### 56121 results for qubit oscillator frequency

Contributors: Poletto, S., Chiarello, F., Castellano, M. G., Lisenfeld, J., Lukashenko, A., Cosmelli, C., Torrioli, G., Carelli, P., Ustinov, A. V.

Date: 2008-09-08

We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations....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. ... We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations.

Data types:

Contributors: Makhlin, Yuriy, Shnirman, Alexander

Date: 2003-12-22

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....We analyze the dissipative dynamics of a two-level quantum system subject to low-**frequency**, e.g. 1/f noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-**frequency** noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent **oscillations** under the influence of both low- and high-**frequency** fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing....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

**density matrix). The term in Fig. F:2ordera gives...F:qbFIG. F:qb. The simplest Josephson charge**

**qubit**’s**qubit**... We analyze the dissipative dynamics of a two-level quantum system subject to low-

**frequency**, e.g. 1/f noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-

**frequency**noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent

**oscillations**under the influence of both low- and high-

**frequency**fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing.

Data types:

Contributors: Bertet, P., Chiorescu, I., Semba, K., Harmans, C. J. P. M, Mooij, J. E.

Date: 2004-05-03

(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**....We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%)....(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 ... We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%).

Data types:

Contributors: Xia, K., Macovei, M., Evers, J., Keitel, C. H.

Date: 2008-10-14

To study the fidelity of the population transfer between | a and | s , in Fig. subfig:a2s15P, we choose the anti-symmetric state as the initial condition. Applying a continuous-wave TDMF with Rabi **frequency** Ω 0 = 15 γ 0 and detuning δ = 0 , the symmetric state | s reaches its maximum population of 0.90 at time 0.1 γ 0 -1 . After this maximum, the population continues to **oscillate** between | a and | s due to the applied field. This **oscillation** is damped by an overall decay as we include damping with a rate γ 0 . The corresponding concurrence is shown in in Fig. subfig:a2s15C. The concurrence **oscillates** at twice the **frequency** of the population **oscillation**, since both | s and | a are maximally entangled. The local maximum values of the entanglement occur at times n π / 2 δ 2 + Ω 0 2 where either | s or | a is occupied....Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system....In Fig. subfig:figg2sP, we show results for the population of | s for continuous driving. It can be seen that the population reaches a maximum value, but afterwards exhibits rapid **oscillations** at **frequency** 2 2 Ω 0 , while the amplitude of the subsequent maxima in the population decays as an exponential function exp - γ 0 + γ 12 t until the system approaches its stationary state. This result can be understood by reducing the system to a two-state system only involving in the states | s and | g . The numerical results can be well fitted by the solution of this two-state approximation,...(Color online) Time evolution of the population (solid red line) in the antisymmetric state | a . The idea is to prepare the anti-symmetric state while the two **qubits** are non-degenerate, and only afterwards render the two **qubits** degenerate. For this, the parameters are chosen such that γ 12 = 0.9986 γ 0 , λ = 50 γ 0 , δ = 50 γ 0 , Ω 0 = 50 γ 0 . The two **qubit** transition **frequencies** are adjusted via time-dependent bias fluxes, such that the **frequency** difference Δ t (dashed black line) changes from 18 γ 0 to zero as a cosine function during the time period 120 γ 0 -1 to 160 γ 0 -1 . The driving field (dash-dotted blue line) is turned off from its initial value Ω 0 in the period 165 γ 0 -1 to 175 γ 0 -1 ....Our model system consists of two flux **qubits** coupled to each other through their mutual inductance M and to a reservoir of harmonic **oscillators** modeled as an LC circuit, see Fig. fig:system....An example for this is shown in Fig. fig:figPadtn. Initially, the two **qubits** have a **frequency** difference Δ t = 0 = Δ 0 = 18 γ 0 . Applying a continuous TDMF Ω during 0 ≤ γ 0 t ≤ 165 allows to populate the antisymmetric state, as can be seen in Fig. fig:figPadtn. After a certain time ( γ 0 t = 120 in our example), the bias fluxes are continuously adjusted such that the two **qubits** become degenerate, Δ γ 0 t ≥ 160 = 0 . It can be seen from Fig. fig:figPadtn that a preparation fidelity for the antisymmetric state in the degenerate two-**qubit** system of about F = 0.94 is achieved. Finally, the TDMF is switched off as well in the time period 165 ≤ γ 0 t ≤ 175 , demonstrating that it is not required to preserve the population in the antisymmetric state. It should be noted that this scheme does not rely on a delicate choice and control of parameters, as it is the case, e.g., for state preparation via special-area pulses....fig:BellSCRAPdtn(Color online) Creation of superpositions of Bell states controlled by the static detuning δ 0 . The populations ρ ± in states | Φ + (dashed red line) and | Φ - (dash-dotted green line) exhibit periodic **oscillations** as a function of δ 0 . The maximum concurrence C is larger than 0.95 (solid blue line)....fig:BellSCRAP(Color online) Creation of Bell states | Φ ± using the SCRAP technique from the ground state. The solid red line denotes population ρ + in | Φ + , while the dashed green line shows population ρ - in | Φ - . The concurrence (dash-dotted blue line) has a maximum value of 0.94 . The black dash double dotted line indicates the applied SCRAP pulse Rabi **frequency**, while the thick blue line shows the time-dependent Stark detuning....fig:SCRAPg2s(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for γ 12 = 0.9986 γ 0 , and λ = 50 γ 0 . The solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi **frequency** and detuning required for SCRAP....fig:systemTwo superconducting flux **qubits** interacting with each other through their mutual inductance M and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the **qubit** transition **frequencies** around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath....where Φ 0 = h / 2 e is the flux quantum, the system in Fig. fig:system can be described by the total Hamiltonian H = H Q + H B . The two-**qubit** Hamiltonian H Q in two-level approximation and rotating wave approximation is given by ... Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system.

Data types:

Contributors: Lisenfeld, Juergen, Mueller, Clemens, Cole, Jared H., Bushev, Pavel, Lukashenko, Alexander, Shnirman, Alexander, Ustinov, Alexey V.

Date: 2009-09-18

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....Superconducting

**qubits**often show signatures of coherent coupling to microscopic two-level fluctuators (TLFs), which manifest themselves as avoided level crossings in spectroscopic data. In this work we study a phase

**qubit**, in which we induce Rabi

**oscillations**by resonant microwave driving. When the

**qubit**is tuned close to the resonance with an individual TLF and the Rabi driving is strong enough (Rabi

**frequency**of order of the

**qubit**-TLF coupling), interesting 4-level dynamics are observed. The experimental data shows a clear asymmetry between biasing the

**qubit**above or below the fluctuator's level-splitting. Theoretical analysis indicates that this asymmetry is due to an effective coupling of the TLF to the external microwave field induced by the higher

**qubit**levels....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

**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**

**qubit**’s**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**. ... Superconducting

**qubits**often show signatures of coherent coupling to microscopic two-level fluctuators (TLFs), which manifest themselves as avoided level crossings in spectroscopic data. In this work we study a phase

**qubit**, in which we induce Rabi

**oscillations**by resonant microwave driving. When the

**qubit**is tuned close to the resonance with an individual TLF and the Rabi driving is strong enough (Rabi

**frequency**of order of the

**qubit**-TLF coupling), interesting 4-level dynamics are observed. The experimental data shows a clear asymmetry between biasing the

**qubit**above or below the fluctuator's level-splitting. Theoretical analysis indicates that this asymmetry is due to an effective coupling of the TLF to the external microwave field induced by the higher

**qubit**levels.

Data types:

Contributors: Rosenband, Till

Date: 2012-03-01

(color) Numerically optimized free-evolution period T for some of clock protocols considered here, when the **oscillator** noise has an Allan deviation of 1 Hz....The stability of several clock protocols based on 2 to 20 entangled atoms is evaluated numerically by a simulation that includes the effect of decoherence due to classical **oscillator** noise. In this context the squeezed states discussed by Andr\'{e}, S{\o}rensen and Lukin [PRL 92, 239801 (2004)] offer reduced instability compared to clocks based on Ramsey's protocol with unentangled atoms. When more than 15 atoms are simulated, the protocol of Bu\v{z}ek, Derka and Massar [PRL 82, 2207 (1999)] has lower instability. A large-scale numerical search for optimal clock protocols with two to eight **qubits** yields improved clock stability compared to Ramsey spectroscopy, and for two to three **qubits** performance matches the analytical protocols. In the simulations, a laser local **oscillator** decoheres due to flicker-**frequency** (1/f) noise. The **oscillator** **frequency** is repeatedly corrected, based on projective measurements of the **qubits**, which are assumed not to decohere with one another....(color) Probability (P) of measuring each basis state as a function of the atom-**oscillator** phase difference ( φ ). Shown are the various protocols for two and five atoms. Each differently colored curve corresponds to a basis state that ψ 1 is projected onto after free evolution. Vertical text near the curves’ peaks indicates the optimized phase estimate ( φ E s t ). In the simulations, the **frequency** corrections are φ E s t / 2 π T . Shaded in the background is the Gaussian distribution whose variance φ 2 represents the atom-**oscillator** phase differences that occur in the simulation. Also listed is the optimized probe period T , squeezing parameter κ where applicable, and long-term **frequency** variance of the clock extrapolated to 1 second. For long-term averages of n seconds, the variance is f 2 / n ....Numerical simulations of the clock protocols considered here are summarized in Figure figPerf. Ramsey’s protocol defines the standard quantum limit (SQL), and it is evident that entangled states of two or more **qubits** can reduce clock instability, although the GHZ states yield no gain for the noise model considered here, as has been noted previously . The spin-squeezed states suggested by André et al. yield the best performance for 3 to 15 **qubits**, and improve upon the SQL variance by a factor of N -1 / 3 . For more **qubits**, the protocol of Bu...(color online) Long-term statistical variance of entangled clocks that contain different numbers of **qubits**, compared to the standard quantum limit (SQL). The most stable clocks found by the large-scale search are shown as black points. Each point is based on several hours of runtime on NISTxs computing cluster, where typically 2000 processor cores were utilized in parallel. Also shown is the simulated performance of analytically optimized clock protocols. Approximately 15 **qubits** are required to improve upon the SQL by a factor of two. ... The stability of several clock protocols based on 2 to 20 entangled atoms is evaluated numerically by a simulation that includes the effect of decoherence due to classical **oscillator** noise. In this context the squeezed states discussed by Andr\'{e}, S{\o}rensen and Lukin [PRL 92, 239801 (2004)] offer reduced instability compared to clocks based on Ramsey's protocol with unentangled atoms. When more than 15 atoms are simulated, the protocol of Bu\v{z}ek, Derka and Massar [PRL 82, 2207 (1999)] has lower instability. A large-scale numerical search for optimal clock protocols with two to eight **qubits** yields improved clock stability compared to Ramsey spectroscopy, and for two to three **qubits** performance matches the analytical protocols. In the simulations, a laser local **oscillator** decoheres due to flicker-**frequency** (1/f) noise. The **oscillator** **frequency** is repeatedly corrected, based on projective measurements of the **qubits**, which are assumed not to decohere with one another.

Data types:

Contributors: Zhirov, O. V., Shepelyansky, D. L.

Date: 2007-10-10

We study numerically the behavior of **qubit** coupled to a quantum dissipative driven **oscillator** (resonator). Above a critical coupling strength the **qubit** rotations become synchronized with the **oscillator** phase. In the synchronized regime, at certain parameters, the **qubit** exhibits tunneling between two orientations with a macroscopic change of number of photons in the resonator. The life times in these metastable states can be enormously large. The synchronization leads to a drastic change of **qubit** radiation spectrum with appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev {\it et al.} Nature {\bf 449}, 588 (2007)]....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. ... We study numerically the behavior of **qubit** coupled to a quantum dissipative driven **oscillator** (resonator). Above a critical coupling strength the **qubit** rotations become synchronized with the **oscillator** phase. In the synchronized regime, at certain parameters, the **qubit** exhibits tunneling between two orientations with a macroscopic change of number of photons in the resonator. The life times in these metastable states can be enormously large. The synchronization leads to a drastic change of **qubit** radiation spectrum with appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev {\it et al.} Nature {\bf 449}, 588 (2007)].

Data types:

Contributors: Catelani, G., Schoelkopf, R. J., Devoret, M. H., Glazman, L. I.

Date: 2011-06-04

A further test of the theory presented in Sec. sec:th_s is provided by the measurement of the **qubit** resonant **frequency**. In the semiclassical regime of small E C , the **qubit** can be described by the effective circuit of Fig. fig1(b), with the junction admittance Y J of Eq. ( YJ), Y C = i ω C , and Y L = 1 / i ω L [the inductance is related to the inductive energy by E L = Φ 0 / 2 π 2 / L ]. As discussed in Ref. ...As a second example of a strongly anharmonic system, we consider here a flux **qubit**, i.e., in Eq. ( Hphi) we assume E J > E L and take the external flux to be close to half the flux quantum, Φ e ≈ Φ 0 / 2 . Then the potential has a double-well shape and the flux **qubit** ground states | - and excited state | + are the lowest tunnel-split eigenstates in this potential, see Fig. fig:fl_q. The non-linear nature of the sin ϕ ̂ / 2 **qubit**-quasiparticle coupling in Eq. ( HTle) has a striking effect on the transition rate Γ + - , which vanishes at Φ e = Φ 0 / 2 due to destructive interference: for flux biased at half the flux quantum the **qubit** states | - , | + are respectively symmetric and antisymmetric around ϕ = π , while the potential in Eq. ( Hphi) and the function sin ϕ / 2 in Eq. ( wif_gen) are symmetric. Note that the latter symmetry and its consequences are absent in the environmental approach in which a linear phase-quasiparticle coupling is assumed....The transmon low-energy spectrum is characterized by well separated [by the plasma **frequency** ω p , Eq. ( pl_fr)] and nearly degenerate levels whose energies, as shown in Fig. fig:trans, vary periodically with the gate voltage n g . Here we derive the asymptotic expression (valid at large E J / E C ) for the energy splitting between the nearly degenerate levels. We consider first the two lowest energy states and then generalize the result to higher energies....Schematic representation of the transmon low energy spectrum as function of the dimensionless gate voltage n g . Solid (dashed) lines denotes even (odd) states (see also Sec. sec:cpb). The amplitudes of the **oscillations** of the energy levels are exponentially small, see Appendix app:eosplit; here they are enhanced for clarity. Quasiparticle tunneling changes the parity of the **qubit** sate. The results of Sec. sec:semi are valid for transitions between states separated by energy of the order of the plasma **frequency** ω p , Eq. ( pl_fr), and give, for example, the rate Γ 1 0 . For the transition rates between nearly degenerate states of opposite parity, such as Γ o e 1 , see Appendix app:eorate....As an application of the general approach described in the previous section, we consider here a weakly anharmonic **qubit**, such as the transmon and phase **qubits**. We start with the the semiclassical limit, i.e., we assume that the potential energy terms in Eq. ( Hphi) dominate the kinetic energy term proportional to E C . This limit already reveals a non-trivial dependence of relaxation on flux. Note that assuming E L ≠ 0 we can eliminate n g in Eq. ( Hphi) by a gauge transformation. In the transmon we have E L = 0 and the spectrum depends on n g , displaying both well separated and nearly degenerate states, see Fig. fig:trans. The results of this section can be applied to the single-junction transmon when considering well separated states. The transition rate between these states and the corresponding **frequency** shift are dependent on n g . However, since E C ≪ E J this dependence introduces only small corrections to Γ n n - 1 and δ ω ; the corrections are exponential in - 8 E J / E C . By contrast, the leading term in the rate of transitions Γ e ↔ o between the even and odd states is exponentially small. The rate Γ e ↔ o of parity switching is discussed in detail in Appendix app:eorate....Potential energy (in units of E L ) for a flux **qubit** biased at Φ e = Φ 0 / 2 with E J / E L = 10 . The horizontal lines represent the two lowest energy levels, with energy difference ϵ ̄ given in Eq. ( e0_eff)....As low-loss non-linear elements, Josephson junctions are the building blocks of superconducting **qubits**. The interaction of the **qubit** degree of freedom with the quasiparticles tunneling through the junction represent an intrinsic relaxation mechanism. We develop a general theory for the **qubit** decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the **qubit** properties in devices such as the phase and flux **qubits**, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and non-equilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible non-equilibrium quasiparticle distribution. We also derive expressions for the shift in **qubit** **frequency** in the presence of quasiparticles....which has the same form of the Hamiltonian for the single junction transmon [i.e., Eq. ( Hphi) with E L = 0 ] but with a flux-dependent Josephson energy, Eq. ( EJ_flux). Therefore the spectrum follows directly from that of the single junction transmon (see Fig. fig:trans) and consists of nearly degenerate and well separated states. The energy difference between well separated states is approximately given by the flux-dependent **frequency** [cf. Eq. ( pl_fr)]...(a) Schematic representation of a **qubit** controlled by a magnetic flux, see Eq. ( Hphi). (b) Effective circuit diagram with three parallel elements – capacitor, Josephson junction, and inductor – characterized by their respective admittances. ... As low-loss non-linear elements, Josephson junctions are the building blocks of superconducting **qubits**. The interaction of the **qubit** degree of freedom with the quasiparticles tunneling through the junction represent an intrinsic relaxation mechanism. We develop a general theory for the **qubit** decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the **qubit** properties in devices such as the phase and flux **qubits**, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and non-equilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible non-equilibrium quasiparticle distribution. We also derive expressions for the shift in **qubit** **frequency** in the presence of quasiparticles.

Data types:

Contributors: Oxtoby, Neil P., Gambetta, Jay, Wiseman, H. M.

Date: 2007-06-24

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 extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise)....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 ... The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise).

Data types:

Contributors: Ginossar, Eran, Bishop, Lev S., Girvin, S. M.

Date: 2012-07-19

In this book chapter we analyze the high excitation nonlinear response of the Jaynes-Cummings model in quantum optics when the **qubit** and cavity are strongly coupled. We focus on the parameter ranges appropriate for transmon **qubits** in the circuit quantum electrodynamics architecture, where the system behaves essentially as a nonlinear quantum **oscillator** and we analyze the quantum and semi-classical dynamics. One of the central motivations is that under strong excitation tones, the nonlinear response can lead to **qubit** quantum state discrimination and we present initial results for the cases when the **qubit** and cavity are on resonance or far off-resonance (dispersive)....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 ... In this book chapter we analyze the high excitation nonlinear response of the Jaynes-Cummings model in quantum optics when the **qubit** and cavity are strongly coupled. We focus on the parameter ranges appropriate for transmon **qubits** in the circuit quantum electrodynamics architecture, where the system behaves essentially as a nonlinear quantum **oscillator** and we analyze the quantum and semi-classical dynamics. One of the central motivations is that under strong excitation tones, the nonlinear response can lead to **qubit** quantum state discrimination and we present initial results for the cases when the **qubit** and cavity are on resonance or far off-resonance (dispersive).

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