Nonlinear Dynamical Casimir Effect at Weak Nonstationarity

Nonlinear dynamical Casimir eﬀect at weak nonstationarity

Dmitrii A. Trunin∗1,2

1Moscow Institute of Physics and Technology, 141700, Institutskiy pereulok, 9, Dolgoprudny, Russia 2Institute for Theoretical and Experimental Physics, 117218, B. Cheremushkinskaya, 25, Moscow, Russia

August 18, 2021

Abstract We show that nonlinearities significantly affect particle production in the dynamical Casimir effect. To that end, we derive the effective Hamiltonian and resum leading loop corrections to the particle flux in a massless scalar field theory with time-dependent Dirichlet boundary conditions and quartic self-interaction. To perform the resummation, we assume small deviations from the equilibrium and employ a kind of rotating wave approximation. In addition, we confine our analysis to a quantum mechanical reduction and O(N) generalization of the effective Hamiltonian. In both cases, loop contributions to the number of created particles are comparable to the tree-level value. arXiv:2108.07747v1 [hep-th] 17 Aug 2021

∗[email protected] Contents

1 Introduction 1

2 Field quantization and Bogoliubov coeﬃcients3

3 Eﬀective Hamiltonian6

4 Reduction to quantum mechanics8

5 Large N generalization9

6 Discussion and Conclusion 12

A Alternative derivation of the created particle number 13

B Full expression for the interacting Hamiltonian 14

1 Introduction

The dynamical Casimir effect (DCE) describes particle production by nonuniformly accelerated mirrors [1–6] and aptly illustrates many prominent features of the quantum field theory. On the one hand, this effect is similar to the celebrated Hawking [7–9] and Unruh [10–12] effects. Essentially, all these phenomena are related to a change in the system’s quantum state due to the interactions with strong external fields; the only difference is the nature of the interactions and external fields. On the other hand, the DCE is amenable to a plethora of theoretical and experimental approaches, most of which are concisely described in reviews [13, 14]. In particular, recently, this effect was experimentally measured in a system of superconducting quantum circuits [15–17]. The mentioned properties make the DCE a perfect model for the study of nonstationary quantum field theory. Most of the research on the DCE, including experimental implementations, focuses on a sim- plified two-dimensional model with a free massless scalar field and perfectly reflecting boundary conditions (we set ~ = c = 1): 2 2 ∂t − ∂x φ(t, x) = 0, φ[t, L(t)] = φ[t, R(t)] = 0, (1.1) where mirror trajectories L(t) and R(t) are stationary in the asymptotic past and future, i.e., L(t) ≈ L± + β±t and R(t) ≈ L± + Λ± + β±t as t → ±∞. Here, Λ± > 0 is the distance between the mirrors, and β± is their velocity in the corresponding limits. Note that for physically meaningful trajectories L(t) < R(t) and |L˙ (t)| < 1, |R˙ (t)| < 1 for all t. Usually one also makes a Lorentz boost, shifts the origin and conveniently sets L(t < 0) = 0, R(t < 0) = Λ− without loss of generality. Due to the nonstationarity of the model (1.1), its Hamiltonian cannot be diagonalized once and forever, so the notion of a particle is different in the limits t → ±∞. In general, the modes and creation/annihilation operators that diagonalize the free Hamiltonian in the asymptotic past (in-region) and future (out-region) are related by a generalized Bogoliubov (or canonical) transformation [18–20]: out X ∗ in in ∗ fn = αnkfk − βnk(fk ) , k (1.2) out X in ∗ in † an = αknak + βkn(ak ) , k

1 out with nonzero Bogoliubov coeﬃcients αnk and βnk. Hence, the number of the out-particles (fn modes) created during the nonstationary evolution, in the Heisenberg picture, is given by the following expression (see AppendixA for an alternative derivation):

out † out X 2 X ∗ ∗ X X ∗ ∗ ∗ Nn = hin|(an ) an |ini = |βkn| + (αknαln + βknβln) nkl + αknβlnκkl + αknβlnκkl. k k,l k,l k,l (1.3) in † in Here, we introduce the short notations for the level density nkl = hin|(ak ) al |ini and anomalous in in quantum average (correlated pair density) κkl = hin|ak al |ini and schematically denote the initial quantum state (possibly mixed) as |ini. In the Gaussian approximation, i.e., in the absence of interactions and nonlinearities, particle creation in the DCE is related only to mixing of positive- and negative-frequency modes. Indeed, in assume that the initial quantum state of the system coincides with the vacuum, an |ini = 0 for all n. In this case, both nkl = 0 and κkl = 0, so the identity (1.3) signiﬁcantly simpliﬁes:

free X 2 Nn = |βkn| . (1.4) k

This identity also approximately holds at low temperatures: in this case, nkl is exponentially suppressed and κkl is zero, so they give negligible contributions to the right-hand side of (1.3). Thus, on the tree level, particle creation is fully determined by the Bogoliubov coeﬃcient βkn. For this reason, identity (1.4) is widely considered as the essence of the DCE, and the vast majority of literature — including seminal papers [1–6], reviews [13–15], and textbooks [18–20] — is purely free dedicated to the calculation of Nn and derivative quantities. Nevertheless, real-world systems are rarely free from interactions and nonlinearities. In such systems, approximation (1.4) is not applicable even if the initial values of nkl and κkl are small. In fact, in nonstationary situations, these quantities can receive nonnegligible loop corrections1:

† in † in nkl(t, t0) = hin|U (t, t0)(ak ) al U(t, t0)|ini, (1.5) † in in κkl(t, t0) = hin|U (t, t0)ak al U(t, t0)|ini, (1.6) where U(t, t0) denotes the evolution operator in the interaction picture, and t0, t set the moments when the interaction term is adiabatically switched on and off. In this case, identity (1.3) does not reduce to (1.4). Furthermore, loop corrections can be significant even if interactions are apparently weak. As was recently discovered, in many nonstationary interacting systems, nkl and/or κkl receive secularly (n) an (n) bn growing loop corrections, i.e., nkl ∼ (λt) and κkl ∼ (λt) with some constant powers an > 0 and bn > 0 in every order of the perturbation theory in λ. Such corrections indefinitely grow in the limit t → ∞ and are not suppressed even if the coupling constant goes to zero, λ → 0. For instance, the secular growth of loop corrections was observed in the de Sitter [28–33] and Rindler [34] spaces, strong electric [35–37] and scalar [38–40] fields, gravitational collapse [41], and nonstationary quantum mechanics [42, 43]. A short review of the origin and consequences of the secularly growing loops can be found in [44]. The DCE was also recently shown to possess secularly growing loop corrections [45,46]. Unfor- tunately, this analysis was restricted to the first few loops. At the same time, due to the secular

1Of course, in nonstationary systems, loop corrections should be calculated using a nonstationary (Schwinger- Keldysh) diagrammatic technique [21–27].

2 growth, high and low order loop corrections in this model are comparable at large evolution times. Therefore, a definitive conclusion about the destiny of nkl and κkl can be made only after a resummation of the leading contributions from all loops. In other words, the number of created particles (1.3) and other observable quantities in the nonlinear DCE have physical meaning only nonperturbatively. In this paper, we estimate a nonperturbative contribution to the number of created particles in the nonlinear scalar DCE: 2 2 3 ∂t − ∂x φ(t, x) = λφ (t, x), φ[t, L(t)] = φ[t, R(t)] = 0. (1.7) For simplicity, we restrict ourselves to small deviations from stationarity and assume that the initial state of the field coincides with the vacuum. First, this approximation simplifies the calculation of the Bogoliubov coefficients. Second, it allows us to use the method developed in [43] and resum the perturbative series of loop corrections to nkl and κkl. As a result, we obtain that the nonperturbative loop contributions to (1.3) in the approximation mentioned above are comparable to the tree-level approximation (1.4). This paper is organized as follows. In Sec.2, we discuss the quantization of a free scalar field and calculate the Bogoliubov coefficients in the model (1.1). In Sec.3, we derive the effective Hamiltonian of the nonlinear model (1.7). In Sec.4, we discuss a reduction to a quantum mechanical problem and calculate level density and anomalous quantum average in the low-particle limit. In Sec.5, we evaluate these quantities in a large N generalization of the model (1.7). Finally, we discuss the results and conclude in Sec.6. Besides, we explain the physical meaning of Nn in AppendixA and present the full expression for the interacting Hamiltonian of the nonlinear DCE in AppendixB.

2 Field quantization and Bogoliubov coeﬃcients

Consider a quantization of the free model (1.1) with L(t < 0) = 0 and R(t < 0) = Λ−, i.e., initially resting mirrors. As usual, we expand the operator of the quantized ﬁeld in the mode functions: ∞ X in in φ(t, x) = an fn (t, x) + h.c. . (2.1) n=1 in in † Here, operators an and (an ) satisfy the standard bosonic commutation relations, and mode func- in tions fn are expressed in terms of auxiliary functions G(z) and F (z): i f in(t, x) = √ e−iπnG(t+x) − e−iπnF (t−x) , (2.2) n 4πn which solve the generalized Moore’s equations [1]: G [t + L(t)] − F [t − L(t)] = 0,G [t + R(t)] − F [t − R(t)] = 2 (2.3) and comply the initial conditions G(z ≤ Λ−) = F (z ≤ 0) = z/Λ−. Note that due to stationarity of the motion of mirrors in the asymptotic future, functions G(z) and F (z) periodically grow as 2Λ+ 2Λ+ z → +∞: G(z + ∆zG) = G(z) + 2 and F (z + ∆zF ) = F (z) + 2 with ∆zG = and ∆zF = . 1−β+ 1+β+ This property can be easily inferred from the geometric method for constructing modes [46–49]. Moreover, in this limit, the ﬁrst Moore’s equation implies a simple relation: 1 + β 2L F (z) = G + z + + , as z → +∞. (2.4) 1 − β+ 1 − β+

3 Let us return to mode functions (2.2). First, they can be straightforwardly shown to solve the free equation of motion and form a complete orthonormal basis with respect to the Klein-Gordon inner product: Z R(t) ∗ ∗ (u, v) = −i [u∂tv − v ∂tu] dx. (2.5) L(t)

Second, these functions have definite positive frequency with respect to the Killing vector ξ = ∂t: in 1 πnt πnx fn (t, x) = √ exp −i sin , as t < 0, (2.6) πn Λ− Λ− and diagonalize the free Hamiltonian at the past infinity: ∞ Z R(t) 1 1 X πn 1 H (t) = dx (∂ φ)2 + (∂ φ)2 = (ain)†ain + , as t < 0. (2.7) free 2 t 2 x Λ n n 2 L(t) n=1 − Due to these reasons, functions (2.2) are usually referred to as in-modes. At the same time, at the future infinity in-modes contain both positive and negative frequencies: ∞ X 1 πkt˜ πkt˜ πkx˜ f in(t, x) = √ α exp −i + β exp i sin , as t → +∞, (2.8) n nk ˜ nk ˜ ˜ k=1 πk Λ Λ Λ ˜ where we change to the coordinates t˜= γ+(t − β+x),x ˜ = γ+(x − β+t − L+) and denote Λ = γ+Λ+, p 2 γ+ = 1/ 1 − β+ for shortness. The coefficients in the decomposition (2.8) are nothing but the Bogoliubov coefficients determined by the inner products (2.5):

in out in out ∗ αnk = fn , fk , βnk = − fn , (fk ) , (2.9) of in-modes (2.2) and out-modes: 1 πnt˜ πnx˜ f out(t, x) = √ exp −i sin , as t → +∞. (2.10) n πn Λ˜ Λ˜ For arbitrary mirror motion, functions G(z), F (z) and the Bogoliubov coefficients are very difficult to compute. Nevertheless, this computation significantly simplifies if we restrict ourselves to weak deviations from stationarity, i.e., consider trajectories of the form L(t) = l(t), R(t) = Λ− + r(t), 1, and exclude motions that induce constructive interference. In this case, G(z) and inverse function g(y) = G−1(y) are approximately linear: G(z) ≈ 2 z + O(), g(y) ≈ ∆zG y + O(). ∆zG 2 Keeping in mind that these functions periodically grow with z or y, we expand their derivatives into the Fourier series: ∞ ∆z 2π Z G 2π 0 X i nz 1 0 −i nz G (z) = G e ∆zG ,G = G (z)e ∆zG dz, n n ∆z n=−∞ G 0 ∞ (2.11) X 1 Z 2 g0(y) = g eiπny, g = g0(y)e−iπnydy, n n 2 n=−∞ 0 and find high order Fourier coefficients to be small, Gn ∼ and gn ∼ for all n 6= 0. In addition, ∗ ∗ these coefficients are symmetric, G−n = Gn and g−n = gn, since the original functions are real. These properties allow us to expand complex exponents from (2.2) into simple plane waves:

∆z 1 Z G 2π g n −iπnG(z) −i ∆z kz n+k 2 e e G dz = δn,−k + (1 − δn,−k) + O , (2.12) ∆zG 0 g0 n + k

4 hence, 2π 2π −iπnG(z) −i nz X gn−k n −i kz e ≈ e ∆zG + e ∆zG . (2.13) g0 n − k k6=n Combining this identity with the relation (2.3), changing to the (t,˜ x˜) coordinates and calculating inner products (2.9), we straightforwardly obtain the Bogoliubov coefficients (we single out the Kronecker delta δn,k to make the -expansion explicit): √ g nk α = δ + n−k (1 − δ ) + O 2 , nk n,k g n − k n,k √0 (2.14) gn+k nk 2 βnk = − + O . g0 n + k We emphasize that these approximate identities are valid only for relatively small frequencies, n 1/, where particle wavelength is much larger than the characteristic mirror displacement, λ¯ ∼ Λ˜/πn Λ.˜ We also expect that at high frequencies, interactions between the field and the mirror can be approximated by a quasistationary process (at least if we assume no constructive interference). Hence, the corrections to the stationary Bogoliubov coefficients (i.e., identity transformation) in this case are approximately zero, αn6=k ≈ 0 and βnk ≈ 0 for n 1/ or k 1/. We remind that identities (2.14) are valid only for weakly nonstationary motions. Notable examples of such a motion include “broken” trajectories with a small final velocity : L(t) = tθ(t),R(t) = Λ + tθ(t), Λ 2Λ 1 − (−1)n (2.15) g = , g = − , 0 1 − n6=0 1 − 2iπn or resonant oscillations with destructive interference2, e.g.: πqt πqt L(t) = Λ sin θ(t),R(t) = Λ + Λ sin θ(t), Λ Λ (2.16) q 2iqn 1 − (−1)n g = Λ, g ≈ πΛ, g ≈ Λ , 0 ±q 2 n6=0,±q n2 − q2 2

3 where q = 2, 4, 6, ··· . The derivation of coefficients gn and βnk for the motions (2.15) and (2.16) can be found in [46,51]. In strongly nonstationary situations, the relations between the mirror trajectories, functions G(z) and g(y), the Bogoliubov coefficients, and the number of created particles are more compli- cated. The most notable examples of such situations include resonant oscillations with a constructive interference [5,6,50–54] and an asymptotic light-like mirror motion [3,4,55–61]. Nevertheless, recall that in the present paper, we confine our study to weakly nonstationary motions, for which approximate identities (2.14) can be safely used.

2We emphasize that this is a very special case of resonant oscillations; in fact, the interference is constructive for the majority of oscillation amplitudes, frequencies, and dephasing angles [50, 51]. 3 In comparison, the direct calculation of βnk for “broken” trajectories (2.15) yields the following expression: √ n+k iπk r 2 n 1 − (−1) e k gn+k nk 2 βnk = 2 2 = − + O , iπ (n + k) − (k) 2 n g0 n + k which conﬁrms the validity of approximations (2.14). The apparently troubling phase factor eiπk is signiﬁcant only at very large mode numbers, k ∼ 1/, where its prefactor is proportional to O 2.

5 3 Eﬀective Hamiltonian

The full Hamiltonian of the nonlinear DCE, model (1.7), is very complex. Therefore, we need a reasonable approximation to simplify the calculations and estimate the number of created particles. First, we assume small deviations from stationarity, i.e., βnk ∼ αn6=k ∼ O() with 1. In the leading order in , this approximation forbids almost all processes that violate the energy conservation law. Second, we consider small coupling constants and large evolution times4, λ → 0, t → ∞, λΛ+t = const, i.e., keep only the leading secularly growing contributions to the evolution operator. This limit is similar to the rotating-wave approximation in quantum optics [62–64] and suppresses rapidly oscillating parts of the eﬀective Hamiltonian. We also note that in this limit, the Bogoliubov coeﬃcients are approximately constant. Both these approximations are inspired by a quantum mechanical toy model of the nonlinear DCE [43]. First of all, let us calculate the free Hamiltonian in the suggested limit. Substituting mode decomposition (2.8) into identity (2.7), rotating to the (t,˜ x˜) coordinates, integrating overx ˜ and neglecting rapidly oscillating and constant contributions, we obtain the following approximate identity:

∞ X ωk h i H = (α∗ α + β∗ β ) a† a + (α β + β α ) a a + h.c. free 2 mk nk mk nk m n mk nk mk nk m n m,n,k=1 ∞ ∞ √ (3.1) X 1 † X ωn + ωk gn+k nk 2 ≈ ωnanan − anak + h.c. + O , 2 2 g0 n + k n=1 n,k=1

out ˜ as t → +∞. Here, we denote the frequency of the fn mode as ωn = πn/Λ. In the last line, we also substitute the approximate Bogoliubov coefficients (2.14) and keep only two leading terms in the -expansion. Note that the Hamiltonian of the massive variant of the model (1.1) also has the form (3.1) r 2 with a frequency ω = πn + m2 ≈ πn + Λ˜ m2, as m → 0. At the same time, the Moore’s n Λ˜ Λ˜ 2πn approach [1], which we used to calculate the Bogoliubov coefficients in Sec.2, does not work in the massive model due to the lack of conformal invariance. Hence, in the massive theory, approximation (2.14) and the second identity in (3.1) are valid only at relatively small evolution times, t˜ 1/m2Λ.˜ Now we employ the same method to calculate the interacting Hamiltonian at the future infinity:

(0) (1) 2 Hint = δHfree + Hint + Hint + O , as t → +∞, (3.2)

(0) (1) 0 1 where Hint and Hint are the normal-ordered quartic interaction terms proportional to and , respectively: ˜ ∞ (0) λΛ X 3δk+l,m+n + 6δk,mδl,n 4δk,l+m+n H = √ a† a†a a − √ a† a a a + h.c. , (3.3) int 32π2 k l m n k l m n k,l,m,n=1 klmn klmn

4This limit is equivalent to the limit λ → 0, t˜→ ∞, λΛ˜t˜= const, if the velocity of the mirrors does not approach the speed of light in the future inﬁnity.

6 ˜ ∞ (1) λΛ X 1 h √ † † Hint = 2 (3g−k−l+m+n + 12δk,mg−l+n) (1 − δk+l,m+n) akal aman 32π g0 k,l,m,n=1 klmn † (3.4) − (4g−k+l+m+n + 12δk,lgm+n) (1 − δk,l+m+n) akalaman i + gk+l+m+n akalaman + h.c. , and δHfree is the normal-ordered quadratic term, which can be absorbed into the renormalization of frequency in the free Hamiltonian (3.1):

∞ ˜ " ∞ ∞ ∗ # X Λ 2 1 † X gn+k 1 X gn+k 1 † † 2 δHfree = δm anan − √ anak − √ anak + O . (3.5) 2π n g0 g0 n=1 k=1 nk k=1 nk In the last identity, we single out the correction to the physical mass (which is, essentially, the one-loop correction): n 3λ XUV 1 3λ δm2 ≈ = log (n ) , (3.6) 2π n 2π UV n=1 and introduce the ultraviolet cutoff nUV. In what follows, we assume that this correction is canceled by the corresponding tree-level counterterm5. Furthermore, we believe that this assumption does not affect our calculations. First, they are devoted to infrared rather than ultraviolet contributions. Second, we are interested in the evolution of the quantum state, which is not directly related to the mass of the field. For these reasons, in the following sections, we assume zero physical mass. (0) † † Note that the Kronecker deltas in Hint establish energy conservation in the scattering (a a aa) and decay (a†aaa and a†a†a†a) processes. This is a consequence of the emergent time translation symmetry in the limit → 0. There are no substantial loop corrections in this limit. Conversely, higher-order parts of the interacting Hamiltonian violate the energy conservation law6 and allow (1) usually forbidden processes. For instance, the aaaa term in the Hint describes a simultaneous production of four correlated particles. However, in what follows, we restrict ourselves to the leading nontrivial order in , i.e., assume that the “forbidden” processes occur only once or twice during the evolution of the system. We also remind that in this section, we truncate the -expansion of the interacting Hamilto- nian (3.2) after the first order term. The full expression, which is expressed through the exact Bogoliubov coefficients and do not require the limit → 0, can be found in AppendixB. Unfortunately, the interacting Hamiltonian (3.2) is too complex even in the limit → 0 and λ → 0, t → ∞, λΛ+t = const. So in the following sections we further simplify it. In Sec.4, we consider a reduction to quantum mechanics, i.e., discard all high-frequency modes and leave only the fundamental one, n = 1. In Sec.5, we consider a large N generalization of the model (1.7). In both of these models, particle decays are suppressed in favor of scattering. This feature allows us to resum the series in λ and calculate nonperturbative loop corrections to the level density, anomalous quantum average, and the number of created particles.

5However, note that the loop correction is positive for λ > 0, i.e., the physical mass can be zero only if the bare 2 mass is tachyonic, m0 < 0. 6Essentially, this violation is related to the energy exchange with the external world, which is possible due to the presence of mirrors.

7 4 Reduction to quantum mechanics

As explained in the previous section, the limit → 0 establishes an approximate energy conservation, i.e., restricts the energy exchange with the external world. Hence, we expect this limit to imply a small number of created particles. Furthermore, we expect that created particles mainly populate the most easily excited, lowest-energy mode with n = 1. In this case, we can ignore all modes with n > 1 and reduce the interacting Hamiltonian (3.2) to a quantum-mechanical problem: ˜ QM λΛ 9 † † g2 † g4 2 Hint = 2 a a aa − a aaa + aaaa + h.c. + O , (4.1) π 32 2g0 32g0 where we denote a ≡ a1 for shortness. Recall that Fourier coefficients are small, g2 ∼ g4 ∼ , in the limit → 0. Keeping in mind the normal-ordered structure of this Hamiltonian and assuming the vacuum initial state, we straightforwardly obtain the evolved quantum state in the limit λ → 0, t → +∞ and → 0: Z t 0 0 ˜ QM |Ψ(t)i = T exp −i Hint(t )dt |ini ≈ exp −itHint |ini t0 ∗ (4.2) 1 g 27 4 4 ˜˜ † 2 = |ini + exp −i 2 λΛt − 1 a |ini + O . 216 g0 4π The leading contribution to this expression is ensured by multiple scattering of virtual particles (i.e., by powers of the a†a†aa term). Identity (4.2) readily implies the approximate level density and anomalous quantum average: 2 |g |2 27 † 4 2 ˜˜ 4 n(t) = hΨ(t)|a a|Ψ(t)i = 2 sin 2 λΛt + O , (4.3) 243 g0 8π 8 g g∗ 9 27 2 4 ˜˜ ˜˜ 3 κ(t) = hΨ(t)|aa|Ψ(t)i = − 2 5 − 6 exp −i 2 λΛt + exp −i 2 λΛt + O . 405 g0 8π 4π (4.4) Substituting (4.3), (4.4), and (2.14) into Eq. (1.3) and truncating all summations at n = 1, we find the total number of particles created in the fundamental mode: |g |2 2 |g |2 27 2 4 2 ˜˜ 3 (4.5) N1 = 2 + 2 sin 2 λΛt + O . g0 243 g0 8π The first and second terms on the right-hand side of (4.5) describe the tree-level and loop-level contributions, respectively. Note that the loop contribution is always positive and has the same order in as the tree-level approximation. Moreover, at very large times, t˜ 1/λΛ,˜ the oscillating part can be replaced with the average value: 2 2 |g2| 1 |g4| N1 ≈ 2 + 2 . (4.6) g0 243 g0 Therefore, nonlinearities enhance the number of particles created in the fundamental mode and measured at large evolution times. Furthermore, the loop contribution significantly surpasses the tree-level approximation in the case |g4| |g2|, e.g., in resonant oscillations (2.16) with q = 4. We emphasize that the loop corrections to the number of created particles are related to a change in the initial quantum state, which becomes possible due to the violation of the energy conservation law in a background field. The leading correction, equation (4.2), describes a state with four correlated particles. Note that the full series in (4.2) is obtained by a unitary evolution from a pure state, although unitarity is violated in any fixed order in .

8 5 Large N generalization

Now let us consider the O(N) symmetric, large N generalization of the model (1.7):

λ ∂2 − ∂2 φi = (φjφj)φi, φi[t, L(t)] = φi[t, R(t)] = 0, (5.1) t x N where i = 1, ···,N with N 1, and we assume the summation over the repeated upper indices. The eﬀective Hamiltonian of this model in the limit λ → 0, t → ∞ and → 0 coincides with the Hamiltonian (3.2) after the appropriate change in the operator products:

† † i † i † j j i † j † i j i † j † j i 3akal aman → (ak) (al) aman + (ak) (al ) aman + (ak) (al ) aman, † i † i j j i i j j akalaman → (ak) alaman, akalaman → akalaman, (5.2) † i † i i i 3akal → (N + 2)(ak) al, 3akal → (N + 2)akal, and coupling constant, λ → λ/N. Similarly to the low-particle approximation of Sec.4, the large N limit suppresses particle decays, and the → 0 limit suppresses the production of correlated quadruplets. Hence, in this case, the evolved quantum state is also straightforwardly calculated:

∗ 1 X gp+q Ap,q(t) |Ψ(t)i ≈ |ini − √ (ai )†(ai )†|ini N g pq p q p,q 0 ∗ (5.3) 1 X g Bk,l,m,n(t) 1 k+l+m+n √ i † i † j † j † 2 + (ak) (al) (am) (an) |ini + O + O 2 , N 4g0 N k,l,m,n klmn where

−iτCk,l,m,n X δp+q,k+l+m+n e + iτCk,l,m,n − 1 A (t) = δ p,q klm q,n C2 k,l,m,n k,l,m,n ! 1 e−iτDp,q − 1 e−iτCk,l,m,n − 1 iτ + 2 − 2 − , n(k + l + m) Dp,q (Dp,q − Ck,l,m,n) Ck,l,m,n (Dp,q − Ck,l,m,n) Dp,qCk,l,m,n (5.4) e−iτCk,l,m,n − 1 Bk,l,m,n(t) = . (5.5) 2Ck,l,m,n

λΛ˜t˜ For shortness, we rescale the time, τ = 4π2 , introduce coeﬃcients Ck,l,m,n = Dk,l + Dm,n and Hp+q−1 1 Pn 1 Dp,q = p+q + pq , and denote the Harmonic numbers as Hn = k=1 k ∼ log n + γ, where γ is the Euler-Mascheroni constant. Note that the contribution of the states with more than four particles is suppressed by the powers of 1/N. Substituting the ﬁnal state (5.3) into Eqs. (1.5) and (1.6), we obtain the future asymptotics of the resummed quantum averages: ∗ δi,j 2 X gp,k,l,mgq,k,l,m 1 nij (t) ≈ √ B∗ (t)B (t) + O 3 + O , (5.6) pq N pq g2 klm p,k,l,m q,k,l,m N 2 k,l,m 0 ∗ ij δi,j 1 gp+q h i 2 1 κpq(t) ≈ − √ Ap,q(t) + Aq,p(t) + O + O 2 , (5.7) N pq g0 N

9 10

0 0 20 40 60 80

2 Figure 1: Numerically calculated function Nn(τ)/ , n = 1, 2, 4, 8, 16 (from top to bottom) for the resonant motion (2.16) with q = 2 and = 0.02. The ultraviolet cutoﬀ is set to nUV = 64. For illustrative purposes, we also set N = 1; in this case, the “initial” values of functions coincide with free the normalized tree-level contribution, Nn(0) = Nn /N. and the number of created particles:

2 2 ∗ X |gn+k| nk X |gn+k+l+m| Bn,k,l,m(t)Bn,k,l,m(t) Nn(t) ≈ N 2 2 + 2 2 g0 (n + k) g0 nklm k k,l,m (5.8) 2 X |gn+k| Re [An,k(t) + Ak,n(t)] 1 + 2 + O 3 + O . g2 n + k N 2 k 0

free The first term in Eq. (5.8) describes the tree-level contribution, Nn ; the second and third terms appear only in the nonlinear, interacting theory. Let us also explicitly evaluate Nn(t) for a particular physically meaningful mirror motion. As an example of such a motion, we consider resonant oscillations with q = 2 frequency and destructive interference (2.16). We approximately calculate the particle number for relatively small, λΛ˜t˜ 1, ˜ and large, λΛt˜ nUV/ log(nUV), evolution times, where nUV defines the effective ultraviolet cutoff. At intermediate times, these asymptotics are connected by a smooth curve, which can be calculated numerically (Fig.1). At relatively small evolution times, λΛ˜t˜ 1, the oscillating functions in (5.8) can be ex- panded in a series. The next-to-the-leading term in this series, which originates from the two-loop correction to the quantum averages, quadratically grows with time (compare with [45,46]):

2 2 2 free τ X |gn+k+l+m| 1 2 X |gn+k| δn,l+m+p + δk,l+m+p Nn ≈ Nn + 2 − τ 2 2 g0 nklm g0 (n + k)lmp k,l,m k,l,m,p (5.9) 2 log2 n 1 ∼ N − 2τ 2 + ··· , for 1 n . n n2

loop free In the last line, we discard the overall factor and ﬁt the loop contribution, Nn = Nn − Nn , from the numerically calculated dependence (Fig.2a). We emphasize that the total number of

10 5 10 15 20 25 30 5 0.00

-0.01 4

-0.02 3 -0.03

-0.04 2

-0.05 1 -0.06

-0.07 0 5 10 15 20 25 30

(a) (b)

loop 2 loop 2 Figure 2: Numerically estimated functions Nn (τ)/(τ) , as τ → 0 (a), and Nn (τ)/ , as τ → ∞ (b), for the resonant motion (2.16) with q = 2 and = 0.02. The ultraviolet cutoff is set to nUV = 64. The dots denote the calculated values, the lines denote the fitting curves. Note that the first graph starts from n = 4; the values at n < 4 are positive and much larger than the values at n ≥ 4. created particles is always positive, although the loop contribution is negative for n ≥ 4 and small evolution times (see Fig.1). ˜ At large evolution times, λΛt˜ nUV/ log(nUV), we can replace the oscillating functions with their average values:

2 X |gn+k+l+m| 1 1 N ≈ N free + n n g2 nklm C2 k,l,m 0 n,k,l,m 2 " # X |gn+k| δn+k,l+m+p+q 4 Dn,k + Cl,m,p,q 2δk,q + 2δn,q + − (5.10) g2 (n + k)lmp q(l + m + p) D2 C2 C2 k,l,m,p,q 0 n,k l,m,p,q l,m,p,q 2 log2 n 1 ∼ N + C + ··· , for 1 n , n n where we again discard the overall factor and fit the loop contribution from the numerically calculated dependence (Fig.2b). The factor C in front of the loop contribution is positive and slightly changes with the ultraviolet cutoff, C ∼ log log(nUV). For reasonable values of nUV, this factor has the order of C ∼ 100. It is worth mentioning that Eqs. (5.9) and (5.10) are valid only for not very high mode numbers, n 1/, where the Bogoliubov coefficients can be approximated by Eqs. (2.14). We also note that the number of created particles is proportional to a small factor, 2, and goes to zero at high frequencies, Nn → 0 as n → ∞. This qualitatively validates the low-particle approximation of Sec.4. Finally, we emphasize that the nonperturbative loop contribution to (5.8) has the same order in as the tree-level expression, although it is suppressed by the inverse number of degrees of free- dom, 1/N. Hence, at large evolution times, nonlinearities enhance the rate of particle production in the DCE. This enhancement is most evidently manifested at small mode numbers, i.e., at low particle energies.

11 6 Discussion and Conclusion

In this work, we have shown that nonlinearities (i.e., interactions between the quantum fields) significantly affect particle creation in a weakly nonstationary scalar DCE at long evolution times. First of all, we calculated the Bogoliubov coefficients and derived the effective Hamiltonian of the quantized field. Then we employed this Hamiltonian to evaluate the nonperturbative (in the coupling constant λ) correction to the tree-level number of created particles. For simplicity, we considered the low particle and the large N limits, which suppress particle decays and the production of correlated quadruplets in favor of scattering processes. In both cases, we obtained that the tree-level and loop-level contributions to the number of created particles have the same order in the small parameter , which defines the relative mirror displacement. We emphasize that in this paper, we work in the interaction picture. In other words, we separate the free, Eq. (3.1), and interacting, Eq. (3.2), Hamiltonians and assume that we exactly † know how the free Hamiltonian evolves the field operator φ(t) = Ufree(t, t0)φ(t0)Ufree(t, t0). This knowledge is contained in the Bogoliubov coefficients. Then we approximately calculate the final quantum state, i.e., evolve the initial quantum state with the interacting Hamiltonian. This information is contained in the resummed level density and anomalous quantum average. Note that the quantum state continues evolving even when the Bogoliubov coefficients stop changing, i.e., free evolution is essentially finished. This approach is inspired by similar problems from the high energy physics [28–44]. Note that we restricted ourselves to the calculation of the number of created particles, Eq. (1.3) with the resummed level density (1.5) and anomalous quantum average (1.6). Nevertheless, our analysis can be straightforwardly applied to the calculation of the resummed Keldysh propagator, which is also expressed through the quantum averages (1.5) and (1.6). Using this propagator, one can easily obtain other observable quantities (e.g., see AppendixA). We also note that the reduced quantum mechanical Hamiltonian (4.1) is very similar to the Hamiltonian of a quantum anharmonic oscillator with a time-dependent frequency [43]. This problem can be solved both for small and large deviations from stationarity, and nonperturbative loop corrections in this case qualitatively coincide with (4.5). Moreover, this problem is sometimes used as a toy model of the DCE in a three-dimensional cavity [5, 65–67]. Unfortunately, the calculations of [43] for strongly nonstationary oscillator cannot be directly extended to strongly nonstationary one-dimensional DCE: in this case, energy is rapidly pumped into the high-frequency modes, and low-particle approximation (4.1) is no longer valid [5]. Hence, at large deviations from stationarity, one must consider the full Hamiltonian (3.2). Our work can be extended in several possible directions. First, our approach can be applied to a nonlinear generalization of the circuit DCE [15–17]. The Hamiltonian of such a generalized system is similar to Eqs. (3.2) or (4.1) (e.g., compare them with the Hamiltonians derived in [68–73]), so we expect a similar amplification of the created particle number. Moreover, the number of particles created in the circuit DCE can be directly measured, which opens up a possibility of an experimental verification of our results. Second, the expression for the effective Hamiltonian is easily generalized to more realistic models of the DCE, including massive quantum fields, semitransparent mirrors, and resonant oscillations with constructive interference. In general, the strategy is similar to Secs.2 and3: one calculates the Bogoliubov coefficients for the free field and substitutes them into the Hamiltonian (B.1). Nevertheless, without a small parameter , the exponentiation of the effective Hamiltonian is a much harder task. At large N, this obstacle can be possibly circumvented by an alternative approach to the calculation of Nn proposed in [43].

12 Third, it is promising to extend the analysis of the present paper to other nonequilibrium quantum systems with a “nonkinetic” behavior of loop corrections, such as light self-interacting scalar fields in de Sitter space [32,33,74–77] or upper Rindler wedge [34]. Finally, two-dimensional O(N) and CP N models on a finite interval are known to possess nontrivial classical solutions with a negative energy [78–84]. In these models, the naive vacuum, an|ini = 0, is unstable at some values of the model parameters. In principle, nonequilibrium techniques (similar to those developed in the present paper) might confirm this instability and explicitly trace the evolution from the initial quantum state to the true vacuum. An example of such a calculation in a different setup can be found in [40]. Furthermore, it is promising to generalize the predictions of [78–85], which were obtained in a stationary approximation, to time- dependent boundary conditions. The calculations of Sec.5 may be considered as a starting point for such generalizations.

Acknowledgments

I would like to thank Emil Akhmedov for the fruitful discussions and sharing his ideas, Alexander Gorsky for the useful comments, and Elena Bazanova for the proofreading of the text. This work was supported by the Russian Ministry of Education and Science and by the grant from the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

A Alternative derivation of the created particle number

Correlation functions are the most fundamental objects in the nonstationary quantum field theory [44]. First, they have a clear physical meaning even in time-dependent backgrounds where the notion of a particle is ill-defined. Second, all directly observable quantities are derived from correlation functions. For example, the exact Hamiltonian of the nonlinear DCE (1.7) can be obtained from the exact Keldysh propagator: R(t) Z 1 1 λ 2 H (t) = dx ∂ ∂ GK (x , x ) + ∂ ∂ GK (x , x ) + GK (x , x ) , (A.1) full 1 2 t1 t2 1 2 2 x1 x2 1 2 4 1 2 L(t) x1→x2 where we denote x = (t, x) for shortness. At the tree level, the Keldysh propagator is defined as an anticommutator of quantum fields:

K 1 G (x1, x2) = in {φ(x1), φ(x2)} in free 2 ∞ (A.2) X 1 ∗ = δ + n f in(x ) f in(x ) + κ f in(x )f in(x ) + h.c. , 2 k,l kl k 1 l 2 kl k 1 l 1 k,l=1 where we employ the mode decomposition (2.1) and use short notations for the level density and anomalous quantum average. In the interacting theory, the Keldysh propagator preserves the form (A.2) with exact quantum averages (1.5) and (1.6) if the diﬀerence between the times is much smaller than their average, t1 − t2 (t1 + t2)/2. This allows us to estimate the future asymptotics of the exact Hamiltonian in the interacting theory: ∞ X πm 1 H ≈ δ + n (α α∗ + β β∗ ) + κ (α β + β α ) + h.c. + O λΛ˜ , full ˜ 2 k,l kl km lm km lm kl km lm km lm k,l,m=1 2Λ (A.3)

13 where we take the limit t → +∞, substitute the asymptotics of the in-modes (2.8), rotate to the (t,˜ x˜) coordinates and integrate overx ˜. Note that the contribution of the nonlinear part is negligible if we consider small coupling constants, λ 1/Λ˜ 2. Finally, using the properties of the Bogoliubov coeﬃcients:

X ∗ ∗ X αikαjk − βikβjk = δi,j, (αikβjk − βikαjk) = 0 (A.4) k k

∗ and symmetry of the level density, nkl = nlk, and anomalous quantum average, κkl = κlk, we obtain the following expression for the exact Hamiltonian:

∞ X πm 1 H ≈ + N + O λΛ˜ , (A.5) full ˜ 2 m m=1 Λ where Nm is given by equation (1.3) with level density (1.5) and anomalous quantum average (1.6). The divergent sum P∞ πm describes the standard static Casimir energy and does not depend on m=1 2Λ˜ the mirrors motion at intermediate times. The second sum describes excitations above stationarity. This conﬁrms that Nm has the meaning of the particle number created in a linear or nonlinear DCE at large evolution times.

B Full expression for the interacting Hamiltonian

The full expression for the interacting Hamiltonian of theory (1.7) in the limit λ → 0, t → +∞ has the following form: ∞ λΛ˜ X 1 H = √ Ai,j,k,l , (B.1) int 32π2 ijkl m,n,p,q i,j,k,l,m,n,p,q=1 where

i,j,jk,l † † † h ∗ ∗ Am,n,p,q = 3amanapaq + 6a(nδm),(paq) αmiαnjαpkαql (δi+j,k+l + δi,kδj,l + δi,lδj,k) ∗ ∗ ∗ ∗ − 4αmiαnjαpkβql δi+j+l,k + 2αmiβnjαpkβql (δi+l,j+k + δi,jδk,l + δi,kδj,l) ∗ ∗ ∗ ∗ + 2αmiβnjαpkβql (δi+k,j+l + δi,jδk,l + δi,lδj,k) − 4αmiβnjβpkβql δi+k+l,j ∗ ∗ i + βmiβnjβpkβql (δi+j,k+l + δi,kδj,l + δi,lδj,k)

† h ∗ ∗ − 4amanapaq + 6δm,(napaq) αmiαnjαpkαql δi,j+k+l + βmiβnjβpkβql δi,j+k+l ∗ ∗ − 3αmiαnjαpkβql (δi+l,j+k + δi,jδk,l + δi,kδj,l) + 3βmiαnjαpkβql δi+j+k,l ∗ ∗ i − 3αmiαnjβpkβql (δi+k,j+l + δi,jδk,l + δi,lδj,k) − 3βmiαnjβpkβql (δi+j,k+l + δi,kδj,l + δi,lδj,k) h − 4amanapaq αmiαnjαpkβql δi+j+k,l − 3αmiαnjβpkβql (δi+j,k+l + δi,kδj,l + δi,lδj,k) i + αmiβnjβpkβql δi,j+k+l + h.c.

1 Here, we introduce the short notation for the symmetrized quantities, e.g., A(m,n) = 2! (Amn + Anm). We also neglect rapidly oscillating and constant terms that do not contribute to (1.5) or (1.6) in 1 R t+T 0 0 the limit in question. In other words, we average the Hamiltonian, Hint(t) −→ T t Hint(t )dt , and neglect the subleading contributions in the limit T → +∞.

14 It is worth mentioning that the Kronecker deltas of the form δi+j,k+l appear only in a massless theory. In a massive two-dimensional theory, these deltas are multiplied by the time-dependent q sin[(ωi+ωj −ωk−ωi+j−k)T ] ˜ 2 2 function, , where ωk = (πk/Λ) + m . This function goes to one only if (ωi+ωj −ωk−ωi+j−k)T i = k, i = l, or m → 0; otherwise, it vanishes at large evolution times, T 1/m2Λ.˜ This behavior illustrates the well-known fact that elastic scattering of massive particles in a ﬂat two-dimensional spacetime reduces to a simple momentum exchange. Finally, substituting the approximate Bogoliubov coeﬃcients (2.14) into this identity and dis- carding the subleading in terms, we straightforwardly obtain Eq. (3.2).

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