Filter Results
59137 results
• Reconstructed process matrix for a) Û and b) two repetitions of Û . The map E i j produces a matrix E k , l for each element i j . Hence elements of the matrix E are labeled by m = 4 i - 1 + k , n = 4 j - 1 + l , where the factor 4 results from the size of the two-qubit state space. For example, the 11 00 ( i = 1 , j = 4 ) element of an input density matrix is mapped to E 11 00 , a 4 × 4 block of E given by m ∈ 1 4 and n ∈ 13 16 . The position of each peak is in agreement with the theoretical prediction.... \centering Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305, U.S.A. ... Hybrid qubit storage in the 9 B e + 2s 2 S 1 / 2 hyperfine levels. The states are labeled using the total angular momentum quantum numbers F and M F . 1 , 0 are the qubit states used for single qubit gates and transport, and 1 G , 0 G are used for two-qubit gates. For detection, the 1 , - 1 and 2 , 2 states are used. At the applied magnetic field ( B ≃ 0.011964 T), the frequencies for transitions between pairs of states with the same F are well resolved.... a) Schematic of the qubit ion trajectories (solid red and dotted blue lines) and gate operations used to implement Û . The single qubit rotations are “ π / 2 ” ≡ R π / 2 0 (eq. eq:rotation). The two-qubit gate implements G ̂ = D 1 i i 1 . b) Full sequence used to perform process tomography on Û and Û 2 . This sequence is repeated 350 times for each setting of preparation/analysis.
Data Types:
• Image
• (Color online) Upper panel: Rabi frequency | Ω R | in units of c = n p h g c 2 / L for μ = - 100 ϵ L and κ = 5 ϵ L , where ϵ L = 2 m L 2 -1 . A large photon linewidth κ has been chosen to highlight the essential features. The crosses denote the Rabi frequency and damping determined numerically from Eq. ( eq:Dgamma2). Solid green and red lines correspond to the solutions for the limits Ω | μ | , see Eq. ( eq:GammaOmega_metallic), respectively. Lower panel: The ratio between Rabi frequency and damping, Ω R / Γ R , determines the fidelity of qubit rotations.... These functions are plotted in Fig.  fig:plots. The Rabi frequency is, as expected, exponentially suppressed in the length of the CR. However, as the photon frequency Ω approaches the critical value | μ | , the prefactor 1 - Ω / | μ | qubit state for a time t * = π / 4 Ω R . In the presence of damping, the fidelity of such an operation can be estimated as... (Color online) Upper panel: A semiconductor nanowire (along the x axis) hosting Majorana fermions is embedded in a microwave stripline cavity (along the y axis). The red lines show the amplitude of the electric field E → r → . Dark blue (light yellow) sections of the wire indicate topologically nontrivial (trivial) regions. MBSs (stars) exist at the edges of nontrivial (topological superconductor, TS) regions. The MBSs γ 1 and γ 2 can be braided using a T -junction . Lower panel: Band structure of the individual sections of the wire. The four MBSs γ 1 , 2 , 3 , 4 encode one logical qubit. The central MBSs γ 2 and γ 3 are tunnel-coupled ( t c ) to a topologically trivial, gapped central region (CR, light yellow) with length L . All energies are small compared to the induced gap Δ .
Data Types:
• Image
• Single-qubit-gate parameters used for programming the quantum processor to produce the states in Figure 2 of the main text.... Components for universal computation. Circuit diagrams for arbitrary unitary transformations on a, one and b, two qubits. The operations for each circuit are implemented from left to right with each line representing one qubit. Part a also indicates the decomposition employed for R θ φ . The dashed box in part b contains the three degrees of freedom α , β , δ that determine the two-qubit operation’s local equivalence class. The brackets highlight the decomposition of the two-qubit operation U as described in the text: U = C ⊗ D ⋅ V ⋅ A ⊗ B .... Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA
Data Types:
• Image
• Tabular Data
• Structured bath/weak coupling: We now turn to the structured spectral density given by Eq.( eq:density). The main features of the corresponding system [Eq.( eq:system)] can already be understood by analyzing only the coupled two-level-harmonic oscillator system (without damping, i.e. Γ = 0 ). For ε = 0 this system exhibits two characteristic frequencies, close to Ω and Δ , associated with the transitions 1 and 2 in Fig.  fig:correlation_weak(c). These should also show up in the correlation function C ω ; and indeed Fig.  fig:correlation_weak(a) displays a double-peak structure with the peak separation somewhat larger than Δ - Ω , due to level repulsion. The coupling to the bath will in general lead to a broadening of the resonances and an enhancement of the repulsion of the two energies. Due to the very small coupling ( α = 0.0006 ) peak positions of C ω in Fig.  fig:correlation_weak can with very good accuracy be derived from a second order perturbation calculation for the coupled two-level-harmonic oscillator system, yielding the following transition frequencies [depicted in inset (c) of Fig.  fig:correlation_weak]: ω 1 , + - ω 0 , + = Ω - g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 0.987 Ω and ω 0 , - - ω 0 , + = Δ 0 + g 22 Δ 0 / Δ 2 0 - Ω 2 ≈ 1.346 Ω . With the two peaks we associate two different dephasing times, τ Ω and τ Δ , as shown in inset (a) and (b) of Fig.  fig:correlation_weak.... Spin-spin correlation function as a function of frequency for experimentally relevant parameters discussed in Ref. : α = 0.0006 , Δ 0 = 4 GHz, ε = 0 (this is the so-called “idle state”), Ω = 3 GHz, Γ = 0.02 , and ω c = 8 GHz. The sum rule is fulfilled with an error of less than 1 %. (a) Blow up of the peak region reveals a double peak; (b) blow up of the larger peak, (c) term scheme of a two level system coupled to an harmonic oscillator, drawn for Δ 0 ≫ Ω ; ( α = 0.0006 corresponds to g / Ω ≈ 0.06 .) -0.9cm... Stronger coupling to bath: Figure  fig:times_nobias(b) shows τ Δ , τ Ω , and τ w for a larger coupling strength of α = 0.01 . Figure  fig:correlation_strong(a) shows one of the calculated correlation functions. Note that the stronger coupling α leads to a larger separation, or “level repulsion”, between the Δ - and Ω -peaks than in Fig.  fig:correlation_weak. The inset of Fig.  fig:times_nobias(b) shows the renormalized tunneling matrix element Δ ∞ as a function the initial matrix element Δ 0 . Very importantly, for Δ 0 Ω , Δ increases during the flow, whereas for Δ 0 Ω , it decreases . This behavior can be understood from the fact that f ω l in Eq.( eq:flow1) changes sign at ω = Δ : If the weight of J ω under the integral in ( eq:flow1) is larger for ω > Δ 0 , which is the case if Δ Δ 0 ]. Note also, that the upward renormalization towards larger Δ ∞ in the inset of Fig.  fig:times_nobias(b) is stronger than the downward one towards smaller values, i.e., the renormalization is not symmetric with respect to Δ 0 = Ω . The reason for this asymmetry lies in the fact that f ω l has a larger weight for ω Δ . Also τ Δ and even τ w = 1 / J Δ ∞ in Fig.  fig:times_nobias(b) show an asymmetric behavior with a steep increase at Δ 0 ≈ Ω : dephasing times for Δ 0 > Ω are larger than for Δ 0 Ω than for Δ 0 Ω , dephasing times can be significantly enhanced (as compared to Δ 0 oscillator in (b).... Spin-spin correlation function for the structured bath [Eq.( eq:density)] as a function of frequency . The maximum height of the middle peak in (b) is ≈ 7.2 . -0.9cm
Data Types:
• Image
• Left: Schematic illustration of an experimental arrangement for measuring the phase dependence of the population of the excited state | 1 : (a) The microwave field couples the ground state ( | 0 ) to the excited state ( | 1 ). A third level, | 2 , which can be coupled to | 1 optically, is used to measure the population of | 1 via fluorescence detection. (b) The microwave field is turned on adiabatically with a switching time-constant τ s w , and the fluorescence is monitored after a total interaction time of τ . Right: Illustration of the Bloch-Siegert Oscillation (BSO): (a) The population of state | 1 , as a function of the interaction time τ , showing the BSO superimposed on the conventional Rabi oscillation. (b) The BSO oscillation (amplified scale) by itself, produced by subtracting the Rabi oscillation from the plot in (a). (c) The time-dependence of the Rabi frequency. Inset: BSO as a function of the absolute phase of the field with fixed τ .
Data Types:
• Image
• Power out of the resonator as a function of frequency for different input powers at zero flux bias.... V in, rms / V out, rms versus V out, rms 2 at Φ a = 0 and the low power resonance frequency for zero flux bias (f=8.2423Ghz). The circles show the experimental data, and the line shows the expected theoretical result for parameters determined by fits to figure 3 and figure 4.... Measured resonance frequency as a function of flux bias.
Data Types:
• Image
• To initial entangled state | I 0 of the two qubits with control field mode in its coherent state, its evolution involves on the various parameters in Eq.( t0-1). It depends on the number N in an extremely nonlinear and intricate way. The kind perplexity makes it hard to study the Rabi model, never the less, it also provide the opportunity to preserve the entanglement of the two qubits by careful choice of the appropriate parameters. Because the coherent state has the probability of Poisson distribution, which will be approximated by a Gauss distribution if the average number | α | 2 is large enough. The intricacy of the Rabi model could be utilized to make the quantity Y ~ N , + 2 L ~ N , + 4 = B N in Eq.( t0-1) extremely small when N is in the neighborhood of | α | 2 by some selection of the appropriate parameters, which will guaranty the initial state of the qubits unchanging. This is shown in Fig.( fig8). It can be easy to see that whenever we select the parameters appropriate, for example, the parameters as | α | 2 = 55 ,   a = - 0.6 ,   β = 0.5599 ,   ω 0 ω = 0.24 , the Bell state | I 0 = | 1 , 0 = 1 2 | ↑ ↑ - | ↓ ↓ have the probability about 1 - 0.005 = 99.5 100 to remain unchanged.... Figs.( fig11-3) show the general trait for the qubit remaining in its initial states | 1 , ± for different parameter a = 0.2 ,   0 ,   - 0.2 . Obviously, P N 1 t is influenced by four parameters β , a , N , ω 0 ω . In Ref., it is shown that the coupling strength β ranges from 0.01 to 1 for the application of adiabatic approximation (weak coupling will not be discussed here). From Eqs.( e0)-( t0),( p1- omega00), we see that the parameter a will come to action apparently whenever β ≈ 0.01 - 0.6 . As stated before, the qubits is equivalent to a two qubits system and the non-equal-energy-level parameter a represents the coupling strength between the two qubits. This shows the coupling of the two qubits changes their dynamics considerately in the range of β ≈ 0.1 - 0.6 for the adiabatic approximation method to be applied, and this is our limit on the coupling parameter β... Schematic diagram of P N 1 t with the four parameters as N = 2 ,   ω 0 ω = 0.25 ,   β = 0.2 , a = 0.2 ,   0 ,   - 0.2 from the top to bottom respectively. The apparent difference in these three figures strongly implies that the parameter a influences the qubits dynamically.... The parameter a is connected with the inter-qubit coupling strength κ as a = ± κ in symmetric and asymmetric transition cases respectively. Study also shows that the parameter a negative is favorable for | T α t approaching zero, as Fig.( fig8) exhibits. So the inter-qubit coupling is in favor of preservation of the initial entanglement, especial with asymmetrical transition case ( a < 0 ).
Data Types:
• Image
• These environmentally-induced Rabi oscillations are a clear signature of the non-Markovian behavior produced by the RLC environment, and are completely absent in the RL environment because the energy from the qubits is quickly dissipated without being temporarily stored. In the RL environment the decay in time of ρ 11 t has the characteristic non-oscillatory Markovian behavior. These environmentally-induced Rabi oscillations are generic features of circuits with resonances in the real part of the admittance. The frequency of the Rabi oscillations Ω R a = π κ Ω 3 / 2 Γ is independent of the resistance since Ω R a ≈ Ω π L 2 C / L 1 2 C 0 , and has the value of Ω R a = 2 π f R a ≈ 360 × 10 6  rad/sec in Fig.  fig:seven. This effect is similar to the so-called circuit quantum electrodynamics which has been of great experimental interest recently ... In Fig.  fig:four, T 1 is plotted for the phase qubit as a function of the qubit frequency ω 01 in the case of spectral densities describing an RLC [Eq. ( eqn:spectral-density-isolation)] or Drude [Eq. ( eqn:sd-drude)] isolation network at fixed temperatures T = 0 (main figure) and T = 50 mK (inset), with J i n t ω = 0 corresponding to R 0 ∞ . In the limit of low temperatures k B T / ℏ ω 01 ≪ 1 , the relaxation time becomes... fig:four (Color-online) T 1 (in seconds) as a function of qubit frequency ω 01 . The solid (red) curves describe the phase qubit with RLC isolation network (Fig.  fig:three) with parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = ∞ and L 0 = 0 . The dashed curves correspond to an RL isolation network with the same parameters, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.... As an illustration of the qualitative results discussed in this section, we show in Fig.  fig:six the frequency shift (renormalization) of the phase qubit with RLC isolation network described in Fig.  fig:three. We make the identification E = ω 01 and δ E = δ ω 01 . Near resonance ω 01 ≈ Ω , we find a frequency renormalization of about 2 % which is due to the term δ E r e s .... (Color-online) Schematic drawing of a phase qubit with an RLC isolation circuit. The phase qubit is shown inside the solid (red) box, the RLC isolation circuit is shown inside the dashed box to the left, and the internal admittance circuit is shown inside the dashed box to the right.... fig:six Renormalization of energy splitting for the phase qubit with RLC isolation network (Fig.  fig:three) for the parameters R = 50  ohms, L 1 = 3.9 nH, L = 2.25 pH, C = 2.22 pF, and qubit parameters C 0 = 4.44 pF, R 0 = 5000  ohms and L 0 = 0 , T = 0 , and Ω = 141 × 10 9 rad/sec.... (Color-online) Flux qubit measured by a dc-SQUID gray (blue) line. The qubit corresponds to the inner SQUID loop with critical current I c and capacitance C J for both Josephson junctions denoted by the large × symbol. The inner SQUID is shunted by a capacitance C s , and environmental resistance R and is biased by a ramping current I b . The dc-SQUID loop has junction capatitance C 0 and critical current I c 0 .... fig:five (Color-online) T 1 (in nanoseconds) as a function of qubit frequency ω 01 . The solid (red) curves describe a phase qubit with RLC isolation network (Fig.  fig:three) with same parameters of Fig.  fig:two except that R 0 = 5000  ohms. The dashed curves correspond to an RL isolation network with the same parameters of the RLC network, except that C = 0 . Main figure ( T = 0 ), inset ( T = 50 mK) with Ω = 141 × 10 9 rad/sec.
Data Types:
• Image