Analogue tachyons in SNAIL transmission lines

Tachyons are hypothetical particles with imaginary mass that travel faster than light. However, methods to experimentally verify whether tachyons exist are lacking. Here, we propose a novel scheme to create analogue tachyons using a transmission line composed of superconducting nonlinear asymmetric inductive elements and to detect them by controlling the wavenumber in order to extend their lifetime. In particular, we numerically demonstrate the exotic property of tachyons where their velocity increases with decreasing energy. Our proposal offers a promising approach to understanding tachyon condensation, which is crucial for elucidating the origins of the Universe, in a realizable laboratory system.


Introduction
A tachyon is a particle with an imaginary mass that travels faster than light [1,2]; a distinctive property is that a tachyon's velocity increases as its energy decreases.In modern particle physics, tachyons are understood rather as an instability in a system; the negative squared mass of a particle can be interpreted through an unstable potential local maximum in the Klein-Gordon (KG) field equation.The tachyon field then only exists in an unstable state, and can spontaneously reduce its energy by 'rolling down' to the local minimum of the potential.This results in the field quanta obtaining a real mass and becoming stable, a phenomenon known as tachyon condensation [3].This process is related to spontaneous symmetry breaking, which occurs during an assumed inflationary period [4,5] of the early Universe.The study of tachyons also provides valuable insights into the generation of matter in the early Universe.
The typical potential for a system undergoing spontaneous symmetry breaking leading to tachyon condensation is represented by the wine-bottle potential: V(φ) = µ|φ| 2 + λ|φ| 4 , where µ and λ are constants (λ > 0).The generation of unstable states that induce spontaneous symmetry breaking can be achieved by externally controlling the system's potential using parameters such as µ and λ.Specifically, when µ is varied from a positive to a negative value, an unstable state results that induces spontaneous symmetry breaking, which is essential for creating analogue tachyons in laboratory systems.
In the present work, we propose a method to create and observe tachyons in a feasible experimental system that provides a foundation for studying tachyon condensation.Our approach involves using a transmission line with superconducting nonlinear asymmetric inductive elements, known as SNAILs [44], to generate tachyons.We achieve this by controlling the external magnetic field such that the potential of a SNAIL switches from a stable state to unstable tachyon state, resulting in a phase transition; tachyons exist as an unstable state at a local maximum in the potential.Furthermore, we clarify the properties of tachyons proposed by Aharonov et al [45] based on the dispersion relation.We confirm the existence of long-lived tachyon modes numerically and discuss the potential observability of analogue tachyons in our system.In the following section 2, we introduce the SNAIL unit cell transmission line circuit and show through Kirchhoff 's laws that it can be modeled approximately by a discrete, third-order nonlinear KG equation.Section 3 presents the results of approximate analytical and exact numerical solutions to these equations, verifying the existence of tachyon modes and wavepackets, with details concerning the definition of the (superluminal) tachyon velocity for the discrete transmission line given in appendix.We finish with some concluding remarks in section 4.

Model
The generation of tachyons requires a potential with tunable coefficients for the quadratic field term which can be adjusted from positive to negative while the fourth-order term remains positive.We can create such a tunable potential by utilizing SNAILs [44] with an adjustable external magnetic field.SNAILs are superconducting loops consisting of one Josephson junction with critical current αI c in parallel with multiple identical Josephson junctions with critical current I c in series.This proposal uses the simplest SNAIL consisting of three junctions with controllable α as shown in figure 1.
The two junctions on the left side of a SNAIL are connected in series, resulting in equal Josephson currents through each junction and equal phase differences across them.
From Kirchhoff 's current conservation law, the current I s through a SNAIL is given by I s = I r + I l = αI c sin ϕ r + I c sin ϕ l , where I r(l) and ϕ r(l) represent the current and the phase difference through the junction on right (left) side, respectively.Through the flux quantization condition giving the relation ϕ r − ϕ l = 2πΦ S /Φ 0 ≡ ϕ s , with Φ S the magnetic flux through a SNAIL and Φ 0 the magnetic flux quantum, a SNAIL has only one degree of freedom, namely the variable ϕ; the current I s is then rewritten as I s = αI c sin ϕ + I c sin {(ϕ − ϕ s )/2}.Unlike a single JJ, SNAILs have the capability to introduce odd-order nonlinearities in the potential, in addition to the even-order nonlinearities [44].The coefficients for these nonlinear terms are controlled by α and ϕ s .Notably, when we set ϕ s to zero, the odd-order nonlinearities vanish.
Assuming ϕ s = 0 to obtain the desired potential for spontaneous symmetry breaking, the SNAIL potential is derived as , where E J = Φ 0 I c /(2π) and we set U(0) = 0. Performing a Taylor expansion around ϕ = 0, we obtain which is of the same form as the potential mentioned above.In particular, utilizing a SNAIL, the respective ϕ 2 and ϕ 4 coefficients in the potential are µ = E J (α + 1/2)/2!and λ = −E J (α + 1/8)/4!.These coefficients are controlled by the parameter α as indicated in figure 2. Since a phase transition occurs when λ > 0, α must be less than −1/8.When −1/2 ⩽ α ⩽ −1/8, and the coefficient of ϕ 2 is positive (µ ⩾ 0), the potential has a stable point at ϕ = 0 (figures 2(c)-(e)).On the other hand, when α < −1/2, the coefficient of ϕ 2 is negative (µ < 0), and an unstable state is created, where a local maximum exists at ϕ = 0 (figures 2(a) and (b)).Therefore, tachyon condensation can be induced by tuning the parameter α in the range greater than −1/2 and less than −1/8.We now describe how to control the parameter α related to the critical current of the Josephson junction on the right side of SNAIL.The critical current can be effectively tuned as α(Φ ext ) = 2 cos(πΦ ext /Φ 0 ) by replacing the single junction on the right side of a SNAIL unit cell with a dc SQUID which consists of two identical Josephson junctions in a loop, and with an external magnetic flux Φ ext threading the dc SQUID loop.In order to introduce an external magnetic field only in the region of the dc SQUIDs, we can either vertically design the SQUIDs and apply a lateral external flux or cover the entire circuit with a superconducting sheet, creating a hole solely in the dc SQUID area [46].
To obtain a tachyon field, we consider a one-dimensional transmission line with SNAIL unit cells as shown in figure 1 [47].Kirchhoff 's current conservation law leads to where I n , ϕ n , and V n are respectively the current through the nth inductor with inductance L, the phase difference of the nth SNAIL with Josephson capacitance C J , and the voltage applied to the nth SNAIL.The current I s n through the nth SNAIL is given by the sum of the Josephson current and displacement current  which is derived from dq n /dt, where q n is the charge in the JJ of the nth SNAIL.The quantized flux condition is given as ϕ n − ϕ n+1 = 2π (L/Φ 0 )I n for the loop including the nth inductor, the nth and (n + 1)th SNAILs.Together with the Josephson relation, equation (2) leads to the discrete double sine-Gordon equation, which reduces to the discrete third-order nonlinear KG equation: ).This equation is the basis for the following numerical simulation dynamics of ϕ n .In existing systems proposed for creating tachyon modes, the phase transition has been controlled by the temperature.On the other hand, in our SNAIL transmission line, the mass of the KG equation (m KG = ω J α + 1/2) can be controlled by α as a function of the external magnetic flux, leading to the phase transition.A Josephson device is sensitive to external magnetic fields because of the interference effect of the Josephson phase difference and has good magnetic field controllability in practice, making our system suitable for generating unstable states, i.e. analogue tachyonic states.

Tachyons
We begin this section by first reviewing the properties of long-lived tachyon modes revealed by Aharonov et al in their original analogue tachyon dynamics investigation [45].They noted that a chain of coupled, inverted pendulums corresponds to the KG equation with an imaginary mass and showed that superluminal (i.e.tachyon) modes with wavenumbers larger than a critical wavenumber k c indeed exist in a long-lived state without collapsing, despite being in an inverted (unstable) pendulum state.From the dispersion relation c , the frequency is real indicating the presence of an oscillatory solution, and its derived group velocity is superluminal since v g = dω/dk = v 2 0 k/ω > v 0 when k c < |k|, with v 0 being the limiting velocity.A similar dispersion relation was obtained in [48,49], although they do not discuss tachyons.In these papers, pumping and dissipation were introduced to achieve the dispersion relation, but it is difficult to control the dissipation in general.The controllability provided by the external magnetic flux in our system surpasses that of their system.In addition, our system includes a λϕ 4 model with symmetry breaking, which allows us to explore tachyon condensation, whereas their phase transmission mechanism based on dynamical instability does not allow for spontaneous symmetry breaking.
The extended lifetime of the tachyon modes at k c < |k| is due to a cooperative phenomenon in the dynamics of the coupled pendulums; a single inverted pendulum can only continue to fall once it starts to fall, whereas neighboring coupled inverted pendulums can collectively prevent a pendulum from falling by supporting it-the more pendulums the longer they can remain inverted.However, it is not always possible to save a pendulum from falling.Clearly, too many pendulums on the verge of falling cannot be prevented from doing so through the support of other pendulums.In fact, when the wave number |k| < k c , many pendulums are contained in one wavelength, making it impossible to prevent their falling.With such a wavenumber, the frequency becomes imaginary (ω 2 < 0) and the system has a divergent solution described as ϕ = exp(ikx) exp(ωt).Therefore, if the critical wavenumber can be controlled, we can eliminate the unstable divergence mode in the infinite pendulum number limit, while make observations of a long-lived tachyon mode possible with a large but finite number of pendulums.Now let us examine the fundamental properties of tachyons proposed by Aharonov et al and consider a method for observing tachyons in our system.In particular, we investigate the nature of the unstable modes that prevent observation.We describe the method to obtain the long-lived modes by controlling the mode wavenumbers, hence facilitating their observability.

The dispersion relation of our system can be expressed as
, similar to that discussed by Aharonov et al where the critical wavenumber, k c = r |α + 1/2|/a > 0 and v 0 = aω 0 , with a being the unit cell length.This implies that there exist long-lived unstable tachyon modes if the wavenumber is larger than the critical wavenumber (k c < |k|), while the field evolves towards the stable state around the local minimum of the potential when |k| < k c .
Figure 3 shows the time evolution of a single tachyon mode based on equation (4). Figure 3(a) demonstrates that the tachyon field is unstable when |k| < k c .The process in the early stages of evolution can be analytically determined by solving the KG equation with an imaginary mass and is expressed as ϕ(x, t) = A sin(kx) cosh(v 0 k 2 c − k 2 t), which is in good agreement with the numerical simulation.In contrast, the tachyon field exhibits long-lived oscillations around ϕ = 0 when k c < |k|, even though it is in an unstable state as shown in figure 3(b).These simulations support the analytical results of Aharonov et al [45].
Bearing in mind the above single-mode results, let us now investigate the behavior of wavepackets composed of multiple modes, including both long-lived metastable and short-lived unstable modes, in order to address the observability of tachyon wavepackets.Figure 4(a) depicts the time evolution of an initial Gaussian wave packet.Initially localized around ϕ n = 0, the Gaussian peak splits into two peaks that propagate towards both sides due to momentum conservation.The peaks eventually evolve towards the local minima of the potential and start to oscillate around them.This is reminiscent of tachyon condensation but does not prove the existence of tachyons, which is the basic premise of such condensation.4).The dashed line represents the critical wavenumber kc.Initially, the wave packet separates into a low wavenumber component, which broadens at rest, and two high wavenumber components that move away in space as shown in (a) ( 1)-( 3).As time passes, the modes with smaller wavenumbers are excited shown in (b), the central part with small wavenumber components are growing as shown in (a) ( 4)-( 6), leading to decay.As time progresses further, nonlinear effects become increasingly important, giving rise to a shock-wave phenomenon around the large peaks.This leads to the emergence of two prominent outer oscillations show in (a) ( 7)- (11).For tachyon observation, it is essential to create long-lived wave packets that contain as few immediately unstable modes below k c as possible.In fact, spectral analysis shows that the fraction of such modes increases with the time evolution of the wave packet as shown in figure 4(b).To reduce the number of immediately unstable modes, k c was made as small as possible by controlling the magnetic field and arranging the circuit parameters to make α and r small, respectively.Furthermore, an initial Gaussian wave packet with variance w was prepared by introducing a carrier wave and making the central (carrier) wavenumber large compared to k c + 2/w.
We numerically confirm the existence of a long-lived tachyon by measuring the propagation velocity of an initial Gaussian wave packet with exponentially suppressed short-lived unstable modes.Despite the nonlinear mode mixing inherent in our system that generates such modes during wavepacket propagation, the tachyon wave packet propagates along the transmission line for an extended duration that is sufficient to facilitate observation.The velocity of the tachyon can be measured by a time-of-flight technique based on voltage observations at a fixed position through v g = ∆x/∆t, where ∆x is the probed length the wavepacket propagates and ∆t corresponds to the travel time of the wavepacket.Figure 5(a) shows the frequency of the wave packet as a function of its velocity.The normalized group velocity is given by dividing the group velocity obtained in the numerical simulation by the speed of light in the discrete transmission line (See appendix for the details of the speed of light definition).It exhibits the counterintuitive properties of tachyons expressed as ω = im KG / (v g /v 0 ) 2 − 1 derived from the dispersion relation of tachyons shown in figure 5(b).This numerical verification of tachyons' unique dynamical properties in a realizable transmission line comprising SNAIL unit cell elements is the main result of this Letter.Note that there is no superluminal signaling, since the group velocity is faster than the speed of light v 0 in the transmission line, but in this case, the group velocity is not interpreted as a means of information transfer; the peak of a propagating Gaussian wave packet is not causally related to the wave packet peak at an earlier time [50].Assuming the following feasible circuit parameters: L ≈ 10 −8 H, C J ≈ 10 −13 F, and ω 0 ≈ 2π × 5 × 10 9 Hz [51,52], the bandwidth of the Gaussian packet is ∼ 10 9 Hz for w = 30a and the time-of-flight is 200a/(ω 0 a) ∼ 10 −8 s for n = 200.These values are sufficient for experimental observability.

Concluding remarks
We have proposed a scheme to create and detect tachyon modes in an experimentally feasible transmission line comprising SNAIL unit cells.Superluminal propagation has been already observed as evanescent modes and tunneling modes in various systems [53][54][55].In these cases, the superluminality was manifested by introducing an imaginary wavenumber associated with an imaginary mass in the system.However, the systems considered in these papers did not allow for the imaginary mass to be controlled directly.In addition, while these earlier works focused on superluminarity, the unique property of a tachyon such as the relativistic energy and velocity relationships, remained unexplored and unverified.
In this paper, we have realized superluminal tachyons by introducing the instability through the adjustment of the potential energy in a SNAIL transmission line, which corresponds to incorporating an effective imaginary mass required for the existence of tachyons.The direct artificial control of the imaginary mass is one distinctive feature of our study.We have numerically reproduced the analytical prediction made by Aharonov et al [45] concerning the fundamental properties of tachyons, and numerically confirmed the mysterious properties of tachyons that travel faster the smaller is their energy for parameters that are similar to those in existing experiments [51,52].Furthermore, we have demonstrated the feasibility of generating long-lived tachyons, which offers a significant advantage over previous research efforts.This development provides the stage for experimental tachyon verification.
We have also obtained supporting evidence for tachyon condensation regarded as a phase transition due to the unstable tachyon field evolving towards a stable local minimum of the field potential.These results provide a basis for understanding tachyon condensation, which is key to the evolution of the early, inflationary Universe.and the group velocity is given by The frequency of a tachyon is a function of its group velocity and can be expressed when α < −1/2 as follows, which is the same expression as the relationship between the frequency and the group velocity in the continuum approximation.In figure 5(a) of the main text, the group velocity obtained from the numerical simulations is normalized by the speed of light as described above.In particular, this normalization is achieved by performing the numerical simulation with α = −1/2 for the same wave packet.

Figure 1 .
Figure 1.Schematic diagram of a transmission line comprising identical inductors with inductance L and SNAIL unit cells.The terms In, Vn, and ϕn represent the current through the nth inductor, voltage applied to the nth SNAIL, and phase difference of the nth SNAIL, respectively.

Figure 3 .
Figure 3.The numerical results of the time evolution of the phase difference ϕn (solid lines) at site n for the single tachyon modes, in the potential indicated by the curved surface.We set the initial condition as ϕn(0) = A sin( kn) and dϕn/d t|t =0 = 0, where k = ak, and the periodic boundary condition as ϕ0( t) = ϕL( t) with L = 500, r = 0.1, α = −1.0,and A = 0.1.(a) k = 2π/L < kc and (b) k = 20π/L > kc, where kc = akc.In the (a) case (k < kc), the initial wave decays to the stable local minimum of the potential.On the other hand, in the (b) case (kc < k), the wave remains at the unstable local maximum of the potential.

Figure 4 .
Figure 4. (a) The time evolution of an initial Gaussian wave packet ϕn(0) = A exp[−(n − n0) 2 /(2w 2 )] and dϕn/d t|t =0 = 0, with L = 1000, r = 0.1, α = −0.6where A = 0.1, w = 30, and n0 = 500.The numbers (1)-(11) indicate snapshots at 40 time unit increments from t = 0 to t = 400, and the horizontal dotted line represents the local minimum of the potential ϕ + min .(b) The spectrum of the waves for time instants (1)-(4).The dashed line represents the critical wavenumber kc.Initially, the wave packet separates into a low wavenumber component, which broadens at rest, and two high wavenumber components that move away in space as shown in (a) (1)-(3).As time passes, the modes with smaller wavenumbers are excited shown in (b), the central part with small wavenumber components are growing as shown in (a) (4)-(6), leading to decay.As time progresses further, nonlinear effects become increasingly important, giving rise to a shock-wave phenomenon around the large peaks.This leads to the emergence of two prominent outer oscillations show in (a) (7)-(11).

Figure 6 .
Figure 6.The wavenumber dependence of the group velocity for different values of α in the continuum field appproximation (a) and for the original discrete circuit equations (b), where k = ak and vg = vg/v0.The lines represent the group velocities in descending order for α = −0.7,−0.6, −0.5, −0.4,−0.3.The bold line represents the group velocity in the massless KG equation for small ϕ magnitude with α = −0.5, corresponding to the speed of light.While the speed of light is independent of the wavenumber in the continuum approximation (a), it exhibits a wavenumber dependence for the original discrete circuit equations (b).

) which is indicated by the solid line in figure 7 . 1 ,
The circles display the results of a numerical simulation based on equation(3) in the main text by initializing a Gaussian wave packet.The group velocity is determined by a time-of-flight technique based on transient voltage measurements at two different fixed positions.The numerical results match well to the analytical solution.We can define the speed of light in our discrete transmission line by considering the group velocity when α = −1/2 as v procedure used in the continuum equation.It should be noted that in the discrete massless KG equation with α = −1/2, the group velocity depends on the wavenumber as shown by the bold line in figure6(b).In short, the speed of light has the wavenumber dependence in the discrete transmission line.If α > −1/2, the group velocity is slower than the speed of light.However, if α < −1/2, the presence of a superluminal tachyon mode is indicated, similar to the continuum field approximation case.When we normalize the group velocity by dividing it by the speed of light vthe frequency and the group velocity becomes ω = ω 0 kc ṽ2 g −

Figure 7 .
Figure 7.The relationship between the frequency and the group velocity.The solid line and circles represent the analytical and numerical results, respectively.