### 57595 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...**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

**qubits**, the interaction splits coherent superposition of the single-**qubit**...**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...**frequencies**. Dashed line is the spectrum for interacting **qubits**. Interaction...**oscillations** of the two **qubits**. For non-interacting **qubits** **oscillating**...**qubits**, the non-vanishing **qubit** bias just shifts the **frequency** position...**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**...**qubit**-**qubit** interaction ν ≃ κ ≪ Δ suppresses and subsequently splits the...two-**qubit** system. The zero-**frequency** peak reflects dynamics of transitions...**qubit**-**qubit** interaction shifts all the frequencies. In particular, for...**frequencies** $\Omega_1\pm \Omega_2$. Additional nonlinearity introduced...**qubit** ( j = 1 , 2 ), ν is the **qubit**-**qubit** interaction energy, and σ ’s...**qubit**-**qubit** interaction shifts all the **frequencies**. In particular, for...**qubit**-**qubit** interaction ν ≃ Δ ≫ κ shifts the ω ≃ 2 Ω peak to higher frequencies...**qubit**-**qubit** interaction are finite, the bias splits each of the linear...**qubits** measured continuously by a mesoscopic detector with arbitrary non-linearity...**Qubit**-**qubit** interaction shifts all but the lower-frequency linear peak...**oscillations** in the individual **qubits**, while smaller peaks are non-linear...**qubit**-detector coupling is taken to be λ = 0.15 t 0 ....**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**...**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...**qubits**. Small **qubit** bias ε 1 = ε 2 ≡ ε (solid line) creates transitions...**qubits** with the strength ν of the **qubit**-**qubit** interaction. The **qubit**-detector ... 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)....**qubit** and resonator with the exception of some state-dependent mean-field...**qubit** to end up in the excited state at the final time as a function of...**qubit** as a result of the sweep through the avoided crossing....**qubit**’s minimum gap Δ . Middle: P as a function of k B T / Δ for four ...**qubit** gap Δ . The different panels correspond to different values of the...** 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**-oscillator system with the

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

**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...

**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...

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

**qubit**-

**oscillator**system with the

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

**qubit**'s occupation probabilities at the final time in a number of different...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...

**qubit**as a result of the swee...

**qubit**at the final time. The excitations in the oscillator are in some...

**qubit**with the coupled

**qubit**-

**oscillator**system the single avoided crossing...

**Qubit**’s final excited-state probability P as a function of temperature...

**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

**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**...**qubits** coupled to a harmonic oscillator. The j th ( j = 1 , 2 , 3 ) **qubit**...**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....**qubit**-oscillator coupling strength is denoted by g or λ ....**qubit**-oscillator coupling strength needed for generating the nonclassical...**qubit**'s **frequency** is far larger than the **oscillator**'s **frequency**....**qubits**. The eigenstates (plotted with n no more than 2 ) that have the...**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**...**qubits** (i.e., ℏ w 0 / Δ = 0.1 and ϵ = 0 ): (a,d) λ / ℏ w 0 = 0.5 , (b,...**qubit**'s frequency and (ii) the **qubit**'s frequency is far larger than the...**qubits**: ℏ w 0 / E q = 0.01 . The rescaled energy E k / ℏ w 0 with k = ...**qubits** ultrastrongly coupled to a harmonic oscillator...** oscillator’s** state with three high-

**frequency**

**qubits**(i.e., ℏ w 0 / Δ =...

**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 ...Single-**qubit** lasing and cooling at the Rabi frequency...**qubit** driven to perform Rabi oscillations and coupled to a slow electromagnetic...**qubit**, where due to symmetry the linear coupling to the noise sources ...**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...**qubit** coupled to an LC-oscillator (Fig. fig:systema) with Hamiltonian...**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** (Fig. fig:systema) with Hamiltonian...**qubit** cools the **oscillator**. This behavior persists at the symmetry point...**qubit** driving frequency is blue detuned, δ ω = ω d - Δ E > 0 , we find...**qubit** levels, which in resonance leads to one-**qubit** lasing. In experiments...**qubit**’s energy splitting (in the GHz range), ω T ≪ Δ E . The **qubit** is ...**qubit** cools the oscillator. This behavior persists at the symmetry point...**qubit** is coupled inductively to an LC **oscillator**. b) In an equivalent ...**qubit** is coupled to a mechanical resonator....**frequency** is tuned to resonance with the **oscillator** the latter can be ...**qubit**-oscillator coupling is quadratic and decoherence effects are minimized...**qubit** and a bi-stability with lasing behavior of the **oscillator**; for red...**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** 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...**frequency** **oscillator** ( f o s c = 3.06 GHz) near the **qubit** and microwave...**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** cycles. Then the **oscillator** and **qubit** charge expec...**oscillator** circuit, we extract **frequency** splitting features analogous ...**qubit**. In addition, **qubit** is constantly driven by a microwave field at...**qubit** PSD. However it is important to note that the **qubit** dynamics such...**qubit** (Cooper-pair box coupled) to an RLC **oscillator** model is performed...**qubit** transition frequency ( f q u b i t ≈ 3.49GHz) and the diagonally...**qubit**, characterisation, **frequency** spectrum...**qubit** geometry is transformed to a classical electrical circuit model ...**frequency**. Therefore, it is possible to probe the **qubit** energy level structure...**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 driven at f m w = 5.00 GHz. An increase in bias noise power (...**qubit** energy level structure by using the power increase in the oscillator...**oscillator** spectrum. We also observe a **frequency** splitting when the **qubit**...**qubit**, characterisation, frequency spectrum...**qubit** capacitances C J and C g , (the Josephson junction capacitance and ... 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...**qubit** is coupled inductively to an LC **oscillator**. b) A charge **qubit** is...**qubit** is close to the **oscillator**’s**frequency**. In contrast, in the present...**qubit**-**oscillator** coupling is quadratic and decoherence effects are minimized...**qubit**, e.g., a Josephson charge **qubit**, is coupled to a nano-mechanical...**qubit** (see Fig. fig:systemb). In this case σ z stands for the charge ...**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...

**qubit**’s relaxation rate, Γ 0 at the one-photon resonance, Ω R = ω T for...

**oscillation**amplitude. When the

**qubit**driving

**frequency**is blue detuned...

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

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

**qubit**levels, which in resonance leads to one-

**qubit**lasing. In experiments...

**qubit**coupled to a low-

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

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

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

**qubit**and lasing behavior of the

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

**qubit**’s relaxation rate. Above the saturation threshold for n ̄ > n 0 ...

**qubit**to perform Rabi

**oscillations**with Rabi

**frequency**in resonance with...

**qubit**is coupled inductively to an LC oscillator. b) A charge

**qubit**is...

**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

**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**...**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...**qubit** frequency ω 01 for spectral densities describing an RLC (Eq. eqn...**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** junction, such as defects in the oxide barrier, quasiparticle tunneling...**qubit** parameters C 0 = 4.44 pF, R 0 = 5000 ohms and L 0 = 0 . The dashed...**qubit** with an RLC isolation circuit....**qubit** frequency ω 01 . The solid (red) curves describes an RLC isolation...**qubit** **frequency** ω 01 from 0.1 Ω to 2 Ω at fixed low temperature can produce...**qubit** is at least two times larger than the resonance frequency of the...**qubit** frequency is on resonance with the isolation circuit, an oscillatory...**qubit** and its environment is modeled via the Caldeira-Leggett formulation...**qubit** **frequency** is on resonance with the isolation circuit, an oscillatory...**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** junction; b) it is used as a measurement tool....**qubit** as shown in Fig. fig:one....**qubit** and its environment self-generate Rabi **oscillations** of characteristic...**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: Korotkov, Alexander N.

Date: 2010-04-01

**oscillations** via low-**frequency** noise correlation. The idea is to measure...**oscillations**....**qubit** Rabi oscillations are known to be non-decaying (though with a fluctuating...**frequency** Ω coinciding with the Rabi **frequency** Ω R ....**qubits**....zero-**frequency** detector noise S a a 0 and cross-noise S a b 0 on the phase...**qubit** Rabi **oscillations** are known to be non-decaying (though with a fluctuating...**qubit** by two detectors, biased stroboscopically at the Rabi frequency....**qubit** is continuously monitored in the weak-coupling regime. In this paper...low-**frequency** noise depends on the relative phase between the two combs...**qubit** measured by two QPC detectors, which are biased by combs of short...**qubit** by two detectors, biased stroboscopically at the Rabi **frequency**. ... The **qubit** Rabi **oscillations** are known to be non-decaying (though with a fluctuating phase) if the **qubit** is continuously monitored in the weak-coupling regime. In this paper we propose an experiment to demonstrate these persistent Rabi **oscillations** via low-**frequency** noise correlation. The idea is to measure a **qubit** by two detectors, biased stroboscopically at the Rabi **frequency**. The low-**frequency** noise depends on the relative phase between the two combs of biasing pulses, with a strong increase of telegraph noise in both detectors for the in-phase or anti-phase combs. This happens because of self-synchronization between the persistent Rabi **oscillations** and measurement pulses. Almost perfect correlation of the noise in the two detectors for the in-phase regime and almost perfect anticorrelation for the anti-phase regime indicates a presence of synchronized persistent Rabi **oscillations**. The experiment can be realized with semiconductor or superconductor **qubits**.

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Contributors: Ashhab, S.

Date: 2013-12-25

**qubit** and a single oscillator...**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...**qubit**'s frequency; away from this limit one obtains a finite-width transition...**qubit** frequency Δ . One can see clearly that moving in the vertical direction...**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**’s reduced density matrix) and the correlation function C = σ z s...**qubit**-**oscillator** entanglement on the coupling strength just above the ...**qubit** state. Each one of these sets has a structure that is similar to...**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 approaches N for all the lines as we approach the critical...single-**qubit** case is simple in principle. In the limit ℏ ω 0 / Δ → 0 , ... 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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