Periodic modulation

We can now compute the vacuum persistence amplitude by numerically solving Eqns. (6.21) and (6.22) using a 4^th-order Runge-Kutta method, and from 1 − |h0|0i|² extract the probability to excite quanta in the system. Inspired by Chapter 3,⁷ we will consider a time-dependent resonance frequency of the form

Ω²(s) = Ω²₀



1 + [α1cos ν1s + α2cos ν2s] e⁻[^s2/(2σ2)]^N

, (6.38)

where α_i and ν_i are the amplitudes and frequencies of the modulation. Also, σ denotes the duration of the interval in which we modulate the medium, and N is the order of super-Gaussian used (in order to have sharp turn-on/off times). An example of this can be seen in Fig. 6.3(a) for the medium parameters Ω₀ = 3 and g = 2, which is modulated using α₁ = α₂ = 0.1 for the duration σ = 100 at frequencies ν₁ = 1 and ν₂ = 0.8. Here we have chosen N = 10.

This modulation of the medium resonance frequency produces vacuum radiation, the spectrum of which can be seen in Fig. 6.3(b) (solid blue). At a maximum rela-tive modulation amplitude of 0.1, it can be argued to be slightly non-perturbarela-tive.

Interestingly, the character of the spectrum is identical to the result of the one found in Chapter 3 [Fig. 3.4, p. 67], where we discussed frequency mixing processes for vacuum radiation at a perturbative level. Note that the spectrum has multiple resonances at mixed modulation frequencies.

As can be seen in Fig. 6.3(b) (dashed green), when we increase the modulation amplitude to α₁ = α₂ = 0.3, further resonances become visible in the spectrum.

This is not captured qualitatively by 2^nd-order perturbation theory, they are higher-order mixing processes, similar to a nonlinear optics spectrum [53]. Note that for this choice of modulation frequencies (ν₁ and ν₂), we access mostly resonances with the ω− branch. For example, quanta are excited when ω−(k) = ν₁/2. This is the case for both amplitudes chosen. Physically, this means that we are only driving

7In particular Eq. (3.49) along with Eq. (3.66).

Figure 6.3: (a) Modulation of the medium resonance frequency. Here we use Ω = 3 and g = 2, and modulated using α₁ = α₂ = 0.1 for σ = 100 with N = 10, with ν₁ = 1 and ν₂ = 0.8. (b) Vacuum radiation spectrum produced by a time-dependent medium. Here two different amplitudes were used, α₁ = α₂ = 0.1 (solid blue) and α₁ = α₂ = 0.3 (dashed green). The mixing processes have been labelled accordingly.

For this choice of ν₁ and ν₂, we mainly access resonances related to the ω− branch and have therefore omitted the polariton frequency label unless necessary. In other words, the quanta is excited when for instance ω₋(k) = ν₁/2.

the photon-like degree of freedom. It is interesting to note that we would expect a larger response from the matter-like degree of freedom, as we are modulating the matter parameters.

6.5.1.1 Classical parametric resonances

Differently to a nonlinear system, the quantum behaviour of the macroscopic QED system is solely determined by the classical solution. This follows from its linear-ity, as we discussed in Chapter 3. We should therefore be able to understand the resonances discussed in the previous section in the context of classical parametric resonances. This becomes apparent especially if we consider the classical equations of motion in Eq. (6.12), which can be rewritten as

¨

xk+ k²xk= −g ˙R,

R + Ω¨ ²[1 + f (t)] R = g ˙x_k, (6.39)

where we have ignored the equation of motion for y_k as it is uncoupled to matter (and its solution is simply that of a harmonic oscillator). Here, f (t) contains the time-dependence of the resonance frequency of the medium. The formalism of this chapter lends itself to the stability analysis commonly performed when considering parametric resonances.

As is outlined in a book by Arnol’d [152], we are interested in the stability of the matrix M that connects the system at time t = 0 to the later time t = τ , where τ is the period of the modulation, i.e. we look for the matrix M such that

ξ(τ ) = Mξ(0),

where ξ is given by Eq. (6.13). Interestingly, this matrix M is given by

M = E(τ ) F (τ ) G(τ ) H(τ )

!

, (6.40)

which has already been used extensively. Suppose that we consider a single-frequency modulation such that f (t) = α cos(νt). Then the equation of motion in Eq. (6.39) is the same at times t = 0 and t = 2π/ν. Therefore τ = 2π/ν.

From Ref. [152], we learn that this map is stable if all eigenvalues of M are complex, and can therefore be reduced to a rotation, whereas it is unstable if at least one of the eigenvalues is real. As was noted in Ref. [147], the eigenvalues of M are given by exp (±iω±τ ) for a general period τ . This follows because ±iω± are the eigenvalues of

0 I

−A² −B

!

and from Eq. (6.17) we find that exp (±iω±τ ) are the eigenvalues of M. Thus, we see that at least one of the eigenvalues of M is real when exp (±i2πω±/ν) = ±1, that is, when

ω± = nν 2

for n = Z. From this we find that we expect quantum vacuum radiation to be emitted when

ω± = ν/2, ν, 3ν/2, 2ν, 5ν/2, ...

at decreasing amplitude. Interestingly, it is not straightforward to analyse the two-frequency modulation in Eq. (6.38) within this framework. This is because the two-frequency modulation is not strictly periodic in either τ = 2π/ν₁ or τ = 2π/ν₂. 6.5.1.2 Fock-Darwin spectrum

Whilst it is possible to understand the spectrum of emitted quantum vacuum radi-ation in terms of classical parametric resonances, another interpretradi-ation might be more natural. In this chapter, we have shown that the dynamics of macroscopic quantum electrodynamics can be directly linked to the dynamics of a harmonically trapped particle in a magnetic field. It follows that it must be possible to decompose any wavefunction into a weighted superposition of the energy state wavefunctions.

Now, the energy state space of a trapped particle in a magnetic field is well-studied, and its spectrum is often called the Fock-Darwin spectrum [144, 145]. As is also pointed out in Ref. [147], the states have energies

Emn=

 m + 1

2

 ω++

 n + 1

2

 ω−,

where m and n are positive integers. Therefore, after the modulation of the reso-nance frequency the final state wavefunction can be decomposed as

Ψ(q) =X

m,n

c_mnΨ_mn(q)e^−iEmn,

where Ψ_mn(q) is the wavefunction for the state with energy E_mn. Here we only have two indices m and n, as compared to three in Eq. (6.24) as we ignore the y_k degree of freedom, seeing that it is decoupled from the rest of the system. From this we can conclude that quantum vacuum radiation must fulfil

∆E = energy supplied, mω₊+ nω−= energy supplied,

Figure 6.4: Spectrum of a quench.

where m and n are the number of ω₊–polaritons and ω−–polaritons respectively.