### 54073 results for qubit oscillator frequency

Contributors: Rastelli, G., Vanevic, M., Belzig, W.

Date: 2014-03-18

We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by $2\pi $ occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the **qubit**. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux **qubit**. In particular, when the number of Josephson junctions is increased, several low-energy modes can have **frequencies** lower than the **qubit** **frequency**. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of **oscillations** between the two counterpropagating current states at short times, followed by **oscillation**-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime....**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 . ... We analyze the coherent dynamics of a fluxonium device (Manucharyan et al 2009 Science 326 113) formed by a superconducting ring of Josephson junctions in which strong quantum phase fluctuations are localized exclusively on a single weak element. In such a system, quantum phase tunnelling by $2\pi $ occurring at the weak element couples the states of the ring with supercurrents circulating in opposite directions, while the rest of the ring provides an intrinsic electromagnetic environment of the **qubit**. Taking into account the capacitive coupling between nearest neighbors and the capacitance to the ground, we show that the homogeneous part of the ring can sustain electrodynamic modes which couple to the two levels of the flux **qubit**. In particular, when the number of Josephson junctions is increased, several low-energy modes can have **frequencies** lower than the **qubit** **frequency**. This gives rise to a quasiperiodic dynamics, which manifests itself as a decay of **oscillations** between the two counterpropagating current states at short times, followed by **oscillation**-like revivals at later times. We analyze how the system approaches such a dynamics as the ring's length is increased and discuss possible experimental implications of this non-adiabatic regime.

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Contributors: Ella, Lior, Buks, Eyal

Date: 2012-10-25

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 ....We present a theory describing the semiclassical dynamics of a superconducting flux **qubit** inductively coupled to a nanomechanical **oscillator**. Focusing on the influence of the **qubit** on the mechanical element, and on the nonlinear phenomena displayed by this device, we show that it exhibits retardation effects and self-excited **oscillations**. These can be harnessed for the generation of non-classical states of the mechanical **oscillator**. In addition, we find that this system shares several fundamental properties with cavity optomechanical systems, and elucidate the analogy between these two classes of devices....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 : ... We present a theory describing the semiclassical dynamics of a superconducting flux

**qubit**inductively coupled to a nanomechanical

**oscillator**. Focusing on the influence of the

**qubit**on the mechanical element, and on the nonlinear phenomena displayed by this device, we show that it exhibits retardation effects and self-excited

**oscillations**. These can be harnessed for the generation of non-classical states of the mechanical

**oscillator**. In addition, we find that this system shares several fundamental properties with cavity optomechanical systems, and elucidate the analogy between these two classes of devices.

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Contributors: Bertet, P., Chiorescu, I., Burkard, G., Semba, K., Harmans, C. J. P. M., DiVincenzo, D. P., Mooij, J. E.

Date: 2005-12-18

(a) Measurement of T 1 versus I b at the flux-noise insensitive point ϵ = 0 . (b) Measurement of T e c h o (circles), t 2 (squares) and of the **qubit** **frequency** (triangles), as a function of ϵ for I b = I b * (top) and I b = 0 μ A (bottom). The dotted line is a fit to the formula for ν q ; the solid black line is the prediction of equation eq:tauphi for T = 70 m K and Q = 150 . (c) Value of ϵ for which t 2 is maximum (full squares) compared to the theoretical ϵ m I b (full line). fig4...To obtain the coupling constants g 1 and g 2 , we performed extensive spectroscopic measurements of the **qubit**, as a function of both I b and Φ x . We applied a pre-bias current pulse I b p l through the SQUID while sending a long microwave pulse, followed by a regular measurement pulse at a value I m (see figure fig2a). We measured the SQUID switching probability as a function of the microwave **frequency**, and recorded the position of the **qubit** resonance as a function of I b p l and Φ x . The data are shown in figure fig2a for various values of I b p l . We observe that for each bias current, a specific value of external flux Φ x 0 I b p l realizes the optimal point condition. Fitting all the curves with the formula ν q = Δ 2 + λ I b p l + 2 I p Φ x - Φ 0 / 2 / h 2 , we obtain the **qubit** parameters M = 6.5 p H , Δ = 5.5 G H z , I p = 240 n A , and also λ I b which is shown in figure fig2b together with a parabolic fit. Decoupling occurs at I b * = 180 ± 20 n A and not at I b = 0 because of a 4 % asymmetry of the SQUID junctions. We also measured the parameters of the SQUID **oscillator** by performing resonant activation measurements and fitting the dependence of the resonant activation peak as a function of I b and Φ x . We found a maximum plasma **frequency** ν p = 3.17 G H z , C s h = 7.5 ± 2 p F and L = 100 ± 20 p H , consistent with design values. The width of the peak also gives us an estimate for the **oscillator** quality factor, Q = 120 ± 30 ....We have studied the dephasing of a superconducting flux-**qubit** coupled to a DC-SQUID based **oscillator**. By varying the bias conditions of both circuits we were able to tune their effective coupling strength. This allowed us to measure the effect of such a controllable and well-characterized environment on the **qubit** coherence. We can quantitatively account for our data with a simple model in which thermal fluctuations of the photon number in the **oscillator** are the limiting factor. In particular, we observe a strong reduction of the dephasing rate whenever the coupling is tuned to zero. At the optimal point we find a large spin-echo decay time of $4 \mu s$....The **qubit** is inductively coupled to a SQUID detector with a mutual inductance M (large loop in figure fig1a), and to an on-chip antenna allowing us to apply microwave pulses. The readout scheme and the experimental setup have been described elsewhere . The average persistent current in the **qubit** loop, with a sign depending on its state | k ( k = 0 , 1 ), generates a flux which modifies its critical current I C ∼ 1 μ A to a value I C | k ; a bias current pulse of amplitude I m chosen so that I C | 0 **oscillator** of **frequency** ν p = 1 / 2 π L + L J C s h called the plasma mode (note that the junction capacitance is much smaller than C s h ). We can write its hamiltonian H p = h ν p a a , where a ( a ) is the annihilation (creation) operator. The total current flowing through the SQUID is thus I b + i , with i = δ i 0 a + a being the operator for the current in the plasma mode and δ i 0 the rms fluctuations of the current in the **oscillator** ground state δ i 0 = h ν p / 2 L + L J . The SQUID circuit is connected to the output voltage of our waveform generator E via an impedance Z i n , and to the input of a room-temperature amplifier through Z o u t which define the **oscillator** quality factor Q = 2 π ν p / κ . Z i n and Z o u t take into account low-temperature low-pass filters , and on-chip 8 k Ω thin-film gold resistors thermalized by massive heat-sinks. The resulting impedance seen from the plasma mode is estimated to be 9 k Ω at low **frequencies** and of order 500 Ω at G H z **frequencies**. The measurements were performed at a base temperature T b = 30 m K ....Despite the fluctuators, we were able to induce Rabi **oscillations** by applying microwave pulses at the middle **frequency** of the split line. An example is shown in figure fig3b at the optimal point. The **oscillations** decay non-exponentially and display a clear beating. Nevertheless, by driving the **qubit** strongly enough, we could observe well-behaved **oscillations** for hundreds of nanoseconds (see inset of figure fig3b). We measured the energy relaxation time T 1 by applying a π pulse followed after a delay D t by a measurement pulse (see figure fig3c). At the optimal point, we found that T 1 = 4 μ s . To quantify the dephasing further we also applied the spin-echo sequence , depicted in figure fig4a. Spin-echo measurements are particularly relevant for our purpose, because the photon noise in the plasma mode occurs at a relatively high **frequency** set by κ ≃ 130 M H z . In such conditions, this noise affects the spin-echo damping time T e c h o as strongly as Ramsey experiments so that T e c h o is also given by formula eq:tauphi ; on the other hand the spin-echo experiment is not sensitive to the low-**frequency** noise responsible for the **qubit** line splitting. The results are shown in figure fig3d at the optimal point, by a set of curves obtained at different delays Δ t between the π pulse and the last π / 2 pulses. Fitting the decay of the echo amplitude as a function of the delay between the two π / 2 pulses with an exponential, we find T e c h o = 3.9 ± 0.1 μ s . Compared with previous experiments on flux-**qubits** , the long Rabi and spin-echo times were obtained by reducing the mutual inductance M , and biasing the **qubit** at the optimal point....Our flux-**qubit** consists of a micron-size superconducting aluminum loop intersected by four Josephson junctions fabricated by standard electron-beam lithography and shadow evaporation techniques (see figure fig1a ; note that compared to earlier designs , we added a fourth junction to restore the **qubit**-SQUID coupling symmetry ). When the magnetic flux threading the loop Φ x sets the total phase across the junctions γ q close to π , the loop has two low-energy eigenstates, ground state | 0 and excited state | 1 . The flux-**qubit** is characterized by the minimum energy separation h Δ between | 0 and | 1 , and the persistent current I p . In the basis of the energy eigenstates at the bias point γ q = π , the **qubit** hamiltonian reads H q = - h / 2 Δ σ z + ϵ σ x , where ϵ ≡ I p / e γ q - π / 2 π . The energy separation is E 1 - E 0 ≡ h ν q = h Δ 2 + ϵ 2 . Note that d ν q / d ϵ = 0 when the **qubit** is biased at ϵ = 0 so that it is to first order insensitive to noise in ϵ , in particular to noise in the flux Φ x . This is similar to the doubly optimal point demonstrated in the quantronium experiment ....(a) **Qubit** line shape at the optimal point. The solid line is a fit assuming a double lorentzian shape. (b) Rabi **oscillations** (**frequency** 100 M H z ) at the optimal point. The inset shows well-behaved **oscillations** with nearly no damping during the first 100 n s . (c) Measurement of T 1 at the optimal point ; the dashed grey line is an exponential fit of a time constant 4 μ s . (d) Spin-echo pulse sequence and signal at the optimal point. fig3...(a) SEM picture of the sample. The flux **qubit** is the small loop containing four Josephson junctions in a row ; the SQUID is constituted by the outer loop containing the two large junctions. The bar equals 1 μ m . (b) Measuring circuit diagram. The SQUID, represented by its Josephson inductance L J , is shunted by an on-chip capacitor C s h through superconducting lines of inductance L , forming the plasma mode. fig1...We studied the variation of T 1 as a function of the bias current at the flux-insensitive point ϵ = 0 . This required us to adjust the flux at the value Φ x 0 I b . Results are shown in figure fig4a. We observed a clear maximum of T 1 for I b = I b * . This demonstrates that at least part of the **qubit** relaxation occurs by dissipation in the measuring circuit. We then investigated the dependence of T e c h o and t 2 on ϵ for I b = I b * (figure fig4b top, full circles and full squares). As expected, we observe a sharp maximum for T e c h o at ϵ = 0 and a shallow one for t 2 . However, at a different bias current I b = 0 μ A , the maximum of T e c h o and t 2 is clearly shifted towards ϵ **oscillator** temperature of T = 70 m K and a quality factor of Q = 150 , which yields a mean photon number n ̄ = 0.15 , the dephasing time τ φ predicted by equation eq:tauphi closely matches the spin-echo measurements both for I b = I b * and I b = 0 μ A (see the solid line in figure fig4b). We stress that even at such small n ̄ the photon number fluctuations can strongly limit the **qubit** coherence. This suggests that increasing the plasma **frequency** could lead to significant improvement....(a) Top : Principle of the spectroscopy experiments : a bias current pulse of amplitude I b p l **qubit** resonance **frequency**. The **qubit** state is finally measured by a short bias current pulse as discussed in . Bottom : **Qubit** spectroscopy for I b p l varying between 0 μ A to 0.6 μ A with steps of 0.1 μ A (bottom to top). The curves were offset by 100 M H z for clarity. The solid curves are fits to the formula for ν q . (b) Curve λ I b deduced from the spectroscopy curves as explained in the text. The solid line is a parabolic fit to the data. The decoupling condition is satisfied at I b * = 180 ± 20 n A . (c) Calculated **frequency** shift δ ν 0 I b ϵ for the parameters of our sample. The white scale corresponds to -20 M H z , the black to + 40 M H z . Along the dotted line ϵ m I b , δ ν 0 = 0 . ... We have studied the dephasing of a superconducting flux-**qubit** coupled to a DC-SQUID based **oscillator**. By varying the bias conditions of both circuits we were able to tune their effective coupling strength. This allowed us to measure the effect of such a controllable and well-characterized environment on the **qubit** coherence. We can quantitatively account for our data with a simple model in which thermal fluctuations of the photon number in the **oscillator** are the limiting factor. In particular, we observe a strong reduction of the dephasing rate whenever the coupling is tuned to zero. At the optimal point we find a large spin-echo decay time of $4 \mu s$.

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Contributors: Abovyan, Gor A., Kryuchkyan, Gagik Yu.

Date: 2013-05-22

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 ]....We analyze the dynamics of a superconducting **qubit** and the phenomenon of multiorder Rabi **oscillations** in the presence of a time-modulated external field. Such a field is also presented as a bichromatic field consisting of two spectral components, which are symmetrically detuned from the **qubit** resonance **frequency**. This approach leads to obtaining qualitative quantum effects beyond those for the case of monochromatic excitation of **qubits**. We calculate Floquet states and quasienergies of the composite system "superconducting **qubit** plus time-modulated field" for various resonant regimes. We analyze the dependence of quasienergies from the amplitude of an external field, demonstrating the zeros of difference between quasienergies. We show that, as a rule, populations of **qubit** states exhibit aperiodic **oscillations**, but we demonstrate the specific important regimes in which dynamics of populations becomes periodically regular....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. ... We analyze the dynamics of a superconducting **qubit** and the phenomenon of multiorder Rabi **oscillations** in the presence of a time-modulated external field. Such a field is also presented as a bichromatic field consisting of two spectral components, which are symmetrically detuned from the **qubit** resonance **frequency**. This approach leads to obtaining qualitative quantum effects beyond those for the case of monochromatic excitation of **qubits**. We calculate Floquet states and quasienergies of the composite system "superconducting **qubit** plus time-modulated field" for various resonant regimes. We analyze the dependence of quasienergies from the amplitude of an external field, demonstrating the zeros of difference between quasienergies. We show that, as a rule, populations of **qubit** states exhibit aperiodic **oscillations**, but we demonstrate the specific important regimes in which dynamics of populations becomes periodically regular.

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Contributors: Forney, Anne M., Jackson, Steven R., Strauch, Frederick W.

Date: 2009-09-08

As a further test of this method, we compare the error p E = 1 - p 1 T for this two-**frequency** pulse with that of a single-**frequency** pulse. This is displayed in Fig. error1. The single-**frequency** pulse is seen to have an error that scales as Ω 1 2 / Δ 2 . We see that the two-**frequency** pulse does a significantly better job compared to the single **frequency** pulse, and the error scales as Ω 1 4 / 16 Δ 4 , with **oscillations** of **frequency** Δ ....This provides an excellent approximation to the beating observed in Fig. spin. Note that the perturbation, which is proportional to Ω / Δ , happens to vanish precisely when the unperturbed **oscillation** reaches its maximum ( t = 2 π / Ω .) Thus, it is likely that additional effects limit this approach to a 0 2 transition. By extending the matrix to 15 states and higher orders in perturbation theory, one finds a state 3 population proportional to Ω 2 / Δ 2 ....Error of 0 1 transition using square pulses. The upper points (squares) are the error of a pulse using a single **frequency** with ω 1 = ω 01 . The lower points (dots) are the error of an optimized two-**frequency** pulse with ω 1 = ω 01 + Ω 1 2 / 2 Δ and ω 2 = ω 12 . The upper dashed curve is 3 Ω 2 / 4 Δ 2 , while the lower dashed curve is Ω 4 / 16 Δ 4 (see text)....where the dimensionless coefficients c α and d α are varied to obtain the best transition. These coefficients correct for the reduction in Rabi **frequency** and the ac Stark shift discussed previously, and depend on the pulse shape parameter α . For N s = 4 and α = 2 , we find that c α = 2 = 0.58 and d α = 2 = 1.245 are required. The error using Gaussian pulses with and without the Stark shift correction is displayed in Fig. error2. We see that the single-**frequency** pulse is not effective without these corrections. To incorporate the two-**frequency** pulse, we numerically optimize for A 2 and φ , and find that it provides a significant advantage. Note, however, that the two-**frequency** square pulse outperforms all of the Gaussian pulses for small pulse times....Numerically optimized Ω 2 as a function of the primary Rabi **frequency** Ω 1 . The dashed curve is the approximation Ω 2 ≈ Ω 1 2 / 2 Δ (see text)....**Qubit**, quantum computing, superconductivity, Josephson junction....To illustrate this method, we consider a particular example. Fig. rabi1 shows the result of a numerical simulation of the time-dependent Schrödinger equation for a phase **qubit** with ω 0 / 2 π = 6 GHz and N s = 4 subject to a control field with A 1 = 0.02 ℏ ω 0 , A 2 = 0.0035 ℏ ω 0 , φ = 11.44 , ω 1 = ω 01 + Ω 01 2 / 2 ω 01 - ω 12 , and ω 2 = ω 12 . The values of A 2 and φ were found by a numerical search to optimize the 0 1 transition, providing a significant improvement over the A 2 = 0 dynamics. This search was inspired by the general arguments given in Ref. , and demonstrates that the use of two **frequencies** can improve the control of this quantum system....Superconducting quantum circuits, such as the superconducting phase **qubit**, have multiple quantum states that can interfere with ideal **qubit** operation. The use of multiple **frequency** control pulses, resonant with the energy differences of the multi-state system, is theoretically explored. An analytical method to design such control pulses is developed, using a generalization of the Floquet method to multiple **frequency** controls. This method is applicable to optimizing the control of both superconducting **qubits** and qudits, and is found to be in excellent agreement with time-dependent numerical simulations....Three-level Rabi **Oscillation**. The probability p 1 t = | a 1 t | 2 to be in state 1 is shown as a function of time. The solid curve is a numerical simulation using an optimized control field with A 1 = 0.02 ℏ ω 0 , A 2 = 0.0035 ℏ ω 0 , and φ = 11.44 rad, while the dots are calculations using the two-mode Floquet formalism. The dashed curve is a numerical simulation with A 2 = 0 . Here, the system parameters were chosen to be ω 0 / 2 π = 6 GHz and N s = 4 . Other relevant parameters are Ω 1 / 2 π = 86 MHz, Ω 2 / 2 π = 15 MHz, ω 1 / 2 π = 5.785 GHz, ω 01 / 2 π = 5.77 GHz, and ω 2 / 2 π = ω 12 / 2 π = 5.5 GHz....First, in Fig. rabi2, we show the numerically optimized Ω 2 = A 2 x 01 / ℏ as a function of the bare Rabi **frequency** Ω 1 = A 1 x 01 / ℏ for a phase **qubit** with ω 0 / 2 π = 6 GHz and N s = 4 , comparable to recent experiments ; other parameters can be found in Fig. rabi1. For this system, the anharmonicity is Δ / 2 π ≈ 260 MHz. We see that the analytical result...Error of 0 1 transition using Gaussian pulses. The upper dotted curve is the error of a pulse using a single **frequency** with ω 1 = ω 01 . The dashed curve is the error of a single-**frequency** pulse with ω 1 = ω 01 + d 2 Ω 1 2 / 2 Δ . The lower solid curve is the error of an optimized two-**frequency** pulse with ω 2 = ω 12 (see text). ... Superconducting quantum circuits, such as the superconducting phase **qubit**, have multiple quantum states that can interfere with ideal **qubit** operation. The use of multiple **frequency** control pulses, resonant with the energy differences of the multi-state system, is theoretically explored. An analytical method to design such control pulses is developed, using a generalization of the Floquet method to multiple **frequency** controls. This method is applicable to optimizing the control of both superconducting **qubits** and qudits, and is found to be in excellent agreement with time-dependent numerical simulations.

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Contributors: Ojanen, Teemu, Heikkila, Tero T.

Date: 2005-01-04

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 ....We investigate the measurements of two-state quantum systems (**qubits**) at finite temperatures using a resonant harmonic **oscillator** as a quantum probe. The reduced density matrix and **oscillator** correlators are calculated by a scheme combining numerical methods with an analytical perturbation theory. Correlators provide us information about the system impedance, which depends on the **qubit** state. We show in detail how this property can be exploited in the **qubit** measurement....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. ... We investigate the measurements of two-state quantum systems (**qubits**) at finite temperatures using a resonant harmonic **oscillator** as a quantum probe. The reduced density matrix and **oscillator** correlators are calculated by a scheme combining numerical methods with an analytical perturbation theory. Correlators provide us information about the system impedance, which depends on the **qubit** state. We show in detail how this property can be exploited in the **qubit** measurement.

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Contributors: Green, T. J., Biercuk, M. J.

Date: 2014-08-12

We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic **oscillator** modes. Such systems are particularly susceptible to residual **qubit**-**oscillator** entanglement at the conclusion of a gate period which reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete phase shifts in the field moderating the **qubit**-**oscillator** interaction - easily implemented with modern synthesizers - is sufficient to both ensure multiple **oscillator** modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting **qubits** and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach allows multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols....Fig:F3 Laser amplitude noise filter functions for 3 excited **oscillator** modes k = 1 , 2 , 3 . The effect of the noise on the coupling to each mode is suppressed to third order for k = 1 ( F 1 ω ∝ ω 3 as ω → 0 ), second order for k = 2 ( F 2 ω ∝ ω 2 ), and first order for k = 3 ( F 3 ω ∝ ω ) ....Fig:F2 a) Raman laser geometry for a MS gate applied to 2 ions in a 5 ion chain in which only transverse ( x -direction) phonon modes are excited. The red (r) and blue (b) Raman fields have **frequencies** ω r / b and phases φ r / b . b) Detuning diagram for 5 excited TP modes, ω ~ k = ω 0 ± ω k ( + for ω b and - for ω r ), where ω 0 is the hyperfine **qubit** level splitting. c) Closed paths, α k , ≡ | δ k | α k t , 0 ≤ t ≤ τ , (normalized by | δ k | ) for the detunings shown in b), generated by 7 discrete phase shifts....Fig Fig:F2. shows closed mode trajectories representing the complete decoupling of a pair of 171 Yb + hyperfine **qubits** from five excited transverse phonon modes , using phase-shifts derived from Eq. eq:decouplingcond. In this illustrative example, we set the laser **frequencies** so that two modes have commensurate detunings and choose τ s = 2 π / δ 1 , 5 to match the period of the associated phase space evolution. In this way, a sequence of only n = 7 phase shifts is required to decouple the **qubits** from all 5 modes, rather than the more general sequence of n = 31 phase shifts. Assuming equivalent physical parameters to recent demonstrations of multimode decoupling using optimized amplitude modulation , the resulting phase-modulate gate has duration τ ∼ 140 μ s which compares favorably with the reported value of τ = 190 μ s, while obviating considerations of nonlinear amplitude responses in optical modulators and rf amplifiers. Faster gate times may be achieved, at the expense of a greater number of phase shifts, by allowing τ s to vary arbitrarily....for k = 1 , . . . , M . Each of time-parameterized functions α k t , 0 ≤ t ≤ τ , defines a set of N phase space trajectories α k μ t = f k μ α k t , for μ = 1 , . . . , N , associated with the k -th **oscillator** mode (Fig. Fig:F1). These trajectories vary in extent and orientation, according to the complex coupling constant f k μ . However, by satisfying the condition ( eq:deccond2) all trajectories are closed at t = τ ....In Fig. Fig:F3 we plot F k ω calculated for specific, but arbitrarily chosen, orders associated with each of three modes, k = 1 , 2 , 3 . By increasing the level of concatenation for specific modes we are able to improve **qubit**-**oscillator** decoupling through the suppression of low-**frequency** amplitude fluctuations while simultaneously ensuring all modes are efficiently decoupled. In general, D k depends on the initial **qubit** state | φ 0 and the effective temperature, in addition to the **frequency** of the k -th mode ω k . Here, where we consider only the collective zero temperature limit and the particular initial state | φ 0 = | 11 z . ... We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic **oscillator** modes. Such systems are particularly susceptible to residual **qubit**-**oscillator** entanglement at the conclusion of a gate period which reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete phase shifts in the field moderating the **qubit**-**oscillator** interaction - easily implemented with modern synthesizers - is sufficient to both ensure multiple **oscillator** modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting **qubits** and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach allows multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols.

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Contributors: Zorin, Alexander B., Makhlin, Yuriy

Date: 2011-01-13

The readout of a coupled **qubit** is based on the shift in the plasma **frequency** (and thus of the switching curve) due to different Josephson inductances in two **qubit** states. The inductance values depend on the type of **qubit** and its parameters. During the readout the **qubit** state is encoded in the resulting **oscillations** of the PDBA by tuning the control parameters (such as the drive **frequency** and amplitude, i.e., ξ and P ) to a point with the maximal difference (contrast) between the two switching curves. High contrast is reached when the shift in the plasma **frequency** exceeds the width of the switching curve. In an ideal arrangement, this contrast reaches 100%: P s w = 0 and 1 for two **qubit** states. For the PDBA the contrast reaches values comparable to those for the JBA with similar circuit parameters (for example, about 0.3% in **frequency** sensitivity for the parameters of Fig. fig:switch at low T , that is sufficient for reliable readout of the charge-phase **qubit** shown in Fig. fig:EqvSchm). Further optimization of the PDBA parameters is possible....We propose a threshold detector with an operation principle, based on a parametric period-doubling bifurcation in an externally pumped nonlinear resonance circuit. The ac-driven resonance circuit includes a dc-current-biased Josephson junction ensuring parametric **frequency** conversion (period-doubling bifurcation) due to its quadratic nonlinearity. A sharp onset of **oscillations** at the half-**frequency** of the drive allows for detection of small variations of an effective inductance and, therefore, the read-out of the quantum state of a coupled Josephson **qubit**. The bifurcation characteristics of this circuit are compared with those of the conventional Josephson bifurcation amplifier, and its possible advantages are discussed....The PDB circuit (see Fig. fig:EqvSchm) comprises a dc-current-biased Josephson junction with the critical current I c , capacitance C including the self-capacitance of the junction with, possibly, a contribution of an external capacitance, the linear shunting conductance G , as well as an attached **qubit**, presented here as a charge-phase **qubit** . The circuit is driven by a harmonic signal I ac = I A cos 2 ω t at a **frequency** close to the double **frequency** of small-amplitude plasma **oscillations** ω p , i.e., ω ≈ ω p ....Electric circuit diagram of the period-doubling bifurcation detector with microwave-based readout. The resonator is formed by the inductance of a non-linear Josephson junction (large crossed box), biased at a non-zero phase value ϕ 0 , and the capacitance C . The linear losses are accounted for by the conductance G . The resonator is coupled to a charge-phase **qubit** formed by a superconducting single electron transistor with capacitive gate (left) and attached to the Josephson junction. The **qubit** operation at the optimal point for an arbitrary bias I 0 is ensured by a proper value of the external magnetic control flux Φ c , applied to the **qubit** loop, and the gate charge Q g on the **qubit** island....Typical switching curves (switching probability P s w = 1 - e - Γ τ o during some observation time τ o vs. ξ ) are shown in Fig. fig:switch for various temperatures for a set of typical circuit parameters. Note that the position and the width of the switching curve (see inset) saturates at low temperatures. This effect is not a manifestation of the real quantum tunneling, but is rather linked to the fact that activation in the rotating frame of the first harmonic Eqs. ( A-and-alpha, eq:uv), i.e., the low-**frequency** noise in that frame, is given in the laboratory frame by the noise at a finite **frequency** ω , cf. Eq.( eq:Teff) and above....yield the range of **frequency** detunings, ξ - **oscillating** state with a finite amplitude A + given by Eq. ( A-stat). For ξ < ξ - the parametric resonance curve is multivalued with the stable trivial A = 0 and nontrivial A + solutions, while the solution A - is unstable. Taking into account higher (e.g., ∝ ) terms in Eqs. ( dot-A– dot-alpha) ensures that A + ξ and A - ξ merge, limiting both the amplitude A + and the the range of bistability in ξ ; for stronger drive even higher nonlinearities become important. The shape of the resonance curve, calculated numerically from Eqs. ( dot-A– dot-alpha), is shown in Fig. fig:res-curve for several values of the drive amplitude 3 P just above the excitation threshold. However, for further considerations of the threshold behavior the higher nonlinearities are not crucial, and below we neglect the -terms....(Color online) Intensity A 2 of **oscillations** of the Josephson phase at **frequency** ω versus **frequency** detuning ξ for two amplitudes of the pumping signal (at **frequency** 2 ω ). Dashed lines show unstable states. When the detuning approaches a bifurcation point ((i) or (ii)), PDB occurs (vertical arrows from or to the zero state, respectively). For comparison, a typical resonance curve of a JBA is sketched in the inset....This Hamiltonian for the slow variables can be obtained from the Hamiltonian for the physical quantities . Figure velocity-plot shows a contour plot of the absolute value of the velocity | v | = u ̇ 2 + v ̇ 2 1 / 2 = A ̇ 2 + A 2 α ̇ 2 1 / 2 in the case of a multivalued stationary solution. One can see the darker S -shaped narrow valley, where the motion is slow along the curvilinear s -axis. The black spots in this area show the stationary solutions, which are the stable focus at zero, A = 0 , the stable foci A + and A + * corresponding to equal-amplitude **oscillations** with a mutual phase shift of π , and the unstable saddles A - and A - * (also with a mutual π -shift). For weak dissipation these ‘saddle points’ are the lower points of the barriers separating the basins of attraction of the foci in the landscape of H . Thus the most probable escape path from the zero state is along the S -shaped valley. ... We propose a threshold detector with an operation principle, based on a parametric period-doubling bifurcation in an externally pumped nonlinear resonance circuit. The ac-driven resonance circuit includes a dc-current-biased Josephson junction ensuring parametric **frequency** conversion (period-doubling bifurcation) due to its quadratic nonlinearity. A sharp onset of **oscillations** at the half-**frequency** of the drive allows for detection of small variations of an effective inductance and, therefore, the read-out of the quantum state of a coupled Josephson **qubit**. The bifurcation characteristics of this circuit are compared with those of the conventional Josephson bifurcation amplifier, and its possible advantages are discussed.

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Contributors: De, Amrit, Joynt, Robert

Date: 2012-11-17

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....We use a quasi Hamiltonian formalism to describe the dissipative dynamics of a circuit QED **qubit** that is affected by several fluctuating two level systems with a 1/f noise power spectrum. The **qubit**-resonator interactions are described by the Jaynes Cummings model. We argue that the presence of pure dephasing noise in such a **qubit**-resonator system will also induce an energy relaxation mechanism via a fluctuating dipole coupling term. This random modulation of the coupling is seen to lead to rich physical behavior. For non-Markovian noise, the coupling can either worsen or alleviate decoherence depending on the initial conditions. The magnetization noise leads to behavior resembling the collapse and revival of Rabi **oscillations**. For a broad distribution of noise couplings, the **frequency** of these **oscillations** depends on the mean noise strength. We describe this behavior semi-analytically and find it to be independent of the number of fluctuators. This phenomenon could be used as an in situ probe of the noise characteristics....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 . ... We use a quasi Hamiltonian formalism to describe the dissipative dynamics of a circuit QED **qubit** that is affected by several fluctuating two level systems with a 1/f noise power spectrum. The **qubit**-resonator interactions are described by the Jaynes Cummings model. We argue that the presence of pure dephasing noise in such a **qubit**-resonator system will also induce an energy relaxation mechanism via a fluctuating dipole coupling term. This random modulation of the coupling is seen to lead to rich physical behavior. For non-Markovian noise, the coupling can either worsen or alleviate decoherence depending on the initial conditions. The magnetization noise leads to behavior resembling the collapse and revival of Rabi **oscillations**. For a broad distribution of noise couplings, the **frequency** of these **oscillations** depends on the mean noise strength. We describe this behavior semi-analytically and find it to be independent of the number of fluctuators. This phenomenon could be used as an in situ probe of the noise characteristics.

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Contributors: Tufarelli, Tommaso, Ferraro, Alessandro, Kim, M. S., Bose, Sougato

Date: 2011-09-09

We consider a generic **oscillator** network in an unknown state—possibly being an eigenstate of some simulated anharmonic model, or an intermediate state of a quantum computation. Regardless the previous dynamics, we suppose that, from a certain time t = 0 , the **oscillators** interact only harmonically. In addition, a single **qubit** can interact with a single (fixed) **oscillator** via a tunable bilinear coupling (see Fig. network). Such a network-probe system is then let evolve for a certain period of time, allowing part of the information about the network state to be transfered into the **qubit**. Afterwards, only the **qubit** is measured. Repeating the procedure it is possible to reconstruct the state of the whole network, by solely tuning the profile of the interaction strength....Sketch of a **qubit** tunably coupled to a linear chain of **oscillators** with constant nearest-neighbour couplings. chain...sec_exp The reconstruction scheme described here is based on a rather ubiquitous dynamics. Essentially, it requires a harmonic coupling between the **oscillators** and a bilinear **qubit**-**oscillator** interaction. An experimental platform that is particularly mature for our purposes is given by a chain of ions in a linear trap. There the harmonic dynamic is provided by the Coulomb interaction, as recently demonstrated in Ref. . In addition, the required **qubit**-**oscillator** coupling is standard : two electronic levels of one ion provide the **qubit**, whereas the coupling is realized via a standing laser wave . For example, consider a linear chain of N = 8 ions, with ω n ∼ ω , J n , m = K n m ∼ δ m , n + 1 0.2 ω and take an interaction time t ∼ 100 2 π / ω . Then | g s | max ∼ 0.08 ω allows to reconstruct states with χ ξ having support in | ξ n | 2 . Notice that a modest motional quality factor Q 5 × 10 3 is required (see Fig. chainplots), and it is sufficient to vary the laser power on a time scale of the eigenfrequencies ν k (typically of the order of MHz) in order to realize the desired profile of g s . We stress again that only one ion needs to be illuminated in order to reconstruct the motional state of the entire chain....In the normal modes representation of the **oscillator** network, the **qubit** is interacting with those modes b k such that G k ≠ 0 . Each mode b k behaves as a simple harmonic **oscillator** of **frequency** ν k , and does not interact with the others [see Eqs. ( HINT) and ( H_normal)]. If the assumptions (A1) and (A2) are verified, the **qubit** interacts with all the normal modes, and can distinguish each mode by its **frequency**. normalmod...We introduce a scheme to reconstruct arbitrary states of networks composed of quantum **oscillators**--e.g., the motional state of trapped ions or the radiation state of coupled cavities. The scheme uses minimal resources, in the sense that it i) requires only the interaction between one-**qubit** probe and one constituent of the network; ii) provides the reconstructed state directly from the data, avoiding any tomographic transformation; iii) involves the tuning of only one coupling parameter. In addition, we show that a number of quantum properties can be extracted without full reconstruction of the state. The scheme can be used for probing quantum simulations of anharmonic many-body systems and quantum computations with continuous variables. Experimental implementation with trapped ions is also discussed and shown to be within reach of current technology....linearchain Let us give a concrete example of a quantum **oscillator** network where the presented ideas can be applied. Consider a linear chain of N **oscillators**, each having the same local **frequency** ω , and where only the nearest-neighbours interact with a coupling strength J n , n + 1 = K n , n + 1 = J , constant along the chain. The J ’s are often referred to as hoppings. We assume that the **qubit** is tunably coupled to the first **oscillator** of the chain. The system is sketched in Fig. chain:...which is different from zero for any k ∈ 1 . . . N . Therefore, also assumption (A1) is verified. Thus, the quantum state of a linear chain of **oscillators** with constant nearest-neighbour couplings can be fully reconstructed, by using a single **qubit** coupled to one end of the chain. As an example, Fig. chainplots shows some quantities of interest for the reconstruction protocol of a linear chain of N = 8 **oscillators**....chainplots Reconstruction protocol for a linear chain of N = 8 **oscillators**. In all plots we consider nearest-neighbour couplings J n m = K n m = J δ m , n + 1 , with J = 0.2 ω , decoherence rates κ k = κ = 10 -6 ω temperature T = 200 ω and a coupling noise strength ϵ = 10 -5 ω . Note that we are requiring a motional quality factor Q ∼ ω κ N ∼ 5 × 10 3 . We assumed the decoherence of the **qubit** to be negligible. Plot (a) shows the determinant of the matrix M , which we require to be different from zero in our protocol. We see that M becomes invertible for t 50 / ω . In plots (b) and (c), we have considered a phase-space region | η k | ≤ 2 in the normal modes basis, and the total interaction time has been fixed to t = 100 × 2 π / ω . Plot (b) shows a portion of the interaction strength profile required to obtain the displacement parameters β = -1 - 1 . . . - 1 , which allows us to reconstruct the phase space point η = 2 2 . . . 2 , lying at the boundary of the considered region. With our choice of parameters, it is necessary to access maximal coupling strengths of | g | m a x ≃ 0.08 ω to reach this phase space point. In plot (c), the effect of decoherence on the measured value of the characteristic function is shown. We fixed η 2 = η 3 = . . . = η N = 2 and I m η 1 = 0 . The plot shows the quantity e - f as a function of R e η 1 . We see that near the boundary of the considered phase-space region the quantity measured via the **qubit** expectation values corresponds to about 20-23 % of the actual value of the characteristic function [see chi-deco]. Plot (d) shows the maximum achievable phase-space resolution along the directions that diagonalize V δ β [we take the square roots of its eigenvalues λ k as a measure of the phase-space accuracy]. The asymptotical behaviour for large t is given by λ k ≃ | G k | ϵ t , as expected. As a final remark, we note that the considered range of interaction times is consistent with Eq. ( master-condition), which gives a necessary condition for the validity of the master equation used. Indeed, from Eq. ( master-condition) one can estimate that our treatment is valid for t ≪ t max , where t max = min k 1 κ k N k ω K ∼ 2 × 10 3 2 π / ω ....Graphical representation of the Hamiltonian of HTOT. A **qubit** (two-level system) is tunably coupled [ g t ] to a single constituent ( a 1 ) of an **oscillator** network. The constituents of the network ( a j , with j = 1 , . . . , N ) are in turn harmonically coupled ( J n , m , K n , m ). network...We note that, in the new representation, the **qubit** interacts with all the modes b k such that G k ≠ 0 . Fig. normalmod shows a graphical representation of the Hamiltonian ( HTOT), in terms of the normal modes of the **oscillator** network. ... We introduce a scheme to reconstruct arbitrary states of networks composed of quantum **oscillators**--e.g., the motional state of trapped ions or the radiation state of coupled cavities. The scheme uses minimal resources, in the sense that it i) requires only the interaction between one-**qubit** probe and one constituent of the network; ii) provides the reconstructed state directly from the data, avoiding any tomographic transformation; iii) involves the tuning of only one coupling parameter. In addition, we show that a number of quantum properties can be extracted without full reconstruction of the state. The scheme can be used for probing quantum simulations of anharmonic many-body systems and quantum computations with continuous variables. Experimental implementation with trapped ions is also discussed and shown to be within reach of current technology.

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