Simulating superluminal propagation of Dirac particles using trapped ions

Quantum field theory in curved spacetime [1, 2, 3], as a semi-classical approach to study the movement of quantized matter fields in a fixed gravitational background, has predicted many striking phenomena such as the famous Hawking radiation and cosmological particle production. However, confirmation of those extremely weak quantum effects in real gravity experiments remains highly unlikely with present technology [4, 5]. Following Unruh’s pioneering work [6], much attention has turned into sound waves travelling against the classical [7] and quantum fluids [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], or light in nonlinear optical platforms [20, 21, 22, 23] to effectively emulate quantum field theory in curved spacetime. Remarkably, these relevant analogue experiments among many others [24, 25] has brought us a better understanding of the relationship between space-time structure and quantum theory [26, 27, 28].

Furthermore, recent developments of precise and flexible quantum control also facilitate a quantum simulation of related relativistic phenomena [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. As a representative example, intriguing quantum dynamics related to a Dirac particle moving in curved spacetime has attracted increasingly investigations via the particularly promising trapped-ion platform [41, 42, 43, 44]. The exceptional controllability of quantum simulators makes it feasible to emulate a Dirac particle in "exotic" curved spacetimes [43]. As a particularly important example, the Alcubierre warp-drive, consistent with Einstein’s field equations, can result in faster-than-light propulsion of a spaceship [45]. Compared to the extreme difficulty in realizing such a time-machine model (i.e. creating warp bubbles) in the actual world [46, 47, 48], observation of analogue phenomenon with well-controllable quantum simulators in the laboratory should be more accessible. Besides, merging of fundamental concepts from different fields when implementing such novel quantum simulation, including gravitation, quantum squeezing and quantum entanglement, would provide a fruitful way to reveal unique features of quantum effects in both the exotic curved spacetimes [43] and the basic light-matter interaction models [42].

In this Letter, we propose a trapped-ion quantum simulation of a Dirac particle propagating with a superluminal velocity caused by the emergent Alcubierre warp drive spacetime [45]. The Hamiltonian from the Dirac equation in the Alcubierre (1+1)-dimensional universe is mapped onto a spin-boson interaction quantum model, which can be further realized by a combination of sideband drives and a periodic modulation of the trapping potential. Remarkably, the tunability of system parameters in the well-controlled trapped-ion platform allows us to access the crossover from flat to curved spacetimes. Using exact numerical simulations, we demonstrate that this platform is able to observe analogue superluminal travel of Dirac particles in the tilted light cones, as well as the Zitterbewegung effect of massive Dirac particles incorporating the tilt of the Dirac cone. The extension of the present scheme to simulate the Dirac equation in more general exotic curved spacetimes would make it possible to explore intriguing phenomena arising from the interplay of quantum field theory and general relativity in well-controllable quantum experiments in the laboratory.

We start by first introducing the Alcubierre metric [45, 49], which is given by

where $v_{s}(t)\equiv dx_{s}(t)/dt$ is the velocity associated with a certain trajectory $x_{s}(t)$. In the (1+1)-dimensional case, the light cones at a point in the $t$-$x$ plane are specified by the curves emerging from the point with $ds^{2}=0$, namely

If $v_{s}(t)=0$, the the spacetime is flat. Otherwise the corresponding light cones are tipped over and the particle travelling inside the light cone can have a velocity faster than the light speed in the flat spacetime, which is theoretically consistent with the framework of general relativity [45].

Extension of the Dirac equation into curved spacetimes successfully merges quantum mechanics with the general relativity. In the (1+1)-dimensional spacetime with signature $(+,-)$, the Dirac equation reads [50, 41]

where the Greek letter $\mu$ and the Latin letter $a$ denote the coordinates of curved spacetime and local rest frame respectively, $m$ is the mass of the quantized field, $g$ is the determinant of the metric tensor, and $e^{\mu}_{\,(a)}$ is the vielbein allowing the constant Dirac matrices $\gamma^{a}$ to act at each spacetime point. Here, we choose the chiral representation such that $\gamma^{0}=\sigma_{x}$, $\gamma^{1}=-i\sigma_{y}$, with ${\bm{\sigma}}$ the Pauli matrices. In the specific case of Alcubierre metric, the Dirac equation can be transformed into the Schrödinger equation of the following form [51]

from which the Hamiltonian $H$ that governs the above dynamical evolution can be written as

Here we introduce the operators $\hat{X}$ and $\hat{P}$

to rescale the spatial coordinate $x$ of the simulated Dirac particle by a dimensionless factor $A$. Note that $\hat{X}$ and $\hat{P}$ will be the observables of the trapped ion in our proposed simulation platform. We remark that the second term of the Hamiltonian in Eq.(5), which is linearly dependent on the momentum $\hat{P}$, is an evidence of relativistic physics [52]. Based on the standard commutation relation $[\hat{X},\hat{P}]=i\hbar$, the operators $\hat{X}$ and $\hat{P}$ can be further mapped to a bosonic field of frequency $\nu$

where $m_{0}$ is a constant with the dimension of mass. By substituting Eq.(19) into Eq. (5) and applying the Hadamard transformation (i.e. $\sigma_{x}\to\sigma_{z}$, $\sigma_{z}\to\sigma_{x}$), the Hamiltonian can be rewritten as

To simulate the above Hamiltonian in Eq.(20) with the tunable parameter $v_{s}$, we consider a setup of a single ion trapped above a linear surface electrode radio-frequency trap. The radial motional mode of the trapped ion with a frequency $\nu$ models a harmonic oscillator and its two internal ground hyperfine levels with the transition frequency $\omega_{0}$ plays the role of an effective spin-1/2, denoted as $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$. As an example, the present scheme may employ a trapped ²⁵Mg⁺ ion hyperfine qubit, with an out-of-phase radial motional mode frequency $\nu\approx 2\pi\times 5.9$ MHz [53]. The qubit states are chosen as $\left|\downarrow\right\rangle\equiv\left|F=3,m_{F}=1\right\rangle$ and $\left|\uparrow\right\rangle\equiv\left|F=2,m_{F}=1\right\rangle$ [53, 54], where $F$ is the total angular momentum and $m_{F}$ is its projection along the quantization axis.

The spin-harmonic oscillator coupling of the form in Eq.(20) can be realized by implementing the Mølmer$-$Sørensen interaction [55] via simultaneous blue and red sideband drives at frequencies $\omega_{b}$ and $\omega_{r}$ respectively, which in real experiments can be generated by using oscillating near-field magnetic field gradients [56]. In addition, we introduce a periodic modulation of the single ion’s trapping potential of the strength $\Omega_{p}(t)$ at frequency $\nu$ to obtain the second term $\sim cv_{s}(t)\hat{P}$ in Eq.(20). This can be experimentally implemented by applying an oscillating potential directly to the radio-frequency trapping electrodes, as demonstrated in Ref.[53]. Such a trapped-ion platform can be characterised by the following system Hamiltonian as [51]

where the frequencies of the blue and red sideband drives satisfy the condition $\omega_{b}-\nu=\omega_{r}+\nu=\omega_{0}-\Delta$. By making a displacement transformation $a\to a+\eta$ and $a^{\dagger}\to a^{\dagger}+\eta$ ($\eta\ll 1$) and then moving to the interaction picture with respect to $H_{0}=\hbar(\omega_{0}-\Delta)/2\sigma_{z}+\hbar\nu a^{\dagger}a$, the effective Hamiltonian can be approximated as [51]

under the conditions of $\{|\Delta|,|\Omega_{p}(t)|,|\Omega_{0}|\}\ll\nu\ll\omega_{0}$. Therefore, it can be seen that the Hamiltonian in Eq.(31) as implemented in the trapped-ion platform is equivalent to the Hamiltonian in Eq.(20) that governs the Dirac equation in the Alcubierre (1+1)-dimensional universe with the parameter correspondence as $v_{s}(t)\leftrightarrow 4\eta\Omega_{p}(t)/\Omega_{0}$, $c\leftrightarrow{(\hbar/2m_{0}\nu)^{1/2}}\Omega_{0}/A$, and $m\leftrightarrow-\hbar\Delta/2c^{2}$.

We remark that the experimental tunability of the parameters $\{\Omega_{0},\Omega_{p},\Delta\}$ in the well-controllable trap-ion platform allows access to the crossover from flat to curved spacetimes, namely the tilt of the Dirac equation can be tuned by choosing appropriate values of the parameter $\Omega_{p}$, see Figure 1. The angle of the simulated light cone is given by $\theta=\arctan\left(v_{s}+1\right)-\arctan\left(v_{s}-1\right)$, which decreases for a larger value of $v_{s}$, i.e. the velocity of the Dirac particle would be constrained in a smaller range. In the limit case of extremely large $v_{s}$, $\arctan\left(v_{s}\pm 1\right)\rightarrow\pi/2$ and the angle of the light cone $\theta\to 0$, the trajectory of the trapped ion can be seen as a counterpart of the closed timelike curves.

To show the effective velocity of the the simulated Dirac particle, we solve the following Heisenberg equation of the trapped ion system [51]

where $\hat{x}(t)=U^{\dagger}(t)\hat{x}U(t)$ represents the position operator of the Dirac particle, and $\hat{\phi}_{z}(t)=U^{\dagger}(t)\sigma_{z}U(t)$ with $U(t)$ the evolution operator from the Hamiltonian $H(t)$ in Eq.(5). We remark that Eq.(21) is consistent with the Alcubierre metric, since the two eigenvalues $\pm 1$ of $\hat{\phi}_{z}(t)$ correspond to the two opposite directions of the velocity. The mechanical degrees of freedom of the trapped ion $\hat{X}$ is related to the spatial coordinate of the simulated Dirac particle $\hat{x}$ by $\hat{X}\equiv A\hat{x}$ (see Eq.6), thus the velocity $d\langle\hat{X}\rangle/dt$ is in the range of $Ac(v_{s}\pm 1)$.

As an illustrative example, we choose the initial state of the trapped ion as $\left|\psi_{0}\right\rangle=\mathcal{N}\pi^{-1/4}\int dX\exp(-X^{2}/2)\left|X\right\rangle\otimes\left|s\right\rangle$, where $\left|X\right\rangle$ and $\left|s\right\rangle$ represent the spatial and internal degrees of freedom of the trapped ion respectively. Without loss of generality, the spatial wavefunction satisfies a Gaussian distribution with $\mathcal{N}$ the normalization factor. In Figure 2 (a) and (b), we show the trajectory $\langle\hat{x}(t)\rangle$ (in the unit of $c$) of the simulated massless Dirac particle (i.e. $m=0$) as a function of the evolution time. According to Eq. (21), the trajectory of the evolving states from the initial spin states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ (namely the two eigenstates of $\langle\sigma_{z}\rangle=\pm 1$) specifies the shape of the simulated light cone. It can be seen that, corresponding to massless Dirac particles, the light cones for different values of the parameter $v_{s}$ are consistent with the analytical results from classical general relativity theory [43]. In particular, the simulated light cones start to tip over as the parameter $v_{s}$ increases, which indicates that the spacetime becomes more curved. These numerical simulation results demonstrate that the quantum simulation using trapped-ions can confirm the superluminal propagation of Dirac particles in the Alcubierre warp drive spacetime.

In the cases of initial states with $\left|s\right\rangle=\left|\uparrow\right\rangle$ or $\left|\downarrow\right\rangle$, the velocity is well defined, and thus the wavepacket of the simulated Dirac particle remains localized, as shown by the blue line and green circles in Figure 2 (c). In contrast, if we start from a coherent superposition state $\left|s\right\rangle=(\left|0\right\rangle+\left|1\right\rangle)/\sqrt{2}$, the wavepacket of the simulated Dirac particle has two peaks which correspond to the initial velocities $c(v_{s}\pm 1)$ respectively, resulting in an increase of the variance of the trajectory, see the orange line in Figure 2 (c).

In addition to the analogue superluminal propagation, we also find that for massive Dirac particles with $m\neq 0$, the Zitterbewegung effect induces an oscillatory behaviour in the trajectory of the trapped ion. This effect is caused by the quantum superpositions of electron and positron solutions of the free wavepackets [52, 44] and may exist in both flat and curved spacetime [52, 42]. The equation of motion for the expectation value of the position operator $\hat{x}(t)$ in this case can be obtained as [51]

where $H_{\text{flat}}=Ac\hat{P}\sigma_{z}-mc^{2}\sigma_{x}$ represents the Dirac Hamiltonian in the flat spacetime. From Eq.(12), one can see that the oscillatory behaviour is induced by the mass $m$ but independent of the parameter $v_{s}$, and therefore the Zitterbewegung effect would persist with the superluminal propagation. In Figure 3 (a), we show that the Zitterbewegung effect can manifest for both constant and time-dependent $v_{s}$ in the trapped-ion simulator. As an example, we consider a time-dependent spacetime by choosing $v_{s}$ such that both the initial and final velocities are subluminal, which is derived from a trajectory $x_{s}(t)$ of the simulated Dirac particle. This result is shown by the grey dashed line in Fig.3 (a), which clearly exhibits the Zitterbewegung effect.

In Figure 3 (b-c), we present the properties of the wavepacket of the simulated Dirac particle during its evolution. As shown in Figure 3 (b), we do not observe any squeezing of the density profile caused by the simulated gravity in the trapped-ion system, which represents a different phenomenon from the one in Ref.[42]. We remark that the qualitative behavior of the variance of the position is determined by the mass $m$ and is independent of the parameter $v_{s}$. The wavepackets at $t=0.3$ ms and $t=1.5$ ms are shown in Figure 3(c), which suggests that the wavepacket of the simulated Dirac particle is stretched and the oscillatory feature becomes more prominent as the system evolves, due to the Zitterbewegung effect.

Conclusion.— To summarize, we propose a scheme to implement a trapped-ion quantum simulation of a Dirac particle moving in the Alcubierre (1+1)-dimensional universe, which is a particularly important example of extreme curved spacetime leading to superluminal propagation in consistence with general relativity. We show that the flexibility of control parameters in such a platform allows one to observe counterintuitive effects including the superluminal velocity and the Zitterbewegung effect of the simulated Dirac particle in the Alcubierre curved spacetime. Our work demonstrates the feasibility to explore interesting and counterintuitive features of exotic curved spacetime by quantum simulation of Dirac particles in the corresponding spacetime geometry. Furthermore, the analogy between particles in the setting of semiclassical quantum field theory and basic light-matter interaction quantum models would further inspire the ideas to investigate intriguing phenomena of general relativity and quantum field theory in a variety of quantum platforms that are available in laboratories.

Acknowledgements.— We thank Dr. Dongxiao Li and Prof. Jianwei Cui for very helpful discussions. This work is supported by National Natural Science Foundation of China (Grant No. 11690032), and the Open Project Program of Wuhan National Laboratory for Optoelectronics (No. 2019WNLOKF002).