PROCESO DE ENSEÑANZA APRENDIZAJE

Building upon the Gross-Pitaevskii equations (3.30) or (3.32) to account for the quantum and thermal fluctuations of the quantum field ˆΨLP is a very arduous task

Figure 3.7: Mean field OPO phase diagram. Blue dashed-stable pump-only states. Blue solid-unstable pump-only states. Purple-Population of signale mode. Black vertical-Upper and lower threshold, same lines as in Figure 3.6.

tackle this problem in different regions, with the Path Integral Monte Carlo methods for the thermal equilibrium state [95], diagrammatic techniques [96, 97] density ma- trix renormalization group techniques for the both the ground state and the temporal dynamics [98] amongst the most representative examples.

In this section we will review some phase space techniques originally devel- oped in quantum optics and recently employed in the study of quantum fluids of atoms and photons. The crux is to represent the state of the quantum field as a quasi-probability distribution function on an appropriate classical phase space, and describe the time-evolution of the field in terms of a Fokker-Planck partial differ- ential equation (FPE) which can be subsequently mapped onto a stochastic partial differential equation. An introductory treatment of the quantum field phase space representations can be found in [99].

A crucial feature is the positivity of the quasi-probability distribution func- tion and the time-evolution be described by a FPE in view of efficient numerical simulations [100] ∂P(r, t) ∂t =− M X i=1 ∂ ∂ri [Fi(r, t)P(r, t)] + 1 2 M X i,j=1 ∂2 ∂ri∂rj [Di,j(r, t)P(r, t)], (3.43)

quasi-probability distribution P(r, t) on a M-dimensional space with real spatial co-ordinates ri. The probability distributions for complex quantities are straight-

forwardly formulated by treating the real and imaginary parts as independent real variables. For a positive definite diffusion matrix D, the FPE can be mapped onto a system ofM stochastic differential equations according to the rules of Ito calculus [101] as below

dri=Fi(r, t)dt+dWi (3.44)

with a Wiener noise subject to

dWidWj =Di,j(r, t)dt. (3.45)

These equations can be efficiently simulated by considering the statistical average over many different realizations of the Brownian motion.

In most cases of actual interest this is unfortunately not possible either due to a non positive-definite diffusion matrixDor due to the presence of additional terms with higher order partial derivatives of P in the right-hand side of (3.43): for the system of interacting bosons we are here investigating, non-positive diffusion terms appear in the time-evolution of P and Q representations, while third order derivative terms feature in the time-evolution of the Wigner function [102].

In the following we will concentrate on the Wigner representation that has proven to be most useful in calculations of practical interest. For simplicity, we restrict ourselves to the coherent pumping case for which there exists a self-contained Hamiltonian description [82]. In order to extend to the incoherent pumping case we are in need of some modelling for the relaxation mechanisms taking place in the experimental device: the first endeavour in that direction has been recently reported in [103]. Typical applications of the Wigner representation require the quantum field discretisation on addimensional discrete lattice ofNdpoints confined inside an integration box of sideL. In this configuration, the Wigner distribution is a function of the Nd complex amplitudes ψi=ψ(ri) of the field at the lattice positions ri. Its

time-evolution is governed by the FPE-like equation

∂W ∂t = − X i ∂ ∂ψi [Fi{ψ}W{ψ}]− X i ∂ ∂ψ_i∗[Fi{ψ}W{ψ}] + κLP ∆V ∂2W{ψ} ∂ψ∗_i∂ψi +i gLP 4∆V2 ∂2 ∂ψ∗_i∂ψi ∂ ∂ψ_i∗(ψ ∗ iW{ψ})− ∂ ∂ψi (ψiW{ψ}) (3.46)

where ∆V = (L/N)d is the volume of the elementary cell of the discrete lattice. The drift force term on the ri site Fi{ψ} = Fi{ψ}(r = ri) involves a

deterministic evolution of the field very similar to the right-hand side of (3.30) or (3.32).

We define the derivatives with respect to the complex field variable ψas

∂ ∂ψ = 1 2 ∂ ∂Re[ψ]−i ∂ ∂Im[ψ] , ∂ ∂ψ∗ = 1 2 ∂ ∂Re[ψ]+i ∂ ∂Im[ψ] .

The second-order derivative term is always positive and can be straightforwardly mapped onto a noise term with local spatial correlations. On the other hand, the third order derivative terms can not be included in a standard stochastic differential equation of the form (3.45).

The main practical interest of the Wigner representation arises from the dif- ferent scaling of the various terms featuring in (3.46) within the dilute gas limit

ψ → ∞, gLP → 0 at a constant interaction energy gLP|ψ|2. In this limit, the

presence of noise causes a statistical fluctuation of the fieldψ around its mean field value, on the order of ∆ψ∝1/√∆V. The characteristic magnitude of the third order derivative term is roughly estimated accordingly to be on the order ofgLP/∆V. This

estimate is compared to the diffusion term which is of the order ofκLP. Provided

κLP gLP/∆V, (3.47)

we anticipate that accurate results for a nonequilibrium gas of photons can be ob- tained by truncated Wigner calculations. In this procedure we completely neglect third order derivative terms and as a result the stochastic partial differential equation can be written in the form:

dψ(r, t) =F{ψ}(r)dt+

r

κLP

2∆VdW(r, t), (3.48)

involving a complex Gaussian noise termdW of zero mean that satisfies

hdWi(r, t)dWj∗(r0, t)i=dtδr,r0. (3.49)

Under the condition (3.47), the non-classical correlations due to the third- order derivative term decay fast owing to the losses and the classical noise associated withκLP: in particular, the magnitude of the classical noise is the “appropriate” one

how the truncated Wigner approach has been proven able to capture the quantum fluctuations at least at the level of Bogoliubov theory.