Testing alternative theories of Quantum Mechanics with Optomechanics, and effective modes for Gaussian linear

Introduction

Overview

The search for alternatives to QM is fueled by two issues: the measurement problem and the compatibility of QM with the theory of general relativity. However, if we wish to probe the state of the particles with a measuring device (a concept in QM with no concrete definition), we no longer use deterministic linear evolution equations.

Introduction to collapse models

For example, we have shown that the LISA pathfinder space experiment sets the best limits for CSL and DP at low frequencies. Theorists also set rough lower bounds by imposing that the CSL and DP models must be strong enough to match the fast rate at which the gauges collapse the wave function.

Alternatives to quantum gravity

We showed that for the Hamiltonian in Eq. 1.12), we can define an effective Heisenberg picture that depends on the initial or final state of the system. With a causal prescription for adding measurements to NLQM and a mapping that allows us to exploit the tools of quantum feedback, we calculated the signature of the Schroedinger-Newton theory with the causal-conditional prescription.

Overview of contributions to the theory of optomechanics

Note that we have extended the definition of the boundary condition|Φ(x)i to include a time dependence: |Φ(t,x)i. The environment consists of the input optical field âin(t) and its time-evolved counterpart âout(t), and the thermal bath field ˆbin(t) and its time-evolved counterpart bôut(t).

Figure 1.1: An optomechanical system with n degrees of freedom interacts with n effective input modes, ˆ A (1) in through ˆ A (n)in , which evolve into n effective output modes, ˆ A (1) out through ˆ A (n) out

LISA pathfinder appreciably constrains collapse models

Introduction

We note that Sa has steadily decreased by about a factor of 1.5 since the start of science operations in LISA pathfinder [4], and has continued to decrease significantly since the results were published in June 2016 [22]. We estimated ρ and a using weighted averages of the densities and lattice constants, respectively, of the materials in the alloy from which the test masses were made.

Constraining the collapse models

Discussion

Adler investigates the measurement process of latent imaging in photography and sets a lower limit of λCSL s−1[1]. Note that a lower bound of about 10−17s−1, proposed by Ghirardi, Pearle, and Rimini [16], is also sometimes considered.

Acknowledgments

Note that the second term in parentheses in Eq. 3.42) gives the zero point fluctuations of the oscillator asT →0. For example, according to the Everett interpretation, all limit states are the initial state of the universe.

Figure 2.1: Upper and lower bounds on the CSL collapse rate λ CSL obtained from laboratory experiments operating at different frequencies

Measurable signatures of quantum mechanics in a classical space-

Introduction

In light of this argument, one can go back to an interpretation of quantum mechanics where the wave function does not reduce. In section 3.3 we remind the reader that in nonlinear quantum mechanics the density matrix formalism cannot be used to describe thermal fluctuations.

Free dynamics of an optomechanical setup under the Schroedinger-

However, unlike the Heisenberg picture, the equations of motion depend on the boundary quantum state of the system under analysis. Note that the equation of motion for ζ(t) is particularly easy to solve in the case of the quadratic Hamiltonian given by Eq.

Figure 3.1: Left Panel: according to standard quantum mechanics, both the vector (h xˆ i , h pˆ i) and the uncertainty ellipse of a Gaussian state for the center of mass of a macroscopic object rotate clockwise in phase space, at the same frequency ω = ω C

Nonlinear quantum optomechanics with classical noise

Thermal fluctuations increase the uncertainty in the movement of the center of mass to the point that in realistic experiments the total displacement of the test mass will be much larger than ∆xzp. Nevertheless, after separating classical and quantum uncertainties, we will show that Eq. 3.7) remains valid as long as the quantum (and not total) uncertainty of the test mass is much smaller than ∆xzp.

Figure 3.3: Two ways of forming the same Gaussian density matrix. In the left panel, we have an ensemble of coherent states parameterized by a complex amplitude α, which is Gaussian distributed

Measurements in nonlinear quantum optomechanics

Both prescriptions are equivalent in linear quantum mechanics, but become different in nonlinear quantum mechanics. the state in a particular state, |fi, of the observer associated with that device. Dx p(fcl(ω)= x(ω)) ×p0←ξ(x(ω)), (3.92) kup(fcl(ω)= x(ω)) is the probability that fcl at frequency ω is equal to tox( ω), andξ(x(ω)) is the measured eigenvalue of the observable ˆb2 given that the classical thermal force is given by x.

Figure 3.4: The two prescriptions, pre-selection (top) and post-selection (bottom), that can be used to calculate measurement probabilities

Signatures of classical gravity

Finally, we include the classical noise by taking the ensemble average over different realizations of the classical thermal force, fcl(ω). The equality follows from Eq. 3.76) without taking into account the classic thermal noise, which we will include at the end of the calculation.

Figure 3.5: A depiction of the predicted signatures of semi-classical gravity. The pre-selection measurement prescription’s signature is a narrow and tall Lorentzian peak, while the post-selection measurement prescription’s signature is a shallow but wide

Feasibility analysis

As shown in Fig.3.7(a), numerical simulations of the minimum measurement time needed to decide between white noise and a spectrum of the shape Sh, fit well. In Fig.3.7(b), we show that numerical simulations of the minimum measurement time needed to decide between white noise and a spectrum of the Sd shape fit well.

Table 3.2: The probabilities of the different outcomes of the likelihood ratio test on a particular measurement data stream with an estimator following either of the two distributions shown in Fig

Conclusions

However, testing ex post selection will be much more challenging, although feasible with state-of-the-art experimental parameters. In particular, we need cryogenic temperatures and a low-frequency, high-Q torsional pendulum made of a high-ωSN material. 3.125) contains a scaling of the minimum measurement time required to reliably test posterior selection with these experimental parameters.

Figure 3.10: Minimum measurement time required to distinguish between the Schroedinger-Newton theory with the post-selection measurement prescription and standard quantum mechanics in such a way that the probabilities of indecision and of making an error a

Appendix: Conservation of energy in the SN theory

This result can be linked to the continuity equation (satisfied by the SN theory): 3.135) Integrating over all variables except xi (which we denote byx,i), we obtain.

The efficient methods we have developed in this article simplify the dynamics structure as shown in the figure. Modes of the environment orthogonal to these effective modes interact with each other and with nothing else.

Appendix: More details on calculating B(ω) ˆ

Different interpretations of quantum mechanics make different

Introduction

In this paper, we look for interpretations of quantum mechanics that do not violate the no-signal condition when applied to NLQM. More importantly, how to write down a general nonlinear modification of quantum mechanics is still an open question.

Multiple measurements in sQM and the no-signaling condition

Uˆ(t1,t0) |Ψinii, (4.2) where ÎA ( ÎB) is the identity operator acting on Alice's (Bob's) particle, Û(t,z) is the common time evolution operator for both Alice and Bob's particles from time z to t. For example, both Alice and Bob may agree that a particular choice of Alice's metric could be associated with sending a 0 bit, while another choice could be associated with a 1 bit.

Figure 4.1: A spacetime diagram showing multiple measurement events. Event C describes the preparation of an ensemble of identical 2-particle states |Ψ ini i by Charlie

Ambiguity of Born’s rule in NLQM

Consequently, the linear Schrödinger equation is i~∂t|ψi=. is formally identical to Eq. Heuristically, in the context of Eq. 4.16) is linked to a time-evolution operator, which we denote by ˆUφ(T). The subscript emphasizes that the time-evolution operator is associated with the boundary condition|ψ(T)i = |φi.

The no-signaling condition in NLQM

If we choose all limit states as the initial state of the universe, then we restore Everett's interpretation. The boundary condition associated with the nonlinear time evolution operator of each region is the time-evolved initial state of the experiment conditioned by the measurement events presented in the legend at the top of the figure.

Figure 4.2: Assignment of boundary conditions after two measurements according to the causal-conditional prescription

Conclusions

The rest of the effective modes are assumed to have an arbitrary interaction between them. Note that the RHS of the above equation can be shown to be of the same form as that of Rvin Eq.

Figure 4.5: A general configuration of measurements, labeled by M j where 1 ≤ j ≤ n, occurring before an event B, which describes Bob performing a measurement.

Measurable signatures of a causal theory of quantum mechanics

Introduction

In particular, we will argue that once we establish the initial state of a system, its followed evolution under a large class of nonlinear quantum mechanics theories (NLQM) (of which Equation 5.1 is a part) is equivalent to evolution. under a certain quantum feedback scheme. Since quantum feedback can be causal, we show that adding measurements to Eq. 5.1) does not necessarily mean that the theory violates the condition of non-signaling.

NLQM is formally equivalent to quantum feedback

We first show that for Eq. 5.2), the unmonitored dynamics of the wave function, after we fix its initial state, is the same as the dynamics of a wave function evolving under a time-dependent linear Hamiltonian. If we take αi(t) equal to φ(xi,t, ψ(t)) and Oˆi equal to ˆAlso the evolution of the initial state under Eq. 5.2) is identical to the evolution of the same initial state according to Eq. Therefore, after fixing an initial state, we can use the tools of time-dependent quantum mechanics to examine Eq.

Figure 5.1: Showing how causal NLQM and causal feedback are equivalent in a simple example

An example of continuously monitored optomechanical systems

State that Alice and Bob test the center of mass of the joint quantum state of mass in time. Third, we project the outgoing light into some states that match Alice and Bob's measurement results at the time.

Figure 5.2: Alice and Bob’s optomechanical setups. Note that i = A , B.

Signature of SN with the causal-conditional prescription

6.4, where the degrees of freedom of the system interact only with the effective modes ˆA(1)in through ˆA(n)in a beam-splitting type interaction that switches the states of the system and the modes of the effective environment. Our formalism generally says nothing about the interplay of the system of interest with the environment.

Effective modes for linear Gaussian optomechanics. I. Simplify-

Introduction

In this article, we show that these effective modes only interact with system modes and that no other environmental degrees of freedom interact with these modes or system modes. We also present a general method for constructing a set of effective modes that summarize the interaction of a system of interest with the environment.

Effective modes for an optomechanical setup driven by pulsed blue-

Moreover, if we are only interested in obtaining the equations of motion for the effective modes, then we only need to know β. Fortunately, in the case of the simplified setup discussed in [3], the analysis can be performed analytically because the effective modes given by Eq.

Effective modes for general setups

The only difference is that the mechanical ladder controls would have to be replaced by compressed ladder controls. where at some point after the experiment began, ãoutand ãin is defined in Eq. 6.12), and ˆb1through ˆbna are the ladder operators corresponding to the degrees of freedom of the optomechanical system in question. Note that since the lower half of Âin must be the Hermitian conjugate of the upper half, Sv must be of the form.

Conclusions

Finally, we found that the utility of our proposed effective modes can be limited to simple configurations because the modes are a linear combination of annihilation and creation rate operators. As a result, even when the original modes of the environment are in a vacuum, the ground state of the effective modes may be a complicated vacuum squeezed with many modes.

Figure 6.4: A diagram showing a hypothetical beam-splitter interaction that swaps the quantum states of the system degrees of freedom, ˆ b 1 through ˆ b n , with that of the effective modes it interacts with: ˆ A ( in 1 ) through ˆ A in (n)

Appendix: Constructing effective modes that simplify the dynamics

You are unitary, that implies. whereφu is a phase factor that we are free to choose, and k ®hk is the 2-norm of the vectorh®. φv is another phase factor that we are free to choose. We have fixed ˆAin and ˆAout to a phase factor, but there are two more constraints we must satisfy.

Appendix: An alternative proof based on group theory

Appendix: Constructing effective modes that simplify the dynamics

This can be directly seen by forming the covariance matrix of one of the effective degrees of freedom of the system, say ˆsi,1, sˆi.2. Furthermore, most information about the optomechanical system leaks into only one mode of the environment, such as N1 N2.

Effective modes for linear Gaussian optomechanics. II. Simpli-

Introduction

In this article we show that an optomechanical system with degrees of freedom is only intertwined with environmental modes. Although we cannot make general statements about the entanglement structure of an optomechanical system with its optical bath, we show that the correlation structure can be reduced to a 'chain'.

Setup and notation

However, in general we cannot limit our analysis to these effective modes, because the system may be confused with effective modes with which it does not interact. A system with degrees of freedom is coupled only to ineffective optical modes, which in turn are coupled only to other effective optical modes.

Figure 7.1: General optomechanical setup with n degrees of freedom interacting with m bosonic environment fields.

Entanglement structure

In particular, each of the effective modes described by wsys forms a two-mode pressed state with a mode in wenv. Following a procedure similar to that of section I of the supplementary information of [4], we can analytically calculate the system's steady-state covariance matrix from Eq.

Figure 7.3: Two-mode squeezing between the modes ˆ s i and ˆ e i for 1 ≤ i ≤ n. The bottom two graphs show the phase space distribution of both of these modes.

Applications

N1 (N2) is the logarithmic negativity of the first (second) diagonalizing symplectic system state of Eq. 7.102) with the effective environmental state it is correlated with. Due to the simple correlation structure between ˆsi and ˆei, given by Eq. 7.31), any observable system can be estimated with the same accuracy: 1/νi.