Quantum vacuum effects in non-relativistic quantum field theory

Nonlinearities in the dispersion relations associated with different interactions designs, boundary conditions and the existence of a physical cut-off scale can alter the quantum vacuum energy of a nonrelativistic system nontrivially. As a material realization of this, we consider a 1D-periodic rotating, interacting non-relativistic setup. The quantum vacuum energy of such a system is expected to comprise two contributions: a fluctuation-induced quantum contribution and a repulsive centrifugal-like term. We analyze the problem in detail within a complex Schoedinger quantum field theory with a quartic interaction potential and perform the calculations non-perturbatively in the interaction strength by exploiting the nonlinear structure of the associated nonlinear Schroedinger equation. Calculations are done in both zeta-regularization, as well as by introducing a cut-off scale. We find a generic, regularization-independent behavior, where the competition between the interaction and rotation can be balanced at some critical ring-size, where the quantum vacuum energy has a maxima and the force changes sign. The inclusion of a cut-off smoothes out the vacuum energy at small distance but leaves unaltered the long distance behavior. We discuss how this behavior can be tested with ultracold-atoms.


I. INTRODUCTION
In quantum field theory, the canonical quantization scheme does not fix the order of non-commuting operators in the Hamiltonian, leaving a residual divergent "zero-point energy" contribution to the energy density (in natural units): with ω n representing the frequencies of the quantum fluctuations.Wick's normal ordering is then used to enforce a specific order of operators' products, resulting in the subtraction of this infinite shift from the vacuum expectation value (vev) of the Hamiltonian that will then vanish.This has the consequence that the quantum vacuum, so defined, does not carry energy, linear or angular momentum.Such a procedure is usually justified by saying that a constant shift in the energy cannot be measured, although this view is not entirely tenable as any finite energy is, in principle, measurable due to its gravitational effect.In relativistic quantum field theory, a better justification follows from the fact that the expectation value of the Hamiltonian in the noninteracting vacuum (i.e., in absence of external fields or interactions) must vanish for the Hamiltonian, a generator of the Poincaré group, to satisfy the correct commutation rules.Then, the usual notion of a noninteracting vacuum as a state devoid of energy follows, justifying the use of normal ordering [1,2].
Even without calling gravity into question [3], a variety of quantum vacuum phenomena, most notably the Casimir effect [4], clearly demonstrates some level of inadequacy of the above definition of an empty physical vacuum tout court.In the original version of the Casimir effect, for example, this was evident owing to the imposition of boundary conditions on the quantum fluctuations of the electromagnetic field in the presence of perfectly conducting, parallel plates, resulting in an attractive force between the plates.More general (and realistic) situations are not different, as boundary conditions result from quantum fields existing in interaction with other fields, and modify the spectrum of the quantum fluctuations, thus changing the zero-point energy.
These arguments converge into Casimir's definition of the energy of the quantum vacuum E vac as the difference between the zero-point energies in the presence, E [∂B], and in absence, E [∅], of boundaries: Such a definition is compatible with the vanishing of the vev of the Hamiltonian in the noninteracting vacuum (i.e., no boundary) and gives a calculable recipe (within any regularization scheme) of the quantum vacuum energy in response to changes in external conditions [2,5,6].This view on the complexity of the vacuum has been vindicated during the past quarter of a century by many successful experiments starting with [7,8] (see also Ref. [9] for a recent additional list of examples of applications to nanophotonics, nanomechanics, and chemistry).
A less explored question concerns the quantum va-arXiv:2309.07454v1[hep-th] 14 Sep 2023 cuum energy in non-relativistic systems (see, for some discussions, Refs.[10][11][12][13][14][15]). The answer might seem simple, since in a non-relativistic context there is no issue associated with antiparticles or the ordering of the operators, suggesting that the zero-point energy can be safely ignored.However, this is not the case in general.Even from the vantage point of the original Casimir effect, the story remains subtle because the quantum vacuum energy emerges from deformations of the electromagnetic quantum fluctuations, and no simple nonrelativistic limit can be taken: the photon is massless and propagates at the speed of light.However, in a non-relativistic set-up one can imagine emergent degrees of freedom, constrained by boundaries, and how these could give rise to non-trivial quantum vacuum phenomena and a number of works have explored such question, particularly in the context of quantum liquids and Bose-Einstein condensation where (1) contributes to the zero temperature thermodynamic potential (on top of the classical ground state contribution); see, for example, Refs.[16][17][18][19][20][21][22][23][24][25][26].
There are at least two reasons why in a non-relativistic setting the situation is far from obvious.The first is that any time we are in the presence of interactions and nontrivial boundary conditions, the frequencies ω n in (1) develop a non-trivial dependence on the ground state of the system.This can be seen using the background field method (see Ref. [11]), although computing the frequencies within this framework becomes a hard task.Earlier calculations relying on a perturbative expansion around small coupling exist [27][28][29] and more recently Ref. [30] has developed a way to compute the quantum vacuum energy for a relativstic 1 + 1 dimensional scalar field theory without relying on expansions in powers of the interaction strength (see also Refs.[31][32][33][34]).The second reason has to do with the regularization.In the relativistic case, the quantum vacuum energy emerges from the summation of the entire spectrum as in (1); this summation is divergent and must be regularized.A subtlety with this is due to the existence of a physical cut-off that may alter the spectral sum in (1).Within a lattice approach this should be possible (see Ref. [12]), however it is not at all obvious how to do this within an effective field theory approach.It is certainly an interesting question to ask whether any remnant of the quantum vacuum energy remains in the non-relativistic limit.
When the model under consideration is nonlinear, the difference in the dispersion relation due to the presence of interactions, the presence of external forcing (e.g., rotation), a physical cut-off scale and boundary conditions are all factors that together conjure to induce intricate behaviors in the quantum vacuum energy.Here we look at the above questions within the paradigmatic nonlinear Schrödinger equation.Our approach to compute the quantum vacuum energy exploits the integrability structure of the nonlinear Schrödinger equation associated with our problem.The calculations are done both using zeta-regularization including the contribution from the whole spectrum, as well as a more physical regularization scheme where the spectral sums are modulated by a frequency dependent window-function that suppresses the contribution of the high-energy modes, leaving a dependence on a physical cross-over scale.As we shall see the two methods lead to compatible results, with the only expected consequence of the cut-off being that of regularizing the vacuum energy at short distance.In conclusion, we will describe how our predictions can, in principle, be measured experimentally with cold-atom rings.

II. NON-RELATIVISTIC SCHRÖDINGER MODEL
We shall consider a system of non-relativistic interacting bosons, described by a complex Schrödinger quantum field Φ = (ϕ 1 + iϕ 2 )/ √ 2, with ϕ 1 , ϕ 2 ∈ R, confined to a 1D ring of radius R, rotating with constant angular velocity Ω.We assume that the periodicity of the ring is externally broken by the presence of a barrier that we describe by imposing Dirichlet boundary conditions at one point on the ring.The Lagrangian density is where 0 ≤ φ ≤ 2π, x = Rφ, ̇= d/dt and ′ = d/dφ.We adopt units of ℏ = 1.Expression (3) represents the Lagrangian density of an observer co-rotating with the ring.In this reference frame, boundary conditions for the co-rotating observer are time-independent [35][36][37].The following nonlinear Schrödinger equation can be derived from (3): The normal mode decomposition can be carried out by looking for stationary solutions of the form This allows us to write the original equation ( 4) as To solve (6) we decompose f p as that leads to The above system of equations can be solved analytically, first obtaining α ′ in terms of ρ from Eq. ( 9), (C is an integration constant) and then substituting α ′ in Eq. ( 8); this gives rise to a cubic nonlinear equation in ρ that can be solved in terms of Jacobi elliptic functions.Imposition of the boundary conditions selects the solution as a Jacobi sn function and leads to the following quantization conditions for the eigenfrequencies.The procedure is straightforward but lengthy.For completeness we give all the details in the Appendix and refer the reader to Refs.[38][39][40][41][42] for further details on elliptic equations.The solution can be written as with the normalization factor A n expressed in terms of elliptic integrals of first and second kind, K(z) and E(z) respectively, The momentum q n and the elliptic modulus k n are quantized according to the following relations where (13) comes from the periodicity of the solution and ( 14) is derived from the first integral of the equation of motion.Finally, the eigenfrequencies are given by Details on how to derive Eqs. ( 11), ( 12), ( 13), ( 14) and ( 15) are given in the Appendix.

III. QUANTUM VACUUM ENERGY AND SPECTRAL ASYMPTOTICS
In the following, we illustrate how to compute the quantity (1) for the present case.A non-renormalized expression for the quantum vacuum energy can be writ-  15).Here ΩRml/ℏ = 0.5 and λ = ml/ℏ 2 = 0.5 throughout.
ten as follows (see Ref. [10,43]): where s ∈ C is a complex-valued regularization parameter and µ is a renormalization scale with dimension of energy.The index r is a reminder that ( 16) refers to the co-rotating frame.The eigenvalues ω n are given in terms of the nonlinear, coupled algebraic equations ( 13), ( 14) and (15).The regularization of ( 16) is done by finding a representation that converges in some region of the complex-s plane, followed by analytical continuation to the physical value s → 0. Here, we use the spectral asymptotics of the eigenvalues and express ( 16) as where and The quantity ω (a) n represents the asymptotic expansion of the eigenvalues ω n as a function of the quantum number n.If the asymptotic expansion includes all terms up to O(1/n 2 ) as we shall do here, then ∆ ∼ O(1/n 2 ), i.e. (19), and thus converges for s → 0 (in formula (19) we have already set s → 0).Such a procedure simply confines the divergences to E r ˜(s) that will need explicit regularization.The first step of the process is to obtain the asymptotic behavior of the eigenvalues.This can be obtained nu-merically, but it is not difficult to find its analytical form.Since the left hand side of Eq.( 14) converges to zero for n → ∞, while the right hand side, as a function of k n , goes to zero only in the limit k n → 0, while decreasing monotonically for increasing k n > 0, the right hand side of ( 14) for small k n gives the relevant limit to capture the large n asymptotic behavior k 2 n ≈ 2λmR/(πn 2 ).This result used in conjunction with ( 13) and ( 15) allows us to readily extract the leading asymptotic behavior of ω n : where η 2 = 8mR 2 and ρ 2 = 3λ/(8πR) − mR 2 Ω 2 /2.Fig. 1 shows a comparison between the eigenvalues computed numerically and their asymptotic counterpart.The largen scaling of the eigenvalues is consistent with Weyl's law that in the present case predicts a leading large-n behavior of the ω n scaling as n 2 and independent of λ [44].
Using (20), and the Chowla-Selberg representation (See Refs.[45,46]), from which the limit s → 0 can be taken to arrive at the following regularized expression Thus, the total quantum vacuum energy in the corotating frame is given by E r = E r ˜+ ∆.To get the energy in the laboratory frame E s one can use E s − E r = ΩL, where L = −∂E r /∂Ω is the angular momentum [31,35,36,47]: The resulting force Ignoring for the time being the contributions from ∆, the above expression comprises a contribution proportional to −λ/R that vanishes for λ → 0 and scales as the inverse of the ring size: this is an attractive "Casimir-like" contribution.The other contribution E I = 1 2 I R Ω 2 is proportional to the moment of inertia of the ring with radius R. The vanishing behavior for λ → 0 and Ω → 0 is consistent with the fact that the quantum vacuum energy should vanishes in the absence of interactions and boundary condtions.The angular velocity appears as the square of Ω, and this is again consistent with the fact that our model does not include parity breaking terms, thus the energy should be symmetric wrt Ω ↔ −Ω.The force vanishes at the critical radius with its sign changing from negative-attractive for R < R crit to positive-repulsive for R > R crit .Interestingly, also the way the force scales with the ring size changes with the angular velocity: it scales linearly in the regime of fast rotation, while it scales as the inverse square of the ring size for slow rotation.The symbol "≈" in (26) indicates that the contribution of ∆ has been ignored.Units of ℏ are restored in the numerics.Fig. 2 shows the quantum vacuum energy (panels a and c) as a function of radius R and rotation strength Ω respectively, while the two lower panels (b and d) show the corresponding force associated with each dataset from (a) and (c).The grey shaded region shows the parameter regime where the force is repulsive.Fig. 3 shows heatmaps of Eq. ( 26) in the (R, Ω) and (R, λ) parameter spaces, (a) and (b) respectively.In panel (a) the interaction strength is λml/ℏ 2 = 10 while the rotation strength is Ωml 2 /ℏ = 5 in (b).The solid blue lines in both panels show the border between the repulsive regime and the causality limit defined by ΩRml/ℏ = 1.The red dashed line indicates where the force changes sign, obtained from Eq. (26).The red data point in each panel corresponds to the point (R • , Ω • ) in (a) and (R • , λ • ) in (b) where Eq. ( 26) and the causality limit coincide, and The point defined by Eq. ( 27) in Fig. 3 shows the maximum rotation strength where repulsive solutions are obtained, then the model of Eq. ( 4) is expected to support a causal repulsive force in the region Ω c < Ω < R −1 (ml/ℏ) and R > R • .Likewise for panel (b) the causal repulsive regime is defined between 0 < λ < λ c and 0 < R < R • .An analysis, qualitatively similar to Fig. 3(b), can be done for the (R, Ω) parameter space with constant λ.
A subtle point has to do with how the above results will change in the presence of a cut-off scale associated with a minimal length scale (e.g., the inter-atomic separation scale).We address this question by modifying the regularization procedure to include a frequency dependent window function.This is implemented by defining and the residual ∆ as Here, we choose the window function as follows: with the argument of the exponential set by the leading large-n asymptotics of the spectrum.The cut-off scale ℓ c determines how high-frequency modes are suppressed.The limit ℓ c → 0 of (28-29) returns the non-regularized expression for E r ˜discussed earlier.While the choice of σ n (ℓ c ) is arbitrary, (30) allows to write (28) as where θ 3 is the following Jacobi thetanull function [48], This choice of regularization has the advantage that the first term of (31) corresponds to the fully resummed result and the effect of the cut-off is encoded in the latter two terms of (31).Proving the consistency of the two approaches, with and without the cut-off, requires care since in the limit ℓ c → 0 the theta function diverges and requires regularization.The theta function can be regularized by requiring that the cut-off dependent contribution in (31) vanishes in the limit of ℓ c → 0, corresponding to a subtraction of the divergent contribution.To compute the finite ℓ c → 0 limit of (31) we use the modular transformation along with the small x expansion of the theta function [48], leading to Using this expression in (31) and removing the divergent part, consistently with the regularization of the theta function, gives the expected fully resummed result.The corrections due to the cut-off near R ∼ R crit to the fully resummed result can be estimated assuming that ℓ c ≪ R crit and can be computed including higher order corrections in the expansion of the theta function: where σ = exp (−π/x).Using (35) in (31) implies that corrections to the fully resummed result are exponentially small, that is the behavior of the vacuum energy is robust against the inclusion of a cut-off smaller than the critical radius for large enough ring size.For small R, we expect the cut-off to regulate the diverging 1/R behavior of leading term in Eq. (31).Expanding the theta functions in (31) for ℓ c /R 2 large, gives at leading order which can be contrasted with the ρ 2 ∼ R −2 behavior of the vacuum energy as obtained by full resummation.

IV. CONCLUSIONS
The behavior of the quantum vacuum energy of an interacting non-relativistic system is far from trivial.Here we have looked at an example of this using a nonlinear Schrödinger quantum field theory and computed the quantum vacuum energy and force without resorting to any perturbative expansion in the coupling constant, simply relying on the exact integrability of the nonlinear problem.The novel results are summarized in the "phase diagram" of Fig. 3 which shows how the fluctuation-induced force as a function of rotation and interaction strength separates into a noncausal region plus an attractive-repulsive region.This behavior arises from the stabilization between an attractive Casimir-like component and a repulsive centrifugal one.
An interesting potential connection seems evident between our quantum field theoretical set-up and the area of ultracold atoms.A possibly relevant example is the set-up of Ref. [49], which consists a 23 Na BEC confined in a ring of size R ∼ 20µm.Considering a quasi-1D approximation the interaction strength λ can be expressed in terms of the scattering length a s and the transverse length scale l as λ = g/(πl 2 ) (here g = 4πℏ 2 a s /m defines the atomic interaction with a s = 50a 0 for 23 Na) [50].Taking l ∼ 2µm to ensure that l ≪ R and assuming a condensate N = 2 × 10 3 atoms (species other than 23 Na, e.g. 87Rb, can have larger atom numbers and different scattering lengths), allows us to arrive at a dimensionless interaction strength λml/ℏ 2 ∼ 4a s N/l, which, using the above parameter values, gives 4a s N/l ∼ 10, a value close to that used in Fig. (3-a).The force F s can also be estimated in a similar manner, using Eq. ( 25) and the above definitions we obtain a dimensionless force Using a rotation speed of Ω∼2π×25Hz from the experiment of Ref. [51], we obtain F s ml 3 /ℏ 2 ∼ 0.1, modest but potentially large enough to be observable in a future experiment.Using these values a ring of size R ∼ 20µm would fall in the causal repulsive region of Fig. 3(a), and Ωml 2 /ℏ ∼ 0.2 favoring lower rotation frequencies.In this work Dirichlet boundary conditions have been used, which could be simulated using a weak-link as realized in the BEC ring experiments of Refs.[52,53].
The physical system described in this work has poten-tial applications in atomtronics [54], facilitating an additional opportunity to explore the fundamental physics associated with the quantum vacuum.Extensions to systems of fermions [55] or with multiply-connected geometries [56] offers additional avenues to explore the effects described in this work in uncharted scenarios.
where we have defined The first integral can be obtained by multiplying both sides of (A.43) by ρ ′ and integrating where H is an integration constant.Finally, multiplying both sides for ρ 2 and changing variables, gives The above equation corresponds to the differential equation of an undamped quadratic anharmonic oscillator, whose canonic form can be obtained by differentiating with respect to and dividing by 2s ′ , Following Ref. [38] (see also Refs.[39][40][41][42]), we can write Eq. (A.47) with α 1 , α 2 , α 3 being roots of RHS polynomial, and The advantage of expressing Eq. (A.47) as Eq.(A.49) is that the latter is one of the standard nonlinear ordinary differential equation, whose solutions can be expressed in terms of Jacobi elliptic functions as: Summarizing, we have that, together with the normalization condition, give us 4 independent equations for 4 variables (α 1 , α 2 , H, ω).In practice, the above set of equations defines the quantization condition of the eigenvalues ω p .It is possible to show that the system admits a unique solution but it is easier to proceed in an alternative and faster way.Using the condition F 1 = 0 directly in equation (A.43) leads to ρ ′′ + F 2 ρ + F 3 ρ 3 = 0. (A.69)

FIG. 1 .
FIG. 1. (color online) Comparison between(15) and the eigenvalues computed numerically.Panel (a) shows the absolute difference between the asymptotic and exact nonlinear eigenvalues, while (b) shows individual datasets for fixed R. Coloured data in (b) were obtained numerically, while the grey dashed lines were calculated from Eq.(15).Here ΩRml/ℏ = 0.5 and λ = ml/ℏ 2 = 0.5 throughout.

FIG. 2 .
FIG. 2. (color online) Quantum vacuum energy and force.(a) and (c) show the quantum vacuum energy, Eq. (24) for the same λ (colour groups) and varying Ω (panel a) or R (panel c) (see text labels).(b) and (d) show the corresponding force, Eq. (25) for each data set.The light grey shading indicates the parameter regions where the force changes sign from attractive to repulsive.Throughout the paper, the quantity l represents a generic unit-scale.

2 FIG. 3 .
FIG. 3. (color online) Quantum induced-force heatmaps.(a) shows Eq. (25) in the limit ∆ = 0 in the (Ω, R) parameter space for fixed λml/ℏ 2 = 10.The solid blue line indicates the border between the repulsive and noncausal regions, while the dashed red line indicates the point at which the force changes sign.(b) shows the magnitude of the force in the (λ, R) parameter space for fixed Ωml 2 /ℏ = 5.