1 Introduction

The Hawking effect [1, 2], describing particle creation by quantum vacuum fluctuations in the presence of a black hole event horizon, can be mimicked in the laboratory, as argued by Unruh [3]. Since his seminal work, and with increasing experimental capabilities, a veritable explosion of interest occurred in what was dubbed analogue gravity [4], involving, for example, Bose-Einstein condensates [5,6,7,8,9,10,11,12,13,14,15,16,17,18], light in nonlinear media [19,20,21], magnets [22, 23], or superconducting circuits [24,25,26,27,28].

Gauge-gravity duality [29, 30] represents a powerful tool enabling the description of quantum gravity by certain classes of large-N gauge theories. Recently, it was conjectured that the Lyapunov exponent \(\lambda _L\) characterizing chaos in thermal quantum many-body systems has a rigorous bound, \(\lambda _L\le 2\pi \,T/\hbar \) [31]. This bound can also be derived from the eigenstate thermalization hypothesis [32,33,34,35], and is saturated when the quantum system is an exact holographic dual to a black hole [36,37,38]. Thus the MSS bound may capture an essential feature of gauge-gravity duality. Conversely, the bound might suggest that a quantum system with given \(\lambda _{L}\) has a minimal temperature \(T\ge \hbar \lambda _{L}/2\pi \). Thus a system with \(\lambda _{L}\ne 0\), although possessing zero temperature classically, may exhibit thermality in the semiclassical regime [39], that is \(\lambda _{L}\ne 0, T\ne 0\), when \(\hbar \ne 0\). This recalls a salient property of black holes: While being completely black classically, they semiclassically radiate at the Hawking temperature.

Experimental verification of the saturation of the MSS bound on the speed of scrambling in astrophysical black holes is challenging, and therefore studies on the information-scrambling physics of black hole have been confined to the theoretical realm see, e.g., Refs. [31, 36, 37] and citations therein. In order to render this bound accessible to an experimental investigation, we here propose to conduct its verification in an analogue gravity system [4] which can elegantly capture features of black holes. It has been shown that the quantum dynamics of the IHO exhibits thermal behavior with a temperature depending on the Lyapunov exponent \(\lambda _L\) [40, 41], which saturates the MSS bound [39]. The IHO model has, therefore, been used to explore the MSS bound on system temperature from a different angle, revealing its relation to the analogue of Hawking radiation [39], and to study the scattering outside a black hole horizon quantum mechanically [40, 41]. To the best of our knowledge, to both identify and probe experimentally systems displaying the fastest possible scrambling processes, as implied by saturating the MSS bound, and how to extract the corresponding Lyapunov exponent are still open issues. To fill this gap, we propose below an implementation of the IHO with a trapped ion. Specifically, we provide recipes for determining the Lyapunov exponent by measuring the IHO’s OTOC [31, 42], as well as for observing the analogue of Hawking radiation due to scattering at the IHO potential, and demonstrate that Hawking temperature and Lyapunov exponent indeed fulfill \(\lambda _L= 2\pi \,T/\hbar \).

Trapped ions, featuring a unique level of fidelity in preparation, control, and readout of quantum states, have been extensively used to simulate relativistic quantum physics, such as Zitterbewegung [43,44,45], the Klein paradox [46, 47], and (analogue) cosmological particle creation [48,49,50,51]. We here show that our proposal furnishes an important aspect of black hole quantum physics within current experimental reach for trapped ions.

2 Thermal behavior of a semiclassical inverted harmonic oscillator and Hawking radiation

In this section, we introduce the thermal behavior of a semiclassical IHO and Hawking radiation of black hole, and show the formal analogy between them.

2.1 Thermal behavior of a semiclassical IHO

The one-dimensional IHO, with Hamiltonian (\(\alpha >0\))

$$\begin{aligned} H=\frac{{p}^2}{2m}-\frac{\alpha }{2}{x}^2, \end{aligned}$$
(1)

yields the classical trajectory \(x(t)=c_1\,e^{\sqrt{\alpha /m}t}+c_2\,e^{-\sqrt{\alpha /m}t}\) (see discussion below for the relation of the IHO “particle” to the actual ion). This deterministic evolution is exponentially sensitive to the initial condition specified by \(c_1\) and \(c_2\), and the Lyapunov exponent \(\lambda _L=\sqrt{\alpha /m}\). In the quantum version of the IHO, the OTOC for momentum and position operators is given by:

$$\begin{aligned} C(t)=-\langle [{\hat{x}}(t),{\hat{p}}(0)]^2\rangle \sim \hbar ^2e^{2\lambda _Lt}. \end{aligned}$$
(2)

Note that the OTOC (2) for momentum operator, \({\hat{p}}\), and position operator, \({\hat{x}}\), actually does not depend on the averaging procedure \(\langle \dots \rangle \) since the commutation relation \([{\hat{x}}(t), {\hat{p}}(0)]\) is a c number. Applying the Lyapunov exponent bound [31], \(\lambda _L\le 2\pi \,T/\hbar \), to the quantum IHO predicts the existence of a lower bound on temperature \(T\ge \hbar \lambda _L/2\pi \), as conjectured in [39]. For the classical IHO (\(\hbar \rightarrow 0\)), which is both nonthermal and deterministic, this inequality becomes an equality as \(T:=0\). Conversely, in the semiclassical regime, following [39], the classical Lyapunov exponent \(\lambda _L\) acquires a quantum correction \(\mathcal{\,O}(\hbar )\), so that the right-hand side of the inequality becomes \(T\ge \frac{\hbar }{2\pi }[\lambda _L+\mathcal{\,O}(\hbar )]=\frac{\hbar }{2\pi }\lambda _L+\mathcal{\,O}(\hbar ^2)\). This suggests that, at least on the semiclassical level, an effective temperature of \(\mathcal{\,O}(\hbar )\) is induced, which closely resembles the situation of black hole thermodynamics: While black holes are classical solutions of general relativity, semiclassically, they create a thermal bath of radiation [1, 2].

To establish the correspondence of black hole radiation and quantum behavior of the IHO, we consider scattering off the IHO potential. Defining the light-cone-type operators \({\hat{u}}^\pm =({\hat{p}}^\prime \pm {\hat{x}}^\prime )/\sqrt{2}\) with \({\hat{p}}^\prime ={\hat{p}}/\sqrt{m}\) and \({\hat{x}}^\prime =\sqrt{\alpha }{\hat{x}}\), we can represent (1) in the \(u^\pm \) basis as \(H_\pm =\mp \,i\hbar \lambda _L\big (u^\pm \partial _{u^\pm }+\frac{1}{2}\big )\), respectively. The in- and outgoing energy eigenstates are of the form \(\frac{1}{\sqrt{2\pi \hbar \lambda _L}}(\pm \,u^\pm )^{\pm \,i\frac{\varepsilon }{\hbar \lambda _L}-\frac{1}{2}}\Theta (\pm \,u^\pm )\), where \(\Theta (u^\pm )\) is Heaviside step function, and \(\varepsilon \) is energy. Incoming and outgoing states are connected by a Mellin transform, which gives the scattering matrix (S matrix) in the form [40, 41]

$$\begin{aligned} S= & {} \frac{1}{\sqrt{2\pi }}\exp \bigg [-i\frac{\varepsilon }{\hbar \lambda _L}\log \hbar \lambda _L\bigg ]\Gamma \bigg (\frac{1}{2}-i\frac{\varepsilon }{\hbar \lambda _L}\bigg ) \nonumber \\&\times \left( \begin{array}{cccc} e^{-i\frac{\pi }{4}}e^{-\frac{\pi \,\varepsilon }{2\hbar \lambda _L}} &{}\quad e^{i\frac{\pi }{4}}e^{\frac{\pi \,\varepsilon }{2\hbar \lambda _L}} \\ e^{i\frac{\pi }{4}}e^{\frac{\pi \,\varepsilon }{2\hbar \lambda _L}} &{}\quad e^{-i\frac{\pi }{4}}e^{-\frac{\pi \,\varepsilon }{2\hbar \lambda _L}} \end{array} \right) , \end{aligned}$$
(3)

where \(\Gamma \) is the Gamma function. From the S matrix, the transmission and reflection probability are respectively given by, using \(|\Gamma \big (\frac{1}{2}-i\frac{\varepsilon }{\hbar \lambda _L}\big )|^2=\pi {{\,\mathrm{sech}\,}}(\frac{\pi \varepsilon }{\hbar \lambda _L})\),

$$\begin{aligned} |{\mathcal T}(\varepsilon )|^2=\frac{1}{1+e^{-2\pi \,\varepsilon /\hbar \lambda _L}},~~|{\mathcal R}(\varepsilon )|^2=\frac{1}{1+e^{2\pi \,\varepsilon /\hbar \lambda _L}}, \end{aligned}$$
(4)

corresponding to a thermal distribution, with a temperature given by \(T=\hbar \lambda _L/2\pi \).

As illustrated in Fig. 1, we can understand the transmission coefficient above by the quantum mechanical tunneling process through a barrier of the particle with negative energy \(\varepsilon \) moving towards the inverted harmonic potential from the left \((x\rightarrow 0)\), with a tunneling probability \(|{\mathcal {T}}(\varepsilon )|^2\). Similarly, the incoming positive energy particle \((\varepsilon >0)\) from the right may be reflected by the potential quantum-mechanically, with probability \(|{\mathcal R}(\varepsilon )|^2\).

Note that both phenomena, of quantum tunneling of a negative-energy particle, and of quantum-mechanical reflection for a positive-energy particle, result in an analogue of Hawking radiation. Although they both result from a particle moving from left to right, the final localization regions for the quantum-mechanical tunneling and quantum-mechanical reflection particles are different. For the negative-energy case, the particle moves from left to right with initial positive momentum p and position \(x<0\), as shown in (b) of Fig. 1, its quantum tunneling eventually leads to positive momentum and the particle lives in the region \(x>0\). On the other hand, for the positive-energy case, its quantum-mechanical reflection eventually leads to negative momentum and the particle lives in the region \(x<0\). However, from the perspective of energy transportation, these two cases are equivalent: Quantum tunneling in the negative-energy case implies that the negative-energy particle is removed from the left region \((x<0)\), and thus the energy in this region increases by \(-\varepsilon >0\) by comparison with the classical process (where the negative-energy particle is reflected back to the left region by the IHO potential). For the positive-energy case, quantum reflection implies that the particle carrying positive energy \((\varepsilon >0)\) comes into the left region, and thus the energy in the left region, likewise, increases by \(\varepsilon >0\). In our paper, following Ref. [39], we focus on the negative-energy case.

We finally note that although the analogue-radiation direction due to IHO scattering seems to be opposite to the conventionally used picture for black hole radiation, where radiation is directed from left to right, here for both the IHO and black hole scattering cases the energy radiation which occurs is directed opposite to the incident particle flux. In this regard, we point out that the analogue-radiation direction of the IHO case in fact depends on the choice of the direction of incident particle. We can also observe analogue radiation for the IHO scattering case where the energy-radiation flux is directed from left to right by choosing the incident particle flux direction correspondingly (see for more details at the end of the next subsection).

2.2 Chaotic motion and Hawking radiation

As we discussed in the above, when a particle moves in a IHO potential, its classical trajectory would be exponentially sensitive to the initial condition. This similar phenomenon may also occur in the background of a black hole. As shown in Refs. [52, 53], the classical Hamiltonian for a chargeless and massless particle in a region very close to the horizon possesses the same form as (1) as far as radial motion is concerned. The authors discovered that the particle’s radial motion is unstable very near to the horizon, i.e., the radial motion is exponentially sensitive to the initial conditions. The corresponding Lyapunov exponent of the particle’s motion is found to be the surface gravity \(\kappa \) of the black hole. Specifically, it is given by \(\lambda _L=\kappa =\frac{1}{2}\sqrt{g^\prime (r_0)f^\prime (r_0)}\), for the metric \(ds^2=-f(r)dt^2+dr^2/g(r)+r^2(d\theta ^2+\sin ^2\theta \,d\phi ^2)\), where \(g^\prime (r_0)\) and \(f^\prime (r_0)\) are derivatives with respect to r, evaluated at the horizon \(r=r_0\). Note that here the behavior of the particle is treated classically. This implies that classically IHO motion may capture the chaotic characteristic of a test particle moving near the horizon of a spherically symmetric static black hole. Furthermore, this IHO Hamiltonian, at the quantum level, may provide thermality to the system and the temperature is indeed found to be given by the Hawking expression [52, 53].

Fig. 1
figure 1

a Scattering off an effective potential approximately equal to an inverse harmonic potential, outside a black hole, where \(r_*\) is the “tortoise coordinate”, and the event horizon is located at \(r_*\rightarrow -\infty \). Blue and red arrows schematically denote reflection (R) and transmission (T) processes, and the black arrow denotes the incoming (I) particle current. b Scattering via the inverse harmonic potential leads to classical trajectories of incoming particles (solid lines) and particle tunneling (broken lines) near the hyperbolic fixed point \((x, p)=(0, 0)\) in phase space. The dotted lines are the separatrices \((\varepsilon =0)\), and \(\beta =1/T\). Note here that the direction of the incoming particle for the IHO case in (b) is chosen to be consistent with Ref. [39], and is opposite for the black hole case in (a). However, these two cases share effectively the same essential physics, as discussed in the text at the end of Sect. 2.1 and 2.2

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Hawking radiation [1, 2] is a quantum phenomenon occurring in the presence of a black hole horizon. It is associated with the creation of pairs of particles from quantum vacuum fluctuations in curved spacetime, and implies that a black hole is not completely black when treated semiclassically. Although there are numerous variants of (re-)deriving Hawking’s result [1, 2, 54,55,56,57,58,59,60], ranging, e.g., from Hawking himself calculating the Bogoliubov coefficients between the quantum scalar field modes of the “in” and “out” vacuum states [1, 2], to an open quantum system approach [60], the temperature of Hawking radiation is invariably obtained, only depending on the surface gravity at the black hole horizon. In particular, Parikh and Wilzcek [55] presented a short and direct derivation of Hawking radiation as a tunneling process through a classically forbidden region. Recently, Surojit and Bibhas [53] derived in detail the relation between IHO mechanics and near-horizon local instability, and adopted the concept of quantum mechanical tunneling to calculate the tunneling probability of particle at the vicinity of the horizon. For a general static spherically symmetric metric \(ds^2=-f(r)dt^2+dr^2/g(r)+r^2(d\theta ^2+\sin ^2\theta \,d\phi ^2)\), one can follow the Ref. [53] to obtain the IHO-version Hamiltonian for the outgoing particle, and calculate the corresponding tunneling probability,

$$\begin{aligned} \Gamma \sim \exp \bigg [-\frac{4\pi }{\sqrt{g^\prime (r_0)f^\prime (r_0)}}\varepsilon \bigg ]:=\exp [-\varepsilon /T]. \end{aligned}$$
(5)

This tunneling rate implies Hawking radiation with a temperature \(T=\sqrt{g^\prime (r_0)f^\prime (r_0)}/4\pi \).

We conclude, from the above analysis of both scattering off an IHO potential and tunneling through a black hole effective potential, that IHO and black hole share the property that while classically not behaving as if being in thermal equilibrium with a bath, thermality is acquired when quantum mechanics is taken into account. This formal analogy forms the basis of our proposal to observe analogue Hawking radiation in an experimentally feasible quantum IHO system, to which we now proceed.

We have displayed the map** between IHO radiation and black hole radiation in Fig. 1. Note that for the IHO potential scattering, we consider that the incoming particle with negative energy \(\varepsilon <0\) has positive momentum (\(p>0\)) and moves from left (\(x<0\)) to right (\(x>0\)). The negative-energy particle, as a result of quantum tunneling, can penetrate the IHO potential and emerges at the right (\(x>0\)), and thus the negative-energy particle is removed from the left region. Equivalently, the energy in the left region increases by \(-\varepsilon >0\) as the analogue of Hawking radiation. The IHO potential is an approximation of the effective potential hill of black hole under certain conditions pointed out in [61]. Usually, in black hole scattering mechanics, the effective potential scatters the incident mode moving from the right of the effective potential to its left, and its transmitted part and the reflected part respectively appear at the left and right of the effective potential after the scattering. The reflected part is known as the Hawking radiation. Although the analogue “Hawking” radiation of the IHO and the Hawking radiation of real black holes, as discussed in the above, have different radiation directions, they share effectively the same essential physics: Due to quantum mechanics, radiation emerges eventually.

Let us note that the incident particles may be assumed to move towards the IHO potential from either the left or from the right, and at the same time the analogue-radiation direction depends on our choice of direction of the incident particle flux. As a result of the spatial symmetry of the IHO Hamiltonian, no matter in which direction the incident particle moves towards the IHO potential, the same mechanism appears: The analogue energy radiation always occurs opposite to the direction of the incident particle and different incident directions of particle will cause different energy-radiation directions. Therefore, if we consider the incident particle moving from the right of the IHO potential to the left, we can obtain the same radiation mechanism because of the spatial symmetry of the IHO Hamiltonian. However, in this latter case, the direction of the analogue radiation is from left to right, perfectly matching with the traditional case of black-hole Hawking radiation. Specifically, a positive-energy particle has negative momentum and moves from the right of the IHO potential to its left. It would go through the potential since \(\varepsilon >0\) and will be reflected by the potential quantum-mechanically (as for the analogue of Hawking radiation). Alternatively, a negative-energy particle has negative momentum and moves from the right of the IHO potential to its left. It will be reflected by the potential since \(\varepsilon <0\), and would go through the potential as a result of quantum tunneling (as for the analogue of Hawking radiation). For both cases, the energy in the region to the right of the IHO potential increases by \(|\varepsilon |\).

3 Observing OTOC and analogue of Hawking radiation in trapped ions

Fig. 2
figure 2

Transmission/reflection probability and OTOC. a The transmission and reflection probability in Eq. (7) as function of ion displacement x, with fixed width \(\Delta =\hbar \omega \), \(\xi =0.01\), initial energy \(\varepsilon _0=-0.5\hbar \omega \), and at time \(t=2.86\,\upmu {\mathrm {s}}\); b the transmission and reflection probability as a function of time t, with ion displacement \(x=1.35\,{\mathrm {nm}}\); c the fidelity of the prepared initial state (6) for various truncation levels L of the Fock basis. Everywhere, the oscillator frequency is taken to be \(\omega _0/2\pi =10\,{\mathrm {MHz}}\); d the OTOC in Eq. (8) for various pure initial Fock superposition states with driving strength \(\xi =0.01\), and \(x_0=\sqrt{\hbar /2m\omega }\) is vacuum position uncertainty; e the OTOC in Eq. (8) for different Lyapunov exponent parameters (driving strengths) \(\xi \); the initial pure state is \(|\Psi _\text {in}\rangle =(|0\rangle +|1\rangle )/\sqrt{2}\)

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We propose using a trapped ion to experimentally implement the IHO. In our scenario, we monitor one of the oscillation directions of the ion, with associated momentum and position operators in the lab frame denoted as \({\hat{P}}\) and \({\hat{X}}\), respectively. In particular, the ion’s oscillation frequency \(\omega _0\) can be periodically modulated by an applied voltage on the trap electrodes [62]. The corresponding Hamiltonian is given by \(H=\frac{{\hat{P}}^2}{2M}+\frac{1}{2}M\omega ^2(t){\hat{X}}^2\), where \(\omega (t)=\omega _0[1-\xi \cos (\omega _mt+\phi )]\) denotes the time-dependent frequency, \(\xi \) is the modulation depth, \(\phi \) is the initial phase of the modulation, and M is the mass of the ion. Imposing \(0<\xi \ll 1\), and \(\omega _m=2\omega _0\), the Hamiltonian, in rotating-wave approximation, can be rewritten in the form of the IHO in Eq. (1), with \(m=2M/\xi \), and \(\alpha =\frac{1}{4}m\omega _0^2\xi ^2=m\omega ^2\) (see the Appendix for a detailed derivation). The momentum and position operators have been redefined by \({\hat{p}}=-i\sqrt{\frac{m\omega \hbar }{2}}({\hat{a}}e^{i\phi /2}-{\hat{a}}^\dagger \,e^{-i\phi /2})\), and \({\hat{x}}=\sqrt{\frac{\hbar }{2m\omega }}({\hat{a}}e^{i\phi /2}+{\hat{a}}^\dagger \,e^{-i\phi /2})\), respectively. Note that the phase \(\phi \) is tunable by appropriately choosing the initial phase of driving.

From the perspective of a concrete experiment, we will in the following simulate the analogue of Hawking radiation and observe the chaotic dynamics during the “analogue black hole evaporation” in the trapped-ion IHO system.

3.1 Observing the analogue of Hawking radiation

To observe the Hawking radiation associated to the S matrix (3), we need to prepare a suitable incident wave packet. Specifically, we prepare the initial state of the ion’s motion as a superposition of energy states, \(|\Psi _\text {in}\rangle =\int \,d\varepsilon \,f(\varepsilon )|\varepsilon \rangle \). When the distribution \(f(\varepsilon )\) is assumed to be a Gaussian centered at an energy \(\varepsilon _0\) with small width \(\Delta \), i.e., \(f(\varepsilon )=\frac{1}{\sqrt{2\pi ^{3/2}\Delta }}\exp \big [-\frac{(\varepsilon -\varepsilon _0)^2}{2\Delta ^2}\big ]\), the incident wave packet can be written as [63]

$$\begin{aligned} \Psi _\text {in}(x,t)=i{|x|^{-1/2}}e^{-i\varepsilon _0t-i\Phi _0} F\big (t+\log [\sqrt{2}|x|]\big ), \end{aligned}$$
(6)

where \(F(z)=(\Delta /\pi ^{1/2})^{1/2}\exp \{-z^2\Delta ^2/2\}\), \(\Phi _0=\varepsilon _0\log \sqrt{2}|x|+x^2/2+\varphi _0/2+\pi /4\), with \(\varphi _0=\arg \Gamma (\frac{1}{2}-i\varepsilon _0)\). Generating arbitrary harmonic-oscillator states has been experimentally demonstrated recently [64]. To prepare this wave packet in experiment, it is convenient to represent it in a phononic Fock basis, \(|\Psi _\text {in}\rangle =\sum ^\infty _{n=0}\langle \,n|\Psi _\text {in}\rangle |n\rangle \). We note that the overlap \(\langle \,n|\Psi _\text {in}\rangle \) is nonzero only when n is even. Using the experimental techniques of Refs. [64, 65], the incident wave packet (6) can in principle be prepared experimentally. In Fig. 2, we plot the fidelity \(F_\text {in}=\sum ^L_{n=0}|\langle \,n|\Psi _\text {in}\rangle |^2\) for preparing the incident wave packet, with L being the truncation level in Fock space. For the initial state (6), we derive an approximate analytical expression for the transmitted and reflected probability densities as follows [63]:

$$\begin{aligned} |\Psi _R|^2= & {} |{\mathcal R}(\varepsilon _0)|^2\frac{1}{|x|}\left| F\left( t-\log [\sqrt{2}|x|]-\varphi ^\prime (\varepsilon _0)\right) \right| ^2, \nonumber \\ |\Psi _T|^2= & {} |{\mathcal T}(\varepsilon _0)|^2\frac{1}{x}\left| F\left( t-\log [\sqrt{2}x]-\varphi ^\prime (\varepsilon _0)\right) \right| ^2. \end{aligned}$$
(7)

Here, \(\varphi ^\prime =d\varphi /d\varepsilon \), with \(\varphi (\varepsilon )=\arg \Gamma (\frac{1}{2}-i\varepsilon )\). We have taken the units for mass, length, time, and energy as 2m, \((\hbar /2m\omega )^{1/2}\), \(\omega ^{-1}\), and \(\hbar \omega \), respectively. Integrating Eq. (7) over space, we obtain the energy-dependent transmission and reflection probabilities in Eq. (4), which displays the thermal radiation characteristics of the semiclassical black hole.

In an experiment, the quantum mechanical ion motion can be detected by observing the evolution of its internal level populations according to the Hamiltonian \(H_{\mathrm {I}}=\hbar \Omega \sigma _y({\hat{a}}e^{i\phi /2}+{\hat{a}}^\dagger \,e^{-i\phi /2})\) [66]. This interaction can be implemented, in the Lamb–Dicke limit, by driving the ion with both blue and red sidebands, \(H_b=\hbar \eta \Omega _b({\hat{a}}^\dagger \sigma ^+e^{i\phi _b}+{\hat{a}}\sigma ^-e^{-i\phi _b})\) and \(H_r=\hbar \eta \Omega _r({\hat{a}}\sigma ^+e^{i\phi _r}+{\hat{a}}^\dagger \sigma ^-e^{-i\phi _r})\), respectively, and thus with the total Hamiltonian \(H_b+H_r =\hbar \Omega ({\hat{a}}\,e^{i\phi _-}+{\hat{a}}^\dagger \,e^{-i\phi _-})(\sigma ^+e^{i\phi _+}+\sigma ^-e^{-i\phi _+})\). Here, we set \(\eta \Omega _b=\eta \Omega _r=\Omega \), \(2\phi _\pm =\phi _r\pm \phi _b\) by tuning the amplitude and phase of the applied driving field for the sidebands, and \(\sigma ^\pm =(\sigma _x\pm \,i\sigma _y)/2\). To measure the scattering state, we first need to prepare the internal levels of the ion as \(|+\rangle =1/\sqrt{2}(|\uparrow \rangle +|\downarrow \rangle )\), and then apply the Raman-induced interaction coupling \(H_{\mathrm {I}}\). We then record the probability \(P_\downarrow (t)\) of occupation in \(|\downarrow \rangle \). The expected signal is \(P_\downarrow (t)=\int \big (1-\sin (2\Omega \,tx)\big )|\Psi (x)|^2dx\) and the scattering probability \(|\Psi (x)|^2\) in Eq. (7) is then obtained by applying an inverse Fourier transform to \(P_\downarrow (t)\).

Taking the initial energy, \(\varepsilon _0<0\), we plot transmission and reflection probabilities in Fig. 2. Negative \(\epsilon _0\) implies that the particle classically does not penetrate the barrier when coming from the left; Fig. 2 however clearly shows that both nonzero transmission and reflection probability exist simultaneously. The transmission probability, after integrating over position, satisfies a thermal distribution with temperature \(T=\hbar \lambda _L/2\pi \).

The analogy between IHO thermal behavior and Hawking radiation physics of black hole clearly indicates that the IHO can elegantly capture features of gravitational systems—black holes. The IHO therein plays the role of “analogue black hole” and is in thermal equilibrium with its “analogue Hawking radiation”. To check whether the MSS bound can indeed be identically saturated, we then have to extract the Lyapunov exponent of the same thermal IHO system.

3.2 Measuring OTOC

To diagnose the quantum chaotic motion of the IHO, we propose to measure its OTOC using the protocol proposed in Ref. [67] which does not require the reversal of time evolution. Here, the OTOC is defined by \(C(t)={\mathrm {Tr}}[\rho _0W^\dagger (t)V^\dagger \,W(t)V]\), where \(W(t)=U^\dagger (t, 0)WU(t, 0)\) and \(V=V(0)\) are operators evaluated in the Heisenberg picture at times 0 and t, respectively [31, 42].

Note that the MSS bound is universal, and is therefore believed to be a fundamental property of quantum systems, since it is always valid regardless of the choice of operators \({\hat{W}}\) and \({\hat{V}}\) and the details of the quantum-system Hamiltonian [31, 35, 68]. To extract the Lyapunov exponent \(\lambda _L\), we can choose appropriate operators \({\hat{W}}\) and \({\hat{V}}\) for the experimental convenience. In our scenario, we assume the operator V to be a projection operator onto an initially pure state, \(V=\rho _0\) [67]; then, the OTOC can be reduced to \(C(t)=|\langle \,W(t)\rangle |^2\). To measure the OTOC, we therefore need to prepare a pure initial state for the ion motional degree of freedom. For our experimental proposal, we use for concreteness a single \(^9{\mathrm {Be}}^+\) ion trapped in a strong radio-frequency Paul trap with a pseudopotential trap frequency \(\omega _0/2\pi =10\,{\mathrm {MHz}}\). The motional ground state can be prepared by a two-stage laser cooling process. By Doppler cooling, all the motional modes of the ion can be cooled down to near the Doppler limit (with an average phonon number \({\bar{n}}\) of typically a few to ten quanta), through driving the \(^2S_{1/2}\) to \(^2P_{3/2}\) dipole transition. We can further cool the motional mode of interest to its ground state by using sideband cooling with stimulated Raman transitions between the ion’s two hyperfine ground states, i.e., between \(^2S_{1/2}(F=2, m_F=2)\) and \(^2S_{1/2}(F=1, m_F=1)\), denoted by \(|\downarrow \rangle \) and \(|\uparrow \rangle \), respectively, and which are separated by \(\sim 1.25\,{\mathrm {GHz}}\) [69, 70]. This spin initialization and ground state cooling allow us to prepare the ion in the \(|S=\downarrow , n=0\rangle \) state eventually. By applying appropriate laser pulse sequences on blue and red sidebands or the carrier, we can create arbitrary quantum superpositions of Fock states [64, 65, 71, 72].

For the proposed experiment, we assume that the perturbation operator W is represented by an ion displacement measurement, \({\hat{x}}=\sqrt{\frac{\hbar }{2m\omega }}({\hat{a}}+{\hat{a}}^\dagger )\), and the initial pure state is prepared as, \(| \Psi _\text {in}\rangle =\frac{1}{\sqrt{2}}(|n\rangle +|n+1\rangle )\), [64, 65]. We then readily find that the OTOC is given by

(8)

where \(U(t, 0)=\exp [-iHt/\hbar ]\). Measuring OTOC then reduces to determining the directly accessible ion-motional quadratures [44, 73, 74]. We find that the OTOC exhibits exponential growth, \(\propto e^{2\lambda _Lt}+(\text {terms growing more slowly} \text {than}\,e^{2\lambda _Lt})\), with a Lyapunov exponent \(\lambda _L=\sqrt{\alpha /m}\). Figure 2 shows the thus obtained OTOC for various pure initial states and Lyapunov exponents.

From the above results, we obtain a mathematical relation between the temperature T and the Lyapunov exponent \(\lambda _L\) which indeed suggests that the conjectured MSS bound [31] is satisfied by the ion’s motion in our analogue gravity setup.

In the above investigation, we have chosen a certain pair of operators of W and V to measure the OTOC. One can of course choose another set of Hermitian operators. However, for the IHO system the exponential growth of all the obtained OTOCs should be of the same form \(C(t)=c_0+c_{1e}^{\lambda _{Lt}}+\dots \), as the Lyapunov exponent \(\lambda _L\) does not depend on the chosen operators [31, 68, 75]. For example, we can also choose the operator W as the momentum operator \({\hat{p}}=-i\sqrt{{\frac{m\omega \hbar }{2}}}({\hat{a}}-{\hat{a}}^\dagger )\) and perform the same analysis as in the above. We can then find that the corresponding OTOC is \(C(t)=\frac{(n+1)m\hbar \omega }{2}\sinh ^2(\lambda _Lt)\propto \,e^{2\lambda _{Lt}}+\text {(terms growing more slowly than }e^{2\lambda _{Lt}})\), and the Lyapunov exponent \(\lambda _L=\sqrt{\alpha /m}\) that can be extracted remains identical.

Rewriting Eq. (1) with phononic annihilation and creation operators, we find \(H=-\frac{1}{2}\hbar \lambda _L({\hat{a}}^{\dagger 2}+{\hat{a}}^2)\), which corresponds to a squeezing operation [76]. We can thus alternatively interpret the analogue of Hawking radiation (pair creation process) via a squeezed state of the ion’s motion. The phonon number increases as \(\langle {\hat{n}}\rangle =\sinh ^2(\lambda _Lt)\), and the squeezed vacuum state population distribution is restricted to even states, \(P_{2n}=(2n)!\tanh ^{2n}(\lambda _{Lt})/(2^nn!)^2\cosh (\lambda _Lt)\) [77]. The squeezed state can be detected by the evolution of the ion’s internal levels under a Jaynes–Cummings type interaction [66]. The population distribution is shown in Fig. 3, and compared with a slightly thermal state close to the vacuum. The increased population of higher Fock states are evidence of squeezing; only even states populated corresponds to the creation of pairs of phonons.

Fig. 3
figure 3

Phonon number distribution \(P_{n}\) for a squeezed state (yellow) and thermal state (light blue) at very low temperatures (average phonon number \({\bar{n}}=0.02\)). The squeezing parameter is assumed to be \(\lambda _Lt\approx 0.88\). The vacuum squeezed state is created by the action of the Hamiltonian (1) on the phonon vacuum. Although there are no phonons at the beginning, phonons are created during the evolution, as a result of the analogue of Hawking radiation

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4 Conclusion

In summary, we propose using well established quantum optical tools for trapped ions to verify the MSS bound. Chaotic quantum motion of the ion can be accessed by measuring the OTOC, which exhibits to leading order exponential growth, \(\propto e^{2\lambda _Lt}\), with Lyapunov exponent \(\lambda _L=\sqrt{\alpha /m}\), and analogue Hawking radiation temperature \(T=\hbar \lambda _L/2\pi \).

Numerous theoretical approaches employed the IHO concept to establish links between chaos in quantum mechanics and general relativity see, e.g., [39,40,41, 61, 78, 79]. Our IHO implementation proposal paves the way to experimentally explore these theoretical concepts in a highly controllable quantum optical environment.