### 54077 results for qubit oscillator frequency

Contributors: Singh, Mandip

Date: 2014-07-01

To estimate the **oscillation** **frequencies**, consider a flux-**qubit**-cantilever made of niobium which is a type-II superconductor of transition temperature 9.26 ~ K . Consider niobium has a square cross-section with thickness t = 0.5 μ m, l = 6 μ m, w = 4 μ m ( A = l × w ). For these dimensions the mass of the cantilever is 3.64 × 10 -14 Kg and moment of inertia is I m ≃ 7.28 × 10 -25 Kgm 2 . The critical current of Josephson junction I c = 5 μ A, capacitance C = 0.1 pF and self inductance L = 100 pH are of the same order as described in Ref . The quantity β L = 2 π L I c / Φ o ≃ 1.52 . Consider intrinsic **frequency** of the cantilever is ω i = 2 π × 12000 rad/s. For an equilibrium angle θ 0 = θ n + = cos -1 n Φ o / B x A there exists a single global potential energy minimum. If we consider n = 0 and B x = 5 × 10 -2 T the global potential energy minimum is located at ( 0 , π / 2 ). For parameters described above ω φ ≃ 2 π × 7.99 × 10 10 rad/s, ω δ = 2 π × 25398.1 rad/s, κ = 0.012 A. The eigen **frequencies** of the flux-**qubit**-cantilever are ω X ≃ 2 π × 7.99 × 10 10 rad/s and ω Y = 2 π × 21122.5 rad/s. A contour plot of potential energy of the flux-**qubit**-cantilever, indicating a two dimensional global minimum located at ( 0 , π / 2 ) and two local minima, is shown in Fig. fig2. Even if we consider intrinsic **frequencies** to be zero the restoring force is still nonzero due to a finite coupling constant. For ω i = 0 , the angular **frequencies** are ω φ ≃ 2 π × 7.99 × 10 10 rad/s, ω δ = 2 π × 22384.5 rad/s, ω X ≃ 2 π × 7.99 × 10 10 rad/s and ω Y = 2 π × 17382.8 rad/s. The **frequencies** can be increased by increasing external magnetic field and by decreasing the dimensions of the cantilever that reduces mass and moment of inertia....In this paper a macroscopic quantum **oscillator** is introduced that consists of a flux **qubit** in the form of a cantilever. The magnetic flux linked to the flux **qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling is controlled through an external magnetic field. The ground state of the introduced flux-**qubit**-cantilever corresponds to a quantum entanglement between magnetic flux and the cantilever displacement....fig1 A schematic of a flux-**qubit**-cantilever. A part of the flux **qubit** (larger loop) is in the form of a cantilever. External magnetic field B x controls the coupling between the flux **qubit** and the cantilever. An additional magnetic flux threading a DC-SQUID (smaller loop) that consists of two Josephson junctions adjusts the tunneling amplitude. DC-SQUID can be shielded from the effect of B x ....The potential energy of the flux-**qubit**-cantilever corresponds to a symmetric double well potential (i.e. two global minima and m Φ o **qubit**-cantilever is biased at a half of a flux quantum, Φ o / 2 . A contour plot indicating a two dimensional symmetric double well potential is shown in Fig. fig3. Consider the nonseparable ground states of the left and the right well are | α L and | α R , respectively. The barrier height between the wells of the double well potential which is less than 2 E j reduces when ω i is increased. The barrier height controls the tunneling between potential wells and it can also be tuned through an external magnetic flux applied to a DC-SQUID of the flux-**qubit**-cantilever. When tunneling between wells is introduced the ground state of flux-**qubit**-cantilever is | Ψ E = | α L + | α R / 2 . The state | Ψ E is an entangled state of distinct magnetic flux and distinct cantilever deflection states. The state | Ψ E can be realised by cooling the flux-**qubit**-cantilever to its ground state....fig3 A contour plot indicating location of two dimensional potential energy minima forming a symmetric double well potential for cantilever equilibrium angle θ 0 = cos -1 Φ o / 2 B x A , ω i = 2 π × 12000 ~ r a d / s , B x = 5 × 10 -2 T. The contour interval in units of **frequency** ( E / h ) is ∼ 4 × 10 11 Hz....Consider a schematic of a flux-**qubit**-cantilever shown in Fig. fig1 where a part of a superconducting loop of a flux **qubit** forms a cantilever. The larger loop is interrupted by a smaller loop consisting of two Josephson junctions - a DC Superconducting Quantum Interference Device (DC-SQUID). The Josephson energy that is constant for a single Josephson junction can be varied by applying a magnetic flux to a DC-SQUID loop. However, for calculations a flux **qubit** with a single Josephson junction is considered throughout this paper. The external magnetic flux applied to the cantilever is Φ a = B x A cos θ , where B x is the magnitude of an uniform external magnetic field along x -axis and area vector A → substends an angle θ with the magnetic field direction ( x -axis). Consider the cantilever **oscillates** about an equilibrium angle θ 0 with an intrinsic **frequency** of **oscillation** ω i i.e. the **frequency** in absence of magnetic field. The external magnetic flux applied to the flux-**qubit**-cantilever depends on the cantilever deflection therefore, the flux **qubit** whose potential energy depends on an external flux is coupled to the cantilever degrees of freedom. The potential energy of the flux-**qubit**-cantilever corresponds to a two dimensional potential V Φ θ and the Hamiltonian of the flux-**qubit**-cantilever interrupted by a single Josephson junction is...fig2 A contour plot indicating location of a two dimensional global potential energy minimum at ( 0 , π / 2 ) and local minima for cantilever equilibrium angle θ 0 = π / 2 , ω i = 2 π × 12000 rad/s, B x = 5 × 10 -2 T. The contour interval in units of **frequency** ( E / h ) is ∼ 3.9 × 10 11 Hz. ... In this paper a macroscopic quantum **oscillator** is introduced that consists of a flux **qubit** in the form of a cantilever. The magnetic flux linked to the flux **qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling is controlled through an external magnetic field. The ground state of the introduced flux-**qubit**-cantilever corresponds to a quantum entanglement between magnetic flux and the cantilever displacement.

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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....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) 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**....(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....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....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. ... 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.

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Contributors: Meier, Florian, Loss, Daniel

Date: 2004-08-26

(color online). Rabi **oscillations** for a squbit-fluctuator system. The probability p 1 t to find the squbit in state | 1 is obtained from numerical integration of Eq. ( eq:mastereq) (solid line) and the analytical solution Eq. ( eq:pdyn-3) (dashed line) which is valid for b x / J ≫ 1 . For weak decoherence of the fluctuator, γ / J 1 , damped single-**frequency** **oscillations** are restored. The fluctuator leads to a reduction of the first maximum in p 1 t to ∼ 0.8 [(a) and (b)] and ∼ 0.9 [(c) and (d)], respectively. The parameters are (a) b x / J = 1.5 , γ / J = 0.5 ; (b) b x / J = 1.5 , γ / J = 1.5 ; (c) b x / J = 3 , γ / J = 0.5 ; (d) b x / J = 3 , γ / J = 1.5 ....Coherent Rabi **oscillations** between quantum states of superconducting micro-circuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi **frequency** is comparable to the coupling energy of micro-circuit and fluctuator. For larger Rabi **frequencies**, transitions to the second excited state of the superconducting micro-circuit become dominant in suppressing the Rabi **oscillation** visibility. We also calculate the probability for Bogoliubov quasi-particle excitations in typical Rabi **oscillation** experiments....exhibits single-**frequency** **oscillations** with reduced visibility [Fig. Fig2(b)]. Part of the visibility reduction can be traced back to leakage into state | 2 . More subtly, off-resonant transitions from | 1 to | 2 induced by the driving field lead to an energy shift of | 1 , such that the transition from | 0 to | 1 is no longer resonant with the driving field, which also reduces the visibility. For b x / 2 ℏ Δ ω = 1 / 3 , corresponding to b x / h = 150 ~ M H z in Ref. ...Energy shifts induced by AC driving field. – Exploring the visibility reduction at a time-scale of 10 n s requires Rabi **frequencies** | b x | / h 100 M H z . We show next that, in this regime, transitions to the second excited squbit state lead to an oscillatory behavior in p 1 t with a visibility smaller than 0.7 . For characteristic parameters of a phase-squbit, the second excited state | 2 is energetically separated from | 1 by ω 21 = 0.97 ω 10 . Similarly to | 0 and | 1 , the state | 2 is localized around the local energy minimum in Fig. Fig2(a). For adiabatic switching of the AC current, transitions to | 2 can be neglected as long as | b x | ≪ ℏ Δ ω = ℏ ω 10 - ω 21 ≃ 0.03 ℏ ω 10 . However, for b x comparable to ℏ Δ ω , the applied AC current strongly couples | 1 and | 2 because 2 | φ ̂ | 1 ≠ 0 , where φ ̂ is the phase operator. For typical parameters, b x / ℏ Δ ω ranges from 0.05 to 1 , depending on the irradiated power . Taking into account the second excited state of the phase-squbit, the squbit Hamiltonian in the rotating frame is ... Coherent Rabi **oscillations** between quantum states of superconducting micro-circuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi **frequency** is comparable to the coupling energy of micro-circuit and fluctuator. For larger Rabi **frequencies**, transitions to the second excited state of the superconducting micro-circuit become dominant in suppressing the Rabi **oscillation** visibility. We also calculate the probability for Bogoliubov quasi-particle excitations in typical Rabi **oscillation** experiments.

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Contributors: Yoshihara, Fumiki, Nakamura, Yasunobu, Yan, Fei, Gustavsson, Simon, Bylander, Jonas, Oliver, William D., Tsai, Jaw-Shen

Date: 2014-02-06

Parameters in calculations and measurements in units of GHz. In the first column, cal: δ ω Ω R 0 stands for the calculation to study the shift of the resonant **frequency**, and cal: Γ R s t δ ω m w stands for the calculation to study the decay of Rabi **oscillations** due to quasistatic flux noise. “Optimal" in the last column means that at each ε m w , ω m w is chosen to minimize dephasing due to quasistatic flux noise....We infer the high-**frequency** flux noise spectrum in a superconducting flux **qubit** by studying the decay of Rabi **oscillations** under strong driving conditions. The large anharmonicity of the **qubit** and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi **frequencies** up to 1.7 GHz were achieved, approaching the **qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide **frequency** range decreases with increasing **frequency** up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins....(Color online) Rabi **oscillation** curves with different Rabi **frequencies** Ω R measured at different static flux bias ε . At each Ω R , δ ω m w is chosen to minimize dephasing due to quasistatic flux noise. The red lines are the fitting curves. In the measurements shown in the middle and bottom panels, only parts of the **oscillations** are monitored so that we can save measurement time while the envelopes of Rabi **oscillations** are captured. The inset is a magnification of the data in the bottom panel together with the fitting curve....In the Rabi **oscillation** measurements, a microwave pulse is applied to the **qubit** followed by a readout pulse, and P s w as a function of the microwave pulse length is measured. First, we measure the Rabi **oscillation** decay at ε = 0 , where the quasistatic noise contribution is negligible. Figure GRfR1p5(d) shows the measured 1 / e decay rate of the Rabi **oscillations** Γ R 1 / e as a function of Ω R 0 . For Ω R 0 / 2 π up to 400 MHz, Γ R 1 / e is approximately 3 Γ 1 / 4 , limited by the energy relaxation, and S Δ Ω R 0 is negligible. For Ω R 0 / 2 π from 600 MHz to 2.2 GHz, Γ R 1 / e > 3 Γ 1 / 4 . A possible origin of this additional decoherence is fluctuations of ε m w , δ ε m w : Ω R 0 is first order sensitive to δ ε m w , which is reported to be proportional to ε m w itself. Next, the decay for the case ε ≈ Δ is studied. To observe the contribution from quasistatic flux noise, the Rabi **oscillation** decay as a function of ω m w is measured, where the contribution from the other sources is expected to be almost constant. Figure GRfR1p5(b) shows Γ R 1 / e at ε / 2 π = 4.16 GHz as a function of δ ω m w while keeping Ω R / 2 π between 1.5 and 1.6 GHz. Besides the offset and scatter, the trend of Γ R 1 / e agrees with that of the simulated Γ R s t . This result indicates that numerical calculation properly evaluates δ ω m w minimizing Γ R s t . Finally, the decay for the case ε ≈ Δ as a function of ε m w , covering a wide range of Ω R , is measured (Fig. Rabis)....(Color online) Power spectrum density of flux fluctuations S n φ ω extracted from the Rabi **oscillation** measurements in the first ( ε / 2 π = 4.16 GHz) and second cooldowns. The PSDs obtained from the spin-echo and energy relaxation measurements in the second cooldown are also plotted. The black solid line is the 1/ f spectrum extrapolated from the FID measurements in the second cooldown. The purple dashed line is the estimated Johnson noise from a 50 Ω microwave line coupled to the **qubit** by a mutual inductance of 1.2 pH and nominally cooled to 35 mK. The pink dotted line is a Lorentzian, S n φ m o d e l ω = S h ω w 2 / ω 2 + ω w 2 , and the orange solid line is the sum of the Lorentzian and the Johnson noise. Here the parameters are S h = 3.6 × 10 -19 r a d -1 s and ω w / 2 π = 2.7 × 10 7 H z ....Josephson devices, decoherence, Rabi **oscillation**, $1/f$ noise...(Color online) (a) Numerically calculated shift of the resonant **frequency** δ ω (black open circles) and the Bloch–Siegert shift δ ω B S (blue line). (b) Numerically calculated decay rate Γ R s t (black open circles) and Rabi **frequency** Ω R (red solid triangles) as functions of the detuning δ ω m w from ω 01 . The purple solid line is a fit based on Eq. ( fRfull). The measured 1/ e decay rates Γ R 1 / e at ε / 2 π = 4.16 GHz for the range of Rabi **frequencies** Ω R / 2 π between 1.5 and 1.6 GHz (blue solid circles) are also plotted. (c) Calculated Rabi **frequency** Ω R , based on Eq. ( fRfull), as a function of ε for the cases (i) ω m w = ω 01 + δ ω (black solid line) and (ii) ω m w / 2 π = 6.1 GHz (red dashed line). The upper axis indicates ω 01 , corresponding to ε in the bottom axis. (d) The measured 1 / e decay rate of the Rabi **oscillations**, Γ R 1 / e , at ε = 0 and as a function of Ω R 0 . The red solid line indicates 3 4 Γ 1 obtained independently....The condition, ∂ Ω R / ∂ ε = 0 , is satisfied when ε = 0 or δ ω m w = δ ω - Ω R 0 2 / ω 01 . For Ω R 0 / 2 π = 1.52 GHz and ω 01 / 2 π = 6.400 GHz, the latter condition is calculated to be δ ω m w / 2 π = - 295 MHz, slightly different from the minimum of Γ R s t seen in Fig. G R f R 1 p 5 (b). The difference is due to the deviation from the linear approximation in Eq. ( fRfull), Ω R 0 ∝ ε m w / ω 01 . Figure GRfR1p5(c) shows the calculation of Ω R as a function of ε , based on Eq. ( fRfull). The Rabi **frequency** Ω R 0 at the shifted resonance decreases as ε increases, while Ω R , for a fixed microwave **frequency** of ω m w / 2 π = 6.1 GHz, has a minimum of approximately ω 01 / 2 π = 6.4 GHz. Here in the first order, Ω R is insensitive to the fluctuation of ε ....In Fig. GRfR1p5(a), δ ω as a function of Ω R 0 is plotted together with the well-known Bloch–Siegert shift, δ ω B S = 1 4 Ω R 0 2 ω 01 , obtained from the second-order perturbation theory. Fixed parameters for the calculation are Δ / 2 π = 4.869 and ε / 2 π = 4.154 GHz ( ω 01 / 2 π = 6.400 GHz). We find that δ ω B S overestimates δ ω when Ω R 0 / 2 π 800 MHz. The deviation from the Bloch–Siegert shift is due to the component of the ac flux drive that is parallel to the ** qubit’s** energy eigenbasis; this component is not averaged out when Ω R is comparable to ω m w . ... We infer the high-

**frequency**flux noise spectrum in a superconducting flux

**qubit**by studying the decay of Rabi

**oscillations**under strong driving conditions. The large anharmonicity of the

**qubit**and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi

**frequencies**up to 1.7 GHz were achieved, approaching the

**qubit**'s level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide

**frequency**range decreases with increasing

**frequency**up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins.

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

Date: 2002-08-07

Phase **qubit** coupled to a tank circuit....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.

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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...(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....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**....(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...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....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....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: Vierheilig, Carmen, Bercioux, Dario, Grifoni, Milena

Date: 2010-10-22

We consider a **qubit** coupled to a nonlinear quantum **oscillator**, the latter coupled to an Ohmic bath, and investigate the **qubit** dynamics. This composed system can be mapped onto that of a **qubit** coupled to an effective bath. An approximate mapping procedure to determine the spectral density of the effective bath is given. Specifically, within a linear response approximation the effective spectral density is given by the knowledge of the linear susceptibility of the nonlinear quantum **oscillator**. To determine the actual form of the susceptibility, we consider its periodically driven counterpart, the problem of the quantum Duffing **oscillator** within linear response theory in the driving amplitude. Knowing the effective spectral density, the **qubit** dynamics is investigated. In particular, an analytic formula for the **qubit**'s population difference is derived. Within the regime of validity of our theory, a very good agreement is found with predictions obtained from a Bloch-Redfield master equation approach applied to the composite **qubit**-nonlinear **oscillator** system....model We consider a composed system built of a **qubit**, -the system of interest-, coupled to a nonlinear quantum **oscillator** (NLO), see Fig. linearbath. To read-out the **qubit** state we couple the **qubit** linearly to the **oscillator** with the coupling constant g ¯ , such that via the intermediate NLO dissipation also enters the **qubit** dynamics....Jeff The effective spectral density follows from Eqs. ( gl20) and ( chilarger). It reads: J s i m p l J e f f ω e x = g ¯ 2 γ ω e x n 1 0 4 2 Ω 1 | ω e x | + Ω 1 M γ 2 Ω 1 2 2 n t h Ω 1 + 1 2 n 1 0 4 + 4 M Ω 2 | ω e x | - Ω 1 2 . As in case of the effective spectral density J e f f H O , Eq. ( linearspecdens), we observe Ohmic behaviour at low **frequency**. In contrast to the linear case, the effective spectral density is peaked at the shifted **frequency** Ω 1 . Its shape approaches the Lorentzian one of the linear effective spectral density, but with peak at the shifted **frequency**, as shown in Fig. CompLorentz....Schematic representation of the complementary approaches available to evaluate the **qubit** dynamics: In the first approach one determines the eigenvalues and eigenfunctions of the composite **qubit** plus **oscillator** system (yellow (light grey) box) and accounts afterwards for the harmonic bath characterized by the Ohmic spectral density J ω . In the effective bath description one considers an environment built of the harmonic bath and the nonlinear **oscillator** (red (dark grey) box). In the harmonic approximation the effective bath is fully characterized by its effective spectral density J e f f ω . approachschaubild...mapping The main aim is to evaluate the ** qubit’s** evolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the

**oscillator**and the Ohmic bath are put together, as depicted in Figure approachschaubild, to form an effective bath. The effective Hamiltonian...The transition

**frequencies**in Eqs. ( rc1) and ( rc2) coincide, and in Figs. Plowg and Flowg there is no deviation observed when comparing the three different approaches....where the trace over the degrees of freedom of the bath and of the

**oscillator**is taken. In Fig. approachschaubild two different approaches to determine the

**qubit**dynamics are depicted. In the first approach, which is elaborated in Ref. [...Corresponding Fourier transform of P t shown in Fig. CompNLLP. The effect of the nonlinearity is to increase the resonance

**frequencies**with respect to the linear case. As a consequence the relative peak heights change. CompNLLF...Schematic representation of the composed system built of a

**qubit**, an intermediate nonlinear

**oscillator**and an Ohmic bath. linearbath ... We consider a

**qubit**coupled to a nonlinear quantum

**oscillator**, the latter coupled to an Ohmic bath, and investigate the

**qubit**dynamics. This composed system can be mapped onto that of a

**qubit**coupled to an effective bath. An approximate mapping procedure to determine the spectral density of the effective bath is given. Specifically, within a linear response approximation the effective spectral density is given by the knowledge of the linear susceptibility of the nonlinear quantum

**oscillator**. To determine the actual form of the susceptibility, we consider its periodically driven counterpart, the problem of the quantum Duffing

**oscillator**within linear response theory in the driving amplitude. Knowing the effective spectral density, the

**qubit**dynamics is investigated. In particular, an analytic formula for the

**qubit**'s population difference is derived. Within the regime of validity of our theory, a very good agreement is found with predictions obtained from a Bloch-Redfield master equation approach applied to the composite

**qubit**-nonlinear

**oscillator**system.

Files:

Contributors: Liberti, G., Zaffino, R. L., Piperno, F., Plastina, F.

Date: 2005-11-21

tau0 The tangle as a function of α in the symmetric case W = 0 for different values of the **qubit** tunnelling amplitude D . One can appreciate that the result of Eq. ( tangl) is indeed reached asymptotically....Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between th...pot The lower adiabatic potential for D = 10 and α = 2 . The dashed line refers to the symmetric, W = 0 , case (dashed line), while the solid line refers to W = 1 . The case of frozen **qubit** ( W = D = 0 ) would have given a pair of independent parabolas instead of the adiabatic potentials U l , u of Eq. ( udq)....As we have shown, the procedure is easily extended to the asymmetric case and this is important since the entanglement changes dramatically for any finite (however small) value of the asymmetry in the **qubit** Hamiltonian. As mentioned in section sect2 above, this is due to the fact the this term modifies the symmetry properties of the Hamiltonian, so that the form of the ground state changes radically and the same occurs to the reduced **qubit** state. For example, for a large enough interaction strength, the **qubit** state is a complete mixture if W = 0 , while it becomes the lower eigenstate of σ z if W 0 . As a result, for large α , there is much entanglement if W = 0 , while the state of the system is factorized and thus τ = 0 if W 0 . This is seen explicitly in Fig. ( tau10). Furthermore, from the comparison of Figs. ( tau10), ( tau01), and ( tau0), one can see that, with increasing α , the tangle increases monotonically in the symmetric case, while it reaches a maximum before going down to zero if W 0 . This is due to the fact that, in the first case, the ground state of the system becomes a Schrödinger cat-like entangled superposition, approximately given by — 12 { — + —- - — - —+ } , for 1 , schroca where | φ ± are the two coherent states for the **oscillator** defined in Eq. ( due1), centered in Q = ± Q 0 , respectively, and almost orthogonal if α ≫ 1 . In the presence of asymmetry, on the other hand, the **oscillator** localizes in one of the wells of its effective potential and this implies that, for large α , the ground state is given by just one of the two components superposed in Eq. ( schroca). This is, clearly, a factorized state and therefore one gets τ = 0 . Since τ is zero for uncoupled sub-systems (i.e., for very small values of α ), weather W = 0 or not, and since, for W 0 , it has to decay to zero for large α , it follows that a maximum is present in between. In fact, for intermediate values of the coupling, there is a competition between the α -dependences of the two non zero components of the Bloch vector. In particular, the length | b → | is approximately equal to one for both small and large α ’s, see Figs. ( asx)-( asz), but the vector points in the x direction for α ≪ 1 and in the z direction for α ≫ 1 . The maximum of the tangle in the asymmetric case occurs near the point in which b x ≈ b z . For the symmetric case, we were also able to derive analytically the sharp increase of the entanglement at α = 1 . This behavior appears to be reminiscent of the super-radiant transition in the many **qubit** Dicke model, which, in the adiabatic limit, shows exactly the same features described here, and which can be described along similar lines. Finally, we would like to comment on the relationship of this work with those of Refs. and . The approach proposed by Levine and Muthukumar, Ref. , employs an instanton description for the effective action. This has been applied to obtain the entropy of entanglement in the symmetric case, in the same critical limit described above. It turns out that this description is equivalent to a fourth order expansion of the lower adiabatic potential U l . This approximation, although retaining all the distinctive qualitative features discussed above, gives slight quantitative changes in the results. Concerning the asymmetric case, our results for the ground state entanglement appear similar to those found by Costi and McKenzie in Ref. , where the interaction of a **qubit** with an ohmic environment was numerically analyzed. It turns out that, for a bath with finite band-width, the entanglement displays a behavior analogous to that reported in Figs. ( tau10)-( tau01), when plotted with respect to the value of the impedance of the bath. Here, instead, we concentrated on the dependence of the tangle on the coupling strength between the **qubit** and the environmental **oscillator**. Unfortunately, the coupling strength is not easily related to the coefficient of the spectral density used in Ref. , and therefore one cannot make a precise comparison between the two results. At least qualitatively, however, we can say that the ground state quantum correlations induced by the coupling with an ohmic environment are already present when the **qubit** is coupled to a single **oscillator** mode. 99 weiss U. Weiss, Quantum Dissipative Systems, 2 nd ed., World Scientific 1999. yuma see, e.g., Yu. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). levine G. Levine and V. N. Muthukumar, Phys. Rev. B 69, 113203 (2004). martinis R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Phys. Rev. Lett. 93 077003 (2005); P. R. Johnson, W. T. Parsons, F. W. Strauch, J. R. Anderson, A. J. Dragt, C. J. Lobb, and F. C. Wellstood, Phys. Rev. Lett. 94, 187004 (2005). pino E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett. 88, 228304 (2002); G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett. 94, 167002 (2005) hines A.P. Hines, C.M. Dawson, R.H. McKenzie and G.J. Milburn, Phys. Rev. A 70, 022303 (2004). blais A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature 431, 162 (2004); A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). prb03 F. Plastina and G. Falci, Phys. Rev. B 67, 224514 (2003). costi T.A. Costi and R.H. McKenzie, Phys. Rev. A 68, 034301 (2003). ent1 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002); T. J. Osborne, and M. A. Nielsen Phys. Rev. A 66, 032110 (2002). ent2 G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003); L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. Lett. 93, 250404 (2004). ent3 T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Phys. Rev. Lett. 94, 147208 (2005). ent4 N. Lambert, C. Emary, and T. Brandes, Phys. Rev. Lett. 92, 073602 (2004). crisp M.D. Crisp, Phys. Rev. A 46, 4138 (1992). Irish E.K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab, Phys. Rev. B 72, 195410 (2005). Rungta V. Coffman, J. Kundu, and W.K. Wootters, Phys. Rev. A 61, 052306 (2000); T. J. Osborne, Phys. Rev. A 72, 022309 (2005), see also quant-ph/0203087. Wallraff A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M.H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. Lett. 95, 060501 (2005). Nakamura Y. Nakamura, Yu.A. Pashkin and J.S. Tsai, Phys. Rev. Lett. 87, 246601 (2001). armour A.D. Armour, M.P. Blencowe and K.C. Schwab, Phys. Rev. Lett. 88, 148301 (2002). Grajcar M. Grajcar, A. Izmalkov and E. Ilxichev, Phys. Rev. B 71, 144501 (2005). Chiorescu I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M. Harmans and J.E. Mooij, Nature 431, 159 (2004)....wf Normalized ground state wave function for the **oscillator** in the lower adiabatic potential, for D = 10 and α = 2 and with W = 0 (dashed line) and W = 0.1 (solid line)....We discuss the ground state entanglement of a bi-partite system, composed by a **qubit** strongly interacting with an **oscillator** mode, as a function of the coupling strenght, the transition **frequency** and the level asymmetry of the **qubit**. This is done in the adiabatic regime in which the time evolution of the **qubit** is much faster than the **oscillator** one. Within the adiabatic approximation, we obtain a complete characterization of the ground state properties of the system and of its entanglement content. ... We discuss the ground state entanglement of a bi-partite system, composed by a **qubit** strongly interacting with an **oscillator** mode, as a function of the coupling strenght, the transition **frequency** and the level asymmetry of the **qubit**. This is done in the adiabatic regime in which the time evolution of the **qubit** is much faster than the **oscillator** one. Within the adiabatic approximation, we obtain a complete characterization of the ground state properties of the system and of its entanglement content.

Files:

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

Date: 2011-06-04

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. ...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....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....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 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)]...(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. ... 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.

Files:

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

Date: 2007-10-10

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)]....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 ....(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 ....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) 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 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)].

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