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• (color online) (a) Rabi oscillations in the qubit population P vs. Rabi pulse length Δ t (blue dots) and fit with unit visibility (red line). (b) Measured Rabi frequency ν R a b i vs. pulse amplitude ϵ s (blue dots) and linear fit.... (color online) Measurement response φ (blue lines) and theoretical prediction (red lines) vs. time. At t = 6 μ s (a) a π pulse, (b) a 2 π pulse, and (c) a 3 π pulse is applied to the qubit. In each panel the dashed lines correspond to the expected measurement response in the ground state φ , in the saturated state φ = 0 , and in the excited state φ .... The extracted qubit population P is plotted versus Δ t in Fig.  fig:rabioscillationsa. We observe a visibility of 95 ± 6 % in the Rabi oscillations with error margins determined from the residuals of the experimental P with respect to the predicted values. Thus, in a measurement of Rabi oscillations in a superconducting qubit, a visibility in the population of the qubit excited state that approaches unity is observed for the first time. Moreover, we note that the decay in the Rabi oscillation amplitude out to pulse lengths of 100 n s is very small and consistent with the long T 1 and T 2 times of this charge qubit, see Fig.  fig:rabioscillationsa and Ramsey experiment discussed below. We have also verified the expected linear scaling of the Rabi oscillation frequency ν R a b i with the pulse amplitude ϵ s ∝ n s , see Fig.  fig:rabioscillationsb.... In our circuit QED architecture , a Cooper pair box , acting as a two level system with ground and excited states and level separation E a = ℏ ω a = E e l 2 + E J 2 is coupled capacitively to a single mode of the electromagnetic field of a transmission line resonator with resonance frequency ω r , see Fig.  fig:setupa. As demonstrated for this system, the electrostatic energy E e l and the Josephson energy E J of the split Cooper pair box can be controlled in situ by a gate voltage V g and magnetic flux Φ , see Fig.  fig:setupa. In the resonant ( ω a = ω r ) strong coupling regime a single excitation is exchanged coherently between the Cooper pair box and the resonator at a rate g / π , also called the vacuum Rabi frequency . In the non-resonant regime ( Δ = ω a - ω r > g ) the capacitive interaction gives rise to a dispersive shift g 2 / Δ σ z in the resonance frequency of the cavity which depends on the qubit state σ z , the coupling g and the detuning Δ . We have suggested that this shift in resonance frequency can be used to perform a quantum non-demolition (QND) measurement of the qubit state . With this technique we have recently measured the ground state response and the excitation spectrum of a Cooper pair box .... (color online) (a) Measured Ramsey fringes (blue dots) observed in the qubit population P vs. pulse separation Δ t using the pulse sequence shown in Fig.  fig:setupb and fit of data to sinusoid with gaussian envelope (red line). (b) Measured dependence of Ramsey frequency ν R a m s e y on detuning Δ a , s of drive frequency (blue dots) and linear fit (red line).... (color online) (a) Simplified circuit diagram of measurement setup. A Cooper pair box with charging energy E C and Josephson energy E J is coupled through capacitor C g to a transmission line resonator, modelled as parallel combination of an inductor L and a capacitor C . Its state is determined in a phase sensitive heterodyne measurement of a microwave transmitted at frequency ω R F through the circuit, amplified and mixed with a local oscillator at frequency ω L O . The Cooper pair box level separation is controlled by the gate voltage V g and flux Φ . Its state is coherently manipulated using microwaves at frequency ω s with pulse shapes determined by V p . (b) Measurement sequence for Rabi oscillations with Rabi pulse length Δ t , pulse frequency ω s and amplitude ∝ n s with continuous measurement at frequency ω R F and amplitude ∝ n R F . (c) Sequence for Ramsey fringe experiment with two π / 2 -pulses at ω s separated by a delay Δ t and followed by a pulsed measurement.... In the experiments presented here, we coherently control the quantum state of a Cooper pair box by applying to the qubit microwave pulses of frequency ω s , which are resonant with the qubit transition frequency ω a / 2 π ≈ 4.3 G H z , through the input port C i n of the resonator, see Fig.  fig:setupa. The microwaves drive Rabi oscillations in the qubit at a frequency of ν R a b i = n s g / π , where n s is the average number of drive photons within the resonator. Simultaneously, we perform a continuous dispersive measurement of the qubit state by determining both the phase and the amplitude of a coherent microwave beam of frequency ω R F / 2 π = ω r / 2 π ≈ 5.4 G H z transmitted through the resonator . The phase shift φ = tan -1 2 g 2 / κ Δ σ z is the response of our meter from which we determine the qubit populat... We have determined the coherence time of the Cooper pair box from a Ramsey fringe experiment, see Fig.  fig:setupc, when biased at the charge degeneracy point where the energy is first-order insensitive to charge noise . To avoid dephasing induced by a weak continuous measurement beam we switch on the measurement beam only after the end of the second π / 2 pulse. The resulting Ramsey fringes oscillating at the detuning frequency Δ a , s = ω a - ω s ∼ 6 M H z decay with a long coherence time of T 2 ∼ 500 n s , see Fig.  fig:Ramseya. The corresponding qubit phase quality factor of Q ϕ = T 2 ω a / 2 ∼ 6500 is similar to the best values measured so far in qubit realizations biased at such an optimal point . The Ramsey frequency is shown to depend linearly on the detuning Δ a , s , as expected, see Fig.  fig:Ramseyb. We note that a measurement of the Ramsey frequency is an accurate time resolved method to determine the qubit transition frequency ω a = ω s + 2 π ν R a m s e y .
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• model We consider a composed system built of a qubit, -the system of interest-, coupled to a nonlinear quantum oscillator (NLO), see Fig. linearbath. To read-out the qubit state we couple the qubit linearly to the oscillator with the coupling constant g ¯ , such that via the intermediate NLO dissipation also enters the qubit dynamics.... Jeff The effective spectral density follows from Eqs. ( gl20) and ( chilarger). It reads: J s i m p l J e f f ω e x = g ¯ 2 γ ω e x n 1 0 4 2 Ω 1 | ω e x | + Ω 1 M γ 2 Ω 1 2 2 n t h Ω 1 + 1 2 n 1 0 4 + 4 M Ω 2 | ω e x | - Ω 1 2 . As in case of the effective spectral density J e f f H O , Eq. ( linearspecdens), we observe Ohmic behaviour at low frequency. In contrast to the linear case, the effective spectral density is peaked at the shifted frequency Ω 1 . Its shape approaches the Lorentzian one of the linear effective spectral density, but with peak at the shifted frequency, as shown in Fig. CompLorentz.... Schematic representation of the complementary approaches available to evaluate the qubit dynamics: In the first approach one determines the eigenvalues and eigenfunctions of the composite qubit plus oscillator system (yellow (light grey) box) and accounts afterwards for the harmonic bath characterized by the Ohmic spectral density J ω . In the effective bath description one considers an environment built of the harmonic bath and the nonlinear oscillator (red (dark grey) box). In the harmonic approximation the effective bath is fully characterized by its effective spectral density J e f f ω . approachschaubild... mapping The main aim is to evaluate the qubit’s evolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the oscillator and the Ohmic bath are put together, as depicted in Figure approachschaubild, to form an effective bath. The effective Hamiltonian... The transition frequencies in Eqs. ( rc1) and ( rc2) coincide, and in Figs. Plowg and Flowg there is no deviation observed when comparing the three different approaches.... where the trace over the degrees of freedom of the bath and of the oscillator is taken. In Fig. approachschaubild two different approaches to determine the qubit dynamics are depicted. In the first approach, which is elaborated in Ref. [... Corresponding Fourier transform of P t shown in Fig. CompNLLP. The effect of the nonlinearity is to increase the resonance frequencies with respect to the linear case. As a consequence the relative peak heights change. CompNLLF... Schematic representation of the composed system built of a qubit, an intermediate nonlinear oscillator and an Ohmic bath. linearbath
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• So far we constructed slow composite objects paying attention only to the oscillations of the solid lines in the diagrams and assuming very slow dashed lines, i.e., neglected the higher-frequency noise. In fact, one can construct another slow object shown in Fig. F:slowc, if the respective oscillations of the solid lines are compensated by the dashed line from this vertex. In other words, in the frequency domain, one constrains the frequency of the dashed line to be Δ E (or - Δ E , depepnding on the direction of the spin flip at the vertex). The dashed lines from such objects pair up, and the integral w.r.t. their relative position is dominated by small separations, δ t ∼ 1 / Δ E . Thus one finds the slow object of Fig. F:slowd, two vertices linked by a dashed line at frequency Δ E ; it describes the relaxational contribution to dephasing exp - t / 2 T 1 , where... Dissipative dynamics of a Josephson charge qubit. The simplest Josephson charge qubit is the Cooper-pair box shown in Fig. F:qb . It consists of a superconducting island connected by a dc-SQUID (effectively, a Josephson junction with the coupling E J Φ x = 2 E J 0 cos π Φ x / Φ 0 tunable via the magnetic flux Φ x ; here Φ 0 = h c / 2 e ) to a superconducting lead and biased by a gate voltage V g via a gate capacitor C g . The Josephson energy of the junctions in the SQUID loop is E J 0 , and their capacitance C J 0 sets the charging-energy scale E C ≡ e 2 / 2 C g + C J , C J = 2 C J 0 . At low enough temperatures single-electron tunneling is suppressed and only even-parity states are involved. Here we consider low-capacitance junctions with high charging energy E C ≫ E J 0 . Then the number n of Cooper pairs on the island (relative to a neutral state) is a good quantum number; at certain values of the bias V g ≈ V d e g = 2 n + 1 e / C two lowest charge states n and n + 1 are near-degenerate, and even a weak E J mixes them strongly. At low temperatures and operation frequencies higher charge states do not play a role. The Hamiltonian reduces to a two-state model,... FIG.  F:slow. a. Double vertices with low- ω tails, which appear in the evaluation of dephasing. b. Examples of clusters built out of them . c. A low- ω object with a high-frequency dashed line. The relaxation process in e also contributes to dephasing as shown in d.... In the diagrams the horizontal direction explicitly represents the time axis. The solid lines describe the unperturbed (here, coherent) evolution of the qubit’s 2 × 2 density matrix ρ ̂ , exp - i L 0 t θ t , where L 0 is the bare Liouville operator (this translates to 1 / ω - i L 0 in the frequency domain). The vertices are explicitly time-ordered; each of them contributes the term ζ Y σ x τ z / 2 , with the bath operator Y t and the Keldysh matrix τ z = ± 1 for vertices on the upper/lower time branch. Averaging over the fluctuations should be performed; for gaussian correlations it pairs the vertices as indicated by dashed lines in Fig. F:2order, each of the lines corresponding to a correlator Y Y . Fig. F:2order shows contributions to the second-order self-energy Σ ↑ ↓ ← ↑ ↓ 2 (here i j = ↑ ↓ label four entries of the qubit’s density matrix). The term in Fig. F:2ordera gives... F:qbFIG. F:qb. The simplest Josephson charge qubit
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• where the basis comprises the Fock, or number, states of the field. We note that is an eigenstate of the annihilation operator labelled by its eigenvalue α . In this scenario, although the qubit initially exhibits Rabi oscillations analogous to those in the classical case, these apparently “decay”, and then subsequently revive . An example is shown in figure  rabirev. Such collapse and revival of Rabi oscillations of a qubit is widely recognised as a characteristic of a qubit coupled to a quantum field mode. It is understood both theoretically  and experimentally  that the apparent decay of qubit coherence is due to entanglement with the field mode, generated by the coherent evolution of the coupled quantum systems. This is illustrated in figure  rabirev through a plot of the qubit entropy, S t = - T r ( ρ t l n ρ t ) where ρ t is the reduced density matrix of the qubit resulting from a trace over the field (or vice versa, given the initial system state is pure). Clearly there is a sharp rise in entropy sympathetic with the initial collapse. The qubit then disentangles from the field at the “attractor time” , half way to the revival. The revival arises through oscillatory re-entanglement of the qubit and field, as seen through the subsequent entropy oscillations that coincide with the revival. Obviously there is no entanglement between the qubit and the field in the classical limit, because the field is classical, so the qubit entropy is zero for all times.... (Color online) Qubit inversion, σ z , as a function of dimensionless time ω t for the resonant cases of Rabi oscillations (dotted; light grey/magenta) in a classical field and collapse and revival (dark grey/red heavy line) in a quantum field. For the classical field case ν / ω = 8 is used, corresponding to the same dominant Rabi frequency (see ) as in the quantum field case, which uses a coherent state ( coh) with α = 15 and λ / ω = 1 . Also shown (in green/light grey solid line) is the qubit entropy (in Nats), which indicates the degree of entanglement with the field.... (Color online) Results (dark grey/red) showing qubit inversion, σ z , for different values of dissipation (and therefore drive) applied to the field. These illustrate collapse and revival, suppression of collapse and an approach to complete Rabi oscillations as dissipation and drive are increased. For each individual QSD run shown, the qubit entropy in Nats (light grey/green) is superimposed.... Further insight into the classical limit can be obtained from the phase space behaviour of the field. It is well known that in the pure quantum limit  the qubit-field entanglement correlates distinct and localised (coherent-like) states of the field with different qubit amplitudes. Thus, when there is no entanglement at the “attractor time”, the interaction of the atom with the field mode generates a macroscopically distinct superposition of states in the oscillator—a Schrödinger cat state. As one would expect, and in order to render the field behaviour classical, the introduction of decoherence suppresses this phenomenon. We illustrate this in  by providing two animations of the dynamics of the Wigner function and atomic inversion for the parameters of Fig.  rabirev, one undamped and one with dissipation of γ / ω = 0.01 .... The detuning is defined as Δ = ω 0 - ω and the Rabi oscillation frequency as Ω R = Δ 2 + ν 2 , so for the case of the field on resonance with the qubit (zero detuning, Δ = 0 ) the Rabi frequency is simply Ω R = ν . It is set by the amplitude of the field, not its frequency. In this resonant case, a qubit initially in state oscillates fully to at frequency Ω R . In the language of atomic and optical physics, the atomic inversion—in qubit language σ z —satisfies σ z = cos Ω R t . In the absence of any decoherence acting on the qubit, these Rabi oscillations persist and are a well known characteristic of a qubit resonantly coupled to an external classical field. An example is shown in figure rabirev.... It is well known that there are three timescales in the collapse and revival situation . For fields with large n = n ̄ , the Rabi time (or period) is given by t R = 2 π Ω R -1 = π / λ n ̄ , the collapse time that sets the Gaussian decay envelope of the oscillations by t c = 2 / λ and the (first) revival time that determines when the oscillations reappear, such as in the example of figure rabirev, by t r = 2 π n ̄ / λ . For the coherent state ( coh) the average photon, or excitation, number is n ̄ = | α | 2 . Note that the different dependencies of the times on n ̄ (which corresponds to the—e.g. electric—field strength of the coherent field state) allow a sort of “classical limit” to be taken. As n ̄ is increased, there are more Rabi periods packed in before the collapse—so this appears more like persistent Rabi oscillations—and the revival is pushed out further in time. However, the collapse (and revival) still occur eventually, and in any case the reason there are more Rabi oscillations before the collapse is due to the inverse scaling of t R with n ̄ , so the actual Rabi period is shortened as n ̄ is increased. The classical limit we consider in this work is quite different. We shall consider a fixed n ̄ , so the Rabi period of the qubit does not change in our various examples. We’ll show how the transition from quantum (collapse and revival) to classical (continuous Rabi oscillations) can be effected by introducing decoherence to the quantum field. Our work complements the dissipative, small- n , short-time study of Kim et al. , who show that such fields are sufficiently classical to provide Ramsey pulses to Rydberg atoms.... (Color online) Results (dark grey/red) showing qubit inversion, σ z , for different values of dissipation (and therefore drive) applied to the field. These plots focus on the parameter range where the revival of Rabi oscillations begins to emerge. For each individual QSD run shown, the qubit entropy in Nats (light grey/green) is superimposed.... In our approach, there are two ways in which the qubit state could become mixed. Firstly, it could entangle with the field , as happens in the pure quantum limit to generate the collapse. Such entanglement can be inferred from the qubit entropy in a single run of QSD (for which the full qubit-field state is pure). Secondly, the qubit could remain pure in an individual QSD run, but, when averaged over an ensemble, show mixture. For the classical limit of the top left plot of figure  fig2, we have calculated that both of these effects are small for times in excess of the collapse time. The qubit-field entanglement remains very close to zero for all times in a single QSD run, as shown in the entropy plot presented. Independent QSD runs have been made and these show that the qubit mixture is still very small at the collapse time. Therefore the persistence of good Rabi oscillations well beyond the collapse time and all the way out to the revival time, as illustrated in the top left plot of figure  fig2, provides a clear signature of the classical limit of the field. In this limit, the quantum field state is a localized lump in phase space (like a coherent state), following the expected classical trajectory and suffering negligible back-reaction from the qubit. However, the field coherence time is so short as to prevent entanglement with the qubit developing, unlike in the quantum limit . The resultant qubit Rabi oscillations are thus like those due to a classical field, and not like those that arise from entanglement with a single Fock state (which is a delocalized ring in phase space).
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• 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.... 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.
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• Fig. ( fig:spurious) shows the comparison of the numerical result of a qubit dephased by 14 spins with and without a spurious resonator, which is also dephased by the same group of spins. We can see that the Rabi oscillation becomes somewhat irregular with reduced amplitude while the long-time slow decay still persists without the sign of fading away. This explains the experimental observation that decoherence time is not reduced by the coupling to the spurious resonator is because the time has already passed T φ and the Rabi oscillations have entered the featureless slow decay regime. Notice that the irregularity starts to appear only after t ∼ 2 π / g , which suggests the cause is due to the smearing of the beats.... Numerical simulation for the Rabi oscillation of one qubit coupled with 14 spins at T = 200 mK and T = 10 mK. The left column are the real time Rabi oscillations, and the right column are their Fourier transforms. The dotted lines in the right column are the exact Fourier transform obtained in Eq. ( eq:FTsz) for the limit of infinite number of spins. T φ , calculated using Eq. ( eq:t_phi), are 84ns and 165ns respectively. Notice that the small bumps in the oscillation envelope in the T = 10 mK are due to the effects of finite spins and some spins are frozen.... Time evolution of the qubit decohered by many spins undergoing Rabi oscillation by a coherent microwave source, σ z t . The Rabi frequency 2 α / 2 π = 200 MHz, and δ Ω is generated from a special case where all spin energy splittings are chosen for simplification to be ω k / 2 π = 1 GHz and ∑ k A k 2 / 2 π = 50 MHz at the temperature T = 150 mK. For Ω / 2 π = 10 GHz, these parameters correspond to δ Ω / Ω 0.005 . Notice that the amplitude of the oscillation envelope is already down to 50 % at t ∼ 20 ns, but it reaches 25 % only after t = 80 ns, while an exponential decay should reach 25 % around t ∼ 40 ns.... Now the width of the peak ω ' = 0 , i.e. ω = 2 α , is controlled by the Gaussian function with width 2 δ Ω , as shown in Fig. ( fig:sz_w). When δ Ω → 0 , the spectrum becomes a delta function at the Rabi frequency ω = 2 α . At high frequency the Fourier spectrum is dominated by the Gaussian term, which means that σ z t has a Gaussian decay in the short time limit. The singularity at the Rabi frequency ω = 2 α implies a very slow decay in the long time limit. The result agrees with the that obtained for Eq. ( eq:sz) in the limit when α ≫ δ Ω , the oscillation envelope of σ z t is given by σ z 0 1 + 2 t δ Ω 2 / α 2 - 1 4 , where the evolution begins with a Gaussian (quadratic) damping then changes to a slow power law decay of ∼ α / δ Ω 2 t .... Simulation of the spectral probing of resonance on the qubit-spurious resonator system. In both graphs, the vertical axes are the value of σ z at the steady state, and the horizontal axes are the driving microwave frequencies. The spurious resonator here has an energy splitting at 10 GHz. The left graph shows when the qubit energy is detuned from the resonator at 9.95 GHz, the most visible peak is the qubit and the resonator peak is barely visible. When the qubit energy is tuned close to the resonator frequency, as shown in the right graph, level repulsion takes place. Notice that the spectral peaks here are clearly seperated even though δ Ω and δ Ω r e s are both greater than g .... Rabi oscillations of a qubit dephased and relaxed by a many-spin system. The parameters are the same as those used in the T = 10 mK graph in Fig. ( fig:tdep). The only additional parameter, the relaxation time T 1 , calculated from Fermi’s golden rule is 1 μ s. The solid lines are the numerical results, and the dashed lines are the approximations in Eq. ( eq:relax_fit) and Eq. ( eq:relax_ft_fit).... which is the ideal Rabi oscillation, because every spin is frozen to its own ground state. But as soon as we turn up the temperature, when the value B is allowed to fluctuate, the oscillation now has a decay pattern(see in Fig. ( fig:sz)). Notice that the oscillation lasts much longer than an ordinary exponential decay but has a rather drastic decrease of amplitude in the beginning.... The simulation of both dephasing and relaxation present can be done easily in our program. The newly added spins remain non-interactive among themselves. Fig. ( fig:relax) shows a simulation result with the relaxation time T 1 ≫ T φ . In the real time evolution graph we can clearly see the three stages of the decay process, which starts with a fast Gaussian decay followed by slow decay, and then later the exponential decay finally takes over. Similar behavior has also been observed in the result produced by a qubit under the direct influence of 1/f noise. In the graph of the Fourier transform of the same data, the original sharp peak in the Rabi frequency is now smeared. Since this problem cannot be solved analytically, we made the approximation of multiplying the oscillating part of Eq. ( eq:sz) with an exponential factor e - t / 2 T 1 , so that it becomes... Effect on Rabi oscillations caused by a dephased spurious resonator. The solid line represents the case where the qubit is coupled to a spurious resonator, and the dotted line is not. The parameters for both are 2 α / 2 π = 100 MHz and δ Ω / 2 π = 145 MHz. Those for the coupled resonator are g / 2 π = 15 MHz and δ Ω r e s / 2 π = 30 MHz. Both qubit and spurious resonator couple to the same 14 spins. Notice that because δ Ω > h , the initial reduction of amplitude ends even before one period of Rabi oscillation.... Fourier spectrum σ ~ z ω near the Rabi frequency ω = 2 α . We can use the width of the peak to estimate the decoherence time of the Rabi oscillation. δ Ω > and δ Ω frequency ω ≫ 2 α + 2 δ Ω the tail of the peak is dominated by the Gaussian term, therefore we can expect a Gaussian decay of σ z t when t ≪ ℏ / δ Ω . While the frequency ω is very near 2 α , the peak goes like ω - 2 α -1 / 2 , which implies a very slow decay at the long time limit.
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• This amplitude describes the tunneling transition in the presence of a time-modulated external field that shifts the energetic levels. It is interesting to compare this result with the analogous one for the case of an external monochromatic field. It is known that in the latter case the amplitude of the transition → sup with parameters satisfying the resonance does not depend on time intervals, while the amplitude Eq. ( amplitud) contains time-dependent periodic oscillations at the modulation frequency. In Fig. TransProb1 and TransProb2 we depict the corresponding probabilities of the tunneling transition in dependence on dimensionless time for two resonant conditions: N = 1 and 2 . As we see, the transition amplitudes are not constants and are periodic in time, while for the case of a one-monochromatic driving field these quantities have constant values.... The function γ N t is an increasing function in time but it grows also periodically due to its "linear+periodic" structure. Therefore, the dynamics of populations Eq. ( pop) seems to be aperiodic in time. Indeed, the typical results for the phase function as well as the populations are depicted in Figs. pop3, Population1 and Population2. The dynamics of populations for the case of a weak external field is shown in Figs. pop3 for two resonance regimes. In Fig. pop3(a) we compare two curves of the occupation probabilities for N = 1 (solid curve) and for N = 2 (dashed curve). We can see here fast oscillations of the population for the regime N = 1 and slow oscillations for the case of N = 2 (for consideration in details, see the curve corresponding to the case N = 2 for large time intervals in Fig. pop3(b)). The results for the second-order resonance regime are also demonstrated in Fig. pop3(c) for the other parameter Δ / δ . Analyzing these results, we note that dynamics of populations strongly depends on the value of the ratio Δ / δ . It can be seen from the formulas Eqs. ( JmeanApp) and ( fiApp) that population behavior shown in Fig. pop3(b) for N = 2 is mainly governed by the linear in time term in the phase function Eq. ( fiApp); thus, we can see that the dynamics looks like cosinusoidal oscillations. The periodic in time part Φ N t only slightly modulated these oscillations. This part of the phase function increases with increasing the parameter Δ / δ that leads to increasing the role of periodic modulations giving rise to a nontrivial time dependence of occupation probability [see, Fig. pop3(c) for the case N = 2 ].... The typical results for Rabi oscillations with regular, periodic dynamics are depicted in Fig. Periodic for the N = 2 resonance condition. Here, the parameters A / ω 0 and two used parameters, Δ / δ = 401 [see Figs. Periodic(a)] and Δ / δ = 31 [see Figs. Periodic(b)], satisfy the periodicity condition Eq. ( periodrelation). We compare the results shown in Fig. Periodic(a) with the result depicted in Fig. pop3(c). Both results are obtained for the second-order resonance condition and for the same parameter A / ω 0 = 10 -1 ; however, using the parameter Δ / δ satisfying the condition of periodicity Eq. ( periodrelation) in Fig. Periodic(a) leads to the periodic dynamics of the populations. These regimes in which quantum dynamics of occupation probabilities becomes periodically regular can be useful, for example, in applications where one is dealing with logic operations on qubits.... which can be realized on a properly designed superconducting circuit. In particular, a simple design of the charge qubit with tunable effective Josephson coupling can be shown schematically (see Fig. circuit) as... circuit A charge qubit with tunable effective Josephson coupling. It is controlled by V g gate voltage and Φ x magnetic field.
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• A typical example of QT is shown in Fig.  fig1. It shows two main properties of the evolution: the oscillator spends a very long time at some average level n = n - and then jumps to another significantly different value n + . At the same time the polarization vector of qubit ξ → defined as ξ → = T r ρ ̂ σ → also changes its orientation direction with a clear change of sign of ξ x from ξ x > 0 to ξ x qubit polarization ξ = | ξ → | is very close to unity showing that the qubit remains mainly in a pure state. The drops of ξ appear only during transitions between metastable states. Special checks show that an inversion of ξ x by an additional pulse (e.g. from ξ x > 0 to ξ x oscillator to a corresponding state (from n - to n + ) after time t m ∼ 1 / λ . Thus we have here an interesting situation when a quantum flip of qubit produces a marcoscopic change of a state of detector (oscillator) which is continuously coupled to a qubit (we checked that even larger variation n ± ∼ n p is possible by taking n p = 40 ). In addition to that inside a metastable state the coupling induces a synchronization of qubit rotation phase with the oscillator phase which in its turn is fixed by the phase of driving field. The synchronization is a universal phenomenon for classical dissipative systems . It is known that it also exists for dissipative quantum systems at small effective values of ℏ . However, here we have a new unusual case of qubit synchronization when a semiclassical system produces synchronization of a pure quantum two-level system.... (color online) Top panels: the Poincaré section taken at integer values of ω t / 2 π for oscillator with x = â + â / 2 , p = â - â / 2 i (left) and for qubit polarization with polarization angles θ φ defined in text (right). Middle panels: the same quantities shown at irrational moments of ω t / 2 π . Bottom panels: the qubit polarization phase φ vs. oscillator phase ϕ ( p / x = - tan ϕ ) at time moments as in middle panels for g = 0.04 (left) and g = 0.004 (right). Other parameters and the time interval are as in Fig.  fig1. The color of points is blue/black for ξ x > 0 and red/gray for ξ x < 0 .... (color online) Bistability of qubit coupled to a driven oscillator with jumps between two metastable states. Top panel shows average oscillator level number n as a function of time t at stroboscopic integer values ω t / 2 π ; middle panel shows the qubit polarization vector components ξ x (blue/black) and ξ z (green/gray) at the same moments of time; the bottom panel shows the degree of qubit polarization ξ . Here the system parameters are λ / ω 0 = 0.02 , ω / ω 0 = 1.01 , Ω / ω 0 = 1.2 , f = ℏ λ n p , n p = 20 and g = 0.04 .... The phenomenon of qubit synchronization is illustrated in a more clear way in Fig.  fig2. The top panels taken at integer values ω t / 2 π show the existence of two fixed points in the phase space of oscillator (left) and qubit (right) coupled by quantum tunneling (the angles are determined as ξ x = ξ cos θ , ξ y = ξ sin θ sin φ , ξ z = ξ sin θ cos φ ). A certain scattering of points in a spot of finite size should be attributed to quantum fluctuations. But the fact that on enormously long time (Fig.  fig1) the spot size remains finite clearly implies that the oscillator phase ϕ is locked with the driving phase ω t inducing the qubit synchronization with ϕ and ω t . The plot at t values incommensurate with 2 π / ω (middle panels) shows that in time the oscillator performs circle rotations in p x plane with frequency ω while qubit polarization rotates around x -axis with the same frequency. Quantum tunneling gives transitions between two metastable states. The synchronization of qubit phase φ with oscillator phase ϕ is clearly seen in bottom left panel where points form two lines corresponding to two metastable states. This synchronization disappears below a certain critical coupling g c where the points become scattered over the whole plane (panel bottom right). It is clear that quantum fluctuations destroy synchronization for g < g c . Our data give g c ≃ 0.008 for parameters of Fig.  fig1.... (color online) Right panel: dependence of average qubit polarization components ξ x and ξ z (full and dashed curves) on g , averaging is done over stroboscopic times (see Fig. fig1) in the interval 100 ≤ ω t / 2 π ≤ 2 × 10 4 ; color is fixed by the sign of ξ x averaged over 10 periods (red/gray for ξ x 0 ; this choice fixes also the color on right panel). Left panel: dependence of average level of oscillator in two metastable states on coupling strength g , the color is fixed by the sign of ξ x on right panel that gives red/gray for large n + and blue/black for small n - ; average is done over the quantum state and stroboscopic times as in the left panel; dashed curves show theory dependence (see text)). Two QT are used with initial value ξ x = ± 1 . All parameters are as in Fig.  fig1 except g .... (color online) Dependence of number of transitions N f between metastable states on rescaled qubit frequency Ω / ω 0 for parameters of Fig.  fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice is as in Figs.  fig2, fig3).... (color online) Dependence of average level n ± of oscillator in two metastable states on the driving frequency ω (average and color choice are the same as in right panel of Fig.  fig3); coupling is g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig.  fig1.
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• 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).
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