PHYSICAL REVIEW X 7, 041012 (2017)

Quantum Sensors for the Generating Functional of Interacting Quantum Field Theories

A. Bermudez,1,2,* G. Aarts,1 and M. Müller1 1Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom 2Instituto de Física Fundamental, IFF-CSIC, Madrid E-28006, Spain (Received 24 May 2017; revised manuscript received 7 August 2017; published 19 October 2017) Difficult problems described in terms of interacting quantum fields evolving in real time or out of equilibrium abound in condensed-matter and high-energy physics. Addressing such problems via controlled experiments in atomic, molecular, and optical physics would be a breakthrough in the field of quantum simulations. In this work, we present a quantum-sensing protocol to measure the generating functional of an interacting quantum field theory and, with it, all the relevant information about its in- or out-of-equilibrium phenomena. Our protocol can be understood as a collective interferometric scheme based on a generalization of the notion of Schwinger sources in quantum field theories, which make it possible to probe the generating functional. We show that our scheme can be realized in crystals of trapped ions acting as analog quantum simulators of self-interacting scalar quantum field theories.

DOI: 10.1103/PhysRevX.7.041012 Subject Areas: Atomic and Molecular Physics, Quantum Information

I. INTRODUCTION could mimic the relativistic QFTs that appear in high-energy physics, as occurs for the AQS of a Klein-Gordon QFT with Some of the most complicated problems of theoretical Bose-Einstein condensates [10,11]. Note, however, that the physics arise in the study of quantum systems with a large, most versatile AMO quantum simulators to date [3,4] do not sometimes even infinite, number of coupled degrees of work directly in the continuum, but on a physical lattice that is freedom (d.o.f.). These complex problems arise in our effort provided either by additional laser dipole forces for neutral to understand certain observations in condensed-matter [1] atoms [3] or by the interplay of Coulomb repulsion and or high-energy physics [2], which one tries to model with the electromagnetic oscillating forces for singly ionized atoms unifying language of quantum field theories (QFTs). More recently, the field of atomic, molecular, and optical (AMO) [4]. Therefore, the relevant symmetries of the high-energy physics is providing experimental setups [3,4] that aim at QFT, such as Lorentz invariance, must emerge as one takes targeting similar problems. The approach is, however, the continuum or low-energy limit in the AMO quantum rather different. These AMO setups can be microscopically simulator. This occurs trivially for free fermionic QFTs [12], designed to behave with great accuracy according to a which underlies the schemes for the AQS of Dirac QFTs particular model of interest. Hence, it is envisioned that one with ultracold atoms in optical lattices [14]. There are also will be capable of answering open questions about a many- proposals for interacting QFTs, such as the DQS of self- body model described through a QFT by preparing, evolv- interacting Klein-Gordon fields [15], the analog [16] and ing, and measuring the experimental system, what has been digital [17] quantum simulators of coupled Fermi-Bose called a quantum simulation [5,6]. fields, and an ultracold atom AQS of Dirac fields with Either in the form of piecewise time evolution by concat- self-interactions or coupled to scalar bosonic fields [18]. enated unitaries [7], i.e., digital quantumsimulation (DQS),or In the interacting case, as discussed in Refs. [15,18], continuous time evolution by always-on couplings [8],i.e., renormalization techniques must be employed to set the analog quantum simulation (AQS), the main focus in this field right bare parameters in such a way that a QFT with the has been typically placed on the quantum simulation of required Lorentz symmetry and free of ultraviolet diver- condensed-matter problems [3,4,9]. Nonetheless, some theo- gences is obtained in the continuum limit. This is the retical works have also addressed how quantum simulators standard situation in lattice-field theories [19], where the continuum limit is obtained by letting the lattice spacing a → 0, removing thus the natural UV cutoff of the lattice, *[email protected] while maintaining a finite renormalized mass or gap m describing the physical mass of the particles in the Published by the American Physical Society under the terms of corresponding QFT. This requires setting the bare param- the Creative Commons Attribution 4.0 International license. eters close to a critical point of the lattice model, where Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, the dimensionless correlation length, measured in lattice and DOI. units, diverges ξ~ → ∞. In this case, the mass m ∼ 1=ξ~a can

2160-3308=17=7(4)=041012(18) 041012-1 Published by the American Physical Society A. BERMUDEZ, G. AARTS, and M. MÜLLER PHYS. REV. X 7, 041012 (2017)

2 remain constant even for vanishingly small lattice spacings. 1 μ m0 2 L ¼ ∂μϕðxÞ∂ ϕðxÞ − ϕðxÞ − VðϕÞ; ð1Þ Therefore, the experience gained in the classical numerical 2 2 simulation of interacting QFTs on the lattice will be of the ∂ ∂ ∂ μ ∂μ ημν∂ ’ utmost importance for the progress of quantum simulators where μ ¼ = x , ¼ ν with Minkowski s metric of high-energy physics problems. η ¼ diagð1; −1; …; −1Þ, and we use Einstein’s summation In a more direct connection to open problems in high- criterion for repeated indexes. Here, m0 is the bare mass of energy physics, e.g., the phase diagram of quantum chromo- the scalar boson, and VðϕÞ describes its self-interaction 4 dynamics [20], we note that there have been a number of through nonlinearities, e.g., λϕ orR cosðβϕÞ. In these units, proposals for the DQS [21–23] and AQS [24–27] of gauge in order to make the action S ¼ dDxL dimensionless, theories. As noted above, previous knowledge from lattice the scalar field must have classical mass dimensions dϕ D − 2 =2, while the couplings have d 2 2 gauge theories has been essential to come up with schemes ¼ð Þ m0 ¼ for the quantum simulation of Abelian [22,24] and non- and dλ ¼ð4 − DÞ. Abelian [23] QFTs of the gauge sector, as well as Abelian Let us now introduce the so-called Schwinger sources [25,26] and non-Abelian [27] QFTs of gauge fields coupled [29], which are classical background fields that generate to Dirac fields. Starting from the simpler QFTs discussed excitations of the quantum field. For the real scalar QFT above, this body of work constitutes a well-defined long- [2], it suffices to introduce a classical scalar background term road map for the implementation of relevant models of field JðxÞ and modify the Lagrangian according to high-energy physics in AMO platforms [28]. In this work, we address the question of devising a general measurement L → LJ ¼ L þ JðxÞϕðxÞ; ð2Þ strategy to extract the properties of an interacting QFT, which could be adapted to these different quantum simu- where the sources have mass dimension dJ ¼ðD þ 2Þ=2. lators. One possibility would be to mimic the high-energy The normalized generating functional is obtained from scattering experiments in particle accelerators by preparing the vacuum-to-vacuum propagator, after removing proc- wave packets and measuring the outcome after a collision, as esses where particles are spontaneously created or annihi- proposed in the context of DQS [15]. In this work, we lated in the absence of the Schwinger sources. This can be explore a different possibility that would allow the quantum expressed as simulator to extract the complete information about an n R o D ϕ interacting QFT. We introduce a scheme that is capable Z½JðxÞ ¼ hΩjT ei d xJðxÞ HðxÞ jΩi; ð3Þ of measuring the generating functional of the QFT [2].In particular, this functional can be used to extract the Feynman where we introduce the ground state of the interacting propagator, such that one can also make predictions QFT Ω and use the time-ordering symbol T. Additionally, about different scattering experiments. In addition, other j i the field operators are expressed in the Heisenberg relevant properties of the interacting QFT can also be picture of the interacting QFT in the absence of directly extracted from such a functional. Moreover, our Schwinger sources. This is achieved by defining ϕ x scheme is devised for analog quantum simulators, such R R Hð Þ¼ D H − D H that the resource requirements are lower than those of a Tfei d x gϕðxÞTfe i d x g, where the integral in the DQS using a fault-tolerant quantum computing hardware. evolution operator includes integration over time, and 1 1 2 II. SENSORS FOR QUANTUM FIELD THEORIES H π 2 ∇ϕ 2 m0 ϕ 2 V ϕ ¼ 2 ðxÞ þ 2 ðxÞ þ 2 ðxÞ þ ð Þð4Þ In this section, we introduce a scheme to measure the generating functional of a QFT directly in the continuum. is the Hamiltonian density associated to the QFT under For the sake of concreteness, we present our results by study [Eq. (1)]. Here, πðxÞ¼∂tϕðxÞ is the conjugate focusing on a real scalar QFT, and we comment on momentum fulfilling the equal-time canonical commuta- generalizations to other QFTs at the end of the section. tion relations with the scalar field ½ϕðt; xÞ; πðt; yÞ ¼ iδdðx − yÞ. A. Self-interacting Klein-Gordon QFT, Schwinger The normalized generating functional, hereafter sources, and the generating functional simply referred to as the generating functional, Let us consider a self-interacting real Klein-Gordon contains all the relevant information about the QFT. In ðnÞ QFT, which is described by the bosonic scalar field particular, any n-point Feynman propagator G ¼ Ω T ϕ ϕ Ω Z operator ϕðxÞ, where x ¼ðt; xÞ is a point in the h j f Hðx1Þ HðxnÞgj i can be obtained from ½J D ¼ðd þ 1Þ-dimensional Minkowski space-time with by functional differentiation: coordinates xμ, μ ∈ f0;…;dg, and we set ℏ ¼ c ¼ k ¼ 1. B δnZ½JðxÞ The Lagrangian density that governs the dynamics of the ðnÞ … − n G ðx1; ;xnÞ¼ð iÞ δ δ : ð5Þ scalar field is Jðx1Þ JðxnÞ J¼0

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Note that we are using the normalized generating functional QFTs on the lattice, we do not need a continuum but a Eq. (3), such that the factor Z−1ð0Þ in the propagator Eq. (5) countable set of ancillary spins or qubits (see Fig. 1). disappears as Zð0Þ¼1. Through the Gell-Mann–Low The classical background field Eq. (2), which was theorem [30], one can express the generating functional, introduced by Schwinger as a mathematical artifact in and thus any n-point propagator of the interacting QFT, order to calculate the generating functional of the interact- in terms of Feynman diagrams. Accordingly, Z½J becomes ing QFT Eq. (3), has now been promoted onto a quantum- a fundamental tool in the theoretical study of interacting mechanical source that may also have its own dynamics QFTs. The question that we address in the following described by a generic Hamiltonian Hσ. Hence, Eq. (4) section is if such a functional can also become an must be substituted by observable in some experiment. Note that we are not X referring to susceptibilities expressed in terms of retarded α α H → HJ ¼ H þ Hσ − J ðxÞσ ðxÞϕðxÞ: ð7Þ Green’s functions, which are typically measured in linear- α response experiments. We are instead looking for a scheme that allows one to measure the complete generating func- The main idea is that these quantum sources will not only tional, out of which one could calculate any time-resolved act as generators of excitations in the quantum field, but Feynman propagator, or obtain predictions of any type of also as quantum probes capable of measuring the generat- scattering experiment. The generating functional does ing functional of the interacting QFT. We discuss below a indeed contain all the relevant information about a QFT. particular measurement protocol to achieve this goal. The use of quantum-mechanical two-level systems as B. Z2 Schwinger sources sensors for measuring physical quantities with high pre- The proposed scheme promotes the classical Schwinger cision, such as electric or magnetic fields or oscillator fields Eq. (2) to quantum-mechanical Z2 Schwinger frequencies, is a well-developed technique in AMO physics sources. In particular, we consider the suð2Þ Lie algebra, [31]. In the two most standard cases, the two-level system σα σ0 I σβ can get excited (i.e., Rabi probe) or gain a relative phase and define the operators , where ¼ , and f gβ¼1;2;3 (i.e., Ramsey probe) as a consequence of its coupling to are the well-known Pauli matrices. The Z2 Schwinger field now reads the physical quantities that need to be measured. In many X situations of experimental relevance, one uses a single JðxÞ → JαðxÞσαðxÞ; ð6Þ quantum sensor and maintains its quantum coherence for α ever-increasing periods of time to improve the sensitivity of the measurement apparatus. In the context of QFTs, ever where JαðxÞ are classical background fields, and σαðxÞ can since the pioneering work of Unruh [32], Rabi-type probes be interpreted as the operators of an ancillary two-level based on a single particle with discrete energy levels have system (i.e., spin-1=2, qubit) that is attached to every space- been routinely considered as detectors of quantum fields time coordinate. We advance, however, that for AQS of [33]. These type of detectors have also been considered in a

FIG. 1. Schematic representation of the Schwinger sensors for the generating functional. We represent a quantum scalar field ϕðxÞ in a D ¼ 2 þ 1 space-time, which is discretized into a d ¼ 2 spatial lattice, while letting the time coordinate continuous. Inset: Zoom of a small space region, where the field at each point (red circles) is coupled to the fields at neighboring points (small springs) and can be excited by its coupling (wavy lines) to generalized quantum-mechanical Schwinger sources (green circles with arrows). These Z2 Schwinger sources will also function as quantum sensors for the generating functional.

041012-3 A. BERMUDEZ, G. AARTS, and M. MÜLLER PHYS. REV. X 7, 041012 (2017) quantum-simulation context [11,34]. In contrast, Ramsey- picture with respect to H0ðxÞ¼H þ Hσ, with H being the type probes have been mainly considered for the quantum Hamiltonian of the sourceless interacting QFT Eq. (4).In simulation of condensed-matter problems (see, e.g., the distant future, the time-evolution operator becomes Ref. [35]). On the other hand, in the context of high- n R o n R o − D H D ϕ energy physics, Ramsey probes remain largely unexplored i d x 0ðxÞ þi d xJðxÞ HðxÞPHðxÞ UJ ¼ T e T e ; ð9Þ (an exception is Ref. [36], which discusses interferometric R R measurements of string tension and the Wilson-loop D D i d xHσ ðxÞ −i d xHσ ðxÞ operator in gauge theories). In this work, we partially fill where PHðxÞ¼Tfe gPðt0; xÞTfe g, 1 0 3 this gap by showing that the interacting relativistic QFT and Pðt0; xÞ¼2 ½σ ðxÞ − σ ðxÞ is an orthogonal projector Eq. (7) with suð2Þ Schwinger sources of the Ramsey type, onto the ground state of the Z2 sensor localized at α i.e., setting J ðxÞ¼JðxÞðδα;0 − δα;3Þ=2, can function as a coordinate ðt0; xÞ. Finally, the observable information quantum sensor for the QFT generating functional. This about the generating functional is encoded in the expect- particular choice of the generalized Schwinger sources ation value of a spin-parity operator: guarantees a differential coupling of the field to the internal Y P Ψ † σ1 x Ψ states of the sensors, such that their relative phase will ½JðxÞ ¼ h ðt0ÞjUJ ðt0; ÞUJj ðt0Þi: ð10Þ depend on the time evolution of the quantum scalar field. In x the following, we show that a collective interferometric scheme can exploit this differential coupling to probe the We consider a simple Hamiltonian for the quantum sensors, full generating functional. 0 Hσ ¼ δε½σ ðt0; xÞ − Pðt0; xÞ; ð11Þ

C. Quantum sensors for the generating functional where δε is the energy density associated to the transition In addition to exploiting the quantization of energy levels frequency ω0 between the two levels of the sensor, and the quantum coherence, quantum sensors based on which has a natural realization in quantum-simulation ensembles of two-level systems can make use of entangle- AMO platforms. For this particular choice, one finds that ment to increase their sensitivity [37], or to gain informa- the expectation value of theR parity operator evolves accord- tion about equal-time density-density correlations from D δε ϕ ing to P½J¼hΩjTfei d x½ þJðxÞ HðxÞgjΩiþc:c:, and their short-time dynamics [38], which can be of interest thus encodes the generating functional Eq. (3), in the quantum simulation of condensed-matter problems. R In our case, entanglement is not used to increase the 1 P i dDxδεZ sensitivity, but is also an ingredient of paramount impor- ½JðxÞ ¼ 2 e ½JðxÞ þ c:c: ð12Þ tance to map the whole information of the relativistic QFT, which is encoded in the generating functional Z½J, into the Let us recapitulate the results obtained so far. By intro- α ensemble of Z2 Schwinger sources or sensors. Let us ducing a quantum sensor σ ðxÞ at each space-time coor- describe in detail the protocol. dinate, thus upgrading the standard Schwinger sources to We consider an initial state in the remote past as the Z2 fields Eq. (6), we have constructed a parity Ramsey tensor product of the interacting QFT ground state jΩi and interferometer capable of encoding the generating func- the quantum sensor state with all spins pointing down j0σi. tional of an interacting QFT in its time evolution Eq. (12) This assumes that the state of the quantum field is for a particular set of background sources fulfilling α adiabatically prepared in the remote past by starting from J ðxÞ¼JðxÞðδα;0 − δα;3Þ=2. the noninteracting ground state and switching on the self- interactions VðϕÞ sufficiently slowly, i.e., adiabatically, D. Simplified sensors for Feynman propagators while the Schwinger-source couplings remain switched off. One then applies a fully entangling operation to the sensors In this section, we show that the protocol can be generating the so-called Greenberger-Horne-Zeilinger simplified considerably if one is interested in only n-point (GHZ) states [39], which are multipartite generalizations Feynman propagators Eq. (5). Such propagators contain a of the Einstein-Podolski-Rosen states [40]. This leads to lot of information relevant to the typical scattering experi-   ments and other types of real-time nonequilibrium phe- 1 Y Y nomena. For even n, these Feynman propagators can be Ψ Ω ⊗ ffiffiffi σ0 x σ1 x 0 j ðt0Þi¼j i p ðt0; Þþ ðt0; Þ j σi: inferred from the Ramsey parity signal by functional 2 x x differentiation, ð8Þ n R δ P J x 1 D δε ½ ð Þ i d x ðnÞ … At this instant of time t0, the Schwinger-source couplings ¼ e G ðx1; ;xnÞþc:c:; δJðx1ÞδJðxnÞ 0 2 are switched on. The time-evolution operator of the full J¼ sourced QFT Eq. (7) can be expressed in the interaction ð13Þ

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0 0 where we assume that an approximate remote-past to parity signal for a time longer than jx1 − x2j with the distant-future time evolution is obtained by setting following sets of Schwinger sources: Jð1Þ ¼ð0;0Þ, 0 0 − − ∀ 1 … ð2Þ ð3Þ ð4Þ t t0 > maxfjxi xj j; i; j ¼ ; ;ng. J ¼ðJ1;0Þ, J ¼ðJ1;J2Þ,andJ ¼ð0;J2Þ. Using these To estimate such functional derivatives, the required sets of infinitesimal Schwinger sources, one Schwinger field JðxÞ that must be experimentally applied can reconstruct the discretization of the two functional would be a comb of n-point-like sources, derivatives required to calculate the single-particle Feynman propagator in Eq. (5) for n ¼ 2. Therefore, the Xn D Feynman propagator can then be inferred from JðxÞ¼ Jiδ ðx − xiÞ; ð14Þ i¼1 X4 −1 mP JðmÞ ð Þ ½ i2ω0ðt−t0Þ where J are the strengths of infinitesimal field-sensor ≈−e Δðx1 − x2Þþc:c:; ð17Þ i J1J2 m¼1 couplings at the particular space-time coordinates xi. Since the Schwinger field Eq. (14) is applied only to a subset of J J ≪ 2 the quantum sensors located at Xs ¼fx1; …; xng, which where we assume that 1; 2 m0,andwherem0 is the already requires addressability, we may also consider that bare mass of the QFT Eq. (1). According to this expression, the initial entangling operation may involve only that we can infer the real [imaginary] part of the propagator particular subset. In that case, one can simplify the required by measuring at τ ¼ 2πr=ω0 [τ ¼ð2r þ 1Þπ=2ω0], where initial state Eq. (8) to r ∈ Z.   Let us now advance on the results of the following section, 1 Y Y where we discuss an implementation of this sensing scheme Ψ Ω ⊗ ffiffiffi σ0 x σ1 x 0 j ðt0Þi¼j i p ðt0; Þþ ðt0; Þ j σi: using AMO quantum simulators of QFTs. In this case, the 2 x∈ x∈ Xs Xs quantum sensors can also have spurious couplings to other ð15Þ quantum or classical fields, e.g., environmental electromag- netic fields, which cannot be switched on or off, but instead The fact that a GHZ state of all the spins is no longer act continuously during the probing protocol. Accordingly, required is a very important simplification, and also makes the parity oscillations will also get damped as a function of the protocol more robust as one considers the degrading the probing time with a characteristic dephasing time T2. effect of external sources of noise. Additionally, we do not Assuming that evolution of the field-sensor mixed state require one to measure the full spin parity Eq. (10), but only can be described in the Markovian regime, which is the Y case in many AMO platforms, the effects of the noise on the † 1 P Ψ σ x Ψ inω0ðt−t0Þ ½JðxÞ ¼ h ðt0ÞjUJ ðt0; ÞUJj ðt0Þi: ð16Þ time evolution amounts to substituting e → x∈ ω − −f − Xs ein 0ðt t0Þe ðfxjgÞðt t0Þ=T2 in the previous expressions, where fðfx gÞ is a particular function of the number and positions Note that mobile sensors might be available depending j of the probes. In some situations, as occurs for the trapped- on the particular implementation. In that case, we do not ion crystals [41] we describe below, these spurious couplings require a quantum sensor at every space-time coordinate, areP mainly due to global fields, and fðfxjgÞ ¼ but onlyP n sensors located at the corresponding points 1 2 2 n 0 d ð j Þ ¼ n , such that the visibility of the Ramsey parity Hσ ¼ 1 δε½σ ðt0; xÞ − Pðt0; xÞδ ðx − x Þ. j¼ j signal decays faster as the number of quantum sensors To infer the value of the functional derivative of the increases, limiting the advantage of this type of entangled parity signal Eq. (16), one needs to apply different sets of quantum sensors in other contexts [42]. This sets a constraint JðmÞ instantaneous sources Eq. (14), which we label by ¼ on the proposed protocol, as only space-time coordinates ðmÞ ðmÞ 0 0 2 ðJ1 ; …; Jn Þ, with m ¼ 1; …;M. For each of these sets − τ ≪ fulfilling maxfjxi xj jg < T2=n could be probed. of sources, one would then measure the corresponding To overcome this limitation, and given that the protocol parity oscillations P½JðmÞ. Finally, by adding and sub- already requires single-probe addressability, one may tracting these parities according to a given prescription encode the sensors in a decoherence-free subspace by obtained by the discretization of the functional derivatives, considering an entangled Neel-type initial state, one can infer an estimate of the n-point Feynman propa-   gators via Eq. (13). 1 Y Y Ψ Ω ⊗ ffiffiffi σ1 x σ1 x 0 To be more concrete, let us consider the important j ðt0Þi ¼ j i p ðt0; Þ ðt0; Þ j σi; 2 x∈ x∈ case of the single-particle two-point Feynman propagator Xe Xo ð2Þ Δðx1 − x2Þ¼G ðx1;x2Þ. In this case, our protocol ð18Þ requires creating a simple EPR pair between two distant quantum sensors at x1, x2 [Eq. (15)]. Additionally, we have where Xo ¼fx1; x3; …; xn−1g and Xe ¼fx2; x4; …; xng. to consider M ¼ 4 different measurements of the Ramsey Additionally, the Schwinger sources Eq. (6) must be

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δ 2 ðnÞ ρ T ϕ ϕ modified to JαðxÞ¼JðxÞ α;3= , and the SchwingerP field finite temperature, G ¼ Trð T f Hðx1Þ HðxnÞgÞ. n J −1 iþ1 Eq. (14) must become staggered, JðxÞ¼ i¼1 ið Þ Such a functional can be inferred from the spin-parity D δ ðx − xiÞ, which requires alternating field-sensor cou- oscillations of the quantum sensors, provided that the initial plings. In this case, the parity signals for each of the state is ρðt0Þ¼ρT ⊗ jΨðt0ÞihΨðt0Þj, where the initial state P Ψ entangledQ Neel-type initial states, ½J¼h ðt0Þj for the sensors corresponds to the GHZ state of Eq. (8).In † 1 U x∈ σ ðt0; xÞU jΨ ðt0Þi, lead to the following func- this case, we are assuming that the self-interacting Gibbs J Xs J ρ tional derivatives, state T can be prepared dissipatively in the distant past, while the GHZ spin state is prepared in analogy to the δnP J x 1 T ¼ 0 case. Since the distant-future time-evolution oper- ½ ð Þ ðnÞ … ¼ G ðx1; ;xnÞc:c:; ð19Þ ator is still given by Eq. (9), one can directly prove that the δJðx1ÞδJðx Þ 2 n J¼0 finite-T spin-parity evolves as − which directly yield the real (þ) and imaginary ( ) parts  Y  of the n-point propagator. The prescription to evaluate P ρ † σ1 x T½J¼Tr ðt0ÞUJ ðt0; ÞUJ the functional derivatives would be similar to the one x described above. In the ideal case, we assume that R P 1 i dDxδε −1 j 2 0 ¼ e Z ½Jþc:c:; ð21Þ fðfxjgÞ ¼ ½ jð Þ ¼ , such that no decoherence will 2 T affect the parity signals. In practice, as we discuss in more detail below, there will be nonglobal components of the and thus encodes the desired finite-T generating functional. source-field coupling and other sources of noise that will From this expression, one can directly reproduce the degrade the visibility of the parity oscillations, limiting previous results for the Feynman propagators, which the possible space-time points of the propagators that now correspond to finite-T time-ordered Green’s functions. can be measured. We also comment on a different strategy This can be generalized to initial states that are diagonal in to combat the effect of decoherence by combining the the energy eigenbasis of the interacting QFT, but not measurement scheme for the propagators Eq. (17) with necessarily distributed according to the Boltzmann weights. dynamical decoupling techniques (i.e., concatenated spin- Let us now discuss the generalization of these ideas echo sequences) [43]. In the impulsive regime where the to other QFTs, such as N-component scalar fields ϕ N Schwinger sources are switched on or off very fast f aðxÞga¼1, which can be used to model the scalar [Eq. (14)], the spin echoes will only refocus the decohering Higgs sector in theP standard model via an OðNÞ Klein- effect of the much slower fluctuating fields, but will not λ ϕ2 2 Gordon QFT with ½ a aðxÞ interactions. Measuring the affect the signal that we aim to measure. most generic generating functional of this QFT would require the same sensors but with couplings to each of the E. Finite temperature and other interacting QFTs field components that can be switched on or off independ- ently [i.e., different Schwinger functions Jα x N ]. So far, we have focused on a self-interacting bosonic f að Þga¼1 However, for the symmetry-broken phase, it may suffice QFT at T ¼ 0. As we mention in the Introduction, the connection to open problems in high-energy physics, to use a single source coupled to one component which is such as the phase transition between the hadron gas and singled out (Higgs component vs Goldstone modes). For quark-gluon plasma in quantum chromodynamics, would the gauge-field sector of the standard model, the quantum require considering finite-T regimes and other QFTs that sensors need to be coupled to each gauge potential a ∈ 1 … include fermionic matter at finite densities coupled to fAμðxÞg, where a f ; ;Ngg depends on the number gauge fields. The question that we thus address in this of generators of the gauge group; e.g., for the electromag- netic field in 3 þ 1 dimensions it suffices to consider four section is whether the sensing scheme for the generating α 3 functional can be applied to finite temperatures and different source fields fJμðxÞgμ¼0 that can be switched on generalized to other QFTs. or off independently. The situation gets more complicated Let us start by discussing the finite-T regime in the self- for the matter sector of the standard model, since these interacting Klein-Gordon QFT Eq. (1). The generating may require using also fermionic quantum sensors instead, functional in this case becomes whose combined action together with standard sources  n R o could play the role of the usual Grassmann Schwinger D ϕ Z ½JðxÞ ¼ Tr ρ T ei d xJðxÞ HðxÞ ; ð20Þ fields that appear in the generating functional. We leave this T T possibility for a future work, and instead comment on the R R possibility of using the protocol to measure generating −β ddxH −β ddxH where ρT ¼ e =Trðe Þ is the Gibbs state of functionals where the Schwinger sources are coupled to the QFT with Hamiltonian H [Eq. (4)] at temperature fermion bilinears, e.g., in the form of currents. This will be T ¼ 1=β. By functional differentiation, and using Eq. (5), of relevance for transport and linear response theory, in one recovers the correct n-point Feynman propagators at which transport properties can be extracted from real-time

041012-6 QUANTUM SENSORS FOR THE GENERATING FUNCTIONAL … PHYS. REV. X 7, 041012 (2017) P 2 2 2 correlators using Kubo relations. One example is the where ½∇ϕðxÞ ¼ α½ϕðx þ auαÞ − ϕðxÞ =a is the sum electrical conductivity, which is of interest for a wide range of forward differences along the axes with unit vectors uα. of systems, from graphene to the quark-gluon plasma. The spatial lattice serves as a regulator for the QFT, as the high-energy modes are cut off by the finite lattice spacing, such that only momenta below the cutoff are allowed, III. APPLICATION TO QUANTUM p ≤ Λ 2π SIMULATORS OF QFTS j j c ¼ =a. As noted in the Introduction, taking the continuum limit removes the cutoff Λc → ∞, and one has In this section, we argue that AMO quantum simulators to be careful with the UV divergences that appear in loop are an ideal scenario in which to apply our protocol to integrals when VðϕÞ ≠ 0 [2,19]. In this case, the bare mass measure the generating functional of a QFT. By exploiting m0 no longer coincides with the physical mass m of the quantum entanglement and coherence, the quantum sim- particles, but becomes instead a cutoff-dependent param- ulator can function as a nonperturbative gadget that eter m0ðΛcÞ through a so-called renormalization process calculates the Feynman propagator, and thus the corre- that we discuss in more detail below. sponding Feynman diagrams, to all orders in the interaction Let us now introduce the lattice Z2 Schwinger sources α parameters. According to the Introduction, we need to put Eq. (6) by attaching a spin-1=2 quantum sensor σx to each our findings in the generic context of lattice-field theories, lattice point x ∈ Λl, and defining a lattice Schwinger field α which is addressed in Sec. III A. In Secs. III B and III C,we JxðtÞ. Accordingly, we have to supplement the above discuss the direct connection of these lattice-field theory Hamiltonian of the lattice-field theory with concepts to AMO quantum simulators based on crystals X X d α α of trapped atomic ions. After outlining this connection, we H → HJ ¼ H þ Hσ − a JxðtÞϕðxÞσx; ð23Þ describe in detail renormalization and the continuum limit α x∈Λl of a generic scalar field theory in sec. III D, making connections to the trapped-ion implementation that offer where the dynamics of the sensors is governed by a practical view of this abstract topic. X d 0 Hσ ¼ a δεðσx − PxÞ; ð24Þ x∈Λ A. QFT and quantum sensors on the lattice l

In the following section, we focus on the AQS of and Px projects onto the ground state of the sensor at interacting QFTs since, in principle, these simulators can lattice site x ∈ Λl. Considering a Ramsey-type scheme, α be scaled up to the large sizes required to take the continuum Jx ¼ JxðtÞðδα;0 − δα;3Þ=2, the time-evolution operator limit without the need of quantum-error correction. From Eq. (9) on the lattice can be expressed as this perspective, we must consider lattice-field theories in n R P o t 0 d ϕ 0 x real time, where it is only the d-dimensional space that is − − þi dt a JxðtÞ Hðt ; ÞPx T iðt t0ÞH0 T t0 x∈Λl Λ Zd x∶ ∈ Z ∀ α UJðt; t0Þ¼ fe g e ; discretized on a lattice l ¼ a N ¼f xα=a N; ¼ 1 … Z ; ;dg,witha being the lattice spacing and N ¼ ð25Þ f1; 2; …;Ng [44]. However, we note that the scheme could be generalized to DQS, which could address the continuum where H0 ¼ H þ Hσ describes the uncoupled evolution of limit by exploiting quantum-error correction to minimize the self-interacting lattice field and the Z2 sensors. the accumulated Trotter errors and gate imperfections for Considering an initial maximally entangled state for the increasing system sizes. lattice sensors, Once again, we focus on the self-interacting scalar QFT,   ϕ x Y Y such that the field operator ð Þ and its canonically 1 0 1 π x ∂ ϕ x jΨðt0Þi¼jΩi ⊗ pffiffiffi σx þ σx j0σi; ð26Þ conjugate momentum ð Þ¼ t ð Þ are defined only 2 d x∈Λl x∈Λl for x ∈ Λl, and fulfill ½ϕðxÞ; πðyÞ ¼ iδx;y=a , which become the standard commutation relations ½ϕðxÞ; πðyÞ ¼ we find that the correspondingQ spin-parity observable iδdðx − yÞ in the continuum limit a → 0. To put the QFT † 1 P½J; a¼hΨðt0ÞjU x∈Λ σxU jΨðt0Þi can be expressed Eq. (4) on a lattice [19], we need to discretize the spatial J l J as derivatives of the Hamiltonian density, and substitute P integrals by Riemann sums, such that the Hamiltonian of d 1 iðt−t0Þ a δε P x∈Λl Z the lattice-field theory reads ½J; a¼2 e ½J; aþc:c:; ð27Þ   X 1 1 m2 d π x 2 ∇ϕ x 2 0 ϕ x 2 V ϕ where we introduce the lattice-generating functional: H ¼ a 2 ð Þ þ 2 ½ ð Þ þ 2 ð Þ þ ð Þ ; x∈Λl n R P o t 0 d 0 0 i dt a Jxðt ÞϕHðt ;xÞ Z Ω T t0 x∈Λl Ω ð22Þ ½J; a¼h j e j i: ð28Þ

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ðnÞ … is the single-particle Feynman propagator, and we introduce The corresponding Feynman propagators Gx1;…;xn ðt1; ; − π π ×d tnÞ can be obtained by functional differentiation, as we the Brillouin zone BZ ¼½ ð =aÞ; ð =aÞÞ . From this describe in the continuum version Eq. (13), where one must expression, the corresponding propagator in momentum consider again t − t0 > maxfjti − tjj; ∀ i; j ¼ 1; …;ng to space, approximate the remote-past to distant-future conditions. We recall that a set JðmÞ of pointlike sources Eq. (14) would Δ Pi be required, such that one can reconstruct the discretization 0ðp; aÞ¼ 0 2 2 2 a 2 ; ð30Þ ðp Þ − m0 − α½ sinð2 pαÞ of the functional derivatives by the set of measured parities. a This lattice version offers a very vivid image of our quantum-sensing apparatus as a piano (see Fig. 2). Let us has a well-defined continuum limit. Removing the lattice label the jΛlj lattice sites with an integer that maps each site cutoff, this propagator coincides with that of the free scalar Δ 2 − 2 to a particular key of a piano. The list JðmÞ, which describes Klein-Gordon QFT lima→0 0ðp; aÞ¼i=ðp m0Þ [2], 2 0 2 2 the sequence of pulses that couple the sensors to the field where p ¼ðp Þ − p . Note that the pole of the propagator 2 Eq. (14), can be interpreted as a piano score that indicates at p ¼ 0, which determines the physical mass m of the scalar the sequence of keys (sensors) that must be pressed particle, coincides in this case with the bare mass m0 of the (coupled to the field) at different instants of time to produce original field theory Eq. (1). a melody (spin-parity) that encodes the relevant informa- As we note below Eq. (22), the situation is more involved tion about the Feynman propagators. when VðϕÞ ≠ 0, since the bare parameters of the theory Following Ref. [19], the lattice-generating functional must depend on the cutoff to cure the UV divergences. The Eq. (28) in the noninteractingR R P limit, VðϕÞ¼0, becomes particular cutoff dependence of the bare parameters is Z − 1 0 0 2d Δ − determined by requiring that the physical observables at 0½J;a¼expf 2 dx dy x;y∈Λl a JðxÞ 0ðx y; aÞ JðyÞg, where the length scale of interest are not modified when the number of high-energy modes, describing fluctuations at Z much smaller length scales, is increased in the continuum 0 X − dp ieipðx yÞ limit Λ−1 → 0. Since a (or Λ−1) is a length (or inverse Δ0ðx − y; aÞ¼ P c c 2π 0 2 − 2 − 2 a 2 p∈BZ ðp Þ m0 α½a sinð2 pαÞ energy) scale, and hence not dimensionless, taking the continuum limit should always be understood in the sense ð29Þ that ξ=a → ∞. Here, ξ sets the relevant length scale of

FIG. 2. Schematic representation of the quantum sensing for Feynman propagators. The different indexes for the lattice sites x ∈ Λs, s ðmÞ as well as the corresponding Z2 sensors labeled by x , are mapped onto the keys of a piano. The set of pulse sequences J that couple the sensors to the field Eq. (14) corresponds to a piano score that indicates the sequence of keys (sensors) that must be pressed (coupled to the field) at different instants of time to produce a melody (spin-parity measurement) that encodes the relevant information about the Feynman propagators.

041012-8 QUANTUM SENSORS FOR THE GENERATING FUNCTIONAL … PHYS. REV. X 7, 041012 (2017) interest in such a way that physical quantities become are interested in linear and zigzag crystal configurations, independent of the underlying lattice structure. which are routinely obtained in linear Paul traps [53] and, We discuss this point in more detail below, but let us first more recently, also in a combination of a Paul trap and an introduce a particular AMO platform that can be used as optical lattice [54], which we refer to as a subwavelength an AQS of a self-interacting scalar QFT on the lattice. Paul trap. In addition, the recent experiments showing the Regarding the lattice counterpart of the sensing protocols crystallization of ion rings in segmented ring traps [55] for other continuum QFTs discussed in Sec. II E, a similar could also explore different crystal configurations. approach to the one presented in this section would hold In the harmonic-crystal approximation, one considers for N-component scalar fields and fermion fields with small vibrationsP around the equilibrium positions r r0 e bilinear sources. On the other hand, extending our sensing i ¼ i þ α qi;α α and obtains a model of coupled protocol to lattice gauge fields is an open question that harmonic oscillators that leads to the vibrational normal deserves further studies, especially in view of the recent modes of the Coulomb crystal [56]. This approximation, progress towards the quantum simulation of lattice gauge however, cannot account for the motional dynamics of the theories [21–28]. ions close to a structural transition between different crystalline structures. In particular, when ωy ≫ ωx; ωz,a B. Trapped-ion quantum simulators of the λϕ4 QFT structural change between a linear ion chain and a zigzag ω ω The possibility of trapping atomic ions by electromag- ladder occurs as one lowers x below a critical value c via netic fields has allowed us to test the predictions of a second-order phase transition [57]. This phase transition quantum mechanics at the single-atom level [45,46]. can be understood by an effective Landau model [58], After the seminal work by Cirac and Zoller [47],itwas which identifies the transverse zigzag distortion where understood that operating with several ions would allow for neighboring ions vibrate in antiphase as a soft mode. For ωx < ωc, the transverse phonons condense in a differ- quantum information processing [48], turning trapped ions Z into a very promising route towards quantum-error correc- ent ladder structure by spontaneously breaking a 2 tion [49]. Prior to the development of a large-scale fault- inversion symmetry. Not only is this theory in accordance tolerant quantum computer based on trapped ions, one may with previous static predictions [57], but it also serves as exploit the experimental setup for quantum simulations the starting point for studies of nonequilibrium dynamics of [50]. As we argue in the Introduction, with few notable the crystal across the phase transition [59]. exceptions of DQS [51], the experimental emphasis has An effective low-energy theory for the linear-to-zigzag been placed on the quantum simulation of condensed- transition can be derived as follows, both for ion rings [60] matter problems [4]. However, as we discuss in this section, and inhomogeneous linear crystals [61]. Let us rewrite the r0 r~0 trapped ions also have the potential of becoming useful equilibrium positions as i ¼ a i ,wherea is a relevant AQS of relativistic QFTs in a high-energy physics context. length scale in the problem. For the subwavelength Paul The motion of a system of N trapped atomic ions of mass traps or for ring traps, a is the uniform lattice spacing, whereas for linear Paul traps where the crystals are inho- ma and charge e can be described by the Hamiltonian mogeneous, a ¼ðe2=m ω2Þ1=3 is simply a length scale with   0 a z XN X the order of magnitude of the average lattice spacing. We 1 2 1 2 2 Hm ¼ p α þ maωαr α know from the previous discussion that the low-energy 2m i 2 i i¼1 α¼x;y;z a physics will be governed by excitations around the soft 2 X X π e0 1 zigzag mode, which corresponds to momentum ks ¼ =a in þ ; ð31Þ aringtrap(seeFig.3). In analogy with other problems 2 jri − rjj i j≠i in condensed matter, see, e.g., Ref. [62], one puts a cutoff around this momentum, considering only low-energy where we introduce the position r α and momentum p α i i excitations that should capture the long-distance physics. operators fulfilling ½riα;pjβ¼iδi;jδα;β, and the effective This amounts to rewriting the zigzag distortion as qj;x ¼ trapping frequencies fωαgα in the pseudopotential ¼x;y;z iksjaδ δ 2 2 e qj, where qj is a displacement that is slowly varying approximation [45]. Here, e ¼ e =4πε0 is expressed in 0 on the scale of the lattice spacing which contains only the terms of the vacuum permittivity ε0, and we set ℏ ¼ 1, modes near k . This can be generalized to situations without which is customary in AMO physics since energies are then s the periodicity of the ring by simply defining given by the frequency of the electromagnetic radiation used to excite a particular transition observed in spectro- −1 jδ scopic measurements. qj;x ¼ð Þ qj: ð32Þ As a result of the competition between the Coulomb δ ≈ δ r~0 − r~0 ∂ δ repulsion and the trap confinement, the ions can self- A gradient expansion qj qi þð j i Þ iþ1 qi,where assemble in Coulomb crystals of different geometries when ∂jδqi ¼ðδqi − δqjÞ fulfills j∂jδqij ≪ δqi due to its slowly the temperatures get sufficiently low [52]. In this article, we varying condition, yields the following Hamiltonian:

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FIG. 3. Effective λϕ4 QFT for ion crystals and quantum sensing scheme. Left: In the vicinity of the linear-to-zigzag structural phase transition of a trapped-ion crystal, the transverse zigzag vibrations yield the soft mode that contains the universal properties of the transition. Right: The slowly varying envelope of the zigzag distortion Eq. (32) allows us to develop a gradient expansion that leads to the effective λϕ4 QFT. By exploiting two electronic levels of the ions, and a state-dependent dipole force Eq. (49), one can infer the generating functional of the QFT.

  X ~ dimensionless, we start by defining the following lattice ≈ ma ∂ δ 2 ki ∂ δ 2 ki δ 2 ui δ 4 Hm 2 ð t qiÞ þ 2 ð iþ1 qiÞ þ 2 qi þ 4 qi : operators, i ð33Þ 1 ϕ~ ðxÞ¼ δq ; π~ðxÞ¼m ∂ δq ; ð37Þ a i a t i Here, we introduce a local spring constant and self- ~ interaction coupling for each transverse displacement, which show the desired commutation relations ½ϕðxÞ;π~ðyÞ ¼ δ   i x;y=a. The lattice Hamiltonian Eq. (33) can then be 1 3 expressed as H ¼ H0 þ V, where we introduce k ¼ m ω2 1− κζ ð3Þ ;u¼ m ω2κζ ð5Þ; ð34Þ m i a x 2 i i 4a2 a x i   X π~ 2 ~ 3 P ðxÞ kia ∇ϕ~ 2 ζ −1 i − −1 l n−1 r~0 − r~0 −n κ H0 ¼ a 2 þ 2 ½ ðxÞ ð38Þ where iðnÞ¼ l≠i½ð Þ ð Þ j i l j ,and ¼ ∈Λ maa 2 2 3 x l e0=maωxa is a dimensionless constant. Additionally, the spring constants between neighboring displacements are ~ ~ ~ and use the operator ∇ϕðxÞ¼½ϕðx þ auxÞ − ϕðxÞ=a.This X 1 part can be rewritten in terms of ð−1Þlþiþ κ k~ m ω2 : 35   i ¼ a x 2 r~0 − r~0 ð Þ X π~ 2 l≠i j i l j cx ðxÞ ~ 2 H0 ¼ a þ K ½∇ϕðxÞ ; ð39Þ 2 K L;x x∈Λl L;x The Hamiltonian in Eq. (33) already resembles the lattice-field theory of a D ¼ 1 þ 1 Klein-Gordon QFT where we introduce an effective sound velocity, Eq. (22), where the underlying ion crystal plays the role sffiffiffiffiffiffi of the d ¼ 1 lattice: ~ ki cx ¼ a ; ð40Þ Λ ∶ r~0 ∀ 1 … ma l ¼fx x=a ¼ i ; i ¼ ; ;Ng: ð36Þ which has the correct dimensions ½c ¼½k~ a21=2 × We thus specialize to D ¼ 1 þ 1 dimensions, in which, as x i −1=2 ML2T−2 M−1 1=2 LT−1 noted below Eq. (1), the engineering dimension of the ½ma ¼ð × Þ ¼ . Additionally, we also introduce the so-called stiffness or Luttinger parameter, scalar field is dϕ ¼ðD − 2Þ=2 ¼ 0. In order to define the correct scalar field operators, one has to pay special qffiffiffiffiffiffiffiffiffiffi 2 ~ attention to the different system of units in Eqs. (22) KL;x ¼ a kima; ð41Þ and (33). Essentially, we need to identify the speed of sound that will play the role of the effective speed of light which appears in the theory of bosonization and controls the in the relativistic QFT. Since the scalar field must be power-law decay of correlations in Luttinger liquids [1].

041012-10 QUANTUM SENSORS FOR THE GENERATING FUNCTIONAL … PHYS. REV. X 7, 041012 (2017) qffiffiffiffiffiffiffiffiffiffi homogeneous bulk of the crystal in a linear Paul trap. In Reintroducing Planck’s constant, K ¼ða2=ℏÞ k~ m , L;x i a these cases, we can set the corresponding natural units this Luttinger parameter turns out to be dimensionless, c ¼ 1, such that the low-energy Hamiltonian governing the ~ 2 2 2 1=2 2 −2 2 −2 1=2 KL;x ¼ð½kia =½ℏ =maa Þ ¼ðML T =ML T Þ ¼ 1. linear-to-zigzag instability in ion crystals becomes equiv- In order to arrive at the standard definition of a λϕ4 QFTon the alent to the D ¼ 1 þ 1 lattice Klein-Gordon field Eq. (22) lattice Eq. (22), we perform an additional rescaling of with quartic interactions: the lattice-field operators that preserves the commutation   X 1 1 2 λ relations: π 2 ∇ϕ 2 m0 ϕ 2 ϕ 4 Hm ¼ a 2 ðxÞ þ 2½ ðxÞ þ 2 ðxÞ þ 4! ðxÞ : pffiffiffiffiffiffiffiffiffi 1 x∈Λl ϕ x K ϕ~ x ; π x pffiffiffiffiffiffiffiffiffi π~ x : ð Þ¼ L;x ð Þ ð Þ¼ ð Þ ð42Þ ð46Þ KL;x

This leads to the desired lattice-field theory, Note that with all these definitions, we have made sure that the classical mass dimensions dϕ ¼ 0, while the couplings X cx 2 2 have d 2 ¼ 2 ¼ dλ. Finally, we also note that in numerical π ∇ϕ m0 H0 ¼ a 2 ð ðxÞ þ½ ðxÞ Þ; ð43Þ x∈Λl lattice simulations and formal renormalization group (RG) calculations, one typically defines dimensionless couplings 1 1 2 2 2 ~ 2 which yields theR desired QFT of a þ free massless scalar m~ 0 ¼ m0a and λ ¼ λa . In the so-called lattice units, the 2 2 boson H0 ¼ dxðcx=2ÞðπðxÞ þ½∂xϕðxÞ Þ in the con- lattice constant disappears from the above Hamiltonian, tinuum limit a → 0. Note that, as a consequence of the such that taking the continuum limit corresponds to inhomogeneous lattice spacing in a linear Paul trap, all these modifying the dimensionless couplings. As we remark at parameters have inhomogeneities around the edges of the the end of the previous section, taking the continuum limit ion chain, while they become constants for ring traps and does not require changing the actual inter-ion distance, subwavelength Paul traps, where the lattice spacing is but is instead achieved by setting these dimensionless homogeneous. couplings close to a critical point where ξ=a → ∞, and In addition to these terms, the remaining part of the physical quantities become independent of the underlying lattice Hamiltonian Eq. (33) yields a mass term and a self- lattice structure. interaction of the scalar field According to this discussion, trapped-ion crystals can be 4   used as AQSs of the lattice λϕ QFT Eq. (46), where the X m2 λ fields [Eq. (42)] are proportional to the zigzag displacement 0;x ϕ 2 x ϕ 4 V ¼ a 2 ðxÞ þ 4! ðxÞ : ð44Þ Eq. (32) via Eq. (37). The proportionality parameter as ∈Λ x l well as the bare mass and self-interaction strength of the QFT are expressed in terms of microscopic parameters Here, we introduce the bare mass and coupling strength, Eqs. (34) and (35) via Eqs. (40), (41), and (45). We note 6 3 that this approach differs from a noncanonical transforma- 2 kia λ uia m0;x ¼ ; x ¼ 2 ; ð45Þ tion introduced in Ref. [63], which yields a similar KL;x KL;x Hamiltonian Eq. (46) after a particular rescaling of Eq. (33). However, an effective Planck constant ℏ depend- 2 λ ML2T−2 which fulfill ½am0;x¼½a x¼ after taking into ing on the model parameters must be introduced to account the lattice spacing a from the lattice sum in maintain the required commutation relations. This leads Eq. (44), and thus display the expected units of energy. to important differences in the renormalization with respect ’ 2 0 λ 0 4 In Landau s mean-field theory m0;x < , x > signals a to the standard approach for the λϕ theory, which we phase transition where hϕðxÞi ≠ 0 is achieved by sponta- discuss below. neously breaking the Z2 symmetry ϕðxÞ → −ϕðxÞ in the As advanced in the Introduction, the usefulness of an effective lattice-field theory Eq. (46). This is exactly in AQS does not only depend on the accuracy with which it agreement with previous estimates of the linear-to-zigzag behaves according to the model of interest, e.g., a self- phase transition in ion crystals, both in homogeneous and interacting scalar QFT, but also on the measurement inhomogeneous cases. However, the mean-field approach strategies to extract the relevant properties of this simulated predicts a wrong scaling behavior in the vicinity of the model. In our trapped-ion scenario, the position of the ions critical point, which could be tested experimentally with the hrii is routinely measured by driving a transition between protocol we present in this work. two electronic levels and collecting the spontaneously In the context of relativistic QFTs Eq. (1), we should emitted photons in a camera [45], such that the expectation recover Lorentz invariance in the continuum limit. This value hϕðxÞi could be inferred from the above relations. can be achieved for the whole ion crystal in ring traps This can be used to locate the critical point of the Z2 phase or subwavelength Paul traps, or by restricting to the transition when a vacuum expectation value hϕðxÞi ≠ 0 is

041012-11 A. BERMUDEZ, G. AARTS, and M. MÜLLER PHYS. REV. X 7, 041012 (2017)

X Ω developed, which would require an accurate measurement L iðΔk ·r −Δω tÞ H − ¼ P e L i L þ H:c:; ð48Þ of the zigzag ion positions [53]. However, in the symmetry- l i 2 i i broken phase, the zigzag crystal will also experience micromotion (i.e., additional fast oscillations synchronous where we introduce a two-photon Rabi frequency ΩL with the driving fields of the Paul trap) that go beyond the and the wave-vector (frequency) difference of the laser pseudopotential approximation, such that other spectro- beams, ΔkL ¼ ΔkL;1 − ΔkL;2 (ΔωL ¼ ΔωL;1 − ΔωL;2). scopic observables can be modified with respect to the After expressing the ionP position operator in terms of static situation [53]. In the present context, the pseudopo- 0 the vibrations r ¼ r þ αq αeα, one sees directly that if tential approximation is used to derive Eq. (46), and it i i i; the overlapping beams propagate along ΔkLjjex, the would thus be safer for the accuracy of the AQS to perform radiation will couple to the desired zigzag distortion experiments in the symmetry-unbroken phase, where these Eq. (32). Moreover, a Taylor expansion in the Lamb- standard fluorescence measurements cannot be used to Dicke regime, jhΔkL · exqi;xij ≪ 1, shows that the leading- determine the properties of the QFT. For instance, if one is order contribution from the laser-ion interaction Eq. (48), interested in simulating a massive scalar particle with the for jΩLj ≪ ΔωL ∼ ωx, will be a state-dependent dipole AQS, one would like to know how the bare mass gets force that excites the zigzag distortion when the internal renormalized as a consequence of the self-interactions, or state of the ions is in j0 i, namely, how a collection of massive scalar particles would scatter i X off each other due to these interactions. In the following H − g t P δq ; section, we show that the protocol to measure the generat- l i ¼ ið Þ i i i ing functional Z½J; a [Eq. (28)], which contains the Ω Δk e −1 i Δω information about all of these properties, has an exper- giðtÞ¼ Lð L · xaÞð Þ sin Lt: ð49Þ imental realization that is feasible with state-of-the art control over trapped-ion crystals. We also assume that the laser beams can be split into individual addressing beams that couple to any ion in the crystal, such that ΩL → ΩL;iðtÞ can be controlled individu- C. Trapped-ion sensors for the generating functional ally, e.g., switched on or off, by controlling the intensity of The Hamiltonian Eq. (31) describes the motional d.o.f. each of the addressing beams as achieved experimentally in of a collection of trapped ions. Additionally, the ions have Ref. [66]. Using this expression in combination with an internal atomic structure with its own independent Eqs. (46) and (47), we arrive at the desired lattice-field dynamics. We can exploit such internal d.o.f. as the theory with Z2 Schwinger sources [Eq. (23)], namely, quantum sensors introduced in Sec. II (see Fig. 3). X X We consider external laser beams that only couple to a H ¼ H þ H − aJαðtÞϕðxÞσα; ð50Þ 0 1 J m in x x pair of such internal states fj ii; j iig, which have a α x∈Λl transition of frequency ω0. The HamiltonianP governing N ω σ0 − this internal dynamics is simply Hin ¼ i¼1 0ð i PiÞ, where we introduce the source fields σ0 I 0 0 1 1 0 0 where i ¼ ¼j iih ijþj iih ij, and Pi ¼j iih ij is the projector onto the lowest-energy internal state. This α JxðtÞ δ − δ pgiffiffiffiffiffiffiffiffiffiðtÞ JxðtÞ¼ 2 ð α;0 α;3Þ;JxðtÞ¼ : ð51Þ Hamiltonian is directly equivalent to the quantum-sensor KL;x Hamiltonian Eq. (24) using the crystal as the underlying lattice [Eq. (36)], such that Note that, by using the dimensional analysis of the previous X ω section, the source fields also have the desired mass 0 0 2 H ¼ aδεðσ − P Þ; δε ¼ : ð47Þ dimensions dJ ¼ . in x x a x∈Λl Provided that one can prepare the initial state of the ions according to Eq. (8), and measure the parity observable in In order for these electronic levels to act as quantum Eq. (10), it becomes possible to implement the protocol sensors of the QFT generating functional, we need to presented in Sec. II, inferring the generating functional induce a coupling of the form Eq. (23), such that these Z½J; a of the particular trapped-ion λϕ4 QFT. For the probes act as the Z2 Schwinger sources introduced in T ¼ 0 case, the state preparation would rely on an adiabatic Sec. II. We consider the so-called state-dependent dipole evolution that starts far away from the structural phase forces [64], which can be obtained from a pair of laser transition and utilizes laser cooling to prepare a state very beams of frequency ωL;1, ωL;2 that couple the internal state close to the vacuum of the transverse vibrations. Then, j0ii off-resonantly to an auxiliary excited state from the the trap parameters would be adiabatically modified by atomic level structure. Using selection rules [65], and approaching the critical point of the linear-to-zigzag working in the far off-resonant regime, the laser-ion transition, but remaining in the symmetry-preserved phase. coupling can be expressed as a crossed-beam ac-Stark shift, For the T ≠ 0 case, one would perform laser cooling

041012-12 QUANTUM SENSORS FOR THE GENERATING FUNCTIONAL … PHYS. REV. X 7, 041012 (2017) directly in the interacting regime, during a time that is large requires allowing the bare couplings of the theory fgig, 2 enough so that the motional d.o.f. thermalize. Then, the e.g., fm0; λg in Eq. (46), to flow with the lattice cutoff internal state has to be prepared in a GHZ state, which can fgiðΛcÞg in such a way that one stays on the “line of be accomplished using gates mediated by the phonons that constant physics.” In the AQS, this would mean that the are not involved in the structural phase transition [67].We value of the microscopic parameters, which control the remark that the high fidelities already achieved in the effective lattice parameters such as the bare mass, have to experimental preparation of large GHZ states [41] make be tuned to particular values in order to obtain the trapped ions a very promising AMO setup for the imple- renormalized physical mass of the particles at the scale mentation of this proposed protocol. of interest, which will be independent of the cutoff and Before closing this section, let us note that the simplified different from the bare mass. The renormalization group is protocols of Sec. II to measure any Feynman propagator essential to understand this flow and, with it, the nature of could also be implemented in this trapped-ion scenario, such a continuum limit [69]. provided that one has the aforementioned addressability in At the UV limit fgið∞Þg, the resulting QFT must belong the laser-ion couplings [66]. In such a case, the state in to the so-called critical surface; i.e., the couplings must lie Eq. (15) or (18) could be prepared along similar lines, and at the domain of attraction of a fixed point of a trans- the required switching of instantaneous sources to estimate formation that changes the cutoff scale. To preserve the the functional derivatives Eq. (13) would also be available. physics at the length scale of interest, one has to fix a one- The measurement corresponds to a multispin correlation parameter set of field theories with different cutoffs function of the type that is routinely measured through the fgiðΛcÞg that connects to such a well-defined UV limit. state-dependent fluorescence of the ions [45]. Prior to This is achieved by specifying the relevant couplings driving the cycling transition and collecting the emitted r Λ ∈ Λ fgi ð cÞg fgið cÞg that take us away from the critical photons, one should apply a global single-qubit rotation by surface as one moves from the UV towards the infrared driving the so-called carrier transition [45]. We note that the Λc → 0, approaching thus the length scale of interest. The measurements have to be repeated for different values of difficulty lies in identifying the possible RG fixed points the field-sensor couplings to infer the propagators via the and relevant couplings of a particular field theory. In this discretized derivatives Eq. (17). During these additional regard,P the scalar QFT Eq. (1) with self-interactions repetitions, one must avoid slow drifts in the microscopic V ϕ ϕ2n 2 ! 4 ð Þ¼ ng2n =ð nÞ and D ¼ yields a very instruc- trapped-ion parameters. An advantage in this regard is that tive scenario where the RG machinery can be developed in our proposal focuses on the propagator of the vibrations, perturbation theory [69,70]. Typically, one starts from the which will be 1–2 orders of magnitude faster than experi- so-called Gaussian fixed point, where g2 ð∞Þ¼0, and ments on the propagation of spin excitations in effective n shows that it suffices to consider the flow of g2ðΛ Þ and spin-spin models with trapped ions [68], where analogous c g4ðΛcÞ to understand the continuum limit. This follows measurements are typically done. from simple dimensional analysis, since the so-called anomalous dimensions of the fields vanish at this fixed D. Renormalization and the continuum limit point, allowing one to realize that the higher-order cou- Λ As advanced in the sections above, using the lattice- plings fg2nð cÞgn>2 are all irrelevant, i.e., decrease as one generating functional Z½J; a [Eq. (28)] to learn about the moves towards the IR. At one loop in perturbation theory, continuum QFT Eq. (1) requires letting a → 0 and remov- g2ðΛcÞ remains a relevant coupling, while g4ðΛcÞ becomes −1 ing the lattice cutoff Λc ∝ a → ∞. This continuum limit irrelevant. Therefore, unless a different RG fixed point must be performed without affecting the physical observ- exists, the lattice regularization of the scalar QFT in D ¼ 4 ables at the length scale of interest. Note also that the only has a trivial noninteracting continuum limit. Using the Schwinger sources should be spaced at the same physical so-called ε expansion, which allows for noninteger dimen- distance as the “continuum limit" is taken. For instance, in sions D ¼ 4 − ε, it is possible to find a nontrivial fixed the context of the trapped-ion quantum simulator Eq. (46), point that would allow for an interacting and massive QFT such an observable will be the parity operator Eq. (12), in the continuum, the so-called Wilson-Fisher fixed point at which encodes the information about the Feynman propa- finite g2ð∞Þ ≠ 0, g4ð∞Þ ≠ 0. However, this fixed point gators Eq. (5) and thus the physical mass m of the scalar exists only for ε > 0 and thus D<4, suggesting the particles. In this case, the relevant length scale for the scalar triviality of the lattice scalar QFT in D ¼ 4 [70,71]. fields Eq. (42) is set by the envelope of the zigzag distortion To go beyond the perturbative RG, numerical lattice Eq. (32), which varies on a much larger scale than the simulations based on Monte Carlo methods become very lattice spacing. In the generic situation, we can safely send useful [19]. The general strategy of lattice-field theory a → 0 without altering the long-wavelength phenomena, simulations is to set the bare lattice couplings in the vicinity but we must ensure that our calculations will not suffer of a quantum critical point, where the correlation length from possible UV divergences as further high-energy ξ → ∞ diverges, and one expects to recover the universal modes are included by this process. In practice, this features of the QFT in the continuum limit a → 0. In our

041012-13 A. BERMUDEZ, G. AARTS, and M. MÜLLER PHYS. REV. X 7, 041012 (2017) context, g2ðΛcÞ and g4ðΛcÞ must be set in the vicinity of the of two-level quantum sensors. Generalizing the notion of Z2 quantum phase transition, which should be controlled Schwinger fields to serve simultaneously as sources and by the scale-invariant fixed point of the RG transformation. probes of the excitations of a quantum field, we exploit the The renormalized mass can be extracted from the numerical entanglement of the quantum sensors to show that a computation of propagators, whereas the renormalized collective Ramsey-type response of the sensors contains interactions can be obtained from susceptibilities. This all the information about the QFT generating functional. approach corroborates the triviality of the lattice scalar QFT This, in turn, encodes in a compressed manner the relevant in D ¼ 4 in nonperturbative regimes [72]. information of the interacting QFT (i.e., approximating In the D ¼ 2 limit, which is the case of interest for the functional derivatives by combining several measured trapped-ion quantum simulator Eq. (46), the need of responses can be used to decompress any n-point nonperturbative schemes is even more compelling. In this Feynman propagator, and thus any possible scattering or case, applying the above perturbative RG calculation nonequilibrium real-time process). We argue that this around the Gaussian fixed point would show that all of protocol finds a very natural realization in AQS on the Λ 4 the couplings fg2nð cÞgn≥1 are relevant [69], thus ques- lattice, and we consider a trapped-ion realization of the λϕ tioning the validity of the truncation implicit in Eq. (33) that QFT as a realistic example where experimental techniques is used to derive the effective QFT Eq. (46) from the can be applied to implement the generalized Schwinger 1 1 microscopic Hamiltonian Eq. (31). In fact, in þ sources, and infer the generating functional from reso- dimensions, the field operators for a free scalar QFT have nance-fluorescence images. In this case, by performing nonvanishing anomalous dimensions even in the absence of experiments in the vicinity of the linear-to-zigzag structural interactions, such that the simple dimensional analysis phase transition, the trapped-ion AQS can, in principle, around the Gaussian fixed point is no longer valid. In this address nonperturbative questions regarding the nature of case, the tools of conformal field theory would be required the fixed point that controls the QFT obtained in the to understand the RG flow of perturbations around the continuum limit. Recently, Jordan et al. [76] have shown scale-invariant fixed point of a free scalar boson, as occurs that the algorithm for DQS of scattering in scalar QFTs [15] for the sine-Gordon model [1]. However, the particular can be modified to obtain also the generating functional. perturbations of our self-interacting scalar QFT do not have Moreover, they argue that a particular instance of the simple conformal or scaling dimensions, and thus do not generating functional, for a certain functional dependence allow for a simple analytical approach. Accordingly, the of the source fields, cannot be efficiently estimated with existence of nonperturbative numerical methods becomes even more relevant in this situation. Recent results for this any classical algorithm. It would be very interesting to 4 study if similar complexity arguments can be carried over 1 þ 1 scalar λϕ QFT, based on either Monte Carlo [73] onto our AQS, which we believe presents an opportunity to and real-space renormalization group [74] methods on the lattice or Hamiltonian truncation methods in a finite volume measure the generating functional using current trapped- [75], have shown that the continuum limit of this QFT is ion technology. controlled by a nontrivial fixed point corresponding to the In general, understanding real-time dynamics of quantum Ising conformal field theory. These works show the power fields, either in or out of equilibrium, is required in a wide of the lattice approach to solve nonperturbative questions of range of physical applications. One important question in relativistic theories is far-from-equilibrium dynamics and the continuum QFT, such as the precise location of the Z2 2 thermalization [77]. This is relevant, e.g., for the end of quantum phase transition, i.e., the critical value of λ=m inflation and preheating in the early universe [78],orforthe where the scalar field acquires a vacuum expectation value. evolution during the early stages of heavy-ion collisions, At a fundamental level, they also imply that perturbations resulting in the quark-gluon plasma created at the Large fg2 ðΛ Þg 2 around this fixed point, which are generated n c n> Hadron Collider at CERN, or the Relativistic Heavy Ion in the implicit RG process of looking into long-wavelength Collider at BNL [79]. In the former, the efficiency of particle phenomena, are irrelevant. This justifies thus the validity of production and transport of energy across different length our truncation leading to Eq. (46). The hope of this article scales determines the reheating temperature, whereas in is to show that, exploiting the proposed protocol to infer the latter case the very creation of a thermal quark-gluon the full generating functional Z½J; a of a QFT, trapped-ion plasma depends on the ability of the highly nonequilibrium AQS working in the vicinity of the linear-to-zigzag initial gluon fields to thermalize [80]. Since this dynamics structural transition will serve as an alternative nonpertur- takes place manifestly in real time, it is often treated within bative tool to explore such QFT questions. classical approximations that are valid only for highly occupied modes [81]. It would hence be of the utmost IV. CONCLUSIONS AND OUTLOOK interest to study similar, yet simplified, dynamical questions In this work, we present a protocol to infer the normal- relevant for these situations using the protocol outlined in our ized generating functional of a QFT by measuring a work and analyze, e.g., the role of nonthermal fixed points particular interferometric observable through a collection [82] using quantum dynamics in real time.

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In thermal equilibrium, the information about spectral [12] Provided that one takes care of the so-called fermion functions and other real-time correlators is also of interest. doublers [13] that appear in the naive discretization of a Even though they are related [83] to standard Euclidean fermionic Dirac field. correlation functions computable in lattice-field theory, the [13] H. B. Nielsen and M. Ninomiya, A No-Go Theorem analytical continuation from Euclidean to real time (or from for Regularizing Chiral Fermions, Nucl. Phys. B185,20 (1990). Matsubara to real frequency) is a nontrivial process. The [14] A. Bermudez, L. Mazza, M. Rizzi, N. Goldman, M. interest here lies, e.g., in quasiparticle properties and other Lewenstein, and M. A. Martin-Delgado, Wilson Fermions spectral features, such as thermal masses and widths, or in and Axion Electrodynamics in Optical Lattices, Phys. Rev. more ambitious questions related to hydrodynamic struc- Lett. 105, 190404 (2010); L. Mazza, A. 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