### 62592 results for qubit oscillator frequency

Contributors: Jordan, Andrew N., Buttiker, Markus

Date: 2005-05-02

**qubit** oscillation frequency. We take Γ = Γ x = Γ z = .07 Δ / ℏ . S i j...**frequency**. The low **frequency** part describes the incoherent relaxation ...**frequency** part describes the out of phase, coherent **oscillations** of the...**frequency** in Fig. combo(b,c,d) for different values of ϵ . These correlators...**qubit**. As the quantum measurement is taking place, the current outputs...**frequency** (describing incoherent relaxation) to negative at the **qubit** **oscillation** **frequency** (describing out of phase, coherent **oscillations**)...**frequency** and at **qubit** **oscillation** **frequency**. We take Γ = Γ x = Γ z = ...**qubit** oscillation frequency (describing out of phase, coherent oscillations...dot **qubit**. As the quantum measurement is taking place, the current outputs...**frequency**, while the second term has a peak at ω = Ω , with width 3 Γ ... We investigate the advantages of using two independent, linear detectors for continuous quantum measurement. For single-shot quantum measurement, the measurement is maximally efficient if the detectors are twins. For weak continuous measurement, cross-correlations allow a violation of the Korotkov-Averin bound for the detector's signal-to-noise ratio. A vanishing noise background provides a nontrivial test of ideal independent quantum detectors. We further investigate the correlations of non-commuting operators, and consider possible deviations from the independent detector model for mesoscopic conductors coupled by the screened Coulomb interaction.

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Contributors: Mao, Wenjin, Averin, Dmitri V., Plastina, Francesco, Fazio, Rosario

Date: 2004-08-21

**frequencies** $\Omega_1$ and $\Omega_2$, the mixing manifests itself as ...detector-**qubit** coupling are: δ 1 = 0.12 t 0 , δ 2 = 0.09 t 0 , λ = 0.08...**qubits**, the interaction splits coherent superposition of the single-**qubit**...non-interacting **qubits**. Small **qubit** bias ε 1 = ε 2 ≡ ε (solid line) creates...**frequencies**. Dashed line is the spectrum for interacting **qubits**. Interaction...**oscillations** of the two **qubits**. For non-interacting **qubits** **oscillating**...**frequencies**. **Qubit**-**qubit** interaction shifts all but the lower-frequency...**qubits**, the non-vanishing **qubit** bias just shifts the **frequency** position...th **qubit** ( j = 1 , 2 ), ν is the **qubit**-**qubit** interaction energy, and σ...**oscillations** in one **qubit** . Similarly to that case, the maximum of the...**frequencies** Δ 1 ± Δ 2 [see Eq. ( e20)]. Further increase of ε (dashed ...two-**qubit** Hamiltonian ( e2) are different, and show up as six finite-**frequency**...two-**qubit** system. The zero-**frequency** peak reflects dynamics of transitions...**frequencies** $\Omega_1\pm \Omega_2$. Additional nonlinearity introduced...**qubit**-**qubit** interaction shifts all the frequencies. In particular, for... of quadratic **qubit**-detector coupling is taken to be λ = 0.15 t 0 ....bias and the **qubit**-**qubit** interaction are finite, the bias splits each ...Fin...**qubit**-**qubit** interaction shifts all the **frequencies**. In particular, for.......**qubits** measured continuously by a mesoscopic detector with arbitrary non-linearity...**oscillations** in the individual **qubits**, while smaller peaks are non-linear...Stronger **qubit**-**qubit** interaction ν ≃ Δ ≫ κ shifts the ω ≃ 2 Ω peak to ...**frequencies**. **Qubit**-**qubit** interaction shifts all but the lower-**frequency**...lower-**frequency** liner peak down and all other peaks up in **frequency**. Parameters...**qubits**. Solid line is the spectrum in the case of non-interacting **qubits**...unbiased **qubits** with the strength ν of the **qubit**-**qubit** interaction. The...**qubit**-**qubit** interaction ν ≃ Δ ≫ κ shifts the ω ≃ 2 Ω peak to higher **frequencies**...**qubits**. For non-interacting **qubits** oscillating with frequencies $\Omega...**qubits**...**oscillations** in two, in general interacting, **qubits** measured continuously ... We develop a theory of coherent quantum **oscillations** in two, in general interacting, **qubits** measured continuously by a mesoscopic detector with arbitrary non-linearity and discuss an example of SQUID magnetometer that can operate as such a detector. Calculated spectra of the detector output show that the detector non-linearity should lead to mixing of the **oscillations** of the two **qubits**. For non-interacting **qubits** **oscillating** with **frequencies** $\Omega_1$ and $\Omega_2$, the mixing manifests itself as spectral peaks at the combination **frequencies** $\Omega_1\pm \Omega_2$. Additional nonlinearity introduced by the **qubit**-**qubit** interaction shifts all the **frequencies**. In particular, for identical **qubits**, the interaction splits coherent superposition of the single-**qubit** peaks at $\Omega_1=\Omega_2$. Quantum mechanics of the measurement imposes limitations on the height of the spectral peaks.

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Contributors: Ashhab, S.

Date: 2014-10-02

**oscillator** **frequency**: ℏ ω / Δ = 0.2 (top), 1 (middle) and 5 (bottom)....**up**-converted into excitations in the **qubit** as a result of the sweep through... the qubit’s minimum gap Δ . Middle: P as a function of k B T / Δ for ...** qubit’s** minimum gap Δ , the initial excitation of the low-

**frequency**

**oscillator**...

**oscillator**could represent an external mode that is strongly coupled to...

**qubit**) when this system interacts with one harmonic

**oscillator**mode that...

**qubit**and

**oscillator**

**frequencies**, their coupling strength and the temperature...

**oscillator**

**frequency**is ℏ ω / Δ = 0.2 . The sweep rate is chosen such ...

**Qubit’s**final excited state probability P obtained from the semiclassical...the

**qubit**to end

**up**in the excited state at the final time as a function...

**qubit**) when this system interacts with one harmonic oscillator mode that...

**qubit**, e.g. an ionic oscillation mode in a molecule, or it could represent...

**qubit**with the coupled

**qubit**-

**oscillator**system the single avoided crossing...minimum qubit gap Δ . The different panels correspond to different values...

**oscillator**. Here we take ℏ ω / Δ = 0.2 . The different lines correspond...

**frequency**of the harmonic

**oscillator**, â and â † are, respectively, the...

**qubit**-

**oscillator**system with the

**qubit**bias conditions varied according...

**qubit**'s occupation probabilities at the final time in a number of different...Top: Qubit’s final excited-state probability P as a function of temperature...coupled qubit-oscillator system with the qubit bias conditions varied ...low-

**frequency**

**oscillator**). In the semiclassical calculation, there is ...

**oscillator**

**frequency**continues up to ℏ ω / Δ = 20 ). This relation does...

**qubit**, e.g. an ionic

**oscillation**mode in a molecule, or it could represent...

**no**point in time where the

**qubit**and

**oscillator**are resonant with each...

**oscillator**can result in exciting the

**qubit**at the final time. The excitations...

**qubit**with the coupled

**qubit**-

**oscillator**system the single avoided crossing...online) Qubit’s final excited state probability P obtained from the semiclassical...

**qubit**and oscillator frequencies, their coupling strength and the temperature ... We analyze the dynamics and final populations in a Landau-Zener problem for a two level system (or

**qubit**) when this system interacts with one harmonic

**oscillator**mode that is initially set to a finite-temperature thermal equilibrium state. The harmonic

**oscillator**could represent an external mode that is strongly coupled to the

**qubit**, e.g. an ionic

**oscillation**mode in a molecule, or it could represent a prototypical uncontrolled environment. We analyze the

**qubit**'s occupation probabilities at the final time in a number of different regimes, varying the

**qubit**and

**oscillator**

**frequencies**, their coupling strength and the temperature. In particular we find some surprising non-monotonic dependence on the coupling strength and temperature.

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Contributors: Shen, Li-Tuo, Chen, Rong-Xin, Wu, Huai-Zhi, Yang, Zhen-Biao

Date: 2013-11-05

**qubits** and an oscillator within the ultrastrong coupling regime. We apply...**qubit**-**qubit** entanglement in the ground state vanishes if the **qubit**-oscillator...**qubits** coupled to a harmonic **oscillator**. The j th ( j = 1 , 2 , 3 ) **qubit**...**qubits**, ultrastrongly coupled, harmonic **oscillator**...three identical qubits coupled to a harmonic oscillator. The j th ( j = 1 ...The qubits’ entropy S versus λ / ℏ w 0 with: (a) ℏ w 0 / Δ = 0.1 , (b)...**qubits**, ultrastrongly coupled, harmonic oscillator...**qubit**'s **frequency** is far larger than the **oscillator**'s **frequency**, and analyze...**qubit**'s **frequency** is far larger than the **oscillator**'s **frequency**....high-frequency qubits (i.e., ℏ w 0 / Δ = 0.1 and ϵ = 0 ): (a,d) λ / ℏ ...**qubit**-oscillator coupling strength needed for generating the nonclassical...**oscillator**'s **frequency** is far larger than each **qubit**'s **frequency** and (...high-**frequency** **oscillator**: ℏ w 0 / E q = 10 . The rescaled energy E k ...**qubit**-**qubit** entanglement in the ground state vanishes if the **qubit**-**oscillator**... 0 , where the qubit-oscillator coupling strength is denoted by g or λ...**qubit**'s frequency and (ii) the **qubit**'s frequency is far larger than the...**qubits** ultrastrongly coupled to a harmonic oscillator...** oscillator’s** state with three high-

**frequency**

**qubits**(i.e., ℏ w 0 / Δ =... C for the qubits 2 and 3 versus λ / ℏ w 0 : (a) ℏ w 0 / Δ = 0.1 , (b)...

**qubits**and an

**oscillator**within the ultrastrong coupling regime. We apply...

**oscillator**with

**frequency**w 0 , where the

**qubit**-

**oscillator**coupling strength...high-

**frequency**

**qubits**: ℏ w 0 / E q = 0.01 . The rescaled energy E k / ...

**qubit**-oscillator system under the conditions of various system parameters ... We study the system involving mutual interaction between three

**qubits**and an

**oscillator**within the ultrastrong coupling regime. We apply adiabatic approximation approach to explore two extreme regimes: (i) the

**oscillator**'s

**frequency**is far larger than each

**qubit**'s

**frequency**and (ii) the

**qubit**'s

**frequency**is far larger than the

**oscillator**'s

**frequency**, and analyze the energy-level spectrum and the ground-state property of the

**qubit**-

**oscillator**system under the conditions of various system parameters. For the energy-level spectrum, we concentrate on studying the degeneracy in low energy levels. For the ground state, we focus on its nonclassical properties that are necessary for preparing the nonclassical states. We show that the minimum

**qubit**-

**oscillator**coupling strength needed for generating the nonclassical states of the Schr\"{o}dinger-cat type in the

**oscillator**is just one half of that in the Rabi model. We find that the

**qubit**-

**qubit**entanglement in the ground state vanishes if the

**qubit**-

**oscillator**coupling strength is strong enough, for which the entropy of three

**qubits**keeps larger than one. We also observe the phase-transition-like behavior in the regime where the

**qubit**'s

**frequency**is far larger than the

**oscillator**'s

**frequency**.

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Contributors: Hauss, Julian, Fedorov, Arkady, Hutter, Carsten, Shnirman, Alexander, Schön, Gerd

Date: 2007-01-02

**qubits** play the role of two-level atoms, while **oscillators** of various ...**flux** **qubit** is coupled inductively to an LC oscillator. b) In an equivalent...Single-**qubit** lasing and cooling at the Rabi frequency...**qubit** driven to perform Rabi oscillations and coupled to a slow electromagnetic... the qubit and the oscillator, the latter is metal coated and charged ...**frequency** and line-width of the resonator are ω T / 2 π = 6 MHz and κ ...**qubit**-**oscillator** coupling is quadratic and decoherence effects are minimized...than the qubit’s energy splitting (in the GHz range), ω T** . **Δ E** . **The qu...**oscillator**. For this previously unexplored regime of **frequencies** we study...**oscillations**, and the Rabi **frequency** Ω R is tuned close to resonance with...**qubit** driving **frequency** is blue detuned, δ ω = ω d - Δ E > 0 , we find...**qubit** and a bi-stability with lasing behavior of the oscillator; for red... qubit coupled to **an **LC oscillator, but our analysis applies equally to...**qubit** coupled to an LC-**oscillator** (Fig. fig:systema) with Hamiltonian...**qubit** cools the **oscillator**. This behavior persists at the symmetry point... qubit (see Fig. fig:systemb). In this case** . **z stands for the charge...of the qubit levels, which in resonance leads to one-qubit lasing. In ... **qubit** are Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , and Γ 0 / 2 π = 125 kHz, the...**qubit** is coupled inductively to an LC **oscillator**. b) In an equivalent ...**qubit** cools the oscillator. This behavior persists at the symmetry point...**qubit** is coupled to a mechanical resonator....**frequency** is tuned to resonance with the **oscillator** the latter can be ...**qubit** and a bi-stability with lasing behavior of the **oscillator**; for red...**qubit**-oscillator coupling is quadratic and decoherence effects are minimized...**charge** **qubit** is coupled to a mechanical resonator....**qubit** are Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , and Γ 0 / 2 π = 125 kHz, the...**qubit** driven to perform Rabi **oscillations** and coupled to a slow electromagnetic ... For a superconducting **qubit** driven to perform Rabi **oscillations** and coupled to a slow electromagnetic or nano-mechanical **oscillator** we describe previously unexplored quantum optics effects. When the Rabi **frequency** is tuned to resonance with the **oscillator** the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the **qubit** and a bi-stability with lasing behavior of the **oscillator**; for red detuning the **qubit** cools the **oscillator**. This behavior persists at the symmetry point where the **qubit**-**oscillator** coupling is quadratic and decoherence effects are minimized. There the system realizes a "single-atom-two-photon laser".

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Contributors: Griffith, E. J., Ralph, J. F., Greentree, Andrew D., Clark, T. D.

Date: 2005-10-04

**qubit** PSD. Likewise the **qubit** Rabi **frequency** is found to be stronger in...**oscillator** as a probe through the backreaction effect of the **qubit** on ...**oscillator** **frequency**....**qubit** (Cooper-pair box coupled) to an RLC oscillator model is performed...**qubit** on the oscillator circuit, we extract frequency splitting features...**oscillator** circuit, we extract **frequency** splitting features analogous ...**qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed... **qubit**. In addition, **qubit** is constantly driven by a microwave field at...The... / C q ....**qubit**, characterisation, **frequency** spectrum...weakly in the **qubit** PSD. Likewise the **qubit** Rabi frequency is found to...**frequency**. Therefore, it is possible to probe the **qubit** energy level structure...but only weakly in the qubit PSD. Likewise the qubit Rabi frequency is...**qubit** characterization and coupling schemes. In addition we find this ...**oscillator** **frequencies**, (1.36GHz and 3.06GHz)....**oscillator** energies. Firstly, the **oscillator** resonant **frequency** is set...**frequency** **oscillator** of 3.06GHz which can excite this **qubit**. In addition... **qubit** is coupled to a many level harmonic oscillator, investigated for... the qubit PSD. Likewise the qubit Rabi frequency is found to be stronger...**oscillator** cycles. Then the **oscillator** and **qubit** charge expectation values.......stronger in the **qubit** PSD. However it is important to note** that** the **qubit**...**frequency** (3.06GHz) bias **oscillator**. This higher **frequency** bias field ...**qubit**, characterisation, frequency spectrum...**frequency** bias **oscillator** of 3.06GHz.... the qubit PSD. However **it **is important to note that the qubit dynamics...**qubit** energy levels, with one and two photon transitions (3.49GHz and ...**oscillator** period, note that γ differs for the two bias **oscillator** **frequencies** ... A theoretical spectroscopic analysis of a microwave driven superconducting charge **qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed. By treating the **oscillator** as a probe through the backreaction effect of the **qubit** on the **oscillator** circuit, we extract **frequency** splitting features analogous to the Autler-Townes effect from quantum optics, thereby extending the analogies between superconducting and quantum optical phenomenology. These features are found in a **frequency** band that avoids the need for high **frequency** measurement systems and therefore may be of use in **qubit** characterization and coupling schemes. In addition we find this **frequency** band can be adjusted to suit an experimental **frequency** regime by changing the **oscillator** **frequency**.

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Contributors: Hauss, Julian, Fedorov, Arkady, André, Stephan, Brosco, Valentina, Hutter, Carsten, Kothari, Robin, Yeshwant, Sunil, Shnirman, Alexander, Schön, Gerd

Date: 2008-06-07

**qubit** coupled to a low-frequency tank circuit with particular emphasis...three-junction flux qubit is coupled inductively to an LC oscillator. ...the qubit: Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125 kHz, the resonator...**qubit** is coupled inductively to an LC **oscillator**. b) A charge **qubit** is...) A charge qubit is coupled to a mechanical resonator....**qubit** is close to the **oscillator**’s**frequency**. In contrast, in the present...rate of** the **qubit is close to** the **oscillator’s frequency. In contrast,...**qubit**-**oscillator** coupling is quadratic and decoherence effects are minimized...**qubit** is coupled to a slow LC **oscillator** with **frequency** ( ω T / 2 π ∼ ...**qubits** are much lower than the **oscillator**’s**frequency**....** qubit’s** relaxation rate, Γ 0 at the one-photon resonance, Ω R = ω T for...

**qubit**is driven to perform Rabi

**oscillations**, with Rabi

**frequency**in resonance...

**qubit**cools the

**oscillator**. This behavior persists at the symmetry point...charge qubit (see Fig. fig:systemb). In this case σ z stands for the ...

**oscillation**amplitude. When the

**qubit**driving

**frequency**is blue detuned...

**qubit**near resonance. Here m is the number of photons of the driving field...

**qubit**: Δ / 2 π = 1 GHz, ϵ = 0.01 Δ , Γ 0 / 2 π = 125 kHz, the resonator...

**qubit**and lasing behavior of the oscillator ("single-atom laser"). For...

**qubit**is driven to perform Rabi oscillations, with Rabi frequency in resonance...situations where

**the**qubit, e.g., a Josephson charge qubit, is coupled...

**qubit**coupled to a low-

**frequency**tank circuit with particular emphasis... decoherence effects

**in**the qubit.... driven qubit near resonance. Here m is the number of photons of the driving...

**qubit**cools the oscillator. This behavior persists at the symmetry point...

**qubit**and lasing behavior of the

**oscillator**("single-atom laser"). For...

**qubit**-oscillator coupling is quadratic and decoherence effects are minimized...

**qubit**to perform Rabi

**oscillations**with Rabi

**frequency**in resonance with... inversion of

**the**qubit levels, which

**in**resonance leads to one-qubit ... Superconducting qubits coupled to electric or nanomechanical resonators display effects previously studied in quantum electrodynamics (QED) and extensions thereof. Here we study a driven

**qubit**coupled to a low-

**frequency**tank circuit with particular emphasis on the role of dissipation. When the

**qubit**is driven to perform Rabi

**oscillations**, with Rabi

**frequency**in resonance with the

**oscillator**, the latter can be driven far from equilibrium. Blue detuned driving leads to a population inversion in the

**qubit**and lasing behavior of the

**oscillator**("single-atom laser"). For red detuning the

**qubit**cools the

**oscillator**. This behavior persists at the symmetry point where the

**qubit**-

**oscillator**coupling is quadratic and decoherence effects are minimized. Here the system realizes a "single-atom-two-photon laser".

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Contributors: Mitra, Kaushik, Lobb, C. J., de Melo, C. A. R. Sá

Date: 2007-12-06

reaching the qubit junction; b) it is used as a measurement tool....**qubit** caused by an isolation circuit with a resonant frequency. The coupling...**qubit** as a function of time ρ 11 t , with ρ 11 t = 0 = 1 for R = 50 ohms...**Qubits**...of **qubit** frequency ω 01 . The solid (red) curves describes an RLC isolation...**oscillations** in phase **qubits** is shown in Fig. fig:one, which correponds...**qubit** caused by an isolation circuit with a resonant **frequency**. The coupling...**frequency** of the **qubit** is at least two times larger than the resonance...**frequency** of the **qubit** is much smaller than the resonance **frequency** of... plotted versus qubit frequency ω 01 for spectral densities describing...and **qubit** parameters C 0 = 4.44 pF, R 0 = 5000 ohms and L 0 = 0 . The...**qubit** **frequency** ω 01 for spectral densities describing an RLC (Eq. eqn...**qubit** parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves...**qubit** is two orders of magnitude larger than the typical ohmic regime,...**qubit** with an RLC isolation circuit....possible and increasing the qubit frequency ω 01 from 0.1 Ω to 2 Ω at ...**qubit** **frequency** ω 01 from 0.1 Ω to 2 Ω at fixed low temperature can produce...phase **qubit** with an RLC isolation circuit....**qubit** is at least two times larger than the resonance frequency of the...**qubit** frequency is on resonance with the isolation circuit, an oscillatory... of the qubit as shown in Fig. fig:one....**qubit** and its environment is modeled via the Caldeira-Leggett formulation...**qubit** **frequency** is on resonance with the isolation circuit, an oscillatory...forth to the qubit and induces Rabi-oscillations with an effective time...**qubit** and induces Rabi-**oscillations** with an effective time dependent decay...**frequency** of the Rabi **oscillations** Ω R a = π κ Ω 3 / 2 Γ is independent...**qubit** and its environment self-generate Rabi **oscillations** of characteristic...and **qubit** parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves...**qubit** **frequency** ω 01 . The solid (red) curves describes an RLC isolation ... We study decoherence effects in a dc SQUID phase **qubit** caused by an isolation circuit with a resonant **frequency**. The coupling between the SQUID phase **qubit** and its environment is modeled via the Caldeira-Leggett formulation of quantum dissipation/coherence, where the spectral density of the environment is related to the admittance of the isolation circuit. When the **frequency** of the **qubit** is at least two times larger than the resonance **frequency** of the isolation circuit, we find that the decoherence time of the **qubit** is two orders of magnitude larger than the typical ohmic regime, where the **frequency** of the **qubit** is much smaller than the resonance **frequency** of the isolation circuit. Lastly, we show that when the **qubit** **frequency** is on resonance with the isolation circuit, an oscillatory non-Markovian decay emerges, as the dc SQUID phase **qubit** and its environment self-generate Rabi **oscillations** of characteristic time scales shorter than the decoherence time.

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Contributors: Ashhab, S.

Date: 2013-12-25

**qubit** and a single oscillator...multi-**qubit** case to that in the single-**qubit** case approaches N for all...**qubit**-oscillator correlation, we gain insight into the nature of the transition...multi-qubit case to that in the single-qubit case approaches N for all...**qubit**-**oscillator** correlations (which are finite only above the critical...single-qubit case is simple in principle. In the limit ℏ ω 0 / Δ → 0 ,...**qubit**'s frequency; away from this limit one obtains a finite-width transition...single-qubit case, whereas the other lines correspond to the multi-qubit...to each qubit state. Each one of these sets has a structure that is similar...the qubit frequency Δ . One can see clearly that moving in the vertical...with ρ q being the qubit’s reduced density matrix) and the correlation...**qubit**-**oscillator** correlations change more slowly when the coupling strength...**qubit**, and a single harmonic **oscillator**. The system experiences a sudden...**qubit**-**oscillator** coupling strength is varied and increased past a critical...single-**qubit** case, whereas the other lines correspond to the multi-**qubit**...**oscillator** **frequency** ℏ ω 0 and the coupling strength λ , both measured...**oscillator**. For consistency with Ref. , we define it as...**qubits** now have a larger total spin (when compared to the single-**qubit**...**qubit** **frequency** Δ . One can see clearly that moving in the vertical direction...**oscillator** field and its squeezing and the **qubit**-**oscillator** correlation...**oscillator**'s **frequency** is much lower than the **qubit**'s **frequency**; away ...**qubit**-**oscillator** entanglement on the coupling strength just above the ...**qubit**, and a single harmonic oscillator. The system experiences a sudden...**qubit**-oscillator coupling strength is varied and increased past a critical ... We consider the phase-transition-like behaviour in the Rabi model containing a single two-level system, or **qubit**, and a single harmonic **oscillator**. The system experiences a sudden transition from an uncorrelated state to an increasingly correlated one as the **qubit**-**oscillator** coupling strength is varied and increased past a critical point. This singular behaviour occurs in the limit where the **oscillator**'s **frequency** is much lower than the **qubit**'s **frequency**; away from this limit one obtains a finite-width transition region. By analyzing the energy-level structure, the value of the **oscillator** field and its squeezing and the **qubit**-**oscillator** correlation, we gain insight into the nature of the transition and the associated critical behaviour.

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Contributors: Saito, S., Meno, T., Ueda, M., Tanaka, H., Semba, K., Takayanagi, H.

Date: 2005-08-19

its transmission** to** the **qubit** and the efficiency of the coupling between...**frequency** f M W 1 of 10.25 GHz. (b) One-photon Rabi **oscillations** of P ...single-**frequency** microwave pulses. (a) Spectroscopic data of the **qubit**...**oscillations** by using single-**frequency** microwave pulses with each **frequency**...**oscillations** ∝ exp - t p / T d cos Ω R a b i t p . The Rabi **frequencies**...**qubit** at a distance of 20 μ m so that the **qubit** could be strongly driven...**qubit** states when the sum of the two microwave frequencies or the difference...**qubit**...**frequencies** or the difference between them matches the **qubit** Larmor **frequency**...**qubit** measurement system. On-chip components are shown in the dashed box...**qubit**. Each set of the dots represents the resonant frequencies f r e ...Then a **qubit** operation is performed by applying a microwave pulse** to** **the q**...**oscillations**. (c) [(d)] Rabi **frequencies** as a function of V M W 1 , which...**oscillation** fits. Both the **qubit** Larmor **frequency** f q b and the microwave...**qubit** has been achieved by using two-frequency microwave pulses. We have...**qubit** Larmor frequency. We have also observed multi-photon Rabi oscillations...**frequency** range of microwaves for controlling the **qubit** and offers a high...**qubit** and offers a high quality testing ground for exploring nonlinear...**qubit** has been achieved by using two-**frequency** microwave pulses. We have...**oscillations** stemming from parametric transitions between the **qubit** states...fixes the **qubit** Larmor frequency f q b . The **qubit** is thermally initialized... is larger than the **qubit** transition energy ℏ ω q b ....**qubit** with two-**frequency** microwave pulses....**qubit** Larmor frequency f q b and the microwave frequency f M W 1 are 10.25...**qubit** (inner loop) and a dc-SQUID (outer loop). The loop sizes of the **qubit** and SQUID are 10.2 × 10.4 μ m 2 and 12.6 × 13.5 μ m 2 , respectively...**oscillations** corresponding to one- to four-photon resonances by applying...two-**frequency** microwave pulses. (a) [(b)] Two-photon Rabi **oscillations** ... Parametric control of a superconducting flux **qubit** has been achieved by using two-**frequency** microwave pulses. We have observed Rabi **oscillations** stemming from parametric transitions between the **qubit** states when the sum of the two microwave **frequencies** or the difference between them matches the **qubit** Larmor **frequency**. We have also observed multi-photon Rabi **oscillations** corresponding to one- to four-photon resonances by applying single-**frequency** microwave pulses. The parametric control demonstrated in this work widens the **frequency** range of microwaves for controlling the **qubit** and offers a high quality testing ground for exploring nonlinear quantum phenomena.

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