### 54090 results for qubit oscillator frequency

Contributors: Everitt, M. J., Munro, W. J., Spiller, T. P.

Date: 2007-10-10

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....We explore the quantum-classical crossover in the behaviour of a quantum field mode. The quantum behaviour of a two-state system - a **qubit** - coupled to the field is used as a probe. Collapse and revival of the **qubit** inversion form the signature for quantum behaviour of the field and continuous Rabi **oscillations** form the signature for classical behaviour of the field. We demonstrate both limits in a single model for the full coupled system, for states with the same average field strength, and so for **qubits** with the same Rabi **frequency**....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). ... We explore the quantum-classical crossover in the behaviour of a quantum field mode. The quantum behaviour of a two-state system - a **qubit** - coupled to the field is used as a probe. Collapse and revival of the **qubit** inversion form the signature for quantum behaviour of the field and continuous Rabi **oscillations** form the signature for classical behaviour of the field. We demonstrate both limits in a single model for the full coupled system, for states with the same average field strength, and so for **qubits** with the same Rabi **frequency**.

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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: Ku, Li-Chung, Yu, Clare C.

Date: 2004-08-31

The **qubit**-TLS system starts in its ground state at t = 0 . A microwave π or 3 π pulse (from 0 to 5 ns) puts the **qubit**-TLS system in the **qubit** excited state | 1 that is a superposition of the two entangled states | ψ ' 1 ≡ | 0 e + | 1 g / 2 and | ψ ' 2 ≡ | 0 e - | 1 g / 2 . After the microwaves are turned off, the occupation probability starts **oscillating** coherently. Values of g T L S indicated in the figure are normalized by ℏ ω 10 . The rest of the parameters are the same as in Fig. 1. (a) No energy decay of the excited TLS, i.e., τ p h = ∞ . Coherent **oscillations** with various values of g T L S . (b) **Oscillations** following a π -pulse with τ p h = 40 ns and various values of g T L S . (c) **Oscillations** following a π -pulse and a 3 π -pulse with τ p h = 40 ns and g T L S = 0.004 . The dip in the dot-dash line is one and a half Rabi cycles....(a)-(b): Rabi **oscillations** in the presence of 1/f noise in the **qubit** energy level splitting. Dotted curves show the Rabi **oscillations** without the influence of noise. Panel (c) shows the two noise power spectra S f ≡ | δ ω 10 f / ω 10 | 2 of the fluctuations in ω 10 that were used to produce the solid curves in panels (a) and (b). Rabi **frequency** f R = 0.1 GHz....where the **qubit** energy levels ( | 0 and | 1 ) are the basis states and the noise is produced by a single TLS. Our calculations are oriented to the experimental conditions and the results are shown in Fig. fig:RTN. In Fig. fig:RTNa-c the characteristic fluctuation rate t T L S -1 = 0.6 GHz. Panel fig:RTNa shows that the **qubit** essentially stays coherent when the level fluctuations are small ( δ ω 10 / ω 10 = 0.001 ). Panel fig:RTNa shows that when the level fluctuations increase to 0.006, the Rabi **oscillations** decay within 100 ns. The Rabi relaxation time also depends on the Rabi **frequency** as panel fig:RTNc shows. The faster the Rabi **oscillations**, the longer they last. This is because the low-**frequency** noise is essentially constant over several rapid Rabi **oscillations** . Alternatively, one can explain it by the noise power spectrum S I f . Since the noise from a single TLS is a random process characterized by a single characteristic time scale t T L S , it has a Lorentzian power spectrum...Experimentally, the two TLS decoherence mechanisms (resonant interaction and low-**frequency** level fluctuations) can both be active at the same time. We have calculated the Rabi **oscillations** in the presence of both of these decoherence sources by using the **qubit**-TLS Hamiltonian in eq. ( eq:ham) with a fluctuating ω 10 t that is generated in the same way and with the same amplitude as in Figure fig:RTNb. We show the result in Fig. fig:RTN_decay. By comparing Fig. fig:RTN_decay with Fig. 2b, we note that adding level fluctuations reduces the Rabi amplitude and renormalizes the Rabi **frequency**. The result in Fig. fig:RTN_decay is closer to what is seen experimentally ....Solid line represents Rabi **oscillations** in the presence of both TLS decoherence mechanisms: resonant interaction between the TLS and the **qubit**, and low **frequency** **qubit** energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and the energy decay time for the TLS is τ p h = 10 ns, the same as in Fig. 2b. The size of the **qubit** level fluctuations is the same as in Fig. fig:RTNb. The dotted line shows the unperturbed Rabi **oscillations**....We do not expect Rabi **oscillations** to be sensitive to noise at **frequencies** much greater than the **frequency** of the Rabi **oscillations** because the higher the **frequency** f , the smaller the noise power and because the Rabi **oscillations** will tend to average over the noise. Rabi dynamics are sensitive to the noise at **frequencies** comparable to the Rabi **frequency**. In addition, the characteristic fluctuation rate plays an important role in the rate of relaxation of the Rabi **oscillations**. It has been shown that t T L S -1 can be thermally activated for TLS in a metal-insulator-metal tunnel junction. If the thermally activated behavior applies here, the decoherence time τ R a b i should decrease as temperature increases. In Fig. fig:RTNd, the characteristic fluctuation rate has been lowered to 0.06 GHz (which is much lower than ω 10 / 2 π ≈ 10 GHz). The noise still causes **qubit** decoherence but affects the **qubit** less than in Fig. fig:RTNc. Fig. fig:RTN shows that the noise primarily affects the Rabi amplitude rather than the phase....Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**....Rabi **oscillations** of a resonantly coupled **qubit**-TLS system with ε T L S = ℏ ω 10 . There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) P 1 is the occupation probability in the **qubit** state | 1 ; (b) P 0 g is the occupation probability in the state | 0 , g ; (c) P 0 e is the occupation probability of the state | 0 , e ; (d) P 1 g is the occupation probability of the state | 1 , g ; and (e) P 1 e is the occupation probability of the state | 1 , e . Notice the beating with **frequency** 2 η . Throughout the paper, ω 10 / 2 π = 10 GHz. Parameters are chosen mainly according to the experiment in Ref. : η / ℏ ω 10 = 0.0005 , g q b / ℏ ω 10 = 0.01 , and g T L S = 0 . The dotted line in panel (a) shows the usual Rabi **oscillations** without resonant interaction, i.e. η = 0 ....Solid lines show the Rabi **oscillation** decay due to **qubit** level fluctuations caused by a single fluctuating two level system trapped inside the insulating tunnel barrier. The TLS produces random telegraph noise in I o that modulates the **qubit** energy level splitting ω 10 . (a) The level fluctuation δ ω 10 / ω 10 = 0.001 . The characteristic fluctuation rate t T L S -1 = 0.6 GHz. The Rabi **frequency** f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (b) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (c) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.5 GHz. (d) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.06 GHz, and f R = 0.5 GHz. Note that the scales of the horizontal axes in (a)-(c) are the same. They are different from that in (d)....We first consider the case of strong driving with g T L S = 0 and with the TLS in resonance with the **qubit**, i.e. ε T L S = ℏ ω 10 . If there is no coupling between the **qubit** and the TLS, then the four states of the system are the ground state | 0 , g , the highest energy state | 1 , e , and two degenerate states in the middle | 1 , g and | 0 , e . If the **qubit** and the TLS are coupled with coupling strength η , the degeneracy is split by an energy 2 η . Figure fig:QB4L shows the coherent **oscillations** of the resonant **qubit**-TLS system. We define a projection operator P ̂ 1 ≡ | 1 , g 1 , g | + | 1 , e 1 , e | so that P ̂ 1 corresponds to the occupation probability of the **qubit** to be in state | 1 as in the phase-**qubit** experiment. Instead of being sinusoidal like typical Rabi **oscillations** (the dotted curve), the occupation probability P 1 exhibits beating (Fig. 1a) because the two entangled states that are linear combinations of | 1 , g and | 0 , e have a small energy splitting 2 η , and this small splitting is the beat **frequency**. Without any source of decoherence, the resonant beating will not decay. Thus far the beating phenomenon has not yet been experimentally verified. The lack of experiment ... Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**.

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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: 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: Johansson, G., Tornberg, L., Shumeiko, V. S., Wendin, G.

Date: 2006-02-24

Resonant circuits for read-out: a) A lumped element LC-**oscillator** coupled to a driving source and a radio-**frequency** detector through a transmission line. b) The radio-**frequency** single-electron transistor measuring the charge of a charge **qubit** (SCB). The current through the SET determines the dissipation in the resonant circuit. The dissipation is determined by measuring the amplitude of the reflected signal. c) Setup for measuring the quantum capacitance of the charge **qubit**. The **qubit** capacitance influences the resonance **frequency** of the **oscillator**. The capacitance is measured by determining the phase-shift of the reflected signal....up to a constant phase depending on the length of the transmission line. Here Q is the resonator’s quality factor, which for the circuitry in Fig. ResonantCircuitsFig a) is determined by the characteristic impedance on the transmission line Z 0 through Q = ω 0 L C 2 / C c 2 Z 0 . Since there is no dissipation in the **oscillator** we have | Γ ω | = 1 . Driving the **oscillator** at the bare resonance **frequency** ω d = ω 0 the phase-difference between the ground and excited state of the **qubit** will be...The quantum capacitance of the Cooper-pair box is related to the parametric capacitance of small Josephson junctions which is a dual to the Josephson inductance. The origin of the quantum capacitance of a single-Cooper-pair box (SCB) can be understood as follows. Assume that we put a constant voltage V m on the measurement capacitance of the SCB, i.e. we put a voltage source between the open circles in Fig. ResonantCircuitsFigc. The amount of charge on the measurement capacitance q m g / e V m V g will be a nonlinear function of the voltage V m as well as the gate voltage V g and whether the **qubit** is in the ground or excited state. We may define an effective (differential) capacitance...Single-contact flux **qubit** inductively coupled to a linear **oscillator**....We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time....At the charge degeneracy point the effective capacitance of the SCB in the ground and excited state differs by 2 C Q m a x . Imbedding the SCB in a resonant circuit as shown in Fig. ResonantCircuitsFig a) and c) we can detect the corresponding change in the **oscillators** resonance **frequency** ω 0 g / e = 1 / L C ± C Q m a x = ω 0 1 ∓ C Q m a x / 2 C , where ω 0 = 1 / L C is the bare resonance **frequency**. The voltage reflection amplitude Γ ω = V o u t ω / V d ω seen from the driving side of the transmission line can for a high quality **oscillator** be written...Circuit diagrams and 2-level energy spectrum of two basic JJ-**qubit** designs: the SCB charge **qubit** with LC-**oscillator** readout (left), and persistent-current flux **qubit** with SQUID **oscillator** readout (right). For the charge **qubit**, the control variable ϵ on the horizontal axis of the energy spectrum (middle) represents the external gate voltage (induced charge), and the splitting is given by the Josephson tunneling energy mixing the charge states. For the flux **qubit**, the variable ϵ represents the external magnetic flux. In both cases, the energy of the **qubit** can be "tuned" and the working point controlled. Away from the origin (asymptotically) the levels represent pure charge states (zero or one Cooper pair on the SCB island) or pure flux states (left or right rotating currents in the SQUID ring)....LagrangianSubsection The circuit for performing read-out through the quantum capacitance is presented in figure fig:circuit. A Josephson charge **qubit** is capacitatively coupled to a harmonic **oscillator**, which is coupled to a transmission line. Through this line, all measurement on the **qubit** is performed. We model the line as a semi-infinite line of LC-circuits in series. The working point of the Josephson junction can be chosen using the bias V g . In writing down the Lagrangian we are free to chose any quantities as our coordinates as long as they give a full description of our circuit. Since we are treating a system including a Josephson junction, the phases Φ i t = ∫ t d t ' V i t ' across the circuit elements are natural coordinates, as discussed by Devoret in ref. ...Double-well potential and energy levels of the flux **qubit** ( f q = π ). ... We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time.

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Contributors: Berg, J. W. G. van den, Nadj-Perge, S., Pribiag, V. S., Plissard, S. R., Bakkers, E. P. A. M., Frolov, S. M., Kouwenhoven, L. P.

Date: 2012-10-26

Utilizing the large difference in g -factors between the two dots we have achieved coherent control of both **qubits**. Here, we probe the **qubits** using similar microwave **frequencies**, but different magnetic fields. The insets in figure fig4 show the Rabi **oscillations** obtained for each of the **qubits**. The **frequency** of the Rabi **oscillations** (see insets of figure fig4) for the **qubit** corresponding to g -factor 48, was 96 MHz. For the other dot, with a corresponding g -factor of 36, a lower Rabi **frequency** of 47 MHz was achieved. This slower Rabi **oscillation** is consistent with a weaker coupling of the microwave electric field to this dot....Due to the strong spin-orbit interaction in indium antimonide, orbital motion and spin are no longer separated. This enables fast manipulation of **qubit** states by means of microwave electric fields. We report Rabi **oscillation** **frequencies** exceeding 100 MHz for spin-orbit **qubits** in InSb nanowires. Individual **qubits** can be selectively addressed due to intrinsic dierences in their g-factors. Based on Ramsey fringe measurements, we extract a coherence time T_2* = 8 +/- 1 ns at a driving **frequency** of 18.65 GHz. Applying a Hahn echo sequence extends this coherence time to 35 ns....fig4(color online) Main panel: Two well separated EDSR peaks for the spin-orbit **qubit** in each of the two dots. The microwave driving **frequency** is 20.9 GHz. Insets: Rabi **oscillations** for the corresponding EDSR peaks. Linear slopes (attributed to PAT) of 0.6 and 0.9 fA/ns respectively are subtracted to flatten the average. The Rabi data on the left was obtained at 31.2 mT B -field and 20.9 GHz driving **frequency**. From the fit (as in Fig. fig2) a Rabi **frequency** of 96 ± 2 MHz is obtained. On the right the field was 41.2 mT and driving **frequency** 21 GHz. The Rabi **frequency** obtained from the fit is 47 ± 3 MHz....fig1(color online) (a) Electron microscope image of the device consisting of an InSb nanowire contacted by source and drain electrodes, lying across a set of fine gates (numbered, 60 nm pitch) as well as a larger bottom gate (labeled BG). (b) Current through the device when applying microwaves with the double dot in spin blockade configuration (a vertical linecut near 0 mT has been subtracted to suppress resonances at constant **frequency**). When the microwave **frequency** matches the Larmor **frequency** (resonance highlighted by dashed blue box), blockade is lifted and current increases. (c) Schematic illustration of Pauli spin blockade on which read-out depends. Only anti-parallel states (right) can occupy the same dot, allowing current through the device. A parallel configuration (left) leads to a suppression of the current....To demonstrate coherent control over the **qubit** we apply microwave bursts of variable length. First, the **qubit** is initialized into a spin blocked charge configuration. This is accomplished by idling inside the bias triangle (Fig. fig2(a)). In order to prevent the electron from tunneling out of the dot during its subsequent manipulation, the double dot is maintained in Coulomb blockade in the same charge configuration. While in the Coulomb blockade regime, a microwave burst is applied. The double dot is then again quickly brought back to the spin blockade configuration by pulsing the plunger gates. By applying such microwave bursts (schematically depicted in figure fig2(b)), we perform a Rabi measurement. If the manipulation has flipped the electron spin-orbit state, the blockade is lifted and an electron can move from the first to the second dot and exit again through the outgoing lead. By continuously repeating the pulse sequence and measuring the (DC) current through the double dot, we measure the Rabi **oscillations** associated with the rotation of the spin-orbit state (Fig. fig2(c))....fig3(color online) (a) Ramsey experiment; an initial π / 2 -pulse rotates the spin to the xy-plane. After some delay a 3 π / 2 pulse is applied, restoring spin blockade or (partially) lifting it, depending on the phase of the pulse. (b) Decay of the Ramsey fringe contrast with increasing delay time τ for different driving **frequencies**. Solid line is a fit to exp - τ / T 2 * 2 at a driving **frequency** of 18.65 GHz, giving T 2 * = 8 ± 1 ns. (c) A Hahn echo sequence (top), extends the decay of the fringe contrast, to 30 ns in this case. (d) Decay of the fringe contrast in the Hahn echo sequence for different microwave **frequencies**. Solid line is a fit to exp - τ / T e c h o 3 for driving **frequency** 18.65 GHz, yielding T e c h o = 34 ± 2 ns....fig2(color online) (a) Bias triangle in which spin blockade was observed for a negative bias of -5 mV. This is the ( 2 m + 1 , 2 n + 1 ) → ( 2 m , 2 n + 2 ) transition (transition A, see ). (b) Sequence used for measuring Rabi **oscillations**. Pulses are applied to gates 2 and 4 to move the double dot along the detuning axis between Coulomb blockade (CB) and spin blockade (SB) configurations. In CB a microwave burst is applied via gate BG to rotate the spin. (c) Rabi **oscillation** obtained at a driving **frequency** of 18.65 GHz and source power of 11 (bottom) to 17 (top) dBm. Dashed lines are fits to a cos f R τ b u r s t + φ τ b u r s t - d + b , giving Rabi **frequencies** f R of 54 ± 1 ; 67 ± 1 ; 84 ± 1 and 104 ± 1 MHz. Linear slopes, attributed to photon assisted tunneling, of 0.5, 0.6, 0.4, and 0.6 fA/ns (top to bottom) were subtracted. d = 0.5 for the bottom trace and 0.4 for the others. Curves are offset by 0.5 pA for clarity. Inset: Rabi **frequencies** set out against driving amplitudes, including a linear fit through 0....When the delay time between the first and final pulse in the Ramsey sequence is increased, the **qubit** starts to dephase. The loss of phase coherence leads to decay of the Ramsey fringe contrast, as shown in figure fig3(b). By fitting the experimental data to exp - τ / T 2 * 2 we extract a dephasing time of T 2 * = 8 ± 1 ns, obtained at a driving **frequency** of 18.65 GHz. Other driving **frequencies** of 7.9 GHz and 31.91 GHz resulted in similar T 2 * values of 6 ± 1 and 9 ± 1 respectively. To extend the coherence of the **qubit**, we employ a Hahn echo technique : halfway between two π / 2 pulses an extra pulse is applied to flip the state over an angle π . Doing so partially refocuses the dephasing caused by the nuclear magnetic field, which varies slowly compared to the electron spin dynamics . From figure fig3(c), where the total delay has been extended to τ = 30 ns, it is clear that this technique can maintain contrast of the Ramsey fringes for considerably longer times. An increase in the coherence time to T e c h o = 35 ± 1 ns is obtained from the decay of the contrast (figure fig3(d)) for a driving **frequency** of 18.65 GHz. Similar values of 34 ± 2 and 32 ± 1 ns were obtained at driving **frequencies** 7.9 and 31.91 GHz respectively....The Rabi experiment demonstrates rotation of the **qubit** around a single axis. However, in order to be able to prepare the **qubit** in any arbitrary superposition, it is necessary to achieve rotations around two independent axes. We demonstrate such universal control by means of a Ramsey experiment, where the axis of **qubit** rotation is determined by varying the phase of the applied microwave bursts, as illustrated at the top of figure fig3(a). As in the Rabi experiment, the microwave bursts are applied while the dots are kept in Coulomb blockade, to maintain a well defined charge state and prevent the electrons from tunneling out during manipulation. In the Ramsey sequence an initial microwave burst rotates the state by π / 2 to the xy-plane of the Bloch sphere. We take this rotation axis to be the x-axis. A second burst is then applied after some delay τ , making a 3 π / 2 rotation. By varying the phase of this pulse with respect to the initial π / 2 pulse, we can control the axis of the second rotation (see figure fig3(a)). For example, if the two bursts are applied with the same phase, in total a 2 π rotation will have been made. This restores a spin blockade configuration, thus leading to a suppression of the current. A second burst with a phase π , however, would rotate the **qubit** in the opposite direction, ending up along the -direction on the Bloch sphere. For this case spin blockade is thus fully lifted and current increases to a maximum. ... Due to the strong spin-orbit interaction in indium antimonide, orbital motion and spin are no longer separated. This enables fast manipulation of **qubit** states by means of microwave electric fields. We report Rabi **oscillation** **frequencies** exceeding 100 MHz for spin-orbit **qubits** in InSb nanowires. Individual **qubits** can be selectively addressed due to intrinsic dierences in their g-factors. Based on Ramsey fringe measurements, we extract a coherence time T_2* = 8 +/- 1 ns at a driving **frequency** of 18.65 GHz. Applying a Hahn echo sequence extends this coherence time to 35 ns.

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Contributors: Chudzicki, Christopher, Strauch, Frederick W.

Date: 2010-08-10

Entanglement distribution rate R (in units of 1 / T ) as function of the number of nodes N in a quantum network. Three distribution schemes are shown: the massively parallel (MP) and **qubit**-compatible schemes on the hypercube of dimension d (each with N = 2 d ), and the complete graph of size N . Each network was chosen to to have a coupling of Ω 0 / 2 π = 20 MHz with a bandwidth ω max - ω min / 2 π = 2 GHz ....Parallel state transfer on programmable quantum networks. Each node is an **oscillator** with a tunable **frequency**. Each line (solid or dashed) indicates a coupling between **oscillators**. Solid lines indicate couplings between **oscillators** with the same **frequency**; dashed lines indicate couplings between **oscillators** with different **frequencies**. High fidelity state transfer occurs for large detuning. (a) Hypercube network with d = 3 , programmed into two subcubes (red and blue squares). Each node is labeled by a bit-string of length d = 3 , here with the first m = 1 bits indicating the subcube. In the **qubit**-compatible scheme (QC), one entangled pair is sent on each subcube, as indicated by the arrow for the inner (red) square. (b) In the massively parallel scheme (MP) scheme, multiple entangled pairs are sent between every node of each subcube, as indicated by the arrows for the inner (red) square. (b) Completely connected network with N = 8 , programmed into N / 2 = 4 two-site networks....We study the routing of quantum information in parallel on multi-dimensional networks of tunable **qubits** and **oscillators**. These theoretical models are inspired by recent experiments in superconducting circuits using Josephson junctions and resonators. We show that perfect parallel state transfer is possible for certain networks of harmonic **oscillator** modes. We further extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is both optimal and robust in the presence of dissipation and finite bandwidth....The fidelity of entanglement transfer on the hypercube as a function of the detuning parameter η = 2 Ω 0 / Δ ω . The **qubit** curves are, from top to bottom (for small η ), numerical simulations for dimension d = 2 6 . Each network is split into M = 2 subcube channels with one sender and receiver per channel, with entanglement being sent in the same direction on each channel. Also shown also is the lower bound of Eq. ( ES:fid) for the **oscillator** network with m = 1 ....ES:fid was derived for entanglement transfer on **oscillator** networks. However, as long as only one sender-receiver pair uses each channel at a time, numerical calculations, shown in Fig. fig2, indicate that **qubit** networks behave similarly. For this reason we call the parallel state transfer protocol discussed so far the “**qubit**-compatible” (QC) protocol. There are some notable differences between **qubits** and **oscillators**, namely **qubits** do better on average, but do not exhibit perfect state transfer....**Qubit**, entanglement, quantum computing, superconductivity, Josephson junction....These three distribution rates are plotted as a function of the number of nodes in Fig. fig:ESplot, where we have fixed the bandwidth appropriate to recent superconducting **qubit** experiments . For the hypercube schemes, the massively parallel protocol is more than quadratically better than the **qubit**-compatible scheme. Entanglement transfer on the complete graph quickly fails due to significant cross-talk for N ≈ 20 . One might expect that this is due to the large number of couplings, but it is actually due to the finite bandwidth of the network. It is clear that studying extended coupling schemes such as the cavity grid is an important task....Parallel state transfer on the hypercube. Christandl et al. showed that one can perform perfect state transfer from corner-to-corner of a d -dimensional hypercube in constant time T = π / 2 Ω 0 . Here we analyze how this result can be extended to the transfer of quantum states in parallel, by splitting the cube into subcubes. Specifically, by tuning the **frequencies** of each node, the d -dimensional hypercube can be broken up into 2 m subcubes each of dimension d - m , as shown in Fig. fig1(a) for d = 3 and m = 1 . These subcubes can be made to act as good channels between their antipodal nodes by separating the **oscillator** **frequencies** for each channel from adjacent channels by an amount Δ ω . For fixed couplings, there is still the potential for cross-talk between channels, which we now analyze. ... We study the routing of quantum information in parallel on multi-dimensional networks of tunable **qubits** and **oscillators**. These theoretical models are inspired by recent experiments in superconducting circuits using Josephson junctions and resonators. We show that perfect parallel state transfer is possible for certain networks of harmonic **oscillator** modes. We further extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is both optimal and robust in the presence of dissipation and finite bandwidth.

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Contributors: van Heck, B., Hyart, T., Beenakker, C. W. J.

Date: 2014-07-10

An alternative layout that has only Coulomb couplings needs three rather than two islands, forming a tri-junction as in Fig. fig_layoutb. A tri-junction pins a Majorana zero-mode , which can be Coulomb-coupled to each of the other three Majoranas . The tri-junction also binds higher-lying fermionic modes, separated from the zero mode by an excitation energy E M . This is the minimal design for a fully flux-controlled Majorana **qubit**. In Fig. fig_qubit we have worked it out in some more detail for the quantum spin-Hall insulator....The conservation of fermion parity on a single superconducting island implies a minimum of two islands for a Majorana **qubit**, each containing a pair of Majorana zero-modes. The minimal circuit that can operate on a Majorana **qubit** would then have the linear layout of Fig. fig_layouta. While the couplings between Majoranas on the same island are flux-controlled Coulomb couplings, the inter-island coupling is via a tunnel barrier, which would require microscopic control by a gate voltage....Topological **qubit** formed out of four Majorana zero-modes, on either two or three superconducting islands. Dashed lines indicate flux-controlled Coulomb couplings, as in the Cooper pair box of Fig. fig_transmon. In the linear layout (panel a) the coupling between Majoranas on different islands is via a tunnel barrier (thick horizontal line), requiring gate voltage control. By using a tri-junction (panel b) all three couplings can be flux-controlled Coulomb couplings....Energy spectrum of the top-transmon circuit of Fig. fig_qubit, obtained from numerical diagonalization of the Hamiltonian Htilde for E J = 300 GHz, E C = 5 GHz, Φ max = h / 4 e . The junction asymmetry was d = 0.1 , so that E J Φ m a x ≃ 30 GHz. In panel (a), the lowest eight energy levels for E M = 5 GHz are shown as a function of the induced charge q i n d 1 . They correspond to the eight eigenstates | σ , τ | f , where σ = ± 1 labels the excited/ground state of the charge **qubit**, τ = ± 1 labels the even/odd parity state of the topological **qubit**, and f = 0 , 1 the occupation number of the fermionic state in the constriction. As indicated by the colored arrows, the ground and excited state of the charge **qubit** are separated by an energy Ω 0 ± 2 Δ + ≃ 27.5 ± 1.7 GHz, depending on the state of the topological **qubit**. The inset shows the weak charge dispersion of the ground state doublet ( Δ m a x ≃ 120 MHz). In panel (b), the same energy levels are shown as a function of the tunnel coupling E M for a fixed value of q ind 1 = 0 . For a proper operation of the circuit it is required that the states f = 1 with an excited fermionic mode are well separated from both ground and excited states of the charge **qubit**. We have highlighted between grey panels a large energy window 3 G H z E M 8 G H z where this requirement is met....The basic building block of the transmon, shown in Fig. fig_transmon, is a Cooper pair box (a superconducting island with charging energy E C ≪ Josephson energy E J ) coupled to a microwave transmission line (coupling energy ℏ g ). The plasma **frequency** ℏ Ω 0 ≃ 8 E J E C is modulated by an amount Δ + cos π q i n d / e upon variation of the charge q i n d induced on the island by a gate voltage V . Additionally, there is a q i n d -dependent contribution Δ - cos π q i n d / e to the ground state energy. The charge sensitivity Δ ± ∝ exp - 8 E J / E C can be adjusted by varying the flux Φ enclosed by the Josephson junction, which modulates the Josephson energy E J ∝ cos 2 π e Φ / h . In a typical device , a variation of Φ between Φ m i n ≈ 0 and Φ m a x h / 4 e changes Δ ± by several orders of magnitude, so the charge sensitivity can effectively be switched on and off by increasing the flux by half a flux quantum....We construct a minimal circuit, based on the top-transmon design, to rotate a **qubit** formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana **qubit**. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi **oscillations** of the Majorana **qubit** can be monitored via **oscillations** in the resonance **frequency** of the microwave cavity that encloses the Cooper pair box....Schematic of a Cooper pair box in a transmission line resonator (transmon) containing a pair of Majorana zero-modes at the edge of a quantum spin-Hall insulator. This hybrid device (top-transmon) can couple charge **qubit** and topological **qubit** by variation of the flux Φ through a Josephson junction....Hya13 the initialization of the ancillas also requires that k B T ≪ Δ m a x , so the Coulomb coupling Δ m a x cannot be much smaller than 10 G H z . There is no such requirement for the simpler circuit of Fig. fig_qubit, because no ancillas are needed for the nontopological rotation of a Majorana **qubit**. This is one reason, in addition to the smaller number of Majoranas, that we propose this circuit for the first generation of experiments on Majorana **qubits**....Top-transmon circuit to rotate the **qubit** formed out of four Majorana zero-modes at the edge of a quantum spin-Hall insulator. One of the Majoranas ( γ B ) is shared by three superconductors at a constriction. The topological **qubit** is rotated by coupling it to a Cooper pair box in a transmission line resonator (transmon). The coupling strength is controlled by the magnetic flux Φ through a pair of split Josephson junctions. The diagrams at the top indicate how the Coulomb couplings of pairs of Majoranas are switched on and off: they are off (solid line) when Φ = 0 and on (dashed line) when Φ = Φ m a x h / 4 e . This single-**qubit** rotation does not have topological protection, it serves to characterize the coherence times of the Majorana **qubit**....Implementation of the braiding circuit of Ref. Hya13 in a quantum spin-Hall insulator. The two T-junctions are formed by a pair of constrictions. The flux-controlled braiding protocol requires four independently adjustable magnetic fluxes. The Majorana **qubit** formed out of zero-modes γ A , γ B , γ C , γ D is flipped at the end of the operation, as can be measured via a shift of the resonant microwave **frequency**. This braiding operation has topological protection. ... We construct a minimal circuit, based on the top-transmon design, to rotate a **qubit** formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana **qubit**. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi **oscillations** of the Majorana **qubit** can be monitored via **oscillations** in the resonance **frequency** of the microwave cavity that encloses the Cooper pair box.

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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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