### 63697 results for qubit oscillator frequency

Contributors: Chiorescu, I., Bertet, P., Semba, K., Nakamura, Y., Harmans, C. J. P. M., Mooij, J. E.

Date: 2004-07-30

In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits or **qubits**. Single-**qubit** operations, direct coupling between two **qubits**, and the realization of a quantum gate have been reported. However, complex manipulation of entangled states - such as the coupling of a two-level system to a quantum harmonic **oscillator**, as demonstrated in ion/atom-trap experiments or cavity quantum electrodynamics - has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux **qubit** (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic **oscillator**. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi **oscillations** of the coupled system....**Oscillator** relaxation time. a, Rabi **oscillations** between the | 01 and | 10 states (during pulse 2 in the inset) obtained after applying a first pulse (1) in resonance with the **oscillator** transition. Here, the interval between the two pulses is 1 ns. The continuous line represents a fit using an exponentially decaying sinusoidal **oscillation** plus an exponential decay of the background (due to the relaxation into the ground state). The **oscillation**’s decay time is τ c o h = 2.9 ns, whereas the background decay time is ∼ 4 ns. b, The amplitude of Rabi **oscillations** as a function of the interval between the two pulses (the vertical bars represent standard error bars estimated from the fitting procedure, see a). Owing to the **oscillator** relaxation, the amplitude decays in τ r e l ≈ 6 ns (the continuous line represents an exponential fit)....Generation and control of entangled states. a, Spectroscopic characterization of the energy levels (see Fig. 1b inset) after a π (upper scan) and a 2 π (lower scan) Rabi pulse on the **qubit** transition. In the upper scan, the system is first excited to | 10 from which it decays towards the | 01 excited state (red sideband at 3.58 GHz) or towards the | 00 ground state ( F L = 6.48 GHz). In the lower scan, the system is rotated back to the initial state | 00 wherefrom it is excited into the | 10 or | 11 states (see, in dashed, the blue sideband peak at 9.48 GHz for 13 dB more power). b, Coupled Rabi oscillations: the blue sideband is excited and the switching probability is recorded as a function of the pulse length for different microwave powers (plots are shifted vertically for clarity). For large microwave powers, the resonance peak of the blue sideband is shifted to 9.15 GHz. When detuning the microwave excitation away from resonance, the Rabi oscillations become faster (bottom four curves). These oscillations are suppressed by preparing the system in the | 10 state with a π pulse and revived after a 2 π pulse (top two curves in Fig. 3b) c, Coupled Rabi oscillations: after a π pulse on the **qubit** resonance ( | 00 → | 10 ) we excite the red sideband at 3.58 GHz. The switching probability shows coherent oscillations between the states | 10 and | 01 , at various microwave powers (the curves are shifted vertically for clarity). The decay time of the coherent oscillations in a, b is ∼ 3 ns....Rabi **oscillations** at the **qubit** symmetry point Δ = 5.9 GHz. a, Switching probability as a function of the microwave pulse length for three microwave nominal powers; decay times are of the order of 25 ns. For A = 8 dBm, bi-modal beatings are visible (the corresponding **frequencies** are shown by the filled squares in b). b, Rabi **frequency**, obtained by fast Fourier transformation of the corresponding **oscillations**, versus microwave amplitude. In the weak driving regime, the linear dependence is in agreement with estimations based on sample design. A first splitting appears when the Rabi **frequency** is ∼ ν p . In the strong driving regime, the power independent Larmor precession at **frequency** Δ gives rise to a second splitting. c, This last aspect is obtained in numerical simulations where the microwave driving is represented by a term 1 / 2 h F 1 cos Δ t and a small deviation from the symmetry point (100 MHz) is introduced in the strong driving regime (the thick line indicates the main Fourier peaks). Radiative shifts 20 at high microwave power could account for such a shift in the experiment....In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits or **qubits**. Single-**qubit** operations, direct coupling between two **qubits**, and the realization of a quantum gate have been reported. However, complex manipulation of entangled states - such as the coupling of a two-level system to a quantum harmonic **oscillator**, as demonstrated in ion/atom-trap experiments or cavity quantum electrodynamics - has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux **qubit** (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic **oscillator**. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi oscillations of the coupled system....Qubit - SQUID device and spectroscopy a, Atomic force micrograph of the SQUID (large loop) merged with the flux **qubit** (the smallest loop closed by three junctions); the **qubit** to SQUID area ratio is 0.37. Scale bar, 1 μ m . The SQUID (**qubit**) junctions have a critical current of 4.2 (0.45) μ A. The device is made of aluminium by two symmetrically angled evaporations with an oxidation step in between. The surrounding circuit shows aluminium shunt capacitors and lines (in black) and gold quasiparticle traps 3 and resistive leads (in grey). The microwave field is provided by the shortcut of a coplanar waveguide (MW line) and couples inductively to the **qubit**. The current line ( I ) delivers the readout pulses, and the switching event is detected on the voltage line ( V ). b, Resonant **frequencies** indicated by peaks in the SQUID switching probability when a long microwave pulse excites the system before the readout pulse. Data are represented as a function of the external flux through the **qubit** area away from the **qubit** symmetry point. Inset, energy levels of the **qubit** - oscillator system for some given bias point. The blue and red sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96 GHz and -2.90 GHz, respectively, to the central continuous line (numerical fit). These values are close to the oscillator resonance ν **p **at 2.91 GHz (solid circles) and we attribute the small differences to the slight dependence of ν **p **on **qubit** state. c, The plasma resonance (circles) and the distances between the **qubit** peak (here F L = 6.4 GHz) and the red/blue (up/down triangles) sidebands as a function of an offset current I b o f f through the SQUID. The data are close to each other and agree well with the theoretical prediction for ν **p **versus offset current (dashed line)....Generation and control of entangled states. a, Spectroscopic characterization of the energy levels (see Fig. 1b inset) after a π (upper scan) and a 2 π (lower scan) Rabi pulse on the **qubit** transition. In the upper scan, the system is first excited to | 10 from which it decays towards the | 01 excited state (red sideband at 3.58 GHz) or towards the | 00 ground state ( F L = 6.48 GHz). In the lower scan, the system is rotated back to the initial state | 00 wherefrom it is excited into the | 10 or | 11 states (see, in dashed, the blue sideband peak at 9.48 GHz for 13 dB more power). b, Coupled Rabi **oscillations**: the blue sideband is excited and the switching probability is recorded as a function of the pulse length for different microwave powers (plots are shifted vertically for clarity). For large microwave powers, the resonance peak of the blue sideband is shifted to 9.15 GHz. When detuning the microwave excitation away from resonance, the Rabi **oscillations** become faster (bottom four curves). These **oscillations** are suppressed by preparing the system in the | 10 state with a π pulse and revived after a 2 π pulse (top two curves in Fig. 3b) c, Coupled Rabi **oscillations**: after a π pulse on the **qubit** resonance ( | 00 → | 10 ) we excite the red sideband at 3.58 GHz. The switching probability shows coherent **oscillations** between the states | 10 and | 01 , at various microwave powers (the curves are shifted vertically for clarity). The decay time of the coherent **oscillations** in a, b is ∼ 3 ns....Oscillator relaxation time. a, Rabi oscillations between the | 01 and | 10 states (during pulse 2 in the inset) obtained after applying a first pulse (1) in resonance with the oscillator transition. Here, the interval between the two pulses is 1 ns. The continuous line represents a fit using an exponentially decaying sinusoidal oscillation plus an exponential decay of the background (due to the relaxation into the ground state). The oscillation’s decay time is τ c o h = 2.9 ns, whereas the background decay time is ∼ 4 ns. b, The amplitude of Rabi oscillations as a function of the interval between the two pulses (the vertical bars represent standard error bars estimated from the fitting procedure, see a). Owing to the oscillator relaxation, the amplitude decays in τ r e l ≈ 6 ns (the continuous line represents an exponential fit)....**Qubit** - SQUID device and spectroscopy a, Atomic force micrograph of the SQUID (large loop) merged with the flux **qubit** (the smallest loop closed by three junctions); the **qubit** to SQUID area ratio is 0.37. Scale bar, 1 μ m . The SQUID (**qubit**) junctions have a critical current of 4.2 (0.45) μ A. The device is made of aluminium by two symmetrically angled evaporations with an oxidation step in between. The surrounding circuit shows aluminium shunt capacitors and lines (in black) and gold quasiparticle traps 3 and resistive leads (in grey). The microwave field is provided by the shortcut of a coplanar waveguide (MW line) and couples inductively to the **qubit**. The current line ( I ) delivers the readout pulses, and the switching event is detected on the voltage line ( V ). b, Resonant **frequencies** indicated by peaks in the SQUID switching probability when a long microwave pulse excites the system before the readout pulse. Data are represented as a function of the external flux through the **qubit** area away from the **qubit** symmetry point. Inset, energy levels of the **qubit** - **oscillator** system for some given bias point. The blue and red sidebands are shown by down- and up-triangles, respectively; continuous lines are obtained by adding 2.96 GHz and -2.90 GHz, respectively, to the central continuous line (numerical fit). These values are close to the **oscillator** resonance ν p at 2.91 GHz (solid circles) and we attribute the small differences to the slight dependence of ν p on **qubit** state. c, The plasma resonance (circles) and the distances between the **qubit** peak (here F L = 6.4 GHz) and the red/blue (up/down triangles) sidebands as a function of an offset current I b o f f through the SQUID. The data are close to each other and agree well with the theoretical prediction for ν p versus offset current (dashed line)....Oscill...Rabi oscillations at the **qubit** symmetry point Δ = 5.9 GHz. a, Switching probability as a function of the microwave pulse length for three microwave nominal powers; decay times are of the order of 25 ns. For A = 8 dBm, bi-modal beatings are visible (the corresponding **frequencies** are shown by the filled squares in b). b, Rabi **frequency**, obtained by fast Fourier transformation of the corresponding oscillations, versus microwave amplitude. In the weak driving regime, the linear dependence is in agreement with estimations based on sample design. A first splitting appears when the Rabi **frequency** is ∼ ν p . In the strong driving regime, the power independent Larmor precession at **frequency** Δ gives rise to a second splitting. c, This last aspect is obtained in numerical simulations where the microwave driving is represented by a term 1 / 2 h F 1 cos Δ t and a small deviation from the symmetry point (100 MHz) is introduced in the strong driving regime (the thick line indicates the main Fourier peaks). Radiative shifts 20 at high microwave power could account for such a shift in the experiment....Coherent dynamics of a flux **qubit** coupled to a harmonic **oscillator**...Qub ... In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits or **qubits**. Single-**qubit** operations, direct coupling between two **qubits**, and the realization of a quantum gate have been reported. However, complex manipulation of entangled states - such as the coupling of a two-level system to a quantum harmonic **oscillator**, as demonstrated in ion/atom-trap experiments or cavity quantum electrodynamics - has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux **qubit** (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic **oscillator**. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi **oscillations** of the coupled system.

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

Date: 2012-05-10

Figure fig:Ttrans shows the numerically obtained coherence times and whether the decay is predominantly Gaussian or Markovian. For the large **oscillator** damping γ = ϵ , the conditions for the validity of the (Markovian) Bloch-Redfield equation stated at the end of Sec. sec:analytics-g2 hold. Then we observe a good agreement of the numerically obtained T ⊥ * and Eq. ...Non-Markovian **qubit** decoherence during dispersive readout...Figure fig:Ttrans shows the numerically obtained coherence times and whether the decay is predominantly Gaussian or Markovian. For the large oscillator damping γ = ϵ , the conditions for the validity of the (Markovian) Bloch-Redfield equation stated at the end of Sec. sec:analytics-g2 hold. Then we observe a good agreement of the numerically obtained T ⊥ * and Eq. ...(Color online) Dephasing time for purely quadratic **qubit**-**oscillator** coupling ( g 1 = 0 ), resonant driving at large **frequency**, Ω = ω 0 = 5 ϵ , and various values of the **oscillator** damping γ . The driving amplitude is A = 3.5 γ , such that always n ̄ = 6.125 . Filled symbols mark Markovian decay, while stroked symbols refer to Gaussian shape. The solid line depicts the value obtained for γ = ϵ in the Markov limit. The corresponding numerical values are connected by a dashed line which serves as guide to the eye....(Color online) Typical time evolution of the **qubit** operator σ x (solid line) and the corresponding purity (dashed) for Ω = ω 0 = 0.8 ϵ , g 1 = 0.02 ϵ , γ = 0.02 ϵ , and driving amplitude A = 0.06 ϵ such that the stationary photon number is n ̄ = 4.5 . Inset: Purity decay shown in the main panel (dashed) compared to the decay given by Eq. P(t) together with Eq. Lambda(t) (solid line)....Figure fig:timeevolution depicts the time evolution of the **qubit** expectation value σ x which exhibits decaying **oscillations** with **frequency** ϵ . The parameters correspond to an intermediate regime between the Gaussian and the Markovian dynamics, as is visible in the inset....(Color online) Dephasing time for purely linear **qubit**-**oscillator** coupling ( g 2 = 0 ), resonant driving, Ω = ω 0 , and **oscillator** damping γ = 0.02 ϵ . The amplitude A = 0.07 ϵ corresponds to the mean photon number n ̄ = 6.125 . Filled symbols and dashed lines refer to predominantly Markovian decay, while for Gaussian decay, stroked symbols and solid lines are used....We study **qubit** decoherence under generalized dispersive readout, i.e., we investigate a **qubit** coupled to a resonantly driven dissipative harmonic **oscillator**. We provide a complete picture by allowing for arbitrarily large **qubit**-**oscillator** detuning and by considering also a coupling to the square of the **oscillator** coordinate, which is relevant for flux **qubits**. Analytical results for the decoherence time are obtained by a transformation of the **qubit**-**oscillator** Hamiltonian to the dispersive frame and a subsequent master equation treatment beyond the Markov limit. We predict a crossover from Markovian decay to a decay with Gaussian shape. Our results are corroborated by the numerical solution of the full **qubit**-**oscillator** master equation in the original frame....Figure fig:timeevolution depicts the time evolution of the **qubit** expectation value σ x which exhibits decaying oscillations with **frequency** ϵ . The parameters correspond to an intermediate regime between the Gaussian and the Markovian dynamics, as is visible in the inset. ... We study **qubit** decoherence under generalized dispersive readout, i.e., we investigate a **qubit** coupled to a resonantly driven dissipative harmonic **oscillator**. We provide a complete picture by allowing for arbitrarily large **qubit**-**oscillator** detuning and by considering also a coupling to the square of the **oscillator** coordinate, which is relevant for flux **qubits**. Analytical results for the decoherence time are obtained by a transformation of the **qubit**-**oscillator** Hamiltonian to the dispersive frame and a subsequent master equation treatment beyond the Markov limit. We predict a crossover from Markovian decay to a decay with Gaussian shape. Our results are corroborated by the numerical solution of the full **qubit**-**oscillator** master equation in the original frame.

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

Date: 2008-09-08

We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations....The measurement process that we used to observe coherent **oscillations** consists of several steps shown in Fig. fig:3(a). Each step is realized by applying a combination of magnetic fluxes Φ x and Φ c as indicated by numbers in Fig. fig:2(b). The first step in our measurement is the initialization of the system in a defined flux state (1). Starting from a double well at Φ x ≅ Φ 0 / 2 with high barrier, the potential is tilted by changing Φ x until it has only a single minimum (left or right, depending on the amplitude and polarity of the applied flux pulse). This potential shape is maintained long enough to ensure the relaxation to the ground state. Afterwards the potential is tuned back to the initial double-well state (2). The high barrier prevents any tunneling and the **qubit** is thus initialized in the chosen potential well. Next, the barrier height is lowered to an intermediate level (3) that preserves the initial state and allows to use just a small-amplitude Φ c flux pulse for the subsequent manipulation. The following Φ c -pulse transforms the potential into a single well (4). The Φ c -pulse duration Δ t is in the nanosecond range. The relative phase of the ground and the first excited states evolves depending on the energy difference between them. Once Φ c -pulse is over, the double well is restored and the system is measured in the basis | L | R (5). The readout of the **qubit** flux state is done by applying a bias current ramp to the dc SQUID and recording its switching current to the voltage state....(a) The measured double SQUID flux Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for **qubit** manipulation at which the **qubit** potential has a shape as indicated in the insets....Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control flux amplitude Φ c 3 . Circles are the experimentally observed **oscillation** **frequencies** for the corresponding pulse amplitudes....We experimentally demonstrate the coherent oscillations of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states oscillate at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations....The flux pattern is repeated for 10 2 - 10 4 times in order to evaluate the probability P L = L | Ψ f i n a l 2 of occupation of the left state at the end of the manipulation. By changing the duration Δ t of the manipulation pulse Φ c , we observed coherent **oscillations** between the occupations of the states | L and | R shown in Fig. fig:4(a). The **oscillation** **frequency** could be tuned between 6 and 21 GHz by changing the pulse amplitude Δ Φ c . These **oscillations** persist when the potential is made slightly asymmetric by varying the value Φ x 1 . As it is shown in Fig. fig:4(b), detuning from the symmetric potential by up to ± 2.9 m Φ 0 only slightly changes the amplitude and symmetry of the **oscillations**. When the **qubit** was initially prepared in | R state instead of | L state we observed similar **oscillations**....Calculated energy spacing of the first (solid line), second (dashed line) and third (dotted line) energy levels with respect to the ground state in the single well potential, plotted vs. the control **flux **amplitude Φ c 3 . Circles are the experimentally observed oscillation frequencies for the corresponding pulse amplitudes....Assuming identical junctions and negligible inductance of **the** smaller loop ( l ≪ L ), **the** system dynamics is equivalent to **the** motion of a particle with **the** Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is **the** spatial coordinate of **the** equivalent particle, p is **the** relative conjugate momentum, M = C Φ b 2 is **the** effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are **the** normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β **l **shape is used for **qubit** initialization and readout. **The** single well, or more exactly **the** two lowest energy states | 0 and | 1 in this well, is used for **the** coherent evolution of **the** **qubit**....Assuming identical junctions and negligible inductance of the smaller loop ( l ≪ L ), the system dynamics is equivalent to the motion of a particle with the Hamiltonian H = p 2 2 M + Φ b 2 L 1 2 ϕ - ϕ x 2 - β ϕ c cos ϕ , where ϕ = Φ / Φ b is the spatial coordinate of the equivalent particle, p is the relative conjugate momentum, M = C Φ b 2 is the effective mass, ϕ x = Φ x / Φ b and ϕ c = π Φ c / Φ 0 are the normalized flux controls, and β ϕ c = 2 I 0 L / Φ b cos ϕ c , with Φ 0 = h / 2 e and Φ b = Φ 0 / 2 π . For β **qubit** initialization and readout. The single well, or more exactly the two lowest energy states | 0 and | 1 in this well, is used for the coherent evolution of the **qubit**....(a) Schematic of the flux **qubit** circuit. (b) The control flux Φ c changes the potential barrier between the two flux states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control flux Φ x on the potential symmetry....Coherent oscillations in a superconducting tunable flux **qubit** manipulated without microwaves...**The** investigated circuit, shown in Fig. fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of **the** two JJs embedded in **the** dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. **The** **qubit** is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to **the** large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. **The** readout of **the** **qubit** flux is performed by measuring **the** switching current of an unshunted dc SQUID, which is inductively coupled to **the** **qubit** . **The** circuit was manufactured by Hypres using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. **The** dielectric material used for junction isolation is SiO 2 . **The** whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between **the** loops. **The** JJs have dimensions of 3 × 3 μ m 2 and **the** entire device occupied a space of 230 × 430 μ m 2 . All **the** measurements have been performed at a sample temperature of 15 mK. **The** currents generating **the** two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at **the** 1K-pot stage of a dilution refrigerator. To generate **the** flux Φ c , a bias-tee at room temperature was used to combine **the** outputs of a current source and a pulse generator. For biasing and sensing **the** readout dc SQUID, we used superconducting wires and metal powder filters at **the** base temperature, as well as attenuators and low-pass filters with a cut-**off** **frequency** of 10 kHz at **the** 1K-pot stage. **The** chip holder with **the** powder filters was surrounded by one superconducting and two cryoperm shields....The **oscillation** **frequency** ω 0 depends on the amplitude of the manipulation pulse Δ Φ c since it determines the shape of the single well potential and the energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger **oscillation** **frequency** as shown in Fig. fig:4(a). In Fig. fig:5, we plot the energy spacing between the ground state and the three excited states (indicated as E k - E 0 / h with k=1,2,3) versus the flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using the experimental parameters. In the same figure, we plot the measured **oscillation** **frequencies** for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. The fact that a small asymmetry in the potential does not change the **oscillation** **frequency**, as shown in Fig. fig:4(b), is consistent with the interpretation as the energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in the controlling flux Φ x ....Probability to measure the state in dependence of the pulse duration Δ t for the **qubit** initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated oscillation **frequency**, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount....**The** oscillation **frequency** ω 0 depends on **the** amplitude of **the** manipulation pulse Δ Φ c since it determines **the** shape of **the** single well potential and **the** energy level spacing E 1 - E 0 . A pulse of larger amplitude Δ Φ c generates a deeper well having a larger level spacing, which leads to a larger oscillation **frequency** as shown in Fig. fig:4(a). In Fig. fig:5, we plot **the** energy spacing between **the** ground state and **the** three excited states (indicated as E k - E 0 / h with k=1,2,3) versus **the** flux Φ c 3 = Φ c 2 + Δ Φ c obtained from a numerical simulation of our system using **the** experimental parameters. In **the** same figure, we plot **the** measured oscillation frequencies for different values of Φ c (open circles). Excellent agreement between simulation (solid line) and data strongly supports our interpretation. **The** fact that a small asymmetry in **the** potential does not change **the** oscillation **frequency**, as shown in Fig. fig:4(b), is consistent with **the** interpretation as **the** energy spacing E 1 - E 0 is only weakly affected by small variations of Φ x . This provides protection against noise in **the** controlling flux Φ x ....Probability to measure the state in dependence of the pulse duration Δ t for the **qubit** initially prepared in the state, and for (a) different pulse amplitudes Δ Φ c , resulting in the indicated **oscillation** **frequency**, and (b) for different potential symmetry by detuning Φ x from Φ 0 / 2 by the indicated amount....**The** flux pattern is repeated for 10 2 - 10 4 times in order to evaluate **the** probability P L = L | Ψ f i n a **l **2 of occupation of **the** left state at **the** end of **the** manipulation. By changing **the** duration Δ t of **the** manipulation pulse Φ c , we observed coherent oscillations between **the** occupations of **the** states | L and | R shown in Fig. fig:4(a). **The** oscillation **frequency** could be tuned between 6 and 21 GHz by changing **the** pulse amplitude Δ Φ c . These oscillations persist when **the** potential is made slightly asymmetric by varying **the** value Φ x 1 . As it is shown in Fig. fig:4(b), detuning from **the** symmetric potential by up to ± 2.9 m Φ 0 only slightly changes **the** amplitude and symmetry of **the** oscillations. When **the** **qubit** was initially prepared in | R state instead of | L state we observed similar oscillations....(a) The measured double SQUID **flux **Φ in dependence of Φ x , plotted for two different values of Φ c and initial preparation in either potential well. (b) Position of the switching points (dots) in the Φ c - Φ x parameter space. Numbered tags indicate the working points for **qubit** manipulation at which the **qubit** potential has a shape as indicated in the insets....(a) Schematic of the flux **qubit** circuit. (b) The control **flux **Φ c changes the potential barrier between the two **flux **states | L and | R , here Φ x = 0.5 Φ 0 . (c) Effect of the control **flux **Φ x on the potential symmetry....The investigated circuit, shown in Fig. fig:1(a), is a double SQUID consisting of a superconducting loop of inductance L = 85 pH, interrupted by a small dc SQUID of loop inductance l = 6 pH. This dc SQUID is operated as a single Josephson junction (JJ) whose critical current is tunable by an external magnetic field. Each of the two JJs embedded in the dc SQUID has a critical current I 0 = 8 μ A and capacitance C = 0.4 pF. The **qubit** is manipulated by changing two magnetic fluxes Φ x and Φ c , applied to the large and small loops by means of two coils of mutual inductance M x = 2.6 pH and M c = 6.3 pH, respectively. The readout of the **qubit** flux is performed by measuring the switching current of an unshunted dc SQUID, which is inductively coupled to the **qubit** . The circuit was manufactured by Hypres using standard Nb/AlO x /Nb technology in a 100 A/cm 2 critical current density process. The dielectric material used for junction isolation is SiO 2 . The whole circuit is designed gradiometrically in order to reduce magnetic flux pick-up and spurious flux couplings between the loops. The JJs have dimensions of 3 × 3 μ m 2 and the entire device occupied a space of 230 × 430 μ m 2 . All the measurements have been performed at a sample temperature of 15 mK. The currents generating the two fluxes Φ x and Φ c were supplied via coaxial cables including 10 dB attenuators at the 1K-pot stage of a dilution refrigerator. To generate the flux Φ c , a bias-tee at room temperature was used to combine the outputs of a current source and a pulse generator. For biasing and sensing the readout dc SQUID, we used superconducting wires and metal powder filters at the base temperature, as well as attenuators and low-pass filters with a cut-off **frequency** of 10 kHz at the 1K-pot stage. The chip holder with the powder filters was surrounded by one superconducting and two cryoperm shields. ... We experimentally demonstrate the coherent **oscillations** of a tunable superconducting flux **qubit** by manipulating its energy potential with a nanosecond-long pulse of magnetic flux. The occupation probabilities of two persistent current states **oscillate** at a **frequency** ranging from 6 GHz to 21 GHz, tunable via the amplitude of the flux pulse. The demonstrated operation mode allows to realize quantum gates which take less than 100 ps time and are thus much faster compared to other superconducting **qubits**. An other advantage of this type of **qubit** is its insensitivity to both thermal and magnetic field fluctuations.

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Contributors: Bertet, P., Chiorescu, I., Semba, K., Harmans, C. J. P. M, Mooij, J. E.

Date: 2004-05-03

(a) Principle of the detection scheme. After the Rabi pulse, a microwave pulse at the plasma **frequency** resonantly enhances the escape rate. The bias current is maintained for 500 n s above the retrapping value. (b) Resonant activation peak for different Rabi angle. Each curve was offset by 5 % for lisibility. The Larmor **frequency** was f q = 8.5 ~ G H z . Pulse 2 duration was 10 ~ n s . (c) Resonant activation peak without (full circles) and after (open circles) a π pulse. The continuous line is the difference between the two switching probabilities. (d) Rabi oscillation measured by DC current pulse (grey line, amplitude A = 40 % ) and by resonant activation method with a 5 ~ n s RAP (black line, A = 62 % ), at the same Larmor **frequency**. fig4...We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi oscillations with high visibility (65%)....We show the two peaks corresponding to θ 1 = 0 (curve P s w 0 , full circles) and θ 1 = π (curve P s w π , open circles) in figure fig4c. They are separated by f p 0 - f p 1 = 50 ~ M H z and have a similar width of 90 ~ M H z . This is an indication that the π pulse efficiently populates the excited state (any significant probability for the **qubit** to be in 0 would result into broadening of the curve P s w π ), and is in strong contrast with the results obtained with the DCP method (figure fig2b). The difference between the two curves S f = P s w 0 - P s w π (solid line in figure fig4c) gives a lower bound of the excited state population after a π pulse. Because of the above mentioned asymmetric shape of the resonant activation peaks, it yields larger absolute values on the low- than on the high-**frequency** side of the peak. Thus the plasma **oscillator** non-linearity increases the sensitivity of our measurement, which is reminiscent of the ideas exposed in . On the data shown here, S f attains a maximum S m a x = 60 % for a **frequency** f 2 * indicated by an arrow in figure fig4c. The value of S m a x strongly depends on the **microwave** pulse duration and power. The optimal settings are the result of a compromise between two constraints : a long **microwave** pulse provides a better resonant activation peak separation, but on the other hand the pulse should be much shorter than the **qubit** damping time T 1 , to prevent loss of excited state population. Under optimized conditions, we were able to reach S m a x = 68 % ....(a) Principle of the detection scheme. After the Rabi pulse, a microwave pulse at the plasma **frequency** resonantly enhances the escape rate. The bias current is maintained for 500 n s above the retrapping value. (b) Resonant activation peak for different Rabi angle. Each curve was offset by 5 % for lisibility. The Larmor **frequency** was f q = 8.5 ~ G H z . Pulse 2 duration was 10 ~ n s . (c) Resonant activation peak without (full circles) and after (open circles) a π pulse. The continuous line is the difference between the two switching probabilities. (d) Rabi **oscillation** measured by DC current pulse (grey line, amplitude A = 40 % ) and by resonant activation method with a 5 ~ n s RAP (black line, A = 62 % ), at the same Larmor **frequency**. fig4...We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%)....(insert) Typical resonant activation peak (width 40 ~ M H z ), measured after a 50 ~ n s microwave pulse. Due to the SQUID non-linearity, it is much sharper at low than at high **frequencies**. (figure) Center **frequency** of the resonant activation peak as a function of the external magnetic flux (squares). It follows the switching current modulation (dashed line). The solid line is a fit yielding the values of the shunt capacitor and stray inductance given in the text. fig3...(a) Rabi **oscillations** at a Larmor **frequency** f q = 7.15 ~ G H z (b) Switching probability as a function of current pulse amplitude I without (closed circles, curve P s w 0 I ) and with (open circles, curve P s w π I b ) a π pulse applied. The solid black line P t h 0 I b is a numerical adjustment to P s w 0 I b assuming escape in the thermal regime. The dotted line (curve P t h 1 I b ) is calculated with the same parameters for a critical current 100 n A smaller, which would be the case if state 1 was occupied with probability unity. The grey solid line is the sum 0.32 P t h 1 I b + 0.68 P t h 0 I b . fig2...We then measure the effect of the **qubit** on the resonant activation peak. The principle of the experiment is sketched in figure fig4a. A first microwave pulse at the Larmor **frequency** induces a Rabi rotation by an angle θ 1 . A second microwave pulse of duration τ 2 = 10 n s is applied immediately after, at a **frequency** f 2 close to the plasma **frequency**, with a power high enough to observe resonant activation. In this experiment, we apply a constant bias current I b through the SQUID ( I b = 2.85 μ A , I b / I C = 0.85 ) and maintain it at this value 500 ~ n s after the microwave pulse to keep the SQUID in the running state for a while after switching occurs. This allows sufficient voltage to build up across the SQUID and makes detection easier, similarly to the plateau used at the end of the DCP in the previously shown method. At the end of the experimental sequence, the bias current is reduced to zero in order to retrap the SQUID in the zero-voltage state. We measured the switching probability as a function of f 2 for different Rabi angles θ 1 . The results are shown in figure fig4b. After the microwave pulse, the **qubit** is in a superposition of the states 0 and 1 with weights p 0 = c o s 2 θ 1 / 2 and p 1 = s i n 2 θ 1 / 2 . Correspondingly, the resonant activation signal is a sum of two peaks centered at f p 0 and f p 1 with weights p 0 and p 1 , which reveal the Rabi **oscillations**....Detection of a persistent-current **qubit** by resonant activation...We show the two peaks corresponding to θ 1 = 0 (curve P s w 0 , full circles) and θ 1 = π (curve P s w π , open circles) in figure fig4c. They are separated by f p 0 - f p 1 = 50 ~ M H z and have a similar width of 90 ~ M H z . This is an indication that the π pulse efficiently populates the excited state (any significant probability for the **qubit** to be in 0 would result into broadening of the curve P s w π ), and is in strong contrast with the results obtained with the DCP method (figure fig2b). The difference between the two curves S f = P s w 0 - P s w π (solid line in figure fig4c) gives a lower bound of the excited state population after a π pulse. Because of the above mentioned asymmetric shape of the resonant activation peaks, it yields larger absolute values on the low- than on the high-**frequency** side of the peak. Thus the plasma **oscillator** non-linearity increases the sensitivity of our measurement, which is reminiscent of the ideas exposed in . On the data shown here, S f attains a maximum S m a x = 60 % for a **frequency** f 2 * indicated by an arrow in figure fig4c. The value of S m a x strongly depends on the microwave pulse duration and power. The optimal settings are the result of a compromise between two constraints : a long microwave pulse provides a better resonant activation peak separation, but on the other hand the pulse should be much shorter than the **qubit** damping time T 1 , to prevent loss of excited state population. Under optimized conditions, we were able to reach S m a x = 68 % ....A typical resonant activation peak is shown in the insert of figure fig3. Its width depends on the **frequency**, ranging between 20 and 50 ~ M H z . This corresponds to a quality factor between 50 and 150 . The peak has an asymmetric shape, with a very sharp slope on its low-**frequency** side and a smooth high-**frequency** tail, due to the SQUID non-linearity. We could qualitatively recover these features by simple numerical simulations using the RCSJ model . The resonant activation peak can be unambiguously distinguished from environmental resonances by its dependence on the magnetic flux threading the SQUID loop Φ s q . Figure fig3 shows the measured peak **frequency** for different fluxes around Φ s q = 1.5 Φ 0 , together with the measured switching current (dashed line). The solid line is a numerical fit to the data using the above formulae. From this fit we deduce the following values C s h = 12 ± 2 p F and L = 170 ± 20 p H , close to the design. We are thus confident that the observed resonance is due to the plasma **frequency**....(a) AFM picture of the SQUID and **qubit** loop (the scale bar indicates 1 ~ μ m ). Two layers of Aluminium were evaporated under ± ~ 20 ~ ∘ with an oxidation step in between. The Josephson junctions are formed at the overlap areas between the two images. The SQUID is shunted by a capacitor C s h = 12 ~ p F connected by Aluminium leads of inductance L = 170 ~ p H (solid black line). The current is injected through a resistor (grey line) of 400 ~ Ω . (b) DCP measurement method : the microwave pulse induces the designed Bloch sphere rotations. It is followed by a current pulse of duration 20 ~ n s , whose amplitude I b is optimized for the best detection efficiency. A 400 ~ n s lower-current plateau follows the DCP and keeps the SQUID in the running-state to facilitate the voltage pulse detection. (c) Larmor **frequency** of the **qubit** and (insert) persistent-current versus external flux. The squares and (insert) the circles are experimental data. The solid lines are numerical adjustments giving the tunnelling matrix element Δ , the persistent-current I p and the mutual inductance M . fig1...(a) Rabi oscillations at a Larmor **frequency** f q = 7.15 ~ G H z (b) Switching probability as a function of current pulse amplitude I without (closed circles, curve P s w 0 I ) and with (open circles, curve P s w π I b ) a π pulse applied. The solid black line P t h 0 I b is a numerical adjustment to P s w 0 I b assuming escape in the thermal regime. The dotted line (curve P t h 1 I b ) is calculated with the same parameters for a critical current 100 n A smaller, which would be the case if state 1 was occupied with probability unity. The grey solid line is the sum 0.32 P t h 1 I b + 0.68 P t h 0 I b . fig2...A typical resonant activation peak is shown in the insert of figure fig3. Its width depends on the **frequency**, ranging between 20 and 50 ~ M H z . This corresponds to a quality factor between 50 and 150 . The peak has an asymmetric shape, with a very sharp slope on its low-**frequency** side and a smooth high-**frequency** tail, due to the **SQUID** non-linearity. We could qualitatively recover these features by simple numerical simulations using the RCSJ model . The resonant activation peak can be unambiguously distinguished from environmental resonances by its dependence on the magnetic flux threading the **SQUID** loop Φ s q . Figure fig3 shows the measured peak **frequency** for different fluxes around Φ s q = 1.5 Φ 0 , together with the measured switching current (dashed line). The solid line is a numerical fit to the data using the above formulae. From this fit we deduce the following values C s h = 12 ± 2 p F and L = 170 ± 20 p H , close to the design. We are thus confident that the observed resonance is due to the plasma **frequency**....The parameters of our **qubit** were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the SQUID and the **qubit** by fitting the **qubit** “step" appearing in the SQUID’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi **oscillation** experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the microwave **frequency** to the **qubit** resonance and measured the switching probability as a function of the microwave pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the **qubit**. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these **oscillations**....The parameters of our **qubit** were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the **SQUID** and the **qubit** by fitting the **qubit** “step" appearing in the **SQUID**’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi oscillation experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the **microwave** **frequency** to the **qubit** resonance and measured the switching probability as a function of the **microwave** pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the **qubit**. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these os...Finally, we fixed the **frequency** f 2 at the value f 2 * and measured Rabi **oscillations** (black curve in figure fig4d). We compared this curve to the one obtained with the DCP method in exactly the same conditions (grey curve). The contrast is significantly improved, while the dephasing time is evidently the same. This enhancement is partly explained by the rapid 5 ~ n s RAP (for the data shown in figure fig4d) compared to the 30 ~ n s DCP. But we can not exclude that the DCP intrinsically increases the relaxation rate during its risetime. Such a process would be in agreement with the fact that for these bias conditions, T 1 ≃ 100 ~ n s , three times longer than the DCP duration....The parameters of our **qubit** were determined by fitting spectroscopic measurements with the above formulae. For Δ = 5.855 ~ G H z , I p = 272 ~ n A , the agreement is excellent (see figure fig1c). We also determined the coupling constant between the **SQUID** and the **qubit** by fitting the **qubit** “step" appearing in the **SQUID**’s modulation curve (see insert of figure fig1c) and found M = 20 ~ p H . We first performed Rabi oscillation experiments with the DCP detection method (figure fig1b). We chose a bias point Φ x , tuned the **microwave** **frequency** to the **qubit** resonance and measured the switching probability as a function of the **microwave** pulse duration τ m w . The observed oscillatory behavior (figure fig2a) is a proof of the coherent dynamics of the **qubit**. A more detailed analysis of its damping time and period will be presented elsewhere ; here we focus on the amplitude of these oscillations. ... We present the implementation of a new scheme to detect the quantum state of a persistent-current **qubit**. It relies on the dependency of the measuring Superconducting Quantum Interference Device (SQUID) plasma **frequency** on the **qubit** state, which we detect by resonant activation. With a measurement pulse of only 5ns, we observed Rabi **oscillations** with high visibility (65%).

Files:

Contributors: Meier, Florian, Loss, Daniel

Date: 2004-08-26

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

Files:

Contributors: Grajcar, M., Izmalkov, A., Il'ichev, E., Wagner, Th., Oukhanski, N., Huebner, U., May, T., Zhilyaev, I., Hoenig, H. E., Greenberg, Ya. S.

Date: 2003-03-31

Our technique is similar to rf-SQUID readout. The **qubit** loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant **frequency** ω T . Then both amplitude v and phase shift χ (with respect to the bias current I b ) of the tank voltage will strongly depend on (A) the shift in resonant **frequency** due to the change of the effective **qubit** inductance by the tank flux, and (B) losses caused by field-induced transitions between the two **qubit** states. Thus, the tank both applies the probing field to the **qubit**, and detects its response....In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of **the** **qubit**’s quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, **the** effective **qubit** temperature has been verified [Fig. fig:Temp_dep(b)] to be **the** same as **the** mixing chamber’s (after Δ has been determined at low T ), which is often difficult to confirm independently....(a) Quantum energy levels of **the** **qubit** vs external flux. The dashed lines represent **the** classical potential minima. (**b**) Phase **qubit** coupled to a tank circuit....Low-**frequency** measurement of the tunneling amplitude in a flux **qubit**...(a) Quantum energy levels of the **qubit** vs external flux. The dashed lines represent the classical potential minima. (b) Phase **qubit** coupled to a tank circuit....-dependence of ϵ t is adiabatic. However, it does remain valid if the full (Liouville) evolution operator of the **qubit** would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to the Hamiltonian dynamics ( eq01), since the fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from the low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...-dependence of ϵ t is adiabatic. However, it does remain valid if **the** full (Liouville) evolution operator of **the** **qubit** would contain standard Bloch-type relaxation and dephasing terms (which indeed are not probed) in addition to **the** Hamiltonian dynamics ( eq01), since **the** fluctuation–dissipation theorem guarantees that such terms do not affect equilibrium properties. Normalized dip amplitudes are shown vs T in Fig. fig:Temp_dep(b) together with tanh Δ / k B T , for Δ / h = 650 MHz independently obtained above from **the** low- T width. The good agreement strongly supports our interpretation, and is consistent with Δ being T...In conclusion, we have observed resonant tunneling in a macroscopic superconducting system, containing an Al flux **qubit** and a Nb tank circuit. The latter played dual control and readout roles. The impedance readout technique allows direct characterization of some of the ** qubit’s** quantum properties, without using spectroscopy. In a range 50 ∼ 200 mK, the effective

**qubit**temperature has been verified [Fig. fig:Temp_dep(b)] to be the same as the mixing chamber’s (after Δ has been determined at low T ), which is often difficult to confirm independently....Our technique is similar to rf-SQUID readout. The

**qubit**loop is inductively coupled to a parallel resonant tank circuit [Fig. fig:schem(b)]. The tank is fed a monochromatic rf signal at its resonant frequency ω T . Then both amplitude v and phase shift χ (with respect to

**the**bias current I b ) of

**the**tank voltage will strongly depend

**on**(A)

**the**shift in resonant frequency due to

**the**change of

**the**effective

**qubit**inductance by

**the**tank flux, and (B) losses caused by field-induced transitions between

**the**two

**qubit**states. Thus,

**the**tank both applies

**the**probing field to

**the**

**qubit**, and detects its response....We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity....Δ is

**the**tunneling amplitude. At bias ϵ = 0

**the**

**two**lowest energy levels of

**the**

**qubit**anticross [Fig. fig:schem(a)], with a gap of 2 Δ . Increasing ϵ slowly enough,

**the**

**qubit**can adiabatically transform from Ψ l to Ψ r , staying in

**the**ground state E - . Since d E - / d Φ x is

**the**persistent loop current,

**the**curvature d 2 E - / d Φ x 2 is related to

**the**

**qubit**’s susceptibility. Hence, near degeneracy

**the**latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al

**3JJ**

**qubit**....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From

**the**lower to

**the**upper curve (at f x = 0 )

**the**temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for

**the**Δ following from Fig. fig:Bias_dep;

**the**overall scale κ is a fitting parameter. The data indicate a saturation of

**the**effective

**qubit**temperature at 30 mK. (c) Full dip width at half

**the**maximum amplitude vs temperature. The horizontal line fits

**the**low- T ( < 200 mK) part to a constant;

**the**sloped line represents

**the**T 3 behavior observed empirically for higher T ....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From the lower to the upper curve (at f x = 0 ) the temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (b) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for the Δ following from Fig. fig:Bias_dep; the overall scale κ is a fitting parameter. The data indicate a saturation of the effective

**qubit**temperature at 30 mK. (c) Full dip width at half the maximum amplitude vs temperature. The horizontal line fits the low- T ( < 200 mK) part to a constant; the sloped line represents the T 3 behavior observed empirically for higher T ....(a) Tank phase shift vs flux bias near degeneracy and for V d r = 0.5 ~ μ V. From

**the**lower to

**the**upper curve (at f x = 0 )

**the**temperature is 10, 20, 30, 50, 75, 100, 125, 150, 200, 250, 300, 350, 400 mK. (

**b**) Normalized amplitude of tan χ (circles) and tanh Δ / k B T (line), for

**the**Δ following from Fig. fig:Bias_dep;

**the**overall scale κ is a fitting parameter. The data indicate a saturation of

**the**effective

**qubit**temperature at 30 mK. (c) Full dip width at half

**the**maximum amplitude vs temperature. The horizontal line fits

**the**low- T ( < 200 mK) part to a constant;

**the**sloped line represents

**the**T 3 behavior observed empirically for higher T ....Δ is the tunneling amplitude. At bias ϵ = 0 the two lowest energy levels of the

**qubit**anticross [Fig. fig:schem(a)], with a gap of 2 Δ . Increasing ϵ slowly enough, the

**qubit**can adiabatically transform from Ψ l to Ψ r , staying in the ground state E - . Since d E - / d Φ x is the persistent loop current, the curvature d 2 E - / d Φ x 2 is related to the

**susceptibility. Hence, near degeneracy the latter will have a peak, with a width given by | ϵ | Δ . We present data demonstrating such behavior in an Al 3JJ**

**qubit**’s**qubit**. ... We have observed signatures of resonant tunneling in an Al three-junction

**qubit**, inductively coupled to a Nb LC tank circuit. The resonant properties of the tank

**oscillator**are sensitive to the effective susceptibility (or inductance) of the

**qubit**, which changes drastically as its flux states pass through degeneracy. The tunneling amplitude is estimated from the data. We find good agreement with the theoretical predictions in the regime of their validity.

Files:

Contributors: Makhlin, Yuriy, Shnirman, Alexander

Date: 2003-12-22

FIG. F:slow. a. Double vertices with low- ω tails, which appear in the evaluation of dephasing. b. Examples of clusters built out of them . c. A low- ω object with a high-frequency dashed line. The relaxation process in e also contributes to dephasing as shown in d....So far we constructed slow composite objects paying attention only to the **oscillations** of the solid lines in the diagrams and assuming very slow dashed lines, i.e., neglected the higher-**frequency** noise. In fact, one can construct another slow object shown in Fig. F:slowc, if the respective **oscillations** of the solid lines are compensated by the dashed line from this vertex. In other words, in the **frequency** domain, one constrains the **frequency** of the dashed line to be Δ E (or - Δ E , depepnding on the direction of the spin flip at the vertex). The dashed lines from such objects pair up, and the integral w.r.t. their relative position is dominated by small separations, δ t ∼ 1 / Δ E . Thus one finds the slow object of Fig. F:slowd, two vertices linked by a dashed line at **frequency** Δ E ; it describes the relaxational contribution to dephasing exp - t / 2 T 1 , where...We analyze the dissipative dynamics of a two-level quantum system subject to low-**frequency**, e.g. 1/f noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-**frequency** noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent oscillations under the influence of both low- and high-**frequency** fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing....So far we constructed slow composite objects paying attention only to the oscillations of the solid lines in the diagrams and assuming very slow dashed lines, i.e., neglected the higher-frequency noise. In fact, one can construct another slow object shown in Fig. F:slowc, if the respective oscillations of the solid lines are compensated by the dashed line from this vertex. In other words, in the frequency domain, one constrains the frequency of the dashed line to be Δ E (or - Δ E , depepnding on the direction of the spin flip at the vertex). The dashed lines from such objects pair up, and the integral w.r.t. their relative position is dominated by small separations, δ t ∼ 1 / Δ E . Thus one finds the slow object of Fig. F:slowd, two vertices linked by a dashed line at frequency Δ E ; it describes the relaxational contribution to dephasing exp - t / 2 T 1 , where...FIG. F:slow. a. Double vertices with low- ω tails, which appear in the evaluation of dephasing. b. Examples of clusters built out of them . c. A low- ω object with a **high**-**frequency** dashed line. The relaxation process in e also contributes to dephasing as shown in d....In the diagrams the horizontal direction explicitly represents the time axis. The solid lines describe the unperturbed (here, coherent) evolution of the ** qubit’s** 2 × 2 density matrix ρ ̂ , exp - i L 0 t θ t , where L 0 is the bare Liouville operator (this translates to 1 / ω - i L 0 in the

**frequency**domain). The vertices are explicitly time-ordered; each of them contributes the term ζ Y σ x τ z / 2 , with the bath operator Y t and the Keldysh matrix τ z = ± 1 for vertices on the upper/lower time branch. Averaging over the fluctuations should be performed; for gaussian correlations it pairs the vertices as indicated by dashed lines in Fig. F:2order, each of the lines corresponding to a correlator Y Y . Fig. F:2order shows contributions to the second-order self-energy Σ ↑ ↓ ← ↑ ↓ 2 (here i j = ↑ ↓ label four entries of the

**density matrix). The term in Fig. F:2ordera gives...F:qbFIG. F:qb. The simplest Josephson charge qubit...Dissipative dynamics of a Josephson charge qubit. The simplest Josephson charge qubit is the Cooper-pair box shown in Fig. F:qb . It consists of a superconducting island connected by a dc-SQUID (effectively, a Josephson junction with the coupling E J Φ x = 2 E J 0 cos π Φ x / Φ 0 tunable via the magnetic flux Φ x ; here Φ 0 = h c / 2 e ) to a superconducting lead and biased by a gate voltage V g via a gate capacitor C g . The Josephson energy of the junctions in the SQUID loop is E J 0 , and their capacitance C J 0 sets the charging-energy scale E C ≡ e 2 / 2 C g + C J , C J = 2 C J 0 . At low enough temperatures single-electron tunneling is suppressed and only even-parity states are involved. Here we consider low-capacitance junctions with high charging energy E C ≫ E J 0 . Then the number n of Cooper pairs on the island (relative to a neutral state) is a good quantum number; at certain values of the bias V g ≈ V d e g = 2 n + 1 e / C two lowest charge states n and n + 1 are near-degenerate, and even a weak E J mixes them strongly. At low temperatures and operation frequencies higher charge states do not play a role. The Hamiltonian reduces to a two-state model,...Dephasing of**

**qubit**’s**qubits**by transverse low-

**frequency**noise...In the diagrams the horizontal direction explicitly represents the time axis. The solid lines describe the unperturbed (here, coherent) evolution of the qubit’s 2 × 2 density matrix ρ ̂ , exp - i L 0 t θ t , where L 0 is the bare Liouville operator (this translates to 1 / ω - i L 0 in the frequency domain). The vertices are explicitly time-ordered; each of them contributes the term ζ Y σ x τ z / 2 , with the bath operator Y t and the Keldysh matrix τ z = ± 1 for vertices on the upper/lower time branch. Averaging over the fluctuations should be performed; for gaussian correlations it pairs the vertices as indicated by dashed lines in Fig. F:2order, each of the lines corresponding to a correlator Y Y . Fig. F:2order shows contributions to the second-order self-energy Σ ↑ ↓ ← ↑ ↓ 2 (here i j = ↑ ↓ label four entries of the qubit’s density matrix). The term in Fig. F:2ordera gives...Dissipative dynamics of a Josephson charge

**qubit**. The simplest Josephson charge

**qubit**is the Cooper-pair box shown in Fig. F:qb . It consists of a superconducting island connected by a dc-SQUID (effectively, a Josephson junction with the coupling E J Φ x = 2 E J 0 cos π Φ x / Φ 0 tunable via the magnetic flux Φ x ; here Φ 0 = h c / 2 e ) to a superconducting lead and biased by a gate voltage V g via a gate capacitor C g . The Josephson energy of the junctions in the SQUID loop is E J 0 , and their capacitance C J 0 sets the charging-energy scale E C ≡ e 2 / 2 C g + C J , C J = 2 C J 0 . At low enough temperatures single-electron tunneling is suppressed and only even-parity states are involved. Here we consider low-capacitance junctions with high charging energy E C ≫ E J 0 . Then the number n of Cooper pairs on the island (relative to a neutral state) is a good quantum number; at certain values of the bias V g ≈ V d e g = 2 n + 1 e / C two lowest charge states n and n + 1 are near-degenerate, and even a weak E J mixes them strongly. At low temperatures and operation

**frequencies**higher charge states do not play a role. The Hamiltonian reduces to a two-state model,...FIG. F:slow. a. Double vertices with low- ω tails, which appear in the evaluation of dephasing. b. Examples of clusters built out of them . c. A low- ω object with a high-

**frequency**dashed line. The relaxation process in e also contributes to dephasing as shown in d....We analyze the dissipative dynamics of a two-level quantum system subject to low-

**frequency**, e.g. 1/f noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-

**frequency**noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent

**oscillations**under the influence of both low- and high-

**frequency**fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing....F:qbFIG. F:qb. The simplest Josephson charge

**qubit**... We analyze the dissipative dynamics of a two-level quantum system subject to low-

**frequency**, e.g. 1/f noise, motivated by recent experiments with superconducting quantum circuits. We show that the effect of transverse linear coupling of the system to low-

**frequency**noise is equivalent to that of quadratic longitudinal coupling. We further find the decay law of quantum coherent

**oscillations**under the influence of both low- and high-

**frequency**fluctuations, in particular, for the case of comparable rates of relaxation and pure dephasing.

Files:

Contributors: Singh, Mandip

Date: 2014-07-01

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

Files:

Contributors: Rosenband, Till

Date: 2012-03-01

(color) Numerically optimized free-evolution period T for some of clock protocols considered here, when the **oscillator** noise has an Allan deviation of 1 Hz....(color) Numerically optimized free-evolution period T for some of clock protocols considered here, when the oscillator noise has an Allan deviation of 1 Hz....The stability of several clock protocols based on 2 to 20 entangled atoms is evaluated numerically by a simulation that includes the effect of decoherence due to classical **oscillator** noise. In this context the squeezed states discussed by Andr\'{e}, S{\o}rensen and Lukin [PRL 92, 239801 (2004)] offer reduced instability compared to clocks based on Ramsey's protocol with unentangled atoms. When more than 15 atoms are simulated, the protocol of Bu\v{z}ek, Derka and Massar [PRL 82, 2207 (1999)] has lower instability. A large-scale numerical search for optimal clock protocols with two to eight **qubits** yields improved clock stability compared to Ramsey spectroscopy, and for two to three **qubits** performance matches the analytical protocols. In the simulations, a laser local **oscillator** decoheres due to flicker-**frequency** (1/f) noise. The **oscillator** **frequency** is repeatedly corrected, based on projective measurements of the **qubits**, which are assumed not to decohere with one another....Numerical simulations of the clock protocols considered here are summarized in Figure figPerf. Ramsey’s protocol defines the standard quantum limit (SQL), and it is evident that entangled states of two or more qubits can reduce clock instability, although the GHZ states yield no gain for the noise model considered here, as has been noted previously . The spin-squeezed states suggested by André et al. yield the best performance for 3 to 15 qubits, and improve upon the SQL variance by a factor of N -1 / 3 . For more qubits, the protocol of Bu...(color) Probability (P) of measuring each basis state as a function of the atom-**oscillator** phase difference ( φ ). Shown are the various protocols for two and five atoms. Each differently colored curve corresponds to a basis state that ψ 1 is projected onto after free evolution. Vertical text near the curves’ peaks indicates the optimized phase estimate ( φ E s t ). In the simulations, the **frequency** corrections are φ E s t / 2 π T . Shaded in the background is the Gaussian distribution whose variance φ 2 represents the atom-**oscillator** phase differences that occur in the simulation. Also listed is the optimized probe period T , squeezing parameter κ where applicable, and long-term **frequency** variance of the clock extrapolated to 1 second. For long-term averages of n seconds, the variance is f 2 / n ....Numerical simulations of the clock protocols considered here are summarized in Figure figPerf. Ramsey’s protocol defines the standard quantum limit (SQL), and it is evident that entangled states of two or more **qubits** can reduce clock instability, although the GHZ states yield no gain for the noise model considered here, as has been noted previously . The spin-squeezed states suggested by André et al. yield the best performance for 3 to 15 **qubits**, and improve upon the SQL variance by a factor of N -1 / 3 . For more **qubits**, the protocol of Bu...Numerical test of few-**qubit** clock protocols...(color online) Long-term statistical variance of entangled clocks that contain different numbers of **qubits**, compared to the standard quantum limit (SQL). The most stable clocks found by the large-scale search are shown as black points. Each point is based on several hours of runtime on NISTxs computing cluster, where typically 2000 processor cores were utilized in parallel. Also shown is the simulated performance of analytically optimized clock protocols. Approximately 15 **qubits** are required to improve upon the SQL by a factor of two....(color) Probability (P) of measuring each basis state as a function of the atom-oscillator phase difference ( φ ). Shown are the various protocols for two and five atoms. Each differently colored curve corresponds to a basis state that ψ 1 is projected onto after free evolution. Vertical text near the curves’ peaks indicates the optimized phase estimate ( φ E s t ). In the simulations, the frequency corrections are φ E s t / 2 π T . Shaded in the background is the Gaussian distribution whose variance φ 2 represents the atom-oscillator phase differences that occur in the simulation. Also listed is the optimized probe period T , squeezing parameter κ where applicable, and long-term frequency variance of the clock extrapolated to 1 second. For long-term averages of n seconds, the variance is f 2 / n . ... The stability of several clock protocols based on 2 to 20 entangled atoms is evaluated numerically by a simulation that includes the effect of decoherence due to classical **oscillator** noise. In this context the squeezed states discussed by Andr\'{e}, S{\o}rensen and Lukin [PRL 92, 239801 (2004)] offer reduced instability compared to clocks based on Ramsey's protocol with unentangled atoms. When more than 15 atoms are simulated, the protocol of Bu\v{z}ek, Derka and Massar [PRL 82, 2207 (1999)] has lower instability. A large-scale numerical search for optimal clock protocols with two to eight **qubits** yields improved clock stability compared to Ramsey spectroscopy, and for two to three **qubits** performance matches the analytical protocols. In the simulations, a laser local **oscillator** decoheres due to flicker-**frequency** (1/f) noise. The **oscillator** **frequency** is repeatedly corrected, based on projective measurements of the **qubits**, which are assumed not to decohere with one another.

Files:

Contributors: Xia, K., Macovei, M., Evers, J., Keitel, C. H.

Date: 2008-10-14

Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system....In Fig. subfig:figg2sP, we show results for the population of | s for continuous driving. It can be seen that the population reaches a maximum value, but afterwards exhibits rapid **oscillations** at **frequency** 2 2 Ω 0 , while the amplitude of the subsequent maxima in the population decays as an exponential function exp - γ 0 + γ 12 t until the system approaches its stationary state. This result can be understood by reducing the system to a two-state system only involving in the states | s and | g . The numerical results can be well fitted by the solution of this two-state approximation,...(Color online) Time evolution of the population (solid red line) in the antisymmetric state | a . The idea is to prepare the anti-symmetric state while the two **qubits** are non-degenerate, and only afterwards render the two **qubits** degenerate. For this, the parameters are chosen such that γ 12 = 0.9986 γ 0 , λ = 50 γ 0 , δ = 50 γ 0 , Ω 0 = 50 γ 0 . The two **qubit** transition **frequencies** are adjusted via time-dependent bias fluxes, such that the **frequency** difference Δ t (dashed black line) changes from 18 γ 0 to zero as a cosine function during the time period 120 γ 0 -1 to 160 γ 0 -1 . The driving field (dash-dotted blue line) is turned off from its initial value Ω 0 in the period 165 γ 0 -1 to 175 γ 0 -1 ....Robust creation of entangled states of two coupled flux **qubits** via dynamic control of the transition **frequencies**...fig:systemTwo superconducting** flux **qubits interacting with each other through their mutual inductance M and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the qubit transition frequencies around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath....fig:BellSCRAPdtn(Color online) Creation of superpositions of Bell states controlled by the static detuning δ 0 . The populations ρ ± in states | Φ + (dashed red line) and | Φ - (dash-dotted green line) exhibit periodic **oscillations** as a function of δ 0 . The maximum concurrence C is larger than 0.95 (solid blue line)....fig:systemTwo superconducting flux **qubits** interacting with each other through their mutual inductance M and damped to a common reservoir modeled as an LC circuit. The individual bias fluxes are varied dynamically in order to control the **qubit** transition **frequencies** around the optimum point. In the figure, crosses indicate Josephson junctions, whereas the bottom circuit loop visualizes the bath....where Φ 0 = h / 2 e is the flux quantum, the system in Fig. fig:system can be described by the total Hamiltonian H = H Q + H B . The two-**qubit** Hamiltonian H Q in two-level approximation and rotating wave approximation is given by ...Our model system consists of two flux qubits coupled to each other through their mutual inductance M and to a reservoir of harmonic oscillators modeled as an LC circuit, see Fig. fig:system....(Color online) Time evolution of the population (solid red line) in the antisymmetric state | a . The idea is to prepare the anti-symmetric state while the two qubits are non-degenerate, and only afterwards render the two qubits degenerate. For this, the parameters are chosen such that γ 12 = 0.9986 γ 0 , λ = 50 γ 0 , δ = 50 γ 0 , Ω 0 = 50 γ 0 . The two qubit transition frequencies are adjusted via time-dependent bias fluxes, such that the frequency difference Δ t (dashed black line) changes from 18 γ 0 to zero as a cosine function during the time period 120 γ 0 -1 to 160 γ 0 -1 . The driving field (dash-dotted blue line) is turned off from its initial value Ω 0 in the period 165 γ 0 -1 to 175 γ 0 -1 ....In Fig. subfig:figg2sP, we show results for the population of | s for continuous driving. It can be seen that the population reaches a maximum value, but afterwards exhibits rapid oscillations **at **frequency 2 2 Ω 0 , while the amplitude of the subsequent maxima **in **the population decays as an exponential function exp - γ 0 + γ 12 t until the system approaches its stationary state. This result can be understood by reducing the system to a two-state system only involving **in **the states | s and | g . The numerical results can be well fitted by the solution of this two-state approximation,...where Φ 0 = h / 2 e is the flux quantum, the system **in **Fig. fig:system can be described by the total Hamiltonian H = H Q + H B . The two-qubit Hamiltonian H Q **in **two-level approximation and rotating wave approximation is given by ...To study the fidelity of the population transfer between | a and | s , in Fig. subfig:a2s15P, we choose the anti-symmetric state as the initial condition. Applying a continuous-wave TDMF with Rabi **frequency** Ω 0 = 15 γ 0 and detuning δ = 0 , the symmetric state | s reaches its maximum population of 0.90 at time 0.1 γ 0 -1 . After this maximum, the population continues to **oscillate** between | a and | s due to the applied field. This **oscillation** is damped by an overall decay as we include damping with a rate γ 0 . The corresponding concurrence is shown in in Fig. subfig:a2s15C. The concurrence **oscillates** at twice the **frequency** of the population **oscillation**, since both | s and | a are maximally entangled. The local maximum values of the entanglement occur at times n π / 2 δ 2 + Ω 0 2 where either | s or | a is occupied....Our model system consists of two flux **qubits** coupled to each other through their mutual inductance M and to a reservoir of harmonic **oscillators** modeled as an LC circuit, see Fig. fig:system....An example for this is shown in Fig. fig:figPadtn. Initially, the two **qubits** have a **frequency** difference Δ t = 0 = Δ 0 = 18 γ 0 . Applying a continuous TDMF Ω during 0 ≤ γ 0 t ≤ 165 allows to populate the antisymmetric state, as can be seen in Fig. fig:figPadtn. After a certain time ( γ 0 t = 120 in our example), the bias fluxes are continuously adjusted such that the two **qubits** become degenerate, Δ γ 0 t ≥ 160 = 0 . It can be seen from Fig. fig:figPadtn that a preparation fidelity for the antisymmetric state in the degenerate two-**qubit** system of about F = 0.94 is achieved. Finally, the TDMF is switched off as well in the time period 165 ≤ γ 0 t ≤ 175 , demonstrating that it is not required to preserve the population in the antisymmetric state. It should be noted that this scheme does not rely on a delicate choice and control of parameters, as it is the case, e.g., for state preparation via special-area pulses....To study the fidelity of the population transfer between | a and | s , **in **Fig. subfig:a2s15P, we choose the anti-symmetric state as the initial condition. Applying a continuous-wave TDMF with Rabi frequency Ω 0 = 15 γ 0 and detuning δ = 0 , the symmetric state | s reaches its maximum population of 0.90 **at **time 0.1 γ 0 -1 . After this maximum, the population continues to oscillate between | a and | s due to the applied field. This oscillation is damped by an overall decay as we include damping with a rate γ 0 . The corresponding concurrence is shown **in ****in **Fig. subfig:a2s15C. The concurrence oscillates **at **twice the frequency of the population oscillation, since both | s and | a are maximally entangled. The local maximum values of the entanglement occur **at **times n π / 2 δ 2 + Ω 0 2 where either | s or | a is occupied....An example for this is shown **in **Fig. fig:figPadtn. Initially, the two qubits have a frequency difference Δ t = 0 = Δ 0 = 18 γ 0 . Applying a continuous TDMF Ω during 0 ≤ γ 0 t ≤ 165 allows to populate the antisymmetric state, as can be seen **in **Fig. fig:figPadtn. After a certain time ( γ 0 t = 120 **in **our example), the bias fluxes are continuously adjusted such that the two qubits become degenerate, Δ γ 0 t ≥ 160 = 0 . It can be seen from Fig. fig:figPadtn that a preparation fidelity for the antisymmetric state **in **the degenerate two-qubit system of about F = 0.94 is achieved. Finally, the TDMF is switched off as well **in **the time period 165 ≤ γ 0 t ≤ 175 , demonstrating that it is not required to preserve the population **in **the antisymmetric state. It should be noted that this scheme does not rely on a delicate choice and control of parameters, as it is the case, e.g., for state preparation via special-area pulses....fig:BellSCRAP(Color online) Creation of Bell states | Φ ± using the SCRAP technique from the ground state. The solid red line denotes population ρ + in | Φ + , while the dashed green line shows population ρ - in | Φ - . The concurrence (dash-dotted blue line) has a maximum value of 0.94 . The black dash double dotted line indicates the applied SCRAP pulse Rabi frequency, while the thick blue line shows the time-dependent Stark detuning....fig:BellSCRAP(Color online) Creation of Bell states | Φ ± using the SCRAP technique from the ground state. The solid red line denotes population ρ + in | Φ + , while the dashed green line shows population ρ - in | Φ - . The concurrence (dash-dotted blue line) has a maximum value of 0.94 . The black dash double dotted line indicates the applied SCRAP pulse Rabi **frequency**, while the thick blue line shows the time-dependent Stark detuning....fig:SCRAPg2s(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for γ 12 = 0.9986 γ 0 , and λ = 50 γ 0 . The solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi **frequency** and detuning required for SCRAP....fig:SCRAPg2s(Color online) Robust populating the symmetric state from the ground state via the SCRAP technique for γ 12 = 0.9986 γ 0 , and λ = 50 γ 0 . The solid black line shows the population of the desired symmetric state, while the thick solid blue line and the dash-dotted red line are the time-dependent Rabi frequency and detuning required for SCRAP. ... Coherent control and the creation of entangled states are discussed in a system of two superconducting flux **qubits** interacting with each other through their mutual inductance and identically coupling to a reservoir of harmonic **oscillators**. We present different schemes using continuous-wave control fields or Stark-chirped rapid adiabatic passages, both of which rely on a dynamic control of the **qubit** transition **frequencies** via the external bias flux in order to maximize the fidelity of the target states. For comparison, also special area pulse schemes are discussed. The **qubits** are operated around the optimum point, and decoherence is modelled via a bath of harmonic **oscillators**. As our main result, we achieve controlled robust creation of different Bell states consisting of the collective ground and excited state of the two-**qubit** system.

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