Superradiant Phase Transitions and the Standard Description of Circuit QED

Superradiant Phase Transitions and the Standard Description of Circuit QED

Oliver Viehmann,¹Jan von Delft,¹and Florian Marquardt²

1Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universita¨t, Theresienstraße 37, 80333 Mu¨nchen, Germany

2Institut for Theoretical Physics, Universita¨t Erlangen-Nu¨rnberg, Staudtstraße 7, 91058 Erlangen, Germany (Received 29 March 2011; published 8 September 2011)

We investigate the equilibrium behavior of a superconducting circuit QED system containing a large number of artificial atoms. It is shown that the currently accepted standard description of circuit QED via an effective model fails in an important aspect: it predicts the possibility of a superradiant phase transition, even though a full microscopic treatment reveals that a no-go theorem for such phase transitions known from cavity QED applies to circuit QED systems as well. We generalize the no-go theorem to the case of (artificial) atoms with many energy levels and thus make it more applicable for realistic cavity or circuit QED systems.

DOI:10.1103/PhysRevLett.107.113602 PACS numbers: 42.50.Pq, 03.67.Lx, 64.70.Tg, 85.25.!j

Recent years have seen rapid progress in fabrication and experimental control of superconducting circuit QED sys- tems, in which a steadily increasing number of artificial atoms interact with microwaves [1–4]. These develop- ments set the stage to study collective phenomena in circuit QED. An interesting question in that context is whether a system with many artificial atoms undergoes an equilib- rium phase transition as the coupling of artificial atoms and electromagnetic field is increased (at zero temperature).

Phase transitions of this type have been intensely discussed for cavity QED systems [5–10] and are known as super- radiant phase transitions (SPTs) [6]. However, in cavity QED systems with electric dipole coupling their existence is doubted due to a no-go theorem [8]. Recently, it has been claimed that SPTs are possible in the closely related circuit QED systems with capacitive coupling [10–12]. This would imply that the no-go theorem of cavity QED does not apply and challenges the well-established analogy of circuit and cavity QED.

Here, we show in a full microscopic analysis that circuit QED systems are also subject to the no-go theorem. We argue that such an analysis is necessary since the standard description of circuit QED systems by an effective model (EM) is deficient in the regime considered here. A toy model is used to illustrate this failure of an EM. Finally, we close a possible loophole of the no-go theorem by generalizing it from two-level to multilevel (artificial) atoms. Thus, our work restores the analogy of circuit and cavity QED and rules out SPTs in these systems under realistic conditions that have not been covered before.

Dicke Hamiltonian in cavity and circuit QED.—Both circuit QED systems and cavity QED systems with N (artificial) atoms (Fig. 1) are often described by the Dicke Hamiltonian [13] (@¼1)

HD¼!a^yaþ! 2

X^N

k¼1

!^k_zþ "

ffiffiffiffi pNX^N

k¼1

!^k_xða^yþaÞ

þ#ða^yþaÞ²: (1)

The (artificial) atoms are treated as two-level systems with energy splitting ! between ground state jgik ¼ ð⁰₁Þk and excited statejeik¼ ð¹0Þk (!^k_x;!^k_zare Pauli matrices). In the case of circuit QED, we assume Cooper-pair boxes as artificial atoms, which justifies the two-level approxima- tion. Our main results, though, hold for any charge-based artificial atoms (capacitive coupling) [14]. Further, a^y generates a photon of energy !. Matter and field couple with a strength ". The # term, often neglected in other contexts, will become crucial below. In cavity QED,HD

derives from minimal coupling of atoms and electromag- netic field. For an atom (nelectrons) at a fixed position,

H⁰cav¼Xⁿ

i¼1

½p_i!eAðr_iÞ'²

2m þV_intðr₁;. . .;r_nÞ: (2) The pAandA² terms in the analogN-atom Hamiltonian yield the " and # term in HD, respectively. In circuit QED, HD arises from a widely used EM for a charge- based artificial atom in a transmission line resonator [15],

H⁰_cir¼4E_CX

$

ð$!$"Þ²j$ih$j!E_J 2

X

$

ðj$þ1ih$jþH:c:Þ: Here, $ counts the excess Cooper pairs on the island,E_J andE_C¼e²=½2ðC_GþC_JÞ' are the Josephson energy and

FIG. 1 (color online). Cavity QED system withN atoms (a) and circuit QED system withN Cooper-pair boxes as artificial atoms (b).

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0031-9007=11=107(11)=113602(5) 113602-1 !2011 American Physical Society

the charging energy of the Cooper-pair box, andC_GandC_J are the coupling capacitance and the capacitance of the Josephson junction. Moreover,$" ¼C_GðV_GþVÞ=2e,V_G is an external gate voltage andVthe quantum voltage due to the electromagnetic field in the resonator. The Cooper- pair box is assumed to be at its degeneracy point [15]. As it is described by macroscopic quantities (likeE_C) and only 1 degree of freedom ($),H^cir₀ is an EM for a Cooper-pair box in a transmission line. Starting either fromH⁰cav or H⁰_cir, one obtains HD using the following approxima- tions: TheN(artificial) atoms are identical, noninteracting two-level systems with ground and excited states jgi and jei which are strongly localized compared to the wavelength of the single considered field mode [i.e., Aðr^k_iÞ (A)A₀!ða^yþaÞ, where j!j¼1, and Vðr^kÞ (V )V₀ða^yþaÞ].

Superradiant phase transitions and no-go theorem.—In the limit N ! 1, HD undergoes a second order phase transition at a critical coupling strength [6–8]

"²_c ¼!!

4

"

1þ4#

!

#

: (3)

This phase transition was discovered forHD with#¼0 and termed SPT [6]; see [9] for recent studies. At"_c, the atoms polarize spontaneously, hP

k!^k_zi=N!!1, and a macroscopic photon occupation arises, ha^yai=N!0. A gapless excitation signals the critical point [Fig.2(a)].

In cavity QED systems, however,"_c cannot be reached if the#term is not neglected [8]. That is because"and# are not independent of each other. Let us define a parame-

ter%via#¼%"²=!. Then Eq. (3) becomes"²_cð1!%Þ ¼

!!=4, and criticality requires %<1. With A₀ ¼

1=pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2&₀!V (Vis the volume of the cavity) one finds

"_cav ¼!jffiffiffiffiffiffiffiffiffiffiffiffi!*dj 2&₀! p

ffiffiffiffi N V s

; #_cav¼ n 2&₀!

e² 2m

N V; (4)

where d¼hgjeP_n

i¼1r_ijei and %_cav!j!*dj²¼ne²=2m.

But the Thomas-Reiche-Kuhn sum rule (TRK) ([16], Sec. A)

X

l

ðE_l!E_gÞj!*hgjeXⁿ

i¼1

r_ijlij² ¼ne²

2m (5)

for the Hamiltonian H⁰¼P_n

i¼1p²_i=2mþV_intðr₁;. . .;r_nÞ of an uncoupled atom with spectrumfE_l;jligimplies!j!* dj² +ne²=2m, consequently %_cav ,1. This is known as the no-go theorem for SPTs [8,10]. Notice that %_cav determines how strongly !j!*dj² exhausts the TRK.

We remark that a direct dipole-dipole coupling between atoms (omitted here) can lead to a ferroelectric phase transition, which, however, occurs only at very high atomic densities [17].

Surprisingly, the no-go theorem was recently argued not to apply in circuit QED [10]. Indeed, the standard EM of circuit QED yields

"_cir¼ eC_G

C_GþC_J ffiffiffiffiffiffiffiffi

!N Lc s

; #_cir¼ C²_G 2ðC_GþC_JÞ

!N Lc ; (6) where Ldenotes the length of the transmission line reso- nator,cits capacitance per unit length, and we have used V₀ ¼ ð!=LcÞ¹⁼² [15]. Here %_cir¼E_J=4E_C<1 is easily possible [1]. According to this argument, a SPT should be observable in a circuit QED system.

Effective models and superradiant phase transitions.—

The EM has proved to be a very successful description of circuit QED whose predictions have been confirmed in numerous experiments. However, the circuit QED setups operated so far contained only few artificial atoms. It is not obvious that an EM also provides a good description of circuit QED systems withN-1atoms and, thus, a proper starting point to study SPTs in circuit QED. We now present a toy model illustrating how an EM similar to the one in circuit QED can erroneously predict a SPT.

The toy model consists ofNharmonic oscillator poten- tials with frequency !, each trapping n noninteracting fermions of mass m and charge e, which all couple to a bosonic mode with frequency![Fig.2(b)].

This toy model can be viewed as a very simplified description of (artificial) atoms with n microscopic con- stituents inside a resonator. It is governed by the Hamiltonian

Htm¼!a^yaþX^N

k¼1

Xⁿ

i¼1

ðp^k_i !eAÞ²

2m þm!²ðx^k_iÞ² 2 ; (7) where we assume againAðx^k_iÞ (A¼A₀ða^yþaÞ. SinceA couples only to the center of mass coordinate of the kth oscillator,Htmcan be diagonalized ([16], Sec. B):

0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.5 1.0 1.5 2.0 2.5

FIG. 2 (color online). (a) Excitation energies&_þand&_!of the Dicke HamiltonianHDversus coupling"(in units of!¼!),

for %¼#!="²¼0;0:8;1;1:2. For %¼0, &_! vanishes at

"¼0:5, thus signaling a SPT. Only%,1 is compatible with

the TRK sum rule. For these%,&_!! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1!1=%

p and remains

finite for all ". The excitations &_.ð"Þof H_tm correspond to

%¼1. (b) Toy model of an (artificial) atom. The oval line indicates the degree of freedom in the simplified effective model.

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113602-2

Htm¼&_.

"

a^y_.a_.þ1 2

# þ^nNX^!1

i¼1

!

"

b^y_ib_iþ1 2

#

;

2&²_.ð"Þ ¼!²þ4#!þ!²

. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð!²þ4#!!!²Þ²þ16"²!!

q

: (8)

Here, a^y_. generate excitations that mix photon field with collective center of mass motion, theb^y_i excite the remain- ing degrees of freedom,"¼A₀!d ffiffiffiffi

pN

and#¼"²=!. As

d¼hnjexjn!1i¼e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n=2m!

p , the TRK is exhausted.

Note that &_.ð"Þ are also the relevant excitation energies of H_D for N! 1, as can be shown using methods of Ref. [9] ([16], Sec. B), and demanding &_!¼0 yields Eq. (3). One sees that&_.ð"Þis real and nonzero for all"

and that the ground state energy is an analytic function of"

[Fig.2(a)]. Hence, no phase transition is possible.

Let us now consider an EM for the toy model. Similar to the standard EM of circuit QED, we focus on the fermion with the highest energy in thekth harmonic oscillator and treat it as a two-level system with jg_ki¼jn!1ik

and je_ki¼jnik [Fig. 2(b)]. Accounting only for one fermion per ‘‘atom,’’ that is, expanding H^EMtm ¼

!a^yaþP_N

k¼1ðp^k!eAÞ²=2mþm!²ðx^kÞ²=2 in the basis fjn!1ik;jnikg, yields a Dicke Hamiltonian with"_EM ¼"

and #_EM¼#=n¼Ne²A²₀=2m. Crucially, only "_EM de- pends on n. This allows "_EM to be increased at constant

#_EM; therefore, %_EM¼1=n can be <1 and a SPT is possible. This failure of the EM can be interpreted as follows. The relation "¼"_EM /d/pffiffiffinreveals that the coupling of an ‘‘atom’’ to the bosonic mode is fully cap- tured by the EM and grows with atom sizen. However, in a proper description of the system, increasing the coupling by increasingnunavoidably also increases#in proportion ton: all fermions of all atoms couple to the bosonic mode and each causes an A² term. This is lost in the EM with only 1 degree of freedom per atom. Interestingly,%_EM<1 only if n >1, i.e., as long as the effective description actually neglects degrees of freedom.

Microscopic description of circuit QED.—This example suggests not to rely on the standard description for inves- tigating SPTs in circuit QED. Although the dipole coupling of field and qubit states might be fully represented by"_cir,

#_cir could still underestimate the A² terms of all charged particles in the Cooper-pair boxes. Instead, let us describe a circuit QED system withN artificial atoms by a minimal- coupling Hamiltonian that accounts for all microscopic degrees of freedom:

Hmic¼!a^yaþX^N

k¼1

X^nk

i¼1

ðp^k_i!q^k_iAÞ²

2m^k_i þV_intðr^k₁;. . .;r^k_nkÞ: As we allow arbitrary charges q^k_i and massesm^k_i and an arbitrary interaction potentialV_intof then_kconstituents of thekth artificial atom,Hmic most generally captures the coupling of N arbitrary (but mutually noninteracting)

objects to the electromagnetic field. We subject it to the same approximations that led fromH⁰_cir, the EM of circuit QED, toHD. For identical artificial atomsfn_k; q^k_i; m^k_ig ! fn; q_i; m_ig. The Hamiltonian of an uncoupled artificial atom then reads H_mic⁰ ¼P_n

i¼1p²_i=2m_iþV_intðr₁;. . .;r_nÞ. Its qubit states jgi and jei, which in the standard EM are superpositions of the charge statesj$i, are among the eigenstatesfjligofH_mic⁰ . ExpandingHmicin thefjgik;jeikg basis and takingAðr^k_iÞ (Agives the Dicke Hamiltonian HD with parameters generalizing those of cavity QED [Eq. (4)],

"^mic_cir ¼!pjffiffiffiffiffiffiffiffiffiffiffiffi2&!*₀!dj ffiffiffiffi N V s

; #^mic_cir ¼ 1 2&₀!

"Xⁿ

i¼1

q²_i 2m_i

#N

V; (9) where d¼hgjP_n

i¼1q_ir_ijei. This microscopic description of circuit QED facilitates the same line of argument which in Ref. [8] allowed the conclusion that there is no SPT in cavity QED: Criticality [Eq. (3)] requires !j!*dj²>

P_n

i¼1q²_i=2m_i, which is ruled out by TRK forH_mic⁰ , X

l

ðE_l!E_gÞj!*hgjXⁿ

i¼1

q_ir_ijlij²¼Xⁿ

i¼1

q²_i

2m_i: (10) Hence, the no-go theorem of cavity QED applies to circuit QED as well. This result confirms the analogy of cavity and circuit QED also with respect to SPTs. It has been obtained under the same approximations that led from the standard description of circuit QED,H⁰_cir, toHDwith"_cir and#_cir. The discrepancy of the predictions of the micro- scopic and the standard description of circuit QED thus shows the limitations of the validity of the latter. This might be important for future circuit QED architectures with many artificial atoms in general, even for applications not related to SPTs. We emphasize, though, that our con- clusion neither forbids SPTs in circuit QED systems with inductively coupling flux qubits [18] nor is it at odds with the great success of the standard description for few-atom systems: there, the deficiency of #_cir does not manifest itself qualitatively as the # term in HD mimics slightly renormalized system parameters!~ and".~

Possible loophole in the no-go theorem.—Although the two-level approximation for the anharmonic spectrum of (artificial) atoms is well justified in many cases, one might argue that higher levels should be taken into account in this context. Indeed, a SPT does not require !(!, and thereby does not single out a particular atomic transition.

For a more profound reason for dropping the two-level assumption, consider the elementary question of how the presence ofN mutually noninteracting atoms shifts a res- onator’s frequency!. This situation is described byHmic. It can be rewritten as Hmic¼!a^yaþP_N

k¼1ðH_mic^k þ H^k_pAþH^k_A2Þ, whereH^k_pAandH^k_A2 are thepAandA² terms due to thekth atom ([16], Sec. C). Let us perturba- tively calculate the frequency shift '!¼'!_pAþ'!_A2

caused byP

H^k_pAandP

H^k_A2 ([16], Sec. C). To this end, PRL107,113602 (2011) P H Y S I C A L R E V I E W L E T T E R S week ending

9 SEPTEMBER 2011

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take!/!^k_mfor allm; k, where!^k_mis themth excitation energy ofH_mic^k . Remarkably, it turns out ([16], Sec. C) that '!_pA (<0) and '!_A² (>0) cancel almost exactly due to the TRK. The total frequency shift is small, '!0 ð!=!^k_mÞ². As a SPT equates to'!¼ !!, the significance of bothpAandA² terms for its existence becomes clear.

The pA terms cause a strong negative shift and favor a SPT, the A² terms do the opposite. This means, most crucially, that one must not unequally truncate pA and A² terms for assessing the possibility of a SPT by an approximate Hamiltonian. Dropping theA² terms inHD

(#¼0) leads to the prediction of a SPT. In contrast,HD

with#!0fully incorporates theA²terms ofHmic. But, due to the two-level approximation, it has only one matrix element of thepAterms per atom, thereby possibly under- estimating the tendency towards a SPT. To exclude SPTs in cavity and circuit QED, a generalization of the no-go theorem to (artificial) atoms with more than two energy levels is necessary.

Generalized no-go theorem.—Let us consider N! 1 identical atoms coupled to a field mode with frequency!.

The atomic Hamiltonians H_mic^k may have an arbitrary spectrum f!_l;jl_ki¼jlikg, with !₀ ¼0 and ( excited states (Fig.3).

Withd_l;l0 ¼!*hljP_n

i¼1q_ir_ijl⁰i, the full Hamiltonian of the system reads

Hmic¼!a^yaþ#ða^yþaÞ²þX^N

k¼1

X⁽

l;l⁰¼0

ð!_l'_l;l0jl_kihl_kj þiA₀ð!_l0 !!_lÞd_l;l0ða^yþaÞjl_kihl⁰_kjÞ: (11) We now follow a strategy similar to that of Refs. [9]: We derive a generalized Dicke HamiltonianHGD having the same low-energy spectrum asHmicfor a small density of atoms,N=V’0, usingA₀ /V^!1=2as small parameter. We then check whether HGD has a gapless excitation if the density is increased, which would signal a SPT and mark the breakdown of the analogy ofHGDandHmic.

Expanding the eigenstates and eigenenergies ofHmicas jEi /P₁

s¼0A^s₀jEsi and E /P

ss⁰A^s₀^þs0hEsjHmicjEs⁰i, we note that contributions from all d_l!0;l0!0 terms may be neglected: they are smaller than those retained by a factor of at least one power of A₀ (forsþs⁰>1) or)=N/1 (for sþs⁰ +1), where )¼P

kP

l>0jhl_kjE0ij² is the number of atomic excitations in jE0i, which is/N for

low-lying eigenstates ([16], Sec. D). We thus defineHGD

by settingd_l!0;l0!0!0inHmic. Up to a constant, we find ([16], Sec. D)

HGD¼!a~ ^yaþX⁽

l¼1

!_lb^y_lb_lþX⁽

l¼1

"~_lðb^y_l þb_lÞða^yþaÞ; (12) by introducingb^y_l ¼p¹ffiffiffi_N P_N

k¼1jl_kih0_kjas collective excita- tion, omitting the energy of the ‘‘dark’’ collective excitations ([16], Sec. D), and removing the # term by a Bogolyubov transformation yieldingffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !!!~ ¼

!²þ4#!

p and "_l!"~_l¼ ffiffiffi_!

~

!

p "_l, with "_l¼

A₀!_ljd_0;lj ffiffiffiffi

pN

. For dilute excitations, the b_l are bosonic,

½b_l; b^y_l0' ¼'_l;l0 [19]. The system undergoes a SPT if an eigenfrequency &_i of HGD can be pushed to zero by increasing the couplings "_l. We cannot calculate the&_i’s explicitly, but we will show that the assumption &_i¼0 contradicts the TRK. An&_isolves the characteristic equa- tion ([16], Sec. D)

"Y⁽

l⁰¼1

ð!²_l0!&²Þ#"

ð!~²!&²Þ!4 ~!X⁽

l¼1

!_l"~²_l

!²_l!&²

#

¼0: (13)

If&_iwere zero, this would imply

!

4NA²₀ ¼X⁽

l¼1

!_ljd_0lj²!Xⁿ

i¼1

q²_i

2m_i (14)

and contradict the TRK forH_mic⁰ [Eq. (10)], which ensures that the right-hand side is negative even if the entire atomic spectrum is incorporated. This result is irrespective of the details of the atomic spectra. Note that for #¼0, the negative term on the right-hand side of Eq. (14) vanishes, and one recovers the SPT for critical couplings "_lc with P(

l¼1"²_lc=!_l¼!=4. This resembles Eq. (3) with#¼0.

Experimental evidence for our conclusions could be gained by probing the shifted resonator frequency of a suitable circuit QED system. Consider a sample containing N artificial atoms with "= ffiffiffiffi

pN

¼2*1120 MHz and

!=2*¼!=2*¼3 GHz. If %_cir¼E_J=4E_C¼0:1, as predicted by the standard theory, there should be signatures of criticality forN ¼174[according to Eq. (3)], and the resonator frequency should be close to zero. But even if we assume %¼1, the minimal value compatible with the TRK (that corresponds to ideal two-level atoms), we find the lowest excitation&_!to be still at &_!(2*12 GHz.

We have verified that these phenomena are insensitive to small fluctuations of the atomic parameters ([16], Sec. E;

see also [18]) and hence experimentally observable.

We thank S. M. Girvin, A. Wallraff, J. Fink, A. Blais, J. Siewert, D. Esteve, J. Keeling, P. Nataf, and C. Ciuti for discussions. Support by NIM, the Emmy-Noether program, and the SFB 631 of the DFG is gratefully acknowledged.

FIG. 3 (color online). Situation of the generalized no-go theo- rem. Many multilevel (artificial) atoms couple to the photon field. Transitions between excited atomic states are irrelevant for the low-energy spectrum of the system.

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[16] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.107.113602 for fur- ther explanation and details of the calculations.

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EPAPS: Supplementary Information for

“Superradiant Phase Transitions and the Standard Description of Circuit QED”

Oliver Viehmann¹, Jan von Delft¹, and Florian Marquardt²

1Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany

2Institut for Theoretical Physics, Universit¨at Erlangen-N¨urnberg, Staudtstraße 7, 91058 Erlangen, Germany

We provide intermediate steps for the derivation of some important statements and equa- tions of the main text (Secs. A-D). Furthermore, we discuss the influence of disorder in the parameters of artificial atoms on a possible experimental verification of our results (Sec. E).

For clarity, formulas contained in the main text are typeset in blue.

A. Thomas-Reiche-Kuhn sum rule. We derive the TRK [1] for the Hamiltonian H_mic⁰ =

�n

i=1

p²_i

2mi +Vint(r1, . . . ,rn), (S1) yielding Eq. (9) of the main text; Eq. (4) follows as a special case. The derivation of the TRK is based upon the identities

�n i=1

q²_i 2mi

=−i�

�·

�n i=1

qiri,�·

�n i^�=1

qi^�pi^�

2mi^�

�,

�n i=1

qipi

mi

=i� H_mic⁰ ,

�n i=1

qiri

�, (S2)

for a real unit vector �. We denote the eigenspectrum of H_mic⁰ by {El,|l�}. It comprises a ground state|g�of energyEg. The TRK follows by combining the commutators of Eqs. (S2):

�n

i=1

q_i² 2mi

=�g|�

�·

�n i=1

qiri,� 2 ·�

H_mic⁰ ,

�n i^�=1

qi^�ri^�

��|g� (S3a)

=�

l

(El−Eg)|�·�g|

�n

i=1

qiri|l�|². (S3b)

1

B. Diagonalization of HD and Htm. It is demonstrated that the diagonalization of both the Dicke Hamiltonian H^D for N → ∞ and the Hamiltonian H^tm describing the toy model can be reduced to the diagonalization of special cases ofH^GD, which appears in the context of the generalized no-go theorem. The characteristic equation ofH^GD, that will be derived in Sec. D of these supplementary notes, is solvable for the special cases and yields the diagonal forms ofH^DandH^tm.

Diagonalization of H^D. First, we focus on the Dicke Hamiltonian H^D=ωa^†a+Ω

2

�N

k=1

σ_z^k+ λ

√N

�N

k=1

σ_x^k(a^†+a) +κ(a+a^†)² (S4a)

= ˜ωa^†a+Ω 2

�N k=1

σ_z^k+ λ˜

√N

�N k=1

σ_x^k(a^†+a) +C (S4b)

with ˜ω=√

ω²+ 4κω, ˜λ=�

ω/ωλ, and˜ C= (˜ω−ω)/2. The Hamiltonian (S4b) was diagonal- ized by means of a Holstein-Primakoff transformation in Refs. [2]. We employ here a closely related approach developed in [3], which is more convenient for a generalization beyond the two-level approximation and was also used in the derivation of H^GD. We drop C, set the energy of the atomic ground states to zero, introduce the operators

a^†_k=|ek��gk|, b^†_qj = 1

√N

�N k=1

e^iqjk|ek��gk|, (S5) whereqj= 2π(j/N) andj ∈{0,1, . . . , N−1}, and obtain forN → ∞

H^�D= ˜ωa^†a+Ω

�N k=1

a^†_kak+ ˜λ(b^†_q0+bq0)(a^†+a) (S6a)

= ˜ωa^†a+Ω

N�−1 j=0

b^†_qjbqj + ˜λ(b^†_q0+bq0)(a^†+a). (S6b) In the limit of dilute excitations, that is applicable as long as the excitation energies of the system are finite, the bqj obey bosonic commutation relations. Note that only the j = 0 collective mode couples to the radiation field. Thej�= 0 modes are ‘dark’ and will be omitted in the following. We writebinstead ofbq0 and arrive at

H^��D= ˜ωa^†a+Ωb^†b+ ˜λ(b^†+b)(a^†+a), (S7) which corresponds toH^GD(Eqs. (11) and (S28)) withµ= 1. Later we will derive a character- istic equation for the eigenfrequencies ofH^GD(Eqs. (12) and (S32b)). Forµ= 1 this equation has the solutions

2�²_±=ω²+ 4κω+Ω²±�

(ω²+ 4κω−Ω²)²+ 16λ²ωΩ. (S8)

2

Diagonalization ofH^tm. Now we considerH^tm (Eq. (6)). The coupling of the electromag- netic field and a single harmonic oscillator ‘atom’ is described by

H⁰tm=

�n i=1

(pi−eA)²

2m +mΩ²x²_i

2 . (S9)

Note that we drop the index k numbering the atoms in H^tm for a moment. As usual, we assumeA(r)≈A=A0(a^†+a) in the region where the atoms are located. It is convenient to make the canonical transformation ˜xi=−pi/(mΩ) and ˜pi=mΩxi. This yields

H⁰tm=

�n i=1

�p²_i

2m+mΩ²x²_i 2

�

+eA0Ω(a^†+a)

�n i=1

xi+ne²A²₀

2m (a^†+a)², (S10) where we have written xi and pi instead of ˜xi and ˜pi to keep notation simple. Succes- sively introducing relative and center-of-mass coordinates,{x1, p1, x2, p2}→{x˜1,p˜1, X1, P1}, {X1, P1, x3, p3}→{x˜2,p˜2, X2, P2},. . ., leads to

Htm⁰ =

n−1�

i=1

�p˜²_i 2µi

+µiΩ²x˜²_i 2

� + P˜²

2M +MΩ²X²

2 +eA0Ω(a^†+a)nX+ne²A²₀

2m (a^†+a)². (S11) Here, X =Xn = ¹_n�n

j=1xj andP =Pn =�n

j=1pj are the center-of-mass coordinates of all particles in the harmonic oscillator atom, and M =nm. The relative coordinates are given by ˜xi= (1/i�i

j=1xj)−xi+1and ˜pi= 1/(i+ 1)(�i

j=1pj−ipj+1), andµi=mi/(i+ 1). Note that the electromagnetic field couples only to the center of mass. With this preliminary work done, one can write the full Hamiltonian as

H^tm= ˜ωa^†a+Ω

�N k=1

c^†_kck+ ˜γ

�N k=1

(c^†_k+ck)(a^†+a) +Ω

N(n−1)�

i=1

(b^†_ibi+1

2) +C^�. (S12) The operatorc^†_k excites the center-of-mass degree of freedom of thekth atom, and the gener- ators for theN(n−1) relative coordinates are denoted byb^†_i. We have introduced

γ=eA0

�nΩ

2m, κ=nNe²A²₀

2m , (S13)

and removed theκ-term by means of ˜ω=√

ω²+ 4κω and ˜γ=�

ω/ωγ˜ as before (C^� = (˜ω− ω+NΩ)/2). The first three terms are again a special case ofH^GD withΩl=Ωand ˜λl= ˜γfor alll, andµ=N. Hence, their eigenvalues follow from the roots of the characteristic equation forH^GD (Eq. (S32b)), simplified by the present conditions. They can be explicitly calculated and are the frequencies of the normal modes of field and center-of-mass coordinates. We find N −1 eigenfrequencies being equal to Ω and represent the generators of the corresponding collective excitations also byb^†_i. Only two eigenfrequencies�_± are nondegenerate,

2�²_±=ω²+ 4κω+Ω²±�

(ω²+ 4κω−Ω²)²+ 16Nγ²ωΩ (S14a)

=ω²+ 4κω+Ω²±�

(ω²+ 4κω−Ω²)²+ 16λ²ωΩ. (S14b)

3

We have definedλ=√

Nγ. Since the dipole momentdof the transition from the ground state of an atom to its first excited state is given byd=�n|ex|n−1�=e�

n/2mΩ, we can rewrite λ=A0Ωd√

N andκ=λ²/Ω. Denoting the generators of the �_±-modes bya_±, we arrive at H^tm=�_±(a^†_±a_±+1

2) +

nN−1�

i=1

Ω(b^†_ibi+1 2)−ω

2. (S15)

C. Shift of the resonator frequency due to the pA- and A²-terms. Consider a system ofN mutually noninteracting objects (e.g. atoms) with Hamiltonians

H_mic^k =

nk

�

i=1

(p^k_i)²

2m^k_i +Vint(r^k₁, . . . ,r^k_nk) (S16) coupled to a field mode of frequencyω. It is described by

H^mic=ωa^†a+

�N

k=1

�H_mic^k +H^kpA+H^kA²

�, (S17)

where H_mic^k = �nk

i=1(p^k_i)²/2m^k_i +Vint(r^k₁, . . . ,r^k_nk), H^kpA = −�nk

i=1q_i^kAp^k_i/m^k_i, and H^kA² =

�nk

i=1(q^k_i)²A²/2m^k_i. We denote the eigenspectrum of H_mic^k by {E_m^k_k,|mk�^k} and the pho- ton states by |l� and calculate the shifts δωpA and δωA² of the resonator frequency due to

�H^kpA and �

H^kA² using the first nonzero terms in a perturbation series for the energy of |0, . . . ,0, l�. We take ω � (E_m^k_k −E₀^k) =: Ω^k_mk for mk �= 0 and A(r^k_i) ≈ A. With d^k_mk,0=k�mk|�nk

i=1q_i^kr_i^k|0�^k, we find for thejth terms∆E_pA^j and∆E_A^j2 in the perturbation series for the perturbations�

H^kpA and� H^kA²

∆E_pA¹ = 0 (S18a)

∆E_pA² =−A²₀

�N k=1

�

mk�=0

Ω^k_mk|�·d^k_mk,0|²



 (l+ 1) 1 +_Ω^ωk

mk

+ l

1−Ω^ωk_mk



 (S18b)

≈ −A²₀

�N k=1

�

mk�=0

Ω^k_mk|�·d^k_mk,0|²

�

(2l+ 1)−

� ω Ω^k_mk

�

+ (2l+ 1)

� ω Ω^k_mk

�2�

(S18c)

∆E_A¹2 =A²₀(2l+ 1)

�N k=1

nk

�

i=1

(q_i^k)²

2m^k_i (S18d)

Therefore,

δωpA=−2A²₀

�N k=1

�

mk�=0

Ω^k_mk|�·d^k_mk,0|²�

1 + ω² (Ω^k_mk)²

� (S19a)

δωA² =2A²₀

�N k=1

nk

�

i=1

(q_i^k)²

2m^k_i . (S19b)

4

Figure S1: Situation of the generalized no-go theorem. Atomic spectra are drawn black, eigenenergies of the free electromagnetic field blue. (a) Structure of a low-energy state|E⁰�of the uncoupled system.

ForN→ ∞, the numbers of excited atomsξand of photonsχin|E⁰�are small compared toN,ξ�N andχ�N. (b) Structure of a component ofE¹. The coupling has induced one atomic transition and created or annihilated one photon (shown is an excitation of the second atom and the creation of a photon). The state|E¹�is the sum of all such states. Their amplitude in the eigenstate of the coupled system is smaller than the amplitude of|E0� by a factor∝A0 ∝V^−1/2. In general,|Es�represents the sum over all states obtained from|E⁰�viasatomic transitions andscreations or annihilations of a photon. They contribute to the eigenstate of the coupled system by an amplitude∝A^s0.

ThepA-terms cause a negative and theA²-terms a positive frequency shift. Note that δωpA

andδωA² almost cancel due to the TRK (applied for eachk). The resulting total frequency shiftδω=δωpA+δωA² is suppressed by∼(ω/Ω^k_mk)² as compared withδωpA andδωA².

D. The generalized Dicke HamiltonianHGD. In this section, we derive the Hamil- tonian H^GD (Eq. (11)) from H^mic (in the form of Eq. (10)) for N → ∞ and show how to obtain and evaluate its characteristic equation.

According to our strategy formulated in the main text, we start from low atomic densities and expand the eigenstates|E�ofH^micin powers ofA0∝V^−1/2,

|E� ∝

�∞ s=0

A^s₀|E^s�, (S20)

where|E^s�stands for a sum over components that each describestransitions from|E⁰�both in its atomic and in its photonic part and hence has weight∝A^s₀ (Fig. S1). The corresponding eigenenergies can be written as E ∝�

ss^�A^s+s₀ ^��E^s|H^mic|E^s��. We are interested only in the low-energy spectrum of H^mic. Thus, we assume that the number of atomic excitations ξ =

�

k

�

l>0|�lk|E⁰�|² and the number of photonsχ =�E⁰|a^†a|E⁰� in the uncoupled eigenstates

|E⁰� are small compared to N, ξ � N and χ � N. We now calculate E by dropping all s+s^�≥2 terms and show that for the low-energy spectrum ofH^micall matrix elements that induce transitions in-between excited atomic states are irrelevant. To that end, we write

H^mic= ˜ωa^†a+

�N k=1

�µ l,l^�=0

�Ωlδl,l^�|lk��lk|+iA0

�ω˜

ω(Ωl^�−Ωl)dl,l^�(a^†+a)|lk��l^�_k|�

+C, (S21)

5

with ˜ω=√

ω²+ 4κω andC= (˜ω−ω)/2, and we define H= ˜ωa^†a+

�N k=1

�µ l=1

Ωl|lk��lk| (S22a)

Hcpl= (a^†+a)

�N k=1

�µ l=1

A0

�ω˜ ωΩl�

id0,l|0k��lk|−idl,0|lk��0k|�

(S22b)

∆H= (a^†+a)

�N k=1

�µ l>l^�≥1

A0

�ω˜

ω(Ωl−Ωl^�)�

idl^�,l|l^�_k��lk|−idl,l^�|lk��l^�_k|�

. (S22c)

Accordingly,

E∝ �E|H^mic|E� (S23a)

∝ �E⁰|H|E⁰�+A0�

�E⁰|Hcpl|E¹�+�E¹|Hcpl|E⁰�+�E⁰|∆H|E¹�+�E¹|∆H|E⁰��

. (S23b) Let us now compare the contributions of Hcpl and ∆H to E. The photonic parts of Hcpl

and ∆H are equal and need not be further considered. We write �Hcpl� = �E⁰|Hcpl|E¹�+

�E¹|Hcpl|E⁰�and�∆H�=�E⁰|∆H|E¹�+�E¹|∆H|E⁰�, and we find

�∆H�

�Hcpl�=

�µ

l>l^�≥1(Ωl−Ωl^�)Im�

dl^�,l�N k=1

��E⁰|l^�_k��lk|E¹�+�E¹|l^�_k��lk|E⁰���

�µ

l=1ΩlIm� d0,l�N

k=1

��E⁰|0k��lk|E¹�+�E¹|0k��lk|E⁰��� (S24) Since N → ∞, the number of nonzero terms in the k-sums is decisive. For given l, l^�, the sum over k in the numerator has at most ξ nonzero terms, whereas the sum over k in the denominator has at least N −ξ nonzero terms. Hence, we drop ∆H, which represents the matrix elements of H^mic connecting the excited states of an atom, and keep Hcpl as the relevant coupling part ofH^mic. We reintroduce theκ-term and call the resulting Hamiltonian generalized Dicke Hamiltonian H^GD,

H^GD=ωa^†a+κ(a^†+a)²+

�N k=1

�µ l=1

�Ωl|lk��lk|−ΩlA0(a^†+a)�

idl,0|lk��0k|+ H.c.��

. (S25) It has the same low-energy spectrum asH^mic. Paralleling our treatment ofH^D, we introduce

a^†_k,l=|lk��0k|, b^†_qj,l= 1

√N

�N k=1

e^iqjk|lk��0k|, (S26) whereqj= 2π(j/N) andj∈{0,1, . . . , N−1}as before. With�N

k=1a^†_k,lak,l=�N−1

j=0 b^†_qj,lbqj,l, Eq. (S25) becomes

H^GD =ωa^†a+κ(a^†+a)²+

�µ l=1

�Ωl N�−1

j=0

b^†_qj,lbqj,l−A0Ωl

√N(a^†+a)�

idl,0b^†_q0,l+ H.c.��

. (S27) The operators bqj,l are bosonic in the limit of dilute excitations (ξ � N)[3]. The j > 0 collective modes do not couple to the electromagnetic field. Again, we drop the energy of

6

these ‘dark’ modes, writeblinstead ofbq0,l, defineλ=A0Ωl|d0l|√

N, and remove theκ-term by substituting ω → ω˜ = √

ω²+ 4κω and λl → λ˜l =�

ω/ωλ˜ l and adding C = (˜ω−ω)/2.

This gives

H^GD = ˜ωa^†a+

�µ

l=1

Ωlb^†_lbl+

�µ

l=1

λ˜l(b^†_l +bl)(a^†+a)+C. (S28)

In order to find the eigenfrequencies ofH^GD, we introduce canonical coordinates by means of x= 1

√2˜ω(a^†+a), px=i

�ω˜

2(a^†−a), yl= 1

√2Ωl

(b^†_l +bl), pl=i

�Ωl

2 (b^†_l−bl), (S29) and defineX^T= (x, y1, . . . , yµ),P^T= (px, p1, . . . , pµ), andgl= 2˜λl√

˜

ωΩl. This yields H^GD= P^TP

2 +1

2X^TΩ²X−1 2

�ω+

�µ l=1

Ωl) (S30)

where

Ω²=







˜

ω² g1 · · · gµ

g1 Ω²₁ ... . ..

gµ Ω²_µ





 (S31)

The orthogonal matrixGthat diagonalizesΩ² induces a point transformation to the normal modes ˜X=GX and ˜P=GP. The eigenvalues �²_i of Ω² are the squared eigenfrequencies of the system. They solve the characteristic equation

0=��^µ

l^�=1

(Ω²_l�−�²)��

(˜ω²−�²)−

�µ l=1

g²_l Ω²_l −�²

� (S32a)

=��^µ

l^�=1

(Ω²_l�−�²)��

(˜ω²−�²)−4˜ω

�µ

l=1

Ωl˜λ²_l Ω²_l −�²

�. (S32b)

None of them can be zero since this would imply ω

4N A²₀ =

�µ

l=1

Ωl|d0l|²−

�n

i=1

q²_i 2mi

. (S33)

We have usedω˜ =√

ω²+ 4κω,˜λl=�

ω/˜ωλl,λl=A0Ωl|d0l|√

N, andκ=N A²₀�n

i=1q²_i/2mi. However, the left side of Eq. (S33) is positive, whereas its right side is negative according to the TRK forH_mic⁰ (Eqs. (S3) or Eq. (9) of the main text).

7