Probing the strongly driven spin-boson model in a superconducting quantum circuit

Quantum two-level systems interacting with the surroundings are ubiquitous in nature. The interaction suppresses quantum coherence and forces the system towards a steady state. Such dissipative processes are captured by the paradigmatic spin-boson model, describing a two-state particle, the “spin”, interacting with an environment formed by harmonic oscillators. A fundamental question to date is to what extent intense coherent driving impacts a strongly dissipative system. Here we investigate experimentally and theoretically a superconducting qubit strongly coupled to an electromagnetic environment and subjected to a coherent drive. This setup realizes the driven Ohmic spin-boson model. We show that the drive reinforces environmental suppression of quantum coherence, and that a coherent-to-incoherent transition can be achieved by tuning the drive amplitude. An out-of-equilibrium detailed balance relation is demonstrated. These results advance fundamental understanding of open quantum systems and bear potential for the design of entangled light-matter states.


Supplementary note 1: Generalized master equation for the driven spin-boson model
The spin-boson model describes a two-level system -the qubit -interacting with an environment of quantum harmonic oscillators, the so-called heat bath.
The total Hamiltonian of the model reads where σ j are Pauli spin operators and a † i and a i are bosonic creation and annihilation operators, respectively. The angular frequency ∆ is the bare frequency splitting at zero bias. Within the noninteracting-blip approximation (NIBA), the time evolution of the qubit's population difference P (t) = σ z (t) is governed by the following generalized master equation (GME) [1-3] P (t) = t t 0 dt K − (t, t ) − K + (t, t )P (t ) . (2) In the presence of a time dependent bias described by ε(t) = ε 0 +ε p cos(ω p t)+ε d cos(ω d t), where the subscripts "p" and "d" denote probe and drive, respectively, the exact NIBA kernels are where the total dynamical phase has the form ζ tot (t, t ) = t t dt ε(t ) .
Averaging over a period 2π/ω d yields an effective description of the drive by means of the following NIBA kernels [3], which we use for our calculations K + (t, t ) = h + (t − t ) cos ζ(t, t ) , with the functions h ± (t) reading The dynamical phase entering the averaged NIBA kernels in Supplementary Equations (6)-(7) accounts now exclusively for the static bias and the probe field, whereas the drive is taken into account, in an effective description, by the Bessel functions J 0 in the functions of h ± (t).
The functions Q and Q in Supplementary Equations (3)- (4) and (8) Q (t) = α ln(1 + ω 2 c t 2 ) + 4α ln where we have introduced the thermal frequency ω β = ( β) −1 and where Γ(x) is the Euler Gamma function. In the limit ω c k B T (or ω c ω β ), neglecting the ratio ω β /ω c and using Γ(1 + ix)Γ(1 − ix) = πx/ sinh(πx), we get the so-called scaling limit forms These expressions are accurate in every regime, provided that the cutoff frequency is large with respect to the other frequency scales involved. For ω c t 1, these functions assume the approximated forms Especially at high temperature, ω β ∼ ∆, the cutoff operated by the real part Q (t) in the kernels, becomes of purely exponential form on a short time scale, see Supplementary Equation (17) below. Now, this means that, at strong coupling, the kernels go to zero on a rather short time, where the short time behavior of Q , neglected in Supplementary Equation (16), is relevant.
An insight into the different behaviors shown by the two driven setups in Fig. 3 of the main text, is provided by considering the memory time of the kernels K ± . To this end, consider the long-time limit of Q(t) in Supplementary Equations (15)-(16). Specifically, for ω β t = tk B T / 1, the real part of Q(t) acquires the form This form implies that, at fixed, finite temperature, τ env decreases as the coupling α is increased.
Moreover, in the above limit, the bath force operator F (t) of the quantum Langevin equation for the spin-boson model is delta-correlated, as F (t)F (0) ∝ d 2 dt 2 Q(|t|), where the average is taken with respect to the thermal state of the bath (see Ref. [1] for details). As a consequence, on the time scale dictated by the limit (17) the bath is a white noise source.
, coming from the left, is scattered by the qubit placed at the center of the transmission line. The proportionality constant f Z has dimensions of flux whereas ε p is an angular frequency. The scattering at the qubit position results in the transmitted field to the right, V transm (t), and a reflected field to the left, V refl (t). The flux difference across the qubit is the left of the qubit the following equations where Z = l/c is the characteristic impedance of the transmission line. Similarly, to the right of the qubit, where we set V (0 + , t) ≡ V R (t) and I(0 Using the conservation of the current, I L (t) = I R (t), and the relation V from Supplementary Equations (18)-(21) we get We identify the flux difference across the qubit with the population difference of the localized eigenstates of the flux operatorΦ = f σ z , namely we set δΦ(t) ≡ f σ z (t) = f P (t), where f is the proportionality constant with dimensions of flux, as described in the main text.
Let P as (t) = lim t→∞ P (t) be the asymptotic, nonequilibrium population difference. For periodic driving with period 2π/ω p , the time derivativeṖ as (t) can be expanded as the Fourier seriesṖ where p m = ω p 2π The transmission T at frequency ω p (m = 1) is defined as the following ratio between transmitted and input voltages where N = f /f Z and where, in passing from the first to the second line, we used Supplementary Equations (22) and (23). Real and imaginary parts of the transmission are therefore given by and respectively.
Supplementary Note 3: Linear response to a weak probe -closed expression for the transmission In the regime of linear response to an applied monochromatic probe driving, namely for small ratio ε p /ω p , and within the effective description of the pump drive introduced in the Supplementary Note 1, the asymptotic population difference P as (t) is monochromatic [3,5]. It can be thus expressed as the truncated Fourier sum where the superscript (1) denotes first order with respect to the ratio ε p /ω p . Here χ is the linear susceptibility [5] and P 0 is the asymptotic value of P (t) in absence of probe driving.
Within the NIBA, by substituting the expression (28) for P as (t) in the GME (2), setting the upper integration limit to t → ∞, which is valid for times much larger than the kernels' memory time, and expanding the kernels in Fourier series, we get the following closed, linear response (superscripts (0,1) denote the order in ε p /ω p ).
The kernels k ± m and v + , whose approximate forms (perturbative in ε p /ω p ) enter Supplementary Equation (30), are defined by where the pump drive-averaged kernels K ± (t, t ) have been introduced in Supplementary Equations (6)-(7). Expansion of the Bessel functions entering the kernels K ± (t, t ) to lowest order in ε p /ω p by means of J n (x) ∼ (x/2) n , yields the following explicit expressions for the kernels in and The linear susceptibility χ is related to the coefficient p  Thus, from Supplementary Equation (30), by simplifying the notation, we get Here Note that, within the present linear response treatment, the transmission is independent of the probe amplitude ε p , cf. Supplementary Equation (29). Note also that the notation for the kernels H ± reflects the same symmetry with respect to the static bias ε 0 which holds for K ± .
Finally, the forward/backward rates introduced in the main text, describe the incoherent tunneling between the individual localized (flux) states.

Supplementary Note 4: Approximate form of the susceptibility
Whenever the condition ω p τ env 1 is fulfilled, it is possible to expand the kernels K + (iω p ) and H ± (ω p ) [see Supplementary Equation (40)] with respect to ω p τ env . To first order Now, the NIBA prediction for the stationary probability difference P 0 in the absence of the In the presence of the pump driving, within the present effective description of the pump drive (see the Supplementary Note 1), the expression for P 0 is generalized as follows The effective bias ε eff depends on the static bias ε 0 . As a result, in the limit ω p τ env 1, by substituting the expressions in Supplementary Equation (42) into Supplementary Equation (39) we obtain where Supplementary Equation 41), and where At the symmetry point ε eff = ε 0 = 0 so that from Supplementary Equations (41) and (45) we get lim where the functions h ± (t) have been defined in Supplementary Equations (8)-(9).
Note that W(−x) = W * (x). In Supplementary Figure 3 As a consequence, we are not able to extract via fit to data the parameters that characterize the coupling regime of Device III, as done for devices I and II. Moreover, the spectrum at the symmetry point for the undriven Device III appears almost featureless in the measured range of probe frequencies, as can be seen in Fig. 2(f) Moreover, at zero static bias, ε 0 = 0, both H − and K − (0) vanish. The resulting expression for the linear susceptibility is Consider the case of zero static bias, ε 0 = 0. The imaginary part χ of the susceptibility is characterized by a peak centered at a frequency ω * and of FWHM 2γ. In the coherent regime, occurring when ω * > γ, the dynamics of P (t) displays damped oscillations with renormalized oscillation frequency Ω = (ω * ) 2 − γ 2 and damping rate γ. The transition to the incoherent regime is determined by the condition ω * = γ. The incoherent regime, which is realized for ω * < γ, is described by an exponential decay of P (t) with rate γ r , the relaxation rate, given in this case by the position of the peak.
As an illustration, let us consider the three different dissipation regimes mentioned above, namely i) coherent, ii) coherent-incoherent transition, and iii) incoherent. We calculate by Supplementary Equation (39) the imaginary part of χ as a function of the probe frequency and compare the resulting dynamics, namely damped oscillations or incoherent decay with parameters defined by ω * , γ, and γ r , with the dynamics obtained from direct integration of the GME (2) in the static, unbiased case, ε(t) = 0. Results are shown in Supplementary Figures 6-8.
On the basis of the considerations made above, we are able to establish a phase diagram for the nondriven spin-boson model, i.e., to assign a dynamical behavior (coherent/incoherent) to the points of the coupling-temperature parameter space, by studying χ (ω p ) and specifically the condition for the coherent-incoherent transition ω * = γ, where ω * is the position of the peak of χ (ω p ) and 2γ its FWHM.
Such phase diagram, derived within the NIBA, is shown in Fig. 1(c)