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(Color online) Numerical (solid lines) and analytical (dashed lines) dependence of zero-frequency detector noise S a a 0 and cross-noise S a b 0 on the phase shift ϕ between the bias voltage combs for several values of the pulse width δ t a , b , and also for the harmonic biasing. Almost complete noise anticorrelation at ϕ = ± π indicate persistent Rabi oscillations.... Analyzed system: a double-quantum-dot qubit measured by two QPC detectors, which are biased by combs of short voltage pulses with frequency Ω coinciding with the Rabi frequency Ω R .
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Figure   fig:FreqVsAmpl(a) shows linecuts of the experimental (black dots) and numerical (full red lines) chevrons. The linecuts are taken at the frequency ω F C corresponding to the maximum-visibility sideband oscillations, indicated by the full and dashed vertical lines in Fig. 3. The agreement between the experiments and simulations is excellent. In particular, the decay rate of the oscillations can be explained by the separately measured loss of the qubit and cavity and roughly corresponds to κ + γ 1 / 2 , where γ 1 is the bare transmon relaxation rate. This is expected for oscillations between states | e 0 and | g 1 . It also indicates that for these powers, the visibility loss can be completely attributed to damping. The lack of experimental points at pulse widths < 30 n s is a technical limit of the present configuration of our electronics that can be improved in future experiments.x x... (color online) (a) Schematic of energy levels in a combined qubit-resonator system, showing first-order red sideband transition. (b) Optical microscope image with inset showing expanded view of one of the qubits. The terminations of the flux-bias lines for both qubits are visible, and they are used for both dc bias and FC signals. (c) Schematic of qubit-cavity layout and signal paths.... We used a sample consisting of two asymmetric transmon qubits capacitively coupled to the voltage antinodes of a coplanar waveguide resonator [Fig. fig:schem(b, c)]. The cavity had a bare fundamental resonance frequency ω r / 2 π = 8.102 G H z and decay rate κ / 2 π = 0.37 M H z . Qubit-state measurements were performed in the high-power limit . The qubits, labeled Q1 and Q2, were designed to be identical, with mutual inductances to their bias lines of 1 p H for Q2 and 2 p H for Q1. The qubits were excited by microwave pulses sent through the resonator, and the flux lines were used for dc flux biasing of the qubits as well as the high-speed flux modulation pulses for exciting sideband transitions. The dc flux lines included cryogenic filters before connecting to a bias-T for joining to the ac flux line, which had 20 / 6 / 10 d B of attenuation at the 4 K / 0.7 K / 0.03 K plates. The distribution of cold attenuators and the flux-bias mutual inductances were chosen as a compromise to allow for a sufficient flux amplitude for high-speed modulation of the qubit energy levels with negligible Joule heating of the refrigerator while avoiding excessive dissipation coupled to the qubits from the flux-bias lines.... (color online) Spectroscopy vs. flux for Q2 showing g-e (solid blue points) and e-f (hollow red points) transition frequencies. Blue and red lines correspond to numerical fits. Heavy black line shows bare cavity resonance frequency. Vertical dashed line indicates flux bias point for sideband measurements described in subsequent figures along with ac flux drive amplitude, 2 Δ Φ = 70.9  m Φ 0 , corresponding to 2 Δ ω g e / 2 π = 572  MHz, used in Figs. 3(c), 4(c).... Figure   fig:FreqVsAmpl(d) shows the sideband oscillation frequency Ω / 2 π extracted from the experimental linecuts (blackxdots) as a function of the flux-modulation amplitude Δ Φ . As expected from Eq. ( eq:H:t), whose prediction is given by the solid black line, the dependence of Ω with Δ Φ is linear at low amplitude and deviates at larger amplitudes. Beyond this simple model with only two transmon levels, quantitative agreement is found between the measured data and numerical simulations (full red line). For the numerical simulations, the link between the theoretical flux modulation amplitude Δ Φ and applied power is made by taking advantage of the linear dependence of Ω with Δ Φ at low power. Because of this, it is possible to convert the experimental flux amplitude from arbitrary units to m Φ 0 using only the lowest drive amplitude for calibration.... (color online) (a-c) Experimental data showing sideband oscillations as a function of pulse duration vs. flux-drive frequency. The amplitude of the flux pulse is reduced by (a) 10 d B , (b) 4 d B relative to (c). (d-f) Corresponding numerical simulations of sideband oscillations vs. drive frequency. Vertical white lines running through each plot indicate the frequency slices used in Fig. fig:FreqVsAmpl.... (color online) (a),(b),(c) Sideband oscillations corresponding to the white slices in Fig. fig:chevron(a-c). Experimental points correspond to black dots; numerical simulations (not fits) indicated by red lines. (d) Sideband oscillation frequency vs. flux drive amplitude (lower horizontal axis) or corresponding frequency modulation amplitude (upper horizontal axis). The dashed line shows a linear fit to the low frequency data points, while the red solid line indicates the theoretical dependence from the numerical simulations. The full black line shows the analytical sideband oscillation frequency from Eq. ( eq:H:t).
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In our sample geometry, the qubit is spatially separated from the rest of the circuitry. The qubit is enclosed by a superconducting quantum interference device (SQUID) that is inductively coupled to the qubit. The SQUID functions as a detector for the qubit state: the switching current of the SQUID is sensitive to the flux produced by the current in the qubit. The qubit is also enclosed by a larger loop containing on–chip capacitors that provide a well–defined electromagnetic environment for the SQUID and filtering of the measurement leads. The lead inductance L and capacitance C in the outer loop constitute an LC oscillator [see Fig.  fig1(b)] with resonance frequency ω r = 2 π ν r = 1 / L C . The LC oscillator is described by a simple harmonic oscillator Hamiltonian: H o s c = ℏ ω r a † a + 1 / 2 , where a † ( a ) is the plasmon creation (annihilation) operator. The qubit is coupled to the LC oscillator via the mutual inductance M , giving an interaction Hamiltonian H I = h λ σ z a † + a , where the coupling constant is h λ = M I p ℏ ω r / 2 L . The total system is thus described by a Jaynes-Cummings type of Hamiltonian H = h / 2 ϵ σ z + Δ σ x + ℏ ω r a † a + 1 / 2 + h λ σ z a † + a . We denote the state of the system by | Q , i , with the qubit either in the ground ( Q = g ) or excited ( Q = e ) state, and the oscillator in the Fock state ( i = 0 , 1 , 2 , ). The parameters of the system can readily be engineered during fabrication; the qubit gap is determined by α and the junction resistance, the oscillator plasma frequency is fully determined by L and C , and the coupling between the qubit and the oscillator can be tuned by M and L .... fig1 (a) SEM micrograph of the sample. The qubit and the detector SQUID enclosing it are the small square loops in the lower center picture. The square plates at the top of the picture are the top plates of the on-chip capacitors separated by an insulator from the large bottom plate. (b) A close up of the qubit and the SQUID. The qubit dimension is 10.2 × 10.4   μ m 2 . (c) Equivalent circuit of the sample. The Josephson junctions are indicated by crosses: three in the inner qubit loop and two in the SQUID. The LC mode is indicated by the dashed line. The inductance and capacitances are calculated from the geometry to be L = 140  pH and C = 10  pF, and the qubit LC oscillator mutual inductance to be M = 5.7  pH. The current and voltage lines are filtered through a series combination of copper powder filters and lossy coaxial cables at mixing chamber temperature and on–chip resistors ( R V = 3 k Ω and R I = 1 k Ω ).... fig4 (a) Rabi oscillations when a 2 ns long pulse with frequency ν e x = 4.35 GHz and amplitude A M W ∝ 10 P o s c / 20 is inserted between the π pulse and the shift pulse. (b) Measured Rabi oscillations at different drive powers (symbols), and a fit (solid curve) to ∑ n = 0 3 a n cos n + 1   Ω R t exp - Γ t with a 0 , , a 3 and Ω R as the only fitting parameters ( Γ is fixed from a fit to the -100 dBm curve). (c) The weights of the different frequency components a 0 , , a 3 obtained from the fit as a function of the drive amplitude (The red arrows show the position of the curves in (b). The error-bars indicate the errors obtained from the fitting procedure.... fig2 (a) Spectroscopic characterization of the qubitoscillator system showing the LC oscillator at ν r = 4.35  GHz and the qubit dispersion around the gap of Δ = 2.1  GHz. (b) A close up of the region around 4.35 GHz (indicated by the red square in (a) showing an avoided crossing. The lines are guides for the eye. (c) Schematic of the pulse sequence used to obtain the spectroscopy in (b): the qubit operating point is fixed at 10.5 GHz and via the MW line a shift pulse of variable height moves the operating point to the vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to the shift pulse. Here the second MW pulse is phase shifted by 180 ∘ . The phase shifted second pulse damps the oscillations in the LC circuit , and is crucial in terms of resolving the relatively weak qubit signal in this region. After the MW pulses the qubit state is measured by applying a measurement pulse to the SQUID (green curve). The spectroscopy in Fig. 2(a) was obtained with the same scheme, but without the second phase shifted pulse.... Next we investigate the dynamics of the coupled system in the time domain. We performed a measurement cycle where we first excited the qubit and then brought the qubit and the oscillator into resonance where the exchange of a single energy quantum between the qubit and oscillator manifests itself as the vacuum Rabi oscillation | e , 0 ↔ | g , 1 (see Fig. 3). Figure. 3(c) is a schematic of the pulse sequence: We started by fixing the qubit operating point far from the resonance point [point 3 in Fig.  fig3(c)] and prepared the qubit in the excited state by employing a π pulse. The π pulse was followed by a shift pulse, which brought the qubit into resonance with the oscillator for the duration of the shift pulse. After the shift pulse the qubit and the oscillator were brought back into off–resonance and the measurement pulse was applied to detect the state of the qubit. It is important to note that the rise time of the shift pulse, τ r i s e = 0.8  ns, is adiabatic with respect to both the qubit and the oscillator, τ r i s e > 1 / E 2 π / ω r , but non-adiabatic with respect to the coupling of the two systems, τ r i s e oscillations is thus different from that of normal Rabi oscillations where the system is driven by an external classical field and oscillates between two energy eigenstates. Also, in the normal Rabi oscillations the Rabi frequency is determined by the drive amplitude whereas the vacuum Rabi oscillation frequency is determined only by the system‘s intrinsic parameters. The observed Rabi oscillations are in excellent agreement with those calculated numerically [solid line in Fig. 3(a)]. The numerical calculation uses the total Hamiltonian and incorporates the effects of the decoherence of the qubit and the oscillator. The calculation was performed with the known qubit and LC oscillator parameters (obtained from spectroscopy and qubit experiments: qubit dephasing rate Γ φ = 0.1  GHz, qubit relaxation rate Γ e = 0.2  MHz, Δ = 2.1  GHz, ω r / 2 π = 4.35  GHz) and by treating the coupling constant and oscillator dephasing and relaxation rates as fitting parameters. From the fit we extracted the coupling constant λ = 0.2  GHz, oscillator dephasing rate Γ φ = 0.3  GHz, and relaxation rate Γ e = 0.02  GHz. The coupling constant extracted from the fit agrees well with that calculated with the mutual inductance λ = 0.216  GHz.... fig3 (a) Vacuum Rabi oscillations (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when the system is in resonance. (c) The qubit energy level diagram and pulse sequence for the vacuum Rabi measurements. The π pulse (4.6 ns long at -16 dBm) on the qubit brings the system from state 1 to 2 and the shift pulse changes the flux in the qubit by Φ s h i f t , which, in turn, changes the operating point from 2 to 3 where the system undergoes free evolution between | e , 0 and | g , 1 at the vacuum Rabi frequency Ω R until the shift pulse ends and the system returns to the initial operating point where the state is measured to be either in 2 or 4.
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The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-frequency bath. The parameters are the same as in Fig. 1. ... The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the frequency ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz. ... The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-frequency bath in different values of Ω0, the other parameters are the same as in Fig. 1. ... The response functions of the Ohmic bath in (a) low and (c) medium frequencies and effective bath in (b) low and (d) medium frequencies. The parameters are the same as in Fig. 1. The cut-off frequencies for the two cases are taken according to Fig. 2. ... The sketch map on the low-, medium-, and high-frequency baths.
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Sketch of the flux qubit (blue) coupled to a dc-SQUID. The interaction is characterised by the linear coupling g 1 , which depends linearly on the SQUID bias current I b , and the quadratic coupling g 2 . The SQUID with Josephson inductance L J is shunted by a capacitance C . The frequency shift of the resulting harmonic oscillator (green) can be probed by external resonant ac-excitation A cos Ω a c t via the transmission line (black), in which the quantum fluctuations ξ i n q m t are also present.... eq:in-out-total1. In an experiment, this can be achieved by lock-in techniques which we mimic in the following way : First, we focus on the associated spectrum ξ o u t ω depicted in figure  fig:qubit-osc-phase-spectrum(b). It reflects the qubit dynamics in terms of two sidebands around the central peak related to the oscillator frequency, here chosen as Ω = 10 ω q b . The dissipative influence of the environment, modelled by a transmission line (see figure fig:setup), is reflected in a broadening of this peak. The corresponding oscillator bandwidth is given as 2 α Ω , where α denotes the dimensionless damping strength; see app:QME. Here, we recall that the oscillator is driven resonantly by the external driving signal A cos Ω a c t , that is, Ω = Ω a c . In the time domain, the sidebands correspond to the phase-shifted signal ξ o u t t = A cos Ω t - ϕ e x p t with slowly time-dependent phase ϕ e x p t . In order to obtain this phase ϕ e x p t , we select the spectral data from a frequency window of size 2 Δ Ω centred at the oscillator frequency Ω , which means that ξ o u t ω is multiplied with a Gaussian window function exp - ω - Ω 2 / Δ Ω 2 . We choose for the window size the resonator bandwidth, Δ Ω = α Ω , which turns out to suppress disturbing contributions from the low-frequency qubit dynamics. Finally, we centre the clipped spectrum at zero frequency and perform an inverse Fourier transform to the time domain. If the phase shift φ e x t was constant, one could use a much smaller measurement bandwidth. Then the outcome of the measurement procedure would correspond to homodyne detection of a quadrature defined by the phase shift and yield a value ∝ cos φ e x p .... Time-resolved measurement of coherent qubit oscillations at the degeneracy point ϵ = 0 . The full qubit-oscillator state was simulated with the quantum master equation  eq:blochredfield with N = 10 oscillator states and the parameters Ω = Ω a c = 10 ω q b , g 1 = 0.1 ω q b , g 2 = 0.01 ω q b , A = 1.0 ω q b . The dimensionless oscillator dissipation strength is α = 0.12 . The resonator bandwidth is given by 2 α Ω = 2.4 ω q b . (a) Lock-in amplified phase ϕ e x p t (dashed green lines), compared to the estimated phase ϕ t (solid red line) of the outgoing signal ξ o u t t . Here, ϕ t ∝ σ x t [cf.  equation  eq:dr-osc-phase], which is corroborated by the inset showing that σ x t performs oscillations with (angular) frequency ω q b . (b) Power spectrum ξ o u t ω for the resonantly driven oscillator (blue solid line). The sidebands stemming from the qubit dynamics are visible at frequencies Ω ± ω q b . In order to extract the phase information, we apply a Gaussian window function with respect to the frequency window of half-width Δ Ω = 1.2 ω q b , which turns out to be the optimal value for the measurement bandwidth.... We consider a superconducting flux qubit coupled to a SQUID  as sketched in figure  fig:setup. The SQUID is modelled as a harmonic oscillator, which gives rise to the Hamiltonian... In figure  fig:qubit-osc-fid(a) we depict the fidelity defect δ F = 1 - F between ϕ e x p t and ϕ t as a function of the oscillator frequency Ω = Ω a c for different quadratic coupling coefficients g 2 . As expected, the overall fidelity is rather insufficient for small oscillator frequency Ω oscillator bandwidth is too small to resolve the qubit dynamics, i.e., if ω q b oscillator frequencies, we again observe an increase of the fidelity defect, which occurs the sooner the smaller g 2 . This latter effect, which is only visible for the smallest value of g 2 in figure  fig:qubit-osc-fid(a), is directly explained by a reduced maximum angular visibility of the phase ϕ t ∝ g 2 / Ω . Thus, figure  fig:qubit-osc-fid(a) provides a pertinent indication for the validity frame of our central relation ... If the qubit is only weakly coupled to the oscillator, and if the latter is driven only weakly, the qubit’s time evolution is rather coherent (see section  sec:sn on qubit decoherence). For this scenario, figure  fig:qubit-osc-phase-spectrum(a) depicts the time-dependent phase ϕ t computed with the measurement relation... (a) Fidelity defect δ F = 1 - F for the phases ϕ t and ϕ e x p t and (b) time-averaged trace distance D ̄ between the density operators of a qubit with finite coupling to the oscillator and a reference qubit without oscillator. Both quantities are depicted for various coupling strengths g 2 in dependence of the oscillator frequency Ω . All other parameters are as in figure  fig:qubit-osc-phase-spectrum.
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(a) Spectrum of device B. The spectral line at 2 is the resonator, whereas the qubit tunnel coupling is Δ = 5.4 . (b) Rabi frequency vs bias current , measured at f = 5.4 and = 0 and for two different microwave drive amplitudes . Similar to device A, the Rabi frequency depends strongly on , and scales linearly with drive amplitude. The black lines are fits to Eqs. ( eq:drive, eq:Rabi), using the same coupling parameters for both sets of data. Note that the range of in fig:RabiLongT1(b) is several times larger than in fig:Rabi(a).... We have investigated two devices with similar layouts but slightly different parameters, both made of aluminum. Device A was designed and fabricated at MIT Lincoln Laboratory and device B was designed and fabricated at NEC. Figure fig:Sample(b) shows a spectroscopy measurement of device A versus applied flux, with the qubit flux detuning defined as = Φ + Φ 0 / 2 and Φ 0 = h / 2 e . The qubit frequency follows = Δ 2 + ε 2 , where the tunnel coupling Δ = 2.6 is fixed by fabrication and the energy detuning ε = 2 I P / h is controlled by the applied flux Φ ( I P is the persistent current in the qubit loop). The resonator frequency is around 2.3 and depends only weakly on and . In addition, there are features visible at frequencies corresponding to the sum of the qubit and resonator frequencies, illustrating the coherent coupling between the two systems .... (a) Rabi frequency of qubit A, measured vs at = 0 . The driving field seen by the qubit contains two components: one is due to direct coupling to the antenna, the other is due to the coupling mediated by the resonator. (b) Rabi traces for a few of the data points in panel (a). The microwaves in the antenna have the same amplitude and frequency for all traces. (c) Direct coupling between the antenna and the qubit, extracted from measurements similar to the one shown in panel (a). The coupling depends only weakly on frequency. (d) Microwave current in the resonator, induced by a fixed microwave amplitude in the antenna. The black line is a fit to the square root of a Lorentzian, describing the oscillation amplitude of a harmonic oscillator with f r = 2.3 and Q = 100 .... Figures  fig:Rabi(c) and fig:Rabi(d) show how the two drive components depend on microwave frequency, measured by changing the static flux detuning to increase the qubit frequency [see... (a) Circuit diagram of the qubit and the oscillator. The qubit state is encoded in currents circulating clockwise or counterclockwise in the qubit loop (blue arrow), while the mode of the harmonic oscillator is shown by the red arrows. (b) Spectrum for device A, showing the qubit and the harmonic oscillator. In addition, the two-photon qubit ( / 2 ) and the qubit+resonator ( + ) transitions are visible. (c) Flux induced in the qubit loop by the dc bias current . The black lines are parabolic fits. (d) First-order coupling between the qubit and the ground state of the harmonic oscillator, showing that the coupling is tunable by adjusting . The coupling is zero at = * , which is slightly offset from = 0 due to fabricated junction asymmetry. The derivative ε is calculated from the curves in panel (c). The qubit parameters are: I P = 175 for device A and I P = 180 for device B. The resonators have quality factors Q ≈ 100 . The right-hand axis is calculated using = 2.2 and C e f f = 2 C = 14 for both samples.... Having determined the coupling coefficients, we turn to analyzing how the presence of the resonator influences the qubit’s driven dynamics. Figure fig:Rabi(a) shows the extracted Rabi frequency of qubit A as a function of , measured at = 2.6 . We find that changes by a factor of five over the range of the measurement, which is surprising since both the amplitude and the frequency of the microwave current in the antenna are kept constant. The data points were obtained by fitting Rabi oscillations to decaying sinusoids, a few examples of Rabi traces for different values of are shown in... (a) Decay envelopes of the Rabi and rotary-echo sequences for device B, measured with = 65 . The solid lines are fits to eq:f. (b) Decay times for Rabi and rotary echo, extracted from fits similar to the ones shown in panel (a). The dashed line shows the upper limit set by qubit energy relaxation. The dotted line marks the position for the decay envelope shown in panel (a). (c,d) Schematic diagrams describing the two pulse sequences in (a) and (b). For rotary echo, the phase of the microwaves is rotated by 180 ∘ during the second half of the sequence.... To further investigate how the presence of the resonator affects the qubit dynamics at large detunings, we performed measurements on device B. Figure fig:RabiLongT1(a) shows a spectrum of that device, where the qubit and the resonator mode ( f r = 2 ) are clearly visible. This device has a larger tunnel coupling ( Δ = 5.4 ), which allows us to operate the qubit at large frequency detuning from the resonator while still staying at ε d c = 0 , where the qubit, to first order, is insensitive to flux noise . The qubit-resonator detuning corresponds to several hundred linewidths of the resonator, which is the regime of most interest for quantum information processing .... Figure  fig:RabiLongT1(b) shows the Rabi frequency vs bias current of device B, measured at f = 5.4 and for two different values of the microwave drive current . Similarly to
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The profile of coherent quantum oscillations in an unbiased qubit dephased by the non-Gaussian noise with characteristic amplitude v 0 = 0.15 Δ and correlation time τ = 300 Δ -1 obtained by direct simulation of qubit dynamics with noise. Solid line is the exponential fit of the oscillation amplitude at large times. Dashed line is the initial 1 / t decay caused by effectively static distribution of v .... The rate γ of exponential qubit decoherence at long times t ≫ τ for ε = 0 and (a) Gaussian and (b) a model of the non-Gaussian noise with characteristic amplitude v 0 and correlation time τ . Solid lines give analytical results: Eq. ( e7) in (a) and Eq. ( e16) in (b). Symbols show γ extracted from Monte Carlo simulations of qubit dynamics. Note different scales for γ in parts (a) and (b). Inset in (b) shows schematic diagram of qubit basis states fluctuating under the influence of noise v t .
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An important feature of the qubit relaxation in the presence of driving is that the stationary distribution over the qubit states differs from the thermal Boltzmann distribution. If the oscillator-mediated decay is the dominating qubit decay mechanism, the qubit distribution is determined by the ratio of the transition rates Γ e and Γ g . One can characterize it by effective temperature T e f f = ℏ ω q / k B ln Γ e / Γ g . If the term in curly brackets in the numerator of Eq. ( eq:resonant_power_spectrum) is dominating, T e f f ≈ 2 T , but if the field parameters are varied so that this term becomes comparatively smaller T e f f increases, diverges, and then becomes negative, approaching -2 T . Negative effective temperature corresponds to population inversion. The evolution of the effective temperature with the intensity of the modulating field is illustrated in Fig.  fig:effect_temp.... Apart from the proportionality to r a 2 , the attractor dependence of Γ 1 is also due to the different curvature of the effective potentials around the attractors. For weak oscillator damping κ ≪ ν a , the parameter ν a in Eq. ( eq:resonant_power_spectrum) is the frequency of small-amplitude vibrations about attractor a . It sets the spacing between the quasienergy levels, the eigenvalues of the rotating frame Hamiltonian H S r close to the attractor. The function Re  N + - ω has sharp Lorentzian peaks at ω = ± ν a with halfwidth κ determined by the oscillator decay rate. The dependence of ν a on the control parameter β is illustrated in Fig.  fig:nu_a.... Left panel: Squared scaled attractor radii r a 2 as functions of the dimensionless field intensity β for the dimensionless friction κ / | δ ω | = 0.3 . Right panel: The effective frequencies ν a / | δ ω | for the same κ / | δ ω . Curves 1 and 2 refer to small- and large amplitude attractors.... The scaled decay rate factors for the excited and ground states, curves 1 and 2, respectively, as functions of scaled difference between the qubit frequency and twice the modulation frequency; Γ 0 = ℏ C Γ r a 2 / 6 γ S . Left and right panels refer to the small- and large-amplitude attractors, with the values of β being 0.14 and 0.12, respectively. Other parameters are κ / | δ ω = 0.3 , n ̄ = 0.5 .... The scaled decay rates Γ e , g as functions of detuning ω q - 2 ω F are illustrated in Fig.  fig:decay_spectra. Even for comparatively strong damping, the spectra display well-resolved quasi-energy resonances, particularly in the case of the large-amplitude attractor. As the oscillator approaches bifurcation points where the corresponding attractor disappears, the frequencies ν a become small (cf. Fig.  fig:nu_a) and the peaks in the frequency dependence of Γ e , g move to ω q = 2 ω F and become very narrow, with width that scales as the square root of the distance to the bifurcation point. We note that the theory does not apply for very small ω q - 2 ω F | where the qubit is resonantly pumped; the corresponding condition is m ω 0 Δ q δ C r e s 2 r a 2 / ℏ ω q 2 ≪ T 1 T 2 -1 + ω q - 2 ω F 2 T 1 T 2 . For weak coupling to the qubit, Γ e ≪ κ , it can be satisfied even at resonance.... Picot08, and was increasing with the driving strength on the low-amplitude branch (branch 1 in the left panel of Fig.  fig:nu_a), in qualitative agreement with the theory. It is not possible to make a direct quantitative comparison because of an uncertainty in the qubit relaxation rates noted in Ref. ... The effective scaled qubit temperature T e f f * = k B T e f f / ℏ ω q as function of the scaled field strength β in the region of bistability for the small- and large-amplitude attractors, left and right panels, respectively; ω q - 2 ω F / | δ ω | = - 0.2 and 0.1 in the left and right panels; other parameters are the same as in Fig.  fig:decay_spectra.... The decay rate of the excited state of the qubit Γ e ∝ R e ~ N + - ω q - 2 ω F sharply increases if the qubit frequency ω q coincides with 2 ω F ± ν a , i.e., ω q - 2 ω F resonates with the inter-quasienergy level transition frequency. This new frequency scale results from the interplay of the system nonlinearity and the driving and is attractor-specific, as seen in Fig.  fig:nu_a. In the experiment, for ω q close to 2 ω 0 , the resonance can be achieved by tuning the driving frequency ω F and/or driving amplitude F 0 . This quasienergy resonance destroys the QND character of the measurement by inducing fast relaxation.
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Figure Fig::SpectrumVSg shows the quasienergy spectrum against the coupling strength g . For simplicity, we study the unbiased case ε = 0 , which implies m = L = 0 and hence gaps with Ω 0 , 0 n , K = | Δ 0 L K 0 α e - α 2 | ≡ Ω K . Thus, for g = 0 and Δ ≠ 0 , the twofold degeneracy of the unperturbed case is lifted by a gap of width Δ 0 . For g ≠ 0 , the gap size is further determined by the Laguerre polynomial, so that additional degeneracies can occur at the zeros of L K 0 α . When choosing the driving amplitude A such that Δ 0 = 0 the twofold degeneracy is kept for arbitrary g and K . Because the dressing by the Bessel function does not depend on g or the oscillator level, we reach the remarkable conclusion that the coherent destruction of tunneling (CDT), predicted for a driven qubit , might occur also for a qubit-oscillator system in the ultrastrong coupling limit. In Fig. Fig::DressedOsc, the dressed oscillation frequencies are plotted against the dimensionless coupling g / Ω . Next to an exponential decay, they exhibit zeros that depend through the Laguerre polynomial characteristically on the oscillator quantum number K . Hence, because the qubit’s dynamics involves several oscillator levels, we predict that suppression of tunneling cannot be reached by just tuning the coupling g . The dynamics. To prove the statements above, we calculate the survival probability of the qubit P ↓ ↓ t : = ↓ | ρ ̂ r e d t | ↓ , where ρ ̂ r e d is obtained by tracing out the oscillator degrees of freedom from the density operator of the qubit-oscillator system:... In Fig. Fig::Dynam1(c) we are with g / Ω = 1.0 already deep in the ultrastrong coupling regime. The frequency Ω 1 is now different from zero, and additionally Ω 3 appears. The lowest peak belongs to the frequencies Ω 0 , Ω 2 , and Ω 4 , which are equal for g / Ω = 1.0 , see Fig. Fig::DressedOsc. A complete population inversion again takes place. Our results are confirmed by numerical calculations. For g = 0.5 , 1.0 , the latter yield additionally fast oscillations with Ω and ω ex . Furthermore, Ω 1 is shifted in Fig. Fig::Dynam1(c) slightly to the left, so that concerning the survival probability the analytical and numerical curves get out of phase for longer times. To include also the oscillations induced by the driving and the coupling to the quantized modes, connections between the degenerate subspaces need to be included in the calculation of the eigenstates of the full Hamiltonian .... (Color online) Quasienergy spectrum of the qubit-oscillator system against the static bias ε for weak coupling g / ω ex = 0.05 . Further parameters are Δ / ω ex = 0.2 , Ω / ω ex = 2 , A / ω ex = 2.0 . The first six oscillator states are included. Numerical calculations are shown by red (light gray) triangles, analytical results in the region of avoided crossings by black dots. A good agreement between analytics and numerics is found. Blue (dark gray) squares represent the case Δ = 0 . Fig::QuasiEnEpsAnaDfinite... (Color online) Size of the avoided crossing Ω K against the dimensionless coupling strength g / Ω for an unbiased qubit ( ε = 0 ). Further, Δ / Ω = 0.4 , ω ex / Ω = 5.3 and A / Ω = 8.0 . Ω K vanishes at the zeros of the Laguerre polynomial L K 0 α . The dashed lines (a), (b), (c) represent g / Ω = 0.1 , 0.5 , 1.0 , respectively, as considered in Fig. Fig::Dynam1. Fig::DressedOsc... (Color online) Coherent destruction of tunneling in a driven qubit-oscillator system. The same parameters as in Fig. Fig::Dynam1 are used except that A / Ω = 12.7 , which leads to Δ 0 = 0 . Three coupling strengths are examined: g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). The analytical calculations (black, dashed lines) predict complete localization for all three cases. Also the numerics (red curves) shows strong localization for short timescales with fast oscillations overlaid. For long times this localization vanishes (see inset in (a)). Fig::DynamCDT... While tuning the coupling g to a zero of a Laguerre polynomial corresponding to a dominant oscillator mode yields a reduction of tunneling, tuning the driving amplitude A to a zero of a Bessel function can yield almost complete localization at short times. As already noticed in Fig. Fig::SpectrumVSg, this phenomenon is independent of the coupling strength g . We choose in Fig. Fig::DynamCDT the driving amplitude A , so that Δ 0 = 0 . This is the same condition as found for CDT in a driven qubit . Analogously, our analytical solution now predicts localization for arbitrary coupling strength g . All dressed oscillation frequencies Ω K vanish. However, third-order corrections in Δ will give small contributions to Δ 0 . Hence, a numerical exact solution yields oscillations of P ↓ ↓ t with a long period. On a short timescale and for ω ex ≫ Δ also the numerical solution appears to be strongly localized, while for long times, the inset in Fig. Fig::DynamCDT (a) shows complete population inversion for the numerics. In conclusions, we developed a powerful formalism to investigate analytically a qubit-oscillator system in the ultrastrong coupling and extreme driving regime, a situation which is in close experimental reach and offers excellent control possibilities. Our approach relies on perturbation theory with respect to a single parameter only, the qubit tunneling matrix element Δ , and thus goes beyond the driven Jaynes-Cummings model, with no rotating-wave approximation being applied. We acknowledge financial support under DFG Program SFB631. We thank Sigmund Kohler for helpful remarks.... (Color online) Dynamics of the qubit for ε = 0 , Δ / Ω = 0.4 , ω ex / Ω = 5.3 , A / Ω = 8.0 , and temperature ℏ Ω k B T -1 = 10 . The graphs show the Fourier transform F ν of the survival probability P ↓ ↓ t (see the insets). We study the different coupling strengths indicated in Fig. Fig::DressedOsc, g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). Analytical results are shown by black curves, numerics by dashed orange curves. Fig::Dynam1... Additional crossings occur independent of ε if driving and oscillator frequency are commensurable, Ω / ω ex = j / N with integers j , N > 0 , resulting in infinite many degenerate states. We avoid such a situation by choosing incommensurable frequencies or high values for j and N , so that only high-photon processes are affected.. Note that for L ≠ 0 there are always L nondegenerate levels. For L > 0 those are the first L spin-down states (positive slope), while for L < 0 the first L spin-up states (negative slope). At finite Δ avoided crossings occur in the energy spectrum at the sites of the resonances (red triangles and black dots in Fig. Fig::QuasiEnEpsAnaDfinite). To explain the origin of these avoided crossings we express H ̂ in the basis ( CoupledEigenstates) yielding the off-diagonal elements
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(Color online) (a) Nonlinear response A of the detector coupled to the qubit prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig.  fig2. The quadratic qubit-detector coupling induces a global frequency shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the qubit for the same parameters as in a). fig3... (Color online) (a) Asymptotic population difference P ∞ of the qubit states, and (b) the corresponding detector response A as a function of the external frequency ω e x for the same parameters as in Fig.  fig2. fig4... (Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external frequency ω e x . The parameters are the same as in Fig.  fig2. fig5... For a fixed value of g , the shift between the two cases of the opposite qubit states is given by the frequency gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the qubit is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line).... (Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving frequency ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1... Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing oscillator. A key property is its nonlinearity which generates multiphoton transitions at frequencies ω e x close to the fundamental frequency Ω . In order to see this, one can consider first the undriven nonlinear oscillator with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small oscillation state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the oscillation, leading to oscillations in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing oscillator in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig.  fig1(a) (decoupled from the qubit), together with the corresponding quasienergy spectrum [Fig.  fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi frequency, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as... (Color online) Nonlinear response A of the detector as a function of the external driving frequency ω e x in the presence of a finite coupling g = 0.0012 Ω to the qubit (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig.  fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2... Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired oscillator frequency Ω , where the coupling term is considered as a perturbation to the SQUID ( g oscillator to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig.  fig0.
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