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53370 results
For maximum generality, we first define a minimal model needed to describe the splitting of Fig.  fig:Splitting. To this end, we restrict ourselves to the lowest two states of the phase qubit circuit (the qubit subspace) and disregard the longitudinal coupling ∝ τ z . Within the rotating wave approximation (RWA) the Hamiltonian reads... (color online) (a) Analytically obtained transition spectrum of the Hamiltonian ( eq:H4Levels) in the minimal model for Ω q / h = 40  MHz and v ⊥ / h = 25  MHz. Dashed-dotted lines show the transition frequencies while the gray-scale intensity of the thicker lines indicates the weight of the respective Fourier-components in the probability P . The system shows a symmetric response as a function of the detuning δ ω . Two of the four lines are double degenerate. (b) The same as (a) but including the second order Raman process with Ω f v = v ⊥ Ω q / Δ . The two degenerate transitions in (a) split and the symmetry of the response is broken. Inset: Schematic representation of the structure of the Hamiltonian ( eq:H4Levels). We denote the ground and excited states of the qubit as and and those of the TLF as and . Arrows indicate the couplings between qubit and fluctuator v ⊥ and to the microwave field Ω q and Ω f v .... The sample investigated in this study is a phase qubit , consisting of a capacitively shunted Josephson junction embedded in a superconducting loop. Its potential energy has the form of a double well for suitable combinations of the junction’s critical current (here, I c = 1.1 μ A) and loop inductance (here, L = 720  pH). For the qubit states, one uses the two Josephson phase eigenstates of lowest energy which are localized in the shallower of the two potential wells, whose depth is controlled by the external magnetic flux through the qubit loop. The qubit state is controlled by an externally applied microwave pulse, which in our sample is coupled capacitively to the Josephson junction. A schematic of the complete qubit circuit is depicted in Fig.  fig:Splitting(a). Details of the experimental setup can be found in Ref. ... (color online) (a) Schematic of the phase qubit circuit. (b) Probability to measure the excited qubit state (color-coded) vs. bias flux and microwave frequency, revealing the coupling to a two-level defect state having a resonance frequency of 7.805 GHz (indicated by a dashed line).... superconducting qubits, Josephson junctions, two-level fluctuators, microwave spectroscopy, Rabi oscillations ... (color online) (a) Experimentally observed time evolution of the probability to measure the qubit in the excited state, P t , vs. driving frequency; (b) Fourier-transform of the experimentally observed P t . The resonance frequency of the TLF is indicated by vertical lines. (c) Time evolution of P and (d) its Fourier-transform obtained by the numerical solution of Eq. ( eq:master_eq) as described in the text, taking into account also the next higher level in the qubit. (As the anharmonicity Δ / h ∼ 100  MHz in our circuit is relatively small, this required going beyond the second order perturbation theory and analyze the 6-level dynamics explicitly). The qubit’s Rabi frequency Ω q / h is set to 48 MHz.... Spectroscopic data taken in the whole accessible frequency range between 5.8 GHz and 8.1 GHz showed only 4 avoided level crossings due to TLFs having a coupling strength larger than 10 MHz, which is about the spectroscopic resolution given by the linewidth of the qubit transition. In this work, we studied the qubit interacting with a fluctuator whose energy splitting was ϵ f / h = 7.805  GHz. From its spectroscopic signature shown in Fig.  fig:Splitting(b), we extract a coupling strength v ⊥ / h ≈ 25  MHz. The coherence times of this TLF were measured by directly driving it at its resonance frequency while the qubit was kept detuned. A π pulse was applied to measure the energy relaxation time T 1 , f ≈ 850  ns, while two delayed π / 2 pulses were used to measure the dephasing time T 2 , f * ≈ 110  ns in a Ramsey experiment. To read out the resulting TLF state, the qubit was tuned into resonance with the TLF to realize an iSWAP gate, followed by a measurement of the qubit’s excited state.... where δ ω = ϵ q - ϵ f . The level structure and the spectrum of possible transitions in the Hamiltonian ( eq:H4Levels) is illustrated in Fig.  fig:Transitionsa. The transition frequencies in the rotating frame correspond to the frequencies of the Rabi oscillations observed experimentally.... Figure  fig:DataRabi(a) shows a set of time traces of P taken for different microwave drive frequencies. Each trace was recorded after adjusting the qubit bias to result in an energy splitting resonant to the chosen microwave frequency. The Fourier transform of this data, shown in Fig.  fig:DataRabi(b), allows us to distinguish several frequency components. Frequency and visibility of each component depend on the detuning between qubit and TLF. We note a striking asymmetry between the Fourier components appearing for positive and negative detuning of the qubit relative to the TLF’s resonance frequency, which is indicated in Figs.  fig:DataRabi(a,b) by the vertical lines at 7.805 GHz. We argue below that this asymmetry is due to virtual Raman-transitions involving higher levels in the qubit.
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The measurement process that we used to observe coherent oscillations consists of several steps shown in Fig.  fig:3(a). Each step is realized by applying a combination of magnetic fluxes Φ x and Φ c as indicated by numbers in Fig.  fig:2(b). The first step in our measurement is the initialization of the system in a defined flux state (1). Starting from a double well at Φ x ≅ Φ 0 / 2 with high barrier, the potential is tilted by changing Φ x until it has only a single minimum (left or right, depending on the amplitude and polarity of the applied flux pulse). This potential shape is maintained long enough to ensure the relaxation to the ground state. Afterwards the potential is tuned back to the initial double-well state (2). The high barrier prevents any tunneling and the qubit is thus initialized in the chosen potential well. Next, the barrier height is lowered to an intermediate level (3) that preserves the initial state and allows to use just a small-amplitude Φ c flux pulse for the subsequent manipulation. The following Φ c -pulse transforms the potential into a single well (4). The Φ c -pulse duration Δ t is in the nanosecond range. The relative phase of the ground and the first excited states evolves depending on the energy difference between them. Once Φ c -pulse is over, the double well is restored and the system is measured in the basis | L | R (5). The readout of the qubit flux state is done by applying a bias current ramp to the dc SQUID and recording its switching current to the voltage state.... (a) The measured double SQUID flux Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for qubit manipulation at which the qubit potential has a shape as indicated in the insets.... Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control flux amplitude Φ c 3 . Circles are the experimentally observed oscillation frequencies for the corresponding pulse amplitudes.... The oscillation frequency ω 0 depends on the amplitude of the manipulation pulse Δ Φ c since it determines the shape of the single well potential and the energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger oscillation frequency as shown in Fig.  fig:4(a). In Fig.  fig:5, we plot the energy spacing between the ground state and the three excited states (indicated as E k - E 0 / h with k=1,2,3) versus the flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using the experimental parameters. In the same figure, we plot the measured oscillation frequencies for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. The fact that a small asymmetry in the potential does not change the oscillation frequency, as shown in Fig.  fig:4(b), is consistent with the interpretation as the energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in the controlling flux Φ x .... The flux pattern is repeated for 10 2 - 10 4 times in order to evaluate the probability P L = L | Ψ f i n a l 2 of occupation of the left state at the end of the manipulation. By changing the duration Δ t of the manipulation pulse Φ c , we observed coherent oscillations between the occupations of the states | L and | R shown in Fig.  fig:4(a). The oscillation frequency could be tuned between 6 and 21 GHz by changing the pulse amplitude Δ Φ c . These oscillations persist when the potential is made slightly asymmetric by varying the value Φ x 1 . As it is shown in Fig.  fig:4(b), detuning from the symmetric potential by up to ± 2.9 m Φ 0 only slightly changes the amplitude and symmetry of the oscillations. When the qubit was initially prepared in | R state instead of | L state we observed similar oscillations.... Probability to measure the state in dependence of the pulse duration Δ t for the qubit initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated oscillation frequency, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount.... Assuming identical junctions and negligible inductance of the smaller loop ( l ≪ L ), the system dynamics is equivalent to the motion of a particle with the Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is the spatial coordinate of the equivalent particle, p is the relative conjugate momentum, M = C Φ b 2 is the effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are the normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β qubit initialization and readout. The single well, or more exactly the two lowest energy states | 0 and | 1 in this well, is used for the coherent evolution of the qubit.... (a) Schematic of the flux qubit circuit. (b) The control flux Φ c changes the potential barrier between the two flux states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control flux Φ x on the potential symmetry.... The investigated circuit, shown in Fig.  fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of the two JJs embedded in the dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. The qubit is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to the large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. The readout of the qubit flux is performed by measuring the switching current of an unshunted dc SQUID, which is inductively coupled to the qubit . The circuit was manufactured by Hypres  using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. The dielectric material used for junction isolation is SiO 2 . The whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between the loops. The JJs have dimensions of 3 × 3 μ m 2 and the entire device occupied a space of 230 × 430 μ m 2 . All the measurements have been performed at a sample temperature of 15 mK. The currents generating the two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at the 1K-pot stage of a dilution refrigerator. To generate the flux Φ c , a bias-tee at room temperature was used to combine the outputs of a current source and a pulse generator. For biasing and sensing the readout dc SQUID, we used superconducting wires and metal powder filters at the base temperature, as well as attenuators and low-pass filters with a cut-off frequency of 10 kHz at the 1K-pot stage. The chip holder with the powder filters was surrounded by one superconducting and two cryoperm shields.
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Generation and control of entangled states. a, Spectroscopic characterization of the energy levels (see Fig. 1b inset) after a π (upper scan) and a 2 π (lower scan) Rabi pulse on the qubit transition. In the upper scan, the system is first excited to | 10 from which it decays towards the | 01 excited state (red sideband at 3.58 GHz) or towards the | 00 ground state ( F L = 6.48  GHz). In the lower scan, the system is rotated back to the initial state | 00 wherefrom it is excited into the | 10 or | 11 states (see, in dashed, the blue sideband peak at 9.48 GHz for 13 dB more power). b, Coupled Rabi oscillations: the blue sideband is excited and the switching probability is recorded as a function of the pulse length for different microwave powers (plots are shifted vertically for clarity). For large microwave powers, the resonance peak of the blue sideband is shifted to 9.15 GHz. When detuning the microwave excitation away from resonance, the Rabi oscillations become faster (bottom four curves). These oscillations are suppressed by preparing the system in the | 10 state with a π pulse and revived after a 2 π pulse (top two curves in Fig. 3b) c, Coupled Rabi oscillations: after a π pulse on the qubit resonance ( | 00 → | 10 ) we excite the red sideband at 3.58 GHz. The switching probability shows coherent oscillations between the states | 10 and | 01 , at various microwave powers (the curves are shifted vertically for clarity). The decay time of the coherent oscillations in a, b is ∼ 3  ns.... Oscillator relaxation time. a, Rabi oscillations between the | 01 and | 10 states (during pulse 2 in the inset) obtained after applying a first pulse (1) in resonance with the oscillator transition. Here, the interval between the two pulses is 1 ns. The continuous line represents a fit using an exponentially decaying sinusoidal oscillation plus an exponential decay of the background (due to the relaxation into the ground state). The oscillation’s decay time is τ c o h = 2.9  ns, whereas the background decay time is ∼ 4  ns. b, The amplitude of Rabi oscillations as a function of the interval between the two pulses (the vertical bars represent standard error bars estimated from the fitting procedure, see a). Owing to the oscillator relaxation, the amplitude decays in τ r e l ≈ 6  ns (the continuous line represents an exponential fit).... Qubit - SQUID device and spectroscopy a, Atomic force micrograph of the SQUID (large loop) merged with the flux qubit (the smallest loop closed by three junctions); the qubit to SQUID area ratio is 0.37. Scale bar, 1 μ m . The SQUID (qubit) junctions have a critical current of 4.2 (0.45)  μ A. The device is made of aluminium by two symmetrically angled evaporations with an oxidation step in between. The surrounding circuit shows aluminium shunt capacitors and lines (in black) and gold quasiparticle traps 3 and resistive leads (in grey). The microwave field is provided by the shortcut of a coplanar waveguide (MW line) and couples inductively to the qubit. The current line ( I ) delivers the readout pulses, and the switching event is detected on the voltage line ( V ). b, Resonant frequencies indicated by peaks in the SQUID switching probability when a long microwave pulse excites the system before the readout pulse. Data are represented as a function of the external flux through the qubit area away from the qubit symmetry point. Inset, energy levels of the qubit - oscillator system for some given bias point. The blue and red sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96  GHz and -2.90  GHz, respectively, to the central continuous line (numerical fit). These values are close to the oscillator resonance ν p at 2.91 GHz (solid circles) and we attribute the small differences to the slight dependence of ν p on qubit state. c, The plasma resonance (circles) and the distances between the qubit peak (here F L = 6.4  GHz) and the red/blue (up/down triangles) sidebands as a function of an offset current I b o f f through the SQUID. The data are close to each other and agree well with the theoretical prediction for ν p versus offset current (dashed line).... Rabi oscillations at the qubit symmetry point Δ = 5.9  GHz. a, Switching probability as a function of the microwave pulse length for three microwave nominal powers; decay times are of the order of 25 ns. For A = 8  dBm, bi-modal beatings are visible (the corresponding frequencies are shown by the filled squares in b). b, Rabi frequency, obtained by fast Fourier transformation of the corresponding oscillations, versus microwave amplitude. In the weak driving regime, the linear dependence is in agreement with estimations based on sample design. A first splitting appears when the Rabi frequency is ∼ ν p . In the strong driving regime, the power independent Larmor precession at frequency Δ gives rise to a second splitting. c, This last aspect is obtained in numerical simulations where the microwave driving is represented by a term 1 / 2 h F 1 cos Δ t and a small deviation from the symmetry point (100 MHz) is introduced in the strong driving regime (the thick line indicates the main Fourier peaks). Radiative shifts 20 at high microwave power could account for such a shift in the experiment.
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Phase qubit coupled to a tank circuit.... In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig.  fig1). The method was successfully applied to a three-junction qubit in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. 
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(a) Principle of the detection scheme. After the Rabi pulse, a microwave pulse at the plasma frequency resonantly enhances the escape rate. The bias current is maintained for 500 n s above the retrapping value. (b) Resonant activation peak for different Rabi angle. Each curve was offset by 5 % for lisibility. The Larmor frequency was f q = 8.5 ~ G H z . Pulse 2 duration was 10 ~ n s . (c) Resonant activation peak without (full circles) and after (open circles) a π pulse. The continuous line is the difference between the two switching probabilities. (d) Rabi oscillation measured by DC current pulse (grey line, amplitude A = 40 % ) and by resonant activation method with a 5 ~ n s RAP (black line, A = 62 % ), at the same Larmor frequency. fig4... The parameters of our qubit were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the SQUID and the qubit by fitting the qubit “step" appearing in the SQUID’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi oscillation experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the microwave frequency to the qubit resonance and measured the switching probability as a function of the microwave pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the qubit. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these oscillations.... (insert) Typical resonant activation peak (width 40 ~ M H z ), measured after a 50 ~ n s microwave pulse. Due to the SQUID non-linearity, it is much sharper at low than at high frequencies. (figure) Center frequency of the resonant activation peak as a function of the external magnetic flux (squares). It follows the switching current modulation (dashed line). The solid line is a fit yielding the values of the shunt capacitor and stray inductance given in the text. fig3... (a) Rabi oscillations at a Larmor frequency f q = 7.15 ~ G H z (b) Switching probability as a function of current pulse amplitude I without (closed circles, curve P s w 0 I ) and with (open circles, curve P s w π I b ) a π pulse applied. The solid black line P t h 0 I b is a numerical adjustment to P s w 0 I b assuming escape in the thermal regime. The dotted line (curve P t h 1 I b ) is calculated with the same parameters for a critical current 100 n A smaller, which would be the case if state 1 was occupied with probability unity. The grey solid line is the sum 0.32 P t h 1 I b + 0.68 P t h 0 I b . fig2... We then measure the effect of the qubit on the resonant activation peak. The principle of the experiment is sketched in figure fig4a. A first microwave pulse at the Larmor frequency induces a Rabi rotation by an angle θ 1 . A second microwave pulse of duration τ 2 = 10 n s is applied immediately after, at a frequency f 2 close to the plasma frequency, with a power high enough to observe resonant activation. In this experiment, we apply a constant bias current I b through the SQUID ( I b = 2.85 μ A , I b / I C = 0.85 ) and maintain it at this value 500 ~ n s after the microwave pulse to keep the SQUID in the running state for a while after switching occurs. This allows sufficient voltage to build up across the SQUID and makes detection easier, similarly to the plateau used at the end of the DCP in the previously shown method. At the end of the experimental sequence, the bias current is reduced to zero in order to retrap the SQUID in the zero-voltage state. We measured the switching probability as a function of f 2 for different Rabi angles θ 1 . The results are shown in figure fig4b. After the microwave pulse, the qubit is in a superposition of the states 0 and 1 with weights p 0 = c o s 2 θ 1 / 2 and p 1 = s i n 2 θ 1 / 2 . Correspondingly, the resonant activation signal is a sum of two peaks centered at f p 0 and f p 1 with weights p 0 and p 1 , which reveal the Rabi oscillations.... We show the two peaks corresponding to θ 1 = 0 (curve P s w 0 , full circles) and θ 1 = π (curve P s w π , open circles) in figure fig4c. They are separated by f p 0 - f p 1 = 50 ~ M H z and have a similar width of 90 ~ M H z . This is an indication that the π pulse efficiently populates the excited state (any significant probability for the qubit to be in 0 would result into broadening of the curve P s w π ), and is in strong contrast with the results obtained with the DCP method (figure fig2b). The difference between the two curves S f = P s w 0 - P s w π (solid line in figure fig4c) gives a lower bound of the excited state population after a π pulse. Because of the above mentioned asymmetric shape of the resonant activation peaks, it yields larger absolute values on the low- than on the high-frequency side of the peak. Thus the plasma oscillator non-linearity increases the sensitivity of our measurement, which is reminiscent of the ideas exposed in . On the data shown here, S f attains a maximum S m a x = 60 % for a frequency f 2 * indicated by an arrow in figure fig4c. The value of S m a x strongly depends on the microwave pulse duration and power. The optimal settings are the result of a compromise between two constraints : a long microwave pulse provides a better resonant activation peak separation, but on the other hand the pulse should be much shorter than the qubit damping time T 1 , to prevent loss of excited state population. Under optimized conditions, we were able to reach S m a x = 68 % .... A typical resonant activation peak is shown in the insert of figure fig3. Its width depends on the frequency, ranging between 20 and 50 ~ M H z . This corresponds to a quality factor between 50 and 150 . The peak has an asymmetric shape, with a very sharp slope on its low-frequency side and a smooth high-frequency tail, due to the SQUID non-linearity. We could qualitatively recover these features by simple numerical simulations using the RCSJ model . The resonant activation peak can be unambiguously distinguished from environmental resonances by its dependence on the magnetic flux threading the SQUID loop Φ s q . Figure fig3 shows the measured peak frequency for different fluxes around Φ s q = 1.5 Φ 0 , together with the measured switching current (dashed line). The solid line is a numerical fit to the data using the above formulae. From this fit we deduce the following values C s h = 12 ± 2 p F and L = 170 ± 20 p H , close to the design. We are thus confident that the observed resonance is due to the plasma frequency.... Finally, we fixed the frequency f 2 at the value f 2 * and measured Rabi oscillations (black curve in figure fig4d). We compared this curve to the one obtained with the DCP method in exactly the same conditions (grey curve). The contrast is significantly improved, while the dephasing time is evidently the same. This enhancement is partly explained by the rapid 5 ~ n s RAP (for the data shown in figure fig4d) compared to the 30 ~ n s DCP. But we can not exclude that the DCP intrinsically increases the relaxation rate during its risetime. Such a process would be in agreement with the fact that for these bias conditions, T 1 ≃ 100 ~ n s , three times longer than the DCP duration.... (a) AFM picture of the SQUID and qubit loop (the scale bar indicates 1 ~ μ m ). Two layers of Aluminium were evaporated under ± ~ 20 ~ ∘ with an oxidation step in between. The Josephson junctions are formed at the overlap areas between the two images. The SQUID is shunted by a capacitor C s h = 12 ~ p F connected by Aluminium leads of inductance L = 170 ~ p H (solid black line). The current is injected through a resistor (grey line) of 400 ~ Ω . (b) DCP measurement method : the microwave pulse induces the designed Bloch sphere rotations. It is followed by a current pulse of duration 20 ~ n s , whose amplitude I b is optimized for the best detection efficiency. A 400 ~ n s lower-current plateau follows the DCP and keeps the SQUID in the running-state to facilitate the voltage pulse detection. (c) Larmor frequency of the qubit and (insert) persistent-current versus external flux. The squares and (insert) the circles are experimental data. The solid lines are numerical adjustments giving the tunnelling matrix element Δ , the persistent-current I p and the mutual inductance M . fig1
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So far we constructed slow composite objects paying attention only to the oscillations of the solid lines in the diagrams and assuming very slow dashed lines, i.e., neglected the higher-frequency noise. In fact, one can construct another slow object shown in Fig. F:slowc, if the respective oscillations of the solid lines are compensated by the dashed line from this vertex. In other words, in the frequency domain, one constrains the frequency of the dashed line to be Δ E (or - Δ E , depepnding on the direction of the spin flip at the vertex). The dashed lines from such objects pair up, and the integral w.r.t. their relative position is dominated by small separations, δ t ∼ 1 / Δ E . Thus one finds the slow object of Fig. F:slowd, two vertices linked by a dashed line at frequency Δ E ; it describes the relaxational contribution to dephasing exp - t / 2 T 1 , where... Dissipative dynamics of a Josephson charge qubit. The simplest Josephson charge qubit is the Cooper-pair box shown in Fig. F:qb . It consists of a superconducting island connected by a dc-SQUID (effectively, a Josephson junction with the coupling E J Φ x = 2 E J 0 cos π Φ x / Φ 0 tunable via the magnetic flux Φ x ; here Φ 0 = h c / 2 e ) to a superconducting lead and biased by a gate voltage V g via a gate capacitor C g . The Josephson energy of the junctions in the SQUID loop is E J 0 , and their capacitance C J 0 sets the charging-energy scale E C ≡ e 2 / 2 C g + C J , C J = 2 C J 0 . At low enough temperatures single-electron tunneling is suppressed and only even-parity states are involved. Here we consider low-capacitance junctions with high charging energy E C ≫ E J 0 . Then the number n of Cooper pairs on the island (relative to a neutral state) is a good quantum number; at certain values of the bias V g ≈ V d e g = 2 n + 1 e / C two lowest charge states n and n + 1 are near-degenerate, and even a weak E J mixes them strongly. At low temperatures and operation frequencies higher charge states do not play a role. The Hamiltonian reduces to a two-state model,... FIG.  F:slow. a. Double vertices with low- ω tails, which appear in the evaluation of dephasing. b. Examples of clusters built out of them . c. A low- ω object with a high-frequency dashed line. The relaxation process in e also contributes to dephasing as shown in d.... In the diagrams the horizontal direction explicitly represents the time axis. The solid lines describe the unperturbed (here, coherent) evolution of the qubit’s 2 × 2 density matrix ρ ̂ , exp - i L 0 t θ t , where L 0 is the bare Liouville operator (this translates to 1 / ω - i L 0 in the frequency domain). The vertices are explicitly time-ordered; each of them contributes the term ζ Y σ x τ z / 2 , with the bath operator Y t and the Keldysh matrix τ z = ± 1 for vertices on the upper/lower time branch. Averaging over the fluctuations should be performed; for gaussian correlations it pairs the vertices as indicated by dashed lines in Fig. F:2order, each of the lines corresponding to a correlator Y Y . Fig. F:2order shows contributions to the second-order self-energy Σ ↑ ↓ ← ↑ ↓ 2 (here i j = ↑ ↓ label four entries of the qubit’s density matrix). The term in Fig. F:2ordera gives... F:qbFIG. F:qb. The simplest Josephson charge qubit
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fig:dqdqpc Schematic of an isolated DQD qubit and capacitively coupled low-transparency QPC between source (S) and drain (D) leads.... In simple dyne detection (see schematic in Fig.  fig:rfcircuit), the output signal V o u t t is amplified, and mixed with a local oscillator (LO). The LO for homodyne detection of the amplitude quadrature is V L O t ∝ cos ω 0 t , where the LO frequency is the same as the signal of interest (or very slightly detuned). The resulting low-frequency beats due to mixing the signal with the LO are easily detected.... Equivalent circuit for continuous monitoring of a charge qubit coupled to a classical L C oscillator with inductance L and capacitance C . We consider the charge-sensitive detector that loads the oscillator circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local oscillator, L O , and then measured. fig:rfcircuit... The choice e > 0 corresponds to defining current in terms of the direction of electron flow. That is, in the opposite direction to conventional current. The DQDs are occupied by a single excess electron, the location of which determines the charge state of the qubit. The charge basis states are denoted | 0 and | 1 (see Fig.  fig:dqdqpc). We assume that each quantum dot has only one single-electron energy level available for occupation by the qubit electron, denoted by E 1 and E 0 for the near and far dot, respectively.... The two conjugate parameters we use to describe the oscillator state are the flux through the inductor, Φ t , and the charge on the capacitor, Q t . The dynamics of the oscillator are found by analyzing the equivalent circuit of Fig.  fig:rfcircuit using the well-known Kirchhoff circuit laws. Doing this we find that the classical system obeys the following set of coupled differential equations... Consider the equivalent circuit of Fig.  fig:rfcircuit. The oscillator circuit consisting of an inductance L and capacitance C terminates the transmission line of impedance Z T L = 50 Ω . The voltages (potential drops) across the oscillator components can be written as
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Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert Oscillation (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi oscillation. (b) The BSO oscillation (amplified scale) by itself, produced by subtracting the Rabi oscillation from the plot in (a). (c) The time-dependence of the Rabi frequency. Inset: BSO as a function of the absolute phase of the field with fixed τ .
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quantum degenerate regime We now move to exploring what happens when the detuning between the qubit and the cavity is reduced such that the anharmonicity of quantum ladder of states becomes much larger than the corresponding linewidth κ (see Fig.  gino:fig:context). In order to describe the response of the system to external drive in this regime it is important to take into account the quantum dynamics on the lower anharmonic part of the ladder. When the system is initialized in the ground state, there is a range of drive strengths for which the system will remain blockaded from excitations out of the ground state. However, since the anharmonicity of the JC ladder decreases with excitation number, the transition frequency for excitations between adjacent levels ultimately approaches the bare cavity frequency. Qualitatively, when the excitation level n is such that the anharmonicity becomes smaller than the linewidth κ , we expect the state dynamics to be semiclassical, similar to a driven-damped harmonic oscillator... gino:chirp_figure (Color) Readout control pulse (a) Time trace of the drive amplitude: a fast initial chirp frequency chirp( 10 n s ) can selectively steer the initial state, while the qubit is detuned from the cavity ( ω q - ω c / 2 π ≈ 2 g ). It is followed by a slow displacement to increase contrast and lifetime of the latching state, while the qubit is resonant with the cavity ( κ / 2 π = 2.5 MHz ). The drive amplitude ramp is limited so that the photon blockade photon blockadeis not broken, but the contrast is enhanced by additional driving at the highest drive amplitude. (b) A diagram of transition frequencies shows how the drive frequency chirps through the JC ladder frequencies of the (+) manifold, and how the manifold changes due to the time dependent qubit frequency. (c) Wave packet snapshots at selected times (indicated by bullet points on panel (b)) of the chirping drive frequency of panel (b) conditioned on the initial state of the qubit. (d) The temporal evolution of the reduced density matrix | ρ m n | (the x , y axes denote the quantum numbers m , n of the cavity levels) of the cavity with the control pulse (a) when the qubit initial state is superposition 1 2 | 0 + | 1 . The resonator enters a mesoscopic state of superposition around t = t c due to the entanglement with the qubit and the quantum state sensitivity of the protocol. At later times the off-diagonal parts of this superposition dephase quickly due to the interaction with the environment and the state of the system is being completely projected around t = 3 t c .... strong!driving( t qubit being detuned. Due to the interaction with the qubit, the cavity behaves as nonlinear oscillator with its set of transition frequencies depending on the state of the qubit (see the two distinct sets of lines in Fig.  gino:chirp_figure(b)). The cavity responds with a ringing behavior which is different for the two cases (see Fig.  gino:chirp_figure(c)). The ringing due to the pulse effectively maps the | ↓ and | ↑ to the dim and bright state basins, respectively (see Fig. 3(c)). Since κ t c ≪ 1 , an initial superposition α | ↑ + β | ↓ maps into a coherent superposition of the dim and bright states. Next, (2) a much weaker long pulse transfers the initially created bright state (for initial | ↑ ) to even brighter and longer lived states ( t c t c effects a projection of the pointer state. In designing such a pulse sequence we have the following physical considerations: (a) the initial fast selective chirp... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift dispersive regimeof the second qubit; (c) for the model extended to one transmon qubit  koch charge-insensitive 2007, tuned below the cavity, / 2 π = 7 G H z E C / 2 π = 0.2 G H z , E J / 2 π = 30 G H z , g / 2 π = 0.29 G H z . (For the given parameters, δ 01 / 2 π = - 0.5 G H z , δ 12 / 2 π = - 0.7 G H z , defining δ i j = E j - E i - , with E i the energy of the i th transmon level.) In all panels, the transition frequency asymptotically returns to the bare cavity frequency. In (a) the frequencies within the σ z = ± 1 manifolds are (nearly) symmetric with respect to the bare cavity frequency. For (b), if the state of one (‘spectator’) qubit is held constant, then the frequencies are asymmetric with respect to flipping the other (‘active’) qubit. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon.... The solution of eqn  gino:eq:classic is plotted in Fig.  gino:fig:densclass for the same parameters as in Fig.  gino:fig:latch000b. For weak driving the system response approaches the linear response of the dispersively shifted cavity. Above the lower critical amplitude ξ C 1 the frequency response bifurcates, and the JC oscillator enters a region of bistability... Qubit state measurement in circuit QED Circuit QEDcan operate in different parameter regimes and relies on different dynamical phenomena of the strongly coupled transmon-resonator system strong!coupling. The dispersive readout is the least disruptive to the qubit state and it is realized where the cavity and qubit are strongly detuned. The high power readout operates in a regime where the system response can be described using a semi-classical model and yields an relatively high fidelity fidelitywith simple measurement protocol. When the cavity and qubit are on resonance (the quantum degenerate regime quantum degenerate regime) it is theoretically predicted that the photon blockade photon blockade can also be used to realize a high fidelity readout. gino:fig:context... matt-pc in Fig.  gino:fig:latch000, where we show the average heterodyne amplitude a as a function of drive frequency and amplitude. Despite the presence of 4 qubits in the device, the fact that extensions beyond a two-level model would seem necessary since higher levels of the transmons... Solution to the semiclassical equation  gino:eq:classic, using the same parameters as Fig.  gino:fig:latch000b. (a) Amplitude response as a function of drive frequency and amplitude. The region of bifurcation bifurcationis indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr oscillator (Duffing oscillator) Duffing oscillator, constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability bistability Kerr region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the frequency dependence of the classical solutions (solid line). For comparison, the response from the full quantum simulation of Fig.  gino:fig:latch000b is also plotted (dashed line) for the same parameters. (c) Cut through (a) for driving at the bare cavity frequency, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), showing intracavity amplitude on a linear scale. gino:fig:densclass... Transmitted heterodyne amplitude a as a function of drive detuning (normalized by the dispersive shift dispersive regime χ = g 2 / δ ) and drive amplitude (normalized by the amplitude to put n = 1 photon in the cavity in linear response, ξ 1 = κ / 2 ). Dark colors indicate larger amplitudes. (a) Experimental data  matt-pc, for a device with cavity at 9.07 G H z and 4 transmon qubits transmonat 7.0 , 7.5 , 8.0 , 12.3 G H z . All qubits are initialized in their ground state, and the signal is integrated for the first 400 n s ≃ 4 / κ after switching on the drive. (b) Numerical results for the JC model of eqn  gino:eq:master, with qubit fixed to the ground state and effective parameters δ / 2 π = - 1.0 G H z , g / 2 π = 0.2 G H z , κ / 2 π = 0.001 G H z . These are only intended as representative numbers for circuit QED Circuit QEDand were not optimized against the data of panel (a). Hilbert space is truncated at 10,000 excitations (truncation artifacts are visible for the strongest drive), and results are shown for time t = 2.5 / κ .... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift
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A typical example of QT is shown in Fig.  fig1. It shows two main properties of the evolution: the oscillator spends a very long time at some average level n = n - and then jumps to another significantly different value n + . At the same time the polarization vector of qubit ξ → defined as ξ → = T r ρ ̂ σ → also changes its orientation direction with a clear change of sign of ξ x from ξ x > 0 to ξ x qubit polarization ξ = | ξ → | is very close to unity showing that the qubit remains mainly in a pure state. The drops of ξ appear only during transitions between metastable states. Special checks show that an inversion of ξ x by an additional pulse (e.g. from ξ x > 0 to ξ x oscillator to a corresponding state (from n - to n + ) after time t m ∼ 1 / λ . Thus we have here an interesting situation when a quantum flip of qubit produces a marcoscopic change of a state of detector (oscillator) which is continuously coupled to a qubit (we checked that even larger variation n ± ∼ n p is possible by taking n p = 40 ). In addition to that inside a metastable state the coupling induces a synchronization of qubit rotation phase with the oscillator phase which in its turn is fixed by the phase of driving field. The synchronization is a universal phenomenon for classical dissipative systems . It is known that it also exists for dissipative quantum systems at small effective values of ℏ . However, here we have a new unusual case of qubit synchronization when a semiclassical system produces synchronization of a pure quantum two-level system.... (color online) Top panels: the Poincaré section taken at integer values of ω t / 2 π for oscillator with x = â + â / 2 , p = â - â / 2 i (left) and for qubit polarization with polarization angles θ φ defined in text (right). Middle panels: the same quantities shown at irrational moments of ω t / 2 π . Bottom panels: the qubit polarization phase φ vs. oscillator phase ϕ ( p / x = - tan ϕ ) at time moments as in middle panels for g = 0.04 (left) and g = 0.004 (right). Other parameters and the time interval are as in Fig.  fig1. The color of points is blue/black for ξ x > 0 and red/gray for ξ x < 0 .... (color online) Bistability of qubit coupled to a driven oscillator with jumps between two metastable states. Top panel shows average oscillator level number n as a function of time t at stroboscopic integer values ω t / 2 π ; middle panel shows the qubit polarization vector components ξ x (blue/black) and ξ z (green/gray) at the same moments of time; the bottom panel shows the degree of qubit polarization ξ . Here the system parameters are λ / ω 0 = 0.02 , ω / ω 0 = 1.01 , Ω / ω 0 = 1.2 , f = ℏ λ n p , n p = 20 and g = 0.04 .... The phenomenon of qubit synchronization is illustrated in a more clear way in Fig.  fig2. The top panels taken at integer values ω t / 2 π show the existence of two fixed points in the phase space of oscillator (left) and qubit (right) coupled by quantum tunneling (the angles are determined as ξ x = ξ cos θ , ξ y = ξ sin θ sin φ , ξ z = ξ sin θ cos φ ). A certain scattering of points in a spot of finite size should be attributed to quantum fluctuations. But the fact that on enormously long time (Fig.  fig1) the spot size remains finite clearly implies that the oscillator phase ϕ is locked with the driving phase ω t inducing the qubit synchronization with ϕ and ω t . The plot at t values incommensurate with 2 π / ω (middle panels) shows that in time the oscillator performs circle rotations in p x plane with frequency ω while qubit polarization rotates around x -axis with the same frequency. Quantum tunneling gives transitions between two metastable states. The synchronization of qubit phase φ with oscillator phase ϕ is clearly seen in bottom left panel where points form two lines corresponding to two metastable states. This synchronization disappears below a certain critical coupling g c where the points become scattered over the whole plane (panel bottom right). It is clear that quantum fluctuations destroy synchronization for g < g c . Our data give g c ≃ 0.008 for parameters of Fig.  fig1.... (color online) Right panel: dependence of average qubit polarization components ξ x and ξ z (full and dashed curves) on g , averaging is done over stroboscopic times (see Fig. fig1) in the interval 100 ≤ ω t / 2 π ≤ 2 × 10 4 ; color is fixed by the sign of ξ x averaged over 10 periods (red/gray for ξ x 0 ; this choice fixes also the color on right panel). Left panel: dependence of average level of oscillator in two metastable states on coupling strength g , the color is fixed by the sign of ξ x on right panel that gives red/gray for large n + and blue/black for small n - ; average is done over the quantum state and stroboscopic times as in the left panel; dashed curves show theory dependence (see text)). Two QT are used with initial value ξ x = ± 1 . All parameters are as in Fig.  fig1 except g .... (color online) Dependence of number of transitions N f between metastable states on rescaled qubit frequency Ω / ω 0 for parameters of Fig.  fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice is as in Figs.  fig2, fig3).... (color online) Dependence of average level n ± of oscillator in two metastable states on the driving frequency ω (average and color choice are the same as in right panel of Fig.  fig3); coupling is g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig.  fig1.
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