{'fraction_non_alphanumeric': 0.03395644824500348, 'fraction_numerical': 0.011733121209108084, 'mean_word_length': 4.610878661087866, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Spatial correlations of field observables in two half-spaces separated by a movable perfect mirror

We consider a system of two cavities separated by a reflecting boundary of finite mass that is free to move, and bounded to its equilibrium position by a harmonic potential. This yields an effective mirror-field interaction, as well as an effective interaction between the field modes mediated by the movable boundary. Two massless scalar fields are defined in each cavity. We consider the second-order interacting ground state of the system, that contains virtual excitations of both mirror's degrees of freedom and of the scalar fields. We investigate the correlation functions between field observables in the two cavities, and find that the squared scalar fields in the two cavities, in the interacting ground state, are anti-correlated. We discuss the dependence of the correlation on the distance of the two points considered from the mirror's average position, and on its mass and oscillation angular frequency. These results show a sort of communication between the two half-spaces separated by the movable mirror, mediated by its position fluctuations. Observability of this new phenomenon is also discussed.

We consider a system of two cavities separated by a reflecting boundary of finite mass that is free to move, and bounded to its equilibrium position by a harmonic potential. This yields an effective mirror-field interaction, as well as an effective interaction between the field modes mediated by the movable boundary. Two massless scalar fields are defined in each cavity. We consider the secondorder interacting ground state of the system, that contains virtual excitations of both mirror's degrees of freedom and of the scalar fields. We investigate the correlation functions between field observables in the two cavities, and find that the squared scalar fields in the two cavities, in the interacting ground state, are anti-correlated. We discuss the dependence of the correlation on the distance of the two points considered from the mirror's average position, and on its mass and oscillation angular frequency. These results show a sort of communication between the two halfspaces separated by the movable mirror, mediated by its position fluctuations. Observability of this new phenomenon is also discussed.

I. INTRODUCTION
In quantum field theory, the presence of a dielectric or reflecting boundary affects the structure of the field modes and can change the expectation values of several local field observables, for example the field energy density, spatial field correlations or dispersion interactions [1][2][3][4]. This has relevant observable consequences, for examples changes in the spontaneous emission rate of an atom and atom-surface Casimir-Polder interactions [1,5,6]. In the case of an infinite flat, perfectly reflecting and static mirror, it is well known that the boundary separates the space into two independent half spaces. In such cases, the mirror is usually treated as an assigned boundary condition on the field operators. In this paper, we will show that the situation significantly changes if the boundary has a finite mass and it is allowed to move, because correlations between field quantities evaluated in the two half-spaces exist. Inclusion of the motion of a boundary is at the basis of quantum optomechanics [7,8] and dynamical Casimir effect [9,10]. In such cases, the mirror's motion is prescribed from the outside, and it is usually included in the formalism in the form of a time-dependent boundary condition on the relevant field operators; the mechanical degrees of freedom are not dynamical variables of the system. On the contrary, another type of systems investigated in the literature treats the mirror as a quantum system with its mechanical degrees of freedom [11,12], thus allowing typical quantum effects such as quantum fluctuations of its position. As we shall discuss below, in this paper we adopt this second point of view by including the wall's mechanical degrees of freedom in the Hamiltonian description of our system.
A model Hamiltonian describing a quantum field in the presence of a movable boundary of finite mass, bound to its equilibrium position by a harmonic potential, has been introduced in [11,12]. This effective Hamiltonian has been used to evaluate the field energy density change near the fluctuating boundary, as well as corrections to the Casimir force [13,14] and radiation pressure effects on a two-mirror system [15]. These quantum effects become more and more relevant the smaller the mirror mass is [13], and in modern quantum optomechanical experiments it can be reduced even to values of the order of 10 −21 Kg [8,16]. Internal degrees of freedom of a boundary have been considered also in the case of a moving dielectric membrane [17], within microscopic models of the boundary [18,19] and in connection with dissipative dynamics of a movable particle coupled to a field [20].
In this paper, we consider a system of two cavities separated by a movable perfectly reflecting wall of a finite mass and bound to its equilibrium position by a harmonic potential. Two massless one-dimensional scalar fields are considered in the two cavities, each of them interacting with the movable mirror separating the two cavities. We first generalize the mirror-field interaction Hamiltonian mentioned above to the present two-cavity case, and then evaluate by second-order perturbation theory the interacting ground state of the system, that contains virtual excitations of the fields in the cavities and of the mirror (phonons). Successively, we compute expectation values on the ground state of relevant correlation functions of local field observables in the two cavities. By evaluating the correlation between points in the two half spaces separated by the wall, we find that a non-vanishing spatial anti-correlation between the squared fields in the two cavities exists and evaluate its distance dependence. This does not happen in the case of a fixed wall. We discuss the dependence of such correlation from the relevant physical parameters, in particular mass and binding frequency of the movable wall, as well observability of this arXiv:2204.06886v1 [quant-ph] 14 Apr 2022 phenomenon.
This paper is organized as follows. In section II we introduce our two-cavity system and its Hamiltonian, describing an effective wall-field interaction, and obtain by perturbation theory the interacting ground state of the system. In section III we evaluate the expectation value on this state of relevant field quantities, specifically the spatial correlation function between points in the two cavities of the squared field; we find that an anti-correlation exists and we discuss its main physical features. Section IV is devoted to our conclusive remarks.

II. THE TWO-CAVITY SYSTEM
Our physical system is constituted by two ideal cavities, separated by a movable perfectly conducting mirror of mass m, and bounded to its equilibrium position x = L 0 by a harmonic potential of frequency ω 0 . The two fixed mirrors have positions x = 0 and x = 2L 0 . Two massless scalar quantum fields are defined in the two half-spaces separated by the movable mirror. For simplicity, we here consider a one-dimensional model. The situation is shown in Fig. 1. The mechanical degrees of freedom of the movable mirror are treated quantum mechanically, and the mirror is described as a quantum harmonic oscillator of frequency ω 0 ; we label with b and b † its annnihilation and creation operators, respectively. Also, we respectively label with a p and a † p the bosonic annihilation and creation operators of the massless scalar field on the left cavity (0 < x < L 0 ), and with c r and c † r those for the cavity on the right (L 0 < x < 2L 0 ); all these operators refer to the field modes relative to the equilibrium position L 0 of the movable wall. For a single cavity, and in the hypothesis of small displacement of the movable wall from its equilibrium position, the system can be described on the basis of an effective interaction. This Hamiltonian was first introduced by Law for a single cavity with a movable mirror, in terms of field modes relative to the equilibrium position of the movable boundary [11,12]. The Law Hamiltonian, describing the effective mirror-field interaction, is linear in the mirror coordinates and bilinear in the field coordinates, and yields an effective interaction between the field modes, mediated by the movable mirror. Our first step is to generalize the Law Hamiltonian to our two-cavity scenario. This is quite immediate: we quantize the two scalar fields in the two half-spaces with respect to the equilibrium position x = L 0 of the mobile wall, and take into account that if a movable-wall's displacement in one direction makes shorter one cavity, it makes longer the other cavity. For some general preliminary considerations of this two-cavity system, see also [21]. Our Hamiltonian including the effective interaction of the movable wall with the two scalar fields is then of the form H = H 0 + H I , with the unperturbed term and the interaction term H I = H 1 I + H 2 I , where H 1 I and H 2 I are respectively the effective interaction Hamiltonians between the movable wall and the scalar field in the left (1) and right (2) half-spaces. These interaction terms are given by where N is the normal ordering operator, and the coupling constants are given by with ω k = ck. The coupling constants C 1 kj and C 2 kj have an opposite sign since, when the wall moves from its equilibrium position, the sign of its displacement is opposite for the two cavities. This Hamiltonian holds for small displacements around the equilibrium position, and it is the immediate generalization of the Hamiltonian introduced in [12] to our two-cavity case. A generic eigenstate of the unperturbed Hamiltonian H 0 will be labeled as where the first item n wall denotes the number of excitations in the mirror (phonons), while {n 1 } and {n 2 } denote, respectively, a complete set of occupation numbers of the field modes of the first and of the second cavity. The unperturbed ground state is thus |g = |0; {0}; {0} , with unperturbed energy E g = 0.
We can now obtain the first-and second-order corrections to the ground state, due to the mirror-fields interac-tion, by stationary perturbation theory. The first-order correction is which contains states with one mirror excitation and a pair of quanta in one of the two cavities. These are virtual mirror and field excitations, of course.
The second-order correction is given by the sum of several terms, that, for convenience of notations, we arrange in the following form |g |g while |g (2) 2w are terms with two mirror's excitations, that do not contribute to the expectation values we will evaluate in the next section, and thus we do not explicitly write them down.
The normalized corrected ground state assumes thus the form where Λ is a normalization factor obtained, in secondorder perturbation theory, from In the next section, we will evaluate the expectation value on the corrected ground state (11) of correlation functions between field observables evaluated at different points of the two cavities.

III. CORRELATION FUNCTIONS OF FIELD OBSERVABLES IN THE TWO CAVITIES
The operators of the massless 1D scalar fields in the two cavities are where x 1 is a point in the region (0, L 0 ) (left cavity) and x 2 in the region (L 0 , 2L 0 ) (right cavity). The possible values of k are those allowed by the boundary conditions. All quantities in (13,14) are referred to the equilibrium position of the movable wall [12]. We aim to evaluate relevant correlations between field observables in the two cavities over the interacting ground state (11). Using (6,7,11), and the expressions above of the field operators, it is immediate to see that g|φ(x 1 )φ(x 2 )|g − g|φ(x 1 )|g g|φ(x 2 )|g = 0. (15) It is easy to see that the vanishing of the correlation function (15) is valid at any order of perturbation theory, as a consequence of the fact that the effective Hamiltonians (2) are quadratic in the field annihilation and creation operators. This means that the field operators in the two cavities are not correlated. We can now evaluate the correlation of the squared field, that is a very important local field observable, being also related to the field energy density and to the interaction of the field with test bodies (as we will discuss later on in this section) (16) After a lengthy algebraic calculation, we finally find where the first term comes from first-order processes with a pair of virtual photons emitted in one of the two cavities, while the second term (inside square brackets) comes from second-order processes with two pairs of virtual photons, one pair in each cavity. In order to explicitly evaluate the correlation (17), we now consider the continuum limit by taking L 0 → ∞, thus recovering the case of a single trembling mirror in the vacuum space separating two infinite half-spaces. In this limit, the sums over k in (17) become integrals over the continuous variables k i = ω i /c, (i = p, q, r, s), by using k → (L 0 /2π) dk; we also rescale these variables with respect to k 0 = ω 0 /c: k p = pk 0 , k q = qk 0 , k r = rk 0 , k s = sk 0 ,with p, q, r, s dimensionless continuous variables. We also introduce the two scaled (dimensionless) distances from the movable wall After some algebra, the correlation function (17) becomes sin(pd 1 ) sin(qd 1 ) 1 + p + q drds sin(rd 2 ) sin(sd 2 ) 1 + r + s + dpdqdrds sin(pd 1 ) sin(qd 1 ) sin(rd 2 ) sin(sd 2 ) (1 + p + q)(p + q + r + s) All integrals in (19) extend from 0 to ∞, even if we can eventually introduce an upper ultraviolet cutoff frequency to take into account the properties of a real metal. In the first term, the four integrals are factorized, and the double integrals can be obtained analytically using drds sin(rd) sin(sd) q + r + s = 1 2 where f (x) = ci(x) sin(x) + si(x) cos(x) and g(x) = −ci(x) cos(x) − si(x) sin(x) are the auxiliary functions of the sine-integral and cosine-integral functions [22]. The second integral cannot be factorized, and after some algebraic manipulation it can be cast in the following form The integral (21) must be evaluated numerically. For simplicity, we take d 1 = d 2 = d, that is the two points considered are at the same distance from the mirror's equilibrium position, on the opposite side. In this case, taking into account Eqs. (19), (20) and (21), we obtain the following expression for the correlation function C(d) of the squared fields The integral over v in (22), to be evaluated numerically, with the variable change u = vd can be written as (23) In the symptotic limit d >> 1, the numerical evaluation gives I(d) 1.8/d 3 . Thus, in this limit, taking also into account that f (x) ∼ 1/x and g(x) ∼ 1/x 2 for x 1 [22], we get and, in terms of the unscaled distance y = d/k 0 = dc/ω 0 from the wall, (y k −1 0 ). Equations (22), (24), and (25) are the main results of this paper. They show that the squared fields at the two sides of the reflecting mirror are correlated, even if they are independent fields and there is not any direct interaction between them. This correlation is entirely ascribable to their mutual interaction with the movable wall, that allows a sort of communication between the two halfspaces, even in the vacuum state. The negative sign of C(y) in (25) shows that the squared field are indeed anti-correlated; this sign can be understood from a physical point of view, by considering that the two fieldmirror coupling constants C 1 kj anf C 2 kj have an opposite sign, and thus when the position of the mirror fluctuates in one direction, the modes of one cavity are blue-shifted while those of the other cavity are red-shifted. From Eq. (25) we notice that, at large distance from the movable wall, the (anti)correlation between the squared field scales with the distance from the wall's equilibrium position as y −3 . It also depends as 1 mω 2 0 on the mass and oscillation frequency of the movable wall. This means that the smaller mirror's mass and oscillation frequency are, the larger the effect we find is. In modern optomechanics experiments, a typical oscillation frequency is of the order of 10 5 Hz, and a mass as low as 10 −21 Kg can be achieved [8].
Observability of the spatial correlation of the squared field could be obtained exploiting the dispersion interaction of the fluctuating field with a polarizable object A, that in the far-distance regime is usually proportional to the square of the field at the polarizable-body position r A , in the form where the average value is taken on the dressed ground state of the system and α A is the static polarizability of the polarizable body A [23]. The effect we find should therefore yield a correlation between the dispersion interaction between the mirror and two polarizable bodies A and B (ground-state atoms or molecules, for example, in the electromagnetic case), placed at the opposite sides of the movable reflecting wall, as well as an atom-atom dispersion interaction mediated by the trembling mirror. A joint measurement of the dispersion force on A and B, and their correlation as a function of the atoms-mirror distance, should allow to observe the spatial correlation between the squared field in Eqs. (22) and (25). This is also motivated by the fact that the movable mirror acts as a third body interacting with the field, and thus it is capable of modifying the radiation-mediated interacting energy between the two polarizable bodies A and B [24,25]. Another possible experimental observation could result from three-body effects on the atom-atom dispersion interaction between two atoms (or, in general, polarizable bodies) A and B at one side of the wall due to a third atom C placed on the other side of the wall. In fact, the three-body component of the dispersion force on one atom, let say atom A, is related to a sort of interference between the virtual field energy densities due to the two other atoms B and C [24], or to the virtual field correlation at the location of A and B, dressed by C [26]. For a fixed wall, atom C cannot influence the interaction between A and B, while in the present case of a movable wall it can: thus, a measurement of the A-B interaction should keep trace of the presence of C, even if it is placed on the opposite side of the perfectly reflecting movable wall. Finally, we wish to mention that our results have some resemblance to the case of Hawking radiation from a fluctuating event horizon due to quantum gravity effects, where a fuzzy event horizon of a black hole is present and radiation is emitted by the fluctuating boundary [27,28]; also, our results could indicate possibility that a fluctuating event horizon could allow a correlation between physical observables defined in the internal and external regions of the horizon. We hope to investigate this intriguing aspect in the near future.

IV. CONCLUSION
We have considered a system of two cavities separated by a reflecting mirror that can fluctuate around its equilibrium position, with two massless one-dimensional scalar fields defined in the two half-spaces separated by the movable wall. An effective fields-mirror interaction exists, as well an effective interaction between the field modes mediated by the movable mirror. We have calculated the interacting ground state of this system at the second order of the fields-mirror effective interaction, that contains both field and mirror virtual excitations. We have shown that a nonvanishing anti-correlation between the squared fields evaluated at the two sides of the mirror exists, and discussed its dependence on the distance from the mirror's equilibrium position, and on the mass and oscillation frequency of the movable mirror. Possible observability of this new effect has been also discussed. Generalization to the three-dimensional case and to the quantum electromagnetic field case will be the subject of future publications.