Contributors:Johansson, G., Tornberg, L., Shumeiko, V. S., Wendin, G.
Resonant circuits for read-out: a) A lumped element LC-oscillator coupled to a driving source and a radio-frequency detector through a transmission line. b) The radio-frequency single-electron transistor measuring the charge of a charge qubit (SCB). The current through the SET determines the dissipation in the resonant circuit. The dissipation is determined by measuring the amplitude of the reflected signal. c) Setup for measuring the quantum capacitance of the charge qubit. The qubit capacitance influences the resonance frequency of the oscillator. The capacitance is measured by determining the phase-shift of the reflected signal.... up to a constant phase depending on the length of the transmission line. Here Q is the resonator’s quality factor, which for the circuitry in Fig. ResonantCircuitsFig a) is determined by the characteristic impedance on the transmission line Z 0 through Q = ω 0 L C 2 / C c 2 Z 0 . Since there is no dissipation in the oscillator we have | Γ ω | = 1 . Driving the oscillator at the bare resonance frequency ω d = ω 0 the phase-difference between the ground and excited state of the qubit will be... The quantum capacitance of the Cooper-pair box is related to the parametric capacitance of small Josephson junctions which is a dual to the Josephson inductance. The origin of the quantum capacitance of a single-Cooper-pair box (SCB) can be understood as follows. Assume that we put a constant voltage V m on the measurement capacitance of the SCB, i.e. we put a voltage source between the open circles in Fig. ResonantCircuitsFigc. The amount of charge on the measurement capacitance q m g / e V m V g will be a nonlinear function of the voltage V m as well as the gate voltage V g and whether the qubit is in the ground or excited state. We may define an effective (differential) capacitance... Single-contact flux qubit inductively coupled to a linear oscillator.... At the charge degeneracy point the effective capacitance of the SCB in the ground and excited state differs by 2 C Q m a x . Imbedding the SCB in a resonant circuit as shown in Fig. ResonantCircuitsFig a) and c) we can detect the corresponding change in the oscillators resonance frequency ω 0 g / e = 1 / L C ± C Q m a x = ω 0 1 ∓ C Q m a x / 2 C , where ω 0 = 1 / L C is the bare resonance frequency. The voltage reflection amplitude Γ ω = V o u t ω / V d ω seen from the driving side of the transmission line can for a high quality oscillator be written... Circuit diagrams and 2-level energy spectrum of two basic JJ-qubit designs: the SCB charge qubit with LC-oscillator readout (left), and persistent-current flux qubit with SQUID oscillator readout (right). For the charge qubit, the control variable ϵ on the horizontal axis of the energy spectrum (middle) represents the external gate voltage (induced charge), and the splitting is given by the Josephson tunneling energy mixing the charge states. For the flux qubit, the variable ϵ represents the external magnetic flux. In both cases, the energy of the qubit can be "tuned" and the working point controlled. Away from the origin (asymptotically) the levels represent pure charge states (zero or one Cooper pair on the SCB island) or pure flux states (left or right rotating currents in the SQUID ring).... LagrangianSubsection The circuit for performing read-out through the quantum capacitance is presented in figure fig:circuit. A Josephson charge qubit is capacitatively coupled to a harmonic oscillator, which is coupled to a transmission line. Through this line, all measurement on the qubit is performed. We model the line as a semi-infinite line of LC-circuits in series. The working point of the Josephson junction can be chosen using the bias V g . In writing down the Lagrangian we are free to chose any quantities as our coordinates as long as they give a full description of our circuit. Since we are treating a system including a Josephson junction, the phases Φ i t = ∫ t d t ' V i t ' across the circuit elements are natural coordinates, as discussed by Devoret in ref. ... Double-well potential and energy levels of the flux qubit ( f q = π ).
In the absence of high frequency excitation ( f = 0 ) we obtain from Eqs. ( ksi, GammaT) (or from Eqs. ( ksi0, GammaT0) a purely inductive response ( Γ T = γ T , P Z = 1 ), which describes the adiabatic evolution of the qubit in the ground state . The experimental study of the adiabatic evolution provides us with information about the energy gap between the two levels . Therefore, the difference of the low frequency response of the persistent current qubit with and without high frequency excitation can provide additional information about the qubit’s damping rates, Γ and Γ Z (see Eqs. ksi, GammaT). As an illustration of the method we show in Fig. fig1 how the dependence of the tank phase χ on the flux changes with the application of the high frequency excitation. The graphs have been calculated for zero high frequency detuning ( δ = 0 ) from Eq. ( ksi0) for the following parameters: ω / 2 π = ω T / 2 π = 6 MHz, k = 0.03 , Q T = 2000 ; E J = 1.32 × 10 -22 J , E C = 2.14 × 10 -24 J , L q = 40 pH, I q = 280 nA, Δ / 2 π = 1 GHz, Γ Z / 2 π = 0.1 MHz, Γ / 2 π = 4 MHz, and T = 10 mK. At a relatively low power of the irradiation the form of the curves remains unchanged, with the amplitudes of the dips being conditioned by the factor P 0 in Eq. ( ksi0) (Fig. fig1a). At higher powers the second term in the curled brackets of Eq. ( ksi0), which describes the influence of Rabi oscillations, comes into play. As a result, the forms of the phase curves are changed drastically (Fig. fig1b).... The dependence of the tank phase on the flux when under the influence of high frequency irradiation. f = F / h is the power of irradiation in frequency units.
Contributors:Alexander N. Korotkov
Solid lines: synchronization degree D (and in-phase current quadrature 〈X〉) as functions of F for several values of the detection efficiency ηeff. Dashed and dotted lines illustrate the effects of the energy mismatch (ε≠0) and the frequency mismatch (Ω≠Ω0).
(Color online) The reduced dynamics of qubit governed by J A I ω vs by J A F ω in different values of ω c . Top four figures a, b, c, and d plot the evolutions of the diagonal element ρ 11 t beginning at the initial state ρ 1 0 , where Δ = 10 12 Hz, ϵ = 0.01 Δ . Bottom four figures e, f, g, and h plot the evolutions of off-diagonal element ρ 12 t beginning at the initial state ρ 2 0 , where ϵ = 10 12 Hz, Δ = 0.01 ϵ , T=300 k, η ' = 0.004 . The ρ 1 0 and ρ 2 0 are supposed in the text. Here, and in the following, we denote the results from the ESDFs J I ω , and J F ω with I and F in the Legend... (Color online) The values of I m W ω when choosing the cut-off frequency ω c different values. It is shown that the absolute values of I m W ω decreases with the increase of the cut-off frequency of the bath ω c... (Color online) The reduced dynamics of qubit in model B governed by J B I ω vs by J B F ω in different values of ω c . Here, λ = 1 , κ 1 = 1 , η ' = 0.0035 , and Γ = 52 Δ , and the values of other parameters are the same as in Fig. 2.... (Color online) The reduced dynamics of the qubit in model C versus in model D governed by the ESDFs J C I / F ω and J D I / F ω . Top two figures a and b plot the evolutions of the off-diagonal ρ 12 t . Bottom two figures plot the evolutions of the diagonal elements ρ 11 t . Here, κ 1 = κ 2 = λ = 1 , η ' = 0.0035 , and ω c = 7 Δ , Ω 0 = 10 Δ , and the values of the other parameters are the same as in Fig 2.... Spectral density function; Finite cut-off frequency;
Decorerence and Relaxation times
Contributors:Naaman, O., Aumentado, J., Friedland, L., Wurtele, J. S., Siddiqi, I.
In our experiment, the Josephson oscillator was formed by an Al/AlO x /Al Josephson tunnel junction (Fig. circuitc) placed in the middle of a high- Q Nb half-wave coplanar waveguide resonator with a characteristic impedance Z 0 = 50 Ω , and symmetrically coupled to the 50 Ω environment via capacitors C c ≈ 10 fF. The measurements were performed in a dilution refrigerator at T =20 mK; the experimental setup is shown schematically in Fig. circuita. Microwave excitation was applied to the resonator using an HP8780A vector signal generator, frequency-modulated with a triangle waveform to provide a linear, phase-continuous frequency chirp with a 50 MHz span. Typical chirp rates, α = d ω / d t , ranged from α / 2 π = 10 11 Hz/s to 10 13 Hz/s. The transmitted microwave signal, V out , was amplified and then demodulated (1.9 MHz IF bandwidth) to find its amplitude and phase. The amplitude of current oscillations in the resonator is given by I = 2 | V out | / Z 0 × Q / π .... Having characterized its parameters, we proceed to excite the oscillator with a chirped microwave drive. Fig. phasec shows the resonator response to a downwards frequency chirp at the rate of α / 2 π = 5 × 10 11 Hz/s. At low power (blue) the amplitude of oscillations initially increases as the chirp passes through ω 0 . However, as the chirp progresses towards lower frequencies, the resonator decouples from the drive and rings down to rest. For a stronger drive amplitude (red) the response of the resonator changes dramatically: as the chirp passes through ω 0 , the resonator phase becomes locked to the drive and its amplitude grows with time. A threshold for phase-locking can be seen in Fig. AR, which shows the normalized amplitude of current oscillations, I / I 0 , as a function of frequency and drive power for a fixed chirp rate. In region 1, at low powers, no phase locking is observed and the junction current grows only in the vicinity of ω 0 . Above a critical drive power, P c in Fig. AR, the resonator remains phase locked and its amplitude continues to grow up to a deterministic maximum, set either by damping (region 2) or by I 0 (region 3). The existence of a threshold for phase-locking, which obeys a universal scaling law, was first observed in the context of non-neutral plasmas ; our work is the first observation of this transition in a micro-electronic circuit operating at GHz frequency and mK temperature... From the measured width of the phase-locking threshold we can estimate the potential sensitivity of the JCA, where as a benchmark we consider detecting a 1 % variation in I 0 , a typical signal associated with the transition between the ground and excited state of a superconducting ‘Quantronium’ qubit . The discrimination power of the device can be estimated from the fractional change in critical current, Δ I 0 / I 0 , that will shift the ‘s-curve’ by an amount equal to the threshold width: Δ I 0 / I 0 = 2 Δ V d c r / 3 V d c r from equation ( Vc). For the data in the inset of Fig. threshold, and defining the width Δ V d c r for 0.27 < P lock < 0.73 , we find Δ I 0 / I 0 = 9.6 × 10 -3 . The 1 % variation in I 0 (6.1 nA for our device) can be resolved with ∼ 46 % contrast; the ultimate sensitivity of the JCA requires an understanding of the dependence of the threshold width on noise, both classical and quantum, which is currently being pursued. For this chirp rate of α / 2 π = 2 × 10 12 Hz/s, a single measurement can be accomplished in less than 10 μ s. Significantly faster chirp rates and shorter measurement times are in principle possible.... The dynamics associated with operating the amplifier with a frequency modulated (FM) drive versus an amplitude modulated (AM) drive are quite different. In Fig. phaseb,d we show the calculated in-phase and quadrature phase components of I / I 0 for AM and FM drives, respectively. In the AM case, for detuning ω oscillation attractor (A) to a high-amplitude oscillation one (B), with significant phase oscillations associated with the switching as shown in Fig. phaseb. In fact, for large enough detuning, the junction’s phase can exceed δ = π , which can induce state mixing when measuring a quantum system. Trajectories for the chirped drive are shown in Fig. phased, where the two traces correspond to the experimental data shown in Fig. phasec. For sub-critical drives (blue), the system builds amplitude as the chirp frequency crosses ω 0 , but remains in the low-oscillation state and eventually relaxing back to zero amplitude. For super-critical drive (red), the system smoothly follows the evolution of the high-amplitude attractor without spurious oscillations of the phase. Another feature of the JCA is that we can potentially perform quantum measurement with very few photons (near ω 0 ), but “latch” and record the signal with a large number of photons at a final frequency ω < ω 0 , so that minimal fidelity is lost due to noise in the measurement electronics.... phase Microwave transmission and phase space trajectories. a, Transmitted power P out in steady state, measured with a network analyzer. Input powers are -155.5 dBm (blue) to -123 dBm at 2.5 dB step. The system exhibits bifurcation above a critical power P c = - 148 dBm ( 1.58 × 10 -18 W). b, Simulated in-phase and quadrature components of current oscillations for detuning ω - ω 0 / 2 π = 0.5 MHz and drive power ramped in time from -126 dBm to -120 dBm. A and B label the low- and high-amplitude attractors, θ is the phase of the oscillations. c, Response of the resonator to a chirped drive, α / 2 π = 5 × 10 11 Hz/s, at P in = - 126 dBm (blue) and -120 dBm (red). d, Simulated quadratures of the current in a chirped resonator with the same parameters as the data in c. The simulation was terminated at ω / 2 π = 1.6106 GHz.... AR Microwave response of a chirped non-linear resonator. Normalized amplitude of current oscillations in the resonator as a function of drive power and frequency, with the frequency swept downwards at a rate of 5 × 10 11 Hz/s. The arrow indicates the critical power P c for phase-locking. In region 1 where P in is sub-critical, the oscillation amplitude grows in the vicinity of ω 0 and rings back to zero below ω 0 . Above P c , the oscillator phase-locks and its amplitude grows until it reaches a maximum set either by damping (region 2) or by I 0 (region 3).... threshold Critical drive scaling and phase-locking probability curve. Comparison of the critical drive voltage V d obtained from the experiment ( ) to equation ( Vc) with (solid line) and without (dashed line) dissipation. Also shown are values for the critical drive from simulations of the fully non-linear equation of motion for the equivalent R L C circuit ( + ), for lossless resonator (blue) and for Q = 2.75 × 10 4 (red). Inset: probability of phase-locking for drive amplitudes near threshold at α / 2 π = 2 × 10 12 Hz/s, evaluated over a 0.5 MHz frequency band centered at 1.6095 GHz. Solid line is a sigmoidal fit.... We first measure the oscillator in steady state (Fig. phasea). From the microwave transmission, P out , at low power in the linear regime (Fig. phasea, blue curve), we measure ω 0 / 2 π = 1.61564 GHz and infer a quality factor Q = 27500 ± 1000 . The Q is limited by coupling to the 50 Ω environment and is within 10% of its predicted value Q = π / 4 Z 0 2 ω 0 2 C c 2 . When the drive power exceeds a critical value P c = - 148 dBm, the oscillator response bifurcates into two branches . From the measured P c we estimate the junction critical current I 0 = 0.61 ± 0.04 μ A; this value is consistent with room temperature dc resistance measurements on co-fabricated junctions. The stated uncertainties are dominated by possible cryogenic variations in the attenuation of the coaxial lines attached to the resonator. Additionally, the observed power at which bifurcation occurs at different drive frequencies agrees well with theory using the measured values of P c and Q .
(a) Probability distribution of frequencies for ε = 0 , Ω = 5 , and J 0 = 0.05 g . The inset shows the comparison between the probability distribution of frequency and that of the case of increasing N 10 times (i.e., Ω = 5 × 10 , J 0 = 0.05 g / 10 ) with other parameters unchanged. (b) Probability distribution of frequencies for the parameters used in Fig. fig2(b). The inset shows the comparison between the probability distribution of frequencies and that of the case of increasing N 10 times (i.e., Ω = 1 × 10 , J 0 = 0.05 g / 10 ) with other parameters unchanged.... (a) Time evolution of S 0 z for different values of Ω = 2 , J 0 = 0.1 g (green solid curve), Ω = 2 × 10 , J 0 = 0.1 g / 10 (blue dashed curve) and Ω = 2 × 100 , J 0 = 0.1 g / 100 (red dotted curve). The initial qubit state is | ψ 0 = | 1 and the other parameter is ε = 0 . (b) Time evolution of S 0 z for different values of Ω = 1 , J 0 = 0.05 g (green solid curve), Ω = 1 × 10 , J 0 = 0.05 g / 10 (blue dashed curve) and Ω = 1 × 100 , J 0 = 0.05 g / 100 (red dotted curve). The initial qubit state is | ψ 0 = | 1 and the other parameter is ε = 0.5 g . The inset shows the time evolution of the case of Ω = 1 and J 0 = 0.05 g in a larger time interval.... (a) Entropy E t for different initial qubit states: | ψ 0 = | ϕ 1 + | ϕ 2 / 2 (pink dot-dashed curve), | ψ 0 = | 1 (blue dashed curve), | ψ 0 = | ϕ 1 or | ψ 0 = | ϕ 2 (red solid curve). Other parameters are ε = g , J 0 = 0.01 g , Ω = 20 . (b) Entropy E t for (i) pink dot-dashed curve: J 0 = 0.03 g , ε = 0 , (ii) green dashed curve: J 0 = 0.01 g , ε = 0 , (iii) red solid curve: J 0 = 0.01 g , ε = g , and (iv) blue dotted curve: J 0 = 0.01 g , ε = 3 g . The qubit initial state is | ψ 0 = | ϕ 1 or | ψ 0 = | ϕ 2 , and other parameter is Ω = 20 .... If n = | P 2 - P 1 | is chosen to be 3 , then other pairs of integers P ' 1 and P ' 2 with m = | P ' 2 - P ' 1 | = 17 correspond to the same frequency κ for the parameters ε = 0.5 g and J 0 = 0.05 g used in Fig. fig2(b). Thus, summing the probability distributions with all possible combinations of P 1 and P 2 that correspond to the same frequency κ leads to the final frequency probability distribution.... In this section, we present and discuss the results we obtain. We first study the dynamics of the central spin inversion. Figures fig1(a) and fig1(b) show the time evolution of S 0 z for different values of Ω in the case of resonance (i.e., detuning ε = 0 ). Figure fig2(a) shows the time evolution of S 0 z also in resonance (i.e., detuning ε = 0 ) but with a different value of the system-environment coupling strength J 0 from that of Fig. fig1(a). For the parameters chosen in Figs. fig1(a), fig1(b) and fig2(a), the driving strength g is much larger than the coupling strength, i.e., g ≫ J 0 . As a result, the self-Hamiltonian is dominant over the interaction with the environment. The eigenstates of the self-Hamiltonian are | + = | 1 + | 0 / 2 and | - = | 1 - | 0 / 2 , separated by a large Rabi frequency g . The main influence of the environment on the central spin is to destroy the initial phase relation between the states | + and | - . This leads to the decay of S 0 z , i.e., Rabi oscillation decay. From Figs. fig1(a) and fig2(a), we can see that as expected, increasing the value of J 0 results in the increase of the decay rate of the Rabi oscillations. Apart from the coupling constant J 0 , the important factor Ω , Eq. ( eq:Omega), also reflects the influence of the environment on the central spin as shown in Fig. fig1(b). We can see from Figs. fig1(a), fig1(b) and fig2(a) that increasing the factor Ω and increasing the coupling constant J 0 have similar effects. One can observe that the larger the value of Ω is, the stronger the decay of the amplitude of the Rabi oscillations will be. As in the case of the spin-boson model discussed in Ref. , the central spin inversion does not oscillate around the value of S 0 z = 0 . Its oscillations are, however, biased (shifted) a little bit toward the positive value of the initial S 0 z 0 . With the increase of the value of Ω or J 0 , this effect is enhanced.... Figure fig3(a) shows the frequency probability distribution for the case of ε = 0 , J 0 = 0.05 g , and Ω = 5 . It is obvious that the main frequencies are located near g which is approximately the Rabi frequency. This is also the case for Figs. fig1(a), fig1(b) and fig2(a) where the detuning ε = 0 . The contribution of many different frequencies, a consequence of interacting with the AF bath,... Time evolution of S 0 z for different values of (a) Ω = 2 and (b) Ω = 30 . The initial qubit state is | ψ 0 = | 1 and other parameters are ε = 0 and J 0 = 0.05 g .... The environmental conditions affect the dynamics of the central spin and the completeness of the collapse. Figure fig3(b) shows the frequency probability distribution for the central spin inversion with parameters used in Fig. fig2(b). The distribution has a left-hand-side cutoff at κ m i n = g and the center of the distribution is located at κ c = ε 2 + g 2 . One can easily understand this from the frequency relation Eq. ( kappa_freq). It is also obvious from Eq. ( kappa_freq) that the center of the distribution shifts toward larger frequencies with the increase of the detuning [this can also be seen from the comparison between Figs. fig3(a) and fig3(b)]. If the detuning ε exists, it is possible that the shape of the frequency probability distribution becomes approximately Gaussian, which then results in well-defined collapse and revival behavior regions . If the detuning is zero, only a half side of the distribution exists [see Fig. fig3(a)] and the central spin inversion will never show the (complete) collapse and revival behaviors. The frequency probability distribution is determined also by the coupling strength J 0 and the important factor Ω . With the increase of J 0 or Ω , the width of frequency distribution increases and the probability decreases. As a result, the decay of the Rabi oscillations is enhanced.
Contributors:Yu.A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D.V. Averin, T. Tilma, F. Nori, J.S. Tsai
Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks.
... EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the oscillationfrequencies of both qubits in the case of zero coupling (Em=0).
... Schematic diagram of the two-coupled-qubit circuit. Black bars denote Cooper pair boxes.
... Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of each qubit, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant.
... Solid-state qubits
Contributors:Brown, K. R., Ospelkaus, C., Colombe, Y., Wilson, A. C., Leibfried, D., Wineland, D. J.
Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway,
Boulder, CO 80305, USA
... Motional exchange between two ions in separate trapping potentials at approximately the single-quantum level. The probability, P | ↑ a , of measuring ion a in spin state | ↑ a at the end of the experimental sequence is plotted with error bars (s.e.m.) against the time, τ , for which the ions interact. P | ↑ a oscillates with period 2 τ e x = 437 4 μ s as a quantum exchanges between the ions.... We sweep the trapping wells through resonance by varying the static potentials that are applied to the trap electrodes. A plot of the resulting mode frequencies, determined as above, is given in Fig. fig:avoided_crossingb, c, showing a minimum of δ f = 3.0 5 kHz, in agreement with theory.... Micrograph of the ion trap, showing radio-frequency (RF) and d.c. electrodes, and gaps between electrodes (darker areas). The lower part of the figure indicates the simulated potential along the trap x axis. Two trapping wells are separated by 40 μ m, with ion positions marked by red spheres. The d.c. electrodes are sufficient to control the axial frequency and the position of each ion independently. Here both frequencies are ∼ 4 MHz and the potential barrier between the two ions is ∼ 3 meV.... Motional spectroscopy of two coupled ions near the avoided crossing. a, Decreases in collected fluorescence occur at values of excitation frequency, f r f , corresponding to the ion mode frequencies. With τ p = 960 μ s, the splitting on resonance is resolved. b, c, Mode frequencies (b) and mode frequency splitting, δ f (c), for the axial normal modes of two ions separated by 40 μ m. Error bars are smaller than the size of the points. The data were acquired over a 1-h period, and slow variations in ambient potentials gave rise to the fluctuations.... A signature of coupling between the ions is the splitting between the two axial normal mode frequencies. As the trap potential is tuned into the resonance condition, this splitting, δ f , reaches a theoretical minimum δ f = Ω e x / π = 3.1 kHz. A plot of the mode frequencies will therefore show an avoided crossing. We measure the mode frequencies by applying a nearly resonant oscillating potential pulse to one of the trap electrodes. We then illuminate both ions with laser radiation resonant with the 2 S 1 / 2 - 2 P 3 / 2 cycling transition at 313 nm. A decrease in the resulting fluorescence indicates that a mode of the ions’ motion has been resonantly excited. For pulse lengths τ p ≫ 1 / δ f , we resolve the two modes (Fig. fig:avoided_crossinga).... As a final experiment, we demonstrate energy exchange at approximately the single-quantum level. Ideally, the experiment takes the following form. The ions are tuned to the resonance condition throughout and are initially Doppler cooled. Ion a is Raman-cooled, sympathetically cooling ion b and thereby preparing the state | 0 a | ↓ a | 0 b . To create a single motional quantum, we drive ion a with a blue-sideband Raman π pulse (of duration 10 μ s, which is much less than τ e x ), creating the state | 1 a | ↑ a | 0 b . The system oscillates between | 1 a | ↑ a | 0 b and | 0 a | ↑ a | 1 b with period 2 τ e x . After a time τ , we drive ion a with another blue-sideband π pulse, conditionally flipping the spin from | 1 a | ↑ a to | 0 a | ↓ a , dependent on the presence of a motional quantum in ion a . The final internal state probability will be given by P | ↑ a τ = sin 2 Ω e x τ . In practice, contrast in the oscillations (Fig. fig:quantum_exchange) is significantly reduced by incomplete cooling, motional decoherence and decoherence due to imperfect Raman sideband pulses.... Energy swapping between two ions in separate trapping potentials at the level of a few quanta. The mean occupation, n a , of ion a is plotted with error bars (s.e.m.) for various durations, τ , that the ion motional frequencies remain on resonance. The blue curve represents a fit to theory with four free parameters: the two initial mean quantum numbers, the exchange time and the heating rate. Energy exchanges between the ions at 155(1)- μ s intervals. The linearly increasing trend in n a is due to ion heating at a rate of 1,885(10) quanta per second. Uncertainties represent standard errors of the fit parameters.... We estimate that ion a is cooled initially to n a = 0.3 1 . Although we were unable to measure the initial temperature of ion b directly, comparison of the contrast and temporal behavior of our exchange data (Fig. fig:quantum_exchange) with simulations indicates that n b 0.6 . Motional decoherence results from heating and from trap frequency instability over the time required to acquire the data. Raman sideband pulses suffer from variations in laser intensity and fluctuations in the sideband coupling caused by thermal spread in the y and z motional states (Debye-Waller factors ). For ω 0 / 2 π = 5.56 MHz, we observe oscillations with period 2 τ e x = 437 4 μ s. The 2% disagreement with the prediction (447 μ s) of equations eqn:interaction_strength and eqn:sol is probably due to uncertainty in the ion separation and the difficulty of maintaining the exact resonance condition.
Contributors:Shevchenko, S. N., Oelsner, G., Greenberg, Ya. S., Macha, P., Karpov, D. S., Grajcar, M., Hubner, U., Omelyanchouk, A. N., Il'ichev, E.
In a first set of experiments the bigger qubit was studied. A driving signal at the fifth harmonic of the resonator ω d = 5 ω r with a constant power was chosen. (We note that considering the fifth harmonic instead of the third does not change any point in the theoretical description of the system except of putting different values for the driving frequency.) The transmission was measured at small detunings δ ω r from the resonators fundamental mode, while the energy bias ε 0 of the qubit was varied. The experimental results are shown in Fig. Fig:expic1 (a) together with simulations using Eq. ( trans) Fig. Fig:expic1 (b). It can be seen that a good agreement between the two was achieved for a relaxation rate Γ 1 / 2 π = 4 MHz and a pure dephasing Γ φ / 2 π = 200 MHz.... (Color online) Normalized transmission amplitude through the resonator at a probing frequency ω p = ω r for different driving amplitudes ranging from about -131 dBm in (a) to -117 dBm in (g) in 2 dBm steps at the input of the resonator. The transmission is plotted as a function of the energy bias ε 0 of the second qubit and the driving frequency detuning δ ω d = ω d - 3 ω r . The driving frequency is changed only around the third harmonic in the order of its linewidth (about 6 κ ). The presented results correspond to a symmetric power dependence around the center frequency of the third harmonic, since the resonator acts as bandpass filter. Each plot is split into an experimental (positive detuning) and a theoretical (negative detuning) part. For the calculations the figures were split into regions for which Eq. ( trans) was used with a certain index k in Eq.( need_Ref). Several features were highlighted with black rectangles to interrelate theory and experiment (see text).... (Color online) (a) Measured normalized transmission amplitude t for the first qubit (see text) while a strong driving signal is applied. The results are plotted in dependence on the detuning between qubitfrequency and driving signal, (controlled by the qubit bias ε 0 ), and the probing frequency detuning δ ω r = ω p - ω r . The amplification and the attenuation of the transmitted signal is found in agreement with the resonance conditions, ℏ ω p = Δ E ˜ , Eq. ( (i)). (b) Normalized transmission calculated for the same parameters as in (a) following Eq. ( trans) together with Eq. ( a).... We find a quantitative agreement between the theoretical predictions and the experiment for a relaxation rate Γ 1 2 / 2 π = 6 MHz and a pure dephasing Γ φ 2 / 2 π = 100 MHz of the second qubit. As examples we highlight several regions with black rectangles. Direct resonances between the Rabi levels are marked in plot (a) for k = 1 and in (d) for k = 2 . Note, that for k = 1 no amplification is observed since the driving frequency is chosen below the qubit gap. In Fig. Fig:expic2 (d) the power dependence of the resonance lines for k = 2 show a similar dependence as in Ref. [... (Color online) Bare, dressed, and doubly-dressed energy levels. The bare qubit’s energy levels, ± Δ E / 2 , are shown in (a); when they are matched by the driving frequency ω d , the qubit is resonantly excited. (At higher values of the bias ε 0 , the bare-qubit multiphoton excitation should be studied.) The position of the resonance, ℏ ω d = Δ E , is described by the avoided crossing of the dressed-state levels; the dressed and averaged energy levels, ± Δ E ˜ / 2 = ± ℏ Ω R / 2 , are plotted in panel (b). When the dressed energy levels are matched by the second (probe) signal, ℏ ω p = Δ E ˜ , a resonance interaction of the coalesced system is expected, Eq. ( (i)). Also a resonance condition is given by the two-photon process ( (ii)). This is visualized as the avoided crossings of the doubly-dressed states, plotted in panel (c).... Oelsner12]: Δ / h = 3.7 GHz, ω r / 2 π = 2.59 GHz and ω d / 2 π = 3 ω r / 2 π = 7.77 GHz, A d / h = 7 GHz, g 1 / 2 π = 0.8 MHz, and ω p = ω r . Figure Fig:En_levels can be seen as the graphical description of dressing the dressed qubit, which can be considered as the mesoscopic tunable analogue of the atomic systems, as in Ref. [... In order to test our model in the strong driving regime, where Eq. ( need_Ref0) is replaced by Eq.( need_Ref), we analyze the response of the system as a function of the driving amplitude. Here we consider the case where the qubit gap is higher than the driving frequency. For this purpose the smaller qubit was used. The transmission at the fundamental mode, ω p = ω r , was measured while changing the frequency of the driving signal around the third harmonic frequency, and consequently the driving amplitude at the qubit. The results are shown in Fig. Fig:expic2, together with calculated data following from Eq. ( trans). Several sharp lines of amplification (dark) and damping (light) were experimentally observed. The number of lines increases with increasing power and each of the lines corresponds to a resonance condition between the dressed states and the probing signal. To understand their origin, calculated transmission data was added into each plot. We split these theoretical plots into regions and for each of them use Eq. ( trans) with different index k in Eq. ( need_Ref). For high powers we note that the levels of one step of the dressed ladder are equivalent to the ones of the stairs above or below. In that way, we assume that Eq. ( trans) is valid also for those regions where resonances between the lower level of one stair and the higher level of the lower stair occur. To account for these interaction we replace the splitting of the dressed states Δ E ˜ k by ω d - Δ E ˜ k . This makes it possible to relate each of the resonance lines to one index k and to an interaction directly between the Rabi levels or levels of different stairs.
The readout of a coupled qubit is based on the shift in the plasma frequency (and thus of the switching curve) due to different Josephson inductances in two qubit states. The inductance values depend on the type of qubit and its parameters. During the readout the qubit state is encoded in the resulting oscillations of the PDBA by tuning the control parameters (such as the drive frequency and amplitude, i.e., ξ and P ) to a point with the maximal difference (contrast) between the two switching curves. High contrast is reached when the shift in the plasma frequency exceeds the width of the switching curve. In an ideal arrangement, this contrast reaches 100%: P s w = 0 and 1 for two qubit states. For the PDBA the contrast reaches values comparable to those for the JBA with similar circuit parameters (for example, about 0.3% in frequency sensitivity for the parameters of Fig. fig:switch at low T , that is sufficient for reliable readout of the charge-phase qubit shown in Fig. fig:EqvSchm). Further optimization of the PDBA parameters is possible.... The PDB circuit (see Fig. fig:EqvSchm) comprises a dc-current-biased Josephson junction with the critical current I c , capacitance C including the self-capacitance of the junction with, possibly, a contribution of an external capacitance, the linear shunting conductance G , as well as an attached qubit, presented here as a charge-phase qubit . The circuit is driven by a harmonic signal I ac = I A cos 2 ω t at a frequency close to the double frequency of small-amplitude plasma oscillations ω p , i.e., ω ≈ ω p .... Electric circuit diagram of the period-doubling bifurcation detector with microwave-based readout. The resonator is formed by the inductance of a non-linear Josephson junction (large crossed box), biased at a non-zero phase value ϕ 0 , and the capacitance C . The linear losses are accounted for by the conductance G . The resonator is coupled to a charge-phase qubit formed by a superconducting single electron transistor with capacitive gate (left) and attached to the Josephson junction. The qubit operation at the optimal point for an arbitrary bias I 0 is ensured by a proper value of the external magnetic control flux Φ c , applied to the qubit loop, and the gate charge Q g on the qubit island.... Typical switching curves (switching probability P s w = 1 - e - Γ τ o during some observation time τ o vs. ξ ) are shown in Fig. fig:switch for various temperatures for a set of typical circuit parameters. Note that the position and the width of the switching curve (see inset) saturates at low temperatures. This effect is not a manifestation of the real quantum tunneling, but is rather linked to the fact that activation in the rotating frame of the first harmonic Eqs. ( A-and-alpha, eq:uv), i.e., the low-frequency noise in that frame, is given in the laboratory frame by the noise at a finite frequency ω , cf. Eq.( eq:Teff) and above.... yield the range of frequency detunings, ξ - oscillating state with a finite amplitude A + given by Eq. ( A-stat). For ξ < ξ - the parametric resonance curve is multivalued with the stable trivial A = 0 and nontrivial A + solutions, while the solution A - is unstable. Taking into account higher (e.g., ∝ ) terms in Eqs. ( dot-A– dot-alpha) ensures that A + ξ and A - ξ merge, limiting both the amplitude A + and the the range of bistability in ξ ; for stronger drive even higher nonlinearities become important. The shape of the resonance curve, calculated numerically from Eqs. ( dot-A– dot-alpha), is shown in Fig. fig:res-curve for several values of the drive amplitude 3 P just above the excitation threshold. However, for further considerations of the threshold behavior the higher nonlinearities are not crucial, and below we neglect the -terms.... (Color online) Intensity A 2 of oscillations of the Josephson phase at frequency ω versus frequency detuning ξ for two amplitudes of the pumping signal (at frequency 2 ω ). Dashed lines show unstable states. When the detuning approaches a bifurcation point ((i) or (ii)), PDB occurs (vertical arrows from or to the zero state, respectively). For comparison, a typical resonance curve of a JBA is sketched in the inset.... This Hamiltonian for the slow variables can be obtained from the Hamiltonian for the physical quantities . Figure velocity-plot shows a contour plot of the absolute value of the velocity | v | = u ̇ 2 + v ̇ 2 1 / 2 = A ̇ 2 + A 2 α ̇ 2 1 / 2 in the case of a multivalued stationary solution. One can see the darker S -shaped narrow valley, where the motion is slow along the curvilinear s -axis. The black spots in this area show the stationary solutions, which are the stable focus at zero, A = 0 , the stable foci A + and A + * corresponding to equal-amplitude oscillations with a mutual phase shift of π , and the unstable saddles A - and A - * (also with a mutual π -shift). For weak dissipation these ‘saddle points’ are the lower points of the barriers separating the basins of attraction of the foci in the landscape of H . Thus the most probable escape path from the zero state is along the S -shaped valley.