### 63610 results for qubit oscillator frequency

Contributors: Green, T. J., Biercuk, M. J.

Date: 2014-08-12

We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic **oscillator** modes. Such systems are particularly susceptible to residual **qubit**-**oscillator** entanglement at the conclusion of a gate period which reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete phase shifts in the field moderating the **qubit**-**oscillator** interaction - easily implemented with modern synthesizers - is sufficient to both ensure multiple **oscillator** modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting **qubits** and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach allows multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols....Fig:F3 Laser amplitude noise filter functions for 3 excited **oscillator** modes k = 1 , 2 , 3 . The effect of the noise on the coupling to each mode is suppressed to third order for k = 1 ( F 1 ω ∝ ω 3 as ω → 0 ), second order for k = 2 ( F 2 ω ∝ ω 2 ), and first order for k = 3 ( F 3 ω ∝ ω ) ....Fig:F3 Laser amplitude noise filter functions for 3 excited oscillator modes k = 1 , 2 , 3 . The effect of the noise on the coupling to each mode is suppressed to third order for k = 1 ( F 1 ω ∝ ω 3 as ω → 0 ), second order for k = 2 ( F 2 ω ∝ ω 2 ), and first order for k = 3 ( F 3 ω ∝ ω ) ....Fig Fig:F2. shows closed mode trajectories representing the complete decoupling of a pair of 171 Yb + hyperfine qubits from five excited transverse phonon modes , using phase-shifts derived from Eq. eq:decouplingcond. In this illustrative example, we set the laser frequencies so that two modes have commensurate detunings and choose τ s = 2 π / δ 1 , 5 to match the period of the associated phase space evolution. In this way, a sequence of only n = 7 phase shifts is required to decouple the qubits from all 5 modes, rather than the more general sequence of n = 31 phase shifts. Assuming equivalent physical parameters to recent demonstrations of multimode decoupling using optimized amplitude modulation , the resulting phase-modulate gate has duration τ ∼ 140 μ s which compares favorably with the reported value of τ = 190 μ s, while obviating considerations of nonlinear amplitude responses in optical modulators and rf amplifiers. Faster gate times may be achieved, at the expense of a greater number of phase shifts, by allowing τ s to vary arbitrarily....Fig:F2 a) Raman laser geometry for a MS gate applied to 2 ions in a 5 ion chain in which only transverse ( x -direction) phonon modes are excited. The red (r) and blue (b) Raman fields have **frequencies** ω r / b and phases φ r / b . b) Detuning diagram for 5 excited TP modes, ω ~ k = ω 0 ± ω k ( + for ω b and - for ω r ), where ω 0 is the hyperfine **qubit** level splitting. c) Closed paths, α k , ≡ | δ k | α k t , 0 ≤ t ≤ τ , (normalized by | δ k | ) for the detunings shown in b), generated by 7 discrete phase shifts....In Fig. Fig:F3 we plot F k ω calculated for specific, but arbitrarily chosen, orders associated with each of three modes, k = 1 , 2 , 3 . By increasing the level of concatenation for specific modes we are able to improve qubit-oscillator decoupling through the suppression of low-frequency amplitude fluctuations while simultaneously ensuring all modes are efficiently decoupled. In general, D k depends on the initial qubit state | φ 0 and the effective temperature, in addition to the frequency of the k -th mode ω k . Here, where we consider only the collective zero temperature limit and the particular initial state | φ 0 = | 11 z ....Fig Fig:F2. shows closed mode trajectories representing the complete decoupling of a pair of 171 Yb + hyperfine **qubits** from five excited transverse phonon modes , using phase-shifts derived from Eq. eq:decouplingcond. In this illustrative example, we set the laser **frequencies** so that two modes have commensurate detunings and choose τ s = 2 π / δ 1 , 5 to match the period of the associated phase space evolution. In this way, a sequence of only n = 7 phase shifts is required to decouple the **qubits** from all 5 modes, rather than the more general sequence of n = 31 phase shifts. Assuming equivalent physical parameters to recent demonstrations of multimode decoupling using optimized amplitude modulation , the resulting phase-modulate gate has duration τ ∼ 140 μ s which compares favorably with the reported value of τ = 190 μ s, while obviating considerations of nonlinear amplitude responses in optical modulators and rf amplifiers. Faster gate times may be achieved, at the expense of a greater number of phase shifts, by allowing τ s to vary arbitrarily....for k = 1 , . . . , M . Each of time-parameterized functions α k t , 0 ≤ t ≤ τ , defines a set of N phase space trajectories α k μ t = f k μ α k t , for μ = 1 , . . . , N , associated with the k -th **oscillator** mode (Fig. Fig:F1). These trajectories vary in extent and orientation, according to the complex coupling constant f k μ . However, by satisfying the condition ( eq:deccond2) all trajectories are closed at t = τ ....for k = 1 , . . . , M . Each of time-parameterized functions α k t , 0 ≤ t ≤ τ , defines a set of N phase space trajectories α k μ t = f k μ α k t , for μ = 1 , . . . , N , associated with the k -th oscillator mode (Fig. Fig:F1). These trajectories vary in extent and orientation, according to the complex coupling constant f k μ . However, by satisfying the condition ( eq:deccond2) all trajectories are closed at t = τ ....Phase-modulated decoupling and error suppression in **qubit**-**oscillator** systems...In Fig. Fig:F3 we plot F k ω calculated for specific, but arbitrarily chosen, orders associated with each of three modes, k = 1 , 2 , 3 . By increasing the level of concatenation for specific modes we are able to improve **qubit**-**oscillator** decoupling through the suppression of low-**frequency** amplitude fluctuations while simultaneously ensuring all modes are efficiently decoupled. In general, D k depends on the initial **qubit** state | φ 0 and the effective temperature, in addition to the **frequency** of the k -th mode ω k . Here, where we consider only the collective zero temperature limit and the particular initial state | φ 0 = | 11 z . ... We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic **oscillator** modes. Such systems are particularly susceptible to residual **qubit**-**oscillator** entanglement at the conclusion of a gate period which reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete phase shifts in the field moderating the **qubit**-**oscillator** interaction - easily implemented with modern synthesizers - is sufficient to both ensure multiple **oscillator** modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting **qubits** and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach allows multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols.

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Contributors: de Groot, P. C., Lisenfeld, J., Schouten, R. N., Ashhab, S., Lupascu, A., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-08-06

Coupled **qubit** system and transitions. a, Optical micrograph of the sample, showing two flux qubits colored in blue and red. The inset shows part of each **qubit** loop, both containing four Josephson tunnel junctions. Overlapping the **qubit** loops, in light-grey, are the SQUID-based **qubit**-state detectors. In the top right and bottom left are the two antennas from which the qubits are driven. b, Energy level diagram of the coupled **qubit** system. Arrows of the same color indicate transitions of the same **qubit** and are degenerate in **frequency**. c, Pulse sequence used for the coherent excitation of the qubits. The first pulse is resonant with **qubit** 1. The second pulse, applied from both antennas simultaneously with independent amplitudes and phases, is resonant with **qubit** 2. After the second pulse the state of both qubits is read out. d, The normalized transition strengths of the four transitions in b as a function of the net driving amplitudes a 1 / a 1 + a 2 for ϕ 2 - ϕ 1 = 0 . For ϕ 2 - ϕ 1 = π the dashed and solid lines are interchanged. The black dotted lines indicate the locations of the darkened transitions....Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**....Coupled **qubit** system and transitions. a, Optical micrograph of the sample, showing two flux **qubits** colored in blue and red. The inset shows part of each **qubit** loop, both containing four Josephson tunnel junctions. Overlapping the **qubit** loops, in light-grey, are the SQUID-based **qubit**-state detectors. In the top right and bottom left are the two antennas from which the **qubits** are driven. b, Energy level diagram of the coupled **qubit** system. Arrows of the same color indicate transitions of the same **qubit** and are degenerate in **frequency**. c, Pulse sequence used for the coherent excitation of the **qubits**. The first pulse is resonant with **qubit** 1. The second pulse, applied from both antennas simultaneously with independent amplitudes and phases, is resonant with **qubit** 2. After the second pulse the state of both **qubits** is read out. d, The normalized transition strengths of the four transitions in b as a function of the net driving amplitudes a 1 / a 1 + a 2 for ϕ 2 - ϕ 1 = 0 . For ϕ 2 - ϕ 1 = π the dashed and solid lines are interchanged. The black dotted lines indicate the locations of the darkened transitions....Driving from a single antenna. Measurement of the state of the qubits, represented by switching probabilities P s w , 1 and P s w , 2 , after applying a pulse of duration τ 1 resonant with **qubit** 1, followed by a pulse of duration τ 2 resonant with **qubit** 2. a, P s w , 1 , showing coherent oscillations of **qubit** 1 induced by pulse 1. The white solid and dashed lines indicate a π - and 2 π -rotation respectively. For pulse 2, **qubit** 1 only shows relaxation. b, P s w , 2 , showing coherent oscillations induced by pulse 2. After an odd number of π -rotations on **qubit** 1, the oscillation **frequency** is higher than after an even number of π -rotations. For superposition states of **qubit** 1, a beating pattern of the two oscillations is observed. c-f, Level occupations Q of the four different levels. Note that a value of 0.2 has been added to Q 11 to improve visibility....Transition strength tuning and darkened transitions. a-c, Rabi **frequency** dependence on φ 2 - φ 1 for three different amplitude-ratios. The color scale represents the Fourier component of P s w , 2 τ 2 . **Qubit** 1 is prepared with a π / 2 -rotation. Markers X 0 and X 1 indicate the conditions for a darkened transition on 00 ↔ 01 and 10 ↔ 11 respectively. d-f P s w , 2 versus the durations τ 1 and τ 2 . The white solid and dashed lines indicate a π - and 2 π -rotation of **qubit** 1, respectively. The driving conditions are as marked by Y left arrow (d), X 0 (e) and X 1 (f)....Driving from a single antenna. Measurement of the state of the **qubits**, represented by switching probabilities P s w , 1 and P s w , 2 , after applying a pulse of duration τ 1 resonant with **qubit** 1, followed by a pulse of duration τ 2 resonant with **qubit** 2. a, P s w , 1 , showing coherent **oscillations** of **qubit** 1 induced by pulse 1. The white solid and dashed lines indicate a π - and 2 π -rotation respectively. For pulse 2, **qubit** 1 only shows relaxation. b, P s w , 2 , showing coherent **oscillations** induced by pulse 2. After an odd number of π -rotations on **qubit** 1, the **oscillation** **frequency** is higher than after an even number of π -rotations. For superposition states of **qubit** 1, a beating pattern of the two **oscillations** is observed. c-f, Level occupations Q of the four different levels. Note that a value of 0.2 has been added to Q 11 to improve visibility. ... Controlled manipulation of quantum states is central to studying natural and artificial quantum systems. If a quantum system consists of interacting sub-units, the nature of the coupling may lead to quantum levels with degenerate energy differences. This degeneracy makes **frequency**-selective quantum operations impossible. For the prominent group of transversely coupled two-level systems, i.e. **qubits**, we introduce a method to selectively suppress one transition of a degenerate pair while coherently exciting the other, effectively creating artificial selection rules. It requires driving two **qubits** simultaneously with the same **frequency** and specified relative amplitude and phase. We demonstrate our method on a pair of superconducting flux **qubits**. It can directly be applied to the other superconducting **qubits**, and to any other **qubit** type that allows for individual driving. Our results provide a single-pulse controlled-NOT gate for the class of transversely coupled **qubits**.

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Contributors: Neto, O. P. de Sa, de Oliveira, M. C., Caldeira, A. O.

Date: 2011-04-27

When the atomic Hamiltonian is diagonalized, the first two energy levels as a function of the gate charge n g ≡ C g V g / 2 e are described in fig. fig2, where the vertical axis represents the energy and the horizontal represents the gate charge which is limited by the gate voltage. Changing the basis through a rotation, σ z → σ x and σ x → - σ z , and going to the rotating frame with the field frequency, ω , through R f = exp i ω t σ z + a a , gives...with Ω t = ω - E J ℏ c o s π Φ x t Φ 0 , as given by the second order terms from Eq. ( 10). From Eq. ( 11) it is possible to understand that as the qubit is brought closer to resonance with the resonator field, it will imprint an accumulated phase on it, given by Im θ ± t conditioned on the qubit state | ± . In Fig. fig4 we depict the numerical results for those two conditioned accumulated phases. We see that practically only when the qubit is in the state | - the phase in α is changed. With an appropriate accumulation of -3 π , as shown in Fig. fig4a, it is possible to create a state if the qubit is initially in the | - state. We have consistently checked that this approximation is indeed very good, not only for small α , if we respect a balance between the field intensity and the operation time. Moreover, we observed that around the time of optimal phase accumulation, t o p = 7.5 ns , the real part of θ ± t , related to damping or amplification, is negligible, ( ≈ 10 -3 - 10 -4 ), so that in Eq. ( 11), f ± t = 0 at the instant of measurement as shown in the Appendix, and shall not be considered from now on. To generate the superposition of coherent states for the field, we note that due to the low temperature of the system it is easy to prepare the qubit state initially in . Obviously the decoupled coherent state of the resonator and the qubit state may be written as . If the aforementioned pulse is applied to the qubit, coupling it to the resonator field through the evolution given by Eq. ( 10), we shall have U(t 0+ t,t 0)—0 — Pulse —- —- + —+ —2 14, or [ which is an entangled state between the qubit and the resonator field. Consequently, the resonator field can be left in an odd or even superposition of coherent states depending on the detection of the qubit state with a single electron transistor . Field state decoherence and qubit relaxation probing Certainly, the exact preparation of such a state is compromised by external noise. In contrast to experiments with microwave fields and Rydberg atoms, dissipative effects are most noticeable for the qubit states, which can flip from the ground state to the excited one and vice versa due to thermal effects and inductive coupling of the qubit to the external circuit. While the relaxation time of the qubits are of the order of 10 -6 s , the relaxation of the resonator field is of the order of 10 -3 s and can, in principle, be neglected. If compared with the time of the pulses for the accumulated phase of the field, Δ t ≈ 7.5 × 10 -9 s , the relaxation of the atom is negligible as well, but certainly will be important if any further manipulation is to be executed. So in fact the effects of dissipation are more relevant after the pulse is applied, i.e., when Φ x t = 0 , and will affect the probability to detect a given qubit state, and consequently the generation of an appropriate field. To understand the effects of noise in the system, we couple the qubit two-state to a bath of harmonic oscillators in an adaptation of the standard spin-boson model with Ohmic dissipation , through the Hamiltonian H = H S + H A R + H R . Here H S is given by Eq. ( s), presented in reference , with the convention at the degeneracy point , and . Eq. ( s) can be conveniently rewriten as...Phase of the resonator coherent field, due to the time dependent interaction with the **qubit** prepared in a ) the ground state, and b ) the excited state. The pulse Φ x t oscillates for a half period ( Δ t ≈ 7.5 ns ) with frequency 8 π × 10 6 Hz ....We demonstrate how a superposition of coherent states can be generated for a microwave field inside a coplanar transmission line coupled to a single superconducting charge **qubit**, with the addition of a single classical magnetic pulse for chirping of the **qubit** transition **frequency**. We show how the **qubit** dephasing induces decoherence on the field superposition state, and how it can be probed by the **qubit** charge detection. The character of the charge **qubit** relaxation process itself is imprinted in the field state decoherence profile....(a) Schematic setup, with the central transmission line (resonator) capacitively coupled to the source and drain, and capacitively coupled to a SQUID**-type** **qubit**. (b) Variable energy levels of the **qubit** ( 0 < n g < 1 ), with an external classical magnetic flux Φ x t . The resonator field is always blue detuned from that transition....Cavity quantum electrodynamics in superconducting circuits offer a exquisite playground for quantum information processing, and has provided the first coherent coupling between an “artificial atom”, the charge qubit, and a field mode of a resonator . Mappings of qubit states , and also tests for fundamental problems, such as the Purcell effect and photon number state resolving have also been achieved. The setup employed in all those remarkable experiments is shown in Fig. fig1, where a niobium transmission line resonator is capacitively coupled to a source (on the left) and to a drain (on the right). The resonator is also capacitively coupled to the charge states of a SQUID . The advantage of employing a SQUID is that the charge states can be addressed and manipulated in such a way to be set close or far from resonance with a given resonator field mode by an externally applied classical magnetic flux. By considering only the ground and the first excited states near the charge degeneracy point, the superconducting device can be well approximated by a two level system (Fig. 1b), here addressed as a qubit. In this regime the Hamiltonian describing a quantized electromagnetic field mode coupled to the charge qubit is given by...Generation of Superposition States and Charge-**Qubit** Relaxation Probing in a Circuit...(a) Schematic setup, with the central transmission line (resonator) capacitively coupled to the source and drain, and capacitively coupled to a SQUID-type **qubit**. (b) Variable energy levels of the **qubit** ( 0 < n g < 1 ), with an external classical magnetic flux Φ x t . The resonator field is always blue detuned from that transition....with Ω t = ω - E J ℏ c o s π Φ x t Φ 0 , as given by the second order terms from Eq. ( 10). From Eq. ( 11) it is possible to understand that as the **qubit** is brought closer to resonance with the resonator field, it will imprint an accumulated phase on it, given by Im θ ± t conditioned on the **qubit** state | ± . In Fig. fig4 we depict the numerical results for those two conditioned accumulated phases. We see that practically only when the **qubit** is in the state | - the phase in α is changed. With an appropriate accumulation of -3 π , as shown in Fig. fig4a, it is possible to create a state if the **qubit** is initially in the | - state. We have consistently checked that this approximation is indeed very good, not only for small α , if we respect a balance between the field intensity and the operation time. Moreover, we observed that around the time of optimal phase accumulation, t o p = 7.5 ns , the real part of θ ± t , related to damping or amplification, is negligible, ( ≈ 10 -3 - 10 -4 ), so that in Eq. ( 11), f ± t = 0 at the instant of measurement as shown in the Appendix, and shall not be considered from now on. To generate the superposition of coherent states for the field, we note that due to the low temperature of the system it is easy to prepare the **qubit** state initially in . Obviously the decoupled coherent state of the resonator and the **qubit** state may be written as . If the aforementioned pulse is applied to the **qubit**, coupling it to the resonator field through the evolution given by Eq. ( 10), we shall have U(t 0+ t,t 0)—0 — Pulse —- —- + —+ —2 14, or [ which is an entangled state between the **qubit** and the resonator field. Consequently, the resonator field can be left in an odd or even superposition of coherent states depending on the detection of the **qubit** state with a single electron transistor . Field state decoherence and **qubit** relaxation probing Certainly, the exact preparation of such a state is compromised by external noise. In contrast to experiments with microwave fields and Rydberg atoms, dissipative effects are most noticeable for the **qubit** states, which can flip from the ground state to the excited one and vice versa due to thermal effects and inductive coupling of the **qubit** to the external circuit. While the relaxation time of the **qubits** are of the order of 10 -6 s , the relaxation of the resonator field is of the order of 10 -3 s and can, in principle, be neglected. If compared with the time of the pulses for the accumulated phase of the field, Δ t ≈ 7.5 × 10 -9 s , the relaxation of the atom is negligible as well, but certainly will be important if any further manipulation is to be executed. So in fact the effects of dissipation are more relevant after the pulse is applied, i.e., when Φ x t = 0 , and will affect the probability to detect a given **qubit** state, and consequently the generation of an appropriate field. To understand the effects of noise in the system, we couple the **qubit** two-state to a bath of harmonic **oscillators** in an adaptation of the standard spin-boson model with Ohmic dissipation , through the Hamiltonian H = H S + H A R + H R . Here H S is given by Eq. ( s), presented in reference , with the convention at the degeneracy point , and . Eq. ( s) can be conveniently rewriten as...Probabilities of charge **qubit** detections and consequently of postselected | 0 L , or | 1 L field states. The decoherence of the preselected field state is given by 2 P 0 t - 1 ....Phase of the resonator coherent field, due to the time dependent interaction with the **qubit** prepared in a ) the ground state, and b ) the excited state. The pulse Φ x t **oscillates** for a half period ( Δ t ≈ 7.5 ns ) with **frequency** 8 π × 10 6 Hz ....When the atomic Hamiltonian is diagonalized, the first two energy levels as a function of the gate charge n g ≡ C g V g / 2 e are described in fig. fig2, where the vertical axis represents the energy and the horizontal represents the gate charge which is limited by the gate voltage. Changing the basis through a rotation, σ z → σ x and σ x → - σ z , and going to the rotating frame with the field **frequency**, ω , through R f = exp i ω t σ z + a a , gives...Cavity quantum electrodynamics in superconducting circuits offer a exquisite playground for quantum information processing, and has provided the first coherent coupling between an “artificial atom”, the charge **qubit**, and a field mode of a resonator . Mappings of **qubit** states , and also tests for fundamental problems, such as the Purcell effect and photon number state resolving have also been achieved. The setup employed in all those remarkable experiments is shown in Fig. fig1, where a niobium transmission line resonator is capacitively coupled to a source (on the left) and to a drain (on the right). The resonator is also capacitively coupled to the charge states of a SQUID . The advantage of employing a SQUID is that the charge states can be addressed and manipulated in such a way to be set close or far from resonance with a given resonator field mode by an externally applied classical magnetic flux. By considering only the ground and the first excited states near the charge degeneracy point, the superconducting device can be well approximated by a two level system (Fig. 1b), here addressed as a **qubit**. In this regime the Hamiltonian describing a quantized electromagnetic field mode coupled to the charge **qubit** is given by ... We demonstrate how a superposition of coherent states can be generated for a microwave field inside a coplanar transmission line coupled to a single superconducting charge **qubit**, with the addition of a single classical magnetic pulse for chirping of the **qubit** transition **frequency**. We show how the **qubit** dephasing induces decoherence on the field superposition state, and how it can be probed by the **qubit** charge detection. The character of the charge **qubit** relaxation process itself is imprinted in the field state decoherence profile.

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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....(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....Dynamics of relaxation, decoherence and entropy of a

**qubit**in anisotropic photonic crystals ... 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.

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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 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....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: Yu.A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D.V. Averin, T. Tilma, F. Nori, J.S. Tsai

Date: 2005-01-01

Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks.
...Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of** each** qubit, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant.
...Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks.
...EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the **oscillation** **frequencies** of both **qubits** in the case of zero coupling (Em=0).
...EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the oscillation frequencies of both qubits in the case of zero coupling (Em=0).
...Coherent manipulation of coupled Josephson charge **qubits**...Schematic diagram of the two-coupled-**qubit** circuit. Black bars denote Cooper pair boxes.
...Schematic diagram of the two-coupled-qubit circuit. Black bars denote Cooper pair boxes.
...Probe current **oscillations** in the first (a) and the second (b) **qubit** when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of each **qubit**, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant.
...We have analyzed and measured the quantum coherent dynamics of a circuit containing two-coupled superconducting charge **qubits**. Each **qubit** is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two **qubits** are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation of the **qubit** circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each **qubit** by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup, the measured pulse-induced probe currents are proportional to the probability for each **qubit** to have an extra Cooper pair after the manipulation. As expected from theory and observed experimentally, the measured pulse-induced current in each probe has two **frequency** components whose position on the **frequency** axis can be externally controlled. This is a result of the inter-**qubit** coupling which is also responsible for the avoided level crossing that we observed in the **qubits**’ spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape, the system remains entangled most of the time reaching maximally entangled states at certain instances....Solid-state **qubits** ... We have analyzed and measured the quantum coherent dynamics of a circuit containing two-coupled superconducting charge **qubits**. Each **qubit** is based on a Cooper pair box connected to a reservoir electrode through a Josephson junction. Two **qubits** are coupled electrostatically by a small island overlapping both Cooper pair boxes. Quantum state manipulation of the **qubit** circuit is done by applying non-adiabatic voltage pulses to the common gate. We read out each **qubit** by means of probe electrodes connected to Cooper pair boxes through high-Ohmic tunnel junctions. With such a setup, the measured pulse-induced probe currents are proportional to the probability for each **qubit** to have an extra Cooper pair after the manipulation. As expected from theory and observed experimentally, the measured pulse-induced current in each probe has two **frequency** components whose position on the **frequency** axis can be externally controlled. This is a result of the inter-**qubit** coupling which is also responsible for the avoided level crossing that we observed in the **qubits**’ spectra. Our simulations show that in the absence of decoherence and with a rectangular pulse shape, the system remains entangled most of the time reaching maximally entangled states at certain instances.

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

Date: 2012-01-19

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...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...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 ....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 ....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...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....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 $\ket{2_{s},0_{a}}$....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 ....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...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 e frequency conversion ....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 ....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....Coherent **frequency** conversion in a superconducting artificial atom with two internal degrees of freedom ... 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: 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

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....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)....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....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....Entanglement of Atomic **Qubits** using an Optical **Frequency** Comb...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....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....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....(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....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....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....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....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....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 ):...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....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.

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

Date: 2005-07-21

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

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Contributors: Gambetta, J. M., Houck, A. A., Blais, Alexandre

Date: 2010-09-22

A superconducting **qubit** with Purcell protection and tunable coupling...In this letter we will present a three island device that has the properties of a **qubit** (two levels, arbitrary control and measurement) and has the ability to independently tune both the resonance **frequency** and coupling strength g , whilst still exhibiting exponential suppression of the charge noise and maintaining an anharmonicity equivalent to that of the transmon and phase **qubit**. This tunable coupling **qubit** (TCQ) can be tuned from a configuration which is totally Purcell protected from the resonator g = 0 (in a DFS) to a position which couples strongly to the resonator with values comparable to those realized for the transmon. Furthermore, we show that in the DFS position a strong measurement can be performed. The TCQ only needs to be moved from the DFS position when single and two **qubit** gates are required and as such in the off position all multi-**qubit** coupling rates are zero. That is, the TCQ in the circuit QED architecture (see Fig. Fig:TCQ A) is its own tunable coupler to any other TCQ, going beyond the nearest neighbour tunable couplers presented in Refs. and ....Fig:Tune(color online) Matrix element of the collective Cooper pair number operator (A) and transition energy (B) of the dark state as a function of the energy ratio E J + / E C for n ' g + = n ' g - = 0 , E I = - E C and E J - is numerically solved to ensure that only the coupling strength is tuned (blue) and **frequency** (red). Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model....We present a superconducting **qubit** for the circuit quantum electrodynamics architecture that has a tunable coupling strength g. We show that this coupling strength can be tuned from zero to values that are comparable with other superconducting **qubits**. At g = 0 the **qubit** is in a decoherence free subspace with respect to spontaneous emission induced by the Purcell effect. Furthermore we show that in the decoherence free subspace the state of the **qubit** can still be measured by either a dispersive shift on the resonance **frequency** of the resonator or by a cycling-type measurement....Since charge fluctuations are one of the leading sources of noise in superconducting circuits we want to ensure that quantum information in the TCQ is not destroyed by charge noise. Following Ref. , the dephasing time T φ for the **qubit** and m level will scale as 1 / | ε q m | where ε q m is the the peak to peak value for the charge dispersion of the 0 - 1 and 0 - m transition respectively. The dispersion in the energy levels arises from the gate charges n ' g α and the fact the the potential is periodic. This can not be predicted with the coupled anharmonic **oscillator** model and as such is investigated numerically. We expect, that like the transmon, this will exponentially decrease with the ratio of E J / E C as in this limit the effects of tunneling from one minima to the next becomes exponentially suppressed. This is confirmed in Fig. Fig:Levels B where the have plotted | ε q m | / E q m (the numerical maximum and minimum of the energy level over n ' g α ) as a function of E J / E C for E I = E C and E I = 0 (transmon limit). That is, the TCQ has the same charge noise immunity as the transmon....Currently the most successful superconducting **qubits** are the flux , phase , and transmon as these **qubits** are essentially immune to offset charge (charge noise) by design. The transmon receives its charge noise immunity by operating at a point in parameter space where the energy level variations with offset charge are exponentially suppressed. This suppression has experimentally been observed and resulted in these **qubits** being approximately T 1 limited ( T 2 ≈ 2 T 1 ) in the circuit quantum electrodynamics (QED) architecture . In this architecture the **qubits** are coupled to a coplanar waveguide resonator through a Jaynes-Cummings Hamiltonian operated in the dispersive regime . This resonator acts as the channel to control, couple, and readout the state of the **qubit** (see Fig. Fig:TCQ A)....Since charge fluctuations are one of** the **leading sources of noise in superconducting circuits we want to ensure that quantum information in** the **TCQ is not destroyed by charge noise. Following Ref. , the dephasing time T φ **for** the qubit and m level will scale as 1 / | ε q m | where ε q m is** the **the peak to peak value **for** the charge dispersion of** the **0 - 1 and 0 - m transition respectively. The dispersion in** the **energy levels arises from** the **gate charges n ' g α and** the **fact** the **the potential is periodic. This can not be predicted with** the **coupled anharmonic oscillator model and as such is investigated numerically. We expect, that like** the **transmon, this will exponentially decrease with** the **ratio of E J / E C as in this limit** the **effects of tunneling from one minima to** the **next becomes exponentially suppressed. This is confirmed in Fig. Fig:Levels B where** the **have plotted | ε q m | / E q m (the numerical maximum and minimum of** the **energy level over n ' g α ) as a function of E J / E C **for** E I = E C and E I = 0 (transmon limit). That is, the TCQ has** the **same charge noise immunity as** the **transmon....To achieve tuning of g ~ + we modify the original circuit and replace the Josephson junctions by SQUIDs with Josephson energy E J ± 1 and E J ± 2 (this is hinted at in Fig. Fig:TCQ B). In making this replacement the only change in the above theory is the replacement E J ± → E J ± m a x cos π Φ x ± / Φ 0 1 + d 2 tan 2 π Φ x / Φ 0 with E J ± m a x = E J ± 1 + E J ± 2 , d = E J ± 1 - E J ± 2 / E J ± m a x and Φ x ± is the external flux applied to each SQUID which we assume to be independent (this is not required but simplifies our argument). This independent control allows us to change E J ± independently which in-turn allows independent control on g ~ + and ω q . To illustrate this we consider the symmetric case and plot in Fig. Fig:Tune the normalized coupling strength g ~ + ℏ / 2 e 2 V r m s β (A) and ω ~ q (B) as a function of the ratio E J + / E C when E I = - E C and E J - is numerically solved to ensure that only the coupling rate (blue) and **frequency** (red) vary respectively for both the full numerical (solid) and effective model (dashed). In the full numerical model g ~ + = 2 e 2 V r m s 1 | ( β + n + + β - n - | 0 / ℏ . Here the independent control is clearly observed. Note that while our numerical investigation was only for the symmetric case independent tunable g ~ + (from zero to large values) and ω q will still occur when the device is not symmetric. There is just a different condition on E J ± for the required tuning....In this letter we will present a three island device that has** the **properties of a qubit (two levels, arbitrary control and measurement) and has** the **ability to independently tune both** the **resonance frequency and coupling strength g , whilst still exhibiting exponential suppression of** the **charge noise and maintaining an anharmonicity equivalent to that of** the **transmon and phase qubit. This tunable coupling qubit (TCQ) can be tuned from a configuration which is totally Purcell protected from** the **resonator g = 0 (in a DFS) to a position which couples strongly to** the **resonator with values comparable to those realized **for** the transmon. Furthermore, we show that in** the **DFS position a strong measurement can be performed. The TCQ only needs to be moved from** the **DFS position when single and two qubit gates are required and as such in** the **off position all multi-qubit coupling rates are zero. That is, the TCQ in** the **circuit QED architecture (see Fig. Fig:TCQ A) is its own tunable coupler to any other TCQ, going beyond** the **nearest neighbour tunable couplers presented in Refs. and ....Currently** the **most successful superconducting qubits are** the **flux , phase , and transmon as these qubits are essentially immune to offset charge (charge noise) by design. The transmon receives its charge noise immunity by operating at a point in parameter space where** the **energy level variations with offset charge are exponentially suppressed. This suppression has experimentally been observed and resulted in these qubits being approximately T 1 limited ( T 2 ≈ 2 T 1 ) in** the **circuit quantum electrodynamics (QED) architecture . In this architecture** the **qubits are coupled to a coplanar waveguide resonator through a Jaynes-Cummings Hamiltonian operated in** the **dispersive regime . This resonator acts as** the **channel to control, **couple**, and readout** the **state of** the **qubit (see Fig. Fig:TCQ A)....To achieve tuning of g ~ + we modify** the **original circuit and replace** the **Josephson junctions by SQUIDs with Josephson energy E J ± 1 and E J ± 2 (this is hinted at in Fig. Fig:TCQ B). In making this replacement** the **only change in** the **above theory is** the **replacement E J ± → E J ± m a x cos π Φ x ± / Φ 0 1 + d 2 tan 2 π Φ x / Φ 0 with E J ± m a x = E J ± 1 + E J ± 2 , d = E J ± 1 - E J ± 2 / E J ± m a x and Φ x ± is** the **external flux applied to each SQUID which we assume to be independent (this is not required but simplifies our argument). This independent control allows us to change E J ± independently which in-turn allows independent control on g ~ + and ω q . To illustrate this we consider** the **symmetric case and plot in Fig. Fig:Tune** the **normalized coupling strength g ~ + ℏ / 2 e 2 V r m s β (A) and ω ~ q (B) as a function of** the **ratio E J + / E C when E I = - E C and E J - is numerically solved to ensure that only** the **coupling rate (**blue) and** frequency (red) vary respectively **for** both** the **full numerical (solid) and effective model (dashed). In** the **full numerical model g ~ + = 2 e 2 V r m s 1 | ( β + n + + β - n - | 0 / ℏ . Here** the **independent control is clearly observed. Note that while our numeri...Fig:Levels(color online) A) Eigenenergies of the TCQ Hamiltonian as a function of E I / E C for E J ± = 50 E C . Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model. B) Charge dispersion | ε q m | as a function of the ratio E J / E C for E I = - E C (solid lines) and E I = 0 (dashed lines)....Fig:Tune(color online) Matrix element of the collective Cooper pair number operator (A) and transition energy (B) of the dark state as a function of the energy ratio E J + / E C for n ' g + = n ' g - = 0 , E I = - E C and E J - is numerically solved to ensure that only the coupling strength is tuned (blue) and frequency (red). Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic **oscillator** model....where ω ~ ± = ω ± + δ ~ ± - δ ± / 2 + δ + + δ - J 2 / 2 μ 2 ± μ / 2 ∓ η / 2 , δ ~ ± = δ + + δ - 1 + η 2 / μ 2 / 4 ± η δ + - δ - / 2 μ and δ ~ c = 2 J 2 δ + + δ - / μ 2 with μ = 4 J 2 + η 2 and the tilde indicating the diagonalized frame. The coupling has induced a conditional anharmonicity δ ~ c , it is this anharmonicity that makes this system different to two coupled **qubits**, it ensures that E 11 is not equal to E 01 + E 10 . Here we have introduced the notation that superscript i j refers to i excitations in the dark mode ( ` ` + " ) and j excitations in the bright mode ( ` ` - " ). The choice of these names will become clearer latter. The dotted lines in Fig. Fig:Levels A are the predictions from this effective model, which agree well with the full numerics. Thus from the effective model the anharmonicities are all around E C provided | J | > | η | . That is the TCQ has not lost any anharmonicity in comparison to the transmon or phase **qubit** and with simple pulse shaping techniques arbitrary control of the lowest three levels will be possible . We will now introduce the notation that the **qubit** is formed by the space | 0 = | 00 ~ , | 1 = | 10 ~ and | m = | 01 ~ is the measurement state....where ω ~ ± = ω ± + δ ~ ± - δ ± / 2 + δ + + δ - J 2 / 2 μ 2 ± μ / 2 ∓ η / 2 , δ ~ ± = δ + + δ - 1 + η 2 / μ 2 / 4 ± η δ + - δ - / 2 μ and δ ~ c = 2 J 2 δ + + δ - / μ 2 with μ = 4 J 2 + η 2 and** the **tilde indicating** the **diagonalized frame. The coupling has induced a conditional anharmonicity δ ~ c , it is this anharmonicity that makes this system different to two coupled qubits, it ensures that E 11 is not equal to E 01 + E 10 . Here we have introduced** the **notation that superscript i j refers to i excitations in** the **dark mode ( ` ` + " ) and j excitations in** the **bright mode ( ` ` - " ). The choice of these names will become clearer latter. The dotted lines in Fig. Fig:Levels A are** the **predictions from this effective model, which agree well with** the **full numerics. Thus from** the **effective model** the **anharmonicities are all around E C provided | J | > | η | . That is** the **TCQ has not lost any anharmonicity in comparison to** the **transmon or phase qubit and with simple pulse shaping techniques arbitrary control of** the **lowest three levels will be possible . We will now introduce** the **notation that** the **qubit is formed by** the **space | 0 = | 00 ~ , | 1 = | 10 ~ and | m = | 01 ~ is** the **measurement state. ... We present a superconducting **qubit** for the circuit quantum electrodynamics architecture that has a tunable coupling strength g. We show that this coupling strength can be tuned from zero to values that are comparable with other superconducting **qubits**. At g = 0 the **qubit** is in a decoherence free subspace with respect to spontaneous emission induced by the Purcell effect. Furthermore we show that in the decoherence free subspace the state of the **qubit** can still be measured by either a dispersive shift on the resonance **frequency** of the resonator or by a cycling-type measurement.

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