### 63697 results for qubit oscillator frequency

Contributors: Shahriar, M. S., Pradhan, Prabhakar

Date: 2002-12-19

Fundamental limitation on **qubit** operations due to the Bloch-Siegert Oscillation...Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert **Oscillation** (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi **oscillation**. (b) The BSO **oscillation** (amplified scale) by itself, produced by subtracting the Rabi **oscillation** from the plot in (a). (c) The time-dependence of the Rabi **frequency**. Inset: BSO as a function of the absolute phase of the field with fixed τ ....We show that if the Rabi **frequency** is comparable to the Bohr **frequency** so that the rotating wave approximation is inappropriate, an extra oscillation is present with the Rabi oscillation. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-**frequency** transitions. We present a scheme for observing this effect in an atomic beam....We show that if the Rabi **frequency** is comparable to the Bohr **frequency** so that the rotating wave approximation is inappropriate, an extra **oscillation** is present with the Rabi **oscillation**. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-**frequency** transitions. We present a scheme for observing this effect in an atomic beam....Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert Oscillation (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi oscillation. (b) The BSO oscillation (amplified scale) by itself, produced by subtracting the Rabi oscillation from the plot in (a). (c) The time-dependence of the Rabi **frequency**. Inset: BSO as a function of the absolute phase of the field with fixed τ . ... We show that if the Rabi **frequency** is comparable to the Bohr **frequency** so that the rotating wave approximation is inappropriate, an extra **oscillation** is present with the Rabi **oscillation**. We discuss how the sensitivity of the degree of excitation to the phase of the field may pose severe constraints on precise rotations of quantum bits involving low-**frequency** transitions. We present a scheme for observing this effect in an atomic beam.

Files:

Contributors: Shevchenko, S. N., Ashhab, S., Nori, Franco

Date: 2011-10-17

Inverse Landau-Zener-Stuckelberg problem for **qubit**-resonator systems...Superconducting **qubit**, quantum capacitance, nanomechanical
resonator, Landau-Zener transition, Stuckelberg **oscillations**, interferometry.%
...Superconducting **qubit**, quantum capacitance, nanomechanical
resonator, Landau-Zener transition, Stuckelberg oscillations, interferometry.%
...We display the direct LZS interferometry in Fig**. **Fig:Dw, where the resonator’**s **frequency shift Δ ω N R was calculated with Eqs. ( DwNR_2) and ( CQ2). Figure Fig:Dw demonstrates that our formalism is valid for a description of the experimentally measurable quantities: the quantum capacitance or the resonant frequency shift , (see also Appendix C). Such a description allows to correctly find the position of the resonance peaks in the interferogram and to demonstrate the sign-changing behavior of the quantum capacitance, which relates to the measurable quantities. The appearance of the interferogram depends on several factors: the values of the qubit parameters, the model for the dissipative environment (such as Eqs. ( T1, T2) and the parameters α and B ), the value of the bias current (which distorts the shape of the resonances, as demonstrated in Ref. [...In Section III, we formulate the inverse problem. There, we are interested in the influence of the NR’s state (its position) on the ** qubit’s** state. We graphically demonstrate the formulation of the problem for the direct and inverse interferometry in Fig. Scheme0. There, the two-level system represents a

**qubit**with control parameter ε 0 ; the parabola represents the resonator’s potential energy as a function of the displacement x . Thus, in the first part of our work (Sec. II) we deal with the direct problem, where the influence of the

**state on the resonator is studied....(Color online) Schematic diagram of a split-junction charge**

**qubit**’s**qubit**coupled to a nanomechanical resonator. The charge

**qubit**(shown in red) is biased

**by**the magnetic flux Φ and the dc+ μ w voltage, V C P B + V M W , to which it is coupled through the capacitance C C P B . The

**qubit**is coupled to the NR (shown in green) through the capacitance C N R . The NR is biased

**by**a large dc voltage V N R ; its state is controlled and measured

**by**applying the dc and rf voltages between the gate and the NR, V G N R and V R F , through the capacitance C G N R . The NR’s motion is described

**by**the displacement at the midpoint x . Capacitances form the island (Cooper-pair box) with the total capacitance C Σ , voltage V I and charge -2 e n ....The idea of the measurement procedure, presented in Fig

**.**Fig2, could be as follows. Driving the qubit in a wide range of parameters is done first to plot the interferogram as in Fig

**.**Fig2(a) and/or (d). Then a region of high sensitivity, where small changes in the qubit bias result in large changes in the final state, is chosen. Examples of such high-sensitivity regions are shown in Fig

**.**Fig2(b) and/or (e)....(Color online) LZS interferometry probed via the resonator’s

**frequency**shift Δ ω N R . (a) The

**frequency**shift versus the energy bias ( n g ) and the driving amplitude ( n μ ). Arrows show the values of n μ and n g at which the graphs (b) and (c) are plotted as functions of n g and n μ , respectively. The upper curves were shifted for clarity. The parameters for calculations were taken close to the ones of Ref. [ LaHaye09]: ω N R / 2 π = 58 MHz, E J 0 / h = 13 GHz, E C / h = 14 GHz, ω / 2 π = 4 GHz, k B T / h = 2 GHz, α = 0.005 , B = 0.2 , and the proportionality coefficient β defined by the

**qubit**-NR coupling constant λ from Ref. [ LaHaye09]: ℏ λ 2 / π E J 0 = β ⋅ E C ω N R / π E J 0 = 1.6 kHz.... to the

**qubit**, measure its state at the end of the pulse and extract the resonator’s position x from the measured

**state, see Fig. Fig2(c) and (f), where ε 0 (which parametrically depends on x ) is plotted as a function of the**

**qubit**’s**occupation probability....The split-junction charge qubit (also called Cooper-pair box and shown in red in Fig**

**qubit**’s**.**Fig:scheme) consists of a small island between two Josephson junctions. The state of the qubit is controlled by the magnetic flux Φ and the gate voltage V C P B + V M W . Here V C P B is the dc voltage used to tune the energy levels of the qubit and V M W = V μ sin ω t is the microwave signal used to drive and manipulate the energy-level occupations. The Cooper-pair box is described in the two-level approximation by the Hamiltonian in the charge representation (see e.g. Ref. [...(Color online) Schematic representation of the formulated problems for direct and inverse interferometry. The red curves on the left represent the bias-dependent energy levels of the

**qubit**, and the green parabola on the right shows the potential energy of the (classical) resonator. In the direct problem, the resonator is used to probe the state of the

**qubit**. In the inverse problem, the response of the

**qubit**to external driving is used to infer the state of the resonator....(Color online) Slow-passage and fast-passage LZS interferometry of a

**qubit**. (a) and (d): the time-averaged upper-level occupation probabilities, defined in the adiabatic ( P + ¯ ) and diabatic ( P ¯ u p ) bases, as functions of the bias ε 0 and driving amplitude A . The parameters are the same as for Fig. Fig:Dw except for the

**frequency**: (a) ω / 2 π = 6.5 GHz Δ / h . (b) and (e): Cross-sections for the respective dependencies of the upper-level occupation probabilities as functions of the bias along the horizontal dashes shown in red and green in (a) and (d). (c) and (f): Inverse graphs, which show the dependence of the bias on the upper-level occupation probabilities (assuming that ε 0 lies on the right-hand side of the resonance peak)....(Color online) Slow-passage and fast-passage LZS interferometry of a

**qubit**. (a) and (d): the time-averaged upper-level occupation probabilities, defined in the adiabatic ( P + ¯ ) and diabatic ( P ¯ u p ) bases, as functions of the bias ε 0 and driving amplitude A . The parameters are the same as for Fig. Fig:Dw except for the frequency: (a) ω / 2 π = 6.5 GHz Δ / h . (b) and (e): Cross-sections for the respective dependencies of the upper-level occupation probabilities as functions of the bias along the horizontal dashes shown in red and green in (a) and (d). (c) and (f): Inverse graphs, which show the dependence of the bias on the upper-level occupation probabilities (assuming that ε 0 lies on the right-hand side of the resonance peak)....The split-junction charge

**qubit**(also called Cooper-pair box and shown in red in Fig. Fig:scheme) consists of a small island between two Josephson junctions. The state of the

**qubit**is controlled by the magnetic flux Φ and the gate voltage V C P B + V M W . Here V C P B is the dc voltage used to tune the energy levels of the

**qubit**and V M W = V μ sin ω t is the microwave signal used to drive and manipulate the energy-level occupations. The Cooper-pair box is described in the two-level approximation by the Hamiltonian in the charge representation (see e.g. Ref. [...We display the direct LZS interferometry in Fig. Fig:Dw, where the resonator’s

**frequency**shift Δ ω N R was calculated with Eqs. ( DwNR_2) and ( CQ2). Figure Fig:Dw demonstrates that our formalism is valid for a description of the experimentally measurable quantities: the quantum capacitance or the resonant

**frequency**shift , (see also Appendix C). Such a description allows to correctly find the position of the resonance peaks in the interferogram and to demonstrate the sign-changing behavior of the quantum capacitance, which relates to the measurable quantities. The appearance of the interferogram depends on several factors: the values of the

**qubit**parameters, the model for the dissipative environment (such as Eqs. ( T1, T2) and the parameters α and B ), the value of the bias current (which distorts the shape of the resonances, as demonstrated in Ref. [...where F A = ∂ C G N R / ∂ x ⋅ Δ V ⋅ V A . From the other side (left side of the NR in Fig

**.**Fig:scheme) the voltage difference is defined by the island’

**s**voltage V I . The respective force is...Sillanpaa06]: E J 0 / h = 12.5 GHz, E C / h = 24 GHz, ω / 2 π = 4 GHz, k B T / h = 1 GHz, and also we have taken α = 0.005 , B = 0.5 . We note that besides the difference in the parameters, in Fig

**.**Fig:Dw the frequency shift Δ ω was plotted, while in Fig

**.**Fig:CQ the quantum capacitance C Q was shown. Both figures were calculated by numerically solving the Bloch equation....(Color online) Scheme showing how the charge

**qubit**can be described as an effective capacitance coupled either to the NR or to L C R resonator. (a) To the left, the charge

**qubit**(CPB) is shown to be described as the capacitance 2 C J controlled

**by**the voltage V C P B and coupled through the coupling capacitance C N R to a measuring circuitry. This is described as the effective capacitance C e f f as shown to the right. (b) The effective capacitance is coupled to the NR, which can be used to model our system shown in Fig. Fig:scheme. (c) The effective capacitance is coupled to the electric L C R tank circuit....(Color online) Schematic diagram of a split-junction charge

**qubit**coupled to a nanomechanical resonator. The charge

**qubit**(shown in red) is biased by the magnetic flux Φ and the dc+ μ w voltage, V C P B + V M W , to which it is coupled through the capacitance C C P B . The

**qubit**is coupled to the NR (shown in green) through the capacitance C N R . The NR is biased by a large dc voltage V N R ; its state is controlled and measured by applying the dc and rf voltages between the gate and the NR, V G N R and V R F , through the capacitance C G N R . The NR’s motion is described by the displacement at the midpoint x . Capacitances form the island (Cooper-pair box) with the total capacitance C Σ , voltage V I and charge -2 e n ....Stückelberg

**oscillations**, described by Eq. ( Pp2), are demonstrated in Fig. PIPII(b) for 0

**oscillations**, the higher the sensitivity. This is related to the period of the Stückelberg

**oscillations**, which decreases with increasing A / ω . Here we also note that P + I I ε 0 is not a symmetric function, and the period of the Stückelberg

**oscillations**is smaller for ε 0 0 . Therefore, using negative values of ε 0 results in slightly higher sensitivity than what is shown in Fig. PIPII(d)....(Color online) LZS interferometry probed via the resonator’s frequency shift Δ ω N R . (a) The frequency shift versus the energy bias ( n g ) and the driving amplitude ( n μ ). Arrows show the values of n μ and n g at which the graphs (b) and (c) are plotted as functions of n g and n μ , respectively. The upper curves were shifted for clarity. The parameters for calculations were taken close to the ones of Ref. [ LaHaye09]: ω N R / 2 π = 58 MHz, E J 0 / h = 13 GHz, E C / h = 14 GHz, ω / 2 π = 4 GHz, k B T / h = 2 GHz, α = 0.005 , B = 0.2 , and the proportionality coefficient β defined

**by**the

**qubit**-NR coupling constant λ from Ref. [ LaHaye09]: ℏ λ 2 / π E J 0 = β ⋅ E C ω N R / π E J 0 = 1.6 kHz....(Color online) Scheme showing how the charge

**qubit**can be described as an effective capacitance coupled either to the NR or to L C R resonator. (a) To the left, the charge

**qubit**(CPB) is shown to be described as the capacitance 2 C J controlled by the voltage V C P B and coupled through the coupling capacitance C N R to a measuring circuitry. This is described as the effective capacitance C e f f as shown to the right. (b) The effective capacitance is coupled to the NR, which can be used to model our system shown in Fig. Fig:scheme. (c) The effective capacitance is coupled to the electric L C R tank circuit....We consider theoretically a superconducting

**qubit**- nanomechanical resonator (NR) system, which was realized by LaHaye et al. [Nature 459, 960 (2009)]. First, we study the problem where the state of the strongly driven

**qubit**is probed through the

**frequency**shift of the low-

**frequency**NR. In the case where the coupling is capacitive, the measured quantity can be related to the so-called quantum capacitance. Our theoretical results agree with the experimentally observed result that, under resonant driving, the

**frequency**shift repeatedly changes sign. We then formulate and solve the inverse Landau-Zener-Stuckelberg problem, where we assume the driven

**qubit**'s state to be known (i.e. measured by some other device) and aim to find the parameters of the

**qubit**'s Hamiltonian. In particular, for our system the

**qubit**'s bias is defined by the NR's displacement. This may provide a tool for monitoring of the NR's position. ... We consider theoretically a superconducting

**qubit**- nanomechanical resonator (NR) system, which was realized by LaHaye et al. [Nature 459, 960 (2009)]. First, we study the problem where the state of the strongly driven

**qubit**is probed through the

**frequency**shift of the low-

**frequency**NR. In the case where the coupling is capacitive, the measured quantity can be related to the so-called quantum capacitance. Our theoretical results agree with the experimentally observed result that, under resonant driving, the

**frequency**shift repeatedly changes sign. We then formulate and solve the inverse Landau-Zener-Stuckelberg problem, where we assume the driven

**qubit**'s state to be known (i.e. measured by some other device) and aim to find the parameters of the

**qubit**'s Hamiltonian. In particular, for our system the

**qubit**'s bias is defined by the NR's displacement. This may provide a tool for monitoring of the NR's position.

Files:

Contributors: Du, Lingjie, Yu, Yang

Date: 2010-12-13

(Color online). (a) and (b). Schematic energy diagram of Rabi **oscillation** induced interference. (a) describes the transition from state | 1 to | 0 . (b) describes the transition from state | 0 to | 1 . (c), (d) and (e). The interference pattern of population in state | 0 obtained from Eqs. (45), (46), and (47), respectively. The parameters used here are ω ~ / 2 π = 2 GHz, Ã / ω ~ = 0.9 , Γ 01 / 2 π = 0.000008 GHz and the temperature is 20 mK. Other parameters of the **qubit** are identical with Fig. 4 (a)....(Color online). (a). Schematic energy diagram of a flux **qubit**. The dotted curve represents the strong driving field A cos ω t . The field through the tunnel coupling Δ forms a LZS interference, exchanging photons **with **the **qubit**. (b). Quantum tunnel coupling exists between states | 0 and | 1 . The interaction between a **qubit** and an electromagnetic system (such as the environment bath or a single-mode electromagnetic field) would form new couplings between the two states....(Color online). (a). Schematic energy diagram of a flux **qubit**. The dotted curve represents the strong driving field A cos ω t . The field through the tunnel coupling Δ forms a LZS interference, exchanging photons with the **qubit**. (b). Quantum tunnel coupling exists between states | 0 and | 1 . The interaction between a **qubit** and an electromagnetic system (such as the environment bath or a single-mode electromagnetic field) would form new couplings between the two states....Electromagnetically induced interference at superconducting **qubits**...(Color online). The stationary population of relaxation induced interference. The pattern is obtained from Eq. (34). (a). The characteristic **frequency** ω c / 2 π = 0.05 GHz with the temperature 20 mK. Features of population inversion and periodical modulation are notable. (b). The characteristic **frequency** ω c / 2 π = 6 GHz with the temperature 20 mK. (c). The characteristic **frequency** ω c / 2 π = 0.05 GHz with the temperature 2 × 10 -5 mK. In above figures, the driving **frequency** ω / 2 π = 0.6 GHz....(Color online). (a). Schematic energy diagram of a strongly driven flux **qubit** interacting with a weak single-mode field. The green solid curve represents the weak field, forming effective coupling between states | 0 and | 1 ....We study electromagnetically induced interference at superconducting **qubits**. The interaction between **qubits** and electromagnetic fields can provide additional coupling channels to **qubit** states, leading to quantum interference in a microwave driven **qubit**. In particular, the interwell relaxation or Rabi **oscillation**, resulting respectively from the multi- or single-mode interaction, can induce effective crossovers. The environment is modeled by a multi-mode thermal bath, generating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the **qubits** and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic **frequency** of the bath. In addition, we demonstrate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi **oscillation**, Rabi **oscillation** induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the **qubit**, with more controllable parameters including the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation, in not only the flux **qubit** but also the systems with no crossover structure, e.g. phase **qubits**....(Color online). (a) and (b). Schematic energy diagram of Rabi oscillation induced interference. (a) describes the transition from state | 1 to | 0 . (b) describes the transition from state | 0 to | 1 . (c), (d) and (e). The interference pattern of population in state | 0 obtained from Eqs. (45), (46), and (47), respectively. The parameters used here are ω ~ / 2 π = 2 GHz, Ã / ω ~ = 0.9 , Γ 01 / 2 π = 0.000008 GHz and the temperature is 20 mK. Other parameters of the **qubit** are identical **with **Fig. 4 (a)....(Color online). Calculated final **qubit** population versus energy detuning and microwave amplitude. (a). The stationary interference pattern in the weak relaxation situation. The parameters we used are the driving **frequency** ω / 2 π = 0.6 GHz, the dephasing rate Γ 2 / 2 π = 0.06 GHz, the couple tunneling Δ / 2 π = 0.013 GHz, φ 2 α = 0.0002 GHz, the temperature is 20 mK, and the characteristic **frequency** ω c / 2 π = 0.05 GHz. The periodical patterns of RII can be seen, although not clear. (b). The stationary interference pattern in the strong relaxation situation with α φ 2 = 0.02 GHz and ω c / 2 π = 0.05 GHz. Since the relaxation strength is stronger, the periodical interference patterns are more notable. (c). The stationary interference pattern in the weak relaxation situation with α φ 2 = 0.000002 GHz and ω c / 2 π = 6 GHz. (d). The stationary interference pattern in the strong relaxation situation with α φ 2 = 0.0002 GHz and ω c / 2 π = 6 GHz. (e). The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic **frequency** ω c / 2 π = 0.05 GHz. α φ 2 = 0.0002 GHz. (f). The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic **frequency** ω c / 2 π = 6 GHz. α φ 2 = 0.000002 GHz. (g). The unsaturated interference pattern in the strong relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic **frequency** ω c / 2 π = 0.05 GHz. α φ 2 = 0.02 GHz. (h). The unsaturated interference pattern in the strong relaxation situation. The dynamics time t = 0.5 μ s. The characteristic **frequency** ω c / 2 π = 6 GHz. α φ 2 = 0.0002 GHz. The other parameters used in these Figures are the same with those in Fig. 4 (a)....We study electromagnetically induced interference at superconducting **qubits**. The interaction between **qubits** and electromagnetic fields can provide additional coupling channels to **qubit** states, leading to quantum interference in a microwave driven **qubit**. In particular, the interwell relaxation or Rabi oscillation, resulting respectively from the multi- or single-mode interaction, can induce effective crossovers. The environment is modeled by a multi-mode thermal bath, generating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the **qubits** and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic **frequency** of the bath. In addition, we demonstrate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi oscillation, Rabi oscillation induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the **qubit**, with more controllable parameters including the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation, in not only the flux **qubit** but also the systems with no crossover structure, e.g. phase **qubits**....(Color online). (a). Schematic energy diagram of a strongly driven flux **qubit** interacting **with **a weak single-mode field. The green solid curve represents the weak field, forming effective coupling between states | 0 and | 1 ....(Color online). The stationary population of relaxation induced interference. The pattern is obtained from Eq. (34). (a). The characteristic frequency ω c / 2 π = 0.05 GHz **with **the temperature 20 mK. Features of population inversion and periodical modulation are notable. (b). The characteristic frequency ω c / 2 π = 6 GHz **with **the temperature 20 mK. (c). The characteristic frequency ω c / 2 π = 0.05 GHz **with **the temperature 2 × 10 -5 mK. In above figures, the driving frequency ω / 2 π = 0.6 GHz....(Color online). Calculated final **qubit** population versus energy detuning and microwave amplitude. (a). The stationary interference pattern in the weak relaxation situation. The parameters we used are the driving frequency ω / 2 π = 0.6 GHz, the dephasing rate Γ 2 / 2 π = 0.06 GHz, the couple tunneling Δ / 2 π = 0.013 GHz, φ 2 α = 0.0002 GHz, the temperature is 20 mK, and the characteristic frequency ω c / 2 π = 0.05 GHz. The periodical patterns of RII can be seen, although not clear. (b). The stationary interference pattern in the strong relaxation situation **with **α φ 2 = 0.02 GHz and ω c / 2 π = 0.05 GHz. Since the relaxation strength is stronger, the periodical interference patterns are more notable. (c). The stationary interference pattern in the weak relaxation situation **with **α φ 2 = 0.000002 GHz and ω c / 2 π = 6 GHz. (d). The stationary interference pattern in the strong relaxation situation **with **α φ 2 = 0.0002 GHz and ω c / 2 π = 6 GHz. (e). The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic frequency ω c / 2 π = 0.05 GHz. α φ 2 = 0.0002 GHz. (f). The unsaturated interference pattern in the weak relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic frequency ω c / 2 π = 6 GHz. α φ 2 = 0.000002 GHz. (g). The unsaturated interference pattern in the strong relaxation situation. The system dynamics time t = 0.5 μ s. The characteristic frequency ω c / 2 π = 0.05 GHz. α φ 2 = 0.02 GHz. (h). The unsaturated interference pattern in the strong relaxation situation. The dynamics time t = 0.5 μ s. The characteristic frequency ω c / 2 π = 6 GHz. α φ 2 = 0.0002 GHz. The other parameters used in these Figures are the same **with **those in Fig. 4 (a). ... We study electromagnetically induced interference at superconducting **qubits**. The interaction between **qubits** and electromagnetic fields can provide additional coupling channels to **qubit** states, leading to quantum interference in a microwave driven **qubit**. In particular, the interwell relaxation or Rabi **oscillation**, resulting respectively from the multi- or single-mode interaction, can induce effective crossovers. The environment is modeled by a multi-mode thermal bath, generating the interwell relaxation. Relaxation induced interference, independent of the tunnel coupling, provides deeper understanding to the interaction between the **qubits** and their environment. It also supplies a useful tool to characterize the relaxation strength as well as the characteristic **frequency** of the bath. In addition, we demonstrate the relaxation can generate population inversion in a strongly driving two-level system. On the other hand, different from Rabi **oscillation**, Rabi **oscillation** induced interference involves more complicated and modulated photon exchange thus offers an alternative means to manipulate the **qubit**, with more controllable parameters including the strength and position of the tunnel coupling. It also provides a testing ground for exploring nonlinear quantum phenomena and quantum state manipulation, in not only the flux **qubit** but also the systems with no crossover structure, e.g. phase **qubits**.

Files:

Contributors: Agarwal, S., Rafsanjani, S. M. Hashemi, Eberly, J. H.

Date: 2012-01-13

Entanglement dynamics between the qubits. (a) ω 0 = 0.1 ω , β = 0.16 , (b) ω 0 = 0.15 ω , β = 0.16 , (c) ω 0 = 0.1 ω , β = 0.2 and (d) ω 0 = 0.15 ω , β = 0.2 . For all the figures, α = 3 ....With recent advances in the area of circuit QED, it is now possible to engineer systems** for **which the qubits are coupled to the oscillator so strongly, or are so far detuned from the oscillator, that the RWA cannot be used to describe the system’s evolution correctly . The parameter regime** for **which the coupling strength is strong enough to invalidate the RWA is called the ultra-strong coupling regime . Niemczyk, et al. and Forn-Díaz, et al. have been able to experimentally achieve ultra-strong coupling strengths and have demonstrated the breakdown of the RWA. Motivated by these experimental developments and the importance of understanding collective quantum behavior, we investigate a two-qubit TC model beyond the validity regime of RWA. The regime of parameters we will be concerned with is the regime where the qubits are quasi-degenerate, i.e., with frequencies much smaller than the oscillator frequency, ω 0 ≪ ω , while the coupling between the qubits and the oscillator is allowed to be an appreciable fraction of the oscillator frequency. In this parameter regime, the dynamics of the system can neither be correctly described under the RWA, nor can the effects of the counter rotating terms be taken as a perturbative correction to the dynamics predicted within the RWA by including higher powers of β . For illustration, systems are shown in Fig. f.model** for **which the RWA is valid, or breaks down, because the condition ω 0 ≈ ω is valid, or is violated. The regime that we will be interested in, for which ω 0 ≪ ω , is shown on the right....Here, within the adiabatic approximation, we extend the examination to the two-**qubit** case. Qualitative differences between the single-**qubit** and the multi-**qubit** cases are highlighted. In particular, we study the collapse and revival of joint properties of both the **qubits**. Entanglement properties of the system are investigated and it is shown that the entanglement between the **qubits** also exhibits collapse and revival. We derive what we believe are the first analytic expressions for the individual revival signals beyond the RWA, as well as analytic expression for the collapse and revival dynamics of entanglement. In the quasi-degenerate regime, the invalidity of the RWA in predicting the dynamical evolution will clearly be demonstrated in Sec. s.collapse_rev (see Figs. f.collapse_revival_double and f.collapse_revival_single)....Numerical and analytical evaluation of the entanglement dynamics between the two qubits for ω 0 = 0.15 ω , β = 0.16 and α = 3 . Entanglement between the qubits exhibits collapse and revival. The analytic expression agrees well with the envelope of the numerically evaluated entanglement evolution....Here, within the adiabatic approximation, we extend the examination to the two-qubit case. Qualitative differences between the single-qubit and the multi-qubit cases are highlighted. In particular, we study the collapse and revival of joint properties of** both **the qubits. Entanglement properties of the system are investigated and it is shown that the entanglement between the qubits also exhibits collapse and revival. We derive what we believe are the first analytic expressions** for **the individual revival signals beyond the RWA, as well as analytic expression** for **the collapse and revival dynamics of entanglement. In the quasi-degenerate regime, the invalidity of the RWA in predicting the dynamical evolution will clearly be demonstrated in Sec. s.collapse_rev (see Figs. f.collapse_revival_double and f.collapse_revival_single)....When squared, the probability shows two **frequencies** of **oscillation**, 2 Ω N ω and 2 2 Ω N ω . Since three new basis states are involved, we could expect three **frequencies**, but two are equal: | E N + - E N 0 | = | E N - - E N 0 | . This is in contrast to the single-**qubit** case where only one Rabi **frequency** determines the evolution . We show below in Fig. f.col_rev the way differences between one and a pair of **qubits** can be seen....If the average excitation of the **oscillator**, n ̄ = α 2 , is large one can evaluate the above sum approximately (see Appendix) and obtain analytic expressions and graphs of the evolution, as shown in Fig. f.col_rev. As expected, because of the double **frequency** in ( eqnP(t)), the revival time for S t 2 ω 0 is half the revival time for S t ω 0 . Thus there are two different revival sequences in the time series. Appropriate analytic formulas, e.g., ( a.Phi), agree well with the numerically evaluated evolution even for the relatively weak coherent excitation, α = 3 ....Entanglement dynamics between the **qubits**. (a) ω 0 = 0.1 ω , β = 0.16 , (b) ω 0 = 0.15 ω , β = 0.16 , (c) ω 0 = 0.1 ω , β = 0.2 and (d) ω 0 = 0.15 ω , β = 0.2 . For all the figures, α = 3 ....Numerical and analytical evaluation of the entanglement dynamics between the two **qubits** for ω 0 = 0.15 ω , β = 0.16 and α = 3 . Entanglement between the **qubits** exhibits collapse and revival. The analytic expression agrees well with the envelope of the numerically evaluated entanglement evolution....Collapse and revival dynamics** for **P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-qubit case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even** for **the single qubit case....The three potential wells corresponding to the states | 1 , 1 | N 1 (left), | 1 , 0 | N 0 (middle) and | 1 , - 1 | N -1 (right). The factor Δ X z p is the zero point fluctuation of a harmonic **oscillator**. For an **oscillator** of mass M and **frequency** ω the zero point fluctuation is given by Δ X z p = ℏ / 2 M ω ....Collapse and revival dynamics for P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-qubit case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even for the single qubit case....Tavis-Cummings model beyond the rotating wave approximation: Quasi-degenerate **qubits**...Collapse and revival dynamics for ω 0 = 0.1 ω , β = 0.16 and α = 3 . The first two panels show analytic evaluations of (a.) one-**qubit** and (b.) two-**qubit** probability dynamics, and (c.) shows that the two-**qubit** analytic formula matches well to the corresponding numerical evolution. In each case the initial state is a product of a coherent **oscillator** state with the lowest of the S x states. Note the breakup in the main revival peak of the two-**qubit** numerical evaluation, which comes from the ω - 2 ω beat note, not included in the analytic calculation, and not present for a single **qubit**....Collapse and revival dynamics for ω 0 = 0.1 ω , β = 0.16 and α = 3 . The first two panels show analytic evaluations of (a.) one-qubit and (b.) two-qubit probability dynamics, and (c.) shows that the two-qubit analytic formula matches well to the corresponding numerical evolution. In each case the initial state is a product of a** coherent ****oscillator** state with the lowest of the S x states. Note the breakup in the main revival peak of the two-qubit numerical evaluation, which comes from the ω - 2 ω beat note, not included in the analytic calculation, and not present for a single qubit....Collapse and revival dynamics for P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-**qubit** case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even for the single **qubit** case....With recent advances in the area of circuit QED, it is now possible to engineer systems for which the **qubits** are coupled to the **oscillator** so strongly, or are so far detuned from the **oscillator**, that the RWA cannot be used to describe the system’s evolution correctly . The parameter regime for which the coupling strength is strong enough to invalidate the RWA is called the ultra-strong coupling regime . Niemczyk, et al. and Forn-Díaz, et al. have been able to experimentally achieve ultra-strong coupling strengths and have demonstrated the breakdown of the RWA. Motivated by these experimental developments and the importance of understanding collective quantum behavior, we investigate a two-**qubit** TC model beyond the validity regime of RWA. The regime of parameters we will be concerned with is the regime where the **qubits** are quasi-degenerate, i.e., with **frequencies** much smaller than the **oscillator** **frequency**, ω 0 ≪ ω , while the coupling between the **qubits** and the **oscillator** is allowed to be an appreciable fraction of the **oscillator** **frequency**. In this parameter regime, the dynamics of the system can neither be correctly described under the RWA, nor can the effects of the counter rotating terms be taken as a perturbative correction to the dynamics predicted within the RWA by including higher powers of β . For illustration, systems are shown in Fig. f.model for which the RWA is valid, or breaks down, because the condition ω 0 ≈ ω is valid, or is violated. The regime that we will be interested in, for which ω 0 ≪ ω , is shown on the right....From ( e.rev_time), the non-monotonic dependence of the revival time on the coupling strength, β , and oscillator excitation strength, α , is clear. Note that the asymptotic formula ( e.asymp_Laguerre) works best for | α | ≫ 1 and equation ( e.rev_time) is not restricted by the constraint | α β | < 1 . In Fig. f.col_rev_big_alpha, we plot S t ω 0** for **α = 10 and various values of β . The revival times predicted by ( e.rev_time) are denoted by vertical lines and are seen to have excellent agreement with the numerically evaluated revival signals. We see from Fig. f.col_rev_big_alpha that by increasing β from 0.1 to 0.15 , the revival time increases. On the other hand, further increase of the coupling strength from 0.15 to 0.25 results in a decrease of the revival time. This clearly demonstrates the non-monotonic dependence of the revival time on the coupling strength and also highlights the departure from the formula derived in Sec. s.collapse_rev under the constraint | α β | < 1 ....There is only one revival sequence for the single **qubit** system as a consequence of having only one Rabi **frequency** in the single **qubit** case. The analytic and numerically exact evolution of P 1 2 , - 1 2 α t is plotted in Fig. f.collapse_revival_single. The single revival sequence is evident from the figure. A discussion on the multiple revival sequences for the K -**qubit** TC model, within the parameter regime where the RWA is valid, can be found in ....The Tavis-Cummings model for more than one **qubit** interacting with a common **oscillator** mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the **frequencies** of the **qubits** are much smaller than the **oscillator** **frequency** and the coupling strength is allowed to be ultra-strong. The application of the adiabatic approximation, introduced by Irish, et al. (Phys. Rev. B \textbf{72}, 195410 (2005)), for a single **qubit** system is extended to the multi-**qubit** case. For a two-**qubit** system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intra-manifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-**qubit** dynamics that are different from the single **qubit** case, including calculations of **qubit**-**qubit** entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived. ... The Tavis-Cummings model for more than one **qubit** interacting with a common **oscillator** mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the **frequencies** of the **qubits** are much smaller than the **oscillator** **frequency** and the coupling strength is allowed to be ultra-strong. The application of the adiabatic approximation, introduced by Irish, et al. (Phys. Rev. B \textbf{72}, 195410 (2005)), for a single **qubit** system is extended to the multi-**qubit** case. For a two-**qubit** system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intra-manifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-**qubit** dynamics that are different from the single **qubit** case, including calculations of **qubit**-**qubit** entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived.

Files:

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

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

Date: 2006-04-04

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

Files:

Contributors: Wirth, T., Lisenfeld, J., Lukashenko, A., Ustinov, A. V.

Date: 2010-10-05

(Color online) Shift of the resonance **frequency** of the SQUID resonator by 30 MHz due to the **qubit** changing its magnetic flux by approximately Φ 0 . (a) In the linear regime. (b) SQUID driven in the non-linear regime. Note the larger signal amplitude compared to the linear regime....Figure fig:4 shows Rabi oscillations** of**** the **qubit measured for different driving powers

**of**

**qubit microwave driving. As it is expected,**

**the****the**frequency

**of**Rabi oscillations increases approximately linearly with

**driving field amplitude**

**the****. The**measured energy relaxation time

**of**

**tested qubit is rather short and is**

**the****of**order

**of**T 1 = 5 ns. This time is it not limited by

**chosen type**

**the****of**readout but rather determined by

**intrinsic coherence**

**the****of**

**qubit itself. We verified this fact by measuring**

**the****same qubit with**

**the****conventional SQUID switching current method, which yielded very similar T 1 . The observed short coherence time is likely to be caused by**

**the****dielectric loss in**

**the****silicon oxide forming**

**the****insulating dielectric layer around**

**the****qubit Josephson junction ....Dispersive readout scheme for a Josephson phase**

**the****qubit**...The SQUID resonator frequency shift induced by

**qubit is shown in detail in Fig. fig:3(a). It displays two traces**

**the****of**

**normalized reflected signal amplitude versus**

**the****applied microwave frequency in**

**the****vicinity**

**the****of**

**qubit-state switching. Here,**

**the****the**resonance was located at around 1.9 GHz where

**SQUID has higher sensitivity to**

**the****flux**

**the****. The**amplitude

**of**

**reflected signal drops at**

**the****resonance frequency. For this measurement, very low microwave power**

**the****of**-120 dBm was applied to SQUID to stay in

**linear regime, giving rise to**

**the****Lorentzian shape**

**the****of**

**resonance dips. Taking into account**

**the****line width**

**the****of**4 MHz and

**dependence**

**the****of**

**resonance frequency on**

**the****flux, we achieve a flux resolution**

**the****of**2-3 m Φ 0

**of**

**detector at operating frequency**

**the****of**1.9 GHz. As

**two qubit states differ by magnetic flux**

**the****of**

**order**

**the****of**Φ 0 , this allows for a very weak inductive coupling between SQUID and qubit for future experiments. Fig. fig:3 (b) shows

**same frequency range as above, but now**

**the****power**

**the****of**

**input signal is larger, -115 dBm, driving**

**the****SQUID into**

**the****nonlinear regime. This is revealed by**

**the****shape**

**the****of**

**dips**

**the****. The**advantage

**of**

**non-linear regime is**

**the****sharper edge on**

**the****low frequency side which allows for an even better flux resolution**

**the****of**about 0.5-0.7 m Φ 0 ....The SQUID resonator

**frequency**shift induced by the

**qubit**is shown in detail in Fig. fig:3(a). It displays two traces of the normalized reflected signal amplitude versus the applied microwave

**frequency**in the vicinity of the

**qubit**-state switching. Here, the resonance was located at around 1.9 GHz where the SQUID has higher sensitivity to the flux. The amplitude of the reflected signal drops at the resonance

**frequency**. For this measurement, very low microwave power of -120 dBm was applied to SQUID to stay in the linear regime, giving rise to the Lorentzian shape of the resonance dips. Taking into account the line width of 4 MHz and the dependence of the resonance

**frequency**on the flux, we achieve a flux resolution of 2-3 m Φ 0 of the detector at operating

**frequency**of 1.9 GHz. As the two

**qubit**states differ by magnetic flux of the order of Φ 0 , this allows for a very weak inductive coupling between SQUID and

**qubit**for future experiments. Fig. fig:3 (b) shows the same

**frequency**range as above, but now the power of the input signal is larger, -115 dBm, driving the SQUID into the nonlinear regime. This is revealed by the shape of the dips. The advantage of the non-linear regime is the sharper edge on the low

**frequency**side which allows for an even better flux resolution of about 0.5-0.7 m Φ 0 ....Figure fig:4 shows Rabi

**oscillations**of the

**qubit**measured for different driving powers of the

**qubit**microwave driving. As it is expected, the

**frequency**of Rabi

**oscillations**increases approximately linearly with the driving field amplitude. The measured energy relaxation time of the tested

**qubit**is rather short and is of order of T 1 = 5 ns. This time is it not limited by the chosen type of readout but rather determined by the intrinsic coherence of the

**qubit**itself. We verified this fact by measuring the same

**qubit**with the conventional SQUID switching current method, which yielded very similar T 1 . The observed short coherence time is likely to be caused by the dielectric loss in the silicon oxide forming the insulating dielectric layer around the

**qubit**Josephson junction ....(Color online) Microwave

**frequency**applied to the SQUID vs. externally applied flux. The measurement points show the position of a dip in the reflected signal amplitude for two different directions of the flux sweep....(Color online) Coherent oscillations of the qubit for different driving powers, from bottom to top: -18 dBm, -15 dBm, -12 dBm, -9 dBm and -6 dBm. Curves are offset by 0.1 for better visibility....The the position of a dip in the amplitude of the reflected pulse is plotted in Fig. fig:2 as a function of microwave

**frequency**and applied SQUID flux bias Φ S . Data points indicate the dependence of the tank circuit resonance

**frequency**on the applied bias flux. The larger (red) circles correspond to the flux swept from negative to positive values, while the smaller (blue) dots stand for the flux swept in opposite direction. During the flux sweep, due to the crosstalk between Φ S and Φ Q flux lines approximately one flux quantum Φ 0 enters or leaves the

**qubit**loop, which gives rise to abrupt shift of the dip

**frequency**at specific flux bias values. The resonance

**frequency**shift at a bias flux of -0.35 Φ 0 is about 22 MHz, which is larger than the tank circuit’s resonance line width of about 4 MHz. This scheme is thus capable of single-shot detection of the

**qubit**flux state....(Color online) Scheme of the measurement setup. The SQUID with shunt capacitor C 1 coupled to the

**qubit**. The pulsed microwave signal is applied via a cryogenic circulator, and the reflected signal is amplified by a cryogenic amplifier....(Color online) Shift of the resonance

**frequency**of the SQUID resonator by 30 MHz due to the qubit changing its magnetic flux by approximately Φ 0 . (a) In the linear regime. (b) SQUID driven in the non-linear regime. Note the larger signal amplitude compared to the linear regime....On chip, there are two magnetic flux lines, one for flux Φ S biasing

**SQUID, see Fig. fig:1, and another for flux Φ Q biasing**

**the****qubit**

**the****. The**qubit is controlled by microwave pulses which are applied via a separate line (not shown) attenuated at several low temperature stages

**. The**SQUID flux bias line is equipped with a current divider and filter at

**1 K stage, and a powder filter at**

**the****sample holder. By taking**

**the****crosstalk**

**the****of**

**two flux coils into account, we can independently change**

**the****flux that is seen by**

**the****qubit and**

**the****flux that is seen by**

**the****SQUID. This sample was designed with a large mutual inductance between qubit loop and dc-SQUID which allowed us to independently characterize**

**the****sample by**

**the****conventional switching-current technique**

**the****. The**dispersive readout results presented below were obtained without applying any dc-bias to

**readout SQUID....The**

**the****position**

**the****of**a dip in

**amplitude**

**the****of**

**reflected pulse is plotted in Fig. fig:2 as a function**

**the****of**microwave frequency and applied SQUID flux bias Φ S . Data points indicate

**dependence**

**the****of**

**tank circuit resonance frequency on**

**the****applied bias flux**

**the****. The**larger (red) circles correspond to

**flux swept from negative to positive values, while**

**the****smaller (blue) dots stand for**

**the****flux swept in opposite direction. During**

**the****flux sweep, due to**

**the****crosstalk between Φ S and Φ Q flux lines approximately one flux quantum Φ 0 enters or leaves**

**the****qubit loop, which gives rise to abrupt shift**

**the****of**

**dip frequency at specific flux bias values**

**the****. The**resonance frequency shift at a bias flux

**of**-0.35 Φ 0 is about 22 MHz, which is larger than

**tank circuit’s resonance line width**

**the****of**about 4 MHz. This scheme is thus capable

**of**single-shot detection

**of**

**qubit flux state....We couple the**

**the****qubit**to a capacitively shunted dc-SQUID which forms a tank circuit having a resonance

**frequency**around 2 GHz. It is connected to a microwave line by a coupling capacitor C 0 shown in Fig. fig:1. Our sample was fabricated in a standard niobium-aluminium trilayer process. Measurement of the amplitude and phase of a reflected microwave pulse allows one to determine the shift of the resonance

**frequency**of the SQUID-resonator and by doing this deduce the magnetic flux of the

**qubit**state....superconducting

**qubits**, phase

**qubit**, dispersive readout, SQUID...We present experimental results on a dispersive scheme for reading out a Josephson phase

**qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the

**qubit**. We detect the flux state of the

**qubit**by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the

**qubit**state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple

**qubits**using a single microwave line by employing

**frequency**-division multiplexing....We couple

**qubit to a capacitively shunted dc-SQUID which forms a tank circuit having a resonance frequency around 2 GHz. It is connected to a microwave line by a coupling capacitor C 0 shown in Fig. fig:1. Our sample was fabricated in a standard niobium-aluminium trilayer process. Measurement**

**the****of**

**amplitude and phase**

**the****of**a reflected microwave pulse allows one to determine

**shift**

**the****of**

**resonance frequency**

**the****of**

**SQUID-resonator and by doing this deduce**

**the****magnetic flux**

**the****of**

**qubit state....(Color online) Scheme of the measurement setup. The SQUID with shunt capacitor C 1 coupled to the qubit. The pulsed microwave signal is applied via a cryogenic circulator, and the reflected signal is amplified by a cryogenic amplifier....(Color online) Coherent**

**the****oscillations**of the

**qubit**for different driving powers, from bottom to top: -18 dBm, -15 dBm, -12 dBm, -9 dBm and -6 dBm. Curves are offset by 0.1 for better visibility....On chip, there are two magnetic flux lines, one for flux Φ S biasing the SQUID, see Fig. fig:1, and another for flux Φ Q biasing the

**qubit**. The

**qubit**is controlled by microwave pulses which are applied via a separate line (not shown) attenuated at several low temperature stages. The SQUID flux bias line is equipped with a current divider and filter at the 1 K stage, and a powder filter at the sample holder. By taking the crosstalk of the two flux coils into account, we can independently change the flux that is seen by the

**qubit**and the flux that is seen by the SQUID. This sample was designed with a large mutual inductance between

**qubit**loop and dc-SQUID which allowed us to independently characterize the sample by the conventional switching-current technique. The dispersive readout results presented below were obtained without applying any dc-bias to the readout SQUID. ... We present experimental results on a dispersive scheme for reading out a Josephson phase

**qubit**. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the

**qubit**. We detect the flux state of the

**qubit**by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the

**qubit**state measurement time down to 25 microseconds, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple

**qubits**using a single microwave line by employing

**frequency**-division multiplexing.

Files:

Contributors: Kim, Mun Dae

Date: 2008-09-02

The values of g / h for the main fidelity maxima ( n = 1 ) obtained from numerical calculation and from the RWA of Eq. ( g) for various coupling J and** qubit** energy gap ω 0 . For small ω 0 and large J the oscillations are far from the Rabi oscillation. Here, the unit of all numbers is GHz....The scheme for CNOT gate operation in this study uses the non-Rabi oscillations for | 10 and | 11 states which are commensurate with the Rabi oscillation for | 00 and | 01 states. In Fig. TwoRabi we display the numerical results obtained from the Hamiltonian in Eqs. ( tilH0) and ( tilH1), which show such commensurate mode oscillations. The initial state, | ψ 0 = | 00 + | 10 / 2 , is driven by an oscillating field with the resonant frequency ω = ω 0 < ω 1 ....Commensurate Quantum Oscillations in Coupled

**Qubits**...In Fig. PC we show the Rabi-type

**oscillation**for strongly coupled

**qubits**. While the P 00 ( P 01 ) is reversed from 0.5 (0) to 0 (0.5) at Ω t = (odd) π , we can observe that the probabilities P 10 and P 11 remain their initial values 0.5 and 0, respectively. In this case the parameters need not satisfy the commensurate condition of Eq. ( condition) for the CNOT gate operation....The scheme for CNOT gate operation in this study uses the non-Rabi

**oscillations**for | 10 and | 11 states which are commensurate with the Rabi

**oscillation**for | 00 and | 01 states. In Fig. TwoRabi we display the numerical results obtained from the Hamiltonian in Eqs. ( tilH0) and ( tilH1), which show such commensurate mode

**oscillations**. The initial state, | ψ 0 = | 00 + | 10 / 2 , is driven by an

**oscillating**field with the resonant

**frequency**ω = ω 0 < ω 1 ....The commensurate

**oscillations**of resonant and non-resonant modes enable the high fidelity CNOT gate operation by finely tuning the

**oscillating**field amplitude for any given values of

**qubit**energy gap and coupling strength between

**qubits**. While for a sufficiently strong coupling the CNOT gate can be achieved for any given parameter values, for a weak coupling a relation between the parameters should be satisfied for the fidelity maxima. For a sufficiently weak coupling compared to the

**qubit**energy gap, J / ℏ ω 0 ≪ 1 , we have α 1 ≈ 1 and β 1 ≈ 0 , resulting in the expression for g in Eq. ( ga). For J / ℏ ω 0 ≪ 1 , Eq. ( ga) immediately gives rise to the relation g / J ≪ 1 and thus g / ℏ ω 0 ≪ 1 after some manipulation. This means that for a weak coupling J / ℏ ω 0 ≪ 1 the numerical results are well fit with the RWA as shown in Table table, because the RWA is good for g / ℏ ω 0 ≪ 1 . As a result, the high performance CNOT gate operation can be achieved as shown in Fig. dF....Let us consider a concrete example for comprehensive understanding. For superconducting flux

**qubits**, g = m B is the coupling between the amplitude B of the magnetic microwave field and the magnetic moment m , induced by the circulating current, of the

**qubit**loop. In order to adjust the value of g , actually we need to vary the microwave amplitude B , because the

**qubit**magnetic moment is fixed at a specified degeneracy point. The Rabi-type

**oscillation**occurs between the transformed states | 0 = | + | / 2 and | 1 = | - | / 2 . The states of

**qubits**can be detected by shifting the magnetic pulse adiabatically . Since these

**qubit**states are the superposition of the clockwise and counterclockwise current states, | and | , the averaged current of

**qubit**states vanishes at the degeneracy point in Fig. weak(a). Thus, one can apply a finite dc magnetic pulse to shift the

**qubits**slightly away from the degeneracy point to detect the

**qubit**current states....The commensurate oscillations of resonant and non-resonant modes enable the high fidelity CNOT gate operation by finely tuning the oscillating field amplitude for any given values of qubit energy gap and coupling strength between qubits. While for a sufficiently strong coupling the CNOT gate can be achieved for any given parameter values, for a weak coupling a relation between the parameters should be satisfied for the fidelity maxima. For a sufficiently weak coupling compared to the qubit energy gap, J / ℏ ω 0 ≪ 1 , we have α 1 ≈ 1 and β 1 ≈ 0 , resulting in the expression for g in Eq. ( ga). For J / ℏ ω 0 ≪ 1 , Eq. ( ga) immediately gives rise to the relation g / J ≪ 1 and thus g / ℏ ω 0 ≪ 1 after some manipulation. This means that for a weak coupling J / ℏ ω 0 ≪ 1 the numerical results are well fit with the RWA as shown in Table table, because the RWA is good for g / ℏ ω 0 ≪ 1 . As a result, the high performance CNOT gate operation can be achieved as shown in Fig. dF....(Color online) (a) Energy levels E ρ ρ ' of coupled qubits, where ρ , ρ ' ∈ 0 1 . E s s ' with s , s ' ∈ are shown as thin dotted lines. The distance between two degeneracy points corresponds to the coupling strength between two qubits. (b) Occupation probabilities of | ρ ρ ' states during Rabi-type oscillations at the lower degeneracy point where E = E . Here we use the parameter values such that coupling strength J / h = 0.6GHz,

**qubit**energy gap ω 0 / 2 π =4GHz, and Rabi frequency Ω 0 / 2 π = 600 MHz. The initial state is chosen as ψ 0 = | 00 + | 10 / 2 and the CNOT gate is expected to be achieved at Ω t = (odd) π ....Let us consider a concrete example for comprehensive understanding. For superconducting flux qubits, g = m B is the coupling between the amplitude B of the magnetic microwave field and the magnetic moment m , induced by the circulating current, of the qubit loop. In order to adjust the value of g , actually we need to vary the microwave amplitude B , because the qubit magnetic moment is fixed at a specified degeneracy point. The Rabi-type oscillation occurs between the transformed states | 0 = | + | / 2 and | 1 = | - | / 2 . The states of qubits can be detected by shifting the magnetic pulse adiabatically . Since these qubit states are the superposition of the clockwise and counterclockwise current states, | and | , the averaged current of qubit states vanishes at the degeneracy point in Fig. weak(a). Thus, one can apply a finite dc magnetic pulse to shift the qubits slightly away from the degeneracy point to detect the qubit current states....The values of g / h for the main fidelity maxima ( n = 1 ) obtained from numerical calculation and from the RWA of Eq. ( g) for various coupling J and

**qubit**energy gap ω 0 . For small ω 0 and large J the

**oscillations**are far from the Rabi

**oscillation**. Here, the unit of all numbers is GHz....We study the coupled-

**qubit**oscillation driven by an oscillating field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi oscillation, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate oscillations, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of oscillating field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and

**qubit**energy gap in experiments....(Color online) (a) Commensurate

**oscillations**of occupation probability of coupled-

**qubit**states with the initial state, | ψ 0 = | 00 + | 10 / 2 for g / h = 0.265 GHz. The non-resonant

**oscillation**modes ( P 10 and P 11 ) are commensurate with the resonant modes ( P 00 and P 01 ). At Ω t = (odd) π , P 10 and P 11 recover their initial values, thus the CNOT gate operation is achieved. Here Ω 0 = g / ℏ , J / h = 0.5 GHz, and ω 0 / 2 π = 4.0GHz. (b) Higher order commensurate modes for smaller g / h = 0.122 GHz with the same J and ω 0 ....Figure weak(a) shows the energy levels E s s ' as a function of κ b , where we choose κ a such that | E s s ' - E - s s ' | ≫ t q a and thus t q a can be negligible. In the figure there are two degeneracy points; lower degeneracy point where E = E and upper degeneracy point where E = E . By adjusting the variable κ b , the coupled-

**qubit**states can be brought to one of these degeneracy points. Here the distance between these degeneracy points is related to the coupling strength between two

**qubits**....(Color online) Rabi-type

**oscillations**of occupation probabilities of | ρ ρ ' states for strongly coupled

**qubits**with the initial state ψ 0 = | 00 + | 10 / 2 . Here the parameters are J / h = 5 GHz, ω 0 / 2 π =4GHz, and Ω 0 / 2 π = 600 MHz at the degeneracy point where E = E in Fig. weak(a)....We study the coupled-

**qubit**

**oscillation**driven by an

**oscillating**field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi

**oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate

**oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of

**oscillating**field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and

**qubit**energy gap in experiments....(Color online) Rabi-type oscillations of occupation probabilities of | ρ ρ ' states for strongly coupled qubits with the initial state ψ 0 = | 00 + | 10 / 2 . Here the parameters are J / h = 5 GHz, ω 0 / 2 π =4GHz, and Ω 0 / 2 π = 600 MHz at the degeneracy point where E = E in Fig. weak(a)....(Color online) (a) Energy levels E ρ ρ ' of coupled

**qubits**, where ρ , ρ ' ∈ 0 1 . E s s ' with s , s ' ∈ are shown as thin dotted lines. The distance between two degeneracy points corresponds to the coupling strength between two

**qubits**. (b) Occupation probabilities of | ρ ρ ' states during Rabi-type

**oscillations**at the lower degeneracy point where E = E . Here we use the parameter values such that coupling strength J / h = 0.6GHz,

**qubit**energy gap ω 0 / 2 π =4GHz, and Rabi

**frequency**Ω 0 / 2 π = 600 MHz. The initial state is chosen as ψ 0 = | 00 + | 10 / 2 and the CNOT gate is expected to be achieved at Ω t = (odd) π ....(Color online) (a) Commensurate oscillations of occupation probability of coupled-

**qubit**states with the initial state, | ψ 0 = | 00 + | 10 / 2 for g / h = 0.265 GHz. The non-resonant oscillation modes ( P 10 and P 11 ) are commensurate with the resonant modes ( P 00 and P 01 ). At Ω t = (odd) π , P 10 and P 11 recover their initial values, thus the CNOT gate operation is achieved. Here Ω 0 = g / ℏ , J / h = 0.5 GHz, and ω 0 / 2 π = 4.0GHz. (b) Higher order commensurate modes for smaller g / h = 0.122 GHz with the same J and ω 0 ....In Fig. weak(b) we show the resonant and non-resonant oscillations, when κ b is adjusted to the lower degeneracy point where E = E . Then a microwave with resonant frequency ω = ω 0 gives rise to the Rabi oscillation between two states | 00 and | 01 , while the states | 10 and | 11 experience a non-resonant oscillation. The controlled-NOT gate operation requires that the target qubit flips for a specific state of control qubit such that | 00 → | 01 while | 10 → | 10 . However, for example, at Ω t = π in Fig. weak(b) the states | 11 and | 10 also evolves during the transition from | 00 to | 01 . Thus we cannot expect a good CNOT gate operation in this case. ... We study the coupled-

**qubit**

**oscillation**driven by an

**oscillating**field. When the period of the non-resonant mode is commensurate with that of the resonant mode of the Rabi

**oscillation**, we show that the controlled-NOT (CNOT) gate operation can be demonstrated. For a weak coupling the CNOT gate operation is achievable by the commensurate

**oscillations**, while for a sufficiently strong coupling it can be done for arbitrary parameter values. By finely tuning the amplitude of

**oscillating**field it is shown that the high fidelity of the CNOT gate can be obtained for any fixed coupling strength and

**qubit**energy gap in experiments.

Files:

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

Date: 2007-12-12

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

Files:

Contributors: Zueco, David, Reuther, Georg M., Kohler, Sigmund, Hänggi, Peter

Date: 2009-07-20

**Oscillator** **frequency** shift as function or the **qubit** splitting ϵ = ω + Δ for the spin state | ↓ obtained (a) within RWA, Eq. shiftRWA, and (b) beyond RWA, Eq. wrnonRWA. The lines mark the analytical results, while the symbols refer to the numerically obtained splitting between the ground state and the first excited state in the subspace of the **qubit** state | ↓ ....We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**....**Qubit**-**oscillator** dynamics in the dispersive regime: analytical theory beyond rotating-wave approximation...**Oscillator** **frequency** shift as function or the **qubit** splitting ϵ = ω + Δ for the spin state | ↓ obtained (a) within RWA, Eq. shiftRWA, and (b) beyond RWA, Eq. wrnonRWA. The lines mark the analytical results, while the symbols refer to the numerically obtained splitting between the ground state and the first excited state in the subspace **of** the **qubit** state | ↓ ....HRabi in the subspace of the **qubit** state | ↓ , where σ z | ↓ = - | ↓ . The results are depicted in Fig. fig:wr. ... We generalize the dispersive theory of the Jaynes-Cummings model beyond the frequently employed rotating-wave approximation (RWA) in the coupling between the two-level system and the resonator. For a detuning sufficiently larger than the **qubit**-**oscillator** coupling, we diagonalize the non-RWA Hamiltonian and discuss the differences to the known RWA results. Our results extend the regime in which dispersive **qubit** readout is possible. If several **qubits** are coupled to one resonator, an effective **qubit**-**qubit** interaction of Ising type emerges, whereas RWA leads to isotropic interaction. This impacts on the entanglement characteristics of the **qubits**.

Files:

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

Date: 2010-01-26

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

Files: