Absence of spin-boson quantum phase transition for transmon qubits Kuljeet Kaur,1 Th´eoS´epulcre,2 Nicolas Roch,2 Izak Snyman,3 Serge Florens,2 and Soumya Bera1 1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India 2Univ. Grenoble Alpes, CNRS, Institut N´eel,F-38000 Grenoble, France 3Mandelstam Institute for Theoretical Physics, School of Physics, University of the Witwatersrand, Johannesburg, South Africa Superconducting circuits are currently developed as a versatile platform for the exploration of many-body physics, both at the analog and digital levels. Their building blocks are often idealized as two-level qubits, drawing powerful analogies to quantum spin models. For a charge qubit that is capacitively coupled to a transmission line, this analogy leads to the celebrated spin-boson de- scription of quantum dissipation. We put here into evidence a failure of the two-level paradigm for realistic superconducting devices, due to electrostatic constraints which limit the maximum strength of dissipation. These prevent the occurence of the spin-boson quantum phase transition for trans- mons, even up to relatively large non-linearities. A different picture for the many-body ground state describing strongly dissipative transmons is proposed, showing unusual zero point fluctuations. Quantum computation has been hailed as a promising siderations impose strong constraints on the underlying avenue to tackle a large class of unsolved problems, from models. Third, many-body effects beyond the RWA ap- physics and chemistry [1] to algorithmic complexity [2], proximation are notoriously difficult to simulate due to following an original proposition from Feynman [3], long an exponentially large Hilbert space, for instance in the before the technological and conceptual tools were devel- ultra-strong coupling regime of dissipation [21{23]. To oped to make such ideas tangible. While a general pur- overcome this challenge, we will develop new analytical pose digital quantum computer could theoretically out- and numerical techniques for the quantitative analysis perform classical hardware for some exponentially hard of quantum circuits involving many strongly interacting tasks, building such a complex quantum machine is at degrees of freedom (including charge offsets), a problem present still out of reach. For this reason, analog quan- that is raising increasing interest [9, 20, 24] due to poten- tum simulation has been put forward as a crucial mile- tial applications ranging from hardware-protected qubits stone [4], aiming at the design of fully controllable exper- [25, 26] to quantum optics with metamaterials [27]. imental devices mimicking the features of difficult quan- Having set the stage, we propose to examine the poten- tum problems of interest. This route has met tremendous tial implementation of many-body physics in realistic su- success in the past, with the realization of Kondo impuri- perconducting circuits from the perspective of quantum ties in quantum dots [5], the simulation of artificial solids dissipation [28, 29], stressing that our findings will apply in optical lattices [6], and is gaining momentum with new to a wider range of simulation platforms. Precedent stud- tools from superconducting circuits [7{13]. Ironically, ies of this problem [30{35] relied on the two-level approx- while Feynman anticipated quantum simulators [3], he imation, which is only valid for strong non-linearities, a often warned in his lectures (where analogy was used as regime that is very hard to investigate experimentally, a powerful teaching method) that there is no such thing due to high sensitivity to external noise sources. We as a perfect analogue, and that some interesting physics study the circuit of Fig.1, composed of a superconduct- can emerge when the analogy breaks down [14]. Explor- ing qubit containing a junction with Josephson energy EJ ing the limitations of realistic superconducting qubits in and capacitance CJ = Cs +Cg where Cs is a shunt capac- simulating dissipative two-level systems is the main pur- itance and Cg is a gate capacitance, that is capacitively pose of this Letter. In the process we will unveil the coupled via Cc to a transmission line characterized by unique many-body physics of such simulators. lumped element inductance L and capacitance C. (The In this context, three important challenges must be ad- transmission line may be designed from an array of lin- dressed. First, most quantum computation/simulation ear Josephson elements [20]). All nodes are grounded via capacitances Cg. A DC charge offset controlled by volt- arXiv:2010.01016v1 [cond-mat.mes-hall] 2 Oct 2020 protocols assume that superconducting qubits behave as idealized spin 1/2 degrees of freedom, which is clearly age Vg is included on qubit node 0, appearing as dimen- invalid at strong driving [15], and generally questionable sionless charge ng = VgCg=2e. The circuit Lagrangian in the many-body regime due to proliferation of quan- reads [36] (working in units of ~ = 2e = 1): tum states. We will propose here a microscopic approach 1 _ _ 1 L = Φ~ |CΦ~ − Φ~ |1=LΦ~ + E cos(Φ ) − n Φ_ ; (1) that takes into account the full Josephson potential of 2 2 J 0 g 0 the qubit. Second, quantitative modelling of designs in- volving several qubits or resonators demands incorpo- where Φ~ = (Φ0; Φ1;:::) is a vector of dimensionless node rating the full capacitance network, even in the linear fluxes labeled according to Fig.1. C and 1=L are the ca- regime [16{20]. We will see that such electrokinetic con- pacitance and inductance matrices read from Fig1, that 2 2 define a generalized eigenvalue problem 1=LP = CP! , chain) [28, 29]. It is crucial to note that the spectral func- ! being the diagonal matrix of the system eigenfrequen- tion J(!) and the charging energy Ec are not mutually cies. Noting that [P |CP ; !2] = 0, implying that we independent. For the circuit of Fig.1, we find J(!) = | p 2 2 2 2 can take P CP diagonal, we normalize the columns of 2πα! 1 − ! =!P =(1 + ! =!J )θ(!P − !), dissipation | ~ −1~ 2p P such that P CP = 11. In the new basis φ = P Φ, strength α = (1=2π)[Cc=(Cc + CJ )] L=Cg, (angular) the Lagrangian is in normal modes form: p plasma frequency of the line !P = 1p= L(C + Cg=4), 1 and RC cutoff of the junction !J = 1= LCeff (a deriva- ~ | 2 2 ~ X X _ L = φ (−@ − ! )φ + EJ cos P0kφk − ng P0kφk; tion of these parameters is given in the Supplementary 2 t k k Materials). For realistic parameters describing the cir- (2) cuit of Fig.1, the linear behavior of J(!) is cut-off at a where the dispersion relation !k and the coupling P0k are scale !J well below the plasma scale !P , thus limiting given in Supplementary Materials. The qubit degree of the ohmic range of dissipation. freedom is recovered via the change of variables: Due to the mutual dependence between Ec and J(!) P arising from electrostatics, the dissipation strength α has ' = k P0kφk n = N0=P00 ; ; (3) an upper bound of order Ec=Min(!P ; j!J j). To show 'm = φm nm = Nm − (P0m=P00)N0 this, we parametrize the spectral function as J(!) = where ~n (resp. N~ ) is the vector of charges conjugate to 2πα! exp (−!=!c), without specifying the origin of the ~ cutoff !c. Eq. (5) gives α = [2Ec − 1=(4CJ + 4Cc)]=!c. ~' (resp. φ). Since !0 = 0, the change of variables does not generate any diamagnetic (or `A2') term [17, 18]. Using Ec and !c as independent parameters imposes: L L α 6 αmax = 2Ec=!c: (6) C C g 0 c 1 2 3 Such electrostatic constraint must be fulfilled for any mi- V g croscopic model, and in the Supplementary Materials, we Cs EJ Cg Cg Cg provide a similar bound for the circuit of Fig.1, showing C C that only the prefactor depends on the shape of the cut- off function. Clearly, the maximum value of dissipation qubit transmission line αmax is attained for Cc ! 1, namely when the qubit be- comes wire coupled to the transmission line (see Fig.1). The simple constraint (6) has profound consequences for FIG. 1. Microscopic electrokinetic model for a realistic circuit of dissipative superconducting transmon. The qubit, located the dissipative quantum mechanics of realistic supercon- at node 0, is characterized by Josephson energy EJ , shunt ducting qubits. Indeed, reaching the ultrastrong coupling capacitance Cs, and is capacitively coupled via Cc to a trans- regime (α ' 1) where many-body effects are most promi- mission line. All nodes are shunted to the ground via the nent implies Ec ' !c. For Cooper pair boxes (CPB) with capacitance Cg, and each lumped element in the line is char- EJ Ec, one obtains EJ !c, and ohmic dissipation acterized by its inductance L and self-capacitance C. Charge dominates the qubit dynamics, since the first transition offsets are modeled by a DC voltage source Vg. at energy EJ is well within the linear regime of J(!). However, for transmons with EJ Ec, the first transi- Once quantized in terms of creation/annihilation op- p tion located at 8EcEJ !c now lies in the tails of the erators, we obtain the Hamiltonian : cutoff function J(!). Hence ultra-strong coupling physics X y X y and the spin-boson quantum phase transition [28] are not H^ = !ka^ a^ + (^n − ng) igk(^a − a^ ) k k k k possible for a capacitively coupled transmon qubit, shed- k k ding light on previous experimental attempts [20, 24], + 4E (^n − n )2 − E cos'; ^ (4) c g J and extending also predictions for simpler systems of which we nickname the \transmon-boson model", as it transmons coupled to single cavities [37, 38]. One can generalizes the ubiquitous spin-boson model describing thus wonder at which range of non-linearity the two-level p quantum dissipation [28, 29].