Heat Bath in a Quantum Circuit

We discuss the concept and realization of a heat bath in solid state quantum systems. We demonstrate that, unlike a true resistor, a finite one-dimensional Josephson junction array or analogously a transmission line with non-vanishing frequency spacing, commonly considered as a reservoir of a quantum circuit, does not strictly qualify as a Caldeira–Leggett type dissipative environment. We then consider a set of quantum two-level systems as a bath, which can be realized as a collection of qubits. We show that only a dense and wide distribution of energies of the two-level systems can secure long Poincare recurrence times characteristic of a proper heat bath. An alternative for this bath is a collection of harmonic oscillators, for instance, in the form of superconducting resonators.

The question of thermalization in closed quantum systems and the nature of thermal reservoirs are topics of considerable interest [1][2][3][4][5][6][7].However, experimental realizations, in particular in solid-state domain are largely missing [4,8].In this paper we compare different types of reservoirs that can be realized in the context of superconducting quantum circuits.An ideal heat bath is a resistor [9][10][11][12][13][14][15], which can be realized in a straightforward way.But mainly because of the compatibility of the fabrication processes, the circuit QED community typically prefers to mimic resistors or simply to produce highimpedance environments by arrays of Josephson junctions or superconducting cavities [16][17][18][19][20][21][22][23][24].The advantages of a physical resistor in form of metal film is that it has a truly gapless and smooth absorption spectrum, and on the practical side its temperature can be probed by a standard thermometer [25].A one-dimensional Josephson junction array, on the contrary, although acting as a high impedance environment [26,27], presents welldefined resonances in its absorption spectrum up to the plasma frequency and purely capacitive behavior above it, and cannot thus be considered rigorously as a resistor.Experiments on multimode cavities support our conclusion as they exhibit periodic recoveries of the qubit coupled to them [28].In order to realize a Caldeira-Leggett type true reservoir [29,30] out of superconducting elements, we propose an ensemble of qubits or LCresonators with a distribution of energies among them.
We start by an elementary classical analysis of a onedimensional Josephson junction array (see Fig. 1 (a)), which in a linearized form can be presented by a chain of parallel LC elements for the junctions, and a ground capacitance C g between two of them as in Fig. 1 (b).Assuming a long array, we can write for voltage V (k) on island k and current I(k) through the corresponding junction Here , ω is the angular frequency of driving, ω p = 1/ √ LC is the plasma frequency of the junction, and I(k) is the current through the k:th junction.One can solve these equations with different terminations of the array.One finds the dispersion relation of angular frequencies ω n for infinite impedance at where, for an array of N junctions ω n,0 = (n − 1/2)π/(N LC g ) for a shorted termination and ω n,0 = nπ/(N LC g ) for an open line [31].This is the functional dependence of the dispersion relation used in fitting the data, e.g., in Refs.[22,24], and it is depicted in Fig. 1 (c) for two different values of C/C g , one for pure LC transmission line C/C g = 0, and the other for C/C g = 100.
Figure 1 (d)-(f) shows the modulus of the frequency dependent impedance of an array calculated numerically for C/C g = 100.We conclude that such an array can hardly be considered to be a resistor.Resonant absorption at frequencies corresponding to Eq. ( 2) is presented in experiments as well [22,24].At frequencies above ω p there are no more modes and the impedance is purely capacitive with impedance Z(ω) = (iω CC g ) −1 asymptotically at high frequencies (see Fig. 1 (f)).
We next analyze the energy exchange between the system (here a qubit) and a reservoir to assess whether the latter qualifies as a thermal bath.In general, an ideal array presents a reactive element that cannot dissipate the energy.Such a conclusion can be drawn for instance by analyzing the population of a qubit coupled to the array.To be concrete, we follow the model in Refs.[28,32], and consider a qubit with energy ℏΩ coupled to a bath of N states with energy of the j:th one equal to ℏω j .The Hamiltonian of the whole system and bath is given by b † i bi .The parameters γ i represent the coupling of the qubit with each state in the environment for the perturbation, which reads in the interaction picture with respect to Ĥ0 The basis that we use is formed of the states of the system and environment as {|0⟩ = |1000...0⟩, |1⟩ = |0100...0⟩, ..., |i⟩ = |0 0...1 (i:th) ...0⟩}, where the first entrance refers to the qubit and from the second on to each of the N states in the bath.In what follows we apply this model to both a multimode cavity and spins as environment.We choose the initial state of the whole system (qubit and environment) as |ψ I (0)⟩ ≡ |0⟩.This corresponds to the ground state of the environment (zero temperature, T = 0) but with the qubit excited.We solve the Schrödinger equation iℏ∂ t |ψ I (t)⟩ = VI (t)|ψ I (t)⟩ in the interaction picture to find the time evolution of the state of the whole system, |ψ Returning first to a Josephson junction array, or a finite transmission line, we may write the (angular) frequencies of the multimode resonator as ω k = k∆ω (exactly for an LC transmission line, and approximately for the array well below ω p , see Eq. ( 2)), where the spacing ∆ω is given by the length of the line or array as discussed above for the latter.Furthermore, we assume the standard coupling as γ k = g √ k, where g is the coupling constant arising, e.g., from the capacitance between the qubit and the resonator [28].This model, with the system depicted in Fig. 2 (a), demonstrates in the absence of true dissipative elements almost periodic exchange of energy between the qubit and the cavity shown in Fig. 2 (b), where the excited state population of the qubit p e ≡ |C 0 | 2 is depicted against the normalized time Ωt.In this numerical example we chose ∆ω = 0.01Ω, and included N = 300 states in the calculation.This energy spacing mimics approximately the experiment of Ref. [28].We can see that the revivals are not full, and the energy of the qubit is distributed over many states with energies in the neighborhood of ℏΩ.Zooming in to the short time regime as in Fig. 2 (c), we observe exponential decay of the population over eight orders of magnitude.A closer analysis of the dynamics yields that indeed the decay in short times is exponential, with a decay rate Γ = 2π g 2 ℏ 2 Ω ∆ω 2 , following the numerical result of Fig. 2 (c).The other important feature in the dynamics is naturally the periodic recoveries of p e (t).The first repopulation demonstrates a sharp peak that sets abruptly on at time t = 2π/∆ω.We may associate this with the time of flight of a photon with frequency Ω through the transmission line and reflected back.In practical circuits this recovery time falls into very short, nanosecond regime, meaning that the transmission line acts as a bath only for times shorter than this.In Ref. [28] similar results as in Fig. 2 (b) were obtained using the input-output theory [33].The results are robust against different terminations of the line.
As is well known, a set of reactive elements can, however, effectively approximate a dissipative element in the spirit of Caldeira and Leggett [29].We will next discuss the conditions of forming a heat bath in solid state quan-tum context without actual dissipative building blocks.In particular we focus on a collection of coupled quantum two-level systems (TLSs), which can in practice be formed of Josephson junction based qubits [34], or of unknown structural defects in superconducting circuits [35].A set of harmonic oscillators in form of superconducting cavities would provide an alternative realization of a Caldeira-Leggett environment.Here we focus on TLSs.Returning to the archetypal setup, where a central qubit couples to an ensemble of these TLSs, we observe the dynamics of this qubit when initially set to its excited state.We use the same model as above, but now with different distributions of energies and couplings of the TLSs.For the sake of clarity of the argument, all the TLSs are again set initially to their ground state, mimicking a zero temperature environment.As we have shown in another context [32], a broad distribution of energies of the TLSs secures exponential decay of the qubit population in time.This can be seen also analytically, for instance, by standard means resumming in all orders of perturbation assuming a large number of uniformly distributed TLS energies.The distribution of energies and couplings of the TLSs is an essential condition for absorbing the energy of the qubit to this bath without recoveries over any practical timescales.In this case, the qubit decays exponentially as Here Γ 0 = 2πν 0 Λ 2 0 /N with ν 0 the density of TLSs around Ω, and Λ 2 0 = N i=1 γ 2 i /ℏ 2 .In general, for any distribution of energies and couplings, we find that the qubit amplitude C 0 (t) in the excited state is governed by the integro-differential equation We see immediately that for the case where all the TLSs have the same energy as the qubit, ω k ≡ Ω for all k, the qubit does not decay, even when the couplings γ i are fully random, but it oscillates with population |C 0 (t)| 2 = cos 2 (Λ 0 t), i.e. the Poincare recovery time is π/Λ 0 .We can generalize the conclusion above for a bath where ω k = (1 − r)Ω for arbitrary positive r, meaning detuned equal-energy TLSs in the environment.In this case, Eq. ( 6) leads to D(t) − irΩ Ḋ(t) + Λ 2 0 D(t) = 0, where D(t) = Ċ0 (t).C 0 (t) satisfies the initial conditions C 0 (0) = 1, Ċ0 (0) = 0 and C0 (0) = −Λ 2 0 .We then have the oscillatory solution Figure 3 (a) shows the numerically calculated results of p e (t) for N = 10 7 TLSs and for different choices of parameters following closely the analytical results given above.For a uniform distribution of TLS energies in the range [0, 2ℏΩ] the decay is exponential as described above, whereas for TLSs with identical energies there are periodic revivals, in quantitative agreement with the analytic result.These results serve as a warning sign for models where bath spins are assumed to have equal energies.In Fig. 3 (b) and (c) we monitor numerically the long time behavior of p e (t) under the same conditions as in the main frame, but with N = 10 5 and N = 3000 TLSs with distributed energies and couplings.We see that there are no revivals over this long period of time in both cases, and the long time population follows closely the prediction p e (t → ∞) = 4Ω/(N πΓ 0 ) indicated by the horizontal lines [36].
Two possible realizations of such reactive baths can be immediately envisioned.The one that corresponds to our analysis here is that of a qubit coupled to TLS environment with variable energies: with modern qubits as TLSs the couplings and energies can be varied almost arbitrarily [34].One can envison to couple hundreds, perhaps even thousands of such artificial TLSs to a qubit.FIG. 3. A qubit coupled to a reservoir of N = 10 7 two-level systems in (a).The central qubit is coupled to each TLS via coupling constants γi that have a uniform distribution between 0 and its maximum level, corresponding to the overall relaxation rate Γ0 = 0.03.The dark blue line corresponds to the evolution of pe ≡ |C0| 2 in the environment of TLSs with uniform distribution of energies in the range 0 < ωi < 2Ω leading to nearly exponential decay.The oscillatory qubit populations of the other curves correspond to uniform environments with ωi = (1 − r)Ω for all i, with r = 0, 0.25 for grey and red lines, respectively.These dynamics follow that given by Eq. ( 7 A simpler choice could be an ensemble of superconducting resonators with the same idea: here the tunability is more limited and instead of TLSs, these resonators work as harmonic oscillators.In summary, it is possible to form a thermal bath on a chip avoiding recurrences [37] over any practical time scale in the spirit of Caldeira and Leggett [29] using just reactive elements.However, a one-dimensional array of Josephson junctions or alternatively a transmission line exhibits periodic recoveries on nanosecond time scales in practical physical circuits for two reasons: first, the energy distribution is not dense and, equally importantly the coupling is not random but essentially equal (∝ √ i) to each state i.Such an environment is thus a heat bath only if it has significant intrinsic dissipation, valid typically for N > 10 5 [18,20], or if it is terminated by a resistive element [38]; in this case the termination itself is the bath.A way around to achieve a true bath is to form a network of harmonic oscillators or TLSs with distributed parameters and couple it to the quantum system.

FIG. 1 .
FIG. 1. Basic properties of a one-dimensional Josephson junction array.(a) An array with N junctions, terminated by impedance ZL.Current is I and voltage V .Junctions can be replaced by superconducting interference devices (SQUIDs) acting as tunable junctions.(b) Equivalent circuit for a uniform array with junctions linearized as inductors L. Junction capacitance is C and the stray "ground" capacitance of each island is Cg.(c) Dispersion relation for modes in the array for two cases, C = 0 (black line, LCg) and C = 100Cg (green line, CLCg) for an array with N = 3000.Here we assume an open ended array (ZL = ∞).The (angular) frequencies are scaled by the plasma frequency ωp = 1/ √ LC of each junction.(d) Modulus of the impedance of the CLCg array as a function of frequency, and (e) a zoom out of it for lower frequencies (red line), together with that of the linear LCg array as well (blue line).(f) At frequencies ω ≫ ωp, the CLCg array behaves as a capacitor with effective capacitance CCg.

FIG. 2 . 2 ℏ 2 Ω ∆ω 2 ,
FIG. 2. A qubit coupled to a linear Josephson junction array or a transmission line.(a)A schematic presentation of the circuit.(b) Time-dependent population pe(t) of the qubit after initialization to the excited state.The transmission line is assumed to be initially in the ground state.Coupling parameter between the qubit and the line is g = 0.001.We have chosen ∆ω = 0.01Ω, corresponding to typically either N = 10 4 − 10 5 junctions or 1 m long transmission line, close to that in Ref.[28].The value of the impedance ZL has almost no effect on pe(t).(c) Initially the qubit decays exponentially with decay rate Γ = 2π g 2 ℏ 2 FIG.3.A qubit coupled to a reservoir of N = 10 7 two-level systems in (a).The central qubit is coupled to each TLS via coupling constants γi that have a uniform distribution between 0 and its maximum level, corresponding to the overall relaxation rate Γ0 = 0.03.The dark blue line corresponds to the evolution of pe ≡ |C0| 2 in the environment of TLSs with uniform distribution of energies in the range 0 < ωi < 2Ω leading to nearly exponential decay.The oscillatory qubit populations of the other curves correspond to uniform environments with ωi = (1 − r)Ω for all i, with r = 0, 0.25 for grey and red lines, respectively.These dynamics follow that given by Eq. (7) quantitatively.(b) and (c) show the population in a similar distributed bath of N = 10 5 and N = 3000 TLSs, respectively, over a time period of Ωt = 3 × 10 5 .The horizontal lines are the analytical long time predictions given in the text.