Filter Results
56056 results
model We consider a composed system built of a qubit, -the system of interest-, coupled to a nonlinear quantum oscillator (NLO), see Fig. linearbath. To read-out the qubit state we couple the qubit linearly to the oscillator with the coupling constant g ¯ , such that via the intermediate NLO dissipation also enters the qubit dynamics.... Jeff The effective spectral density follows from Eqs. ( gl20) and ( chilarger). It reads: J s i m p l J e f f ω e x = g ¯ 2 γ ω e x n 1 0 4 2 Ω 1 | ω e x | + Ω 1 M γ 2 Ω 1 2 2 n t h Ω 1 + 1 2 n 1 0 4 + 4 M Ω 2 | ω e x | - Ω 1 2 . As in case of the effective spectral density J e f f H O , Eq. ( linearspecdens), we observe Ohmic behaviour at low frequency. In contrast to the linear case, the effective spectral density is peaked at the shifted frequency Ω 1 . Its shape approaches the Lorentzian one of the linear effective spectral density, but with peak at the shifted frequency, as shown in Fig. CompLorentz.... Schematic representation of the complementary approaches available to evaluate the qubit dynamics: In the first approach one determines the eigenvalues and eigenfunctions of the composite qubit plus oscillator system (yellow (light grey) box) and accounts afterwards for the harmonic bath characterized by the Ohmic spectral density J ω . In the effective bath description one considers an environment built of the harmonic bath and the nonlinear oscillator (red (dark grey) box). In the harmonic approximation the effective bath is fully characterized by its effective spectral density J e f f ω . approachschaubild... mapping The main aim is to evaluate the qubit’s evolution described by q t . This can be achieved within an effective description using a mapping procedure. Thereby the oscillator and the Ohmic bath are put together, as depicted in Figure approachschaubild, to form an effective bath. The effective Hamiltonian... The transition frequencies in Eqs. ( rc1) and ( rc2) coincide, and in Figs. Plowg and Flowg there is no deviation observed when comparing the three different approaches.... where the trace over the degrees of freedom of the bath and of the oscillator is taken. In Fig. approachschaubild two different approaches to determine the qubit dynamics are depicted. In the first approach, which is elaborated in Ref. [... Corresponding Fourier transform of P t shown in Fig. CompNLLP. The effect of the nonlinearity is to increase the resonance frequencies with respect to the linear case. As a consequence the relative peak heights change. CompNLLF... Schematic representation of the composed system built of a qubit, an intermediate nonlinear oscillator and an Ohmic bath. linearbath
Data Types:
• Image
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.
Data Types:
• Image
Data Types:
• Image
• Tabular Data
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.
Data Types:
• Image
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 .
Data Types:
• Image
Data Types:
• Image