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  • In the absence of high frequency excitation ( f = 0 ) we obtain from Eqs. ( ksi,  GammaT) (or from Eqs. ( ksi0,  GammaT0) a purely inductive response ( Γ T = γ T , P Z = 1 ), which describes the adiabatic evolution of the qubit in the ground state  . The experimental study of the adiabatic evolution provides us with information about the energy gap between the two levels . Therefore, the difference of the low frequency response of the persistent current qubit with and without high frequency excitation can provide additional information about the qubit’s damping rates, Γ and Γ Z (see Eqs.  ksi, GammaT). As an illustration of the method we show in Fig.  fig1 how the dependence of the tank phase χ on the flux changes with the application of the high frequency excitation. The graphs have been calculated for zero high frequency detuning ( δ = 0 ) from Eq. ( ksi0) for the following parameters: ω / 2 π = ω T / 2 π = 6  MHz, k = 0.03 , Q T = 2000 ; E J = 1.32 × 10 -22 J , E C = 2.14 × 10 -24 J , L q = 40  pH, I q = 280  nA, Δ / 2 π = 1  GHz, Γ Z / 2 π = 0.1  MHz, Γ / 2 π = 4  MHz, and T = 10  mK. At a relatively low power of the irradiation the form of the curves remains unchanged, with the amplitudes of the dips being conditioned by the factor P 0 in Eq. ( ksi0) (Fig.  fig1a). At higher powers the second term in the curled brackets of Eq. ( ksi0), which describes the influence of Rabi oscillations, comes into play. As a result, the forms of the phase curves are changed drastically (Fig.  fig1b).... The dependence of the tank phase on the flux when under the influence of high frequency irradiation. f = F / h is the power of irradiation in frequency units.
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  • quantum degenerate regime We now move to exploring what happens when the detuning between the qubit and the cavity is reduced such that the anharmonicity of quantum ladder of states becomes much larger than the corresponding linewidth κ (see Fig.  gino:fig:context). In order to describe the response of the system to external drive in this regime it is important to take into account the quantum dynamics on the lower anharmonic part of the ladder. When the system is initialized in the ground state, there is a range of drive strengths for which the system will remain blockaded from excitations out of the ground state. However, since the anharmonicity of the JC ladder decreases with excitation number, the transition frequency for excitations between adjacent levels ultimately approaches the bare cavity frequency. Qualitatively, when the excitation level n is such that the anharmonicity becomes smaller than the linewidth κ , we expect the state dynamics to be semiclassical, similar to a driven-damped harmonic oscillator... gino:chirp_figure (Color) Readout control pulse (a) Time trace of the drive amplitude: a fast initial chirp frequency chirp( 10 n s ) can selectively steer the initial state, while the qubit is detuned from the cavity ( ω q - ω c / 2 π ≈ 2 g ). It is followed by a slow displacement to increase contrast and lifetime of the latching state, while the qubit is resonant with the cavity ( κ / 2 π = 2.5 MHz ). The drive amplitude ramp is limited so that the photon blockade photon blockadeis not broken, but the contrast is enhanced by additional driving at the highest drive amplitude. (b) A diagram of transition frequencies shows how the drive frequency chirps through the JC ladder frequencies of the (+) manifold, and how the manifold changes due to the time dependent qubit frequency. (c) Wave packet snapshots at selected times (indicated by bullet points on panel (b)) of the chirping drive frequency of panel (b) conditioned on the initial state of the qubit. (d) The temporal evolution of the reduced density matrix | ρ m n | (the x , y axes denote the quantum numbers m , n of the cavity levels) of the cavity with the control pulse (a) when the qubit initial state is superposition 1 2 | 0 + | 1 . The resonator enters a mesoscopic state of superposition around t = t c due to the entanglement with the qubit and the quantum state sensitivity of the protocol. At later times the off-diagonal parts of this superposition dephase quickly due to the interaction with the environment and the state of the system is being completely projected around t = 3 t c .... strong!driving( t qubit being detuned. Due to the interaction with the qubit, the cavity behaves as nonlinear oscillator with its set of transition frequencies depending on the state of the qubit (see the two distinct sets of lines in Fig.  gino:chirp_figure(b)). The cavity responds with a ringing behavior which is different for the two cases (see Fig.  gino:chirp_figure(c)). The ringing due to the pulse effectively maps the | ↓ and | ↑ to the dim and bright state basins, respectively (see Fig. 3(c)). Since κ t c ≪ 1 , an initial superposition α | ↑ + β | ↓ maps into a coherent superposition of the dim and bright states. Next, (2) a much weaker long pulse transfers the initially created bright state (for initial | ↑ ) to even brighter and longer lived states ( t c t c effects a projection of the pointer state. In designing such a pulse sequence we have the following physical considerations: (a) the initial fast selective chirp... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift dispersive regimeof the second qubit; (c) for the model extended to one transmon qubit  koch charge-insensitive 2007, tuned below the cavity, / 2 π = 7 G H z E C / 2 π = 0.2 G H z , E J / 2 π = 30 G H z , g / 2 π = 0.29 G H z . (For the given parameters, δ 01 / 2 π = - 0.5 G H z , δ 12 / 2 π = - 0.7 G H z , defining δ i j = E j - E i - , with E i the energy of the i th transmon level.) In all panels, the transition frequency asymptotically returns to the bare cavity frequency. In (a) the frequencies within the σ z = ± 1 manifolds are (nearly) symmetric with respect to the bare cavity frequency. For (b), if the state of one (‘spectator’) qubit is held constant, then the frequencies are asymmetric with respect to flipping the other (‘active’) qubit. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon.... The solution of eqn  gino:eq:classic is plotted in Fig.  gino:fig:densclass for the same parameters as in Fig.  gino:fig:latch000b. For weak driving the system response approaches the linear response of the dispersively shifted cavity. Above the lower critical amplitude ξ C 1 the frequency response bifurcates, and the JC oscillator enters a region of bistability... Qubit state measurement in circuit QED Circuit QEDcan operate in different parameter regimes and relies on different dynamical phenomena of the strongly coupled transmon-resonator system strong!coupling. The dispersive readout is the least disruptive to the qubit state and it is realized where the cavity and qubit are strongly detuned. The high power readout operates in a regime where the system response can be described using a semi-classical model and yields an relatively high fidelity fidelitywith simple measurement protocol. When the cavity and qubit are on resonance (the quantum degenerate regime quantum degenerate regime) it is theoretically predicted that the photon blockade photon blockade can also be used to realize a high fidelity readout. gino:fig:context... matt-pc in Fig.  gino:fig:latch000, where we show the average heterodyne amplitude a as a function of drive frequency and amplitude. Despite the presence of 4 qubits in the device, the fact that extensions beyond a two-level model would seem necessary since higher levels of the transmons... Solution to the semiclassical equation  gino:eq:classic, using the same parameters as Fig.  gino:fig:latch000b. (a) Amplitude response as a function of drive frequency and amplitude. The region of bifurcation bifurcationis indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr oscillator (Duffing oscillator) Duffing oscillator, constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability bistability Kerr region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the frequency dependence of the classical solutions (solid line). For comparison, the response from the full quantum simulation of Fig.  gino:fig:latch000b is also plotted (dashed line) for the same parameters. (c) Cut through (a) for driving at the bare cavity frequency, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), showing intracavity amplitude on a linear scale. gino:fig:densclass... Transmitted heterodyne amplitude a as a function of drive detuning (normalized by the dispersive shift dispersive regime χ = g 2 / δ ) and drive amplitude (normalized by the amplitude to put n = 1 photon in the cavity in linear response, ξ 1 = κ / 2 ). Dark colors indicate larger amplitudes. (a) Experimental data  matt-pc, for a device with cavity at 9.07 G H z and 4 transmon qubits transmonat 7.0 , 7.5 , 8.0 , 12.3 G H z . All qubits are initialized in their ground state, and the signal is integrated for the first 400 n s ≃ 4 / κ after switching on the drive. (b) Numerical results for the JC model of eqn  gino:eq:master, with qubit fixed to the ground state and effective parameters δ / 2 π = - 1.0 G H z , g / 2 π = 0.2 G H z , κ / 2 π = 0.001 G H z . These are only intended as representative numbers for circuit QED Circuit QEDand were not optimized against the data of panel (a). Hilbert space is truncated at 10,000 excitations (truncation artifacts are visible for the strongest drive), and results are shown for time t = 2.5 / κ .... (Color) Symmetry breaking. State-dependent transition frequency ω n , q = 2 π E n + 1 , q - E n , q versus photon number n , where E n , q denotes energy of the system eigenstate with n photons and qubit state q : (a) for the JC model, parameters as in Figs.  gino:fig:latch000 and  gino:fig:densclass; (b) for the model extended to 2 qubits, δ 1 / 2 π = - 1.0 G H z , δ 2 / 2 π = - 2.0 G H z , g 1 / 2 π = g 2 / 2 π = 0.25 G H z . Here, χ 2 denotes the 0-photon dispersive shift
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  • To initial entangled state | I 0 of the two qubits with control field mode in its coherent state, its evolution involves on the various parameters in Eq.( t0-1). It depends on the number N in an extremely nonlinear and intricate way. The kind perplexity makes it hard to study the Rabi model, never the less, it also provide the opportunity to preserve the entanglement of the two qubits by careful choice of the appropriate parameters. Because the coherent state has the probability of Poisson distribution, which will be approximated by a Gauss distribution if the average number | α | 2 is large enough. The intricacy of the Rabi model could be utilized to make the quantity Y ~ N , + 2 L ~ N , + 4 = B N in Eq.( t0-1) extremely small when N is in the neighborhood of | α | 2 by some selection of the appropriate parameters, which will guaranty the initial state of the qubits unchanging. This is shown in Fig.( fig8). It can be easy to see that whenever we select the parameters appropriate, for example, the parameters as | α | 2 = 55 ,   a = - 0.6 ,   β = 0.5599 ,   ω 0 ω = 0.24 , the Bell state | I 0 = | 1 , 0 = 1 2 | ↑ ↑ - | ↓ ↓ have the probability about 1 - 0.005 = 99.5 100 to remain unchanged.... Figs.( fig11-3) show the general trait for the qubit remaining in its initial states | 1 , ± for different parameter a = 0.2 ,   0 ,   - 0.2 . Obviously, P N 1 t is influenced by four parameters β , a , N , ω 0 ω . In Ref., it is shown that the coupling strength β ranges from 0.01 to 1 for the application of adiabatic approximation (weak coupling will not be discussed here). From Eqs.( e0)-( t0),( p1- omega00), we see that the parameter a will come to action apparently whenever β ≈ 0.01 - 0.6 . As stated before, the qubits is equivalent to a two qubits system and the non-equal-energy-level parameter a represents the coupling strength between the two qubits. This shows the coupling of the two qubits changes their dynamics considerately in the range of β ≈ 0.1 - 0.6 for the adiabatic approximation method to be applied, and this is our limit on the coupling parameter β... Schematic diagram of P N 1 t with the four parameters as N = 2 ,   ω 0 ω = 0.25 ,   β = 0.2 , a = 0.2 ,   0 ,   - 0.2 from the top to bottom respectively. The apparent difference in these three figures strongly implies that the parameter a influences the qubits dynamically.... The parameter a is connected with the inter-qubit coupling strength κ as a = ± κ in symmetric and asymmetric transition cases respectively. Study also shows that the parameter a negative is favorable for | T α t approaching zero, as Fig.( fig8) exhibits. So the inter-qubit coupling is in favor of preservation of the initial entanglement, especial with asymmetrical transition case ( a < 0 ).
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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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  • (a) Amplitude of the r f tank voltage as a function of the d c magnetic bias of the qubit, f x d c , for a microwave driving frequency ω d = 2 π × 14.125 GHz and various amplitudes of the r f current driving the tank circuit. (b) The height of the peaks (solid squares) and dips (solid circles) as a function of the voltage amplitude of the unloaded tank circuit V T 0 . The voltage V T 0 is equal to, e.g., V T taken at f x d c = 0.02 where the effect of the interaction between the tank circuit and the qubit is negligible. The height saturates near 200  nV. Fig:spectr_ampl... (a) The energy levels of the qubit as a function of the energy bias of the qubit ε f x = 2 Φ 0 I p f x . The sinusoidal current in the tank coil, indicated by the wavy line, drives the bias of the qubit. The starting point of the cooling (heating) cycles is denoted by blue (red) dots. The resonant excitation of the qubit due to the high-frequency driving, characterized by Ω R 0 , is indicated by two green arrows and by the Lorentzian depicting the width of this resonance. The relaxation of the qubit is denoted by the black dashed arrows. The inset shows a schematic of the qubit coupled to an LC circuit. The high frequency driving is provided by an on-chip microwave antenna. (b) SEM picture of the superconducting flux qubit prepared by shadow evaporation technique.... Color lines: experimental data. Black lines: numerical solution of Eqs. ( Eq:master) for Ω R 0 = 2 π × 0.5  GHz, Γ R = 1.0 ⋅ 10 8 s -1 , Γ ϕ * = 5.0 ⋅ 10 9 s -1 . The central dip is due to the quadratic coupling term in ( eq:H_RWA) which causes a shift of the oscillator frequency .... (a) Spectral density of the voltage noise in the LC circuit, S V ω , measured when the low-frequency r f -driving is switched off while the high-frequency qubit driving is fixed at ω d = 2 π × 6  GHz. S V is shown as a function of frequency (vertical axis) and normalized magnetic flux in the qubit f x d c (horizontal axis). (b) The integral, ∫ S V ω d ω , evaluated using the data from panel (a) as a function of f x d c . The integral is directly proportional to the number of quanta (effective temperature) of the tank circuit. At optimal dc bias f x d c the effective temperature T of the oscillator is lower than the temperature T 0 away from the resonance, T 0 - T / T 0 = 0.08 , corresponding to an 8 % cooling. (c) Spectral density S V ω measured at three different values of f x d c corresponding to damping (blue), amplification (red), and away from the resonance (black). These values of f x d c are marked, respectively, by blue, red, and black arrows in panel (a).
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  • Frequency spectrum for a particle coupled to a bath of N c = 20 modes with α k = 0.1 and the level spacing ℏ δ ω / 2 V = 2 (diamonds), 0.15 (circles), and 0.08 (squares).... Particle in a double-well potential coupled to N harmonic oscillators. States localized around the two potential minima are denoted by | L and | R , respectively.... The kernel G t for the bath with N c = 10 modes (dotted) and N c = 20 modes (solid curve) and the coupling strength α k = 0.1 (top) and α k = 0.2 (bottom). The frequencies of the modes are assumed equidistant, ω k = k δ ω .... eq:nuRenorm, while the low energy modes determine the details of the particle dynamics. The choice of the cutoff frequency ω c is nonessential provided it is much larger than the frequency of particle tunneling, ω c ≫ 2 V / ℏ (see Fig.  fig:antiadiabatic and Appendices app:1 and app:2). In this case, the system is described by the Hamiltonian in Eq. ... eq:alpha k, respectively (see inset of Fig.  fig:plSofTjjChain). The non-adiabatic dynamics of the qubit is shown in Fig.  fig:plSofTjjChain obtained by numerical simulation of Eq.... Embedding the system into a superconducting loop makes a flux qubit which enables the study of coherent quantum dynamics between few quantum states, provided the system is sufficiently decoupled from external environment, see Fig.  fig:fig1(a). One of the first examples was the flux qubit made of a superconducting loop with a few Josephson junctions biased with an external magnetic flux. In such system, the two distinguishable macroscopic states with supercurrents circulating in opposite directions exhibit coherent oscillations due to mutual coupling via quantum tunneling characterized by a quantum amplitude V . A qubit made of a one-dimensional homogeneous chain of Josephson junctions has been studied recently. Similarly, the quantum phase slips in superconducting nanowires of finite size can also exhibit coherent quantum dynamics when the wire is embedded in a superconducting loop threaded by external magnetic flux.... Non-adiabatic dynamics of a phase-slip qubit made of a Josephson junction chain with a weak element. Parameters are C ̄ / C = 0.1 , C 0 / C = 0.05 , Z J / R q = 0.18 , N = 100 , and ℏ ω p / V ̄ = 3 . Inset: Dispersion ω k (circles, left axis) and the coupling constants α k 2 (squares, right axis) of the modes in the chain.... eq:nuRenorm with the sum including all the modes (see Appendix app:1). In this case the dynamics corresponds simply to coherent oscillations shown Fig.  fig:plSofT(a). As the density of the modes is increased, several frequencies Ω m start to contribute, with amplitudes R m shown in Fig.  fig:plRmOm. In the weak coupling regime which we consider, the particle still oscillates between the two minima with the frequency 2 V / ℏ corresponding to the fast oscillations in Figs.  fig:plSofT(b) and (c). The amplitude of these oscillations initially decays as the bath modes are populated and the energy is transferred from the particle to the bath. The decay time is τ d ∝ ℏ 2 / V 2 ∑ k α k 2 ω k 2 / ∑ k α k 2 . However, after time τ r = 2 π / δ ω , the populated bath modes start to feed energy back to the particle and revivals of oscillations take place. From that point on, we have two different behaviors depending on the ratio τ d / τ r . For τ d τ r , the dynamics of a particle has a form of a quasiperiodic beating instead of a decay. Reducing τ d ≪ τ r , the dynamics exhibits again a decay after a revival of the oscillation amplitude. For a dense continuum of bath modes ( N c ∞ , δ ω 0 ) the revival time is infinite, τ r ∞ . In this case the bath cannot feed significant amounts of energy back to the particle and one recovers exponentially damped oscillations characteristic for Ohmic dissipation.... where c 0 = exp - ∑ k ≥ 1 α k 2 . Thus, there is a single pole 2 V / ℏ at low-frequencies (see Fig.  fig:plRmOm for ℏ δ ω / 2 V = 2 ) whereas the other poles are relevant only at higher frequencies ∼ δ ω . In the time domain, Eq. ... (colors online) Bath energy spectrum in the non-adiabatic regime where several discrete modes have frequencies smaller or comparable to the tunnelling frequency 2 V / ℏ .
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  • In a first set of experiments the bigger qubit was studied. A driving signal at the fifth harmonic of the resonator ω d = 5 ω r with a constant power was chosen. (We note that considering the fifth harmonic instead of the third does not change any point in the theoretical description of the system except of putting different values for the driving frequency.) The transmission was measured at small detunings δ ω r from the resonators fundamental mode, while the energy bias ε 0 of the qubit was varied. The experimental results are shown in Fig.  Fig:expic1 (a) together with simulations using Eq. ( trans) Fig.  Fig:expic1 (b). It can be seen that a good agreement between the two was achieved for a relaxation rate Γ 1 / 2 π = 4  MHz and a pure dephasing Γ φ / 2 π = 200  MHz.... (Color online) Normalized transmission amplitude through the resonator at a probing frequency ω p = ω r for different driving amplitudes ranging from about -131  dBm in (a) to -117  dBm in (g) in 2  dBm steps at the input of the resonator. The transmission is plotted as a function of the energy bias ε 0 of the second qubit and the driving frequency detuning δ ω d = ω d - 3 ω r . The driving frequency is changed only around the third harmonic in the order of its linewidth (about 6 κ ). The presented results correspond to a symmetric power dependence around the center frequency of the third harmonic, since the resonator acts as bandpass filter. Each plot is split into an experimental (positive detuning) and a theoretical (negative detuning) part. For the calculations the figures were split into regions for which Eq. ( trans) was used with a certain index k in Eq.( need_Ref). Several features were highlighted with black rectangles to interrelate theory and experiment (see text).... (Color online) (a) Measured normalized transmission amplitude t for the first qubit (see text) while a strong driving signal is applied. The results are plotted in dependence on the detuning between qubit frequency and driving signal, (controlled by the qubit bias ε 0 ), and the probing frequency detuning δ ω r = ω p - ω r . The amplification and the attenuation of the transmitted signal is found in agreement with the resonance conditions, ℏ ω p = Δ E ˜ , Eq. ( (i)). (b) Normalized transmission calculated for the same parameters as in (a) following Eq. ( trans) together with Eq. ( a).... We find a quantitative agreement between the theoretical predictions and the experiment for a relaxation rate Γ 1 2 / 2 π = 6  MHz and a pure dephasing Γ φ 2 / 2 π = 100  MHz of the second qubit. As examples we highlight several regions with black rectangles. Direct resonances between the Rabi levels are marked in plot (a) for k = 1 and in (d) for k = 2 . Note, that for k = 1 no amplification is observed since the driving frequency is chosen below the qubit gap. In Fig.  Fig:expic2 (d) the power dependence of the resonance lines for k = 2 show a similar dependence as in Ref. [... (Color online) Bare, dressed, and doubly-dressed energy levels. The bare qubit’s energy levels, ± Δ E / 2 , are shown in (a); when they are matched by the driving frequency ω d , the qubit is resonantly excited. (At higher values of the bias ε 0 , the bare-qubit multiphoton excitation should be studied.) The position of the resonance, ℏ ω d = Δ E , is described by the avoided crossing of the dressed-state levels; the dressed and averaged energy levels, ± Δ E ˜ / 2 = ± ℏ Ω R / 2 , are plotted in panel (b). When the dressed energy levels are matched by the second (probe) signal, ℏ ω p = Δ E ˜ , a resonance interaction of the coalesced system is expected, Eq. ( (i)). Also a resonance condition is given by the two-photon process ( (ii)). This is visualized as the avoided crossings of the doubly-dressed states, plotted in panel (c).... Oelsner12]: Δ / h = 3.7 GHz, ω r / 2 π = 2.59 GHz and ω d / 2 π = 3 ω r / 2 π = 7.77 GHz, A d / h = 7 GHz, g 1 / 2 π = 0.8 MHz, and ω p = ω r . Figure  Fig:En_levels can be seen as the graphical description of dressing the dressed qubit, which can be considered as the mesoscopic tunable analogue of the atomic systems, as in Ref. [... In order to test our model in the strong driving regime, where Eq. ( need_Ref0) is replaced by Eq.( need_Ref), we analyze the response of the system as a function of the driving amplitude. Here we consider the case where the qubit gap is higher than the driving frequency. For this purpose the smaller qubit was used. The transmission at the fundamental mode, ω p = ω r , was measured while changing the frequency of the driving signal around the third harmonic frequency, and consequently the driving amplitude at the qubit. The results are shown in Fig.  Fig:expic2, together with calculated data following from Eq. ( trans). Several sharp lines of amplification (dark) and damping (light) were experimentally observed. The number of lines increases with increasing power and each of the lines corresponds to a resonance condition between the dressed states and the probing signal. To understand their origin, calculated transmission data was added into each plot. We split these theoretical plots into regions and for each of them use Eq. ( trans) with different index k in Eq. ( need_Ref). For high powers we note that the levels of one step of the dressed ladder are equivalent to the ones of the stairs above or below. In that way, we assume that Eq. ( trans) is valid also for those regions where resonances between the lower level of one stair and the higher level of the lower stair occur. To account for these interaction we replace the splitting of the dressed states Δ E ˜ k by ω d - Δ E ˜ k . This makes it possible to relate each of the resonance lines to one index k and to an interaction directly between the Rabi levels or levels of different stairs.
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  • (Color online) The transition probability as a function of r.f frequency ω . Initial bath states with different polarizations are considered. The resonant frequency varies with the bath polarization as ω = ω 0 + g N P B / 2 . In plotting the above we have taken N = 20 , ω 0 = 100 , ω 1 = 10 and g / ℏ = 1 . The frequencies are in the units of MHz.... (Color online) Concurrence with time. The time-dependent concurrence, for the Bell-states | B 1 = 1 2 | ↑ ↓ + | ↓ ↑ | B 2 = 1 2 | ↑ ↑ + | ↓ ↓ , | B 3 = 1 2 | ↑ ↑ - | ↓ ↓ , is plotted for two values of Δ ω = ω - ω 0 . The loss of entanglement is slowest at the resonant frequency δ ω = 0 , for all the states, and it increases as the r.f frequency shifts away from ω 0 . The bath is unpolarized with uniform coupling strengths between the qubits and bath-spins. In plotting the above we have taken N = 20 , ω = 100 , ω 1 = 10 and g / ℏ = 1 . The frequencies are in the units of MHz.... (Color online) Concurrence with time. The time-dependent concurrence for an initial singlet shared between two non-interacting qubits is plotted for three different cases, (i) when the r.f frequencies are tuned to their resonance values i.e., δ ω 1 2 = 0 , (ii) the r.f frequencies are slightly away from resonance and (iii) much away from their resonant frequencies. In the above δ ω 1 2 = ω 0 1 2 - ω 1 2 , where ω 0 1 2 , are the fields at the local sites of the qubits. The initial states of the baths are unpolarized each consiting of N = 20 spins and the local fields experienced by the qubits are ω 0 1 = 100 , ω 0 1 = 110 respectively.... (Color online) Probability distribution of the resonant frequencies given in Eq. resf. The bath consists of N = 20 spins. The distributions for the qubit-bath couplings are normalized such that ∑ k g k = g , where g / ℏ = 20 MHz. In plotting the above we have set ω = ω 0 .... (Color online) The z -component of qubit polarization with time. The plot shows the decay of Rabi oscillations with time. The bath is unpolarized consisting of N = 20 spins. Two different distributions for the qubit-bath couplings are considered, where ∑ k g k = g for both the distributions. In plotting the above we have taken ω 0 = 10 3 , ω 1 = 10 and g / ℏ = 40 , which are in the units of MHz
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  • (Color online) Time evolution of the coherence σ x t versus the time t multiplied by the qubit energy spacing Δ . (a) The case of weak interaction between the bath and the qubit, where the parameters of the low-frequency Lorentzian-type spectrum are α / Δ 2 = 0.01 ,   λ = 0.09 Δ  (red solid curve); while for the high-frequency Ohmic bath with Drude cutoff the parameters are α o h = 0.01 , ω c = 10 Δ (green dashed-dotted curve). (b)The case of strong interaction between the bath and the qubit, where the parameters of the low-frequency bath are α / Δ 2 = 0.1 ,   λ = 0.3  (red solid line), and for the high-frequency Ohmic bath are  α o h = 0.1 , ω c = 10 Δ (green dashed-dotted line). These results show that the decay rate for the low-frequency bath is shorter than for the high-frequency Ohmic bath. This means that the coherence time of the qubit in the low-frequency bath is longer than in the high-frequency noise case, demonstrating the powerful temporal memory of the low-frequency bath. Also, our results reflect the structure of the solution with branch cuts  . The oscillation frequency for the low-frequency noise is ω 0 > Δ , in spite of the strength of the interaction. This can be referred to as a blue shift. However, in an Ohmic bath, the oscillation frequency is ω 0 frequency noises.... (Color online) The spectral density J ω of the low- and high-frequency baths. (a) The case of weak interaction between the bath and the qubit, where the parameters of the low-frequency Lorentzian-like spectrum are α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve), while for the high-frequency Ohmic bath with Drude cutoff the parameters are α o h = 0.01 and ω c = 10 Δ (green dashed-dotted curve). (b) The case of strong interaction between the bath and the qubit, where the parameters of the low-frequency bath are α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve) and the parameters of the high-frequency Ohmic bath are α o h = 0.1 and ω c = 10 Δ (green dashed-dotted curve). The characteristic energy of the isolated qubit is indicated by a vertical blue dotted line. Here, and in the following figures, the energies are shown in units of Δ .... (Color online) The effective decay γ τ / γ 0 , versus the time interval τ between successive measurements, for a strong coupling between the qubit and the bath. The time-interval τ is multiplied by the qubit energy difference Δ . The curves in (a) correspond to the case of a low-frequency bath with parameters α / Δ 2 = 0.1 and λ = 0.3 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.1 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ .... (Color online) The effective decay γ τ / γ 0 , versus the time interval τ between consecutive measurements, for a weak coupling between the qubit and the bath. In the horizontal axis, the time-interval τ is multiplied by the qubit energy difference Δ . The curves in (a) correspond to the case of a low-frequency bath with parameters α / Δ 2 = 0.01 and λ = 0.09 Δ (red solid curve). (b) corresponds to the case of an Ohmic bath with parameters α o h = 0.01 and ω c = 10 Δ (red solid curve). The green dashed-dotted curves are the results under the RWA when the same parameters are used. Note how different the RWA result is in (b), especially for any short measurement interval τ .
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  • To show the performances of the optimal effective current operator I ~ φ 4 , Fig. fig:Ip depicts the numerical data of I ~ p 2 , I ~ p 4 and I p vs. β on the cases of α 3 = 0.6 and α 3 = 0.8 . There is no doubt that I ~ φ 4 perfectly achieves the results in a high precision regardless of α 3 ; even when the inductance has a non-negligible size ( β ≃ 1 ), it can also correctly predict the profiles of the I p - β curves. These curves resemble their classical counterparts I q in Fig. fig:classical(a), which infers us that the shifts of the classical potential minima introduced by a large inductance also take significant roles in the quantum regime. Compared to I p and I ~ p 4 , I ~ p 2 without full O β corrections fails to describe I p when the influences imposed by the inductance become notable, e.g. β > 0.01 , which also emphasizes that the O β effects dominate in this region. In fact, the inductive energy term on O β in H ~ e f f 1 tends to make itself minimized averagely in a relatively large β region which forces I ~ p 2 to rise too pronouncedly to approximate the real value I p . As mentioned above, the vacuum fluctuations of the LC oscillator bring in the O β 1 / 2 effects and, thus, reduce the effective sizes of the junctions. Therefore, the currents are expected to decline when β is small enough to make the O β effects negligible, which is also confirmed by the inset of Fig. fig:Ip. When α 3 = 0.8 , since the net O β effects also depress the currents ( see I p when β > 0.01 ), I p and I ~ p 4 both monotonously decrease in the whole region. On the other hand, lacking full O β effects, I ~ p 2 | α 3 = 0.8 increases in the large β region, so there exists a minimum at β ≃ 0.01 in the corresponding curve when the O β 1 / 2 and O β effects strike a balance. For α 3 = 0.6 , minima are also found to show the balances between the opposite O β 1 / 2 and O β effects. Both of those two types of minima support our previous conclusion that β ≃ 10 -3 ∼ 10 -2 is the watershed to distinguish the region dominated by the vacuum fluctuations. Figure fig:err demonstrates the β -dependence of the errors which Δ ~ and I ~ p 4 bear. The linear fitting indicating that these errors are approximately on O β 1.5 sufficiently verifies our analytic conclusions.... (a)circuit of an inductive flux qubit with the phase difference φ across the loop inductance L , the reduced applied external flux φ X = 2 π Φ X / Φ 0 with Φ 0 the flux quantum, and the phase difference φ k across the k th junction characterized via the critical current I C k and the capacitance C k for k =1,2 and 3; (b) transformation between the current and voltage sources, the arrow and the plus/minus symbols indicate the directions of the current and voltage sources, respectively.... Energy diagram of flux qubit with a loop inductance. When the inductance-free flux qubit and the LC oscillator interact with each other in a perturbation condition, the lowest eigenstates in the dressed state manifold M 0 denoted with the dashed-line box are well separated from the ones in other manifolds M 1 , M 2 , ⋯ due to the large shifting caused by the LC-photon energy ℏ ω L C .... Therefore, the effective Hamiltonian H ~ e f f 3 / 2 has taken account of four corrections of different types to the unperturbed one H ̂ 0 . Its complicated expression indicates that treating the LC-oscillator as a three-level system does not stand as an easy task on the derivations and analysis. First of all, unlike common perturbation situations where two subsystems couple with each other via a weak linear interaction, the Josephson junctions exhibiting as nonlinear inductances keep the interaction H ̂ i n t in Eq.( eq:H_int) split into the couplings of different strengths. For inst... Figure fig:delta plots 2 Δ and 2 Δ ~ as functions of β based on α 3 = 0.6 and 0.8 . When β is small enough, e.g., β oscillator actually reduce the effective sizes of the Josephson junctions, thus suppressing the barriers and enhancing the interactions between these two flux states. On the other hand, the self-biased inductive effects like - L I 2 / 2 on O β increase the barriers and slow down the current direction switching speed. As a numerical o r d e r prediction, we have those two characteristic factors equal as γ 1 2 ≃ β and get a critical value β ∼ 10 -3 agreeing with the data of the inset. As β becomes larger, a clear tunnel rate damping means that the self-biased effects grow up to a non-negligible level. When β > 1 , 2 Δ is more than one order of magnitude smaller than its inductance-free value, and the effective result 2 Δ ~ decays more excessively than 2 Δ does; in this situation, a small Δ means that two flux states of the flux qubit interact with each other weakly and slowly, rendering that the whole system fails to act as a useful qubit in a larger α 3 such as α 3 = 0.8 , but α 3 = 0.6 only makes the flux qubit slow down which may benefit the design on it with a large loop inductance. It is a pleasure that when the effective Hamiltonian on O β fails to calculate the inductive effects that involve higher excited levels of the oscillator, the three-phase system with a set of traditional design parameters may no longer perform as a good qubit.... Tunnel splitting of the flux qubit, in units of E J 0 , vs. the reduced inductance β . Parameter α 3 is selected to be equal to two typical values 0.6 and 0.8 while others are α 1 = 1 , α 2 = 1 , g = 80 and f = 0.5 . The inset with the same symbols re-scales the range of β and draws the percent changes of the numerical results to the corresponding values on β = 0 .... The schematic circuit for the 3jj flux qubit with a loop inductance is demonstrated in Fig. fig:circuit(a), where the 3rd junction is a little smaller than those two others; representing the relative sizes, the parameter α k as
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