Quantum Flutter: A Dance of an Impurity and a Hole in a Quantum Wire

[Clockwise from Top left]: Charles J. M. Mathy, Eugene Demler, Mikhail B. Zvonarev.

Authors: 
Charles J. M. Mathy^1,2, Mikhail B. Zvonarev^2,3,4, Eugene Demler²

Affiliation:
¹ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts, USA
³Université Paris-Sud, Laboratoire LPTMS, UMR8626, Orsay, France,
⁴CNRS, Orsay, France.

What happens when a particle moves through a medium at a velocity comparable to the speed of sound? The consequences lie at the heart of several striking phenomena in physics. In aerodynamics, for example, an object experiencing winds close to the speed of sound may experience a vibration that grows with time called flutter, which can ultimately have dramatic consequences such as the destruction of aeroplane wings or the iconic Tacoma Narrows bridge collapse. Other examples of physics induced by fast motion include acoustic shock waves and Cerenkov radiation. What we addressed in our work [1] was the effect of fast disturbances in strongly interacting quantum systems of many particles, in a case where the particles are effectively restricted to move in one dimension known as a quantum wire.

When the interactions between particles are weak, quantum systems can sometimes be described by a simple hydrodynamic equation. For example, the Gross-Pitaevskii Equation (GPE) describes the evolution of a weakly coupled gas of bosons at low temperatures when it forms a Bose-Einstein condensate. The GPE is analogous to equations found in hydrodynamics, which explains why one can see analogs of classical hydrodynamical effects such as shock waves and solitons in these systems [2,3]. But what if the interactions are too strong and such an approximation breaks down?

We have found a model that shows interesting physics induced by supersonic motion which goes beyond a hydrodynamical description [1]. The system is a one-dimensional gas of hardcore bosons known as a Tonks-Girardeau gas (TG) [4]. We start the system in its ground state and inject a supersonic impurity that interacts repulsively with the background particles. We obtained exact results on what happens next using an approach from mathematical physics called the Bethe Ansatz approach, coupled with large-scale computing resources [5]. Thus we track the impurity velocity as a function of time and find two main surprising features. Firstly, the impurity does not come to a complete stop, instead it only sheds part of its momentum and keeps on propagating at a reduced velocity forever (Fig 1a). Secondly, the impurity velocity oscillates a function of time, a phenomenon we call quantum flutter as it arises from nonlinear interactions of a fast particle with its environment (see Fig. 1b).

Figure 1: Impurity momentum evolution and quantum flutter:
a, Schematic picture of our setup. Top: We start with a one-dimensional gas of hardcore bosons of mass m known as a Tonks Girardeau (TG) gas (red arrows), in its ground state. We then inject an impurity also of mass m with finite momentum Q (green arrow). Middle: The impurity loses part of its momentum by creating a hole around itself (sphere) and emitting a sound wave in the background gas (blue arrow). However it retains a finite momentum Q_sat after this process and carries on propagating without dissipation. Bottom: legend of the different characters in the story.
b, Time evolution of the expected momentum of the impurity, <P̂_↓(t)>. The momentum decays to a finite value Q_sat, and shows oscillations around Q_sat at a frequency we call ω_osc. The background gas has density ρ_↑, and we define a Fermi momentum k_F = πρ_↑, a Fermi energy E_F = k_F²/(2m) where m is the mass of the particles, and a Fermi time t_F = 1/E_F. Inset: zoom into the plot of <P̂_↓(t)> showing the oscillations we call quantum flutter.
c, Time evolution of the density of the background gas in the impurity frame. More precisely, shown is the density-density correlation function G_↓↑(x,t) = <ρ_↓(0,t) ρ_↑(x,t)> in units of ρ_↑/L where L is the system size, and the position along the wire is written in units of the interparticle distance ρ_↑^-1 in the background gas. Here ρ_↑ is the density of the background gas, and ρ_↓ the density of the impurity. G_↓↑(x,t) is effectively the density of the background gas with respect to the impurity position. We see the formation of the correlation hole around x ρ_↑ = 0 (blue valley), and the emission of the sound wave (red ridge). Underneath a schematic illustration of the dynamics is given: the blue arrow represents the emitted sound wave, the sphere is the hole, and the green arrow the impurity (see a). Inside the correlation hole the impurity and hole are dancing, meaning that they are oscillating with respect to each other, the phenomenon we denote as quantum flutter.

Using the exact methods just mentioned we were able to look in detail at the dynamical processes underlying quantum flutter. The time evolution of the impurity in the gas of bosons can be broken down into several steps. First the impurity carves out a depletion of the gas around itself, called a correlation hole. It expels the background density into a sound wave that carries away a large part of the momentum of the impurity, but not of all it (fig 1c). In fact the impurity retains part of its momentum and no longer sheds momentum because of kinematic constraints: there are no sound waves it can emit in the background gas while conserving momentum and energy.

After formation of the correlation hole, the impurity momentum starts to oscillate. When the dynamics of a quantum system shows a feature that is periodic in time, typically the frequency of the feature corresponds to an energy difference between two states of the system. Examples include light emission of an atom, or spin precession in response to a magnetic field, which underlies Nuclear Magnetic Resonance. In our case, the two states are an exciton and a polaron. The exciton corresponds to the impurity binding to a hole, since if the impurity repels the background gas, it is attracted to a hole (i.e. a missing particle in the background). The polaron is an impurity dressed due to interactions with the background particles, which affects its properties such as its effective mass: it becomes heavier as it carries a cloud of displaced background particles around it [6]. Thus we arrive at the following picture, as shown schematically in Fig. 2: first the impurity causes the emission of a sound wave in the background gas and creation of a hole close to it. It can bind to this hole and form an exciton, or not bind to it and form a polaron instead. In fact the impurity does both in the sense that it forms a quantum superposition of a polaron and an exciton. This quantum superposition leads to oscillations in the impurity velocity, a phenomenon called quantum beating, which is analogous to Larmor precession of a spin in a magnetic field. The difference here is that the two states that are beating, the exciton and polaron, are strongly entangled many-particle states. That we observe long-lived quantum coherence effects in a system composed of infinitely many particles is surprising. Namely, typically such systems exhibit decoherence, such that if one puts a particle in a quantum superposition of two states, the superposition decays because of interactions with other particles.

Figure 2: Origin of quantum flutter:
a, The quantum flutter oscillations originate from the formation of a superposition of entangled states of the impurity with its environment. After the impurity is injected in the system is creates a hole around itself. It can then bind to this hole and form an exciton, or not bind to it and form a state that is dressed with its environment called a polaron. In fact the system forms a coherent superposition of these two possibilities, which then leads a quantum beating and oscillations in the impurity momentum with a frequency given by the energy difference between these two possibilities.
b, Comparison between the frequency ω_osc of oscillations in the impurity momentum, and the energy difference between the polaron E(Pol(0)) and the exciton E(Exc(0)) (the zero between brackets refers to the exciton and polaron having momentum zero). The x-axis denotes the interaction strength between the impurity and the background particles: the interaction between a background particle at position x_i and the impurity at position x_↓ is a contact interaction of the form g δ(x_i - x_↓), and one defines the dimensionless interaction parameter γ = m g/ρ_↑. ℏω_osc and E(Pol(0))-E(Exc(0)) are in quantitative agreement, which motivates the interpretation of quantum flutter as quantum beating between exciton and polaron.

To see quantum flutter in the laboratory directly, one can use methods from the field of ultracold atoms, in which neutral atoms are cooled and trapped using a combination of lasers and magnetic fields. The trapping potential can be chosen to restrict the atoms to move along 1D tubes, and effectively behave like a TG gas [7,8]. The interaction between the particles can be tuned using a Feshbach resonance. Impurity physics in one-dimensional TG gases has already been studied [9,10,11]. The only added ingredient needed for quantum flutter is to create impurities at finite velocities, which can be done using two-photon Raman processes. Quantum flutter can be observed by measuring the expected impurity velocity as a function of time. Thus cold atom experiments could confirm our predictions, and one could vary different parameters of the model so see how robust quantum flutter is. Our preliminary calculations suggest that quantum flutter survives within a certain window of varying all the parameters in the theory such as the interaction between background particles, the relative mass of the impurity and the background particles, and the form of the interactions.

In summary, we have found an example of a system of many particles where injecting a supersonic impurity leads to the spontaneous formation of a long-lived quantum superposition state which travels through the system at a finite velocity. The question of which systems allow transport of quantum coherent states is important for quantum computing applications [12], and has surfaced in recent studies of quantum effects in biology [13]. Thanks to the advent of exact methods and the development of precise experiments in the study of many-particle quantum dynamics, we expect to see progress being made on this question in the near future.

References
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