Dissipation in circuit quantum electrodynamics: lasing and cooling of a low-frequency oscillator

Superconducting qubits coupled to electric or nanomechanical resonators display effects previously studied in quantum electrodynamics (QED) and extensions thereof. Here we study a driven qubit coupled to a low-frequency tank circuit with particular emphasis on the role of dissipation. When the qubit is driven to perform Rabi oscillations, with Rabi frequency in resonance with the oscillator, the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the qubit and lasing behavior of the oscillator ("single-atom laser"). For red detuning the qubit cools the oscillator. This behavior persists at the symmetry point where the qubit-oscillator coupling is quadratic and decoherence effects are minimized. Here the system realizes a"single-atom-two-photon laser".


INTRODUCTION
Recent experiments on quantum state engineering with superconducting circuits realized concepts originally introduced in the field of quantum optics, as well as extensions thereof, e.g., to the regime of strong coupling [1,2,3,4,5,6,7,8], and prompted substantial theoretical activities [9,10,11,12,13,14,15,16,17,18,19]. Josephson qubits play the role of two-level atoms while electric or nanomechanical oscillators play the role of the quantized radiation field. In most QED or circuit QED experiments the atom or qubit transition frequency is near resonance with the oscillator. In contrast, in the experiments of Refs. [1], with setup shown in Fig. 1a), the qubit is coupled to a slow LC oscillator with frequency (ω T /2π ∼ MHz) much lower than the qubit's level splitting (∆E/2π ∼ 10 GHz). The idea of this experiment is to drive the qubit to perform Rabi oscillations with Rabi frequency in resonance with the oscillator, Ω R ≈ ω T . In this situation the qubit should drive the oscillator and increase its oscillation amplitude. When the qubit driving frequency is blue detuned, the driving creates a population inversion of the qubit, and the system exhibits lasing behavior ("single-atom laser"); for red detuning the qubit cools the oscillator [18]. A similar strategy for cooling of a nanomechanical resonator via a Cooper pair box qubit has been recently suggested in Ref. [20]. The analysis of the driven circuit QED system shows that these properties depend strongly on relaxation and decoherence effects in the qubit.
In the experiments of Ref. [1] an enhancement of the oscillator due to the driving was observed. However, first a) x dc 1 2 3 driv.
T T FIG. 1: a) In the setup of Ref. [1] an externally driven threejunction flux qubit is coupled inductively to an LC oscillator. b) A charge qubit is coupled to a mechanical resonator.
attempts to explain the effect theoretically did not resolve several issues [21,22]. In the experiments, in order to minimize decoherence effects, the Josephson flux qubit was biased near the flux degeneracy point. At this symmetry point also the coupling to the oscillator is tuned to zero, and the enhancement should vanish. Uncontrolled small deviations from the symmetry point might lead to the observed effect [1,21], but this explanation has not been supported by experiments. Here we explore an alternative, namely that a quadratic coupling to the oscillator near the resonance condition Ω R ≈ 2ω T is responsible for the observed enhancement. In the following we will consider both linear as well as quadratic coupling, which dominate away from the symmetry point and at this point, respectively. The second unresolved problem is the magnitude of the effect. The experiments [1] showed an increase by a factor 4 − 5 in the amplitude, i.e., 16 − 25 in the number of oscillator quanta. The theory of Ref. [21], valid in the perturbative regime, predicts a much weaker effect. We obtain a strong effect as follows [18]: due to a detuning of the qubit driving a population inversion is created at the Rabi frequency, and the system becomes a "single-atom laser" at the resonance Ω R ≈ ω T , or a "single-atom-two-photon laser" for Ω R ≈ 2ω T [23,24]. In both cases the lasing threshold is reached for realistic system parameters, and the number of quanta in the oscillator is increased considerably.
A related situation, called "dressed-state lasing", had been studied before in quantum optics [25,26]. The present scenario differs from that one in so far as the resonator modes are coupled to the low-frequency Rabi oscillations rather than to the high-frequency Mollow transitions. The Rabi frequency can be readily tuned to resonance with the oscillator, which should facilitate reaching the lasing threshold and a proper lasing state. A similar idea has been explored in Ref. [27] in connection with coupling of atoms.
In experiments with the same setup as shown in Fig. 1a) but in a different parameter regime the mechanisms of Sisyphus cooling and amplification has recently been demonstrated [7]. Due to the resonant highfrequency driving of the qubit, depending on the detuning, the oscillator is either cooled or amplified with a tendency towards lasing. The Sisyphus mechanism is most efficient when the relaxation rate of the qubit is close to the oscillator's frequency. In contrast, in the present paper we concentrate on the "resolved sub-band" regime where the dissipative transition rates of the qubits are much lower than the oscillator's frequency.
Also in situations where the qubit, e.g., a Josephson charge qubit, is coupled to a nano-mechanical oscillator (Fig. 1b) it either cools or amplifies the oscillator. On one hand, this may constitute an important tool on the way to ground state cooling. On the other hand, this setup provides a realization of what is called a SASER [28].
Lasing and cooling of the oscillator has also been observed in a slightly different setup, when the ac-driven qubit is replaced by a driven superconducting singleelectron transistor biased near the Josephson quasiparticle cycle [29,30,31]. When the SSET is coupled to a nanomechanical or electric oscillator it can be used to either cool the oscillator [32,33,34,35,36,37] or to produce laser-like behavior. The latter has recently been observed in experiments [38].

The Hamiltonian
The systems to be considered are shown in Fig. 1. A qubit is coupled to an oscillator and driven to perform Rabi oscillations. To be specific we first analyze the flux qubit coupled to an electric oscillator ( Fig. 1a) with Hamiltonian The first two terms describe the qubit, with Pauli matrices σ x,z operating in the flux basis of the qubit. The energy bias between the flux states ǫ(Φ dc x ) is controlled by an external DC magnetic flux, and ∆ is the tunneling amplitude between the basis states. The resulting level spacing ∆E ≡ √ ǫ 2 + ∆ 2 typically lies in the range of several GHz. The third term accounts for the driving of the qubit by an applied AC magnetic flux with amplitude Ω R0 and frequency ω d . The last two terms describe the oscillator with frequency ω T = 1/ √ L T C T , which for the experiments of of Ref. [1] lies in the range of several 10 MHz, as well as the qubit-oscillator interaction. We estimate the coupling constant g ≈ M I p I T,0 to be of the order of 10 MHz. Here M is the mutual inductance, I p the magnitude of the persistent current in the qubit, and I T,0 = ω T /2L T the amplitude of the vacuum fluctuation of the current in the LC oscillator.
After transformation to the eigenbasis of the qubit, which is the natural basis for the description of the dissipation, the Hamiltonian reads with tan ζ = ǫ/∆ and ∆E ≡ √ ǫ 2 + ∆ 2 . Because of the large difference of the energy scales between the qubit and the oscillator, ∆E ≫ ω T , it is tempting, in the spirit of the usual rotating wave approximation (RWA), to drop the transverse coupling term −g cos ζ σ x a + a † of Eq. (2). However, near the symmetry point (where sin ζ = 0) the longitudinal coupling is weak. Therefore, we retain the transverse coupling, but transform it by employing a Schrieffer-Wolff transformation, U S = exp (iS), with generator S = (g/∆E) cos ζ a + a † σ y , into a second-order longitudinal coupling. On the other hand, since ω d ∼ ∆E, we can drop within RWA the longitudinal driving term − Ω R0 cos (ω d t) sin ζ σ z . The Hamiltonian then reads A further unitary transformation with U R = exp (−iω d σ z t/2) brings the Hamiltonian to the ro- Here δω ≡ ω d − ∆E/ is the detuning. After diagonalization of the qubit terms we obtaiñ Here Ω R = Ω 2 R0 cos 2 ζ + δω 2 and tan β = δω/(Ω R0 cos ζ).
Finally we employ a second RWA. While the first one dropped terms oscillating with frequencies of order ∆E/ , the second one assumes the Rabi frequency Ω R and the oscillator frequency ω T to be fast. In the interaction representation with respect to the non-interacting Hamiltonian,H 0 = ( Ω R /2)σ z + ω T a † a, we then obtaiñ We kept both single-photon and two-photon interactions with g 1 = g sin ζ cos β and g 2 = (g 2 /∆E) cos 2 ζ cos β, although within RWA only one of them survives: the single-photon term for Ω R ∼ ω T , or the two-photon term for Ω R ∼ 2ω T . The last term of (6) with g 3 = −(g 2 /∆E) cos 2 ζ sin β is the ac-Stark effect, causing a qubit state dependent frequency shift of the oscillator [39]. In what follows we will assume that the qubit is kept near the symmetry point, i.e., ǫ ≪ ∆ and cos ζ ≃ 1.

Transition rates in the rotation frame
The transformation to "dressed states" in the rotating frame modifies the relaxation, excitation and decoherence rates as compared to the standard results. To illustrate these effects and justify the treatment of the dissipation in latter sections we first consider a driven qubit (ignoring the coupling to the oscillator) coupled to a bath observableX B , In the absence of driving, Ω R0 = 0, and for regular power spectra of the fluctuating bath observables we can proceed using Golden rule type arguments. The transverse noise is responsible for the relaxation and excitation with rates while longitudinal noise produces a pure dephasing with rate [40] Γ Here b ⊥ ≡ b x + ib y , and we introduced the ordered correlation function X 2 , as well as the power spectrum, i.e., the symmetrized cor- (8) and (9) also define the relaxation rate 1/T 1 = Γ 1 = Γ ↓ + Γ ↑ and the total dephasing rate 1/T 2 = Γ ϕ = Γ 1 /2 + Γ * ϕ which appear in the Bloch equations.
To account for the driving with frequency ω d it is convenient to transform to the rotating frame via a unitary transformation U R = exp (−iω d σ z t/2). Within RWA the transformed Hamiltonian reduces tõ The RWA cannot be used in the second line of (10) since the fluctuationsX B contain potentially all frequencies, including those of order ±ω d which can compensate fast oscillations.
Diagonalizing the first line of (10) one obtains with Ω R = Ω 2 R0 + δω 2 and tan β = δω/Ω R0 . From here Golden-rule arguments yield the relaxation and excitation rates in the rotating frame [40] as well as the "pure" dephasing ratẽ We note the effect of the frequency mixing, and due to the diagonalization the effects of longitudinal and transverse noise on relaxation and decoherence get mixed. In addition we note that the rates also depend on the fluctuations' power spectrum at the Rabi frequency. For a sufficiently regular power spectrum of fluctuations with ω ≈ ±∆E/ , we can ignore the effect of detuning and the small shifts by ±Ω R as compared to the high frequency ω d ≈ ∆E/ . We further assume that Ω R ≪ k B T / . In this case we find the simple relations where the rates in the lab frame are given by Eq. (8) and the new rate, Γ ν ≡ 1 2 b 2 z S X (Ω R ), depends on the power spectrum at the Rabi frequency.
To proceed we concentrate on the regime relevant for our system. Near the symmetry point of the qubit we have b z ≈ 0. At low temperatures, k B T ≪ ∆E ≈ ω d , we can neglect Γ ↑ as it is exponentially small. Thus we are left with where The ratio of up-and down-transitions depends on the detuning and can be expressed by an effective temperature. Right on resonance, where β = 0, we haveΓ ↑ =Γ ↓ , corresponding to infinite temperature or a classical drive. For "blue" detuning, β > 0, we findΓ ↑ >Γ ↓ , i.e., negative temperature. This leads to a population inversion of the qubit, which is the basis for the lasing behavior which will be described below. In a more careful analysis, not making use of the "white noise" approximation, we obtain for β = 0 For instance, for Ohmic noise and low bath temperature this reduces toΓ ↓ /Γ ↑ ∼ 1 + 2Ω R /ω d . This corresponds to an effective temperature of order 2 ω d /k B ∼ 2∆E/k B , which by assumption is high but finite. The infinite temperature threshold is crossed toward negative temperatures at weak blue detuning when the condition is satisfied. To illustrate how the population inversion is created for blue detuning we show in Fig. 2 the level structure, i.e., the formation of dressed states, of a near-resonantly driven qubit. For the purpose of this explanation the driving field is quantized. This level structure was described first by Mollow [26]. The picture also illustrates how for blue detuning a pure relaxation process, Γ ↓ = Γ 0 , in the laboratory frame predominantly leads to an excitation process,Γ ↑ , in the rotating frame.

The Liouville equation in the rotating frame
Within the approximation leading to the rates (15) the Liouville equation governing the dynamics of the density matrix in the rotating frame can presented in a simple Lindblad form. Start in the lab frame the dissipation is accounted for by two damping terms, The qubit's dissipation is described by with rates given by (8) and (9). The resonator damping, with strength parametrized by κ, can be written as [41] Here is the thermal distribution function of photons in the resonator. The transformations to the rotating frame and RWA described above transform the Liouville equation to a new form, which in our approximation again has a Lindblad form. In the interaction representation it iṡ whereL while the oscillator damping term is not affected by the transformation. Although at low temperatures in the lab frame the relaxation processes dominate (see Eq. 17), the transformation to the rotating frame introduces excitation and pure dephasing processes.

THE SINGLE-QUBIT LASER
In the following we will consider two resonance situations, Ω R ∼ ω T or Ω R ∼ 2ω T , when either the oneor the two-photon interactions dominate, and investigate the effects of blue or red detuning, δω ≡ ω d − ∆E/ , of the qubit driving frequency. We also study the effects of detuning of the Rabi frequency Ω R relative to that of the oscillator.

One-photon interaction
When the Rabi frequency is in resonance with the oscillator, Ω R ≈ ω T , the Hamiltonian (6) in RWA reduces to From here we can proceed in the frame of the standard semiclassical approach [41,42] of laser physics with the following main steps: In the absences of fluctuations the system is described by Maxwell-Bloch equations for the classical variables α = a , α * = a † , s ± = σ ± and s z = σ z , which can be derived from the Hamiltonian (25) if all correlation functions are assumed to factorize. Next the qubit variables can be adiabatically eliminated as long as κ, g 1 ≪Γ 1 ,Γ ϕ , which leads to a closed equation of motion for α. If we finally account for fluctuations, e.g., due to thermal noise in the resonator, α becomes a stochastic variable obeying a Langevin equation [41], Here C ≡ 2g 2 1 , s st z = −D 0 / 1 + |α 2 |/ñ 0 is the stationary value of the population difference between the qubit levels, and D 0 = Γ ↓ −Γ ↑ /Γ 1 is the normalized difference between the rates withΓ 1 =Γ ↑ +Γ ↓ . We further introduced the photon saturation number n 0 =Γ ϕΓ1 /4g 2 1 andñ 0 ≡ n 0 (1 + δΩ 2 /Γ 2 ϕ ), and the total dephasing rateΓ ϕ =Γ 1 /2 +Γ * ϕ . The detuning of the Rabi frequency enters in combination with a frequency renormalization, δΩ ≡ Ω R − ω T + g 3 |α| 2 . The Langevin force due to thermal noise in the oscillator satisfies ξ(t)ξ * (t ′ ) = κN th δ(t − t ′ ) and ξ(t)ξ(t ′ ) = 0. Noise originating from the qubit can be neglected provided the thermal noise is strong, κN th ≫ g 2 1 /Γ ϕ .
while noise arising from the qubit can be neglected if κN th ≫ g 2 2n /Γ ϕ . The detuning of the Rabi frequency for two-photon interaction is given by δΩ ≡ Ω R − 2ω T + g 3 |α| 2 .

Results obtained from the Langevin equation
If one neglects the frequency shifts of the oscillator, i.e., for g 3 = 0, the Fokker-Planck equations corresponding to the Langevin equations (26) and (28) have exact analytic solutions [42]. Also for g 3 = 0 the Eqs. (26) and (28) written asα = −f (n)α/2 + ξ(t) can be transformed to equations for the average number of photons |α| 2 =n in the formṅ = − nRe[f (n)] + κN th . In the steady state, forn ≫ 1 they can be approximated bȳ nRe[f (n)] = κN th . The results of this analysis are shown in Fig. 3. To demonstrate both the one-photon and the two-photon effects we have assumed a small deviation from the symmetry point, ǫ = 0.01∆. The two-photon resonance (the outer one) persist even for ǫ = 0, while the one-photon resonance (the inner one) vanishes there.
We observe that the solution shows bistability bifurcations (see below). As a result we see in in Fig. 3 sharp drops ofn for both resonances as only the lowest stable value is plotted .
We can estimate the asymptotic solutions analytically. In the one-photon case, assumingn ≫ñ 0 , we obtain The blue curves are obtained from the Langevin equations (26) and (28). They show the bistability with the solid curve denoting stable solutions, while the dashed curve denotes the unstable solution. The red curve is obtained from a numerical solution of the master equation (23). The driving amplitude is taken as ΩR0/2π = 5MHz. The parameters of the qubit: ∆/2π = 1 GHz, ǫ = 0.01∆, Γ0/2π = 125 kHz, the resonator: ωT /2π = 6MHz, κ/2π = 1.7 kHz, N th = 5, and the coupling: from Eq. (26)n This result holds independent of whether the second contribution due to the qubit is larger or smaller than the thermal number N th as long asn ≫ñ 0 . In the twophoton case, assumingn ≫ñ 0 , we obtain from Eq.(28)

Solution of the master equation
We also solved the full master equation (23) numerically, which provides access not only to the average number of photons in the oscillator,n, but also to the whole distribution function P (n). To reach convergence with a limited number of photon basis states (n ≤ 100) we assumed a low thermal number, N th = 5 and a relatively high relaxation constant of the oscillator κ/2π = 1.7 kHz. In Fig. 4 the solutions of the Langevin equations (26) and (28) and those of the master equation (23) is compared.
In Fig. 5 the Fano factor F = n 2 − n 2 / n of the photon number distribution is presented. We observe two phenomena. First, in the regime of the lasing without bistability the Fano factor is reduced as compared to that of the thermal state. In the bistable regime it is increased due to the switching between the two solutions.
In Fig. 6 the distribution function, P (n), for the number of photons in the oscillator is plotted both for the The distribution function, P (n), obtained by numerically solving the master equation (23). Blue curve: cooling regime of the one-photon resonance with ΩR0 = 2π × 5MHz and δω = −2π×3.2MHz. Red curve: lasing regime of the twophoton resonance with ΩR0 = 2π×5MHz δω = 2π×11.7MHz. We observe a peak in the P (n) distribution between n = 20 and 30 as a result of the lasing behavior. Black curve: thermal distribution with N th = 5. The parameters are as in Fig. 4.
cooling and enhancement regime. For comparison also the thermal (Bose-Einstein) distribution is plotted. The enhancement and cooling effects described here rely crucially on the transition rates, i.e., the dissipative effects as described by the Liouville equations. In order to demonstrate this dependence we show in Fig. 7 the dependence of the average photon number on the qubit's relaxation rate at the one-photon resonance. We note a non-monotonic dependence on the qubit's relaxation rate. Above the saturation threshold forn > n 0 the pumping rate is limited by Γ 0 , leading to a roughly linear growth of the photon number with increasing Γ 0 , consistent with Eq. (29). At the saturation threshold for n ∼ n 0 the effective coupling is determined by g 1 and the photon number becomes insensitive to small variations of Γ 0 . Finally, forn < n 0 , an increase of Γ 0 predominantly increases the dephasing rateΓ ϕ . As can be seen from Eq. (26) this destroys the coherent coupling between qubit and oscillator and the photon number decreases towards N th . In Figure 7 we plot both the results of a numerical solution of the master equation and the Langevin approximation. We find good agreement between both (except for the bistability in the corresponding parameter regions).

RESULTS AND DISCUSSION
We summarize our main conclusions. Our results for the number of photonsn are plotted in Fig. 3 as a function of the detuning δω of the driving frequency and driving amplitude Ω R0 . It exhibits sharp structures along two curves corresponding to the one-and two-photon resonance conditions, Ω R = ω T − g 3n and Ω R = 2ω T − g 3n . Blue detuning, δω > 0, induces a strong population inversion of the qubit levels, which in resonance leads to one-qubit lasing. In experiments the effect can be measured as a strong increase of the number of photons in the resonator above the thermal values. Red detuning produces a one-qubit cooler with resulting photon numbers substantially below the thermal value.
The bistability of the solution of the Langevin description is illustrated in Fig. 4. In the range of bistability we expect a telegraph-like noise corresponding to the random switches between the two solutions.
Potentially useful applications of the considered scheme are the lasing behavior and the creation of a highly non-thermal population of the oscillator as well as the cooling. Within the accuracy of our approach we estimate that a population of ordern = 1 can be reached for optimal detuning. A more detailed analysis is required to determine the precise cooling limit.
So far we described an LC oscillator coupled to a flux qubit. But our analysis equally applies for a nanomechanical resonator coupled capacitively to a Josephson charge qubit (see Fig. 1b). In this case σ z stands for the charge of the qubit and both the coupling to the oscillator as well as the driving are capacitive, i.e., involve σ z . To produce the capacitive coupling between the qubit and the oscillator, the latter could be metalcoated and charged by the voltage source V x . The dc component of the gate voltage V g puts the system near the charge degeneracy point where the dephasing due to the 1/f charge noise is minimal. Rabi driving is induced by an ac component of V g . Realistic experimental parameters are expected to be very similar to the ones used in the examples discussed above, except that a much higher quality factor of the resonator (∼ 10 5 ) and a much higher number of quanta in the oscillator can be reached. This number will easily exceed the thermal one, thus a proper lasing state with Poisson statistics, appropriately named SASER [28], is produced. One should then observe the usual line narrowing with line width given by κN th /(4n) ∼ κ 2 N th /Γ 1 . Experimental observation of this line-width narrowing would constitute a confirmation of the lasing/sasing.