Relativistic quantum mechanics with trapped ions

The n=0 state is equivalent to a single covering of the sphere, or skyrmion, with topological charge ν=1.

“This is equivalent to a single covering of the sphere, or skyrmion, with a topological charge 8 ν = 1.”

State n=2 performs a double covering of the sphere.

“moving on to state n = 2, which performs a double covering of the sphere.”

The model can be rewritten as a simple JC model with a detuning, H = c√2(σ+ a + a† σ-) + mc^2 σ_z.

“we may rewrite the previous model as a simple JC model with a detuning, H = c mc p x + ix p x -ix -mc = c √ 2(σ + a + a † σ -) + mc 2 σ z .”

In the axial gauge A(x,y)=Bx, the momentum along the y-direction is a constant of motion ([p_y,H]=0).

“In this setup, the momentum along the y-direction becomes a constant of motion, [ p y , H ] = 0.”

The model has a ladder of discrete energy eigenstates E ±n = ±√(2c^2 n^2 + m^2 c^4), the Landau levels.

“This model has a ladder of discrete energy eigenstates, E ±n = ± √ 2c 2 n 2 + m 2 c 4 ,(23) the so-called Landau levels.”

In the trapped-ion simulation, the Zitterbewegung frequency can be estimated as ω_ZB ≈ 2 sqrt(N η^2 Ω^2 + Ω^2).

“the Zitterbewegung frequency can be estimated in the simulation as”

• AssumptionThe authors state this as an assumption: In the trapped-ion simulation, the position of the ion can be identified with the position of the simulated Dirac particle.

The Zitterbewegung amplitude can be estimated as R_ZB = η h^2 Ω / sqrt(4η^2 Ω^2 p_0^2 + h^2 Ω^2).

“Additionally, one can estimate the Zitterbewegung amplitude to be”

• AssumptionThe authors state this as an assumption: In the trapped-ion simulation, the position of the ion can be identified with the position of the simulated Dirac particle.

The double covering is done via an extended singularity of the angle φ at radius r=1, which is when the spin points North.

“This is done via an extended singularity of the angle φ at radius r = 1, which is when the spin points North.”

The number of coverings increases linearly with the number of quanta, ν = ±n.

“In general, we have verified that the number of coverings increases linearly with the number of quanta, ν = ±n, leading to a family of topological defects with unprecedented richness.”

The simulated plots show rapid suppression of Klein tunneling for increasing transverse momentum p_y.

“These plots evidence the rapid suppression of Klein tunneling for increasing p y .”

At p_y = 1, almost all of the wave function gets reflected.

“we find that already at p y = 1 almost all of the wave function gets reflected.”

Bidimensional relativistic scattering contains interesting novel features, such as entanglement between transmitted/reflected wave packets and the transverse momentum.

“In addition, we showed that it contains interesting novel features, such as entanglement between transmitted/reflected wave packets and the transverse momentum.”

The normal modes for a three-ion system with equal mass and charge are Q_cm = (x1+x2+x3)/√3, Q_r = -(x1-x3)/√2, Q_3 = (x1-2x2+x3)/√6.

“The normal modes for a three-ion system with equal mass and charge are Q cm = 1 √ 3 (x 1 + x 2 + x 3 ), P cm = 1 √ 3 ( p 1 + p 2 + p 3 ),(30) Q r = - 1 √ 2 (x 1 -x 3 ), p r = - 1 √ 2 ( p 1 -p 3 ),(31) Q 3 = 1 √ 6 (x 1 -2x 2 + x 3 ), P 3 = 1 √ 6 ( p 1 -2 p 2 + p 3 ). (32 )”

For relative momentum equal to zero, there is little Klein tunneling associated with asymptotic freedom.

“For relative momentum equal to zero, there is little Klein tunneling (c) associated with asymptotic freedom.”

Bidimensional relativistic scattering for an x-dependent potential can be solved with the help of the 1D case.

“To summarize, we have described the bidimensional relativistic scattering for an x-dependent potential, showing that this case can be solved with the help of the 1D case.”

For = -1 and finite relative momentum p_r = 2, the two fermions have a larger Klein tunneling than for zero relative momentum p_r = 0.

“On the other hand, for = -1 (another spinor initial state), we observe that for finite relative momentum, pr = 2 (d), the two fermions have a larger Klein tunneling than for zero relative momentum, pr = 0 (c).”

By considering the Hamiltonian H_ABM,sim, one simulates an equivalent dynamics to two free Dirac equations with a potential in the relative coordinate, H_ABM.

“By considering the Hamiltonian H ABM,sim = 2η cm cm ˜ cm (σ x,1 -σ x,3 )P cm + 2η r r ˜ r (σ x,1 + σ x,3 ) p r + h σ y,1 + h σ y,3 + hη cm 3 cm σ x,2 Q cm + h 3 σ z,2 ,(33) we simulate an equivalent dynamics to two free Dirac equations with a potential in the relative coordinate, H ABM = cσ x,1 p 1 + cσ x,3 p 3 + mc 2 σ y,1 + mc 2 σ y,3 + V 0 (x 1 -x 3 ) 2 = c(σ x,1 -σ x,3 ) pr + c(σ x,1 + σ x,3 )”

For = +1 and p_r = 0, the dynamics is quasifree, besides a small Klein tunneling appearance.

“For = +1 and pr = 0 the dynamics is quasifree, besides a small Klein tunneling appearance.”

The mass term mc^2 σ_j can be implemented by a carrier interaction between two internal levels of the ion.

“For implementing the second kind of coupling, mc 2 σ j , one may consider the carrier interaction between two internal levels of the ion.”

The spin-orbit couplings σ_i p_i can be implemented by a combination of Jaynes-Cummings and anti-Jaynes-Cummings interactions.

“In general, one may implement the first kind of couplings, σ i p i , by a combination of Jaynes-Cummings (JC, also named red sideband) and anti-Jaynes-Cummings (AJC, also named blue sideband) interactions.”

A quantum simulation of the Dirac equation in trapped ions is a useful playground to analyze different regimes and to verify in a table-top experiment predicted phenomena such as Zitterbewegung or the Klein paradox.

“A quantum simulation of the Dirac equation is a useful playground to analyze different regimes and to verify in a table-top experiment predicted phenomena such as Zitterbewegung [9,10] or the Klein paradox [12,13].”

• cites [9] sourceNot checked yet
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In trapped-ion quantum simulations of relativistic quantum mechanics, the values of the speed of light and rest energy can be controlled at will by changing the laser intensities.

“An important property of quantum simulations of relativistic quantum mechanics with trapped ions is that the values of the speed of light and rest energy in equations ( 9) may be controlled at will just by changing the laser intensities, ˜ and .”

The experimental simulation of the Klein paradox relied on the manipulation of two trapped ions.

“The experimental simulation of the Klein paradox relied on the manipulation of two trapped ions [13].”

• cites [13] sourceNot checked yet

A Dirac particle may tunnel through a steep barrier and propagate indefinitely even if the potential extends to arbitrarily long distances.

“A Dirac particle may tunnel through a steep barrier and propagate indefinitely regardless of the fact that the potential may be extended to arbitrarily long distances.”

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The momentum-dependent group velocity leads to a spreading of wave packets in position space.

“which in turn leads to a spreading of wave packets in position space.”

The reconstruction formula evidences conditional evolution and entanglement between the x-coordinate transmitted and reflected wave packets and the y-component wave packet.

“This reconstruction formula evidences the conditional evolution mentioned before, and in particular, the entanglement between the x-coordinate transmitted and reflected wave packets and the ycomponent wave packet.”

The theoretical proposal for simulating the Klein paradox in 1+1 dimensions was based on the Hamiltonian H = cσ_x p_x + mc^2 σ_z + αx.

“The theoretical proposal [12] of the simulation of the Klein paradox in 1 + 1 dimensions was based on the Hamiltonian”

• cites [12] sourceNot checked yet

An actual experimental simulation of 2+1 Klein scattering may have important value because, without memory, computational power and narrow boundary constraints, it allows independent and perhaps more reliable verification of the predictions.

“These are the reasons why an actual experimental simulation of the 2 + 1 Klein scattering may have such an important value: without these constraints (memory, computational power and narrow boundary conditions), it allows for an independent and perhaps more reliable verification of our predictions.”

The plots in figure 1 can be interpreted as entanglement between the transmitted and reflected wave packets in the x-coordinate and the transverse momentum along y.

“The plots in figure 1 admit another interpretation, based on the entanglement between the transmitted and reflected wave packets in the x-coordinate and the transverse momentum along the y-direction.”

Electron-positron pair production along x is conditional on the specific value of each momentum wave packet component along y.

“In the language of quantum field theory, the electron-positron pair production along the x-direction will be conditional to the specific value of each momentum wave packet component along the y-direction.”

The positive energy electron may become a negative energy electron upon collision with the barrier and propagate freely throughout the barrier wherever V > 2mc^2.

“The reason for this is that the positive energy electron may become, upon collision with the barrier, a negative energy electron and as such propagate freely throughout the barrier, wherever the potential barrier is V 2mc 2 .”

Based on the behavior at fixed p_y, the dynamics of arbitrary wave packets can be reconstructed fully in position space.

“Based on the behavior at fixed values of p y , we can reconstruct the dynamics of arbitrary wave packets, fully in position space.”

The trapped-ion Hamiltonian for the 2+1 Klein dynamics coincides with the 2+1 Dirac-Klein Hamiltonian under the stated analogies when ion 2 is initialized in the eigenstate + of σ_x,2.

“which coincides with equation ( 15) when one performs the analogies 2η cm cm ˜ cm = 2η st st ˜ st ↔ c, h ↔ mc 2 and η cm ˜ 0 / cm ↔ α, when ion 2 is initialized in the eigenstate + of σ x,2 .”

For each p_y, the 2+1 problem reduces to a 1+1 Klein Hamiltonian.

“Accordingly, the problem, for each p y , is reduced to a Klein Hamiltonian in 1 + 1 dimensions,”

For fixed p_y, cσ_y p_y + mc^2 σ_z can be expressed as an effective Pauli matrix with effective mass coefficient m c^2 = sqrt(p_y^2 c^2 + m^2 c^4).

“One can realize that cσ y p y + mc 2 σ z , with p y fixed, can be expressed as a new Pauli matrix with an effective mass coefficient, with mc 2 = p 2 y c 2 + m 2 c 4 .”

Such topologically nontrivial models, when embedded on surfaces with boundaries, present robust topologically protected edge states that may transport charge or spin even when the bulk is insulating.

“The intriguing result is that such models, when embedded on surfaces with boundaries, present robust topologically protected 'edge states', that is, states that may transport charge or spin even when the bulk is placed in an insulating state (e.g. with a large mass).”

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The size of the states and the associated Hamiltonian matrix force the use of efficient schemes such as finite differences or Fourier representations and Trotter methods.

“The size of the states and the size of the associated Hamiltonian matrix force us to adopt particularly efficient schemes in the simulation, such as using finite differences or Fourier representations for the Hamiltonian and Trotter methods for the unitary evolution.”

The spreading is most relevant for small values of p_x, i.e. when the particle crashes against the potential barrier.

“This spreading is most relevant for small values of p x , i.e. when the particle crashes against the potential barrier.”

This would amount to a phenomenological simulation of the phase transition between the quark-gluon plasma and the hadronic confined phases that took place in the early universe.

“This would amount to a phenomenological simulation of the phase transition between the quark-gluon plasma and the hadronic confined phases that took place in the early universe.”

A possible quantum simulation of the Dirac particle in a magnetic field, including its topological features, can be done in trapped ions.

“We showed a possible quantum simulation of this case, including its topological features in trapped ions.”

The spinor degrees of freedom can be encoded in long-lived electronic states of Ca+ ions, as was done before in quantum simulations of relativistic quantum mechanics [10,13].

“The spinor degrees of freedom can be encoded in long-lived electronic states of this ion, as was done before in quantum simulations of relativistic quantum mechanics [10,13].”

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In this model, Klein tunneling is interpreted as pair creation of a new quark and a new antiquark, allowing the original pair to split up without violating color neutrality.

“In this model, we interpret this Klein tunneling as a pair creation of a new quark and a new antiquark: when the original quark and antiquark go very far away from each other, there is energy available for creating a new antiquark (that gets bound to the original quark) and a new quark (that gets bound to the original antiquark), such that the original pair can split up without violating the color”

• AssumptionRests on an assumption: The usual explanation for Klein tunneling in quantum field theory is the creation of a particle/antiparticle pair.
• AssumptionRests on an assumption: In QCD, the splitting of a hadron into new hadrons is well known and takes place, for example, in jet emission in high-energy colliders.

An experiment with three ions can simulate an analogue of the MIT bag model [47,48] for two Dirac particles in 1+1 dimensions.

“To sum up, we have shown that an experiment with three ions can simulate an analogue of the well-known MIT bag model [47,48] for two Dirac particles in 1 + 1 dimensions.”

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Three trapped ions together with two motional modes suffice to implement the dynamics of two Dirac particles coupled by a confining potential, like in bag models of nuclear physics.

“Finally, we have shown that three trapped ions together with two motional modes suffice to implement the dynamics of two Dirac particles that are coupled by a confining potential, like in the bag models of nuclear physics.”

Such experiments would allow us to study signatures of asymptotic freedom and confinement.

“Despite their simplicity, such experiments would allow us to study signatures of asymptotic freedom and confinement.”

This may indicate that for large enough relative momentum, the Dirac fermion pair has enough energy to create another quark-antiquark pair and escape from each other while keeping color neutrality.

“This may also indicate that for large enough relative momentum, the Dirac fermion pair has enough energy to create another quark-antiquark pair and escape from each other while keeping color neutrality (d).”

Those experiments would easily reach simulation regimes that are beyond current numerical simulation possibilities.

“In addition, those experiments would easily reach simulation regimes that are beyond current numerical simulation possibilities.”

This more involved dynamics may be related to confinement taking place when the two fermions try to separate.

“This may be related to confinement taking place when the two fermions try to separate, as seen in figure 4(b).”

When V_0 x_r^2 > 2mc^2, one would expect Klein tunneling to take place.

“Additionally, when V 0 x2 r 2mc 2 , one would expect Klein tunneling to take place.”

Trapped-ion physics provides a flexible platform to simulate a number of interesting cases and effects in relativistic quantum mechanics.

“In this paper, we have shown that trapped-ion physics provides a flexible platform to simulate a number of interesting cases and effects in relativistic quantum mechanics.”

A Dirac particle in a magnetic field can establish interesting analogies with an equivalent quantum optical model, namely the JC model with detuning.

“We have introduced a relativistic quantum mechanics model-a Dirac particle in a magnetic field-that could shed light and establish interesting analogies with an equivalent quantum optical model, namely the JC model with detuning.”