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Cited by 141

The DOI matches its Crossref record; 36 of the 36 references it cites were found in a registry or an index.

Coherent Oscillations inside a Quantum Manifold Stabilized by Dissipation

Touzard, S., Grimm, A., Leghtas, Z., Mundhada, S. O., Reinhold, P., Axline, C., Reagor, M., Chou, K., Blumoff, J., Sliwa, K. M., Shankar, S., Frunzio, L., Schoelkopf, R. J., Mirrahimi, M., Devoret, M. H.·2018·Physical Review X
DOI 10.1103/physrevx.8.021005
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Abstract

New experiments present experimental evidence for the dynamical quantum Zeno effect, in which a stabilizing friction leads to a continuous phase change among degenerate states in a quantum superposition.

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Figures

Figure 1 Quantum Zeno dynamics and its implementation. (a) Conceptual representation of the experiment. The quantum state of a harmonic oscillator is represented here by a point in a 2D plane (not to be confused with phase space). The dark blue circle represents a cross-section of the Bloch sphere of a two-state manifold in the larger Hilbert space of the oscillator. The quantum Zeno effect observed in our experiment corresponds to motion along the circle. A weak excitation drive is applied to the oscillator and the resulting trajectory has both a component along the circle and out of it. The non-linear dissipation and drive (orange) cancels the movement outside the circle while being blind to the position of the quantum state on the circle. (b) Schematics of the experimental device. The quantum manifold is stabilized within the Hilbert space of the fundamental mode of an aluminium post cavity (cyan, storage in the text). This resonator is coupled to two Josephson junctions on sapphire (yellow for the reservoir and crimson for the transmon qubit, see text), which are read out by stripline resonators (grey). Three couplers (brown) bring microwave drives into the system and carry signals out of it.
Figure 1 Quantum Zeno dynamics and its implementation. (a) Conceptual representation of the experiment. The quantum state of a harmonic oscillator is represented here by a point in a 2D plane (not to be confused with phase space). The dark blue circle represents a cross-section of the Bloch sphere of a two-state manifold in the larger Hilbert space of the oscillator. The quantum Zeno effect observed in our experiment corresponds to motion along the circle. A weak excitation drive is applied to the oscillator and the resulting trajectory has both a component along the circle and out of it. The non-linear dissipation and drive (orange) cancels the movement outside the circle while being blind to the position of the quantum state on the circle. (b) Schematics of the experimental device. The quantum manifold is stabilized within the Hilbert space of the fundamental mode of an aluminium post cavity (cyan, storage in the text). This resonator is coupled to two Josephson junctions on sapphire (yellow for the reservoir and crimson for the transmon qubit, see text), which are read out by stripline resonators (grey). Three couplers (brown) bring microwave drives into the system and carry signals out of it.CC-BY · From the PDF
Figure 2 Experimental protocol and Wigner tomography result. (a) Sequence of different drives outlined in fig 1. The storage is either initialized in a cat state N ( | α ∞ 〉 ± |-α ∞ 〉 ) with optimal control pulses or in a coherent state |± α ∞ 〉 with a displacement D( ± α ∞ ). The drives stabilizing the manifold are turned on in 24 ns (purple and yellow). They are on for a duration ∆ t during which the storage drive (cyan) can be turned on to induce the parity oscillations. The drives are left on for another 500 ns and then turned off in 24 ns after which the Wigner function is measured. (b) In the left column is the Wigner functions of the storage cavity after initialization in an even or odd cat state ( | α ∞ | 2 = 3) . The right column shows the corresponding Wigner functions after a quarter of an oscillation. The colormaps are averaged raw data of the Wigner function measurement (see text) and the orange circles are cuts along Re( α )=0. The grey solid lines are theoretical curves corresponding to even or odd cats (left column, lines 1 and 2 respectively) and parityless cats (right column, lines 1 and 2). The only fit parameter in the theory is the renormalization of the amplitude by a factor 0.87 on the left, and 0.65 on the right. These factors account for the fidelity of the parity measurement and the decay of the fringes of the cat states during the stabilization.
Figure 2 Experimental protocol and Wigner tomography result. (a) Sequence of different drives outlined in fig 1. The storage is either initialized in a cat state N ( | α ∞ 〉 ± |-α ∞ 〉 ) with optimal control pulses or in a coherent state |± α ∞ 〉 with a displacement D( ± α ∞ ). The drives stabilizing the manifold are turned on in 24 ns (purple and yellow). They are on for a duration ∆ t during which the storage drive (cyan) can be turned on to induce the parity oscillations. The drives are left on for another 500 ns and then turned off in 24 ns after which the Wigner function is measured. (b) In the left column is the Wigner functions of the storage cavity after initialization in an even or odd cat state ( | α ∞ | 2 = 3) . The right column shows the corresponding Wigner functions after a quarter of an oscillation. The colormaps are averaged raw data of the Wigner function measurement (see text) and the orange circles are cuts along Re( α )=0. The grey solid lines are theoretical curves corresponding to even or odd cats (left column, lines 1 and 2 respectively) and parityless cats (right column, lines 1 and 2). The only fit parameter in the theory is the renormalization of the amplitude by a factor 0.87 on the left, and 0.65 on the right. These factors account for the fidelity of the parity measurement and the decay of the fringes of the cat states during the stabilization.CC-BY · From the PDF
Figure 3 Characterization of the oscillations. (a) Evolution of the measured parity as a function of time. The initial cat states are even, with ¯ n = | α ∞ | 2 = 2 , 3 , 5 (circles, squares, diamonds). The storage drive is either off (black markers) or on (coloured markers) with various strengths given in units of a chosen base strength ϵ 0 . Simulations are shown as solid lines. The minimum of each experimental curve is emphasized for each drive strength (full marker with black contour). (b) A fit of the data gives the frequency Ω and the time constant τ of the decaying oscillations. The former is plotted as a function of the relative drive strength ϵ/ϵ 0 (top panel). The case ¯ n =2 is fitted with a linear function (dashed line). Based on this, we make predictions for ¯ n =3, 5 (solid lines). The bottom panel shows the characteristic decay time of the oscillations τ , normalized by the decay time of the non-driven case τ 0
Figure 3 Characterization of the oscillations. (a) Evolution of the measured parity as a function of time. The initial cat states are even, with ¯ n = | α ∞ | 2 = 2 , 3 , 5 (circles, squares, diamonds). The storage drive is either off (black markers) or on (coloured markers) with various strengths given in units of a chosen base strength ϵ 0 . Simulations are shown as solid lines. The minimum of each experimental curve is emphasized for each drive strength (full marker with black contour). (b) A fit of the data gives the frequency Ω and the time constant τ of the decaying oscillations. The former is plotted as a function of the relative drive strength ϵ/ϵ 0 (top panel). The case ¯ n =2 is fitted with a linear function (dashed line). Based on this, we make predictions for ¯ n =3, 5 (solid lines). The bottom panel shows the characteristic decay time of the oscillations τ , normalized by the decay time of the non-driven case τ 0CC-BY · From the PDF
Figure 4 Gate on cardinal points of the Bloch sphere of the protected manifold. An arbitrary cat state N ( cos( θ 2 ) ∣ ∣ C + α ∞ 〉 +sin( θ 2 ) e iφ ∣ ∣ C -α ∞ 〉) is represented by a point on a sphere. Six initial states are chosen, corresponding to the cardinal points ( θ, φ ) = (0 , 0) , ( π, 0) , ( ± π/ 2 , 0) , ( π/ 2 , ± π/ 2) , with | α ∞ | 2 = 3 , and their equivalent Pauli operators are measured. The markers corresponding to each initial state are respectively red and blue circles, grey up/down triangles, and black up/down triangles. The initial octahedron formed by those points (left) is either transformed under the action of the identity (upper right) or a rotation of π/ 2 around the X axis (lower right).
Figure 4 Gate on cardinal points of the Bloch sphere of the protected manifold. An arbitrary cat state N ( cos( θ 2 ) ∣ ∣ C + α ∞ 〉 +sin( θ 2 ) e iφ ∣ ∣ C -α ∞ 〉) is represented by a point on a sphere. Six initial states are chosen, corresponding to the cardinal points ( θ, φ ) = (0 , 0) , ( π, 0) , ( ± π/ 2 , 0) , ( π/ 2 , ± π/ 2) , with | α ∞ | 2 = 3 , and their equivalent Pauli operators are measured. The markers corresponding to each initial state are respectively red and blue circles, grey up/down triangles, and black up/down triangles. The initial octahedron formed by those points (left) is either transformed under the action of the identity (upper right) or a rotation of π/ 2 around the X axis (lower right).CC-BY · From the PDF
Figure S1 Experimental setup. See text of first section.
Figure S1 Experimental setup. See text of first section.CC-BY · From the PDF
Figure S2 Temperature dependent phase-flip error. Normalized leakage from the initial coherent state | α ∞ 〉 to the state of opposite phase as a function of time. The dashed red, yellow and blue curves give the theoretical behavior at zero temperature for ¯ n = 2, 3 and 5 respectively. The corresponding markers show the experimentally observed values. Each of the latter curves is well reproduced by simulation (solid black lines) for a thermal population of 1.5% for ¯ n = 2 and 3 and 2% for ¯ n = 5 in the reservoir mode. The data and the corresponding simulations are taken at a fixed frequency for the microwave pumps. As the frequency matching condition is not optimized for each ¯ n , the theory curve for ¯ n = 3 has a lower leakage rate than ¯ n = 5.
Figure S2 Temperature dependent phase-flip error. Normalized leakage from the initial coherent state | α ∞ 〉 to the state of opposite phase as a function of time. The dashed red, yellow and blue curves give the theoretical behavior at zero temperature for ¯ n = 2, 3 and 5 respectively. The corresponding markers show the experimentally observed values. Each of the latter curves is well reproduced by simulation (solid black lines) for a thermal population of 1.5% for ¯ n = 2 and 3 and 2% for ¯ n = 5 in the reservoir mode. The data and the corresponding simulations are taken at a fixed frequency for the microwave pumps. As the frequency matching condition is not optimized for each ¯ n , the theory curve for ¯ n = 3 has a lower leakage rate than ¯ n = 5.CC-BY · From the PDF
Figure S3 Cavity Stark shift. We do a spectroscopy of the storage cavity while applying the pump with different strengths. We send a 17 µ s square pulse and measure the overlap of the resulting storage state with the Fock state | 0 〉 . The frequency of the storage pulse is modulated around the value corresponding to the bare frequency of the storage cavity (given by zero probe detuning). The x-axis shows the amplitude of the pump. The zero means that no tone is sent while 1 corresponds to the maximum amplitude that our control electronics can deliver. The right panel is a zoom of the anti-crossing corresponding to the frequency matching condition being met.
Figure S3 Cavity Stark shift. We do a spectroscopy of the storage cavity while applying the pump with different strengths. We send a 17 µ s square pulse and measure the overlap of the resulting storage state with the Fock state | 0 〉 . The frequency of the storage pulse is modulated around the value corresponding to the bare frequency of the storage cavity (given by zero probe detuning). The x-axis shows the amplitude of the pump. The zero means that no tone is sent while 1 corresponds to the maximum amplitude that our control electronics can deliver. The right panel is a zoom of the anti-crossing corresponding to the frequency matching condition being met.CC-BY · From the PDF
Figure S4 Frequency of the reservoir drive versus amplitude of the pump tone. We play the pump tone and the reservoir drive during 100 µ s and plot the overlap of the resulting storage state with | 0 〉 for different frequencies of the reservoir drive and amplitudes for the pump tone. The reservoir drive frequency is expressed by its detuning from the Stark-shifted frequency of the reservoir at the frequency matching condition. The right panel is a zoom on the anti-crossing taking place at that particular point.
Figure S4 Frequency of the reservoir drive versus amplitude of the pump tone. We play the pump tone and the reservoir drive during 100 µ s and plot the overlap of the resulting storage state with | 0 〉 for different frequencies of the reservoir drive and amplitudes for the pump tone. The reservoir drive frequency is expressed by its detuning from the Stark-shifted frequency of the reservoir at the frequency matching condition. The right panel is a zoom on the anti-crossing taking place at that particular point.CC-BY · From the PDF

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Metadata

Authors
Touzard, S., Grimm, A., Leghtas, Z., Mundhada, S. O., Reinhold, P., Axline, C., Reagor, M., Chou, K., Blumoff, J., Sliwa, K. M., Shankar, S., Frunzio, L., Schoelkopf, R. J., Mirrahimi, M., & Devoret, M. H.
Venue
Physical Review X
Publisher
American Physical Society (APS)
Published
2018-04-04
Volume
8
Issue
2
ISSN
2160-3308
DOI
10.1103/physrevx.8.021005
Cited by
141
References
36
Language
en
Licence
creativecommons.org/licenses/by/4.0/ (vor)
Funders
  • Yale University
  • National Science Foundation (MRSEC DMR-1119826, DGE-1122492)
  • Army Research Office (W911NF-14-1-0011, W911NF-16-1-034, W911NF-14-1-0563)
  • Air Force Office of Scientific Research (FA9550-15-1-0015)
  • YINQE

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