Thermal-Light-Induced Coherent Dynamics in Atoms and Molecules -- an Exact Quantum Mechanical Treatment
TThermal-Light-Induced Coherent Dynamics in Atoms and Molecules –an Exact Quantum Mechanical Treatment
David Avisar † , Asaf Eilam, and A.D. Wilson-Gordon Department of Chemistry, Bar-Ilan University, Ramat Gan 5290002, Israel (Dated: September 22, 2018)The question of whether sunlight induces coherent dynamics in biological systems is under de-bate. Here we show, on the basis of an exact fully quantum mechanical treatment, that thermal lightinduces excited-state coherences in matter similar to those induced by a coherent state. We demon-strate the phenomenon on a V -type model system and a two-state Born-Oppenheimer molecularsystem. Remarkably, wavepacket-like dynamics is induced in the excited molecular potential-energysurface. PACS numbers: 33.80.-b, 42.50.-p, 42.50.Ct, 42.50.Md
Coherence is a fundamental and central theme in quan-tum mechanics and is manifested in light-matter interac-tion through a variety of unique phenomena [1], includingquantum control [2], storage and retrieval of information[3], and quantum state reconstruction [4, 5]. In recentyears, there has been a lively debate on whether sunlight-induced chemical reactions in biological systems evolvecoherently and consequently exhibit extremely efficientenergy transfer. Experimental evidence for vibrationaland electronic coherence in such systems has been re-ported in several studies [6–13]. However, it has been ar-gued that since such studies employ coherent laser pulsesrather than sunlight (so-called thermal or chaotic light)to initiate material dynamics, the results obtained can-not establish conclusively that sunlight-induced biologi-cal processes evolve coherently [14].Theoretical studies of the excitation of atomic andmolecular systems by thermal light have reached contra-dictory conclusions as to whether the induced dynamicsis coherent. Most of these studies are approximate insome sense, as they involve either semiclassical or per-turbative treatments, and some of them rely on the pres-ence of an intrinsic coupling between the material excitedstates, such as vacuum- or decay-induced coherence [15–22]. Others, treat the material system only following anassumed coherent excitation [23]. None of these studiesemploy an exact and fully quantum mechanical treatmentof the interaction of thermal light with matter withoutsuch limiting constraints. Interestingly, the entanglement(correlations) between separated material sites has alsobeen studied in this context [24, 25].Since the interaction of thermal light with matter isa fundamental issue of enormous potential implications,the need for a definite conclusion on whether thermallight can induce coherent material dynamics is of pri-mary importance and significance. In order to reach adefinite conclusion, we treat the problem starting frombasic considerations which, to the best of our knowledge,has never been done before.We employ an exact (non-perturbative) dynamicaltreatment for the interaction of a material system (atomic and molecular) with a thermal state of the elec-tromagnetic field, where both the material and light sub-systems are quantized. Concretely, we treat the light-matter system as a bipartite composite system whosesubsystems interact via a Jaynes-Cummings-type [26] in-teraction model. We thus express the initial system byits density matrix, ρ (0) = ρ F (0) ⊗ ρ M (0) (where F and M stand for ‘field’ and ‘material’, respectively), and fullypropagate it in time according to ρ ( t ) = U ( t ) ρ (0) U † ( t ),where U ( t ) is the system propagator. In the case we con-sider, the material subsystem is initially in its groundstate, while the field is in a single-mode statistical mix-ture of Fock states.As we describe below in detail, we apply the interactionmodel to a three-level V -type system as a basic model,and, for the first time to the best of our knowledge,to a two-state Born-Oppenheimer molecular system forwhich we propagate the full density matrix, representedin the electronic ⊗ bond-coordinate ⊗ photon-Fock prod-uct space. To conform with the conditions of a naturalprocess in our simulations, we consider the weak-fieldregime and initial average photon number characteristicof sunlight (about 500nm wavelength at 6000 K, [27]).We find that thermal light induces excited-state coher-ent dynamics in both the atomic and molecular subsys-tems, as it also does in the regime of strong field andhigh average photon number. We compare this resultwith that obtained for the interaction of the materialsystems with a coherent-state. Remarkably, we find thatboth scenarios exhibit almost identical excited-state co-herence features. On the other hand, in contrast to acoherent-state, thermal light does not induce coherencebetween the material ground and excited states.As a prototype for a molecular system, we consider athree-level V -type model system interacting with a singlemode of the radiation field. The Jaynes-Cummings-type a r X i v : . [ qu a n t - ph ] M a y system Hamiltonian is H = (cid:88) i ω i | i (cid:105)(cid:104) i | + ω (ˆ a † ˆ a + )+ (cid:88) i (cid:54) = g λ i (cid:0) | i (cid:105)(cid:104) g | ˆ a + | g (cid:105)(cid:104) i | ˆ a † (cid:1) , (1)where we have invoked the dipole and rotating-wave ap-proximations, and set (cid:126) = 1. The index i stands forthe ground and two excited states of the V system, g, e and f , respectively. The operators ˆ a † and ˆ a are the pho-ton creation and annihilation operators, respectively, op-erating on the photon states {| n (cid:105)} . The parameter λ i is the matter-field interaction constant. Note that thismodel excludes any intrinsic coupling between the ex-cited states.The material system is initially in its ground state ρ M (0) = | g (cid:105)(cid:104) g | . For the field we consider two possibleinitial states: a coherent state and a thermal state. A co-herent state takes the form | α (cid:105) = e − | α | (cid:80) ∞ n =0 α n √ n ! | n (cid:105) ,where | α | = ¯ n is the field average photon number. Thecorresponding density matrix is ρ F (0) = | α (cid:105)(cid:104) α | . For thethermal state ρ F (0) = (cid:80) n p n | n (cid:105)(cid:104) n | , with p n = ¯ n n (1+¯ n ) n +1 ,and the average photon number ¯ n , given by the Boltz-mann distribution [27]. Thus, the initial state of thecomposite system is ρ (0) = ρ F (0) ⊗ ρ M (0), and the stateof the system at any time in its evolution is given by ρ ( t + ∆ t ) = e − iH ∆ t ρ ( t ) e iH ∆ t . (2)The state of each subsystem is obtained by tracing ρ ( t )over the space of the other subsystem; that is, ρ A ( t ) = Tr B [ ρ ( t )] . (3)In the supplemental material, we give an analytical ex-pression for ρ ( t ) for the V system, as well as for the co-herence element of the material subsystem for the fullyresonant case where ω − ω e,f = 0. Furthermore, we com-pare the analytical and numerical calculations in orderto justify the numerical simulations presented here.We now compare the V system dynamics induced bya coherent state with that induced by a thermal state.For both scenarios, the energy separation between the | e (cid:105) and | f (cid:105) states is set to 250 cm − , and the field fre-quency is tuned between them. We set λ e,f = 10 − , and¯ n = 0 .
008 (obtained for 495.9 nm and temperature 6000K for the field mode). We construct the initial statesof the light with the first ten Fock states. In Fig. 1,we show the material excited states populations (toppanel), the real and imaginary parts of the material ex-cited states coherence element, ρ M,fe ( t ) (middle panel),and the trace of the material (and field) reduced den-sity matrix, Tr [ ρ M,F ( t )], and of its square, Tr (cid:2) ρ M,F ( t ) (cid:3) (bottom panel), as obtained for the interaction with thecoherent state. In Fig. 2, we present the same dynam-ical measures for the interaction of the V system with thermal light; in the lower panel, however, we show onlyTr (cid:2) ρ M ( t ) (cid:3) and Tr (cid:2) ρ F ( t ) (cid:3) which are generally differentfor a mixed state. −3 T r[ | i 〉 〈 i | ρ ( t )] −1−0.500.51 x 10 −3 ρ M , f e ( t ) Tr[|f 〉 〈 f| ρ (t)]Tr[|e 〉 〈 e| ρ (t)]ReImTr[ ρ M,F (t)]Tr[ ρ M,F2 (t)]
FIG. 1: (color online). Populations (top), coherences (middle)and partial traces (bottom) obtained for the interaction of a coher-ent state with a three-level V -type system. Figures 1 and 2 show striking similarity between thematerial populations and, most remarkably, the excitedstate coherence for the coherent and thermal light sce-narios. In this context, it is interesting to refer to thesimilarity in the interaction of a coherent and thermalstates with a two-level system as noted by Cummings[28]. In contrast, Tr (cid:2) ρ M ( t ) (cid:3) and Tr (cid:2) ρ F ( t ) (cid:3) show dif-ferent behavior. In the mixed state scenario (interactionwith the thermal light), the two oscillate out-of-phase (asif purity is exchanged between the subsystems), while inthe pure scenario (interaction with the coherent state)both are identical, as is well known for a pure system.We note that throughout the interaction of the thermallight with the V system, ρ F ( t ) remains diagonal as weshow analytically in the supplemental material.Another major difference between thermal and coher-ent state of the light is that while the latter induces co-herence between the material ground and excited states,the former does not. (We derive this result analytically inthe supplemental material). We explain this phenomenon −3 T r[ | i 〉 〈 i | ρ ( t )] −1−0.500.51 x 10 −3 ρ M , f e ( t ) Tr[|f 〉 〈 f| ρ (t)]Tr[|e 〉 〈 e| ρ (t)]ReImTr[ ρ M2 (t)]Tr[ ρ F2 (t)] FIG. 2: (color online). Populations (top), coherences (middle)and partial traces (bottom) obtained for the interaction of thermallight with a three-level V -type system. in the following way. According to our model, the inter-action of a single Fock state, say | n (cid:105) , with the mate-rial ground-state | g (cid:105) produces the general superposition a | g, n (cid:105) + ( b | e (cid:105) + c | f (cid:105) ) | n − (cid:105) , which in the material sub-space (that is, after tracing over the field subspace) cor-responds to a mixed state of the material ground and ex-cited states. However, the material excited-states are ina coherent superposition with coherences between them.Thermal light is a statistical mixture of single Fock statesand, therefore, induces only excited-state coherences inthe material subsystem. The absence of coherence be-tween the ground and excited states is related to thevanishing of the material dipole moment in this scenarioas pointed out in [29]. In contrast, since a coherent stateis a coherent superposition of Fock states, its interactionwith the material ground-state will generate coherencebetween the material ground and excited states (sincethey share common photon states), and also among theexcited states themselves. Note that a Schr¨odinger cat(coherent) state [27] is expected to behave just like athermal state, as noted elsewhere [30].The material excited-state coherence, induced by thethermal light, is obtained through an exact quantum mechanical treatment without any intrinsic coupling be-tween the excited states. Such couplings were introducedin a number of previous studies, and were necessary toallow the material excited-state coherence. Our exacttreatment proves such coherence is obtained naturallyvia the interaction with thermal light, similarly to thatobtained in the interaction with a coherent state. In thesupplemental material we show that similar dynamicalcharacteristics are obtained for the interaction of a the V system with two-mode thermal (and coherent) state.A realistic molecular system is by far more significantfor understanding matter interaction with thermal light.We therefore apply a similar fully exact quantum me-chanical treatment to a molecular system of two Born-Oppenheimer potential energy surfaces (PESs). Specif-ically, the field is represented in Fock space, and themolecular system is spanned by the two-dimensional elec-tronic space and the continuous coordinate space for thenuclei separation. Thus the state of the system is givenby the tensor product of these three spaces.The Hamiltonian of the entire system is H mol = (cid:88) i = g,e H i ( r ) | i (cid:105)(cid:104) i | + ω (ˆ a † ˆ a + )+ λ (cid:0) | e (cid:105)(cid:104) g | ˆ a + | g (cid:105)(cid:104) e | ˆ a † (cid:1) , (4)where H i ( r ) = V i ( r ) + T is the nuclear Hamiltonian ofthe ground and excited electronic states ( | g (cid:105) and | e (cid:105) ),with the PES V i ( r ) and the kinetic energy operator T = − m ∇ . The parameter m is the molecular reducedmass. We denote the (vibrational) eigenstates of H i ( r )by ψ i,ν ( r ), where ν is the vibrational quantum number.In writing the Hamiltonian of Eq. 4 we invoke the dipole,Condon, and rotating-wave approximations.We examine the molecular dynamics induced byexcitation with both thermal and coherent states,as before. The initial molecular (ground) state is ρ M (0) = | g, ψ g, (cid:105)(cid:104) g, ψ g, | , and the initial state of thefield ρ F (0) is either a coherent or thermal state, asdescribed above. The initial state of the system is ρ (0) = ρ F (0) ⊗ ρ M (0), and its state at any time in thecourse of interaction is ρ ( t + ∆ t ) = e − iH mol ∆ t ρ ( t ) e iH mol ∆ t . (5)It is convenient to employ the “split operator” method[31] in propagating ρ , and split the kinetic en-ergy term of the Hamiltonian from the rest of theterms. Thus, for each time-step the propagator is e − iH mol ∆ t ≈ e − iT ∆ t e − i (cid:101) H mol ∆ t , where (cid:101) H mol is the fullHamiltonian of Eq. 4 without T . In practice, we sim-ulate the interaction of thermal light with a two-stateBorn-Oppenheimer model system of two one-dimensionalMorse-type PESs, V i ( r ) = D i (1 − e − b i ( r − r i ) ) + T i (thepotentials parameters, based on these of the Li molecule,are given in the supplemental material). The frequencyof the exciting field is tuned between the third and fourthvibrational eigenstates of the excited state. The initialstate ρ F (0) = (cid:80) n p n | n (cid:105)(cid:104) n | is constructed with the firstseven Fock states. We set λ = 10 − , and ¯ n = 0 . ρ M,ee ( t ). In the leftcolumn we show ρ M,ee ( t ) in the coordinate representa-tion, while in the right column we show its projectiononto the first 15 excited vibrational eigenstates. The ˚ A −5 ˚ A −3 ˚ A −4 ˚ A −4 ˚ A ˚ A −5 ν (vib. quantum number) FIG. 3: (color online). Snapshots of ρ M,ee ( t ) for thermal lightinteracting with two-state Born-Oppenheimer molecule. The leftcolumn shows ρ M,ee ( t ) in coordinate space (with the time specifiedin femtoseconds), and the right column shows its projection ontothe vibrational eigenstates of the molecular excited-state. clear observation that emerges from these snapshots isthat thermal light induces excited-state vibrational co-herent dynamics. An almost identical coherence fea- ture is obtained by the interaction of a coherent statewith the molecular system (not shown here). The spa-tial representation indicates wavepacket-like dynamics,unlike that reported recently on the basis of a semi-classical treatment [20]. (When the field is resonantwith a vibrational eigenstate of the excited-state, thestate created, both by the interaction with coherent andthermal states, is the corresponding vibrational eigen-states). The difference between the interaction withcoherent and thermal state is that the former also in-duces coherence between the ground and excited elec-tronic states, while the latter does not, as discussed abovefor the V system. To have a quantitative measure of thewavepacket-like dynamics, we calculate the excited-statecorrelation function C ( t ) = Tr [ ρ M,gg (0) ρ M,ee ( t )] and itsFourier-transform σ ( ω ), shown in the supplemental ma-terial. The spectral signature is characterized by the har-monics of the energy difference between the excited-statevibrational eigenstates ω ex ≈ . V -type system and in atwo-state Born-Oppenheimer molecular systems. Theinduced coherence is similar to that induced by a co-herent state. Remarkably, the thermal light is shown toinduce wavepacket-like dynamics in the molecular sys-tem. Unlike a coherent state, however, the thermal lightdoes not induce coherence between the material groundand excited states. Most significantly, and in contrastto previous studies, the generation of the excited-statecoherences in our model requires no intrinsic couplingbetween the excited states. In general, thermal light ex-cludes coherence between the states it couples directly,and permits coherence between material states which arenot coupled directly by the field or by any other couplingmechanism. Interestingly, a Schr¨odinger cat (coherent)state is expected to induce coherences similar to thermallight. On the basis of recent studies [21], we expect thatthe interaction of the system with environment would noteliminate the induced coherence. In addition, based onour two-mode calculations, we expect that a continuous-mode of thermal light, interacting with matter, shouldinduce similar type of coherences.The results we present in this Letter may suggest asignificant role that a statistical source of modes canhave in coherent dynamics, as was pointed out in asimilar context [23]. Consequently, our results mayhave direct impact on the understanding the dynamicalcharacteristics of sunlight-induced biological processes. † Author to whom correspondence should be addressed,[email protected] [1] M. O. Scully and M. S. Zubariry,
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SUPPLEMENTARY MATERIALAnalytical dynamical expressions for the interactionof the V system In the interaction picture, the Hamiltonian for the V -type three-level system, assuming ω − ω e,f = 0 and acommon parameter λ , is given by H I = λ (cid:88) i = e,f (cid:0) | i (cid:105)(cid:104) g | ˆ a + | g (cid:105)(cid:104) i | ˆ a † (cid:1) (6)Employing Taylor expansion, we get the following ma-trix for the propagator e − iH I t in the three-dimensionalmaterial space e − iH I t = ˆ C f ( t ) ˆ C fe ( t ) ˆ S fg ( t )ˆ C ef ( t ) ˆ C e ( t ) ˆ S eg ( t )ˆ S gf ( t ) ˆ S ge ( t ) ˆ C g ( t ) , (7)with the matrix elements given byˆ C f ( t ) = ˆ C e ( t ) = 12 (cid:104) cos( √ a ˆ a † λt ) + 1 (cid:105) ˆ C fe ( t ) = ˆ C ef ( t ) = 12 (cid:104) cos( √ a ˆ a † λt ) − (cid:105) ˆ S fg ( t ) = ˆ S eg ( t ) = − i √ √ a ˆ a † λt ) √ ˆ a ˆ a † ˆ a ˆ S gf ( t ) = ˆ S ge ( t ) = − i √ √ a † ˆ aλt ) √ ˆ a † ˆ a ˆ a † ˆ C g ( t ) = cos( √ a † ˆ aλt ) . (8)The operations of each of the propagator matrix elementson Fock states are thenˆ C f ( t ) | n (cid:105) = 12 (cid:104) cos( (cid:112) n + 1) λt ) + 1 (cid:105) | n (cid:105) ≡ C f,n ( t ) | n (cid:105) ˆ C fe ( t ) | n (cid:105) = 12 (cid:104) cos( (cid:112) n + 1) λt ) − (cid:105) | n (cid:105) ≡ C fe,n ( t ) | n (cid:105) ˆ S fg ( t ) | n (cid:105) = − i √ √ nλt ) | n − (cid:105) ≡ S fg,n ( t ) | n − (cid:105) ˆ S gf ( t ) | n (cid:105) = − i √ (cid:112) n + 1) λt ) | n + 1 (cid:105) ≡ S gf,n ( t ) | n + 1 (cid:105) ˆ C g ( t ) | n (cid:105) = cos( √ nλt ) | n (cid:105) ≡ C g,n ( t ) | n (cid:105) . (9)For the initial state ρ (0) = | g (cid:105)(cid:104) g | ⊗ ρ F (0) of the com-posite system, the density matrix at any time t of theevolution, in the interaction picture, is ρ I ( t ) = − ˆ S fg ρ F (0) ˆ S ge − ˆ S fg ρ F (0) ˆ S ge ˆ S fg ρ F (0) ˆ C g − ˆ S fg ρ F (0) ˆ S ge − ˆ S fg ρ F (0) ˆ S ge ˆ S fg ρ F (0) ˆ C g − ˆ C g ρ F (0) ˆ S ge − ˆ C g ρ F (0) ˆ S ge ˆ C g ρ F (0) ˆ C g , (10)where we have omitted, for convenience, the explicit timedependence of the matrix elements of Eq. 7.First we derive an expression for the coherence elementbetween the ground and excited state for the case of a co-herent state of the field. We then show that this elementis identically zero for the thermal state of the field. Letus consider the element − ˆ C g ρ F (0) ˆ S ge . For the coherentstate of the field we plug ρ F (0) = | α (cid:105)(cid:104) α | into this elementand trace over the field space. As a result, we get ρ M,gf ( t ) = Tr F (cid:104) − ˆ C g ( t ) | α (cid:105)(cid:104) α | ˆ S ge ( t ) (cid:105) = − (cid:88) m (cid:104) m | ˆ C g ( t ) | α (cid:105)(cid:104) α | ˆ S ge ( t ) | m (cid:105) = −(cid:104) α | ˆ S ge ( t ) ˆ C g ( t ) | α (cid:105) = ie −| α | (cid:88) n α n ( α ∗ ) n +1 (cid:112) n !( n + 1)! sin (cid:104)(cid:112) n + 1) λt (cid:105) × cos( √ nλt ) . (11)If we plug into − ˆ C g ρ F (0) ˆ S ge the expression for the ther-mal state of the field, (cid:80) n p n | n (cid:105)(cid:104) n | , and trace over thefield space, the coherence element ρ M,gf ( t ) is identicallyzero (as well as for the eg element): ρ M,gf ( t ) = Tr F (cid:34) − ˆ C g ( t ) (cid:88) n p n | n (cid:105)(cid:104) n | ˆ S ge ( t ) (cid:35) = − (cid:88) n,m p n (cid:104) m | ˆ C g ( t ) | n (cid:105)(cid:104) n | ˆ S ge ( t ) | m (cid:105) = − (cid:88) n p n (cid:104) n | ˆ S ge ( t ) ˆ C g ( t ) | n (cid:105) = 0 (12)The coherence element ρ M,fe ( t ) is obtained by tracingover the field subspace of the matrix element ρ I,fe ( t ): ρ M,fe ( t ) = Tr F (cid:104) − ˆ S fg ( t ) ρ F (0) ˆ S ge ( t ) (cid:105) = − (cid:88) m (cid:104) m | ˆ S fg ( t ) ρ F (0) ˆ S ge ( t ) | m (cid:105) . (13)For a coherent state, we plug ρ F (0) = | α (cid:105)(cid:104) α | (with | α (cid:105) = e − | α | (cid:80) ∞ n =0 α n √ n ! | n (cid:105) ) into Eq.13 and get the fol-lowing expression for the excited-state coherence element ρ M,fe ( t ) = 12 e −| α | (cid:88) n | α | n n ! sin ( √ nλt )= 12 e − ¯ n (cid:88) n ¯ n n n ! sin ( √ nλt ) . (14)For the case of thermal field, with ρ F (0) = (cid:80) n p n | n (cid:105)(cid:104) n | ,we get ρ M,fe ( t ) = 12 (cid:88) n p n sin ( √ nλt )= 12 (cid:88) n ¯ n n (1 + ¯ n ) n +1 sin ( √ nλt ) . (15)In addition, we can also show that during the interac-tion of the thermal light with the V system, the reduceddensity matrix of the light, ρ F ( t ), remains diagonal. Toshow that we trace ρ I ( t ) of Eq. 10 over the material sub-space, and use the the equalities of Eq. 9. The resultobtained is ρ F ( t ) = Tr M [ ρ I ( t )] = (cid:88) i (cid:104) i | ρ I ( t ) | i (cid:105) = − S fg ( t ) ρ F (0) ˆ S ge ( t ) + ˆ C g ( t ) ρ F (0) ˆ C g ( t )= (cid:88) n p n sin ( √ nλt ) | n − (cid:105)(cid:104) n − | + (cid:88) n p n cos ( √ nλt ) | n (cid:105)(cid:104) n | , (16)which represents a diagonal reduced density matrix ρ F ( t ).We use the above analytical expressions for the ex-cited states coherence to compare with our numerical re-sults and verify our numerical simulations for the den-sity matrix dynamics. In Fig. 4 we show the real partof the excited state coherence element of the density ma-trix ρ M,fe ( t ), obtained for the field initially in a coherentstate. In blue is the result obtained from the numericalpropagation, whereas in red is the result obtained usingEq. 14. In Fig.(5), we show the comparison of the an- t(fs) ρ M , f e ( t ) NumericalAnalytical
FIG. 4: (color online.) Real part of ρ M,fe ( t ) obtained numer-ically, after Eq. 2 of the main text (blue) and analytically, afterEq. 14, (red) for the interaction of a coherent state with a V sys-tem. alytical (following Eq. 15) and numerical calculation forthe excited state coherence obtained for the thermal field.Note the remarkable similarity of the results obtained forthe coherent state of the light and the thermal state!Figures (4) and (5) confirm unequivocally our numericalpropagation for the density matrix. t(fs) ρ M , f e ( t ) NumericalAnalytical
FIG. 5: (color online.) Real part of ρ M,fe ( t ) obtained numer-ically, after Eq. 2 of the main text (blue) and analytically, afterEq. 15, (red) for the interaction of the thermal light with a V sys-tem. Interaction of V system with two-mode field. Here we show that, similar to the interaction of asingl-mode states of the light, the interaction with two-mode states of a coherent state and thermal state, induceexcited-state coherence. −3 T r[ | i 〉 〈 i | ρ ( t )] −3 ρ M , f e ( t ) ReImTr[|f 〉 〈 f| ρ (t)]Tr[|e 〉 〈 e| ρ (t)]Tr[ ρ M,F (t)]Tr[ ρ M,F2 (t)]
FIG. 6: (color online.) Populations (top), coherences (middle)and partial traces (bottom) obtained for the interaction of a two-mode coherent state with V system. In Fig. 6 we show the dynamical measures, as discussedin the main text, of the interaction of a V system withtwo-mode of a coherent state. In Fig. 7 we show the samedynamical measures obtained for the interaction of a V system with two-mode thermal state of the light −3 T r[ | i 〉 〈 i | ρ ( t )] −3 ρ M , f e ( t ) Tr[|f 〉 〈 f| ρ (t)]Tr[|e 〉 〈 e| ρ (t)]ReImTr[ ρ M2 (t)]Tr[ ρ F2 (t)] FIG. 7: (color online.) Populations (top), coherences (middle)and partial traces (bottom) obtained for the interaction of a two-mode theraml light with V system. Potentials parameters
Below are the potentials parameters used for the two-state Born-Oppenheimer molecular dynamics. The PESsare essentially the X and A states of the Li molecule,but only with the A state shifted in energy [1]. TABLE I:
The parameters, in atomic units, for PESs used for themolecular simulations. V g V e D b r T Spectrum of the excited-state coherent dynamics