The two terms you probably want to google are mirrorless and large-sensor compact. These cameras will typically cost roughly the same as a dSLR entry-level kit, and will offer some serious tools for photography you won't find in a low-end compact camera, such as full manual mode, a flash hotshoe, and RAW capability. And a larger sensor.
Mirrorless cameras offer an interchangeable lens mount and (often, but not always) a comparable sensor size to most dSLRs, but use the data on the sensor for preview and autofocus, simplifying the lightpath and mechanisms to make simpler, more compact camera bodies.
Whether the system overall is more portable depends on the size of the sensor, since there is a tradeoff between sensor size and lens size. For systems like the Sony e-mount, or Fuji X, which use APS-C sized (1.5x crop) sensors, their lenses are equivalently sized to the ASP-C lenses for dSLRs; the same holds true for full-frame mirrorless and dSLR (i.e., Sony E-mount full frame and equivalent Canon/Nikon dSLR full frame lenses are roughly the same size). The main savings is in the body size. For micro four-thirds cameras (4/3"-format, 2x crop) or Nikon 1 (1"-format, 3x crop) can have smaller-than-dSLR lenses, and are more portable, but the smaller sensors inherently have less dynamic range and high iso noise performance.
Large-sensored compacts are similar to mirrorless but have fixed lenses instead of interchangeable lens mounts. Here, the lens and camera body can be optimized for the specific sensor-and-lens combination. So the combination can actually be smaller than an equivalent mirrorless setup (e.g., the LX-100 vs. the GX85 + 12-35/2.8).
Here, too, there is a tradeoff game between the lens size and the sensor. The larger the sensor in the camera, typically the more limited the focal length range of the lens will be to keep the lens small. Full frame and APS-C sized sensors tend to have prime (fixed focal length) lenses (e.g., Sony's RX1 and Fuji'x X100/X70 cameras). And you generally don't get a zoom lens until you're down to 4/3"- (Canon G1X, Panasonic LX-100) or 1"-format (Sony RX100, Canon G3/5/7/9X) sensors.
A Note About Image Quality
Is it really possible to combine compact camera portability with professional SLR quality?
This all depends on your definition of "professional SLR quality". If you mean "SLR quality" in technical terms of the sensors and processors in dSLRs, absolutely--sensors and processors are shared among dSLRs, mirrorless, and compacts. In fact, there's one mirrorless camera (Hasselblad X1D) that has a medium-format sensor larger than the ones you can get in any Canon or Nikon dSLR.
But if you mean "professional quality" as in the type of image a professional photographer can make, then you're looking at the wrong thing. The camera is just a tool. It's up to the photographer, not the camera, to make a gorgeous/arresting image that says something meaningful.
A person with enough talent and drive to put them in the profession, and all the post-processing and shooting skill, experience, and practice that being a professional requires can probably shoot rings around anything that you as a newb could produce, no matter the cameras used. (See DigitalRev's Youtube series: "Pro Photographer, Cheap Camera".) Professional work has been done with iPhones.
Owning a "dSLR-quality" camera won't guarantee you beautiful professional-quality images. You still have to learn how to use it, and to develop your own artistic vision and creativity. And beautiful, gorgeous photos typically take a lot of vision, creativity, time, and effort. Possibly also money.
How willing are you to spend the money, time, and effort? How creative is the vision in your mind's eye? Spending money on a book, class, seminar, or airline ticket to something cool to shoot might improve your image quality far more than a new camera.