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  • Schematic of a qubit continuously measured by a detector with output signal I t .... A particular realization of the evoltion of ρ 11 t due to continuous measurement for ε / H = 1 , α = 0.1 and η = 1 . Notice the fluctuation of both the phase and the asymmetry of oscillations.... The situation changes as the coupling between the detector and qubit increases, α 1 . The strong influence of measurement destroys quantum oscillations, and the Quantum Zeno effect develops, so that for α ≫ 1 the qubit performs random jumps between two localized states (see Fig.  I(t)b). In this case the properly averaged detector current follows pretty well the evolution of the qubit (however, the unsuccessful tunneling “attempts” still cannot be directly resolved), and the spectral density of I t can be calculated using the classical theory of telegraph noise leading to the Lorentzian shape of S I ω . Figure transitiona shows the gradual transformation of the spectral density with the increase of the coupling α for a symmetric qubit, ε = 0 , and an ideal detector, η = 1 . The results for an asymmetric qubit, ε / H = 1 , are shown in Fig.  transitionb.... has an obvious relation to the average square of the detector current variation due to oscillations in the measured system. Notice, however, that this integral is twice as large as one would expect from the classical harmonic signal, since one half of the peak height comes from nonclassical correlation between the qubit evolution and the detector noise. Classically, Eq. ( integral) would be easily understood if the signal was not harmonic but rectangular-like, which is obviously not the case. Actually, the detector current shows neither clear harmonic nor rectangular signal distinguishable from the intrinsic noise contribution. Figure I(t)a shows the simulation of ρ 11 evolution (thick line) together with the detector current I t . Since I t contains white noise, it necessarily requires some averaging. Thin solid, dotted, and dashed lines show the detector current averaged with different time constants τ a : τ a Ω / 2 π = 0.3 , 1, and 3, respectively. For weak averaging the signal is too noisy, while for strong averaging individual oscillation periods cannot be resolved either, so quantum oscillations can never be observed directly by a continuous measurement (although they can be calculated using Eqs. ( Bayes1)–( Bayes2)). This unobservability is revealed in the relatively low peak height of the spectral density of the detector current.... The curves in Fig.  transition as well as the dashed curves in Fig.  M-C are calculated using the conventional master equation approach which gives the same results for the detector spectral density as the Bayesian formalism (we will prove this later). In the conventional approach we should assume no correlation between the detector noise and the qubit evolution (the last term in Eq. ( 3contrib) is absent) while the correlation function K z ̂ τ should be calculated considering z t not as an ordinary function but as an operator. Then the calculation of z ̂ t z ̂ t + τ can be essentially interpreted as follows. The first (in time) operator z ̂ t collapses the qubit into one of two eigenstates which correspond to localized states, then during time τ the qubit performs the evolution described by conventional Eqs. ( conv1)–( conv2), and finally the second operator z ̂ t + τ gives the probability for the qubit to be measured in one of two localized states. (Of course, this procedure can be done purely formally, without any interpretation.) Notice that there is complete symmetry between states “1” and “2” even for ε ≠ 0 (in particular, in the stationary state ρ 11 = ρ 22 = 1 / 2 ), so the evolution after the first collapse can be started from any localized state leading to the same contribution to the correlation function. In this way we obviously get K z ̂ τ = ρ 11 τ - ρ 22 τ where ρ i i is the solution of Eqs. ( conv1)–( conv2) with the initial conditions ρ 11 0 = 1 and ρ 12 0 = 0 .... The detector current spectral density S I ω for η = 1 and different coupling α with (a) symmetric ( ε = 0 ) and (b) asymmetric ( ε / H = 1 ) qubit.... Figure envir shows the numerically calculated spectral density S I ω of the detector current for a nonideal detector, η = 0.5 (dashed lines) and for an ideal detector but extra coupling of the qubit to the environment at temperature T = H (solid lines). The rates γ 1 and γ 2 are chosen according to Eqs. ( enveqv1) and ( enveqv2). For the symmetric qubit, ε = 0 , the results of two models practically coincide. In contrast, the solid and dashed lines for ε = 2 H significantly differ from each other at low frequencies, while the spectral peak at ω ∼ Ω is fitted quite well.... Using Eqs. ( Bayes1)–( Bayes3) and the Monte-Carlo method (similar to Ref. ) we can calculate in a straightforward way the spectral density S I ω of the detector current I t . Solid lines in Fig.  M-C show the results of such calculations for the ideal detector, η = 1 , and weak coupling between the qubit and the detector, α = 0.1 , where α ≡ ℏ Δ I 2 / 8 S 0 H ( α is 8 times less than the parameter C introduced in Ref. ). One can see that in the symmetric case, ε = 0 , the peak at the frequency of quantum oscillations is 4 times higher than the noise pedestal, S I Ω = 5 S 0 while the peak width is determined by the coupling strength α (see Fig.  transition below). In the asymmetric case, ε ≠ 0 , the peak height decreases (Fig.  M-C), while the additional Lorentzian-shape increase of S I ω appears at low frequencies. The origin of this low-frequency feature is the slow fluctuation of the asymmetry of ρ 11 oscillations (Fig.  asym). In case ε = 0 the amplitude of ρ 11 oscillations is maximal (see thick line in Fig.  I(t)a), hence there is no such asymmetry and the low-frequency feature is absent, while the spectral peak at the frequency of quantum oscillation is maximally high.
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  • Consider the schematic of a superconducting(SC) qubit coupled to a resonator , as shown in fig. fig:sch. At the metal insulator interface there exist a number of spins which randomly flip at different instances of time. The Hamiltonian describing the entire system is as follows... Two extreme cases of the Bloch vector’s dissipative dynamics. (a) TLS induced decoherence in the Markovian ( γ > g z ), intermediate and non-Markovian noise regimes with negligible coupling to the resonator. (b) Collapse and revival type behavior of Rabi oscillations induced by n = 10 photons in the coherent state and in the absence of coupling to the TLS. Note the different initial conditions ( η o ) in each case.... In the case of g j = 0.1 one has a mixture of Markovian and mostly intermediate noise sources. This leads to an oscillatory type of behavior where small oscillations are superposed on top of a smoothly decaying function. Now, if g j = 1 , then one is entirely in the non-Markovian noise regime, and the decay of the Bloch vector is strongly oscillatory. However, now the decay of the Bloch vector is far more strongly affected by the resonator due to cos g z λ t 2 type terms in Eqs. nx1- nz2. The initial collapse of Rabi oscillations now mixes with TLSs induced oscillations with characteristic frequencies of Ω j . A broad distribution of Ω j (due to γ j ) results in the complicated beating of the Bloch vector seen in Fig. fig:BV_L01 (for g j = 1 ). However now, the subsequent revival of Rabi oscillations (similar to that of Fig. fig:nvst2-b for g j = 1 ) will not be seen at longer times as this behavior will be suppressed by e - ∑ γ j t due to the presence of multiple fluctuators. For this to be visible we have to increase the resonator coupling strength, which is what is done for the next set of calculations.... Mean value, g z , as a function of the location of the peak values in the fourier spectrum (see Fig. fig:Rabi_FFT d-f) frequency for various n . These calculations were carried out for seven TLS with ς = 0.1 MHz and κ = 0.01 .... Schematic showing a superconducting(SC) qubit, coupled to a resonator, under the influence of several fluctuating two level systems present in the SQUID’s metal-insulator interface.... The dependency of the frequency of these CR oscillations on g z and on n is examined more closely nest. In Fig. fig:g_vs_w, the frequency at which the peak in the Fourier spectra occurs, ω p e a k , (corresponding to the first set of oscillations) is shown as a function of g z and n . As expected ω p e a k varies linearly with g j and as n is increased ω p e a k ’s dependence on g z becomes more discernable.... The resultant temporal dynamics and the respective Fourier transforms are shown in Fig. fig:Rabi_FFT. The first peak in the Fourier spectrum corresponds to the initial set of oscillations while the second less prominent peak is due to the revived secondary set of oscillations. The width of these peaks corresponds to the various frequency components of these oscillations. As seen in the figure, the collapse and revival type phenomenon is more apparent if ς is small. For a larger ς , the visibility of this effect diminishes due to the superposition of the collapse and revival type oscillations with more widely varying frequencies. This washing out effect is also apparent in the Fourier spectra, particularly in the significant broadening and lowering of the secondary peak. However, we also see that this washing out effect ,with increasing ς , can be countered to some extent by increasing n .
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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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  • In the opposite, strong driving case, Δ n > Γ r / 2 , there are coherent oscillations between the two levels. The appropriate description is then to say that doublets of new eigenstates, are formed which are split in energy by 2 Δ n , see Fig.  Fig:Strong_Driving_Cooling. The doublets are defined for n ≥ 1 as ψ n ± ≡ ( and for n = 0 we have a single state ψ n = 0 ≡ .... Another driving-induced heating process, not characteristic for quantum optics, is due to the fact that in solid state systems there is strong noise at low frequencies ( 1 / f noise). Thus, processes like the one shown in Fig.  Fig:Spin_Heating become relevant. This process excites the qubit with the rate... Qubit heating induced by the applied drive.... Another way to achieve AC cooling is by applying radio frequency voltage bias to the gates. In Fig.  Fig:SQUIDsys, apply a driving voltage V x = V 0 cos ω d t on the resonator and another driving voltage V g = - C x / C g V x on the CPB. The ac voltage V x generates resonant coupling between the mechanical resonator and the CPB when ω a c = E J - ω 0 , which corresponds to the first red sideband coupling in quantum optics. The voltage V x also generates an oscillating charge bias on the CPB with δ N g x = C x V x / 2 e ; however, it is balanced by the bias V g , which prevents harmful ac pumping of the CPB.... While the Hamiltonians ( Eq:Spin_Hamiltonian_pumped) and ( Eq:Spin_Hamiltonian_V_rotated) look similar, there are two important differences. One, already discussed, is the fact that the pumping frequency ω J in ( Eq:Spin_Hamiltonian_V_rotated) is fundamentally noisy, while ω d in ( Eq:Spin_Hamiltonian_pumped) can be made coherent. The second (very important) difference is that in ( Eq:Spin_Hamiltonian_pumped) the pumping is applied to σ z only, while in ( Eq:Spin_Hamiltonian_V_rotated) it couples to σ z and σ y . Both these facts hinder the cooling. Indeed, the coupling to σ y gives a direct matrix element E J , R / 4 between the states and . This interaction repels the levels and we must choose E J , R ≪ 4 ω 0 so that the resonant detuning as in Fig.  Fig:Stair_Cooling is possible. In addition, the noise of the transport voltage translates into the line width for the transition equal to Γ ϕ = 2 π α t r k B T / ℏ , where α t r ≡ R / R Q . The fluctuations of the transport voltage are not screened by the ratio of capacitances as it happens for the gate charge. Therefore α t r ≈ 10 -2 . Because of these additional constraints the applicability of the DC cooling scheme is limited to higher frequency/quality factor resonators. For an estimate, consider an oscillator with ω 0 = 2 π × 1   G H z ≈ 5   μ e V ≈ 50 mK at temperature T = 50 mK. We then obtain Γ ϕ ≈ 0.3   μ e V , which significantly exceeds Γ r . Hence, we have to substitute Γ r by Γ ϕ in all formulas. For the Josephson coupling in the right junction we take E J , R = 2   μ eV. Then, instead of Eq. ( Eq:Delta), we find Δ ≈ E J , R λ / 2 E J , L ≈ 2 ⋅ 10 -3   μ eV (we assume E J , L ≈ 50   μ eV). The cooling rate can again be represented as A - n , where A - ≈ 2 Δ 2 / Γ ϕ ≈ 2 ⋅ 10 -5 μ eV. Thus, cooling becomes possible only if Q > ω 0 / A - ≈ 2.5 ⋅ 10 5 .... The direct coupling between the bath and the oscillator gives the dissipative rates between the oscillator states : Γ n → n - 1 ≈ g 2 X ω = ω 0 2 n / ℏ 2 and Γ n → n + 1 ≈ g 2 X ω = - ω 0 2 n + 1 / ℏ 2 . In addition, the oscillator can relax via the virtual excitations of the qubit. The corresponding processes are shown in Fig.  Fig:Add_Diss.... Dissipative processes due to the presence of the qubit: a) n → n - 1 ; b) n → n + 1 . The spectra of the oscillator and the qubit are superimposed.
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  • This means that at ω = Δ E there is resonance, P ¯ + = 1 2 , and P + t is an oscillating function with the frequency x 0 . This is illustrated in Fig. P(t)a. The width of the peak at ω = Δ E of the P ¯ + – ω curve at the half-maximum (i.e., at P + = 1 / 4 ) is approximately 2 x 0 (see the upper panels of Fig. P(w)).... Dependence of the probability P ¯ + on the frequency ω for different x 0 at Γ φ = Γ r e l a x = 0 and at x o f f = 0 (solid line) and x o f f = 0.2 Δ (dashed line). (Only the first few resonant peaks are plotted; the others, which are very narrow, are not shown at the graphs.) Inset: enlargement of the low-frequency region.... Finally, we illustrate the multiphoton resonant excitations of the interferometer-type charge qubit. Making use of the numerical solution of the master equation (Sec. II), we find the time-averaged probability P ¯ + plotted in Fig. P(w)_2. The position of the multiphoton resonant peaks is defined by the relation ω = Δ E / K , where Δ E = Δ E δ D C is supposed to be fixed. Alternatively, when the δ D C component of the phase is changed and the frequency ω is fixed, a similar graph can be plotted with resonances at δ D C = δ D C K defined by the relation Δ E δ D C K = K ω .... Dependence of the probability P ¯ + on the frequency ω for the phase-biased charge qubit at n g = 0.95 , E J 1 / E C = 12.4 , E J 2 / E C = 11 , Γ φ / E C = 5 ⋅ 10 -4 , Γ r e l a x / E C = 10 -4 , δ A C = 0.2 π , δ D C = π + 0.2 π .... Now making use of the numerical solution of Eqs. ( eq1– eq3) for the Hamiltonian ( Ham_Rabi), we study the dependence of the time-averaged probability P ¯ + on frequency ω and amplitude x 0 . For small amplitudes, x 0 ≪ Δ E , there are resonant peaks in the P ¯ + – ω dependence at ω ≃ Δ E / K , as described in Sec. III.A and illustrated in Fig. P(w). With increasing amplitude x 0 , the resonances shift to higher frequencies. For x o f f = 0 , the resonances appear at "odd" frequencies ( K = 1 , 3 , 5 , . . . ) only, as it was studied in Ref. 9. For x o f f ≠ 0 there are also resonances at "even" frequencies ( K = 2 , 4 , . . . ), which is demonstrated in Fig. P(w). We note that Fig. P(w) is plotted for the ideal case of the absence of decoherence and relaxation, Γ φ = Γ r e l a x = 0 , when the resonant value is P ¯ + = 1 / 2 . The effect of finite dephasing, Γ φ ≠ 0 , and relaxation, Γ r e l a x ≠ 0 , is to decrease the resonant values of P ¯ + and to widen the peaks for Γ φ > Γ r e l a x . Thus from the comparison of the theoretically calculated resonant peaks with the experimentally observed ones, the dephasing Γ φ and the relaxation rates Γ r e l a x  can be obtained .... Time dependence of upper-level occupation probabilities. (a) Rabi oscillations in P + with the period T R = 2 π / x 0 , (b) LZ transition in P ↑ (see Sec. III.B), (c) and (d) P + probability evolution in the case of periodically swept parameters at x o f f = 0 and x o f f ≠ 0 . Here Γ φ = Γ r e l a x = 0 ; T = 2 π / ω .
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  • The qubit-TLS system starts in its ground state at t = 0 . A microwave π or 3 π pulse (from 0 to 5 ns) puts the qubit-TLS system in the qubit excited state | 1 that is a superposition of the two entangled states | ψ ' 1 ≡ | 0 e + | 1 g / 2 and | ψ ' 2 ≡ | 0 e - | 1 g / 2 . After the microwaves are turned off, the occupation probability starts oscillating coherently. Values of g T L S indicated in the figure are normalized by ℏ ω 10 . The rest of the parameters are the same as in Fig. 1. (a) No energy decay of the excited TLS, i.e., τ p h = ∞ . Coherent oscillations with various values of g T L S . (b) Oscillations following a π -pulse with τ p h = 40 ns and various values of g T L S . (c) Oscillations following a π -pulse and a 3 π -pulse with τ p h = 40 ns and g T L S = 0.004 . The dip in the dot-dash line is one and a half Rabi cycles.... (a)-(b): Rabi oscillations in the presence of 1/f noise in the qubit energy level splitting. Dotted curves show the Rabi oscillations without the influence of noise. Panel (c) shows the two noise power spectra S f ≡ | δ ω 10 f / ω 10 | 2 of the fluctuations in ω 10 that were used to produce the solid curves in panels (a) and (b). Rabi frequency f R = 0.1 GHz.... where the qubit energy levels ( | 0 and | 1 ) are the basis states and the noise is produced by a single TLS. Our calculations are oriented to the experimental conditions and the results are shown in Fig.  fig:RTN. In Fig.  fig:RTNa-c the characteristic fluctuation rate t T L S -1 = 0.6 GHz. Panel fig:RTNa shows that the qubit essentially stays coherent when the level fluctuations are small ( δ ω 10 / ω 10 = 0.001 ). Panel fig:RTNa shows that when the level fluctuations increase to 0.006, the Rabi oscillations decay within 100 ns. The Rabi relaxation time also depends on the Rabi frequency as panel fig:RTNc shows. The faster the Rabi oscillations, the longer they last. This is because the low-frequency noise is essentially constant over several rapid Rabi oscillations . Alternatively, one can explain it by the noise power spectrum S I f . Since the noise from a single TLS is a random process characterized by a single characteristic time scale t T L S , it has a Lorentzian power spectrum... Experimentally, the two TLS decoherence mechanisms (resonant interaction and low-frequency level fluctuations) can both be active at the same time. We have calculated the Rabi oscillations in the presence of both of these decoherence sources by using the qubit-TLS Hamiltonian in eq. ( eq:ham) with a fluctuating ω 10 t that is generated in the same way and with the same amplitude as in Figure fig:RTNb. We show the result in Fig.  fig:RTN_decay. By comparing Fig.  fig:RTN_decay with Fig. 2b, we note that adding level fluctuations reduces the Rabi amplitude and renormalizes the Rabi frequency. The result in Fig.  fig:RTN_decay is closer to what is seen experimentally .... Solid line represents Rabi oscillations in the presence of both TLS decoherence mechanisms: resonant interaction between the TLS and the qubit, and low frequency qubit energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and the energy decay time for the TLS is τ p h = 10 ns, the same as in Fig. 2b. The size of the qubit level fluctuations is the same as in Fig.  fig:RTNb. The dotted line shows the unperturbed Rabi oscillations.... We do not expect Rabi oscillations to be sensitive to noise at frequencies much greater than the frequency of the Rabi oscillations because the higher the frequency f , the smaller the noise power and because the Rabi oscillations will tend to average over the noise. Rabi dynamics are sensitive to the noise at frequencies comparable to the Rabi frequency. In addition, the characteristic fluctuation rate plays an important role in the rate of relaxation of the Rabi oscillations. It has been shown that t T L S -1 can be thermally activated for TLS in a metal-insulator-metal tunnel junction. If the thermally activated behavior applies here, the decoherence time τ R a b i should decrease as temperature increases. In Fig.  fig:RTNd, the characteristic fluctuation rate has been lowered to 0.06 GHz (which is much lower than ω 10 / 2 π ≈ 10 GHz). The noise still causes qubit decoherence but affects the qubit less than in Fig.  fig:RTNc. Fig.  fig:RTN shows that the noise primarily affects the Rabi amplitude rather than the phase.... Rabi oscillations of a resonantly coupled qubit-TLS system with ε T L S = ℏ ω 10 . There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) P 1 is the occupation probability in the qubit state | 1 ; (b) P 0 g is the occupation probability in the state | 0 , g ; (c) P 0 e is the occupation probability of the state | 0 , e ; (d) P 1 g is the occupation probability of the state | 1 , g ; and (e) P 1 e is the occupation probability of the state | 1 , e . Notice the beating with frequency 2 η . Throughout the paper, ω 10 / 2 π = 10 GHz. Parameters are chosen mainly according to the experiment in Ref. : η / ℏ ω 10 = 0.0005 , g q b / ℏ ω 10 = 0.01 , and g T L S = 0 . The dotted line in panel (a) shows the usual Rabi oscillations without resonant interaction, i.e. η = 0 .... Solid lines show the Rabi oscillation decay due to qubit level fluctuations caused by a single fluctuating two level system trapped inside the insulating tunnel barrier. The TLS produces random telegraph noise in I o that modulates the qubit energy level splitting ω 10 . (a) The level fluctuation δ ω 10 / ω 10 = 0.001 . The characteristic fluctuation rate t T L S -1 = 0.6 GHz. The Rabi frequency f R = 0.1 GHz. The dotted lines show the usual Rabi oscillations without any noise source. (b) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.1 GHz. The dotted lines show the usual Rabi oscillations without any noise source. (c) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.5 GHz. (d) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.06 GHz, and f R = 0.5 GHz. Note that the scales of the horizontal axes in (a)-(c) are the same. They are different from that in (d).... We first consider the case of strong driving with g T L S = 0 and with the TLS in resonance with the qubit, i.e. ε T L S = ℏ ω 10 . If there is no coupling between the qubit and the TLS, then the four states of the system are the ground state | 0 , g , the highest energy state | 1 , e , and two degenerate states in the middle | 1 , g and | 0 , e . If the qubit and the TLS are coupled with coupling strength η , the degeneracy is split by an energy 2 η . Figure fig:QB4L shows the coherent oscillations of the resonant qubit-TLS system. We define a projection operator P ̂ 1 ≡ | 1 , g 1 , g | + | 1 , e 1 , e | so that P ̂ 1 corresponds to the occupation probability of the qubit to be in state | 1 as in the phase-qubit experiment. Instead of being sinusoidal like typical Rabi oscillations (the dotted curve), the occupation probability P 1 exhibits beating (Fig. 1a) because the two entangled states that are linear combinations of | 1 , g and | 0 , e have a small energy splitting 2 η , and this small splitting is the beat frequency. Without any source of decoherence, the resonant beating will not decay. Thus far the beating phenomenon has not yet been experimentally verified. The lack of experiment
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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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  • fig:epsartTwo-photon transitions. Large Ω c hinders two-photon transitions, leading to a lower resonant peak in the measured spectrum as marked with arrows. The lower inset shows the energy levels in the qubit-TLS coupled system. The upper inset shows the spectrum density versus Ω c . The outside peaks are due to the stationary population from to 1 2 ( ). The middle peak is due to the two-photon transitions from to .... fig:epsart Spectroscopy and coherent oscillations. (a) Spectroscopy of the qubit versus the flux bias with a splitting at f = 16.572 GHz due to the coupling of the qubit-TLS. (b) Usual Rabi oscillation with the damping time T R ≈ 81.5 ns at f = 16.728 GHz (arrow in the left) where the effect of the TLS is negligible. (c) Coherent oscillation at the avoided crossing (arrow in the right) shows quantum beating due to the interference of Rabi oscillations in the coupled system. In (b) and (c), the red dots are the experimental results and the solid lines are the theoretical results.... fig:epsartFrequencies Ω R in P φ . (a) and (b), Four frequencies with different weight (indicated by the color) in P 1 g and P 1 e versus Ω m , respectively. (c) Rabi oscillations with the microwave power, at the top of the fridge, increasing from -13 dBm to -1 dBm with a step of 1 dBm from bottom to top. Curves are shifted vertically for clarity. Quantum beating becomes more clear as the amplitude of microwave increases. (d) Frequencies (dots), obtained by the Fourier transformations of the corresponding Rabi oscillations in (c), versus the microwave amplitude. The color lines are the two frequencies in P 1 obtained from the theoretical analysis. (e) Frequencies in P 1 induced by the two-photon transitions versus Ω m with three different Ω c / 2 π : 15.0 MHz, 26.5 MHz, and 50.0 MHz.
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  • Realization to carry out transmission impedance measurement of a solid-state quantum circuit. The impedance Z ω represents the coupled oscillator-qubit system. The impedance Z ω is dependent on the quantum state of the system which can be probed by a microwave pulse.... (a) Real part of Z ω t 0 at temperatures T = ℏ ω 0 / 10 k b (blue), T = ℏ ω 0 / 2 k b (red) and T = ℏ ω 0 / k b (green). The initial state at t = t 0 is prepared so that the oscillator is in the thermal state and the qubit state is up (the lower energy qubit state). (b) Same as (a) but the initial state is prepared so that the qubit state is down.... The real and imaginary parts of the impedance Z of the system are shown in Figs.  kuva1 and kuva2 for three different temperatures T of the bath. The coupling between the oscillator and the qubit results in multiple peaks in strong contrast to a single oscillator. Moreover, the impedance of the coupled system is now strongly dependent on temperature. In the low temperature limit there are only two peaks in R e Z ω , corresponding to the vacuum Rabi splitting. They have been experimentally observed recently for example in Ref. ... Diagrammatic representation of the correlator ( mix). First the initial state develops to the moment t , then follows external vertex B and the vertex corrections. The last step is the propagation in the frequency space to the other external vertex A .... The impedance of a slightly off-resonant oscillator-qubit system is plotted in Fig.  nonres. The vacuum Rabi splitting, corresponding to the transitions I and II, is sensitive even to a slight detuning and thus the positions of these peaks follow the frequency ω q b of the qubit. On the other hand, the positions of the peaks corresponding to the transitions III and IV follow rather the frequency ω 0 of the oscillator. With small detuning, the height of the impedance peaks is only slightly altered.... Possible realization of the studied system. The resonator circuit is coupled to the qubit represented by σ . In practice σ could be for example a Cooper-pair box with a capacitive coupling to the oscillator.... When the oscillator-qubit system is not in thermal equilibrium, the susceptibility changes radically (see Fig.  ensim). The detailed-balance condition, which relates emission and absorption processes, is violated. For example, one can have net emission of energy in some frequencies in addition to the absorption. In those frequencies the susceptibility takes negative values which lead to negative impedance peaks as in Fig.  ensim(b). The negative peaks are signs of spontaneous emission of energy and relaxation towards the equilibrium state.... As the system relaxes towards the equilibrium state, the impedance settles to the equilibrium pattern. In Fig.  toinen we plot the temporal evolution of two nonequilibrium impedances. They correspond to the initial states where the oscillator is in thermal equlibrium and the qubit is prepared either up or down. The susceptibility changes slowly compared to ω 0 -1 and can be considered as quasistatic. After the time t Q / ω 0 the susceptibilities in both cases are nearly equal, reflecting the uncertainty about the state of the qubit (Fig.  toinen). When γ ≪ κ , the oscillator dissipation yields the dominant time scale for the relaxation of the qubit near resonance.... Equilibrium impedance of a resonant and a slightly off-resonant oscillator-qubit system at T = ℏ ω 0 / 2 k b . The blue curve represents the exactly resonant case, the red curve corresponds to the case ω q b = 1.01 ω 0 and the green curve to the case ω q b = 0.99 ω 0 .... Our system under study consists of a qubit coupled resonantly to a harmonic oscillator (Fig.  skeema), both coupled to a bosonic heat bath at a finite temperature. This model has a wide range of applications in solid-state physics as well as in quantum optics and has raised considerable attention lately. In the literature there exists various propositions to realize this system. In solid-state physics the harmonic oscillator is realized by a resonator circuit and the qubit, for example, by a Josephson charge or flux qubit. The heat bath corresponds to the electromagnetic environment of the circuit. The connection of these systems to cavity QED has been explained and studied in Refs. 
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  • We have found that when described using a semiclassical approximation, the dynamics of a flux qubit coupled to a nanomechanical oscillator via a coupling of the form ℏ g / 2 a + a † σ z influences the mechanics in a manner similar to that found in cavity optomechanics: In the linear regime, we have shown that the mechanical dissipation coefficient and resonance frequency are renormalized with expressions given in ( eq:gmre_general), ( eq:qubit_response_func) and ( eq:stokes_anti_stokes). These expressions reveal that the condition for resonance in this system is that the mechanical frequency equals the Rabi frequency of the qubit, and that the resolved sideband limit is at ω m ≫ γ ̄ b , where γ ̄ b is defined in ( eq:gamma_approx_def). They also show that in this case the response has a richer structure, with an additional peak at ω m = γ ̄ a . Considering the possibility of multi-photon driving of the qubit, we have shown that the Stokes and anti-Stokes sidebands of the qubit response exhibit a Bessel-ladder behavior , as shown in Fig.  fig:Correction-to-dissipation.... fig:Real_and_ImThe imaginary (a) and real (b) parts of the qubit response function, χ z - i ω m , as given by  eq:qubit response func in the blue-detuned ( δ qubit decay times γ 1 and γ 2 . The blue dots in (a) corresponds to the maximum at ω = γ ̄ a , and the red dots to the maximum at ω = Ω ̄ R . Solid line: γ 1 = 0.001 , γ 2 = 0.01 . Dashed line: γ 1 = 0.05 . γ 2 = 0.1 , Dot-dashed line: γ 1 = 0.1 , γ 2 = 0.5 .... fig:qubit_response_fullThe response function χ z - i ω m as given in  eq:qubit response func, as a function of δ and ω . The vertical axis corresponds to the absolute value, the color to the phase, and the contours to the imaginary part. For all δ qubit adds delay, thus decreasing the effective dissipation coefficient of the oscillator. For δ > 0 , the opposite is true. Note that here, in addition to the resonant peak, another smaller peak appears at ω m = γ ̄ a (see also Fig.  fig:Real_and_Im)... fig:schematic (a) The rf-SQUID, which is operated as a flux qubit, has a vibrating arm whose position of center of mass u = α + α ∗ alters the flux through the SQUID loop. Concurrently, the circulating current in the SQUID, in the presence of a magnetic field, leads to a Lorentz force acting on the beam. (b) The double well potential of the circulating current near the half-flux quantum biasing point of the SQUID. This potential leads to two localized circulating current states and , which span the qubit’s Hilbert space. (c) An illustration of dynamics of the qubit coupled to the mechanical element in the semiclassical picture and in the rotating frame, for the linear regime. Here a sinusoidal oscillation of the beam leads to a response of the z-component of the qubit, which is proportional to the circulating current. This response, given in ( eq:qubit_response_func) and ( eq:stokes_anti_stokes), leads to a renormalization of the mechanical dissipation coefficient and resonance frequency.... In what follows we analyze the dynamics of a flux qubit with a vibrating arm that functions as a mechanical oscillator  (see Fig.  fig:schematic). We assume that this system is described by the Hamiltonian ... Extending our analysis to the nonlinear regime for a blue-detuned qubit, we have shown that the system exhibits self-excited oscillations when the coupling g , whose strength is controlled by an external magnetic field, is increased beyond g c which is given in ( eq:g_crit). We have found the amplitude of the limit cycle close to criticality in ( eq:nonzero_FP), and calculated numerically its behavior for general g , as shown in Fig.  fig:Bifurcation-curves-for.... eq:qubit response func, as a function of δ and ω . The vertical axis corresponds to the absolute value, the color to the phase, and the contours to the imaginary part. For all δ qubit adds delay, thus decreasing the effective dissipation coefficient of the oscillator. For δ > 0 , the opposite is true. Note that here, in addition to the resonant peak, another smaller peak appears at ω m = γ ̄ a (see also Fig.  fig:Real_and_Im)... the Rabi frequency of the qubit, defined here to have the same sign as δ . We consider small deviations from the equilibrium point of the qubit-oscillator system found by setting the time derivatives in ( eq:EOM_semiclassical) to zero. In the linear regime, a periodic oscillation of the mechanical amplitude α t = α e q + α 0 e - i ω m t around its equilibrium position will lead to a response s z t = s z , e q + χ z - i ω m α 0 e - i ω m t (see Fig.  fig:schematic). This response, when fed back to the mechanical amplitude equation ( subeq:EOM_semiclassical_a), will lead to a renormalization of γ m and ω m  :
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