### 54077 results for qubit oscillator frequency

Contributors: Xian-Ting Liang

Date: 2008-09-03

The evolutions of reduced density matrix elements ρ12 (below) and ρ11 (up) in SB and SIB models in low-**frequency** bath. The parameters are the same as in Fig. 1.
...The spectral density functions Johm(ω) (b) and Jeff(ω) (a) versus the **frequency** ω of the bath modes, where Δ=5×109Hz,λκ=1,ξ=0.01,Ω0=10Δ,T=0.01K,Γ=2.6×1011Hz.
...The evolutions of reduced density matrix elements of ρ12 (below) and ρ11 (up) in SIB model in medium-**frequency** bath in different values of Ω0, the other parameters are the same as in Fig. 1.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO....The response functions of the Ohmic bath in (a) low and (c) medium **frequencies** and effective bath in (b) low and (d) medium **frequencies**. The parameters are the same as in Fig. 1. The cut-off **frequencies** for the two cases are taken according to Fig. 2.
...The sketch map on the low-, medium-, and high-**frequency** baths.
... Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** in spin-boson (SB) and spin-intermediate harmonic **oscillator** (IHO)-bath (SIB) models. The cases that the environment baths with low and medium **frequencies** are investigated. It is shown that the **qubits** in SB and SIB models have the same decoherence and relaxation as the baths with low **frequencies**. However, the **qubits** in the two models have different decoherence and relaxation as the baths with medium **frequencies**. The decoherence and relaxation of the **qubit** in SIB model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO.

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Contributors: Zorin, A. B., Chiarello, F.

Date: 2009-08-27

(Color online) (a) The **qubit** **frequency** as a function of parameter β L for fixed L = 50 pH and several values of capacitance C = 0.1 , 0.3, 1.0 and 3.0 pF (from top to bottom), corresponding to the values of the ratio E L / E c ≈ 1.7 × 10 4 , 5.1 × 10 4 , 1.7 × 10 5 and 5.1 × 10 5 . (b) Anharmonicity parameter δ as a function of parameter β L for the same as in (a) inductance L and capacitance values (from top to bottom)....(Color online) Position of the lowest six levels (solid lines) in the potential Eq. ( U-phi) for φ e = π as a function of parameter β L for typical values of L and C , yielding E J / E c ∼ E L / E c ≈ 5.1 × 10 4 . With an increase of β L , the spectrum crosses over from that of the harmonic **oscillator** type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the **oscillator**-type states in two separate wells. The spectrum in the central region β L ≈ 1 is strongly anharmonic. The dashed line shows the bottom energy of the potential U φ φ e = π , which in the case of β L > 1 is equal to - Δ U ≈ - 1.5 E L β L - 1 2 / β L (in other words, Δ U is the height of the energy barrier in the right inset) . The dotted (zero-level) line indicates the energy in the symmetry point φ = 0 , i.e. at the bottom of the single well ( β L ≤ 1 ) or at the top of the energy barrier ( β L > 1 ). The black dot shows the critical value β L c at which the ground state energy level touches the top of the barrier separating the two wells....We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling....where Î is the operator of supercurrent circulating in the **qubit** loop. The dependence of the reverse inductance L J Φ e = Φ 0 / 2 n calculated numerically in the two lowest quantum states ( n = 0 and 1) for L = 50 pH and the same set of capacitances C as in Fig. **frequency**-anharmonicity is shown in Fig. inductance-L01. One can see that the ratio of the geometrical to Josephson inductances L / L J takes large and very different values that can be favorably used for the dispersive readout, ensuring a sufficiently large output signal. Note that for β L 1 the inductance L J n = 1 changes the sign to positive....Figure f-shift shows this relative **frequency** shift versus parameter β L . One can see that for the rather conservative value of dimensionless coupling κ = 0.05 , the relative **frequency** shift can achieve the easily measured values of about 10 % . The efficiency of the dispersive readout can be improved in the non-linear regime with bifurcation . With our device this regime can be achieved in the resonance circuit including, for example, a Josephson junction (marked in the diagram in Fig. 1 by a dashed cross). Due to the high sensitivity of the amplitude (phase) bifurcation to the threshold determined by the effective resonance **frequency** of the circuit, one can expect a readout with high fidelity even at a rather weak coupling of the **qubit** and the resonator (compare with the readout of quantronium in Ref. ). Further improvement of the readout can be achieved in the QED-based circuit including this **qubit** ....Such a large, positive anharmonicity is a great advantage of the quartic potential **qubit** allowing manipulation within the two basis **qubit** states | 0 and | 1 not only when applying resonant microwave field, ν μ w ≈ ν 10 , but also when applying control microwave signals with large **frequency** detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic **qubit** **frequency** ν 10 = Δ E 0 / h and the anharmonicity factor δ computed from the Schrödinger equation for the original potential Eq. ( U-phi) in the range 0.9 ≤ β L ≤ 1.02 are shown in Fig. **frequency**-anharmonicity. One can see that the significant range in the tuning of the **qubit** **frequency** within the range of sufficiently large anharmonicity ( ∼ 20 - 50 % ) is attained at a rather fine (typically ± 1 - 2 % ) tuning of β L around the value β L = 1 . Such tuning of β L is possible in the circuit having the compound configuration shown in Fig. 1b. For values of β L > 1 , the symmetric energy potential has two minima and a barrier between them. The position of the ground state level depends on β L and the ratio of the characteristic energies E J / E c = β L E L / E c . The value of β L at which the ground state level touches the top of the barrier sets the upper limit β L c for the quartic **qubit** (marked in Fig. levels-beta by solid dot). At β L > β L c , the **qubit** energy dramatically decreases and the **qubit** states are nearly the symmetric and antisymmetric combinations of the states inside the two wells (see the right inset in Fig. 2). Although the **qubit** with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses , its readout can hardly be accomplished in a dispersive fashion....(a) Electric diagram of the **qubit** coupled to a resonant circuit and (b) possible equivalent compound (two-junction SQUID) circuit of the Josephson element included into the **qubit** loop. Capacitance C includes both the self-capacitance of the junction and the external capacitance. Due to inclusion in the resonant circuit of a Josephson junction JJ’, the resonator may operate in the nonlinear regime, enabling a bifurcation-based readout....(Color online) The resonance **frequency** shift in the circuit due to excitation of the **qubit** with the inductance value L = 50 pH and the set of capacitances C , decreasing from top to bottom. The dimensionless coupling coefficient κ = 0.05 ....(Color online) The values of the Josephson inductance of the quartic potential **qubit** in the ground (solid lines) and excited (dashed lines) states calculated for the geometric inductance value L = 50 pH and the set of capacitances C , increasing from top to bottom for both groups of curves. ... We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling.

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Contributors: Fedorov, A., Feofanov, A. K., Macha, P., Forn-Díaz, P., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-04-09

fig:rabi_sym_2(color online). (a) MW **frequency** vs f ϵ . The dotted white line is obtained from Eq. ( total H1) with I p = 400 nA and Δ = ν o s c . The observed vacuum Rabi splitting is maximal due to fully transverse coupling of the **qubit** to the **oscillator** η = π / 2 . (b) Vacuum Rabi **oscillations** for different values of f ϵ . In the experiment f ϵ was controlled by the amplitude of the current pulse I ϵ while Δ was tuned to ν o s c by changing the external magnetic field B . The inset shows ν R extracted from data (red circles) and estimated from Eq. ( Rabi freq general) (blue line). The color indicates the switching probability of the SQUID minus 0.5....fig:spectrum1(color online). (a) Schematic representation of the control and measurement pulses to perform spectroscopy. (b) Diagram of Landau-Zener transitions transferring the excitation of the **oscillator** to the **qubit**. (c) MW **frequency** vs f ϵ (controlled by the amplitude of the current pulse I ϵ ). The color indicates the switching probability of the SQUID minus 0.5. The white dotted line is obtained from Eq. ( total H1) with Δ = 2.04 GHz, I p = 420 nA. The vacuum Rabi splitting of 180 MHz corresponds to the effective **qubit**-**oscillator** coupling strength reduced by sin η ....A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible....fig:rabi_sym(color online). Vacuum Rabi **oscillations** (a) and MW **frequency** (b) vs magnetic f α . In the experiment the **qubit** was kept in its symmetry point ( ϵ = 0 ) by appropriately adjusting the amplitude of the current pulse I ϵ while Δ was changed by f α with use of external magnetic field B (a) or by applying the current pulse I α for fixed B (b). The color scale shows the switching probability of the SQUID minus 0.5. (c) **Frequency** of the vacuum Rabi **oscillations** extracted from data (a) and theoretical estimation (blue line) from Eq. ( Rabi **frequency**) as a function of f α . The minimum in ν R determines the bare **qubit**-**oscillator** coupling 2 g and corresponds to the resonance conditions Δ = ν o s c . (d) Single trace of the vacuum Rabi **oscillations** for Δ ≃ ν o s c ....fig:scheme(color online). (a) Circuit schematics: the tunable gap flux **qubit** (green) coupled to a lumped element superconducting LC **oscillator** (red) and controlled by the bias lines I ϵ , I ϵ , d c , I α (black). The SQUID (blue) measures the state of the **qubit**. The gradiometer loop (emphasized by a dashed line) is used to trap fluxoids. (b) Scanning Electron Micrograph (SEM) of the sample. (c) Energy diagram of the **qubit**-**oscillator** system. The minimum of energy splitting of the **qubit** Δ is reached at the symmetry point when one fluxoid is trapped in the gradiometer loop and the difference in magnetic fluxes f ϵ Φ 0 is 0 controlled by I ϵ and I ϵ , d c . By controlling the flux f α Φ 0 with I α and uniform field B one can tune Δ in resonance with **oscillator** **frequency** ν o s c . ... A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible.

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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.

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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.

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Contributors: Leyton, V., Thorwart, M., Peano, V.

Date: 2011-09-26

(Color online) (a) Nonlinear response A of the detector coupled to the **qubit** prepared in its ground state | ↓ (orange solid line) and in its excited state | ↑ (black dashed line) for the same parameters as in Fig. fig2. The quadratic **qubit**-detector coupling induces a global **frequency** shift of the response by δ ω e x = 2 g . (b) Discrimination power D ω e x of the detector coupled to the **qubit** for the same parameters as in a). fig3...(Color online) (a) Asymptotic population difference P ∞ of the **qubit** states, and (b) the corresponding detector response A as a function of the external **frequency** ω e x for the same parameters as in Fig. fig2. fig4...(Color online) (a) Relaxation rate Γ of the nonlinear quantum detector, (b) the measurement time T m e a s , and (c) the measurement efficiency Γ m e a s / Γ as a function of the external **frequency** ω e x . The parameters are the same as in Fig. fig2. fig5...For a fixed value of g , the shift between the two cases of the opposite **qubit** states is given by the **frequency** gap δ ω e x ≃ 2 g . Figure fig3 (a) shows the nonlinear response of the detector for the two cases when the **qubit** is prepared in one of its eigenstates: | ↑ (orange solid line) and | ↓ (black dashed line)....(Color online) (a) Amplitude A of the nonlinear response of the decoupled quantum Duffing detector ( g = 0 ) as a function of the external driving **frequency** ω e x . (b) The corresponding quasienergy spectrum ε α . The labels N denote the corresponding N -photon (anti-)resonance. The parameters are α = 0.01 Ω , f = 0.006 Ω , T = 0.006 Ω , and γ = 1.6 × 10 -4 Ω . fig1...We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed....Before turning to the quantum detection scheme, we discuss the dynamical properties of the isolated detector, which is the quantum Duffing **oscillator**. A key property is its nonlinearity which generates multiphoton transitions at **frequencies** ω e x close to the fundamental **frequency** Ω . In order to see this, one can consider first the undriven nonlinear **oscillator** with f = 0 and identify degenerate states, such as | n and | N - n (for N > n ), when δ Ω = α N + 1 / 2 . For finite driving f > 0 , the degeneracy is lifted and avoided quasienergy level crossings form, which is a signature of discrete multiphoton transitions in the detector. As a consequence, the amplitude A of the nonlinear response signal exhibits peaks and dips, which depend on whether a large or a small **oscillation** state is predominantly populated. The formation of peaks and dips goes along with jumps in the phase of the **oscillation**, leading to **oscillations** in or out of phase with the driving. A typical example of the nonlinear response of the quantum Duffing **oscillator** in the deep quantum regime containing few-photon (anti-)resonances is shown in Fig. fig1(a) (decoupled from the **qubit**), together with the corresponding quasienergy spectrum [Fig. fig1(b)]. We show the multiphoton resonances up to a photon number N = 5 . The resonances get sharper for increasing photon number, since their widths are determined by the Rabi **frequency**, which is given by the minimal splitting at the corresponding avoided quasienergy level crossing. Performing a perturbative treatment with respect to the driving strength f , one can get the minimal energy splitting at the avoided quasienergy level crossing 0 N as...(Color online) Nonlinear response A of the detector as a function of the external driving **frequency** ω e x in the presence of a finite coupling g = 0.0012 Ω to the **qubit** (black solid line). The blue dashed line indicates the response of the isolated detector. The parameters are the same as in Fig. fig1 and ϵ = 2.2 Ω and Δ = 0.05 Ω , in correspondence to realistic experimental parameters . fig2...Notice that g and α depend on the external flux ϕ e x , i.e., they are tunable in a limited regime with respect to the desired **oscillator** **frequency** Ω , where the coupling term is considered as a perturbation to the SQUID ( g **oscillator** to dominate. The dependence of the dimensionless ratios α / Ω and g / Ω is shown in Fig. fig0. ... We introduce a detection scheme for the state of a **qubit**, which is based on resonant few-photon transitions in a driven nonlinear resonator. The latter is parametrically coupled to the **qubit** and is used as its detector. Close to the fundamental resonator **frequency**, the nonlinear resonator shows sharp resonant few-photon transitions. Depending on the **qubit** state, these few-photon resonances are shifted to different driving **frequencies**. We show that this detection scheme offers the advantage of small back action, a large discrimination power with an enhanced read-out fidelity, and a sufficiently large measurement efficiency. A realization of this scheme in the form of a persistent current **qubit** inductively coupled to a driven SQUID detector in its nonlinear regime is discussed.

Files:

Contributors: Kofman, A. G., Zhang, Q., Martinis, J. M., Korotkov, A. N.

Date: 2006-06-02

The first-**qubit** **oscillation** **frequency** f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma **frequency** for the second **qubit**: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 ....The circuit schematic of a flux-biased phase **qubit** and the corresponding potential profile (as a function of the phase difference δ across the Josephson junction). During the measurement the state | 1 escapes from the “left” well through the barrier, which is followed by **oscillations** in the “right” well. This dissipative evolution leads to the two-**qubit** crosstalk....The **oscillating** term in Eq. ( 3.11a) describes the beating between the **oscillator** and driving force **frequencies**, with the difference **frequency** increasing in time, d t ~ 2 / d t = α t - t c , and amplitude of beating decreasing as 1 / t ~ (see dashed line in Fig. f4a). Notice that F 0 = 1 / 4 , F ∞ = 1 , and the maximum value is F 1.53 = 1.370 , so that E 0 is the long-time limit of the **oscillator** energy E 2 , while the maximum energy is 1.37 times larger:...The second **qubit** energy E 2 (in units of ℏ ω l 2 ) in the **oscillator** model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving **frequency** f d (see Fig. f3) is in resonance with ω l 2 / 2 π = 8.91 GHz....mcd05, a short flux pulse applied to the measured **qubit** decreases the barrier between the two wells (see Fig. f0), so that the upper **qubit** level becomes close to the barrier top. In the case when level | 1 is populated, there is a fast population transfer (tunneling) from the left well to the right well. Due to dissipation, the energy in the right well gradually decreases, until it reaches the bottom of the right well. In contrast, if the **qubit** is in state | 0 the tunneling essentially does not occur. The **qubit** state in one of the two potential minima (separated by almost Φ 0 ) is subsequently distinguished by a nearby SQUID, which completes the measurement process....Now let us consider the effect of dissipation in the second **qubit**. ...We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability....Dots: Rabi **frequencies** R k , k - 1 / 2 π for the left-well transitions at t = t c , for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Dashed line shows analytical dependence 1.1 k GHz....2.16 Figure f2 shows the **qubit** potential U δ for N l = 10 (corresponding to φ = 4.842 ), N l = 5 ( φ = 5.089 ), and N l = 1.355 ( φ = 5.308 ); the last value corresponds to the bias during the measurement pulse (see below). The **qubit** levels | 0 and | 1 are, respectively, the ground and the first excited levels in the left well....Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [**oscillator** model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation **frequency** of the two-**qubit** imaginary-swap quantum gate....3.17 in the absence of dissipation in the second **qubit** ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account the mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs. f4(a) and f7 shows that in both models the **qubit** energy remains small before a sharp increase in energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than in the **oscillator** model (the position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of the **qubit** may be to a much lower energy than for the **oscillator**; (c) After the sharp increase, the energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot the initial energy maximum; (d) The model now explicitly describes the **qubit** escape (switching) to the right well [Figs. f7(b) and f7(c)]; in contrast to the **oscillator** model, the escape may happen much later than initial energy increase; for example, in Fig. f7(b) the escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns. ... We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability.

Files:

Contributors: Hausinger, Johannes, Grifoni, Milena

Date: 2010-09-08

Figure Fig::SpectrumVSg shows the quasienergy spectrum against the coupling strength g . For simplicity, we study the unbiased case ε = 0 , which implies m = L = 0 and hence gaps with Ω 0 , 0 n , K = | Δ 0 L K 0 α e - α 2 | ≡ Ω K . Thus, for g = 0 and Δ ≠ 0 , the twofold degeneracy of the unperturbed case is lifted by a gap of width Δ 0 . For g ≠ 0 , the gap size is further determined by the Laguerre polynomial, so that additional degeneracies can occur at the zeros of L K 0 α . When choosing the driving amplitude A such that Δ 0 = 0 the twofold degeneracy is kept for arbitrary g and K . Because the dressing by the Bessel function does not depend on g or the **oscillator** level, we reach the remarkable conclusion that the coherent destruction of tunneling (CDT), predicted for a driven **qubit** , might occur also for a **qubit**-**oscillator** system in the ultrastrong coupling limit. In Fig. Fig::DressedOsc, the dressed **oscillation** **frequencies** are plotted against the dimensionless coupling g / Ω . Next to an exponential decay, they exhibit zeros that depend through the Laguerre polynomial characteristically on the **oscillator** quantum number K . Hence, because the ** qubit’s** dynamics involves several

**oscillator**levels, we predict that suppression of tunneling cannot be reached by just tuning the coupling g . The dynamics. To prove the statements above, we calculate the survival probability of the

**qubit**P ↓ ↓ t : = ↓ | ρ ̂ r e d t | ↓ , where ρ ̂ r e d is obtained by tracing out the

**oscillator**degrees of freedom from the density operator of the

**qubit**-

**oscillator**system:...We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling....In Fig. Fig::Dynam1(c) we are with g / Ω = 1.0 already deep in the ultrastrong coupling regime. The

**frequency**Ω 1 is now different from zero, and additionally Ω 3 appears. The lowest peak belongs to the

**frequencies**Ω 0 , Ω 2 , and Ω 4 , which are equal for g / Ω = 1.0 , see Fig. Fig::DressedOsc. A complete population inversion again takes place. Our results are confirmed by numerical calculations. For g = 0.5 , 1.0 , the latter yield additionally fast

**oscillations**with Ω and ω ex . Furthermore, Ω 1 is shifted in Fig. Fig::Dynam1(c) slightly to the left, so that concerning the survival probability the analytical and numerical curves get out of phase for longer times. To include also the

**oscillations**induced by the driving and the coupling to the quantized modes, connections between the degenerate subspaces need to be included in the calculation of the eigenstates of the full Hamiltonian ....(Color online) Quasienergy spectrum of the

**qubit**-

**oscillator**system against the static bias ε for weak coupling g / ω ex = 0.05 . Further parameters are Δ / ω ex = 0.2 , Ω / ω ex = 2 , A / ω ex = 2.0 . The first six

**oscillator**states are included. Numerical calculations are shown by red (light gray) triangles, analytical results in the region of avoided crossings by black dots. A good agreement between analytics and numerics is found. Blue (dark gray) squares represent the case Δ = 0 . Fig::QuasiEnEpsAnaDfinite...(Color online) Size of the avoided crossing Ω K against the dimensionless coupling strength g / Ω for an unbiased

**qubit**( ε = 0 ). Further, Δ / Ω = 0.4 , ω ex / Ω = 5.3 and A / Ω = 8.0 . Ω K vanishes at the zeros of the Laguerre polynomial L K 0 α . The dashed lines (a), (b), (c) represent g / Ω = 0.1 , 0.5 , 1.0 , respectively, as considered in Fig. Fig::Dynam1. Fig::DressedOsc...(Color online) Coherent destruction of tunneling in a driven

**qubit**-

**oscillator**system. The same parameters as in Fig. Fig::Dynam1 are used except that A / Ω = 12.7 , which leads to Δ 0 = 0 . Three coupling strengths are examined: g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). The analytical calculations (black, dashed lines) predict complete localization for all three cases. Also the numerics (red curves) shows strong localization for short timescales with fast

**oscillations**overlaid. For long times this localization vanishes (see inset in (a)). Fig::DynamCDT...While tuning the coupling g to a zero of a Laguerre polynomial corresponding to a dominant

**oscillator**mode yields a reduction of tunneling, tuning the driving amplitude A to a zero of a Bessel function can yield almost complete localization at short times. As already noticed in Fig. Fig::SpectrumVSg, this phenomenon is independent of the coupling strength g . We choose in Fig. Fig::DynamCDT the driving amplitude A , so that Δ 0 = 0 . This is the same condition as found for CDT in a driven

**qubit**. Analogously, our analytical solution now predicts localization for arbitrary coupling strength g . All dressed

**oscillation**

**frequencies**Ω K vanish. However, third-order corrections in Δ will give small contributions to Δ 0 . Hence, a numerical exact solution yields

**oscillations**of P ↓ ↓ t with a long period. On a short timescale and for ω ex ≫ Δ also the numerical solution appears to be strongly localized, while for long times, the inset in Fig. Fig::DynamCDT (a) shows complete population inversion for the numerics. In conclusions, we developed a powerful formalism to investigate analytically a

**qubit**-

**oscillator**system in the ultrastrong coupling and extreme driving regime, a situation which is in close experimental reach and offers excellent control possibilities. Our approach relies on perturbation theory with respect to a single parameter only, the

**qubit**tunneling matrix element Δ , and thus goes beyond the driven Jaynes-Cummings model, with no rotating-wave approximation being applied. We acknowledge financial support under DFG Program SFB631. We thank Sigmund Kohler for helpful remarks....(Color online) Dynamics of the

**qubit**for ε = 0 , Δ / Ω = 0.4 , ω ex / Ω = 5.3 , A / Ω = 8.0 , and temperature ℏ Ω k B T -1 = 10 . The graphs show the Fourier transform F ν of the survival probability P ↓ ↓ t (see the insets). We study the different coupling strengths indicated in Fig. Fig::DressedOsc, g / Ω = 0.1 (a), 0.5 (b) and 1.0 (c). Analytical results are shown by black curves, numerics by dashed orange curves. Fig::Dynam1...Additional crossings occur independent of ε if driving and

**oscillator**

**frequency**are commensurable, Ω / ω ex = j / N with integers j , N > 0 , resulting in infinite many degenerate states. We avoid such a situation by choosing incommensurable

**frequencies**or high values for j and N , so that only high-photon processes are affected.. Note that for L ≠ 0 there are always L nondegenerate levels. For L > 0 those are the first L spin-down states (positive slope), while for L < 0 the first L spin-up states (negative slope). At finite Δ avoided crossings occur in the energy spectrum at the sites of the resonances (red triangles and black dots in Fig. Fig::QuasiEnEpsAnaDfinite). To explain the origin of these avoided crossings we express H ̂ in the basis ( CoupledEigenstates) yielding the off-diagonal elements ... We introduce an approach to studying a driven

**qubit**-

**oscillator**system in the ultrastrong coupling regime, where the ratio $g/\Omega$ between coupling strength and

**oscillator**

**frequency**approaches unity or goes beyond, and simultaneously for driving strengths much bigger than the

**qubit**energy splitting (extreme driving). Both

**qubit**-

**oscillator**coupling and external driving lead to a dressing of the

**qubit**tunneling matrix element of different nature: the former can be used to suppress selectively certain

**oscillator**modes in the spectrum, while the latter can bring the

**qubit**'s dynamics to a standstill at short times (coherent destruction of tunneling) even in the case of ultrastrong coupling.

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.

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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.

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