### 54077 results for qubit oscillator frequency

Contributors: Greenberg, Ya. S., Izmalkov, A., Grajcar, M., Il'ichev, E., Krech, W., Meyer, H. -G.

Date: 2002-08-07

Phase **qubit** coupled to a tank circuit....Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range....In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-junction **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range.

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Contributors: Chen, Yu, Sank, D., O'Malley, P., White, T., Barends, R., Chiaro, B., Kelly, J., Lucero, E., Mariantoni, M., Megrant, A.

Date: 2012-09-09

(Color online) (a) Schematic representation of **qubit** projective measurement, where a current pulse allows a **qubit** in the excited state | e to tunnel to the right well ( R ), while a **qubit** in the ground state | g stays in the left well ( L ). (b) Readout circuit, showing lumped-element L R - C R readout resonator inductively coupled to the **qubit**, with Josephson junction effective inductance L J and capacitance C , with loop inductance L . **Qubit** control is through the differential flux bias line ( F B ). The readout resonator is capacitively coupled through C c to the readout line, in parallel with the other readout resonators. The readout line is connected through a cryogenic circulator to a low-noise cryogenic amplifier and to a room temperature microwave source. (c) Photomicrograph of four-**qubit** sample. F B 1 - 4 are control lines for each **qubit** and R R is the resonator readout line. Inset shows details for one **qubit** and its readout resonator. Scale bar is 50 μ m in length; fig.setup...(Color online) (a) Phase of signal reflected from readout resonator, as a function of the probe microwave **frequency** (averaged 900 times), for the **qubit** in the left ( L , blue) and right ( R , red) wells. Dashed line shows probe **frequency** for maximum visibility. (b) Reflected phase as a function of **qubit** flux bias, with no averaging. See text for details. fig.phase...(Color online) Setup for **frequency**-multiplexed readout. Multiplexed readout signals I p and Q p from top FGPA-DAC board are up-converted by mixing with a fixed microwave tone, then pass through the circulator into the **qubit** chip. Reflected signals pass back through the circulator, through the two amplifiers G 1 and G 2 , and are down-converted into I r and Q r using the same microwave tone, and are then processed by the bottom ADC-FPGA board. Data in the shadowed region are the down-converted I r and Q r spectra output from the ADC-FPGA board; probe signals from the FPGA-DAC board have the same **frequency** spectrum. D C indicates the digital demodulation channels, each processed independently and sent to the computer. fig.measure...With the bias points chosen for each **qubit**, we demonstrated the **frequency**-multiplexed readout by performing a multi-**qubit** experiment. To minimize crosstalk, we removed the coupling capacitors between **qubits** used in Ref. 6. In this experiment, we drove Rabi **oscillations** on each ** qubit’s** | g ↔ | e transition and read out the

**qubit**states simultaneously. We first calibrated the pulse amplitude needed for each

**qubit**to perform a | g → | e Rabi transition in 10 ns. The drive amplitude was then set to 1, 2/3, 1/2 and 2/5 the calibrated Rabi transition amplitude for

**qubits**Q 1 to Q 4 respectively, so that the Rabi period was 20 ns, 30 ns, 40 ns and 50 ns for

**qubits**Q 1 to Q 4 . We then drove each

**qubit**separately using an on-resonance Rabi drive for a duration τ , followed immediately by a projective measurement and

**qubit**state readout. This experiment yielded the measurements shown in Fig. fig.rabi(a)-(d) for

**qubits**Q 1 - Q 4 respectively....We introduce a

**frequency**-multiplexed readout scheme for superconducting phase

**qubits**. Using a quantum circuit with four phase

**qubits**, we couple each

**qubit**to a separate lumped-element superconducting readout resonator, with the readout resonators connected in parallel to a single measurement line. The readout resonators and control electronics are designed so that all four

**qubits**can be read out simultaneously using

**frequency**multiplexing on the one measurement line. This technology provides a highly efficient and compact means for reading out multiple

**qubits**, a significant advantage for scaling up to larger numbers of

**qubits**....(Color online) (a)-(d) Rabi

**oscillations**for

**qubits**Q 1 - Q 4 respectively, with the

**qubits**driven with 1, 2/3, 1/2 and 2/5 the on-resonance drive amplitude needed to perform a 10 ns Rabi | g → | e transition. (e) Rabi

**oscillations**measured simultaneously for all the

**qubits**, using the same color coding and drive amplitudes as for panels (a)-(d). fig.rabi...We demonstrated the multiplexed readout using a quantum circuit comprising four phase

**qubits**and five integrated resonators, shown in Fig. fig.setup(c). The design of this chip is similar to that used for a recent implementation of Shor’s algorithm,, but here the

**qubits**were read out off a single line using microwave reflectometry, replacing the SQUID readout used Ref 6. . This dramatically simplifies the chip design and significantly reduces the footprint of the quantum circuit. We designed the readout resonators so that they resonated at

**frequencies**of 3-4 GHz (far de-tuned from the

**qubit**| g ↔ | e transition

**frequency**of 6-7 GHz), with loaded resonance linewidths of a few hundred kilohertz. This allows us to use

**frequency**multiplexing, which has been successfully used in the readout of microwave kinetic inductance detectors as well as other types of

**qubits**. Combined with custom GHz-

**frequency**signal generation and acquisition boards, this approach provides a compact and efficient readout scheme that should be applicable to systems with 10-100

**qubits**using a single readout line, with sufficient measurement bandwidth for microsecond-scale readout times....The calibration of the readout process was done in two steps. We first optimized the microwave probe

**frequency**to maximize the signal difference between the left and right well states. This was performed by measuring the reflected phase φ as a function of the probe

**frequency**, with the

**qubit**prepared first in the left and then in the right well. In Fig. fig.phase(a), we show the result with the

**qubit**flux bias set to 0.15 Φ 0 , where the difference in L J in two well states was relatively large. The probe

**frequency**that maximized the signal difference was typically mid-way between the loaded resonator

**frequencies**for the

**qubit**in the left and right wells, marked by the dashed line in Fig. fig.phase(a). We typically obtained resonator

**frequency**shifts as large as ∼ 150 kHz for the

**qubit**between the two wells, as shown in Fig. fig.phase(a), significantly larger than the resonator linewidth....With the probe

**frequency**set in the first step, the flux bias was then set to optimize the readout. As illustrated in Fig. fig.phase(b), the optimization was performed by measuring the resonator’s reflected phase as a function of

**qubit**bias flux, at the optimal probe

**frequency**, 3.70415 GHz in this case. The

**qubit**was initialized by setting the flux to its negative “reset” value (position I), where the

**qubit**potential has only one minimum. The flux was then increased to an intermediate value Φ , placing the

**qubit**state in the left well, and the reflection phase measured with a 5 μ s microwave probe signal (blue data). The flux was then set to its positive reset value (position V), then brought back to the same flux value Φ , placing the

**qubit**state in the right well, and the reflection phase again measured with a probe signal (red data). Between the symmetry point III ( Φ = 0.5 ) and the regions with just one potential minimum ( Φ ≤ 0.1 or Φ ≥ 0.9 ), the

**qubit**inductance differs between the left and right well states, which gives rise to the difference in phase for the red and blue data measured at the same flux. This difference increases for the flux bias closer to the single-well region, which can give a signal-to-noise ratio as high as 30 at ambient readout microwave power. The optimal flux bias was then set to a value where the readout had a high signal-to-noise ratio (typically > 5), but with a potential barrier sufficient to prevent spurious readout-induced switching between the potential wells. Several iterations were needed to optimize both the probe

**frequency**and flux bias....With each

**qubit**individually characterized, we then excited and measured all four

**qubits**simultaneously, as shown in Fig. fig.rabi(e). There is no measurable difference between the individually-measured Rabi

**oscillations**in panels (a)-(d) compared to the multiplexed readout in panel (e). ... We introduce a

**frequency**-multiplexed readout scheme for superconducting phase

**qubits**. Using a quantum circuit with four phase

**qubits**, we couple each

**qubit**to a separate lumped-element superconducting readout resonator, with the readout resonators connected in parallel to a single measurement line. The readout resonators and control electronics are designed so that all four

**qubits**can be read out simultaneously using

**frequency**multiplexing on the one measurement line. This technology provides a highly efficient and compact means for reading out multiple

**qubits**, a significant advantage for scaling up to larger numbers of

**qubits**.

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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%).

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Contributors: Vierheilig, Carmen, Bercioux, Dario, Grifoni, Milena

Date: 2010-10-22

We consider a **qubit** coupled to a nonlinear quantum **oscillator**, the latter coupled to an Ohmic bath, and investigate the **qubit** dynamics. This composed system can be mapped onto that of a **qubit** coupled to an effective bath. An approximate mapping procedure to determine the spectral density of the effective bath is given. Specifically, within a linear response approximation the effective spectral density is given by the knowledge of the linear susceptibility of the nonlinear quantum **oscillator**. To determine the actual form of the susceptibility, we consider its periodically driven counterpart, the problem of the quantum Duffing **oscillator** within linear response theory in the driving amplitude. Knowing the effective spectral density, the **qubit** dynamics is investigated. In particular, an analytic formula for the **qubit**'s population difference is derived. Within the regime of validity of our theory, a very good agreement is found with predictions obtained from a Bloch-Redfield master equation approach applied to the composite **qubit**-nonlinear **oscillator** system....model We consider a composed system built of a **qubit**, -the system of interest-, coupled to a nonlinear quantum **oscillator** (NLO), see Fig. linearbath. To read-out the **qubit** state we couple the **qubit** linearly to the **oscillator** with the coupling constant g ¯ , such that via the intermediate NLO dissipation also enters the **qubit** dynamics....Jeff The effective spectral density follows from Eqs. ( gl20) and ( chilarger). It reads: J s i m p l J e f f ω e x = g ¯ 2 γ ω e x n 1 0 4 2 Ω 1 | ω e x | + Ω 1 M γ 2 Ω 1 2 2 n t h Ω 1 + 1 2 n 1 0 4 + 4 M Ω 2 | ω e x | - Ω 1 2 . As in case of the effective spectral density J e f f H O , Eq. ( linearspecdens), we observe Ohmic behaviour at low **frequency**. In contrast to the linear case, the effective spectral density is peaked at the shifted **frequency** Ω 1 . Its shape approaches the Lorentzian one of the linear effective spectral density, but with peak at the shifted **frequency**, as shown in Fig. CompLorentz....Schematic representation of the complementary approaches available to evaluate the **qubit** dynamics: In the first approach one determines the eigenvalues and eigenfunctions of the composite **qubit** plus **oscillator** system (yellow (light grey) box) and accounts afterwards for the harmonic bath characterized by the Ohmic spectral density J ω . In the effective bath description one considers an environment built of the harmonic bath and the nonlinear **oscillator** (red (dark grey) box). In the harmonic approximation the effective bath is fully characterized by its effective spectral density J e f f ω . approachschaubild...mapping The main aim is to evaluate the ** qubit’s** evolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the

**oscillator**and the Ohmic bath are put together, as depicted in Figure approachschaubild, to form an effective bath. The effective Hamiltonian...The transition

**frequencies**in Eqs. ( rc1) and ( rc2) coincide, and in Figs. Plowg and Flowg there is no deviation observed when comparing the three different approaches....where the trace over the degrees of freedom of the bath and of the

**oscillator**is taken. In Fig. approachschaubild two different approaches to determine the

**qubit**dynamics are depicted. In the first approach, which is elaborated in Ref. [...Corresponding Fourier transform of P t shown in Fig. CompNLLP. The effect of the nonlinearity is to increase the resonance

**frequencies**with respect to the linear case. As a consequence the relative peak heights change. CompNLLF...Schematic representation of the composed system built of a

**qubit**, an intermediate nonlinear

**oscillator**and an Ohmic bath. linearbath ... We consider a

**qubit**coupled to a nonlinear quantum

**oscillator**, the latter coupled to an Ohmic bath, and investigate the

**qubit**dynamics. This composed system can be mapped onto that of a

**qubit**coupled to an effective bath. An approximate mapping procedure to determine the spectral density of the effective bath is given. Specifically, within a linear response approximation the effective spectral density is given by the knowledge of the linear susceptibility of the nonlinear quantum

**oscillator**. To determine the actual form of the susceptibility, we consider its periodically driven counterpart, the problem of the quantum Duffing

**oscillator**within linear response theory in the driving amplitude. Knowing the effective spectral density, the

**qubit**dynamics is investigated. In particular, an analytic formula for the

**qubit**'s population difference is derived. Within the regime of validity of our theory, a very good agreement is found with predictions obtained from a Bloch-Redfield master equation approach applied to the composite

**qubit**-nonlinear

**oscillator**system.

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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.

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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)].

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Contributors: Wirth, T., Lisenfeld, J., Lukashenko, A., Ustinov, A. V.

Date: 2010-10-05

The the position of a dip in the amplitude of the reflected pulse is plotted in Fig. fig:2 as a function of microwave **frequency** and applied SQUID flux bias Φ S . Data points indicate the dependence of the tank circuit resonance **frequency** on the applied bias flux. The larger (red) circles correspond to the flux swept from negative to positive values, while the smaller (blue) dots stand for the flux swept in opposite direction. During the flux sweep, due to the crosstalk between Φ S and Φ Q flux lines approximately one flux quantum Φ 0 enters or leaves the **qubit** loop, which gives rise to abrupt shift of the dip **frequency** at specific flux bias values. The resonance **frequency** shift at a bias flux of -0.35 Φ 0 is about 22 MHz, which is larger than the tank circuit’s resonance line width of about 4 MHz. This scheme is thus capable of single-shot detection of the **qubit** flux state....(Color online) Scheme of the measurement setup. The SQUID with shunt capacitor C 1 coupled to the **qubit**. The pulsed microwave signal is applied via a cryogenic circulator, and the reflected signal is amplified by a cryogenic amplifier....(Color online) Shift of the resonance **frequency** of the SQUID resonator by 30 MHz due to the **qubit** changing its magnetic flux by approximately Φ 0 . (a) In the linear regime. (b) SQUID driven in the non-linear regime. Note the larger signal amplitude compared to the linear regime....We couple the **qubit** to a capacitively shunted dc-SQUID which forms a tank circuit having a resonance **frequency** around 2 GHz. It is connected to a microwave line by a coupling capacitor C 0 shown in Fig. fig:1. Our sample was fabricated in a standard niobium-aluminium trilayer process. Measurement of the amplitude and phase of a reflected microwave pulse allows one to determine the shift of the resonance **frequency** of the SQUID-resonator and by doing this deduce the magnetic flux of the **qubit** state....The SQUID resonator **frequency** shift induced by the **qubit** is shown in detail in Fig. fig:3(a). It displays two traces of the normalized reflected signal amplitude versus the applied microwave **frequency** in the vicinity of the **qubit**-state switching. Here, the resonance was located at around 1.9 GHz where the SQUID has higher sensitivity to the flux. The amplitude of the reflected signal drops at the resonance **frequency**. For this measurement, very low microwave power of -120 dBm was applied to SQUID to stay in the linear regime, giving rise to the Lorentzian shape of the resonance dips. Taking into account the line width of 4 MHz and the dependence of the resonance **frequency** on the flux, we achieve a flux resolution of 2-3 m Φ 0 of the detector at operating **frequency** of 1.9 GHz. As the two **qubit** states differ by magnetic flux of the order of Φ 0 , this allows for a very weak inductive coupling between SQUID and **qubit** for future experiments. Fig. fig:3 (b) shows the same **frequency** range as above, but now the power of the input signal is larger, -115 dBm, driving the SQUID into the nonlinear regime. This is revealed by the shape of the dips. The advantage of the non-linear regime is the sharper edge on the low **frequency** side which allows for an even better flux resolution of about 0.5-0.7 m Φ 0 ....superconducting **qubits**, phase **qubit**, dispersive readout, SQUID...We present experimental results on a dispersive scheme for reading out a Josephson phase **qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the **qubit**. We detect the flux state of the **qubit** by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the **qubit** state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple **qubits** using a single microwave line by employing **frequency**-division multiplexing....Figure fig:4 shows Rabi **oscillations** of the **qubit** measured for different driving powers of the **qubit** microwave driving. As it is expected, the **frequency** of Rabi **oscillations** increases approximately linearly with the driving field amplitude. The measured energy relaxation time of the tested **qubit** is rather short and is of order of T 1 = 5 ns. This time is it not limited by the chosen type of readout but rather determined by the intrinsic coherence of the **qubit** itself. We verified this fact by measuring the same **qubit** with the conventional SQUID switching current method, which yielded very similar T 1 . The observed short coherence time is likely to be caused by the dielectric loss in the silicon oxide forming the insulating dielectric layer around the **qubit** Josephson junction ....(Color online) Coherent **oscillations** of the **qubit** for different driving powers, from bottom to top: -18 dBm, -15 dBm, -12 dBm, -9 dBm and -6 dBm. Curves are offset by 0.1 for better visibility....(Color online) Microwave **frequency** applied to the SQUID vs. externally applied flux. The measurement points show the position of a dip in the reflected signal amplitude for two different directions of the flux sweep....On chip, there are two magnetic flux lines, one for flux Φ S biasing the SQUID, see Fig. fig:1, and another for flux Φ Q biasing the **qubit**. The **qubit** is controlled by microwave pulses which are applied via a separate line (not shown) attenuated at several low temperature stages. The SQUID flux bias line is equipped with a current divider and filter at the 1 K stage, and a powder filter at the sample holder. By taking the crosstalk of the two flux coils into account, we can independently change the flux that is seen by the **qubit** and the flux that is seen by the SQUID. This sample was designed with a large mutual inductance between **qubit** loop and dc-SQUID which allowed us to independently characterize the sample by the conventional switching-current technique. The dispersive readout results presented below were obtained without applying any dc-bias to the readout SQUID. ... We present experimental results on a dispersive scheme for reading out a Josephson phase **qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the **qubit**. We detect the flux state of the **qubit** by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the **qubit** state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple **qubits** using a single microwave line by employing **frequency**-division multiplexing.

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Contributors: Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Majer, J., Girvin, S. M., Schoelkopf, R. J.

Date: 2005-02-27

(color online) (a) Rabi **oscillations** in the **qubit** population P vs. Rabi pulse length Δ t (blue dots) and fit with unit visibility (red line). (b) Measured Rabi **frequency** ν R a b i vs. pulse amplitude ϵ s (blue dots) and linear fit....(color online) Measurement response φ (blue lines) and theoretical prediction (red lines) vs. time. At t = 6 μ s (a) a π pulse, (b) a 2 π pulse, and (c) a 3 π pulse is applied to the **qubit**. In each panel the dashed lines correspond to the expected measurement response in the ground state φ , in the saturated state φ = 0 , and in the excited state φ ....The extracted **qubit** population P is plotted versus Δ t in Fig. fig:rabioscillationsa. We observe a visibility of 95 ± 6 % in the Rabi **oscillations** with error margins determined from the residuals of the experimental P with respect to the predicted values. Thus, in a measurement of Rabi **oscillations** in a superconducting **qubit**, a visibility in the population of the **qubit** excited state that approaches unity is observed for the first time. Moreover, we note that the decay in the Rabi **oscillation** amplitude out to pulse lengths of 100 n s is very small and consistent with the long T 1 and T 2 times of this charge **qubit**, see Fig. fig:rabioscillationsa and Ramsey experiment discussed below. We have also verified the expected linear scaling of the Rabi **oscillation** **frequency** ν R a b i with the pulse amplitude ϵ s ∝ n s , see Fig. fig:rabioscillationsb....In our circuit QED architecture , a Cooper pair box , acting as a two level system with ground and excited states and level separation E a = ℏ ω a = E e l 2 + E J 2 is coupled capacitively to a single mode of the electromagnetic field of a transmission line resonator with resonance **frequency** ω r , see Fig. fig:setupa. As demonstrated for this system, the electrostatic energy E e l and the Josephson energy E J of the split Cooper pair box can be controlled in situ by a gate voltage V g and magnetic flux Φ , see Fig. fig:setupa. In the resonant ( ω a = ω r ) strong coupling regime a single excitation is exchanged coherently between the Cooper pair box and the resonator at a rate g / π , also called the vacuum Rabi **frequency** . In the non-resonant regime ( Δ = ω a - ω r > g ) the capacitive interaction gives rise to a dispersive shift g 2 / Δ σ z in the resonance **frequency** of the cavity which depends on the **qubit** state σ z , the coupling g and the detuning Δ . We have suggested that this shift in resonance **frequency** can be used to perform a quantum non-demolition (QND) measurement of the **qubit** state . With this technique we have recently measured the ground state response and the excitation spectrum of a Cooper pair box ....(color online) (a) Measured Ramsey fringes (blue dots) observed in the **qubit** population P vs. pulse separation Δ t using the pulse sequence shown in Fig. fig:setupb and fit of data to sinusoid with gaussian envelope (red line). (b) Measured dependence of Ramsey **frequency** ν R a m s e y on detuning Δ a , s of drive **frequency** (blue dots) and linear fit (red line)....(color online) (a) Simplified circuit diagram of measurement setup. A Cooper pair box with charging energy E C and Josephson energy E J is coupled through capacitor C g to a transmission line resonator, modelled as parallel combination of an inductor L and a capacitor C . Its state is determined in a phase sensitive heterodyne measurement of a microwave transmitted at **frequency** ω R F through the circuit, amplified and mixed with a local **oscillator** at **frequency** ω L O . The Cooper pair box level separation is controlled by the gate voltage V g and flux Φ . Its state is coherently manipulated using microwaves at **frequency** ω s with pulse shapes determined by V p . (b) Measurement sequence for Rabi **oscillations** with Rabi pulse length Δ t , pulse **frequency** ω s and amplitude ∝ n s with continuous measurement at **frequency** ω R F and amplitude ∝ n R F . (c) Sequence for Ramsey fringe experiment with two π / 2 -pulses at ω s separated by a delay Δ t and followed by a pulsed measurement....In a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes....In the experiments presented here, we coherently control the quantum state of a Cooper pair box by applying to the **qubit** microwave pulses of **frequency** ω s , which are resonant with the **qubit** transition **frequency** ω a / 2 π ≈ 4.3 G H z , through the input port C i n of the resonator, see Fig. fig:setupa. The microwaves drive Rabi **oscillations** in the **qubit** at a **frequency** of ν R a b i = n s g / π , where n s is the average number of drive photons within the resonator. Simultaneously, we perform a continuous dispersive measurement of the **qubit** state by determining both the phase and the amplitude of a coherent microwave beam of **frequency** ω R F / 2 π = ω r / 2 π ≈ 5.4 G H z transmitted through the resonator . The phase shift φ = tan -1 2 g 2 / κ Δ σ z is the response of our meter from which we determine the **qubit** populat...We have determined the coherence time of the Cooper pair box from a Ramsey fringe experiment, see Fig. fig:setupc, when biased at the charge degeneracy point where the energy is first-order insensitive to charge noise . To avoid dephasing induced by a weak continuous measurement beam we switch on the measurement beam only after the end of the second π / 2 pulse. The resulting Ramsey fringes **oscillating** at the detuning **frequency** Δ a , s = ω a - ω s ∼ 6 M H z decay with a long coherence time of T 2 ∼ 500 n s , see Fig. fig:Ramseya. The corresponding **qubit** phase quality factor of Q ϕ = T 2 ω a / 2 ∼ 6500 is similar to the best values measured so far in **qubit** realizations biased at such an optimal point . The Ramsey **frequency** is shown to depend linearly on the detuning Δ a , s , as expected, see Fig. fig:Ramseyb. We note that a measurement of the Ramsey **frequency** is an accurate time resolved method to determine the **qubit** transition **frequency** ω a = ω s + 2 π ν R a m s e y . ... In a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes.

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Contributors: Shahriar, M. S., Pradhan, Prabhakar

Date: 2002-12-19

Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert **Oscillation** (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi **oscillation**. (b) The BSO **oscillation** (amplified scale) by itself, produced by subtracting the Rabi **oscillation** from the plot in (a). (c) The time-dependence of the Rabi **frequency**. Inset: BSO as a function of the absolute phase of the field with fixed τ ....We show that if the Rabi **frequency** is comparable to the Bohr **frequency** so that the rotating wave approximation is inappropriate, an extra **oscillation** is present with the Rabi **oscillation**. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-**frequency** transitions. We present a scheme for observing this effect in an atomic beam. ... We show that if the Rabi **frequency** is comparable to the Bohr **frequency** so that the rotating wave approximation is inappropriate, an extra **oscillation** is present with the Rabi **oscillation**. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-**frequency** transitions. We present a scheme for observing this effect in an atomic beam.

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Contributors: Zueco, David, Reuther, Georg M., Kohler, Sigmund, Hänggi, Peter

Date: 2009-07-20

**Oscillator** **frequency** shift as function or the **qubit** splitting ϵ = ω + Δ for the spin state | ↓ obtained (a) within RWA, Eq. shiftRWA, and (b) beyond RWA, Eq. wrnonRWA. The lines mark the analytical results, while the symbols refer to the numerically obtained splitting between the ground state and the first excited state in the subspace of the **qubit** state | ↓ ....We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**....HRabi in the subspace of the **qubit** state | ↓ , where σ z | ↓ = - | ↓ . The results are depicted in Fig. fig:wr. ... We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**.

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