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Results in Physics

Abrupt energy flipping in a continuously observed Rabi driven quantum bit

Gilbert Reinisch

Published Sep 10, 2021• 5 citations
Publisher

Abstract

In the presence of a periodic external drive, the mean value of the energy of a two-level quantum system (or qubit) displays the well-known Rabi oscillatory dynamics that is usually described within the rotating wave approximation. In order to calculate its standard deviation, we get rid of this latter and obtain a huge ratio (of order unity) of standard deviation over mean value. This is due to the large-amplitude high-frequency oscillations of the internal as well as of the overall phase of the two-level system. In presence of continuous measurement of, say, the ground state energy that yields the quantum Zeno effect, the system remains frozen in its ground state as long as the dynamics of the driven system allows its energy to stay within its Rabi standard deviation. When reaching the boundary, the energy suddenly flips over to the excited level in order to stay further within its standard deviation. We model this energy flip by use of parametrized quantum transitions and show that it can be defined as the result of action quantization. We check our theory by recovering the so-called “quantum jumps” published by Minev et al. (2019).

DOI
10.1016/j.rinp.2021.104761

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In Nature, 570, 200 (2019), Minev and co-authors’ experiment shows how to deterministically “catch and reverse a quantum jump mid-flight” in a continuously-observed Rabi-stimulated qubit. Its interpretation is in debate (La Recherche, 555, 40, (2020)). We show that the quantum Zeno effect (QZE) of continuous measurement —by use of photon emission from a 3rd high-rate monitored ancilla level— can be described by an action-angle canonical transformation of the original Hamiltonian dynamical system (HDS) theory of QZE. Then energy whose mean value yields the well-known resonant Rabi harmonic dynamics is actually defined by large-amplitude high-frequency oscillations of the internal as well as of the overall phase of the two-level system. By making use of their standard deviation, we show that the separatrix crossing of the HDS trajectory yields the quantized action nh where n = 1, 2, 3 .... Therefore, the jump dynamics observed in Minev et al. experiment belongs to a series of discrete quantum jumps: it corresponds in this experiment to n = 3.

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In the presence of a periodic external drive, the mean value of the energy of a two-level quantum system (or qubit) displays the well-known Rabi oscillatory dynamics that is usually described within the rotating wave approximation. In order to calculate its standard deviation, we get rid of this latter and obtain a huge ratio (of order unity) of standard deviation over mean value. This is due to the large-amplitude high-frequency oscillations of the internal as well as of the overall phase of the two-level system. In presence of continuous measurement of, say, the ground state energy that yields the quantum Zeno effect, the system remains frozen in its ground state as long as the dynamics of the driven system allows its energy to stay within its Rabi standard deviation. When reaching the boundary, the energy suddenly flips over to the excited level in order to stay further within its standard deviation. We model this energy flip by use of parametrized quantum transitions and show that it can be defined as the result of action quantization. We check our theory by recovering the so-called “quantum jumps” published by Minev et al. (2019).
In Nature, 570, 200 (2019), Minev and co-authors’ experiment shows how to deterministically “catch and reverse a quantum jump mid-flight” in a continuously-observed Rabi-stimulated qubit. Its interpretation is in debate (La Recherche, 555, 40, (2020)). We show that the quantum Zeno effect (QZE) of continuous measurement —by use of photon emission from a 3rd high-rate monitored ancilla level— can be described by an action-angle canonical transformation of the original Hamiltonian dynamical system (HDS) theory of QZE. Then energy whose mean value yields the well-known resonant Rabi harmonic dynamics is actually defined by large-amplitude high-frequency oscillations of the internal as well as of the overall phase of the two-level system. By making use of their standard deviation, we show that the separatrix crossing of the HDS trajectory yields the quantized action nh where n = 1, 2, 3 .... Therefore, the jump dynamics observed in Minev et al. experiment belongs to a series of discrete quantum jumps: it corresponds in this experiment to n = 3.
We show that the energy spectrum of a driven two-level quantum system-or driven quantum bit (qubit)-can be mathematically defined by its two "state energies" as the time derivatives of its large-amplitude, high-frequency internal as well as overall phases. While the mean values of these latter do fit of course Rabi's well-known harmonic (weak drive) as well as anharmonic (strong drive) dynamics, the standard deviations of these both state energies may become comparable to their mean values themselves. This remarkable property, which we check by use of the quantum expectation value of the variance operator, defines a large statistical time-dependent range of available values for the qubit energy in the presence of an external perturbation. We take the example of a recent experiment [Z. Minev et al. Nature, 570, 200 (2019)] and recover by use of the present theory the measured "time-of-flight" values.
We consider the quantum Rabi model as an open system that is subject to dissipation, dephasing, and sinusoidal qubit driving. One can change to an interaction picture where the qubit-driving term disappears at the expense of changing the free energy of the qubit which becomes time dependent. If the driving frequency is large with respect to the rest of the parameters with the exception of the driving strength, then one can obtain an effective Hamiltonian that accurately describes the dynamics of the system. The driving has two effects: the qubit-transition frequency is changed and the qubit has reduced dephasing. The driving strength can be chosen so that the qubit-transition frequency is reduced, made equal to zero, or even made negative so that the excited and ground states of the qubit are interchanged. Therefore, sinusoidal qubit driving offers another method to control the qubit-transition frequency and to reduce qubit dephasing. Adjusting the driving strength allows one to consider a qubit with degenerate energy levels. Not taking dissipation into account, the evolution operator of the qubit-harmonic oscillator system is given by a linear combination of the orthogonal projectors onto the eigenstates of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ followed by the evolution operator of a forced harmonic oscillator, the harmonic oscillator can be prepared in such a way that it is always found in a Schr\"odinger cat state, and the transition probability of the qubit can exhibit a collapse-revival behavior. In addition, the Born-Markov-secular master equation is deduced and the effects of dissipation are presented. In particular, smaller ultrastrong-coupling values are preferable over larger ultrastrong-coupling values and deep strong-coupling values in order to have long-lived, easily distinguishable Schr\"odinger cat states because the decoherence rate is inversely proportional to the square of the coupling. Finally, the qubit-harmonic oscillator system can be prepared in highly entangled states that are stable under dissipation.
In an externally driven multilevel quantum system observation that the NEXT jump has not yet happened affects its future development. In previous work [Phys. Rev. A36, 929 (1987)] it was shown that this class of measurement makes it possible to observe remarkably long dark intervals -- or intermittency -- in the atomic fluorescence of an atom with 3 or more levels. Those calculations were carried out when the driven oscillations or Rabi flopping between the ground state and a strongly fluorescing state were fast compared to its lifetime. In systems with solid state Qubits the accessible parameter space is generally limited to the regime where oscillations are slower than the lifetime. In this paper we evaluate intermittency in atomic transitions, due to measurements with a null result, in this limit. During the dark periods the wave function of the continuously measured multilevel system is coherent.
The model of a two-level system (TLS) coupled to a harmonic oscillator has found a widespread application in physics and chemistry. In this thesis we focus on the field of circuit quantum electrodynamics where the TLS stands for a quantum bit (qubit), which is the basic building unit of a potential quantum computer, and the oscillator represents a transmission-line resonator, which can be used, e.g., to store information contained on the qubit or for communication between several qubits. Furthermore, the oscillator can model a superconducting quantum interference device (SQUID) determining the state of the qubit. In first experimental realizations the coupling between the qubit and the oscillator was small compared to characteristic frequencies of the two devices. In such a situation the Jaynes-Cummings model provides a realistic and completely analytical description of the energy spectrum and the dynamics of the setup. However, it relies on a rotating-wave approximation (RWA) which neglects counter-rotating terms in the full qubit-oscillator Hamiltonian. These counter-rotating terms become important under increase of the coupling strength, as it has been achieved in recent experiments. We present two analytical approaches to take these additional contributions into account: The first one treats the qubit-oscillator Hamiltonian perturbatively in the coupling strength and predicts a frequency shift in its energy spectrum, the Bloch-Siegert shift. The second one considers the qubit's tunneling splitting as small parameter and thus treats the problem to all orders in the coupling allowing to enter the ultrastrong coupling regime, where the coupling strength becomes comparable to the qubit's and/or oscillator's frequency. For both cases we give a thorough analysis of the qubit's dynamics. In order to perform operations the qubit is usually probed by an external driving force. We model this situation by coupling the TLS to a classical oscillator and examine the resulting time dependent Hamiltonian using a combination of Floquet theory and Van Vleck perturbation theory. Thus, we provide an analysis of the qubit's energy spectrum and dynamics which is nonperturbative in the driving amplitude. We investigate effects like coherent destruction of tunneling and driving-inducing tunneling oscillations. By coupling the driven TLS to a quantized oscillator we give for the first time an analytical treatment of the qubit-oscillator system being simultaneously exposed to both ultrastrong coupling and extreme driving. We observe interesting phenomena in the dynamics like the suppression of specific frequencies under a variation of the coupling strength. To mimic environmental influences on the qubit we apply the Caldeira-Leggett master equation approach to the driven TLS and to the qubit-oscillator system and provide for both cases analytical expressions for the relaxation and dephasing rates.
Environmental influences on the dynamics of a coupled qubit-oscillator system are studied analytically in a regime where deviations from the exactly solvable Jaynes–Cummings model are important. We investigate the case of a quasi-degenerate qubit within the ultra-strong coupling regime for which the qubit frequency is much smaller than the frequency of the oscillator, and the coupling between the qubit and the oscillator is large, both of which invalidate the usually employed rotating wave approximation. In contrast to the standard quantum optics master equation, we explicitly take the qubit-oscillator coupling into account while microscopically deriving a dressed state master equation. Using the derived master equation, we discuss a spectroscopic technique which can be used to probe the dressed energy-level structure of the qubit-oscillator system.