### 54077 results for qubit oscillator frequency

Contributors: Bertet, P., Chiorescu, I., Semba, K., Harmans, C. J. P. M, Mooij, J. E.

Date: 2004-05-03

(a) Principle of the detection scheme. After the Rabi pulse, a microwave pulse at the plasma **frequency** resonantly enhances the escape rate. The bias current is maintained for 500 n s above the retrapping value. (b) Resonant activation peak for different Rabi angle. Each curve was offset by 5 % for lisibility. The Larmor **frequency** was f q = 8.5 ~ G H z . Pulse 2 duration was 10 ~ n s . (c) Resonant activation peak without (full circles) and after (open circles) a π pulse. The continuous line is the difference between the two switching probabilities. (d) Rabi **oscillation** measured by DC current pulse (grey line, amplitude A = 40 % ) and by resonant activation method with a 5 ~ n s RAP (black line, A = 62 % ), at the same Larmor **frequency**. fig4...The parameters of our **qubit** were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the SQUID and the **qubit** by fitting the **qubit** “step" appearing in the SQUID’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi **oscillation** experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the microwave **frequency** to the **qubit** resonance and measured the switching probability as a function of the microwave pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the **qubit**. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these **oscillations**....We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%)....(insert) Typical resonant activation peak (width 40 ~ M H z ), measured after a 50 ~ n s microwave pulse. Due to the SQUID non-linearity, it is much sharper at low than at high **frequencies**. (figure) Center **frequency** of the resonant activation peak as a function of the external magnetic flux (squares). It follows the switching current modulation (dashed line). The solid line is a fit yielding the values of the shunt capacitor and stray inductance given in the text. fig3...(a) Rabi **oscillations** at a Larmor **frequency** f q = 7.15 ~ G H z (b) Switching probability as a function of current pulse amplitude I without (closed circles, curve P s w 0 I ) and with (open circles, curve P s w π I b ) a π pulse applied. The solid black line P t h 0 I b is a numerical adjustment to P s w 0 I b assuming escape in the thermal regime. The dotted line (curve P t h 1 I b ) is calculated with the same parameters for a critical current 100 n A smaller, which would be the case if state 1 was occupied with probability unity. The grey solid line is the sum 0.32 P t h 1 I b + 0.68 P t h 0 I b . fig2...We then measure the effect of the **qubit** on the resonant activation peak. The principle of the experiment is sketched in figure fig4a. A first microwave pulse at the Larmor **frequency** induces a Rabi rotation by an angle θ 1 . A second microwave pulse of duration τ 2 = 10 n s is applied immediately after, at a **frequency** f 2 close to the plasma **frequency**, with a power high enough to observe resonant activation. In this experiment, we apply a constant bias current I b through the SQUID ( I b = 2.85 μ A , I b / I C = 0.85 ) and maintain it at this value 500 ~ n s after the microwave pulse to keep the SQUID in the running state for a while after switching occurs. This allows sufficient voltage to build up across the SQUID and makes detection easier, similarly to the plateau used at the end of the DCP in the previously shown method. At the end of the experimental sequence, the bias current is reduced to zero in order to retrap the SQUID in the zero-voltage state. We measured the switching probability as a function of f 2 for different Rabi angles θ 1 . The results are shown in figure fig4b. After the microwave pulse, the **qubit** is in a superposition of the states 0 and 1 with weights p 0 = c o s 2 θ 1 / 2 and p 1 = s i n 2 θ 1 / 2 . Correspondingly, the resonant activation signal is a sum of two peaks centered at f p 0 and f p 1 with weights p 0 and p 1 , which reveal the Rabi **oscillations**....We show the two peaks corresponding to θ 1 = 0 (curve P s w 0 , full circles) and θ 1 = π (curve P s w π , open circles) in figure fig4c. They are separated by f p 0 - f p 1 = 50 ~ M H z and have a similar width of 90 ~ M H z . This is an indication that the π pulse efficiently populates the excited state (any significant probability for the **qubit** to be in 0 would result into broadening of the curve P s w π ), and is in strong contrast with the results obtained with the DCP method (figure fig2b). The difference between the two curves S f = P s w 0 - P s w π (solid line in figure fig4c) gives a lower bound of the excited state population after a π pulse. Because of the above mentioned asymmetric shape of the resonant activation peaks, it yields larger absolute values on the low- than on the high-**frequency** side of the peak. Thus the plasma **oscillator** non-linearity increases the sensitivity of our measurement, which is reminiscent of the ideas exposed in . On the data shown here, S f attains a maximum S m a x = 60 % for a **frequency** f 2 * indicated by an arrow in figure fig4c. The value of S m a x strongly depends on the microwave pulse duration and power. The optimal settings are the result of a compromise between two constraints : a long microwave pulse provides a better resonant activation peak separation, but on the other hand the pulse should be much shorter than the **qubit** damping time T 1 , to prevent loss of excited state population. Under optimized conditions, we were able to reach S m a x = 68 % ....A typical resonant activation peak is shown in the insert of figure fig3. Its width depends on the **frequency**, ranging between 20 and 50 ~ M H z . This corresponds to a quality factor between 50 and 150 . The peak has an asymmetric shape, with a very sharp slope on its low-**frequency** side and a smooth high-**frequency** tail, due to the SQUID non-linearity. We could qualitatively recover these features by simple numerical simulations using the RCSJ model . The resonant activation peak can be unambiguously distinguished from environmental resonances by its dependence on the magnetic flux threading the SQUID loop Φ s q . Figure fig3 shows the measured peak **frequency** for different fluxes around Φ s q = 1.5 Φ 0 , together with the measured switching current (dashed line). The solid line is a numerical fit to the data using the above formulae. From this fit we deduce the following values C s h = 12 ± 2 p F and L = 170 ± 20 p H , close to the design. We are thus confident that the observed resonance is due to the plasma **frequency**....Finally, we fixed the **frequency** f 2 at the value f 2 * and measured Rabi **oscillations** (black curve in figure fig4d). We compared this curve to the one obtained with the DCP method in exactly the same conditions (grey curve). The contrast is significantly improved, while the dephasing time is evidently the same. This enhancement is partly explained by the rapid 5 ~ n s RAP (for the data shown in figure fig4d) compared to the 30 ~ n s DCP. But we can not exclude that the DCP intrinsically increases the relaxation rate during its risetime. Such a process would be in agreement with the fact that for these bias conditions, T 1 ≃ 100 ~ n s , three times longer than the DCP duration....(a) AFM picture of the SQUID and **qubit** loop (the scale bar indicates 1 ~ μ m ). Two layers of Aluminium were evaporated under ± ~ 20 ~ ∘ with an oxidation step in between. The Josephson junctions are formed at the overlap areas between the two images. The SQUID is shunted by a capacitor C s h = 12 ~ p F connected by Aluminium leads of inductance L = 170 ~ p H (solid black line). The current is injected through a resistor (grey line) of 400 ~ Ω . (b) DCP measurement method : the microwave pulse induces the designed Bloch sphere rotations. It is followed by a current pulse of duration 20 ~ n s , whose amplitude I b is optimized for the best detection efficiency. A 400 ~ n s lower-current plateau follows the DCP and keeps the SQUID in the running-state to facilitate the voltage pulse detection. (c) Larmor **frequency** of the **qubit** and (insert) persistent-current versus external flux. The squares and (insert) the circles are experimental data. The solid lines are numerical adjustments giving the tunnelling matrix element Δ , the persistent-current I p and the mutual inductance M . fig1 ... We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%).

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Contributors: Hausinger, Johannes, Grifoni, Milena

Date: 2010-07-30

Population difference for zero static bias. Further parameters are Δ / Ω = 0.5 , ℏ β Ω = 10 and g / Ω = 1.0 . The adiabatic approximation and VVP are compared to numerical results. The first one only covers the longscale dynamics, while VVP also returns the fast **oscillations**. With increasing time small differences between numerical results and VVP become more pronounced. Fig::P_e=0_D=0.5_g=1.0...As a first case, we consider in Fig. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0 a weakly biased **qubit** ( ε / Ω = 0.5 ) being at resonance with the **oscillator** ( Δ b = Ω ). For a coupling strength of g / Ω = 1.0 , we find a good agreement between the numerics and VVP. The adiabatic approximation, however, conveys a slightly different picture: Looking at the time evolution it reveals collapse and rebirth of **oscillations** after a certain interval. This feature does not survive for the exact dynamics. Like in the unbiased case, the adiabatic approximation gives only the first group of **frequencies** between the quasidegenerate subspaces, and thus yields a wrong picture of the dynamics. In order to cover the higher **frequency** groups, we need again to go to higher-order corrections by using VVP. For the derivation of our results we assumed that ε is a multiple of the **oscillator** **frequency** Ω , ε = l Ω . In this case we found that the levels E ↓ , j 0 and E ↑ , j + l 0 form a degenerate doublet, which dominates the long-scale dynamics through the dressed **oscillations** **frequency** Ω j l . For l being not an integer those doublets cannot be identified unambiguously anymore. For instance, we examine the case ε / Ω = 1.5 in Fig. Fig::PF_e=1.5_D=0.5_g=1.0. Here, it is not clear which levels should be gathered into one subspace: j and j + 1 or j and j + 2 . Both the dressed **oscillation** **frequencies** Ω j 1 and Ω j 2 influence the longtime dynamics....Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole **frequency** range. The lowest **frequency** peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed **oscillation** **frequency** Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify **frequencies** Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the **frequency** Ω 3 0 . This first gr...We examine a two-level system coupled to a quantum **oscillator**, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\Omega$ between coupling strength and **oscillator** **frequency** approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail, because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the **qubit** tunneling matrix element $\Delta$ we are able to enlarge the regime of applicability of existing analytical treatments, including in particular also the finite bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the **qubit** for an **oscillator** at low temperature. We consider the coupling strength $g$ to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many **frequencies** are contributing to the dynamics. They are gathered into groups whose spacing depends on the **qubit**-**oscillator** detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with $g$, but by a non-trivial dressing of the tunneling matrix element, which can be used to suppress specific **frequencies** through a variation of the coupling....Population difference and Fourier spectrum for a biased **qubit** ( ε / Ω = 0.5 ) at resonance with the **oscillator** Δ b = Ω in the ultrastrong coupling regime ( g / Ω = 1.0 ). Concerning the time evolution VVP agrees well with numerical results. Only for long time weak dephasing occurs. The inset in the left-hand figure shows the adiabatic approximation only. It exhibits death and revival of **oscillations** which are not confirmed by the numerics. For the Fourier spectrum, VVP covers the various **frequency** peaks, which are gathered into groups like for the unbiased case. The adiabatic approximation only returns the first group. Fig::PF_e=Sqrt0.5_D=Sqrt0.5_g=1.0...respectively. Concerning the population difference, we see a relatively good agreement between the numerical calculation and VVP for short timescales. In particular, VVP also correctly returns the small overlaid **oscillations**. For longer timescales, the two curves get out of phase. The adiabatic approximation only can reproduce the coarse-grained dynamics. The fast **oscillations** are completely missed. To understand this better, we turn our attention to the Fourier transform in Fig. Fig::F_e=0_D=0.5_g=1.0. There, we find several groups of **frequencies** located around ν / Ω = 0 , ν / Ω = 1.0 , ν / Ω = 2.0 and ν / Ω = 3.0 . This can be explained by considering the transition **frequencies** in more detail. We have from Eq. ( VVEnergies)...with ζ k , j l = 1 8 ε ↓ , k 2 - ε ↓ , j 2 + ε ↑ , j + l 2 - ε ↑ , k + l 2 being the second-order corrections. For zero bias, ε = 0 , the index l vanishes. The term k - j Ω determines to which group of peaks a **frequency** belongs and Ω j 0 its relative position within this group. The latter has Δ as an upper bound, so that the range over which the peaks are spread within a group increases with Δ . The dynamics is dominated by the peaks belonging to transitions between the same subspace k - j = 0 , while the next group with k - j = 1 yields already faster **oscillations**. To each group belong theoretically infinite many peaks. However, under the low temperature assumption only those with a small **oscillator** number play a role. For the used parameter regime, the adiabatic approximation does not take into account the connections between different manifolds. It therefore covers only the first group of peaks with k - j = 0 , providing the long-scale dynamics. For ε = 0 , the dominating **frequencies** in this first group are given by Ω 0 0 = | Δ e - α / 2 | , Ω 1 0 = | Δ 1 - α e - α / 2 | and Ω 2 0 = | Δ L 2 0 α e - α / 2 | , where Ω 0 0 and Ω 2 0 coincide. A small peak at Ω 3 0 = | Δ L 3 0 α e - α / 2 | can also be seen. Notice that for certain coupling strengths some peaks vanish; like, for example, choosing a coupling strength of g / Ω = 0.5 makes the peak at Ω 1 0 vanish completely, independently of Δ , and the Ω 0 0 and Ω 2 0 peaks split. The JCM yields two **oscillation** peaks determined by the Rabi splitting and fails completely to give the correct dynamics, see the left-hand graph in Fig. Fig::F_e=0_D=0.5_g=1.0. Now, we proceed to an even stronger coupling, g / Ω = 2.0 , where we also expect the adiabatic approximation to work better. From Fig. Fig::EnergyVSg_e=0_W=1_D=0.5 we noticed that at such a coupling strength the lowest energy levels are degenerate within a subspace. Only for **oscillator** numbers like j = 3 , we see that a small splitting arises. This splitting becomes larger for higher levels. Thus, only this and higher manifolds can give significant contributions to the long time dynamics; that is, they can yield low **frequency** peaks. Also the adiabatic approximation is expected to work better for such strong couplings . And indeed by looking at Figs. Fig::P_e=0_D=0.5_g=2.0 and Fig::F_e=0_D=0.5_g=2.0, we notice that both the adiabatic approximation and VVP agree quite well with the numerics. Especially the first group of Fourier peaks in Fig. Fig::F_e=0_D=0.5_g=2.0 is also covered almost correctly by the adiabatic approximation. The first manifolds we can identify with those peaks are the ones with j = 3 and j = 4 . This is a clear indication that even at low temperatures higher **oscillator** quanta are involved due to the large coupling strength. Also **frequencies** coming from transitions between the energy levels from neighboring manifolds are shown enlarged in Fig. Fig::F_e=0_D=0.5_g=2.0. The adiabatic approximation and VVP can cover the main structure of the peaks involved there, while the former shows stronger deviations. If we go to higher values Δ / Ω 1 , the peaks in the individual groups become more spread out in **frequency** space, and for the population difference dephasing already occurs at a shorter timescale. For Δ / Ω = 1 , at least VVP yields still acceptable results in Fourier space but gets fast out of phase for the population difference....Fourier spectrum of the population difference in Fig. Fig::P_e=0_D=0.5_g=2.0. In the left-hand graph a large **frequency** range is covered. Peaks are located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 etc. Even the adiabatic approximation exhibits the higher **frequencies**. The upper right-hand graph shows the first group close to ν / Ω = 0 . The two main peaks come from Ω 3 0 and Ω 4 0 and higher degenerate manifolds. **Frequencies** from lower manifolds contribute to the peak at zero. The adiabatic approximation and VVP agree well with the numerics. The lower right-hand graph shows the second group of peaks around ν / Ω = 1.0 . This group is also predicted by the adiabatic approximation and VVP, but they do not fully return the detailed structure of the numerics. Interestingly, there is no peak exactly at ν / Ω = 1.0 indicating no nearest-neighbor transition between the low degenerate levels. Fig::F_e=0_D=0.5_g=2.0...Population difference and Fourier spectrum for ε / Ω = 1.5 , Δ / Ω = 0.5 and g / Ω = 1.0 . Van Vleck perturbation theory is confirmed by numerical calculations, while results obtained from the adiabatic approximation deviate strongly. In Fourier space, we find pairs of **frequency** peaks coming from the two dressed **oscillation** **frequencies** Ω j 1 and Ω j 2 . The spacings in between those pairs is about 0.5 Ω . The adiabatic approximation only returns one of those dressed **frequencies** in the first pair. Fig::PF_e=1.5_D=0.5_g=1.0...Fourier transform of the population difference in Fig. Fig::P_e=0_D=0.5_g=1.0. The left-hand graph shows the whole **frequency** range. The lowest **frequency** peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed **oscillation** **frequency** Ω j 0 . Numerical calculations and VVP predict group of peaks located around ν / Ω = 0 , 1.0 , 2.0 , 3.0 . The first group at ν / Ω = 0 is shown in the middle graph. One can identify **frequencies** Ω 0 0 and Ω 2 0 , which fall together, and Ω 1 0 . The small peak comes from the **frequency** Ω 3 0 . This first group of peaks is also covered by the adiabatic approximation. The other groups come from transitions between different manifolds. The adiabatic approximation does not take them into account, while VVP does. A blow-up of the peaks coming from transitions between neighboring manifolds is given in the right-hand graph. In the left-hand graph additionally the Jaynes-Cummings peaks are shown, which, however, fail completely. Fig::F_e=0_D=0.5_g=1.0 ... We examine a two-level system coupled to a quantum **oscillator**, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\Omega$ between coupling strength and **oscillator** **frequency** approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail, because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the **qubit** tunneling matrix element $\Delta$ we are able to enlarge the regime of applicability of existing analytical treatments, including in particular also the finite bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the **qubit** for an **oscillator** at low temperature. We consider the coupling strength $g$ to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many **frequencies** are contributing to the dynamics. They are gathered into groups whose spacing depends on the **qubit**-**oscillator** detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with $g$, but by a non-trivial dressing of the tunneling matrix element, which can be used to suppress specific **frequencies** through a variation of the coupling.

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Contributors: Kim, Mun Dae

Date: 2008-09-02

(Color online) Rabi-type **oscillations** of occupation probabilities of | ρ ρ ' states for strongly coupled **qubits** with the initial state ψ 0 = | 00 + | 10 / 2 . Here the parameters are J / h = 5 GHz, ω 0 / 2 π =4GHz, and Ω 0 / 2 π = 600 MHz at the degeneracy point where E = E in Fig. weak(a)....We study the coupled-**qubit** **oscillation** driven by an **oscillating** field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi **oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate **oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of **oscillating** field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and **qubit** energy gap in experiments....In Fig. PC we show the Rabi-type **oscillation** for strongly coupled **qubits**. While the P 00 ( P 01 ) is reversed from 0.5 (0) to 0 (0.5) at Ω t = (odd) π , we can observe that the probabilities P 10 and P 11 remain their initial values 0.5 and 0, respectively. In this case the parameters need not satisfy the commensurate condition of Eq. ( condition) for the CNOT gate operation....The scheme for CNOT gate operation in this study uses the non-Rabi **oscillations** for | 10 and | 11 states which are commensurate with the Rabi **oscillation** for | 00 and | 01 states. In Fig. TwoRabi we display the numerical results obtained from the Hamiltonian in Eqs. ( tilH0) and ( tilH1), which show such commensurate mode **oscillations**. The initial state, | ψ 0 = | 00 + | 10 / 2 , is driven by an **oscillating** field with the resonant **frequency** ω = ω 0 < ω 1 ....The commensurate **oscillations** of resonant and non-resonant modes enable the high fidelity CNOT gate operation by finely tuning the **oscillating** field amplitude for any given values of **qubit** energy gap and coupling strength between **qubits**. While for a sufficiently strong coupling the CNOT gate can be achieved for any given parameter values, for a weak coupling a relation between the parameters should be satisfied for the fidelity maxima. For a sufficiently weak coupling compared to the **qubit** energy gap, J / ℏ ω 0 ≪ 1 , we have α 1 ≈ 1 and β 1 ≈ 0 , resulting in the expression for g in Eq. ( ga). For J / ℏ ω 0 ≪ 1 , Eq. ( ga) immediately gives rise to the relation g / J ≪ 1 and thus g / ℏ ω 0 ≪ 1 after some manipulation. This means that for a weak coupling J / ℏ ω 0 ≪ 1 the numerical results are well fit with the RWA as shown in Table table, because the RWA is good for g / ℏ ω 0 ≪ 1 . As a result, the high performance CNOT gate operation can be achieved as shown in Fig. dF....Let us consider a concrete example for comprehensive understanding. For superconducting flux **qubits**, g = m B is the coupling between the amplitude B of the magnetic microwave field and the magnetic moment m , induced by the circulating current, of the **qubit** loop. In order to adjust the value of g , actually we need to vary the microwave amplitude B , because the **qubit** magnetic moment is fixed at a specified degeneracy point. The Rabi-type **oscillation** occurs between the transformed states | 0 = | + | / 2 and | 1 = | - | / 2 . The states of **qubits** can be detected by shifting the magnetic pulse adiabatically . Since these **qubit** states are the superposition of the clockwise and counterclockwise current states, | and | , the averaged current of **qubit** states vanishes at the degeneracy point in Fig. weak(a). Thus, one can apply a finite dc magnetic pulse to shift the **qubits** slightly away from the degeneracy point to detect the **qubit** current states....(Color online) (a) Energy levels E ρ ρ ' of coupled **qubits**, where ρ , ρ ' ∈ 0 1 . E s s ' with s , s ' ∈ are shown as thin dotted lines. The distance between two degeneracy points corresponds to the coupling strength between two **qubits**. (b) Occupation probabilities of | ρ ρ ' states during Rabi-type **oscillations** at the lower degeneracy point where E = E . Here we use the parameter values such that coupling strength J / h = 0.6GHz, **qubit** energy gap ω 0 / 2 π =4GHz, and Rabi **frequency** Ω 0 / 2 π = 600 MHz. The initial state is chosen as ψ 0 = | 00 + | 10 / 2 and the CNOT gate is expected to be achieved at Ω t = (odd) π ....The values of g / h for the main fidelity maxima ( n = 1 ) obtained from numerical calculation and from the RWA of Eq. ( g) for various coupling J and **qubit** energy gap ω 0 . For small ω 0 and large J the **oscillations** are far from the Rabi **oscillation**. Here, the unit of all numbers is GHz....(Color online) (a) Commensurate **oscillations** of occupation probability of coupled-**qubit** states with the initial state, | ψ 0 = | 00 + | 10 / 2 for g / h = 0.265 GHz. The non-resonant **oscillation** modes ( P 10 and P 11 ) are commensurate with the resonant modes ( P 00 and P 01 ). At Ω t = (odd) π , P 10 and P 11 recover their initial values, thus the CNOT gate operation is achieved. Here Ω 0 = g / ℏ , J / h = 0.5 GHz, and ω 0 / 2 π = 4.0GHz. (b) Higher order commensurate modes for smaller g / h = 0.122 GHz with the same J and ω 0 ....Figure weak(a) shows the energy levels E s s ' as a function of κ b , where we choose κ a such that | E s s ' - E - s s ' | ≫ t q a and thus t q a can be negligible. In the figure there are two degeneracy points; lower degeneracy point where E = E and upper degeneracy point where E = E . By adjusting the variable κ b , the coupled-**qubit** states can be brought to one of these degeneracy points. Here the distance between these degeneracy points is related to the coupling strength between two **qubits**. ... We study the coupled-**qubit** **oscillation** driven by an **oscillating** field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi **oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate **oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of **oscillating** field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and **qubit** energy gap in experiments.

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### Temperature square dependence of the low **frequency** 1/f charge noise in the Josephson junction **qubits**

Contributors: Astafiev, O., Pashkin, Yu. A., Nakamura, Y., Yamamoto, T., Tsai, J. S.

Date: 2006-04-04

fig:1fFig2 (a) Solid dots show temperature dependence of α 1 / 2 with a fixed bias current (the bias voltage is adjusted to keep the current constant). Open dots show α 1 / 2 derived from the measurement of **qubit** dephasing during coherent **oscillations**. The coherent **oscillations** (solid line) as well as the envelope exp - t 2 / 2 T 2 * 2 with T 2 * = 180 ps (dashed line) are in the inset. (b) Solid dots show temperature dependence of α 1 / 2 for the SET on GaAs substrate....Note that at a fixed bias voltage the average current through the SET increases with temperature (see Fig. fig:1fFig1(a)). However, it has almost no effect on the noise as we confirmed from the measurement of the current noise dependence. Nevertheless, to avoid possible contribution from the current dependent noise we adjust the bias voltage in the next measurements so that the average current is kept nearly constant at the measurement points for different temperatures. Fig. fig:1fFig2(a) shows the temperature dependences of α 1 / 2 for a different sample with a similar geometry taken in the **frequency** range from 0.1 Hz to 10 Hz with a bias current adjusted to about I = 12 ± 2 pA. The straight line in the plot is α 1 / 2 = η 1 / 2 T , which corresponds to T 2 -dependence of α with η ≈ ( 1.3 × 10 -2 e / K ) 2 ....Solid dots in Fig. fig:1fFig1(c) represent α 1 / 2 as a function of temperature. α 1 / 2 saturates at temperatures below 200 mK at the level of 2 × 10 -3 e and exhibits nearly linear rise at temperatures above 200 mK with α 1 / 2 ≈ η 1 / 2 T , where η ≈ 1.0 × 10 -2 e / K 2 (the solid line in Fig. fig:1fFig1(c)). T 2 dependence of α is observed in many samples, though sometimes the noise is not exactly 1 / f , having a bump from the Lorentzian spectrum of a strongly coupled low **frequency** fluctuator. In such cases, switches from the single two-level fluctuator are seen in time traces of the current ....To verify the hypothesis about the common origin of the low **frequency** 1/f noise and the quantum f noise recently measured in the Josephson charge **qubits**, we study temperature dependence of the 1/f noise and decay of coherent **oscillations**. T^2 dependence of the 1/f noise is experimentally demonstrated, which supports the hypothesis. We also show that dephasing in the Josephson charge **qubits** off the electrostatic energy degeneracy point is consistently explained by the same low **frequency** 1/f noise that is observed in the transport measurements....The typical current **oscillation** as a function of t away from the degeneracy point ( θ ≠ π / 2 ) is exemplified in the inset of Fig. fig:1fFig2(a). If dephasing is induced by the Gaussian noise, the **oscillations** decay as exp - t 2 / 2 T 2 * 2 with...We use the **qubit** as an SET and measure the low **frequency** charge noise, which causes the SET peak position fluctuations. Temperature dependence of the noise is measured from the base temperature of 50 mK up to 900 - 1000 mK. The SET is normally biased to V b = 4 Δ / e ( ∼ 1 mV), where Coulomb **oscillations** of the quasiparticle current are observed. Figure fig:1fFig1(a) exemplifies the position of the SET Coulomb peak as a function of the gate voltage at temperatures from 50 mK up to 900 mK with an increment of 50 mK. The current noise spectral density is measured at the gate voltage corresponding to the slope of the SET peak (shown by the arrow), at the maximum (on the top of the peak) and at the minimum (in the Coulomb blockade). Normally, the noise spectra in the two latter cases are **frequency** independent in the measured **frequency** range (and usually do not exceed the noise of the measurement setup). However, the noise spectra taken on the slope of the peak show nearly 1 / f **frequency** dependence (see examples of the current noise S I at different temperatures in Fig. fig:1fFig1(b)) saturating at a higher **frequencies** (usually above 10 - 100 Hz depending on the device properties) at the level of the noise of the measurement circuit. The fact that the measured 1 / f noise on the slope is substantially higher than the noises on the top of the peak and in the blockade regime indicates that the noise comes from fluctuations of the peak position, which can be translated into charge fluctuations in the SET....The solid line in the inset of Fig. fig:1fFig2(a) shows decay of coherent **oscillations** measured at T = 50 mK and the dashed envelope exemplifies a Gaussian with T 2 * = 180 ps. We derive α 1 / 2 from Eq. ( eq:Eq3) and plot it in Fig. fig:1fFig2(a) by open dots as a function of temperature. The low **frequency** integration limit and the high **frequency** cutoff are taken to be ω 0 ≈ 2 π × 25 Hz and ω 1 ≈ 2 π × 5 GHz for our measurement time constant τ = 0.02 s and typical dephasing time T 2 * ≈ 100 ps . ... To verify the hypothesis about the common origin of the low **frequency** 1/f noise and the quantum f noise recently measured in the Josephson charge **qubits**, we study temperature dependence of the 1/f noise and decay of coherent **oscillations**. T^2 dependence of the 1/f noise is experimentally demonstrated, which supports the hypothesis. We also show that dephasing in the Josephson charge **qubits** off the electrostatic energy degeneracy point is consistently explained by the same low **frequency** 1/f noise that is observed in the transport measurements.

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Contributors: Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Majer, J., Girvin, S. M., Schoelkopf, R. J.

Date: 2005-02-27

(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 a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes....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 . ... In a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes.

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Contributors: Hoffman, Anthony J., Srinivasan, Srikanth J., Gambetta, Jay M., Houck, Andrew A.

Date: 2011-08-12

Along this contour of constant **qubit** **frequency**, the **qubit**-cavity coupling strength, g 10 - 00 , changes due to the quantum interference of the two transmon-like halves of the TCQ. In Fig. figure2b, we measure the **frequency** response of the **qubit** while moving along the parameterized contour and can clearly see that the dressed **qubit** **frequency** remains 7.500 G H z . Moreover, in this constant power measurement, the amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. When the coupling is small, little response is seen because the **qubit** cannot be driven. The disappearance of a signal corresponds to the situation where the **qubit**-cavity coupling is tuned through zero....figure1 Energy level diagram for the TCQ showing the hybridized energy levels. The transitions that have a high probability of occurring are indicated by arrows. Considering that the system is primarily in the | 00 or | 10 ˜ states, single-photon transitions leading out of these two states have the maximum transition probabilities and they are indicated by arrows. The red, solid arrows indicate transitions with low coupling strengths and the blue, dashed arrows indicate transitions with high coupling strengths. The levels shown here are for the bare energy levels of the device; there are no effects of coupling to a cavity. In this work, the | 00 and | 10 ˜ states are used as the logical states of the **qubit**....In this work, we are mainly concerned with changing only the coupling strength of the **qubit** to the cavity while keeping the **qubit** **frequency** fixed. Since the flux controls allow for a wide range of coupling strengths and dressed **qubit** transition **frequencies**, it is necessary to find the control subspace that corresponds to constant dressed **qubit** **frequency**. This subspace accounts for any dispersive shifts due to changes in **qubit**-cavity coupling. To accomplish this, we use standard dispersive readout techniques of cQED: monitoring the amplitude and phase of cavity transmission while applying a second spectroscopy tone. Here, though, we keep the spectroscopy tone at a constant **frequency** of 7.500 G H z while sweeping the two control fluxes. When the dressed **qubit** **frequency**, which is a function of the two control fluxes, is resonant with the 7.500 G H z spectroscopy tone, a change in the cavity transmission is measured . Over a wide range of control voltages, it is then possible to extract a contour that corresponds to where the dressed **qubit** **frequency** is 7.500 G H z ; such a contour is shown in Fig. figure2a....figure2 (a) Observed cavity transmission versus the two control voltages with a fixed spectroscopy tone at 7.5 G H z . Both the dressed **qubit** **frequency** and coupling strength are functions of the control voltages. The contour shows where the **qubit** is resonant with the 7.5 G H z tone and is therefore driven between the ground and excited states. (b) Measured dressed **frequency** response of the **qubit** while moving along the 7.5 G H z contour in Fig. 1. The dressed **qubit** **frequency** remains constant at 7.5 G H z . The amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. The point where the signal disappears corresponds to coupling strengths where the **qubit** cannot be driven by the spectroscopy tone. The dotted and dashed lines indicate g 10 - 00 control values where the measurements were performed for Fig. 3....figure3 Rabi **oscillations** for three different **qubit**-cavity coupling strengths and a fixed dressed **qubit** **frequency** of 7.5 G H z . Panels (a), (b), and (c) correspond to the dashed, dot-dash, and dotted lines in Fig. 2, respectively. In (a), a spectroscopy power of -32 dBm is used. To keep the number of **oscillations** approximately the same for the lower **qubit**-cavity coupling strength in (b), the spectroscopy power is increased to -22 dBm. In panel (c), 27 dBm more power than that in (a) is applied and no **oscillations** are observed. Given the measurement noise, we put a bound of 1 / 10 of a Rabi **oscillation**....figure4 (a) Observed Rabi **oscillations** when the **qubit** starts in the g 10 - 00 = 0 state and is simulataneously moved to a large g 10 - 00 state and driven by a 7.5 G H z spectroscopy pulse of varying amplitude. The fast flux pulse is 60 ns in duration and is followed by an identical pulse of the opposite sign so that the total pulse integral is zero; these zero integral pulses help reduce slow transients. (b) Pulsed measurements showing the probability of the **qubit** being in the excited state as a function of delay following a π -pulse. The **qubit** starts in the g 10 - 00 = 0 state and is excited with a π -pulse in the manner described in (a); a pulsing scheme is included as an inset to the figure. The measured T 1 is 1.6 μ s. (c) Hahn echo measurements with the **qubit** starting in the g 10 - 00 = 0 state. Each of the pulses in the Hahn sequence is synchronized with a pair of fast flux pulses. A pulsing scheme is included as an inset to the figure. The measured T 2 time is 1.9 μ s....Using these fast flux bias pulses, we first measure T 1 by applying a π -pulse that is synchronized with the fast flux pulse, and measure the **qubit** excitation probability after a delay. We measure T 2 using a Hahn echo experiment. The **qubit** is returned to the g 10 - 00 state after each pulse in the Hahn echo sequence. The results of these measurements and the pulse schemes are shown in Fig figure4b, c. The measured T 1 and T 2 times are 1.6 and 1.9 μ s , respectively. The times are only slightly shorter than the 1.9 and 2.8 μ s times recorded at high g 10 - 00 without any fast flux pulses....We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures....Time domain measurements provide a more quantitative assessment of any residual coupling at the g 10 - 00 = 0 point. The rate of Rabi driving is proportional to the coupling strength g 10 - 00 and the applied drive amplitude as per the equation Ω R a b i = g 10 - 00 n , where n is the number of drive photons . In Fig figure3, we demonstrate Rabi **oscillations** at three different points on the constant **frequency** contour; these three points are marked on Fig. figure2b. Figure figure3a, b show Rabi **oscillations** at high and medium coupling respectively. In the two panels, the **oscillation** rate is kept nearly constant by increasing the applied rf spectroscopy power by 10 dB to compensate for the reduction in **qubit**-cavity coupling. Fig. figure3c shows the measurement at the g 10 - 00 = 0 point, with 27 dB more rf power than at the high coupling point. No excitation is visible. Given the measurement noise, we should easily be able to detect a tenth of a Rabi **oscillation**; that we see no excitation puts a lower bound on the change in the Rabi rate of a factor of 80. Together with the much higher excitation power, we estimate that the coupling is at least 1500 times smaller at the g 10 - 00 = 0 point compared with the high coupling point. If several **qubits** were in a single cavity, this tuning provides protection against single **qubit** gate errors in one **qubit** while a second **qubit** is driven. ... We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures.

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Contributors: Shevchenko, S. N., Omelyanchouk, A. N., Zagoskin, A. M., Savel'ev, S., Nori, F.

Date: 2007-12-12

which determines the thermally activated escape probability from the local minimum of the potential in Eq. ( eq_U). The classical Rabi-like **oscillations** are displayed in Fig. Figure1. In Fig. Figure1(a) the modulated transient **oscillations** of the phase difference φ are plotted. These **oscillations** result in the **oscillating** behaviour of the energy of the system as shown in Fig. Figure1(b). Averaging over fast **oscillations**, we plot in Fig. Figure1(c) with green solid curve the damped **oscillations** of the energy, analogous to the quantum Rabi **oscillations** . These curves are plotted for the following set of the parameters: η = 0.95 , α = 10 -3 , ϵ = 10 -3 , and γ = γ 0 . For comparison we also plotted the energy averaged over the fast **oscillations** for different parameters, changing one of these parameters and leaving the others the same. The dashed black curve in Fig. Figure1(c) is for the smaller damping, α = 10 -4 ; the solid (violet) line and the dash-dotted (black) line in Fig. Figure1(d) demonstrate the change in the **frequency** and the amplitude of the **oscillations** respectively for ϵ = 2 ⋅ 10 -3 and η = 0.9 . We notice that the effect analogous to the classical Rabi **oscillations** exist in a wide range of parameters....(Color online). The time-averaged probability P ¯ of the upper level to be occupied versus the driving **frequency**. The parameters used here are: η = 0.95 , E J / ℏ ω p = 300 , Γ r e l a x / ℏ ω p = Γ φ / ℏ ω p = 3 ⋅ 10 -4 . Numbers next to the curves stand for ϵ multiplied by 10 3 . Upper inset: the time dependence of the probability P τ . Right inset: the shift of the principal resonance (at ℏ ω ≈ Δ E ), where Δ ω = ω - Δ E / ℏ ....Rabi **oscillations** are coherent transitions in a quantum two-level system under the influence of a resonant perturbation, with a much lower **frequency** dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently [N. Gronbech-Jensen and M. Cirillo, Phys. Rev. Lett. 95, 067001 (2005)] that in phase **qubits** (current-biased Josephson junctions) this effect can be mimicked by classical **oscillations** arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effect. First, while the quantum Rabi **oscillations** can be produced by the subharmonics of the resonant **frequency** (multiphoton processes), the classical effect also exists when the system is excited at the overtones. Second, the shape of the resonance is, in the classical case, characteristically asymmetric; while quantum resonances are described by symmetric Lorentzians. Third, the anharmonicity of the potential results in the negative shift of the resonant **frequency** in the classical case, in contrast to the positive Bloch-Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow to confidently distinguish the bona fide Rabi **oscillations** from their classical Doppelganger....Superconducting phase **qubits** provide a clear demonstration of quantum coherent behaviour in macroscopic systems. They also have a very simple design: a phase **qubit** is a current-biased Josephson junction (see Fig. scheme(a)), and its working states | 0 , | 1 are the two lowest metastable energy levels E 0 , 1 in a local minimum of the washboard potential. The transitions between these levels are produced by applying an RF signal at a resonant **frequency** ω 10 = E 1 - E 0 / ℏ ≡ Δ E / ℏ . The readout utilizes the fact that the decay of a metastable state of the system produces an observable reaction: a voltage spike in the junction or a flux change in a coupled dc SQUID. In the three-level readout scheme (Fig. scheme(b)) both | 0 and | 1 have negligible decay rates. A pulse at a **frequency** ω 21 = E 2 - E 1 / ℏ transfers the probability amplitude from the state | 1 to the fast-decaying state | 2 . Its decay corresponds to a single-shot measurement of the **qubit** in state | 1 . Alternatively, instead of an RF readout pulse one can apply a dc pulse, which increases the decay rate of | 1 ....(Color online). Rabi-type **oscillations** in current-biased Josephson junctions: (a) and (b) show the time-dependence of the phase difference φ and of the energy H , (c) presents the time-dependence of the energy, averaged over the fast **oscillations** with period 2 π / ω . All energies are shifted by their stationary value: H 0 = 1 - 1 - η 2 - η arcsin η . The parameters for the blue, red, and green curves are: η = 0.95 , α = 10 -3 , ϵ = 10 -3 , and γ = γ 0 ; for the other curves in (c) and (d) only one parameter was different from the above, for comparison. Namely: (c) dashed black line α = 10 -4 , (d) solid violet line ϵ = 2 ⋅ 10 -3 , dash-dotted black line η = 0.9 ....(Color online). The time-averaged energy H ¯ - H 0 versus reduced **frequency** for relatively weak (a) and strong (b) driving. Different values of the driving amplitude ϵ (multiplied by 10 3 ) are shown by the numbers next to the curves. The parameters are: η = 0.95 and α = 10 -4 . In (b) the region between the vertical black lines corresponds to the escape from the phase-locked state....When the system is driven close to resonance, ω ≈ Δ E , the upper level occupation probability P τ exhibits Rabi **oscillations**. The damped Rabi **oscillations** are demonstrated in the upper inset in Fig. Figure3, which is analogous to the classical **oscillations** presented in Fig. Figure1. After averaging the time dependent probability, we plot it versus **frequency** in Fig. Figure3 for two values of the amplitude, demonstrating the multiphoton resonances. Figure Figure3 demonstrates the following features of the multiphoton resonances in the quantum case: (a) in contrast to the classical case, the resonances appear only at the subharmonics, at ℏ ω ≈ Δ E / n ; (b) the resonances have Lorentzian shapes (as opposed to the classical asymmetric resonances); (c) with increasing the driving amplitude the resonances shift to the higher **frequencies** – the Bloch-Siegert shift, which has the opposite sign from its classical counterpart. The Bloch-Siegert shift (the shift of the principal resonance at ℏ ω ≈ Δ E ) is plotted numerically in the right inset in Fig. Figure3. Analogous shifts of the positions of the resonances were recently observed experimentally ....In Fig. Figure2 the effect of the driving current on the time-averaged energy of the system is shown for different driving amplitudes: Fig. Figure2(a) for weaker amplitudes, close to the main resonance, to show the asymmetry and negative shift of the resonance; and Fig. Figure2(b) for stronger amplitudes, to show the resonances at γ 0 / 2 and 2 γ 0 (which are also shown closer in the insets). We note that the parametric-type resonance at 2 γ 0 originates from the third-order terms when the solution of the equation for ψ is sought by iterations ; when there are two or more terms responsible for this resonance, the respective resonance may become splitted, which is visible in Fig. Figure2(b) for the lowest curve. An analogous tiny splitting of the resonance was obtained for the driven flux **qubit** in Fig. 4 of Ref. [...Phase **qubit** (a) and its Josephson energy (b). The metastable states and can be used as **qubit** states. ... Rabi **oscillations** are coherent transitions in a quantum two-level system under the influence of a resonant perturbation, with a much lower **frequency** dependent on the perturbation amplitude. These serve as one of the signatures of quantum coherent evolution in mesoscopic systems. It was shown recently [N. Gronbech-Jensen and M. Cirillo, Phys. Rev. Lett. 95, 067001 (2005)] that in phase **qubits** (current-biased Josephson junctions) this effect can be mimicked by classical **oscillations** arising due to the anharmonicity of the effective potential. Nevertheless, we find qualitative differences between the classical and quantum effect. First, while the quantum Rabi **oscillations** can be produced by the subharmonics of the resonant **frequency** (multiphoton processes), the classical effect also exists when the system is excited at the overtones. Second, the shape of the resonance is, in the classical case, characteristically asymmetric; while quantum resonances are described by symmetric Lorentzians. Third, the anharmonicity of the potential results in the negative shift of the resonant **frequency** in the classical case, in contrast to the positive Bloch-Siegert shift in the quantum case. We show that in the relevant range of parameters these features allow to confidently distinguish the bona fide Rabi **oscillations** from their classical Doppelganger.

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Contributors: Cao, Xiufeng, You, J. Q., Zheng, H., Nori, Franco

Date: 2010-01-26

(Color online) Time evolution of the coherence σ x t versus the time t multiplied by the **qubit** energy spacing Δ . (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-type spectrum are α / Δ 2 = 0.01 , λ = 0.09 Δ (red solid curve); while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 , ω c = 10 Δ (green dashed-dotted curve). (b)The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 , λ = 0.3 (red solid line), and for the high-**frequency** Ohmic bath are α o h = 0.1 , ω c = 10 Δ (green dashed-dotted line). These results show that the decay rate for the low-**frequency** bath is shorter than for the high-**frequency** Ohmic bath. This means that the coherence time of the **qubit** in the low-**frequency** bath is longer than in the high-**frequency** noise case, demonstrating the powerful temporal memory of the low-**frequency** bath. Also, our results reflect the structure of the solution with branch cuts . The **oscillation** **frequency** for the low-**frequency** noise is ω 0 > Δ , in spite of the strength of the interaction. This can be referred to as a blue shift. However, in an Ohmic bath, the **oscillation** **frequency** is ω 0 **frequency** noises....(Color online) The spectral density J ω of the low- and high-**frequency** baths. (a) The case of weak interaction between the bath and the **qubit**, where the parameters of the low-**frequency** Lorentzian-like spectrum are α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve), while for the high-**frequency** Ohmic bath with Drude cutoff the parameters are α o h = 0.01 and ω c = 10 Δ (green dashed-dotted curve). (b) The case of strong interaction between the bath and the **qubit**, where the parameters of the low-**frequency** bath are α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve) and the parameters of the high-**frequency** Ohmic bath are α o h = 0.1 and ω c = 10 Δ (green dashed-dotted curve). The characteristic energy of the isolated **qubit** is indicated by a vertical blue dotted line. Here, and in the following figures, the energies are shown in units of Δ ....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between successive measurements, for a strong coupling between the **qubit** and the bath. The time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.1 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ ....We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment....(Color online) The effective decay γ τ / γ 0 , versus the time interval τ between consecutive measurements, for a weak coupling between the **qubit** and the bath. In the horizontal axis, the time-interval τ is multiplied by the **qubit** energy difference Δ . The curves in (a) correspond to the case of a low-**frequency** bath with parameters α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.01 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under the RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ . ... We use a non-Markovian approach to study the decoherence dynamics of a **qubit** in either a low- or high-**frequency** bath modeling the **qubit** environment. This approach is based on a unitary transformation and does not require the rotating-wave approximation. We show that for low-**frequency** noise, the bath shifts the **qubit** energy towards higher energies (blue shift), while the ordinary high-**frequency** cutoff Ohmic bath shifts the **qubit** energy towards lower energies (red shift). In order to preserve the coherence of the **qubit**, we also investigate the quantum Zeno effect in two cases: low- and high-**frequency** baths. For very frequent projective measurements, the low-**frequency** bath gives rise to the quantum anti-Zeno effect on the **qubit**. The quantum Zeno effect only occurs in the high-**frequency** cutoff Ohmic bath, after considering counter-rotating terms. For a high-**frequency** environment, the decay rate should be faster (without frequent measurements) or slower (with frequent measurements, in the Zeno regime), compared to the low-**frequency** bath case. The experimental implementation of our results here could distinguish the type of bath (either a low- or high-**frequency** one) and protect the coherence of the **qubit** by modulating the dominant **frequency** of its environment.

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Contributors: Schmidt, Thomas L., Borkje, Kjetil, Bruder, Christoph, Trauzettel, Bjoern

Date: 2010-02-25

(Color online) Possible experimental setup consisting of a **qubit** and an **oscillator** coupled to an atomic point contact (APC). Electrons tunnel at the APC ( t a ) and a fixed tunnel junction ( t b ), which are both biased with a voltage V . The area enclosed by the junctions (red dashed line) is threaded with a magnetic flux to create an Aharonov-Bohm phase. The **qubit** is realized as a Cooper pair box (CPB, yellow). Its state can be tuned using the gate voltage V g and it couples capacitively to both junctions. The **oscillation** of the nanomechanical resonator (NR, green) modulates the tunneling amplitude t a . As discussed in more detail in Appendix app:phases, this setup can be used to realize the tunneling amplitude ( gamma)....which contain Lorentz and Fano shaped resonances at the characteristic **frequencies** of the system, ω = 0 , Ω , 2 Δ . The complete expression for the noise reads S ω = ∑ X S X ω X where X denotes all combinations of **qubit** and **oscillator** operators contained in the EVM ( chi). As mentioned above, it turns out that all except three of the prefactors S X ω are nonvanishing and are distinguishable combinations of the functions α 1 , 2 ω and β 1 , 2 , 3 ω . Results for S X ω can be found in Appendix app:noise. A plot of the relevant cross-correlations’ prefactors is shown in Fig. NoisePlot. Since the shapes of these functions are rather distinct, the expectation values constituting the EVM can be recovered from the total measurable noise S ω ....In this article, we propose a system which allows the detection of entanglement between an **oscillator** and a **qubit** using an electronic measurement in an atomic point contact (APC). The electronic system is based on a tunneling contact, a readout device which is known to be quantum-limited. We find that the measurement of the current and the symmetrized current noise in this system allows the evaluation of a criterion for entanglement based on the density matrix of the **oscillator**-**qubit** system. This allows for the detection of entanglement in arbitrary pure or mixed states. All elements of the proposed setup have been realized separately in different experiments. Moreover, it has been shown that the current and the noise of an APC can be measured with a high accuracy. Therefore, it should be possible to combine both elements into one functional device as schematically shown in Fig. FigScheme and to measure its current and noise properties....In general, the amplitudes γ j = | γ j | e i δ j ( j = 0 , 1 , 2 ) can be complex. Since the global phase is irrelevant, we set δ 0 = 0 . Finite phases δ 1 , 2 can be realized experimentally by closing the electric circuit using an additional tunnel junction as shown in Fig. FigScheme. Threading the loop with a magnetic flux causes Aharonov-Bohm phases which can be absorbed in the tunneling amplitudes and generally lead to finite phases δ 1 and δ 2 . This is discussed in more detail in Appendix app:phases. The benefits of a controllable δ 1 have been investigated for a system consisting of an APC and an **oscillator**: while for δ 1 = 0 , the current noise only depends on the **oscillator** position x 2 , a finite δ 1 leads to terms proportional to p 2 and thus contains information about the **oscillator** momentum. Similarly, the presence of tunable phases δ 1 , 2 increases the number of measurable **oscillator** and **qubit** properties....(Color online) Schematic density plot of the prefactors S X ω X = x σ x x σ y … of the **frequency**-dependent noise S ω = ∑ X S X ω X as a function of δ 1 and ω ....The state of the **qubit**-**oscillator** system modulates the tunneling amplitude γ of the APC. If the **oscillator** acts as one of the electron reservoirs of the APC as shown in Fig. FigScheme, the tunneling gap depends on the **oscillator** displacement x . For small x one obtains γ ∝ γ 0 + γ 1 x . The same dependence can also be realized for capacitive coupling. The **qubit** can be realized as a Cooper pair box in which case a depletion of the electron reservoirs of the APC depending on the state of the **qubit** leads to an additional term γ 2 σ z in the tunneling amplitude. Irrespective of the concrete realization, to lowest order the combined effect of the **oscillator** and the **qubit** leads to...where δ 1 = arg 1 + c e - i Φ / Φ 0 . Hence, this setup provides a way to obtain a tunneling Hamiltonian with tunable δ 1 . This has been important for the calculation of the noise. The setup can easily be extended to achieve the second tunable phase δ 2 which we used in the calculation of the current. Here, we need a third junction with an amplitude t c = t c 0 , which is decoupled from both **oscillator** and **qubit**, and two magnetic fluxes Φ 1 , 2 . A schematic is shown in Fig. fig:phasesb....Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting **qubits** close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a **qubit** in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an **oscillator** and a **qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state. ... Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting **qubits** close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a **qubit** in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an **oscillator** and a **qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state.

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Contributors: Hayes, D., Matsukevich, D. N., Maunz, P., Hucul, D., Quraishi, Q., Olmschenk, S., Campbell, W., Mizrahi, J., Senko, C., Monroe, C.

Date: 2010-01-13

The parity **oscillation** that is used to calculate the fidelity of the spin state of two ions with respect to the maximally entangled state | χ after performing the entangling gate. The phase φ of the analyzing pulse is scanned by changing the relative phase of the rotation pulses. The offset and lack of full contrast in the parity signal can be attributed to state detection errors....(a) Using a Raman probe duration of 80 μ s , ( N ∼ 6500 ), a **frequency** scan of AO1 shows the resolved carrier and motional sideband transitions of a single trapped ion. The transitions are labeled, Δ n x Δ n y , to indicate the change in the number of phonons in the two transverse modes that accompany a spin flip. The x and y mode splitting is controlled by applying biasing voltages to the trap electrodes. Unlabeled peaks show higher order sideband transitions and transitions to other Zeeman levels due to imperfect polarization of the Raman beams. (b) Ground state cooling of the motional modes via a train of phase-coherent ultra-fast pulses. The red open-circle data points show that after Doppler cooling and optical pumping, both the red and blue sidebands are easily driven. The blue filled-circle data points show that after sideband cooling, the ion is close to the motional ground state, ( n ̄ x , y ≤ 0.03 ), as evidenced by the suppression of the red-sideband transition....Schematic of the experimental setup showing the paths of the pulse trains emitted by a mode-locked Ti:Sapphire (Ti:Sapph) laser, where the optical pulses are **frequency** shifted by AOs. Single **qubit** rotations only require a single pulse train, but to address the motional modes the pulse train is split into two and sent through AOs to tune the relative offset of the two combs. We lock the repetition rate ( ν R ) by first detecting ν R with a photodetector (PD). The output of the PD is an RF **frequency** comb spaced by ν R . We bandpass filter (BP) the RF comb at 12.685 GHz and then mix the signal with a local **oscillator** (LO). The output of the mixer is sent into a feedback loop (PID) which stabilizes ν R by means of a piezo mounted on one of the laser cavity mirrors. When locked, ν R is stable to within 1 Hz for more than an hour. As an alternative, instead of locking the repetition rate of the pulsed laser, an error signal could be sent to one of the AOs to use the relative offset of the two combs to compensate for a change in the comb spacing....We demonstrate the use of an optical **frequency** comb to coherently control and entangle atomic **qubits**. A train of off-resonant ultrafast laser pulses is used to efficiently and coherently transfer population between electronic and vibrational states of trapped atomic ions and implement an entangling quantum logic gate with high fidelity. This technique can be extended to the high field regime where operations can be performed faster than the trap **frequency**. This general approach can be applied to more complex quantum systems, such as large collections of interacting atoms or molecules....High fidelity **qubit** operations through Raman transitions are typically achieved by phase-locking **frequency** components separated by the energy difference of the **qubit** states. This is traditionally accomplished in a bottom-up type of approach where either two monochromatic lasers are phase-locked or a single cw laser is modulated by an acousto-optic (AO) or an electro-optic (EO) modulator. However, the technical demands of phase-locked lasers and the limited bandwidths of the modulators hinder their application to experiments. Here we exploit the large bandwidth of ultrafast laser pulses in a simple top-down approach toward bridging large **frequency** gaps and controlling complex atomic systems. By starting with the broad bandwidth of an ultrafast laser pulse, a spectral landscape can be sculpted by interference from sequential pulses, pulse shaping and **frequency** shifting. In this paper, we start with a picosecond pulse and, through the application of many pulses, generate a **frequency** comb that drives Raman transitions by stimulating absorption from one comb tooth and stimulating emission into another comb tooth as depicted in Fig. fig:energydiagram. Because this process only relies on the **frequency** difference between comb teeth, their absolute position is irrelevant and the carrier-envelope phase does not need to be locked . As an example of how this new technique promises to ease experimental complexities, the control of metastable-state **qubits** separated by a terahertz was recently achieved using cw lasers that are phase-locked through a **frequency** comb , but might be controlled directly with a 100 fs Ti:sapph pulsed laser....For many applications in quantum information, the motional modes of the ion must be cooled and initialized to a nearly pure state. Fig. fig:cooling shows that the pulsed laser can also be used to carry out the standard techniques of sideband cooling to prepare the ion in the motional ground state with near unit fidelity. The set-up also easily lends itself to implementing a two-**qubit** entangling gate by applying two fields whose **frequencies** are symmetrically detuned from the red and blue sidebands . By simultaneously applying two modulation **frequencies** to one of the comb AO **frequency** shifters, we create two combs in one of the beams. When these combs are tuned to drive the red and blue sidebands (in conjunction with the third **frequency** comb in the other beam), the ion experiences a spin-dependent force in a rotated basis as described in Ref. . Ideally, when the fields are detuned from the sidebands by an equal and opposite amount δ = 2 η Ω , a decoupling of the motion and spin occurs at gate time t g = 2 π / δ , and the spin state evolve to the maximally entangled state | χ = . In the experiment, t g = 108 μ s ( N ∼ 8700 pulses)....After Doppler cooling and optical pumping to the state, a single pulse train is directed onto the ion. When the ratio of **qubit** splitting to pulse repetition rate, q , is an integer, pairs of comb teeth can drive Raman transitions as shown by the blue circular data points. However, if the q parameter is a half integer, the **qubit** remains in the initial state as shown by the red square data points....The Rabi **frequency** of these **oscillations** can be estimated by considering the Hamiltonian resulting from an infinite train of pulses. After adiabatically eliminating the excited 2 P 1 / 2 state and performing the rotating-wave approximation, the resonant Rabi **frequency** of Raman transitions between the **qubit** states is given by a sum over all spectral components of the comb teeth as indicated in Fig. fig:energydiagram ( ℏ = 1 ):...To demonstrate coherent control with a pulse train, 171 Y b + ions confined in a linear Paul trap are used to encode **qubits** in the 2 S 1 / 2 hyperfine clock states and , having hyperfine splitting ω 0 / 2 π = 12.6428 GHz. For state preparation and detection we use standard Doppler cooling, optical pumping, and state-dependent fluorescence methods on the 811 THz 2 S 1 / 2 ↔ 2 P 1 / 2 electronic transition . The **frequency** comb is produced by a **frequency**-doubled mode-locked Ti:Sapphire laser at a carrier **frequency** of 802 THz, detuned by Δ / 2 π = 9 THz from the electronic transition. The repetition rate of the laser is ν R = 80.78 MHz, with each pulse having a duration of τ ≈ 1 psec. The repetition rate is phase-locked to a stable microwave **oscillator** as shown in Fig. fig:experiment, providing a ratio of hyperfine splitting to comb spacing of q = 156.5 . An EO pulse picker is used to allow the passage of one out of every n pulses, decreasing the comb spacing by a factor of n and permitting integral values of q . As shown in Fig. fig:pulse:picking, when n = 2 ( q = 313 and ν R = 40.39 MHz), application of the pulse train drives **oscillations** between the **qubit** states of a single ion. However, when n = 3 ( q = 469.5 and ν R = 26.93 MHz), the **qubit** does not evolve....In order to entangle multiple ions, we first address the motion of the ion by resolving motional sideband transitions. As depicted in Fig. fig:experiment, the pulse train is split into two perpendicular beams with wavevector difference k along the x - direction of motion. Their polarizations are mutually orthogonal to each other and to a weak magnetic field that defines the quantization axis . We control the spectral beatnotes between the combs by sending both beams through AO modulators (driven at **frequencies** ν 1 and ν 2 ), imparting a net offset **frequency** of Δ ω / 2 π = ν 1 - ν 2 between the combs. For instance, in order to drive the first upper/lower sideband transition we set | 2 π j ν R + Δ ω | = ω 0 ± ω t , with j an integer and ω t the trap **frequency**. In order to see how the sidebands are spectrally resolved, we consider the following Hamiltonian of a single ion and single mode of harmonic motion interacting with the Raman pulse train:...The Stokes Raman process driven by **frequency** combs is shown here schematically. An atom starting in the state can be excited to a virtual level by absorbing a photon from the blue comb and then driven to the state by emitting a photon into the red comb. Although drawn here as two different combs, if the pulsed laser’s repetition rate or one of its harmonics is in resonance with the hyperfine **frequency**, the absorption and emission can both be stimulated by the same **frequency** comb. Because of the even spacing of the **frequency** comb, all of the comb teeth contribute through different virtual states which result in indistinguishable paths and add constructively. ... We demonstrate the use of an optical **frequency** comb to coherently control and entangle atomic **qubits**. A train of off-resonant ultrafast laser pulses is used to efficiently and coherently transfer population between electronic and vibrational states of trapped atomic ions and implement an entangling quantum logic gate with high fidelity. This technique can be extended to the high field regime where operations can be performed faster than the trap **frequency**. This general approach can be applied to more complex quantum systems, such as large collections of interacting atoms or molecules.

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