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• We see from Eq. ( e62) that for non-interacting qubits, the non-vanishing qubit bias just shifts the frequency position of the liner peaks ( e57) without qualitatively changing their shape. If both the bias and the qubit-qubit interaction are finite, the bias splits each of the linear peaks in two simple Lorentzians bringing the total number of the finite-frequency peaks in the spectrum of the detector output to six as it should be in the generic situation (see, e.g., Fig.  fig3).... Output spectra of the non-linear detector measuring two different unbiased qubits. Solid line is the spectrum in the case of non-interacting qubits. The two larger peaks are the “linear” peaks that correspond to the oscillations in the individual qubits, while smaller peaks are non-linear peaks at the combination frequencies. Dashed line is the spectrum for interacting qubits. Interaction shifts the lower-frequency liner peak down and all other peaks up in frequency. Parameters of the detector-qubit coupling are: δ 1 = 0.12 t 0 , δ 2 = 0.09 t 0 , λ = 0.08 t 0 .... Finite qubit bias should lead to averaging of the two spectra S I ± ( e27) similar to that discussed in the case of non-interacting qubits and illustrated in Fig.  fig4.... The two spectral densities ( e20) correspond to two possible outcomes of measurement: the qubits found in one or the other subspace D ± , the probability of the outcomes being determined by the initial state of the qubits. Each of the spectral densities coincides with the spectral density of the linear detector measuring coherent oscillations in one qubit . Similarly to that case, the maximum of the ratio of the oscillation peak versus noise S 0 for each spectrum S I ± ω is 4. As one can see from Eq. ( e20), this maximum is reached when the measurement is weak: | λ | ≪ | t 0 | , and the detector is “ideal”: arg t 0 λ * =0, and only Γ + or Γ - is non-vanishing. If, however, there is small but finite transition rate between the two subspaces that mixes the two outcomes of measurement, the peak height is reduced by averaging over the two spectral densities ( e20). This situation is illustrated in Fig.  fig4 which shows the output spectra of the purely quadratic detector, when the subspaces D ± are mixed by small qubit bias ε . Since the stationary density matrix ( e14) is equally distributed over all qubit states, the two peaks of the spectral densities ( e20) are mixed with equal probabilities, and the maximum of the ratio of the oscillation peak heights versus noise S 0 for the combined spectrum S I ω is 2. Spectrum shown in Fig.  fig4 for ε = 0.1 Δ 1 (solid line) is close to this limit.... An example of the output spectrum of the non-linear detector measuring unbiased qubits with different tunneling amplitudes is shown in Fig.  fig6. One can see that when the linear and non-linear coefficient of the detector-qubit coupling are roughly similar, the linear peaks are more pronounced than the peaks at combination frequencies. Qubit-qubit interaction shifts all but the lower-frequency linear peak up in frequency and reduces both the amplitudes of the higher-frequency peaks and the distance between them.... Evolution of the output spectrum of the non-linear detector measuring two identical unbiased qubits with the strength ν of the qubit-qubit interaction. The qubit-detector coupling constants δ 1 , 2 are taken to be slightly different to average the spectrum over all qubit states. The three solid curves correspond to ν / Δ = 0.0 , 0.1 , 0.2 . In agreement with Eqs. ( e42) – ( e44), the peak at ω ≃ Δ is at first suppressed and then split in two by increasing ν , while the peak at ω ≃ 2 Δ is not changed noticeably by such a weak interaction. Dashed and dotted lines show the regime of relatively strong interaction: ν / Δ = 0.5 and ν / Δ = 1.0 , respectively, that is described by Eqs. ( e46) and ( e47).... Figure fig5 illustrates evolution of the output spectrum of the non-linear detector measuring identical qubits due to changing interaction strength. We see that this evolution agrees with the analytical description developed above. Weak qubit-qubit interaction ν ≃ κ ≪ Δ suppresses and subsequently splits the spectral peak at ω ≃ Ω while not changing the peak ω ≃ 2 Ω . Stronger qubit-qubit interaction ν ≃ Δ ≫ κ shifts the ω ≃ 2 Ω peak to higher frequencies while moving the two peaks around ω ≃ Ω further apart.... Output spectrum of a nonlinear detector measuring two qubits with “the most general” set of parameters. Six peaks in the spectrum at finite frequencies correspond to six different energy intervals in the energy spectrum of the two-qubit system. The zero-frequency peak reflects dynamics of transitions between energy levels. Detector parameters are: δ 1 = 0.1 , δ 2 = 0.07 , λ = 0.09 (all normalized to t 0 ). In this Figure, and in all numerical plots below we take Γ + | t 0 | 2 = Δ 1 , Γ - = 0 , and assume that the detector tunneling amplitudes are real.... Diagram of a mesoscopic detector measuring two qubits. The qubits modulate amplitude t of tunneling of detector particles between the two reservoirs.... Output spectra of a purely quadratic detector measuring two non-interacting qubits. Small qubit bias ε 1 = ε 2 ≡ ε (solid line) creates transitions that lead to averaging of the two main peaks at combination frequencies Δ 1 ± Δ 2 [see Eq. ( e20)]. Further increase of ε (dashed line) makes additional spectral peaks associated with these transitions more pronounced. The strength of quadratic qubit-detector coupling is taken to be λ = 0.15 t 0 .
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• (color online) Qubit’s final excited state probability P obtained from the semiclassical calculation as a function of temperature k B T and coupling strength g , both measured relative to the minimum qubit gap Δ . The different panels correspond to different values of the harmonic oscillator frequency: ℏ ω / Δ = 0.2 (top), 1 (middle) and 5 (bottom).... (color online) Energy level diagram of a coupled qubit-oscillator system with the qubit bias conditions varied according to the LZ protocol.... We can also see in Fig.  Fig:ExcitationProbability02 that for g / Δ 1 the temperature dependence is non-monotonic. In particular, for low temperatures we obtain the intuitively expected increase in excitation probability with increasing temperature, but this trend reverses for higher temperatures. In order to investigate this feature further, we calculate the qubit’s final excited-state probability as a function of the number n of excitation quanta present in the initial state of the oscillator (Note that this calculation differs from the ones described above in that here we do not use the Boltzmann distribution for the oscillator’s initial state). The results are plotted in Fig.  Fig:ExcitationProbabilityAsFunctionOfInitialOscillatorExcitationNumber. These results explain the non-monotonic dependence on temperature. For intermediate values of g / Δ (e.g. for g / Δ = 1 ), there is a peak at a small but finite excitation number followed by a steady decrease. As the temperature is increased from zero, the qubit’s final excited-state probability samples the probabilities for increasingly high excitation numbers, and a peak at intermediate values of temperature is obtained. Note that for large excitation numbers, the increase in P as a function of n resumes, and this increase will also be reflected in the temperature dependence.... where ω is the characteristic frequency of the harmonic oscillator, â and â † are, respectively, the oscillator’s annihilation and creation operators, and g is the qubit-oscillator coupling strength. The energy level diagram of this problem is illustrated in Fig.  Fig:EnergyLevelDiagram.... Another feature worth noting is the temperature dependence of P close to zero temperature. As can be seen clearly in Figs.  Fig:ExcitationProbability10 and Fig:ExcitationProbability50, the initial increase in P with temperature is very slow, indicating that it probably follows an exponential function that corresponds to the probability of populating the excited states in the harmonic oscillator (and the same dependence is probably present but difficult to see because of the scale of the x axis in Fig.  Fig:ExcitationProbability02). After this initial slow rise, and in particular when k B T ℏ ω , we see a steady rise that in the case of Fig.  Fig:ExcitationProbability02 can be approximated as a linear increase in P with increasing T . Importantly, the slope of this increase can be quite large for intermediate g values. From the results shown in Figs.  Fig:ExcitationProbability02- Fig:ExcitationProbability50, we find that the maximum slope d P / d k B T / Δ m a x = 0.18 × ℏ ω / Δ -0.57 , and results for other parameter values extending up to ℏ ω / Δ = 20 follow this dependence. The implication of this result can be seen clearly in the middle panel of Fig.  Fig:ExcitationProbability02: even when the temperature is substantially smaller than the qubit’s minimum gap Δ , the initial excitation of the low-frequency oscillator (stemming from the finite temperature) can cause a large increase in the qubit’s final excited-state probability. This result is in contrast with the exact result of Ref.  stating that at zero temperature the qubit’s final excited-state probability is given by P L Z regardless of the value of g . The typical temperature scale at which deviations from the LZ formula occur can therefore be much lower than Δ / k B . This result is relevant for adiabatic quantum computing, because it contradicts the expectation that having a minimum gap that is large compared to the temperature might provide automatic protection for the ground state population against thermal excitation. Another point worth noting here is that when ℏ ω qubit and oscillator are resonant with each other, yet the initial thermal excitation of the oscillator can result in exciting the qubit at the final time. The excitations in the oscillator are in some sense up-converted into excitations in the qubit as a result of the sweep through the avoided crossing.... In addition to solving the Schrödinger equation, we have performed semiclassical calculations where we assume that there is no quantum coherence between the different LZ processes. (Note here that when we replace the isolated qubit with the coupled qubit-oscillator system the single avoided crossing is replaced by a complex network of avoided crossings.) Under this approximation, we only need to calculate the occupation probabilities of the different states, and these probabilities change (according to the LZ formula) only at the points of avoided crossing. This approach greatly simplifies the numerical calculations because the locations and gaps for the different avoided crossings can be determined easily (see e.g.~Fig.~ Fig:EnergyLevelDiagram). The results are shown in Fig.  Fig:ExcitationProbabilityFromIncoherentCalculation. The results of this calculation agree generally well with those obtained by solving the Schrödinger equation when ℏ ω / Δ = 1 . For ℏ ω / Δ = 5 , the semiclassical calculation consistently underestimates the excited-state probability, but the overall dependence on temperature and coupling strength is remarkably similar to that shown in Fig.  Fig:ExcitationProbability50. We should note that higher values of ℏ ω (not shown) exhibit more pronounced deviations, with side peaks appearing in the dependence of P on g / Δ . The most striking deviation from the results of the fully quantum calculation is seen in the case ℏ ω / Δ = 0.2 (i.e. the case of a low-frequency oscillator). In the semiclassical calculation, there is a rather high peak at a small value of the coupling strength (and sufficiently high temperatures), and the excited-state probability starts decreasing when the coupling strength g becomes larger than ℏ ω . In the fully quantum calculation, however, the peak is located at a much higher value, somewhere between 0.5 and 1 depending on the temperature.... (color online) Top: Qubit’s final excited-state probability P as a function of temperature k B T and coupling strength g , both measured relative to the qubit’s minimum gap Δ . Middle: P as a function of k B T / Δ for four different values of g / Δ : 0.1 (red solid line), 0.3 (green dashed line), 1 (blue dotted line) and 2 (magenta dash-dotted line). Bottom: P as a function of g / Δ for three different values of k B T / Δ : 1 (red solid line), 3 (green dashed line), and 5 (blue dotted line). In all the panels, the harmonic oscillator frequency is ℏ ω / Δ = 0.2 . The sweep rate is chosen such that P L Z = 0.1 , and this value is the baseline for all of the results plotted in this figure.... (color online) The final excited state probability P as a function of the number of excitation quanta n present in the initial state of the oscillator. Here we take ℏ ω / Δ = 0.2 . The different lines correspond to different values of the coupling strength: g / Δ = 0.1 (red solid line), 0.5 (green dashed line), 1 (blue dotted line) and 2 (magenta dash-dotted line).... The probability for the qubit to end up in the excited state at the final time as a function of temperature and coupling strength is plotted in Figs.  Fig:ExcitationProbability02- Fig:ExcitationProbability50. As expected from known results , the final excited-state occupation probability P remains equal to 0.1 whenever the temperature or the coupling strength is equal to zero. Otherwise, the coupling to the oscillator causes this probability to increase. A common, and somewhat surprising, trend for all values of ℏ ω / Δ is the non-monotonic dependence on the coupling strength g . As the coupling strength is increased from zero to finite but small values, P increases. But when the coupling strength is increased further, P starts decreasing. Based on the results that are plotted in Figs.  Fig:ExcitationProbability02- Fig:ExcitationProbability50, one can expect that in the limit of large g / Δ (and assuming not-very-large values of k B T / Δ ) the excited-state occupation probability will go back to its value in the uncoupled case, i.e.  P = 0.1 . This phenomenon is probably a manifestation of the superradiance-like behaviour in a strongly coupled qubit-oscillator system . In the superradiant regime (i.e. the strong-coupling regime), the ground state is highly entangled exactly at the symmetry point (which corresponds to the bias conditions at t = 0 in the LZ problem), but even small deviations from the symmetry point can lead to an effective decoupling between the qubit and resonator with the exception of some state-dependent mean-field shifts. Indeed the maximum values of P reached in Figs.  Fig:ExcitationProbability10 and Fig:ExcitationProbability50 occur at coupling strength values that are comparable to the expression for the uncorrelated-to-correlated crossover value, namely g ∼ ℏ ω (and we have verified that the near-linear increase in peak location as a function of oscillator frequency continues up to ℏ ω / Δ = 20 ). This relation does not apply in the case ℏ ω / Δ = 0.2 , shown in Fig.  Fig:ExcitationProbability02. In this case, the peak occurs when the coupling strength g is comparable to the minimum gap Δ . It is in fact quite surprising that the excitation peak in the case ℏ ω / Δ = 0.2 occurs at a higher coupling strength than that obtained in the case ℏ ω / Δ = 1 . In order to investigate this point further, we tried values close to ℏ ω / Δ = 1 and found that this value gives a minimum in the peak location (i.e. the peak in P when plotted as a function of g / Δ ).
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• 9 Be + qubit structure and coherent control. (a) Relevant atomic structure of 9 Be + at 4.5 T. Qubit states labeled with (m I ,m J ), the nuclear and total spin projections of the atom along the quantization axis. (Inset) Strobed optical image of ion fluorescence showing hexagonal-close-packed order with spacing ∼ 10 μ m. (b) Schematic block diagram of the microwave system used to drive qubit rotations. PLL = Phase Lock Loop, DRO = Dielectric Resonator Oscillator, CPL = Coupler. 77 MHz PLL reference provided by computer-controlled DDS or the beat note of DDS and a frequency modulated synthesizer during noise injection (see Methods). (c) Qubit transition driven via a square microwave π pulse of τ π ≈ 185 μ s. Readout consists of fluorescence detection immediately before and after application of microwave pulses followed by normalization to account for slow fluctuations in laser intensity (Normalized Counts). Each data point consists of the average of 20 experiments. (d) Rabi oscillations driven on-resonance give a decay time of 30-40 ms. τ π is tunable via an input attenuator. (e) Ramsey fringes measured for two different microwave detunings, 50 kHz and 100 kHz ( π / 2 pulses were driven on-resonance and the microwaves detuned during the free-precession period). Detuning was controlled by switching between pre-programmed frequency output profiles via a TTL input on the DDS.... NIST Time and Frequency Division, Boulder, CO... Pulse sequence performance in the presence of various noise spectra. (a) Decoherence traces as a function of free-precession time for CPMG and UDD and ambient magnetic field noise. Phase errors are manifested as nonzero fluorescence detection. Traces saturate at 0.5 for total phase randomization. Traces for n =4,5 offset by 0.5 units for clarity. Each data point corresponds to 50 averages. Fits utilize ambient noise spectrum (see inset) and two free parameters - overall noise scaling ( α ) and scaling factor for the noise spur at ∼ 153 Hz ( γ ). Extracted fit parameters yield α = 1.46 ± 0.31 and γ = 0.23 ± 0.04 . Individual fit uncertainty is ∼ 0.05-0.07 for each trace. Variations in extracted noise intensities are consistent with direct observation of slow fluctuations in the noise spectrum, including significant changes in the amplitude of the ∼ 153 Hz spur. (Inset) Ambient noise spectrum measured via a solenoid embedded in our NMR magnet. (b) Performance of UDD and CPMG for Ohmic spectrum with 500 Hz cutoff (see inset). UDD outperforms CPMG by increasing factors as the noise intensity is increased. Single fit parameter: noise scaling, α . Extracted noise scaling: α O h m i c = 1.49 ± 0.19 , expected α = 1.65 for this noise intensity. (c) Sequence performance in a 1 / ω noise spectrum with a sharp cutoff. Extracted noise scaling: α 1 / ω = 3.32 ± 0.15 . Insets to (b) and (c)) Smoothed, measured noise on a log-log plot, measured using V N = 0.7 V. Red line represents intended noise envelope up to high-frequency cutoff. Measured noise spectra extending from .01 Hz to 1 MHz used for fitting, but truncated for visibility.
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• (Color online) Energy spectrums for lowest eight levels under the situation with three high-frequency qubits: ℏ w 0 / E q = 0.01 . The rescaled energy E k / ℏ w 0 with k = 1 , 2 , 3 , . . . , 8 versus the rescaled coupling strength λ / ℏ w 0 is plotted: (a) θ = 0 ; (b) θ = π / 6 ; (c) θ = π / 3 .... (Color online) Schematic of four displaced oscillators. The horizontal and vertical axises represent the position and displaced oscillator’s eigenenergy E d o , respectively. Four displaced oscillators are shifted to the left or right from the equilibrium position with a specific constant, where the shift direction is determined by the state of three qubits. The eigenstates (plotted with n no more than 2 ) that have the same value of n are degenerate for the states | A ± 1 (or | A ± 3 ), and have the symmetry divided by the origin point in horizontal axis.... adiabatic approximation, three qubits, ultrastrongly coupled, harmonic oscillator... (Color online) Energy spectrums for lowest eight levels under the situation with a high-frequency oscillator: ℏ w 0 / E q = 10 . The rescaled energy E k / ℏ w 0 with k = 1 , 2 , 3 , . . . , 8 versus the rescaled coupling strength λ / ℏ w 0 is plotted: (a) θ = 0 ; (b) θ = π / 6 ; (c) θ = π / 3 .... (Color online) Schematic of the system with three identical qubits coupled to a harmonic oscillator. The j th ( j = 1 , 2 , 3 ) qubit with one ground ( | g j ) and one excited states ( | e j ) is coupled to the oscillator with frequency w 0 , where the qubit-oscillator coupling strength is denoted by g or λ .... (Color online) The Q function (upside) and the Wigner function (underside) of the oscillator’s state with three high-frequency qubits (i.e., ℏ w 0 / Δ = 0.1 and ϵ = 0 ): (a,d) λ / ℏ w 0 = 0.5 , (b,e) λ / ℏ w 0 = 1 , (c,f) λ / ℏ w 0 = 1.25 .
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• NIST Time and Frequency Division, Boulder, CO, 80305... A smoothed power spectrum with Ohmic scaling, measured for V N = 0.7 V is displayed in the inset to Fig.  fig:ohmicnoisea. The measured spectral peak at the cutoff of the ohmic spectrum with V N = 0.7 V is approximately five orders of magnitude larger than the ambient spectrum at the same frequency. We apply the CPMG and UDD sequences in the presence of this noise power spectrum for V N = 2 V and display the results for various values of n in Fig.  fig:ohmicnoisea. The UDD sequence clearly and dramatically outperforms CPMG for all measured values of n . Error bars are shown for data with n = 6 , and are derived from the statistical variations in the individual experiments that are averaged to produce a single data point , demonstrating that the traces differ with statistical significance. As expected, the coherence time may be extended with the addition of π X pulses. Data show good agreement with theoretical predictions for qubit decoherence using the measured noise spectrum and a single fit parameter, α , the overall noise scaling.... fig:pulsenumbers (Color online) (a-d) Colorscale images of measured qubit decoherence under ambient noise as a function of total free precession time and pulsenumber, n , for various sequences. Colorscale illustrates phase error probability as normalized counts (0.5 = 50 % probability of ending sequence as or ), with light colors corresponding to low qubit errors and black corresponding to full dephasing. Axis of rotation is designated in the figure. π Y rotations driven by a combination of a π X pulse and a net π Z rotation during each free precession period. (e-f) Theoretical fits to qubit decoherence using the filter function as defined in Eq. (2), and the noise spectrum plotted in panel (g). (g) Measured noise spectrum showing 1 / ω 2 field fluctuations ( S β ω ∝ 1 / ω 4 ). Note strong spur at ω ∼ 2 π × 153 Hz. Dashed line is a guide to the eye representing S β ω ∝ 1 / ω 4 scaling.... Qubit rotations are achieved by directly driving the ∼ 124 GHz electron spin-flip transition using a quasi-optical microwave system. The use of this technique eliminates the effects of spontaneous emission which are a limiting parameter in Raman-laser-mediated qubit rotations. Coherent control over this qubit transition was first demonstrated in reference . We generate 124 GHz microwaves using a Gunn diode oscillator ph... The multipulse spin echo sequence was originally developed in the context of NMR systems where inhomogeneous broadening required an ability to refocus the ensemble Bloch vector as it spread out during free precession . However, the same sequence is quite effective at suppressing phase randomization due to homogeneous effects (e.g. β t ) when noise processes are dominated by low-frequency components (e.g. S β ω ∝ 1 / ω ). An n -pulse spin echo sequence has all pulses evenly spaced, with the first and last free precession periods half as long as the interpulse free precession periods (Fig.  fig:pulseseqc). In the case where (the state vector at the leading edge of the first π -pulse) is oriented along the direction of the applied microwave field, the sequence is known as CPMG, after its proponents. For reasons that will be described in a later section, this particular version of the multipulse spin echo sequence is most robust against control error accumulation.... A schematic pulse sequence for an arbitrary, n -pulse dynamical decoupling sequence is shown in Figs. fig:pulseseqa & b. The sequence begins with the application of a π / 2 X -pulse which rotates the Bloch vector to the equatorial plane. Microwaves are detuned during the interpulse period; application of this detuning results in effective Larmor precession and also serves to reduce the Rabi rate, mitigating unwanted rotations due to microwave leakage. Short delays called “Profile Heads (Tails)” between the trailing (leading) edge of a pulse and the application (removal) of the detuning are employed to precisely control the phase evolution of the qubits. For instance, with 100 kHz detuning, and the first interpulse precession period set to an integer multiple of 10 μ s, the Primary Tail may be set to 12.5 μ s such that the Bloch vector rotates from the Y ̂ to the X ̂ axis. Varying the length of the Primary Tail therefore allows a form of process tomography in the equatorial plane to be performed, as will be discussed in Section sec:robustness. The Final Tail is set such that in the absence of dephasing the qubits will be rotated to the dark state at the conclusion of the sequence.... fig:schematic (Color online) Schematic diagram of experimental system. (a) Qubit level diagram showing relevant transitions and energy levels (see text). (Inset a) Microwave resonance transition showing Rabi lineshape for square excitation pulses. Vertical axis is normalized counts. (b) Rabi oscillations for directly driven qubit transition showing τ π ∼ 185 μ s. (c) Effective Larmor precession measured using a Ramsey interferometer. Open markers correspond to DDS detuning during free-precession time of 100 kHz, solid markers correspond to 50 kHz detuning. (d) Schematic of microwave generation system. CPL=coupler. (e) Schematic of system for generating microwave frequency noise. (f) Schematic of trap system showing relative positions of hardware elements. AOM=Acousto-Optic Modulator. Inset (f) Top-view optical micrograph of an ion array showing hexagonal crystal order, obtained by imaging ion fluorescence parallel to the magnetic field axis. (g) Schematic of experimental control system.... fig:robustness (Color online) Robustness of n = 6 spin echo and UDD pulse sequences to systematic rotation errors for a variety of input states, measured for free-precession time 8 ms and ambient noise. (a-b) Colorscale representation of qubit rotation to as a function of microwave reference frequency (centered on f 0 ) and initial state. Dark colors indicate rotation errors. (c) Difference between UDD and echo results for above parameters. White indicates regions where UDD outperforms spin echo. (d-e) Colorscale representation of normalized counts as a function of actual “ π -pulse” rotation angle ( π = 185 μ s) where τ π is manipulated to mimic systematic rotation errors. (f) Difference between UDD and echo results for “ π -pulse” rotations, white shows regions where UDD outperforms echo. Randomly located black dots correspond to laser dropouts during data collection. (g) Bloch sphere representations of several relevant starting states used in colorscale images above.... We apply the CPMG and UDD sequences under ambient noise while varying the pulse number, showing that in both cases we are able to extend the coherence time with n , as predicted by theory (Fig.  fig:pulsenumbers). Qubit decoherence is represented as a colorscale taking values of 0 to 0.5, and corresponds to 1 2 1 - W t . Fully dephased qubits have an equal probability of being measured in either of the two basis states, corresponding to a measured value of 0.5 for error probability.
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• In the example figure (Fig.  fig:qubosc1d2d), the control bias is varied from left to right for a low frequency oscillator circuit (1.36GHz). For each bias point the simulation is reinitialised, the stochastic time evolution of the system density matrix is simulated over 1500 oscillator cycles. Then the oscillator and qubit charge expectation values are extracted to obtain the power spectrum for each component, with a frequency resolution of 4.01MHz. The power spectra for each time series are collated as an image such that the power axis is now represented as a colour, and the individual power spectra are vertical ‘slices’ through the image. The dominant frequency peaks become line traces, therefore illustrating the various avoided crossings, mergeings and intersections. The example figure shows the PSD ‘slice’ at Bias = 0.5187 , the broadband noise is readily apparent and is due to the discontinuous quantum jumps in the qubit. The bias oscillator peak (1.36GHz) is most prominent in the oscillator PSD, as would be expected, but it is also present in the qubit PSD. It should also be noted that most features are present in both the qubit and oscillator, including the noise which is generated by the quantum jumps and the quantum state diffusion processes. Interestingly, the qubit PSD is significantly stronger than the oscillator PSD, however, a larger voltage is generated by the smaller charge due to the extremely small island capacitance, V q = q / C q .... fig:mwRamp (Color online) Oscillator PSD as a function of the applied microwave drive frequency f m w , for microwave amplitudes A m w = 0.0050 (A) and A m w = 0.0100 (B). It is important to notice that there are now two frequency axes per plot, a drive (H) and a response (V). Of particular interest is the magnified section which shows clearly the distinct secondary splitting in the sub-GHz regime. This occurs due to a high frequency interaction seen in the upper plots, where the lower Rabi sideband of the microwave drive passes through the high frequency oscillator signal. The maximum splitting occurs when the Rabi amplitude is a maximum, hence this is observed for a very particular combination of bias and drive, which is beneficial for charactering the qubit. Most importantly, this would not be observed with a conventional low frequency oscillator configuration as the f m w - f o s c separation would be too large for the Rabi frequency. ( κ = 5 × 10 -5 ).... Fig.  fig:mwRamp is presented in a similar manner as Fig.  fig:BiasRamp. However there are now two frequency axes: the horizontal axis represents the frequency of the applied microwave drive field, and the vertical axis is the frequency response. It should be remembered that the microwave frequency axis is focused near the qubit transition frequency ( f q u b i t ≈ 3.49GHz) and the diagonally increasing line is now the microwave frequency.... Autler Townes effect, charge qubit, characterisation, frequency spectrum... fig:QubitOscEnergy A two level qubit is coupled to a many level harmonic oscillator, investigated for two different oscillator energies. Firstly, the oscillator resonant frequency is set to 1.36GHz, this more resembles the conventional configuration such that the fundamental component of the oscillator does not drive the qubit. However, we also investigate the use of a high frequency oscillator of 3.06GHz which can excite this qubit. In addition, qubit is constantly driven by a microwave field at 3.49GHz to generate Rabi oscillations and in this paper we examine the relation between these three fields.... fig:qubosc1d2d (Color online) Oscillator and Qubit power spectra slices for Bias = 0.5187, using the low frequency oscillator circuit f o s c = 1.36 GHz. The solid lines overlay the energy level separations found in Fig.  fig:EnergyLevel. ( κ = 5 × 10 -5 ). As one would expect, the bias oscillator peak at 1.36GHz is clearly observed in the oscillator PSD, but only weakly in the qubit PSD. Likewise the qubit Rabi frequency is found to be stronger in the qubit PSD. However it is important to note that the qubit dynamics such as the Rabi oscillations are indeed coupled to the bias oscillator circuit and so can be extracted. In addition, it is recommended to compare the layout of the most prominent features with Fig.  fig:BiasRamp.... fig:BiasRamp (Color online) Oscillator PSD as a function of bias, for microwave amplitudes A m w = 0.0025 (A) and A m w = 0.0050 (B). The red lines track the positions (in frequency) of significant power spectrum peaks (+10dB to +15dB above background), the overlaid black and blue lines are the qubit energy and microwave transition (Fig.  fig:EnergyLevel). Unlike Fig.  fig:qubosc1d2d, in these figures the 3.06GHz oscillator circuit can now drive the qubit (Fig.  fig:EnergyLevel) and so creates excitations which mix with the microwave driven excitations creating a secondary splitting centred on f m w - f o s c (430MHz). This feature contains the Rabi frequency information in the sidebands of the splitting, but now in a different and controllable frequency regime. In addition, the intersection of the two differently driven excitations (illustrated in the magnified sections), opens the possibility of calibrating the biased qubit against a fixed engineered oscillator circuit, using a single point feature. ( κ = 5 × 10 -5 ).... In a previous paper , a method was proposed by which the energy level structure of a charge qubit can be obtained from measurements of the peak noise in the bias/control oscillator, without the need of extra readout devices. This was based on a technique originally proposed for superconducting flux qubits but there are many similarities between the two technologies. The oscillator noise peak is the result of broadband noise caused by quantum jumps in the qubit being coupled back to the oscillator circuit. This increase in the jump rate becomes a maximum when the Rabi oscillations are at peak amplitude, this should only occur when the qubit is correctly biased and the microwave drive is driving at the transition frequency. Therefore by monitoring this peak as a function of bias, we can associate a bias position with a microwave frequency equal to that of the energy gap, hence constructing the energy diagram (Fig.  fig:EnergyLevel).... fig:Jumps (Color online) (A) Oscillator power spectra when the coupled qubit is driven at f m w = 5.00 GHz. An increase in bias noise power ( f o s c = 1.36 GHz) can be observed when Rabi oscillations occur, the more frequent quantum jump noise couples back to the oscillator. (B) Bias noise power peak position changes as a function of f m w , the microwave drive frequency. Therefore, it is possible to probe the qubit energy level structure by using the power increase in the oscillator which is already in place, eliminating the need for additional measurement devices. However, it should be noted that the surrounding oscillator harmonics may mask the microwave driven peak. ( κ = 1 × 10 -3 ).
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• The systems considered are shown in Fig.  fig:system. To be specific we first analyze the Rabi driven flux qubit coupled to an LC-oscillator (Fig.  fig:systema) with Hamiltonian... Average number of photons in the resonator as function of the driving detuning δ ω and amplitude Ω R 0 . Peaks at δ ω > 0 correspond to lasing, dips at δ ω qubit are Δ / 2 π = 1  GHz, ϵ = 0.01 Δ , and Γ 0 / 2 π = 125  kHz, the frequency and line-width of the resonator are ω T / 2 π = 6 MHz and κ / 2 π = 1.7  kHz, the coupling constant is g / 2 π = 3.3  MHz and the temperature of the resonator T = 10  mK. The inset shows the bistability of the photon number for Ω R 0 / 2 π = 7 MHz. The dashed line represents the unstable solution.... So far we described a flux qubit coupled to an LC oscillator, but our analysis applies equally to a nano-mechanical resonator capacitively coupled to a Josephson charge qubit (see Fig.  fig:systemb). In this case σ z stands for the charge of the qubit, and both the coupling to the oscillator and the driving are capacitive, i.e., involve σ z . To produce capacitive coupling between the qubit and the oscillator, the latter is metal coated and charged by a voltage source . 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 , is produced. One should then observe the usual line narrowing with line width given by κ N t h / 4 n ̄ ∼ κ 2 N t h / Γ 1 . Experimental observation of this line-width narrowing would constitute a confirmation of the lasing/sasing.... In Fig.  3dphoton we summarize our main results obtained by solving the Langevin (Fokker-Plank) equations . The number of photons n ̄ is plotted as a function of the detuning δ ω of the driving frequency and driving amplitude Ω R 0 . It exhibits sharp extrema along two curves corresponding to the one- and two-photon resonances, Ω R = ω T - 4 g 3 n ̄ and Ω R = 2 ω T - 4 g 3 n ̄ . Blue detuning, δ ω > 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 photon number in the resonator above the thermal values. On the other hand, red detuning produces a one-qubit cooler with photon numbers substantially below the thermal value. Near the resonances we find regions of bi-stability illustrated in the inset of Fig.  3dphoton. In these regions we expect a telegraph-like noise due to random switching between the two solutions.... Several recent experiments on quantum state engineering with superconducting circuits  realized concepts originally introduced in the field of quantum optics and stimulated substantial theoretical activities  . Josephson qubits play the role of two-level atoms, while oscillators of various kinds replace the quantized light field. Motivated by one such experiment , we investigate a Josephson qubit coupled to a slow LC oscillator (Fig.  fig:system a) with eigenfrequency (in the MHz range) much lower than the qubit’s energy splitting (in the GHz range), ω T ≪ Δ E . The qubit is ac-driven to perform Rabi oscillations, and the Rabi frequency Ω R is tuned close to resonance with the oscillator. For this previously unexplored regime of frequencies we study both one-photon (for Ω R ≈ ω T ) and two-photon (for Ω R ≈ 2 ω T ) qubit-oscillator couplings. The latter is dominant at the “sweet" point of the qubit, where due to symmetry the linear coupling to the noise sources is tuned to zero and dephasing effects are minimized . When the qubit driving frequency is blue detuned, δ ω = ω d - Δ E > 0 , we find that the system exhibits lasing behavior; for red detuning the qubit cools the oscillator. Similar behavior is expected in an accessible range of parameters for a Josephson qubit coupled to a nano-mechanical oscillator (Fig.  fig:systemb), thus providing a realization of a SASER  (Sound Amplifier by Stimulated Emission of Radiation).... The systems. a) In the circuit QED setup of Ref.  an externally driven three-junction flux qubit is coupled inductively to an LC oscillator. b) In an equivalent setup a charge qubit is coupled to a mechanical resonator.
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• Average number of photons in the resonator as function of the driving detuning δ ω and amplitude Ω R 0 . Peaks at δ ω > 0 correspond to lasing, while dips at δ ω qubit: Δ / 2 π = 1  GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125  kHz, the resonator: ω T / 2 π = 6 MHz, κ / 2 π = 0.34  kHz, and the coupling: g / 2 π = 3.3  MHz. The bath temperature is T = 10  mK.... Dressed states of a driven qubit near resonance. Here m is the number of photons of the driving field, which is assumed to be quantized.... In experiments with the same setup as shown in Fig.  fig:systema) but in a different parameter regime the mechanisms of Sisyphus cooling and amplification has recently been demonstrated . Due to the resonant high-frequency 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.... Average number of photons n ̄ versus the detuning. The blue curves are obtained from the Langevin equations ( dot alpha) and ( dot alpha2). 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 ( eq:Master_Equation). The driving amplitude is taken as Ω R 0 / 2 π = 5 MHz. The parameters of the qubit: Δ / 2 π = 1  GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125  kHz, the resonator: ω T / 2 π = 6 MHz, κ / 2 π = 1.7  kHz, N t h = 5 , and the coupling: g / 2 π = 3.3  MHz.... So far we described an LC oscillator coupled to a flux qubit. But our analysis equally applies for a nano-mechanical resonator coupled capacitively to a Josephson charge qubit (see Fig.  fig:systemb). 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 metal-coated 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 a c 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 , is produced. One should then observe the usual line narrowing with line width given by κ N t h / 4 n ̄ ∼ κ 2 N t h / Γ ~ 1 . Experimental observation of this line-width narrowing would constitute a confirmation of the lasing/sasing.... Average number of photons in the resonator as function of the qubit’s relaxation rate, Γ 0 at the one-photon resonance, Ω R = ω T for g 3 = 0 and N t h = 5 . The dark blue line shows the numerical solution of the master equation, the light blue solid line represents the solution of the Langevin equation, Eq. ( dot alpha ). The green and red dashed curves represent respectively the saturation number n 0 and the thermal photon number N t h . The parameters are as in Fig.  fig:compar (except for Γ 0 ).... Also in situations where the qubit, e.g., a Josephson charge qubit, is coupled to a nano-mechanical oscillator (Fig.  fig:systemb) 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 .... 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 , and prompted substantial theoretical activities . 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. , with setup shown in Fig.  fig:systema), the qubit is coupled to a slow LC oscillator with frequency ( ω T / 2 π ∼ MHz) much lower than the qubit’s level splitting ( Δ E / 2 π ℏ ∼ 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 . A similar strategy for cooling of a nanomechanical resonator via a Cooper pair box qubit has been recently suggested in Ref. . The analysis of the driven circuit QED system shows that these properties depend strongly on relaxation and decoherence effects in the qubit.... a) In the setup of Ref.  an externally driven three-junction flux qubit is coupled inductively to an LC oscillator. b) A charge qubit is coupled to a mechanical resonator.... The systems to be considered are shown in Fig.  fig:system. 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.  fig:systema) with Hamiltonian
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• The first term has a peak at zero frequency, while the second term has a peak at ω = Ω , with width 3 Γ / 2 , and signal -1 / 3 Γ . Bounding this signal in relation to the noise in the individual twin detectors gives | S 1 , 2 Ω | ≤ 2 / 3 S I . The interesting feature of this correlator is that it changes sign as a function of frequency. The low frequency part describes the incoherent relaxation to the stationary state, while the high frequency part describes the out of phase, coherent oscillations of the z and x degrees of freedom. The measured correlator S z x , as well as S x x , S z z are plotted as a function of frequency in Fig. combo(b,c,d) for different values of ϵ . These correlators all describe different aspects of the time domain destruction of the quantum state by the weak measurement, visualized in Fig. comboa. We note that the cross-correlator changes sign for ϵ = - Δ .... (color online). (a) Time domain destruction of the quantum state by the weak measurement process for ϵ = Δ . The elapsed time is parameterized by color, and (x,y,z) denote coordinates on the Bloch sphere. (b) The measured cross-correlator S z x ω changes sign from positive at low frequency (describing incoherent relaxation) to negative at the qubit oscillation frequency (describing out of phase, coherent oscillations). (c,d) The correlators S x x , S z z have both a peak at zero frequency and at qubit oscillation frequency. We take Γ = Γ x = Γ z = .07 Δ / ℏ . S i j are plotted in units of Γ -1 .... Cross-correlated quantum measurement set-up: Two quantum point contacts are measuring the same double quantum dot qubit. As the quantum measurement is taking place, the current outputs of both detectors can be averaged or cross-correlated with each other.
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• In the preceeding analysis we neglected the effect of the local environment by setting Y i n t ω = 0 . As a result, the low-frequency value of T 1 is substantially larger than obtained in experiment . By modeling the local environment with R 0 = 5000  ohms and L 0 = 0 we obtain the T 1 versus ω 01 plot shown in Fig.  fig:three. Notice that this value of R 0 brings T 1 to values close to 20 ns at T = 0 . The message to extract from Figs.  fig:two and  fig:three is that increasing R 0 as much as possible and increasing the qubit frequency ω 01 from 0.1 Ω to 2 Ω at fixed low temperature can produce a large increase in T 1 .... Schematic drawing of the phase qubit with an RLC isolation circuit.... The circuit used to describe intrinsic decoherence and self-induced Rabi oscillations in phase qubits is shown in Fig.  fig:one, which correponds to an asymmetric dc SQUID . The circuit elements inside the dashed box form an isolation network which serves two purposes: a) it prevents current noise from reaching the qubit junction; b) it is used as a measurement tool.... In the limit of T = 0 , we can solve for c 1 t exactly and obtain the closed form c 1 t = L -1 s + Γ - i ω 01 2 + Ω 2 - Γ 2 s s + Γ - i ω 01 2 + Ω 2 - Γ 2 - κ Ω 4 π i / Γ where L -1 F s is the inverse Laplace transform of F s , and κ = α / M ω 01 × Φ 0 / 2 π 2 ≈ 1 / ω 01 T 1 , 0 . The element ρ 11 = | c 1 t | 2 of the density matrix is plotted in Fig.  fig:four for three different values of resistance, assuming that the qubit is in its excited state such that ρ 11 0 = 1 . We consider the experimentally relevant limit of Γ ≪ ω 01 ≈ Ω , which corresponds to the weak dissipation limit. Since Γ = 1 / 2 C R the width of the resonance in the spectral density shown in Eq. ( eqn:sd-poles) is smaller for larger values of R . Thus, for large R , the RLC environment transfers energy resonantly back and forth to the qubit and induces Rabi-oscillations with an effective time dependent decay rate γ t = - 2 ℜ c ̇ 1 t / c 1 t .... fig:three T 1 (in nanoseconds) as a function of qubit frequency ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = 5000  ohms and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz.... fig:four Population of the excited state of the qubit as a function of time ρ 11 t , with ρ 11 t = 0 = 1 for R = 50  ohms (solid curve), 350  ohms (dotted curve), and R = 550  ohms (dashed curve), and L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 .... fig:two T 1 (in seconds) as a function of qubit frequency ω 01 . The solid (red) curves describes an RLC isolation network with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 GHz.... In Fig.  fig:two, T 1 is plotted versus qubit frequency ω 01 for spectral densities describing an RLC (Eq.  eqn:spectral-density-isolation) or Drude (Eq.  eqn:sd-drude) isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), for J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes T 1 ω 01 = M ω 01 / J ω 01 . From Fig.  fig:two (main plot) several important points can be extracted. First, in the low frequency regime ( ω 01 ≪ Ω ) the RL (Drude) and RLC environments produce essentially the same relaxation time T 1 , R L C 0 = T 1 , R L 0 = T 1 , 0 ≈ L 1 / L 2 R C 0 , because both systems are ohmic. Second, near resonance ( ω 01 ≈ Ω ), T 1 , R L C is substantially reduced because the qubit is resonantly coupled to its environment producing a distinct non-ohmic behavior. Third, for ( ω 01 > Ω ), T 1 grows very rapidly in the RLC case. Notice that for ω 01 > 2 Ω , the RLC relaxation time T 1 , R L C is always larger than T 1 , R L . Furthermore, in the limit of ω 01 ≫ m a x Ω , 2 Γ , T 1 , R L C grows with the fourth power of ω 01 behaving as T 1 , R L C ≈ T 1 , 0 ω 01 4 / Ω 4 , while for ω 01 ≫ Ω 2 / 2 Γ , T 1 , R L grows only with second power of ω 01 behaving as T 1 , R L ≈ 4 T 1 , 0 Γ 2 ω 01 2 / Ω 4 . Thus, T 1 , R L C is always much larger than T 1 , R L for sufficiently large ω 01 . Notice, however, that for parameters in the experimental range such as those used in Fig  fig:two, T 1 , R L C is two orders of magnitude larger than T 1 , R L , indicating a clear advantage of the RLC environment shown in Fig  fig:one over the standard ohmic RL environment. Thermal effects are illustrated in the inset of Fig.  fig:two where T = 50 mK is a characteristic temperature where experiments are performed . The typical values of T 1 at low frequencies vary from 10 -5 s at T = 0 to 10 -6 s at T = 50 mK, while the high frequency values remain essentially unchanged as the thermal effects are not important for ℏ ω 01 ≫ k B T .... These environmentally-induced Rabi oscillations are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the qubits is quickly dissipated without being temporarily stored. These environmentally-induced Rabi oscillations are generic features of circuits with resonances in the real part of the admittance. The frequency of the Rabi oscillations Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6  rad/sec for Fig.  fig:four.
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