Quantum Simulation of the Majorana Equation and Unphysical Operations

J. Casanova,1 C. Sab´ın,2 J. León,2 I. L. Egusquiza,3 R. Gerritsma,4, 5 C. F. Roos,4, 5 J. J. Garc´ıa-Ripoll,2 and E. Solano1, 6 1Departamento de Qu´ımica F´ısica, Universidad del Pa´ısVasco – Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain 2Instituto de F´ısica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain 3Departamento de F´ısica Teórica, Universidad del Pa´ısVasco – Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain 4Institut fürQuantenoptik und Quanteninformation, Osterreichische¨ Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria 5Institut fürExperimentalphysik, UniversitätInnsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria 6IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain We design a quantum simulator for the Majorana equation, a non-Hamiltonian relativistic wave equation that might describe neutrinos and other exotic particles beyond the standard model. The simulation demands the implementation of charge conjugation, an unphysical operation that opens a new front in quantum simulations, including other discrete symmetries as complex conjugation and time reversal. Furthermore, we describe how to implement this general method in trapped ions.

A quantum simulator is a device engineered to repro- the non-Hamiltonian character stems from the simulta- duce the properties of an ideal quantum model. This neous presence of ψ and ψc. The significance of the Ma- still-emerging topical area has generated a remarkable jorana equation rests on the fact that it can be derived exchange of scientific knowledge between apparently un- from first principles in a similar fashion as the Dirac equa- connected subfields of physics. In terms of applications, tion [17, 19]. Both wave equations are Lorentz invariant it allows for the study of quantum systems that can- but the former preserves helicity and does not admit sta- not be efficiently simulated on classical computers [1]. tionary solutions. The Majorana equation is considered While a quantum computer would also implement a uni- a possible model [20] for describing exotic particles in versal quantum simulator [2], only particular systems supersymmetric theories –photinos and gluinos–, or in have been simulated up to now using dedicated quan- grand unified theories, as is the case of neutrinos. In- tum simulators [3]. Still, there is a wealth of successful deed, the discussion of whether neutrinos are Dirac or cases, such as spin models [4, 5], quantum chemistry [6] Majorana particles still remains open [21]. Nevertheless, and quantum phase transitions [7]. The quantum simu- despite the similar naming, this work is neither related lation of fermionic systems [8] and relativistic quantum to the Majorana fermions (modes) in many-body sys- physics have also attracted recent attention, reproduc- tems [22, 23], nor to the Majorana fermions (spinors) in ing dynamics and effects currently out of experimental the Dirac equation [20, 24]. reach. Examples include black holes in Bose-Einstein In order to simulate the Majorana equation, we have to condensates [9], quantum field theories [10, 11] and re- solve a fundamental problem: the physical implementa- cent quantum simulations of relativistic quantum effects tion of antilinear and antiunitary operations in a quan- as Zitterbewegung, Klein paradox and interacting rela- tum simulator. Here, we introduce a mapping [25] by tivistic particles [12–16] in trapped ions. which complex conjugation, an unphysical operation, be- In this paper, we show how the Majorana equation [17] comes a unitary operation acting on an enlarged Hilbert can be simulated in an analog quantum simulator, hav- space. The mapping works in arbitrary dimensions and ing as a key requirement the implementation of complex can be immediately applied on different experimental se- conjugation of the wavefunction. In this manner, we are tups. We show how to simulate the Majorana equation able to propose this and other unphysical operations such in 1+1 dimensions and other unphysical operations us- as charge conjugation and time reversal, constituting a ing two trapped ions. We also give a recipe for measuring novel toolbox of accessible quantum operations in the observables and a roadmap towards more general scenar- general frame of quantum simulations. While quantum ios. In this sense, this work provides a novel toolbox for simulators may soon realize calculations impossible for quantum simulations. classical computers, we show here the possibility of im- There are three discrete symmetries [26] which are cen- plementing quantum dynamics that are impossible for tral to quantum mechanics and our understanding of our quantum world. particles, fields and their interactions: parity, P, time The Majorana equation is a relativistic wave equation reversal, T , and charge conjugation, C. None of these for fermions where the mass term contains the charge operations can be carried out in the real world: P in- conjugate of the spinor, ψc, volves a global change of physical space, while C and i ∂/ψ = mcψ . (1) T are antiunitaries. However, there is no apparent fun- ~ c damental restriction for implementing them in physical µ Here, ∂/ = γ ∂µ and γµ are the Dirac matrices [18], while systems that simulate quantum mechanics. We will fo- 2

FIG. 1. Diagram showing the different steps involved in the quantum simulation of unphysical operations in 1+1 dimensions. cus on the study of antiunitary operations, which can be the Majorana equation for a complex spinor becomes a decomposed into a product of a unitary, UC or UT , and 3+1 Dirac equation with dimensional reduction, py, pz = complex conjugation, Kψ = ψ∗. We consider the map- 0, and a four-component real bispinor ping of the quantum states of an n-dimensional complex 2 i ∂tΨ = c(11 ⊗ σx)px − mc σx ⊗ σy Ψ. (5) Hilbert space, Cn, onto a real Hilbert space [25], R2n, ~ 1 ψ + ψ∗ Here, the dynamics preserves the reality of the bispinor ψ ∈ → Ψ = ∈ . (2) Cn 2 i(ψ∗ − ψ) R2n Ψ and cannot be reduced to a single 1+1 Dirac particle. In general, the complex-to-real map in arbitrary dimen- This mapping can be implemented by means of an auxil- sions transforms a Majorana equation into a higher di- iary two-level system, such that R2n ∈ H2 ⊗ Hn. In this mensional Dirac equation [27]. Since Eq. (5) is a Hamilto- manner, the complex conjugation of the simulated state nian equation, it can be simulated in a quantum system. becomes a local unitary VK acting solely on the ancillary The mapping of wavefunctions into larger spinors also ∗ space, Kψ = ψ → VKΨ = (σz ⊗11)Ψ, and thus physically allows us to explore exotic symmetries and unphysical op- implementable for a wavefunction of arbitrary dimen- erations, otherwise impossible in nature. From Eqs. (3), sions. Furthermore, unitaries and observables can also be (4), and (5), for the 1+1 dimensional case, we can deduce y mapped onto the real space, O → Θ = 11⊗Or −iσ ⊗Oi, that charge conjugation is implemented in the enlarged 1 i where Or = 2 (O + KOK) and Oi = − 2 (O − KOK), space via the unitary operation VC preserving unitarity and Hermiticity. The proposed simulator also accommodates the antiunitary operations C = ψc = Cψ = UC Kψ → VCΨ = −(σz ⊗ σx)Ψ. (6) U K and T = U K. To this end, we have to choose a C T We can do something similar with time reversal, defined particular representation that fixes the unitaries U and C as the change t → (−t). In this case, we expect [19] U , as will be shown below. T i ∂ ψ0(τ) = Hψ0(τ), where the time variable τ = −t We possess now the basic tools to simulate the Majo- ~ τ and the modified spinor ψ0(τ) = T ψ(t). In order to pre- rana equation (1). The expression for the charge con- serve scalar products and distances, the time reversal jugate spinor is given by ψ = Wγ0Kψ, with W a uni- c operator must be an anti-unitary operator and thus de- tary matrix satisfying W−1γµW = − (γµ)T . We illus- composable as the product T = U K. In 1 + 1 dimen- trate now the proposed quantum simulation with the case T sions, imposing that the Hamiltonian be invariant under of 1+1 dimensions. Here, a suitable representation of time reversal, T −1HT , implies that the unitary satisfies charge conjugation is ψ = iσ σ ψ∗, that is W = iσ , c y z y U −1(iσ ∂ )U = −iσ ∂ , with a possible choice being and the Majorana equation reads T x x T x x UT = σz. In other words, in the enlarged simulation space 2 ∗ i~∂tψ = cσxpxψ − imc σyψ , (3) T ψ = UT Kψ → VT Ψ = (σz ⊗ σz)Ψ. (7) where p = −i ∂ is the momentum operator. Note that x ~ x See Fig. 1 for a scheme of the simulated symmetries. As Eq. (3) is not Hamiltonian, (i ∂ ψ 6= Hψ). This is due ~ t mentioned before, quantum simulations of unphysical op- to the presence of a complex conjugate operation in the erations can be straightforwardly extended to higher di- right-hand side of Eq. (3), which is not a linear Hermitian mensions. In this sense, Eqs. (6) and (7) will be valid for operator. Surprisingly, through our mapping (2), wave functions ψ of dimension d as long as we consider  r  ψ1 the complex conjugation of an arbitrary wavefunction as r ψ1  ψ2  VKΨ = (σz ⊗ 11d)Ψ. ∈ C2 → Ψ =  i  ∈ R4, (4) ψ2  ψ1  The proposed protocol for implementing unphysical i ψ2 operations onto a physical setup allows us to deal with 3 situations that are, otherwise, intractable wih conven- the Hamitonian in the interaction picture reads tional quantum simulations. To exemplify the value of † iδt −iδt this novel building block in the quantum simulation tool- H = ~ηrΩ(σx ⊗ 11 − 11 ⊗ σy)(b e + be ), ˜ † box, we consider the case of an advanced experimental +~ηΩ(11 ⊗ σx)i(a − a) (9) quantum simulation, impossible to reproduce with classi- 1/4 p 0 cal computers. We assume that, after a certain evolution where η ≡ ηr3 ≡ ~/4m ν 1 is the Lamb- 0 time, it is crucial to realize an unphysical operation such Dicke parameter and m the ion mass. In the limit of p † ˜ † as charge conjugate or time reversal, before continuing large detuning, we have δ ηrΩ hb bi, ηΩ|ha − ai| the unitary (physical) evolution. With existing tools in and we recover Eq. (5) with the momentum operator † ˜ quantum simulations, we would need to stop the dynam- px = i~(a −a)/2∆ and the equivalences c = 2η∆Ω and 2 2 2 p 0 ics, implement a full quantum tomography of the current mc = 2~ηr Ω /δ with ∆ = ~/4m ν. Introducing the 2 quantum state associated to a huge Hilbert space, apply ratio γ = |mc2/hcp i|, with γ = 2(ηr Ω/δ) , we see it x |hi(a†−a)i|(ηΩ˜/δ) the unphysical operation in a classical computer, encode is possible to tune the numerator and denominator inde- back the modified quantum state into the experimental pendently so as to preserve the dispersive regime, while setup, and then to go ahead with the quantum simula- exploring simultaneously the range from γ ' 0 (ultrarel- tion. Clearly, this task would be impossible with classical ativistic limit) to γ → ∞ (nonrelativistic limit). resources and would become possible with a suitable im- A relevant feature of the Majorana equation in 3+1 di- plementation of our proposed ideas. mensions is the conservation of helicity. A reminiscent of In a recent experiment, the dynamics of a free Dirac the latter in 1+1 dimensions is the observable called here- particle was simulated with a single trapped ion [13]. after as pseudo-helicity Σ = σxpx. This quantity is con- Here, Eq. (5) has a more complex structure requiring served in the 1+1 Majorana dynamics of Eq. (3) but not a different setup. Moreover, the encoded Majorana dy- in the 1+1 Dirac equation. We will use this observable namics requires a systematic decoding via a suitable re- to illustrate measurements on the Majorana wavefunc- verse mapping of observables. We can simulate Eq. (5) tion. The mapping for operators can be simplified if we in two trapped ions, with lasers coupling their internal are only interested in expectation values. Reconstructing states and motional degrees of freedom. The kinetic the complex spinor with the non-square matrix ψ = MΨ part, cpx(11 ⊗ σx), is created with a laser tuned to the and M = 11 i11 , associated with Eqs. (4) and (5), we † blue and red motional sidebands of an electronic transi- have hOiψ = hψ|O|ψi = hΨ|M OM|Ψi =: hO˜iΨ. There- tion [12, 14], focussed on ion 2. The second term, σx ⊗σy, fore, to obtain the pseudo-helicity Σ, we have to measure is derived from detuned red and blue sideband excitations † acting on each ion. The Hamiltonian describing this sit- Σ˜ = M σxpx M = (11 ⊗ σx − σy ⊗ σx) ⊗ px (10) uation reads in the enlarged simulation space. In ion-trap experi- ω ω ments, we can use laser pulses to map information about H = 0 σz + 0 σz + νa†a + ν b†b ~ 2 1 ~ 2 2 ~ ~ r the pseudo-helicity onto the internal states. The appli- h 0 0 i i(qz1−ω1t+φ1) i(qz1−ω1t+φ1) + cation of a state-dependent displacement operation on + ~Ω (e + e )σ1 + H.c. ion 2, U2 = exp(−ik(11 ⊗ σy) ⊗ px/2), generated by res- h 0 0 i i(qz2−ω2t+φ2) i(qz2−ω2t+φ2) + onant blue and red sidebands, followed by a measure- + ~Ω (e + e )σ2 + H.c. ment of 11 ⊗ σz, is equivalent to measuring the observ- h 0 0 i ˜ i(qz2−ωt+φ) i(qz2−ω t+φ ) + 1 1 + ~Ω (e + e )σ2 + H.c. . able [13] A(k) = cos(k px)( 1 ⊗ σz) + sin(k px)( 1 ⊗ σx). Here, k is proportional to the probe time tprobe. Note d that hA(k)i ∝ h(11⊗σx)⊗pxi. Therefore, the first z dk k=0 Here z1,2 = Z ± 2 are the ion positions, measured from term in (10) can be measured by applying a short probe the center of mass, Z, and relative coordinate, z. The pulse to the ions and measuring the initial slope of the 0 phases of the lasers φi for i = 1, 2, (φ, φ ), are controlled observable A(k). To measure the second term in Eq. (10), to perform the interaction term (kinetic term). The fre- we apply the operation U1 = exp(−ik(σx ⊗ 11) ⊗ px/2), quencies of the center of mass and stretch mode are given √ and measure the spin correlation σz ⊗σx. We have, then, † † by ν and νr = 3ν, while a , a, b , and b, are the corre- d dk hσz ⊗ σxi k=0 = 2h(σy ⊗ σx) ⊗ pxi. sponding creation and annihilation operators. Finally, Ω So far, we have presented a complete toolbox of un- ˜ and Ω are the laser Rabi frequencies in the rotating-wave physical operations, C, T , and K. We can combine all approximation. With the adequate choice of parameters, these tools to study dynamical properties of the transformed wavefunctions. To exemplify the kind of experi- ω1 = ω0 + νr − δ ω = ω0 − ν φ1 = π/2 ments that become available, we have studied the scat- ω0 = ω − ν + δ ω0 = ω + ν φ0 = π/2 tering of wavepackets against a linearly growing poten- 1 0 r 0 1 (8) ω2 = ω0 − νr + δ φ = π φ2 = 0 tial, V (x) = αx, with conventional numerical tools. It 0 0 0 ω2 = ω0 + νr − δ, φ = 0, φ2 = 0, is known that repulsive potentials can be penetrated by 4

FIG. 2. Scattering of a fermion against a linearly growing potential (inset). (a) Ordinary Klein process (b) At an instant of time we apply the time reversal operator T causing the particle to retrace its own trajectory. (c) Similar to (b) but now we apply charge conjugation, converting the particle in its antiparticle. (d) Scattering of a Majorana particle, which propagates through the potential. Parameters are m = 0.5, c = 1 and V (x) = x, in dimensionless units.

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