Quantum phase transition and degeneracy of a circuit-QED vacuum Pierre Nataf, Cristiano Ciuti

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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)

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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 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.

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(September 18, 2009) each cell of size a is labeled by the index j and is lo- Circuit (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 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 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, |thei 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  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. 1, with a double well structure. Due to the strong only to the collective mode with the same spa- anharmonicity of its energy spectrum, the fluxonium can tial symmetry and = k. Hence, the eigenstates H k H be approximated as a two-level system, when EJ ELJ . are products of the eigenstates corresponding to the k- ≫ P We call the two first eigenstates of the jth fluxonium as subspaces. The effective Hamiltonian in Eq. (6) has been 0 j and 1 j and we introduce the raising operatorσ ˆ+,j = ˆ | i | i obtained by assuming that the operators bk† are bosonic, † 1 0 j andσ ˆ ,j =σ ˆ+,j = 0 1 j . By using the Pauli i.e. [ˆbk, ˆb† ] 1, an approximation working in the limit | ih | − | ih | k ≃ matrix notation, we have σx,j =σ ˆ† +ˆσ+,j and σy,j = N 1. The excitation spectrum of the collective bosonic +,j ≫ 3 modes depends on the eigenvalues of the matrix k. A infinite coupling limit (where HF can be neglected with M very interesting property of the circuit-QED system here respect to Hres + Hcoupling ), every eigenstate has the considered can be appreciated by inspecting the determi- following form: Πj ζj Ψres where ζj , + and | i ⊗ | i ∈ {− } nant of k: Ψres is a generic quantum state for the transmission line M |resonatori field. In particular, we have analytically found 2 Det( k)= ωkωF (ωkωF 4Ω ). (9) M − k the asymptotic expression for the two ground states G+ and G : | i Remarkably, it vanishes when the vacuum Rabi frequency | −i is equal to the critical coupling value g√2 iko ( a† ) k1.5 sin( π ) ko ± o 2N G CGΠj j Πko e 0 ko Πke 0 ke ± √ωkωF | i≃ |±i ⊗ | i ⊗ | i Ωc = , (10) (11) k 2

Ωk=1 implying that two of the 4 eigenvalues of k are exactly with CG a normalisation constant, g = √ the di- M Nωk=1 zero. This is reminiscent of quantum phase transitions[7] mensionless coupling constant per fluxonium , ko (ke) with Dicke-like Hamiltonians[8], where at the quantum standing for the odd (even) k values for the resonator critical point there is a gapless bosonic excitation. For modes. k Ωk > Ωc , two of the 4 eigenvalues of the matrix k becomes imaginary, manifesting an instability of the nor-M mal, non-degenerate, quantum vacuum phase. Note that, depending on the type of interaction, a system with (ul- 2 2

F 1.5 ω )/

tra)strong coupling light-matter regime does not neces- G 1 ω −

0 ω sarily have a quantum critical point[9]. ( 0.5

Above the critical point, the system can not be any 0 0 0.2 0.4 0.6 0.8 1 1.2 longer described by the effective bosonic Hamiltonian −2 in Eq. (6) and one has to necessarily consider de- 0

H F 10

viations from bosonicity. Indeed, above the critical ω / −4 −2 coupling, other analytical approaches (e.g., expansions ω 10 ) F ω based on the Holstein-Primakoff representation[8]) or ex- ( −4 δ 10 act numerical diagonalizations of the actual Hamiltonian −6 −6 H with fermionic operators are needed. In the limit Splitting 10

N , it can be shown that above the critical point a −8 −8 → ∞ 10 symmetry breaking occurs and that the ground state be- 0 0.2 0.4 0.6 0.8 1 1.2 comes twice degenerate. Indeed, one can define a parity −10 iπνˆ 0 0.2 0.4 0.6 0.8 1 1.2 operator Pˆ = e , whereν ˆ = aˆ† aˆk + σˆ† σˆ+,j Ω /ω k k j +,j k=1 k=1 is the operator counting the totalP numberP of excita- tions. The Hamiltonian commutes with such parity op- erator. In the undercritical normal phase, the vacuum FIG. 2: First 20 energy eigenvalues (normalized to the flux- G has a well definite parity and it is therefore ’inco- onium transition energy) versus dimensionless vacuum Rabi | i coupling for a circuit-QED system with N = 5 fluxonium herent’, i.e. G ak G = 0 and G σx,j G = 0. On the h | | i h | | i atoms, Nm = 3 resonator modes (20 cut-off quanta per mode) other hand, in the overcritical case, the parity symme- and fluxonium frequency ωF = ωk=1. Due to the finite value try is broken and there is a photon and qubit coherence, of N, the transition from a non-degenerate to a twice degener- namely G ak G = αk = 0 and G σx,j G = βj = 0. ate ground state is not abrupt. Upper inset: the difference be- The groundh | state| i is twice6 degenerate,h | with| i a second6 or- tween the considered energy eigenvalues and the ground state energy is plotted. Lower inset: normalized energy difference thogonal ground state such that G′ ak G′ = αk and h | | i − (log scale, over 8 decades) between the first 2 quasi-degenerate G′ σx,j G′ = βj . h | | i − levels versus the dimensionless coupling. For large couplings, For the case of finite number of fluxonium qubits N and the two collective ground states are excellently approximated finite number of modes Nm, we have performed numeri- by the analytical formula in Eq. (11). cal diagonalizations of the circuit-QED Hamiltonian (see Fig. 2). In the limit of very large dimensionless coupling ( Ωk=1 ), we have also derived a simple analytical ωF → ∞ expression for all eigenstates and in particular for the Eq. (11) shows that the two degenerate ground states two degenerate ground states by taking into account an are the product of a ’ferromagnetic’ state for the chain arbitrary number Nm of modes for the transmission line of qubits times coherent states for the different resonator resonator. It is convenient to introduce the x-polarized modes. Importantly, the two orthogonal ground states states (eigenstates ofσ ˆ ), namely + = 1 ( 1 + 0 ) x,j j √2 j j have opposite polarization of the qubits and opposite 1 | i | i | i and j = ( 1 j 0 j ). We have found that, in such phases for the coherent states. Fig. 2 shows the first |−i √2 | i − | i 4

20 numerically calculated eigenvalues with N = 5 and This is the control parameter thanks to which it is possi- Ωk=1 Nm = 3. It is apparent that the transition from a ble to tune the dimensionless coupling . The branch- ωk=1 non-degenerate to a twice degenerate ground state is ing ratio γ is approximately zero for L1 L2 and tends L ≪ not abrupt, as opposed to the thermodynamic limit of to 1 when 1 1. Note that in the resonant case the L2 ≫ Ωc large N. However, even for a relatively small number critical coupling is such that k=1 = 0.5, hence by con- ωk=1 of qubits and finite coupling g, quasi-degeneracy of the trolling the branching ratio it is indeed possible to explore circuit-QED ground state is obtained for coupling slightly all the phase diagram. c larger than the critical value (Ωk/ωk=1 =0.5) predicted We wish to point out that the excitations contained in by the effective bosonic Hamiltonian in Eq. (6), which the quantum ground state cannot give rise to extracavity is valid for N 1. Moreover, we haveH verified in detail ≫ microwave radiation unless a non-adiabatic modulation that above the critical value the corresponding states are of the Hamiltonian parameters is applied[11, 14]. The excellently described by the analytical expression in Eq. quantum vacuum radiation across the quantum phase (11) (for the strongest coupling considered in Fig. 2, the transition is an interesting problem to explore in the overlap between the analytical expression for G and | ±i future. In the opposite limit of adiabatic changes of the numerical counterpart is approximately 95%). the Hamiltonian, thanks to the degeneracy, it may be It is important to point out that in presence of a possible in principle to create Berry phases (eventually large, but finite coupling, HF ’perturbs’ the effect of non-abelian) and conveniently control quantum super- Hres + Hcoupling by lifting the degeneracy of the quan- positions in the quantum ground state subspace, another tum ground state doublet. We have found that for large interesting issue to explore. In conclusion, a chain of flux- couplings the corresponding energy splitting decreases onium atoms inductively coupled to a transmission line exponentially with increasing coupling per fluxonium g resonator proposed here appears to be a very promising and fluxonium number N (see inset in Fig. 2). Clearly, system towards the observation of interesting quantum we recover the result of exact degeneracy in the limit of phase transitions effects and the manipulation of quan- large N. Perturbation theory also allows us to estimate tum vacuua in circuit-QED. the splitting between the two first states when applying We are gratefully indebted to M.H. Devoret for many local noise sources proportional to the operators σy,j and enthusiastic conversations and for suggesting us to con- σz,j . In fact, for the degenerate ground states considered sider a fluxonium qubit as elementary element of the above, we have G σˆy,j G = G σˆy,j G = 0 and h ±| | ±i h ±| | ∓i chain[15]. We are pleased to thank B. Dou¸cot, I. Caru- G σˆz,j G = G σˆz,j G = 0. Moreover, in pres- sotto and S. De Liberato for discussions. enceh ±| of N| ±qubits,i h the±| noise| ∓ effecti is zero up to the N- th order perturbation theory[13]. This implies that, by increasing the number N, the noise-induced degeneracy splitting decreases exponentially with N, an issue that we ∗ have also verified numerically. The degeneracy is instead E-mail: [email protected] 431 not protected with respect to local noise sources propor- [1] A. Wallraff et al. Nature , 162 (2004). 5 tional to σ , though different geometrical arrangements [2] L. S. Bishop et al., Nature Phys. , 105 (2009) x,j [3] L. DiCarlo et al., Nature 460, 240 (2009). in multicavity systems may add further protection. [4] V. E. Manucharyan, J. Koch, L.I. Glazman, M.H. De- As we describe in the following, by a judicious tuning of voret, Science in press; preprint arXiv:0906.0831. the controllable parameters of the system, it is possible to [5] M.H. Devoret, S.M. Girvin, and R.J. Schoelkopf, Ann. explore all the phase diagram with a realistic system. Let Phys. 16, 767 (2007). us suppose that each fluxonium is resonant with the first [6] A.Blais et al., Phys. Rev. A 69, 062320 (2004). [7] See, e.g., Subir Sachdev, Quantum Phase Transitions, cavity mode, i.e., (ωF = ωk=1). In this resonant case, we (Cambridge University Press, 2001). find that the dimensionless vacuum Rabi coupling reads [8] C. Emary, T. Brandes, Phys. Rev. Lett. 90, 166802 (2003) and Phys. Rev. E 67, 066203 (2003). Ω Z k=1 = √N vac µνγ 5.7√Nγ , (12) [9] It is important to point out that in the case of photon ωk=1 r2Zrα ∼ mode interacting via electric dipole coupling to a large collection of two-level systems, one would find an Hop- πa 1 1 EJ sin( 2d ) a field matrix of the form where ν = 4π ϕ01 4 for E 1, µ = πa . For d ∼ LJ ≫ 2d → + Zvac h ωk + 2Dk −iΩk −2Dk −iΩk 0 , we have µ 1. Finally, 2α = e2 = Rk 25.8kΩ is 0 1 ∼ ∼ Hopfield iΩk ω12 −iΩk 0 Lr Mk = , the impedance quantum, while Zr = = 50Ω is the B 2Dk −iΩk −ωk − 2Dk −iΩk C Cr B C standard transmission line impedanceq . @ −iΩk 0 iΩk −ω12 A Finally, we have the branching ratio where ω12 is the two-level transition frequency, Ck is the vacuum Rabi coupling (proportional to the electric dipole Lr 1 L1 4 matrix element) and Dk is a term arising from the square γ = ( ) 3 . (13) L1Lr + L1L2 + L2Lr (L1 + L2) 4 vector potential in the light-matter interaction. In the 5

2 Hopfield Hopfield matrix[10], Dk = Ck /ω12 and Det(Mk )= Lett. 98, 103602 (2007); S. De Liberato et al., arXiv 2 (ωkω12) 6= 0 and no quantum critical point exists, even arXiv:0906.2706. if the coupling can be ultrastrong[11, 12]. [15] We have found that in the case of Cooper pair quan- [10] J. J. Hopfield, Phys. Rev. 112, 1555 (1958). tum boxes capacitively coupled to the resonator a criti- [11] C. Ciuti, G. Bastard, I. Carusotto, Phys. Rev. B 72, cal coupling exists. In that case, a much larger number 115303 (2005). of qubits is required. The corresponding Hopfield-like[9] [12] C. Ciuti, I. Carusotto, Phys. Rev. A 74, 033811 (2006). matrix contains a finite Dk term, which can be made 2 [13] B. Dou¸cot, M. V. Feigel’man, L. B. Ioffe, and A. S. Iose- much smaller than Ωk/ω12. This point is crucial to have levich, Phys. Rev. B 71, 024505 (2005) a quantum critical point. [14] S. De Liberato, C. Ciuti, I. Carusotto, Phys. Rev.