Dephasing effects on stimulated Raman adiabatic passage in tripod configurations
aa r X i v : . [ qu a n t - ph ] S e p Dephasing effects on stimulated Raman adiabatic passage in tripod configurations
C. Lazarou
1, 2 and N. V. Vitanov Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria (Dated: November 5, 2018)We present an analytic description of the effects of dephasing processes on stimulated Ramanadiabatic passage in a tripod quantum system. To this end, we develop an effective two-level model.Our analysis makes use of the adiabatic approximation in the weak dephasing regime. An effectivemaster equation for a two-level system formed by two dark states is derived, where analytic solutionsare obtained by utilizing the Demkov-Kunike model. From these, it is found that the fidelity for thefinal coherent superposition state decreases exponentially for increasing dephasing rates. Dependingon the pulse ordering and for adiabatic evolution the pulse delay can have an inverse effect.
PACS numbers: 32.80.Qk, 33.80.Be, 42.50.Dv, 42.50.Le
I. INTRODUCTION
Stimulated Raman adiabatic passage (STIRAP) [1–3]is a powerful and robust technique for achieving completepopulation transfer in three-state quantum systems. Byusing two pulsed laser fields population is adiabaticallytransferred from an initially populated state ψ , to a tar-get state ψ via an intermediate state ψ . A unique fea-ture of this technique is that the intermediate state isnever populated. This is due to the fact that the sys-tem at all times adiabatically follows a dark state, hencepopulation losses due to spontaneous emission are sup-pressed.Apart from being used for population transfer, STI-RAP can also be used to create coherent superpositionsof states in quantum systems in tripod configurations[2, 4, 5]. The main idea behind this method is the sameas with STIRAP in Λ-configurations, with the exceptionthat the system now adiabatically follows a superpositionof two dark states. The interference of these two statesresults in a coherent superposition between two or threeof the ground states of the tripod system. The exactform for this final state is defined by the geometric phaseacquired by the dark states.In addition to the creation of coherent superpositions[4–6] , STIRAP in tripod configurations can be furtherexploited to implement quantum gates [7–9]. Further-more, adiabatic passage in tripod systems can be usedto engineer non-Abelian gauge potentials for ultracoldatoms [10]. The formation of such potentials is madepossible because during the adiabatic evolution, the sys-tem can acquire non-Abelian phases [11].Since the intermediate state is not populated in theadiabatic limit, spontaneous emission from this state isexpected to have no effect on the fidelity. On the otherhand, the creation of a coherent superposition relies onthe formation of coherent dark states. Thus maintainingcoherence is vital for achieving higher fidelities. Howeverphase relaxation effects induced for example by elasticcollisions or laser phase fluctuations can have an adverseeffect on the fidelity. Previous studies on STIRAP in the presence of dephas-ing [12, 13], have shown that decoherence can lead to pop-ulation losses from the dark state, resulting in a transferefficiency reduction. On the other hand, increasing therelative delay between the two laser pulses increases thetransfer efficiency. This is due to the inverse dependenceof the transition time with respect to the delay. In a re-cent paper by Møller , Madsen and
Mølmer [8], dephas-ing in tripod systems and the effect it has on single qubitgates was considered. Using the Monte Carlo wavefunc-tion method, they were able to show that the system ac-quires complex geometric phases, implying losses whichreduce the gate fidelity.In the present paper, we extend the method used inRef. [13] to study dephasing effects on STIRAP in tripodconfigurations. The method makes use of the adiabaticapproximation in the weak dephasing regime, where wederive an effective two-level master equation for the darkstates. Analytic solutions are obtained when the Stokesand control pulses overlap, whereas for other pulse order-ings we make use of numerical simulations. Dependingon the pulse ordering, similar with or different featuresfrom STIRAP in a Λ-configuration are observed.The paper is organized as follows. In Sec. II we providea brief introduction on STIRAP in tripod configurationsand the master equation is introduced. In Sec. III wepresent the effective two-level model for the dark states,and derive analytic solutions in Sec. IV. In Sec. V wepresent results from numerical simulations. A summaryof the results is given in Sec. V.
II. THE TRIPOD CONFIGURATION
The tripod system is shown in Fig. 1. The threeground states ψ , ψ and ψ are resonantly coupled tothe intermediate level ψ via a pump Ω p ( t ), a StokesΩ s ( t ) and a control pulse Ω c ( t ) respectively. The systemis initially prepared in state ψ , i.e. ψ ( −∞ ) = ψ . Such aconfiguration was part of a proposed scheme for creatingand phase probing coherent superpositions [4]. This was FIG. 1. (Color online) The tripod configuration and the threelaser pulses: Ω p ( t ) (pump), Ω c ( t ) (control) and Ω s ( t ) (Stokes).The relative dephasing for each pair of states is depicted bywavy lines. demonstrated in an experiment by Theuer et al. [5].The scheme we examine here uses an adiabatic passagemethod which is an extension of STIRAP in Λ-systems.The population initially placed in state ψ is either trans-ferred to the other two ground states ψ and ψ or is splitbetween the states ψ and ψ . The final superpositionbetween ψ and ψ , or between ψ ψ , is determined bythe relative distance of the pulses. The conditions for therobust creation of such superpositions is that the systemevolves adiabatically [2, 4]. A. Adiabatic states: Creating coherentsuperpositions
The interaction Hamiltonian in the rotating wave ap-proximation reads [2] H ( t ) = ~ p ( t ) 0 0Ω p ( t ) 0 Ω s ( t ) Ω c ( t )0 Ω s ( t ) 0 00 Ω c ( t ) 0 0 , (1)where we take all transitions to be resonant with therespective laser field. Furthermore, we assume thatthese are the only allowed dipole transitions. This time-dependent Hamiltonian has two dark states Φ ( t ) andΦ ( t ) [2]Φ ( t ) = 1 √ ψ cos θ − √ ψ (sin θ cos φ + i sin φ ) − √ ψ (sin θ sin φ − i cos φ ) , (2a)Φ ( t ) = 1 √ ψ cos θ − √ ψ (sin θ cos φ − i sin φ ) − √ ψ (sin θ sin φ + i cos φ ) , (2b) where the time dependent mixing angles φ ( t ) and θ ( t )read tan( φ ( t )) = Ω c ( t )Ω s ( t ) , (3a)and tan( θ ( t )) = Ω p ( t ) p Ω s ( t ) + Ω c ( t ) . (3b)We note here, that since the two dark states are eigen-states of H ( t ) with zero eigenvalue, i.e. H ( t )Φ , ( t ) = 0,any linear superposition of these two states is also a darkstate [2, 4]. In addition to the two dark states, there arealso two adiabatic states with non-zero time-dependenteigenenergiesΦ ( t ) = 1 √ ψ sin θ + 1 √ ψ + 1 √ ψ cos θ cos φ + 1 √ ψ cos θ sin φ, (4a)Φ ( t ) = 1 √ ψ sin θ − √ ψ + 1 √ ψ cos θ cos φ + 1 √ ψ cos θ sin φ, (4b)with eigenenergies ǫ ( t ) = − ǫ ( t ) = ~ t ) , (5)where Ω( t ) = q Ω p ( t ) + Ω s ( t ) + Ω c ( t ) is the rms Rabifrequency.In the adiabatic limit, the time-dependent eigenstatesare weakly coupled and this is also valid for the two darkstates Φ ( t ) and Φ ( t ). Although they form a pair ofdegenerate states, ǫ ( t ) = ǫ ( t ) = 0, the correspondingdiabatic coupling is always zero, h ˙Φ ( t ) | Φ ( t ) i = 0 , (6)whereas h Φ ( t ) | ˙Φ ( t ) i = −h Φ ( t ) | ˙Φ ( t ) i = i ˙ φ sin θ. (7)Because of this time-dependent energy shift, both darkstates acquire a geometric phase ϑ j [14] ϑ = − ϑ = ϑ g = Z ∞−∞ d t ˙ φ sin θ. (8)Thus when the system starts in state Φ ( t ) or Φ ( t ),it will adiabatically follow this state and remain in thisstate at all times acquiring a net phase ± ϑ g respectivelyΦ ( −∞ ) → e − i ϑ g Φ ( ∞ ) , Φ ( −∞ ) → e i ϑ g Φ ( ∞ ) . (9)As said in the begining of this section, the system isinitially prepared in state ψ . Then in the adiabatic limitand for θ ( −∞ ) = 0, the system state is a symmetricsuperposition of the two dark statesΨ( −∞ ) = ψ = 1 √ ( −∞ ) + Φ ( −∞ )) , (10)and for t → ∞ it readsΨ( ∞ ) = 1 √ (cid:0) e − i ϑ g Φ ( ∞ ) + e i ϑ g Φ ( ∞ ) (cid:1) . (11)The final superposition state in terms of the bare statesdepends on the ordering of the three pulses. Of the manypossible pulse orderings four different pulse sequences areparticularly interesting for each produces a different co-herent state [2]. These pulse orderings are as follows. • The pulses are ordered so that the Stokes pulsestarts first and ends after the pump pulse, whilethe control pulse is delayed with respect to bothof them. For this pulse sequence we have the fol-lowing asymptotic relations: θ ( −∞ ) = θ ( ∞ ) = 0, φ ( −∞ ) = 0 and φ ( ∞ ) = π/
2. Then the final statereads Ψ( ∞ ) = ψ cos ϑ g − ψ sin ϑ g . (12) • A different pulse ordering is arranged so that theStokes pulse comes first, followed by the controlpulse and then the pump pulse. For this case theasymptotic relations are θ ( −∞ ) = 0, θ ( ∞ ) = π/ φ ( −∞ ) = 0 and φ ( ∞ ) = π/
2. Then the final stateis Ψ( ∞ ) = − ψ sin ϑ g − ψ cos ϑ g . (13) • Alternatively one can reverse the order of theStokes and control pulses, i.e. the latter pulse pre-cedes the Stokes pulse. Then for t → ±∞ wehave θ ( −∞ ) = 0, θ ( ∞ ) = π/ φ ( −∞ ) = π/ φ ( ∞ ) = 0. The final coherent state readsΨ( ∞ ) = − ψ cos ϑ g + ψ sin ϑ g . (14) • Finally, the Stokes and control pulses can coincidein time and precede the pump pulse. Then thefollowing asymptotic relations apply: θ ( −∞ ) = 0, θ ( ∞ ) = π/ φ ( −∞ ) = φ ( ∞ ) = π/
4. Because φ ( t ) is constant, we have that ˙ φ ( t ) = 0 and thusthe geometric phase is zero. Hence the final stateis Ψ( ∞ ) = − √ ψ + ψ ) . (15) B. Dephasing
In order to model the effect of dephasing in the tripodsystem, we make use of the master (Liouville) equationi ~ ˙ ρ = [ H ( t ) , ρ ] + D. (16) The dissipator matrix D describes dephasing effects D = − i ~ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ γ ρ , (17)where γ ij = γ ji are the constant relaxation rates and ρ is the density matrix in the bare basis ψ m , i.e. ρ mn = h ψ m | ˆ ρ | ψ n i . The initial conditions are ρ ( −∞ ) = 1 and ρ mn = 0 for mn = 11.The derivation of master equations such as the one inEq. (16) is based on the use of the Born-Markov approx-imation [15]. This imposes restrictions on the spectralproperties of the heat bath with which a quantum systeminteracts [16], while the relative coupling strength mustbe weak. Thus, collisions between atoms or moleculesmust be weak [17], whereas the fluctuating laser phasemust be well approximated by a Markovian process [18].In the following section we derive approximate solu-tions in the weak dephasing limit and for adiabatic evo-lution. Because of this latter assumption we will be usingthe density matrix in the adiabatic basis, defined fromthe following transformation ρ a = R − ρR (18)where ρ a is the density matrix in the adiabatic basisΦ j ( t ), i.e. ρ amn = h Φ m ( t ) | ˆ ρ | Φ n ( t ) i . The rotation matrix R is formed by using the adiabatic states as its columns R ( t ) = (cid:2) Φ T ( t ) , Φ T ( t ) , Φ T ( t ) , Φ T ( t ) (cid:3) , (19)where R − = R † . Then the master equation in the adia-batic basis isi ~ ˙ ρ a = [ H a ( t ) , ρ a ] − i ~ [ R − ˙ R, ρ a ] + R − DR, (20)where H a ( t ) is the Hamiltonian in the adiabatic basis, H a ( t ) = diag(0 , , ǫ ( t ) , ǫ ( t )) . (21)The second term on the right hand side of Eq. (20),i.e. R − ˙ R , describes non-adiabatic interactions (off diag-onal terms), whereas the two diagonal terms ( R − ˙ R ) = − ( R − ˙ R ) are responsible for the geometric phases ac-quired by the two dark states. III. THE TWO-LEVEL APPROXIMATION
As mentioned earlier, the creation of coherent statesis the result of interference effects between the two darkstates (2). This suggests that it is possible to solve themaster equation (16) approximately by deriving an effec-tive two-level master equation. To this end, we make twoapproximations: the adiabatic and the weak dephasingapproximations. These were used before when studyingthe effect of dephasing on STIRAP in Λ-systems [13].
A. The adiabatic approximation
The first of the two approximations that we are usingis that of adiabatic evolution. In order for this to be validwe need large pulse areas [2, 4] so that | ˙ θ ( t ) | ≪ | Ω( t ) | , | ˙ φ ( t ) | ≪ | Ω( t ) | . (22)When these two conditions are satisfied, the system adi-abatically follows a superposition of the two dark states,whereas the adiabatic states Φ ( t ) and Φ ( t ) are not pop-ulated. Because of this the coherences ρ aij related to thesetwo states are negligible. B. The weak dephasing approximation
The second approximation is that of weak dephasing,i.e. the relaxation rates γ ij are much smaller than therms Rabi frequency Ω( t ) γ ij ≪ | Ω( t ) | , ( i, j = 1 , , , i = j ) . (23)Assuming that the adiabatic approximation Eq. (22) isvalid, then for γ ij ≥ | Ω( t ) | we would have that γ ij T ≫ T a characteristic time length for the pulse dura-tions. This latter inequality corresponds to strong de-phasing which would lead to complete incoherent dynam-ics that are governed by rate equations [19]. This justifiesthe use of the weak dephasing approximation Eq. (23).Using both approximations we can now simplify theanalysis by neglecting all coherences that include the twoadiabatic states Φ ( t ) and Φ ( t ), ρ aij ≈ , ( ij = 12 , . (24)With this the density matrix in the adiabatic basis ρ a acquires the following approximate form ρ a ≈ ρ a ρ a ρ a ρ a ρ a
00 0 0 ρ a . (25)Using Eqs. (18), (19), (20) and (25) (see appendix A),we can first show that the population inversion for thebright states, w a ( t ) = ρ a − ρ a , and for the dark states, w a ( t ) = ρ a − ρ a , are both zero at all times, i.e. ρ a = ρ a , ρ a = ρ a . (26)Hence, the populations for the adiabatic states can beparametrized in terms of a single function s ( t ), i.e. ρ a = ρ a = 1 − s ( t )2 , (27a) ρ a = ρ a = 1 + 2 s ( t )2 . (27b) From this we can derive a set of coupled differential equa-tions for s ( t ) and the coherences for the dark states, u ( t ) = √ { ρ a } and v ( t ) = √ { ρ a } ,˙ s ( t ) = − Γ s ( t ) s ( t ) + √ su ( t ) u ( t ) + √ sv ( t ) v ( t ) , (28a)˙ u ( t ) = − Γ u ( t ) u ( t ) + (cid:16) φ sin θ + Ω uv ( t ) (cid:17) v ( t )+ √ su ( t ) s ( t ) , (28b)˙ v ( t ) = − Γ v ( t ) v ( t ) + (cid:16) − φ sin θ + Ω uv ( t ) (cid:17) u ( t )+ √ sv ( t ) s ( t ) . (28c)The initial conditions are s ( t ) = − / u ( t ) = 1 / √ v ( t ) = 0. The derivation of these equations and the ex-pressions for the Rabi frequencies Ω su ( t ), Ω sv ( t ), Ω uv ( t ),and those for the relaxation rates Γ j ( t ), j = s, u, v , areprovided in appendix A. We note here that the effectiverelaxation rates Γ j ( t ) and the Rabi frequencies Ω ij ( t ),depend only on the relaxation rates γ , γ and γ , seeEqs. (A8) and (A9). C. Fidelity for the target coherent superpositionstate
In the analysis that follows we will be using the fidelity F ( t ) for the final coherent state Ψ( ∞ ) [20] F ( t ) = |h Ψ( ∞ ) | ρ ( t ) | Ψ( ∞ ) i| , (29)where Ψ( ∞ ) is one of the four states (12), (13), (14) or(15), and ρ ( t ) is the density matrix in the bare basis. For t → ∞ the fidelity can be expressed in terms of the darkstate populations and coherences. Using Eq. (11) forthe coherent state Ψ( ∞ ), and Eq. (25) for the densitymatrix in the adiabatic basis and for weak dephasing, thefidelity reads F ( ∞ ) = ρ a ( ∞ ) + cos(2 ϑ g )Re { ρ a ( ∞ ) }− sin(2 ϑ g )Im { ρ a ( ∞ ) } , (30)where ϑ g is given by Eq. (8).To this end, and before proceeding with the solutionof Eqs. (28), we introduce one more tool that we will beusing in the following sections. This is the transition time T tr ( ǫ ), and is defined as the time needed for the fidelityto rise from a small value F ( t ǫ ) = ǫ to F ( t − ǫ ) = 1 − ǫ ,i.e. T tr ( ǫ ) = t − ǫ − t ǫ . (31)The importance of the transition time was discussed ona previous work on dephasing effects on STIRAP in Λ-systems [13]. For STIRAP the pulse delay has an inverseeffect on the efficiency, and this is due to the fact thatthe transition time is inversely proportional to the delaytime.We should note that the above definition for the tran-sition time, does not apply when the Stokes pulse endsafter the pump pulse, see Eq. (12). While for the otherthree possible pulse orderings the fidelity is initially zero,i.e. F ( −∞ ) = 0, for the former pulse ordering the fi-delity is F ( −∞ ) = cos ( ϑ g ). For this case, instead ofusing the above definition for the transition time, we willbe using the following one T tr ( ǫ ) = t − ǫ − t ǫ , (32)where the time t ǫ is such that F ( t ǫ ) = (1 + ǫ ) F ( −∞ ) , (33)and t − ǫ remains the same. IV. ANALYTIC SOLUTIONSA. Equal relaxation rates γ ij = γ and Ω s ( t ) = Ω c ( t ) A special case is that when the Stokes and control pulsecoincide, i.e. Ω c ( t ) = Ω s ( t ). Then at all times we have φ ( t ) = π/ φ ( t ) = 0. When taking all relaxationrates equal i.e. γ ij = γ we have thatΩ sv ( t ) = Ω uv ( t ) = 0 , (34)and the equations for s ( t ) and u ( t ) decouple from thatfor v ( t ), i.e.˙ c ( t ) = √ su ( t ) c ( t ) , (35a)˙ c ( t ) = ∆ su ( t ) c ( t ) + √ su ( t ) c ( t ) , (35b)and for v ( t ) we have˙ v ( t ) = − Γ v ( t ) v ( t ) . (35c)The new variables c ( t ) and c ( t ) are c ( t ) = s ( t ) exp (cid:18)Z t Γ s ( t ′ )d t ′ (cid:19) ,c ( t ) = u ( t ) exp (cid:18)Z t Γ s ( t ′ )d t ′ (cid:19) . (36)This parametrization is used in order to emphasize theanalogy to a two-level system. The effective relaxationrates and coupling in terms of the mixing angle θ ( t ) readΓ v ( t ) = γ cos ( θ ) , (37a)Ω su ( t ) = γ (cid:18) (2 θ )4 − cos θ (cid:19) , (37b)∆ su ( t ) = γ (cid:18) (2 θ )4 + cos θ (cid:19) − γ sin θ. (37c) -1-0.8-0.6-0.4-0.200.20.4 -3 -2 -1 0 1 2 3 F r e qu e n c y ( un it s o f T − ) Time (units of T )(a) W su ( t ) D su ( t ) -1-0.8-0.6-0.4-0.200.20.40.60.8 -3 -2 -1 0 1 2 3 F r e qu e n c y ( un it s o f T − ) Time (units of T )(b) ˙ x ( t ) d ( t ) FIG. 2. (Color online) (a) The coupling Ω su ( t ) and the de-tuning ∆ su ( t ) Eq. (35) for Gaussian pulses Eq. (44). (b) Thecoupling ˙ ξ ( t ) and the detuning ∆( t ) Eq. (45) in the adiabaticbasis (43). The delay is τ = T and γ = T − . In Fig. 2(a), we plot the coupling Ω su ( t ) and the detun-ing ∆ su ( t ) for Gaussian pulses Eq. (44). To this end, wenote that since v ( −∞ ) = 0 equation (35c) has the trivialsolution v ( t ) = 0.In order to connect Eqs. (35a) and (35b) with those fora driven two-level system and exploit this to solve them,we make use of the following time-dependent rotation R ( t ) = 2 √ (cid:18) − cos ξ sin ξ sin ξ cos ξ (cid:19) , (38)where the angle ξ ( t ) is ξ ( t ) = 12 arctan √ su ( t )∆ su ( t ) ! . (39)With this equations (35a) and (35b) take the form (cid:18) ˙ c − ( t )˙ c + ( t ) (cid:19) = (cid:18) ǫ − ( t ) ˙ ξ ( t ) − ˙ ξ ( t ) ǫ + ( t ) (cid:19) (cid:18) c − ( t ) c + ( t ) (cid:19) , (40)where the “energies” ǫ ± ( t ) = ∆ su ( t )2 (1 ± sec(2 ξ )) , (41)are the eigenvalues of the matrix W ( t ) = (cid:18) √ su ( t ) √ su ( t ) ∆ su ( t ) (cid:19) . (42)The corresponding eigenstates ψ ± ( t ) are ψ + ( t ) = [sin ξ, cos ξ ] T ,ψ − ( t ) = [ − cos ξ, sin ξ ] T . (43)Multiplying now both sides of Eq. (40) with the imagi-nary unit, we see that it takes the form of a Schr¨odingerequation for a two-level system. The two states ψ − and ψ + , are driven by a laser field with Rabi frequency ˙ ξ ( t ),while decaying with rates ǫ − ( t ) and ǫ + ( t ) respectively.The initial conditions for Eq. (40) can be derived fromEqs. (38), (39), (37c) and (37b), and are c + ( −∞ ) = 1and c − ( −∞ ) = 0.In Fig. 2(b) we plot the “chirped” detuning ∆( t ) = ǫ + ( t ) − ǫ − ( t ) and the coupling ˙ ξ ( t ) for Gaussian pulsesof the form Ω p ( t ) = Ω e − ( t − τ/ /T , Ω s ( t ) = Ω c ( t ) = Ω e − ( t + τ/ /T . (44)The “chirped” detuning ∆( t ) and the coupling ˙ ξ ( t ) are∆( t ) = γf (cid:18) tτT (cid:19) , ˙ ξ ( t ) = τT f (cid:18) tτT (cid:19) , (45)where f ( x ) = 1 − e x (2 + e x ) s e x ) ,f ( x ) = 2 √
21 + 5 cosh( x ) − x ) . (46)From Fig. 2, we see that both ˙ ξ ( t ) and ∆( t ) resemblancethe laser field and the frequency chirp for the Demkov-Kunike (DK) model [21]. This feature is exploited nextto derive analytic solutions for Eqs. (40). To this end, itshould be pointed out that the time-dependent detuningin the original DK model corresponds to real frequencychirp, whereas in the two-level system of Eq. (40) it isan imaginary chirp, i.e. a time-dependent decay rate.
1. Adiabatic following
The method we use to solve Eq. (40), is the one used inRefs. [22, 23] to derive the propagator in three-state sys-tems with pairwise crossings. Starting with the systeminitially in state ψ + ( ∞ ), we assume that it evolves adi-abatically until reaching the crossing point t = 0, where∆(0) = 0. At this point, diabatic transitions will oc-cur and the new system state will be a mixture of thetwo adiabatic states ψ ± ( t ). The effect of the crossing isexpressed in terms of a transition matrix U c (0) = (cid:18) U −− (0) U − + (0) U + − (0) U ++ (0) (cid:19) , (47)where U jj (0) and U ij (0) ( j = i ) are the survival andtransition probability amplitudes, for a two-level systemdriven by a laser pulse ˙ ξ eff ( t ) ≈ ˙ ξ ( t ), in the presence ofa time-dependent spontaneous emission ∆ eff ( t ) ≈ ∆( t ).For times t >
0, i.e. after the crossing the system willevolve adiabatically. With this the final system propaga-tor U ( ∞ , −∞ ) reads U ( ∞ , −∞ ) = U a ( ∞ , U c (0) U a (0 , −∞ ) , (48)where the adiabatic propagator U a ( t f , t i ) is U a ( t f , t i ) = e R tfti ǫ − ( t ) dt e R tfti ǫ + ( t ) dt ! . (49)Upon using the above equations with the initial condition ψ ( −∞ ) = ψ + ( −∞ ), the final two-level state takes theform ψ ( ∞ ) = U − + (0) e R ∞ ǫ − ( t ) dt + R −∞ ǫ + ( t ) dt ψ − ( ∞ )+ U ++ (0) e R ∞−∞ ǫ + ( t ) dt ψ + ( ∞ ) . (50)
2. Diabatic transitions
The survival probability amplitude U ++ (0) and thetransition amplitude U − + (0), are solutions of theSchr¨odinger equation˙ c − ( t ) = ˙ ξ eff ( t ) e R t ∆ eff ( t ′ ) dt ′ c + ( t ) , ˙ c + ( t ) = − ˙ ξ eff ( t ) e − R t ∆ eff ( t ′ ) dt ′ c − ( t ) . (51)As already pointed out earlier and shown in Fig. 2(b),this two-level system resemblance the Demkov-Kunikemodel [21], with the only difference that the real fre-quency chirp for the latter model is replaced by an imag-inary one, i.e. population loss.To proceed with the solution of Eq. (51) with the ini-tial condition c + ( −∞ ) = 1 and c − ( −∞ ) = 0, we utilizea hyperbolic sech function to approximate ˙ ξ ( t )˙ ξ eff ( t ) = A sech (( t − t max ) /T eff ) , (52a)and a hyperbolic tanh function with a constant term for∆( t ) ∆ eff ( t ) = D + B tanh (( t − t max ) /T eff ) . (52b)The time t max refers to the maximum of ˙ ξ ( t ) Eq. (45)and is t max = T τ arctanh (cid:18) (cid:19) . (53)The amplitude A for ˙ ξ eff ( t ) is derived from the condi-tion ˙ ξ ( t max ) = ˙ ξ eff ( t max ), and is A = τ √ T . (54)The effective pulse duration T eff is obtained by requiringthat both ˙ ξ ( t ) and ˙ ξ eff ( t ) have the same pulse area. Withthis T eff reads T eff = T √ πτ arctan(2 √ . (55)Finally, the condition for obtaining D and B , is thatfor t ≈ t max , ∆( t ) ≈ ∆ eff ( t ). After performing a Taylorseries expansion for both ∆( t ) and ∆ eff ( t ) at the vicinityof t max and keeping only terms of first order in t , we have D = − √ γ , (56a)and B = − √ γ π arctan(2 √ . (56b)Solutions for Eq. (51) and for the Rabi frequency˙ ξ eff ( t ) Eq. (52a) and for ∆ eff ( t ) Eq. (52b), can beexpressed in terms of hypergeometric functions, see forexample Refs. [24, 25]. For t → ∞ the probability am-plitudes are U ++ (0) = Γ (cid:0) + δ − β (cid:1) Γ (cid:16) + δ + p β + α (cid:17) × Γ (cid:0) + δ + β (cid:1) Γ (cid:16) + δ − p β + α (cid:17) , (57a) U − + (0) = α Γ (cid:0) + δ − β (cid:1) Γ (cid:16) − β + p β + α (cid:17) × Γ (cid:0) − δ − β (cid:1) Γ (cid:16) − β − p β + α (cid:17) , (57b) where Γ( x ) is the gamma function [26]. The parameters α , β and δ are α = AT eff = 12 π arctan(2 √ , (58a) β = BT eff − √ (2 √ π γT τ , (58b) δ = DT eff − √ √ π γT τ . (58c)
3. Populations and coherences for the dark states
After substituting Eqs. (57a) and (57b) into Eq. (50),and using the inverse rotation R − ( ∞ ) along with Eqs.(36), we obtain s ( ∞ ) and u ( ∞ ). From these we derivethe real part for the coherence ρ a ( ∞ ) of the two-darkstates (2),Re { ρ a ( ∞ ) } = lim t →∞ r U ++ (0) exp (cid:18) − c u γT τ − γt (cid:19) , (59a)and their populations ρ a ( ∞ ) = ρ a ( ∞ ) = 14 + √ U − + (0) exp (cid:18) − c s γT τ (cid:19) , (59b)where c s = 2 .
42 and c u = 0 .
68. The exponential terms inthe above expressions are contributions acquired duringthe adiabatic evolution, see Eq. (50). Their dependenceon the different parameters, i.e. the factor γT /τ can beeasily derived, whereas the two factors c u and c s can becalculated numerically, see appendix B.We should recall here that the imaginary part of ρ a ( ∞ ) is zero, because v ( t ) = 0. Furthermore we notethat in the long time limit the real part is also zero. Then,the fidelity for a coherent state (30), will be proportionalto the population of the dark states. As we can see fromEq. (59b) this will decrease for increasing relaxation rate γ , or will increase for increasing delay τ . This is becauseof the inverse dependence of the transition time with re-spect to τ . The transition time T tr ( ǫ ) for ǫ = 0 . F ( t ) = sin θ = exp (cid:0) tτT (cid:1) (cid:0) tτT (cid:1) , (60)and is T tr (0 .
1) = T τ log(3) . (61)Finally, although Eqs. (59) were obtained within theweak dephasing approximation, they are valid even inthe strong dephasing regime. In this limit dynamics arecompletely incoherent and are governed by rate equations[19]. B. General case with γ ij = γ Although analytic solutions for Eqs. (28) cannot bederived when φ ( t ) is no longer constant, the main fea-tures for the system dynamics can be qualitatively dis-cussed. In order to simplify Eqs. (28), we first note thatfor Gaussian pulses the coupling term ˙ φ sin θ scales withthe relative pulse delay τ , see Eqs. (65) and (66). Onthe other hand the effective coupling terms Ω su ( t ), Ω sv ( t )and Ω uv ( t ) all scale with the relaxation rate γ . Then, inthe weak dephasing regime these latter coupling termsare negligible compared to ˙ φ sin θ . Using this approx-imation the equations for the coherences u ( t ) and v ( t )become ˙ u ( t ) ≈ − Γ u ( t ) u ( t ) + 2 v ( t ) ˙ φ sin θ, ˙ v ( t ) ≈ − Γ v ( t ) v ( t ) − u ( t ) ˙ φ sin θ. (62)Furthermore, we have that | Γ s ( t ) | ≫ | Ω su ( t ) | , | Ω sv ( t ) | . (63)With this the evolution of s ( t ) and consequently that ofthe dark states populations, is dominated by the effectiverelaxation rate Γ s ( t ),˙ s ( t ) ≈ − Γ s ( t ) s ( t ) . (64)Thus, in the weak dephasing regime coherences obeythe dynamics of a two-level system Eq. (62), while thepopulations decay exponentially at a rate Γ s ( t ) (64).Although analytic solutions could be derived for Eq.(62) following a similar method to that of Sec. IV A,the complexity of the final expressions limits their use.Nevertheless, useful conclusions can be drawn by sim-ple inspection of these equations. When the relaxationrate γ is increasing, and for fixed delays, the coherencesand the populations will decay. Taking into accountthat Γ s ( ±∞ ) = Γ u ( ±∞ ) = 0, Γ v ( ±∞ ) = γ and thatΩ ij ( ±∞ ) = 0, we anticipate that in the long time limit, v ( t → ∞ ) = 0, whereas both s ( t ) and u ( t ) acquire a finiteconstant value. Thus in the long time limit coherencesare partially preserved.The effect for different delay times on the system dy-namics is more complicated. This is due to the depen-dence that both the coupling ˙ φ sin( θ ) and the effective re-laxation rates have with respect to the delay time. Tak-ing into account the results from the previous sectionIV A and those for STIRAP in Λ-systems [13], we expectthat the dependence of the fidelity with respect to thepulse delay will reflect the dependence of the transitiontime T tr ( ǫ ) on the pulse delay. V. NUMERICAL SIMULATIONS
We present now results from numerical simulationswith the master equation (16) and for Gaussian pulses. For overlapping control and Stokes pulses, Ω c ( t ) = Ω s ( t ),we use the pulses given in Eq. (44). For the pulse order-ing where the Stokes pulse proceeds and ends after thepump pulse, while the control pulse is delayed relative toboth pulses, the parametrization readsΩ p ( t ) = Ω e − ( t − τ/ /T , Ω s ( t ) = Ω e − ( t − τ/ / (2 T ) , Ω c ( t ) = Ω e − ( t + τ/ / (2 T ) . (65)For the pulse ordering Stokes-control-pump we haveΩ p ( t ) = Ω e − ( t + τ/ /T , Ω s ( t ) = Ω e − ( t − τ/ /T , Ω c ( t ) = Ω e − t /T . (66)For the ordering control-Stokes-pump, the pulses are thesame as above with the only change that the Stokes andcontrol pulses are interchanged, i.e. Ω s ( t ) → Ω c ( t ) andΩ c ( t ) → Ω s ( t ). A. Overlapping Stokes and control pulses
Starting with a pulse ordering where the control andStokes pulse overlap, i.e. Ω s ( t ) = Ω c ( t ), in Fig. 3 we plotthe populations ρ jj ( ∞ ) for the bare states ψ j , and fordifferent relaxation rates γ ij = γ . The validity of theanalytic results of SecIV A is confirmed, where we seethat the results almost coincide. The expected popula-tion damping for the dark states due to losses towardsthe bright states, is evidenced as a population drop forthe states ψ and ψ . This is accompanied from popula-tion rise for the other two states ψ and ψ . We also notethe predicted asymptotic behavior for strong dephasing,where ρ jj ( ∞ ) → / γT ≪ /γ ,i.e. T ≪ t max ≪ /γ , the fidelity (30) reads F ( t max ) = ρ a ( ∞ ) + Re { ρ a ( t max ) } , (67)where ρ a ( ∞ ) is given by Eq. (59b) and Re { ρ a ( t max ) } is derived from Eq. (59a), with the substitutionlim t →∞ e − γt → e − γt max . In Fig. 4, the fidelity is plotted against the relaxation rate γ ij = γ for the Gaussian pulses (44). The dotted line isthe fidelity F ( t max ) for t max = 5 T , where an exponentialdrop for increasing γ is noted. More specifically, it canbe shown that for γT /τ ≪ F ( t max ) ≈ (cid:18) e − csγT τ (cid:19) + 12 e − cuγT τ − γt max . (68) P opu l a ti on s g (units of T − ) r , r r , r FIG. 3. (Color online) Final populations ρ jj ( ∞ ) for the barestates ψ j plotted against the dephasing rate γ ij = γ for Gaus-sian pulses Eq. (44). The peak Rabi frequency is Ω = 50 T − ,and the delay is τ = 1 . T . The dots are numeric results andthe solid lines were obtained from Eqs. (27) and (59b), andwith the aid of Eqs. (18), (19) and (25). On the other hand for strong dephasing γT /τ ≫
1, orin the long time limit t ≫ /γ , the system completelydecoheres and the only contribution to the fidelity is fromthe population ρ a ( ∞ ) Eq. (59b), i.e. F ( ∞ ) = ρ a ( ∞ ) . (69)For this limit the final fidelity is well below unity, seesolid line, and it rapidly drops for increasing dephasingrate.As already noted in Sec. IV A the delay τ is expectedto have an inverse effect on the fidelity. This is due tothe inverse dependence of the transition time Eq. (61)with respect to the delay. This is demonstrated in Fig. 5.The fidelity for different peak Rabi frequencies is plottedagainst the delay for γ ij = γ = 0, Fig. 5(a), and for γ ij = γ = T − , Fig. 5(b). From this we see that foradiabatic evolution the fidelity increases as Eqs. (69)and (59b) predict (dashed). The regime of validity forthese equations increases for increasing pulse areas, andthis is because of the adiabatic condition. B. Stokes-control-pump pulse ordering
When considering a pulse ordering where the Stokesproceeds the control and the pump is delayed furtherfrom the control, Eq. (66), the exact form of the result-ing coherent state (13) will be a function of the delay τ viathe geometric phase ϑ g (8). This means that the inverseeffect that the increasing delay τ has on the fidelity willreflect differently upon different coherent states. This isshown in Fig. 6, where the fidelity for different coherentstates (different τ ) as obtained from numerical simula-tions is plotted against the dephasing rate γ ij = γ . It is F i d e lit y g (units of T − ) F ( ¥ ) F ( T ) FIG. 4. (Color online) The fidelity for the coherent superpo-sition state (15) as a function of the dephasing rate γ ij = γ for the Gaussian pulses (44). The peak Rabi frequency isΩ = 50 T − and the delay is τ = 1 . T . The dots are numericresults, whereas the solid line is the analytic result for strongdephasing or for t ≫ /γ . The dotted line is the fidelity forthe weak dephasing regime (67), and for t max = 5 T . clear that as we increase the delay time and for a given γ , the fidelity for the corresponding stateΨ( ∞ ) = − ψ sin ϑ g ( τ ) − ψ cos ϑ g ( τ ) , (70)will be higher than the fidelity for a state that corre-sponds to smaller τ . This, as already said, is because ofthe inverse dependence of the transition time (31) on thedelay τ , see Fig. 7. As the relaxation rate increases, thefidelity for all the coherent states decreases reaching thestrong dephasing asymptotic F = 0 . C. Stokes proceeds and ends after the pump pulse
In contrast to the previous two pulse orderings, thisone differs in the effect that the delay τ has on the fi-delity. Dephasing has the same effect where the fidelitydrops exponentially for increasing γ ij = γ , see Fig. 8. Inaddition to this increasing the delay will also result in afidelity reduction. The explanation for this is again givenin terms of the transition time and its dependence withrespect to the delay.As already pointed out in Sec. III C the fidelity forthe coherent state (12) and for t → −∞ is F ( −∞ ) =cos ϑ g . As the geometric phase increases, i.e. the de-lay increases, the initial value for the fidelity decreases.Thus, for increasing τ the transition time is expected toincrease too. This is shown in Fig. 9, where the fidelity asa function of time and for different delays in the absenceof decoherence, Fig. 9(a), and the corresponding transi-tion times, Fig. 9(b), are plotted. As we can see in Fig.9(a) the initial value for the fidelity reduces for increasing0 F i d e lit y Delay (units of T )10 30 100 3001000 (a) F i d e lit y Delay (units of T )10 30 100 3001000 F ( ¥ ) (b) FIG. 5. (a) The fidelity for the coherent state (15) plottedagainst the delay time τ for the Gaussian pulses (44), asobtained from simulations with the master equation. Thepeak Rabi frequencies Ω are denoted next to the respectivecurves and γ ij = γ = 0. (b) The same with (a) but for γ ij = γ = T − . The dashed line is the analytic result ob-tained from Eqs. (69) and (59b). delays. For this reason the transition time will increase,Fig. 9(b), and consequently the effect of dephasing onthe fidelity increases. VI. CONCLUSION
In this paper we have studied the effects that dephas-ing has on STIRAP in tripod configurations. We havederived an exact adiabatic solution for the master equa-tion in the case of overlapping Stokes and control pulses,and for weak dephasing. The results were verified withnumerical simulations to provide a very accurate approx-imation for the fidelity dynamics. In the adiabatic limit,population losses and dephasing for the dark states de-pend only on the relative relaxation rates of the three F i d e lit y g (units of T − ) J g = . ( t = . T ) J g = . ( t = . T ) J g = . ( t = . T ) FIG. 6. (Color online) The fidelity for the coherent state (13)as a function of the dephasing rate γ ij = γ for the Gaussianpulses (66). The peak Rabi frequency is Ω = 200 T − andthe delay is τ = 1 . T (solid) τ = 1 . T (dashed) and τ =2 . T (dotted). The results were obtained from a numericalsimulation with the master equation (16). T r a n s iti on T i m e ( un it s o f T ) Delay (units of T ) FIG. 7. The transition time T tr ( ǫ ) (31) for the Gaussianpulses (66). The peak Rabi frequency is Ω = 200 T − and ǫ = 0 .
1. The times t ǫ and t − ǫ , were derived by numericallysolving the equations F ( t ǫ ) = ǫ and F ( t − ǫ ) = 1 − ǫ . bare ground states, but not on those for the intermediatestate.The fidelity exponentially decreases for increasing de-phasing, whereas the pulse delay has an inverse effect.This is due to the fact that the transition time decreasesfor increasing delay. This way dephasing effects are sup-pressed. Using numerical simulations we extended ourstudies to different pulse orderings. For a pulse orderingof the form Stokes-control-pump, or when interchangingthe order of the control and Stokes pulses, similar dynam-ics are observed as with overlapping Stokes and control1 F i d e lit y g (units of T − ) J g = . ( t = . T ) J g = . ( t = . T ) J g = . ( t = . T ) FIG. 8. (Color online) The fidelity for the coherent state (12)as a function of the dephasing rate γ ij = γ for the Gaussianpulses (65). The peak Rabi frequency is Ω = 200 T − andthe delay is τ = 0 . T (solid) τ = 1 . T (dashed) and τ =1 . T (dotted). The results were obtained from a numericalsimulation with the master equation (16). pulses. The fidelity decrease exponentially with the de-phasing, whereas the delay has again an inverse effect.This will reflect differently on each coherent state. Thisis because the geometric phase, which characterizes thefinal superposition, is a function of the delay.For a pulse ordering where the Stokes pulse proceedsand ends after the pump pulse, while the control pulsefollows the delay has a different effect. The reason forthis is that the transition time increases with the delay.Dephasing has again the same effect leading to an expo-nential decrease for the fidelity. ACKNOWLEDGMENTS
This work has been supported by the European Com-mission’s ITN project FASTQUAST and the BulgarianNSF grant No D002-90/08.
Appendix A: Deriving the equations for thetwo-level system
From Eqs. (17), (18), (19), (20), and (25) we can derivethe following equation for the population inversion w = ρ a − ρ a ˙ w ( t ) = (cid:0) γ sin ( θ ) + cos ( θ )Γ ( φ ) (cid:1) w ( t ) , (A1a)whereΓ j ( φ ) = cos ( φ ) γ j + sin ( φ ) γ j . (A1b)Taking into account the initial condition ρ a ( −∞ ) = ρ a ( −∞ ) = 0, or w ( −∞ ) = 0, we see that that the F i d e lit y Time (units of T )(a) J g = . ( t = . T ) J g = . ( t = . T ) J g = . ( t = . T ) T r a n s iti on T i m e ( un it s o f T ) Delay (units of T )(b) FIG. 9. (Color online) (a) The fidelity for the coherent state(12) as a function of time for the Gaussian pulses (65), asobtained from simulations with the master equation. Thepeak Rabi frequency is Ω = 200 T − , whereas the delay is τ = 0 . T (solid) τ = 1 . T (dashed) and τ = 1 . T (dotted).The relaxation rates are γ ij = γ = 0. (b) The transitiontime T tr ( ǫ ) Eq. (32) for ǫ = 0 .
1. The times t ǫ and t − ǫ ,were derived by numerically solving the equations F ( t ǫ ) =(1 + ǫ ) cos ( ϑ g ) and F ( t − ǫ ) = 1 − ǫ . inversion is w ( t ) = 0, i.e. ρ a = ρ a . Making next thesubstitution ρ a = ρ a = (1 − ρ a − ρ a ) / , (A2)we get the following master equation for the effective two-level system spanned by the two dark states (2)˙ ρ akl = −D ijkl ρ aij − i ˙ φ sin( θ )[ σ z , ρ d ] kl − D kl , (A3a)where ρ d is ρ d = (cid:18) ρ a ρ a ρ a ρ a (cid:19) , (A3b)2and σ z is the relevant Pauli matrix. In this equa-tion the Einstein summation convention is used, where i, j, k, l = 1 ,
2. The tensor D ijkl and the matrix D ij re-sulted when using the rotation matrix R ( t ) (19) to trans-form the master equation (16) in the adiabatic basis Eq.(20). At this point their exact analytic form is not impor-tant. Instead some very useful properties that are goingto be used next are listed below( D ij ) ∗ = D ji , (A4a) D = D , D = D , (A4b) D = D , D = D , (A4c) D = D , D = D , (A4d) D = ( D ) ∗ = D = ( D ) ∗ , (A4e)Im {D } = −
12 Im {D } , (A4f)Re {D } = 12 Re {D } , (A4g) D = D = −
14 ( D + D ) , (A4h) D = ( D ) ∗ = − D . (A4i)Having these properties we can now proceed and de-rive the equations for the population inversion w a ( t ) = ρ a − ρ a , the populations ρ a and ρ a , and the coherences u ( t ) = √ { ρ a } and v ( t ) = √ { ρ a } . Starting fromthe inversion w a ( t ), is easy to show that˙ w a ( t ) = ( D − D ) w a ( t ) . (A5)Taking into account the fact that θ ( −∞ ) = 0 and ρ ( −∞ ) = 1, we have that ρ a ( −∞ ) = ρ a ( −∞ ) = 1 / w a ( −∞ ) = 0. From the above equation we havethat at all times w a ( t ) = 0 (A6)and that ρ a = ρ a = 14 − s ( t )2 . (A7)With this result the remaining three equations for s ( t ), u ( t ) and v ( t ) (28) are derived, with the initial conditionsbeing s ( −∞ ) = − / u ( −∞ ) = 1 / √ v ( −∞ ) = 0.The effective relaxation rates and Rabi frequencies readΓ s ( t ) = D + D = 12 sin (2 θ )Γ ( φ ) + 12 cos ( θ ) sin (2 φ ) γ , (A8a)Γ u ( t ) = D + Re {D } = 14 sin (2 θ )Γ ( φ ) + 14 (cid:0) ( θ ) (cid:1) sin (2 φ ) γ , (A8b)Γ v ( t ) = D − Re {D } = cos ( θ )Γ ( φ ) + sin ( θ ) cos (2 φ ) γ , (A8c) andΩ su ( t ) =Re {D } = 14 sin (2 θ )Γ ( φ ) −
14 cos ( θ ) (cid:0) ( θ ) (cid:1) sin (2 φ ) γ , (A9a)Ω sv ( t ) =Im {D } = −
14 cos ( θ ) sin( θ ) sin(4 φ ) γ + 12 cos ( θ ) sin( θ ) sin(2 φ )( γ − γ ) , (A9b)Ω uv ( t ) =Im {D } = −
116 (sin(3 θ ) − θ )) sin(4 φ ) γ −
12 cos ( θ ) sin( θ ) sin(2 φ )( γ − γ ) . (A9c) Appendix B: Adiabatic integrals
In order to obtain s ( ∞ ) and u ( ∞ ), we first use theinverse rotation R − ( ∞ ) (38) on the state ψ ( ∞ ) (50)to get c ( ∞ ) and c ( ∞ ) (36). Using the expressions for c ± ( ∞ ) and Eq. (36) we have for s ( ∞ ) s ( ∞ ) ∝ exp( I s ) (B1a)where I s is I s = (cid:18)Z −∞ ( ǫ + ( t ) − Γ s ( t )) dt + Z ∞ ( ǫ − ( t ) − Γ s ( t )) dt (cid:19) (B1b)and for u ( ∞ ) u ( ∞ ) ∝ exp( I u ) = exp (cid:18)Z ∞−∞ ( ǫ + ( t ) − Γ s ( t )) dt (cid:19) . (B1c)Although the expressions for ǫ ± ( t ) and Γ s ( t ) are verycomplicated, they can all be parametrized in terms ofsingle dimensionless variable x = 4 tτ /T , i.e. ǫ ± ( t ) = γg ± ( x ) , (B2a)andΓ s ( t ) = γg s ( x ) , (B2b)where g − ( x ) = 4( e x − e x )(2 + e x ) (1 + p / (1 + e x ) ) , (B2c) g + ( x ) = (1 − e x )(1 + p / (1 + e x ) )2(2 + e x ) , (B2d)3and g s ( x ) = 2(1 + 2 e x )(2 + e x ) . (B2e)With this the integral for I s in Eq. (B1a), takes theform I s = γT τ Z −∞ ( g + ( x ) − g s ( x )) dx + γT τ Z ∞ ( g − ( x ) − g s ( x )) dx, (B3)where both integrals converge, and can be calculated nu-merically giving I s = − c s γT τ , (B4)where c s = 2 .
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