### 63603 results for qubit oscillator frequency

Contributors: Catelani, G., Schoelkopf, R. J., Devoret, M. H., Glazman, L. I.

Date: 2011-06-04

Potential energy (in units of E L ) for a flux **qubit** biased at Φ e = Φ 0 / 2 with E J / E L = 10 . The horizontal lines represent **the** two lowest energy levels, with energy difference ϵ ̄ given in Eq. ( e0_eff)....As a second example of a strongly anharmonic system, we consider here a flux **qubit**, i.e., in Eq. ( Hphi) we assume E J > E L and take the external flux to be close to half the flux quantum, Φ e ≈ Φ 0 / 2 . Then the potential has a double-well shape and the flux **qubit** ground states | - and excited state | + are the lowest tunnel-split eigenstates in this potential, see Fig. fig:fl_q. The non-linear nature of the sin ϕ ̂ / 2 **qubit**-quasiparticle coupling in Eq. ( HTle) has a striking effect on the transition rate Γ + - , which vanishes at Φ e = Φ 0 / 2 due to destructive interference: for flux biased at half the flux quantum the **qubit** states | - , | + are respectively symmetric and antisymmetric around ϕ = π , while the potential in Eq. ( Hphi) and the function sin ϕ / 2 in Eq. ( wif_gen) are symmetric. Note that the latter symmetry and its consequences are absent in the environmental approach in which a linear phase-quasiparticle coupling is assumed....A further test of** the **theory presented in Sec. sec:th_s is provided **by **the measurement of** the **qubit resonant frequency. In** the **semiclassical regime of small E C , the qubit can be described **by **the effective circuit of Fig. fig1(b), with** the **junction admittance Y J of Eq. ( YJ), Y C = i ω C , and Y L = 1 / i ω L [the inductance is related to** the **inductive energy **by **E L = Φ 0 / 2 π 2 / L ]. As discussed in Ref. ...The transmon low-energy spectrum is characterized by well separated [by the plasma **frequency** ω p , Eq. ( pl_fr)] and nearly degenerate levels whose energies, as shown in Fig. fig:trans, vary periodically with the gate voltage n g . Here we derive the asymptotic expression (valid at large E J / E C ) for the energy splitting between the nearly degenerate levels. We consider first the two lowest energy states and then generalize the result to higher energies....Schematic representation of the transmon low energy spectrum as function of the dimensionless gate voltage n g . Solid (dashed) lines denotes even (odd) states (see also Sec. sec:cpb). The amplitudes of the **oscillations** of the energy levels are exponentially small, see Appendix app:eosplit; here they are enhanced for clarity. Quasiparticle tunneling changes the parity of the **qubit** sate. The results of Sec. sec:semi are valid for transitions between states separated by energy of the order of the plasma **frequency** ω p , Eq. ( pl_fr), and give, for example, the rate Γ 1 0 . For the transition rates between nearly degenerate states of opposite parity, such as Γ o e 1 , see Appendix app:eorate....As an application of the general approach described in the previous section, we consider here a weakly anharmonic **qubit**, such as the transmon and phase **qubits**. We start with the the semiclassical limit, i.e., we assume that the potential energy terms in Eq. ( Hphi) dominate the kinetic energy term proportional to E C . This limit already reveals a non-trivial dependence of relaxation on flux. Note that assuming E L ≠ 0 we can eliminate n g in Eq. ( Hphi) by a gauge transformation. In the transmon we have E L = 0 and the spectrum depends on n g , displaying both well separated and nearly degenerate states, see Fig. fig:trans. The results of this section can be applied to the single-junction transmon when considering well separated states. The transition rate between these states and the corresponding **frequency** shift are dependent on n g . However, since E C ≪ E J this dependence introduces only small corrections to Γ n n - 1 and δ ω ; the corrections are exponential in - 8 E J / E C . By contrast, the leading term in the rate of transitions Γ e ↔ o between the even and odd states is exponentially small. The rate Γ e ↔ o of parity switching is discussed in detail in Appendix app:eorate....Schematic representation of **the** transmon low energy spectrum as function of **the** dimensionless gate voltage n g . Solid (dashed) lines denotes even (odd) states (see also Sec. sec:cpb). The amplitudes of **the** oscillations of **the** energy levels are exponentially small, see Appendix app:eosplit; here they are enhanced for clarity. Quasiparticle tunneling changes **the** parity of **the** **qubit** sate. The results of Sec. sec:semi are valid for transitions between states separated by energy of **the** order of **the** plasma **frequency** ω p , Eq. ( pl_fr), and give, for example, **the** rate Γ 1 0 . For **the** transition rates between nearly degenerate states of opposite parity, such as Γ o e 1 , see Appendix app:eorate....A further test of the theory presented in Sec. sec:th_s is provided by the measurement of the **qubit** resonant **frequency**. In the semiclassical regime of small E C , the **qubit** can be described by the effective circuit of Fig. fig1(b), with the junction admittance Y J of Eq. ( YJ), Y C = i ω C , and Y L = 1 / i ω L [the inductance is related to the inductive energy by E L = Φ 0 / 2 π 2 / L ]. As discussed in Ref. ...which has** the **same form of** the **Hamiltonian for** the **single junction transmon [i.e., Eq. ( Hphi) with E L = 0 ] but with a flux-dependent Josephson energy, Eq. ( EJ_flux). Therefore** the **spectrum follows directly from that of** the **single junction transmon (see Fig. fig:trans) and consists of nearly degenerate and well separated states. The energy difference between well separated states is approximately given **by **the flux-dependent frequency [cf. Eq. ( pl_fr)]...Potential energy (in units of E L ) for a flux **qubit** biased at Φ e = Φ 0 / 2 with E J / E L = 10 . The horizontal lines represent the two lowest energy levels, with energy difference ϵ ̄ given in Eq. ( e0_eff)....As an application of** the **general approach described in** the **previous section, we consider here a weakly anharmonic qubit, such as** the **transmon and **phase **qubits. We start with** the **the semiclassical limit, i.e., we assume that** the **potential energy terms in Eq. ( Hphi) dominate** the **kinetic energy term proportional to E C . This limit already reveals a non-trivial dependence of relaxation on flux. Note that assuming E L ≠ 0 we can eliminate n g in Eq. ( Hphi) **by **a gauge transformation. In** the **transmon we have E L = 0 and** the **spectrum depends on n g , displaying both well separated and nearly degenerate states, see Fig. fig:trans. The results of this section can be applied to** the **single-junction transmon when considering well separated states. The transition rate between these states and** the **corresponding frequency shift are dependent on n g . However, since E C ≪ E J this dependence introduces only small corrections to Γ n n - 1 and δ ω ; the corrections are exponential in - 8 E J / E C . By contrast, the leading term in** the **rate of transitions Γ e ↔ o between** the **even and odd states is exponentially small. The rate Γ e ↔ o of parity switching is discussed in detail in Appendix app:eorate....As low-loss non-linear elements, Josephson junctions are the building blocks of superconducting **qubits**. The interaction of the **qubit** degree of freedom with the quasiparticles tunneling through the junction represent an intrinsic relaxation mechanism. We develop a general theory for the **qubit** decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the **qubit** properties in devices such as the phase and flux **qubits**, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and non-equilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible non-equilibrium quasiparticle distribution. We also derive expressions for the shift in **qubit** **frequency** in the presence of quasiparticles....which has the same form of the Hamiltonian for the single junction transmon [i.e., Eq. ( Hphi) with E L = 0 ] but with a flux-dependent Josephson energy, Eq. ( EJ_flux). Therefore the spectrum follows directly from that of the single junction transmon (see Fig. fig:trans) and consists of nearly degenerate and well separated states. The energy difference between well separated states is approximately given by the flux-dependent **frequency** [cf. Eq. ( pl_fr)]...As a second example of a strongly anharmonic system, we consider here a flux qubit, i.e., in Eq. ( Hphi) we assume E J > E L and take** the **external flux to be close to half** the **flux quantum, Φ e ≈ Φ 0 / 2 . Then** the **potential has a double-well shape and** the **flux qubit ground states | - and excited state | + are** the **lowest tunnel-split eigenstates in this potential, see Fig. fig:fl_q. The non-linear nature of** the **sin **ϕ** ̂ / 2 qubit-quasiparticle coupling in Eq. ( HTle) has a striking effect on** the **transition rate Γ + - , which vanishes at Φ e = Φ 0 / 2 due to destructive interference: for flux biased at half** the **flux quantum** the **qubit states | - , | + are respectively symmetric and antisymmetric around **ϕ** = π , while** the **potential in Eq. ( Hphi) and** the **function sin **ϕ** / 2 in Eq. ( wif_gen) are symmetric. Note that** the **latter symmetry and its consequences are absent in** the **environmental approach in which a linear phase-quasiparticle coupling is assumed....Relaxation and **frequency** shifts induced by quasiparticles in superconducting **qubits**...(a) Schematic representation of a **qubit** controlled by a magnetic flux, see Eq. ( Hphi). (b) Effective circuit diagram with three parallel elements – capacitor, Josephson junction, and inductor – characterized by their respective admittances....The transmon low-energy spectrum is characterized **by **well separated [**by **the plasma frequency ω p , Eq. ( pl_fr)] and nearly degenerate levels whose energies, as shown in Fig. fig:trans, vary periodically with** the **gate voltage n g . Here we derive** the **asymptotic expression (valid at large E J / E C ) for** the **energy splitting between** the **nearly degenerate levels. We consider first** the **two lowest energy states and then generalize** the **result to higher energies. ... As low-loss non-linear elements, Josephson junctions are the building blocks of superconducting **qubits**. The interaction of the **qubit** degree of freedom with the quasiparticles tunneling through the junction represent an intrinsic relaxation mechanism. We develop a general theory for the **qubit** decay rate induced by quasiparticles, and we study its dependence on the magnetic flux used to tune the **qubit** properties in devices such as the phase and flux **qubits**, the split transmon, and the fluxonium. Our estimates for the decay rate apply to both thermal equilibrium and non-equilibrium quasiparticles. We propose measuring the rate in a split transmon to obtain information on the possible non-equilibrium quasiparticle distribution. We also derive expressions for the shift in **qubit** **frequency** in the presence of quasiparticles.

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

Date: 2005-02-27

(color online) (a) Rabi **oscillations** in the **qubit** population P vs. Rabi pulse length Δ t (blue dots) and fit with unit visibility (red line). (b) Measured Rabi **frequency** ν R a b i vs. pulse amplitude ϵ s (blue dots) and linear fit....(color online) Measurement response φ (blue lines) and theoretical prediction (red lines) vs. time. At t = 6 μ s (a) a π pulse, (b) a 2 π pulse, and (c) a 3 π pulse is applied to the **qubit**. In each panel the dashed lines correspond to the expected measurement response in the ground state φ , in the saturated state φ = 0 , and in the excited state φ ....In** the **experiments presented here, we coherently control** the **quantum state of a Cooper pair box by applying to** the **qubit microwave pulses of frequency ω s , which are resonant with** the **qubit transition frequency ω a / 2 π ≈ 4.3 G H z , through** the **input port C **i** n of** the **resonator, see Fig. fig:setupa. The microwaves drive Rabi oscillations in** the **qubit at a frequency of ν R a b **i** = n s g / π , where n s **is**** the **average number of drive photons within** the **resonator. Simultaneously, we perform a continuous dispersive measurement of** the **qubit state by determining both** the **phase and** the **amplitude of a coherent microwave beam of frequency ω R F / 2 π = ω r / 2 π ≈ 5.4 G H z transmitted through** the **resonator . The phase shift φ = tan -1 2 g 2 / κ **Δ σ** z **is**** the **response of our meter from which we determine** the **qubit population. For** the **measurement, we chose a resonator that has a quality factor of Q ∼ 0.7 × 10 4 corresponding to a photon decay rate of κ / 2 π = 0.73 M H z . The resonator **is** populated with n ∼ 1 measurement photons on average, where n **is** calibrated using** the **ac-Stark shift ....The extracted **qubit** population P is plotted versus Δ t in Fig. fig:rabioscillationsa. We observe a visibility of 95 ± 6 % in the Rabi **oscillations** with error margins determined from the residuals of the experimental P with respect to the predicted values. Thus, in a measurement of Rabi **oscillations** in a superconducting **qubit**, a visibility in the population of the **qubit** excited state that approaches unity is observed for the first time. Moreover, we note that the decay in the Rabi **oscillation** amplitude out to pulse lengths of 100 n s is very small and consistent with the long T 1 and T 2 times of this charge **qubit**, see Fig. fig:rabioscillationsa and Ramsey experiment discussed below. We have also verified the expected linear scaling of the Rabi **oscillation** **frequency** ν R a b i with the pulse amplitude ϵ s ∝ n s , see Fig. fig:rabioscillationsb....(color online) (a) Simplified circuit diagram of measurement setup. A Cooper pair box with charging energy E C and Josephson energy E J is coupled through capacitor C g to a transmission line resonator, modelled as parallel combination of an inductor L and a capacitor C . Its state is determined in a phase sensitive heterodyne measurement of a microwave transmitted at **frequency** ω R F through the circuit, amplified and mixed with a local **oscillator** at **frequency** ω L O . The Cooper pair box level separation is controlled by the gate voltage V g and flux Φ . Its state is coherently manipulated using microwaves at **frequency** ω s with pulse shapes determined by V p . (b) Measurement sequence for Rabi **oscillations** with Rabi pulse length Δ t , pulse **frequency** ω s and amplitude ∝ n s with continuous measurement at **frequency** ω R F and amplitude ∝ n R F . (c) Sequence for Ramsey fringe experiment with two π / 2 -pulses at ω s separated by a delay Δ t and followed by a pulsed measurement....In our circuit QED architecture , a Cooper pair box , acting as a two level system with ground and excited states and level separation E a = ℏ ω a = E e l 2 + E J 2 **is** coupled capacitively to a single mode of** the **electromagnetic field of a transmission line resonator with resonance frequency ω r , see Fig. fig:setupa. As demonstrated for this system, the electrostatic energy E e l and** the **Josephson energy E J of** the **split Cooper pair box can be controlled in situ by a gate voltage V g and magnetic flux Φ , see Fig. fig:setupa. In** the **resonant ( ω a = ω r ) strong coupling regime a single excitation **is** exchanged coherently between** the **Cooper pair box and** the **resonator at a rate g / π , also called** the **vacuum Rabi frequency . In** the **non-resonant regime ( Δ = ω a - ω r > g ) the capacitive interaction gives rise to a dispersive shift g 2 / **Δ σ** z in** the **resonance frequency of** the **cavity which depends on** the **qubit state σ z , the coupling g and** the **detuning Δ . We have suggested that this shift in resonance frequency can be used to perform a quantum non-demolition (QND) measurement of** the **qubit state . With this technique we have recently measured** the **ground state response and** the **excitation spectrum of a Cooper pair box ....In the experiments presented here, we coherently control the quantum state of a Cooper pair box by applying to the **qubit** microwave pulses of **frequency** ω s , which are resonant with the **qubit** transition **frequency** ω a / 2 π ≈ 4.3 G H z , through the input port C i n of the resonator, see Fig. fig:setupa. The microwaves drive Rabi **oscillations** in the **qubit** at a **frequency** of ν R a b i = n s g / π , where n s is the average number of drive photons within the resonator. Simultaneously, we perform a continuous dispersive measurement of the **qubit** state by determining both the phase and the amplitude of a coherent microwave beam of **frequency** ω R F / 2 π = ω r / 2 π ≈ 5.4 G H z transmitted through the resonator . The phase shift φ = tan -1 2 g 2 / κ Δ σ z is the response of our meter from which we determine the **qubit** populat...We have determined the coherence time of the Cooper pair box from a Ramsey fringe experiment, see Fig. fig:setupc, when biased at the charge degeneracy point where the energy is first-order insensitive to charge noise . To avoid dephasing induced by a weak continuous measurement beam we switch on the measurement beam only after the end of the second π / 2 pulse. The resulting Ramsey fringes **oscillating** at the detuning **frequency** Δ a , s = ω a - ω s ∼ 6 M H z decay with a long coherence time of T 2 ∼ 500 n s , see Fig. fig:Ramseya. The corresponding **qubit** phase quality factor of Q ϕ = T 2 ω a / 2 ∼ 6500 is similar to the best values measured so far in **qubit** realizations biased at such an optimal point . The Ramsey **frequency** is shown to depend linearly on the detuning Δ a , s , as expected, see Fig. fig:Ramseyb. We note that a measurement of the Ramsey **frequency** is an accurate time resolved method to determine the **qubit** transition **frequency** ω a = ω s + 2 π ν R a m s e y ....(color online) (a) Measured **Ramsey** fringes (blue dots) observed in the **qubit** population P vs. pulse separation Δ t using** the **pulse sequence shown in Fig. fig:setupb and fit of data **to **sinusoid with gaussian envelope (red line). (b) Measured dependence of **Ramsey** **frequency** ν R a m s e y on detuning Δ a , s of **drive** **frequency** (blue dots) and linear fit (red line)....(color online) (a) Simplified circuit diagram of measurement setup. A Cooper pair box with charging energy E C and Josephson energy E J is coupled through capacitor C g **to **a transmission line resonator, modelled as parallel combination of an inductor L and a capacitor C . Its state is determined in a phase sensitive heterodyne measurement of a microwave transmitted at **frequency** ω R F through** the **circuit, amplified and mixed with a local oscillator at **frequency** ω L O . The Cooper pair box level separation is controlled by** the **gate voltage V g and flux Φ . Its state is coherently manipulated using microwaves at **frequency** ω s with pulse shapes determined by V p . (b) Measurement sequence for Rabi oscillations with Rabi pulse length Δ t , pulse **frequency** ω s and amplitude ∝ n s with continuous measurement at **frequency** ω R F and amplitude ∝ n R F . (c) Sequence for **Ramsey** fringe experiment with two π / 2 -pulses at ω s separated by a delay Δ t and followed by a pulsed measurement....(color online) (a) Measured Ramsey fringes (blue dots) observed in** the **qubit population P vs. pulse separation Δ** t** using** the **pulse sequence shown in Fig. fig:setupb and fit of data to sinusoid with gaussian envelope (red line). (b) Measured dependence of Ramsey frequency ν R a** m** s e y on detuning Δ a , s of drive frequency (blue dots) and linear fit (red line)....In our circuit QED architecture , a Cooper pair box , acting as a two level system with ground and excited states and level separation E a = ℏ ω a = E e l 2 + E J 2 is coupled capacitively to a single mode of the electromagnetic field of a transmission line resonator with resonance **frequency** ω r , see Fig. fig:setupa. As demonstrated for this system, the electrostatic energy E e l and the Josephson energy E J of the split Cooper pair box can be controlled in situ by a gate voltage V g and magnetic flux Φ , see Fig. fig:setupa. In the resonant ( ω a = ω r ) strong coupling regime a single excitation is exchanged coherently between the Cooper pair box and the resonator at a rate g / π , also called the vacuum Rabi **frequency** . In the non-resonant regime ( Δ = ω a - ω r > g ) the capacitive interaction gives rise to a dispersive shift g 2 / Δ σ z in the resonance **frequency** of the cavity which depends on the **qubit** state σ z , the coupling g and the detuning Δ . We have suggested that this shift in resonance **frequency** can be used to perform a quantum non-demolition (QND) measurement of the **qubit** state . With this technique we have recently measured the ground state response and the excitation spectrum of a Cooper pair box ....(color online) (a) Measured Ramsey fringes (blue dots) observed in the **qubit** population P vs. pulse separation Δ t using the pulse sequence shown in Fig. fig:setupb and fit of data to sinusoid with gaussian envelope (red line). (b) Measured dependence of Ramsey **frequency** ν R a m s e y on detuning Δ a , s of drive **frequency** (blue dots) and linear fit (red line)....(color online) (a) Rabi oscillations in the **qubit** population P vs. Rabi pulse length Δ t (blue dots) and fit with unit visibility (red line). (b) Measured Rabi **frequency** ν R a b i vs. pulse amplitude ϵ s (blue dots) and linear fit....In a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes....In a Rabi oscillation experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes....We have determined** the **coherence time of** the **Cooper pair box from a Ramsey fringe experiment, see Fig. fig:setupc, when biased at** the **charge degeneracy point where** the **energy **is** first-order insensitive to charge noise . To avoid dephasing induced by a weak continuous measurement beam we switch on** the **measurement beam only after** the **end of** the **second π / 2 pulse. The resulting Ramsey fringes oscillating at** the **detuning frequency Δ a , s = ω a - ω s ∼ 6 M H z decay with a long coherence time of T 2 ∼ 500 n s , see Fig. fig:Ramseya. The corresponding qubit phase quality factor of Q ϕ = T 2 ω a / 2 ∼ 6500 **is** similar to** the **best values measured so far in qubit realizations biased at such an optimal point . The Ramsey frequency **is** shown to depend linearly on** the **detuning Δ a , s , as expected, see Fig. fig:Ramseyb. We note that a measurement of** the **Ramsey frequency **is** an accurate time resolved method to determine** the **qubit transition frequency ω a = ω s + 2 π ν R a** m** s e y ....Approaching Unit Visibility for Control of a Superconducting **Qubit** with Dispersive Readout...(color online) Measurement response φ (blue lines) and theoretical prediction (red lines) vs. time. At t = 6 μ s (a) a π pulse, (b) a 2 π pulse, and (c) a 3 π pulse is applied **to **the **qubit**. In each panel** the **dashed lines correspond **to **the expected measurement response in** the **ground state φ , in** the **saturated state φ = 0 , and in** the **excited state φ . ... In a Rabi **oscillation** experiment with a superconducting **qubit** we show that a visibility in the **qubit** excited state population of more than 90 % can be attained. We perform a dispersive measurement of the **qubit** state by coupling the **qubit** non-resonantly to a transmission line resonator and probing the resonator transmission spectrum. The measurement process is well characterized and quantitatively understood. The **qubit** coherence time is determined to be larger than 500 ns in a measurement of Ramsey fringes.

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Contributors: Oxtoby, Neil P., Gambetta, Jay, Wiseman, H. M.

Date: 2007-06-24

Equivalent circuit for continuous monitoring of a charge **qubit** coupled to a classical L C oscillator with inductance L and capacitance C . We consider the charge-sensitive detector that loads the oscillator circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local oscillator, L O , and then measured. fig:rfcircuit...Model for monitoring of a charge **qubit** using a radio-**frequency** quantum point contact including experimental imperfections...The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise)....In simple dyne detection (see schematic in Fig. fig:rfcircuit), the output signal V o u t t is amplified, and mixed with a local oscillator (LO). The LO for homodyne detection of the amplitude quadrature is V L O t ∝ cos ω 0 t , where the LO frequency is the same as the signal of interest (or very slightly detuned). The resulting low-frequency beats due to mixing the signal with the LO are easily detected....The choice e > 0 corresponds to defining current in terms of the direction of electron flow. That is, in the opposite direction to conventional current. The DQDs are occupied by a single excess electron, the location of which determines the charge state of the **qubit**. The charge basis states are denoted | 0 and | 1 (see Fig. fig:dqdqpc). We assume that each quantum dot has only one single-electron energy level available for occupation by the **qubit** electron, denoted by E 1 and E 0 for the near and far dot, respectively....The two conjugate parameters we use to describe the oscillator state are the flux through the inductor, Φ t , and the charge on the capacitor, Q t . The dynamics of the oscillator are found by analyzing the equivalent circuit of Fig. fig:rfcircuit using the well-known Kirchhoff circuit laws. Doing this we find that the classical system obeys the following set of coupled differential equations...The two conjugate parameters we use to describe the **oscillator** state are the flux through the inductor, Φ t , and the charge on the capacitor, Q t . The dynamics of the **oscillator** are found by analyzing the equivalent circuit of Fig. fig:rfcircuit using the well-known Kirchhoff circuit laws. Doing this we find that the classical system obeys the following set of coupled differential equations...Consider the equivalent circuit of Fig. fig:rfcircuit. The **oscillator** circuit consisting of an inductance L and capacitance C terminates the transmission line of impedance Z T L = 50 Ω . The voltages (potential drops) across the **oscillator** components can be written as...The choice e > 0 corresponds to defining current in terms of the direction of electron flow. That is, in the opposite direction to conventional current. The DQDs are occupied by a single excess electron, the location of which determines the charge state of the qubit. The charge basis states are denoted | 0 and | 1 (see Fig. fig:dqdqpc). We assume that each quantum dot has only one single-electron energy level available for occupation by the qubit electron, denoted by E 1 and E 0 for the near and far dot, respectively....fig:dqdqpc Schematic of an isolated DQD **qubit** and capacitively coupled low-transparency QPC between source (S) and drain (D) leads....In simple dyne detection (see schematic in Fig. fig:rfcircuit), the output signal V o u t t is amplified, and mixed with a local **oscillator** (LO). The LO for homodyne detection of the amplitude quadrature is V L O t ∝ cos ω 0 t , where the LO **frequency** is the same as the signal of interest (or very slightly detuned). The resulting low-**frequency** beats due to mixing the signal with the LO are easily detected....Equivalent circuit for continuous monitoring of a charge **qubit** coupled to a classical L C **oscillator** with inductance L and capacitance C . We consider the charge-sensitive detector that loads the **oscillator** circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local **oscillator**, L O , and then measured. fig:rfcircuit...Consider the equivalent circuit of Fig. fig:rfcircuit. The oscillator circuit consisting of an inductance L and capacitance C terminates the transmission line of impedance Z T L = 50 Ω . The voltages (potential drops) across the oscillator components can be written as...Equivalent circuit for continuous monitoring of a charge qubit coupled to a classical L C oscillator with inductance L and capacitance C . We consider the charge-sensitive detector that loads the oscillator circuit to be a QPC (see Fig. fig:dqdqpc for details). Measurement is achieved using reflection with the input voltage, V i n t , and the output voltage, V o u t t , being separated by a directional coupler. The output voltage is then amplified and mixed with a local oscillator, L O , and then measured. fig:rfcircuit ... The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state **qubits** is important for quantum computation, particularly for the purposes of state preparation and error-correction as well as for readout of computations. Previously this has been achieved for low-**frequency** (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio **frequency** (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge **qubit**, is used to damp a classical **oscillator** circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the **oscillator** (relative to the QPC), where the **oscillator** slaves to the **qubit**. The rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc QPC is shown to be a low-efficiency charge-**qubit** detector, that may nevertheless be higher than the dc-QPC (which is subject to 1/f noise).

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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: Zhirov, O. V., Shepelyansky, D. L.

Date: 2007-10-10

(color online) Dependence of number of transitions N f between metastable states on** rescaled** qubit frequency Ω / ω 0 for parameters of Fig. fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice **is **as in Figs. fig2, fig3)....A typical example of QT is shown in Fig. fig1. It shows two main properties of the evolution: the **oscillator** spends a very long time at some average level n = n - and then jumps to another significantly different value n + . At the same time the polarization vector of **qubit** ξ → defined as ξ → = T r ρ ̂ σ → also changes its orientation direction with a clear change of sign of ξ x from ξ x > 0 to ξ x **qubit** polarization ξ = | ξ → | is very close to unity showing that the **qubit** remains mainly in a pure state. The drops of ξ appear only during transitions between metastable states. Special checks show that an inversion of ξ x by an additional pulse (e.g. from ξ x > 0 to ξ x **oscillator** to a corresponding state (from n - to n + ) after time t m ∼ 1 / λ . Thus we have here an interesting situation when a quantum flip of **qubit** produces a marcoscopic change of a state of detector (**oscillator**) which is continuously coupled to a **qubit** (we checked that even larger variation n ± ∼ n p is possible by taking n p = 40 ). In addition to that inside a metastable state the coupling induces a synchronization of **qubit** rotation phase with the **oscillator** phase which in its turn is fixed by the phase of driving field. The synchronization is a universal phenomenon for classical dissipative systems . It is known that it also exists for dissipative quantum systems at small effective values of ℏ . However, here we have a new unusual case of **qubit** synchronization when a semiclassical system produces synchronization of a pure quantum two-level system....(color online) Top panels: the Poincaré section taken at integer values of ω t / 2 π for **oscillator** with x = â + â / 2 , p = â - â / 2 i (left) and for **qubit** polarization with polarization angles θ φ defined in text (right). Middle panels: the same quantities shown at irrational moments of ω t / 2 π . Bottom panels: the **qubit** polarization phase φ vs. **oscillator** phase ϕ ( p / x = - tan ϕ ) at time moments as in middle panels for g = 0.04 (left) and g = 0.004 (right). Other parameters and the time interval are as in Fig. fig1. The color of points is blue/black for ξ x > 0 and red/gray for ξ x < 0 ....The phenomenon of **qubit** synchronization is illustrated in a more clear way in Fig. fig2. The top panels taken at integer values ω t / 2 π show the existence of two fixed points in the phase space of **oscillator** (left) and **qubit** (right) coupled by quantum tunneling (the angles are determined as ξ x = ξ cos θ , ξ y = ξ sin θ sin φ , ξ z = ξ sin θ cos φ ). A certain scattering of points in a spot of finite size should be attributed to quantum fluctuations. But the fact that on enormously long time (Fig. fig1) the spot size remains finite clearly implies that the **oscillator** phase ϕ is locked with the driving phase ω t inducing the **qubit** synchronization with ϕ and ω t . The plot at t values incommensurate with 2 π / ω (middle panels) shows that in time the **oscillator** performs circle rotations in p x plane with **frequency** ω while **qubit** polarization rotates around x -axis with the same **frequency**. Quantum tunneling gives transitions between two metastable states. The synchronization of **qubit** phase φ with **oscillator** phase ϕ is clearly seen in bottom left panel where points form two lines corresponding to two metastable states. This synchronization disappears below a certain critical coupling g c where the points become scattered over the whole plane (panel bottom right). It is clear that quantum fluctuations destroy synchronization for g < g c . Our data give g c ≃ 0.008 for parameters of Fig. fig1....Synchronization and bistability of **qubit** coupled to a driven dissipative **oscillator**...(color online) Dependence of average level n ± of oscillator in two metastable states on the driving frequency ω (average and color choice are the same as in right panel of Fig. fig3); coupling **is **g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig. fig1....The phenomenon of qubit synchronization **is **illustrated in a more clear way in Fig. fig2. The top panels taken at integer values ω **t** / 2 π show the existence of two fixed points in the phase space of oscillator (left) and qubit (right) coupled by quantum tunneling (the angles are determined as ξ x = ξ cos θ , ξ y = ξ sin θ sin φ , ξ z = ξ sin θ cos φ ). A certain scattering of points in a spot of finite size should be attributed to quantum fluctuations. But the fact **that** on enormously long time (Fig. fig1) the spot size remains finite clearly implies **that** the oscillator phase ϕ **is **locked with the driving phase ω **t** inducing the qubit synchronization with ϕ and ω **t** . The plot at **t** values incommensurate with 2 π / ω (middle panels) shows **that** in time the oscillator performs circle rotations** in p** x plane with frequency ω while qubit polarization rotates around x -axis with the same frequency. Quantum tunneling gives transitions between two metastable states. The synchronization of qubit phase φ with oscillator phase ϕ **is **clearly seen in bottom left panel where points form two lines corresponding to two metastable states. This synchronization disappears below a certain critical coupling g c where the points become scattered over the whole plane (panel bottom right). It **is **clear **that** quantum fluctuations destroy synchronization for g < g c . Our data give g c ≃ 0.008 for parameters of Fig. fig1....(color online) Top panels: the Poincaré section taken at integer values of ω t / 2 π for oscillator with x = â + â / 2 , p = â - â / 2 i (left) and for qubit polarization with polarization angles θ φ defined in text (right). Middle panels: the same quantities shown at irrational moments of ω t / 2 π . Bottom panels: the qubit polarization phase φ vs. oscillator phase ϕ ( p / x = - tan ϕ ) at time moments as in middle panels for g = 0.04 (left) and g = 0.004 (right). Other parameters and the time interval are as in Fig. fig1. The color of points is blue/black for ξ x > 0 and red/gray for ξ x < 0 ....We study numerically the behavior of **qubit** coupled to a quantum dissipative driven **oscillator** (resonator). Above a critical coupling strength the **qubit** rotations become synchronized with the **oscillator** phase. In the synchronized regime, at certain parameters, the **qubit** exhibits tunneling between two orientations with a macroscopic change of number of photons in the resonator. The life times in these metastable states can be enormously large. The synchronization leads to a drastic change of **qubit** radiation spectrum with appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev {\it et al.} Nature {\bf 449}, 588 (2007)]....(color online) Bistability of qubit coupled to a driven oscillator with jumps between two metastable states. Top panel shows average oscillator level number n as a function of time t at stroboscopic integer values ω t / 2 π ; middle panel shows the qubit polarization vector components ξ x (blue/black) and ξ z (green/gray) at the same moments of time; the bottom panel shows the degree of qubit polarization ξ . Here the system parameters are λ / ω 0 = 0.02 , ω / ω 0 = 1.01 , Ω / ω 0 = 1.2 , f = ℏ λ n p , n p = 20 and g = 0.04 ....(color online) Dependence of average level n ± of oscillator in two metastable states on the driving frequency ω (average and color choice are the same as in right panel of Fig. fig3); coupling is g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig. fig1....(color online) Dependence of number of transitions N f between metastable states on** rescaled** qubit frequency Ω / ω 0 for parameters of Fig. fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ±...(color online) Right panel: dependence of average qubit polarization components ξ x and ξ z (full and dashed curves) on g , averaging is done over stroboscopic times (see Fig. fig1) in the interval 100 ≤ ω t / 2 π ≤ 2 × 10 4 ; color is fixed by the sign of ξ x averaged over 10 periods (red/gray for ξ x 0 ; this choice fixes also the color on right panel). Left panel: dependence of average level of oscillator in two metastable states on coupling strength g , the color is fixed by the sign of ξ x on right panel that gives red/gray for large n + and blue/black for small n - ; average is done over the quantum state and stroboscopic times as in the left panel; dashed curves show theory dependence (see text)). Two QT are used with initial value ξ x = ± 1 . All parameters are as in Fig. fig1 except g ....(color online) Bistability of **qubit** coupled to a driven **oscillator** with jumps between two metastable states. Top panel shows average **oscillator** level number n as a function of time t at stroboscopic integer values ω t / 2 π ; middle panel shows the **qubit** polarization vector components ξ x (blue/black) and ξ z (green/gray) at the same moments of time; the bottom panel shows the degree of **qubit** polarization ξ . Here the system parameters are λ / ω 0 = 0.02 , ω / ω 0 = 1.01 , Ω / ω 0 = 1.2 , f = ℏ λ n p , n p = 20 and g = 0.04 ....(color online) Right panel: dependence of average **qubit** polarization components ξ x and ξ z (full and dashed curves) on g , averaging is done over stroboscopic times (see Fig. fig1) in the interval 100 ≤ ω t / 2 π ≤ 2 × 10 4 ; color is fixed by the sign of ξ x averaged over 10 periods (red/gray for ξ x 0 ; this choice fixes also the color on right panel). Left panel: dependence of average level of **oscillator** in two metastable states on coupling strength g , the color is fixed by the sign of ξ x on right panel that gives red/gray for large n + and blue/black for small n - ; average is done over the quantum state and stroboscopic times as in the left panel; dashed curves show theory dependence (see text)). Two QT are used with initial value ξ x = ± 1 . All parameters are as in Fig. fig1 except g ....(color online) Dependence of number of transitions N f between metastable states on rescaled **qubit** **frequency** Ω / ω 0 for parameters of Fig. fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice is as in Figs. fig2, fig3)....(color online) Dependence of number of transitions N f between metastable states on rescaled qubit frequency Ω / ω 0 for parameters of Fig. fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice is as in Figs. fig2, fig3)....(color online) Dependence of average level n ± of **oscillator** in two metastable states on the driving **frequency** ω (average and color choice are the same as in right panel of Fig. fig3); coupling is g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig. fig1. ... We study numerically the behavior of **qubit** coupled to a quantum dissipative driven **oscillator** (resonator). Above a critical coupling strength the **qubit** rotations become synchronized with the **oscillator** phase. In the synchronized regime, at certain parameters, the **qubit** exhibits tunneling between two orientations with a macroscopic change of number of photons in the resonator. The life times in these metastable states can be enormously large. The synchronization leads to a drastic change of **qubit** radiation spectrum with appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev {\it et al.} Nature {\bf 449}, 588 (2007)].

Files:

Contributors: Greenberg, Ya. S., Izmalkov, A., Grajcar, M., Il'ichev, E., Krech, W., Meyer, H. -G.

Date: 2002-08-07

In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-**junction** **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. ...Phase **qubit** coupled to a tank circuit....Method for direct observation of coherent quantum oscillations in a superconducting phase **qubit**...Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range....Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi oscillations in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi oscillations of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range....In our method a resonant tank circuit with known inductance L T , capacitance C T and quality factor Q T is coupled with a target Josephson circuit through the mutual inductance M (Fig. fig1). The method was successfully applied to a three-junction **qubit** in classical regime, when the hysteretic dependence of ground-state energy on the external magnetic flux was reconstructed in accordance to the predictions of Ref. ... Time-domain observations of coherent **oscillations** between quantum states in mesoscopic superconducting systems were so far restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a new method for direct observation of Rabi **oscillations** in a phase **qubit**. The external source, typically in GHz range, induces transitions between the **qubit** levels. The resulting Rabi **oscillations** of supercurrent in the **qubit** loop are detected by a high quality resonant tank circuit, inductively coupled to the phase **qubit**. Detailed calculation for zero and non-zero temperature are made for the case of persistent current **qubit**. According to the estimates for dephasing and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi **frequency** in MHz range.

Files:

Contributors: Chen, Yu, Sank, D., O'Malley, P., White, T., Barends, R., Chiaro, B., Kelly, J., Lucero, E., Mariantoni, M., Megrant, A.

Date: 2012-09-09

(Color online) (a) Schematic representation of **qubit** projective measurement, where a current pulse allows a **qubit** in the excited state | e to tunnel to the right well ( R ), while a **qubit** in the ground state | g stays in the left well ( L ). (b) Readout circuit, showing lumped-element L R - C R readout resonator inductively coupled to the **qubit**, with Josephson junction effective inductance L J and capacitance C , with loop inductance L . **Qubit** control is through the differential flux bias line ( F B ). The readout resonator is capacitively coupled through C c to the readout line, in parallel with the other readout resonators. The readout line is connected through a cryogenic circulator to a low-noise cryogenic amplifier and to a room temperature microwave source. (c) Photomicrograph of four-**qubit** sample. F B 1 - 4 are control lines for each **qubit** and R R is the resonator readout line. Inset shows details for one **qubit** and its readout resonator. Scale bar is 50 μ m in length; fig.setup...We introduce a **frequency**-multiplexed readout scheme for superconducting phase **qubits**. Using a quantum circuit with four phase **qubits**, we couple each **qubit** to a separate lumped-element superconducting readout resonator, with the readout resonators connected in parallel to a single measurement line. The readout resonators and control electronics are designed so that all four **qubits** can be read out simultaneously using **frequency** multiplexing on the one measurement line. This technology provides a highly efficient and compact means for reading out multiple **qubits**, a significant advantage for scaling up to larger numbers of **qubits**....With** the **bias points chosen for each** qubit**, we demonstrated** the **frequency-multiplexed readout by performing a multi-qubit experiment. To minimize crosstalk, we removed** the **coupling capacitors between qubits used in Ref. 6. In this experiment, we drove Rabi oscillations on each** qubit**’s | g ↔ | e transition and read out the** qubit** states simultaneously. We first calibrated** the **pulse amplitude needed for each** qubit** to perform a | g → | e Rabi transition in 10 ns. The drive amplitude was then set to 1, 2/3, 1/2 and 2/5** the **calibrated Rabi transition amplitude for qubits Q 1 to Q 4 respectively, so that** the **Rabi period was 20 ns, 30 ns, 40 ns and 50 ns for qubits Q 1 to Q 4 . We then drove each** qubit** separately using an on-resonance Rabi drive for a duration τ , followed immediately by a projective measurement and** qubit** state readout. This experiment yielded** the **measurements shown in Fig. fig.rabi(a)-(d) for qubits Q 1 - Q 4 respectively....(Color online) (a)-(d) Rabi **oscillations** for **qubits** Q 1 - Q 4 respectively, with the **qubits** driven with 1, 2/3, 1/2 and 2/5 the on-resonance drive amplitude needed to perform a 10 ns Rabi | g → | e transition. (e) Rabi **oscillations** measured simultaneously for all the **qubits**, using the same color coding and drive amplitudes as for panels (a)-(d). fig.rabi...With** the **probe frequency set in** the **first step, the flux bias was then set to optimize** the **readout. As illustrated in Fig. fig.phase(b), the optimization was performed by measuring** the **resonator’s reflected phase as a function of** qubit** bias flux, at** the **optimal probe frequency, 3.70415 GHz in this case. The** qubit** was initialized by setting** the **flux to its negative “reset” value (position I), where the** qubit** potential has only one minimum. The flux was then increased to an intermediate value Φ , placing the** qubit** state in** the **left well, and** the **reflection phase measured with a 5 μ s microwave probe signal (blue data). The flux was then set to its positive reset value (position V), then brought back to** the **same flux value Φ , placing the** qubit** state in** the **right well, and** the **reflection phase again measured with a probe signal (red data). Between** the **symmetry point III ( Φ = 0.5 ) and** the **regions with just one potential minimum ( Φ ≤ 0.1 or Φ ≥ 0.9 ), the** qubit** inductance differs between** the **left and right well states, which gives rise to** the **difference in phase for** the **red and blue data measured at** the **same flux. This difference increases for** the **flux bias closer to** the **single-well region, which can give a signal-to-noise ratio as high as 30 at ambient readout microwave power. The optimal flux bias was then set to a value where** the **readout had a high signal-to-noise ratio (typically > 5), but with a potential barrier sufficient to prevent spurious readout-induced switching between** the **potential wells. Several iterations were needed to optimize both** the **probe frequency and flux bias....We demonstrated the multiplexed readout using a quantum circuit comprising four phase **qubits** and five integrated resonators, shown in Fig. fig.setup(c). The design of this chip is similar to that used for a recent implementation of Shor’s algorithm,, but here the **qubits** were read out off a single line using microwave reflectometry, replacing the SQUID readout used Ref 6. . This dramatically simplifies the chip design and significantly reduces the footprint of the quantum circuit. We designed the readout resonators so that they resonated at **frequencies** of 3-4 GHz (far de-tuned from the **qubit** | g ↔ | e transition **frequency** of 6-7 GHz), with loaded resonance linewidths of a few hundred kilohertz. This allows us to use **frequency** multiplexing, which has been successfully used in the readout of microwave kinetic inductance detectors as well as other types of **qubits**. Combined with custom GHz-**frequency** signal generation and acquisition boards, this approach provides a compact and efficient readout scheme that should be applicable to systems with 10-100 **qubits** using a single readout line, with sufficient measurement bandwidth for microsecond-scale readout times....We used a standard heterodyne detection method to measure** the **reflected signal from each readout resonator, as shown in Fig. fig.measure. Key components include two customized field-programmable gate array (FPGA) boards, one connected to a 14 bit digital-to-analog converter (DAC) for generating arbitrary probe waveforms, with a 1 GS/s digitizing rate (GS/s: gigasample per second), and** the **other connected to a 8 bit analog-to-digital converter (ADC) for data acquisition and processing, also with a 1 GS/s digitizing rate. Probe waveforms were generated by preparing multi-tone signals in both I p and Q p (cosine and sine) probe quadratures for mixer up-conversion, each tone chosen so that after frequency up-conversion in an I Q mixer, it matched** the **resonance frequency f n ( n = 1 - 4 corresponding to** the **readout resonator for qubits 1-4) of** the **readout resonators. The reflected signals were amplified and down-converted with a second I Q mixer; the reflected I r t and Q r t signals comprise** the **same signal tones as** the **probe waveform, but with an additional phase shift that encodes** the **measurement signal, i.e. the phase φ n of** the **reflected tone at frequency f n encodes** the **state of** qubit** n . The phases φ n , n = 1 - 4 , were then evaluated using** the **digital demodulation channel on** the **data acquisition FPGA board. This is performed by digital mixing and integrating of** the **digitized I r t and Q r t signals at** the **resonator frequency f n , each with a separate demodulation channel....The calibration of the readout process was done in two steps. We first optimized the microwave probe **frequency** to maximize the signal difference between the left and right well states. This was performed by measuring the reflected phase φ as a function of the probe **frequency**, with the **qubit** prepared first in the left and then in the right well. In Fig. fig.phase(a), we show the result with the **qubit** flux bias set to 0.15 Φ 0 , where the difference in L J in two well states was relatively large. The probe **frequency** that maximized the signal difference was typically mid-way between the loaded resonator **frequencies** for the **qubit** in the left and right wells, marked by the dashed line in Fig. fig.phase(a). We typically obtained resonator **frequency** shifts as large as ∼ 150 kHz for the **qubit** between the two wells, as shown in Fig. fig.phase(a), significantly larger than the resonator linewidth....Multiplexed dispersive readout of superconducting phase **qubits**...We demonstrated** the **multiplexed readout using a quantum circuit comprising four phase qubits and five integrated resonators, shown in Fig. fig.setup(c). The design of this chip is similar to that used for a recent implementation of Shor’s algorithm,, but here** the **qubits were read out off a single line using microwave reflectometry, replacing** the **SQUID readout used Ref 6. . This dramatically simplifies** the **chip design and significantly reduces** the **footprint of** the **quantum circuit. We designed** the **readout resonators so that they resonated at frequencies of 3-4 GHz (far de-tuned from the** qubit** | g ↔ | e transition frequency of 6-7 GHz), with loaded resonance linewidths of a few hundred kilohertz. This allows us to use frequency multiplexing, which has been successfully used in** the **readout of microwave kinetic inductance detectors as well as other types of qubits. Combined with custom GHz-frequency signal generation and acquisition boards, this approach provides a compact and efficient readout scheme that should be applicable to systems with 10-100 qubits using a single readout line, with sufficient measurement bandwidth for microsecond-scale readout times....The calibration of** the **readout process was done in two steps. We first optimized** the **microwave probe frequency to maximize** the **signal difference between** the **left and right well states. This was performed by measuring** the **reflected phase φ as a function of** the **probe frequency, with the** qubit** prepared first in** the **left and then in** the **right well. In Fig. fig.phase(a), we show** the **result with the** qubit** flux bias set to 0.15 Φ 0 , where** the **difference in L J in two well states was relatively large. The probe frequency that maximized** the **signal difference was typically mid-way between** the **loaded resonator frequencies for the** qubit** in** the **left and right wells, marked by** the **dashed line in Fig. fig.phase(a). We typically obtained resonator frequency shifts as large as ∼ 150 kHz for the** qubit** between** the **two wells, as shown in Fig. fig.phase(a), significantly larger than** the **resonator linewidth....(Color online) (a) Phase of signal reflected from readout resonator, as a function of the probe microwave **frequency** (averaged 900 times), for the **qubit** in the left ( L , blue) and right ( R , red) wells. Dashed line shows probe **frequency** for maximum visibility. (b) Reflected phase as a function of **qubit** flux bias, with no averaging. See text for details. fig.phase...(Color online) Setup for **frequency**-multiplexed readout. Multiplexed readout signals I p and Q p from top FGPA-DAC board are up-converted by mixing with a fixed microwave tone, then pass through the circulator into the **qubit** chip. Reflected signals pass back through the circulator, through the two amplifiers G 1 and G 2 , and are down-converted into I r and Q r using the same microwave tone, and are then processed by the bottom ADC-FPGA board. Data in the shadowed region are the down-converted I r and Q r spectra output from the ADC-FPGA board; probe signals from the FPGA-DAC board have the same **frequency** spectrum. D C indicates the digital demodulation channels, each processed independently and sent to the computer. fig.measure...With the bias points chosen for each **qubit**, we demonstrated the **frequency**-multiplexed readout by performing a multi-**qubit** experiment. To minimize crosstalk, we removed the coupling capacitors between **qubits** used in Ref. 6. In this experiment, we drove Rabi **oscillations** on each ** qubit’s** | g ↔ | e transition and read out the

**qubit**states simultaneously. We first calibrated the pulse amplitude needed for each

**qubit**to perform a | g → | e Rabi transition in 10 ns. The drive amplitude was then set to 1, 2/3, 1/2 and 2/5 the calibrated Rabi transition amplitude for

**qubits**Q 1 to Q 4 respectively, so that the Rabi period was 20 ns, 30 ns, 40 ns and 50 ns for

**qubits**Q 1 to Q 4 . We then drove each

**qubit**separately using an on-resonance Rabi drive for a duration τ , followed immediately by a projective measurement and

**qubit**state readout. This experiment yielded the measurements shown in Fig. fig.rabi(a)-(d) for

**qubits**Q 1 - Q 4 respectively....With the probe

**frequency**set in the first step, the flux bias was then set to optimize the readout. As illustrated in Fig. fig.phase(b), the optimization was performed by measuring the resonator’s reflected phase as a function of

**qubit**bias flux, at the optimal probe

**frequency**, 3.70415 GHz in this case. The

**qubit**was initialized by setting the flux to its negative “reset” value (position I), where the

**qubit**potential has only one minimum. The flux was then increased to an intermediate value Φ , placing the

**qubit**state in the left well, and the reflection phase measured with a 5 μ s microwave probe signal (blue data). The flux was then set to its positive reset value (position V), then brought back to the same flux value Φ , placing the

**qubit**state in the right well, and the reflection phase again measured with a probe signal (red data). Between the symmetry point III ( Φ = 0.5 ) and the regions with just one potential minimum ( Φ ≤ 0.1 or Φ ≥ 0.9 ), the

**qubit**inductance differs between the left and right well states, which gives rise to the difference in phase for the red and blue data measured at the same flux. This difference increases for the flux bias closer to the single-well region, which can give a signal-to-noise ratio as high as 30 at ambient readout microwave power. The optimal flux bias was then set to a value where the readout had a high signal-to-noise ratio (typically > 5), but with a potential barrier sufficient to prevent spurious readout-induced switching between the potential wells. Several iterations were needed to optimize both the probe

**frequency**and flux bias....(Color online) (a)-(d) Rabi oscillations for

**qubits**Q 1 - Q 4 respectively, with the

**qubits**driven with 1, 2/3, 1/2 and 2/5 the on-resonance drive amplitude needed to perform a 10 ns Rabi | g → | e transition. (e) Rabi oscillations measured simultaneously for all the

**qubits**, using the same color coding and drive amplitudes as for panels (a)-(d). fig.rabi...With each

**qubit**individually characterized, we then excited and measured all four

**qubits**simultaneously, as shown in Fig. fig.rabi(e). There is no measurable difference between the individually-measured Rabi

**oscillations**in panels (a)-(d) compared to the multiplexed readout in panel (e). ... We introduce a

**frequency**-multiplexed readout scheme for superconducting phase

**qubits**. Using a quantum circuit with four phase

**qubits**, we couple each

**qubit**to a separate lumped-element superconducting readout resonator, with the readout resonators connected in parallel to a single measurement line. The readout resonators and control electronics are designed so that all four

**qubits**can be read out simultaneously using

**frequency**multiplexing on the one measurement line. This technology provides a highly efficient and compact means for reading out multiple

**qubits**, a significant advantage for scaling up to larger numbers of

**qubits**.

Files:

Contributors: Hua, Ming, Deng, Fu-Guo

Date: 2013-09-30

(color online) (a) The probability distribution of the two quantum Rabi **oscillations** ROT 0 (the blue solid line) and ROT 1 (the green solid line) of two charge **qubits** coupled to a resonator. (b) The probability distribution of the four quantum Rabi **oscillations** in our cc-phase gate on a three-charge-**qubit** system. Here, the blue-solid, green-solid, red-dashed, and Cambridge-blue-dot-dashed lines represent the quantum Rabi **oscillations** ROT 00 ( | 0 1 | 0 2 | 1 3 | 0 a ↔ | 0 1 | 0 2 | 0 3 | 1 a ), ROT 01 ( | 0 1 | 1 2 | 1 3 | 0 a ↔ | 0 1 | 1 2 | 0 3 | 1 a ), ROT 10 ( | 1 1 | 0 2 | 1 3 | 0 a ↔ | 1 1 | 0 2 | 0 3 | 1 a ), and ROT 11 ( | 1 1 | 1 2 | 1 3 | 0 a ↔ | 1 1 | 1 2 | 0 3 | 1 a ), respectively....Eq.( stark) means** the **NSD** qubit** transition and Eq.( kerr) means** the **QSD resonator transition, shown in Fig. fig1(a) and (b), respectively....(color online) Sketch of a coplanar geometry for the circuit QED with three superconducting **qubits**. **Qubits** are placed around the maxima of the electrical field amplitude of R a and R b (not drawn in this figure), and the distance between them is large enough so that there is no direct interaction between them. The fundamental **frequencies** of resonators are ω r j / 2 π ( j = a , b ), the **frequencies** of the **qubits** are ω q i / 2 π ( i = 1 , 2 , 3 ), and they are capacitively coupled to the resonators. The coupling strengths between them are g i j / 2 π . We can use the control line (not drawn here) to afford the flux to tune the transition **frequencies** of the **qubits**....We present a fast quantum entangling operation on superconducting **qubits** assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified **qubit**-state-dependent resonator transition **frequency** and the tunable period relation between a wanted quantum Rabi oscillation and an unwanted one. This operation does not require any kind of drive fields and the interaction between **qubits**. More interesting, the non-computational third excitation states of the charge **qubits** can play an important role in shortening largely the operation time of the entangling gates. All those features provide an effective way to realize much faster quantum entangling gates on superconducting **qubits** than previous proposals....Eq.( stark) means the NSD **qubit** transition and Eq.( kerr) means the QSD resonator transition, shown in Fig. fig1(a) and (b), respectively....(color online) Simulated outcomes for the maximum amplitude value of the expectation about the quantum Rabi oscillation varying with the coupling strength g 2 and the frequency of the second qubit ω 2 . (a) The outcomes for ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a . (b) The outcomes for ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a . Here the parameters of the resonator and the first qubit q 1 are taken as ω a / 2 π = 6.0 GHz, ω q 1 / 2 π = 7.0 GHz, and g 1 / 2 π = 0.2 GHz....and it is shown in Fig. fig4 (a). In the simulation of our SR, we choose the reasonable parameters by considering the energy level structure of a charge **qubit**, according to Ref. . Here ω r a / 2 π = 6.0 GHz. The transition **frequency** of two **qubits** between and are chosen as ω 0 , 1 ; 1 / 2 π = E 1 ; 1 - E 0 ; 1 = 5.0 GHz, ω 1 , 2 ; 1 / 2 π = E 2 ; 1 - E 1 ; 1 = 6.2 GHz, ω 0 , 1 ; 2 / 2 π = E 1 ; 2 - E 0 ; 2 = 6.035 GHz, and ω 1 , 2 ; 2 / 2 π = E 2 ; 2 - E 1 ; 2 = 7.335 GHz. Here E i ; q is the energy for the level i of the **qubit** q , and σ i , i ' ; q + ≡ i q i ' . g i , j ; q is the coupling strength between the resonator R a and the **qubit** q in the transition between the energy levels | i q and | j q ( i = 0 , 1 , j = 1 , 2 , and q = 1 , 2 ). For convenience, we take the coupling strengths as g 0 , 1 ; 1 / 2 π = g 1 , 2 ; 1 / 2 π = 0.2 GHz and g 0 , 1 ; 2 / 2 π = g 1 , 2 ; 2 / 2 π = 0.0488 GHz....We numerically simulate the maximal expectation values (MAEVs) of ROT 0 and ROT 1 based on the Hamiltonian H 2 q , shown in Fig. fig3(a) and (b), respectively. Here, the expectation value is defined as | ψ | e - i H 2 q t / ℏ | ψ 0 | 2 . | ψ 0 and | ψ are the initial and the final states of a quantum Rabi **oscillation**, respectively. The MAEV s vary with the transition **frequency** ω 2 and the coupling strength g 2 . It is obvious that the amplified QSD resonator transition can generate a selective resonance (SR) when the coupling strength g 2 is small enough....in which we neglect** the **direct interaction between** the **two qubits (i.e., q 1 and q 2 ), shown in Fig. fig2. Here σ i + = | 1 i 0 | is** the **creation operator of q i ( i = 1 , 2 ). g i is** the **coupling strength between q i and R a . The parameters are chosen to make q 1 interact with R a in** the **quasi-dispersive regime. That is, the transition frequency of R a is determined by** the **state of q 1 . By taking a proper transition frequency of q 2 (which equals to** the **transition frequency of R a when q 1 is in** the **state | 0 1 ), one can realize** the **quantum Rabi oscillation (ROT) ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a , while ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a occurs with a small probability as q 2 detunes with R a when q 1 is in** the **state | 1 1 . Here** the **Fock state | n a represents** the **photon number n in R a ( n = 0 , 1 ). | 0 i and | 1 i are** the **ground and** the **first excited states of q i , respectively....and it is shown in Fig. fig4 (a). In** the **simulation of our SR, we choose** the **reasonable parameters by considering** the **energy level structure of a charge** qubit**, according to Ref. . Here ω r a / 2 π = 6.0 GHz. The transition frequency of two qubits between and are chosen as ω 0 , 1 ; 1 / 2 π = E 1 ; 1 - E 0 ; 1 = 5.0 GHz, ω 1 , 2 ; 1 / 2 π = E 2 ; 1 - E 1 ; 1 = 6.2 GHz, ω 0 , 1 ; 2 / 2 π = E 1 ; 2 - E 0 ; 2 = 6.035 GHz, and ω 1 , 2 ; 2 / 2 π = E 2 ; 2 - E 1 ; 2 = 7.335 GHz. Here E i ; q is** the **energy for** the **level i of the** qubit** q , and σ i , i ' ; q + ≡ i q i ' . g i , j ; q is** the **coupling strength between** the **resonator R a and the** qubit** q in** the **transition between** the **energy levels | i q and | j q ( i = 0 , 1 , j = 1 , 2 , and q = 1 , 2 ). For convenience, we take** the **coupling strengths as g 0 , 1 ; 1 / 2 π = g 1 , 2 ; 1 / 2 π = 0.2 GHz and g 0 , 1 ; 2 / 2 π = g 1 , 2 ; 2 / 2 π = 0.0488 GHz....The quantum entangling operation based on the SR can also help us to complete a single-step controlled-controlled phase (cc-phase) quantum gate on the three charge **qubits** q 1 , q 2 , and q 3 by using the system shown in Fig. fig2 except for the resonator R b . Here, q 1 and q 2 act as the control **qubits**, and q 3 is the target **qubit**. The initial state of this system is prepared as | Φ 0 = 1 2 2 | 0 1 | 0 2 | 0 3 + | 0 1 | 0 2 | 1 3 + | 0 1 | 1 2 | 0 3 + | 0 1 | 1 2 | 1 3 + | 1 1 | 0 2 | 0 3 + | 1 1 | 0 2 | 1 3 + | 1 1 | 1 2 | 0 3 + | 1 1 | 1 2 | 1 3 | 0 a . In this system, both q 1 and q 2 are in the quasi-dispersive regime with R a , and the transition **frequency** of q 3 is adjusted to be equivalent to that of R a when q 1 and q 2 are in their ground states. The QSD transition **frequency** on R a becomes...The quantum entangling operation based on** the **SR can also help us to complete a single-step controlled-controlled phase (cc-phase) quantum gate on** the **three charge qubits q 1 , q 2 , and q 3 by using** the **system shown in Fig. fig2 except for** the **resonator R b . Here, q 1 and q 2 act as** the **control qubits, and q 3 is** the **target** qubit**. The initial state of this system is prepared as | Φ 0 = 1 2 2 | 0 1 | 0 2 | 0 3 + | 0 1 | 0 2 | 1 3 + | 0 1 | 1 2 | 0 3 + | 0 1 | 1 2 | 1 3 + | 1 1 | 0 2 | 0 3 + | 1 1 | 0 2 | 1 3 + | 1 1 | 1 2 | 0 3 + | 1 1 | 1 2 | 1 3 | 0 a . In this system, both q 1 and q 2 are in** the **quasi-dispersive regime with R a , and** the **transition frequency of q 3 is adjusted to be equivalent to that of R a when q 1 and q 2 are in their ground states. The QSD transition frequency on R a becomes...(color online) (a) The **qubit**-state-dependent resonator transition, which means the **frequency** shift of the resonator transition δ r arises from the state ( | 0 q or | 1 q ) of the **qubit**. (b) The number-state-dependent **qubit** transition, which means the **frequency** shift δ q takes place on the **qubit** due to the photon number n = 1 or 0 in the resonator in the dispersive regime....(color online) (a) The qubit-state-dependent resonator transition, which means the frequency shift of the resonator transition δ r arises from the state ( | 0 q or | 1 q ) of the qubit. (b) The number-state-dependent qubit transition, which means the frequency shift δ q takes place on the qubit due to the photon number n = 1 or 0 in the resonator in the dispersive regime....in which we neglect the direct interaction between the two **qubits** (i.e., q 1 and q 2 ), shown in Fig. fig2. Here σ i + = | 1 i 0 | is the creation operator of q i ( i = 1 , 2 ). g i is the coupling strength between q i and R a . The parameters are chosen to make q 1 interact with R a in the quasi-dispersive regime. That is, the transition **frequency** of R a is determined by the state of q 1 . By taking a proper transition **frequency** of q 2 (which equals to the transition **frequency** of R a when q 1 is in the state | 0 1 ), one can realize the quantum Rabi **oscillation** (ROT) ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a , while ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a occurs with a small probability as q 2 detunes with R a when q 1 is in the state | 1 1 . Here the Fock state | n a represents the photon number n in R a ( n = 0 , 1 ). | 0 i and | 1 i are the ground and the first excited states of q i , respectively....(color online) Simulated outcomes for the maximum amplitude value of the expectation about the quantum Rabi **oscillation** varying with the coupling strength g 2 and the **frequency** of the second **qubit** ω 2 . (a) The outcomes for ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a . (b) The outcomes for ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a . Here the parameters of the resonator and the first **qubit** q 1 are taken as ω a / 2 π = 6.0 GHz, ω q 1 / 2 π = 7.0 GHz, and g 1 / 2 π = 0.2 GHz....(color online) (a) The probability distribution of the two quantum Rabi oscillations ROT 0 (the blue solid line) and ROT 1 (the green solid line) of two charge **qubits** coupled to a resonator. (b) The probability distribution of the four quantum Rabi oscillations in our cc-phase gate on a three-charge-qubit system. Here, the blue-solid, green-solid, red-dashed, and Cambridge-blue-dot-dashed lines represent the quantum Rabi oscillations ROT 00 ( | 0 1 | 0 2 | 1 3 | 0 a ↔ | 0 1 | 0 2 | 0 3 | 1 a ), ROT 01 ( | 0 1 | 1 2 | 1 3 | 0 a ↔ | 0 1 | 1 2 | 0 3 | 1 a ), ROT 10 ( | 1 1 | 0 2 | 1 3 | 0 a ↔ | 1 1 | 0 2 | 0 3 | 1 a ), and ROT 11 ( | 1 1 | 1 2 | 1 3 | 0 a ↔ | 1 1 | 1 2 | 0 3 | 1 a ), respectively....Selective-Resonance-Based Quantum Entangling Operation on **Qubits** in Circuit QED...We present a fast quantum entangling operation on superconducting **qubits** assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified **qubit**-state-dependent resonator transition **frequency** and the tunable period relation between a wanted quantum Rabi **oscillation** and an unwanted one. This operation does not require any kind of drive fields and the interaction between **qubits**. More interesting, the non-computational third excitation states of the charge **qubits** can play an important role in shortening largely the operation time of the entangling gates. All those features provide an effective way to realize much faster quantum entangling gates on superconducting **qubits** than previous proposals....A complex three-qubit gate, such as a Fredkin gate on a three-qubit system can also be constructed with** the **quantum entangling operation based on** the **SR assisted by** the **two resonators R a and R b in a simple way, shown in Fig. fig2. In this system, R b has a different transition frequency with R a , and q 1 couples to both R a and R b simultaneously in** the **quasi-dispersive regime. Let q 2 and q 3 resonate selectively with R a and R b when q 1 is in** the **state | 1 1 , respectively, with g 0 , 1 ; 2 a t = g 0 , 1 ; 3 b t = 1.5 π first (here g i , j ; q k is** the **coupling strength between** the **resonator k and the** qubit** q in** the **transition between** the **energy levels | i q and | j q ), and then let q 2 and q 3 be selectively resonant with R b and R a when q 1 is in** the **state | 1 1 , respectively, with g 0 , 1 ; 3 a t = g 0 , 1 ; 2 b t = 0.5 π , a Fredkin gate can be realized. ... We present a fast quantum entangling operation on superconducting **qubits** assisted by a resonator in the quasi-dispersive regime with a new effect --- the selective resonance coming from the amplified **qubit**-state-dependent resonator transition **frequency** and the tunable period relation between a wanted quantum Rabi **oscillation** and an unwanted one. This operation does not require any kind of drive fields and the interaction between **qubits**. More interesting, the non-computational third excitation states of the charge **qubits** can play an important role in shortening largely the operation time of the entangling gates. All those features provide an effective way to realize much faster quantum entangling gates on superconducting **qubits** than previous proposals.

Files:

Contributors: Chudzicki, Christopher, Strauch, Frederick W.

Date: 2010-08-10

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

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Contributors: 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**.

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