### 54087 results for qubit oscillator frequency

Contributors: Korotkov, Alexander N.

Date: 2000-03-13

Schematic of a **qubit** continuously measured by a detector with output signal I t ....A particular realization of the evoltion of ρ 11 t due to continuous measurement for ε / H = 1 , α = 0.1 and η = 1 . Notice the fluctuation of both the phase and the asymmetry of **oscillations**....The situation changes as the coupling between the detector and **qubit** increases, α 1 . The strong influence of measurement destroys quantum **oscillations**, and the Quantum Zeno effect develops, so that for α ≫ 1 the **qubit** performs random jumps between two localized states (see Fig. I(t)b). In this case the properly averaged detector current follows pretty well the evolution of the **qubit** (however, the unsuccessful tunneling “attempts” still cannot be directly resolved), and the spectral density of I t can be calculated using the classical theory of telegraph noise leading to the Lorentzian shape of S I ω . Figure transitiona shows the gradual transformation of the spectral density with the increase of the coupling α for a symmetric **qubit**, ε = 0 , and an ideal detector, η = 1 . The results for an asymmetric **qubit**, ε / H = 1 , are shown in Fig. transitionb....has an obvious relation to the average square of the detector current variation due to **oscillations** in the measured system. Notice, however, that this integral is twice as large as one would expect from the classical harmonic signal, since one half of the peak height comes from nonclassical correlation between the **qubit** evolution and the detector noise. Classically, Eq. ( integral) would be easily understood if the signal was not harmonic but rectangular-like, which is obviously not the case. Actually, the detector current shows neither clear harmonic nor rectangular signal distinguishable from the intrinsic noise contribution. Figure I(t)a shows the simulation of ρ 11 evolution (thick line) together with the detector current I t . Since I t contains white noise, it necessarily requires some averaging. Thin solid, dotted, and dashed lines show the detector current averaged with different time constants τ a : τ a Ω / 2 π = 0.3 , 1, and 3, respectively. For weak averaging the signal is too noisy, while for strong averaging individual **oscillation** periods cannot be resolved either, so quantum **oscillations** can never be observed directly by a continuous measurement (although they can be calculated using Eqs. ( Bayes1)–( Bayes2)). This unobservability is revealed in the relatively low peak height of the spectral density of the detector current....The curves in Fig. transition as well as the dashed curves in Fig. M-C are calculated using the conventional master equation approach which gives the same results for the detector spectral density as the Bayesian formalism (we will prove this later). In the conventional approach we should assume no correlation between the detector noise and the **qubit** evolution (the last term in Eq. ( 3contrib) is absent) while the correlation function K z ̂ τ should be calculated considering z t not as an ordinary function but as an operator. Then the calculation of z ̂ t z ̂ t + τ can be essentially interpreted as follows. The first (in time) operator z ̂ t collapses the **qubit** into one of two eigenstates which correspond to localized states, then during time τ the **qubit** performs the evolution described by conventional Eqs. ( conv1)–( conv2), and finally the second operator z ̂ t + τ gives the probability for the **qubit** to be measured in one of two localized states. (Of course, this procedure can be done purely formally, without any interpretation.) Notice that there is complete symmetry between states “1” and “2” even for ε ≠ 0 (in particular, in the stationary state ρ 11 = ρ 22 = 1 / 2 ), so the evolution after the first collapse can be started from any localized state leading to the same contribution to the correlation function. In this way we obviously get K z ̂ τ = ρ 11 τ - ρ 22 τ where ρ i i is the solution of Eqs. ( conv1)–( conv2) with the initial conditions ρ 11 0 = 1 and ρ 12 0 = 0 ....The detector current spectral density S I ω for η = 1 and different coupling α with (a) symmetric ( ε = 0 ) and (b) asymmetric ( ε / H = 1 ) **qubit**....Figure envir shows the numerically calculated spectral density S I ω of the detector current for a nonideal detector, η = 0.5 (dashed lines) and for an ideal detector but extra coupling of the **qubit** to the environment at temperature T = H (solid lines). The rates γ 1 and γ 2 are chosen according to Eqs. ( enveqv1) and ( enveqv2). For the symmetric **qubit**, ε = 0 , the results of two models practically coincide. In contrast, the solid and dashed lines for ε = 2 H significantly differ from each other at low **frequencies**, while the spectral peak at ω ∼ Ω is fitted quite well....We consider a two-level quantum system (**qubit**) which is continuously measured by a detector and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the **frequency** of quantum **oscillations** and a Lorentzian-shape increase of the detector noise at low **frequency**. With increasing coupling the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed....Using Eqs. ( Bayes1)–( Bayes3) and the Monte-Carlo method (similar to Ref. ) we can calculate in a straightforward way the spectral density S I ω of the detector current I t . Solid lines in Fig. M-C show the results of such calculations for the ideal detector, η = 1 , and weak coupling between the **qubit** and the detector, α = 0.1 , where α ≡ ℏ Δ I 2 / 8 S 0 H ( α is 8 times less than the parameter C introduced in Ref. ). One can see that in the symmetric case, ε = 0 , the peak at the **frequency** of quantum **oscillations** is 4 times higher than the noise pedestal, S I Ω = 5 S 0 while the peak width is determined by the coupling strength α (see Fig. transition below). In the asymmetric case, ε ≠ 0 , the peak height decreases (Fig. M-C), while the additional Lorentzian-shape increase of S I ω appears at low **frequencies**. The origin of this low-**frequency** feature is the slow fluctuation of the asymmetry of ρ 11 **oscillations** (Fig. asym). In case ε = 0 the amplitude of ρ 11 **oscillations** is maximal (see thick line in Fig. I(t)a), hence there is no such asymmetry and the low-**frequency** feature is absent, while the spectral peak at the **frequency** of quantum **oscillation** is maximally high. ... We consider a two-level quantum system (**qubit**) which is continuously measured by a detector and calculate the spectral density of the detector output. In the weakly coupled case the spectrum exhibits a moderate peak at the **frequency** of quantum **oscillations** and a Lorentzian-shape increase of the detector noise at low **frequency**. With increasing coupling the spectrum transforms into a single Lorentzian corresponding to random jumps between two states. We prove that the Bayesian formalism for the selective evolution of the density matrix gives the same spectrum as the conventional master equation approach, despite the significant difference in interpretation. The effects of the detector nonideality and the finite-temperature environment are also discussed.

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Contributors: Rao, D. D. Bhaktavatsala

Date: 2007-06-19

(Color online) The transition probability as a function of r.f **frequency** ω . Initial bath states with different polarizations are considered. The resonant **frequency** varies with the bath polarization as ω = ω 0 + g N P B / 2 . In plotting the above we have taken N = 20 , ω 0 = 100 , ω 1 = 10 and g / ℏ = 1 . The **frequencies** are in the units of MHz....(Color online) Concurrence with time. The time-dependent concurrence, for the Bell-states | B 1 = 1 2 | ↑ ↓ + | ↓ ↑ | B 2 = 1 2 | ↑ ↑ + | ↓ ↓ , | B 3 = 1 2 | ↑ ↑ - | ↓ ↓ , is plotted for two values of Δ ω = ω - ω 0 . The loss of entanglement is slowest at the resonant **frequency** δ ω = 0 , for all the states, and it increases as the r.f **frequency** shifts away from ω 0 . The bath is unpolarized with uniform coupling strengths between the **qubits** and bath-spins. In plotting the above we have taken N = 20 , ω = 100 , ω 1 = 10 and g / ℏ = 1 . The **frequencies** are in the units of MHz....(Color online) Concurrence with time. The time-dependent concurrence for an initial singlet shared between two non-interacting **qubits** is plotted for three different cases, (i) when the r.f **frequencies** are tuned to their resonance values i.e., δ ω 1 2 = 0 , (ii) the r.f **frequencies** are slightly away from resonance and (iii) much away from their resonant **frequencies**. In the above δ ω 1 2 = ω 0 1 2 - ω 1 2 , where ω 0 1 2 , are the fields at the local sites of the **qubits**. The initial states of the baths are unpolarized each consiting of N = 20 spins and the local fields experienced by the **qubits** are ω 0 1 = 100 , ω 0 1 = 110 respectively....(Color online) Probability distribution of the resonant **frequencies** given in Eq. resf. The bath consists of N = 20 spins. The distributions for the **qubit**-bath couplings are normalized such that ∑ k g k = g , where g / ℏ = 20 MHz. In plotting the above we have set ω = ω 0 ....(Color online) The z -component of **qubit** polarization with time. The plot shows the decay of Rabi **oscillations** with time. The bath is unpolarized consisting of N = 20 spins. Two different distributions for the **qubit**-bath couplings are considered, where ∑ k g k = g for both the distributions. In plotting the above we have taken ω 0 = 10 3 , ω 1 = 10 and g / ℏ = 40 , which are in the units of MHz...An exactly solvable model for the decoherence of one and two-**qubit** states interacting with a spin-bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\hat{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi **oscillations** between the spin-states of a single **qubit** is shown to depend on the size of bath, the distribution of **qubit**-bath couplings and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the **qubit** shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-**qubit** states, can be controlled by tuning the **frequency** of the rotating field. The decay rates of entanglement and purity for all the Bell-states are same when the **qubits** are non-interacting, and different when they are interacting. ... An exactly solvable model for the decoherence of one and two-**qubit** states interacting with a spin-bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\hat{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi **oscillations** between the spin-states of a single **qubit** is shown to depend on the size of bath, the distribution of **qubit**-bath couplings and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the **qubit** shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-**qubit** states, can be controlled by tuning the **frequency** of the rotating field. The decay rates of entanglement and purity for all the Bell-states are same when the **qubits** are non-interacting, and different when they are interacting.

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Contributors: Greenberg, Ya. S., Il'ichev, E., Izmalkov, A.

Date: 2005-07-21

Fast Fourier transform of at different amplitudes G / h of low-**frequency** field....As an example we show below the time evolution of the quantity σ Z t = Z t , obtained from the numerical solution of the equations ( sigmaZ), ( sigmaY), and ( sigmaX) where we take a low **frequency** excitations as G t = G c o s ω L t . The calculations have been performed with initial conditions σ Z 0 = 1 , σ X 0 = σ Y 0 = 0 for the following set of the parameters: F / h = 36 MHz, Δ / h = 1 GHz, Γ / 2 π = 4 MHz, Γ z / 2 π = 1 MHz, ϵ / Δ = 1 , Z 0 = - 1 , δ / 2 π = 6.366 MHz, ω L / Ω R = 1 . As is seen from Fig. fig1 in the absence of low **frequency** signal ( G = 0 ) the **oscillations** are damped out, while if G ≠ 0 the **oscillations** persist....The Fourier spectra of these signals are shown on Fig. fig2 for different amplitudes of low **frequency** excitation. For G = 0 the Rabi **frequency** is positioned at approximately 26.2 MHz, which is close to Ω R = 26.24 MHz. With the increase of G the peak becomes higher. It is worth noting the appearance of the peak at the second harmonic of Rabi **frequency**. This peak is due to the contribution of the terms on the order of G 2 which we omitted in our theoretical analysis....Time evolution of . (thick) G=0, (thin) G / h = 1 MHz. The insert shows the undamped **oscillations** of at G / h = 1 MHz....We have analyzed the interaction of a dissipative two level quantum system with high and low **frequency** excitation. The system is continuously and simultaneously irradiated by these two waves. If the **frequency** of the first signal is close to the level separation the response of the system exhibits undamped low **frequency** **oscillations** whose amplitude has a clear resonance at the Rabi **frequency** with the width being dependent on the damping rates of the system. The method can be useful for low **frequency** Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux and charge **qubits**, etc. As the examples, the application of the method to a nuclear spin and to the readout of a flux **qubit** are briefly discussed....The comparison of analytical and numerical resonance curves calculated for low **frequency** amplitude, G / h = 1 MHz and different dephasing rates, Γ are shown on Fig. fig3. The curves at the figure are the peak-to-peak amplitudes of **oscillations** of Z t calculated from Eq. ( ZOmega) with g ˜ ω = g δ ω + ω L + δ ω - ω L / 2 , where δ ω is Dirac delta function. The point symbols are found from numerical solution of Eqs. ( sigmaZ),( sigmaY),( sigmaX). The widths of the curves depend on Γ (see the insert) and the positions of the resonances coincide with the Rabi **frequency**. A good agreement between numerics and Eq. ZOmega, as shown at Fig. fig3, is observed only for relative small low **frequency** amplitude G / h , for which our linear response theory is valid. ... We have analyzed the interaction of a dissipative two level quantum system with high and low **frequency** excitation. The system is continuously and simultaneously irradiated by these two waves. If the **frequency** of the first signal is close to the level separation the response of the system exhibits undamped low **frequency** **oscillations** whose amplitude has a clear resonance at the Rabi **frequency** with the width being dependent on the damping rates of the system. The method can be useful for low **frequency** Rabi spectroscopy in various physical systems which are described by a two level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux and charge **qubits**, etc. As the examples, the application of the method to a nuclear spin and to the readout of a flux **qubit** are briefly discussed.

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Contributors: Osborn, K. D., Strong, J. A., Sirois, A. J., Simmonds, R. W.

Date: 2007-03-04

Power out of the resonator as a function of **frequency** for different input powers at zero flux bias....V in, rms / V out, rms versus V out, rms 2 at Φ a = 0 and the low power resonance **frequency** for zero flux bias (f=8.2423Ghz). The circles show the experimental data, and the line shows the expected theoretical result for parameters determined by fits to figure 3 and figure 4....Measured resonance **frequency** as a function of flux bias....We have fabricated and measured a high-Q Josephson junction resonator with a tunable resonance **frequency**. A dc magnetic flux allows the resonance **frequency** to be changed by over 10 %. Weak coupling to the environment allows a quality factor of $\thicksim$7000 when on average less than one photon is stored in the resonator. At large photon numbers, the nonlinearity of the Josephson junction creates two stable **oscillation** states. This resonator can be used as a tool for investigating the quality of Josephson junctions in **qubits** below the single photon limit, and can be used as a microwave **qubit** readout at high photon numbers. ... We have fabricated and measured a high-Q Josephson junction resonator with a tunable resonance **frequency**. A dc magnetic flux allows the resonance **frequency** to be changed by over 10 %. Weak coupling to the environment allows a quality factor of $\thicksim$7000 when on average less than one photon is stored in the resonator. At large photon numbers, the nonlinearity of the Josephson junction creates two stable **oscillation** states. This resonator can be used as a tool for investigating the quality of Josephson junctions in **qubits** below the single photon limit, and can be used as a microwave **qubit** readout at high photon numbers.

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Contributors: Kleff, S., Kehrein, S., von Delft, J.

Date: 2003-04-08

Structured bath/weak coupling: We now turn to the structured spectral density given by Eq.( eq:density). The main features of the corresponding system [Eq.( eq:system)] can already be understood by analyzing only the coupled two-level-harmonic **oscillator** system (without damping, i.e. Γ = 0 ). For ε = 0 this system exhibits two characteristic **frequencies**, close to Ω and Δ , associated with the transitions 1 and 2 in Fig. fig:correlation_weak(c). These should also show up in the correlation function C ω ; and indeed Fig. fig:correlation_weak(a) displays a double-peak structure with the peak separation somewhat larger than Δ - Ω , due to level repulsion. The coupling to the bath will in general lead to a broadening of the resonances and an enhancement of the repulsion of the two energies. Due to the very small coupling ( α = 0.0006 ) peak positions of C ω in Fig. fig:correlation_weak can with very good accuracy be derived from a second order perturbation calculation for the coupled two-level-harmonic **oscillator** system, yielding the following transition **frequencies** [depicted in inset (c) of Fig. fig:correlation_weak]: ω 1 , + - ω 0 , + = Ω - g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 0.987 Ω and ω 0 , - - ω 0 , + = Δ 0 + g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 1.346 Ω . With the two peaks we associate two different dephasing times, τ Ω and τ Δ , as shown in inset (a) and (b) of Fig. fig:correlation_weak....We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic **oscillator**. This system is realized in measurements on solid-state Josephson **qubits**. It can be mapped to a spin-boson model with a spectral function with an approximately Lorentzian resonance. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), and calculate correlation functions, dephasing rates, and **qubit** quality factors. We find that these depend strongly on the environmental resonance **frequency** $\Omega$; in particular, quality factors can be enhanced significantly by tuning $\Omega$ to lie below the **qubit** **frequency** $\Delta$....Spin-spin correlation function as a function of **frequency** for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an harmonic **oscillator**, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm...Stronger coupling to bath: Figure fig:times_nobias(b) shows τ Δ , τ Ω , and τ w for a larger coupling strength of α = 0.01 . Figure fig:correlation_strong(a) shows one of the calculated correlation functions. Note that the stronger coupling α leads to a larger separation, or “level repulsion”, between the Δ - and Ω -peaks than in Fig. fig:correlation_weak. The inset of Fig. fig:times_nobias(b) shows the renormalized tunneling matrix element Δ ∞ as a function the initial matrix element Δ 0 . Very importantly, for Δ 0 Ω , Δ increases during the flow, whereas for Δ 0 Ω , it decreases . This behavior can be understood from the fact that f ω l in Eq.( eq:flow1) changes sign at ω = Δ : If the weight of J ω under the integral in ( eq:flow1) is larger for ω > Δ 0 , which is the case if Δ Δ 0 ]. Note also, that the upward renormalization towards larger Δ ∞ in the inset of Fig. fig:times_nobias(b) is stronger than the downward one towards smaller values, i.e., the renormalization is not symmetric with respect to Δ 0 = Ω . The reason for this asymmetry lies in the fact that f ω l has a larger weight for ω Δ . Also τ Δ and even τ w = 1 / J Δ ∞ in Fig. fig:times_nobias(b) show an asymmetric behavior with a steep increase at Δ 0 ≈ Ω : dephasing times for Δ 0 > Ω are larger than for Δ 0 Ω than for Δ 0 Ω , dephasing times can be significantly enhanced (as compared to Δ 0 **oscillator** in (b)....Spin-spin correlation function for the structured bath [Eq.( eq:density)] as a function of **frequency** . The maximum height of the middle peak in (b) is ≈ 7.2 . -0.9cm ... We discuss dephasing times for a two-level system (including bias) coupled to a damped harmonic **oscillator**. This system is realized in measurements on solid-state Josephson **qubits**. It can be mapped to a spin-boson model with a spectral function with an approximately Lorentzian resonance. We diagonalize the model by means of infinitesimal unitary transformations (flow equations), and calculate correlation functions, dephasing rates, and **qubit** quality factors. We find that these depend strongly on the environmental resonance **frequency** $\Omega$; in particular, quality factors can be enhanced significantly by tuning $\Omega$ to lie below the **qubit** **frequency** $\Delta$.

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Contributors: Alexander N. Korotkov

Date: 2005-03-01

We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**....Solid lines: synchronization degree D (and in-phase current quadrature 〈X〉) as functions of F for several values of the detection efficiency ηeff. Dashed and dotted lines illustrate the effects of the energy mismatch (ε≠0) and the **frequency** mismatch (Ω≠Ω0).
... We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**.

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Contributors: Zhang, Jing, Liu, Yu-xi, Zhang, Wei-Min, Wu, Lian-Ao, Wu, Re-Bing, Tarn, Tzyh-Jong

Date: 2011-01-17

(color online) Decoherence suppression by the auxiliary chaotic setup. (a) the evolution of the coherence C x y = S ̂ x 2 + S ̂ y 2 of the state of the **qubit**, where the red asterisk curve and the black triangle curve represent the ideal trajectory without any decoherence and the trajectory under natural decoherence and without corrections; and the green curve with plus signs and the blue solid curve denote the trajectories with I 0 / ω q = 5 and 30 . With these parameters, the dynamics of the Duffing **oscillator** exhibits periodic and chaotic behaviors. τ = 2 π / ω q is a normalized time scale. (b) and (c) are the energy spectra of δ q t with I 0 / ω q = 5 (the periodic case) and 30 (the chaotic case). The energy spectrum S δ q ω is in unit of decibel (dB). (d) the normalized decoherence rates Γ / ω q versus the normalized driving strength I 0 / ω q ....We propose a strategy to suppress decoherence of a solid-state **qubit** coupled to non-Markovian noises by attaching the **qubit** to a chaotic setup with the broad power distribution in particular in the high-**frequency** domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-**frequency** components of our control are generated by the chaotic setup driven by a low-**frequency** field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide **frequency** domain, including low-**frequency** $1/f$, high-**frequency** Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the **qubits** to a Duffing **oscillator** as the chaotic setup. Significantly, the decoherence time of the **qubit** is prolonged approximately $100$ times in magnitude. ... We propose a strategy to suppress decoherence of a solid-state **qubit** coupled to non-Markovian noises by attaching the **qubit** to a chaotic setup with the broad power distribution in particular in the high-**frequency** domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-**frequency** components of our control are generated by the chaotic setup driven by a low-**frequency** field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide **frequency** domain, including low-**frequency** $1/f$, high-**frequency** Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the **qubits** to a Duffing **oscillator** as the chaotic setup. Significantly, the decoherence time of the **qubit** is prolonged approximately $100$ times in magnitude.

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Contributors: Majer, J., Chow, J. M., Gambetta, J. M., Koch, Jens, Johnson, B. R., Schreier, J. A., Frunzio, L., Schuster, D. I., Houck, A. A., Wallraff, A.

Date: 2007-09-13

Controllable effective coupling and coherent state transfer via off-resonant Stark shift. a Spectroscopy of **qubits** versus applied Stark tone power. Taking into account an attenuation of 67 dB before the cavity and the filtering effect of the cavity, 0.77 mW corresponds to an average of one photon in the resonator. The **qubit** transition **frequencies** (starting at ω 1 / 2 π = 6.469 G H z and ω 2 / 2 π = 6.546 G H z ) are brought into resonance with a Stark pulse applied at 6.675 G H z . An avoided crossing is observed with one of the **qubit** transition levels becoming dark as in Figure TransmissionSpec. b Protocol for the coherent state transfer using the Stark shift. The pulse sequence consists of a Gaussian-shaped π pulse (red) on one of the **qubits** at its transition **frequency** ω 1 , 2 followed by a Stark pulse (brown) of varying duration Δ t and amplitude A detuned from the **qubits**, and finally a square measurement pulse (blue) at the cavity **frequency**. The time between the π pulse and the measurement is kept fixed at 130 ns. c Coherent state transfer between the **qubits** according to the protocol above. The plot shows the measured homodyne voltage (average of 3,000,000 traces) with the π pulse applied to **qubit** 1 (green dots) and to **qubit** 2 (red dots) as a function of the Stark pulse length Δ t . For reference, the black dots show the signal without any π pulse applied to either **qubit**. The overall increase of the signal is caused by the residual Rabi driving due to the off-resonant Stark tone, which is also reproduced by the theory. Improved designs featuring different coupling strengths for the individual **qubits** could easily avoid this effect. The thin solid lines show the signal in the absence of a Stark pulse. Adding the background trace (black dots) to these, we construct the curves consisting of open circles, which correctly reproduce the upper and lower limits of the **oscillating** signals due to coherent state transfer. d The **oscillation** **frequency** (red) of the time domain state transfer measurement (c) and the splitting **frequency** (blue) of the continuous wave spectroscopy (a) versus power of the Stark tone. The agreement shows that the **oscillations** are indeed due to the coupling between the **qubits**. StarkSwap...Superconducting circuits are promising candidates for constructing quantum bits (**qubits**) in a quantum computer; single-**qubit** operations are now routine, and several examples of two **qubit** interactions and gates having been demonstrated. These experiments show that two nearby **qubits** can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant **qubits** is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the **qubits** to a quantum bus, which distributes quantum information among the **qubits**. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting **qubits** on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the **qubits** to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the **qubits**. The cavity is also used to perform multiplexed control and measurement of the **qubit** states. This approach can be expanded to more than two **qubits**, and is an attractive architecture for quantum information processing on a chip....Multiplexed control and read-out of uncoupled **qubits**. a Predicted cavity transmission for the four uncoupled **qubit** states. In the dispersive limit ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 ), the **frequency** is shifted by χ 1 σ 1 z + χ 2 σ 2 z . Operating the **qubits** at transition **frequencies** ω 1 / 2 π = 6.617 G H z and ω 2 / 2 π = 6.529 G H z , we find χ 1 / 2 π = - 5.9 M H z and χ 2 / 2 π = - 7.4 M H z . Measurement is achieved by placing a probe at a **frequency** where the four cavity transmissions are distinguishable. The two-**qubit** state can then be reconstructed from the homodyne measurement of the cavity. Rabi **oscillations** of b **qubit** 1 and c **qubit** 2. A drive pulse of increasing duration is applied at the **qubit** transition **frequency** and the response of the cavity transmission is measured after the pulse is turned off. **Oscillations** of quadrature voltages are measured for each of the **qubits** and mapped onto the polarization σ 1 , 2 z . The solid line shows results from a master equation simulation, which takes into account the full dynamics of the two **qubits** and the cavity. The absence of beating in both traces is a signature of the suppression of the **qubit**-**qubit** coupling at this detuning. d The homodyne response (average of 1,000,000 traces) of the cavity after a π pulse on **qubit** 1 (green), **qubit** 2 (red), and both **qubits** (blue). The black trace shows the level when no pulses are applied. The contrasts(i.e. the amplitude of the pulse relative to its ideal maximum value) for these pulses are 60% (green), 61% (green) and 65% (blue). The solid line shows the simulated value including the **qubit** relaxation and the turn-on time of the cavity. The agreement between the theoretical prediction and the data indicates the measured contrast is the maximum observable. From the theoretical calculation one can estimate the selectivity (see text for details) for each π -pulse to be 87% (**qubit** 1) and 94% (**qubit** 2). We note that this figure of merit is not at all intrinsic and that it could be improved by increasing the detuning between the two **qubits** for instance, or using shaped excitation pulses. MultiplexedControl...We can perform coherent state transfer in the time domain by rapidly turning the effective **qubit**-**qubit** coupling on and off. Rather than the slow flux tuning discussed above, we now make use of a strongly detuned rf-drive, which results in an off-resonant Stark shift of the **qubit** **frequencies** on the nanosecond time scale. Figure StarkSwapa shows the spectroscopy of the two **qubits** when this off-resonant Stark drive is applied with increasing power. The **qubit** **frequencies** are pushed into resonance and a similar avoided crossing is observed as in Fig. TransmissionSpecb. With the Stark drive’s ability to quickly tune the **qubits** into resonance, it is possible to observe coherent **oscillations** between the **qubits**, using the following protocol (see Fig. StarkSwapb): Initially the **qubits** are 80 MHz detuned from each other, where their effective coupling is small, and they are allowed to relax to the ground state ↓ ↓ . Next, a π -pulse is applied to one of the **qubits** to either create the state ↑ ↓ or ↓ ↑ . Then a Stark pulse of power P A C is applied bringing the **qubits** into resonance for a variable time Δ t . Since ↑ ↓ and ↓ ↑ are not eigenstates of the coupled system, **oscillations** between these two states occur, as shown in Fig. StarkSwapc. Fig. StarkSwapd shows the **frequency** of these **oscillations** for different powers P A C of the Stark pulse, which agrees with the **frequency** domain measurement of the **frequency** splitting observed in Fig. StarkSwapa. These data are strong evidence that the **oscillations** are due to the coupling between the **qubits** and that the state of the **qubits** is transferred from one to the other. A quarter period of these **oscillations** should correspond to a i S W A P , which would be a universal gate. Future experiments will seek to demonstrate the performance and accuracy of this state transfer....Sample and scheme used to couple two **qubits** to an on-chip microwave cavity. Circuit a and optical micrograph b of the chip with two transmon **qubits** coupled by a microwave cavity. The cavity is formed by a coplanar waveguide (light blue) interrupted by two coupling capacitors (purple). The resonant **frequency** of the cavity is ω r / 2 π = 5.19 G H z and its width is κ / 2 π = 33 M H z , determined be the coupling capacitors. The cavity is operated as a half-wave resonator ( L = λ / 2 = 12.3 m m ) and the electric field in the cavity is indicated by the gray line. The two transmon **qubits** (optimized Cooper-pair boxes) are located at opposite ends of the cavity where the electric field has an antinode. Each transmon **qubit** consists of two superconducting islands connected by a pair of Josephson junctions and an extra shunting capacitor (interdigitated finger structure in the green inset). The left **qubit** (**qubit** 1) has a charging energy of E C 1 / h = 424 M H z and maximum Josephson energy of E J 1 m a x / h = 14.9 G H z . The right **qubit** (**qubit** 2) has a charging energy of E C 2 / h = 442 M H z and maximum Josephson energy of E J 2 m a x / h = 18.9 G H z . The loop area between the Josephson junctions for the two transmon **qubits** differs by a factor of approximately 5 / 8 , allowing a differential flux bias. The microwave signals enter the chip from the left, and the response of the cavity is amplified and measured on the right. c Scheme of the dispersive **qubit**-**qubit** coupling. When the **qubits** are detuned from the cavity ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 ) the **qubits** both dispersively shift the cavity. The excited state in the left **qubit** ↑ ↓ 0 interacts with the excited state in the right **qubit** ↓ ↑ 0 via the exchange of a virtual photon ↓ ↓ 1 in the cavity. SchemePicture...Cavity transmission and spectroscopy of single and coupled **qubits**. a The transmission through the cavity as a function of applied magnetic field is shown in the **frequency** range between 5 GHz and 5.4 GHz. When either of the **qubits** is in resonance with the cavity, the cavity transmission shows an avoided crossing due to the vacuum Rabi splitting. The maximal vacuum Rabi splitting for the two **qubits** is the same within the measurement uncertainty and is ∼ 105 MHz. Above 5.5 GHz, spectroscopic measurements of the two **qubit** transitions are displayed. A second microwave signal is used to excite the **qubit** and the dispersive shift of the cavity **frequency** is measured. The dashed lines show the resonance **frequencies** of the two **qubits**, which are a function of the applied flux according to ω 1 , 2 = ω 1 , 2 m a x cos π Φ / Φ 0 . The maximum transition **frequency** for the first **qubit** is ω 1 m a x / 2 π = 7.8 G H z and for the second **qubit** is ω 2 m a x / 2 π = 6.45 G H z . For strong drive powers, additional resonances between higher **qubit** levels are visible. b Spectroscopy of the two-**qubit** crossing. The **qubit** levels show a clear avoided crossing with a minimal distance of 2 J / 2 π = 26 M H z . At the crossing the eigenstates of the system are symmetric and anti-symmetric superpositions of the two **qubit** states. The spectroscopic drive is anti-symmetric and therefore unable to drive any transitions to the symmetric state, resulting in a dark state. c Predicted spectroscopy at the **qubit**-**qubit** crossing using a Markovian master equation that takes into account higher modes of the cavity. The parameters for this calculation are obtained from the vacuum Rabi splitting and the single **qubit** spectroscopy. TransmissionSpec...In the first measurement we observe strong coupling of each of the **qubits** separately to the cavity. By varying the flux, each of the two **qubits** can be tuned into resonance with the cavity (see Fig. TransmissionSpeca). Whenever a **qubit** and the cavity are degenerate, the transmission is split into two well-resolved peaks in **frequency**, an effect called vacuum Rabi splitting, demonstrating that each **qubit** is in the strong coupling limit with the cavity. Each of the peaks corresponds to a superposition of **qubit** excitation and a cavity photon in which the energy is shared between the two systems. From the **frequency** difference at the maximal splitting, the coupling parameters g 1 , 2 ≈ 105 M H z can be determined for each **qubit**. The transition **frequency** of each of the two **qubits** (see Fig. TransmissionSpeca) can also be measured far from the cavity **frequency** as described below....In this regime, no energy is exchanged with the cavity. However, the **qubits** and cavity are still dispersively coupled, resulting in a **qubit**-state-dependent shift ± χ 1 , 2 of the cavity **frequency** (see Fig. MultiplexedControla) or equivalently an AC Stark shift of the **qubit** **frequencies**. The **frequency** shift χ 1 , 2 can be calculated from the detuning Δ 1 , 2 and the measured coupling strength g 1 , 2 . The last term describes the interaction between the **qubits**, which is a transverse exchange interaction of strength J = g 1 g 2 1 / Δ 1 + 1 / Δ 2 / 2 (See Fig. SchemePicturec). The **qubit**-**qubit** interaction is a result of virtual exchange of photons with the cavity. When the **qubits** are degenerate with each other, an excitation in one **qubit** can be transferred to the other **qubit** by virtually becoming a photon in the cavity (see Fig. MultiplexedControlb). However, when the **qubits** are non-degenerate | ω 1 - ω 2 | ≫ J this process does not conserve energy, and therefore the interaction is effectively turned off. Thus, instead of modifying the actual coupling constant, we control the effective coupling strength by tuning the **qubit** transition **frequencies**. This is possible since the **qubit**-**qubit** coupling is transverse, which also distinguishes our experiment from the situation in liquid-state NMR quantum computation, where an effective switching-off can only be achieved by repeatedly applying decoupling pulses....In addition to acting as a quantum bus, the cavity can also be used for multiplexed read-out and control of the two **qubits**. Here, “multiplexed" refers to acquisition of information or control of more than one **qubit** via a single channel. To address the **qubits** independently, the flux is tuned such that the **qubit** **frequencies** are 88 MHz apart ( ω 1 = 6.617 G H z , ω 2 = 6.529 G H z ), making the **qubit**-**qubit** coupling negligible. Rabi experiments showing individual control are performed by applying an rf-pulse at the resonant **frequency** of either **qubit**, followed by a measurement pulse at the resonator **frequency**. The response (see Fig. MultiplexedControlb and MultiplexedControlc) is consistent with that of a single **qubit** **oscillation** and shows no beating, indicating that the coupling does not affect single-**qubit** operations and read-out. With similar measurements the relaxation times ( T 1 ) of the two **qubits** are determined to be 78 ns and 120 ns, and with Ramsey measurements the coherence times ( T 2 ) are found to be 120 ns and 160 ns. The ability to simultaneously read-out the states of both **qubits** using a single line is shown by measuring the cavity phase shift, proportional to χ 1 σ 1 z + χ 2 σ 2 z (see Eq. Hamiltonian), after applying a π -pulse to one or both of the **qubits**. Figure MultiplexedControld shows the response of the cavity after a π -pulse has been applied on the first **qubit** (green points), on the second **qubit** (red points) or on both **qubits** (blue points). For comparison the response of the cavity without any pulse applied (black points) is shown. Since the cavity **frequency** shifts for the two **qubits** are different ( χ 1 ≠ χ 2 ), so we are able to distinguish the four states ↓ ↓ , ↓ ↑ , ↑ ↓ , ↑ ↑ of the **qubits** with a single read-out line. One can show that this measurement, with sufficient signal to noise and combined with single-**qubit** rotations, should in principle allow for a full reconstruction of the density matrix (state tomography), although not demonstrated in the present experiment. ... Superconducting circuits are promising candidates for constructing quantum bits (**qubits**) in a quantum computer; single-**qubit** operations are now routine, and several examples of two **qubit** interactions and gates having been demonstrated. These experiments show that two nearby **qubits** can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant **qubits** is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the **qubits** to a quantum bus, which distributes quantum information among the **qubits**. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting **qubits** on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the **qubits** to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the **qubits**. The cavity is also used to perform multiplexed control and measurement of the **qubit** states. This approach can be expanded to more than two **qubits**, and is an attractive architecture for quantum information processing on a chip.

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

Date: 2008-06-17

(color online) Decaying **qubit** **oscillations** with initial state | ↑ in a weakly probed CPB with 6 states for α = Z 0 e 2 / ℏ = 0.08 , A = 0.1 E J / e , E C = 5.25 E J and N g = 0.45 , so that E e l = 2.1 E J and ω q b = 2.3 E J / ℏ . (a) Time evolution of the measured difference signal Q ̇ ∝ ξ o u t - ξ i n (in units of 2 e E J / ℏ ) of the full CPB and its lock-in amplified phase φ o u t (**frequency** window Δ Ω = 5 E J / ℏ ), compared to the estimated phase φ h f 0 ∝ σ x 0 in the **qubit** approximation. The inset resolves the underlying small rapid **oscillations** with **frequency** Ω = 15 E J / ℏ in the long-time limit. (b) Power spectrum of Q ̇ for the full CPB Hamiltonian (solid) and for the two-level approximation (dashed)....We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-**frequency** ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge **qubit** realized with a Cooper-pair box, where we focus on monitoring coherent **oscillations**....Although later on we focus on the dynamics of a superconducting charge **qubit** as sketched in Fig. fig:setup, our measurement scheme is rather generic and can be applied to any open quantum system. We employ the system-bath Hamiltonian...eq:7 allows one to retrieve information about the coherent **qubit** dynamics in an experiment. Figure fig:oscillation(a) shows the time evolution of the expectation value Q ̇ t for the initial state | ↑ ≡ | 1 , obtained via numerical integration of the master equation ...CPB in the presence of the ac driving which in principle may excite higher states. The driving, due to its rather small amplitude, is barely noticeable on the scale chosen for the main figure, but only on a refined scale for long times; see inset of Fig. fig:oscillation(a). This already insinuates that the backaction on the dynamics is weak. In the corresponding power spectrum of Q ̇ depicted in Fig. fig:oscillation(b), the driving is nevertheless reflected in sideband peaks at the **frequencies** Ω and Ω ± ω q b . In the time domain these peaks correspond to a signal cos Ω t - φ o u t t . Moreover, non-**qubit** CPB states leads to additional peaks at higher **frequencies**, while their influence at **frequencies** ω Ω is minor. Experimentally, the phase φ o u t t can be retrieved by lock-in amplification of the output signal, which we mimic numerically in the following way : We only consider the spectrum of ξ o u t in a window Ω ± Δ Ω around the driving **frequency** and shift it by - Ω . The inverse Fourier transformation to the time domain provides φ o u t t which is expected to agree with φ h f 0 t and, according to Eq. ...(color online) (a) Fidelity defect δ F = 1 - F and (b) time-averaged trace distance between the driven and the undriven density operator of the CPB for various driving amplitudes as a function of the driving **frequency**. All other parameters are the same as in Fig. fig:oscillation....In order to quantify this agreement, we introduce the measurement fidelity F = φ o u t σ x 0 , where f g = ∫ d t f g / ∫ d t f 2 ∫ d t g 2 1 / 2 with time integration over the decay duration. Thus, the ideal value F = 1 is assumed if φ o u t t and σ x t 0 are proportional to each other, i.e. if the agreement between the measured phase and the unperturbed expectation value σ x 0 is perfect. Figure fig:fidelity(a) depicts the fidelity as a function of the driving **frequency**. As expected, whenever non-**qubit** CPB states are excited resonantly, we find F ≪ 1 , indicating a significant population of these states. Far-off such resonances, the fidelity increases with the driving **frequency** Ω . A proper **frequency** lies in the middle between the **qubit** doublet and the next higher state. In the present case, Ω ≈ 15 E J / ℏ appears as a good choice. Concerning the driving amplitude, one has to find a compromise, because as A increases, so does the phase contrast of the outgoing signal...eq:7, to reflect the unperturbed time evolution of σ x 0 with respect to the **qubit**. Although the condition of high-**frequency** probing, Ω ≫ ω q b , is not strictly fulfilled and despite the presence of higher charge states, the lock-in amplified phase φ o u t t and the predicted phase φ 0 h f t are barely distinguishable for an appropriate choice of parameters as is shown in Fig. fig:oscillation(a). ... We propose a scheme for monitoring coherent quantum dynamics with good time-resolution and low backaction, which relies on the response of the considered quantum system to high-**frequency** ac driving. An approximate analytical solution of the corresponding quantum master equation reveals that the phase of an outgoing signal, which can directly be measured in an experiment with lock-in technique, is proportional to the expectation value of a particular system observable. This result is corroborated by the numerical solution of the master equation for a charge **qubit** realized with a Cooper-pair box, where we focus on monitoring coherent **oscillations**.

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Contributors: Oh, Sangchul, Kim, Jaewan

Date: 2006-05-03

The harmonic **oscillator** plays two competing roles: (i) making two **qubits** entangled by inducing the two-**qubit** coupling I e f f , (ii) having two **qubits** decoherent through entanglement with them, i.e. Γ R t . The role of the imaginary part Γ I t is different from that of the real part Γ R t . While Γ R t makes the two **qubit** decoherent, i.e. decaying the off-diagonal elements, Γ I t causes the two-**qubit** coupling to fluctuate as shown in Fig. Fig2(b). These could be uncovered by examining the reduced density matrix of the two **qubits**...(color online). (a) Maximum concurrence of the two **qubits** (b) entropy of two **qubits** at equilibrium as a function of α and ω 0 / ω c . (c) exp - Γ R ∞ as a function of temperature T and ω 0 / ω c for α = 0.25 ....As ω / λ becomes large, C goes to C i d e a l and S to 0 as depicted in Fig. Fig2. This is explained by means of the Born-Oppenheimer approximation that is based on the assumption of the weak coupling between the two **qubits** and the environment. The **frequency** ω of the harmonic **oscillator** is larger than that of the two **qubits**, θ . The harmonic **oscillator** **oscillates** very fast in comparison with the two **qubits**. So the two **qubits** feel the harmonic **oscillator** stays in the same state. However, the condition of ω ≫ λ is not a unique way to make the two **qubits** entangled maximally. Surprisingly, we find the maximum entanglement of the two **qubits**, C = 1 , at θ t = π / 4 under the condition that ω / λ = 4 n with n = 1 , 2 , as shown in Figs. Fig2(a) and Fig2(d). This is due to the fact that the **frequency** θ for entangling the two **qubits** is commensurate with the **frequency** ω of the harmonic **oscillator**. If the condition of ω / λ = 4 n , n = 1 , 2 , is not met, then the concurrence C does not reach 1 at θ t = π / 4 and 0 at θ t = π / 2 due to Γ t as shown in Fig. Fig2(b). The **oscillation** period of C (red solid line) does not coincide with that of the ideal case C i d e a l (thick black line). This implies that the two **qubit** coupling fluctuates due to Γ I t . For ω / λ **qubits** could not be entangled. Fig. Fig2(c) shows the case of ω = λ ....Fig. Fig2(d) shows two competing roles of the harmonic **oscillator**. Let us define the average concurrence C a v g for a period τ ≡ π / 2 θ by C a v g ≡ 1 τ ∫ 0 τ C t d t . For the first period 0 ≤ θ t **frequencies** ω and θ , S a v g and C a v g show the behavior of the stair case....The environment is characterized by the spectral density function J ω = ∑ j λ j 2 δ ω - ω j . As shown in the inset of Fig. Fig3-(a), let us consider an Ohmic environment with a gap ω 0 and an exponential cutoff function of the cutoff **frequency** ω c...The harmonic **oscillator** remains isolated always from the system, but it induces the indirect interaction between the two **qubits** and thus entanglement between them . For a pure two **qubits**, concurrence, an entanglement measure, reads C i d e a l = 2 | a d - e i 4 θ t b c | . Here the subscript, ideal, stands for the case of the **qubits** in a pure state. If a = b = c = d = 1 / 2 , we have C i d e a l = | sin 2 θ t | as shown in Fig. Fig2....Two circles refer to two **qubits** and an oval is an environment. The coupling between **qubits** and the common environment (solid arrow) induces the indirect interaction between two **qubits** (dashed arrow)....We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus. ... We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus.

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