PLANTEAMIENTO DE PROCEDIMIENTOS
5.4. PROCESOS DE VENTAS Y DEVOLUCIONES
5.4.7. Proceso de venta de mercadería a Sucursal Argentina.
-0.11 -0.09 -0.07 -0.05 0 0.1 0.2 0.3 0.4 F/N λ -0.138 -0.134 -0.13 -0.126 0 0.1 0.2 0.3 F/N λ
Figure5.5.: Top panels show the free versus order parameter for the ex- act solution (blue line) and ansatz (black) for two strengths of Huang-Rhys factor: (a)S=0.1 and (b)S=6. Other parameters:
Ω=0.05g0√N,∆=2g0√N,g= g0,µ=−2.1g0√N,T=0
relative displacement between the vibrational configurations in the electronic ground and excited states, i.e. αc ≃ −αd. Consequently, the effective ˜g is very suppressed and the gradient of bare g(ζ) is steep, implying that at small fields the optimal configuration is that of the normal state. At large fields, large ζ limit, it is energetically favourable to reduce the relative displacement thus increasing the overlap, i.e.αc ≃ αd. Therefore the origin of the discontinuous phase transition asgc
√
Nis increased can be explained as follows: At small bare couplings gc
√
N the effective ˜gc √
N is significantly reduced re- sulting in a weak polarisation of the system. However, at some point it becomes energetically favourable to pay the cost of the vibrational energy by having the same displacements in both electronic states (big overlap), as this in return inflates ˜gc and the polarisation energy. Consequently, a jump of the condensate density is observed.
The non-monotonicity of g(ζ) persists even when the chemical po- tential is faraway from ǫ. However, the first order phase transition is no longer observed. It is because at larger detunings only one so- lution to δ(ζ) is in the allowed region set by ζ ≥ |δ|, and as such the weakly polarised condensate disappears, and only the strongly polarised phase remains. Asµ moves away from ǫ the discountinu- ous boundary between the normal and the strongly polarised state becomes continuous.
5.5 comparison with the exact solution
&
con-
clusion
The comparison between the ansatz (black line) and exact solution (blue) for two different values of the Huang-Rhys parameter is de- picted in figure5.5. At smallS, panel (a), there is a perfect agreement between free energies coming from both approaches suggesting that either the ground state is a displaced coherent state for vibrational
modes, or that the effect of vibrational modes is very small and as such it is not important how these modes are modelled. At large S
the free energies still match in the normal state, but the variational solution deviates in the condensed region, panel (b).
In conclusion, the variational wavefunction which assumes a BCS like state for fermions (two-level system) as well as a coherent state for the photon field and the vibrational modes correctly predicted the shape of the phase diagram including the discontinuous phase transition between a strongly and a weakly polarised condensed states and the normal state over a narrow region close toµ=ǫ. It also provided us a means to understanding the origin of the suppression of the effective light-matter coupling. A variational wavefunction of this form has been shown to work best at small values of Huang-Rhys factor,S<1,
where there is little effect of vibrational modes and so it is likely to be of little consequence how these modes are treated — or indeed the vibrational excitations are described by a displaced coherent state. Regardless of the value of S, the ansatz always matches the exact solution in the normal state.
6
VA R I AT I O N A L M E T H O D : F I N I T E T E M P E R AT U R E
T
he variational approach outlined in the proceeding chapter suc- cessfully predicted the shape of phase boundaries at zero temper- ature, as well as explained the origin of suppression of light-matter coupling strength. Now we wish to extend this analysis to finite tem- peratures. In addition to carrying out an in-depth investigation of the physics, we shall also focus on understanding the drawbacks of our ansatz.We begin by constructing a variational state for the entire system. As before we treat the cavity photon within the mean-field theory. The effects of temperature come via the entropy of the system, leading to a thermal mixture of electronic populations in the ground and excited states as well as a thermal factor in vibrational modes. The variational ansatz for the density matrix can be expressed as
ρ=nhUρtls⊗ρvibU†
i
⊗no⊗ρphoton (6.1)
where the product is over each molecule n. A state of this form is explicitly a mean-field treatment for the photon field, i.e. a product state, however the entanglement between vibrational modes and elec- tronic states is introduced through the unitary polaron transformU=
exph√Sησz(b†−b)/2i [95, 96]. Such a transformation, when acting
on a product state, conditionally displaces the vibrational mode. In turn, the displacement is determined by the state of a two-level sys- tem. By using such a displacement one obtains a state which is effec- tively entangled in the original basis, with the entanglement (or the extent of polaron transform) quantified by a variational parameter η. As before, the variational ρphoton will be a coherent state with field λ.
The density matrix, which accounts for the thermal population of electronic states, has the form
ρtls= 1 ZtlsR e−βE 0 0 eβE ! R−1 (6.2)
where the partition function is Ztls = 2 cosh(βE). In order for the variational state to be as close to the real one as possible, we express the occupation of two-level systems with a variational parameter E. Hence, Ecan be understood as parametrising the occupation of two- state systems. The rotation matrixRis defined as
R= cos(θ) −sin(θ)
sin(θ) cos(θ)
!
(6.3)
withθbeing the rotation angle between the eigenvectors and the basis we work in.
The density matrix for a vibrational state is described by a thermal, coherently displaced state
ρvib =e−α(b−b†) 1 Zvib ∞
∑
n e−βνn|nihn|eα(b−b†) (6.4) with the partition function Zvib = 1−e−βν−1. The quantity α is a variational parameter which describes the vibrational displacement. Similarly to the two-level system case, we assign a variational pa- rameter to the vibrational energy ν. By doing so, we ensure a vari- ational thermal occupation. For comparison with the zero temper- ature variational calculation, the ground state wavefunction can be expressed |GSi = eλa†eα(b−b†)−η√S/2σz(b−b†)(cos(θ)| ↑i+sin(θ)| ↓i), where α = (αc+αd)/2 and η = (αc−αd)/√S in the notation in- troduced in the previous section.6.1 free energy minimisation
Having constructed the variational density matrix, we shall now find optimal values for variational parameters by minimising the free en- ergy. The variational free energy specifies the upper limit for the true free energy, i.e.F6 Fvar, where (¯h=1)
Fvar =Uρ−TSρ (6.5)
=TrHˆρ+kBT·Tr[ρlnρ]. (6.6) Here the internal energy of the system is Uρ, Sρ = −kBTr[ρlnρ]the entropy and the Hamiltonian ˆH = Nωc|˜ λ|2+Nh with the on-site
part h given by Eq. (4.2). Because of the way the vibrational den- sity is currently defined, finding the free energy would require tak- ing a trace of displacement operators. To simplify the analysis, we will instead redefine the vibrational density matrix to explicitly in- clude the displacement operatorsDρvibD†= DZ−1
vib ∑ ∞
n e−βνn|nihn|D† with D = e−α(b†−b). Subsequently, using the property of trace we transfer the displacement operators so as to shift the Hamiltonian
6.1 free energy minimisation 69
Tr[DρD†Hˆ] =Tr[ρD†HDˆ ]. For the entropy term one can employ the
property TrDρD†ln(DρD†) = Tr[ρlnρ]. Thus, from now on ρvib will refer to a thermal stateρvib = Z−1
vib ∑ ∞
n e−βνn|nihn|. Therefore, the energy of the system can be found from
Tr[Hˆρ] =Trhe−α(b†−b)UHeˆ α(b†−b)U†ρi (6.7) =Tr N ˜ ωc|λ|2+ ǫ˜ 2σ z+g√Nλσ+e√Sη(b†−b) +h.c. +Ωb†b−Ω(b†+b) −α+ √ S 2 σ z(η −1) ! +Ω α− √ S 2 σ zη !2 +Ω√S ασz− √ S 2 η ! [ρtls⊗ρvib]⊗ρphoton
which after taking the trace of a product state simplifies to
Tr[Hˆρ] =N ˜ ωcλ2−tanh(βE)cos(2θ) ˜ ǫ 2 +Ω √ Sα(1−η) (6.8) −g√Nλtanh(βE)sin(2θ)e−Sη2 1 2−1−1eβν +Ωα2+Ω 1 eβν−1 − ΩS 4 η(2−η) .
The entropic part of the free energy is determined by considering the contribution of all constituent subsystems, i.e. electronic states, vibra- tional states and the cavity photon mode. The mean-field treatment of the photon field indicates that the cavity state can either be in a vacuum or a coherent state, both of which are pure states, and con- sequently have no entropy, Tr[ρphotonlnρphoton] = 0. Therefore, the entropy of the system isSρ =Stls+Svib with (kB =1)
Stls =−
∑
n Tr[ρtlslnρtls] (6.9) =N(−βEtanh(βE) +ln[2 cosh(βE)]) Svib =−∑
n Tr[ρviblnρvib] (6.10) =N βν eβν−1−ln h 1−e−βνi, see appendix B.In this form, the variational free energy Fvar N = ωcλ˜ 2−tanh(βE)cos(2θ) ˜ ǫ 2+Ω √ Sα(1−η) (6.11) −g√Nλtanh(βE)sin(2θ)e−Sη2 1 2−1−1eβν +Ωα2+ Ω−ν eβν−1− ΩS 4 η(2−η) +Etanh(βE) − 1βln[2 cosh(βE)]−ln[1−e−βν]
needs to be minimised in order to find optimal values for variational parameters: E, ν, λ, θ, η, α. Minimisation with respect to the an- gle θ yields tan(2θ) = −gˆλ/δ, where the notation is defined in ac- cordance with the zero temperature variational approach as well as that outlined in Ref. [90]. The quantity ˆg refers to an effective light- matter coupling strength with the renormalisation coming through an exponential suppression due to vibrational overlaps between both electronic states (as inT =0 case) and also due to the thermal occupa- tion of that mode. The quantityδ is the effective molecular transition frequency shifted by the presence of vibrational modes. These two effective terms are defined as follows
δ = ǫ˜ 2 +Ω √ Sα(1−η) (6.12) and ˆ g=g√Ne−Sη2 1 2+eβν1−1 . (6.13)
Minimisation with respect to E yields the new energy which an ex- citon has in the presence of a photon field, i.e. E = pδ2+ (gˆλ)2. In
terms of this quantity, one can also express angles cos(2θ) = δ/E
and sin(2θ) = −gˆλ/E. Just like in the T = 0 case, the above terms depend on the vibrational displacement and the extent of polaron formation. By differentiating the free energy with respect to both quantities, then combining equations to eliminate αand solve for η, and subsequently for α one obtains optimal values, which in turn depend onδ and ˆg, η= Ω Ω+2 cothβν2 Eκ ( 6.14) α= √ Sδcothβν2κtanh(βE) Ω+2 cothβν2 Eκ (6.15) where we have defined
κ= (gˆλ)
2tanh(βE)