### 56015 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**.

Data types:

Contributors: Baur, M., Filipp, S., Bianchetti, R., Fink, J. M., Göppl, M., Steffen, L., Leek, P. J., Blais, A., Wallraff, A.

Date: 2008-12-23

depending linearly on the drive amplitude ε . Therefore, one would expect that the strong drive at the **qubit** transition **frequency** ω d ≈ ω g e should lead to a square-root dependence of the Autler-Townes and Mollow spectral lines on the drive power P d ∝ ε 2 . However, the Autler-Townes spectral lines show a clear power dependent shift, see Fig. fig:fig3, and the splitting of both pairs of lines scales weaker than linearly with ε ....We measure the Autler-Townes and the Mollow spectral lines according to the scheme shown in Fig. fig:fig1(b). First, we tune the **qubit** to the **frequency** ω g e / 2 π ≈ 4.811 G H z , where it is strongly detuned from the resonator by Δ / 2 π = 1.63 G H z . We then strongly drive the transition | g → | e with a first microwave tone of amplitude ε applied to the **qubit** at the fixed **frequency** ω d = 4.812 G H z . The drive field is described by the Hamiltonian H d = ℏ ε a † e - i ω d t + a e i ω d t where the drive amplitude ε is given in units of a **frequency**. The **qubit** spectrum is then probed by sweeping a weak second microwave signal over a wide range of **frequencies** ω p including ω g e and ω e f . Simultaneously, amplitude T and phase φ of a microwave signal applied to the resonator are measured . We have adjusted the measurement **frequency** to the **qubit** state-dependent resonance of the resonator under **qubit** driving for every value of ε . Figures fig:fig2(a) and (b) show the measurement response T and φ for selected values of ε . For drive amplitudes ε / 2 π > 65 M H z , two peaks emerge in amplitude from the single Lorentzian line at **frequency** ω e f corresponding to the Autler-Townes doublet, see Fig. fig:fig2(a). The signal corresponding to the sidebands of the Mollow triplet is visible at high drive amplitudes ε / 2 π > 730 M H z in phase, see Fig. fig:fig2(b). Black lines in Fig. fig:fig2 are fits of the data to Lorentzians from which the dressed **qubit** resonance **frequencies** are extracted....(a) Extracted splitting **frequencies** of the Mollow triplet sidebands (red dots) and the Autler-Townes doublet (blue dots) as a function of the drive field amplitude. Dashed lines: Rabi **frequencies** obtained with Eq. ( eq:1). Black solid lines: Rabi **frequencies** calculated by numerically diagonalizing the Hamiltonian Eq. ( eq:2) taking into account 5 transmon levels. (b) Zoom in of the region in the orange rectangle in (a). Orange dots: Rabi **frequency** Ω g e vs. drive amplitude ε extracted from time resolved Rabi **oscillation** experiments, lines as in (a). (c) Rabi **oscillation** measurements between states | g and | e with Ω R / 2 π = 50 M H z and 85 M H z ....fig:fig3 Measured Autler-Townes doublet (blue dots) and Mollow triplet sideband **frequencies** (red dots) vs. drive power P d at a fixed drive **frequency** ω d / 2 π = 4.812 G H z . Black solid lines are transition **frequencies** calculated by numerically diagonalizing the Hamiltonian ( eq:2) taking into account the lowest 5 transmon levels....The **frequencies** of the Autler-Townes doublet (blue data points) and of the Mollow triplet sidebands (red data points) extracted from the Lorentzian fits in Fig. fig:fig2(a) and (b) are plotted in Fig. fig:fig3. The splitting of the spectral lines in pairs separated by Ω R and 2 Ω R , respectively, is observed for Rabi **frequencies** up to Ω R / 2 π ≈ 300 M H z corresponding to about 6% of the **qubit** transition **frequency** ω g e ....(a) Simplified circuit diagram of the measurement setup analogous to the one used in Ref. . In the center at the 20 mK stage, the **qubit** is coupled capacitively through C g to the resonator, represented by a parallel LC **oscillator**, and the resonator is coupled to the input and output transmission lines over capacitances C i n and C o u t . Three microwave signal generators are used to apply the measurement ν r f and drive and probe tones ν d r i v e / p r o b e to the input port of the resonator. The transmitted measurement signal is then amplified by an ultra-low noise amplifier at 1.5 K, down-converted with an IQ-mixer and a local **oscillator** (LO) to an intermediate **frequency** at 300K and digitized with an analog-to-digital converter (ADC). (b) Energy-level diagram of a bare three-level system with states | g , | e , | f ordered with increasing energy. Drive and probe transitions are indicated by black and red/blue arrows, respectively. (c) Energy-level diagram of the dipole coupled dressed states with the coherent drive tone. Possible transitions induced by the probe tone between the dressed states and the third **qubit** level ( ν - , f , ν + , f ) and between the dressed states ( ν - , + , ν + , - ) are indicated with blue and red arrows....In the experiments presented here, we use a version of the Cooper pair box , called transmon **qubit** , as our multilevel quantum system. States of increasing energies are labelled | l with l = g , e , f , h , i , … The transition **frequency** ω g e between the ground | g and first excited state | e is approximated by ℏ ω g e ≈ 8 E C E J m a x | cos 2 π Φ / Φ 0 | - E C , where E C / h = 233 M H z is the charging energy and E J m a x / h = 32.8 G H z is the maximum Josephson energy. The transition **frequency** ω g e can be controlled by an external magnetic flux Φ applied to the SQUID loop formed by the two Josephson junctions of the **qubit**. The transition **frequency** from the first | e to the second excited state | f is given by ω e f = ω g e - α , where α ≈ 2 π E C / h is the **qubit** anharmonicity . The **qubit** is strongly coupled to a coplanar waveguide resonator with resonance **frequency** ω r / 2 π = 6.439 G H z and photon decay rate κ / 2 π ≈ 1.6 MHz. A schematic circuit diagram of the setup is shown in Fig fig:fig1(a)....We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives....To confirm the direct relationship between the measured dressed state splitting **frequency** and the Rabi **oscillation** **frequency** of the excited state population we have also performed time resolved measurements of the Rabi **frequency** up to 100 M H z , see Fig. fig:fig4(c). The extracted Rabi **frequencies** (orange data points) are in good agreement with the spectroscopically measured Rabi **frequencies** (blue squares) over the range of accessible ε , as shown in Fig. fig:fig4(b) ... We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives.

Data types:

Contributors: Serban, I., Solano, E., Wilhelm, F. K.

Date: 2007-02-28

Discrimination time as function of the coupling strength between **qubit** and **oscillator**. Here ℏ Ω / k B T = 2 , κ / Ω = 0.025 and ℏ ν / k B T = 1.95 ....In this section we present a different measurement protocol. It is based on the short time dynamics illustrated as follows: for the **qubit** initially in the state 1 / 2 | ↑ + | ↓ the probability distribution of momentum is plotted in Fig. probability (a) and (b)....As one can see for the parameters of Fig. comp, in the WQOC regime the measurement time is longer than the dephasig time. Their difference decreases as we increase Δ due to the onset of the strong coupling plateau in the dephasing rate, approaching the quantum limit where the measurement time becomes comparable to the dephasing time. Note that, for superstrong coupling either between **qubit** and **oscillator** or between **oscillator** and bath, corrections of the order κ / Ω ↓ 2 of the dephasing rate gain importance. These corrections are not treated in our Born approximation. Therefore the regions where the dephasing rate becomes lower than the measurement rate, in violation with the quantum limitation of Ref. , should be regarded as a limitation of our approximation....Dephasing rate dependence on driving: dependence on Δ for different driving strengths F 0 ( κ / Ω = 10 -4 and ν = 2 Ω ). Top inset: dependence of the decoherence rate on F 0 for different values of κ ( Δ / Ω = 5 ⋅ 10 -2 and ν = 2 Ω ). Bottom inset: dependence of the decoherence rate on driving **frequency** ν for different vales of κ ( Δ / Ω = 0.5 ). Here ℏ Ω / k B T = 2 and F ¯ 0 is the dimensionless force F 0 ℏ / k B T m k B T ....Probability density of momentum P p 0 t (a), snapshots of it at different times (b) and expectation value of momentum for the two different **qubit** states (c). Here ℏ Ω / k B T = 2 , Δ / Ω = 0.45 , κ / Ω = 0.025 and ℏ ν / k B T = 1.9 and p ¯ 0 is the dimensionless momentum p 0 / k B T m ....We consider a simplified version of the experiment described in Ref. . The circuit consists of a flux **qubit** drawn in the single junction version, the surrounding SQUID loop, an ac source, and a shunt resistor, as depicted in Fig. circuit. We note here that we later approximate the **qubit** as a two-level system. The **qubit** used in the actual experiment contains three junctions. An analogous but less transparent derivation would, after performing the two-state approximation, lead to the same model, parameterized by the two-state Hamiltonian, the circulating current, and the mutual inductance, in an identical way ....Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost....The dependence on the driving **frequency** has also been analyzed in Fig. antrieb. Here we observe two peaks at Ω ↑ and Ω ↓ . At ν = Ω the classical driven and undamped trajectory ξ t diverges. In terms of the calculation this means that the Floquet modes are not well-defined when the driving **frequency** is at resonance with the harmonic **oscillator** — we have a continuum instead. Physically this means that at t = 0 our **oscillator** has the **frequency** Ω because it has not yet "seen" the **qubit**, and we are driving it at resonance, and by amplifying the **oscillations** of x ̂ which is subject to noise we amplify the noise seen by the **qubit**. The dephasing rate is also expected to diverge. The peaks at Ω ↑ and Ω ↓ show the same effect after the **qubit** and the **oscillator** become entangled. The dephasing rate drops again for large driving **frequencies** to the value obtained in the case without driving....Simplified circuit consisting of a **qubit** with one Josephson junction (phase γ , capacitance C q and inductance L q ) inductively coupled to a SQUID with two identical junctions (phases γ 1 , 2 , capacitance C S ) and inductance L S . The SQUID is driven by an ac bias I B t and the voltage drop is measured by a voltmeter with internal resistance R . The total flux through the **qubit** loop is Φ q and through the SQUID is Φ S ....In Fig. probability one can see that the two peaks corresponding to the two states of the **qubit** split already during the transient motion of p ̂ t , much faster than the transient decay time. If the peaks are well enough separated, a single measurement of momentum gives the needed information about the **qubit** state, and has the advantage of avoiding decoeherence effects resulting from a long time coupling to the environment. Nevertheless the parameters we need to reduce the discrimination time also enhance the decoherence rate. ... Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost.

Data types:

Contributors: Wu, Jing-Nuo, Chen, Hung-Kuang, Hsieh, Wen-Feng, Cheng, Szu-Cheng

Date: 2012-07-02

(Color online) (a) Polarization P z t = 1 2 ρ 10 t + ρ 01 t and (b) Decoherence rate Γ d e c . t = - ρ ̇ 10 t + ρ ̇ 01 t ρ 10 t + ρ 01 t of the **qubit** with **frequency** lying inside ( δ / β < 0 ) and outside ( δ / β = 2 ) the PBG region ....(Color online) Dynamics of (a) the ** qubit’s** excited-state probability P t and (b) relaxation rate Γ r e l a x . t of the

**qubit**with different detuning

**frequencies**δ / β = ω 10 - ω c / β from the band edge

**frequency**ω c of the PhC reservoir....(Color online) (a) A

**qubit**with excited state and ground state . The transition

**frequency**ω 10 is nearly resonant with the

**frequency**range of the PhC reservoir. (b) Directional dependent dispersion relation near band edge expressed by the effective-mass approximation with the edge

**frequency**ω c . (c) Photon DOS ρ ω of the anisotropic PhC reservoir exhibiting cut-off photon mode below the edge

**frequency**ω c ....We study the quantum dynamics of relaxation, decoherence and entropy of a

**qubit**embedded in an anisotropic photonic crystal (PhC) through fractional calculus. These quantum measurements are investigated by analytically solving the fractional Langevin equation. The

**qubit**with

**frequency**lying inside the photonic band gap (PBG) exhibits the preserving behavior of energy, coherence and information amount through the steady values of excited-state probability, polarization

**oscillation**and von Neumann entropy. This preservation does not exist in the Markovian system with

**qubit**

**frequency**lying outside the PBG region. These accurate results are based on the appropriate mathematical method of fractional calculus and reasonable inference of physical phenomena. ... We study the quantum dynamics of relaxation, decoherence and entropy of a

**qubit**embedded in an anisotropic photonic crystal (PhC) through fractional calculus. These quantum measurements are investigated by analytically solving the fractional Langevin equation. The

**qubit**with

**frequency**lying inside the photonic band gap (PBG) exhibits the preserving behavior of energy, coherence and information amount through the steady values of excited-state probability, polarization

**oscillation**and von Neumann entropy. This preservation does not exist in the Markovian system with

**qubit**

**frequency**lying outside the PBG region. These accurate results are based on the appropriate mathematical method of fractional calculus and reasonable inference of physical phenomena.

Data types:

Contributors: Lecocq, Florent, Pop, Ioan M., Matei, Iulian, Dumur, Etienne, Feofanov, A. K., Naud, Cécile, GUICHARD, Wiebke, Buisson, Olivier

Date: 2012-01-19

Energy spectrum. Escape probability P e s c versus **frequency** as a function of current bias (a) and flux bias (b) measured at Φ b = 0.48 Φ 0 and I b = 0 respectively. P e s c is enhanced when the **frequency** matches a resonant transition of the circuit. The microwave amplitude was tuned to keep the resonance peak amplitude at 10%. Dark and bright blue scale correspond to high and small P e s c . The red dashed lines are the transition **frequencies** deduced from the spectrum of the full hamiltonian with C = 510 fF (see text). The green diamond is the initial working point for the measurement of coherent free **oscillations** between the two modes, presented in Fig. **Oscillations**, and the green dotted square is the area where these **oscillations** take place. spectro...Description of the device. (a) A micrograph of the aluminium circuit. The two small squares are the two Josephson junctions (enlarged in the top right inset, 10 μ m 2 area, I c = 713 nA and C = 510 fF) decoupled by a large inductive loop ( L = 629 pH). The width of the two SQUID arms were adjusted to reduce the inductance asymmetry to about 10%. Very narrow current bias lines, with a large 15 nH-inductance, isolate the quantum circuit from the dissipative environment at high **frequencies** . The symmetric and antisymmetric **oscillation** modes are illustrated by blue and red arrows, respectively. (b) and (c) Potentials of the s and a-mode respectively, for the bias working point ( I b = 0 , Φ b = 0.37 Φ 0 ). (d) Schematic energy level diagram indexed by the quantum excitation number of the two modes n s , n a at the same working point. Climbing each vertical ladder one increases the excitation number of the s-mode, keeping the excitation number of the a-mode constant. The two first levels, 0 s , 0 a and 1 s , 0 a , realize a camelback phase **qubit**. fig1...By adding a large inductance in a dc-SQUID phase **qubit** loop, one decouples the junctions' dynamics and creates a superconducting artificial atom with two internal degrees of freedom. In addition to the usual symmetric plasma mode ({\it s}-mode) which gives rise to the phase **qubit**, an anti-symmetric mode ({\it a}-mode) appears. These two modes can be described by two anharmonic **oscillators** with eigenstates $\ket{n_{s}}$ and $\ket{n_{a}}$ for the {\it s} and {\it a}-mode, respectively. We show that a strong nonlinear coupling between the modes leads to a large energy splitting between states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$. Finally, coherent **frequency** conversion is observed via free **oscillations** between the states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$....We now discuss the coherent properties and measurement contrast in our device. The unexpectedly short coherence time of the a-mode can be explained by the coupling to spurious two-level systems (TLS) . With a junction area of 10 μ m 2 our device suffers from a large TLS density of about 12 TLS/GHz (barely visible in Fig. spectro). Therefore it is very difficult to operate the a-mode in a **frequency** window free of TLS since ν 01 a is only slightly flux dependent. However this is not a real issue as it can be solved easily by reducing the junction area . The minimum linewidth of both a-mode and s-mode, and therefore their coherence times, are limited in our experiment by low **frequency** flux noise. Operating the system at Φ b = 0 will lift this limitation since it is an optimal point with respect to flux noise. The small **oscillation** amplitude in Fig. **Oscillations** has two additionnals origins. First the duration t π of the π -pulse applied for preparation of the state 0 s , 1 a has to fulfill the condition t π -1 **frequency** conversion ....One of the opportunities given by the rich spectrum of this two DoF artificial atom is the observation of a coherent **frequency** conversion process using the x ̂ s 2 x ̂ a coupling of Eq. eq:2. The pulse sequence, similar to other states swapping experiment , is presented in (see Fig. Oscillationsa). At t = 0 , the system is prepared in the state 0 s , 1 a , at the initial working point Φ b = 0.37 Φ 0 (green diamond in Fig. spectrob). Immediately after, a non-adiabatic flux pulse brings the system to the working point defined by Φ i n t , close to the degeneracy point ( ν 02 s ≈ ν 01 a ). After the free evolution of the quantum state during the time Δ t i n t , we measure the escape probility P e s c . Fig. Oscillationsb presents P e s c as a function of Δ t i n t for Φ i n t = 0.515 Φ 0 . The observed **oscillations** have a 815 M H z -characteristic **frequency** (inset of Fig. Oscillationsb) that matches precisely the theoritical **frequency** splitting at this flux bias (red arrow). In Fig. Oscillationsc, we present these **oscillations** as function of Φ i n t . Their **frequency** varies with Φ i n t , showing a typical “chevron” pattern. In the inset of Fig. Oscillationsc, the **oscillation** **frequency** versus Φ i n t is compared to theoretical predictions. The good agreement between theory and experiment is a striking confirmation of the observation of swapping between the quantum states 0 s , 1 a and 2 s , 0 a . Instead of the well known linear coupling x ̂ s x ̂ a between two **oscillators**, which corresponds to a coherent exchange of single excitations between the two systems, here the coupling x ̂ s 2 x ̂ a is non-linear and produces a coherent exchange of a single excitation of the a-mode with a double excitation of the s-mode, i.e. a coherent **frequency** conversion. Starting from the state 0 s , 1 a , an excitation pair 2 s , 0 a is then deterministically produced in about a single nanosecond at the degeneracy point Φ i n t = 0.537 Φ 0 ....Free coherent **oscillations** between states 0 s , 1 a and 2 s , 0 a produced by a nonlinear coupling. (a) Schematic pulse sequence. The energy diagram, without coupling in blue/red and with coupling in black, is represented for both the preparation and interaction steps.(b) Escape probability P e s c versus interaction time Δ t i n t . The inset presents the Fourier transform of these **oscillations** with a clear peak at 815 MHz. The red arrow indicates the theoretically expected **frequency**. (c) P e s c versus interaction time Δ t i n t for different interaction flux Φ i n t close to the resonance condition between ν 02 s and ν 01 a . For clarity the data is numerically processed using 200 MHz high-pass filter(dashed line in inset of (b)). Inset : **oscillation** **frequency** as function of flux. The dashed red line shows the theoretical predictions....The readout of the circuit is performed using switching current techniques. For spectroscopy measurements we apply a microwave pulse field, through the current bias line (see Fig fig1a), followed by the readout nanosecond flux pulse that produces a selective escape depending on the quantum state of the circuit . The energy spectrum versus current bias at Φ b = 0.48 Φ 0 and versus flux bias at I b = 0 are plotted in Fig. spectroa and Fig. spectrob respectively. In the following we will denote ν n m α as the transition **frequency** between the states n α and m α , with the other mode in the ground state. The first transition **frequency** in Fig. spectro is the one of the camelback phase **qubit**, ν 01 s . With a maximum **frequency** at zero-current bias, the system is at an optimal working point with respect to current fluctuations . At higher **frequency**, the second transition of the s-mode is observed with ν 02 s ≈ 2 ν 01 s . In the flux biased spectrum the third transition ν 03 s is also visible. An additional transition is observed at about 14.6 GHz, with a very weak current dependence (Fig. spectroa) but a finite flux dependence (Fig. spectrob). It corresponds to the first transition of the a-mode, ν 01 a . The s-mode transition **frequencies** drop when Φ b / Φ 0 approaches 0.7 which is consistent with the critical flux Φ c / Φ 0 = 1 / 2 + L / 2 π L J = 0.717 for which ω p s → 0 and x a m i n → π / 2 . On the contrary ω p a remains finite when Φ b → Φ c with ω p a → 4 E J / m L J / L . One also observes a large level anti-crossing of about 700 MHz between the two transitions ν 02 s and ν 01 a . Additionally no level anti-crossing is measurable between ν 03 s and ν 01 a ....The circuit is a camelback phase **qubit** with a large loop inductance, i.e a dc-SQUID build by a superconducting loop of large inductance L interrupted by two identical Josephson junctions with critical current I c and capacitance C , operated at zero current bias (see Fig. fig1). As we will see in the following, the presence of a large loop inductance modifies dractically the quantum dynamics of this system. The two phase differences φ 1 and φ 2 across the two junctions correspond to the two degrees of freedom of this circuit, which lead to two **oscillating** modes: the symmetric ( s-) and the anti-symmetric (a-) plasma modes . The s-mode corresponds to the well-known in-phase plasma **oscillation** of the two junctions with the average phase x s = φ 1 + φ 2 / 2 , **oscillating** at a characteristic **frequency** given by the plasma **frequency** of the dc-SQUID, ω p s . The a-mode is an opposite-phase plasma mode related to **oscillations** of the phase difference x a = φ 1 - φ 2 / 2 , producing circulating current **oscillations** at **frequency** ω p a . In previous experiments , the loop inductance L was small compared to the Josephson inductance L J = Φ 0 / 2 π I c . Therefore the two junctions were strongly coupled and the dynamics of the phase difference x a was neglected and fixed by the applied flux. The quantum behavior of the circuit was described by the s-mode only, showing a one-dimensional motion of the average phase x s . Hereafter we will consider a circuit with a large inductance ( L ≥ L J ) that decouples the phase dynamics of the two junctions. This large inductance lowers the **frequency** of the a-mode and the dynamics of the system becomes fully two-dimensional. The a-mode was previously introduced to discuss the thermal and quantum escape of a current-biased dc-SQUID but its dynamics was never observed. We present measurements of the full spectrum of this artificial atom, independent coherent control of both modes and finally we exploit the strong nonlinear coupling between the two DoF to observe a time resolved up and down **frequency** conversion of the system excitations. ... By adding a large inductance in a dc-SQUID phase **qubit** loop, one decouples the junctions' dynamics and creates a superconducting artificial atom with two internal degrees of freedom. In addition to the usual symmetric plasma mode ({\it s}-mode) which gives rise to the phase **qubit**, an anti-symmetric mode ({\it a}-mode) appears. These two modes can be described by two anharmonic **oscillators** with eigenstates $\ket{n_{s}}$ and $\ket{n_{a}}$ for the {\it s} and {\it a}-mode, respectively. We show that a strong nonlinear coupling between the modes leads to a large energy splitting between states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$. Finally, coherent **frequency** conversion is observed via free **oscillations** between the states $\ket{0_{s},1_{a}}$ and $\ket{2_{s},0_{a}}$.

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Contributors: Gul, Yusuf

Date: 2014-12-29

fig3 (Color online) Emergence of **frequency** locking for two-mode JT system shown in spectrum of the lowest five eigenvalues depending on the **frequency** difference Δ . (a) At Δ = 0 Rabi splitting of first energy levels occurs for k = 0.1 / 2 . Interaction between priviledged and disadvantaged mode can be tuned up to Δ = 0.1 in single effective mode. (b) Range of single mode regime extends up to Δ = 0.5 in ultrastrong regime k = 1.0 / 2...fig3 (Color online) Emergence of localization and synchronization transitions in weak, strong and ultrastrong regime. (a) shows snynchronous structure between damped **oscillating** population imbalance and correlation of priviledged mode in weak coupling k = 0.01 / 2 . (b) priviledged mode becomes synchronous with **qubit** and delocalization-localization transition occurs in population imbalance in strong coupling regime k = 0.1 / 2 . (c) presents the photon blockade in priviledged mode and fully trapped regime in population imbalance with k = 1.0 / 2 and γ φ = 0.1...We consider the nonlinear effects in Jahn-Teller system of two coupled resonators interacting simultaneously with flux **qubit** using Circuit QED. Two **frequency** description of Jahn Teller system that inherits the networked structure of both nonlinear Josephson Junctions and harmonic **oscillators** is employed to describe the synchronous structures in multifrequency scheme. Emergence of dominating mode is investigated to analyze **frequency** locking by eigenvalue spectrum. Rabi Supersplitting is tuned for coupled and uncoupled synchronous con?gurations in terms of **frequency** entrainment switched by coupling strength between resonators. Second order coherence functions are employed to investigate self-sustained **oscillations** in resonator mode and **qubit** dephasing. Snychronous structure between correlations of priviledged mode and **qubit** is obtained in localization-delocalization and photon blockade regime controlled by the population imbalance. ... We consider the nonlinear effects in Jahn-Teller system of two coupled resonators interacting simultaneously with flux **qubit** using Circuit QED. Two **frequency** description of Jahn Teller system that inherits the networked structure of both nonlinear Josephson Junctions and harmonic **oscillators** is employed to describe the synchronous structures in multifrequency scheme. Emergence of dominating mode is investigated to analyze **frequency** locking by eigenvalue spectrum. Rabi Supersplitting is tuned for coupled and uncoupled synchronous con?gurations in terms of **frequency** entrainment switched by coupling strength between resonators. Second order coherence functions are employed to investigate self-sustained **oscillations** in resonator mode and **qubit** dephasing. Snychronous structure between correlations of priviledged mode and **qubit** is obtained in localization-delocalization and photon blockade regime controlled by the population imbalance.

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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: 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: J.F. Ralph, T.D. Clark, M.J. Everitt, H. Prance, P. Stiffell, R.J. Prance

Date: 2003-10-20

Power spectral density for the low **frequency** **oscillator** at the resonance point (Φdc=0.00015Φ0) for the three spontaneous decay rates shown in Fig. 2: γ=0.005,0.05,0.5 per cycle. The other parameters are given in the text.
...(a) Close-up of the time-averaged (Floquet) energies of the single photon resonance (500 MHz), solid lines, with the time-independent energies given dotted lines. (b) The output power of the low **frequency** **oscillator** at 300 MHz, as a function of the static magnetic flux bias: γ=0.005 per cycle (solid line), γ=0.05 per cycle (crosses), γ=0.5 per cycle (circles). The other parameters are given in the text.
...We propose a method for characterising the energy level structure of a solid state **qubit** by monitoring the noise level in its environment. We consider a model persistent current **qubit** in a lossy reservoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field....Schematic diagram of persistent current **qubit** [6] inductively coupled to a (low **frequency**) classical **oscillator**. The insert graph shows the time-averaged (Floquet) energies as a function of the external bias field Φx1 for the parameters given in the text.
...Persistent current **qubit** ... We propose a method for characterising the energy level structure of a solid state **qubit** by monitoring the noise level in its environment. We consider a model persistent current **qubit** in a lossy reservoir and demonstrate that the noise in a classical bias field is a sensitive function of the applied field.

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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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