Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis

Abstract

We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low n the engineered Hamiltonians reproduce the √ n scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of n = 6 with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.The control of quantum harmonic oscillators has played a prominent role in the development of quantum state control [1,2]. A single quantum oscillator provides access to a Hilbert space with a dimension which increases rapidly as the oscillator energy increases. It is also an example of a system which has a natural transition from the quantum to the classical regimes. One of the primary methods for performing control and measurement of quantum harmonic oscillator states is by coupling the oscillator to a single spin using a JaynesCummings Hamiltonianwhere Ω and φ are real constants,â † andâ are the creation and annihiliation operators of energy quanta for the oscillator, andσ − = |↓ ↑| withσ + =σ † − . |↑ , |↓ are energy eigenstates of the spin. The Jaynes-Cummings Hamiltonian arises naturally for Cavity-QED systems [3] and can be implemented straightforwardly with trapped ions by using a laser to resonantly drive a motional sideband of an internal state transition [2,4,5]. For an ion starting in one of the basis elements |↓ |n where |n are the energy eigenstates of the oscillator, evolution as a function of the duration t of the Hamiltonian H JC results in Rabi oscillations between the two states |↓ |n ↔ |↑ |n − 1 which can be viewed as a rotation R(θ, φ) = cos(θ/2)Î + i sin(θ/2)(cos(φ)ŝ x − sin(φ)ŝ y ) where θ = Ω √ nt andÎ,ŝ x ,ŝ y are Pauli operators which act in the basis {|↓ |n , |↑ |n − 1 }. These Rabi oscillations can be observed by making projective measurements on the spin states of the ion as a function of the duration of the applied Hamiltonian. An important feature is that the Rabi frequency of the oscillations scales with the matrix element n − 1|â |n = √ n. This has played an important role in the diagnosis of energy distributions of various well-known oscillator states which are not energy eigenstates [1]. The use of coupling to a two-state system isolates pairs of states of the oscillator, which simplifies the dynamical evolution and allows simple prescriptions for creating arbitrary superpositions of states [6][7][8].Although the Jaynes-Cummings Hamiltonian arises naturally in the light-matter interaction, similar physics c...

Adiabatic preparation of squeezed states of oscillators and large spin systems coupled to a two-level system

“…where ν/µ = − tan 2 (θ/2). While lower sideband excitation of a trapped ion can be used to implement the JC model, upper sideband excitation leads to the so-called anti-JC model with interaction ∝ (σ − a + σ + a † ) [37], and simultaneous driving of both sidebands implement ladder operators among squeezed Fock states (Bogoliubov transformed linear combinations of a and a † ) [38] with Hamiltonians that are also explicitly of the form of (21). To obtain similar results as in the present article, we recall that one must verify the existence of the E = 0 product state which will depend on the properties of the ancillary system (e.g., it does not occur for Eq.…”

Sequential Modular Position and Momentum Measurements of a Trapped Ion Mechanical Oscillator

“…The dashed lines in Fig. 4(b) show the expected violations for an ideal experiment, and the solid lines show simulations using the level of motional and qubit dephasing that was observed in previous experiments performed in the same apparatus [30]. Spin decoherence limits the violation at small α, and the sharp drop in violation above α ¼ 2 is caused by motional dephasing.…”

Fluctuations of work in realistic equilibrium states of quantum systems with conserved quantities

“…[22], we consider a system composed of N two-level systems, with energy splitting ω at , coupled with equal strength g to a bosonic field of frequency ω b , i.e., the Nparticle Dicke model [31][32][33]. We write the Hamiltonian describing this system in the form [34][35][36][37]:…”

“…This model was introduced to describe the coupling of atoms to light fields [31]. More recently, it has been implemented in systems of trapped ions [37]. In Eq.…”