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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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Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the double-degeneracy point X for the case EJ1=9.1GHz and EJ2=13.4GHz. Right panels show the corresponding spectra obtained by Fourier transformation. Arrows and dotted lines indicate theoretically expected position of the peaks. ... EJ1 dependence of the spectrum components of Fig. 6. Solid lines: dependence of Ω+ε and Ω−ε obtained from Eq. (6) using EJ2=9.1GHz and Em=14.5GHz and varying EJ1 from zero to its maximum value of 13.4GHz. Dashed lines: dependence of the oscillation frequencies of both qubits in the case of zero coupling (Em=0). ... Schematic diagram of the two-coupled-qubit circuit. Black bars denote Cooper pair boxes. ... Probe current oscillations in the first (a) and the second (b) qubit when the system is driven non-adiabatically to the points R and L, respectively. Right panels show the corresponding spectra obtained by the Fourier transform. Peak position in the spectrum gives the value of the Josephson energy of each qubit, indicated by arrow. In both cases, the experimental data (open triangles and open dots) can be fitted to a cosine dependence (solid lines) with an exponential decay with 2.5ns time constant. ... Solid-state qubits
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The persistent current is not, however, the only phase-dependent quantity characterizing the quantum state of the charge-phase qubit. Another useful quantity is the Josephson inductance of the double junction, which can be probed by small rf oscillations induced in the qubit. Recently, we proposed a transistor configuration of the Cooper pair box (see Fig.  Scheme), in which the macroscopic superconducting loop closing the transistor terminals was inductively coupled to a radio-frequency tank circuit . Similar to the rf-SQUID-based method of measurement of the Josephson junction impedance , this setup makes it possible to measure the rf impedance (more specifically, the Josephson inductance) of the system of two small tunnel junctions connected in series and in doing so, to probe the macroscopic states of the qubit.... (a) The electric circuit diagram of the charge-flux qubit inductively coupled to a tank circuit by mutual inductance M . The macroscopic superconducting loop of inductance L is interrupted by two small Josephson tunnel junctions positioned close to each other and forming a single charge transistor; the capacitively coupled gate polarizes the island of this transistor. The qubit is controlled by charge Q 0 generated by the gate and flux Φ m induced by coil L m . The tank circuit which is either of a parallel (b) or a serial (c) type is driven by a harmonic signal ( I r f or V r f , respectively) of frequency ω r f ≈ ω 0 , the resonant frequency of the uncoupled tank circuit.... Evaluated qubit parameters derived on the assumption E J 0 = 2 E c = 80 μ eV (i.e., I c 0 ≈ 40  nA and 5 E c = Δ A l ≈ 200 μ eV, the energy gap of Al) and j 1 - j 2 = κ 1 - κ 2 = 0.1 . The tank circuit quality factor Q = 100 , frequency ω 0 = 2 π × 100  MHz, L T / C T 1 / 2 = 100 Ω , k 2 Q β L = 20 and temperature T * = 1  K ≫ T ∼ 20  mK. As long as the dephasing rate in the magic points is nominally zero, a 0.1% inaccuracy of the adjustment of the values φ = π and 0 was assumed.... The principle of narrow-band radio-frequency readout of the qubit. (a) The resonance curves of the uncoupled (dotted line) and coupled to the qubit tank circuit biased at operation point A in the excited state (dashed line) and in the ground state (solid line). (b) Driving pulse applied to the tank circuit (top curve) and the response signal of the tank in resonance (the ground qubit state, bottom curve) and off-resonance (excited state, middle curve). A smooth envelope of the driving pulse is used to suppress transient oscillations and has a small effect on the rise time of the response signal. For clarity the curves are shifted vertically.... The small tunnel junctions of the charge-flux qubit are characterized by self-capacitances C 1 and C 2 and the Josephson coupling strengths E J 1 and E J 2 . These junctions with a small central island in-between and a capacitively coupled gate therefore form a single charge transistor connected in our network as the Cooper pair box (see Fig.  Scheme). Critical currents of the junctions are equal to I c 1 , c 2 = 2 π Φ 0 E J 1 , J 2 , where Φ 0 = h / 2 e is the flux quantum, and their mean value I c 0 = 1 2 I c 1 + I c 2 . The design enables magnetic control of the Josephson coupling in the box in a dc SQUID manner. The system therefore has two parameters, the total Josephson phase across the two junctions φ = ϕ 1 + ϕ 2 = 2 π Φ / Φ 0 controlled by flux Φ threading the loop and the gate charge Q 0 set by the gate voltage V g . The geometrical inductance of loop L is assumed to be much smaller than the Josephson inductance of the junctions L J 0 = Φ 0 / 2 π I c 0 ,... Terms composed of diagonal (a) and off-diagonal (b) matrix elements of operators cos χ and sin χ , respectively, calculated for different values of flux Φ e for the given qubit parameters (see caption of Fig. 2).... The terms composed of diagonal (a) and off-diagonal (b) matrix elements of operator V ̂ and entering Eqs. ( Vmod) and ( Vtan) are presented for different values of flux Φ e for the given qubit parameters (see caption of Fig. 2).... is nonzero for any Q 0 and Φ (see the plots of the two items in Fig.  e2ME), only the condition E c ≤ Δ s c / 5 can ensure suppression of these transitions in arbitrary operation point of our qubit. Possibly insufficiently small value of E c was the reason of very short relaxation time (tens of ns) in the recent experiment with a charge qubit by Duty et al. . Their Al Cooper pair box had E c ≈ 0.8 Δ s c and E J ≈ 0.4 E c , so the energy gain in the chosen operation point ( Q 0 = 0.4 e ) was too large, i.e., about 2.2 E c ≈ 1.8 Δ s c > Δ s c (although in the ground state this sample nicely showed the pure Cooper pair characteristic).... For reading out the final state, the qubit dephasing is of minor importance, while the requirement of a sufficiently long relaxation time is decisive. Moreover, the relaxation rate may somewhat increase due to oscillations in the tank induced by a drive pulse (see Fig.  Pulse), which leads to the development of oscillations around a magic point along φ axis, Eq. ( phase_osc). If the frequency of these oscillations is sufficiently low, ω r f ≪ Ω , they result only in a slow modulation of transition frequency Ω . Increase in amplitude of steady oscillations up to φ a ≈ π / 2 (determined by the amplitude of the drive pulse and detuning) yields a large output signal and still ensures the required resolution in the measurement provided the product k 2 Q β L > 1 is sufficiently large. (At larger amplitudes φ a , the circuit operates in a non-linear regime probing an averaged reverse inductance of the qubit whose value, as well as the produced frequency shift δ ω 0 , is smaller .) As points A and B lie on the axis Q 0 = 0 and are both characterized by a sufficiently long relaxation time, reading-out of the qubit state with the rf oscillation span ± π / 2 is preferable in either point. In the case of operation point C , the limited amplitude of the oscillations does not reduce much the relaxation time either. Significant reduction of the relaxation time occurs in the vicinity of point D . Due to this property which is due to the dependence of the transversal coupling strength on φ , Eqs. ( I-2)-( tan-n), the measurement of the Quantronium state using a switching current technique was possible in the middle of segment C D (see Fig. 2), where the maximum values of the circulating current in the excited and ground states were of different sign .... The two lowest energy levels E n q φ , i.e., n = 0 and 1 (see their dependencies on q and φ in Fig.  Eigenenergies), form the basis | 0 | 1 suitable for qubit operation. In this basis, the Hamiltonian ( H0) is diagonal,
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Structured bath/weak coupling: We now turn to the structured spectral density given by Eq.( eq:density). The main features of the corresponding system [Eq.( eq:system)] can already be understood by analyzing only the coupled two-level-harmonic oscillator system (without damping, i.e. Γ = 0 ). For ε = 0 this system exhibits two characteristic frequencies, close to Ω and Δ , associated with the transitions 1 and 2 in Fig.  fig:correlation_weak(c). These should also show up in the correlation function C ω ; and indeed Fig.  fig:correlation_weak(a) displays a double-peak structure with the peak separation somewhat larger than Δ - Ω , due to level repulsion. The coupling to the bath will in general lead to a broadening of the resonances and an enhancement of the repulsion of the two energies. Due to the very small coupling ( α = 0.0006 ) peak positions of C ω in Fig.  fig:correlation_weak can with very good accuracy be derived from a second order perturbation calculation for the coupled two-level-harmonic oscillator system, yielding the following transition frequencies [depicted in inset (c) of Fig.  fig:correlation_weak]: ω 1 , + - ω 0 , + = Ω - g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 0.987 Ω and ω 0 , - - ω 0 , + = Δ 0 + g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 1.346 Ω . With the two peaks we associate two different dephasing times, τ Ω and τ Δ , as shown in inset (a) and (b) of Fig.  fig:correlation_weak.... Spin-spin correlation function as a function of frequency for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an harmonic oscillator, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm... Stronger coupling to bath: Figure  fig:times_nobias(b) shows τ Δ , τ Ω , and τ w for a larger coupling strength of α = 0.01 . Figure  fig:correlation_strong(a) shows one of the calculated correlation functions. Note that the stronger coupling α leads to a larger separation, or “level repulsion”, between the Δ - and Ω -peaks than in Fig.  fig:correlation_weak. The inset of Fig.  fig:times_nobias(b) shows the renormalized tunneling matrix element Δ ∞ as a function the initial matrix element Δ 0 . Very importantly, for Δ 0 Ω , Δ increases during the flow, whereas for Δ 0 Ω , it decreases . This behavior can be understood from the fact that f ω l in Eq.( eq:flow1) changes sign at ω = Δ : If the weight of J ω under the integral in ( eq:flow1) is larger for ω > Δ 0 , which is the case if Δ Δ 0 ]. Note also, that the upward renormalization towards larger Δ ∞ in the inset of Fig.  fig:times_nobias(b) is stronger than the downward one towards smaller values, i.e., the renormalization is not symmetric with respect to Δ 0 = Ω . The reason for this asymmetry lies in the fact that f ω l has a larger weight for ω Δ . Also τ Δ and even τ w = 1 / J Δ ∞ in Fig.  fig:times_nobias(b) show an asymmetric behavior with a steep increase at Δ 0 ≈ Ω : dephasing times for Δ 0 > Ω are larger than for Δ 0 Ω than for Δ 0 Ω , dephasing times can be significantly enhanced (as compared to Δ 0 oscillator in (b).... Spin-spin correlation function for the structured bath [Eq.( eq:density)] as a function of frequency . The maximum height of the middle peak in (b) is ≈ 7.2 . -0.9cm
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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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In this letter we will present a three island device that has the properties of a qubit (two levels, arbitrary control and measurement) and has the ability to independently tune both the resonance frequency and coupling strength g , whilst still exhibiting exponential suppression of the charge noise and maintaining an anharmonicity equivalent to that of the transmon and phase qubit. This tunable coupling qubit (TCQ) can be tuned from a configuration which is totally Purcell protected from the resonator g = 0 (in a DFS) to a position which couples strongly to the resonator with values comparable to those realized for the transmon. Furthermore, we show that in the DFS position a strong measurement can be performed. The TCQ only needs to be moved from the DFS position when single and two qubit gates are required and as such in the off position all multi-qubit coupling rates are zero. That is, the TCQ in the circuit QED architecture (see Fig. Fig:TCQ A) is its own tunable coupler to any other TCQ, going beyond the nearest neighbour tunable couplers presented in Refs. and .... Fig:Tune(color online) Matrix element of the collective Cooper pair number operator (A) and transition energy (B) of the dark state as a function of the energy ratio E J + / E C for n ' g + = n ' g - = 0 , E I = - E C and E J - is numerically solved to ensure that only the coupling strength is tuned (blue) and frequency (red). Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic oscillator model.... Fig:Levels(color online) A) Eigenenergies of the TCQ Hamiltonian as a function of E I / E C for E J ± = 50 E C . Solid lines are from a numerical diagonalization and dashed lines are from the coupled anharmonic oscillator model. B) Charge dispersion | ε q m | as a function of the ratio E J / E C for E I = - E C (solid lines) and E I = 0 (dashed lines).... Since charge fluctuations are one of the leading sources of noise in superconducting circuits we want to ensure that quantum information in the TCQ is not destroyed by charge noise. Following Ref. , the dephasing time T φ for the qubit and m level will scale as 1 / | ε q m | where ε q m is the the peak to peak value for the charge dispersion of the 0 - 1 and 0 - m transition respectively. The dispersion in the energy levels arises from the gate charges n ' g α and the fact the the potential is periodic. This can not be predicted with the coupled anharmonic oscillator model and as such is investigated numerically. We expect, that like the transmon, this will exponentially decrease with the ratio of E J / E C as in this limit the effects of tunneling from one minima to the next becomes exponentially suppressed. This is confirmed in Fig. Fig:Levels B where the have plotted | ε q m | / E q m (the numerical maximum and minimum of the energy level over n ' g α ) as a function of E J / E C for E I = E C and E I = 0 (transmon limit). That is, the TCQ has the same charge noise immunity as the transmon.... Currently the most successful superconducting qubits are the flux , phase , and transmon as these qubits are essentially immune to offset charge (charge noise) by design. The transmon receives its charge noise immunity by operating at a point in parameter space where the energy level variations with offset charge are exponentially suppressed. This suppression has experimentally been observed and resulted in these qubits being approximately T 1 limited ( T 2 ≈ 2 T 1 ) in the circuit quantum electrodynamics (QED) architecture . In this architecture the qubits are coupled to a coplanar waveguide resonator through a Jaynes-Cummings Hamiltonian operated in the dispersive regime . This resonator acts as the channel to control, couple, and readout the state of the qubit (see Fig. Fig:TCQ A).... To achieve tuning of g ~ + we modify the original circuit and replace the Josephson junctions by SQUIDs with Josephson energy E J ± 1 and E J ± 2 (this is hinted at in Fig. Fig:TCQ B). In making this replacement the only change in the above theory is the replacement E J ± → E J ± m a x cos π Φ x ± / Φ 0 1 + d 2 tan 2 π Φ x / Φ 0 with E J ± m a x = E J ± 1 + E J ± 2 , d = E J ± 1 - E J ± 2 / E J ± m a x and Φ x ± is the external flux applied to each SQUID which we assume to be independent (this is not required but simplifies our argument). This independent control allows us to change E J ± independently which in-turn allows independent control on g ~ + and ω q . To illustrate this we consider the symmetric case and plot in Fig. Fig:Tune the normalized coupling strength g ~ + ℏ / 2 e 2 V r m s β (A) and ω ~ q (B) as a function of the ratio E J + / E C when E I = - E C and E J - is numerically solved to ensure that only the coupling rate (blue) and frequency (red) vary respectively for both the full numerical (solid) and effective model (dashed). In the full numerical model g ~ + = 2 e 2 V r m s 1 | ( β + n + + β - n - | 0 / ℏ . Here the independent control is clearly observed. Note that while our numerical investigation was only for the symmetric case independent tunable g ~ + (from zero to large values) and ω q will still occur when the device is not symmetric. There is just a different condition on E J ± for the required tuning.... where ω ~ ± = ω ± + δ ~ ± - δ ± / 2 + δ + + δ - J 2 / 2 μ 2 ± μ / 2 ∓ η / 2 , δ ~ ± = δ + + δ - 1 + η 2 / μ 2 / 4 ± η δ + - δ - / 2 μ and δ ~ c = 2 J 2 δ + + δ - / μ 2 with μ = 4 J 2 + η 2 and the tilde indicating the diagonalized frame. The coupling has induced a conditional anharmonicity δ ~ c , it is this anharmonicity that makes this system different to two coupled qubits, it ensures that E 11 is not equal to E 01 + E 10 . Here we have introduced the notation that superscript i j refers to i excitations in the dark mode ( ` ` + " ) and j excitations in the bright mode ( ` ` - " ). The choice of these names will become clearer latter. The dotted lines in Fig. Fig:Levels A are the predictions from this effective model, which agree well with the full numerics. Thus from the effective model the anharmonicities are all around E C provided | J | > | η | . That is the TCQ has not lost any anharmonicity in comparison to the transmon or phase qubit and with simple pulse shaping techniques arbitrary control of the lowest three levels will be possible . We will now introduce the notation that the qubit is formed by the space | 0 = | 00 ~ , | 1 = | 10 ~ and | m = | 01 ~ is the measurement state.
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This amplitude describes the tunneling transition in the presence of a time-modulated external field that shifts the energetic levels. It is interesting to compare this result with the analogous one for the case of an external monochromatic field. It is known that in the latter case the amplitude of the transition → sup with parameters satisfying the resonance does not depend on time intervals, while the amplitude Eq. ( amplitud) contains time-dependent periodic oscillations at the modulation frequency. In Fig. TransProb1 and TransProb2 we depict the corresponding probabilities of the tunneling transition in dependence on dimensionless time for two resonant conditions: N = 1 and 2 . As we see, the transition amplitudes are not constants and are periodic in time, while for the case of a one-monochromatic driving field these quantities have constant values.... The function γ N t is an increasing function in time but it grows also periodically due to its "linear+periodic" structure. Therefore, the dynamics of populations Eq. ( pop) seems to be aperiodic in time. Indeed, the typical results for the phase function as well as the populations are depicted in Figs. pop3, Population1 and Population2. The dynamics of populations for the case of a weak external field is shown in Figs. pop3 for two resonance regimes. In Fig. pop3(a) we compare two curves of the occupation probabilities for N = 1 (solid curve) and for N = 2 (dashed curve). We can see here fast oscillations of the population for the regime N = 1 and slow oscillations for the case of N = 2 (for consideration in details, see the curve corresponding to the case N = 2 for large time intervals in Fig. pop3(b)). The results for the second-order resonance regime are also demonstrated in Fig. pop3(c) for the other parameter Δ / δ . Analyzing these results, we note that dynamics of populations strongly depends on the value of the ratio Δ / δ . It can be seen from the formulas Eqs. ( JmeanApp) and ( fiApp) that population behavior shown in Fig. pop3(b) for N = 2 is mainly governed by the linear in time term in the phase function Eq. ( fiApp); thus, we can see that the dynamics looks like cosinusoidal oscillations. The periodic in time part Φ N t only slightly modulated these oscillations. This part of the phase function increases with increasing the parameter Δ / δ that leads to increasing the role of periodic modulations giving rise to a nontrivial time dependence of occupation probability [see, Fig. pop3(c) for the case N = 2 ].... The typical results for Rabi oscillations with regular, periodic dynamics are depicted in Fig. Periodic for the N = 2 resonance condition. Here, the parameters A / ω 0 and two used parameters, Δ / δ = 401 [see Figs. Periodic(a)] and Δ / δ = 31 [see Figs. Periodic(b)], satisfy the periodicity condition Eq. ( periodrelation). We compare the results shown in Fig. Periodic(a) with the result depicted in Fig. pop3(c). Both results are obtained for the second-order resonance condition and for the same parameter A / ω 0 = 10 -1 ; however, using the parameter Δ / δ satisfying the condition of periodicity Eq. ( periodrelation) in Fig. Periodic(a) leads to the periodic dynamics of the populations. These regimes in which quantum dynamics of occupation probabilities becomes periodically regular can be useful, for example, in applications where one is dealing with logic operations on qubits.... which can be realized on a properly designed superconducting circuit. In particular, a simple design of the charge qubit with tunable effective Josephson coupling can be shown schematically (see Fig. circuit) as... circuit A charge qubit with tunable effective Josephson coupling. It is controlled by V g gate voltage and Φ x magnetic field.
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As an example, we now describe how we would generate a maximally entangled state with two flux qubits biased at their flux-noise insensitive points, assuming Δ 1 = 5 G H z and Δ 2 = 7 G H z . We fix the bias current in the SQUID to I b = 2 s 0 I 0 and start with the ground state | 0 1 , 0 2 . We first apply a π pulse to qubit 1 thus preparing state | 1 1 , 0 2 . Then we apply a pulse at a frequency Δ 2 - Δ 1 = 2 G H z in the SQUID bias current of amplitude δ s = 0.015 . This results in an effective coupling of strength δ g / 2 = d g / d s δ s / 2 = 50 M H z . A pulse of duration δ t = 5 n s suffices then to generate the state | 0 1 0 2 + | 1 1 1 2 / 2 . We stress that thanks to the large value of the derivative d g / d s , even a small modulation of the bias current of δ I b = 2 I 0 δ s = 15 n A is enough to ensure such rapid gate operation. We performed a calculation of the evolution of the whole density matrix under the complete interaction hamiltonian g t σ x 1 σ x 2 with the parameters just mentioned. We initialized the two qubits in the | 1 1 , 0 2 state at t = 0 ; at t = 10 n s an entangling pulse g t = δ g cos 2 π Δ 1 - Δ 2 t and lasting 20 n s was simulated. The result is shown as a black curve in figure fig2. We plot the diagonal elements of the total density matrix. As expected, ρ 00 , 00 t = ρ 11 , 11 = 0 , and ρ 10 , 10 = 1 - ρ 01 , 01 = cos 2 π δ g / 2 t 2 . We did another calculation for the same qubit parameters but assuming a fixed coupling g 0 = 200 M H z . Following the analysis presented above, we initialized the system in the dressed state | 10 ' and simulated the application of a microwave pulse g t = δ g cos ω M W t at a frequency ω M W = 2.04 G H z taking into account the energy shift of the dressed states. The evolution of the density matrix elements (grey curve in figure fig2) shows that despite the finite value of g 0 , the two qubits become maximally entangled as previously. The evolution is not simply sinusoidal because we plot the density matrix coefficients in the uncoupled state basis. Note also the slightly slower evolution compared to the g 0 = 0 case, consistent with our analysis. This shows that the scheme should actually work for a wide range of experimental parameters.... (Color online) a) Two flux-qubits (shown coupled to their read-out SQUIDs and to their flux control line C i ( i = 1 , 2 )) coupled by a fixed mutual inductance M . b) Parametric coupling scheme : the two flux-qubits are now coupled through a circuit that allows to modulate the coupling constant g through the control parameter λ . fig0... a) Circuit proposed in to implement a tunable coupling between two flux-qubits. The two qubits are directly coupled by a mutual inductance M q q , and also via the dynamical inductance of a DC-SQUID which depends on the bias current I b at fixed flux bias. The total coupling constant g is shown in b) for the same parameters as were considered in as a function of s = I b / 2 I 0 . The dashed line indicates g = 0 . Inset : (dg/ds) as a function of s . fig1... It is straightforward to extend the scheme discussed above to the case of a qubit coupled to a harmonic oscillator of widely different frequency. As an example we consider the circuit studied in which is shown in figure fig3a. A flux qubit is coupled to the plasma mode of its DC SQUID shunted by an on-chip capacitor C s h (resonance frequency ν p ) via the SQUID circulating current J . As discussed in , the coupling between the two systems can be written H I = g 1 I b a + a + g 2 I b a + a 2 σ x . We evaluated g 1 I b for the following parameters : Φ S = 0.45 Φ 0 , I 0 = 1 μ A , qubit-SQUID mutual inductance M = 10 p H , qubit persistent current I p = 240 n A , Δ = 5.5 G H z , ν p = 9 G H z as shown in figure fig3b. At I b = I b * = 0 , the coupling constant g 1 vanishes. It has been shown in that when biased at I b = I b * and at its flux-insensitive point, the flux-qubit could reach remarkably long spin-echo times (up to 4 μ s ). On the other hand, the derivative of g 1 is shown in figure fig3b to be nearly constant with a value d g 1 / d I b ≃ - 4 G H z / μ A . Therefore, inducing a modulation of the SQUID bias current δ i cos 2 π ν p - Δ t with amplitude δ i = 50 n A would be enough to reach an effective coupling constant of 100 M H z . The state of the qubit and of the oscillator are thus swapped in 5 n s for reasonable circuit parameters. This process is very similar to the sideband resonances which have been predicted and observed . However, in order to use these sideband resonances for quantum information processing, the quality factor of the harmonic oscillator must be as large as possible, contrary to the experiments described in where Q ≃ 100 . This can be achieved by superconducting distributed resonators for which quality factors in the 10 6 range have been observed . Employing this harmonic oscillator as a bus allows the extension of the scheme to an arbitrary number of qubits, each of them coupled to the bus via a SQUID-based parametric coupling scheme.... (Color online) Flux qubit parametrically coupled to an LC oscillator via a DC SQUID. a) Electrical scheme : the qubit (blue loop) is inductively coupled to a DC SQUID shunted by a capacitor and thus forming a LC oscillator. b) Dependence of the coupling constant g 1 as a function of the bias current I b . At the current I b * the coupling constant vanishes. c) Derivative d g 1 / d I b as a function of I b . It stays nearly at a constant value on the current range considered. fig3... Calculated evolution of the density matrix under the application of an entangling microwave pulse at the frequency | Δ 1 - Δ 2 | in the SQUID bias current, with Δ 1 = 5 G H z , Δ 2 = 7 G H z , g t = g 0 + δ g cos 2 π Δ 2 - Δ 1 t and δ g = 100 M H z . For the black curve, g 0 = 0 ; for the grey curve, g 0 = 200 M H z . fig2... We will now discuss the physical implementation of the above ideas. Simple circuits based on Josephson junctions, and thus on the same technology as the qubits themselves, allow to modulate the coupling constant at G H z frequency . To be more specific in our discussion, we will focus in particular on the scheme discussed in , and show that the very circuit analyzed by the authors (shown in figure fig1a) can be used to implement our parametric coupling scheme. Two flux-qubits of persistent currents I q , i and energy gaps Δ i ( i = 1 , 2 ) are inductively coupled by a mutual inductance M q q . They are also inductively coupled to a DC-SQUID with a mutual inductance M q s . The SQUID loop (of inductance L ) is threaded by a flux Φ S , and bears a circulating current J . The critical current of its junctions is denoted I 0 . Writing the hamiltonian in the qubit energy eigenstates at the flux-insensitive point, equation (2) in now writes H = - h / 2 Δ 1 σ z 1 + Δ 2 σ z 2 + h g σ x 1 σ x 2 , where g = M q q | I q 1 I q 2 | + M q s 2 | I q 1 I q 2 | R e ∂ J / ∂ Φ s I b / h . In figure fig1b we plot the coupling constant g as a function of the dimensionless parameter s = I b / 2 I 0 for the same parameters as in : I 0 = 0.48 μ A , L = 200 p H , I q 1 = I q 2 = 0.46 μ A , M q q = 0.25 p H , M q s = 33 p H , Φ s = 0.45 Φ 0 . We see that g strongly depends on s . In particular g s 0 = 0 for a specific value s 0 . On the other hand the derivative d g / d s is finite (for instance, d g / d s s 0 = 7 G H z ) as can be shown in the inset of figure fig1b. Biasing the system at s 0 protects it against 1 / f flux-noise in the SQUID loop and noise in the bias current. At GHz frequencies, the noise power spectrum of s is ohmic due to the bias current line dissipative impedance, and has a resonance due to the plasma frequency of the SQUID junctions. This resonance is in the 40 G H z range for typical parameters and should not affect the coupled system dynamics.... We first discuss why the simplest fixed linear coupling scheme as was implemented in the two-qubit experiments fails in that respect. Consider two flux qubits biased at their flux-noise insensitive point γ Q = π ( γ Q being the total phase drop across the three junctions), and inductively coupled as shown in figure fig0a . The uncoupled energy states of each qubit are denoted | 0 i , | 1 i ( i = 1 , 2 ) and their minimum energy separation h Δ i ≡ ℏ ω i . Throughout this article, we will suppose that Δ 1 ≥ Δ 2 . As shown before , the system hamiltonian can be written as H = H q 1 + H q 2 + H I , with H q i = - h / 2 Δ i σ z i ( i = 1 , 2 ) and H I = h g 0 σ x 1 σ x 2 = h g 0 σ 1 + σ 2 + + σ 1 - σ 2 - + σ 1 + σ 2 - + σ 1 - σ 2 + . Here we introduced the Pauli matrices σ x . . z ; i referring to each qubit subspace, the raising (lowering) operators σ i + ( σ i - ) and we wrote the hamiltonian in the energy basis of each qubit. It is more convenient to rewrite the previous hamiltonian in the interaction representation, resulting in H ' I = exp i H q 1 + H q 2 t / ℏ H I exp - i H q 1 + H q 2 t / ℏ . We obtain... While in the scheme proposed by Rigeti et al. quantum gates are realized with a fixed coupling constant g , our scheme relies on the possibility to modulate g by varying a control parameter λ . This gives us the possibility of realizing two-qubit operations with arbitrary fixed qubit frequencies, which is particularly attractive for flux-qubits. We first assume that we dispose of a “black box" circuit realizing this task, as shown in figure fig0b, actual implementation will be discussed later. Our parametric coupling scheme consists in modulating λ at a frequency ω / 2 π close to Δ 1 - Δ 2 or Δ 1 + Δ 2 . Supposing λ t = λ 0 + δ λ cos ω t leads to g t = g 0 + δ g c o s ω t , with g 0 = g λ 0 and δ g = d g / d λ δ λ . Then, if ω is close to the difference in qubit frequencies ω = ω 1 - ω 2 + δ 12 while | δ 12 | < < | ω 1 - ω 2 | , a few terms in the hamiltonian eq1:ham will rotate slowly. Keeping only these terms, we obtain
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The system described by the potential v x of Eq.  sinf. (a) shows the system with the qubit in the initial state + and (b) shows the system after the transition. The shaded rectangle represents the semi-infinite wire. The U-shaped contour indicates the single chiral channel in which electrons propagate. The diagram is only schematic. In an actual realization, the left and right propagating electrons need not be spatially separated, i.e. the U-shape of the contour may be squashed into a line. The distance l between the tip of the wire and the point on the wire closest to the qubit is indicated. The qubit is represented as a double quantum dot with a single electron in it. The dashed circle indicates the range of the potential through which the qubit and the wire interact. In the exploded view, the qubit level spacing ε and the tunneling amplitude γ are indicated. While the range of the potential and the distance l are of similar size in the figure, we will investigate the regime where l is much larger than the range of the potential in the text. f0... with θ ω the unit step function, i.e. θ ω = 1 for ω > 0 and θ ω = 0 for ω qubit interaction potentials V ± , which we denote a . Consider for instance a qubit placed next to a semi-infinite wire. (See Fig.  f0.) Here the size of the scattering region is the distance from the tip of the wire to the point closest to the qubit, which can be much larger that the range of the potential produced by the charge of the qubit. In general l ≥ a .... The general results we obtain will be applied to the case where the electron gas resides in a semi-infinite 1D quantum wire. In the limit of a large Fermi energy, this system is mapped onto Eqs.  h1, disp, and pot through the standard trick of “unfolding”, so that coordinates x and - x refer to the same spatial point, but to “different sides of the road”, i.e. ψ † - x and ψ † x create electrons at the same position but moving in opposite directions. Thus the potential v x is symmetric about x = 0 . The system is depicted in Figure f0. As a simple model for the interaction between the qubit and the electron gas, we will take v x = u x - l + u x + l with... The function 1 F 2 α 4 ; α 2 α + 1 2 ; - l ω 2 that determines the finite ω behavior of the tunneling rate W , as a function of ω for various couplings α , for the potential v x of Eq.  sinf. The dotted curve corresponds to α = 1 / 8 , the dashed curve to α = 1 / 2 , and the solid curve to α = 1 . Oscillations with period π / l result from the resonant creation of a single particle hole pair by the qubit. At larger α the oscillations are washed out due to Fermi-sea shake-up, i.e. the excitation by the qubit of a multitude of particle hole pairs with a broad distribution of energies. f1... Armed with an expression for the closed loop factor that is valid also for small times, we obtain an exact expression for W for the semi-infinite wire system. The finite ω features of W probe the spatial profile of the potential V ± at length scales 1 / ω (in units where v F = 1 ). We study how the ω dependence of W changes as the coupling α between the qubit and the electron gas grows. At weak coupling (small α ), we find that the rate W oscillates as a function of ω , and that the period is π / l . This is due to the resonant excitation of a single particle-hole pair in the Fermi sea. The wavelength of the associated charge density fluctuation is π / ω . The resonance condition is that an integer number of half wavelengths fit into the part of the wire between the tip and the point where the qubit interacts with the wire. At strong coupling (large α ) on the other hand, many particle hole excitations are created. This is known as Fermi sea shake-up. The corresponding density fluctuations have a broad distribution of wavelengths and hence there are no clear resonances. This results in the damping of the oscillations in W ω as α is increased. One of our main results (Eq.  Wl2) is an exact formula for this damping by means of Fermi sea shake-up. The result is illustrated in Fig.  f1. We also analyzed rate W in the ω ≫ 1 / l limit and saw that here the left and right moving electrons contribute to the rate W as if they belong to independent channels. This happens despite the fact that each electron incident on the tip encounters the qubit twice, once before being reflected at the tip, and once afterwards, and no electron relaxation occurs between qubit encounters. The result therefore indicates that processes in which an individual electron wave-packet with width 1 / ω ≪ l is scattered twice, once while incident on the tip and once after being reflected at the tip, make a vanishingly small contribution to the rate W ω .... As shown in Figure f1, the oscillations become damped as α is increased. The damping is a signature of a phenomenon known as Fermi-sea shake-up. At strong coupling (large α ), rather than creating a single particle hole pair, a large number of particle hole pairs are created. This corresponds to many charge density excitations (created by the bosonic operators a q † ) with a broad distribution of wavelengths 1 / q . As a result there is no clear resonance any more, and the oscillations in W ω are washed out.
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