PHYSICAL REVIEW X 1, 021018 (2011)
Quantum Simulation of the Majorana Equation and Unphysical Operations
J. Casanova,1 C. Sabı´n,2 J. Leo´n,2 I. L. Egusquiza,3 R. Gerritsma,4,5 C. F. Roos,4,5 J. J. Garcı´a-Ripoll,2 and E. Solano1,6 1Departamento de Quı´mica Fı´sica, Universidad del Paı´s Vasco - Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain 2Instituto de Fı´sica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain 3Departamento de Fı´sica Teo´rica, Universidad del Paı´s Vasco - Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain 4Institut fu¨r Quantenoptik und Quanteninformation, O¨ sterreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria 5Institut fu¨r Experimentalphysik, Universita¨t Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria 6IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain (Received 22 April 2011; published 9 December 2011) 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. Driven by the need of the simulation, we devise a general method for implementing a number of mathematical operations that are unphysical, including charge conjugation, complex conjugation, and time reversal. Furthermore, we describe how to realize the general method in a system of trapped ions. The work opens a new front in quantum simulations.
DOI: 10.1103/PhysRevX.1.021018 Subject Areas: Quantum Physics, Optics
A quantum simulator is a device engineered to reproduce may soon realize calculations that are impossible for the properties of an ideal quantum model. This still- classical computers, we show here the possibility of im- emerging topical area has generated a remarkable exchange plementing quantum dynamics that do not occur in the real of scientific knowledge between apparently unconnected space-time quantum world. subfields of physics. In terms of applications, it allows for The Majorana equation is a relativistic wave equation the study of quantum systems that cannot be efficiently for fermions where the mass term contains the charge simulated on classical computers [1]. While a quantum conjugate of the spinor, c c, computer could in principle implement a universal quantum simulator [2], only particular systems have been simulated i@6@c ¼ mcc c: (1) up to now using dedicated quantum simulators [3]. Still, 6@ ¼ @ there is a wealth of successful cases, such as spin models Here, and are the Dirac matrices [18], while [4,5], quantum chemistry [6], and quantum phase transi- the non-Hamiltonian character stems from the simulta- c c tions [7]. The quantum simulation of fermionic systems [8] neous presence of and c. The significance of the and relativistic quantum physics have also attracted recent Majorana equation lies in the fact that it can be derived attention, reproducing dynamics and effects that are cur- from first principles in a similar fashion as the Dirac rently out of experimental reach. Examples include black equation [17,19]. Both wave equations are Lorentz invari- holes in Bose-Einstein condensates [9], quantum field theo- ant but the former preserves helicity and does not admit ries [10,11], and recent quantum simulations of relativistic stationary solutions. The Majorana equation is considered quantum effects such as Zitterbewegung and the Klein a possible model [20] for describing exotic particles in paradox [12–16] in trapped ions. supersymmetric theories—photinos and gluinos—or in In this paper, we show how the Majorana equation [17] grand unified theories, as with the case of neutrinos. can be simulated in an analog quantum simulator, having Indeed, the discussion of whether neutrinos are Dirac or the implementation of complex conjugation of the wave Majorana particles still remains open [21]. Despite the function as a key requirement. In this manner, we are able similar naming, however, this work is neither related to to propose a general scheme for implementing this and the Majorana fermions (modes) in many-body systems other unphysical operations, such as charge conjugation [22,23], nor to the Majorana fermions (spinors) in the and time reversal. The implementations constitute a novel Dirac equation [20,24]. toolbox of accessible quantum operations in the general In order to simulate the Majorana equation, we have frame of quantum simulations. While quantum simulators to solve a fundamental problem: the physical implementa- tion of antilinear and antiunitary operations in a quantum simulator. Here, we introduce a mapping [25] by which complex conjugation, an unphysical operation, becomes a unitary operation acting on an enlarged Hilbert Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- space. The mapping works in arbitrary dimensions and bution of this work must maintain attribution to the author(s) and can be immediately applied on different experimental the published article’s title, journal citation, and DOI. setups. We show how to simulate the Majorana equation
2160-3308=11=1(2)=021018(5) 021018-1 Published by the American Physical Society J. CASANOVA et al. PHYS. REV. X 1, 021018 (2011)
1 1 r;i in þ dimensions and other unphysical operations using where c 1;2 are the real and imaginary parts of c 1 and c 2, two trapped ions and their interactions with laser pulses. respectively. Surprisingly, the non-Hermitian Majorana We also give a recipe for measuring observables and a road equation for a complex spinor becomes a 3 þ map toward more general scenarios. In this sense, this work 1-dimensional Dirac equation for a four-component real provides a novel toolbox for quantum simulations. bispinor, There are three discrete symmetries [26] that are central 2 to quantum mechanics and our understanding of particles, i@@t ¼½cð1 xÞpx mc x y ; (5) fields, and their interactions: parity P , time reversal T , and charge conjugation C. None of these operations can be with dimensional reduction, py;pz ¼ 0. Here, the dynam- carried out in the real world: P involves a global change of ics preserves the reality of the bispinor and cannot be physical space, while C and T are antiunitaries. However, reduced to a single 1 þ 1 Dirac particle. In general, the there is no apparent fundamental restriction for implement- complex-to-real map in arbitrary space-time dimensions ing them in physical systems that simulate quantum transforms a Majorana equation into a higher dimensional mechanics. We will focus on the study of antiunitary Dirac equation [27]. Since Eq. (5) is a Hamiltonian equa- operations, which can be decomposed into a product of a tion, it can be simulated in a quantum system. unitary, UC or UT , and complex conjugation, Kc ¼ c . The mapping of wave functions into higher-dimensional We consider the mapping of the quantum states of an spinors also allows us to explore exotic symmetries and n-dimensional complex Hilbert space Cn onto a real unphysical operations, which are otherwise impossible in Hilbert space [25] R2n, nature. From Eqs. (3)–(5), for the 1 þ 1-dimensional case, ! we can deduce that charge conjugation is implemented in 1 c þ c the enlarged space via the unitary operation VC, c 2 Cn ! ¼ 2 R2n: (2) 2 iðc c Þ c c ¼ Cc ¼ UCKc ! VC ¼ ð z xÞ : (6) This mapping can be implemented by means of an auxil- We can perform something similar with time reversal, iary two-level system, such that R2n 2 H 2 H n. In this defined as the change t !ð tÞ. In this case, we expect manner, the complex conjugation of the simulated state 0 0 V [19] i@@ c ð Þ¼Hc ð Þ, where the time variable ¼ t becomes a local unitary K acting solely on the ancillary 0 and the modified spinor c ð Þ¼T c ðtÞ. In order to space, Kc ¼ c ! VK ¼ð z 1Þ , and thus is physically implementable for a wave function of arbitrary preserve scalar products and distances, the time-reversal dimensions. Furthermore, unitaries and observables operator must be an antiunitary operator and thus decom- T U K 1 1 can also be mapped onto the real space, O ! ¼ posable as the product ¼ T .In þ dimensions, 1 O i y O O 1 O KOK O imposing that the Hamiltonian be invariant under time r i, where r ¼ 2 ð þ Þ and i ¼ 1 i O KOK reversal T HT implies that the unitary operator satisfies 2ð Þ, preserving unitarity and Hermiticity. 1 U ði x@xÞU ¼ i x@x, with a possible choice being The proposed simulator also accommodates the antiunitary T T UT ¼ z. In other words, in the enlarged simulation operations C ¼ UCK and T ¼ UT K. To this end, we have to choose a particular representation that fixes the space unitaries UC and UT , as will be shown below. T c ¼ UT Kc ! VT ¼ð z zÞ : (7) We possess now the basic tools to simulate the Majorana equation (1). The expression for the charge-conjugate Figure 1 illustrates a scheme of the simulated symmetries. 0 spinor is given by c c ¼ W Kc , with W a unitary As mentioned before, quantum simulations of unphysical 1 T matrix satisfying W W ¼ ð Þ . We illustrate operations can be straightforwardly extended to higher now the proposed quantum simulation with the case of dimensions. In this sense, Eqs. (6) and (7) will be valid 1 þ 1 dimensions. Here, a suitable representation of charge for wave functions c in a d-dimensional space as long as conjugation is c c ¼ i y z c , for which W ¼ i y, and we consider the complex conjugation of an arbitrary wave the Majorana equation reads function as VK ¼ð z 1dÞ . The proposed protocol for implementing unphysical 2 i@@t c ¼ c xpx c imc y c ; (3) operations within a physical setup allows us to deal with situations that are, otherwise, intractable with conventional where px ¼ i@@x is the momentum operator. In particular, quantum simulations. To exemplify the value of this novel our mapping of Eq. (2) for n ¼ 2 reads building block in the quantum-simulation toolbox, we 0 1 consider a case of an advanced quantum simulation, where c r ! B 1 C after a certain time of unitary (physical) evolution of the c B c r C system, an unphysical operation, such as the charge con- 1 C B 2 C R ; 2 2 ! ¼ B i C 2 4 (4) jugate or time reversal, needs to be performed, before the c 2 @ c A 1 unitary evolution is continued. Such a situation is impos- c i 2 sible to reproduce seamlessly with classical computers.
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FIG. 1. Diagram showing the different steps involved in the proposed quantum simulation of unphysical operations in 1 þ 1 dimensions.
0 With tools that currently exist in quantum simulations, we !1 ¼!0 þ r ; !1 ¼!0 r þ ; !2 ¼!0 r þ ; would need to stop the dynamics, implement a full quan- !0 ! ; ! ! ; !0 ! ; ; tum tomography of the current quantum state 2 ¼ 0 þ r ¼ 0 ¼ 0 þ ¼ 0 0 0 associated to a huge Hilbert space, apply the unphysical ¼0; 1 ¼ =2; 1 ¼ =2; 2 ¼0; 2 ¼0; operation in a classical computer, encode back the modi- fied quantum state into the experimental setup, and then to (8) proceed with the quantum simulation. Clearly, this task would be impossible with classical resources but could the Hamitonian in the interaction picture reads as be accomplished with a suitable implementation of our y i t i t proposed ideas. H ¼ @ r ð x 1 1 yÞðb e þ be Þ In a recent experiment, the dynamics of a free Dirac @ ~ 1 i ay a ; particle was simulated with a single trapped ion [13]. þ ð xÞ ð Þ (9) Here, Eq. (5) has a more complex structure requiring a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=4 0 different setup. Moreover, the encoded Majorana dynamics where r3 @=4m 1 is the Lamb-Dicke requires a systematic decoding via a suitable reverse map- m0 parameter and is the ionp mass.ffiffiffiffiffiffiffiffiffiffiffiffi In the limit of large ping of observables. We can simulate Eq. (5) with two y ~ y detuning, we have r hb bi, jha aij, and trapped ions, with lasers coupling their internal states and we recover Eq. (5) with the momentum operator px ¼ motional degrees of freedom. The kinetic part, cpxð1 xÞ, i@ ay a =2 c 2 ~ is created with a laser tuned to the blue and red motional ð Þ and the equivalencespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ and 2 2 2 0 mc ¼ 2@ r = with ¼ @=4m . Introducing the ra- sidebands of an electronic transition [12,14], which is fo- 2 2 2ð r = Þ tio ¼jmc =hcpxij,with ¼ , we see that cused on ion 2. The second term, x y, is derived from jhiðay aÞijð ~ = Þ detuned red and blue sideband excitations stimulated in each it is possible to tune the numerator and denominator inde- ion. The Hamiltonian describing this situation reads as pendently to preserve the dispersive regime, while exploring 0 !0 !0 simultaneously the range from ’ (ultrarelativistic limit) H ¼ @ z þ @ z þ @ aya þ @ byb 2 1 2 2 r to !1(nonrelativistic limit). 3 1 i qz ! t i qz !0 t 0 An interesting feature of the Majorana equation in þ @ e ð 1 1 þ 1Þ e ð 1 1 þ 1Þ þ H:c: þ f½ þ 1 þ g dimensions is the conservation of helicity. Its reminiscent 0 0 iðqz2 !2tþ 2Þ iðqz2 ! tþ Þ þ 1 1 þ @ f½e þ e 2 2 2 þ H:c:g in þ dimensions is an observable called, hereafter, p 0 0 pseudohelicity ¼ x x. This quantity is conserved in @ ~ eiðqz2 !tþ Þ eiðqz2 ! tþ Þ þ H:c: : þ f½ þ 2 þ g the 1 þ 1-dimensional Majorana dynamics of Eq. (3) but not in the 1 þ 1-dimensional Dirac equation. We will use z ¼ Z z=2 Here 1;2 are the positions, of the two ions, this observable to illustrate measurements on the Majorana respectively, measured from their center of mass Z and z wave function. The mapping for operators can be simpli- relative coordinate . The phases of the lasers i, for fied if we are only interested in their expectation values. i 1; 2 ; 0 ¼ ( ), are controlled to simulate the interaction Reconstructing the complex spinor with the nonsquare term (kinetic term). The frequencies of the center-of-mass matrix c ¼ M and M ¼ð1 i1 Þ, associated with vibration and the stretch mode are given by and pffiffiffi Eqs. (4) and (5), we have hOic ¼hc jOjc i¼ 3 ay a by b r ¼ , respectively, and whereas , , , and ,are y ~ h jM OMj i¼: hOi . Therefore, to obtain the pseudo- the corresponding creation and annihilation operators. ~ helicity , we have to measure Finally, and are the laser Rabi frequencies in the rotating-wave approximation. With the adequate choice of ~ My p M 1 p parameters, ¼ x x ¼ð x y xÞ x (10)
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FIG. 2. Scattering of a fermion against a linearly growing potential (inset). (a) Ordinary Klein process. (b) At an instant of time indicated by the dashed line 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 into its antiparticle. (d) Scattering of a Majorana particle, which propagates through the potential. Parameters are m ¼ 0:5, c ¼ 1, and VðxÞ¼x, in dimensionless units. in the enlarged simulation space. In the type of trapped-ion While there are no plane wave solutions in the Majorana experiments discussed above, we can use laser pulses to equation, we can still see a wave packet penetrating the map information about the pseudohelicity onto the internal barrier, showing a counterintuitive insensitivity to the states. The application of a state-dependent displacement presence of it. operation on ion 2, U2 ¼ exp½ ikð1 yÞ px=2 , gen- The previous implementation of discrete symmetries is erated by resonant blue and red sidebands, followed by a valid both for Majorana and Dirac equations. Equally measurement of 1 z, is equivalent to measuring interesting is the possibility of combining both Dirac and the observable [13] AðkÞ¼cosðkpxÞð1 zÞþsinðkpxÞ Majorana mass terms in the same equation [20], i@6@c ¼ ð1 xÞ. Here, k is proportional to the probe time tprobe. mMcc c þ mDcc , which still requires only two ions to d Note that hAðkÞijk 0 /hð1 xÞ pxi. Therefore, the implement experimentally. It also becomes feasible to dk ¼ CP first term in (10) can be measured by applying a short probe have -violating phases in the Dirac mass term m exp i 5 pulse to the ions and measuring the initial slope of the D ð Þ. Furthermore, we could study the dynamics M c M observable AðkÞ. To measure the second term in Eq. (10), of coupled Majorana neutrinos with a term c, where is now a matrix and c ¼ c ðx1;x2Þ is the combination of we apply the operation U1 ¼ exp½ ikð x 1Þ px=2 , two such particles, simulated with three ions and two and measure the spin correlation z x. We have, then, d vibrational modes. dk h z xijk¼0 ¼ 2hð y xÞ pxi. So far, we have presented a complete toolbox for im- In summary, we have introduced a general method to plementing unphysical operations, C, T , and K. We can implement quantum simulations of unphysical operations combine all these tools to study the dynamical properties of and non-Hamiltonian dynamics, such as the Majorana the transformed wave functions. To exemplify the kind of equation, in a Hamiltonian system. experiments that become available, we have studied the The authors acknowledge funding from Basque scattering of wave packets against a linearly growing po- Government Grants No. BFI08.211, IT559-10, and tential, VðxÞ¼ x, with conventional numerical tools. It is No. IT472-10; Spanish MICINN FIS2008-05705, FIS2009- known that repulsive potentials can be penetrated by Dirac 10061, and FIS2009-12773-C02-01; QUITEMAD; the EC particles [18] due to Klein tunneling [14,15]. This is shown Marie-Curie program; and the CCQED and SOLID in Fig. 2(a), where a Dirac particle splits into a fraction of a European projects. particle that bounces back, and a large antiparticle compo- nent that penetrates the barrier. This numerical experiment has been combined with the discrete symmetries and the Majorana equation. In Fig. 2(b) we apply the time-reversal [1] R. Feynman, Simulating Physics with Computers, Int. J. 21 operation after the particle has entered the barrier: All Theor. Phys. , 467 (1982). [2] S. Lloyd, Universal Quantum Simulators, Science 273, momenta are reversed and the wave packet is refocused, 1073 (1996). tracing back exactly its trajectory. In Fig. 2(c) we apply [3] I. Buluta and F. Nori, Quantum Simulators, Science 326, charge conjugation, changing the sign of the charge and 108 (2009). turning a repulsive electric potential into an attractive one, [4] H. Friedenauer, H. Schmitz, J. Glueckert, D. Porras, and which can be easily penetrated by the antiparticle. In T. Scha¨tz, Simulating a Quantum Magnet with Trapped Fig. 2(d), we show the scattering of a Majorana particle. Ions, Nature Phys. 4, 757 (2008).
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