Quantum harmonic oscillator state synthesis by reservoir engineering

Kienzler, Daniel; Lo, H-Y.; Keitch, Ben; Clercq, Ludwig de; Leupold, Florian; Lindenfelser, Frieder; Marinelli, Matteo; Negnevitsky, Vlad; Home, Jonathan

doi:10.1126/science.1261033

Cited by 233 publications

(236 citation statements)

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“…Recent ideas of designing a quantum state by engineering a specific environment [36][37][38][39][40][41][42] or by a periodic modulation of the open system's Hamiltonian [16,43,44] open a path to new forms of control over quantum systems. Combining these two concepts of time-periodic modulations and environment (bath) engineering, here we study the dynamics of open quantum systems where the environment thermodynamic parameters are periodically modulated.…”

Section: A Introductionmentioning

confidence: 99%

Quantum transport under ac drive from the leads: A Redfield quantum master equation approach

Purkayastha

2017

Phys. Rev. B

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Evaluating the time-dependent dynamics of driven open quantum systems is relevant for a theoretical description of many systems, including molecular junctions, quantum dots, cavity-QED experiments, cold atoms experiments and more. Here, we formulate a rigorous microscopic theory of an out-of-equilibrium open quantum system of non-interacting particles on a lattice weakly coupled bilinearly to multiple baths and driven by periodically varying thermodynamic parameters like temperature and chemical potential of the bath. The particles can be either bosonic or fermionic and the lattice can be of any dimension and geometry. Based on Redfield quantum master equation under Born-Markov approximation, we derive a linear differential equation for equal time two-point correlation matrix, sometimes also called single-particle density matrix, from which various physical observables, for example, current, can be calculated. Various interesting physical effects, such as resonance, can be directly read-off from the equations. Thus, our theory is quite general gives quite transparent and easy-to-calculate results. We validate our theory by comparing with exact numerical simulations. We apply our method to a generic open quantum system, namely a double-quantum dot coupled to leads with modulating chemical potentials. Two most important experimentally relevant insights from this are : (i) time-dependent measurements of current for symmetric oscillating voltages (with zero instantaneous voltage bias) can point to the degree of asymmetry in the system-bath coupling, and (ii) under certain conditions, time-dependent currents can exceed time-averaged currents by several orders of magnitude, and can therefore be detected even when the average current is below the measurement threshold. A. IntroductionThe dynamics of a quantum system coupled to an external environment (so-called "open" quantum system) are a result of the intricate interplay between the coherent evolution of the quantum degrees of freedom, the structure of the environment and the form and strength of the system-environment coupling. From quantum cavities [1-3] and superconducting qubits [4][5][6][7][8][9] , through quantum dots [10][11][12][13][14][15], molecular junctions [16][17][18][19][20][21][22][23][24][25][26][27] and cold atoms [28][29][30][31] , to excitons traveling in photosynthetic complexes [32][33][34][35], open quantum systems show dynamics which can be far richer and more surprising than their coherent (environment-free) counterparts.Recent ideas of designing a quantum state by engineering a specific environment [36][37][38][39][40][41][42] or by a periodic modulation of the open system's Hamiltonian [16,43,44] open a path to new forms of control over quantum systems. Combining these two concepts of time-periodic modulations and environment (bath) engineering, here we study the dynamics of open quantum systems where the environment thermodynamic parameters are periodically modulated. Even though there exists formally exact methods of treating such set-ups...

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Section: A Introductionmentioning

confidence: 99%

Quantum transport under ac drive from the leads: A Redfield quantum master equation approach

Purkayastha

2017

Phys. Rev. B

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“…In earlier work, we used this observation to demonstrate reservoir engineering for the creation of squeezed and displaced vacuum states, which can be viewed as cooling to the ground state of a given engineered basis. We also showed how the use ofĤ + provides a simple diagnosis that the ground state had been produced [9]. More recently, we used the predicted √ n scaling of Rabi oscillation frequencies produced byĤ + to reconstruct Schrödinger's cat states of a motional oscillator which were beyond the range of standard methods which use only an energy-eigenstate decomposition [10].…”

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confidence: 99%

Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis

Kienzler¹,

Lo²,

Negnevitsky³

et al. 2017

Phys. Rev. Lett.

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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...

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“…We utilize the novel idea of dissipative squeezing [11][12][13] [see Fig. 1(a)], where the system is allowed to cool towards a squeezed low-energy state.…”

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confidence: 99%

Squeezing of Quantum Noise of Motion in a Micromechanical Resonator

et al. 2015

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A pair of conjugate observables, such as the quadrature amplitudes of harmonic motion, have fundamental fluctuations that are bound by the Heisenberg uncertainty relation. However, in a squeezed quantum state, fluctuations of a quantity can be reduced below the standard quantum limit, at the cost of increased fluctuations of the conjugate variable. Here we prepare a nearly macroscopic moving body, realized as a micromechanical resonator, in a squeezed quantum state. We obtain squeezing of one quadrature amplitude 1.1 AE 0.4 dB below the standard quantum limit, thus achieving a long-standing goal of obtaining motional squeezing in a macroscopic object. DOI: 10.1103/PhysRevLett.115.243601 PACS numbers: 42.50.Wk, 03.65.Ta, 42.50.Ct, 81.07.Oj The motion xðtÞ ¼ X 1 ðtÞ cosðω m tÞ þ X 2 ðtÞ sinðω m tÞ of a harmonic oscillator having the natural oscillating frequency ω m can be described by the quadrature amplitudes X 1 and X 2 which have slow fluctuations. The fluctuations, presented in units of the quantum zero-point fluctuation amplitude x zp , satisfy the Heisenberg uncertainty relation ΔX 1 ΔX 2 ≥ 1. One of the two can be prepared (¼ squeezed) below the value 1, at the expense of increased fluctuations in the other quadrature. In optics, squeezing of laser light was observed in early 1980s [1,2], not long after the possibility was realized.It has been a formidable challenge to obtain squeezing in the motional state of a macroscopic object. The possibility of squeezing in the oscillations of massive gravitational antennae was hypothesized a long time ago [3,4], but technological limitations are too severe for experimental realization. Other motional quantum-mechanical phenomena, on the other hand, have recently been experimentally demonstrated [5,6] in micromechanical resonators. The latter systems are nearly macroscopic in physical size, and therefore they provide an ideal test system for treating the borderline between quantum and classical. Of particular interest for these studies has been the cavity optomechanics setting coupling electromagnetic cavity mode and the oscillator motion [7]. Output of squeezed light [8][9][10] was recently observed, but this does not yet imply that the oscillator mode is squeezed.Here we report the first realization of squeezing of the motional state of a nearly macroscopic body, realized as a micromechanical resonator measuring 15 microns in diameter. We utilize the novel idea of dissipative squeezing [11][12][13] [see Fig. 1(a)], where the system is allowed to cool towards a squeezed low-energy state. This method has the great advantage of being able to create unconditional squeezing in the steady state. This is in contrast with many other plausible methods of squeezing generation [14][15][16][17][18][19][20]. Our approach is closely related to the quantum nondemolition measurements [21][22][23] which, however, are not able to generate true squeezing without feedback. At this point we mention that classical squeezing of thermal noise is routinely observed in mechanical system...

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Quantum harmonic oscillator state synthesis by reservoir engineering

Cited by 233 publications

References 30 publications

Quantum transport under ac drive from the leads: A Redfield quantum master equation approach

Quantum transport under ac drive from the leads: A Redfield quantum master equation approach

Quantum Harmonic Oscillator State Control in a Squeezed Fock Basis

Squeezing of Quantum Noise of Motion in a Micromechanical Resonator

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