week ending PRL 112, 133904 (2014) PHYSICAL REVIEW LETTERS 4 APRIL 2014 Quantum-Limited Amplification via Reservoir Engineering A. Metelmann* and A. A. Clerk Department of Physics, McGill University, 3600 rue University, Montréal, Quebec H3A 2T8, Canada (Received 8 November 2013; published 4 April 2014) We describe a new kind of phase-preserving quantum amplifier which utilizes dissipative interactions in a parametrically coupled three-mode bosonic system. The use of dissipative interactions provides a fundamental advantage over standard cavity-based parametric amplifiers: large photon number gains are possible with quantum-limited added noise, with no limitation on the gain-bandwidth product. We show that the scheme is simple enough to be implemented both in optomechanical systems and in super- conducting microwave circuits. DOI: 10.1103/PhysRevLett.112.133904 PACS numbers: 42.65.Yj, 03.65.Ta, 07.10.Cm, 42.50.Wk Introduction.—The past few years have seen a resur- modes. Unlike a NDPA, our scheme explicitly involves gence of interest in amplifiers working near the funda- dissipative (i.e., non-Hamiltonian) interactions between the mental limits set by quantum mechanics [1], in contexts modes. We show that this approach allows a large gain with varying from quantum information processing in circuits quantum limited noise, but crucially is not limited by a [2–4], to radio astronomy [5], to ultrasensitive force fixed gain-bandwidth product: the gain can be arbitrarily detection (e.g., for gravity wave detection [6]). The large without any corresponding loss of bandwidth. Note standard paradigm for a quantum-limited, phase-preserving that non-Hamiltonian evolution is also utilized in a very amplifier is the nondegenerate parametric amplifier different way in probabilistic amplifiers [27–30], which can (NDPA) [7–10], which is based on a coherent interaction stochastically amplify signals without adding noise. involving three bosonic modes (pump, signal, and idler). Our approach is related to reservoir engineering [31], This interaction simply converts a pump mode photon into where one constructs a nontrivial dissipative reservoir that two photons, one in the signal mode, the other in the idler relaxes a system to a desired target state (e.g., an entangled mode. The result is that weak signals incident on the signal state [32–36]). Here, we instead construct an engineered mode are amplified, with the minimum possible added reservoir which mediates a dissipative amplification proc- noise. There has been remarkable progress in realizing such ess. Our mechanism can also be interpreted as a kind of amplifiers using superconducting circuits [11–18]. This in coherent feedback process [37–40], where the amplifica- turn has enabled a number of breakthroughs, from the tion is the result of an autonomous quantum nondemolition measurement of mechanical motion near the quantum limit (QND) measurement combined with a feedback operation. [19], to the measurement of quantum jumps of a super- Our scheme is simple enough to be realized using existing conducting qubit [20,21] and the implementation of quan- experimental capabilities, either with three-mode optome- tum feedback schemes [22,23]. chanical systems (where a mechanical mode couples to two Despite their many advantages, standard cavity-based electromagnetic cavity modes) [41,42], or with supercon- parametric amplifiers suffer from the limitation of having a ducting circuits [16,43,44]. fixed gain-bandwidth product: as one increases the gain of Model.—While our scheme is amenable to many pos- the amplifier, one also reduces the range of signal frequen- sible realizations, we focus here for concreteness on a cies over which there is amplification. This is a funda- three-mode optomechanical system. Two cavity modes mental consequence of the amplification mechanism, which (frequencies ω1 and ω2), are coupled to a single mechanical involves introducing effective negative damping to the signal mode ωM, cf. Fig. 1. The cavity photons interact with the mode. The consequent reduced damping rate determines mechanical mode via radiation pressure forces, and the the amplification, but also sets the amplification bandwidth system is described by the Hamiltonian Hˆ ¼ Hˆ þ Hˆ . ˆ S diss (see, e.g., [3]). This tradeoff between gain and bandwidth Here, HS is the coherent system Hamiltonian (ℏ ¼ 1), can severely limit the utility of cavity-based parametric X Hˆ ω ˆ ˆ † ˆ † ˆ ω ˆ † ˆ (1) amplifiers in many applications. Traveling-wave parametric S ¼ f j þ gjðb þ b Þgaj aj þ Mb b; amplifiers (TWPAs) [24,25] do not use cavities and are in j¼1;2 principle not limited in the same way. In practice, however, ˆ ˆ good device performance and bandwidth of TWPAs is where b (aj) is the annihilation operator for the mechanical limited by the requirement of phase-matching (though see resonator (cavity j), and gj is the optomechanical coupling Hˆ Ref. [26] for recent progress in the microwave domain). strength for cavity j. diss describes the damping of all In this work, we introduce a new approach for quantum- three modes (each by independent baths), and the laser limited amplification based now on three localized bosonic drives on the two cavity modes; these are treated at the level 0031-9007=14=112(13)=133904(5) 133904-1 © 2014 American Physical Society week ending PRL 112, 133904 (2014) PHYSICAL REVIEW LETTERS 4 APRIL 2014 intracavity entanglement [52], it is optimal in allowing the mechanics to mediate amplifying interactions between the two cavities. Dissipative interactions.—If the mechanical resonator was strongly detuned (in the interaction picture) from the two cavity modes by a frequency Δ, then standard adia- batic elimination of the mechanics would yield the ˆ PA ~ ˆ ˆ ~ 2 NDPA Hamiltonian, H ¼Gd1d2 þH:c:, with G∼G =Δ. FIG. 1 (color online). (a) Schematic showing the optomechan- In contrast, we are interested in the resonant case, where ical realization of the dissipative amplification scheme. Two the induced interactions are more subtle. As the system driven cavities (1,2) are both coupled parametrically to a third is linear, one can exactly solve the Heisenberg-Langevin auxiliary mechanical mode. The mechanics mediates a dissipative equations corresponding to Eq. (2), and use these to derive interaction between modes 1 and 2. Signals incident on either effective equations for the cavity modes with the mechanics cavity are amplified in reflection. (b) Alternate realization, where ω ω eliminated [46]. We first consider the simple limit where two pump modes 1;P, 2;P are used to generate the required γ ≫ κ, G; this results in effectively instantaneous induced interaction Hamiltonian, Eq. (2); this setup could be directly interactions. We also specialize to the ideal case where implemented using superconducting microwave circuits [46]. κ1 ¼ κ2 ≡ κ (see [46] for κ1 ≠ κ2). Introducing the effective coupling rate Γ ¼ 4G2=γ, the resulting Langevin equations of standard input-output theory [3,45], resulting in cavity for the cavity modes are (mechanical) damping rates κj (γ). In what follows, the two cavity modes will play roles κ Γ Γ pffiffiffi pffiffiffi ˆ_ − ð þ Þ ˆ − ˆ † − κ ˆ Γˆ (3a) similar to “signal” and “idler” modes in a NDPA, whereas d1 ¼ 2 d1 2 d2 d1; in þ i bin; the mechanics will be used to mediate an effective inter- κ − Γ Γ ffiffiffi ffiffiffi action between them. To achieve this, we assume a strong ˆ_ ð Þ ˆ ˆ † p ˆ p ˆ † d2 ¼ − d2 þ d1 − κd þ i Γb : (3b) coherent drive on each cavity, detuned to the red (blue) 2 2 2; in in mechanical sideband for cavity 1 (2), (i.e., drive frequencies ˆ ˆ The operators dj;in (bin) describe the quantum and ω 1 2 ¼ ω1 2∓ω Þ. We work in an interaction picture L; = = M thermal noise incident on the two cavities (the mechanics); with respect to the free Hamiltonians, and perform displace- † ω ˆ ˆ 0 Æi Mt ˆ they have zero mean and correlation functionshoinðtÞo ðt Þi ¼ ment transformations: aˆ j ≡ a¯ je þ dj, where a¯ j is the in ˆ † ˆ 0 δ − 0 δ − 0 ¯ T 1 average classical amplitude of cavity j due to the laser drive; hoinðtÞoinðt Þi þ ðt t Þ¼ ðt t Þðno þ Þ, where o ¼ dj, T we take these to be real without loss of generality. Assuming b, and n¯ o is the thermal occupancy of each bath. the standard experimental situation where g are small and The mechanical resonator gives rise to two effects in ˆ j a¯ j are large, we linearize HS, resulting in Eqs. (3). First, it gives rise to an additional positive damping Γ of mode 1, and an additional negative damping Hˆ ˆ ˆ † ˆ † ˆ ˆ ˆ ˆ † ˆ † Hˆ (2) −Γ S ¼G1ðd1b þ d1bÞþG2ðd2b þ d2b Þþ CR: of mode 2; each effect corresponds simply to one of the two terms in Eq. (2). In contrast, the joint action of both Here, Gj ¼ gja¯ j are the many-photon optomechanical interaction terms gives rise to terms in Eqs. (3) reminiscent ˆ ˆ † Hˆ of a NDPA, where d1 is driven by d and vice versa. Note couplings, and CR describe nonresonant interaction proc- 2 crucially the opposite sign of this term in Eq. (3a) versus esses. We focus on the good-cavity limit ωM ≫ κj, γ, where Hˆ Eq. (3b); this difference implies that these terms cannot the effects of CR will be negligible. We will thus start by Hˆ PA dropping Hˆ for transparency; i.e., we make the rotating be derived from an NDPA interaction Hamiltonian . CR dissipative wave approximation (RWA); full results beyond the RWA Instead, they correspond to an effective para- metric interaction. Such terms can be obtained from are presented in the figures and in the Supplemental Lindbladian dissipators in a quantum master equation Material [46].