Magnus Expansion Approach to Parametric Oscillator Systems in a Thermal Bath

1 Introduction

The study of periodically driven systems has experienced renewed interest in recent times. Both in solid state and ultra cold atom systems, strong periodic driving has been used to control nonequilibrium states. In ultra-cold atom systems, periodic lattice driving has been used to realise an effective, synthetic gauge field, see [1]. In solid-state systems, pump–probe experiments [2], on high-T_c superconductors and on graphene have been performed, see [3], [4], [5]. Theoretical studies on light-induced superconductivity were reported in [6], [7], [8], [9], [10], [11].

Remarkably, in both cases, external high-frequency driving is used to control the low-frequency behaviour of each system. The quintessential example for this phenomenon is the Kapitza effect [12]. In the case of the effective synthetic field in an ultra-cold atom system, this process is explicitly described by an approximate, effective low-energy Hamiltonian, which, in contrast to the original, nondriven Hamiltonian, has a synthetic field. In the case of a driven high-T_c superconductor, the near-resonant driving of an optical phonon mode results in a modified response in the low-frequency optical conductivity. Both of these observations exemplify the development of a new field of emergence in driven many-body systems.

In this article, we give a systematic expansion of the emergent low-energy description of a driven system. This discussion applies and extends the Magnus formalism, as discussed in [13], [14], [15], [16]. Our formalism provides a systematic expansion both in the driving amplitude and in the inverse driving frequency and is applicable to closed and open classical systems, to closed quantum systems. We derive explicit, general expressions for the leading terms beyond second order. As a key example, we apply this formalism to a parametrically driven oscillator, coupled to a thermal bath [17], [18], and determine the properties of its steady state. An insightful discussion of parametric oscillators was given in [19], [20], as well as in [21]. We then apply our results to a chain of parametrically driven oscillators. This provides insight in how the dispersion of a system can be controlled via parametric driving.

This article is organised as follows: In Section 2, we describe the dissipatively coupled, parametrically driven oscillator, and give a discussion of its properties using elementary ansatz functions. In Section 3, we develop the Magnus expansion in full generality first and then apply it to the parametric oscillator in Section 4. In Section 5, we discuss the control of the dispersion of a parametrically driven chain of oscillators, and in Section 6, we conclude.

2 Parametric Oscillator

As the key example to which we apply the Magnus expansion, we consider a parametrically driven oscillator, described by the Hamiltonian

with

p and x are the momentum and spatial coordinate of the oscillator, m the mass, and ω₀ the bare oscillator frequency. A is the amplitude of the parametric driving term and ω_m the driving frequency.

We assume that this oscillator is coupled to a thermal bath of temperature T, via a dissipative term. The resulting equations of motion are of the Langevin form:

γ is the damping rate, and ξ describes white noise, with the correlation function 〈ξ(t₁)ξ(t₂)〉=2γk_BTmδ(t₁−t₂), where k_B is the Boltzmann constant. In thermal equilibrium, in the absence of driving, the system is described by the canonical distribution ρ₀(x, p)=exp(−βH₀(x, p))/Z, with β=1/(k_BT). Z is the partition function, which normalises this probability distribution. For this distribution, the variances of x and p are 〈x2〉=xT2 and 〈p2〉=pT2, with pT=mkBT and xT=kBTmω02. Furthermore, we have 〈x〉=〈p〉=〈xp〉=0. We note that for a classical oscillator, x_T and p_T can be used to rescale x and p. With this choice, the temperature does not appear in any of the remaining quantities and simply provides an energy scale for the system. For a quantum mechanical oscillator, this rescaling cannot be performed. Here, an additional regime appears in which quantum fluctuations dominate, for k_BT=ћω₀. A full discussion of the driven, dissipative quantum mechanical oscillator will be given elsewhere. The analysis presented here addresses isolated quantum systems, in addition to dissipative classical systems.

In Figure 1, we depict the time averaged variance 〈x(t)²〉, of the steady state of the driven system, as a function of A and ω_m/ω₀. Here, and in the examples throughout this article, we choose γ/ω₀=0.1. The most striking feature of this plot is the parametric resonance that appears near ω_m≈2ω₀, for small A. This feature is then power broadened for increasing A. In this regime, the magnitude of 〈x(t)²〉 is increased by orders of magnitude, compared to the equilibrium value. In addition to this strong heating effect, there is a regime for large ω_m/ω₀, and large amplitude, for which a reduction of 〈x(t)²〉 is observed. Here, the parametric driving leads to a dynamic stabilisation of the fluctuations of x. It is this counterintuitive and quintessential example of reducing fluctuations via high-frequency driving that we study systematically in this article.

Figure 1:

We depict the time averaged magnitude of 〈x2(t)〉/xT2 in the steady state as a function of driving frequency and driving amplitude. Panels (A) and (B) depict the same data on different scales. The system displays a power broadened instability emerging from ~2ω₀, and a dynamical stabilisation for large ωm and A. In panel (A), we show the comparison to (8), in panel (B) to (7).

In Figure 2, we show the same quantity in the steady state as a function of A, for a fixed value of the driving frequency, ω_m/ω₀=20, to give a clearer insight into the quantitative behaviour. The magnitude of these fluctuations is visibly reduced with increasing driving amplitude. However, eventually this trend of decreasing fluctuations is rapidly reverted, resulting in a steep increase of the fluctuations. As visible from Figure 1, this steep increase is due to the power-broadened parametric instability. The onset of this instability determines the location of the minimal amount of fluctuations that can be achieved with this type of driving. It is therefore imperative to understand the origin of this steep increase of the fluctuations and provide a systematic approach to determine its behaviour.

Figure 2:

We depict the time averaged expectation value of 〈x²〉, in units of xT2, as a function of the driving amplitude A, for the driving frequency ω_m/ω₀=20, and for γ/ω₀=0.1. We compare the numerically obtained result to the prediction in (10). The dashed, vertical line corresponds to (8).

In Figure 3, we show a histogram of the distribution ρ_dr in the steady state, in comparison to the equilibrium distribution ρ₀; we show ρ_dr−ρ₀. The distribution ρ_dr is generated from trajectories of the Langevin equation, which have been low-frequency filtered via xc(t)=∫dsGσ(s−t)x(t), and similarly for p_c(t), derived from p(t). G_σ(s) is a normalised Gaussian, with a time scale σ, for which we choose σ=1/ω₀. In Figure 3, furthermore, we choose A=10 and ω_m/ω₀=20. We observe that the width of the distribution along x-direction is reduced, due to the dynamical stabilisation that is described in the following sections. Along the p-direction, the distribution is only weakly affected. We emphasise that for a quantitative comparison of the driven state to the effective, low-frequency predictions, the exclusion of the high-frequency contributions in the numerics is essential. We elaborate on this point in Appendix A.

Figure 3:

Distribution in phase space of the driven system in the steady state. The trajectories of the time evolution have been smoothed out on a time scale of σ=1/ω₀. We use ω_m/ω₀=20, A=10, and γ/ω₀=0.1. The binning size is Δx/x_T=Δp/p_T=0.01. The reduction of the width of the distribution in the x-direction is clearly visible.

3 Magnus Expansion

We now turn to the Magnus expansion of the system. This expansion provides a time-independent approximation of the low-frequency sector of the system, derived from the original, time-dependent Hamiltonian that describes all frequencies. After deriving general expressions for the Magnus terms beyond second order, we ask the question if and how the key features of the parametric oscillator, the dynamical stabilisation and the instability regime, can be captured within this approach. We note that these features, as they were described in the previous section, might suggest that such an approach might not be possible in a consistent fashion for the fourth-order correction. This is due to the following two observations. We observed, as shown in (9), that the fourth-order correction has a positive prefactor, which results in an addition stabilisation of the oscillator. This term would be derived from a term in an effective Hamiltonian that is of the form ~A4/ωm6, with a positive prefactor. On the other hand, if the instability of (8) is derived from an effective Hamiltonian, it also needs to be derived from a term of the form ~A4/ωm6, but now with a negative prefactor.

Interestingly, as we discuss below, the Magnus expansion provides two types of terms at fourth order. One of them is cut-off independent and features a positive prefactor. The resulting renormalisation due to this term is in agreement with the numerically obtained result. The other term is cut-off dependent. It indicates that the hierarchy of time scales that is required for the Magnus expansion breaks down. We interpret this as a precursor of the instability regime, and indeed find that the scaling for this regime, as shown in (8), is predicted correctly.

4 Magnus Expansion of the Parametric Oscillator

We now apply our results to the case of the parametric oscillator, introduced above. For the Leff(2,2) correction, we use (26) and find

This implies a renormalisation of the oscillator frequency of the form

This coincides with the second-order term that was obtained in (10). At the fourth in the inverse driving frequency, we have

where we applied (29). Interestingly, in addition to a further renormalisation of the oscillator frequency, a renormalisation of the temperature is created:

It is generated because the white-noise dissipative term contains fluctuations at all frequencies, in particular at the driving frequency ω_m. This results in an additional renormalisation of the low-frequency regime, via time averaging, of the system at this higher order. For the example presented here, the magnitude of the renormalisation is small. However, nonlinear systems will in general create nonlinear effective dissipative terms at this order. Finally, we determine the two terms at order A⁴. The cut-off independent term is

This term generates an additional renormalisation of the oscillator frequency, resulting in

We note that this renormalisation at fourth order in A has a positive prefactor, as in the estimate in (9). However, the systematic Magnus expansion gives the correct magnitude of the prefactor.

In Figure 4, we depict the power spectrum Sp(ω) of the momentum p in steady state, as a function of the driving amplitude A, and for the fixed driving frequency ω_m/ω₀=20. The power spectrum is defined via

Figure 4:

Power spectrum S_p(ω) as a function of the driving amplitude A, depicted on a logarithmic scale. For the driving frequency, we use ω_m/ω₀=20. We show the second-order estimate of the effective frequency, ω_eff,2, which refers to (37). In addition, we show the fourth-order estimate ω_eff,4, based on (42).

with p(ω)=(1/Ts)∫dt′exp(−iωt′)p(t′), where T_s is the sampling interval. At A=0, the power spectrum reduces to that of a harmonic oscillator, with a single peak at ω₀. As the driving is turned on, additional peaks appear at nω_m±ω₀, where n is an integer describing the Floquet band. We note that these frequencies are approximately the ones that were used in the ansatz functions in Section 2.1 and Appendix B. With increasing driving amplitude, the effective oscillator frequency increases. We compare this increase to the second-order prediction, given in (37), and the fourth-order prediction (42). The fourth-order estimate describes the oscillator frequency well almost up to the instability, which is reached around A≈180, in this example. We emphasise that the orange bar at A≈180 is numerical data. Here, the magnitude of power spectrum increases rapidly by many orders of magnitude.

The cut-off dependent term is

This term competes with the previously discussed terms that stabilise the oscillator. For simplicity, we only consider the dominant term of the effective frequency (36). We relate the time scale Δt_c to a frequency cut-off via Δt_c=2π/ω_c. We assume to be in the strongly renormalised regime, ωeff2/ω02≈A2ω022ωm2. Therefore, Leff,c(4,6) competes with this renormalisation if

This results in the criterium

If we consider a cut-off frequency chosen as fraction of the driving frequency, and therefore ω_c~ω_m, we recover

which displays the same scaling as in (8). The scaling displayed in (46) can also be motivated by comparing the cut-off frequency ω_c to ω_eff/ω₀~Aω₀/ω_m. Again, this condition indicates that the originally assumed hierarchy of energy scales is no longer valid. This property of the system derives from the cut-off dependent term Leff,c(4,6). While this term in itself cannot be interpreted as a contribution to the effective equation of motion, it can give an insight into the breakdown of the necessary hierarchy of time scales of the system.

5 Parametrically Driven Chain

We apply this formalism to the stabilisation of a chain of oscillators via parametric driving. This, and related mechanisms have been considered in the context of light enhanced superconductivity, with the following motivation. If we imagine a complex order parameter field describing fluctuating superconducting order, a key feature of this system is its phase stiffness. In equilibrium, it controls the superconducting stability and the critical current. The phase stiffness in turn is related to how steeply the dispersion of the system increases with increasing momentum. Therefore, one possible explanation of light enhanced superconductivity might entail stabilising and steepening the dispersion of the system.

We here give the simplest, yet generic, case of a one-dimensional chain of oscillators. The system is described by H=H₀+H_dr(t), with

with i=1, …,N. The driving term is

We therefore have a parametrically driven lattice of oscillators. We Fourier transform the system via xi=1N∑kexp(ikri)xk and pi=1N∑kexp(ikri)pk, and note that [xk1, pk2]=iℏδk1,−k2. The Langevin equations for the system are

with 〈ξk1(t1)ξk2(t2)〉=2γkBTmδk1,−k2δ(t1−t2). In real space, this corresponds to 〈ξ_i(t₁)ξ(t₂)〉=2γk_BTmδ_ijδ(t₁−t₂). The dispersion ω_k,0 is

We observe that (50 and 51) are equivalent to (4 and 5), with the replacement ω₀→ω_k,0. We can therefore apply the results for the single oscillator (42), to each momentum mode and obtain the effective dispersion

The second-order correction, derived from Leff(2,2), contains contributions of the form ~cos(2k). This can be seen by substituting ω_k,0=2ω₀|sin(k/2)|, see (52). This describes coupling to the next-nearest neighbor, induced by the periodic driving, because a next-nearest coupling term of the form ∑ixixi + 2 gives rise to cos(2k) terms in momentum space when Fourier transformed. When substituting (52), in the term that is quadratic in A and quartic in ωm−1, we obtain terms up to ~cos(3k), which corresponds to coupling to the third neighbor. Finally, the term quartic in A contains coupling to the fourth nearest neighbor.

In Figure 5, we show the two point correlation Γ(k, ω) in momentum and coordinate space in the steady state for a one-dimensional chain of parametrically driven oscillators. The two point correlation function is defined by

Figure 5:

Two-point correlation function 𝒢(k, ω), depicted on a logarithmic scale. We use ω_m/ω₀=80 for the driving frequency with driving amplitude A=400. We compare the numerics with the analytical estimates (white lines), respectively, thermal situation (dotted line), second-order estimate (dashed line), and fourth-order estimate (solid line) of the effective frequency, ω_k,eff, which refers to (53).

with

X(k, ω)=1KsTs∫dt′∫dr′x(r′,t′)exp(−ikr′)exp(−iωt′)

where T_s and K_s are sampling time interval and space interval, respectively. Compared with the nondriven situation, the driven dispersion line has a steeper slope, which means that the driving term stiffens the system significantly. We compare the numerics with effective dispersion ω_k,eff, (53). It is clearly seen that at higher k modes, the second-order correction deviates from the numerics while the fourth-order correction describes the numerics precisely.

We also observe that the strongest renormalisation of the dispersion occurs at its upper edge. This includes the onset of the parametric instability. We now have the condition

which is first reached for the maximum of the band. This sets the upper limit for the driving amplitude A that can be used to stabilise the dispersion. However, we note that, depending on the physical system, the range of A might be much more limited. For example, the value of the spring constant between neighboring oscillators might not allow for negative values, meaning that A<1. With this constraint, the magnitude of the renormalisation is small, of the order of ωk,02/ωm2.

6 Conclusions

We have developed a systematic Magnus expansion in the driving term and the inverse driving frequency. In this formalism, we have derived explicit expressions for a system with a driving term with cosine time dependence. This system can be either a quantum mechanical system or a classical system including dissipative terms. The main, conceptual formulas are given in (29, 34, and 35), which are the terms beyond the widely discussed lowest order term in (26). At fourth order in the driving term, we find two contributions, one cut-off independent and one cut-off dependent. The cut-off independent term contributes to the effective Kramers or Hamilton operator, whereas the increasing magnitude of the cut-off dependent term indicates the breakdown of the hierarchy of time scales that was originally assumed. We apply this formalism to a parametrically driven oscillator, coupled to a thermal bath, and to a parametric oscillator chain. We obtain the magnitude of stabilisation that can be achieved for these systems and the onset of the instability. We emphasise that our formalism can be applied to a wide range of driven systems, including nonlinear systems and many-body systems. It will be of particular interest to the emerging field of controlling many-body systems via external driving.