Stabilizing arrays of photonic cat states via spontaneous symmetry breaking
Stabilizing arrays of photonic cat states via spontaneous symmetry breaking
Lebreuilly, José;Aron, Camille;Mora, Christophe
2018-09-27 00:00:00
1 2, 3 1 Jose ´ Lebreuilly, Camille Aron, and Christophe Mora Laboratoire Pierre Aigrain, Ecole Normale Superieur ´ e - PSL Research University, CNRS, Universite´ Pierre et Marie Curie-Sorbonne Universite, ´ Universite´ Paris Diderot-Sorbonne Paris Cite, ´ Paris 75005, France Laboratoire de Physique Theorique ´ , Ecole Normale Superieur ´ e, CNRS, PSL University, Sorbonne Universite, ´ Paris 75005, France Instituut voor Theoretische Fysica, KU Leuven, Belgium The controlled generation and the protection of entanglement is key to quantum simulation and quantum computation. At the single-mode level, protocols based on photonic cat states hold strong promise as they present unprecedentedly long-lived coherence and may be combined with powerful error correction schemes. Here, we demonstrate that robust ensembles of “many-body photonic cat states” can be generated in a Bose- Hubbard model with pair hopping via a spontaneous U (1) symmetry breaking mechanism. We identify a parameter region where the ground state is a massively degenerate manifold consisting of local cat states which are factorized throughout the lattice and whose conserved individual parities can be used to make a register of qubits. This phenomenology occurs for arbitrary system sizes or geometries, as soon as long-range order is established, and it extends to driven-dissipative conditions. In the thermodynamic limit, it is related to a Mott insulator to pair-superfluid phase transition. Introduction. The ability to engineer a large variety of ated by bringing extended bosonic systems to develop a pair- Hamiltonian couplings with sufficient tunability and to pro- superfluid (PSF) order [31, 32], and we propose both equilib- tect quantum coherence is essential for the generation of ex- rium and driven-dissipative routes for the emergence of this otic quantum states [1–3] and for quantum information pro- many-body phase. The intimate connection between PCS and cessing [4, 5]. In various platforms such as ultracold atomic a PSF phase (the bosonic counterpart of the Barden-Cooper- gases or superconducting circuits, the huge timescale separa- Schrieffer, BCS, phase), starts with them sharing identical tion between Hamiltonian and dissipative dynamics is suitable symmetries: they both break U (1) invariance while preserv- for adiabatic preparation schemes where the physics is domi- ing a Z subsymmetry. Until now, generating individual PCS nated by ground-state properties. Adopting a new perspective, has mostly been achieved by explicitly breaking the U (1) quantum reservoir engineering ideas [6–10] that harness dis- symmetry associated with particle number conservation, e.g., sipation as a resource rather than a flaw have opened the gates by shining single- or two-photon coherent sources on photonic to mostly uncharted territory: the nonequilibrium generation cavities [22, 26, 27, 33, 34]. Here, we shall rather capitalize of quantum states. on a purely many-body mechanism, namely a spontaneous In the context of cavity-QED, quantum reservoir engineer- symmetry breaking of U (1) invariance, to achieve the long- ing schemes have been proposed and successfully imple- range PSF order and generate arrays of PCS. mented for the preparation of single qubit states on the Bloch Equilibrium model. We flesh out our proposal in the con- sphere [11], entangled states of distant qubits [12–15], and text of a modified Bose-Hubbard model very recently of the first Mott insulator of light [16–20]. An- h i X X U J y2 y2 2 2 other resounding success for quantum error correction is the H = a a a a + H:c: ; (2) i i i j 2 z preparation of photonic cat states (PCS) [21–23], i hi;ji where the usual single-particle hopping between a site i and its C () / (jij i) ; (1) z nearest neighbors is replaced by two-particle hopping pro- which are macroscopic multi-photon superpositions naturally cesses of amplitude J , and U > 0 accounts for standard on- insensitive to dephasing in the limit of a large , and which site repulsive interactions. Such pair interactions can be read- can be efficiently protected against photon losses via par- ily realized within ultracold atoms [31] or circuit-QED [22] ity measurement [24] and feedback control [25–27]. These platforms. We propose and detail a realistic implementation schemes heavily rely upon the current development of generic for the latter in the Supplemental Material [36]. The effects of ny m nonlinearities of the type a b with n; b 2 N , such as 3- lattice geometry, such as the distinction between a superfluid wave mixing i.e. n + m = 3 [28–30]. order and Bose-Einstein condensation in low dimensions, is A natural extension of this joint endeavour is the prepa- washed away in both our subsequent mean-field description ration of cat states in the many-body context. However, the and exact results. However, for simplicity purposes we have conditions for the emergence of cat states in large multi- in mind a cubic lattice with N and periodic boundary con- sites mode architectures still remains elusive, as hybridization be- ditions. In addition to the global U (1) symmetry, a 7! e a , i i tween neighboring PCS is expected to be detrimental to the corresponding to the conservation of the total number of parti- local nature of those states. In this Letter, we claim that cles N , the Hamiltonian is also symmetric under local discrete loc large quantum registers of PCS can be spontaneously gener- Z transformations, namely a 7! (i)a with (i) = i i arXiv:1809.10634v2 [quant-ph] 6 Mar 2019 2 FIG. 1: (a): Zero-temperature phase diagram in the even-parity sector as a function of the pair-hopping J and the chemical potential in +() 2 units of the interaction U . The fidelity F = max (jh jC ij ) between the Gutzwiller wave function and the closest cat state is cat GW represented in color plot. (b): Wigner quasiprobability distribution [35] of the Gutzwiller wave function j i for three points in the phase GW diagram, respectively located in the Mott Phase (P ), and in the pair-superfluid phase in the weak (P ) and strong (P ) PSF regimes. (c): 1 2 3 pair-superfluid order parameter as a function of J at fixed chemical potential =U = 1:2. The green solid line and the orange circles represent the predictions of the Gutzwiller analysis and the semiclassical result of Eq. (5), respectively. In the inset, the formation of a Mexican hat potential sketches the spontaneous symmetry-breaking mechanism generating the many-body photonic cat states. where (i) can vary from site to site. This latter symmetry ous change via spontaneous symmetry breaking from a Fock corresponds to the conservation of the parity of the particle state in the Mott-insulating phase to a cat state deep into the number at each site. PSF phase. Finally, we emphasize that the protection of the Ground-state phase diagram. The zero-temperature global Z symmetry can survive even in absence of the local loc phase diagram was obtained numerically within a Gutzwiller Z symmetries, as a consequence of a non-zero energy gap mean-field approach (see Suppl. Mat. [36]). We monitored / exp ( C=(U 2J )) separating the even and odd parity both the single-particle and two-particle order parameters, states for J < U=2 (see Suppl. Mat. [36] where we illustrate (1) (2) 2 in particular the robustness of PSF order and many-body PCS ha i and ha i. Another important figure against single-particle hopping). of merit is the fidelity F max jh jC ()ij cat ; GW Many-body cat states. Remarkably, at the special value between the local ground-state wavefunction j i of the GW J = U=2, the gap cancels exactly and one can analytically Gutzwiller ansatz and the closest cat state. The outcome is compute the ground state of the many-body Hamiltonian H displayed in Fig. 1a as a function of the chemical potential N . At = 0, the ground-state manifold is located at zero and the pair-hopping J . For simplicity, we restricted the energy and spanned by the following set of many-body wave- results to the sector with only even local parities, see Suppl. functions, Mat. [36] for a complete picture including the odd parity sector. P P (i) () = C () ; (3) At weak hopping amplitude J , the ground state is analo- gous to the one of the standard single-particle hopping Bose- Hubbard model. It features a series of Mott-insulating regions defined as an extended product state of local PCS. The proof with even integer densities n ha a i = 0; 2; 4; : : : charac- of this exact result is detailed in [36]. The hopping-induced (1) (2) terized by = = 0, reflecting the underlying global locking of the cat states at a common coherent field is the U (1) symmetry. The lobe boundaries at stronger J corre- consequence of a protection against relative dephasing be- spond to a second-order phase transition to a superfluid phase tween the various lattice sites. Importantly, the on-site par- where the U (1) symmetry is spontaneously broken while the ity P (i) = can vary from site to site, yielding an exten- Z symmetry is preserved. Here, given that superfluidity is sively large degeneracy of the ground-state manifold. This only carried by pairs of photons, this translates into a van- latter property is particularly compelling for quantum memory (1) ishing single-particle order parameter = 0 and a non- applications, as the parities at each sites act as an emergent (2) vanishing two-particle order parameter 6= 0. stable register of qubits. Large-scale and versatile entangle- As J is increased towards the special value J U=2, ment between these qubits can then be prepared via arbitrary (2) the PSF order parameter diverges to +1, and the lo- superpositions of these states, which are themselves preserved cal fidelity to a cat state approaches one. The corresponding by the many-body dynamics generated by H . Wigner functions, displayed in Fig. 1, illustrate the continu- At finite chemical potential and J = U=2, the states of Pair Superfluidity a 0, a =0 (a) (c) (2) Eq. (3) are no longer eigenstates but rather follow the sim- P it ple dynamical evolution j (t)i = (e ) . Moreover, definingP as the projector onto the submanifold with a total particle N , the exact ground states within this subspace are Normal/MI -1 Phase P P -2 2 = P () (4) a = a =0 N N (b) (d) 1 and are located at an energy N . The degeneracy associ- cat ated to the local parities is preserved. The distinction between P P and () is nonetheless meaningless in the thermody- 0 0.5 0.5 namic limit (N ! +1). There, () thus accurately sites -1 describes the ground-state physical properties even for 6= 0. -2 0 0 Importantly for realistic implementations, these exact results 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 are valid regardless of the system size and spatial dimension- J/U J/U ality, and PCS are expected already with N = 2 sites (see sites also Ref. [37] where a similar phenomenology was observed FIG. 2: (a)-(b): Phase diagram in the driven-dissipative scenario, for a two-mode system). A detection scheme of the N -particle truncated to the even-parity sector, as a function of the pair-hopping many-body cat states is detailed in the Supplemental J and the detuning = ! =2 ! in units of the interaction at c Material [36]. U . The two-photon loss rate is set to = 10 . The pair- l p Thermodynamic instability and semiclassical analysis. (2) 2 superfluid order parameter = ha i, and the fidelity F = cat We emphasize that the exact solutions are located on the verge +() +() max hC j jC i between the Gutzwiller density matrix GW of an instability. This can be seen by using the coherent state and the closest cat state are represented in color plot in pan- GW (2) ji as a variational ansatz, yielding an energy landscape i els (a) and (b), respectively. (c)-(d): and F are respectively cat 2 4 jj + (U=2 J )jj which is unbounded from below represented for various dissipative rates ; as a function of J em l for J > U=2. On the contrary, for J < U=2 the model is and at fixed detuning = 2U . The results of the Gutzwiller analysis (2) thermodynamically stable as confirmed by our Gutzwiller nu- are displayed in solid lines, and compared for to the semiclas- sical results with (resp. without) the effect of saturation, displayed merical calculations and exact results (see Suppl. Mat. [36] in crosses (resp. black dashed line). All panels feature an interac- for the proof). In this case, a first-order calculation of the tion strength U= = 0:7, and the Rabi coupling is chosen to p R ground-state energy in J U=2 provides a precise estimate maintain a fixed ratio = = 9 between pumping and losses. The em of the density and PSF order parameter close to the instability various curves of panels (c) and (d) correspond to = = 3 10 l p 2 3 3 threshold (blue), 1 10 (orange), 3 10 (yellow), and 1 10 (purple). ha a i ' jha ij ' ; (5) i i J!U=2 U 2J method could be of interest for the implementation and the regardless of the choice of on-site parities P (i). The excellent initialization of quantum registers. agreement of the semiclassical description with Gutzwiller In the remainder of this work, we rather investigate the con- simulations (see Fig. 1c) [45], together with the strict local- nection between many-body PCS and spontaneously-broken parity conservation, further explain why the emergence of PSF phases within a driven-dissipative scenario which pre- many-body cat states occurs throughout a wide region of the serves the initial U (1) symmetry of the model as well as phase diagram, U=3 . J U=2 . the conservation of local parities: to the unitary physics of Beyond semiclassics, we compute the Bogoliubov spec- the two-particle hopping Bose-Hubbard model, we add two- trum of the elementary excitations preserving the local pari- photon decay channels and incoherently pumped two-level ties (the excitations corresponding to a change of local parity systems exchanging pairs of photons. are characterized by the gap ). We find the dispersion law The dynamics are described by the following master equa- E = ( + 2), typical of superfluidity, with a finite k k k tion: sound speed c = 2a 2J at low-momenta and verifying the Landau criterion. Here 4J=z [cos(k a) 1], =1 @ = i [H + H + H ; ] and a is the lattice constant. The validity of the Landau crite- t ph at ph at rion ensure that the physics is dominated by the ground-state 2 + + D[a ]() + D[ ]() ; (6) l p i i manifold of many-body PCS even in presence of small pertur- bations to H . Driven-dissipative model. Many-body systems in a spon- with the photonic Hamiltonian H = H + ! a a . ph 0 c i i i taneously broken phase are naturally sensitive to external per- H is two-photon hopping Bose-Hubbard Hamiltonian pre- turbations breaking the symmetry: in the Supplementary Ma- viously introduced in Eq. (2) and ! is the cavity frequency. terial [36], we show how to prepare a large ensemble of cat H = ! is the Hamitonian of the two-level sys- at at i i i states by simply shining a two-photon coherent drive at a sin- tems which can coherently emit or absorb pairs of photons at a y2 gle dissipative site in the lattice. Such a hardware-efficient Rabi frequency according to H = [ a + R ph at R i i i /U /U (2) cat 0 4 a ]. Finally the Lindblad superoperators in the second with zero temperature, which we show below to proceed from i i (2) line of Eq. (6) account for two-photon losses and an inco- the saturation of emitters. First, the two-photon field does herent pumping of the two-level emitters occuring at rates not diverge and presents an upper bound. Second, the domain (2) and respectively. We used the notation D[X ]() l p of optimal fidelity to PCS, and maximum , is tilted with max y y XX 1=2 X X; . respect to J=U = 1=2. The main function of the pumped two-level systems is to Semiclassical analysis. In order to gain further insight on implement a frequency-dependent incoherent pump injecting the complex driven-dissipative dynamics of our model, we de- photons by pairs [17, 38] at a frequency-dependent rate rived the self-consistent mean-field equation on the order pa- (2) (2) rameter (t) in the semiclassical regimej (t)j 1 (see ( =2) details in Suppl. Mat. [36]). S (!) = ; (7) em em 2 2 (! ! ) + ( =2) at p In agreement with the Gutzwiller results, the PSF order parameter develops a steady-state oscillatory behavior in the 0 2 whose maximum is set by = 4 = . Our (2) (2) i! t p PSF em R U (1)-broken phase, (t) = e . Non-trivial so- scheme is most efficient in the non-saturating regime lutions are found only above the lasing threshold , em 0 2 ( max(n ; 1) , with n the density) where once a em p with their amplitude obeying two-level system has emitted, it is quickly and efficiently pumped back to its excited state, thus maintaining a nearly (2) PSF = : (8) perfect population inversion, and in the weakly dissipative 0 U 2J regime ( , U; J ) where the photonic dynamics are em dominated on short timescales by the Hamiltonian part H . ph The effective detuning ! =2 ! and the order PSF PSF c Single-photon losses are detrimental to our scheme: we as- parameter frequency ! are set by the balance between the PSF sume that these processes occur at a rate
; , such l energy injected in the photonic system and the energy lost by em that there is enough time for the relaxation within each par- dissipation ity sector to take place before any single-photon loss event S (! ) occurs, and we study the physics within this transient regime. em PSF = ; (9) Steady-state phase diagram. After a relaxation period, the 1 + s driven-dissipative dynamics are expected to reach a nonequi- (2) librium steady state. Neglecting single-particle losses (see where the saturation parameter s = 2j j S (! )= em PSF p discussion above), the steady states are non-unique and limits the pump amplification power. The Lorentzian form present a large multiplicity: the state reached after a long of S then yields two distinct frequencies ! , however em PSF evolution depends on the local parities initially imprinted on at most one solution at a time was found to be non-trivial, the system. Here we restrict ourselves to the even-parity sec- physical and dynamically stable. tor, and explore the resulting phase diagram by means of a Noteworthy, observe the strong similarity between Eq. (8) nonequilibrium Gutzwiller mean-field approach (see Suppl. and its equilibrium counterpart in Eq. (5) when identifying the Mat. [36]). The results are presented in Fig. 2 as a function of effective detuning with the chemical potential . The di- PSF (2) the two-photon hopping J and the detuning ! =2 ! at c vergence in can be interpreted as a breakdown of the pho- which, as we will see, plays a role analogous to the equilib- ton blockade at J = U=2: the energy separation 2! be- cav P P rium chemical potential. tween two successive groundstates and of H cav N N +2 Similarly to the zero-temperature equilibrium case, we find does not depend anymore on the total particle number N , as (1) (2) a normal phase with (t) hai(t) = 0 and (t) the two-photon hopping counterbalances perfectly the photon (2) ha i(t) = 0 at weak hopping J . For this particular computa- repulsion. However, as shown in Eq. (9), when increases tion (U = 0:7 ), the resulting phase is not insulating. The the saturation becomes relevant, setting an upper bound to the stabilization of a photonic Mott insulator in the strong pho- (2) (2) order parameter = ( = = ) =2. is max p l p max em ton blockade regime (U ) is discussed in Refs. [17, 38]. (2) achieved at J = U=2 =(2 ) which depends linearly c max At stronger hopping amplitudes J , we find the onset of PSF: on . This explains the tilting of the PSF domain observed in the U (1) symmetry is spontaneously broken, yielding a non- (2) the Gutzwiller computations, as well as J = U=2 at = 0. vanishing two-photon order parameter 6= 0, while the (1) The results of the semiclassical analysis are presented in single-photon order parameter remains zero, = 0, as a Fig. 2c (crosses) for various degrees of saturation, and ac- consequence of the unbroken global Z symmetry. curately reproduce the Gutzwiller predictions (solid lines) Similarly to the equilibrium case, the steady-state (2) within its regime of validity, i.e., when j j 1. As Gutzwiller density matrix is found to be very close to a GW predicted, in the limit of a vanishing photon pumping rate PCS in a wide region of the PSF phase, with fidelities achiev- 0 0 = ! 0 at fixed ratio = , both in the Gutzwiller and ing values over 95% for a ratio U= = 70 (see Fig. 2a-b). p l em em semiclassical results (black dashed lines) predict a diverging Moreover, a scaling analysis presented in Fig. 2c-d indicates (2) that fidelity even reaches unity in the ideal limit of vanishing order parameter ! +1 as well as an instability located max rates ; . We nonetheless highlight two strong differences at J = U=2, even for a non-vanishing detuning . l c m 5 Conclusions. In this work, we developed a comprehen- C. Ciuti, and I. Carusotto, Phys. Rev. A 96, 033828 (2017). [21] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frun- sive theoretical framework for the emergence of photonic cat zio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. 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Fazio, 97, 033603 (2018). 6 [45] In contrast with the standard Bose-Hubbard model with single- than at infinite hopping. particle hopping [40], the regime of validity of the semiclassical description is thus located at finite J around J = U=2 rather Supplementary Material CIRCUIT-QED IMPLEMENTATION OF PAIR HOPPING In this Supplementary Material, we propose a realistic implementation of the Bose-Hubbard model with pair hopping in a superconducting-circuit architecture. In this context, the realization of multi-photon couplings typically involves a combination of nonlinear inducting dipoles and parametric modulations. Note that one could think of simply using Josephson junction couplings to realize coherent transport of pairs of photons while suppressing single-photon hopping: in the single-photon case, a beam splitter between two resonators of different frequencies ! and ! can be realized via a Josephson junction parametrically modulated at the frequency difference ! ! . Instead, 1 2 2 1 modulating at 2(! ! ) would enable only the two-photon hopping processes and leave the single-photon hopping strongly 2 1 off-resonance. However, in addition to providing the desired coupling, the use of four-wave mixing Josephson elements usually yields spurious couplings, some of which often reveal detrimental. In our case, such an approach would lead in particular to the presence of negative self-Kerr and cross-Kerr terms [22, 41, 42], which in the many-body language translate to on-site and nearest neighbor attractive interactions, respectively. Note also that while one can argue that the physics described in the manuscript could also be observed in presence of negative interactions (by evolving adiabatically in the highest-energy manifold in the isolated case instead of the ground-state manifold, or by simply changing the sign of the detuning between the cavities and the emitters in the driven-dissipative case), such a modification of the original model goes beyond the scope of our work. Here instead, we propose an alternative method for the generation of multi-photon transport processes circumventing the effects mentioned above. Beyond many-body cat states, the development of methods to avoid cross-Kerr couplings is of interest for quantum simulation applications. Our approach, described in Fig. 3, is based on the recently developed SNAIL devices [29, 30] (Superconducting Nonlinear Inductive Asymmetric eLement) presenting important three-wave mixing, as well as tunable and possibly cancellable four-wave mixing. The lattice sites are inspired by the design of fluxonium qubits, although they operate in a different parameter range. The flux injected in the qubit allows to tune the strength and the sign of the two-photon interactions U , and to even completely cancel it if desired. We assume U > 0. The qubits are coupled virtually via three-wave 0 0 mixing processes to auxiliary resonators, whose frequency ! = 2! + (with ! ) is close to twice the bare frequency aux c c ! of the main lattice sites. The presence of this auxiliary degree of freedom efficiently prevents cross-Kerr couplings between the main lattice sites. Below, we present the second-quantized many-body Hamiltonian describing the circuit; show that, once auxiliary degrees of freedom are integrated out, it yields the original Hamiltonian [Eq. (2)) in the main text] with an effective pair-photon hopping that we compute; generalize the exact ground state of factorized many-body cat states found at J = U=2 to the full circuit (i.e. including the auxiliary degrees of freedom); derive the second-quantized many-body Hamiltonian starting from a first-quantized description of the underlying micro- scopic circuit; discuss the robustness of this implementation against likely mismatches in the fluxes. Second-quantized description of the circuit Our circuit is modeled by the following many-body Hamiltonian (the derivation from microscopic parameters is given in the next section) X X y 0 y2 y H = ! a a + a a + (2! + ) b b c i c i;j i i i i;j hi;ji h i h i X (1) y2 y2 y y y y 2 2 p (a + a )b + (a + a )b + p (a + a )b + (a + a )b : (10) i;j i;j i j i j i j i;j i j i;j z z hi;ji 8 a) Nonlinear assymetric dipoles b) φ φ = e e c c 2 c 3 U (φ)/E =c (φ-φ ) +c (φ-φ ) SNAIL J 2 min 3 min Auxiliary Lattice site i Lattice site i+1 (2) †2 2 †2 c) resonator -J (a a +a a ) i+1 i i i+1 n=1 Δ≫U n=2 n=2 n=2 U >0 n=2 n=1 n=1 n=1 n=1 n=0 n=0 n=0 n=0 n=0 Lattice site i Lattice site i+1 † 2 †2 † 2 †2 η(b a +a b ) η(b a +a b ) i i i i i i+1 i+1 i FIG. 3: Proposal for implementation of two-photon hopping in a superconducting circuit architecture. (a): Description of the circuit and the emergent photon lattice model. Each lattice site is composed of a qubit with bare frequency ! . The qubit design, involving a small junction shunted with an array of two larger junctions with a flux ' , is inspired from the fluxonium qubit and is chosen in such a way to implement a Kerr nonlinearity U with tunable strength and sign. In particular for ' = , U > 0 accounts for repulsive photon interactions. We suggest to couple the qubit to auxiliary resonators with a bare frequency ! = 2! + close to twice the lattice single-photon energy ! . (b): Instead aux c c of a traditional Josephson junction-based nonlinear intersite coupling, the coupling to auxiliary resonators is implemented via three-wave mixing SNAIL (Superconduction Nonlinear Asymmetric Inductive eLement) devices [29, 30] presenting a phase-dependent energy profile c c 2 c 3 c 5 U (') = E c ('