Sisyphus cooling and amplification by a superconducting qubit
M. Grajcar, S.H.W. van der Ploeg, A. Izmalkov, E. Il'ichev, H.-G. Meyer, A. Fedorov, A. Shnirman, Gerd Schön
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Sisyphus cooling and amplification by a superconducting qubit
M. Grajcar , , S.H.W. van der Ploeg , A. Izmalkov , E. Il’ichev ,H.-G. Meyer , A. Fedorov , A. Shnirman , and Gerd Sch¨on Institute of Photonic Technology, P.O. Box 100239, D-07702 Jena, Germany Department of Experimental Physics, Comenius University, SK-84248 Bratislava, Slovakia Quantum Transport Group, Delft University of Technology, 2628CJ Delft, The Netherlands Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, A-6020 Innsbruck, Austria Institut f¨ur Theoretische Festk¨orperphysik and DFG-Center for Functional Nanostructures (CFN),Universit¨at Karlsruhe, D-76128 Karlsruhe, Germany
Laser cooling of the atomic motion paved the way for remarkable achievements in the fields ofquantum optics and atomic physics, including Bose-Einstein condensation and the trapping of atomsin optical lattices. More recently superconducting qubits were shown to act as artificial two-levelatoms, displaying Rabi oscillations, Ramsey fringes, and further quantum effects . Coupling suchqubits to resonators brought the superconducting circuits into the realm of quantum electro-dynamics (circuit QED). It opened the perspective to use superconducting qubits as micro-coolersor to create a population inversion in the qubit to induce lasing behavior of the resonator .Furthering these analogies between quantum optical and superconducting systems we demonstratehere Sisyphus cooling of a low frequency LC oscillator coupled to a near-resonantly driven super-conducting qubit. In the quantum optics setup the mechanical degrees of freedom of an atom arecooled by laser driving the atom’s electronic degrees of freedom. Here the roles of the two degreesof freedom are played by the LC circuit and the qubit’s levels, respectively. We also demonstratethe counterpart of the Sisyphus cooling, namely Sisyphus amplification.For red-detuned high-frequency driving of the qubit the low-frequency LC circuit performs workin the forward and backward part of the oscillation cycle, always pushing the qubit up in energy,similar to Sisyphus who always had to roll a stone uphill. The oscillation cycle is completed with arelaxation process, when the work performed by the oscillator together with a quantum of energyof the high-frequency driving is released by the qubit to the environment via spontaneous emission.For blue-detuning the same mechanism creates excitations in the LC circuit with a tendency towardslasing and the characteristic line-width narrowing. In this regime “lucky Sisyphus” always rolls thestone downhill. Parallel to the experimental demonstration we analyze the system theoretically andfind quantitative agreement, which supports the interpretation and allows us to estimate systemparameters. The system considered is shown in the inset of Fig. 1. It consists of a three-junction flux qubit , with the two qubit FIG. 1: (a)
The energy levels of the qubit as a function of the energy bias of the qubit ε ( f x ) = 2Φ I p f x . The sinusoidal currentin the tank coil, indicated by the wavy line, drives the bias of the qubit. The starting point of the cooling (heating) cyclesis denoted by blue (red) dots. The resonant excitation of the qubit due to the high-frequency driving, characterized by Ω R ,is indicated by two green arrows and by the Lorentzian depicting the width of this resonance. The relaxation of the qubit isdenoted by the black dashed arrows. The inset shows a schematic of the qubit coupled to an LC circuit. The high frequencydriving is provided by an on-chip microwave antenna. (b) SEM picture of the superconducting flux qubit prepared by shadowevaporation technique. states corresponding to persistent currents of amplitude I p flowing clockwise and counterclockwise. When operatedin the vicinity of the degeneracy point, f x ≡ Φ x / Φ − / ≈
0, where Φ x is the magnetic flux applied to the qubitloop and Φ = h/ e the flux quantum, the Hamiltonian of the qubit in the basis of the persistent current states reads H = − ε ( f x ) σ z −
12 ∆ σ x . (1)Here σ x , σ z are Pauli matrices, ∆ is the tunneling amplitude, and ε ( f x ) = 2Φ I p f x is the energy bias. The energylevels of the isolated qubit, separated by ∆ E ( f x ) = p ε ( f x ) + ∆ are shown in Fig. 1. The qubit is driven by ahigh-frequency field with frequency ω d and coupled via a mutual inductance M to a low-frequency tank circuit withfrequency ω T ≃ π ×
20 MHz ≪ ∆ / ¯ h . Both circuit to the qubit can be included at this stage via their contributionsto the external flux Φ x . Since the eigenfrequency of the tank circuit, ω T , is much lower than the level spacing of thequbit, the oscillations of the current in the tank circuit can be treated in an adiabatic approximation, i.e., the currentin the tank circuit shifts the bias flux of the qubit by Φ x ( t ) = M [ I dc + I rf ( t )].The system mimics the Sisyphus mechanism of damping (cooling) and amplification (heating) known from quantumoptics . This mechanism is illustrated in Fig. 1. Here we describe the damping (cooling, hence marked in blue) fora situation where the driving is red-detuned, ¯ hω d < ∆ E ; the amplification (marked in red) for blue-detuning can bedescribed in an analogous way. The oscillations of the current in the tank circuit, I rf ( t ), lead to oscillations of ε ( f x )around a value determined by the dc component, I dc . In the first part of the cycle, when the qubit is in the groundstate, the current shifts the qubit towards the resonance, ∆ E = ¯ hω d , i.e., the energy of the qubit grows due to workdone by the LC circuit. Once the system reaches the vicinity of the resonance point, the qubit can get excited, theenergy being provided by the high-frequency driving field. With parameters adjusted such that this happens at theturning point of the oscillating trajectory, the qubit in the excited state is now shifted by the current away from theresonance, such that the qubits energy continues to grow. Again the work has to be provided by the LC circuit. Thecycle is completed by a relaxation process which takes the qubit back to the ground state. The maximum effect isachieved when the driving frequency and relaxation rate are of the same order of magnitude. If the relaxation istoo slow the state is merely shifted back and forth adiabatically during many periods of oscillations. Note that thecomplete cycle resembles the ideal Otto-engine thermodynamic cycle .Two types of measurements have been performed on the system. In the first the LC tank circuit is additionallydriven near-resonantly by a low- frequency rf current, and the response of the LC circuit is detected using lock-intechniques. In these measurements we identify the influence of the high-frequency driven qubit on the effective qualityfactor and eigenfrequency of the tank circuit. We associate a reduction of the effective quality factor with cooling andidentify regions in parameter space where the effect is optimized.In the second type of measurements the low-frequency rf -driving is switched off, while the emission of the LCtank circuit is monitored by a spectrum analyzer. This analysis probes the influence of the qubit on the effectivequality factor, and, most importantly, allows us to determine the energy stored in the tank circuit, i.e., its effectivetemperature. Thus we are able to demonstrate cooling or heating of the LC oscillator.We start with a qualitative analysis of the first type of measurement. The LC tank circuit is driven by a currentsource with amplitude I dT . Then the amplitude, V T , of the voltage oscillations across the tank circuit, which ismeasured in the experiment, is V T = ω T L T I T = Q ω T L T I dT , (2)where I T = p h I i = QI dT is the actual amplitude of the rf current in the inductance, and Q is the effective qualityfactor of the tank circuit. It is given by the ratio, Q = 2 πW T /A , between the energy stored in the tank, W T = L T I T / A . The latter is composed of two contributions, A = A T + A Sis . The intrinsic losses of thetank are given by A T = 2 πW T /Q , where Q is the intrinsic quality factor of the tank circuit. The average work doneby the tank on the qubit in one period, A Sis , can be estimated as follows: We consider the optimal situation when theoscillator brings the qubit into the resonance at the turning point of its trajectory (Fig. 1). We assume further thatthe state of the qubit is instantaneously “thermalized”, with equal probabilities to remain in the ground state or toget excited. In the latter case, after leaving the area of resonance the qubit can relax with rate Γ R . The probability ofqubit relaxing during the time 0 < t < T (here T = 2 π/ω T ) within an interval dt is given by dP = exp( − Γ R t ) Γ R dt .Thus we obtain the average value of the work (taking into account the probability 1 / A Sis = M I p I T Z dP [1 − cos( ω T t )]= M I p I T f (Γ R , ω T ) , (3)where f (Γ R , ω T ) ≡ (cid:16) − e − π Γ R /ω T (cid:17) ω T ω T + Γ R . (4) -0.02 -0.01 0.00 0.01 0.02020040060080010001200 V T ( n V ) f dcx a | V T - V T | ( n V ) V T0 (nV) b FIG. 2: (a)
Amplitude of the rf tank voltage as a function of the dc magnetic bias of the qubit, f dcx , for a microwave drivingfrequency ω d = 2 π × .
125 GHz and various amplitudes of the rf current driving the tank circuit. (b) The height of the peaks(solid squares) and dips (solid circles) as a function of the voltage amplitude of the unloaded tank circuit V T . The voltage V T is equal to, e.g., V T taken at f dcx = 0 .
02 where the effect of the interaction between the tank circuit and the qubit is negligible.The height saturates near 200 nV.
This function demonstrates that the optimal situation for damping or amplification is reached when Γ R ∼ ω T . Forthe effective quality factor we obtain Q = Q (cid:18) ± A Sis Q πW T (cid:19) − = Q (cid:18) ± M I p Q ω T f (Γ R , ω T ) πV T (cid:19) − , (5)where ± stand for the damping/amplification. Substituting into Eq. (2) we arrive at V T − V T = ∓ M ω T I p Q f (Γ R , ω T ) /π , (6)where V T = Q ω T L T I dT is the voltage on the tank circuit far from the resonance. Eq. (6) provides the estimate forthe increase/decrease of the tank voltage for large driving currents I dT , such that the qubit spends most of the timeaway from the resonant excitation area. For weaker driving, when the times spend away and within the excitationarea are comparable, the damping effect becomes weaker.Our results of the first type of experiment are shown in Fig. 2. The dips (peaks) correspond to the Sisyphusdamping (amplification) of the tank circuit, i.e., to the decrease (increase) of the effective quality factor Q . Thecentral dip is due to the shift of the oscillator frequency when the qubit is at it’s degeneracy point and is not relatedto the Sisyphus effect. We observe from Fig. 2b that the additional voltage saturates for large driving amplitudes atapproximately 200 nV. Using this value and Eq. (6) we see that indeed the relaxation time is close to the period ofoscillations, 1 / Γ R ≈ . T .The results for the second type of experiment are shown in Fig. 3. The comparison shows that at the bias pointcorresponding to maximum Sisyphus damping the resonant line widens in good agreement with the results for thedriven LC circuit. At the bias point corresponding to maximum amplification the line narrows, showing a tendencytowards lasing behavior. We can extract the quality factors in these two regimes, as well as for the bias point far fromthe resonance where no additional damping occurs. The comparison of the experimental results with Eq. (5) yields aquantitatively reasonable agreement.Finally, the results displayed in Fig. 3 allow us to estimate the efficiency of the cooling or heating in the two regimes.Integrating the power spectra we observe that in the damping regime the number of photons in the LC circuit decreasesby about 8% compared to the undamped case. The cooling effect was observed also for other microwave frequenciesand lies in the range of 6% − Q Sis and temperature T Sis . The standard analysis gives the totalquality factor and temperature, Q − = Q − + Q − and T Q − = T Q − + T Sis Q − , respectively. Here T is the S v ( n V / H z ) Frequency (MHz) c FIG. 3: (a)
Spectral density of the voltage noise in the LC circuit, S V ( ω ), measured when the low-frequency rf -driving isswitched off while the high-frequency qubit driving is fixed at ω d = 2 π × S V is shown as a function of frequency (verticalaxis) and normalized magnetic flux in the qubit f dc x (horizontal axis). (b) The integral, R S V ( ω ) dω , evaluated using the datafrom panel (a) as a function of f dc x . The integral is directly proportional to the number of quanta (effective temperature) of thetank circuit. At optimal dc bias f dc x the effective temperature T of the oscillator is lower than the temperature T away fromthe resonance, ( T − T ) /T = 0 .
08, corresponding to an 8% cooling. (c)
Spectral density S V ( ω ) measured at three differentvalues of f dc x corresponding to damping (blue), amplification (red), and away from the resonance (black). These values of f dc x are marked, respectively, by blue, red, and black arrows in panel (a) . temperature of the oscillator without Sisyphus damping, which in our case is determined mostly by the amplifier.Our data show that Q ∼ Q /
2. Thus the coupling strength of the Sisyphus mechanism is comparable to that of therest of the environment, i.e., Q Sis ≈ Q ≈ T Sis were comparable with T and we have only little cooling.In contrast, in the amplification mode of our experiment the line gets much sharper, which formally can be expressedby choosing Q Sis negative, Q Sis ≈ − T Sis . Its value is not known, but we can get a lower bound for the effectivetemperature from the inequality
T /Q > T /Q . It predicts an increase in temperature and number of photons bya factor ∼
2. The experiment shows a 36% increase, which demonstrates the expected trend, even though somequantitative discrepancy remains.We now outline the theoretical analysis of the problem. Quantizing the oscillations in the tank circuit we arrive atthe Hamiltonian H = − ε (cid:0) f dcx (cid:1) σ z −
12 ∆ σ x − ¯ h Ω R cos( ω d t ) σ z + ¯ hω T a † a + g σ z (cid:0) a + a † (cid:1) . (7)where the coupling constant is g = M I p I T, , and I T, = p ¯ hω T / L T is the amplitude of the vacuum fluctuationof the current in the LC oscillator. The third term of Eq. (7) describes the high-frequency driving with amplitudeΩ R . After transformations to the eigenbasis of the qubit and some approximations appropriate for the consideredsituations (and described in the Appendix) the Hamiltonian reduces to H = −
12 ∆
E σ z + ¯ h Ω R cos ( ω d t ) cos ζ σ x + ¯ hω T a † a + g sin ζ σ z (cid:0) a + a † (cid:1) − g ∆ E cos ζ σ z (cid:0) a + a † (cid:1) , (8)with ∆ E ≡ √ ǫ + ∆ and tan ζ ≡ ε/ ∆. Thus we obtain the effective Rabi frequency Ω R cos ζ and the effectiveconstant of qubit-oscillator linear coupling g sin ζ . As we need both these terms for the Sisyphus cooling the qubitshould be biased neither at the symmetry point nor very far from it. The second order term ∝ g is responsible, e.g.,for the qubit-dependent shift of the oscillator frequency .To account for the effects of dissipation we consider the Liouville equation for the density operator of the systemincluding the two relevant damping terms, ˙ ρ = − i ¯ h [ H, ρ ] + L Q ρ + L R ρ . (9) dc f x FIG. 4: Color lines: experimental data. Black lines: numerical solution of Eqs. (9) for Ω R = 2 π × . R = 1 . · s − ,Γ ∗ ϕ = 5 . · s − . The central dip is due to the quadratic coupling term in (8) which causes a shift of the oscillator frequency . As far as the qubit is concerned we consider spontaneous emission with rate Γ R and pure dephasing with rate Γ ∗ ϕ .Hence, we have L Q ρ = Γ R σ − ρσ + − ρσ + σ − − σ + σ − ρ )+ Γ ∗ ϕ σ z ρσ z − ρ ) . (10)We neglect the excitation rate since the qubit’s energy splitting exceeds the temperature. Thus the standard longi-tudinal relaxation time T is given by T − = Γ R . The rates Γ R and Γ ∗ ϕ may depend on the working point, i.e., on ζ , which in turn is determined by the driving frequency ω d by the condition ¯ hω d ≈ ∆ E . The rates should describethe dissipation around these values of ζ . It should be mentioned that pure dephasing is frequently caused by the 1 /f noise , for which the Markovian description is not applicable. We hope, however, that the main features are stillcaptured provided Γ ∗ ϕ is chosen properly.The resonator damping term can be written in usual form , L R ρ = κ N th + 1) (cid:0) aρa † − a † aρ − ρa † a (cid:1) + κ N th (cid:0) a † ρa − aa † ρ − ρaa † (cid:1) , (11)where κ = ω T /Q characterizes the strength of the resonator damping, and N th = [exp(¯ hω T /k B T ) − − is thethermal average number of photons in the resonator. We solve the master equation (9) numerically in the quasi-classical limit, i.e., for n ≡ h a † a i ≫
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Appendix
After transformation to the eigenbasis of the qubit the Hamiltonian (7) becomes H = −
12 ∆
E σ z − ¯ h Ω R cos ( ω d t ) (sin ζ σ z − cos ζ σ x )+ ¯ hω T a † a + g (sin ζ σ z − cos ζ σ x ) (cid:0) a + a † (cid:1) , with tan ζ = ε/ ∆ and ∆ E ≡ √ ε + ∆ . Because of the large difference of the energy scales between the qubit and theoscillator, ∆ E ≫ ¯ hω T , we can drop within the usual rotating wave approximation (RWA) the longitudinal driving term − ¯ h Ω R cos ( ω d t ) sin ζ σ z . On the other hand, we retain the transverse coupling term, − g cos ζ σ x (cid:0) a + a † (cid:1) , but trans-form it by employing a Schrieffer-Wolff transformation, U S = exp ( iS ), with generator S = ( g/ ∆ E ) cos ζ (cid:0) a + a † (cid:1) σ yy