Oct 3, 2017 - the no-go theorem for atomic systems. We also show some other configurations where the absence of SRPTs cannot be confirmed...

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arXiv:1703.03533v2 [quant-ph] 3 Oct 2017

Several superconducting circuit conﬁgurations are examined on the existence of super-radiant phase transitions (SRPTs) in thermal equilibrium. For some conﬁgurations consisting of artiﬁcial atoms, whose circuit diagrams are however not speciﬁed, and an LC resonator or a transmission line, we conﬁrm the absence of SRPTs in the thermal equilibrium following the similar analysis as the no-go theorem for atomic systems. We also show some other conﬁgurations where the absence of SRPTs cannot be conﬁrmed. I.

INTRODUCTION

A super-radiant phase transition (SRPT), i.e., a spontaneous appearance of (static) coherent amplitude of transverse electromagnetic fields in the thermal equilibrium due to the light-matter interaction, was first proposed theoretically around 1970 [1–3]. It is different from the so-called super-radiance or super-fluorescence [4], i.e., a collective spontaneous emission from many atoms. It is also different from the exciton super-radiance (one-photon super-radiance) [5], i.e., an emission-rate enhancement by spatial broadening of wave-function of an excitation. In contrast to these non-equilibrium phenomena, SRPTs are phase transitions in the thermal equilibrium. Since the first proposals [1–3], its absence (no-go theorem) in atomic systems has been discussed based on the so-called A2 term [6–9], P 2 term [10, 11], gauge-invariance [12–14], and minimal-coupling Hamiltonian [14, 15]. Influences of the longitudinal dipole-dipole interaction have also been discussed recently [16–20]. SRPTs require an ultra-strong light-matter interaction [21, 22], i.e., the interaction strength (vacuum Rabi splitting or absorption/emission rate in single-photon level) must be comparable to or larger than frequencies of electromagnetic waves and of transitions in matters. In recent years, the ultra-strong interactions have been realized experimentally in a variety of systems [23–37]. The presence of the so-called vacuum photons [21] and the Schrödinger-cat-like state [21, 62] are expected in the ground state under the ultra-strong interaction, and recent experiments are indicating a signature of them [37]. However, the coherent amplitude of the electromagnetic fields (expectation value of annihilation operator of a photon) does not appear even in such a ground state, but it is obtained only after a SRPT. Currently, SRPTs are not yet observed experimentally in the thermal equilibrium, while non-equilibrium analogues were proposed theoretically [38] and observed experimentally in cold atoms driven by laser light [39, 40]. Instead of the atomic systems [23–26, 30–32, 35], which are basically described by the minimal-coupling Hamil-

∗

E-mail: [email protected]

tonian [41], the possibility of the thermal-equilibrium SRPTs in superconducting circuits [27–29, 36, 37] has been discussed [42–46]. The existence of a SRPT was proposed for a superconducting circuit with capacitive coupling between (two-level) artificial atoms and a resonator by estimating the A2 term to be relatively small [42]. However, its estimation was doubted through a standard description of superconducting circuit systems [43]. After that, the existence of a SRPT was proposed again for superconducting circuit with three-level artificial atoms as a result of the modification of the sum rule (and then of the A2 term) [44]. In these three works, their Hamiltonians were guessed for standard circuit configurations but without specifying circuit diagrams in detail, although the derivation of exact Hamiltonians is crucial for discussing the possibility of SRPTs. Recently, the absence of SRPTs was confirmed for a superconducting circuit diagram with capacitive coupling between an LC resonator and charge qubits by deriving its Hamiltonian in the standard quantization procedure [45]. Almost at the same time, for a circuit diagram consisting of an LC resonator coupled with Josephson junctions through inductors, the existence of a SRPT was proposed also in the standard quantization procedure [46]. No doubt is raised until now. A remarkable feature of SRPTs is a decrease of the zero-point energy in the whole system due to the lightmatter interaction [46–49]. Chemical reactions [50] and work functions [51] were reported to be modified by the ultra-strong interaction with the vacuum electromagnetic fields. The free energy, i.e., thermodynamic behaviors at finite temperatures, should also be modified as suggested in Ref. [52], while its experimental and theoretical evaluations are still under debate [53, 54]. In the superconducting circuit proposed in Ref. [46], an external magnetic flux bias or π junctions [55] are inevitable for realizing the SRPT in the thermal-equilibrium. The external magnetic flux increases the zero-point energy of the circuit. While the zero-point energy is certainly decreased by the increase in the photon-atom interaction strength, it cannot be lower than the zero-point energy in the absence of the external magnetic flux. It is still open to dispute whether there is a lower bound of the zero-point energy in superconducting circuits. If there exists a superconducting circuit showing a SRPT with-

2 out the external magnetic flux or π junctions, the zeropoint energy should be purely decreased by increasing the strength of the interaction with the transverse electromagnetic fields, and the thermodynamic properties, e.g., the superconducting transition temperature, of the circuit might be modified. In order to find such a circuit structure, in this paper, we show some hopeless circuit configurations where SRPTs are absent even in the presence of an external magnetic flux or π junctions. There is a large number of degrees of freedom in designing circuit structures, and there is not a standard Hamiltonian corresponding to the minimal-coupling one for the atomic systems. In order to rule out a wide range of circuit structures, we treat artificial atoms as a black box, i.e., we do not specify their circuit diagrams. We consider some capacitive- and inductive-coupling configurations between the black box and an LC resonator or a transmission line. The absence of SRPTs in those configurations are confirmed following the similar analysis as the no-go theorem for the atomic systems [14, 15] by deriving Hamiltonians in the flux- [56] or charge-based [57] standard quantization procedure. In the analyses based on the A2 term [6–9, 42–44], on the P 2 term [10, 11, 45], or on the softening of transition frequency [42, 46–48], we must specify circuit diagrams of whole systems in detail. In contrast, in this paper, the artificial atoms are treated as a black box following the no-go theorem [14, 15], but we need to specify only the connection between the black box and a resonator. We also show some other circuit configurations where the absence of SRPTs in the thermal equilibrium cannot be confirmed. The circuit structure proposed in Ref. [46] is certainly included in these configurations. While our analysis does not depend on whether an external magnetic flux or π junctions are absent or not, it does not rule out the possibility of SRPTs without the external magnetic flux and π junctions. This paper is organized as follows. We first review the no-go theorem for atomic systems in Sec. II. Following the similar analysis, in Sec. III, we show the absence of SRPTs in three circuit configurations by deriving Hamiltonians without specifying circuit diagrams of artificial atoms. In Sec. IV, we show some other configurations where the absence of SRPTs cannot be confirmed. The discussion is summarized in Sec. V.

The minimal-coupling Hamiltonian is expressed as [41] ( ) Z ˆ⊥ (r)2 ˆ 2 E ε B(r) 0 ˆ min = dr H + 2 2µ0 +

N X ˆ rj )]2 [pˆj − ej A(ˆ + Vˆ ({rˆj }). 2mj j=1

(1)

Here, the second last term is the kinetic energy of charged particles. N is the number of the particles. rˆj and pˆj are operators of a position and a momentum, respectively, of the j-th particle with a mass mj and a charge ej . They satisfy [ˆ rj , pˆj ′ ] = δj,j ′ i~1. The last term Vˆ represents the Coulomb interaction between the charged particles, and it depends only on the particles’ positions {rˆj }. The first and second terms represent the energies of the transverse ˆ⊥ (r) = −Π(r)/ε ˆ electric field E 0 and the magnetic flux ˆ ˆ ˆ density B(r) = ∇ × A(r), respectively. Here, A(r) is ˆ the vector potential and Π(r) is its conjugate momentum satisfying h i ˆ ˆ ′ ) = i~δ⊥ (r − r ′ ), A(r), Π(r (2) where δ⊥ (r −r ′ ) is the transverse delta function [41]. We rewrite these fields by annihilation and creation operators as r M X ~ ˆ A(r) = ek fk (r) a ˆk + a ˆ†k , (3a) 2ε0 ωk k=1 r M X ~ε0 ωk ˆ fk (r) a ˆk − a ˆ†k . (3b) ek i Π(r) =− 2 k=1

Here, a ˆk annihilates a photon in the k-th mode of the electromagnetic wave with a frequency of ωk . fk (r) is the wavefunction of the k-th mode, ek is the unit vector in its polarization direction, and ε0 is the vacuum permittivity. M is the number of modes. The minimal-coupling Hamiltonian in Eq. (1) is rewritten as ˆ min = H

M X

k=1

+

1 ˆ†k a ˆk + ~ωk a 2

N X ˆ rj )]2 [pˆj − ej A(ˆ + Vˆ ({rˆj }). 2m j j=1

(4)

For simplicity, as discussed in Ref. [14], we apply the long-wavelength approximation (electric-dipole approximation), i.e., the vector potential is rewritten as II.

NO-GO THEOREM FOR ATOMIC SYSTEMS

In this section, we review the no-go theorem of SRPTs in atomic systems described by the minimal-coupling Hamiltonian. It was mainly discussed in Refs. [14, 15] based on the c-number substitution [3, 15, 58, 59], which is also used in the semi-classical analysis of Ref. [46].

ˆ rj ) ≃ A(R ˆ j ), A(ˆ

(5)

where Rj is the rough position of the j-th particle (e.g., position of lattice site). The long-wavelength approximation is justified when the amplitude of the vector potential varies only slightly by the distance rˆj − Rj . In other words, rˆj −Rj is much shorter than the wavelength of the

3 electromagnetic wave in the frequency range of interest. A more general discussion beyond the long-wavelength approximation is shown in Ref. [15]. Expanding the kinetic energy of the charged particles P ˆ rj ) and ˆj ·A(ˆ in Eq. (1) or Eq. (4), we get − N j=1 (ej /mj )p PN 2 ˆ e A(ˆ rj )2 /(2mj ). The former leads to the lightj=1

matter interaction term, and the latter leads to the A2 term [6–9]. The absence of SRPTs by the presence of the A2 term can be confirmed when we specify the atomic systems of interest, especially the shape of Vˆ ({rˆj }). In contrast, the following no-go theorem shows the absence of SRPTs generally in the minimal-coupling Hamiltonian, i.e., without specifying the systems in detail. The thermodynamic properties at a finite temperature T is analyzed by the partition function for β = 1/(kB T ) as i h ˆ (6) Z(T ) = Tr e−β Hmin . As discussed in Refs. [3, 15, 58, 59], we replace the trace over the photonic variables by the integral over the coherent state as Z Y 2 ! h i d αk ˆ′ ¯ Z(T ) = Tr e−β Hmin , (7) π k

{ˆ ak , a ˆ†k }

where the photon operators and vector potential ˆ j ) are replaced by c-numbers as A(R ˆ′ = H min

M X

k=1

+

1 2 ~ωk |αk | + 2

N X [pˆj − ej A(Rj )]2 + Vˆ ({rˆj }). 2m j j=1

(8)

Here, αk ∈ C is an amplitude of a coherent state |αk ik in the k-th mode giving a ˆk |αk ik = αk |αk ik . The c-number vector potential is expressed as r M X ~ A(r) = ek fk (r) (αk + α∗k ) . (9) 2ε0 ωk k=1

The replacement (approximation) performed in Eq. (7) is called the c-number substitution [15, 59], and the analysis based on it is called the semi-classical analysis in Ref. [46], since the photonic operators are treated as the c-numbers. For justifying this c-number substitution, we must consider the thermodynamic limit N → ∞. Further, in the early study by Wang and Hioe [3], they note that this substitution is justified on the following two assumptions: Assumption as N → ∞ of the field operator √ 1: The† limits √ a ˆ/ N and a ˆ / N exist. Assumption 2: The order of the double limit in the exPR ˆ r /r! ponential series limN →∞ limR→∞ r=1 (−β H) can be interchanged.

√ The first assumption implies that αk / N should be of a finite value after the SRPTs in the thermodynamic limit N → ∞. On the other hand, it is hard to check the second assumption for arbitrary systems. Instead, we follow the justification discussed in Ref. [58]. The exact partition function Z(T ) in Eq. (6) and the approximated ¯ ) in Eq. (7) satisfy the following relation [58]: one Z(T ¯ ) ≤ Z(T ) ≤ exp Z(T

M 1 X ~ωk kB T k=1

!

¯ ). Z(T

(10)

From this, the free energy −(kB T /N ) ln Z(T ) per atom satisfies −

M 1 X kB T ¯ ) ln Z(T ~ωk − N N k=1

≤−

kB T kB T ¯ ). (11) ln Z(T ) ≤ − ln Z(T N N

Therefore, in the thermodynamic limit N → ∞, Z(T ) is ¯ ), if systems of interest satisfy well approximated by Z(T Assumption A: lim

N →∞

M X ~ωk

k=1

kB T ¯ ≪ ln Z(T ) . N N

This condition can be checked when we specify atomic systems of interest. It is satisfied for ensemble of twolevel atoms [58], i.e., in the Dicke Hamiltonian. For superconducting circuits, it was checked numerically for the circuit proposed in Ref. [46]. In this paper, we implicitly consider that the systems of interest satisfy Assumptions 1 and 2, or A in the thermodynamic limit N → ∞, while we do not specify the systems in detail. In other words, we cannot discuss the absence of SRPTs in systems that do not satisfy these assumptions, since we cannot rewrite the partition function as Eq. (7) and the following analysis is not justified. The no-go theorem [14] for atomic systems in the long-wavelength approximation is discussed based on the partition function in Eq. (7) described by the minimalcoupling Hamiltonian in Eq. (8) under the c-number substitution. If there exists a state |ψ({αk })i that minimizes ˆ ′ |ψ({αk })i for a non-zero amplithe energy hψ({αk })|H min tude αk 6= 0, the transverse electromagnetic fields get an amplitude spontaneously in the ground state (and also in the thermal equilibrium for T > 0), i.e., the system shows a SRPT. However, the absence of such a super-radiant ground state is confirmed as seen in the following. Here, we introduce a unitary operator N X i ˆc ≡ exp U ej rˆj · A(Rj ) . (12) ~ j=1 Using this, we get

ˆc = pˆj + ej A(Rj ). ˆ † pˆj U U c

(13)

4 Then, since the Coulomb interaction Vˆ does not depend on the momentum {pˆj } of the charged particles, we get M X

1 ˆ atom , +H ~ωk |αk |2 + 2 k=1 (14) ˆ atom is the Hamiltonian of the charged particles where H without the interaction with the transverse electromagnetic fields as ′ ′′ ˆ min ˆc = ˆ min ˆc† H H ≡U U

ˆ atom ≡ H

N X pˆj 2 + Vˆ ({rˆj }). 2mj j=1

(15)

ˆc is a unitary operator, the partition function in Since U Eq. (7) can be rewritten as Z Y 2 ! h i d αk ˆ ′′ ¯ )= Z(T (16) Tr e−β Hmin . π k

Then, the problem is reduced to the minimization of ˆ ′′ |ψ({αk })i for trial state |ψ({αk })i. Since hψ({αk })|H min ˆ Hatom in Eq. (14) is simply the Hamiltonian of the charged particles, the minimum energy is obtained for the following state: ′′ |ψmin,g i = |ψg iatom ⊗ |{αk = 0}iem,

(17)

ˆ atom and |{αk = where |ψg iatom is the ground state of H 0}iem represents a classical state with zero amplitude for all the photonic modes. In this way, the photonic modes do not spontaneously get an amplitude in the ground state (and also in thermal equilibrium). This is the basic logic of the no-go theorem of SRPTs in atomic systems discussed in Refs. [14, 15]. On the other hand, from the minimal-coupling Hamilˆ min in Eq. (4) without the c-number substitutonian H ˆ dip of the length form tion, we can get the Hamiltonian H ˆ min called the velocity form. [16–20, 41], in contrast to H Recovering the vector potential as an operator in the unitary operator as N X i ˆ j ) , ˆ = exp ej rˆj · A(R (18) U ~ j=1 the Hamiltonian of the length form is obtained in the long-wavelength approximation as

ˆ ˆ dip = U ˆ †H ˆ min U (19) H ) ( Z ˆ 2 ˆ ⊥ (r) − Pˆ⊥ (r)]2 B(r) [D ˆ atom +H + = dr 2ε0 2µ0 (20) Z M X 1 1 ˆ ⊥ (r) − dr Pˆ⊥ (r) · D ˆ†k a ˆk + = ~ωk a 2 ε0 k=1 Z 1 ˆ atom . + dr Pˆ⊥ (r)2 + H (21) 2ε0

Here, Pˆ⊥ (r) is the transverse component of the electric P polarization P (r) = j ej rˆj δ(r − rˆj ), while a more general definition is required beyond the long-wavelength approximation (Power-Zienau-Woolley transformation) [17, 19, 41]. The last term in the first line of Eq. (21) represents the light-matter interaction mediated by Pˆ⊥ (r) and the transverse component of the electric displaceˆ ⊥ (r), which corresponds to the conjugate ment field D ˆ ⊥ (r) = −Π(r) ˆ momentum of the vector potential as D in the length form. The second last term in Eq. (21) is called the P 2 term, by which the absence of SRPTs can also been confirmed [10, 11] in the similar manner as the A2 term. ′′ ˆ ′′ is not the exact The ground state |ψmin,g i of H min ˆ dip . However, the absence of ground state |ψdip,g i of H SRPTs itself can be confirmed as discussed above if systems of interest satisfy Assumptions 1 and 2, or A in the thermodynamic limit. When the transverse electric polarization P⊥ (r) = hψg |Pˆ⊥ (r)|ψg iatom gets an amplitude spontaneously in the ground state |ψg iatom of the charged particles, the electric displacement field can be induced as D⊥ (r) = P⊥ (r), while the electric field is basically zero E⊥ = (D⊥ −P⊥ )/ε0 = 0, by simply considering the minimization of the first term in Eq. (20) as the classical analysis in Ref. [46]. Even though the photonic amplitude ˆ can get an amplitude as hψdip,g |Π(r)|ψ dip,g i ≈ −D⊥ (r) ˆ in the ground state of Hdip , we do not call it a SRPT in this paper, because the appearance of the photonic amplitude originates from the system of charged particles ˆ atom not from the light-matter interaction. H While the possibility of SRPTs in atomic systems is still under debate especially beyond the long-wavelength approximation [15, 19, 20], the above logic is basically valid if the c-number substitution performed in Eq. (7) is justified, i.e., if systems of interest satisfy Assumptions 1 and 2, or A in the thermodynamic limit N → ∞. Following this semi-classical analysis, we examine the possibility of SRPTs in some superconducting circuit configurations in the following sections.

III.

CIRCUIT CONFIGURATIONS WHERE SRPTS ARE ABSENT

In this section, we show three superconducting circuit configurations where the absence of SRPTs can be confirmed by the semi-classical analysis explained in the previous section. Once we get an exact Hamiltonian of a circuit, we can examine the possibility of SRPTs following the semi-classical analysis or in other approaches [1, 2, 46–48]. However, in order to discuss a wide range of circuit structures, Hamiltonians of general forms are preferred, such as the minimal-coupling one for atomic systems. Figure 1 shows a map of circuit configurations which we will discuss in this paper and the circuit structure proposed in Ref. [46]. We discuss the three circuit con-

5 The Lagrangian Lblack represents the elements in the black box, and it is described by the flux ψ, its time˙ and others inside the black box. The conderivative ψ, jugate momenta (charges) of φ and ψ are derived, respectively, as ∂L1 ˙ = Cr φ, ∂ φ˙ ∂L1 ∂Lblack ρ≡ = . ˙ ∂ψ ∂ ψ˙ q≡

FIG. 1. Map of circuit conﬁgurations discussed in this paper and the circuit proposed in Ref. [46]. SRPTs are absent in the conﬁgurations depicted in Figs. 2–4, if systems of interest satisfy Assumptions 1 and 2, or A. The absence of SRPTs cannot be conﬁrmed in the conﬁgurations depicted in Figs. 5 and 6. The circuit proposed in Ref. [46] [depicted in Fig. 5(c)] shows a SRPT and is included in the conﬁgurations of Figs. 5(a) and (b).

(23b)

Then, we get a quantized Hamiltonian as 2 ˆ ˆ2 ˆ ρˆ; . . .), ˆ black (ψ, ˆ 1 = qˆ + (φ − ψ) + H H 2Cr 2Lr

(24)

ˆ black is the Hamiltonian of the black box derived where H from Lblack . The operators satisfy the following commutation relations: ˆ qˆ] = i~, [φ, ˆ ρˆ] = i~, [ψ,

FIG. 2. (a) An LC resonator coupled inductively with a black box. (b,c) Examples of circuits with artiﬁcial atoms. This circuit conﬁguration does not show SRPTs by the coupling between the black box and the LC resonator.

(23a)

(25a) (25b)

and the other combinations are commutable. We consider the flux φˆ and the charge qˆ of the LC resonator as canonical variables of a photonic mode. Introducing the annihilation operator a ˆ of a photon and an impedance p Zr = Lr /Cr , they are described as r ~Zr ˆ (ˆ a+a ˆ† ), (26a) φ= 2 r ~ (ˆ a−a ˆ† ). (26b) qˆ = −i 2Zr The resonance frequency is expressed as

figurations depicted in Figs. 2–4 with treating artificial atoms as a black box (without specifying their circuit diagrams). The absence of SRPTs will be confirmed in an inductive-coupling configuration with an LC resonator in Sec. III A (Fig. 2), capacitive-coupling one with an LC resonator in Sec. III B (Fig. 3), and capacitive-coupling one with a transmission line in Sec. III C (Fig. 4). The two configurations depicted in Figs. 5 and 6, where the absence of SRPTs is not confirmed, will be discussed in the next section.

A.

Inductive coupling with an LC resonator

We first consider the circuit configuration depicted in Fig. 2(a) consisting of a black box and an LC resonator with inductance Lr and capacitance Cr . Following the flux-based quantization procedure in Ref. [56], we define two node fluxes φ, ψ and the ground as Fig. 2(a). A Lagrangian of this circuit is written as Cr ˙ 2 (φ − ψ)2 ˙ . . .). L1 = φ − + Lblack (ψ, ψ; 2 2Lr

(22)

1 ωr = √ . Lr Cr

(27)

In Eq. (24), the coupling between the LC resonator and the black box is described by the second term, the inductive energy at Lr . This expression corresponds to ˆ dip of the length form in Eq. (20). the Hamiltonian H ˆ r, Expanding the second term, we get φˆ2 /(2Lr ), −φˆψ/L 2 and ψˆ /(2Lr ) corresponding to the photonic flux energy, the interaction term, and the P 2 term, respectively. The no-go theorem for atomic systems starts from the minimal-coupling Hamiltonian, Eq. (1), where the lightmatter coupling is described by the kinetic term of the charged particles. In order to describe the coupling by a part of the black box as similar as the minimal-coupling Hamiltonian, we transform Eq. (24) by a unitary operator ˆ ˆ1 = e−iˆqψ/~ U .

(28)

Using this operator, we get ˆ ˆ1 = φˆ + ψ, ˆ † φˆU U 1 ˆ † ρˆU ˆ1 = ρˆ − qˆ, U 1

(29a) (29b)

6 and the Hamiltonian is transformed to ˆ1 ˆ1U ˆ †H ˆ 1′ ≡ U H 1

φˆ2 qˆ2 ˆ ρˆ − qˆ; . . .) ˆ black (ψ, + +H 2Cr 2Lr ˆ ρˆ − qˆ; . . .). ˆ black (ψ, = ~ωr (ˆ a† a ˆ + 1/2) + H =

(30a) (30b) (30c)

This Hamiltonian has a similar form as the minimalcoupling Hamiltonian in Eq. (4). Specifying the black box and expanding the capacitive term depending on ρˆ − qˆ, such as (ˆ ρ − qˆ)2 /(2C) for a capacitance C, we get an interaction term −ρˆqˆ/C and the A2 term qˆ2 /(2C). However, the following discussion does not depend on the detail of the black box. Here, we suppose that there are many artificial atoms in the black box, for example, as Figs. 2(b) and 2(c), and the circuit satisfies Assumptions 1 and 2, or A. In the thermodynamic limit (infinite number of artificial atoms; N → ∞), the partition function is written approximately as Z 2 i h d α ˆ′ ¯ Z(T ) = (31) Tr e−β H1′ , π where a ˆ is replaced by a c-number α as

ˆ ρˆ − q; . . .), ˆ 1′ ′ = ~ωr (|α|2 + 1/2) + H ˆ black (ψ, H and the operator qˆ is also replaced by r ~ q = −i (α − α∗ ). 2Zr

(32)

(33)

FIG. 3. (a) An LC resonator coupled capacitively with a black box. (b) Example of circuits with artiﬁcial atoms, which was already discussed in Ref. [45]. This circuit conﬁguration does not show SRPTs by the coupling between the black box and the LC resonator.

B.

Capacitive coupling with an LC resonator

Next, we consider the circuit configuration depicted in Fig. 3(a). An LC resonator couples with a black box through capacitances inside the black box. Following the charge-based quantization procedure in Ref. [57], a Lagrangian is obtained as follows. We define the ground, voltage V , current I, charges q and {ρj } for j = 1, 2, . . . , N as in Fig. 3(a). The voltage V and charge q at capacitance Cr are related as V =

Here, by substituting the c-number also to the unitary operator as ˆ ˆ1c = e−iqψ/~ U ,

the partition function is rewritten as Z 2 h i d α ˆ ′′ ¯ Tr e−β H1′ , Z(T ) = π

(34)

(35)

where 2 ˆ ˆ ˆ; . . .). ˆ′ ′ U ˆ† ˆ ′′′ ≡ U ˆ1c H H A 1c = ~ωr (|α| + 1/2) + Hblack (ψ, ρ 1 (36) In this way, the problem is reduced to the similar one discussed around Eq. (14) for atomic systems. Then, SRPTs originating from the coupling between the LC resonator and the black box are absent in the circuit configuration of Fig. 2(a), if the circuits satisfy Assumptions 1 and 2, or A. In Figs. 2(b) and 2(c), we suppose many flux qubits [60], which basically require an external magnetic flux in each loop consisting of three Josephson junctions for reaching the ideal two-level systems. Even in the presence of the external magnetic fluxes in these loops, the SRPTs are absent, because the Lagrangian is still expressed as Eq. (22), while some phase transitions originating from the black box (not from the coupling with LC resonator) can exist. Of course, the SRPTs are absent also when the external magnetic fluxes are completely absent.

q . Cr

(37)

The current I through inductance Lr , charges {ρj } at coupling capacitances in the black box, and q at Cr are related as I = −q˙ −

N X

ρ˙j .

(38)

j=1

Further, the voltage V and current I are related as ˙ V = Lr I.

(39)

Then, we get an equation of motion as q¨ +

N X j=1

ρ¨j =

q . Lr Cr

(40)

There are some other equations of motion describing the inside of the black box. A Lagrangian giving these equations is in general represented as 2 N X q2 Lr ρ˙ j − + Lblack ({ρj }, {ρ˙ j }; . . .). L2 = q˙ + 2 2Cr j=1

(41)

7 The conjugate momenta are derived as N X ∂L2 ρ˙ j , = Lr q˙ + φ≡ ∂ q˙ j=1 N X ∂L2 ∂Lblack ψj ≡ ρ˙ j + = Lr q˙ + . ∂ ρ˙ j ∂ ρ˙ j j=1

(42a)

(42b)

They satisfy

ˆ = i~, [ˆ q , φ] [ˆ ρj , ψˆj ] = i~,

(43a) (43b)

and other combinations are commutable. The Hamiltonian is obtained as H2 =

φ2 q2 + + Hblack ({ρj }, {ρ˙ j }; . . .), 2Lr 2Cr

(44)

FIG. 4. A transmission line coupled capacitively with a long black box. This circuit conﬁguration does not show SRPTs by the coupling between the black box and the transmission line.

Using a unitary operator

where the Hamiltonian of the black box is defined as N X

∂Lblack ρ˙ j Hblack ({ρj }, {ρ˙ j }; . . .) ≡ ∂ ρ˙ j j=1 − Lblack ({ρj }, {ρ˙ j }; . . .). (45) Let us rewrite this in terms of {ρj }, {ψj }, . . . . From Eqs. (42), we get ∂Lblack = ψj − φ. ∂ ρ˙ j

(46)

In the absence of the LC resonator, we simply get ∂Lblack /∂ ρ˙ j = ψj , and the Hamiltonian is represented as Hblack ({ρj }, {ψj }; . . .). Then, in the presence of the LC resonator, ψj is replaced by ψj − φ in Hblack , and the Hamiltonian in Eq. (44) is rewritten in terms of {ρj }, {ψj }, . . . and in the quantized form as 2 ˆ2 ˆ . . .). ˆ 2 = φ + qˆ + H ˆ black ({ρˆj }, {ψˆj − φ}; H 2Lr 2Cr

(47)

In this case, expanding an inductive energy depending on ψˆj − φˆ in the black box, we get an interaction term and the A2 term, when we specify the black box in detail. In the same manner as the previous subsection, we rewrite φˆ and qˆ by annihilation operator a ˆ as r ~Zr (ˆ a−a ˆ† ), (48a) φˆ = −i 2 r ~ (ˆ a+a ˆ† ). (48b) qˆ = 2Zr Then, we replace a ˆ by a c-number α as ˆ ′ = ~ωr (|α|2 + 1/2) + H ˆ black (ˆ H ρ, ψˆ − φ; . . .), 2 where φˆ is also replaced by r ~Zr (α − α∗ ). φ = −i 2

(49)

we get

N X

ˆ2c = exp i φ ρˆj , U ~ j=1 ˆ2c = ψˆj + φ, ˆ † ψˆj U U 2c

(51)

(52)

and the problem is reduced to the minimization of ˆ′ ˆ ˆ† H ˆ 2′′ = U H 2c 2 U2c

ˆ black ({ρˆj }, {ψˆj }; . . .). = ~ωr (|α|2 + 1/2) + H

(53)

In the same manner as discussed above, the SRPTs due to the coupling between the LC resonator and the black box are absent in the circuit configuration of Fig. 3(a), if systems of interest satisfy Assumptions 1 and 2, or A. Then, for example, the SRPTs are absent in the circuit of Fig. 3(b), where the charge qubits couple capacitively with an LC resonator as already discussed in Ref. [45]. C.

Capacitive coupling with a transmission line

We next consider a transmission line coupled capacitively with a long black box as depicted in Fig. 4. We can derive its Hamiltonian in the similar manner as the previous subsection. In Fig. 4, Ct and Lt are, respectively, capacitance and inductance per unit length, and ∆x is a short length for the discrete description of the transmission line. We define voltage Vj , current Ij , and charges qj and ρj as in Fig. 4. The voltage Vj and the charge qj at the j-th capacitance Ct ∆x is related as Vj =

qj . Ct ∆x

(54)

The current Ij are related with the charges qj and ρj as (50)

Ij = Ij−1 − q˙j − ρ˙ j .

(55)

8 Further, the voltage Vj and the current Ij are related as Vj+1 − Vj = −Lt ∆xI˙j .

(56)

From these relations, we get a difference equation as q¨j + ρ¨j =

qj+1 + qj−1 − 2qj . Lt Ct (∆x)2

(57)

This equation can be obtained by the following Lagrangian: X (qj+1 − qj )2 Lt ∆x (q˙j + ρ˙ j )2 − L3 = 2Ct ∆x j + Lblack ({ρj }, {ρ˙ j }; . . .).

(58)

The conjugate momenta are derived as ∂L3 = Lt ∆x(q˙j + ρ˙ j ), ∂ q˙j ∂Lblack ∂L3 = Lt ∆x(q˙j + ρ˙ j ) + . ψj ≡ ∂ ρ˙ j ∂ ρ˙ j φj ≡

(59a) (59b)

Then, in the same manner as the previous subsection, the Hamiltonian is derived as X Lt ∆x (qj+1 − qj )2 2 H3 = (q˙j + ρ˙ j ) + 2 2Ct ∆x j + Hblack ({ρj }, {ρ˙ j }; . . .), " # 2 X ˆj 2 φ (ˆ q − q ˆ ) j+1 j ˆ3 = H + 2Lt ∆x 2Ct ∆x j

ˆ black ({ρˆj }, {ψˆj − φˆj }; . . .). +H

(60)

(61)

The first two terms are simply the Hamiltonian of the transmission line, in which a photon √ (microwave) propagates with a speed of v = 1/ Lt Ct in the onedimensional system. The boundary conditions of the transmission line do not affect the possibility of SRPTs in the semi-classical analysis relying on the c-number substitution. In order to justify the c-number substitution performed in Eq. (7), let us discuss when the systems with the transmission line satisfy Assumption A. Here, we consider that the transmission line has a length of ℓ. The frequency of the photonic mode is ωk = k(πv/ℓ) for k = 1, 2, . . .. Considering the minimum wavelength λmin where the electromagnetic wave interacts sufficiently with the artificial atoms and is confined sufficiently in the one-dimensional transmission line, the effective number of the photonic modes is determined as M = ℓ/λmin . The free energy per atom is in the same order as the characteristic frequency ωa of the atomic transition, which gives a wavelength of λa = 2πv/ωa . Instead of the limit N → ∞, we consider the limit of the number of atoms in the length of λa as n = N λa /ℓ → ∞. Then, Assumption A is rewritten as 1 ~πv M (M + 1) ≪ ~ωa , N ℓ 2 (λa /λmin )2 ≪ n. 4

FIG. 5. An LC resonator coupled with a black box, where the absence of SRPTs cannot be conﬁrmed by the analysis in this paper. It is because we could not derive a Hamiltonian for (a). For (b) and (d), their Hamiltonians can be derived, but they cannot be transformed as the minimal-coupling Hamiltonian. (c) is the circuit proposed in Ref. [46].

(62) (63)

In this way, the c-number substitution is justified when the number n of atoms in λa is much lager than (λa /λmin )2 . In the same manner as the previous subsections, when the c-number substitution is justified under the above condition, the SRPTs due to the coupling between the transmission line and the black box are absent in the circuit configuration of Fig. 4. IV.

CIRCUIT CONFIGURATIONS WHERE SRPTS CAN EXIST

Next, we show some circuit configurations where the absence of SRPTs cannot be confirmed by the analysis in this paper. In Sec. IV A (Fig. 5), we discuss another inductive-coupling configuration with an LC resonator. In Sec. IV B (Fig. 6), an inductive-coupling configuration with a transmission line is discussed. As shown in Fig. 1, these configurations include also the circuit structures that do not show SRPTs, while the configuration of Fig. 5 includes the circuit proposed in Ref. [46] that shows a SRPT. A.

Another inductive coupling with an LC resonator

Let us first consider the circuit configuration depicted in Fig. 5(a), which is generalized from the capacitivecoupling configuration in Fig. 3(a). We could not derive a Hamiltonian of this configuration in the flux- [56] or charge-based [57] quantization procedure. While other quantization procedures [61] might give a Hamiltonian, it in fact includes the circuit of Fig. 5(c) proposed in Ref. [46], which shows a SRPT in the presence of an external magnetic flux or π junctions. Then, even if we get a Hamiltonian of the circuit configuration in Fig. 5(a),

9 the absence of SRPTs would not be confirmed by the semi-classical analysis. For example, let us consider the configuration in Fig. 5(b), which is less general than Fig. 5(a) but includes the circuit of Fig. 5(c) proposed in Ref. [46]. Following the flux-based procedure, we define the ground and node fluxes φ and {ψj } as in Fig. 5(b). In the same manner as Sec. III A, a Hamiltonian can be derived as N X (φˆ − ψˆj )2 qˆ2 φˆ2 ˆ H4 = + + 2Cr 2Lr j=1 2Lc

ˆ black ({ψˆj }, {ρˆj }; . . .). +H

(64)

Let us derive the black-box Hamiltonian and roughly check the existence of the SRPT for the circuit proposed in Ref. [46] by specifying the detail inside the black box as Fig. 5(c). Each Lc is connected with a Josephson junction with Josephson energy EJ and capacitance CJ . A half of flux quantum Φ0 = h/(2e) is applied to a loop as an external flux bias Φext = Φ0 /2. For this circuit, the black-box Hamiltonian is derived as [46] ˆ Ref. [46] ({ψˆj }, {ρˆj }) = H black

N X j=1

2π ψˆj ρˆj + EJ cos 2CJ Φ0 2

!

.

(65) The sign of the last term (potential energy of the Josephson effect) is positive by the presence of the external flux bias Φext = Φ0 /2. We can intuitively understand the existence of a SRPT by analyzing the minima of the inductive energy: U (φ, ψ) =

N X φ2 (φ − ψj )2 2πψj . (66) + + EJ cos 2Lr j=1 2Lc Φ0

For N Lr > [Φ0 /(2π)]2 /EJ − Lc , this function has two minima at φ = ±φ0 6= 0 (and ψj = ±[1 + Lc/(N Lr )]φ0 6= 0). Since the potential barrier between the two minima becomes infinitely high in the thermodynamic limit N → ∞, the symmetry (superposition of the two minima) in the ground state is broken spontaneously, and we get a coherent amplitude of the flux φ ≈ ±φ0 below a critical temperature. In this way, SRPTs exist in superconducting circuits where the photonic harmonic energy [φ2 /(2Lr ) minimized at φ = 0] and the atomic anharmonic energy [EJ cos(2πψj /Φ0 ) minimized at ψj 6= 0] competes through the coupling term [(φ − ψj )2 /(2Lc) minimized for φ = ψj ]. As we already found a counter-example above, we cannot get the no-go theorem for the Hamiltonian in Eq. (64) derived for the circuit in Fig. 5(b). In contrast to Sec. III A, we cannot relocate the photonic flux ˆ black by unitary transformations, since there are φ into H N coupling terms (φˆ − ψˆj )2 /(2Lc), while the absence of SRPTs can be shown for N = 1 in the same manner as Sec. III A. On the other hand, if we consider the third

term, the inductive energies at Lc , as a part of the blackbox Hamiltonian as ′ ˆ {ψˆj }, {ρˆj }; . . .) ˆ black H (φ;

≡

N X (φˆ − ψˆj )2 j=1

2Lc

ˆ black ({ψˆj }, {ρˆj }; . . .), +H

(67)

the coupling term is certainly included in the black box as 2 ˆ2 ˆ ˆ ˆ′ ˆ 4 = qˆ + φ + H ˆj }; . . .). (68) H black (φ; {ψj }, {ρ 2Cr 2Lr However, we cannot remove the photonic flux φ from the black-box Hamiltonian even under the c-number substitution. For example, by introducing a unitary operator as N X ˆ4c = exp − i φ U ρˆj , (69) ~ j=1

ˆ 4′ under the c-number substitution is the Hamiltonian H transformed to

q2 φ2 ˆ′ ˆ ˆ† H ˆ ′′ (φ; {ψˆj }, {ρˆj }; . . .), (70) U + +H black 4c 4 U4c = 2Cr 2Lr where the black-box Hamiltonian is transformed as ˆ ′′ (φ; {ψˆj }, {ρˆj }; . . .) H black

=

N ˆ 2 X ψj ˆ black ({ψˆj + φ}, {ρˆj }; . . .). +H 2L c j=1

(71)

In this way, the problem cannot be reduced to the minimization of the black-box Hamiltonian without the LC resonator. In other words, the Hamiltonian of the circuit configuration in Fig. 5(b) cannot be expressed as similar as the minimal-coupling Hamiltonian. Then, the absence of SRPTs cannot be confirmed by the same logic as the no-go theorem for atomic systems. This result is consistent with the proposal of a SRPT in Ref. [46]. In the similar manner, for the circuit configuration of Fig. 5(d), where Lr is eliminated, its Hamiltonian is simply derived as Eq. (64) without the second term. The absence of SRPTs cannot be confirmed also in this circuit configuration. B.

Inductive coupling with a transmission line

Finally, let us consider the circuit configuration depicted in Fig. 6. A transmission line couples with a long black box inductively, or we can instead consider small LC resonators coupled through the black box. Following the flux-based procedure, a Lagrangian is obtained as " # ′ X Ct ∆x (φj − ψj−1 )2 (φj − ψj )2 2 ˙ L5 = φj − − 2 2Lt ∆x 2L′t ∆x j + Lblack ({ψj }, {ψ˙ j }; {ψj′ }, {ψ˙ j′ }; . . .).

(72)

10 Hamiltonian could be derived with treating the artificial atoms as a black box. V.

FIG. 6. A transmission line coupled inductively with a long black box. The absence of SRPTs cannot be conﬁrmed by the analysis in this paper.

The conjugate momenta are derived as ∂L5 = Ct ∆xφ˙ j , ∂ φ˙ j ∂Lblack ∂L5 = , ρj ≡ ˙ ∂ ψj ∂ ψ˙ j ∂Lblack ∂L5 = . ρ′j ≡ ∂ ψ˙ j′ ∂ ψ˙ j′ qj ≡

(73a) (73b) (73c)

Then, we get the Hamiltonian as # " ′ 2 X ˆj − ψˆj )2 (φˆj − ψˆj−1 )2 ( φ q ˆ j ˆ5 = + + H 2Ct ∆x 2Lt ∆x 2L′t ∆x j ˆ black ({ψˆj }, {ρˆj }; {ψˆ′ }, {ρˆ′ }; . . .). +H j j

(74)

For this Hamiltonian, we cannot relocate the coupling ˆ black as in the previous sections. For examterms into H ple, using a unitary operator X i ˆ5 = exp − U (75) qˆj ψˆj , ~ j we get

ˆ5 = φˆj + ψˆj , ˆ † φˆj U U 5 † ˆ5 = ρˆj − qˆj , ˆ ρˆj U U 5

(76a) (76b)

SUMMARY

Following the similar analysis as the no-go theorem for atomic systems [14, 15], we examined the possibility of SRPTs in some configurations of superconducting circuits. By deriving Hamiltonians with treating artificial atoms as a black box, we show that three configurations depicted in Figs. 2–4 do not show SRPTs if the systems satisfy Assumptions 1 and 2, or A in the thermodynamic limit, which justify the c-number substitution performed in Eq. (7) and are essential in the no-go theorem for the atomic systems [14, 15]. The absence of SRPTs cannot be confirmed for the circuit configurations in Figs. 5 and 6. It is because, for Fig. 5(a), we could not derive its Hamiltonian with treating artificial atoms as a black box. Concerning Figs. 5(b), (d), and 6, we can derive their Hamiltonians, but they cannot be transformed as the minimal-coupling Hamiltonian. Then, the absence of SRPTs cannot be confirmed in the analysis of this paper. In fact, Figs. 5(a) and (b) includes the circuit in Ref. [46] depicted in Fig. 5(c), where a SRPT in the thermal equilibrium was proposed in the presence of an external magnetic flux or π junctions. The analysis in this paper shows the absence of SRPTs originating from the coupling between the black box and the LC resonator or the transmission line. If the black box includes another resonator or transmission line, we must examine whether it can be reduced to the three circuit configurations in Fig. 2–4 or we must extend the discussion for circuits with multiple resonators or transmission lines. Further, there also remains the possibility of SRPTs in systems that do not satisfy Assumption 1, 2, or A, i.e., those SRPTs cannot be analyzed under the c-number substitution performed in Eq. (7). In order to find SRPTs in the absence of an external magnetic flux or π junctions, we should explore the circuit configurations in Figs. 5 and 6 or others except Figs. 2–4, while the analysis in this paper does not basically depend on whether an external magnetic flux or π junctions exist or not.

and ˆ ˆ ˆ †H U 5 5 U5 " # ′ X (φˆj + ψˆj − ψˆj−1 )2 qˆj 2 φˆj 2 = + + 2Ct ∆x 2Lt ∆x 2L′t ∆x j ˆ black ({ψˆj }, {ρˆj − qˆj }; {ψˆ′ }, {ρˆ′ }; . . .). +H j j

(77)

In this way, the coupling terms inevitably remains in the photonic Hamiltonian as far as we tried. Then, the absence of SRPTs in the transmission line of Fig. 6 cannot be confirmed by the analysis in this paper, while its

ACKNOWLEDGMENTS

M. B. thanks P.-M. Billangeon for fruitful discussions. This work was funded by ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan) and by KAKENHI (Grants No. 26287087, No. JP16H02214, and No. 24-632).

11

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