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A typical example of QT is shown in Fig.  fig1. It shows two main properties of the evolution: the oscillator spends a very long time at some average level n = n - and then jumps to another significantly different value n + . At the same time the polarization vector of qubit ξ → defined as ξ → = T r ρ ̂ σ → also changes its orientation direction with a clear change of sign of ξ x from ξ x > 0 to ξ x qubit polarization ξ = | ξ → | is very close to unity showing that the qubit remains mainly in a pure state. The drops of ξ appear only during transitions between metastable states. Special checks show that an inversion of ξ x by an additional pulse (e.g. from ξ x > 0 to ξ x oscillator to a corresponding state (from n - to n + ) after time t m ∼ 1 / λ . Thus we have here an interesting situation when a quantum flip of qubit produces a marcoscopic change of a state of detector (oscillator) which is continuously coupled to a qubit (we checked that even larger variation n ± ∼ n p is possible by taking n p = 40 ). In addition to that inside a metastable state the coupling induces a synchronization of qubit rotation phase with the oscillator phase which in its turn is fixed by the phase of driving field. The synchronization is a universal phenomenon for classical dissipative systems . It is known that it also exists for dissipative quantum systems at small effective values of ℏ . However, here we have a new unusual case of qubit synchronization when a semiclassical system produces synchronization of a pure quantum two-level system.... (color online) Top panels: the Poincaré section taken at integer values of ω t / 2 π for oscillator with x = â + â / 2 , p = â - â / 2 i (left) and for qubit polarization with polarization angles θ φ defined in text (right). Middle panels: the same quantities shown at irrational moments of ω t / 2 π . Bottom panels: the qubit polarization phase φ vs. oscillator phase ϕ ( p / x = - tan ϕ ) at time moments as in middle panels for g = 0.04 (left) and g = 0.004 (right). Other parameters and the time interval are as in Fig.  fig1. The color of points is blue/black for ξ x > 0 and red/gray for ξ x < 0 .... (color online) Bistability of qubit coupled to a driven oscillator with jumps between two metastable states. Top panel shows average oscillator level number n as a function of time t at stroboscopic integer values ω t / 2 π ; middle panel shows the qubit polarization vector components ξ x (blue/black) and ξ z (green/gray) at the same moments of time; the bottom panel shows the degree of qubit polarization ξ . Here the system parameters are λ / ω 0 = 0.02 , ω / ω 0 = 1.01 , Ω / ω 0 = 1.2 , f = ℏ λ n p , n p = 20 and g = 0.04 .... The phenomenon of qubit synchronization is illustrated in a more clear way in Fig.  fig2. The top panels taken at integer values ω t / 2 π show the existence of two fixed points in the phase space of oscillator (left) and qubit (right) coupled by quantum tunneling (the angles are determined as ξ x = ξ cos θ , ξ y = ξ sin θ sin φ , ξ z = ξ sin θ cos φ ). A certain scattering of points in a spot of finite size should be attributed to quantum fluctuations. But the fact that on enormously long time (Fig.  fig1) the spot size remains finite clearly implies that the oscillator phase ϕ is locked with the driving phase ω t inducing the qubit synchronization with ϕ and ω t . The plot at t values incommensurate with 2 π / ω (middle panels) shows that in time the oscillator performs circle rotations in p x plane with frequency ω while qubit polarization rotates around x -axis with the same frequency. Quantum tunneling gives transitions between two metastable states. The synchronization of qubit phase φ with oscillator phase ϕ is clearly seen in bottom left panel where points form two lines corresponding to two metastable states. This synchronization disappears below a certain critical coupling g c where the points become scattered over the whole plane (panel bottom right). It is clear that quantum fluctuations destroy synchronization for g < g c . Our data give g c ≃ 0.008 for parameters of Fig.  fig1.... (color online) Right panel: dependence of average qubit polarization components ξ x and ξ z (full and dashed curves) on g , averaging is done over stroboscopic times (see Fig. fig1) in the interval 100 ≤ ω t / 2 π ≤ 2 × 10 4 ; color is fixed by the sign of ξ x averaged over 10 periods (red/gray for ξ x 0 ; this choice fixes also the color on right panel). Left panel: dependence of average level of oscillator in two metastable states on coupling strength g , the color is fixed by the sign of ξ x on right panel that gives red/gray for large n + and blue/black for small n - ; average is done over the quantum state and stroboscopic times as in the left panel; dashed curves show theory dependence (see text)). Two QT are used with initial value ξ x = ± 1 . All parameters are as in Fig.  fig1 except g .... (color online) Dependence of number of transitions N f between metastable states on rescaled qubit frequency Ω / ω 0 for parameters of Fig.  fig1; N f are computed along 2 QT of length 10 5 driving periods. Inset shows life time dependence on Ω / ω 0 for two metastable states ( τ + for red/gray, τ - for blue/black, τ ± are given in number of driving periods; color choice is as in Figs.  fig2, fig3).... (color online) Dependence of average level n ± of oscillator in two metastable states on the driving frequency ω (average and color choice are the same as in right panel of Fig.  fig3); coupling is g = 0.04 and g = 0.08 (dashed and full curves). Inset shows the variation of position of maximum at ω = ω ± with coupling strength g , Δ ω ± = ω ± - ω 0 . Other parameters are as in Fig.  fig1.
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A further test of the theory presented in Sec.  sec:th_s is provided by the measurement of the qubit resonant frequency. In the semiclassical regime of small E C , the qubit can be described by the effective circuit of Fig.  fig1(b), with the junction admittance Y J of Eq. ( YJ), Y C = i ω C , and Y L = 1 / i ω L [the inductance is related to the inductive energy by E L = Φ 0 / 2 π 2 / L ]. As discussed in Ref. ... As a second example of a strongly anharmonic system, we consider here a flux qubit, i.e., in Eq. ( Hphi) we assume E J > E L and take the external flux to be close to half the flux quantum, Φ e ≈ Φ 0 / 2 . Then the potential has a double-well shape and the flux qubit ground states | - and excited state | + are the lowest tunnel-split eigenstates in this potential, see Fig.  fig:fl_q. The non-linear nature of the sin ϕ ̂ / 2 qubit-quasiparticle coupling in Eq. ( HTle) has a striking effect on the transition rate Γ + - , which vanishes at Φ e = Φ 0 / 2 due to destructive interference: for flux biased at half the flux quantum the qubit states | - , | + are respectively symmetric and antisymmetric around ϕ = π , while the potential in Eq. ( Hphi) and the function sin ϕ / 2 in Eq. ( wif_gen) are symmetric. Note that the latter symmetry and its consequences are absent in the environmental approach in which a linear phase-quasiparticle coupling is assumed.... The transmon low-energy spectrum is characterized by well separated [by the plasma frequency ω p , Eq. ( pl_fr)] and nearly degenerate levels whose energies, as shown in Fig.  fig:trans, vary periodically with the gate voltage n g . Here we derive the asymptotic expression (valid at large E J / E C ) for the energy splitting between the nearly degenerate levels. We consider first the two lowest energy states and then generalize the result to higher energies.... Schematic representation of the transmon low energy spectrum as function of the dimensionless gate voltage n g . Solid (dashed) lines denotes even (odd) states (see also Sec.  sec:cpb). The amplitudes of the oscillations of the energy levels are exponentially small, see Appendix  app:eosplit; here they are enhanced for clarity. Quasiparticle tunneling changes the parity of the qubit sate. The results of Sec.  sec:semi are valid for transitions between states separated by energy of the order of the plasma frequency ω p , Eq. ( pl_fr), and give, for example, the rate Γ 1 0 . For the transition rates between nearly degenerate states of opposite parity, such as Γ o e 1 , see Appendix  app:eorate.... As an application of the general approach described in the previous section, we consider here a weakly anharmonic qubit, such as the transmon and phase qubits. We start with the the semiclassical limit, i.e., we assume that the potential energy terms in Eq. ( Hphi) dominate the kinetic energy term proportional to E C . This limit already reveals a non-trivial dependence of relaxation on flux. Note that assuming E L ≠ 0 we can eliminate n g in Eq. ( Hphi) by a gauge transformation. In the transmon we have E L = 0 and the spectrum depends on n g , displaying both well separated and nearly degenerate states, see Fig.  fig:trans. The results of this section can be applied to the single-junction transmon when considering well separated states. The transition rate between these states and the corresponding frequency shift are dependent on n g . However, since E C ≪ E J this dependence introduces only small corrections to Γ n n - 1 and δ ω ; the corrections are exponential in - 8 E J / E C . By contrast, the leading term in the rate of transitions Γ e ↔ o between the even and odd states is exponentially small. The rate Γ e ↔ o of parity switching is discussed in detail in Appendix  app:eorate.... Potential energy (in units of E L ) for a flux qubit biased at Φ e = Φ 0 / 2 with E J / E L = 10 . The horizontal lines represent the two lowest energy levels, with energy difference ϵ ̄ given in Eq. ( e0_eff).... which has the same form of the Hamiltonian for the single junction transmon [i.e., Eq. ( Hphi) with E L = 0 ] but with a flux-dependent Josephson energy, Eq. ( EJ_flux). Therefore the spectrum follows directly from that of the single junction transmon (see Fig.  fig:trans) and consists of nearly degenerate and well separated states. The energy difference between well separated states is approximately given by the flux-dependent frequency [cf. Eq. ( pl_fr)]... (a) Schematic representation of a qubit controlled by a magnetic flux, see Eq. ( Hphi). (b) Effective circuit diagram with three parallel elements – capacitor, Josephson junction, and inductor – characterized by their respective admittances.
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Numerical and analytical evaluation of the entanglement dynamics between the two qubits for ω 0 = 0.15 ω , β = 0.16 and α = 3 . Entanglement between the qubits exhibits collapse and revival. The analytic expression agrees well with the envelope of the numerically evaluated entanglement evolution.... Here, within the adiabatic approximation, we extend the examination to the two-qubit case. Qualitative differences between the single-qubit and the multi-qubit cases are highlighted. In particular, we study the collapse and revival of joint properties of both the qubits. Entanglement properties of the system are investigated and it is shown that the entanglement between the qubits also exhibits collapse and revival. We derive what we believe are the first analytic expressions for the individual revival signals beyond the RWA, as well as analytic expression for the collapse and revival dynamics of entanglement. In the quasi-degenerate regime, the invalidity of the RWA in predicting the dynamical evolution will clearly be demonstrated in Sec. s.collapse_rev (see Figs. f.collapse_revival_double and f.collapse_revival_single).... The three potential wells corresponding to the states | 1 , 1 | N 1 (left), | 1 , 0 | N 0 (middle) and | 1 , - 1 | N -1 (right). The factor Δ X z p is the zero point fluctuation of a harmonic oscillator. For an oscillator of mass M and frequency ω the zero point fluctuation is given by Δ X z p = ℏ / 2 M ω .... Collapse and revival dynamics for ω 0 = 0.1 ω , β = 0.16 and α = 3 . The first two panels show analytic evaluations of (a.) one-qubit and (b.) two-qubit probability dynamics, and (c.) shows that the two-qubit analytic formula matches well to the corresponding numerical evolution. In each case the initial state is a product of a coherent oscillator state with the lowest of the S x states. Note the breakup in the main revival peak of the two-qubit numerical evaluation, which comes from the ω - 2 ω beat note, not included in the analytic calculation, and not present for a single qubit.... When squared, the probability shows two frequencies of oscillation, 2 Ω N ω and 2 2 Ω N ω . Since three new basis states are involved, we could expect three frequencies, but two are equal: | E N + - E N 0 | = | E N - - E N 0 | . This is in contrast to the single-qubit case where only one Rabi frequency determines the evolution . We show below in Fig. f.col_rev the way differences between one and a pair of qubits can be seen.... If the average excitation of the oscillator, n ̄ = α 2 , is large one can evaluate the above sum approximately (see Appendix) and obtain analytic expressions and graphs of the evolution, as shown in Fig. f.col_rev. As expected, because of the double frequency in ( eqnP(t)), the revival time for S t 2 ω 0 is half the revival time for S t ω 0 . Thus there are two different revival sequences in the time series. Appropriate analytic formulas, e.g., ( a.Phi), agree well with the numerically evaluated evolution even for the relatively weak coherent excitation, α = 3 .... Entanglement dynamics between the qubits. (a) ω 0 = 0.1 ω , β = 0.16 , (b) ω 0 = 0.15 ω , β = 0.16 , (c) ω 0 = 0.1 ω , β = 0.2 and (d) ω 0 = 0.15 ω , β = 0.2 . For all the figures, α = 3 .... Collapse and revival dynamics for P 1 2 , - 1 2 α t , given ω 0 = 0.15 ω , β = 0.16 and α = 3 . Note the single revival sequence. Also, note that there are no breakups in the revival peaks in contrast to the two-qubit case (Fig. f.collapse_revival_double). The RWA fails to describe the dynamical evolution even for the single qubit case.... With recent advances in the area of circuit QED, it is now possible to engineer systems for which the qubits are coupled to the oscillator so strongly, or are so far detuned from the oscillator, that the RWA cannot be used to describe the system’s evolution correctly . The parameter regime for which the coupling strength is strong enough to invalidate the RWA is called the ultra-strong coupling regime . Niemczyk, et al. and Forn-Díaz, et al. have been able to experimentally achieve ultra-strong coupling strengths and have demonstrated the breakdown of the RWA. Motivated by these experimental developments and the importance of understanding collective quantum behavior, we investigate a two-qubit TC model beyond the validity regime of RWA. The regime of parameters we will be concerned with is the regime where the qubits are quasi-degenerate, i.e., with frequencies much smaller than the oscillator frequency, ω 0 ≪ ω , while the coupling between the qubits and the oscillator is allowed to be an appreciable fraction of the oscillator frequency. In this parameter regime, the dynamics of the system can neither be correctly described under the RWA, nor can the effects of the counter rotating terms be taken as a perturbative correction to the dynamics predicted within the RWA by including higher powers of β . For illustration, systems are shown in Fig. f.model for which the RWA is valid, or breaks down, because the condition ω 0 ≈ ω is valid, or is violated. The regime that we will be interested in, for which ω 0 ≪ ω , is shown on the right.... There is only one revival sequence for the single qubit system as a consequence of having only one Rabi frequency in the single qubit case. The analytic and numerically exact evolution of P 1 2 , - 1 2 α t is plotted in Fig. f.collapse_revival_single. The single revival sequence is evident from the figure. A discussion on the multiple revival sequences for the K -qubit TC model, within the parameter regime where the RWA is valid, can be found in .
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(color online) (a) The probability distribution of the two quantum Rabi oscillations ROT 0 (the blue solid line) and ROT 1 (the green solid line) of two charge qubits coupled to a resonator. (b) The probability distribution of the four quantum Rabi oscillations in our cc-phase gate on a three-charge-qubit system. Here, the blue-solid, green-solid, red-dashed, and Cambridge-blue-dot-dashed lines represent the quantum Rabi oscillations ROT 00 ( | 0 1 | 0 2 | 1 3 | 0 a ↔ | 0 1 | 0 2 | 0 3 | 1 a ), ROT 01 ( | 0 1 | 1 2 | 1 3 | 0 a ↔ | 0 1 | 1 2 | 0 3 | 1 a ), ROT 10 ( | 1 1 | 0 2 | 1 3 | 0 a ↔ | 1 1 | 0 2 | 0 3 | 1 a ), and ROT 11 ( | 1 1 | 1 2 | 1 3 | 0 a ↔ | 1 1 | 1 2 | 0 3 | 1 a ), respectively.... The quantum entangling operation based on the SR can also help us to complete a single-step controlled-controlled phase (cc-phase) quantum gate on the three charge qubits q 1 , q 2 , and q 3 by using the system shown in Fig. fig2 except for the resonator R b . Here, q 1 and q 2 act as the control qubits, and q 3 is the target qubit. The initial state of this system is prepared as | Φ 0 = 1 2 2 | 0 1 | 0 2 | 0 3 + | 0 1 | 0 2 | 1 3 + | 0 1 | 1 2 | 0 3 + | 0 1 | 1 2 | 1 3 + | 1 1 | 0 2 | 0 3 + | 1 1 | 0 2 | 1 3 + | 1 1 | 1 2 | 0 3 + | 1 1 | 1 2 | 1 3 | 0 a . In this system, both q 1 and q 2 are in the quasi-dispersive regime with R a , and the transition frequency of q 3 is adjusted to be equivalent to that of R a when q 1 and q 2 are in their ground states. The QSD transition frequency on R a becomes... (color online) Sketch of a coplanar geometry for the circuit QED with three superconducting qubits. Qubits are placed around the maxima of the electrical field amplitude of R a and R b (not drawn in this figure), and the distance between them is large enough so that there is no direct interaction between them. The fundamental frequencies of resonators are ω r j / 2 π ( j = a , b ), the frequencies of the qubits are ω q i / 2 π ( i = 1 , 2 , 3 ), and they are capacitively coupled to the resonators. The coupling strengths between them are g i j / 2 π . We can use the control line (not drawn here) to afford the flux to tune the transition frequencies of the qubits.... (color online) (a) The qubit-state-dependent resonator transition, which means the frequency shift of the resonator transition δ r arises from the state ( | 0 q or | 1 q ) of the qubit. (b) The number-state-dependent qubit transition, which means the frequency shift δ q takes place on the qubit due to the photon number n = 1 or 0 in the resonator in the dispersive regime.... in which we neglect the direct interaction between the two qubits (i.e., q 1 and q 2 ), shown in Fig. fig2. Here σ i + = | 1 i 0 | is the creation operator of q i ( i = 1 , 2 ). g i is the coupling strength between q i and R a . The parameters are chosen to make q 1 interact with R a in the quasi-dispersive regime. That is, the transition frequency of R a is determined by the state of q 1 . By taking a proper transition frequency of q 2 (which equals to the transition frequency of R a when q 1 is in the state | 0 1 ), one can realize the quantum Rabi oscillation (ROT) ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a , while ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a occurs with a small probability as q 2 detunes with R a when q 1 is in the state | 1 1 . Here the Fock state | n a represents the photon number n in R a ( n = 0 , 1 ). | 0 i and | 1 i are the ground and the first excited states of q i , respectively.... Eq.( stark) means the NSD qubit transition and Eq.( kerr) means the QSD resonator transition, shown in Fig. fig1(a) and (b), respectively.... (color online) Simulated outcomes for the maximum amplitude value of the expectation about the quantum Rabi oscillation varying with the coupling strength g 2 and the frequency of the second qubit ω 2 . (a) The outcomes for ROT 0 : | 0 1 | 1 2 | 0 a ↔ | 0 1 | 0 2 | 1 a . (b) The outcomes for ROT 1 : | 1 1 | 1 2 | 0 a ↔ | 1 1 | 0 2 | 1 a . Here the parameters of the resonator and the first qubit q 1 are taken as ω a / 2 π = 6.0 GHz, ω q 1 / 2 π = 7.0 GHz, and g 1 / 2 π = 0.2 GHz.... and it is shown in Fig. fig4 (a). In the simulation of our SR, we choose the reasonable parameters by considering the energy level structure of a charge qubit, according to Ref. . Here ω r a / 2 π = 6.0 GHz. The transition frequency of two qubits between and are chosen as ω 0 , 1 ; 1 / 2 π = E 1 ; 1 - E 0 ; 1 = 5.0 GHz, ω 1 , 2 ; 1 / 2 π = E 2 ; 1 - E 1 ; 1 = 6.2 GHz, ω 0 , 1 ; 2 / 2 π = E 1 ; 2 - E 0 ; 2 = 6.035 GHz, and ω 1 , 2 ; 2 / 2 π = E 2 ; 2 - E 1 ; 2 = 7.335 GHz. Here E i ; q is the energy for the level i of the qubit q , and σ i , i ' ; q + ≡ i q i ' . g i , j ; q is the coupling strength between the resonator R a and the qubit q in the transition between the energy levels | i q and | j q ( i = 0 , 1 , j = 1 , 2 , and q = 1 , 2 ). For convenience, we take the coupling strengths as g 0 , 1 ; 1 / 2 π = g 1 , 2 ; 1 / 2 π = 0.2 GHz and g 0 , 1 ; 2 / 2 π = g 1 , 2 ; 2 / 2 π = 0.0488 GHz.... We numerically simulate the maximal expectation values (MAEVs) of ROT 0 and ROT 1 based on the Hamiltonian H 2 q , shown in Fig. fig3(a) and (b), respectively. Here, the expectation value is defined as | ψ | e - i H 2 q t / ℏ | ψ 0 | 2 . | ψ 0 and | ψ are the initial and the final states of a quantum Rabi oscillation, respectively. The MAEV s vary with the transition frequency ω 2 and the coupling strength g 2 . It is obvious that the amplified QSD resonator transition can generate a selective resonance (SR) when the coupling strength g 2 is small enough.
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Oscillator frequency shift as function or the qubit splitting ϵ = ω + Δ for the spin state | ↓ obtained (a) within RWA, Eq.  shiftRWA, and (b) beyond RWA, Eq.  wrnonRWA. The lines mark the analytical results, while the symbols refer to the numerically obtained splitting between the ground state and the first excited state in the subspace of the qubit state | ↓ .... HRabi in the subspace of the qubit state | ↓ , where σ z | ↓ = - | ↓ . The results are depicted in Fig.  fig:wr.
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(color online) (a) Rabi oscillations in the qubit population P vs. Rabi pulse length Δ t (blue dots) and fit with unit visibility (red line). (b) Measured Rabi frequency ν R a b i vs. pulse amplitude ϵ s (blue dots) and linear fit.... (color online) Measurement response φ (blue lines) and theoretical prediction (red lines) vs. time. At t = 6 μ s (a) a π pulse, (b) a 2 π pulse, and (c) a 3 π pulse is applied to the qubit. In each panel the dashed lines correspond to the expected measurement response in the ground state φ , in the saturated state φ = 0 , and in the excited state φ .... The extracted qubit population P is plotted versus Δ t in Fig.  fig:rabioscillationsa. We observe a visibility of 95 ± 6 % in the Rabi oscillations with error margins determined from the residuals of the experimental P with respect to the predicted values. Thus, in a measurement of Rabi oscillations in a superconducting qubit, a visibility in the population of the qubit excited state that approaches unity is observed for the first time. Moreover, we note that the decay in the Rabi oscillation amplitude out to pulse lengths of 100 n s is very small and consistent with the long T 1 and T 2 times of this charge qubit, see Fig.  fig:rabioscillationsa and Ramsey experiment discussed below. We have also verified the expected linear scaling of the Rabi oscillation frequency ν R a b i with the pulse amplitude ϵ s ∝ n s , see Fig.  fig:rabioscillationsb.... In our circuit QED architecture , a Cooper pair box , acting as a two level system with ground and excited states and level separation E a = ℏ ω a = E e l 2 + E J 2 is coupled capacitively to a single mode of the electromagnetic field of a transmission line resonator with resonance frequency ω r , see Fig.  fig:setupa. As demonstrated for this system, the electrostatic energy E e l and the Josephson energy E J of the split Cooper pair box can be controlled in situ by a gate voltage V g and magnetic flux Φ , see Fig.  fig:setupa. In the resonant ( ω a = ω r ) strong coupling regime a single excitation is exchanged coherently between the Cooper pair box and the resonator at a rate g / π , also called the vacuum Rabi frequency . In the non-resonant regime ( Δ = ω a - ω r > g ) the capacitive interaction gives rise to a dispersive shift g 2 / Δ σ z in the resonance frequency of the cavity which depends on the qubit state σ z , the coupling g and the detuning Δ . We have suggested that this shift in resonance frequency can be used to perform a quantum non-demolition (QND) measurement of the qubit state . With this technique we have recently measured the ground state response and the excitation spectrum of a Cooper pair box .... (color online) (a) Measured Ramsey fringes (blue dots) observed in the qubit population P vs. pulse separation Δ t using the pulse sequence shown in Fig.  fig:setupb and fit of data to sinusoid with gaussian envelope (red line). (b) Measured dependence of Ramsey frequency ν R a m s e y on detuning Δ a , s of drive frequency (blue dots) and linear fit (red line).... (color online) (a) Simplified circuit diagram of measurement setup. A Cooper pair box with charging energy E C and Josephson energy E J is coupled through capacitor C g to a transmission line resonator, modelled as parallel combination of an inductor L and a capacitor C . Its state is determined in a phase sensitive heterodyne measurement of a microwave transmitted at frequency ω R F through the circuit, amplified and mixed with a local oscillator at frequency ω L O . The Cooper pair box level separation is controlled by the gate voltage V g and flux Φ . Its state is coherently manipulated using microwaves at frequency ω s with pulse shapes determined by V p . (b) Measurement sequence for Rabi oscillations with Rabi pulse length Δ t , pulse frequency ω s and amplitude ∝ n s with continuous measurement at frequency ω R F and amplitude ∝ n R F . (c) Sequence for Ramsey fringe experiment with two π / 2 -pulses at ω s separated by a delay Δ t and followed by a pulsed measurement.... In the experiments presented here, we coherently control the quantum state of a Cooper pair box by applying to the qubit microwave pulses of frequency ω s , which are resonant with the qubit transition frequency ω a / 2 π ≈ 4.3 G H z , through the input port C i n of the resonator, see Fig.  fig:setupa. The microwaves drive Rabi oscillations in the qubit at a frequency of ν R a b i = n s g / π , where n s is the average number of drive photons within the resonator. Simultaneously, we perform a continuous dispersive measurement of the qubit state by determining both the phase and the amplitude of a coherent microwave beam of frequency ω R F / 2 π = ω r / 2 π ≈ 5.4 G H z transmitted through the resonator . The phase shift φ = tan -1 2 g 2 / κ Δ σ z is the response of our meter from which we determine the qubit populat... We have determined the coherence time of the Cooper pair box from a Ramsey fringe experiment, see Fig.  fig:setupc, when biased at the charge degeneracy point where the energy is first-order insensitive to charge noise . To avoid dephasing induced by a weak continuous measurement beam we switch on the measurement beam only after the end of the second π / 2 pulse. The resulting Ramsey fringes oscillating at the detuning frequency Δ a , s = ω a - ω s ∼ 6 M H z decay with a long coherence time of T 2 ∼ 500 n s , see Fig.  fig:Ramseya. The corresponding qubit phase quality factor of Q ϕ = T 2 ω a / 2 ∼ 6500 is similar to the best values measured so far in qubit realizations biased at such an optimal point . The Ramsey frequency is shown to depend linearly on the detuning Δ a , s , as expected, see Fig.  fig:Ramseyb. We note that a measurement of the Ramsey frequency is an accurate time resolved method to determine the qubit transition frequency ω a = ω s + 2 π ν R a m s e y .
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fig:1fFig2 (a) Solid dots show temperature dependence of α 1 / 2 with a fixed bias current (the bias voltage is adjusted to keep the current constant). Open dots show α 1 / 2 derived from the measurement of qubit dephasing during coherent oscillations. The coherent oscillations (solid line) as well as the envelope exp - t 2 / 2 T 2 * 2 with T 2 * = 180 ps (dashed line) are in the inset. (b) Solid dots show temperature dependence of α 1 / 2 for the SET on GaAs substrate.... Note that at a fixed bias voltage the average current through the SET increases with temperature (see Fig.  fig:1fFig1(a)). However, it has almost no effect on the noise as we confirmed from the measurement of the current noise dependence. Nevertheless, to avoid possible contribution from the current dependent noise we adjust the bias voltage in the next measurements so that the average current is kept nearly constant at the measurement points for different temperatures. Fig.  fig:1fFig2(a) shows the temperature dependences of α 1 / 2 for a different sample with a similar geometry taken in the frequency range from 0.1 Hz to 10 Hz with a bias current adjusted to about I = 12 ± 2 pA. The straight line in the plot is α 1 / 2 = η 1 / 2 T , which corresponds to T 2 -dependence of α with η ≈ ( 1.3 × 10 -2 e / K ) 2 .... Solid dots in Fig.  fig:1fFig1(c) represent α 1 / 2 as a function of temperature. α 1 / 2 saturates at temperatures below 200 mK at the level of 2 × 10 -3 e and exhibits nearly linear rise at temperatures above 200 mK with α 1 / 2 ≈ η 1 / 2 T , where η ≈ 1.0 × 10 -2 e / K 2 (the solid line in Fig.  fig:1fFig1(c)). T 2 dependence of α is observed in many samples, though sometimes the noise is not exactly 1 / f , having a bump from the Lorentzian spectrum of a strongly coupled low frequency fluctuator. In such cases, switches from the single two-level fluctuator are seen in time traces of the current .... The typical current oscillation as a function of t away from the degeneracy point ( θ ≠ π / 2 ) is exemplified in the inset of Fig.  fig:1fFig2(a). If dephasing is induced by the Gaussian noise, the oscillations decay as exp - t 2 / 2 T 2 * 2 with... We use the qubit as an SET and measure the low frequency charge noise, which causes the SET peak position fluctuations. Temperature dependence of the noise is measured from the base temperature of 50 mK up to 900 - 1000 mK. The SET is normally biased to V b = 4 Δ / e ( ∼ 1 mV), where Coulomb oscillations of the quasiparticle current are observed. Figure fig:1fFig1(a) exemplifies the position of the SET Coulomb peak as a function of the gate voltage at temperatures from 50 mK up to 900 mK with an increment of 50 mK. The current noise spectral density is measured at the gate voltage corresponding to the slope of the SET peak (shown by the arrow), at the maximum (on the top of the peak) and at the minimum (in the Coulomb blockade). Normally, the noise spectra in the two latter cases are frequency independent in the measured frequency range (and usually do not exceed the noise of the measurement setup). However, the noise spectra taken on the slope of the peak show nearly 1 / f frequency dependence (see examples of the current noise S I at different temperatures in Fig.  fig:1fFig1(b)) saturating at a higher frequencies (usually above 10 - 100 Hz depending on the device properties) at the level of the noise of the measurement circuit. The fact that the measured 1 / f noise on the slope is substantially higher than the noises on the top of the peak and in the blockade regime indicates that the noise comes from fluctuations of the peak position, which can be translated into charge fluctuations in the SET.... The solid line in the inset of Fig.  fig:1fFig2(a) shows decay of coherent oscillations measured at T = 50 mK and the dashed envelope exemplifies a Gaussian with T 2 * = 180 ps. We derive α 1 / 2 from Eq. ( eq:Eq3) and plot it in Fig.  fig:1fFig2(a) by open dots as a function of temperature. The low frequency integration limit and the high frequency cutoff are taken to be ω 0 ≈ 2 π × 25 Hz and ω 1 ≈ 2 π × 5 GHz for our measurement time constant τ = 0.02 s and typical dephasing time T 2 * ≈ 100 ps .
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