### 54077 results for qubit oscillator frequency

Contributors: Thrailkill, Z. E., Lambert, J. G., Ramos, R. C.

Date: 2009-09-20

The second case where ω R 1 > ω R 2 > ω R 3 is shown in Fig. fig3. This is very similar to the first case, however, there is a slight overall increase in the state transfer rate. This is because the detuning is still relative to R2, but the detuning of R3 is now slightly less. As the **qubits** approach resonance with the resonators from below, the dispersive coupling strength becomes slightly larger because the **frequency** of R3 is a little closer to the **frequency** of the **qubits**. The small differences in resonator **frequencies** also cause slight non-uniformities in the high **frequency** **oscillations** or ripples that become more pronounced at lower detuning. This interference is not a factor if the **qubits** are sufficiently detuned. Regardless of these artifacts, it is clear that an array of resonators that is used for memory storage can also be used to dispersively couple **qubits**....In both cases, at large detuning of -2000 MHz, the excitation appears to smoothly **oscillate** into **qubit** 2, as shown in Fig. fig2 and fig3. As the detuning decreases, two things happen: First, the time it takes for the excitation to move into **qubit** 2 gets shorter, indicating an increase in the effective coupling strength between the two **qubits**. Second, small ripples, or **oscillations**, begin to appear. These **oscillations** are due to the detuning becoming small enough that the dispersive limit approximation starts to break down. This result shows us that using three identical resonators in parallel to dispersively couple **qubits** is effectively similar to using only one resonator with three times the coupling strength. In general, an array of n resonators can be replaced by a single resonator with n times the coupling strength. The dispersive coupling can be increased by reducing the detuning, but at the cost of having significant **oscillations** between the **qubits** and resonators....figure2(a) Two **qubits** (Q1, Q2) dispersively coupled to an array of three resonators (R1, R2, R3) via identical coupling capacitors C c . (b),(c) For these two simulations, the system is initialized with a single excitation in Q1, and the two **qubits** maintain equal **frequencies** as they are simultaneously detuned from the resonators. Each vertical cut represents the population in R2 over time at a particular detuning Δ Q , R 2 , of the **qubits** from R2. In (b), the **frequencies** of all three resonators are equal ( ω R 1 = ω R 2 = ω R 3 ). At large detuning ( Δ Q , R 2 = -2000 MHz), the excitation smoothly **oscillates** between the two **qubits** without significant interference from the resonators. As the magnitude of the detuning decreases, the effective coupling between the two **qubits** strengthens, thus the **oscillation** of the excitation becomes more frequent. Also, the direct coupling of the **qubits** to the resonators strengthens, causing the small ripples. In (c), the **frequencies** of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). The excitation **oscillates** slightly faster than in (b) because the small offsets in **frequency** of R1 and R3 increases the coupling bandwidth, resulting in a small increase in coupling between the **qubits** over the same range of detuning. The offset of R1 and R3 from R2 also causes the ripples to be non-uniform at smaller detuning....We consider an array of resonators used as a memory register. To accomplish this, one must be able to transfer an excitation from a **qubit** to a specific resonator, without coupling to the other resonators in the array. This is implemented by designing resonators that are sufficiently detuned from each other. To determine the amount of detuning required to avoid crosstalk between resonators, we examine a single **qubit** coupled to an array of two resonators as shown in Fig. block1. The **qubit** (Q) and resonator 1 (R1) are fixed at the same **frequency** while resonator 2 (R2) is detuned. All **qubit**-resonator couplings are presumed to be identical with a strength of g i j = 110 M H z for all i , j , i.e., all coupling capacitances C c are equal. This value is typical in experiments such as in Ref. . We begin with Q in the excited state and let the system evolve over time while recording the population in R1, as we increase the detuning Δ R 2 , R 1 , of R2. The result of this simulation is shown in Fig. fig1....The result of the simulation is shown in Fig. fig4 where the population in R2 is plotted versus the detuning Δ Q 2 , Q 1 , of Q2 from Q1, in the range of -2000 MHz to 2000 MHz. As per design, at zero detuning, the excitation is transferred after a time of 16 ns. This time can be chosen based on the desired time scale by adjusting the **qubit**-**qubit** and **qubit**-resonator coupling strengths. At large detuning, i.e., beyond ± 1.5 GHz, the coupling between Q1 and R2 becomes dispersive up to about 30 ns, showing that dispersive coupling can be weak enough to isolate the active **qubit**. Q2 can be detuned further to reduce the dispersive coupling if the desired time scale is longer, e.g., to perform operations on Q1. Thus, this simulation shows that a control **qubit** can be effectively used to turn coupling on and off between a **qubit** and an array of resonators....figure3 (a) Placing a control **qubit** (Q2) between the active **qubit** (Q1) and the resonator array (R1, R2, R3) allows coupling to be turned on and off. (b) The system is initialized with an excitation in Q1, Q1 is in-resonance with R2, and the **frequencies** of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). Each vertical cut represents the population in R2 over time at a particular detuning Δ Q 2 , Q 1 , of Q2 from Q1. At zero detuning, the excitation readily **oscillates** in and out of the resonator R2. As Q2 is detuned further away, the coupling between Q1 and R2 becomes weaker, resulting in slower **oscillations** of the excitation. The detuning of the control **qubit** Q2 can be chosen based on the desired time scale, e.g. the time required to manipulate Q1....figure1(a) Schematic of a **qubit** (Q) coupled to an array of two resonators (R1, R2) via identical coupling capacitors C c . The **qubit** is characterized as having capacitance C J and critical current I c , with bias current I b . The resonators are characterized by inductance L i and capacitance C i , for i = 1 , 2 . (b) For this simulation, the system is initialized with a single excitation in Q, and Q is in-resonance with R1. Each vertical cut represents the population in R1 over time at a particular detuning Δ R 2 , R 1 , of R2 from R1. The **qubit**-resonator coupling strength is 110 MHz. At zero detuning, the excitation **oscillates** between the **qubit** and the two resonators; since the two resonators are identical here, each resonator is only half populated. As R2 is detuned, the excitation **oscillates** between Q and R1 with minimal population in R2. After more **oscillations**, R2 will accumulate some population, even at large detuning, which causes the appearance of ripples....Josephson junction-based **qubits** have been shown to be promising components for a future quantum computer. A network of these superconducting **qubits** will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple **qubits**. We explore the use of multiple resonators with different resonant **frequencies** to couple **qubits**. We find that an array of resonators with different **frequencies** can be individually addressed to store and retrieve information, while coupling **qubits** dispersively. We show that a control **qubit** can be used to effectively isolate an active **qubit** from an array of resonators so that it can function within the same **frequency** range used by the resonators....Next, we investigate the behavior of a system consisting of two **qubits** dispersively coupled to an array of three resonators, as shown in Fig. block2. We consider two cases: (1) All three resonators are designed with the same resonant **frequency**, and (2) the resonant **frequencies** of the three resonators are slightly detuned so that ω R 1 > ω R 2 > ω R 3 . We demonstrate how information in the form of an excitation is transferred dispersively from one **qubit** to another through an array of resonators. In both cases, the system is initialized with a single excitation in Q1, and both **qubits** are held in resonance with each other as the magnitude of their detuning from R2 is decreased from, say, -2000 MHz to -400 MHz. ... Josephson junction-based **qubits** have been shown to be promising components for a future quantum computer. A network of these superconducting **qubits** will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple **qubits**. We explore the use of multiple resonators with different resonant **frequencies** to couple **qubits**. We find that an array of resonators with different **frequencies** can be individually addressed to store and retrieve information, while coupling **qubits** dispersively. We show that a control **qubit** can be used to effectively isolate an active **qubit** from an array of resonators so that it can function within the same **frequency** range used by the resonators.

Files:

Contributors: Cooper, K. B., Steffen, Matthias, McDermott, R., Simmonds, R. W., Oh, Seongshik, Hite, D. A., Pappas, D. P., Martinis, John M.

Date: 2004-05-31

(a) Detail of the **qubit** spectroscopy near Δ U / ℏ ω p = 3.55 , showing splittings of strengths S ≈ 44 MHz and 24 MHz. (b) Tunneling probability versus measurement delay time τ D after application of π -pulse. Solid (dashed) line is taken at a well depth of solid (dashed) arrow in (a), corresponding to a resonant (off-resonant) bias. Inset illustrates how the **qubit** probability amplitude first moves to state | 1 g and then **oscillates** between | 1 g and | 0 e . (c) and (d) Tunneling probability (gray scale) versus well depth and τ D for experimental data (c) and numerical simulation (d). The peak **oscillation** periods are observed to correspond to the spectroscopic splittings....Spectroscopy of ω 10 obtained using the current-pulse measurement method, as a function of well depth Δ U / ℏ ω p . For each value of Δ U / ℏ ω p , the grayscale intensity is the normalized tunneling probability, with an original peak height of 0.1 - 0.3 . Insets: A given splitting in the spectroscopy of magnitude S comes from a critical-current fluctuator coupled to the **qubit** with strength h S / 2 . On resonance, the **qubit**-fluctuator eigenstates are linear combinations of the states | 1 g and | 0 e , where | g and | e are the two states of the fluctuator....(a) Room temperature measurement of the fast current pulse. (b) Tunneling probability versus δ I m a x with the **qubit** in state | 0 (solid circles) and in an equal mixture of states | 1 and | 0 (open circles). Fit to data is shown by the solid line. The plateau, being less than 0.5, corresponds to a maximum measurement fidelity of 0.63....(a) Schematic of the **qubit** circuitry. For the **qubit** used in Fig. 2, the Josephson critical-current and junction capacitance are I 0 ≈ 10 μ A and C ≈ 2 pF; in Figs. 3 and 4, each of these values is about 5 times smaller. (b) Potential energy landscape and quantized energy levels for I φ = I d c prior to the state measurement. (c) At the peak of δ I t , the **qubit** well is much shallower and state | 1 rapidly tunnels to the right hand well....We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain. ... We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain.

Files:

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

Date: 2005-07-13

Contrary to the shift produced by the linear coupling term, the sign of this **frequency** shift now depends on ϵ . Since g 2 is negative (see figure fig:couplings), δ ν 0 2 actually has the same sign as ϵ . We also note that the quadratic term has no effect on the **qubit** when ϵ = 0 , since at that point the average flux generated by both **qubit** states | 0 and | 1 averages out to zero so that the SQUID Josephson inductance is unchanged....We will now discuss quantitatively the behaviour of g 1 and g 2 for actual sample parameters : I C = 3.4 μ A , M = 6.5 p H , I p = 240 n A , Δ = 5.5 G H z , ν p = 3.1 G H z , L J = 300 p H , f ' / 2 = 1.45 π . We will restrict ourselves to a range of bias conditions relevant for our conditions, supposing that I b varies between ± 300 n A and that f ' / 2 varies by d f ' = ± 4 ⋅ 10 -3 π around 1.45 π . We chose such an interval for f ' because it corresponds to changing the **qubit** bias point ϵ by ± 2 G H z around 0 . The constants g 1 and g 2 are plotted in figure fig:couplings as a function of I b for two different values of f ' ( g 1 is shown as a full line, g 2 as a dashed line, and the two different values of f ' are symbolized by gray for d f ' = - 2 π 4 ⋅ 10 -3 and black for d f ' = 0 ). It can be seen that the coupling constants only weakly depend on the value of the flux in this range, so that we will neglect this dependence in the following and consider that g 1 and g 2 only depend on the bias current I b . Moreover we see from figure fig:couplings that the approximations made in equation eq:g1g2approx are justified in this range of parameters since g 1 is closely linear in I b and g 2 nearly constant. We also note that g 1 = 0 for I b = 0 . This fact can be generalized to the case where the SQUID-**qubit** coupling is not symmetric and the junctions critical current are dissimilar : in certain conditions these asymmetries can be compensated for by applying a bias current I b * for which g 1 I b * = 0 . At the current I b * , the **qubit** is effectively decoupled from the measuring circuit fluctuations to first order....(a) **qubit** biased by Φ x and SQUID biased by current I b . (b) Simplified electrical scheme : the SQUID-**qubit** system is seen as an inductance L J connected to the shunct capacitor C s h through inductance L s h . Φ a is the flux across the two inductances L J and L s h in series....**Qubit** **frequency** ν q as a function of the bias ϵ for Δ = 5.5 G H z (minimum **frequency** in the figure). The dashed line indicates the phase-noise insensitive bias point ϵ = 0 where d ν q / d ϵ = 0...**Frequency** shift per photon δ ν 0 as a function of I b and ϵ . The white regions correspond to -15 M H z and the black to + 35 M H z . The solid line ϵ m I b indicates the bias conditions for which δ ν 0 = 0 . The dashed line indicates the phase noise insensitive point ϵ = 0 ; the dotted line indicates the decoupling current I b = I b * ....Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed....The hamiltonian eq:qubit_hamiltonian yields a **qubit** transition **frequency** ν q = Δ 2 + ϵ 2 . The corresponding dependence is plotted in figure fig:nuq for realistic parameters. An interesting property is that when the **qubit** is biased at ϵ = 0 (dashed line in figure fig:nuq), it is insensitive to first order to noise in the bias variable ϵ ....We stress that these biasing conditions are non-trivial in the sense that they do not satisfy an obvious symmetry in the circuit. This point is emphasized in figure fig:deltanu0 where we plotted as a dashed line the bias conditions ϵ = 0 for which the **qubit** is insensitive to phase noise (due to flux or bias current noise) ; and as a dotted line the decoupling current conditions I b = I b * for which the **qubit** is effectively decoupled from its measuring circuit. The ϵ m I b line shares only one point with these two curves : the point I b * ϵ which is optimal with respect to flux, bias current, and photon noise. For the rest, the three lines are obviously distinct. This makes it possible to experimentally discriminate between the various noise sources limiting the **qubit** coherence by studying the dependence of τ φ on bias parameters....The flux-**qubit** is a superconducting loop containing three Josephson junctions threaded by an external flux Φ x ≡ f Φ 0 / 2 π . It is coupled to a DC-SQUID detector shunted by an external capacitor C s h whose role is to limit phase fluctuations across the SQUID as well as to filter high-**frequency** noise from the dissipative impedance. The SQUID is threaded by a flux Φ S q ≡ f ' Φ 0 / 2 π . The circuit diagram is shown in figure fig1a. There, the flux-**qubit** is the loop in red containing the three junctions of phases φ i and capacitances C i ( i = 1 , 2 , 3 ). It also includes an inductance L 1 which models the branch inductance and eventually the inductance of a fourth larger junction . The two inductances K 1 and K 2 model the kinetic inductance of the line shared by the SQUID and the **qubit**. The SQUID is the larger loop in blue. The junction phases are called φ 4 and φ 5 and their capacitances C 4 and C 5 . The critical current of the circuit junctions is written I C i ( i = 1 to 5 ). The SQUID loop also contains two inductances K 3 and L 2 which model its self-inductance. The SQUID is connected to the capacitor C s h through superconducting lines of parasitic inductance L s . The phase across the stray inductance and the SQUID is denoted φ A . The whole circuit is biased by a current source I b in parallel with a dissipative admittance Y ω . Since our goal is primarily to determine the **qubit**-plasma mode coupling hamiltonian, we will neglect the admittance Y ω . ... Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed.

Files:

Contributors: Averin, D. V.

Date: 2002-02-05

The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics....Schematic of the Josephson-junction **qubit** structure that enables measurements of the two non-commuting observables of the **qubit**, σ z and σ y , as required in the QND Hamiltonian ( 2). For discussion see text....Spin representation of the QND measurement of the quantum coherent **oscillations** of a **qubit**. The **oscillations** are represented as a spin rotation in the z - y plane with **frequency** Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with **frequency** Ω ≃ Δ . ... The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics.

Files:

Contributors: Mandip Singh

Date: 2015-07-14

A contour plot indicating location of two-dimensional potential energy minima forming a symmetric double well potential when the cantilever equilibrium angle θ0=cos−1[Φo/2BxA], ωi=2π×12000 rad/s, Bx=5×10−2 T. The contour interval in units of **frequency** (E/h) is ∼4×1011 Hz.
...In this paper a macroscopic quantum **oscillator** is proposed, which consists of a flux-**qubit** in the form of a cantilever. The net magnetic flux threading through the flux-**qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling between the cantilever and the magnetic flux is controlled through an external magnetic field. The ground state of the flux-**qubit**-cantilever turns out to be an entangled quantum state, where the cantilever deflection and the magnetic flux are the entangled degrees of freedom. A variant, which is a special case of the flux-**qubit**-cantilever without a Josephson junction, is also discussed....A superconducting-loop-**oscillator** with its axis of rotation along the z-axis consists of a closed superconducting loop without a Josephson Junction. The superconducting loop can be of any arbitrary shape.
...A contour plot indicating location of a two-dimensional global potential energy minimum at (nΦ0=0, θn+=π/2) and the local minima when the cantilever equilibrium angle θ0=π/2, ωi=2π×12000 rad/s, Bx=5.0×10−2 T. The contour interval in units of **frequency** (E/h) is ∼3.9×1011 Hz.
...The potential energy profile of the superconducting-loop-**oscillator** when the intrinsic **frequency** is 10 kHz. (a) For external magnetic field Bx=0, a single well harmonic potential near the minimum is formed. (b) Bx=0.035 T. (c) For Bx=0.045 T, a double well potential is formed.
...A schematic of the flux-**qubit**-cantilever. A part of the flux-**qubit** (larger loop) is projected from the substrate to form a cantilever. The external magnetic field Bx controls the coupling between the flux-**qubit** and the cantilever. An additional magnetic flux threading through a dc-SQUID (smaller loop) which consists of two Josephson junctions adjusts the tunneling amplitude. The dc-SQUID can be shielded from the effect of Bx.
... In this paper a macroscopic quantum **oscillator** is proposed, which consists of a flux-**qubit** in the form of a cantilever. The net magnetic flux threading through the flux-**qubit** and the mechanical degrees of freedom of the cantilever are naturally coupled. The coupling between the cantilever and the magnetic flux is controlled through an external magnetic field. The ground state of the flux-**qubit**-cantilever turns out to be an entangled quantum state, where the cantilever deflection and the magnetic flux are the entangled degrees of freedom. A variant, which is a special case of the flux-**qubit**-cantilever without a Josephson junction, is also discussed.

Files:

Contributors: Strauch, F. W., Dutta, S. K., Paik, Hanhee, Palomaki, T. A., Mitra, K., Cooper, B. K., Lewis, R. M., Anderson, J. R., Dragt, A. J., Lobb, C. J.

Date: 2007-03-02

The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the **oscillation** **frequency** Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 ....Experimental microwave spectroscopy of a Josephson phase **qubit**, scanned in **frequency** (vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 ....Rabi **frequency** Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **frequency** Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment. ... Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.

Files:

Contributors: Saiko, A. P., Fedaruk, R.

Date: 2010-12-10

Energy-level diagram of a **qubit** and transitions created by a bichromatic field at double resonance ( ω 0 = ω , ω 1 = ω r f )....Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account. ... Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account.

Files:

Contributors: Dial, O. E., Shulman, M. D., Harvey, S. P., Bluhm, H., Umansky, V., Yacoby, A.

Date: 2012-08-09

The **oscillations** in these FID experiments decay due to voltage noise from DC up to a **frequency** of approximately 1 / t . As the relaxation time, T 1 is in excess of 100 μ s in this regime, T 1 decay is not an important source of decoherence (Fig. S4). The shape of the decay envelope and the scaling of coherence time with d J / d ϵ (which effectively changes the magnitude of the noise) reveal information about the underlying noise spectrum. White (Markovian) noise, for example, results in an exponential decay of e - t / T 2 * where T 2 * ∝ d J / d ϵ -2 is the inhomogeneously broadened coherence time . However, we find that the decay is Gaussian (Fig. t2stard) and that T 2 * (black line in Fig. t2stare) is proportional to d J / d ϵ -1 (red solid line in Fig. t2stare) across two orders of magnitude of T 2 * . Both of these findings can be explained by quasistatic noise, which is low **frequency** compared to 1 / T 2 * . In such a case, one expects an amplitude decay of the form exp - t / T 2 * 2 , where T 2 * = 1 2 π d J / d ϵ ϵ R M S and ϵ R M S is the root-mean-squared fluctuation in ϵ (Eq. S3). From the ratio of T 2 * to d J / d ϵ -1 , we calculate ϵ R M S = 8 μ V in our device. At very negative ϵ , J becomes smaller than Δ B z , and nuclear noise limits T 2 * to approximately 90ns, which is consistent with previous work . We confirm that this effect explains deviations of T 2 * from d J / d ϵ -1 by using a model that includes the independently measured T 2 , n u c l e a r * and Δ B z (Eq. S1) and observe that it agrees well with measured T 2 * at large negative ϵ (dashed red line in Fig. t2stare)....Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity....Two level quantum systems (**qubits**) are emerging as promising candidates both for quantum information processing and for sensitive metrology . When prepared in a superposition of two states and allowed to evolve, the state of the system precesses with a **frequency** proportional to the splitting between the states. However, on a timescale of the coherence time, T 2 , the **qubit** loses its quantum information due to interactions with its noisy environment. This causes **qubit** **oscillations** to decay and limits the fidelity of quantum control and the precision of **qubit**-based measurements. In this work we study singlet-triplet ( S - T 0 ) **qubits**, a particular realization of spin **qubits** , which store quantum information in the joint spin state of two electrons. We form the **qubit** in two gate-defined lateral quantum dots (QD) in a GaAs/AlGaAs heterostructure (Fig. pulsesa). The QDs are depleted until there is exactly one electron left in each, so that the system occupies the so-called 1 1 charge configuration. Here n L n R describes a double QD with n L electrons in the left dot and n R electrons in the right dot. This two-electron system has four possible spin states: S , T + , T 0 , and T - . The S , T 0 subspace is used as the logical subspace for this **qubit** because it is insensitive to homogeneous magnetic field fluctuations and is manipulable using only pulsed DC electric fields . The relevant low-lying energy levels of this **qubit** are shown in Fig. pulsesc. Two distinct rotations are possible in these devices: rotations around the x -axis of the Bloch sphere driven by difference in magnetic field between the QDs, Δ B z (provided in this experiment by feedback-stabilized hyperfine interactions), and rotations around the z -axis driven by the exchange interaction, J (Fig. pulsesb) . A S can be prepared quickly with high fidelity by exchanging an electron with the QD leads, and the projection of the state of the **qubit** along the z -axis can be measured using RF reflectometery with an adjacent sensing QD (green arrow in Fig. pulsesa)....The device used in these measurements is a gate-defined S - T 0 **qubit** with an integrated RF sensing dot. a The detuning ϵ is the voltage applied to the dedicated high-**frequency** control leads pictured. b, The Bloch sphere that describes the logical subspace of this device features two rotation axes ( J and Δ B Z ) both controlled with DC voltage pulses. c, An energy diagram of the relevant low-lying states as a function of ϵ . States outside of the logical subspace of the **qubit** are grayed out. d, J ϵ and d J / d ϵ in three regions; the 1 1 region where J and d J / d ϵ are both small and S - T 0 **qubits** are typically operated, the transitional region where J and d J / d ϵ are both large where the **qubit** is loaded and measured, and the 0 2 region where J is large but d J / d ϵ is small and large quality **oscillations** are possible. pulses...Ramsey oscilllations reveal low **frequency** enivronmental dynamics. a, The pulse sequence used to measure exchange **oscillations** uses a stabilized nuclear gradient to prepare and readout the **qubit** and gives good contrast over a wide range of J . b, Exchange **oscillations** measured over a variety of detunings ϵ and timescales consistently show larger T 2 * as d J / d ϵ shrinks until dephasing due to nuclear fluctuations sets in at very negative ϵ . c, Extracted values of J and d J / d ϵ as a function of ϵ . d, The decay curve of FID exchange **oscillations** shows Gaussian decay. e, Extracted values of T 2 * and d J / d ϵ as a function of ϵ . T 2 * is proportional to d J / d ϵ -1 , indicating that voltage noise is the cause of dephasing of charge **oscillations**. f, Charge **oscillations** measured in 0 2 . This figure portrays the three basic regions we can operate our device in: a region of low **frequency** **oscillations** and small d J / d ϵ , a region of large **frequency** **oscillations** and large d J / d ϵ , and a region where **oscillations** are fast but d J / d ϵ is comparatively small. t2star...Since we observe J to be approximately an exponential function of ϵ , ( d J / d ϵ ∼ J ), we expect and observe the quality (number of coherent **oscillations**) of these FID **oscillations**, Q ≡ J T 2 * / 2 π ∼ J d J / d ϵ -1 , to be approximately constant regardless of ϵ . However, when ϵ is made very positive and J is large, an avoided crossing occurs between the 1 1 T 0 and the 0 2 T 0 state, making the 0 2 S and 0 2 T 0 states electrostatically virtually identical. Here, as ϵ is increased, J increases but d J / d ϵ decreases(Fig. pulsesd), allowing us to probe high quality exchange rotations and test our charge noise model in a regime that has never before been explored....Spin-echo measurements reveal high **frequency** bath dynamics. a, The pulse sequence used to measure exchange echo rotations. b, A typical echo signal. The overall shape of the envelope reflects T 2 * , while the amplitude of the envelope as a function of τ (not pictured) reflects T 2 e c h o . c, T 2 e c h o and Q ≡ J T 2 e c h o / 2 π as a function of J . A comparison of the two noise models: power law and a mixture of white and 1 / f noise. Noise with a power law spectrum fits over a wide range of **frequencies** (constant β ), but the relative contributions of white and 1 / f noise change as a function of ϵ . d, A typical echo decay is non-exponential but is well fit by exp - τ / T 2 e c h o β + 1 . e, T 2 e c h o varies with d J / d ϵ in a fashion consistent with dephasing due to power law voltage fluctuations. echo...Using a modified pulse sequence that changes the clock **frequency** of our waveform generators to achieve picosecond timing resolution (Fig. S1)), we measure exchange **oscillations** in 0 2 as a function of ϵ and time (Fig. t2stare) and we extract both J (Fig. t2starc) and T 2 * (Fig. t2stard) as a function of ϵ . Indeed, the predicted behavior is observed: for moderate ϵ we see fast **oscillations** that decay after a few ns, and for the largest ϵ we see even faster **oscillations** that decay slowly. Here, too, we observe that T 2 * ∝ d J d ϵ -1 (Fig. t2stard), which indicates that FID **oscillations** in 0 2 are also primarily dephased by low **frequency** voltage noise. We note, however, that we extract a different constant of proportionality between T 2 * and d J / d ϵ -1 for 1 1 and 0 2 . This is expected given that the charge distributions associated with the **qubit** states are very different in these two regimes and thus have different sensitivities to applied electric fields. We note that in the regions of largest d J / d ϵ (near ϵ = 0 ), T 2 * is shorter than the rise time of our signal generator and we systematically underestimate J and overestimate T 2 * (Fig. S1)....The use of Hahn echo dramatically improves coherence times, with T 2 e c h o (the τ at which the observed echo amplitude has decayed by 1 / e ) as large as 9 μ s , corresponding to qualities ( Q ≡ T 2 e c h o J / 2 π ) larger than 600 (Fig. echoc). If at high **frequencies** (50kHz-1MHz) the voltage noise were white (Markovian), we would observe exponential decay of the echo amplitude with τ . However, we find that the decay of the echo signal is non-exponential (Fig. echod), indicating that even in this relatively high-**frequency** band being probed by this measurement, the noise bath is not white. ... Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity.

Files:

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

Date: 2008-06-28

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

Files:

Contributors: Chirolli, Luca, Burkard, Guido

Date: 2009-06-04

The QND character of the **qubit** measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the **qubit** in the state | 0 , then rotate the **qubit** by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the **qubit** in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal **oscillation** with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of **qubit** pulses and **oscillator** driving is depicted in Fig. Fig1b). The conditional probability P 0 | 0 to detect the **qubit** in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig. Fig1c) for Δ = 0 , and in Fig. Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the **qubit** undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the **qubit** state and the **oscillator** state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 ....We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**....(Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the **qubit** around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the **qubit** in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic **oscillator** is driven at resonance with the bare harmonic **frequency** and a strong driving together with a strong damping of the **oscillator** are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6...In Fig. Fig5 we plot the second order correction to the probability to obtain "1" having prepared the **qubit** in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the **qubit** Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig. Fig3. Around the two **qubit**-shifted **frequencies**, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the **qubit**-shifted **frequency**....We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig. Fig7 we study separately the effect of **qubit** relaxation and **qubit** tunneling on the conditional probability P 0 | 0 . In Fig. Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the **qubit** is flipped in the first rotation. This is due to imperfection in the mapping between the **qubit** state and the state of the harmonic **oscillator**, already at the level of a single measurement. The inclusion of a phenomenological **qubit** relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the **oscillation** along τ 2 and washes out the response change when the **qubit** is flipped during the first rotation. This is shown in Fig. Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the **oscillations**, as clearly shown in Fig. Fig7 c)....(Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus **qubit** driving time τ 1 and τ 2 starting with the **qubit** in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The **oscillator** driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7...For driving at resonance with the bare harmonic **oscillator** **frequency** ω h o , the state of the **qubit** is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig. Fig3 for Δ ω = 0 . When matching one of the two **frequencies** ω i the **qubit** state is encoded in the amplitude of the signal, as also clearly shown in Fig. Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between **qubit** state and measurement outcomes and a weak **qubit** measurement....(Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the **qubit** states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the **qubit**-dependent "position" x s are shown in the top panel. Fig2...The combined effect of the quantum fluctuations of the **oscillator** together with the tunneling between the **qubit** states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the **qubit** QND measurement is studied in the regime of strong projective **qubit** measurement and **qubit** relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig. Fig6 and Fig. Fig7 directly to Fig. 4 in Ref. [...(Color online) a) Schematics of the 4-Josephson junction superconducting flux **qubit** surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at **frequency** ϵ 2 + Δ 2 , before and between two measurements prepare the **qubit** in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two **qubit** states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic **oscillator** to a **qubit**-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the **qubit** in the state "0" vs driving time τ 1 and τ 2 , at Rabi **frequency** of 1 ~ G H z . The **oscillator** driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological **qubit** relaxation time T 1 = 10 ~ n s is assumed. Fig1 ... We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**.

Files: