### 54056 results for qubit oscillator frequency

Contributors: Schmidt, Thomas L., Borkje, Kjetil, Bruder, Christoph, Trauzettel, Bjoern

Date: 2010-02-25

(Color online) Possible experimental setup consisting of a **qubit** and an **oscillator** coupled to an atomic point contact (APC). Electrons tunnel at the APC ( t a ) and a fixed tunnel junction ( t b ), which are both biased with a voltage V . The area enclosed by the junctions (red dashed line) is threaded with a magnetic flux to create an Aharonov-Bohm phase. The **qubit** is realized as a Cooper pair box (CPB, yellow). Its state can be tuned using the gate voltage V g and it couples capacitively to both junctions. The **oscillation** of the nanomechanical resonator (NR, green) modulates the tunneling amplitude t a . As discussed in more detail in Appendix app:phases, this setup can be used to realize the tunneling amplitude ( gamma)....which contain Lorentz and Fano shaped resonances at the characteristic **frequencies** of the system, ω = 0 , Ω , 2 Δ . The complete expression for the noise reads S ω = ∑ X S X ω X where X denotes all combinations of **qubit** and **oscillator** operators contained in the EVM ( chi). As mentioned above, it turns out that all except three of the prefactors S X ω are nonvanishing and are distinguishable combinations of the functions α 1 , 2 ω and β 1 , 2 , 3 ω . Results for S X ω can be found in Appendix app:noise. A plot of the relevant cross-correlations’ prefactors is shown in Fig. NoisePlot. Since the shapes of these functions are rather distinct, the expectation values constituting the EVM can be recovered from the total measurable noise S ω ....In this article, we propose a system which allows the detection of entanglement between an **oscillator** and a **qubit** using an electronic measurement in an atomic point contact (APC). The electronic system is based on a tunneling contact, a readout device which is known to be quantum-limited. We find that the measurement of the current and the symmetrized current noise in this system allows the evaluation of a criterion for entanglement based on the density matrix of the **oscillator**-**qubit** system. This allows for the detection of entanglement in arbitrary pure or mixed states. All elements of the proposed setup have been realized separately in different experiments. Moreover, it has been shown that the current and the noise of an APC can be measured with a high accuracy. Therefore, it should be possible to combine both elements into one functional device as schematically shown in Fig. FigScheme and to measure its current and noise properties....In general, the amplitudes γ j = | γ j | e i δ j ( j = 0 , 1 , 2 ) can be complex. Since the global phase is irrelevant, we set δ 0 = 0 . Finite phases δ 1 , 2 can be realized experimentally by closing the electric circuit using an additional tunnel junction as shown in Fig. FigScheme. Threading the loop with a magnetic flux causes Aharonov-Bohm phases which can be absorbed in the tunneling amplitudes and generally lead to finite phases δ 1 and δ 2 . This is discussed in more detail in Appendix app:phases. The benefits of a controllable δ 1 have been investigated for a system consisting of an APC and an **oscillator**: while for δ 1 = 0 , the current noise only depends on the **oscillator** position x 2 , a finite δ 1 leads to terms proportional to p 2 and thus contains information about the **oscillator** momentum. Similarly, the presence of tunable phases δ 1 , 2 increases the number of measurable **oscillator** and **qubit** properties....(Color online) Schematic density plot of the prefactors S X ω X = x σ x x σ y … of the **frequency**-dependent noise S ω = ∑ X S X ω X as a function of δ 1 and ω ....The state of the **qubit**-**oscillator** system modulates the tunneling amplitude γ of the APC. If the **oscillator** acts as one of the electron reservoirs of the APC as shown in Fig. FigScheme, the tunneling gap depends on the **oscillator** displacement x . For small x one obtains γ ∝ γ 0 + γ 1 x . The same dependence can also be realized for capacitive coupling. The **qubit** can be realized as a Cooper pair box in which case a depletion of the electron reservoirs of the APC depending on the state of the **qubit** leads to an additional term γ 2 σ z in the tunneling amplitude. Irrespective of the concrete realization, to lowest order the combined effect of the **oscillator** and the **qubit** leads to...where δ 1 = arg 1 + c e - i Φ / Φ 0 . Hence, this setup provides a way to obtain a tunneling Hamiltonian with tunable δ 1 . This has been important for the calculation of the noise. The setup can easily be extended to achieve the second tunable phase δ 2 which we used in the calculation of the current. Here, we need a third junction with an amplitude t c = t c 0 , which is decoupled from both **oscillator** and **qubit**, and two magnetic fluxes Φ 1 , 2 . A schematic is shown in Fig. fig:phasesb....Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting **qubits** close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a **qubit** in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an **oscillator** and a **qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state. ... Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting **qubits** close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a **qubit** in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an **oscillator** and a **qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state.

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Contributors: Cao, Xiufeng, You, J. Q., Zheng, H., Nori, Franco

Date: 2010-01-26

(Color online) Time evolution of the coherence σ x t versus the time t multiplied by the **qubit** energy spacing Δ . (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-type spectrum are α / Δ 2 = 0.01 , λ = 0.09 Δ (red solid curve); while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 , ω c = 10 Δ (green dashed-dotted curve). (b)The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 , λ = 0.3 (red solid line), and for the high-**frequency** Ohmic bath are α o h = 0.1 , ω c = 10 Δ (green dashed-dotted line). These results show that the decay rate for the low-**frequency** bath is shorter than for the high-**frequency** Ohmic bath. This means that the coherence time of the **qubit** in the low-**frequency** bath is longer than in the high-**frequency** noise case, demonstrating the powerful temporal memory of the low-**frequency** bath. Also, our results reflect the structure of the solution with branch cuts . The **oscillation** **frequency** for the low-**frequency** noise is ω 0 > Δ , in spite of the strength of the interaction. This can be referred to as a blue shift. However, in an Ohmic bath, the **oscillation** **frequency** is ω 0 **frequency** noises....(Color online) The spectral density J ω of the low- and high-**frequency** baths. (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-like spectrum are α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve), while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 and ω c = 10 Δ (green dashed-dotted curve). (b) The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve) and the parameters of the high-**frequency** Ohmic bath are α o h = 0.1 and ω c = 10 Δ (green dashed-dotted curve). The characteristic energy of the isolated **qubit** is indicated by a vertical blue dotted line. Here, and in the following figures, the energies are shown in units of Δ ....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between successive measurements, for a strong coupling between the **qubit** and the bath. The time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.1 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ ....We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between consecutive measurements, for a weak coupling between the **qubit** and the bath. In the horizontal axis, the time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.01 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under the RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ . ... We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment.

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Contributors: de Groot, P. C., Lisenfeld, J., Schouten, R. N., Ashhab, S., Lupascu, A., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-08-06

Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**....Coupled **qubit** system and transitions. a, Optical micrograph of the sample, showing two flux **qubits** colored in blue and red. The inset shows part of each **qubit** loop, both containing four Josephson tunnel junctions. Overlapping the **qubit** loops, in light-grey, are the SQUID-based **qubit**-state detectors. In the top right and bottom left are the two antennas from which the **qubits** are driven. b, Energy level diagram of the coupled **qubit** system. Arrows of the same color indicate transitions of the same **qubit** and are degenerate in **frequency**. c, Pulse sequence used for the coherent excitation of the **qubits**. The first pulse is resonant with **qubit** 1. The second pulse, applied from both antennas simultaneously with independent amplitudes and phases, is resonant with **qubit** 2. After the second pulse the state of both **qubits** is read out. d, The normalized transition strengths of the four transitions in b as a function of the net driving amplitudes a 1 / a 1 + a 2 for ϕ 2 - ϕ 1 = 0 . For ϕ 2 - ϕ 1 = π the dashed and solid lines are interchanged. The black dotted lines indicate the locations of the darkened transitions....Transition strength tuning and darkened transitions. a-c, Rabi **frequency** dependence on φ 2 - φ 1 for three different amplitude-ratios. The color scale represents the Fourier component of P s w , 2 τ 2 . **Qubit** 1 is prepared with a π / 2 -rotation. Markers X 0 and X 1 indicate the conditions for a darkened transition on 00 ↔ 01 and 10 ↔ 11 respectively. d-f P s w , 2 versus the durations τ 1 and τ 2 . The white solid and dashed lines indicate a π - and 2 π -rotation of **qubit** 1, respectively. The driving conditions are as marked by Y left arrow (d), X 0 (e) and X 1 (f)....Driving from a single antenna. Measurement of the state of the **qubits**, represented by switching probabilities P s w , 1 and P s w , 2 , after applying a pulse of duration τ 1 resonant with **qubit** 1, followed by a pulse of duration τ 2 resonant with **qubit** 2. a, P s w , 1 , showing coherent **oscillations** of **qubit** 1 induced by pulse 1. The white solid and dashed lines indicate a π - and 2 π -rotation respectively. For pulse 2, **qubit** 1 only shows relaxation. b, P s w , 2 , showing coherent **oscillations** induced by pulse 2. After an odd number of π -rotations on **qubit** 1, the **oscillation** **frequency** is higher than after an even number of π -rotations. For superposition states of **qubit** 1, a beating pattern of the two **oscillations** is observed. c-f, Level occupations Q of the four different levels. Note that a value of 0.2 has been added to Q 11 to improve visibility. ... Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**.

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Contributors: Lupascu, A., Bertet, P., Driessen, E. F. C., Harmans, C. J. P. M., Mooij, J. E.

Date: 2008-10-03

fig1 (a) **Frequency** of the spectroscopy peaks p 1 (black squares), p 2 (black circles), and p 3 (triangles) versus Φ . The black lines are a fit for the peaks p 1 and p 2 with the expressions for E 01 q b + T L S and E 02 q b + T L S , yielding the following parameters: I p = 331 nA, Δ = 4.512 GHz, ν T L S = 4.706 GHz, and g = 0.104 GHz. The gray line is a plot of E 01 q b + T L S + E 02 q b + T L S / 2 with the above parameters. (b) Spectroscopy for different values of the microwave power P m w at Φ = 3 Φ 0 / 2 . The curves are vertically shifted for clarity....where σ x , y , z T L S are TLS operators and g is the coupling strength. This Hamiltonian is easily diagonalized, yielding the eigenenergies E n q b + T L S ( n = 0 to 3 ) and the transition energies E m n q b + T L S = E n q b + T L S - E m q b + T L S . The continuous lines in In Fig. fig1b are a combined fit of the Φ -dependent transition energies E 01 q b + T L S and E 02 q b + T L S with the **frequency** of the peaks p 1 and p 2 . This fit yields the parameters I p , Δ , ν T L S , and g . The agreement of the model with the data is very good. We note that the good agreement does not justify the specific model for the interaction in Eq. eq_Hamiltonian_interaction, as discussed in more detail below, but it justifies the model of resonant interaction and moreover it provides the value of the coupling g ....We observed the dynamics of a superconducting flux **qubit** coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the **qubit**. The excitation of a two-photon transition to the third excited state of the **qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the **qubit**. The transition **frequency** and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin....fig3 Energy level structure for the **qubit** (qb) coupled to (a) a two level system (TLS) or (b) an harmonic **oscillator** (HO). The dotted lines indicates energy levels for the uncoupled system. Black/gray arrows indicate one-/two-photon transitions starting in the ground state and are labeled by p n , with n the final state of the coupled system....The observation of the two-photon transition brings important additional information about the coupled EQS. We observe (see Fig. fig1a) that the **frequency** of the peak p 3 is the average of the **frequencies** of peaks p 1 and p 2 . This is clearly shown by the gray line in Fig. fig1a, which is a plot of the average of the transition energies E 01 q b + T L S and E 02 q b + T L S , where E 01 q b + T L S and E 02 q b + T L S are given by the best fit to the **frequencies** of p 1 and p 2 . This particular position of the 2-photon peak is consistent with the hypothesis that the EQS is a TLS , but rules out that the EQS is a HO. This can be understood by considering the structure of levels for the coupled **qubit**-TLS and **qubit**-HO cases, as shown in Fig. fig3 a and b respectively. For the latter case the two-photon transition to the third excited state would have a **frequency** significantly lower than the average value of the one-photon transition **frequencies** to the first and second excited states. This conclusion holds for any type coupling between the **qubit** and the HO which is linear in the **oscillator** creation and annihilation operators. Making the distinction between coupled TLSs and HOs is important since HO modes coupled to the **qubit** can appear due to spurious resonances in the electromagnetic circuit used to control and read out the **qubit**....During the experiments we observed important changes of the energy-level structure of the combined **qubit**-TLS system. Our **qubit** sample was used in two experiments, A and B. Between these two experiments our cryostat was warmed up to room temperature. In experiment A a few different configurations of the energy-level structure were observed. In Fig. fig4a we present the spectroscopy data at small power (only lines p 1 and p 2 ) for two such configurations. The measured spectroscopy is in both cases well described by the **qubit**-TLS model (given by Eqs. eq_Hamiltonian_qubit, eq_Hamiltonian_TLS, and eq_Hamiltonian_interaction), but with different **frequency** and coupling of the TLS: ν T L S = 4.706 GHz and g = 0.104 GHz for the first configuration and ν T L S = 4.493 GHz and g = 0.099 GHz for the second configuration. During the first experiment we observed a few changes between such configurations. The change between two configurations was fast on the time scale of a few tens of minutes, which is the time necessary to acquire the data in order to characterize the spectroscopic structure. Each configuration was in turn stable over times of the order of days. We observed for each of these configurations the two-photon transition and Rabi **oscillations** on all the three transitions. In experiment B we observed a similar spectrum (see Fig. fig4b), with a configuration given by I p = 350 nA, Δ = 4.565 GHz, ν T L S = 5.039 GHz, and g = 0.036 GHz. The **frequency** and the coupling of the TLS are significantly different. In contrast to experiment A, the spectrum was stable over all the duration of the experiment (two months). These observations are consistent with other experiments and support the idea that the coupled TLS is of microscopic origin....fig2 Measurements of Rabi **oscillations** at Φ = 3 Φ 0 / 2 , for a **qubit**-TLS configuration given by I p = 331 nA, Δ = 4.47 GHz, ν T L S = 4.39 GHz, and g = 0.099 GHz, for transitions p 1 (squares), p 2 (circles), and p 3 (triangles). (a) Rabi **oscillations** for microwave power P m w = 7 dBm. (b) Rabi **frequency** F Rabi vs P m w . The lines are power law fits for the one- (black) and two-photon (gray) transitions. Only values of F Rabi smaller than 40 MHz are considered for the fit....We now discuss the origin of the peak p 3 in the spectroscopy signal shown in Fig. fig1. Further understanding on this peak is provided by the analysis of Rabi **oscillations**, observed at strong microwave driving. These are shown in Fig. fig2a for three different **frequencies**, corresponding respectively to the peaks p 1 , p 2 , and p 3 . It is interesting to note that the measurement of the Rabi **oscillations** shows that the EQS has coherence times comparable to those of the **qubit** . The microwave amplitude dependence of the Rabi **frequency** is shown in Fig. fig2b for the three different transitions. For low microwave power, we observe a power law behavior with exponent 1.0 for peaks p 1 and p 2 , and 1.8 for peak p 3 . This confirms that p 1 and p 2 are one-photon transitions. We attribute p 3 to a two-photon transition to the third excited state of the coupled system. The value of the exponent of the amplitude dependence, 1.8 , is smaller than the ideal value of 2 . This is consistent with numerical simulations of the driven dynamics. We attribute this difference to the partial excitation of the first two excited states of the coupled system....Spectroscopy is performed by repeating, typically 10 6 times, the following steps : the **qubit** is first prepared in the ground state by energy relaxation. Transitions to excited states are then induced with microwaves at power P m w and **frequency** f m w , applied for a time T m w ; for spectroscopy measurements we take T m w > > T 1 , T 2 . As a final step the driving of the resonant circuit used for readout is switched on and the amplitude V a c is measured. The information on the **qubit** state is provided by the average value of V a c , . In Fig. fig1a the position of the observed spectroscopy peaks for low power is shown as black squares and circles. Away from the symmetry point of the **qubit** ( Φ = 3 Φ 0 / 2 ) the spectrum is similar to the usual flux **qubit** spectrum: we observe a single peak at **frequency** f m w ≈ 2 I p / h Φ - 3 Φ 0 / 2 2 + Δ 2 , corresponding to the transition between the ground and excited states of the **qubit**. However, around Φ = 3 Φ 0 / 2 , we observe two peaks (labeled p 1 and p 2 ) with a Φ dependence characteristic of an anticrossing. This reveals the presence of an EQS with a **frequency** close to the **qubit** parameter Δ . At larger microwave power a third peak is observed (labeled p 3 ) between the peaks p 1 and p 2 . In Fig. fig1b we plot the average as a function of the microwave **frequency** at Φ = 3 Φ 0 / 2 , for increasing microwave power. ... We observed the dynamics of a superconducting flux **qubit** coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the **qubit**. The excitation of a two-photon transition to the third excited state of the **qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the **qubit**. The transition **frequency** and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin.

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Contributors: Plourde, B. L. T., Robertson, T. L., Reichardt, P. A., Hime, T., Linzen, S., Wu, C. -E., Clarke, John

Date: 2005-01-27

(a) SQUID switching probability vs. amplitude of bias current pulse near **qubit** 2 transition. The two curves represent the states corresponding to Φ Q 2 = 0.48 Φ 0 (red) and Φ Q 2 = 0.52 Φ 0 (blue); Φ S is held constant. Each curve contains 100 points averaged 8 , 000 times. (b) I s 50 % vs. Φ S . Each period of **oscillation** contains ∼ 5 , 000 flux values, and each switching current is averaged 8 , 000 times. (c) Dependence of I s 50 % on Φ Q 1 for constant Φ S . (d) **Qubit** flux map. fig:flux-map...We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip....(a) Chip layout. Dark gray represents Al traces, light gray AuCu traces. Pads near upper edge of chip provide two independent flux lines; wirebonded Al jumpers couple left and right halves. Pads near lower edge of chip supply current pulses to the readout SQUID and sense any resulting voltage. (b) Photograph of center region of completed device. Segments of flux lines are visible to left and right of SQUID, which surrounds the two **qubits**. fig:layout...Spectroscopy of **qubit** 2. Enhancement and suppression of I s 50 % is shown as a function of Φ Q 2 and f m relative to measurements in the absence of microwaves. Dashed lines indicate fit to hyperbolic dispersion for 1- and 2-photon **qubit** excitations. The 2-photon fit is one-half the **frequency** of the 1-photon fit. Inset containing ∼ 23 , 000 points is at higher resolution. fig:spectroscopy...Coherent manipulation of **qubit** state. (a) Rabi **oscillations**, scaled to measured SQUID fidelity, as a function of width of 10.0 GHz microwave pulses. (b) Rabi **frequency** vs. 10.0 GHz pulse amplitude; line is least squares fit to the data. (c) Ramsey fringes for **qubit** splitting of 9.95 GHz, microwave **frequency** of 10.095 GHz. (d) Ramsey fringe **frequency** vs. microwave **frequency**. Lines with slopes ± 1 are fits to data. fig:rabi ... We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip.

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Contributors: Huang, Ren-Shou, Dobrovitski, Viatcheslav, Harmon, Bruce

Date: 2005-04-18

Recent experiments on Josephson junction **qubits** have suggested the existence in the tunnel barrier of bistable two level fluctuators that are responsible for decoherence and 1/f critical current noise. In this article we treat these two-level systems as fictitious spins and investigate their influence quantum mechanically with both analytical and numerical means. We find that the Rabi **oscillations** of the **qubit** exhibit multiple stages of decay. New approaches are established to characterize different decoherence times and to allow for easier feature extraction from experimental data. The Rabi **oscillation** of a **qubit** coupled to a spurious resonator is also studied, where we proposed an idea to explain the serious deterioration of the Rabi osillation amplitude....Fig. ( fig:spurious) shows the comparison of the numerical result of a **qubit** dephased by 14 spins with and without a spurious resonator, which is also dephased by the same group of spins. We can see that the Rabi **oscillation** becomes somewhat irregular with reduced amplitude while the long-time slow decay still persists without the sign of fading away. This explains the experimental observation that decoherence time is not reduced by the coupling to the spurious resonator is because the time has already passed T φ and the Rabi **oscillations** have entered the featureless slow decay regime. Notice that the irregularity starts to appear only after t ∼ 2 π / g , which suggests the cause is due to the smearing of the beats....Numerical simulation for the Rabi **oscillation** of one **qubit** coupled with 14 spins at T = 200 mK and T = 10 mK. The left column are the real time Rabi **oscillations**, and the right column are their Fourier transforms. The dotted lines in the right column are the exact Fourier transform obtained in Eq. ( eq:FTsz) for the limit of infinite number of spins. T φ , calculated using Eq. ( eq:t_phi), are 84ns and 165ns respectively. Notice that the small bumps in the **oscillation** envelope in the T = 10 mK are due to the effects of finite spins and some spins are frozen....Time evolution of the **qubit** decohered by many spins undergoing Rabi **oscillation** by a coherent microwave source, σ z t . The Rabi **frequency** 2 α / 2 π = 200 MHz, and δ Ω is generated from a special case where all spin energy splittings are chosen for simplification to be ω k / 2 π = 1 GHz and ∑ k A k 2 / 2 π = 50 MHz at the temperature T = 150 mK. For Ω / 2 π = 10 GHz, these parameters correspond to δ Ω / Ω 0.005 . Notice that the amplitude of the **oscillation** envelope is already down to 50 % at t ∼ 20 ns, but it reaches 25 % only after t = 80 ns, while an exponential decay should reach 25 % around t ∼ 40 ns....Now the width of the peak ω ' = 0 , i.e. ω = 2 α , is controlled by the Gaussian function with width 2 δ Ω , as shown in Fig. ( fig:sz_w). When δ Ω → 0 , the spectrum becomes a delta function at the Rabi **frequency** ω = 2 α . At high **frequency** the Fourier spectrum is dominated by the Gaussian term, which means that σ z t has a Gaussian decay in the short time limit. The singularity at the Rabi **frequency** ω = 2 α implies a very slow decay in the long time limit. The result agrees with the that obtained for Eq. ( eq:sz) in the limit when α ≫ δ Ω , the **oscillation** envelope of σ z t is given by σ z 0 1 + 2 t δ Ω 2 / α 2 - 1 4 , where the evolution begins with a Gaussian (quadratic) damping then changes to a slow power law decay of ∼ α / δ Ω 2 t ....Simulation of the spectral probing of resonance on the **qubit**-spurious resonator system. In both graphs, the vertical axes are the value of σ z at the steady state, and the horizontal axes are the driving microwave **frequencies**. The spurious resonator here has an energy splitting at 10 GHz. The left graph shows when the **qubit** energy is detuned from the resonator at 9.95 GHz, the most visible peak is the **qubit** and the resonator peak is barely visible. When the **qubit** energy is tuned close to the resonator **frequency**, as shown in the right graph, level repulsion takes place. Notice that the spectral peaks here are clearly seperated even though δ Ω and δ Ω r e s are both greater than g ....Rabi **oscillations** of a **qubit** dephased and relaxed by a many-spin system. The parameters are the same as those used in the T = 10 mK graph in Fig. ( fig:tdep). The only additional parameter, the relaxation time T 1 , calculated from Fermi’s golden rule is 1 μ s. The solid lines are the numerical results, and the dashed lines are the approximations in Eq. ( eq:relax_fit) and Eq. ( eq:relax_ft_fit)....which is the ideal Rabi **oscillation**, because every spin is frozen to its own ground state. But as soon as we turn up the temperature, when the value B is allowed to fluctuate, the **oscillation** now has a decay pattern(see in Fig. ( fig:sz)). Notice that the **oscillation** lasts much longer than an ordinary exponential decay but has a rather drastic decrease of amplitude in the beginning....The simulation of both dephasing and relaxation present can be done easily in our program. The newly added spins remain non-interactive among themselves. Fig. ( fig:relax) shows a simulation result with the relaxation time T 1 ≫ T φ . In the real time evolution graph we can clearly see the three stages of the decay process, which starts with a fast Gaussian decay followed by slow decay, and then later the exponential decay finally takes over. Similar behavior has also been observed in the result produced by a **qubit** under the direct influence of 1/f noise. In the graph of the Fourier transform of the same data, the original sharp peak in the Rabi **frequency** is now smeared. Since this problem cannot be solved analytically, we made the approximation of multiplying the **oscillating** part of Eq. ( eq:sz) with an exponential factor e - t / 2 T 1 , so that it becomes...Effect on Rabi **oscillations** caused by a dephased spurious resonator. The solid line represents the case where the **qubit** is coupled to a spurious resonator, and the dotted line is not. The parameters for both are 2 α / 2 π = 100 MHz and δ Ω / 2 π = 145 MHz. Those for the coupled resonator are g / 2 π = 15 MHz and δ Ω r e s / 2 π = 30 MHz. Both **qubit** and spurious resonator couple to the same 14 spins. Notice that because δ Ω > h , the initial reduction of amplitude ends even before one period of Rabi **oscillation**....Fourier spectrum σ ~ z ω near the Rabi **frequency** ω = 2 α . We can use the width of the peak to estimate the decoherence time of the Rabi **oscillation**. δ Ω > and δ Ω **frequency** ω ≫ 2 α + 2 δ Ω the tail of the peak is dominated by the Gaussian term, therefore we can expect a Gaussian decay of σ z t when t ≪ ℏ / δ Ω . While the **frequency** ω is very near 2 α , the peak goes like ω - 2 α -1 / 2 , which implies a very slow decay at the long time limit. ... Recent experiments on Josephson junction **qubits** have suggested the existence in the tunnel barrier of bistable two level fluctuators that are responsible for decoherence and 1/f critical current noise. In this article we treat these two-level systems as fictitious spins and investigate their influence quantum mechanically with both analytical and numerical means. We find that the Rabi **oscillations** of the **qubit** exhibit multiple stages of decay. New approaches are established to characterize different decoherence times and to allow for easier feature extraction from experimental data. The Rabi **oscillation** of a **qubit** coupled to a spurious resonator is also studied, where we proposed an idea to explain the serious deterioration of the Rabi osillation amplitude.

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Contributors: Ralph, J. F., Clark, T. D., Everitt, M. J., Prance, H., Stiffell, P., Prance, R. J.

Date: 2003-06-18

Schematic diagram of persistent current **qubit** inductively coupled to a (low **frequency**) classical **oscillator**. The insert graph shows the time-averaged (Floquet) energies as a function of the external bias field Φ x 1 for the parameters given in the text....We propose a method for characterising the energy level structure of a solid-state **qubit** by monitoring the noise level in its environment. We consider a model persistent-current **qubit** in a lossy resevoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field....Power spectral density for the low **frequency** **oscillator** at the resonance point ( Φ d c = 0.00015 Φ 0 ) for the three spontaneous decay rates shown in Figure 2: γ = 0.005 , 0.05 , 0.5 per cycle. The other parameters are given in the text...(a) Close-up of the time-averaged (Floquet) energies of the single photon resonance (500 MHz) - solid lines - with the time-independent energies given dotted lines. (b) The output power of the low **frequency** **oscillator** at 300 MHz, as a function of the static magnetic flux bias: γ = 0.005 per cycle (solid line), γ = 0.05 per cycle (crosses), γ = 0.5 per cycle (circles). The other parameters are given in the text ... We propose a method for characterising the energy level structure of a solid-state **qubit** by monitoring the noise level in its environment. We consider a model persistent-current **qubit** in a lossy resevoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field.

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Contributors: Wu, Jing-Nuo, Chen, Hung-Kuang, Hsieh, Wen-Feng, Cheng, Szu-Cheng

Date: 2012-07-02

(Color online) (a) Polarization P z t = 1 2 ρ 10 t + ρ 01 t and (b) Decoherence rate Γ d e c . t = - ρ ̇ 10 t + ρ ̇ 01 t ρ 10 t + ρ 01 t of the **qubit** with **frequency** lying inside ( δ / β < 0 ) and outside ( δ / β = 2 ) the PBG region ....(Color online) Dynamics of (a) the ** qubit’s** excited-state probability P t and (b) relaxation rate Γ r e l a x . t of the

**qubit**with different detuning

**frequencies**δ / β = ω 10 - ω c / β from the band edge

**frequency**ω c of the PhC reservoir....(Color online) (a) A

**qubit**with excited state and ground state . The transition

**frequency**ω 10 is nearly resonant with the

**frequency**range of the PhC reservoir. (b) Directional dependent dispersion relation near band edge expressed by the effective-mass approximation with the edge

**frequency**ω c . (c) Photon DOS ρ ω of the anisotropic PhC reservoir exhibiting cut-off photon mode below the edge

**frequency**ω c ....We study the quantum dynamics of relaxation, decoherence and entropy of a

**qubit**embedded in an anisotropic photonic crystal (PhC) through fractional calculus. These quantum measurements are investigated by analytically solving the fractional Langevin equation. The

**qubit**with

**frequency**lying inside the photonic band gap (PBG) exhibits the preserving behavior of energy, coherence and information amount through the steady values of excited-state probability, polarization

**oscillation**and von Neumann entropy. This preservation does not exist in the Markovian system with

**qubit**

**frequency**lying outside the PBG region. These accurate results are based on the appropriate mathematical method of fractional calculus and reasonable inference of physical phenomena. ... We study the quantum dynamics of relaxation, decoherence and entropy of a

**qubit**embedded in an anisotropic photonic crystal (PhC) through fractional calculus. These quantum measurements are investigated by analytically solving the fractional Langevin equation. The

**qubit**with

**frequency**lying inside the photonic band gap (PBG) exhibits the preserving behavior of energy, coherence and information amount through the steady values of excited-state probability, polarization

**oscillation**and von Neumann entropy. This preservation does not exist in the Markovian system with

**qubit**

**frequency**lying outside the PBG region. These accurate results are based on the appropriate mathematical method of fractional calculus and reasonable inference of physical phenomena.

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Contributors: Gul, Yusuf

Date: 2014-12-29

fig3 (Color online) Emergence of **frequency** locking for two-mode JT system shown in spectrum of the lowest five eigenvalues depending on the **frequency** difference Δ . (a) At Δ = 0 Rabi splitting of first energy levels occurs for k = 0.1 / 2 . Interaction between priviledged and disadvantaged mode can be tuned up to Δ = 0.1 in single effective mode. (b) Range of single mode regime extends up to Δ = 0.5 in ultrastrong regime k = 1.0 / 2...fig3 (Color online) Emergence of localization and synchronization transitions in weak, strong and ultrastrong regime. (a) shows snynchronous structure between damped **oscillating** population imbalance and correlation of priviledged mode in weak coupling k = 0.01 / 2 . (b) priviledged mode becomes synchronous with **qubit** and delocalization-localization transition occurs in population imbalance in strong coupling regime k = 0.1 / 2 . (c) presents the photon blockade in priviledged mode and fully trapped regime in population imbalance with k = 1.0 / 2 and γ φ = 0.1...We consider the nonlinear effects in Jahn-Teller system of two coupled resonators interacting simultaneously with flux **qubit** using Circuit QED. Two **frequency** description of Jahn Teller system that inherits the networked structure of both nonlinear Josephson Junctions and harmonic **oscillators** is employed to describe the synchronous structures in multifrequency scheme. Emergence of dominating mode is investigated to analyze **frequency** locking by eigenvalue spectrum. Rabi Supersplitting is tuned for coupled and uncoupled synchronous con?gurations in terms of **frequency** entrainment switched by coupling strength between resonators. Second order coherence functions are employed to investigate self-sustained **oscillations** in resonator mode and **qubit** dephasing. Snychronous structure between correlations of priviledged mode and **qubit** is obtained in localization-delocalization and photon blockade regime controlled by the population imbalance. ... We consider the nonlinear effects in Jahn-Teller system of two coupled resonators interacting simultaneously with flux **qubit** using Circuit QED. Two **frequency** description of Jahn Teller system that inherits the networked structure of both nonlinear Josephson Junctions and harmonic **oscillators** is employed to describe the synchronous structures in multifrequency scheme. Emergence of dominating mode is investigated to analyze **frequency** locking by eigenvalue spectrum. Rabi Supersplitting is tuned for coupled and uncoupled synchronous con?gurations in terms of **frequency** entrainment switched by coupling strength between resonators. Second order coherence functions are employed to investigate self-sustained **oscillations** in resonator mode and **qubit** dephasing. Snychronous structure between correlations of priviledged mode and **qubit** is obtained in localization-delocalization and photon blockade regime controlled by the population imbalance.

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Contributors: Schmidt, Thomas L., Nunnenkamp, Andreas, Bruder, Christoph

Date: 2012-11-09

(Color online) Upper panel: Rabi **frequency** | Ω R | in units of c = n p h g c 2 / L for μ = - 100 ϵ L and κ = 5 ϵ L , where ϵ L = 2 m L 2 -1 . A large photon linewidth κ has been chosen to highlight the essential features. The crosses denote the Rabi **frequency** and damping determined numerically from Eq. ( eq:Dgamma2). Solid green and red lines correspond to the solutions for the limits Ω | μ | , see Eq. ( eq:GammaOmega_metallic), respectively. Lower panel: The ratio between Rabi **frequency** and damping, Ω R / Γ R , determines the fidelity of **qubit** rotations....These functions are plotted in Fig. fig:plots. The Rabi **frequency** is, as expected, exponentially suppressed in the length of the CR. However, as the photon **frequency** Ω approaches the critical value | μ | , the prefactor 1 - Ω / | μ | **qubit** state for a time t * = π / 4 Ω R . In the presence of damping, the fidelity of such an operation can be estimated as...(Color online) Upper panel: A semiconductor nanowire (along the x axis) hosting Majorana fermions is embedded in a microwave stripline cavity (along the y axis). The red lines show the amplitude of the electric field E → r → . Dark blue (light yellow) sections of the wire indicate topologically nontrivial (trivial) regions. MBSs (stars) exist at the edges of nontrivial (topological superconductor, TS) regions. The MBSs γ 1 and γ 2 can be braided using a T -junction . Lower panel: Band structure of the individual sections of the wire. The four MBSs γ 1 , 2 , 3 , 4 encode one logical **qubit**. The central MBSs γ 2 and γ 3 are tunnel-coupled ( t c ) to a topologically trivial, gapped central region (CR, light yellow) with length L . All energies are small compared to the induced gap Δ ....Majorana bound states have been proposed as building blocks for **qubits** on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi **oscillations** between adjacent Majorana bound states. These **oscillations** can be used to implement an additional single-**qubit** gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-**qubit** operations. ... Majorana bound states have been proposed as building blocks for **qubits** on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi **oscillations** between adjacent Majorana bound states. These **oscillations** can be used to implement an additional single-**qubit** gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-**qubit** operations.

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