### 63697 results for qubit oscillator frequency

Contributors: Poletto, S, Chiarello, F, Castellano, M G, Lisenfeld, J, Lukashenko, A, Carelli, P, Ustinov, A V

Date: 2009-10-23

Rabi **oscillation** of the double SQUID manipulated as a phase **qubit** by applying microwave pulses at 19 GHz. The **oscillation** **frequency** changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB....Probability of measuring the state | L as a function of the pulse duration. The coherent **oscillation** shown here has a **frequency** of 14 GHz and a coherence time of approximately 1.2 ns....We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations....Measured Rabi **oscillation** **frequency** versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states....Rabi oscillation of the double SQUID manipulated as a phase qubit by applying microwave pulses at 19 GHz. The oscillation frequency changes from 540 MHz to 1.2 GHz by increasing the power of the microwave signal by 10 dB....We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like oscillation of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor oscillation and microwave-driven Rabi oscillation are rather similar. Due to the higher **frequency** of the Larmor oscillations, the microwave-free **qubit** manipulation allows for much faster coherent operations....Measured oscillation **frequencies** versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit....Probability of measuring the state | L as a function of the pulse duration. The coherent oscillation shown here has a frequency of 14 GHz and a coherence time of approximately 1.2 ns....A tunable rf SQUID manipulated as flux and phase **qubit**...Measured Rabi oscillation frequency versus the normalized amplitude of the microwave signal (solid dots). The dashed line is a linear fit taking into account slightly off-resonance microwave field, while the fit represented by the solid line considers a population of higher excited states....Measurement of the relaxation time T 1 for the double SQUID operated as a phase **qubit**....Measurement of the relaxation time T 1 for the double SQUID operated as a phase qubit....Measured **oscillation** **frequencies** versus amplitude of the short flux pulse (full dots). The solid curve is a numerical simulation using the measured parameters of the circuit. ... We report on two different manipulation procedures of a tunable rf SQUID. First, we operate this system as a flux **qubit**, where the coherent evolution between the two flux states is induced by a rapid change of the energy potential, turning it from a double well into a single well. The measured coherent Larmor-like **oscillation** of the retrapping probability in one of the wells has a **frequency** ranging from 6 to 20 GHz, with a theoretically expected upper limit of 40 GHz. Furthermore, here we also report a manipulation of the same device as a phase **qubit**. In the phase regime, the manipulation of the energy states is realized by applying a resonant microwave drive. In spite of the conceptual difference between these two manipulation procedures, the measured decay times of Larmor **oscillation** and microwave-driven Rabi **oscillation** are rather similar. Due to the higher **frequency** of the Larmor **oscillations**, the microwave-free **qubit** manipulation allows for much faster coherent operations.

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Contributors: Dutta, S. K., Strauch, Frederick W., Lewis, R. M., Mitra, Kaushik, Paik, Hanhee, Palomaki, T. A., Tiesinga, Eite, Anderson, J. R., Dragt, Alex J., Lobb, C. J.

Date: 2008-06-28

F031806Dfosc Rabi oscillation frequency Ω R , 0 1 at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig. F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930 μ A and C 1 = 4.50 p F with ω r f / 2 π = 6.2 G H z ....F031806Dfosc Rabi **oscillation** **frequency** Ω R , 0 1 at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig. F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930 μ A and C 1 = 4.50 p F with ω r f / 2 π = 6.2 G H z ....FDeviceThe dc SQUID phase **qubit**. (a) The **qubit** junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the phase difference γ 1 across the **qubit** junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n differs in energy from m by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c) The photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-**qubit** experiments; the second SQUID was kept unbiased throughout the course of this work....F010206H1 (Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a**) The** escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the qubit; Γ ranges from 0 (white) to 3 × 10 8 1 / s (black). (b**) The** normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828 μ A and C 1 = 4.52 p F , and microwave current I r f = 24.4 n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d)....In** **order** **to** **follow** **experimentally** **the** **shift** **of** **the** **resonance** **condition, it** **was** **necessary** **to** **measure** **Rabi** **oscillations** **for** **different** **detunings** **of** **the** **microwave** **drive. We** **chose** **to** **do** **this** **by** **keeping** **the** **drive** **frequency** **ω** **r** **f fixed** **and** **changing** **the** **level** **spacing** **ω** **0** **1 (through** **I** **b ), because** **the** **power** **transmitted** **by** **the** **microwave** **lines** **had** **a** **nontrivial** **frequency** **dependence. Figure F010206H1(a) shows** **a** **grayscale** **plot** **of** **Rabi** **oscillations** **measured** **from** **such** **an** **experiment, where** **black** **represents** **a** **high** **escape** **rate. Each** **horizontal** **line** **is** **the** **escape** **rate** **versus** **time** **due** **to** **a** **microwave** **current** **of** **6.5** **GHz** **and -11** **dBm, which** **was** **turned** **on** **at** **the** **value** **of** **the** **current** **bias** **I** **b** **indicated** **on** **the** **vertical** **axis. While** **the** **measurements** **were** **performed** **at** **110** **mK, this** **is** **not** **expected** **to** **have** **a** **significant** **impact** **on** **the** **Rabi** **oscillations, as** **the** **temperature** **was** **well** **below** **ℏ** **ω** **0** **1 / k** **B ≈ 325 m** **K ....We present Rabi oscillation measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi oscillations were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed oscillation **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states....F031806DGtot Rabi **oscillations** in the escape rate Γ were induced at I b = 17.746 μ A by switching on a microwave current at t = 0 with a **frequency** of 6.2 GHz (resonant with the 0 → 1 transition) and source powers P S between -12 and -32 dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with I 01 = 17.930 μ A , C 1 = 4.50 p F , T 1 = 17 n s , and T φ = 16 n s ....Figure F010206H1(b) shows that the minimum **oscillation** **frequency** Ω R , 0 1 m i n / 2 π = 540 M H z of the first (experimental) band occurs at I b = 17.624 μ A , for which ω 0 1 / 2 π = 6.4 G H z . This again indicates an ac Stark shift of this transition, which we denote by Δ ω 0 1 ≡ ω r f - ω 0 1 ≈ 2 π × 100 M H z . In addition, the higher levels have suppressed the **oscillation** **frequency** below the bare Rabi **frequency** of Ω 0 1 / 2 π = 620 M H z [calculated with Eq. ( eqf)]....Multilevel effects in the Rabi oscillations of a Josephson phase **qubit**...F032206MNstats (Color online**) The** (a) on-resonance Rabi oscillation frequencies Ω R , 0 1 m i n and Ω R , 0 2 m i n and (b) resonance frequency shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110 m K with a microwave drive of frequency ω r f / 2 π = 6.5 G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10 d B m . Values extracted from data for the 0 → 1 ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736 μ A and C 1 = 4.49 p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system....We fit the escape rates in Fig. F031806DGtot (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted **frequencies** are shown with circles in Fig. F031806Dfosc. To compare to theory, Ω R , 0 1 , calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the **oscillation** **frequencies** of Γ and ρ 11 are equal, even at high power in a multilevel system, will be addressed in Sec. SSummary. In plotting the data, we have introduced a single fitting parameter 117 n A / m W that converts the power P S at the microwave source to the current amplitude I r f at the **qubit**. Good agreement is found over the full range of power....FEnergyGamma Qubit energy-level spectroscopy and tunneling escape rates. (a) Open circles show the resonance frequency of the transition between the ground and first excited states of the qubit, measured at 20 mK. The scatter in the values is indicative of the uncertainty in the measurement. Also plotted are theoretical values of ω 0 1 (solid), ω 1 2 (dashed), and ω 2 3 (dotted) for I 01 = 17.930 μ A and C 1 = 4.50 p F . (b) At high bias, the measured background escape rate (open circles) agrees with the predicted ground-state escape rate Γ 0 (solid line) for the junction parameters given above. Calculated Γ 1 , Γ 2 , and Γ 3 are plotted as dashed, dotted, and dashed-dotted lines. In both plots, the bottom axes show the total bias current I b , while the top axes indicate the normalized barrier height N s , calculated using the extracted junction parameters....We next consider the time dependence of the escape rate for the data plotted in Fig. F032106FN10. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the 0 → 1 transition of the **qubit** junction. The measured escape rate shows Rabi **oscillations** followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase **qubits** and we have found them in several of our devices....We** **fit** **the** **escape** **rates** **in** **Fig. F031806DGtot (and** **additional** **data** **for** **other** **powers** **not** **shown) to** **a** **decaying** **sinusoid** **with** **an** **offset. The** **extracted** **frequencies** **are** **shown** **with** **circles** **in** **Fig. F031806Dfosc. To** **compare** **to** **theory, Ω** **R , 0** **1 , calculated** **using** **the** **rotating** **wave** **solution** **for** **a** **system** **with** **five** **levels, is** **shown** **with** **a** **solid** **line. The** **implied** **assumption** **that** **the** **oscillation** **frequencies** **of** **Γ** **and** **ρ** **11** **are** **equal, even** **at** **high** **power** **in** **a** **multilevel** **system, will** **be** **addressed** **in** **Sec. SSummary. In** **plotting** **the** **data, we** **have** **introduced** **a** **single** **fitting** **parameter** **117 n** **A / m** **W** **that** **converts** **the** **power** **P** **S at** **the** **microwave** **source** **to** **the** **current** **amplitude** **I** **r** **f at** **the** **qubit. Good** **agreement** **is** **found** **over** **the** **full** **range** **of** **power....As I r f increases in Fig. F031806Dfosc, the **oscillation** **frequency** is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase **qubit**. There are two distinct phenomena that affect 0 → 1 Rabi **oscillations** in such a device. To describe...FDeviceThe dc SQUID **phase** qubit. (a**) The** qubit junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the **phase** difference γ 1 across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n differs in energy from m by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c**) The** photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work....F032206MNstats (Color online) The (a) on-resonance Rabi **oscillation** **frequencies** Ω R , 0 1 m i n and Ω R , 0 2 m i n and (b) resonance **frequency** shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110 m K with a microwave drive of **frequency** ω r f / 2 π = 6.5 G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10 d B m . Values extracted from data for the 0 → 1 ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736 μ A and C 1 = 4.49 p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system....We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states....The** **resonant** **oscillation** **frequencies** **Ω** **R , 0** **1** **m** **i** **n and** **Ω** **R , 0** **2** **m** **i** **n in** **Fig. F032206MNstats(a) are** **well** **described** **by** **Eq. ( eqd) over** **the** **full** **range** **of** **I** **r** **f . The** **deviation** **between** **the** **0 → 1 oscillation** **frequency** **and** **the** **values** **expected** **in** **a** **two-level** **system (dotted** **line) increases** **with** **I** **r** **f . Similar** **measurements** **taken** **at** **a** **higher** **I** **b** **show** **a** **smaller** **frequency** **suppression** **over** **a** **similar** **range** **of** **I** **r** **f , as** **expected** **for** **a** **system** **with** **stronger** **anharmonicity....F010206H1 (Color online) Multiphoton, multilevel Rabi **oscillations** plotted in the time and **frequency** domains. (a) The escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the **qubit**; Γ ranges from 0 (white) to 3 × 10 8 1 / s (black). (b) The normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi **frequencies** obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828 μ A and C 1 = 4.52 p F , and microwave current I r f = 24.4 n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d)....Figure F010206H1(b) shows** **that** **the** **minimum** **oscillation** **frequency** **Ω** **R , 0** **1** **m** **i** **n / 2** **π = 540 M** **H** **z** **of** **the** **first (experimental) band** **occurs** **at** **I** **b = 17.624 μ** **A , for** **which** **ω** **0** **1 / 2** **π = 6.4 G** **H** **z . This** **again** **indicates** **an** **ac** **Stark** **shift** **of** **this** **transition, which** **we** **denote** **by** **Δ** **ω** **0** **1 ≡ ω** **r** **f - ω** **0** **1 ≈ 2** **π × 100 M** **H** **z . In** **addition, the** **higher** **levels** **have** **suppressed** **the** **oscillation** **frequency** **below** **the** **bare** **Rabi** **frequency** **of** **Ω** **0** **1 / 2** **π = 620 M** **H** **z [calculated** **with** **Eq. ( eqf)]....For this data set, the level spacing ω 0 1 / 2 π is equal to the microwave **frequency** ω r f / 2 π = 6.5 G H z at I b = 17.614 μ A . The band with the highest current in Fig. F010206H1(b) is centered about I b = 17.624 μ A , suggesting that 0 → 1 Rabi **oscillations** are the dominant process near this bias. For slightly higher or lower I b , the **oscillation** **frequency** increases as Ω R , 0 1 ≈ Ω 01 ′ 2 + ω r f - ω 0 1 2 , in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot....Figure FDevice(a) shows** **the** **circuit** **schematic** **for** **our** **dc** **SQUID** **phase** **qubit. The** **qubit** **junction** **J** **1 (with** **critical** **current** **I** **01** **and** **capacitance** **C** **1 ) is** **shown** **on** **the** **left. It** **is** **isolated** **from** **the** **current** **bias** **source** **I** **b** **by** **geometrical** **inductances** **L** **1** **and** **L** **2** **and** **the** **second** **junction** **J** **2 (with** **I** **02** **and** **C** **2 ). In** **order** **to** **independently** **control** **the** **currents** **in** **the** **two** **arms** **of** **the** **resulting** **dc** **SQUID, a** **current** **source** **I** **f** **applies** **a** **flux** **Φ** **a** **to** **the** **SQUID** **loop** **through** **mutual** **inductance** **M . Good** **isolation** **of** **the** **qubit** **junction** **is** **obtained** **when** **L** **1 / M ≫ 1** **and** **L** **1 / L** **2 + L** **J** **2 ≫ 1 , where** **L** **J** **2** **is** **the** **Josephson** **inductance** **of** **the** **isolation** **junction. ... We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states.

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

Date: 2007-02-28

In this section we present a different measurement protocol. It is based on the short time dynamics illustrated as follows: for the **qubit** initially in the state 1 / 2 | ↑ + | ↓ the probability distribution of momentum is plotted in Fig. probability (a) and (b)....Probability density of momentum P p 0 t (a), snapshots of it at different times (b) and expectation value of momentum for the two different **qubit** states (c). Here ℏ Ω / k B T = 2 , Δ / Ω = 0.45 , κ / Ω = 0.025 and ℏ ν / k B T = 1.9 and p ¯ 0 is the dimensionless momentum p 0 / k B T m ....We consider a simplified version of the experiment described in Ref. . The circuit consists of a flux **qubit** drawn in the single junction version, the surrounding SQUID loop, an ac source, and a shunt resistor, as depicted in Fig. circuit. We note here that we later approximate the **qubit** as a two-level system. The **qubit** used in the actual experiment contains three junctions. An analogous but less transparent derivation would, after performing the two-state approximation, lead to the same model, parameterized by the two-state Hamiltonian, the circulating current, and the mutual inductance, in an identical way ....Discrimination time as function of the coupling strength between qubit and oscillator. Here ℏ Ω / k B T = 2 , κ / Ω = 0.025 and ℏ ν / k B T = 1.95 ....Simplified circuit consisting** of a** qubit with one Josephson junction (phase γ , capacitance C q and inductance L q ) inductively coupled to a SQUID with two identical junctions (phases γ 1 , 2 , capacitance C S ) and inductance L S . The SQUID is driven by an ac bias I B t and the voltage drop is measured by a voltmeter with internal resistance R . The total flux through the qubit loop is Φ q and through the SQUID is Φ S ....Simplified circuit consisting of a **qubit** with one Josephson junction (phase γ , capacitance C q and inductance L q ) inductively coupled to a SQUID with two identical junctions (phases γ 1 , 2 , capacitance C S ) and inductance L S . The SQUID is driven by an ac bias I B t and the voltage drop is measured by a voltmeter with internal resistance R . The total flux through the **qubit** loop is Φ q and through the SQUID is Φ S ....As one can see for the parameters of Fig. comp, in the WQOC regime the measurement time is longer than the dephasig time. Their difference decreases as we increase Δ due to the onset of the strong coupling plateau in the dephasing rate, approaching the quantum limit where the measurement time becomes comparable to the dephasing time. Note that, for superstrong coupling** either** between qubit and oscillator or between oscillator and bath, corrections of the order κ / Ω ↓ 2 of the dephasing rate gain importance. These corrections are not treated in our Born approximation. Therefore the regions where the dephasing rate becomes lower than the measurement rate, in violation **with the** quantum limitation of Ref. , should be regarded as a limitation of our approximation....Discrimination time as function of the coupling strength between **qubit** and **oscillator**. Here ℏ Ω / k B T = 2 , κ / Ω = 0.025 and ℏ ν / k B T = 1.95 ....As one can see for the parameters of Fig. comp, in the WQOC regime the measurement time is longer than the dephasig time. Their difference decreases as we increase Δ due to the onset of the strong coupling plateau in the dephasing rate, approaching the quantum limit where the measurement time becomes comparable to the dephasing time. Note that, for superstrong coupling either between **qubit** and **oscillator** or between **oscillator** and bath, corrections of the order κ / Ω ↓ 2 of the dephasing rate gain importance. These corrections are not treated in our Born approximation. Therefore the regions where the dephasing rate becomes lower than the measurement rate, in violation with the quantum limitation of Ref. , should be regarded as a limitation of our approximation....Dephasing rate dependence on driving: dependence on Δ for different driving strengths F 0 ( κ / Ω = 10 -4 and ν = 2 Ω ). Top inset: dependence of the decoherence rate on F 0 for different values of κ ( Δ / Ω = 5 ⋅ 10 -2 and ν = 2 Ω ). Bottom inset: dependence of the decoherence rate on driving **frequency** ν for different vales of κ ( Δ / Ω = 0.5 ). Here ℏ Ω / k B T = 2 and F ¯ 0 is the dimensionless force F 0 ℏ / k B T m k B T ....Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost....The dependence on the driving **frequency** has also been analyzed in Fig. antrieb. Here we observe two peaks at Ω ↑ and Ω ↓ . At ν = Ω the classical driven and undamped trajectory ξ t diverges. In terms of the calculation this means that the Floquet modes are not well-defined when the driving **frequency** is at resonance with the harmonic **oscillator** — we have a continuum instead. Physically this means that at t = 0 our **oscillator** has the **frequency** Ω because it has not yet "seen" the **qubit**, and we are driving it at resonance, and by amplifying the **oscillations** of x ̂ which is subject to noise we amplify the noise seen by the **qubit**. The dephasing rate is also expected to diverge. The peaks at Ω ↑ and Ω ↓ show the same effect after the **qubit** and the **oscillator** become entangled. The dephasing rate drops again for large driving **frequencies** to the value obtained in the case without driving....In Fig. probability one can see that the two peaks corresponding to the two states of the qubit split already during the transient motion of p ̂ t , much faster than the transient decay time. If the peaks are well enough separated, a single measurement of momentum gives the needed information about the qubit state, and** has the **advantage of avoiding decoeherence effects resulting from a long time coupling to the environment. Nevertheless the parameters we need to reduce the discrimination time also enhance the decoherence rate....The dependence on the** driving **frequency has also been analyzed in Fig. antrieb. Here we observe two peaks **at **Ω ↑ and Ω ↓ . At ν = Ω the classical driven and undamped trajectory ξ t diverges. In terms of the calculation this means that the Floquet modes are not well-defined when the** driving **frequency is **at **resonance **with the** harmonic oscillator — we have a continuum instead. Physically this means that **at **t = 0 our oscillator** has the **frequency Ω because it has not yet "seen" the qubit, and we are** driving **it **at **resonance, and by amplifying the oscillations of x ̂ which is subject to noise we amplify the noise seen by the qubit. The dephasing rate is also expected to diverge. The peaks **at **Ω ↑ and Ω ↓ show the same effect after the qubit and the oscillator become entangled. The dephasing rate drops again for large** driving **frequencies to the value obtained in the case without driving....Probability density of momentum P p 0 t (a), snapshots of it at different times (b) and expectation value of momentum for the two different qubit states (c). Here ℏ Ω / k B T = 2 , Δ / Ω = 0.45 , κ / Ω = 0.025 and ℏ ν / k B T = 1.9 and p ¯ 0 is the dimensionless momentum p 0 / k B T m ....As already mentioned, for the short time, single shot measurement strong, close to resonance** driving **is needed for the rapid separation of the peaks. While the discrimination time is not very sensitive to the change of κ , we observe in Fig. shorttime that one needs relatively strong coupling ( Δ / Ω ∈ 0.03 0.1 ) for the discrimination time to become shorter than the decoherence time. The picture of the dephasing rate is also qualitatively different from the case without** driving **or with far off-resonant driving, since for Δ / Ω ≈ 0.4 , Ω ↑ becomes resonant **with the** driving frequency. In this region our numerical calculation also becomes unstable. Nevertheless, as one can see in Fig. shorttime, the dephasing rate is proportional to κ . Thus, by reducing the damping of the oscillator, one can extend the domain of values of Δ where the measurement can be performed....In Fig. probability one can see that the two peaks corresponding to the two states of the **qubit** split already during the transient motion of p ̂ t , much faster than the transient decay time. If the peaks are well enough separated, a single measurement of momentum gives the needed information about the **qubit** state, and has the advantage of avoiding decoeherence effects resulting from a long time coupling to the environment. Nevertheless the parameters we need to reduce the discrimination time also enhance the decoherence rate....We consider a simplified version of the experiment described in Ref. . The circuit consists **of a **flux qubit drawn in the single junction version, the surrounding SQUID loop, an ac source, and a shunt resistor, as depicted in Fig. circuit. We note here that we later approximate the qubit as a two-level system. The qubit used in the actual experiment contains three junctions. An analogous but less transparent derivation would, after performing the two-state approximation, lead to the same model, parameterized by the two-state Hamiltonian, the circulating current, and the mutual inductance, in an identical way ....Phase-space theory for dispersive detectors of superconducting **qubits** ... Motivated by recent experiments, we study the dynamics of a **qubit** quadratically coupled to its detector, a damped harmonic **oscillator**. We use a complex-environment approach, explicitly describing the dynamics of the **qubit** and the **oscillator** by means of their full Floquet state master equations in phase-space. We investigate the backaction of the environment on the measured **qubit** and explore several measurement protocols, which include a long-term full read-out cycle as well as schemes based on short time transfer of information between **qubit** and **oscillator**. We also show that the pointer becomes measurable before all information in the **qubit** has been lost.

Files:

Contributors: Il'ichev, E., Oukhanski, N., Izmalkov, A., Wagner, Th., Grajcar, M., Meyer, H. -G., Smirnov, A. Yu., Brink, Alec Maassen van den, Amin, M. H. S., Zagoskin, A. M.

Date: 2003-03-20

We plotted S V , t ω for different HF powers P in Fig. fig3. As P is increased, ω R grows and passes ω T , leading to a non-monotonic dependence of the maximum signal on P in agreement with the above picture. This, and the sharp dependence on the tuning of ω H F to the **qubit** **frequency**, confirm that the effect is due to Rabi **oscillations**. The inset shows that the shape is given by the second line of Eq. ( S) for all curves....Measurement setup. The flux **qubit** is inductively coupled to a tank circuit. The DC source applies a constant flux Φ e ≈ 1 2 Φ 0 . The HF generator drives the **qubit** through a separate coil at a **frequency** close to the level separation Δ / h = 868 MHz. The output voltage at the resonant **frequency** of the tank is measured as a function of HF power....We** **plotted** **S** **V , t** **ω** **for** **different** **HF** **powers** **P** **in** **Fig. fig3. As** **P** **is** **increased, ω** **R** **grows** **and** **passes ω** **T , leading** **to** **a** **non-monotonic** **dependence** **of** **the** **maximum** **signal** **on P** **in** **agreement** **with** **the** **above** **picture. This, and** **the** **sharp** **dependence** **on** **the** **tuning** **of** **ω** **H** **F** **to** **the** **qubit** **frequency, confirm** **that** **the** **effect** **is** **due** **to** **Rabi** **oscillations. The** **inset** **shows** **that** **the** **shape** **is** **given** **by** **the** **second** **line** **of** **Eq. ( S) for** **all** **curves....The Al **qubit** inside the Nb pancake coil....(a) Comparing the data to the theoretical Lorentzian. The fitting parameter is g ≈ 0.02 . Letters in the picture correspond to those in Fig. fig3. (b) The Rabi **frequency** extracted from (a) vs the applied HF amplitude. The straight line is the predicted dependence ω R / ω T = P / P 0 . The good agreement provides strong evidence for Rabi **oscillations**....The** **spectral** **amplitude** **of** **the** **tank** **voltage** **for** **HF** **powers** **P** **a irradiated** ****qubit**** **modifying** **the** **tank’s** **inductance** **and** **hence** **its** **central** **frequency, and** **in** **principle** **similarly** **for** **dissipation** **in** **the** ****qubit**** **increasing** **the** **tank’s** **linewidth ; these** **are** **inconsequential** **for** **our** **analysis....The** **Al** ****qubit**** **inside** **the** **Nb** **pancake** **coil....(a) Comparing** **the** **data** **to** **the** **theoretical** **Lorentzian. The** **fitting** **parameter** **is** **g ≈ 0.02 . Letters** **in** **the** **picture** **correspond** **to** **those** **in** **Fig. fig3. (b) The** **Rabi** **frequency** **extracted** **from (a) vs** **the** **applied** **HF** **amplitude. The** **straight** **line** **is** **the** **predicted** **dependence** **ω** **R / ω** **T = P / P** **0 . The** **good** **agreement** **provides** **strong** **evidence** **for** **Rabi** **oscillations....The** **spectral** **amplitude** **of** **the** **tank** **voltage** **for** **HF** **powers** **P** **a** **e** **irradiated** **qubit** **modifying** **the** **tank’s** **inductance** **and** **hence** **its** **central** **frequency, and** **in** **principle** **similarly** **for** **dissipation** **in** **the** **qubit** **increasing** **the** **tank’s** **linewidth ; these** **are** **inconsequential** **for** **our** **analysis....We use a small-inductance superconducting loop interrupted by three Josephson junctions (a 3JJ **qubit**) , inductively coupled to a high-quality superconducting tank circuit (Fig. fig1). This approach is similar to the one in entanglement experiments with Rydberg atoms and microwave photons in a cavity . The tank serves as a sensitive detector of Rabi transitions in the **qubit**, and simultaneously as a filter protecting it from noise in the external circuit. Since ω T ≪ Ω / ℏ , the **qubit** is effectively decoupled from the tank unless it **oscillates** with **frequency** ω T . That is, while wide-band (i.e., fast on the **qubit** time scale) detectors up to now have received most theoretical attention (e.g., ), we use narrow-band detection to have sufficient sensitivity at a single **frequency** even with a small coupling coefficient; cf. above Eq. ( S). The tank voltage is amplified and sent to a spectrum analyzer. This is a development of the Silver–Zimmerman setup in the first RF-SQUID magnetometers , and is effective for probing flux **qubits** . As such, it was used to determine the potential profile of a 3JJ **qubit** in the classical regime ....We** **use** **a** **small-inductance** **superconducting** **loop** **interrupted** **by** **three** **Josephson** **junctions (a** **3JJ** **qubit) , inductively** **coupled** **to** **a** **high-quality** **superconducting** **tank** **circuit (Fig. fig1). This** **approach** **is** **similar** **to** **the** **one** **in** **entanglement** **experiments** **with** **Rydberg** **atoms** **and** **microwave** **photons** **in** **a** **cavity . The** **tank** **serves** **as** **a** **sensitive** **detector** **of** **Rabi** **transitions** **in** **the** **qubit, and** **simultaneously** **as** **a** **filter** **protecting** **it** **from** **noise** **in** **the** **external** **circuit. Since** **ω** **T ≪ Ω / ℏ , the** **qubit** **is** **effectively** **decoupled** **from** **the** **tank** **unless** **it** **oscillates** **with** **frequency ω** **T . That** **is, while** **wide-band (**i**.e., fast** **on** **the** **qubit** **time** **scale) detectors** **up** **to** **now** **have** **received** **most** **theoretical** **attention (e.g., ), we** **use** **narrow-band** **detection** **to** **have** **sufficient** **sensitivity** **at** **a** **single** **frequency** **even** **with** **a** **small** **coupling** **coefficient; cf. above** **Eq. ( S). The** **tank** **voltage** **is** **amplified** **and** **sent** **to** **a** **spectrum** **analyzer. This** **is** **a** **development** **of** **the** **Silver–Zimmerman** **setup** **in** **the** **first** **RF-SQUID** **magnetometers , and** **is** **effective** **for** **probing** **flux** **qubits . As** **such, it** **was** **used** **to** **determine** **the** **potential** **profile** **of** **a** **3JJ** **qubit** **in** **the** **classical** **regime ....Measurement** **setup. The** **flux** ****qubit**** **is** **inductively** **coupled** **to** **a** **tank** **circuit. The** **DC** **source** **applies** **a** **constant** **flux** **Φ** **e ≈ 1** **2** **Φ** **0 . The** **HF** **generator** **drives** **the** ****qubit**** **through** **a** **separate** **coil** **at** **a** **frequency** **close** **to** **the** **level** **separation** **Δ / h = 868** **MHz. The** **output** **voltage** **at** **the** **resonant** **frequency** **of** **the** **tank** **is** **measured** **as** **a** **function** **of** **HF** **power....Under resonant irradiation, a quantum system can undergo coherent (Rabi) **oscillations** in time. We report evidence for such **oscillations** in a _continuously_ observed three-Josephson-junction flux **qubit**, coupled to a high-quality tank circuit tuned to the Rabi **frequency**. In addition to simplicity, this method of_Rabi spectroscopy_ enabled a long coherence time of about 2.5 microseconds, corresponding to an effective **qubit** quality factor \~7000....Under resonant irradiation, a quantum system can undergo coherent (Rabi) oscillations in time. We report evidence for such oscillations in a _continuously_ observed three-Josephson-junction flux **qubit**, coupled to a high-quality tank circuit tuned to the Rabi **frequency**. In addition to simplicity, this method of_Rabi spectroscopy_ enabled a long coherence time of about 2.5 microseconds, corresponding to an effective **qubit** quality factor \~7000....Continuous Monitoring of Rabi Oscillations in a Josephson Flux **Qubit**...The spectral amplitude of the tank voltage for HF powers P a **qubit** modifying the tank’s inductance and hence its central **frequency**, and in principle similarly for dissipation in the **qubit** increasing the tank’s linewidth ; these are inconsequential for our analysis....Without** **an** **HF** **signal, the** **qubit’s** **influence** **at** **ω** **T** **is** **negligible. Thus, the “dark” trace** **in** **Fig. fig3** **is** **a** **quantitative** **measure** **of S** **b ....Without an HF signal, the ** qubit’s** influence at ω T is negligible. Thus, the “dark” trace in Fig. fig3 is a quantitative measure of S b . ... Under resonant irradiation, a quantum system can undergo coherent (Rabi)

**oscillations**in time. We report evidence for such

**oscillations**in a _continuously_ observed three-Josephson-junction flux

**qubit**, coupled to a high-quality tank circuit tuned to the Rabi

**frequency**. In addition to simplicity, this method of_Rabi spectroscopy_ enabled a long coherence time of about 2.5 microseconds, corresponding to an effective

**qubit**quality factor \~7000.

Files:

Contributors: Chiarello, F., Paladino, E., Castellano, M. G., Cosmelli, C., D'Arrigo, A., Torrioli, G., Falci, G.

Date: 2011-10-07

Decay envelopes of the observed **oscillations** (red points), and relative fitting curves (blue continuous lines)....Superconducting **qubit** manipulated by fast pulses: experimental observation of distinct decoherence regimes...The **frequency** Ω / 2 π given by eq. omega for ϕ x = 0 as a function of ϕ c = 2 π Φ c / Φ 0 (blue curve). Dashed orizontal lines mark the range of **oscillation** **frequencies** observed (about 10-20 GHz), corresponding to the range of used top values of the applied pulse (defined by the vertical dashed lines). In the inset it is sketched the flux pulse used for the **qubit** manipulation, changing the potential from the two-well “W” case to the single-well “V” case (in red)....A particular superconducting quantum interference device (SQUID)**qubit**, indicated as double SQUID **qubit**, can be manipulated by rapidly modifying its potential with the application of fast flux pulses. In this system we observe coherent **oscillations** exhibiting non-exponential decay, indicating a non trivial decoherence mechanism. Moreover, by tuning the **qubit** in different conditions (different **oscillation** **frequencies**) by changing the pulse height, we observe a crossover between two distinct decoherence regimes and the existence of an "optimal" point where the **qubit** is only weakly sensitive to intrinsic noise. We find that this behaviour is in agreement with a model considering the decoherence caused essentially by low **frequency** noise contributions, and discuss the experimental results and possible issues....The frequency Ω / 2 π given by eq. omega** for **ϕ x = 0 as a function of ϕ c = 2 π Φ c / Φ 0 (blue curve). Dashed orizontal lines mark the range of oscillation frequencies observed (about 10-20 GHz), corresponding to the range of used top values of the applied pulse (defined by the vertical dashed lines). In the inset it is sketched the flux pulse used** for **the qubit manipulation, changing the potential from the two-well “W” case to the single-well “V” case (in red)....A particular superconducting quantum interference device (SQUID)**qubit**, indicated as double SQUID **qubit**, can be manipulated by rapidly modifying its potential with the application of fast flux pulses. In this system we observe coherent oscillations exhibiting non-exponential decay, indicating a non trivial decoherence mechanism. Moreover, by tuning the **qubit** in different conditions (different oscillation **frequencies**) by changing the pulse height, we observe a crossover between two distinct decoherence regimes and the existence of an "optimal" point where the **qubit** is only weakly sensitive to intrinsic noise. We find that this behaviour is in agreement with a model considering the decoherence caused essentially by low **frequency** noise contributions, and discuss the experimental results and possible issues....Some experimental **oscillations** observed for different pulse height. The measured **frequency** is indicated in the top right part of each plot....The ...Red points are the decay rate γ I (left panel) and γ I I (right panel) obtained by fitting of the experimental decay curves in **oscillations** with eq. envelope. The blue line in the left panel is the fit of the data points with eq. gammaI (Section 3). In the right panel the blue line is the average value of the scattered values of γ I I...Some experimental oscillations observed** for **different pulse height. The measured frequency is indicated in the top right part of each plot. ... A particular superconducting quantum interference device (SQUID)**qubit**, indicated as double SQUID **qubit**, can be manipulated by rapidly modifying its potential with the application of fast flux pulses. In this system we observe coherent **oscillations** exhibiting non-exponential decay, indicating a non trivial decoherence mechanism. Moreover, by tuning the **qubit** in different conditions (different **oscillation** **frequencies**) by changing the pulse height, we observe a crossover between two distinct decoherence regimes and the existence of an "optimal" point where the **qubit** is only weakly sensitive to intrinsic noise. We find that this behaviour is in agreement with a model considering the decoherence caused essentially by low **frequency** noise contributions, and discuss the experimental results and possible issues.

Files:

Contributors: Eugene Grichuk, Margarita Kuzmina, Eduard Manykin

Date: 2010-09-26

A network of coupled stochastic **oscillators** is
proposed for modeling of a cluster of entangled **qubits** that is
exploited as a computation resource in one-way quantum
computation schemes. A **qubit** model has been designed as a
stochastic **oscillator** formed by a pair of coupled limit cycle
**oscillators** with chaotically modulated limit cycle radii and
**frequencies**. The **qubit** simulates the behavior of electric field of
polarized light beam and adequately imitates the states of two-level
quantum system. A cluster of entangled **qubits** can be associated
with a beam of polarized light, light polarization degree being
directly related to cluster entanglement degree. Oscillatory network,
imitating **qubit** cluster, is designed, and system of equations for
network dynamics has been written. The constructions of one-**qubit**
gates are suggested. Changing of cluster entanglement degree caused
by measurements can be exactly calculated....network of stochastic oscillators...network of stochastic **oscillators**...Network of Coupled Stochastic **Oscillators** and One-way Quantum Computations ... A network of coupled stochastic **oscillators** is
proposed for modeling of a cluster of entangled **qubits** that is
exploited as a computation resource in one-way quantum
computation schemes. A **qubit** model has been designed as a
stochastic **oscillator** formed by a pair of coupled limit cycle
**oscillators** with chaotically modulated limit cycle radii and
**frequencies**. The **qubit** simulates the behavior of electric field of
polarized light beam and adequately imitates the states of two-level
quantum system. A cluster of entangled **qubits** can be associated
with a beam of polarized light, light polarization degree being
directly related to cluster entanglement degree. Oscillatory network,
imitating **qubit** cluster, is designed, and system of equations for
network dynamics has been written. The constructions of one-**qubit**
gates are suggested. Changing of cluster entanglement degree caused
by measurements can be exactly calculated.

Files:

Contributors: Omelyanchouk, A. N., Shevchenko, S. N., Zagoskin, A. M., Il'ichev, E., Nori, Franco

Date: 2007-05-12

The average energy H of the system as a function of the driving **frequency** ω . The main peak ( ω 0 ≈ 0.6 ) corresponds to the resonance. The left peak at ω 0 / 2 is the nonlinear effect of the excitation by a subharmonic, similar to a multiphoton process in the quantum case. The right peak at 2 ω 0 is the first overtone and it has no quantum counterpart. Here ϕ e d = π ; ϕ e a = 0.05 ; γ = 10 -3 ....Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** oscillations in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi oscillations, the **frequency** of these pseudo-Rabi oscillations is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$....(a) Driven oscillations around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving **frequency** ω = 0.612 , and the decay rate γ = 10 -3 . Low-**frequency** classical beat oscillations are clearly seen. (b) Low-**frequency** oscillations of the persistent current in the 3JJ loop. (c) Same for the energy of the system....In the presence of the external field ( eq_external) the system will undergo forced **oscillations** around one of the equlibria. For α = 0.8 , which is close to the parameters of the actual devices , the values of the dimensionless **frequencies** become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations of motion ( eq_motion) numerically, we see the appearance of slow **oscillations** of the amplitude and energy superimposed on the fast forced **oscillations** (Fig. fig2), similar to the classical **oscillations** in a phase **qubit** (Fig. 2 in ). The dependence of the **frequency** of these **oscillations** on the driving amplitude shows an almost linear behaviour (Fig. fig3), which justifies the “Pseudo-Rabi” moniker....The key observable difference between the classical and quantum cases, which would allow to reliably distinguish between them, is that the classical effect can also be produced by driving the system at the overtones of the resonance signal, ∼ n ω 0 (Fig. fig4). This effect can be detected using a standard technique for RF SQUIDs . The current circulating in the **qubit** circuit produces a magnetic moment, which is measured by the inductively coupled high-quality tank circuit. For the tank voltage V T we have...In the presence **of** the external field ( eq_external) the system will undergo forced oscillations around one **of** the equlibria. For α = 0.8 , which is close to the parameters **of** the actual devices , the values **of** the dimensionless **frequencies** become ω θ ≈ 0.612 , and ω χ ≈ 0.791 . Solving the equations **of** motion ( eq_motion) numerically, we see the appearance **of** slow oscillations **of** the amplitude and energy superimposed on the fast forced oscillations (Fig. fig2), similar to the classical oscillations in a **phase** **qubit** (Fig. 2 in ). The dependence **of** the **frequency** **of** these oscillations on the driving amplitude shows an almost linear behaviour (Fig. fig3), which justifies the “Pseudo-Rabi” moniker....where τ T = R T C T is the RC-constant of the tank, ω T = L T C T -1 / 2 its resonant **frequency**, M the mutual inductance between the tank and the **qubit**, and I q t the current circulating in the **qubit**. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence of an external RF field is shown in Fig. fig2c. Note that the sign of the current does not change, which is due to the fact that the **oscillations** take place inside one potential well (solid arrow in Fig. fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)...where τ T = R T C T is the RC-constant **of** the tank, ω T = L T C T -1 / 2 its resonant **frequency**, M the mutual inductance between the tank and the **qubit**, and I q t the current circulating in the **qubit**. The persistent current in the 3JJ loop can be determined directly from ( eq_I). Its behaviour in the presence **of** an external RF field is shown in Fig. fig2c. Note that the sign **of** the current does not change, which is due to the fact that the oscillations take place inside one potential well (solid arrow in Fig. fig1), and not between two separate nearby potential minima like in the quantum case. (Alternatively, this would also allow to distinguish between the classical and quantum effects by measuring the magnetization with a DC SQUID.)...The dependence of the pseudo-Rabi **frequency** on the driving amplitude ϕ e a for ω = 0.6 , γ = 10 -3 . The solid line, Ω = 0.35 ϕ e a 2 + ω - 0.63 2 1 / 2 , is the best fit to the calculated data....Pseudo-Rabi oscillations in superconducting flux **qubits** in the classical regime...(Color online) The potential profile of Eq. ( eq_potential) with α = 0.8 , ϕ e d = π . The arrows indicate quantum (solid) and classical (dotted) **oscillations**....A quantitative difference between this effect and true Rabi **oscillations** is in the different scale of the resonance **frequency**. To induce Rabi **oscillations** between the lowest quantum levels in the potential ( eq_potential), one must apply a signal in resonance with their tunneling splitting, which is exponentially smaller than ω 0 . Still, this is not a very reliable signature of the effect, since the classical effect can also be excited by subharmonics, ∼ ω 0 / n , as we can see in Fig. fig4....Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$....(a) Driven oscillations around a minimum **of** the potential profile **of** Fig. fig1 as a function **of** time. The driving amplitude is ϕ **e **a = 0.01 , driving **frequency** ω = 0.612 , and the decay rate γ = 10 -3 . Low-**frequency** classical beat oscillations are clearly seen. (b) Low-**frequency** oscillations **of** the persistent current in the 3JJ loop. (c) Same for the energy **of** the system....(a) Driven **oscillations** around a minimum of the potential profile of Fig. fig1 as a function of time. The driving amplitude is ϕ e a = 0.01 , driving **frequency** ω = 0.612 , and the decay rate γ = 10 -3 . Low-**frequency** classical beat **oscillations** are clearly seen. (b) Low-**frequency** **oscillations** of the persistent current in the 3JJ loop. (c) Same for the energy of the system. ... Nonlinear effects in mesoscopic devices can have both quantum and classical origins. We show that a three-Josephson-junction (3JJ) flux **qubit** in the _classical_ regime can produce low-**frequency** **oscillations** in the presence of an external field in resonance with the (high-**frequency**) harmonic mode of the system, $\omega$. Like in the case of_quantum_ Rabi **oscillations**, the **frequency** of these pseudo-Rabi **oscillations** is much smaller than $\omega$ and scales approximately linearly with the amplitude of the external field. This classical effect can be reliably distinguished from its quantum counterpart because it can be produced by the external perturbation not only at the resonance **frequency** $\omega$ and its subharmonics ($\omega/n$), but also at its overtones, $n\omega$.

Files:

Contributors: Whittaker, J. D., da Silva, F. C. S., Allman, M. S., Lecocq, F., Cicak, K., Sirois, A. J., Teufel, J. D., Aumentado, J., Simmonds, R. W.

Date: 2014-08-08

(Color online) Coupling rate 2 g / 2 π (design A ) as a function of **cavity** **frequency** ω c / 2 π . The solid red (blue) line is the prediction from Eq. ( eq:g) (including L x and C J ’s). The (dotted) dashed line is the prediction for capacitive coupling** with** C = 15 fF ( C = 5 fF). The solid circles were measured spectroscopically (see text). At lowest **cavity** **frequency**, the solid ⋆ results from a fit to the Purcell data, discussed later in section TCQEDC. The gray region highlights where the **phase** **qubit** (design A ) remains stable enough for operation (see text)....QBB Qubit spectroscopy was acquired for **both **designs A and **B** . Like** the **

**cavity**,

**the**

**qubit**

**frequency**f 01 is periodic in

**magnetic flux φ q penetrating**

**the****rf SQUID loop, with a maximum and a minimum operating**

**the****frequency**. However, in this case,

**the**minimum operation

**frequency**is determined by

**shallowest metastable potential well with a | g -state tunneling probability of 50 %, which depends on**

**the****length of time**

**the****metastable potential well configuration is maintained during quantum manipulations . We reset**

**the****the**

**qubit**into a single metastable well (or to a single current branch or “current step”, see Fig. Fig4) and then perform spectroscopy across a region from

**left-most step edge to right-most step edge. During spectroscopy measurements, we apply an offset flux keeping**

**the****the**

**qubit**at its double-well, “readout spot”. An arbitrary waveform generator (Tektronix 5014B) was used to apply fast flux pulses to move

**the**

**qubit**to fixed flux locations, for reset and for scanning many potential well configurations for

**qubit**operation (see Fig. Fig4). At these locations, a spectroscopic microwave tone is applied to

**the**

**qubit**for various

**frequencies**in order to excite

**the**

**qubit**transitions, followed by a fast measurement flux pulse. As described above, any tunneling events are stored at

**readout spot. Fig. Fig5(a) shows an example of**

**the****qubit**spectroscopy for design A . A fit to

**spectroscopic data (that includes**

**the****junction capacitance) gives L q + M = 2.5 nH, L x = 272 pH, I o q ≈ 0.33 μ A, C q ≈ 0.39 pF, and C J = 20 fF in agreement with design values. Here, L q + M and C J were held fixed at values determined by geometry using Fast Henry and**

**the****junction area respectively. The flux periodicity provides a convenient means for evaluating**

**the****resultant flux coupling of**

**the****the**

**qubit**bias coil, M q

**B**= 10.9 pH. The

**qubit**response at

**maximum**

**the****frequency**is shown in Fig. Fig5(b) and shows that at

**deepest well configuration,**

**the****the**

**qubit**is still sufficiently coherent for

**f 01 transition to be spectroscopically distinguishable from**

**the****two-photon transition f 02 / 2 and**

**the****next higher**

**the****qubit**level transition f 12 . This provides a measurement of

**minimum relative anharmonicity, α r = f 12 - f 01 / f 01 = - 0.2 % . Tracking these spectroscopic peaks along**

**the****full spectroscopy provides us with a measure of**

**the****the**

**qubit**anharmonicity across

**full spectroscopic range, as shown later in Fig. Fig9(a) in Sec. TCQEDB. For this demonstration, during**

**the****spectroscopic drive tone,**

**the****the**tunable

**cavity**was rapidly shifted (via a flux

**pulse**through φ c ) to its minimum

**frequency**f c min ≈ 4.8 GHz, in order to place it sufficiently far below all

**the**

**qubit**transition

**frequencies**, providing a “clean” spectroscopic portrait of

**phase**

**the****qubit**. Design A had no visible spectroscopic splittings indicative of spurious two-level systems, while design

**B**showed one over

**spectroscopic range from roughly 5.5**

**the****GHz**to 7.5 GHz . This low occurrence of defects results from

**use of small Josephson junctions ( ≪ 1 μ m 2 )....Tunable-Cavity QED with Phase**

**the****Qubits**...The two possible flux values at the readout spot leads to two possible

**frequencies**for the tunable cavity coupled to the

**qubit**loop. Similar microwave readout schemes have been used with other rf-SQUID phase

**qubits**. For our circuit design, the size of this

**frequency**difference is proportional to the slope d f c / d φ c of the cavity

**frequency**versus flux curve at a particular cavity flux φ c = Φ c / Φ o . The transmission of the cavity can be measured with a network analyzer to resolve the

**qubit**flux (or circulating current) states. The periodicity of the rf SQUID phase

**qubit**can be observed by monitoring the cavity’s resonance

**frequency**while sweeping the

**qubit**flux. This allows us to observe the single-valued and double-valued regions of the hysteretic rf SQUID. In Fig. Fig4(a), we show the cavity response to such a flux sweep for design A . Two data sets have been overlaid, for two different

**qubit**resets ( φ q = ± 2 ) and sweep directions (to the left or to the right), allowing the double-valued or hysteretic regions to overlap. There is an overall drift in the cavity

**frequency**due to flux crosstalk between the

**qubit**bias line and the cavity’s rf SQUID loop that was not compensated for here. This helps to show how the

**frequency**difference in the overlap regions increases as the slope d f c / d φ c increases....(Color online) (a) Cavity spectroscopy (design A ) while sweeping the cavity flux bias with the

**qubit**far detuned, biased at its maximum

**frequency**. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum cavity

**frequency**showing a slot-mode. (c) Line-cut on resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line)....In general, rf SQUID phase

**qubits**have lower T 2 * (and T 2 ) values than transmons, specifically at lower

**frequencies**, where d f 01 / d φ q is large and therefore the

**qubit**is quite sensitive to bias fluctuations and 1/f flux noise . For example, 600 MHz higher in

**qubit**

**frequency**, at f 01 = 7.98 GHz, Ramsey

**oscillations**gave T 2 * = 223 ns. At this location, the decay of on-resonance Rabi

**oscillations**gave T ' = 727 ns, a separate measurement of

**qubit**energy decay after a π -pulse gave T 1 = 658 ns, and so, T 2 ≈ 812 ns, or T 2 ≈ 3.6 × T 2 * , a small, but noticeable improvement over the lower

**frequency**results displayed Fig. Fig6. The current device designs suffer from their planar geometry, due to a very large area enclosed by the non-gradiometric rf SQUID loop (see Fig. Fig1). Future devices will require some form of protection against flux noise , possibly gradiometric loops or replacing the large geometric inductors with a much smaller series array of Josephson junctions ....(Color online) Coupling rate 2 g / 2 π (design A ) as a function of cavity

**frequency**ω c / 2 π . The solid red (blue) line is the prediction from Eq. ( eq:g) (including L x and C J ’s). The (dotted) dashed line is the prediction for capacitive coupling with C = 15 fF ( C = 5 fF). The solid circles were measured spectroscopically (see text). At lowest cavity

**frequency**, the solid ⋆ results from a fit to the Purcell data, discussed later in section TCQEDC. The gray region highlights where the phase

**qubit**(design A ) remains stable enough for operation (see text)....(Color online) (a) Time domain measurements (design A ). Rabi

**oscillations**for

**frequencies**near f 01 = 7.38 GHz. (b) Line-cut on-resonance along the dashed line in (a). The fit (solid line) yields a Rabi

**oscillation**decay time of T ' = 409 ns. (c) Ramsey

**oscillations**versus

**qubit**flux detuning near f 01 = 7.38 GHz. (d) Line-cut along the dashed line in (c). The fit (solid line) yields a Ramsey decay time of T 2 * = 106 ns. With T 1 = 600 ns, this implies a phase coherence time T 2 = 310 ns....(Color online) (a) Relative

**qubit**anharmonicity

**α r**versus

**qubit**

**frequency**ω 01 / 2 π (design A ). The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in (b–d), while the blue line is a theoretical prediction of the relative anharmonicity (including L x , but neglecting C J ) using perturbation theory and no fit parameters. (b) Full dispersive shift 2 χ versus relative detuning Δ 01 / ω 01 for four different

**cavity**frequencies, f c = 6.78, 6.68, 6.58, and 6.48 GHz. Symbols represent the data

**with**lines showing the three-level model predictions. The bold dashed line shows the two-level model prediction when f c = 6.58 GHz. (c) Full dispersive shift 2 χ versus the relative anharmonicity α r . (d) Full dispersive shift 2 χ versus both the relative anharmonicity

**α r**and relative detuning Δ 01 / ω 01 ....The weak additional splitting just below the cavity in Fig. Fig7(d) is from a resonant slot-mode. We can determine the coupling rate 2 g / 2 π between the

**qubit**and the cavity by extracting the splitting size as a function of cavity

**frequency**f c from the measured spectra. Three examples of fits are shown in Fig. Fig7(b–d) with solid lines representing the bare

**qubit**and cavity

**frequencies**, whereas the dashed lines show the new coupled normal-mode

**frequencies**. For design A ( B ), at the maximum cavity

**frequency**of 6.78 GHz (7.07 GHz), we found a minimum coupling rate of 2 g m i n / 2 π = 78 MHz (104 MHz). Notice that the splitting size is clearer bigger in Fig. Fig7(c) than for Fig. Fig7(b) by about 25 MHz. The results for the coupling rate 2 g / 2 π as a function of ω c / 2 π for design A were shown in Fig. Fig2 in section TCQED. Also visible in Fig. Fig7(c–d) are periodic, discontinuous jumps in the cavity spectrum. These are indicative of

**qubit**tunneling events between adjacent metastable energy potential minima, typical behavior for hysteretic rf SQUID phase

**qubits**. Moving away from the maximum cavity

**frequency**increases the flux sensitivity, with the

**qubit**tunneling events becoming more visible as steps. This behavior is clearly visible in Fig. Fig7(c) and was already shown in Fig. Fig4 in Sec. QBA and, as discussed there, provides a convenient way to perform rapid microwave readout of traditional tunneling measurements . Next, we describe dispersive measurements of the phase

**qubit**for design A . These results agree with the tunneling measurements across the entire

**qubit**spectrum....We describe a tunable-cavity QED architecture with an rf SQUID phase

**qubit**inductively coupled to a single-mode, resonant cavity with a tunable

**frequency**that allows for both microwave readout of tunneling and dispersive measurements of the

**qubit**. Dispersive measurement is well characterized by a three-level model, strongly dependent on

**qubit**anharmonicity,

**qubit**-cavity coupling and detuning. A tunable cavity

**frequency**provides a way to strongly vary both the

**qubit**-cavity detuning and coupling strength, which can reduce Purcell losses, cavity-induced dephasing of the

**qubit**, and residual bus coupling for a system with multiple

**qubits**. With our

**qubit**-cavity system, we show that dynamic control over the cavity

**frequency**enables one to avoid Purcell losses during coherent

**qubit**evolutions and optimize state readout during

**qubit**measurements. The maximum

**qubit**decay time $T_1$ = 1.5 $\mu$s is found to be limited by surface dielectric losses from a design geometry similar to planar transmon

**qubits**....We can explore

**coupled**

**the****qubit**-

**cavity**behavior described by Eq. ( eq:H) by performing spectroscopic measurements on either

**the**

**qubit**or

**the****cavity**near

**resonance condition, ω 01 = ω c . Fig. Fig7(a) shows**

**the****qubit**spectroscopy for design A overlaid with

**cavity**spectroscopy for two

**cavity**

**frequencies**, f c = 6.58

**GHz**and 6.78 GHz. Fig. Fig7(d) shows

**cavity**spectroscopy for design

**B**with

**the****cavity**at its maximum

**frequency**of f c m a x = 7.07 GHz while sweeping

**the**

**qubit**flux bias φ q . In

**both**cases, when

**the**

**qubit**

**frequency**f 01 is swept past

**the****cavity**resonance,

**the**inductive coupling generates

**expected spectroscopic normal-mode splitting....(Color online) (a) Cavity spectroscopy (design A ) while sweeping the**

**the****cavity**flux bias

**with**the

**qubit**far detuned, biased at its maximum

**frequency**. The solid line is a fit to the model including the junction capacitance. (b) Zoom-in near the maximum

**cavity**

**frequency**showing a slot-mode. (c) Line-cut on resonance along the dashed line in (b) with a fit to a skewed Lorentzian (solid line)....(Color online) (a) Qubit spectroscopy (design A ). The dashed line is a fit to the model described in the main text. Potential well configurations at various flux biases are sketched

**with**a dot denoting the metastable minimum used for

**qubit**operation. (b) The inset shows that at the deepest well

**with**a minimum anharmonicity ( α r = - 0.2 % ), the

**qubit**is still sufficiently coherent for the f 01 transition to be spectroscopically distinguishable from the f 12 (and f 02 / 2 ) transition, allowing for

**qubit**operations. Tracking these spectroscopic peaks along the full spectroscopy provides us

**with**a measure of the

**qubit**anharmonicity as shown later in Fig. Fig9....In general, rf SQUID phase

**qubits**have lower T 2 * (and T 2 ) values than transmons, specifically at lower

**frequencies**, where d f 01 / d φ q is large and therefore

**the**

**qubit**is quite sensitive to bias fluctuations and 1/f flux noise . For example, 600 MHz higher in

**qubit**

**frequency**, at f 01 = 7.98 GHz, Ramsey oscillations gave T 2 * = 223 ns. At this location,

**the**decay of on-resonance Rabi oscillations gave T ' = 727 ns, a separate measurement of

**qubit**energy decay after a π -pulse gave T 1 = 658 ns, and so, T 2 ≈ 812 ns, or T 2 ≈ 3.6 × T 2 * , a small, but noticeable improvement over

**lower**

**the****frequency**results displayed Fig. Fig6. The current device designs suffer from their planar geometry, due to a very large area enclosed by

**non-gradiometric rf SQUID loop (see Fig. Fig1). Future devices will require some form of protection against flux noise , possibly gradiometric loops or replacing**

**the****large geometric inductors with a much smaller series array of Josephson junctions ....(Color online) (a) Qubit spectroscopy (design A ) overlaid**

**the****with**

**cavity**spectroscopy at two frequencies, f c = 6.58 GHz and 6.78 GHz. (b) Zoom-in of the split

**cavity**spectrum in (a) when f c = 6.78 GHz

**with**corresponding fit lines. (c) Zoom-in of the split

**cavity**spectrum in (a) when f c = 6.58 GHz

**with**corresponding fit lines. (d) Cavity spectroscopy (design B ) while sweeping the

**qubit**flux

**with**f c = 7.07 GHz showing a large normal-mode splitting when the

**qubit**is resonant

**with**the

**cavity**. All solid lines represent the uncoupled

**qubit**and

**cavity**frequencies and the dashed lines show the new coupled normal-mode frequencies. Notice in (d) the additional weak splitting from a slot-mode just below the

**cavity**, and

**in (c**) and (d),

**qubit**tunneling events are visible as abrupt changes in the

**cavity**spectrum....(Color online) (a)

**Qubit**spectroscopy (design A ) overlaid with cavity spectroscopy at two

**frequencies**, f c = 6.58 GHz and 6.78 GHz. (b) Zoom-in of the split cavity spectrum in (a) when f c = 6.78 GHz with corresponding fit lines. (c) Zoom-in of the split cavity spectrum in (a) when f c = 6.58 GHz with corresponding fit lines. (d) Cavity spectroscopy (design B ) while sweeping the

**qubit**flux with f c = 7.07 GHz showing a large normal-mode splitting when the

**qubit**is resonant with the cavity. All solid lines represent the uncoupled

**qubit**and cavity

**frequencies**and the dashed lines show the new coupled normal-mode

**frequencies**. Notice in (d) the additional weak splitting from a slot-mode just below the cavity, and in (c) and (d),

**qubit**tunneling events are visible as abrupt changes in the cavity spectrum....The two possible flux values at

**readout spot leads to two possible**

**the****frequencies**for

**tunable**

**the****cavity**coupled to

**the**

**qubit**loop. Similar microwave readout schemes have been used with other rf-SQUID phase

**qubits**. For our circuit design,

**the**size of this

**frequency**difference is proportional to

**slope d f c / d φ c of**

**the**

**the****cavity**

**frequency**versus flux curve at a particular

**cavity**flux φ c = Φ c / Φ o . The transmission of

**the****cavity**can be measured with a network analyzer to resolve

**the**

**qubit**flux (or circulating current) states. The periodicity of

**rf SQUID phase**

**the****qubit**can be observed by monitoring

**the****cavity**’

**s**resonance

**frequency**while sweeping

**the**

**qubit**flux. This allows us to observe

**single-valued and double-valued regions of**

**the****hysteretic rf SQUID. In Fig. Fig4(a), we show**

**the**

**the****cavity**response to such a flux sweep for design A . Two data sets have been overlaid, for two

**different**

**qubit**resets ( φ q = ± 2 ) and sweep directions (to

**left or to**

**the****right), allowing**

**the****double-valued or hysteretic regions to overlap. There is an overall drift in**

**the**

**the****cavity**

**frequency**due to flux crosstalk between

**the**

**qubit**bias line and

**the****cavity**’

**s**rf SQUID loop that was not compensated for here. This helps to show how

**the**

**frequency**difference in

**overlap regions increases as**

**the****slope d f c / d φ c increases....Next, we carefully explore the size of the dispersive shifts for various cavity and**

**the****qubit**

**frequencies**. In order to capture the maximum dispersive

**frequency**shift experienced by the cavity, we applied a π -pulse to the

**qubit**. A fit to the phase response curve allows us to extract the cavity’s amplitude response time 2 / κ , the

**qubit**T 1 , and the full dispersive shift 2 χ . Changing the cavity

**frequency**modifies the coupling g and the detuning Δ 01 , while changes to the

**qubit**

**frequency**change both Δ 01 and the

**anharmonicity α . In Fig. Fig9(a), we show the phase**

**qubit**’s**anharmonicity as a function of its transition**

**qubit**’s**frequency**ω 01 / 2 π extracted from the spectroscopic data shown in Fig. Fig5 from section QBB for design A . The solid red line is a polynomial fit to the experimental data, used to calculate the three-level model curves in Fig. Fig9(b–d), while the blue line is a theoretical prediction of the relative anharmonicity (including L x , but neglecting C J ) using perturbation theory and the characteristic

**qubit**parameters extracted section QBB. In Fig. Fig9(b–d), we find that the observed dispersive shifts strongly depend on all of these factors and agree well with the three-level model predictions . For comparison, in Fig. Fig9(b), we show the results for the two-level system model (bold dashed line) when f c = 6.58 GHz, which has a significantly larger amplitude for all detunings (outside the “straddling regime”). Notice that it is possible to increase the size of the dispersive shifts for a given | Δ 01 | / ω 01 by decreasing the cavity

**frequency**f c , which increases the coupling rate 2 g / 2 π (as seen in Fig. Fig2 in section TCQED). Also, notice that decreasing the ratio of | Δ 01 | / ω 01 also significantly increases the size of the dispersive shifts, even when the phase

**relative anharmonicity α r decreases as ω 01 increases. Essentially, the ability to reduce | Δ 01 | helps to counteract any reductions in α r . These results clearly demonstrate the ability to tune the size of the dispersive shift through selecting the relative**

**qubit**’s**frequency**of the

**qubit**and the cavity. This tunability offers a new flexibility for optimizing dispersive readout of

**qubits**in cavity QED architectures and provides a way for rf SQUID phase

**qubits**to avoid the destructive effects of tunneling-based measurements....(Color online) (a) Pulse sequence. (b) Rabi

**oscillations**(design A ) for various pulse durations obtained using dispersive measurement at f 01 = 7.18 GHz, with Δ 01 = + 10 g . (c) A single, averaged time trace along the vertical dashed line in (b). (d) Rabi

**oscillations**extracted from the final population at the end of the drive pulse, along the dashed diagonal line in (b). (e) Zoom-in of dashed box in (b) showing Rabi

**oscillations**observed during continuous driving....Data were acquired across

**the**

**qubit**spectrum for several well-separated

**cavity**positions for

**both**design geometries A and

**B**. The results agree well with our model and are summarized in Table tab:table1. For design A , with

**the****cavity**placed at its maximum

**frequency**f c m a x = 6.78 GHz and κ / 2 π = 24 MHz, we find that

**Purcell effect strongly reduces**

**the****combined T 1 = 1 / γ T over a significant portion of**

**the****the**

**qubit**spectrum. However, when f c = 4.9 GHz, near

**minimum**

**the****cavity**

**frequency**, even with significantly stronger coupling g / g m i n ≈ 4 ,

**qubit**lifetimes are relatively large across

**full**

**the****qubit**spectrum with a maximum value of T 1 = 0.72 μ

**, clearly limited by an over-coupled flux bias line ( γ q**

**s****B**). For design

**B**(see Fig. Fig1(c)), we reduced

**coupling to**

**the****bias line by over a factor of 3 and lowered**

**the****maximum**

**the****frequency**of

**the**

**qubit**by over 1

**GHz**in order to take advantage of

**inductive coupling, which improves operation when**

**the****the**

**qubit**is mostly below

**the****cavity**. As seen in Fig. Fig10(b), we find significant improvement with a maximum

**qubit**lifetime of T 1 = 1.5 μ

**, clearly limited by dielectric losses with Q d = 82,400. This value, obta...We can explore the coupled**

**s****qubit**-cavity behavior described by Eq. ( eq:H) by performing spectroscopic measurements on either the

**qubit**or the cavity near the resonance condition, ω 01 = ω c . Fig. Fig7(a) shows

**qubit**spectroscopy for design A overlaid with cavity spectroscopy for two cavity

**frequencies**, f c = 6.58 GHz and 6.78 GHz. Fig. Fig7(d) shows cavity spectroscopy for design B with the cavity at its maximum

**frequency**of f c m a x = 7.07 GHz while sweeping the

**qubit**flux bias φ q . In both cases, when the

**qubit**

**frequency**f 01 is swept past the cavity resonance, the inductive coupling generates the expected spectroscopic normal-mode splitting. ... We describe a tunable-cavity QED architecture with an rf SQUID phase

**qubit**inductively coupled to a single-mode, resonant cavity with a tunable

**frequency**that allows for both microwave readout of tunneling and dispersive measurements of the

**qubit**. Dispersive measurement is well characterized by a three-level model, strongly dependent on

**qubit**anharmonicity,

**qubit**-cavity coupling and detuning. A tunable cavity

**frequency**provides a way to strongly vary both the

**qubit**-cavity detuning and coupling strength, which can reduce Purcell losses, cavity-induced dephasing of the

**qubit**, and residual bus coupling for a system with multiple

**qubits**. With our

**qubit**-cavity system, we show that dynamic control over the cavity

**frequency**enables one to avoid Purcell losses during coherent

**qubit**evolutions and optimize state readout during

**qubit**measurements. The maximum

**qubit**decay time $T_1$ = 1.5 $\mu$s is found to be limited by surface dielectric losses from a design geometry similar to planar transmon

**qubits**.

Files:

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

Date: 2005-11-21

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-Bana...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)....Entanglement of a **qubit** coupled to a resonator in the adiabatic regime...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....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. ... 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.

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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

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 ....(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 ....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 ....(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....(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....Flux **qubit** noise spectroscopy using Rabi oscillations under strong driving conditions...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 ....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) (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....(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....(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 ....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)....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....Josephson devices, decoherence, Rabi

**oscillation**, $1/f$ noise...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....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 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). ... 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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