Quantum phase transition and degeneracy of a circuit-QED vacuum Pierre Nataf, Cristiano Ciuti To cite this version: Pierre Nataf, Cristiano Ciuti. Quantum phase transition and degeneracy of a circuit-QED vacuum. 2009. hal-00418515v1 HAL Id: hal-00418515 https://hal.archives-ouvertes.fr/hal-00418515v1 Preprint submitted on 18 Sep 2009 (v1), last revised 15 Jan 2010 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum phase transition and degeneracy of a circuit-QED vacuum 1 1, Pierre Nataf and Cristiano Ciuti ∗ 1Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, Universit´eParis Diderot-Paris 7 and CNRS, Bˆatiment Condorcet, 10 rue Alice Domont et L´eonie Duquet, 75205 Paris Cedex 13, France We investigate theoretically the quantum vacuum properties of a transmission line resonator inductively coupled to a chain of N superconducting qubits. We derive the quantum Hamiltonian for such circuit-QED system, showing that, due to the type and strength of the interaction, a quantum phase transition occurs in the limit of large N, with a twice degenerate quantum vacuum above a critical coupling. The phase diagram can be fully explored thanks to the controllable ultrastrong coupling of the qubits with the modes of the transmission line resonator. For finite values of N, an energy splitting occurs, which becomes exponentially small with increasing size and coupling. PACS numbers: (September 18, 2009) each cell of size a is labeled by the index j and is lo- Circuit quantum electrodynamics (circuit-QED) is a cated at the position xj . One can effectively model the very fascinating topic for fundamental condensed mat- resonator as a sequence of inductances Lr = alr and ca- ter physics, quantum optics and quantum information. pacitances Cr = acr [6], where lr (cr) is the inductance In superconducting circuit-QED systems, it has been (capacitance) per unit length. In each cell, the Joseph- possible to implement on a chip the celebrated Jaynes- son junction, the capacitances, and the inductances of Cummings model by strongly coupling a superconduct- both resonator and fluxonium contribute to the energy ing qubit to a bosonic mode of a microwave transmission as follows: line resonator[1, 2] and also to perform quantum logi- ˆj 1 ˆj ˆj 2 ˆj 2 cal operations with two qubits[3]. Up to now, manip- ˆ j 1 2 (φr− φr φx) (φx) Hj =4ECr (Nr − ) + − − + ulation of quantum states in such circuit-QED systems 2Lr 2L1 has dealt with excited states. In fact, in the quantum ˆj ˆj 2 (φx φJ ) ˆ j 2 ˆj 2e j + − + 4ECJ (NJ ) EJ cos(φJ + Φext) ,(1) circuits studied up to now, the quantum optical ground 2L2 − ~ state is non-degenerate (e.g., in the Jaynes-Cummings model the ground state is the vacuum of excitations for where the Nˆ and φˆ operators represent the number and the resonator times the ground state of the qubit) and flux operators for the resonator elements and Joseph- no information can be stored or processed by using only son junctions (the index ’r’ stands for resonator; ’J’ the ground state of the circuit-QED system. for Josephson junction). The charging energies in the e2 e2 Hamiltonian are EC = and EC = . By In this letter, we show that by using a chain of qubits r 2Cr J 2CJ embedded in a transmission line resonator, it is possi- ~ applying Kirchoff’s laws and by taking Φext = π 2e , it ble to obtain a quantum phase transition affecting the is possible to rewrite the global Hamiltonian as H = quantum vacuum of the system. We have found that Hres + HF + Hcoupling as follows: a critical coupling occurs in such a circuit-QED system thanks to both the type and ultrastrong strength of the interaction obtainable with inductively coupled qubits. By analytical and numerical calculations, we show that N j j 1 2 above the critical point, the ground state is twice degen- j (ˆϕ ϕˆ − ) H = 4E (Nˆ )2 + E r − r , erate and that it is protected with respect to some types res Cr r Lr 2 of local noise sources. These properties can be achieved Xj=1 N even with a moderate number N of qubits. (ˆϕj )2 H = 4E (Nˆ j )2 + E J + E cos(ˆϕj ) , A sketch of the considered system is depicted in Fig. F CJ J LJ 2 J J 1, namely a chain of N identical artificial two-level atoms Xj=1 N in a one-dimensional transmission line resonator. Each j j 1 j Hcoupling = G(ˆϕ ϕˆ − )ˆϕ , (2) qubit (’fluxonium’ [4]) is made of a Josephson junc- r − r J tion coupled to inductances and an external magnetic Xj=1 flux. This artificial atom is inductively coupled to the where we have introduced the dimensionless fluxes ϕj = transmission line resonator. The fluxonium is known to r 2e φˆj , ϕj = 2e φˆj and the inductance energy con- have controllable parameters and to be free from charge ~ r J ~ J ~ 2 L1+L2 stants are EL = ( ) , EL = r 2e L1Lr + L1L2 +L2Lr J offsets[4]; moreover, the inductive coupling can produce ~ ( )2( L1+Lr ) . The magnitude of the coupling extremely large coupling even with a single qubit[5]. In 2e L1Lr + L1L2 +L2Lr ~ 2 L1 the case of a chain, the Hamiltonian is H = Hj where constant is G = ( ) . j 2e L1Lr + L1L2 +L2Lr P 2 i(ˆσ† σˆ ,j ) and σz,j = 2ˆσ† σˆ ,j 1. Leaving aside − +,j − + +,j + − ~ a constant term, we then have HF = j ωF σˆ+† ,j σˆ+,j , where ~ωF is the energy splitting betweenP the two states 0 and 1 . By considering only the two-level subspace, the| i Josephson| i junction flux has the form j ϕˆ ϕ (ˆσ ,j +ˆσ† )= ϕ σˆx,j . (4) J ≃− 01 + +,j − 01 As it will be clear in the following, it is convenient to ˆ introduce excitation creation operators bk† N 2 ⋆ ˆb† = ∆f (x )ˆσ (5) k rN k j +,j Xj=1 kπ( N+1 +j) − 2 for 1 k N 1, where ∆fk(xj ) = cos( d a) ≤ ≤ − kπ( N+1 +j) − 2 FIG. 1: Description of the system. A chain of N iden- for k odd , and ∆fk(xj ) = sin( d a) for k even. ˆ tical, regularly spaced qubits (”F” stands for fluxonium[4]) Note that the collective operator bk† is a linear superpo- are embedded in a transmission line resonator. Each flux- sition of the excitation operators in each fluxonium with onium is coupled inductively to the transmission line res- an amplitude depending on the spatial dependence of the onator. By properly tuning the external magnetic flux, the flux field of the transmission line resonator. In order to to flux-dependent potential for each fluxonium has a symmetric double well structure with two states, |0i and |1i (with energy get a unitary transformation, it is also necessary to intro- 1 j ~ duce the operator ˆb† = ( 1) σˆ ,j . In the follow- difference ωF ) forming the two-level system artificial atom N √N j − + (parameters used for the inset: EJ /ECJ = 3, EJ /ELJ = 20; ing, we will consider only theP photonic modes 1 k N the third level of the fluxonium is well separated in energy). (equivalent to the first Brillouin zone associated≤ to≤ the periodic spatial spacing of the artificial atoms), because, in the conditions we are considering, the higher order The Hamiltonian Hres describes the transmission line (Bragg) modes are energetically well off-resonant. Hence, resonator with a renormalized inductance per unit of we get the following effective Hamiltonian: L + L + L2L1 1 2 alr length l˜r = lr , accounting for the additional L1+L2 ~ inductances in each fluxonium. By following the quan- = Φˆ † η k Φˆ k (6) H 2 k M tum field treatment in Ref. [6], the position-dependent 1 Xk N flux field can be written as: ≤ ≤ ˆ ˆ ˆ T ~ where Φk = (ˆa k, bk, aˆ†k, bk† ) with the Bogoliubov di- ˆ 1 ωk agonal metric η = diag[1, 1, 1, 1], and the matrix: φ(x)= i fk(x)(ˆak aˆk† ) (3) ωk r 2cr − − − kX1 ≥ ωk iΩk 0 iΩk − − where a† is the bosonic creation operator of a photon iΩk ωF iΩk 0 k k = − . (7) ~ kπa M 0 iΩk ωk iΩk mode with energy ωk = d 8EcELr . The spatial − − − kπx iΩk 0 iΩk ωF profile of the k-th mode is fk(x)=p 2/d sin( ) for k − − − d kπx p odd, while fk(x)= 2/d cos( d ) for k even, d being the The collective vacuum Rabi frequency reads for 1 k length of the one-dimensionalp resonator (in the following, N 1 ≤ ≤ we will consider d = Na). The site-dependent fluxes are − ˆj ˆ 4e kπa 1 ~ω N simply given by the relation φr = φ(xj ). ~ k Ωk = G ~ ϕ01 sin( ) . (8) The Hamiltonian HF describes the sum of the bare 2d ωk r 2dcr energies of the artificial fluxonium atoms. By properly ~ ( and for k = N, ~Ω = G 4e ϕ01 ωN N ). tuning the external magnetic flux, it it possible to obtain N ~ ωN dcr a symmetric flux-dependent potential energy, as shown in Notice that each k-mode of theq resonator is coupled Fig.