abovethat relies to effective selective excitation of two-photonstates, for k = 2.The Wigner function of intracavity mode for time-intervals corresponding to the maximal P = 0 .
61 popu-lation is shown in Fig. 12(b) with the Wigner functionof pure | i state, plotted in Fig. 12(a)) for comparison.It is easy to realize that this Wigner function displays aring signature with the center at x = y = 0 in the phasespace and the good agreement in shape takes place withthe Wigner function of pure | i state.Fig. 13 shows comparative analysis of the contour plotsof the various Wigner functions corresponding to the defi-nite detuning values that determine various resonant one-photon and multi-photon transitions. As we see, for theregime of one-photon excitation, ∆ = 0, on the frequency ω = ω the Wigner function (Fig. 13(a)) displays goodagreement with the Wigner function of pure | i single-photon state. In Fig. 13(b) the result for k = 1 . /γ = − . χ/γ is presented. In this case, thepopulations of vacuum state and single-photon state arecrossing and are approximately equal (see, Fig. 8) thus,the Wigner function seems to be closest to the Wignerfunction of pure superposition state Ψ i = √ ( | i − | i ).For k = 2, the contour plot depicted in Fig. 13(c) cor-responds to the Wigner function of Fig. 12(b). In Fig.13(d) the contour plot for the case of near the resonantthree-photon excitation at k = 3 .
28 is depicted. ThisWigner function shows three-phase symmetry and alsoindicates the regions of quantum interference in the con-tour plot as the negative regions in black. Note thatthree-fold symmetry of the Wigner function and interfer-ence pattern has been demonstrated for the direct three-photon down-conversion in χ (3) media [42] and also indetails for production of three-photon states in [43], [44]. V. CONCLUSION
We have investigated photon-number effects in multi-photon blockade for Kerr nonlinear dissipative resonatordriven by sequence of Gaussian pulses. This model canbe realized at least with two physical systems: a qubitoff-resonantly coupled to a driven cavity or superconduct-ing devices based on Josephson junction. We considerthe cases of strong nonlinearity with respect to the rateof damping of the oscillatory mode. In these regimesthe oscillatory energy levels are well resolved, and spec-troscopic selective excitation of transitions between Fockstates is possible by tuning the frequency of driving field.Comparing the results for continuous wave- andpulsed- operational regimes of Kerr nonlinear resonatorwe demonstrate that the larger photon-number popula-tions of the resonator can be reached if shaped pulses areimplemented. Thus, we have shown that optimized one-photon and multi-photon blockades can be obtained byadequately choosing the duration of pulses and the timeintervals between them.Consideration of the Kerr nonlinear dissipative res-onator under cw driving has been done on the frameworkof the potential solution of the Fokker-Planck equationfor the quasiprobability distribution function in complexP-representation. In this way, we strongly demonstrated -1.6 -0.8 0.0 0.8 1.6-1.6-0.80.00.81.6 (d) X Y -1.6 -0.8 0.0 0.8 1.6-1.6-0.80.00.81.6 (c) X Y -1.6 -0.8 0.0 0.8 1.6-1.6-0.80.00.81.6 (b) X Y -1.6 -0.8 0.0 0.8 1.6-1.6-0.80.00.81.6 (a) X Y FIG. 13: The contour plots of the Wigner functions for thevarious selective excitations: k = 1 (a); k = 1 .
24 (b), wherethe populations of | i and | i are crossing each other (see,Fig. 8); k = 2 (c); k = 3 (d). The negative regions of theWigner functions are indicated in black. The parameters are: χ/γ = 15, the maximum amplitude of pump field Ω /γ = 12,the mean number of reservoir photons N = 0 . τ = 5 . γ − , T = 0 . γ − . that for the case of cw pumping of the Kerr-nonlinear res-onator there are fundamental limits on the populationsof lower photonic-number-states (n=0, 1, 2, 3). Particu-larly, the maximal value of | i state population is limitedby P = 0 . P population overcome 0.8 value.We analyse photon-number effects and investigate indetails the photon-number correlation effects on base ofthe second-order and third-order correlation functionsfor the case of selective excitations of intracavity mode.Specifically we have calculated the Wigner function inphase-space for the regimes of multi-photon excitationsof mode. Acknowledgments
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